Download Nuclear Spin Optical Rotation in Organic Liquids

Nuclear Spin Optical Rotation in
Organic Liquids
Junhui Shi
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Chemistry
Adviser: Michael Romalis
November 2013
UMI Number: 3604505
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Abstract
Nuclear spin induced optical rotation (NSOR) is a novel technique for the detection of nuclear magnetic resonance (NMR) via optical rotation instead of conventional
pick-up coil. Originating from hyperfine interactions between nuclei and orbital electrons, NSOR provides a new method to reveal nuclear chemical environments in different molecules. Previous experiments of NSOR detection have poor signal-to-noise
ratio (SNR), which limits the application of NSOR in chemistry. In this work, based
on a continuous-wave NMR scheme at a low magnetic field (5 G), we employ a multipass cavity and a 405 nm laser to improve the sensitivity of NSOR. By performing
precision measurements of NSOR detection in a range of pure liquid organic chemicals, we demonstrate the capability of NSOR to distinguish 1 H signals in different
chemicals, in agreement with the first-principles quantum mechanical calculations.
The NSOR of
19
F is also measured at low fields with high SNR, showing that heavy
nuclei have higher optical rotation signals than light nuclei.
In addition, in order to obtain NSOR at different chemical sites in the same
molecule via chemical shift, we make efforts to develop a novel scheme based on
liquid-core hollow fiber for the detection of NSOR under high magnetic fields. By
coiling a long liquid-core fiber densely for many loops around a small rod combined
with RF coils, it is possible to measure optical rotation signals inside a narrow-bore
superconducting magnet. Manufactured by filling liquids into capillary tubings, those
liquid-core fibers perform like multimode step-index fibers, and thereby exhibit linear
birefringence and depolarization, significantly reducing the light polarization for the
measurement of optical rotation. According to our attempts, it is possible to suppress
the linear birefringence by filling chiral liquids in hollow fibers, and approach near
single-mode operation by means of launching light beam into the fiber core under
the mode match condition. Although some issues of hollow fibers obstruct the final
iii
measurement of high-frequency NSOR, our work on the liquid-core fiber provides the
basis for future fiber-based NSOR experiments under high magnetic fields.
iv
Acknowledgements
Without help and support from many people that I have worked with, it is impossible for me to finish this PhD thesis. First of all, I would like to thank my advisor
Prof. Michael Romalis for his guidance and help during my past three years. Many
times when I want to give up doing something after facing failures and frustration,
his enthusiasm and insistence inspires me to continue on my work. When I just came
to his lab, I almost had no real expertise on experimental atomic physics. I did not
know the principle and operations of a Lock-in amplifier, why only one side coating
is need in my experiment and how to solder electronic circuits properly... Without
Mike’s patience and encouragement, hardly can I finish my experiments and this thesis. Here, I sincerely acknowledge Mike for his continuous guidance and support in
these years.
I also would like to address my thanks to Prof. William Happer. I benefit a lot
from his kindly comments, interesting talks about scientific knowledge and invaluable
memory of physics history, when we from Mike’s lab had group meeting together
with his lab. In the past two years, I also utilized the superconducting magnet for my
research and learned a lot in Will’s lab. In addition, I greatly acknowledge Prof. Juha
Vaara for his elaborated theoretical calculation as assistance for our experimental
measurements. His theoretical predictions could guide future work on the research
topic in my thesis.
Working in Mike’s lab, I received a lot of help from Justin Brown, Nezih Dural, Marc Smiciklas, Shuguang Li, Dong Sheng, Yunfan (Gerry) Zhang, Oliver Jeong.
Justin’s prompt help, Nezih’s optimism, Marc’s continuous teaching, Shuguang’s carefulness, Dong’s passion for science ... all impress me a lot. There are too many times
when they help me in the basement laboratory than I can remember in details, so I
acknowledge all of them here. In addition, I also benefited a lot from Ben Olsen and
Bart McGuyer from Will’s lab, as they taught me to fill and play with liquid nitrogen
v
and helium for the magnet. I am also grateful to Tom Kornack, Giorgos Vasilakis,
Ioannis Kominis, Ekaterina (Katia) Mehnert, Haifeng Dong, Aaron Kabcenell, Dongwoo Chung, etc, who keep friendly atmosphere when they are working in Mike’s lab.
All through my six years as a graduate student, I also need to acknowledge a lot
people as collaborators and friends as follows. Prof. Herschel Rabitz at Princeton
helped and guided my work in theoretical chemistry in the first two years. It is also a
good opportunity to work with Prof. Feng Shuang at Institute of Intelligent Machines
(Hefei) and Prof. Shian Zhang at East China Normal University, when I spend one
year studying in these two places as a visiting student. Other people and friends
that I have worked with or gotten help from include Tak-san Ho, Xiaojiang Feng,
Rebing Wu, Vincent Beltrani at Princeton; Yaoxiong Wang, Xin Chen at IIM; Prof.
Zhenrong Sun, Hui Zhang, Chenghui Lu, Wenjing Chen at ECNU.
In addition, I need to acknowledge Prof. Steven Bernasek, Prof. Zoltan Soos and
Prof. Annabella Selloni, as they serve as my committee. Also I specially acknowlege
Megan Krause and Sallie Dunner, since they are graduate administrator and provide
a lot of kindly help to assist my graduate study at Princeton. I am also indebted to
Steven Lowe, Claudin Champagne, Darryl Johnson, Catherine Brosowsky, and other
staff of the physics department, for their supports and assistance to ease my lab work.
As to the financial support, here I should acknowledge Department of Chemistry,
Princeton University and National Science Foundation.
Finally, I need to thank my parents and grandparents. Nothing that I have accomplished would have been possible without them. Their supports and love are endless
for me, no matter what difficult I need to face and what choice I will make in my
life.
vi
To my parents and grandparents.
vii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
1 Introduction
1
2 Theory
9
2.1
Natural optical rotation . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Faraday rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3
Nuclear spin induced optical rotation . . . . . . . . . . . . . . . . . .
28
3 Experiments under low magnetic fields
3.1
3.2
34
Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.1.1
Adiabatic passage transfer and CW spin-lock technique . . . .
39
3.1.2
Multipass optical cavity . . . . . . . . . . . . . . . . . . . . .
43
3.1.3
Cell configuration . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.1.4
Coil arrangement . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.1.5
Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
NMR signal calculation, measurement and calibration . . . . . . . . .
54
3.2.1
Theoretical calculation . . . . . . . . . . . . . . . . . . . . . .
56
3.2.2
NMR signal calibration . . . . . . . . . . . . . . . . . . . . . .
58
viii
3.2.3
Polarization decay . . . . . . . . . . . . . . . . . . . . . . . .
59
3.2.4
Measurement of NMR . . . . . . . . . . . . . . . . . . . . . .
60
3.3
Optical rotation constant calculation . . . . . . . . . . . . . . . . . .
64
3.4
Comparison of Signal-to-Noise ratio . . . . . . . . . . . . . . . . . . .
68
3.5
Faraday rotation (Verdet constant) . . . . . . . . . . . . . . . . . . .
70
4 Experimental results and calculations under low fields
73
4.1
NSOR of 1 H in various chemicals . . . . . . . . . . . . . . . . . . . .
4.2
Comparison of
F and 1 H . . . . . . . . . . . . . . . . . . . . . . . .
81
4.3
Theoretical calculation . . . . . . . . . . . . . . . . . . . . . . . . . .
83
19
5 NSOR experiments under high magnetic fields
5.1
5.2
5.3
73
89
Experiment setup at high fields . . . . . . . . . . . . . . . . . . . . .
92
5.1.1
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .
93
5.1.2
Three types of hollow fibers . . . . . . . . . . . . . . . . . . . 100
5.1.3
Pulse sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.1.4
High bandwidth photodiode amplifier . . . . . . . . . . . . . . 107
Polarization in liquid core hollow fibers . . . . . . . . . . . . . . . . . 111
5.2.1
Multimode liquid core fiber . . . . . . . . . . . . . . . . . . . 113
5.2.2
Mode matching . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.3
Bending induced birefringence . . . . . . . . . . . . . . . . . . 122
5.2.4
Mode conversion and bandwidth of multimode fiber . . . . . . 130
Experiments with hollow fibers . . . . . . . . . . . . . . . . . . . . . 136
5.3.1
Transmission increase . . . . . . . . . . . . . . . . . . . . . . . 138
5.3.2
Depolarization and linear birefringence . . . . . . . . . . . . . 140
5.3.3
Chiral liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6 Conclusions
147
ix
Bibliography
149
x
List of Tables
2.1
Character table for group S1 and S2 . . . . . . . . . . . . . . . . . . .
2.2
Change of the magnetic quantum number ∆mF under absorption or
17
emission of photons. . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.1
Parameters of this experiment about cell, coil and flow speed . . . . .
48
3.2
Verdet constant of chemicals from reference . . . . . . . . . . . . . .
71
3.3
Comparison of measured value and referenced value of Verdet constant
72
4.1
Physical properties of chemicals for investigation . . . . . . . . . . . .
75
4.2
Original measured data of NSOR angles . . . . . . . . . . . . . . . .
78
4.3
Measured NSOR, Faraday rotation Verdet constants and reference
value of Verdet constants . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.4
Calculations of NSOR constants via HF, DFT and CCSD methods . .
87
5.1
The cutoff frequency Um for mode HE2m . . . . . . . . . . . . . . . . 123
xi
List of Figures
1.1
Relative NMR receptivities of magnetic nuclei in traditional NMR detections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
First-principle calculations of Nuclear spin induced optical rotation of
1
2.1
2
H,
13
C,
17
O,
14
N and
15
N . . . . . . . . . . . . . . . . . . . . . . . .
4
Rotation of electric field vector of a right-circularly polarized light
beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
The composition of left- and right-circular polarization light . . . . .
13
2.3
Enantiomers of Lactic acid . . . . . . . . . . . . . . . . . . . . . . . .
14
2.4
The dispersion of optical rotation near absorption lines . . . . . . . .
18
2.5
Polarization rotation due to the Faraday effect . . . . . . . . . . . . .
20
2.6
Zeeman splitting and magnetic field induced optical rotation.
22
2.7
Refractive indices difference of n+ and n− as a function of light frequency. 25
2.8
Measurement scheme of Faraday rotation . . . . . . . . . . . . . . . .
2.9
Distant magnetic field and hyperfine interaction induced by a nuclear
. . . .
28
spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.10 Energy shifts due to hyperfine interaction. . . . . . . . . . . . . . . .
32
2.11 Measurement scheme of Nuclear spin induced optical rotation . . . .
33
3.1
Optical absorption coefficients for water from reference . . . . . . . .
35
3.2
Apparatus for measurement of NSOR in organic liquids . . . . . . . .
38
xii
3.3
Polarized liquid flow and adiabatic passage transfer . . . . . . . . . .
41
3.4
Cylindrical-cylindrical mirror cell . . . . . . . . . . . . . . . . . . . .
44
3.5
Geometry of multipass cell and the light spot patterns . . . . . . . .
45
3.6
Two multipass optical pattern in this work . . . . . . . . . . . . . . .
46
3.7
Illustration of a multipass cavity . . . . . . . . . . . . . . . . . . . . .
47
3.8
Picture of sample tube (cell) . . . . . . . . . . . . . . . . . . . . . . .
48
3.9
Drawing of the sample tube . . . . . . . . . . . . . . . . . . . . . . .
49
3.10 Configurations of B0 and B1 coil . . . . . . . . . . . . . . . . . . . . .
51
3.11 Calculated B0 and B0 field distribution . . . . . . . . . . . . . . . . .
52
3.12 Polarimetry for the analysis of polarization rotation angle . . . . . . .
53
3.13 Circuit of two-channel photodiode amplifier with differential output .
55
3.14 Distribution of the magnetic field induced by the nuclear magnetization
along the sample tube . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.15 Polarization decay of 1 H in water . . . . . . . . . . . . . . . . . . . .
61
3.16 On-resonance NMR signal in time and frequency domain . . . . . . .
63
3.17 Off-resonance NMR signal in time and frequency domain . . . . . . .
64
3.18 NMR signal at different supply current I0 . . . . . . . . . . . . . . . .
65
3.19 Dependence of NMR signal and linewidth on RF voltage . . . . . . .
65
3.20 NMR signal-to-noise (S/N) ratio . . . . . . . . . . . . . . . . . . . . .
66
3.21 Comparison of NMR and Optical rotation signal spectrum for water .
67
3.22 Comparison of signal-to-noise (S/N) ratio between the traditional
NMR signal and optical rotation signal . . . . . . . . . . . . . . . . .
69
3.23 Apparatus for Faraday rotation measurement . . . . . . . . . . . . .
72
4.1
1
H NSOR constants in various chemicals . . . . . . . . . . . . . . . .
80
4.2
Calculated NSOR constant of individual 1 H in different groups . . . .
81
4.3
NSOR spectrum of 1 H in C6 H14 and
83
xiii
19
F in C6 F14 . . . . . . . . . . .
4.4
Comparison of first-principle calculations and experimental results of
NSOR constants
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.1
High-field experiment setup with hollow fibers and RF coil . . . . . .
94
5.2
The NMR RF coil and tank circuit in the magnet . . . . . . . . . . .
96
5.3
Toroid coil used for both NSOR and Faraday rotation measurements
97
5.4
Schematic drawing of the fiber connector and liquid port . . . . . . .
99
5.5
Picture of the fiber connector and liquid port . . . . . . . . . . . . . . 100
5.6
Inject laser beam into the liquid core fiber . . . . . . . . . . . . . . . 101
5.7
Photonic crystal fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.8
Flexible capillary tubings . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.9
NMR spectrum of Toluene, measured by a toroid coil. . . . . . . . . . 106
5.10 CPMG sequence
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.11 The performance of CPMG sequence . . . . . . . . . . . . . . . . . . 108
5.12 The long-time performance of CPMG pulse sequence . . . . . . . . . 109
5.13 The model of photodiode and the circuit for photodiode amplifier . . 110
5.14 Picture of the high bandwidth photodiode amplifier . . . . . . . . . . 111
5.15 The model of liquid core hollow fiber . . . . . . . . . . . . . . . . . . 113
5.16 The choices of the normalized propagation constant b . . . . . . . . . 116
5.17 b − V characteristics of weakly guiding fibers . . . . . . . . . . . . . . 117
5.18 Axial intensity distribution of LPlm . . . . . . . . . . . . . . . . . . . 118
5.19 Mode excitation efficiency versus w1 /a . . . . . . . . . . . . . . . . . 122
5.20 A bent liquid core fiber . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.21 Periodic polarization transformation in a linear birefringent fiber . . . 127
5.22 Right-angle 8-configuration and Faraday rotator mirror . . . . . . . . 130
5.23 Dispersion of a multimode fiber. . . . . . . . . . . . . . . . . . . . . . 134
5.24 The frequency spectrum of impulse response h(t) . . . . . . . . . . . 136
5.25 Transmission increment in 5m-long hollow fiber . . . . . . . . . . . . 138
xiv
5.26 Transmission drop when the liquid is flowing . . . . . . . . . . . . . . 140
5.27 The comparison of transmission in a fresh fiber and a reused fiber . . 141
5.28 Measurement of polarization linearity of the light emerging from the
liquid core fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.29 Output light intensity from two channels after periodically modulated
by a rotating λ/2 waveplate . . . . . . . . . . . . . . . . . . . . . . . 143
5.30 The measurement of Faraday rotation in a long coiled fiber at low
frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
xv
Chapter 1
Introduction
Background
As developed for more than half a century, nuclear magnetic resonance (NMR) is one
of the most important spectroscopy tools for the chemical analysis of materials. For
many years, people use a pick-up coil to receive the oscillating induced voltage arising
from the precessing nuclei under a magnetic field [1, 2, 3, 4, 5]. With help of pulsed
excitation and Fourier transform technique, highly-sensitive and multidimensional
NMR is extensively applied to many research areas [6, 7]. In the pick-up coil scheme,
the sensitivity of NMR is proportional to the natural abundance and the square
of gyromagnetic ratio of the nuclei (see the comparison of relative receptivities in
Figure 1.1), therefore the conventional NMR favors the nucleus 1 H, and normally
requires a high magnetic field and bulk volume. In order to overcome these limitations,
spectroscopists developed a lot of substitutional methods to improve the technology of
NMR. For example, high sensitivity magnetometer are applied to sense NMR signal
instead of pick-up coil, which making the signal strength proportional only to the
first power of magnetic field [8, 9, 10]; Optical pumping is implemented to approach
hyperpolarization by transferring angular momentum from photons to the spins and
1
C
Figure 1.1: Relative NMR receptivities (D ) of spin- 12 nuclei in traditional continuousC
wave (CW) NMR detections based on pick-up coil. D is a relative value compared
C
with 13 C, and D is proportional to γ 3 N I(I + 1) for continuous-wave experiments
using optimum radio frequency powers. (γ is the gyromagnetic ratio, N is natural
abundance of the isotopic species concerned, and I is the spin quantum number.)
Adapted from Reference [24].
then increase the spin polarization via nuclei-electron or nuclei-nuclei spin exchange
in condensed matter systems [11, 12, 13, 14] and noble gases [15, 16, 17, 18]. In the
small structured quantum well and nanomaterial, an additional increase in sensitivity
is obtained by optical detection of the photoluminescence polarization arising from
the recombination of electrons [19, 20, 21, 22, 23].
Recently, unlike preceding optical detection methods confined in some specific
systems, a more general optical detection scheme for transparent substance based
on nuclear spin induced optical rotation (NSOR) is proposed and demonstrated by
Savukov et al [25], while previous efforts on shifting NMR frequency via light turns
out to be too small to be detected as far [26, 27, 28]. The first NSOR experiment
is carried out by continuous-wave spin-lock method under a low magnetic field with
prepolarized water and
129
Xe, which shows that rotation of light polarization via
2
nuclear spin precession is pretty small in 1 H in water, but could be enhanced dramatically in samples containing heavy nuclei, e.g.
129
Xe [25]. Subsequent experiments
under high magnetic field demonstrate the chemical-shift-resolved NMR spectrum via
NSOR with aid of pulsed NMR technique [29, 30]. In the pulsed NMR experiments
under high fields in Reference [30], a comparison of traditional NMR (inductive) and
NSOR (optical) of 1 H,
13
C,
19
F and
31
P is investigated, showing different NSOR
receptivities of light and heavy nuclei from traditional NMR receptivities. Besides
experiments, formal theories for NSOR are also developed [31, 32, 33], as well as elaborated first-principle calculations based on ab-initio and density functional methods
[34, 35]. Especially, the first-principle calculations of NSOR [34] predicts that different molecules and groups containing the same nuclei have different optical rotation
constants. As plotted in Figure 1.2, the first-principle calculations of H, C, O and N,
all of which are most important nuclei in organic chemistry, show differences between
NSOR constants in different groups of molecules, paving the way to explore potential
applications of NSOR in chemical analysis.
NSOR essentially originates from the hyperfine interaction between nuclei and
orbital electrons in atoms and molecules [25], as the magnetic field arising from a
precessing nucleus could induce optical rotation in the medium. The magnetic field
arising from the a nuclear spin is composed of a local contact field and a distant dipolar field. The local contact field is only effectively exerted on the orbital electrons in
the same atom, known as hyperfine interactions inside atoms and molecules, resulting in sensitivity in terms of nuclei-electron configuration, while the a distant dipolar
field is much weaker and will be averaged as a macroscopic magnetic field in a bulk
polarized medium. As regular Faraday rotation is induced by a macroscopic magnetic
field (i.e. distant field) when the field is align with the light propagation direction
in the medium, the hyperfine interaction could enhance the Faraday rotation as an
3
Figure 1.2: The first-principle calculation of Nuclear spin induced optical rotation of
Carbon, Hydrogen, Oxygen in different functional groups in molecules, and Nitrogen
in different isotopes. Adapted from Reference [34].
addition to distant field, because the hyperfine interaction in short distance is significantly stronger than distant fields. This extra optical rotation induced by hyperfine
interaction is the key component of NSOR, which differs from regular Faraday rotation, thereby NSOR could be regarded as an enhanced Faraday rotation. Since the
hyperfine interaction is sensitive to the local electron configuration, NSOR provides
a new way to investigate chemical environments of paramagnetic nuclei in atoms and
molecules, as an addition to chemical shifts in traditional NMR.
The promising advantages of NSOR lie in several aspects. First, NSOR increase
for heavier nuclei, because the enhancement arising from the hyperfine interaction
increases with the atomic number generally, while in traditional NMR light nuclei
(especially 1 H) have higher sensitivity. This enhancement was observed in
and
19
129
Xe [25]
F [29]. Second, NSOR provides a new way to distinguish the NSOR of the
same nuclei at different chemical sites, as theoretically demonstrated in Reference
[34]. Due to the poor sensitivity, differences of NSOR between the same nuclei in
4
different molecular sites have not yet been clearly observed in experiments [30]. Third,
as an optical detection of NMR in transparent samples, NSOR provides convenience
for small sample volume detection and site-specific detection in aid of state-of-the-art
manipulations of lasers, as well as improvements of sensitivity by using a multipass or
optical cavity arrangement, because optical rotation is proportional to the length of
optical path. Furthermore, using short laser wavelength (λ) could increase the NSOR
signal as 1/λ2 far from optical resonances for the detection in a long-optical-path
configuration, while operating the laser wavelength near an optical resonance could
also significantly enlarge NSOR signal for short-optical-path configuration, because
optical rotations will increase dramatically under near optical resonance condition.
Our work
As far as now, the signal-to-noise ratio (SNR) of NSOR detection is poor, both
in the low-field CW experiments [25] and in high-field pulsed experiments [29, 30],
especially for the light nuclei 1 H. In order to revolve the distinct 1 H NSOR signal
from different molecules both under low fields and high fields, the SNR of NSOR
should be improved. Our efforts are devoted to improve the sensitivity of NSOR by
elongating the optical path, as NSOR signals are linearly proportional to the length of
optical path in the medium. By using a multipass cavity with 14-pass configuration,
we manage to achieve a 3.15 m optical path with high optical stability in operation.
In addition, we use a 405 nm laser to enhance NSOR signal due to the fact that
optical rotation signal is enhanced at short wavelengths as 1/λ2 , as well as reduce
optical absorption for most chemicals concerned in this work. The NSOR detection
method is based on the low-field (5G) continuous-wave spin-lock technique. Although
the polarization via prepolarization in a permanent magnet in our experiment is
extremely small, and the optical absorption is significant due to the impurities in
5
liquids and reflection loss due to the imperfect antireflection coating on multipass
mirrors, we have achieved unprecedented great SNR of the measurement of 1 H in a
range of organic liquids. We measure 1 H and
19
F NSOR with a SNR greater than
15 after 1000 s of integration, and obtain relatively precision NSOR constant (at low
fields, NSOR constant is averaged for the same nuclei at different chemical sites) for
each chemicals with about 5% uncertainty. According to our measurements, the ratio
of 1 H NSOR to Faraday rotation changes by more than a factor of 2, and the NSOR
constants do not scale with the Verdet constants of the liquids, because the hyperfine
interaction between electrons and nuclei is influenced by the chemical environment.
We find clear distinction between proton signals for different compounds, especially
for water, Methanol and Ethanol. We also obtain the precision NSOR constant of 19 F
in perfluorohexane, which is a factor of 6 larger than that of 1 H in Hexane, because
NSOR signal is enhanced for heavy nuclei. In addition, our experimental results
generally agree with the earlier theoretical predictions [34] and current first-principle
calculations for all the chemicals in this experiment. All of this work is published in
Reference [36], and details of this work are described in Chapter 3 and Chapter 4 in
this thesis.
In order to obtain NSOR at different chemical sites in the same molecule via chemical shift, we make efforts to attempt a novel experimental scheme based on liquid-core
hollow fibers to approach the measurement of NSOR under high magnetic fields, as
well as draw a novel protocol for nanoliter NMR. Filling high-index liquids into fusedsilica capillary tubings makes the liquid-core hollow fibers, while we also explore other
schemes based on photonic crystal fibers and Teflon AF tubings. By coiling a long
fiber densely for many loops around a small rod coupled with RF coils, we find that it
is possible to measure optical rotation signals inside a narrow-bore superconducting
magnet. Because the liquid-core hollow fiber performs like a multimode step-index
fiber, significant linear birefringence and depolarization is observed in those fibers,
6
thereby reducing the fraction of linearly polarized light for the measurement of optical rotation. By understanding the origin of birefringence and depolarization in
multimode fibers, we succeed to suppress the linear birefringence by filling chiral liquids in hollow fibers, and approach near single-mode operation by means of launching
light beam into the fiber core under the mode match condition. In addition, we find
the mode dispersion arising from bending-induced birefringence affects the bandwidth
of optical signal in practice. Our measurement of high-frequency (115 MHz) NSOR
at a 2.7 T magnet has not yet been realized, while we are still looking for methods
to improve the high-bandwidth and polarization-maintaining single-mode operation
of liquid-core hollow fibers. Our attempts on the fiber-based NSOR could provide
benefits for future NSOR and other optical rotation experiments.
Dissertation structure
The main part of this thesis consists of four chapters,
1. In Chapter 2, theories of several optical rotation phenomena, including natural
optical rotation (optical activity) , Faraday rotation and NSOR, are introduced
in details. This chapter elucidates the origins of those circular birefringence
phenomena and the difference among them.
2. Chapter 3 describes the details of experimental setup of the CW spin-lock NMR
at a low magnetic field (5 G) and a multipass cavity, as well as the traditional
NMR detection and NSOR spectra for the measurement of NSOR.
3. Chapter 4 covers the analysis of the NSOR data obtained at the low-field experiment as shown in Chapter 3. In addition, first-principle calculations carried
out by Ikäläinen et al [34, 36] are also included in order to compare with the
experimental results.
7
4. Chapter 5 is mainly about the experimental setup of liquid-core hollow fiber in a
superconducting magnet, as well as the liquid-core fiber theory and experiments.
The theoretical basis of multimode step-index fiber is especially reviewed in
details, because the liquid-core fiber is actually a multimode step-index, thereby
exhibiting the linear birefringence, depolarization and bandwidth limitation.
8
Chapter 2
Theory
Optical rotation is the change of the plane of polarization of a polarized light beam
when it travels through some materials. Optical rotation is a broad research area, as
it occurs under various conditions, e.g. in solutions of chiral molecules, solids with
rotated crystal planes, spin-polarized gases of atoms or molecules, some materials in a
magnetic field or electrical field, and nuclear-spin polarized mediums. Optical rotation
is essentially due to circular birefringence, which rotates the plane of polarization of
light but still keeps it linearly polarized. In this chapter, theories of several types of
optical rotation are reviewed. In Section 2.1, the optical rotation is specifically defined
as natural optical rotation, widely known as optical activity in chemistry, which is a
type of natural phenomenon that occurs in chiral molecules and crystalline structures.
Then the magnetic-field induced optical rotation, named as Faraday rotation, and the
Nuclear Spin induced Optical Rotation (NSOR), will be introduced in Section 2.2 and
Section 2.3 respectively.
2.1
Natural optical rotation
Natural optical rotation, termed as optical activity, was initially discovered in the
quartz crystal and some organic liquids, such as turpentine and sugar solution. It was
9
eventually realized that the source of natural optical activity is a chiral molecular or
crystal structure which arises when the structure has a sufficiently low symmetry that
it is not superposable on its mirror image. These two distinct forms generate optical
rotations of equal magnitude but opposite sense at a given wavelength. According
to early experiments, at given wavelength, the rotation angle is proportional to the
optical path L inside the optically active medium, θ = αλ · L, while the constant αλ
is a specific constant for this material at the given wavelength. When the wavelength
is different, the rotation angle is roughly inversely proportional to the square of the
wavelength λ. More accurate experimental data shows that there exists a empirical
dispersion formula [37] between the rotation constant θ and wavelength λ
θ = Σj
Bj
,
λ2 − λ2j
(2.1)
where λj and Bj could be interpreted from experimental data, and λj is also the
visible or near ultraviolet absorption wavelength, which could be explained by modern
quantum theory.
In optics, the optical rotation in chiral molecules solutions and crystals is explained by circular birefringence, the difference of indices of refraction for left- and
right-circular polarizations in propagation. There are three types of polarization of
light: linear, circular and elliptical polarization. The linear polarization, or plane polarization, of light is a confinement of the electric field vector or magnetic field vector
to a given plane along the direction of propagation, while the elliptical polarization
is the polarization of light such that the tip of the electric field vector describes an
ellipse in any fixed plane intersecting the direction of light propagation. The circular
polarization is a special case of elliptical polarization, in which the electric field of
the passing wave does not change strength but only changes direction in a periodical
rotary manner. Apparently, since the electric vector could be rotated in two manner,
clockwise and counter-clockwise, the circular polarization is further classified as left10
and right-circularly polarization. In most atomic and optical physics literatures, the
left- and right-circularly polarized light is denoted as σ + and σ − light respectively,
which will be used in the following part of this thesis as well. The electric field vector
of a right-circularly polarized light beam is illustrated in Figure 2.1.
z
Figure 2.1: The change of instantaneous electric field vectors of a right-circularly
polarized (σ − ) light beam propagating along +z direction. The electric field vector
rotates clockwise when viewed in the −z direction from the target. Here, the direction
of left and right is defined by pointing one’s left or right thumb of the observer toward
the light source. Adapted from Reference [38].
The linearly polarized light can be regarded as a superposition of left(σ + )- and
right(σ − )-circularly polarized light beams of equal amplitude, and the orientation of
the linear polarization plane is a function of the relative phases of the two components.
As shown in Figure 2.2, the electric component vector of right- and left-circularly
polarized light beam could be written as
~ + = E0 [x̂ · cos(kz − ωt) + ŷ · sin(kz − ωt)], left circular
E
2
E
0
~− =
E
[x̂ · cos(kz − ωt) − ŷ · sin(kz − ωt)], right circular
2
(2.2)
with x̂ and ŷ unit vectors perpendicular to the propagation direction, k = ωn/c, where
c is the light speed, ω is the angular frequency of light, and n is the refractive index
for this wavelength inside the medium. The superposition of these two components
11
~ = x̂·E0 cos(kz −ωt), whose
with equal amplitude is a linearly polarized light beam, E
electric component vector is along the x̂ direction, as shown in Figure 2.2(a). When
the linearly polarized light beam passes through some optically active medium, the
left- and right-circular polarization component undergoes different phase shift. In
other words, the refractive index n+ and n− for these two components are slightly
different, this phenomenon is called circular birefringence. From Equation 2.2, the
phase of σ + and σ − light is ϕ+ = ωt −
ωz
n
c +
and ϕ− = ωt −
ωz
n
c −
. As the light
propagates forward inside the medium by distance d, the phase difference between
these two components is ϕ+ − ϕL− =
ωd
(n+
c
− n− ), and the superposition of them
becomes
ωd
ωd
ωd
~ = E0 cos
(n + n− )/2 − ωt · x̂ · cos
(n − n− )/2 + ŷ · sin
(n − n− )/2
E
c +
c +
c +
(2.3)
which shows that electric vector is still a linearly polarized, but is rotated with an
angle
ωd
(n+
c
− n− )/2, i.e. (ϕ+ − ϕ− )/2, as shown clearly in Figure 2.2(b). In the
perturbative region, which means the frequency of light is far from optical resonance,
the index shift for left- and right-circularly polarized component is quite small and
almost the same but with different sign, while the light intensity will not be affected
by the circular birefringence. In addition, from Equation 2.3, the rotation angle
θ=
ωd
(n+
c
− n− )/2 , so it is proportional to the propagation path distance d in the
medium,
θ=
π
(n − n− ) · d ,
λ +
where λ is the light wavelength, and αλ =
π
(n+
λ
(2.4)
− n− ) is the optical activity con-
stant for the material. A positive optical rotation constant corresponds to clockwise
rotation, while it is anticlockwise for negative rotation constant.
12
(a)
( b)
E
E+
E-
θ
E
EE+
φφ+
Figure 2.2: The composition of left- and right-circular polarization light. (a) A
~+
linearly polarized light beam is decomposed into coherent left-circularly polarized E
~ − components. The propagation direction is out of the
and right-circularly polarized E
plane of the paper. The arrow is the electric field vector. (b) The rotated polarization
plane with angle θ in some optically active medium. Due to circular birefringence,
the left- and right-circular polarization component is rotated with different angles,
leading to a rotation of the linear polarization plane by angle θ = (ϕ+ − ϕ− )/2.
The optical rotation occurs naturally in organic liquids consisting of chiral
molecules and some crystals with structure having a sufficiently low symmetry that
it is not superposable on its mirror image. The physical reason why the refractive
indices are different for left- and right-circularly polarized light lies in the different
responses of mirror-image pairs of chiral molecules or crystal structure. For example,
Lactic acid molecules have a pair of enantiomers that are mirror images of each other,
as shown in Figure 2.3. As an analogy, this pair of molecules can ben viewed as a
pair of helix with different handedness. The helix with left- or right-handedness will
responds differently to left- and right-circularly polarized light during interaction:
only one type of circular polarization could follow one type of helix effectively.
As a consequence, they have different polarizabilities for the light with the same
wavelength and the same circular polarization, hence giving a slight difference in
refractive index and causing optical rotation.
13
COOH
COOH
C
CH3
C
H
H
CH3
HO
OH
(R)-(-)-Lactic acid
(S)-(+)-Lactic acid
Figure 2.3: Lactic acid molecules have a pair of enantiomers: (S)-(+)-lactic acid (left)
and (R)-(–)-lactic acid (right). They are nonsuperposable mirror images of each other.
In order to give a physical explanation for the optical rotation of chiral materials,
some analysis based on classical electromagnetism should be used in the following
part. In a pure electrical field, the electrical polarization of material is normally
expressed as P = N αE. However, for the light, as an electromagnetic wave, there
is an oscillating magnetic field perpendicular to the electrical field, which induces a
spatial variation of electrical field, according to Maxwell equation
−
∂B
= ∇ × E,
∂t
(2.5)
hence producing an extra electrical polarization in the material. The total electrical
polarization could be written as
P = N αE − N β Ḃ.
(2.6)
where N is the number density of molecules N/V , α is the electrical polarizability,
and β is an another material characteristic constant that relates magnetic field and
electrical polarization. Further analysis based on quantum mechanics shows that β is
nonzero for chiral molecules, but zero for achiral molecules. From Equation 2.5, the
induced extra electrical polarization is in a plane parallel to B but perpendicular to
E, while the term αE is parallel to the electrical vector of light. As a result, the total
14
polarization plane is rotated from the direction of the electrical vector of light, thus
rotating the polarization plane of a propagating light beam.
Furthermore, based on the total polarization P in Equation 2.6, from Maxwell
equation,
∇ × B = −µ0
∂
(ε0 E + P) ,
∂t
(2.7)
we can obtain a differential equation for the magnetic component B alone,
2
∇ B = µ 0 ε0
αN
1+
ε0
B̈ + µ0 βN ∇ × B̈,
(2.8)
given that ∇ × ∇ × B = −∇2 B. For a right(+)- or left(−)-circurally polarized light
beam, the time dependance of the electrical and magnetic component vector on time
is
E± = E0 (î · cos φ± ± ĵ · sin φ± )
(2.9a)
B± = B0 (∓î · sin φ± + ĵ · cos φ± )
(2.9b)
where φ± = ωt − k± z, î and ĵ is a pair of orthogonal vector units perpendicular
to the propagation direction. Equation 2.9a is a similar expression as Equation 2.2,
but with a more brief format. From Equation 2.9b, we can obtain
∇ × B± = ∓k± B±
(2.10a)
2
∇2 B± = −k±
B±
(2.10b)
15
B̈± = −ω 2 B±
(2.10c)
Furthermore, after substituting the above three equations of B± into Equation
2.8 , a quadratic equation of k± is shown as
2
k±
αN
± µ0 βN ω 2 k± .
= µ 0 ε0 ω 1 +
ε0
2
(2.11)
√
Since k± = ωn± /c (light speed c = 1/ µ0 ε0 ), where n+ and n− is the refractive index
of right- and left-circularly polarized light, then we can finally get the expression of
refractive index n± by solving Equation 2.11 to first order approximation,
n± ≈ 1 +
ωβN
αN
±
2ε0
2cε0
(2.12)
In the achiral material, β=0, the refractive index is the same for right- and leftcircularly polarized light, therefore there is no natural optical activity. In chiral
material, β 6= 0, ∆n =
ωβN
,
cε0
so an optical rotation is induced after the light beam
passing through a length of L inside the medium,
1
∆θ = N µ0 ω 2 βL.
2
(2.13)
The difference in material constant β of chiral and achiral molecules is essentially
due to the difference in their symmetries [39]. Some perturbative analysis [40] based
on quantum mechanics shows that the induced dipole moment in an electromagnetic
field is
X ωn0 h0 |µ̂| ni · hn |µ̂| 0i
X h0 |µ̂| ni · hn |m̂| 0i
± 2
2
±
µ = Re
Im
E
(t)
−
Ḃ± (t)
2
2
2
2
~
ω
−
ω
~
ω
−
ω
n0
n0
n6=0
n6=0
(2.14)
16
S1
Rz , x, y
Rx , Ry , z
0
A
A00
E
1
1
σh
1
-1
S2
Rx , Ry , Rz
x, y, z
Ag
Au
E i
1 1
1 -1
(a) Character table for group S1 (b) Character table for group S2
Table 2.1: Character table for group S1 and S2 (Group symbol S1 is also termed as Cs ,
and S2 is termed as Ci ). An achiral molecule has either a mirror plane (S1 symmetry)
or an inversion center (S2 symmetry) inside itself. Adapted from Reference [42].
where operator µ̂ is the electrical dipole moment, i.e. µz = −e · z, operator m̂ is the
∂
is the angular momentum
magnetic dipole moment, i.e. mz = − 2me e · lz (lz = −i~ ∂ϕ
operator), ωn0 is the energy difference between energy level |0i and |ni, and ω is the
angular frequency of the electromagnetic field. By comparing Equation 2.14 with
Equation 2.6, we can get
X h0 |µ̂| ni · hn |m̂| 0i
2
β = − Im
,
2
2
~
ω
−
ω
n0
n6=0
(2.15)
which is a sum of the multiplication of electronic dipole and magnetic dipole and
magnetic transition moments divided by quadratic difference of frequency detuning,
called as Rosenfeld Polarizability [41]. Apparently, this expression gives a generally
reasonable explanation for the empirical dispersion formula of rotation constant in
Equation 2.1. Rn0 = Im h0 |µ̂| ni · hn |m̂| 0i is rotational strength of the transition
0 → n. For achiral molecules, there exists a mirror plane or an inversion center inside
the molecule, which corresponds a symmetry group S1 or S2 . The electrical dipole
moment µ̂ is a translational operator, the magnetic dipole moment m̂ a rotational
operator. However, in a S1 or S2 symmetry group, no component of translation and
rotation with the same coordinate variable belongs to the same symmetry species
from group character tables, as shown in Table 2.1, so the product h0 |µ̂| ni · hn |m̂| 0i
in the rotation strength must be zero for any molecular orbital n, which means that
β is zero for achiral molecules, while for chiral molecules it is usually nonzero.
17
Rotation angle, ∆
n0
1
Absorption
Rotation constant
0
a
b
Figure 2.4: (a) The dispersion of optical rotation near an absorption line. (b) The
dispersion of optical rotation of in the vicinity of two absorption bands. Adapted
from Reference [40].
In addition, under a reflection, the operator µ̂ and m̂ have different symmetries:
µ̂ changes its sign, while m̂ does not. As a result, under a reflection, the rotation
strength Rn0 and hence β changes its sign, which actually shows that, for a pair of
enantiomers that is mirror image of each other, the Rosenfeld Polarizability β has
equal but opposite value. That’s why a pair of enantiomers rotate the polarization
plane of light by an equal angle but with opposite directions.
The perturbative result in Equation 2.15 straightforwardly leads to the empirical
dispersion formula in Equation 2.1, based on the fact that
λ2 λ2n0
1
,
=
ωn2 0 − ω 2
c2 λ2 − λ2n0
(2.16)
as the parameter Bj is also dependent on the wavelength that is determined from
experiments. When the optical wavelength is near one of the absorption line of
molecules, i.e. resonant condition, the optical rotation could be thereby enhanced
dramatically. It seems that the rotation angle will reach a critical point at λ =
18
λn0 , however, it will not go to the infinity actually, because the above perturbative
theory break down at this point. In practice, the dispersion curve is more like that
shown as the blue line in Figure 2.4(a), because real energy levels in molecules always
possesses some linewidth. Molecules normally have a lot of absorption bands, hence,
the dispersion curve of optical rotation is typically a superposition of several dispersion
curves. e.g. the optical rotation has a dispersive shape as shown in Figure 2.4 in case
of two absorption bands.
2.2
Faraday rotation
Faraday rotation is the optical rotation in a magnetic field, first discovered by Michael
Faraday in 1845 as the first experimental evidence that light and magnetism are related.. The effect, like optical rotation induced by organic molecules, is due to circular
birefringence, the difference of indices of refraction for left- and right-circular polarizations in propagation. In the presence of an external magnetic field, the atomic or
molecular states in material are weakly perturbed, resulting with splitting of magnetic sub-levels, i.e. Zeeman effect, which causes slightly difference for left- and rightcircular polarization individually, due to the fact that different magnetic sub-levels
only response to either left- or right-circular polarization component. This response
difference induced by magnetic field causes the dispersion in the optical refraction
indices, although it is of the order of a few parts per million normally in nonresonant
condition, far less than the optical rotatory power in chiral materials. In near resonant
condition, that is the light wavelength is in the neighborhood of one absorption line
of the material, the Faraday rotation angle could be enhanced significantly, named as
Macaluso-Corbino Effect, while the light might acquire partial elliptical polarization
due to absorption difference . The Faraday rotation has some applications in measuring instruments, for example, it is used to measure optical rotatory power of materials
19
θ
B
d
E
Figure 2.5: Polarization rotation due to the Faraday effect. The electric field vector
plane (polarization plane E) is rotated by the magnetic field component B that is
parallel to the light beam propagation direction. The rotation angle θ is linearly
proportional to Faraday rotation constant(Verdet constant) v and the optical path
distance d inside the optically active medium. (Adapted from Wikipedia)
for chemical analysis, sense remote magnetic fields, detect the polarization of electron spins in semiconductors in spintronics research, and manufacture high sensitivity
atomic magnetometers based on alkali vapor polarized by optical pumping [43, 44].
In optics, Faraday rotators, based on the large rotatory power in some material, such
as terbium gallium garnet, can be used for amplitude modulation of light, and are
the basis of optical isolators to prevent unwanted feedback light for laser cavities.
As illustrated in Figure 2.5, in Faraday rotation, the plane of polarization is rotated after passing through some medium when there is a magnetic component parallel to the the light propagation direction. The rotation angle of polarization plane
is proportional to the intensity of the magnetic field component, as well as the length
of the optical path in the medium. Hence, the relation between the rotation angle of
the polarization and the magnetic field (not too strong) in a transparent material is
described in a linear expression:
θ = ν · B · d,
20
where, the constant ν is called Verdet constant. A positive Verdet constant corresponds to L-rotation (anticlockwise) when the direction of propagation is parallel to
the magnetic field and to R-rotation (clockwise) when the direction of propagation is
anti-parallel. Thus, if a ray of light is passed through a material and reflected back
through it, the rotation doubles.
The Faraday rotation is chromatic, i.e. Verdet constant has a dispersion relation
in terms of the optical wavelength, empirically expressed as
ν = Σj
Aj
,
λ2 − λ2j
(2.17)
which is similar to the dispersion formula of optical rotations in optically active
medium as shown in Equation 2.1. Therefore, the dispersion dependence of natural
optical rotation, as shown in Figure 2.4, is also adopted to Faraday rotation.
The classical interpretation of Faraday rotation is a typical example shown in
some classical electromagnetism text books, as the motion of electrons is modeled as a
microscopic oscillator coupled with external magnetic field, and then an accumulated
phase difference of left- and right-handed components of a linearly polarized wave
as they traverse the medium could be deduced from classical analysis, as well as
the classical interpretation of Zeeman splitting. However, the classical analysis can
not give quantitative interpretation and related computation for Faraday rotation in
practical material, which have complicated electronic structures. In the following, the
quantum interpretation for Faraday rotation will be reviewed.
In the presence of an external static magnetic field, a spectral line will split into
several components, well known as Zeeman splitting. For example, the Hamiltonian
of an atom in a magnetic field is
−
H = H0 + →
µ · B,
21
(2.18)
F’=0
F=1
(a)
(b)
Figure 2.6: (a) Zeeman splitting of F=1 and F=0 state in the presence of a magnetic
field. F=1 level splits into three Zeeman sublevels M=-1, 0, +1. (b) Different responses of left-(σ + ) and right-(σ − ) circularly polarized light in an F=1→F’=0 atomic
transition with an magnetic field.
→
−
→
−
→
−
−
where the magnetic moment →
µ = µB (gg L + gs S )/~, L the total orbital angular
→
−
momentum, S the total spin angular momentum. In the weak-field case, the magnetic
−
potential energy VM = →
µ ·B has a eigenvalue subset according to the quantum number
→
−
−
combination of l and →
s for each orbital,
VM = µB Bmj gj
(2.19)
where, gj is the Landé g-factor dependent on the state of l and s. mj is the zcomponent of the total angular momentum, which has a (2j + 1) fold degeneracy
→
− →
−
(j = l + s ). Considering the hyperfine interaction, the quantum number of
→
− − →
−
nuclear spin should also be counted in, F = l + →
s + I , and mF is thereby restricted
by all of these quantum number. In the case of F=1 (e.g. l = 0, s=1/2 and I=1/2),
mF could be either +1, 0 or −1, forming a triplet sublevel splitting with energy shift
∆E = mF · gµB B, while there is no splitting for F=0, as shown in Figure 2.6(a).
22
Photon
σ+
σ−
π
Absorption
∆m=+1
∆m=−1
∆m=0
Emission
∆m=−1
∆m=+1
∆m=0
Table 2.2: Change of the magnetic quantum number ∆mF under absorption or emission of photons.
As mentioned in Section 2.1, the left-circular polarization is called σ + light and
right-circular is σ − . This statement is adopted from the physics that photon spin
also possesses angular moment, as the spin angular momentum of polarized light
is considered a circularly polarized light beam directed at a target. The angular
momentum of left-circular polarized photon σ + is +1, while for right-circular polarized
photon σ − it is -1. Now if we consider the optical transition from F=1 to an upper
state F0 =0, due to the presence of the magnetic field, there are three transitions,
mF =-1, 0, +1 to F0 =0. In order to conserve the total angular momentum during
state transitions, as shown in Table 2.2, the transition becomes circular-polarization
selective, mF = −1 to F0 =0 will absorb a σ + photon, while mF = +1 to F0 =0 will
absorb a σ − photon. In addition, due to the magnetic field B, Zeeman sublevel
mF =±1 has a energy shift ±gF µB Bz , so the resonant frequency for σ + and σ − light
is also shifted correspondingly.
Because the refractive index is dispersive in terms of the detuning of light frequency from a transition frequency,
n(ω) ≈ 1 + 2πχ0
γ0
2(ω − ω0 ) + iγ0
(2.20)
where χ0 is the amplitude of the linear susceptibility, and γ0 is the linewidth of energy
level. The above expression of n(ω) leads to a Lorentzian lineshape. For the incoming
linearly polarized light beam with single frequency that could be decomposed into two
equal components of σ + and σ − , the refractive index is slightly shifted by the longitude
component of magnetic field Bz
23
n± ≈ 1 + 2πχ0
γ0
.
2(ω − ω0 ∓ gF µB Bz ) + iγ0
(2.21)
Therefore the difference between the refractive indices n± for σ + and σ − light is
n+ (ω) − n− (ω) = −2πχ0
4gF µB Bz /γ0
2 .
ω−ω0
2
(2gF µB Bz /γ0 ) + 1 − 2i γ0
(2.22)
This results in the dispersive shape shown in Figure 2.7, where only the real part
of refractive indices n± is depicted, since the imaginary part induces elliptical polarization especially under near resonance condition but will not affect the rotation of
polarization plane. Due to the difference of refractive index, after passing through
the medium in magnetic field, the two circular components acquires a relative phase
shift
ϕ=
ωd
· Re(n+ − n− ),
c
(2.23)
where d is the optical path in the medium, then the polarization plane of the original light beam experience an optical rotation with angle θ = ϕ/2, which could be
expressed in terms of frequency detuning ∆ = ω − ω0 ,
θ=
πχ0 ωd
b[1 + b2 − (2∆/γ0 )2 ]
·
c
(2∆/γ0 )2 + [1 + b2 − (2∆/γ0 )2 ]2
(2.24)
where b = 2gF µB Bz /γ0 . The optical rotation takes the dispersive form of Re(n+ −n− ),
as shown in the bottom of Figure 2.7. In summary, the different responses of σ + and
σ − polarized light both in terms of the circular polarization direction and resonant
frequency lead to a difference in refractive indices in the same material.
The above analysis is based on a single transition, F = 1 to F 0 =0, which is
valid when the optical wavelength is near this transition, named as resonant Faraday
24
Re n +
Re n −
ω
Re n + − Re n −
Figure 2.7: (Top) Refractive indices as a function of light frequency for σ + and σ − light
(Real parts of n± is the refractive index, while imaginary parts describes absorption.
Imaginary part of n± causes elliptical polarization, which is not considered in this
thesis). The curves are shifted with respect to one another due to the magnetic field.
(Bottom) Linear optical rotation is proportional to the difference between Re n+ and
Re n− .
rotation or Macaluso-Corbino effect, and thereby these two states dominate over other
states on the contribution to optical rotation. In most cases, we should consider all the
other states for the analysis, since Faraday rotation occurs even when light frequency
is far from resonance of any absorption band of the material. In the following, a more
systematic description of Faraday rotation as an integration of all molecular states in
perturbative treatment following from [45, 38, 31] is shown.
In the microscopic description, the electrical polarization could be written as P =
N αE (comparing with Equation 2.6, the magnetic derivative term is not included,
since there is no natural optical activity in achiral molecules, and only the electrical
polarization part contributes to Faraday rotation). In a oscillating electrical field,
the polarizability α is normally a complex and frequency-dependent tensor. For a
circularly polarized and monochromatic light beam, the Maxwell equation gives
1
n± ≈ 1 + µ0 c2 N ((αxx + αyy ) ± i(αxy − αyx ))
4
25
(2.25)
where αij is the components of polarization tensor α, so the optical rotation could be
expressed as
1
0
θ = µ0 cωLN αxy
2
(2.26)
0
, here L is the optical path, c is the light speed, ω is the light frequency, and αxy =
(αxy − αyx )/2. Perturbative analysis[45, 38, 31] shows that the polarizability
0
αxy
dq
X
2X
ω
=−
ρqj
Im(hq |µx | pi hp |µy | qi)
2
2
~ j=1
ω
−
ω
pq
p6=q
(2.27)
j
where, qj is the subspace of electronic state |qi, e.g. the Zeeman splitting sublevels,
ρqj is the relative population distribution in state |qi. In the absence of external field
as perturbations, the net sum in Equation 2.27 is zero, because the contributions from
transition between a given state |pi and complementary states |qi of the degenerate set
have opposite signs, i.e. asymmetrical distribution, which reveals that the sum across
the whole subspace is zero. However, in the presence of some external field, e.g. a
longitude component of magnetic field B = B0 ẑ along the light propagation direction,
the population and energy level symmetry is broken. The transition frequency is
firstly shifted due to Zeeman splitting. According to the first-order perturbation,
|p0 i = |pi −
0
ωpq =
B0 X 1 up
m |ui
~ u6=p ωpu x
(2.28)
qq
ωpq − (mpp
z − mx )Bz
where mup
x = hr |mz | pi is the z-component of the magnetic dipolar moment. Under
the weak-field condition, the energy shifts are rather small, we can make an approximation
1
1
≈ 2
02
2
ωpq − ω
ωpq − ω 2
qq
2ωpq (mpp
z − mz )B0
1+
2 − ω2)
~(ωpq
26
.
(2.29)
0
which leads to a perturbative expansion for αxy in terms of the weak magnetic field,
0
0
0
(D)
Bz
αxy (B) = αxy + αxy,z
0 (D)
αxy,z =
X 2ωpq
2ω X
qp pq
qq
− 2
(mpp
ρqj ×
z − mz )Im(µx µy )
2 − ω2)
~ j
(ω
pq
p6=q
"
X 1
1
qp pu
qp pu
+ 2
Im(muq
z (µx µy − µy µz ))
ωpq − ω 2 u6=q ωuq
#)
X 1
pu
qp uq
qp uq
Im(mz (µx µy − µy µz ))
+
ωup
u6=p
(2.30)
(2.31)
From Equation 2.26, we can get a linear dependence of the rotation angle on the
magnetic field
1
0 (D)
θ = µ0 cωLN αxy,z
Bz ,
2
(2.32)
which is the general form for Faraday rotation in nonmagnetized medium. The above
analysis assumes that the population in the sublevel set is subject to Boltzmann
distribution in thermal equilibrium. However, the population distribution in equilibrium state could be redistributed through some methods, e.g. polarization under high
magnetic field, polarization by optical pumping, by which the material is actually polarized or magnetized. This externally exerted polarization, as a second mechanism,
also induced Faraday rotation.
The typical value of Faraday rotation is about 1∼20µrad/(G·cm) for most common
transparent liquids, e.g. water, ethanol. By means of increasing magnetic field and
optical path, Faraday rotation is easy to detect, as shown in the Figure 2.8. The
magnetic field can be created by a signal generator, then the induced optical rotation
27
Linear
polarizer
Laser
Sample cell
B
(AC)
B
(AC)
Optical
rotation
polarizing
beam splitter
Photodiode
Faraday Rotation
Photodiode
Figure 2.8: Measurement scheme of Faraday rotation. The magnetic field is in alternative form(AC) driven by a signal generator or amplifier, then the optical rotation
is thereby an AC signal after transduced by photodiode detectors under balanced
operation and finally analyzed by a Lock-in amplifier.
takes the same oscillating frequency after passing through the medium. Following
with a polarization beam splitter that is place at balanced condition (i.e. split the
beam into two equal components), two separate components of the polarized beam
enter a corresponding photodiode detector, finally providing voltage signal for analysis
with help of a Lock-in amplifier. If the light beam passes through the sample cell and
reflects back through it, the rotation doubles. Furthermore, if we put this material
in an optical or multi-pass cavity, in which the light could reflected back and forth
for a lot of times, the rotation could be enhanced linearly by multiplying with the
pass number, while this mechanism also apply for many other magnetic-field induced
optical rotation phenomenon, e.g. Nuclear spin induced optical rotation as described
in this thesis.
2.3
Nuclear spin induced optical rotation
Nuclear spin optical rotation (NSOR) is a recently developed technique for detection of nuclear magnetic resonance via rotation of light polarization in a nuclear
spin-polarized medium. NSOR signals depend on hyperfine interactions with virtual
optical excitations, giving nuclear information as a local chemical environment. The
rotation of light polarization is similar to the Faraday effect caused by a nuclear
28
magnetic field but is enhanced by the hyperfine interaction between nuclear spins
and virtual electronic excitations. In some sense, NSOR is an enhanced version of
Faraday rotation, and it could reveal the strength of nucleus-electron interaction in
various atoms or molecules.
Under an external magnetic field, the nuclear magnetization in a liquid induces a
magnetic field BM that could lead to optical rotation θ = VN SOR LBM , which is proportional to the rotation constant VN SOR and the length of the sample L. Actually,
the induced magnetic field BM can be divided into a short-range field and a distant
dipolar field. The distant dipolar field is an averaged magnetic field over all magnetized nuclear spins, for example, a free electron inside the medium with cylindrical
symmetry can sense a magnetic field
Bd = µ0 M,
(2.33)
whereM is the macroscopic magnetization
M = N mN
(2.34)
Here, mN is the magnetic moment of nuclei in the medium, N is the polarized spin
number, and V is the sample volume. Therefore, in a nuclei-magnetized medium, the
regular Faraday rotation occurs due to the distant magnetic field Bd , i.e.
1
0 (D)
θd = µ0 cωLN αxy,z
Bd .
2
(2.35)
Besides the distant magnetic field inside the medium, a local short-range field
exists when the electron is close to a nuclear spin, especially for an orbital electron
in the vicinity of the nearby nucleus in the same atom or molecule. This short-range
interaction is also actually the intramolecular hyperfine interaction, which consists of
spin dipole-dipole interaction and contact interaction. This contact interaction can be
29
Distant interaction
N
Hyperfine interaction
e
e
N
B
B
B
(b)
(a)
Figure 2.9: (a) Distant magnetic field induced by an ensemble of nuclear spin in the
polarized medium. (b) Hyperfine interaction between an nucleus and orbital-electron
in the same atom or molecule.
enhanced or suppressed depending on the overlap of the wavefunction of the virtual
electron excitation created by the laser and the nuclear spin. The hyperfine interaction
is very sensitive depending on the electron configuration, e.g. the hyperfine interaction
of the hydrogen nuclei 1 H and valance electron in water is negligible, but for
129
Xe
it is considerably larger than the distant magnetic field. The distant interaction and
hyperfine interaction is illustrated in Figure 2.9, where the distant magnetic field is
induced by an ensemble of nuclear spin, while the intramolecular hyperfine interaction
occurs when the electron and nucleus is in the same atom or molecule.
The short-range interaction between the electron and nuclear spin is established
by means of the hyperfine coupling. For an electron of total angular momentum
J = L + S interacting with a nuclear spin I at a distance r, the corresponding
Hamiltonian is
Hhf
X
µ0 ∗
g µB ~
γj hIj i
=
4π
j
L
S
rj (S · rj ) 8
−
+3
+ πSδ(rj ) ,
rj3 rj3
rj5
3
30
(2.36)
where g ∗ is the effective electron g-factor in the system, Ij is the nuclear spin in the
molecule with its gyromagnetic ratio γi , and hIz i stands for the average nuclear spin
magnetization over the ensemble. For a s-orbital electron, the dominant contribution
come from the last term above, which is usually called Fermi contact interaction, since
the wavefunction density of s-orbital electron overlaps with the nucleus at the center.
Due to the fact that hyperfine interaction decays ∼
1
,
r3
only affiliated nuclei-electrons
in the same chemical bond are considered. In addition, because of the rapid tumbling
of molecule in gases and liquids, the intramolecular hyperfine interaction normally
averages out in gases and liquids. The hyperfine interaction, actually the magnetic
dipole interaction, could enhance the Faraday rotation as an addition to distant field.
This extra optical rotation induced by local nuclear spin (or hyperfine interaction) is
the key component of NSOR, which differs from regular Faraday rotation. Similar to
the expression for regular Faraday rotation in Equation 2.32, the hyperfine-interaction
induced an optical rotation
1
0 (N )
hIz i ,
θN = µ0 cωLN αxy,z
2
(2.37)
0 (N )
where αxy,z takes the same expression as Equation 2.31, but mpq
z is replaced by the
hyperfine interaction term αzpq ,
αzpq
X Lz Sz 3z (S · rj ) 8
µ0 ∗
q .
= γ g µB ~
p 3 − 3 +
+
πS
δ(r
)
z
j
5
4π
r
r
r
3
j
j
j
i
(2.38)
A simple illustration hyperfine interaction induced optical rotation induced based
on energy shift is shown in Figure 2.10. In transition 1 S0 →1 P1 , there are two
transitions with different polarization, σ − for mL = 0 → mL = −1, and σ + for
mL = 0 → mL = +1. If there is no energy shifts, there two transitions have the
same transition energy, thereby having the same index property. When the nuclei is
31
-3/2
= -1
=0
-1/2
1/2
= +1
3/2
F=3/2
F=1/2
=0
-1/2
1/2
F=1/2
Figure 2.10: (a) Transition 1 S0 →1 P0 without nuclear spin (I = 0). (b) Hyperfine
splittings due to the nuclear spin I = 1/2. (F = I + J. J(1 S0 ) = 0 and J(1 P1 ) = 1
). The hyperfine splittings induce circular birefringence according to polarizationselective transitions.
magnetic, e.g. I = 1/2, then the hyperfine splitting will shift the transition energy.
According to the hyperfine splitting ∆EHF S =
A
[F (F +1)−J(J +1))−I(I +1)],
2
the 1 P
level is split into two sets F = 1/2 and F = 3/2, with energy shifts from the original
1
P level. The transition mF (F = 1/2) = 1/2 → mF (F = 3/2) = 3/2 only absorbs
σ + light, and its energy is larger than mL (1 S) = 0 → mL (1 P ) = +1. On the other
hand, the σ − transition is split into two channel, mF (F = 1/2) = 1/2 → mF (F =
1/2) = −1/2 and mF (F = 1/2) = 1/2 → mF (F = 3/2) = −1/2. Especially, the
transition energy of mF (F = 1/2) = 1/2 → mF (F = 1/2) = −1/2 becomes smaller,
since F = 1/2 sublevel of 1 P is shifted down. The transitions mF (F = 1/2) = 1/2 →
mF (F = 3/2) = 3/2 and mF (F = 1/2) = 1/2 → mF (F = 1/2) = −1/2 induce
circular birefringence effect, which gives a simple explanation for NSOR.
The hyperfine interaction varies for different nuclear-electron configurations and
atom-molecules systems. Normally, the hyperfine interaction is dominated by Fermi
contact term. For heavier atoms, the contact term enhancement increases with the
atomic number Z. Since NSOR is attributed to both the distant dipolar magnetic field
32
Figure 2.11: Nuclear spin induced optical rotation and its detection.
and the short-range hyperfine interaction between electrons and magnetized nuclei
within short range, it is sensitive to electron-nucleus configuration, which is the basis
for distinguishing 1 H NMR in various organic liquids in this thesis.
The whole NSOR signal is a sum of short-distance hyperfine interaction induced
optical rotation θN and the regular Faraday rotation θd arising from the magnetic field
inside a magnetized medium. The two types of optical rotation θN and θd keep the
same phase when nuclear spins are precessing, while they could not be distinguished in
measurements actually. As to the measurement of NSOR, since its extrinsic behavior
is the same as Faraday rotation, its measurement apparatus also takes the same
form, as shown in Figure 2.11, except that a strong magnetic field should be used
to polarized nuclear spins. Another issue is the weak signal in NSOR, due to the
fact that polarization hIz i is normally quite small, e.g. it is only about 10−5 under
1 Tesla magnetic field, then the induced optical rotation angle is thereby as small
as 10−5 ∼ 10−3 of regular Faraday rotation. As a consequence, several techniques,
such as a long optical path in optical or multipass cavities, short wavelength laser,
near-resonance condition, should be implemented to aid the detection of NSOR.
33
Chapter 3
Experiments under low magnetic
fields
The first Nuclear-Spin-induced Optical Rotation (NSOR) experiment is based on lowfield CW spin-lock technique, first developed by Savukov et al. in Reference [25]. In
this paper, NSOR signal of 1 H water prepolarized by a high field magnet is shown to
be detectable under a low magnetic field, while the signal-to-noise ratio is about 4.5
after averaging 1000 s. However, in order to distinguish the same magnetic nucleus
1
H in different organic liquids, a higher signal-to-noise ratio should be required. In
this chapter, the multipass scheme and a 405 nm laser is implemented in the NSOR
measurement of some organic liquids, which enhances the 1 H NSOR signal-noise-ratio
to be about 16 after averaging 1000 s, then it is adequate to distinguish the NSOR
difference of 1 H in these organic liquids.
There are several NSOR signal improvement methods for choice. The first method
is using lasers with a shorter wavelength(λ) , since the optical rotation is generally
proportional to 1/λ2 when it is far from any optical resonances. For example, comparing with the 770 nm laser, the 405 nm laser could enhance the optical rotation
signal by a factor of 3.6 approximately. In addition, when the laser is close to any
34
optical resonance, although the light transmission will be much smaller, the rotation
constant will dramatically rising. In practice, the light absorption and transmission
should be considered as well. For most transparent pure liquids, i.e. Water, Ethanol
and Toluene, the electronic absorption lies in the UV region ranging from 150 nm to
350 nm, and their absorption linewidths are rather broad, as shown in Figure 3.1.
Therefore, the UV laser could not be used in the optical liquid NMR, due to the
serious absorption in most organic liquids. In the meanwhile, a lot of transparent (in
visible light range) chemicals have almost smallest absorption around 400 nm, hence
a 405 nm laser (this wavelength is commonly accessible in diode lasers) is capable of
passing through long optical path. For example, The absorption coefficient of water
at 405 nm α(λ) =6·10−5 cm−1 , according to the Beer-Lambert law
I
= e−α(λ)·L
I0
its half-attenuation length is L 1 '115m.
2
Figure 3.1: Optical absorption coefficients for water from reference. The lowest absorption region is located around 400nm, which is at the edge between UV and visible
light. (This figure is adapted from http://www.lsbu.ac.uk/water/vibrat.html)
35
In addition, the optical rotation angle is also proportional to the optical path:
θ =V ·L
where θ is the optical rotation angle, V is the rotation constant, and L is the
optical path inside the medium. Therefore, by simply increasing the optical path
length, the rotation signal is linear proportionally enhanced. There exists several
methods to increase optical path : the optical cavity, multipass cavity, long hollow
fiber. Due to the optical robustness, the multipass scheme is used in our NSOR
measurements under low magnetic fields, as well as Faraday rotation measurements.
The multipass scheme will be described in details in Section 3.1.2. The long-hollowfiber based method is also utilized under a high magnetic filed, which will be discussed
in Chapter 5.
In order to get a large NSOR signal, both a long optical path inside sample and
a high spin polarization are preferred. A high spin polarization requires a high field
magnet. Unlike traditional NMR based on a induction coil, NSOR experiment needs
the optical direction, i.e. the sample tube, be perpendicular to the magnetic field. As
far, due to technical limitations of the high field magnet, the effective space with a
homogenous field inside the magnet is quite small. For example, for a superconducting
magnet, the homogenous core size along the transverse direction is typically less than
10 centimeters; for a permanent magnet, the field homogeneousness is much worse. It
means that only a short sample tube could be directly used inside a high field magnet.
The prepolarization method could be used to aid NSOR experiments, which utilizes
a high field magnet (no matter how field homogeneous it is) to polarize the magnetic
nuclei in the liquid sample, and then transfer the polarized sample quickly to another
low magnetic field. By using some current-driven coils, a large area with homogenous
and low field could be created easily and flexibly, and therefore the arrangement of
a long sample tube become possible. In this chapter, the experiment is based on
36
this prepolarization method and the NSOR signal is just measured under a 5 gauss
magnetic field, which does not require any large-core superconducting magnet and
high frequency electronics.
3.1
Experiment setup
The detection of NSOR is based on a CW spin-lock technique [25], as shown in
Figure 3.2, the liquid is circulated continuously by a pump from a reservoir to a
0.85T permanent pre-polarizing magnet, and then to a 22.5 cm long sample glass
tube inside a uniform magnetic field B0 = 5 G generated by a rectangular Helmholtz
coil. As the liquid enters the region of the constant magnetic field B0 , an oscillating
magnetic field is applied at 22 kHz with an amplitude of B1 = 0.2 G perpendicular to
the B0 field, so the nuclear spins are adiabatically transferred to the rotating frame
at the NMR frequency. A 19.5 cm long solenoidal pick-up (induction) coil with 210
turns is wound around the sample tube to measure the traditional CW NMR signal
and determine the polarization for each flowing liquid. With a flow speed of about
50 mL/s we find that the NMR signal drops by less than 30% from the beginning to
the end of the tube, as determined by several small coils wound at different points
of the sample. For water the NMR signal gives a polarization of P = 1.2 × 10−6 ,
corresponding to a pre-polarizing field of 0.36 T, smaller than the field of the magnet
due to polarization loss during flow and inhomogeneous broadening of the NMR
resonance due to B0 magnetic field gradients.
The laser wavelength is chosen to be 405 nm, since NSOR is enhanced at short
wavelengths as 1/λ2 . As shown in Figure 1, the multipass cell consists of two cylindrical mirrors with a small hole in one of the mirrors to let the laser beam enter and
exit the cell. While optical cavities have been used in the past to amplify optical dispersion effects, multipass cells have not been widely used for this purpose, except for
37
Lock-in
Amplifier
0.5Hz
modulation
22KHz
DAQ
PD2
Current
Source
PD1
B0 Coil
PBS
NMR Coil
Laser
Diode
405nm
B0 Coil
B1 Coil
1.5T Magnet
Pump
Liquid
Repository
Figure 3.2: Apparatus for measurement of NSOR in organic liquids. The sample
tube, which is placed inside the multipass cell, is 22.5 cm long with 1.5 cm inner
diameter, while the bottle inside the permanent magnet has 200 mL volume. The
multipass cell consists of two cylindrical mirrors with 40 cm separation and 50 cm
radius of curvature. A 405 nm laser with vertical polarization is incident on the inlet
hole of the left cylindrical mirror. After 14 passes the laser beam exits the same hole
and its polarization is analyzed by a balanced polarimeter. The traditional NMR
signal is measured with a coil wound on the sample tube.
early work in Ref[Silver]. multipass cells have a number of advantages compared to
optical cavities. They do not require locking the frequency of the laser to the optical
cavity resonance and spatial mode matching of the input laser beam to the cavity
standing wave. They also do not require optimization of the power coupling into the
cavity and theoretically can have 100% power transmission. multipass cells with up to
500 light passes have been realized. The number of passes is determined solely by the
distance between the mirrors, their curvature, and the twist angle between their axes
of curvature. To reduce optical losses in our cell, the sample tube end windows have
38
an anti-reflection coating on the outside surfaces. While optical absorption length in
very pure liquids can exceed 100 meters at 405 nm, it is very sensitive to impurities.
Based on optical losses on the mirrors and sample tube windows we estimate ideal
optical transmission to be 94% per pass for very pure water, but in practice found
optical transmission to be about 80% per pass. We adjust the multipass cell to have
14 passes for a total optical path length of 3.15 meters. The number of passes is
determined by counting the number of beam spots on each mirror. In the measurements on water the initial laser intensity of 8 mW was reduced to 0.4 mW after the
multi- pass cell; the transmission was similar for other chemicals studied. Note that
the increase in the photon shot noise by a factor of 4.5 due to light absorption is still
less than the 14-fold increase in the optical rotation signal, demonstrating an increase
of the SNR with the multipass cell.
The optical rotation signal is measured with a balanced polarimeter, consisting
of a polarization beam splitter and two photodiode detectors. The difference of the
photodiode signals and the voltage across the pick-up coil are measured by two lockin amplifiers referenced to 22 kHz. In addition, we modulate the B0 field on and off
resonance at 0.5 Hz to distinguish NMR signals from any backgrounds. For static
liquids the optical rotation noise is limited by photon shot noise, but it increases
by about a factor of 2 during flow, likely due to small bubbles in the liquid. The
signal-to-noise ratio for water is typically about 15 after one hour, while it is larger
for other organic chemicals studied in this experiment.
3.1.1
Adiabatic passage transfer and CW spin-lock technique
When nuclei spins in the liquid are polarized in the permanent magnet, their polarization direction are aligned with the strong magnetic field. But when the liquid is
flowing away from the magnet, the magnetic field is different, which is a superposition of the increasing static magnetic field induced by B0 coil and the fast spatially
39
decaying field of the permanent magnet. Therefore, during the process in which the
liquid is transferred in to the sample tube (cell), the spin polarization direction also
changes. If there is no appropriate method to guide those spins’ precession in the long
transferring precess (takes about 0.1 second; ‘long’ is relative to the spin precession
period 0.05 ms), the spins usually lose their coherence due to the spatial field inhomogeneity, i.e. different spins have various precessing phase and thereby cancel out each
other’s signal. On the other hand, when the spin is precessing in the sample tube,
the maximal oscillating NMR signal is achieved only when the spin is precessing on
a plane containing the axis of induction coil. The method to resolve these two issues
is implementing a continuous-wave RF field B1 , which is perpendicular to the static
magnetic field B0 , as shown in Figure 3.2. Continuous RF field B1 not only help the
spins from the magnet to be adiabatically transferred to the static field B0 region
(also the sample tube region), but also continuously lock the precession of spins when
liquid is flowing inside the sample tube, which is called as continuous wave (CW)
NMR. In the following, the adiabatic passage transfer and CW spin-lock technique
will be described sequentially.
When the spins are transferred from the magnet to the B0 coil region, the superposition of field from the magnet and the field from B0 coil changes from 0.85 T to 5
G. At the same time, the RF field B1 is linearly oscillating at an angular frequency
Ω0 =B0 γ, which is the Larmor frequency only in the presence of field B0 (here γ is
the gyromagnetic ratio of 1 H), and its direction is perpendicular to B0 . This RF field
B1 could be decomposed into two oppositely rotating magnetic fields due to the fact
2B1 cos Ω0 t = B1 eiΩ0 t + B1 e−iΩ0 t . Only one of these two rotating fields has the same
direction as the proton spin precession under field B0 (here we name it as resonant
rotating field ), thereby plays the role in locking the proton spin, while the other one
(anti-resonant rotating field ) is has a reversed rotating direction and will not affect
the spin precession as a result. If we choose a rotating frame with angular frequency
40
(a) Liquid flow diagram.
(b) Spin adiabatic following.
Figure 3.3: (a) The diagram of liquid circulation . The liquid is transfer from a 0.85 T
permanent magnet to the B0 coil region (sample tube) through a plastic (PFA) tubing.
(b) The adiabatic passage following illustrated in the rotating frame of oscillating field
B1 . Axis x0 and z 0 are the coordinates in this rotating frame.
equal to the frequency Ω0 , i.e. this frame is ‘locked’ to the resonant rotating field,
→
−
→
−
then the total effective field in the rotating frame [46] is BE = (Ω0 /γ − Bi ) y 0 + B1 x0 ,
where Bi is the superposition of the field from magnet and the field from B0 coil,
which depends on the location along the liquid flow-in path (see Figure 3.3a), and
B1 is a static field in this rotating frame. As shown in Figure 3.3b, at time ta , when
the spin is inside the permanent magnet, Bi = Ba ∼ 0.85 T, which is much larger
than B1 (∼ 0.2 G), so the effective field BEa is almost equal to Ba , and the spin is
precessing around the direction of BEa . As time goes on from ta to tb and then tc , the
liquid is more and more away from the magnet but closer to the B0 (∼ 5 G) region,
so the z 0 component field is reduced gradually and the effective field BE is slowly
41
approaching B1 . Throughout this process, the spin are following the direction of the
effective magnetic field. At time td , when the spin enters the B0 coil region, the z 0
component field become Ω0 /γ − B0 = 0 , because the Larmor frequency Ω0 = γB0 ;
therefore, at this moment, the effective field is the same as B1 , and the spin is aligned
with B1 in the rotating frame, i.e. the spin is locked to the resonant rotating field B1 ,
while it is perpendicular to B0 and precessing around field B0 at Larmor frequency
Ω0 . As long as the change rate of magnetic field is slower than the spin precession
speed all through this process, the spin could continuously follow the direction of the
effective magnetic field, which is termed as adiabatic passage following [47].
After the liquid undergoes adiabatic passage following along the liquid flow path,
the spins enters the sample tube, in which the field B0 is homogeneous as 5 G. When
the liquid is flowing inside the sample tube, the spins are perpendicular to B0 and
precessing around field B0 at Larmor frequency Ω0 , and it is actually aligned with
the continuous-wave rotating field B1 , termed as CW spin-lock , which is used a
lot in traditional CW NMR experiments. In fact, the spin-lock technique also helps
spins keeps in coherence, because spins are locked to the B1 field with the same
phase. Otherwise, without this locking field, the transverse relaxation (decoherence)
originated from differential precession because of the magnetic field inhomogeneity
will attenuate the NMR signal quickly, although the longitude relaxation time could
be up to several seconds. From our experiment results, the spin-lock technique could
elongate the actual coherence time (T2 ) of spins to be more than one second. At the
same time, known from the speed of liquid, the spin will stay inside the sample tube
for about one second before the liquid flows outside the sample tube. During this
short time, the transverse relaxation could cause the NMR signal loss within 20%
from the measurement of NMR signal at different positions of the sample tube.
42
3.1.2
Multipass optical cavity
Long optical path could increase the NSOR signal proportionally, but simply prolonging the sample tube length is not practical. A simple method to elongate the optical
path without prolonging the sample tube is to use a multipass cavity [48, 49, 50], in
which the light beam could be reflected back and forth for many times. A multipass
optical cavity has several practical advantages for robust detection of optical rotation
compared with optical cavities that have been used in the past to elongate the optical
path.
Multipass optical cavities could be built based on various configuration, e.g. White
cell [48], Herriot spherical mirrors [49], astigmatic mirrors [51], cylindrical mirrors
[52, 50] and circular cavity [53, 54]. Multipass cavity could achieve very long optical
path lengths in a small volume and have been widely used for optical spectroscopy
[55, 56, 57, 58, 59, 53, 54] and atomic magnetometry [60, 61]. In our experiment, for
the convenience of measuring optical rotation signal in a cell containing liquids, we
adopt the cylindrical-mirror multipass cavity, which consists of two cylindrical mirrors
of equal focal lengths separated by some distance, as shown in Figure 3.4. A laser
beam is injected through the hole in one of the mirrors, then the beam is periodically
reflected between the mirrors and finally after some number of passes, exits through
the original input hole. The cylindrical mirrors could get high-density optical pattern,
with hundreds of optical spots, while the Herriott cavity with spherical mirrors is
limited by the number of spots [50]. The number of optical passes is determined solely
by the distance between the mirrors, their curvature, and the twist angle between
their axes of curvature. With fixed mirror curvature and distance, the adjustment
of the twist angle could render various optical patterns with different spot density.
Figure 3.5 adapted from Reference [50] shows the effect of twist angle on optical spot
pattern. Figure 3.6 shows two real images of optical patterns, 14 passes and 32 passes,
on one mirror in our experiment, where the 14 passes patten is used for the NSOR
43
and Faraday rotation measurement. In the 14-pass pattern, there are seven clear light
spots on one mirror, so it is a 14-pass pattern by doubling the spots number, due to
the fact that the laser beam is reflected back and forth between two mirrors. In the
32-pass pattern, sixteen light spots are displayed, so it s a 32-pass pattern. These two
multipass pattern are realized by using two cylindrical mirrors with 40 cm separation
and 50 cm radius of curvature, while their twist angles are different.
Figure 3.4: A multipass cavity is composed of two cylindrical mirror. One mirror
has an on-axis input hole for laser beam. Those two mirrors could be twisted for
some angle with respect to each other, rendering various multipass optical patterns.
Adapted from [50].
The sample tube is placed between these two mirrors, benefiting from the elongated optical path when the laser beam is reflected back and forth in the multipass
cavity. The combination of multipass cavity and the sample tube is illustrated in Figure 3.7. While optical absorption length in very pure liquids can exceed 100 meters
at 405 nm, it is very sensitive to impurities and mirror reflection coefficients. Based
on optical losses on the mirrors and sample tube windows we estimate ideal optical
transmission to be 94% per pass for very pure water, but in practice found optical
transmission to be about 80% per pass. Although high multipass numbers more than
40 could be obtained easily, due to the considerable optical reflection loss and liquid
impurity loss in the current apparatus, the optical transmission is roughly attenuated
44
Figure 3.5: (a) Spot pattern for 26 passes with a 90 deg crossed cylindrical–cylindrical
mirror cell at d/f=0.88. (b) Dense spot pattern of 122 passes created by rotating front
mirror by 9 deg. Adapted from [50].
exponentially according to the pass-number. Higher initial laser intensity (∼100mW)
is feasible, but the optical rotation signal is only proportional to the square root
√
of optical transmission intensity I, which can not compensate the large-numbermultipass optical loss. In addition, the inherent current noise from the diode laser
increases for higher intensity light output. Therefore, in our experiment, we adjust
the multipass cell to have 14 passes as a optimal configuration, determined by counting the number of beam spots on each mirror, with a total optical path length of 3.15
meters. In the measurements on water the initial laser intensity of 8 mW was reduced
to 0.4 mW after the multipass cell due to scattering by impurities; the transmission
was similar for other chemicals studied.
45
(a) 14-pass pattern
(b) 32-pass pattern
Figure 3.6: Two multipass optical patterns. a) 14-pass pattern. As shown, there
are seven clear light spots on one mirror, so it is a 14-pass pattern by doubling the
spots number, due to the fact that the laser beam is reflected back and forth between
two mirrors. b) 32-pass pattern. Sixteen light spots are displayed, so it s a 32-pass
pattern. These two multipass pattern are realized by using two cylindrical mirrors
with 40 cm separation and 50 cm radius of curvature, while their twist angles are
different.
3.1.3
Cell configuration
The cell is a sample tube with two windows at two ends for laser beam passing
through, and liquid samples could flow inside it. As shown in Figure 3.8, the cell
body is a cylindrical tube made of pyrex glass, two glass pipes are connected to it as
two inlets for liquid flowing. Two quartz windows with single-side anti-reflection (AR)
coating are cemented to the left and right ends of the pyrex tube. This type of windows
is actually a beam sampler (BSF10-A, Thorlabs) that has one-side anti-reflection
coating (the side of coating is facing outside of the cell), which could effectively reduce
the light reflection loss on both the air-to-coated-glass and glass-to-liquid interfaces
as the light needs to pass through the window before entering the liquid-filled cell
body, because of the relatively small difference of refractive index between air and
46
Figure 3.7: Illustration of a multipass cavity based on a pair of cylindrical mirrors.
The light-purple cylinder is the cell body filled with liquid. Two cylindrical mirrors
with anti-reflection coating on surface are placed outside this cell. The purple lines
illustrate the light beams which is reflected back and forth between two cylindrical
mirrors.
AR coated glass and between glass and liquid (e.g. water). The spacing between the
inlet and the window is as small as possible, in order to reduce dead volume of flow.
Apparently, a larger cell diameter could provide a larger sample volume, which could
enhance the traditional NMR signal sensed by the induction coil; A large diameter
also aids the setup and adjustment of multipass cavity, since the freedom of laser
beam paths is limited by the cell body in practice, although the multipass cavity is
large enough to realize a high freedom of multipass optical arrangement. However,
considering that the relaxation time of spin polarization in water and other organic
liquids is about 1∼5 seconds, the diameter should be controlled within some range
so that the flow could only stay inside the cell body for an average of one second. In
addition, a smaller cell diameter benefits from homogeneous magnetic field.
In this experiment, the cell is manufactured shown in Figure 3.9, the effective
length of cell is about 20 cm (the liquid turbulence partially mixes polarized and
unpolarized liquid in the dead volume), where the liquid could continuously refreshed.
The NMR induction coil (blue) is wound tightly around the cell body, consisting of 210
turns of copper wire with insulation coatings. Besides the main coil, five short coils
47
Quantity
Cell whole length(L)
22.50cm
Coil whole length
19.50cm
Coil Number(N)
210
Cell inner Diameter
1.50cm
Cell inner cross section(S)
1.77cm2
B0
∼ 5G
Permanent magnet B 0
0.85T
Liquid temperature
∼ 300K
Liquid flow speed
<0.8gpm (60mL/s)
Table 3.1: Parameters of this experiment about cell, coil and flow speed
(Red) are placed at different positions across the cell, which is used to monitor the
polarization decay by measuring the NMR signal at each position. The parameters
of this experiment about cell, coil and flow speed is listed in Table 3.1.
Figure 3.8: Picture of cell (the sample tube). The red-brown layer is the coil. The cell
body is a cylindrical tube made of pyrex glass, two glass pipes are connected to it for
liquid flowing. Two quartz windows with single-side coating are cemented to the left
and right ends of the pyrex tube. This type of windows is actually a beam sampler
purchased from Thorlabs. Since this window only has one-side anti-reflection coating,
it could be used to reduce the light reflection loss between both the liquid-to-glass
and glass-to-air interface.
48
6.4cm
11.3cm
15.0cm
19.2cm
1
2
3
4
5
1.5cm
1.8cm
0.0cm
0
20 turns
6
210 turns (single layer)
1.5cm
19.5cm
1.5cm
Figure 3.9: Drawing of the cell (the sample tube). The effective length is 19.5cm
(exclude dead volumes between the liquid inlet and window at both ends), where the
liquid could continuously refreshed. The NMR induction coil (blue) is wound tightly
around the cell body, consisting of 210 turns of copper wire with insulation coating.
Besides the main coil, five short coils 1-5 (Red) are placed at different positions across
the cell, which is used to monitor the polarization decay by measuring the NMR signal
at each position. Two extra coil mounted 0 and 6 at the ends of the tube (almost on
the window in practice) is used to estimate effective polarization in these dead zones,
as liquid could not flow into these region fluently, but still have some polarization due
to liquid turbulence.
3.1.4
Coil arrangement
A homogeneous magnetic field B0 is crucial for the NMR and therefore the NSOR
measurement, due to the fact that NMR linewidth is limited by the field homogeneousness . In Figure 3.10, when the cell body is placed in the central region horizontally,
the field of B0 is produced by two identical rectangle coils (Red in front and back)
with the same current direction. This coil configuration is called rectangle Helmholtz
coil, usually used to create a region of nearly uniform magnetic field. A stable current source is provided to driven the B0 coil, and the current is about 1.7 A, which
corresponds B 0 =5 G. This current is scanned around resonance while the frequency
of B1 field is fixed. The B1 field is created by another pair of coils, which is shown
as a up-down pair of rectangle coils in Helmholtz configuration. The field intensity
of B1 depends on the uniformness of B0 , it should be at least twice as large as the
deviation of B0 field, in order to achieve a sufficient spin-lock capability. B1 is driven
by an alternative current source I1 , which is an AC power amplifier. Due to the fact
49
that I1 is alternative at frequency ∼22kHz and the B1 coil is inductive, a capacitor
with 0.1µF is connected in series with the coil and current source to form a resonance
circuit with a minimal impedance equal to the resistance of the coil, so that the power
is consumed by the coil with maximal efficiency.
The magnetic field is calculated based on the Biot-Savart law
µ0
B=
4π
Z
C
→
− −
Id l × →
r
,
3
|r|
→
−
where I is the current, l is the coil piece expressed in vector form, r is distance
from the coil piece to the space point where the magnetic field B needs to be calcu−
lated, →
r is the unit vector from from the coil piece to the space point. The optimal
distance between the coil pair is estimated from calculation, and in practice it is then
finely tuned according to the NMR signal measurement. As shown in Figure 3.11,
the B0 and B0 field distribution on two orthogonal cross-sections on the horizontal
direction of the cell. The maximal relative deviation ((Bmax − Bmin )/Bavg ) of B0 is
2%. Actually, the B1 intensity could not be too large, since in practice B1 could be
considered as a deviation of B0 and then the superposition of B0 deviation and B1
contributes to the linewidth of NMR signal. Therefore, B1 is just controlled around
twice as large as the deviation of B0 field. The used magnetic field of B0 is 5.3 G and
B1 is about 0.2 G, which is consistent with the calculated results.
3.1.5
Polarimetry
Polarimetry is used to measure the polarization of polarized light beams. In a polarimeter, a sample cell is placed between a linear polarizer and polarizing beam
splitter (PBS), as shown in Figure 3.12. When the plane of polarized light has a 45◦
with respect to the axis of PBS, which could be easily achieved with aid of a half
(λ/2) wave plate as shown in Figure 3.12, then the polarized beam is split into two
50
Y
X
Z
X = 63.0cm
Y = 7.5cm
Z = 5.0cm
B0~5G
X = 32.2cm
Y = 11.2cm
Z = 22.2cm
B1~0.2G
Figure 3.10: Configurations of B0 and B1 coil. The red coil produces B0 field with
intensity of about 5 gauss, and the blue coil produce the RF field B1 with magnitude
of about 0.2 gauss. Both of them are rectangular Helmholtz coil, thus achieving
homogenous magnetic field in the central region. The cell(22.5cm long) is placed
horizontally at the center region of this coil configuration.
linearly polarized beams that have orthogonal polarization plane to each other. At
this moment, the polarimetry is called at balanced condition.
The polarimetry is first balanced according to the initial polarization plane of the
linearly polarized light without optical rotation, e.g. turning off magnetic field in
Faraday rotation measurements. So the initial intensities of the two output beams is
equal and a half of the whole intensity (I0 ), as given by
π
I0
π
I0 sin2 ( ) = I0 cos2 ( ) = .
4
4
2
(3.1)
When there is an optical rotation induced by the medium via Faraday rotation or
NSOR, after exiting from the sample cell, the beam is split into two separate beams
with individual intensities, given by
I1 = I0 sin2
2
I2 = I0 cos
51
π
4
π
4
−θ
−θ
,
(3.2)
10
10
5.22
10
10
5
5
5
−5
5.20
5.19
0
−5
X [cm]
0
X [cm]
X [cm]
5
5.18
5.17
0
X [cm]
5.21
−1 0 1
Y [cm]
−10
−1 0 1
Z [cm]
5.15
0
0.262
−5
−5
−10
−10
[Gauss]
−1 0 1
Y [cm]
Δ<2%
(a) B0
0.266
0.264
5.16
−10
0.268
−1 0 1
Z [cm]
0.260
0.258
[Gauss]
Δ<5%
(b) B1
Figure 3.11: Calculated B0 and B0 field distribution on two orthogonal cross-sections
on the horizontal direction of the cell. The homogeneity of B0 field is crucial for
the experiment, and the intensity of B1 field should be at least twice as large as the
deviation of B0 field. The maximal relative deviation of B0 is 2%. The measured
magnetic field of B0 is 5.3 G and B1 is about 0.2 G, which is consistent with the
calculated results.
where θ is the rotation angle induced by the medium. In normal condition, Faraday
rotation is rather small, typically ∼10µrad/(G · cm) for organic liquids, while Nuclear
spin induced rotation angle is even smaller because the nuclear spin polarization is
about 10−6 -10−5 under a 1 T magnetic field at room temperature. Thus, we can make
an approximation θ ∼ 0, and thereby lead to
I1 = I0 [1 + sin (2θ)]/2 ∼ I0 (1 + 2θ) /2,
(3.3)
I2 = I0 [1 − sin (2θ)]/2 ∼ I0 (1 − 2θ) /2,
finally giving the expression of the small angle θ,
θ≈
I1 − I2
2(I1 + I2 )
52
(3.4)
Linear λ/2
polarizer plate
Sample cell
polarizing
beam splitter
Photodiode
405nm
Laser
Photodiode
Figure 3.12: Polarimetry for the analysis of polarization rotation angle
In experiments, the optical intensity I1 or I2 is measured as voltage signals V1 and V2 ,
as photons of light are converted to electric currents in photodiodes because of the
photoelectric effect (see below in Figure 5.13), and the electric current is further linearly amplified as voltage signal, Vi =Ri ·Ii , after conversion by a photodiode amplifier
circuits. Therefore, the optical rotation angle is also expressed as
θ≈
V1 − V2
2(V1 + V2 )
(3.5)
where, V1 and V2 is the voltage value from outputs of two channels in a photodiode
amplifier circuit. Here, V1 + V2 = V0 is a fixed value, due to the fact that the sum
of light intensity in two channels is conserved, I1 + I2 = I0 , no matter how the light
beam is split.
The detection of optical signals is carried out by two photodiodes (S9195, Hamamatsu) followed by a two-channel photodiode amplifier circuit, as illustrated in Figure
5.13. Two low-noise operational amplifier (Op Amp) in inverting configuration are
used to convert photoelectric currents from photodiode into voltage signal, according
to
Vf = ie · Rf ,
53
(3.6)
where ie is the photoelectric current and Rf is the feedback resistance, shown as R1
and R2 in Figure 5.13. The feedback resistors (also called amplifier resistor here)
R1 and R2 are set to be the same, so the output amplification factors of these two
photodiode amplifiers are the same. When the polarimetry is placed at balanced
condition, the output voltage V1 ≈V2 , providing the optical rotation signal is very
small (typically 0.1∼10 nrad in this experiment), therefore the balanced condition
could be monitored according to the readings of V1 and V2 . The feedback capacitor
is used to control the frequency bandwidth and achieve a low noise. Because the
frequency of optical rotation signal is about 20 kHz, it is easy to operate the circuit
near shot-noise level. A subsequent differential circuit further gives the differential
signal V1 − V2 , when Ra = Rb .
There are two types of noise source in the measurement of optical rotation. One
p
is called Johnson noise (i.e. thermal noise), nJ = 4kT · Rf , which always exists
as long as any resistance is used in the detection circuits, as well as existing in the
induction coil for traditional NMR detection. The other type of noise is called shot
noise, which is due to the fact that the photon is received as individual particles, so
it is very common in optical detection; In the photodiode amplifier circuit that is
p
used in this experiment, the shot noise is determined by nshot = 2e · Rf · V0 . In this
experiment at low magnetic fields, the shot noise dominates, as Rf (tens of kΩ) and
V0 (∼10 V) is relatively large.
3.2
NMR signal calculation, measurement and calibration
The traditional NMR signal is measured by the induction coil that is wound around
the cell body. When the nuclear spins are periodically precessing around B0 field
direction, the nuclear magnetization creates an alternative magnetic field that is per54
Figure 3.13: The two-channel photodiode amplifier circuit with differential output.
pendicular to the induction coil, so the nuclear spin precession signal is revealed by
the inductive voltage on the induction coil. As shown in Figure 3.2, the alternative
voltage signal from the induction coil is then analyzed by a lock-in amplifier, which
is connected to the same alternative current source for B1 with the same frequency
as a reference source.
Though the field of permanent magnet is known and the liquid is sufficiently polarized inside it, the polarization suffers longitude and transverse decay not only when
it is transferred from the magnet to the sample tube but also when it is flowing inside
the sample tube. Therefore, the actual polarization is smaller than the estimation
from the permanent magnet’s field. Here, the measurement of traditional NMR signal
is used to calculate the spin polarization as a indirect method, since the spin polarization determines NMR signal as well as NSOR signal proportionally. In this part,
the NMR signal calculation, measurement and calibration are described.
55
3.2.1
Theoretical calculation
When the liquid is inside the permanent magnet for sufficiently long time, the spin is
polarized. The energy difference between the spin states is very small in comparison
with the average thermal energy, from the calculation of ∆E = ~ω 0 , where ω 0 is the
Larmor angular frequency of 1 H spin,
ω 0 = −γ · B 0
(3.7)
= 267.52 · 106 rad · s−1 · T −1 × 0.85T
(3.8)
= 2.01 · 108 rad · s−1
(3.9)
This means that the polarization degree is very small at room temperature even the
sample is placed inside a superconducting magnet. the spin polarization obey the
Boltzmann distribution,
ρ=
exp(−~ωr /kB T )
Σα,β exp(−~ωi /kB T )
(3.10)
where ωα = 21 ω 0 (spin up), ωβ = − 12 ω 0 (spin down) and ωr = ωα(β) .
Around temperature T ' 300 K 1 ,
r
with 1, then exp(± k~ω
)'1±
BT
~ωr
,
kB T
~ωr
kB T
= 2.56·10−6 . Since
~ωr
kB T
is small comparing
and thus
1
+ 1.28 · 10−6
2
1
ρβ = − 1.28 · 10−6
2
ρα =
(3.11)
(3.12)
in other words, the polarization difference ρ∆ = ρα − ρβ = 2.56 · 10−6 .
1
Actually, in our experiment, due to the friction of fast flowing liquid, the liquid could be heated
up to 40-55 ◦ C
56
In one liter of water, there are 55.56 mol water molecules, i.e. 2 × 55.56 mol 1 H
spins, hence the polarized spin density is
dpol =
N
= 55.56mol/L × 2 × ρ∆
V
= 0.284mol/m3
(3.13)
(3.14)
The macroscopic magnetization is determined by
M=
N
m
V H
(3.15)
where N/V is the density of unit magnetic moment mH of 1 H. (mH = gS µB S/~ =
14.106 · 10−27 J/T )
So
M=
N
m
V H
(3.16)
= 0.568mol/m3 × 14.106 · 10−27 J/T
(3.17)
= 2.41 · 10−3 J/(T · m3 )
(3.18)
and its induced magnetic field is
Bm = µ 0 M
(3.19)
(SI units)
= 1.2566 · 10−6 H/m × 2.41 · 10−3 J/(T · m3 )
(3.20)
= 3.03 · 10−9 T
(3.21)
From the Lenz‘s induction law,
57
∆Φ
∆t
(3.22)
B · dS
(3.23)
V =N
where the magnetic flux
ZZ
Φ=
so assuming there is no polarization loss, the maximal induced voltage on the induction coil should be (N is loops of coils)
Vmax = N · Bm · S · 2π · f0
(3.24)
= 210 × 3.03 · 10−9 T × 1.767 · 10−4 m2 × 2π · f0
(3.25)
= 16.0µV (when f0 = 22650Hz)
(3.26)
The above calculation is applied for water. In order to calculate other chemicals,
it is more convenient to take in a ratio β = Vmax /dspin = 0.288 V /M , which is a
constant for 1 H NMR chemicals at the same temperature, where dspin is the spin
density of hydrogen nuclei with unit M (e.g. dspin = 111.1M of water).
Actually, from the measurement, the NMR signal of water is 5.45µVrms by using
a Lock-in amplifier, i.e. the voltage amplitude is 7.71µVamp . Hence, the effective
polarization magnetic field is 0.85T ·
7.71µVamp
16.0µVamp
= 0.36T , which means the effective
magnetic field is just 42% comparing with the maximal magnetic field.
3.2.2
NMR signal calibration
For a infinitely long cylinder fulfilled with uniform magnetized material, its magnetic
field distribution along the axis of the cylinder is the same. But for a magnetized
cylinder with a finite length, the field distribution is not uniform at the ends of the
cylinder. First of all, suppose the liquid inside the cell is uniformly magnetized, the
58
L=22.5cm
1
Magnetic field (Normalized)
0.9
0.8
0.7
0.6
0.5
0.4
Integ[1.5, 21.0]=19.37
Correct factor=0.993
0.3
0.2
0.1
0
0
5
10
Position [cm]
15
20
Figure 3.14: Distribution of the magnetic field induced by the nuclear magnetization
along the sample tube. The red color part is the region that the induction coil spans.
calculated magnetic field distribution is shown in Figure 3.14, which shows that the
magnetic field induced at the ends is smaller than at the central region of the cell.
In previous section, the calculation is executed assuming the whole field is uniform
and equal to the field value at the central region, so now the inducted magnetic
field should also be calibrated according to this field distribution. Since the inducted
voltage signal is the sum of all coil turns across the whole sample tube, hence the
calibration could be done by comparing with the integration of magnetic field along
the coil region and the measured inducted voltage. From calculation, the calibration
factor of NMR signal is γc = 0.993, which could be almost ignored.
3.2.3
Polarization decay
From the measurement, the time decay of polarization inside the sample tube exists,
and it is approximately the same for all (1 H) chemicals concerned in this experiment.
Figure 3.15 shows the inducted voltage signal that is sensed by short pick-up coils
59
at five equidistant positions (from position 1 to 5, drawn in Figure 3.9) across the
long induction coil. The liquid flows from position 1 to 5, so the inducted voltage
signal decay at different position in the sample tube exhibits the polarization decay
in time. Position 0 and 6 is located at the ends of the sample tube (almost on the
window), each of which is a dead volume for flowing liquids, hence its induced signal is
significantly smaller because the polarized liquid could not be refreshed sufficiently in
time. The data at the first position (position 1 in Figure 3.9) is smaller than expected,
since the magnetic field induced at the end is smaller than that at the central region
of the magnetized cylinder. However, the data at position 5 exhibits relatively large
signal than expected, it is might because of relatively large polarization at the dead
zone at the right end comparing with left dead zone, as liquid is flowing from 1 to 5
and liquid turbulence is strong at the right end.
3.2.4
Measurement of NMR
In the measurement of NMR signal, the RF field is kept on with fixed frequency
at 22kHz, while the B0 field at 5G is periodically turned on and off in order to
distinguish the NMR signal from other electronic cross talk signal. In addition, since
in practice the B1 field always have some small component along the induction coil
though B1 is designed to be perpendicular to the induction coil, this undesired B1
component causes some Faraday rotation with the same frequency as NSOR signal
but it is much larger. Because Faraday rotation caused by B1 is not affected by static
or slow switching B0 field, the periodically switch of B0 field just marks the NSOR
signal with this switch frequency and distinguish it from the Faraday rotation caused
by B1 .
In the experiment, B0 field is modulated on and off at 0.5 Hz. When B0 is turned
on, the NMR signal reaches its maximal value; When B0 is turned off, there is almost
no induction signal, because the spin-lock condition is immediately violated and then
60
0.12
SNMR/Average NMR
0.1
0.08
0.06
0.04
0.02
0
0
1
2
3
Position
4
5
6
Figure 3.15: Polarization decay of 1 H in water. Each short coil has 20 turns, about
10% of the turns of the long coil across the cell (210turns). The x axis represents the
positions of the short coil. The y axis represents NMR signal of each short coil as a
ratio to the signal of the long coil. The liquid flows from position 1 to 5 fluently (see
Figure 3.9). The position 0 and 6 is located at the ends of the sample tube, each of
which is a dead volume for flowing liquids. Except that the voltage signal at position
1 due to the fact that the magnetic field induced at the end is smaller than that at
the central region of the magnetized cylinder, the signal decay from position 2 to 5.
the spin is not kept precessing around the direction perpendicular to the induction
coil. Therefore, as shown in Figure 3.16, when the driven current I0 for B0 is chosen to
be 1670 mA so that B1 field is on resonance with the Larmor frequency, the inducted
NMR signal looks like a 0.5 Hz square wave because of the simple 0.5 Hz modulation
of B0 . Since the NMR signal is in a square wave form, its FFT spectrum in the
frequency domain becomes a series of peaks located at 0.5 Hz, 1.5 Hz, 2.5 Hz and
so on. For the spectrum of a strict square wave signal, the frequency component at
0.5 Hz is largest, and it is exactly 3 times as large as the component at 1.5 Hz. The
measured spectrum at 0.5 Hz and 1.5 Hz also gives a ratio of almost 3, which is well
consistent with the result for a strict square wave signal.
61
When B0 is not set for resonance, for example, I0 =1780 mA , as shown in Figure
3.17, the NMR signal is not in square wave form although B0 is still modulated on and
off at 0.5 Hz. The NMR signal decays both when B0 is on and off, which means that
the B1 does not lock the spin well when B0 is on, and the spin still have some coherent
precession after B0 is turned off, which is very different from the resonant case. The
difference of peak of bottom signal could identify its equivalent maximal NMR signal
amplitude. The maximal difference is about 1.5µV, while in the resonance case it is
about 5µV from Figure 3.16. The FFT analysis in the frequency domain could further
distinguish these two cases. In the frequency spectrum, the nonresonant NMR signal
has some magnitude of 2nd harmonics at 1 Hz, which is considerable comparing with
the 3rd harmonics at 1.5 Hz, while the 2nd harmonics is negligible in the resonant
case.
By scanning the magnitude of I0 , a whole curve of NMR signal vs I0 could be
displayed, which is used to calculate the linewidth of NMR signal, and then identify
how inhomogeneous the magnetic field is. The data (red dot-line curve) in Figure
3.18 shows the NMR signal component at 0.5 Hz in the frequency spectrum. This red
curve is generally in agreement with Lorentzian function [47] under near resonance
condition, and it is used to calculate the linewidth according to fitting, and then
identify the homogeneousness of the magnetic field. The linewidth could be simply
obtained by measuring the frequency difference of the two points at which their signal
is equal to half of its maximal signal. In this figure, the linewidth is about 150 mA,
corresponding to a linewidth at frequency domain ∆f=1.9 kHz.
The NMR signal and linewidth rely on the magnitude of B1 . The measured NMR
peak signal and linewidth is plotted vs RF voltage amplitude in Figure 3.19, which
shows the dependence of NMR signal and linewidth on B1 , because B1 field is linearly
proportional to the RF voltage Vrf . The linewidth of NMR signal could be estimated
62
Signal (µV)
−60
−65
−70
−75
0
2
FFT/µV
3
4
6
Time (s)
NMR spectrum X: T=110s
8
10
2
1
0
0.5
1
1.5
Frequency/Hz
2
Figure 3.16: NMR signal in time and frequency domain at I0 =1670mA (corresponding
to B0 =5.3 G), at which B1 field is on resonance with the Larmor frequency that is
caused by B0 . The magnitude of B1 is 0.2 G.
by the square sum of B0 field deviation and B1 intensity, due to the fact that B0 and
B1 is perpendicular to each other and B1 could be considered as a deviation of B0 ,
r
∆ω =
γ
( B1 )2 + (γB0 )2 .
2
(3.27)
This expression is roughly demonstrated in Figure 3.19b. Because B0 (∼5 G) is much
larger than B1 , the linewidth could only show slightly quadratic dependence on B1
(Vrf ).
The signal-to-noise (S/N) ratio of traditional NMR signal is quite good. As shown
in Figure 3.20, the S/N ratio is 470 after a 100-seconds measurement. Because the
noise is attributed to the thermal noise and other similar to white noise, S/N is propor63
Signal (µV)
−64
−65
−66
−67
0
2
FFT/µV
1
4
6
Time (s)
NMR spectrum X: T=102s
8
10
0.5
0
0.5
1
1.5
Frequency/Hz
2
Figure 3.17: NMR signal in time and frequency domain at I0 =1780 mA (corresponding to B0 =5.65 G). It is in nonresonant condition. The magnitude of B1 is 0.2 G.
tional to the square root of measurement time
√
√
T . The noise level is 0.07µV/ Hz,
which is very small comparing with the signal peak at 0.5 Hz and 1.5 Hz (the 3rd
harmonic). This high S/N ratio is due to the fact that sample volume is quite large,
although the spin polarization is very small.
3.3
Optical rotation constant calculation
The NSOR angle is proportional to the optical path, spin polarization and spin density. The NSOR rotation constant VN SOR is subjected to
VN SOR · N · L · ρ∗∆ · dspin = θrot ,
64
(3.28)
NMR signal spectra (0.5 Hz Modulation)
Amplitude (µV)
3
Data
Lorentzian
fitting
2
1
0
1000
1200
1400
1600
1800
Current (mA)
2000
2200
2400
Figure 3.18: NMR signal at different supply current I0 . B0 is driven by I0 , and B0 is
proportional to I0 .
300
3.5
250
3
Half Hight width [mA]
NMR signal [µV](θ=−52.0o)
4
2.5
2
150
100
1.5
1
0
200
0.5
1
1.5
Vrf [V] (rms)
2
2.5
3
(a) NMR signal vs RF voltage Vrf .
50
0.5
Linewidth
1
1.5
2
RF Voltage(rms) [V]
2.5
(b) NMR signal linewidth vs RF voltage Vrf .
Figure 3.19: Dependence of NMR signal and linewidth on RF voltage. B1 field is
driven by the RF source and its magnitude is linearly proportional to the RF voltage
Vrf .
65
3
NMR spectrum X: T=100s
4
NMR signal X= 5.16 µV
(0.50, 3.29) (1.50, 1.10)
3.5
S/ N = 470
FFT/µV
3
2.5
2
1.5
1
0.5
0
0.4
0.6
0.8
1
1.2
Frequency/Hz
1.4
1.6
1.8
Figure 3.20: NMR signal-to-noise (S/N) ratio. The S/N ratio is 470 after a 100seconds
to the square root of measurement
√ measurement. This ratio is proportional
√
time T . The noise level is 0.07µV/ Hz, which is very small comparing with the
signal peak at 0.5 Hz and 1.5 Hz.
where N is the multipass number, L is the effective length of a single-pass optical path
(equal to the effective length of the sample tube), dspin is the nuclear spin density,
andρ∗∆ is the actual polarization, which could be calculate from measured NMR signal
indirectly,
ρ∗∆ =
Vnmr /γc
· ρ∆
Vmax
(3.29)
where Vnmr is the measured NMR induction signal, Vmax is the theoretical maximal
NMR signal, γc is the calibration constant, ρ∆ is the polarization assuming there is
no polarization decay after the liquid is polarized inside the permanent magnet. As
a whole, NSOR constant VN SOR could be expressed based on the measured optical
rotation angle θ and NMR induced signal Vnmr :
66
NMR spectrum X: T=12000s
4
NMR signal X= 4.64 µV
(0.50, 2.95) (1.50, 0.95)
2
1
0
0.5
0
1
1.5
Frequency/Hz
X Rotation spectrum
FFT/µV
FFT/µV
0.1
0.2
0.1
0.5
1
Frequency/Hz
0
1.5
0.5
1
Frequency(Hz)
φ Y (10 − 9rad)
(10 − 9rad)
0.5
1
Frequency/Hz
1.5
15
X Rotation angle = 1.09 nrad
(0.50, 0.70) (1.50, 0.69)
<Noise> = 0.52 nrad
φ
X
15
0
1
1.5
Frequency/Hz
Y Rotation spectrum
√
!Noise "=1.2µ V/ Hz
0.3
!Noise "=1.2µ V/ Hz
0.2
5
0.5
0.4
√
0.3
10
2
1
0.4
0
NMR signal Y= 0.09 µV
(0.50, 0.06) (1.50, 0.02)
3
FFT/µV
3
FFT/µV
NMR spectrum Y
4
10
5
0
1.5
Y Rotation angle = 23.45 nrad
(0.50, 14.93) (1.50, 4.90)
<Noise> = 0.52 nrad
0.5
1
Frequency(Hz)
1.5
Figure 3.21: Comparison of NMR and Optical rotation signal spectrum for water. The
lock-in phase is adjusted that the Y component of NMR signal is vanished. Subfigure
a) and b) is the X and Y component of NMR signal. c) and d) is the optical rotation
spectrum expressed in direct voltage signal µV, while e) and f) is the optical rotation
spectrum after the signal is converted to rotation angle unit rad.
VN SOR =
θrot
Vmax
γc
×
×
Vnmr
dspin
N · L · ρ∆
(3.30)
Figure 3.21 shows the comparison of NMR and optical rotation signal spectrum
of 1 H in water. The lock-in phase is adjusted that the Y component of NMR signal
is vanished. Subfigure a) and b) is the X and Y component of NMR signal. c) and
67
d) is the optical rotation spectrum expressed in direct voltage signal µV, while e)
and f) is the optical rotation spectrum after the signal is converted to rotation angle
unit rad. From the figure, we can see that the noise signal-to-noise ratio in optical
rotation spectrum is much smaller than the traditional NMR spectrum. In addition,
the phase of optical rotation and NMR spectrum has a 90◦ phase difference. Because
the traditional NMR detection is based on induction law, so it will produce a 90◦
phase delay, while the optical rotation is has the same phase as the RF field.
The feedback resistor that is used in the photodiode operational amplifier circuit
(see Figure 5.13) is of 93.4 KΩ, and the sum of balanced DC output voltage is 10.4 V,
√
at temperature T=300 K, the shot noise is 0.6 µV/ Hz and the Johnson noise is 2
√
nV/ Hz, so the shot noise dominates in this measurement. The measured noise level
√
of the optical rotation spectrum is about 1.2 µV/ Hz, which is about twice as large
as the shot-noise level. The extra noise is attributed to the optical path fluctuation
due to the liquid flow and dusts. A smaller noise level could be achieved when the
multipass pattern is very stable and the liquid is pure enough.
3.4
Comparison of Signal-to-Noise ratio
68
4
NMR spectrum: T=1000s
Optical rotation spectrum: T=1000s
S/N=1481
3
0.5
Signal/µV
Signal/µV
S/N=10
0.6
2
0.4
0.3
0.2
1
0.1
0
0.4
0.45
0.5
0.55
Frequency/Hz
0
0.4
0.6
0.45
0.5
0.55
Frequency/Hz
0.6
Figure 3.22: Comparison of signal-to-noise (S/N) ratio between the traditional NMR
signal (via pick-up coil) and optical rotation signal. Sample: water. Measurement
time: 1000 seconds.
As far, the NSOR signal has a much worse signal-to-noise ratio than that of the
traditional NMR signal. As shown in Figure 3.22, for a 1000 second measurement
of the same sample 1 H in water and other condition, the S/N of traditional NMR
spectrum is as large as 1481, while it is only about 10 for the optical rotation spectrum. So the S/N ratio of the NSOR measurement is a crucial limitation for its
applications in practice. In the traditional NMR, the signal is larger for nuclei with
higher gyromagnetic ratio, 1 H has the largest gyromagnetic ratio, while other heavier
magnetic nuclei has a much smaller gyromagnetic ratio. Therefore, the traditional
NMR spectroscopy prefers organic molecules, due to its high sensitivity for 1 H, and
1
H is the most common elements in organics molecules. Unlike the traditional NMR
spectroscopy, NSOR signal is enhanced for heavier magnetic nuclei because their hyperfine interaction is roughly proportional to the atomic number Z, while 1 H has a
smallest NSOR signal.
69
3.5
Faraday rotation (Verdet constant)
When a bulk of nuclear spins is precessing, the induced magnetic field BM can be
divided into a local contact field and a distant dipolar field. The local contact interaction can be enhanced or suppressed depending on the overlap of the wavefunction of
the virtual electron excitation created by the laser and the nuclear spin. This effect
could also be treated as hyperfine interaction between the nuclei and electrons, so
a molecular or atomic with a large hyperfine interaction could give a large optical
rotation signal. The distant dipolar field just lead to a regular Faraday rotation,
so if there is no local contact interaction, the magnetic field that is induced by the
processing polarized spin could also cause optical rotation. For water, the NSOR signal is very close to the regular Faraday rotation, which means that the local contact
enhancement is negligible. For other chemicals that are used in this experiment, the
enhancement is considerable. Therefore, NSOR constant is compared with Verdet
constant in regular Faraday rotation to illustrate the NSOR enhancement.
The Verdet constant could be obtained from standard reference data, which is
used to indicate the Faraday rotation in the medium under magnetic field. Since
the unit of Verdet constant is µrad/(G · cm), it is necessary to convert this unit to
µrad/(M · cm), which is used for the NSOR. Based on the relation
B = µ0 · mH ·
where
N
V
N
,
V
(3.31)
is the polarization density with unit mol/m3 , and it is assumed that the
spin polarization is 100%. Therefore, the unit conversion constant (for proton) from
µrad/(G · cm) to µrad/(M · cm) is
σ = µ · mH · mol · 1000 · 10000
= 0.1067
70
(3.32)
Substance
Vd [µrad/(G · cm)]†
Water
8.67
Methanol
6.33
Ethanol
7.37
Propanol
7.89
Isopropanol
8.15
Hexene
9.28
n-Hexane
7.89
Cyclohexane
9.71
Vd [µrad/(M · cm))]
0.93
0.68
0.79
0.84
0.87
0.99
0.84
1.04
Table 3.2: Verdet constant of chemicals (µrad/(M · cm)) from reference. † Cf. International Critical Tables of Numerical Data, Physics, Chemistry and Technology,
Washburn, 1926-1933. In the original reference database, there is no data for 405nm
laser, so the above Verdet constants are interpolated from known value at other wavelengths.
For example, the Verdet constant of water is Vd = 8.67µrad/(G · cm), so it could
be also written in σ · Vd = 0.1067 · 8.67 = 0.925µrad/(M · cm).
Because Verdet constants from reference data [62] is not measured for the wavelength 405nm that is used in this experiment, we can only get interpolated value
based on some empirical formula. It is better to measure the Verdet constant in a
simple experiment setup that is similar to NSOR detection. The measurement apparatus is shown in Figure 3.23. All the instruments, the sample tube, the coil and
other related conditions are the same as used in the NSOR experiment in Figure 3.2,
but it is much simpler, because the regular Faraday rotation could be measured in
static liquids, and the inductive pick-up coil is changed to acting as a magnetic field
coil which creates a uniform field along the axis of the sample tube. The measured
results of Verdet constant is shown in Table 3.3.
71
Lock-in
Amplifier
Function
generator
1KHz
DAQ
PD2
Current
Source
PD1
PBS
Laser
Diode
405nm
Coil
Faraday rotation measurement
Figure 3.23: Apparatus for Faraday rotation measurement. All the instruments,
the sample tube, the coil and other related conditions are the same, but it is much
simpler, because the regular Faraday rotation could be measured in static liquids, and
the induction NMR coil is changed to acting as a magnetic field coil which creates a
uniform field along the axis of the sample tube.
Substance
Vd [µrad/(G · cm)]†
Water
8.67
Methanol
6.33
Ethanol
7.37
Propanol
7.89
Isopropanol
8.15
Hexene
9.28
n-Hexane
7.89
Cyclohexane
9.71
Vd [µrad/(G · cm)](measured)
8.28
5.83
7.05
7.38
7.65
9.15
7.51
7.68
Table 3.3: Comparison of measured value and referenced value of Verdet constant
(µrad/(M · cm))
72
Chapter 4
Experimental results and calculations
under low fields
In this chapter, the data of Nuclear Spin induced Optical Rotation (NSOR) of 1 H in
various pure organic liquids at a low magnetic field (5 Gauss) are shown and analyzed,
as well as the comparison of NSOR of 1 H and
19
F. It is found that experimental
NSOR signals do not scale with the Verdet constant of the chemicals studied, but
provide unique information about the nuclear chemical environment. From the data,
we propose an empirical rule based on electronegativity to distinguish 1 H NSOR
in carbon-hydrogen and oxygen-hydrogen groups among different organic chemicals.
Furthermore, in order to carry out quantitative computations of NSOR in various
chemicals, first-principle calculations are implemented to calculate NSOR constants,
obtaining qualitative agreement with measured data.
4.1
NSOR of 1H in various chemicals
Based on the experiment setup, we make some choices of organic chemicals for investigation. First, the chemicals for consideration should be in liquid states around room
temperature, and its boiling point is above 50◦ C, because the liquid will be heated
73
up to 50◦ C when the pump is circulating liquid flow for continuous polarization via
a magnet. In addition, the viscosity of liquid should be small enough, otherwise the
pump could not drive the liquid with enough power. Second, the chemicals should be
relatively safe, i.e. not too toxic, carcinogenic or corrosive, as the current experiment
setup requires about 1 liter of liquid for continuous running, while the usage of bulk
dangerous liquid might cause some safety issues. Third, the chemicals should have
low absorption coefficients around the laser wavelength 405nm; otherwise the light
transmission is too low for detection after passing through several meters; therefore,
most colorful organic liquids with significant absorptions in the visible region are not
considered, as well as some chemicals having near-UV resonant absorptions. Based
on these criteria, we only consider eight chemicals: water, Ethanol, Methanol, nHexane, Cyclohexane, Hexene, Propanol and Isopropanol, all of which are common
liquids used in chemistry. In these chemicals, the comparison of water, ethanol and
methanol is used to explore the difference of carbon-hydrogen and oxygen-hydrogen,
while ethanol and methanol further have some differences in carbon-hydrogen constitution. The group of n-Hexane, Cyclohexane and Hexene are pure carbon-hydrogen
chemicals with similar structure. Propanol and Isopropanol are also considered, since
they are a pair of isomers with minor difference in alcohol group position. We also try
to measure Toluene and Benzil alcohol, which are aromatic chemicals that normally
having relatively high Verdet constant, however their absorption coefficients are not
low enough, and their high refractive indices further reduce the light transmission at
the interface of liquid and window, thereby giving extremely low signal-noise ratio,
so we have not obtained reliable data for these aromatic chemicals. Basic physical
properties of these chemicals, such as density, refractive index, viscosity and boiling
point, are listed in Table 4.1, where the Perfluorohexane is also shown, as it is used
to measure NSOR of
19
F nuclei in the next section.
74
75
Substance
Formula
Water
H2 O
Ethanol
CH3 CH2 OH
Methanol
CH3 OH
n-Hexane
CH3(CH2)4CH3
Cyclohexane
(CH2 )6
Hexene
CH3 (CH2 )3 CHCH2
Propanol
CH3 CH2 CH2 OH
Isopropanol
CH3 CH(OH)CH3
Toluene
C6 H5 CH3
Benzil alcohol
C6 H5 CH2 OH
Perfluorohexane
C6 F14
1
H(or
19
F) number
2
6
4
14
12
12
8
8
8
8
14
M weight Density(g/ml) Refractive index
18.01
1.000
1.3432(405nm)
46.07
0.789
1.3610
32.04
0.792
1.3314
86.18
0.655
1.3749
84.16
0.779
1.4262
84.16
0.673
1.3879
60.10
0.803
1.3854
60.10
0.786
1.3776
92.14
0.865
1.4960
108.14
1.045
1.5390
338.04
1.681
1.2520
Viscosity(mPa·s)
0.894
1.070
0.544
0.294
0.930
0.510
1.938
1.960
0.560
5.470
0.660
Melting/Boiling point(◦ C)
0.0 / 100
-114 / 78
-98 / 64.7
-95 / 68.7
5.5 / 80.7
-140 / 63
-127 / 97
-89 / 82
-93 / 110.5
-14 / 204
-90 / 56
Table 4.1: Physical properties of chemicals for investigation in this work. Density, Refractive index and viscosity are all reference
data under standard condition, i.e. 25◦ C, 1 atm. Refractive index is reference data measured with 587nm laser (data of water
is measured with 405nm laser).
F
H
19
1
Nucleus
As described in Section 3.1, the optical rotation signal is measured with a balanced
polarimeter shown in Figure 3.12, and thereby obtain the NSOR angle ΦN . The
NSOR constant VN is calculated from the NSOR angle ΦN by dividing by the optical
path length L, the molar density of nuclear spins n, and the nuclear spin polarization
P = hIN i /IN ;VN = ΦN SOR /(nLP ). Here the actual spin polarization P is deduced
from the measurement of traditional NMR signal, as described in Section 3.2. Here,
results of NSOR are normalized to nA = 1 M, L = 1 cm, and fully polarized nuclei.
The original NSOR data, which are rotation angles divided by the traditional
NMR signal in unit of nrad/µV, are shown in Table 4.2, where water is measured
for more times than other chemicals, because its transmission is relatively higher
and it is periodically used to check the long-term stability of the apparatus. The
measured data need to be calibrated by two factors. The first calibration originates
from edge effects of the induction coil when sensing the traditional NMR signal. For
a uniformly magnetized cylinder, the magnetic field in the central part is the most
uniform, but fields at two ends of the cylinder are reduced about a half. Hence the
induced voltage signal on the coil should be calibrated, which has been explained in
Section 3.2. This calibration applied on the induced voltage signal is V’=0.993*V,
which has incorporated the facts that the coil are not extended to the ends of the
cylinder while both ends of the cylinder are not fully polarized. In addition, in the
multi-pass configuration, the light reflection may introduces some part of circular
polarization, i.e. polarization impurity caused by multiple mirror reflections, which
could not be analyzed by the polarization beam splitter, therefore the direct measured
rotation signal is attenuated a bit. Because the measurement of NSOR have the
same multi-pass configuration and other optical components as the regular Faraday
rotation measurement, we can compare the data of multi-pass Faraday rotation with
single-pass Faraday rotation and then obtain the calibration factor. It is found that
measurements of V in a single-pass geometry are larger by 8.3% compared to multi76
pass geometry, so we applied a +8.3% correction to all optical rotation data in the
0
multi-pass geometry, VN =(1+8.30%)·VN . The NSOR were measured several times
for each liquid, with periodic calibration by water NSOR measurements to check the
long-term stability of the apparatus. In addition to the statistical error we assign a
systematic error of 5% to each measurement, which accounts for observed long-term
changes in the signal amplitudes. We also find that our measured V are on average
5% smaller than literature values at 405 nm [62, 63, 64].
NSOR is composed of two parts: the short-distance hyperfine interaction induced
optical rotation θN , and the regular Faraday rotation θd arising from the magnetic
field inside a magnetized medium. Therefore, if there is no short-distance hyperfine
interaction induced optical rotation, or this effect is negligible, the polarized spin
ensemble could still induce a magnetic field in the medium, thereby creating regular
Faraday optical rotation. In order to show the optical rotation enhancement arising
from short-distance hyperfine interactions, we need to compare NSOR to Faraday
rotation. We scale the Verdet constant by the classical magnetic field B = µ0 mN nA
generated in a long cylinder, where mN is the nuclear magnetic moment and nA is
the number of polarized nuclei. All the measured NSOR constants (VN ) and Faraday
rotation Verdet constants (Vd , measured by single-pass configuration), are listed in
Table 4.3, where the standard Faraday rotation Verdet constants (Vr ) are also shown
as a reference. NSOR constants are scaled to 1 M concentration of fully polarized
nuclei and Verdet constants are also expressed in units of µrad/M · cm using the
magnetic field generated by 1 M of polarized nuclei. The single-pass measurement of
Faraday rotation Verdet constants are quite stable with small statistical error, plus
with some small systematic error due to the alignment of the light beam and the
sample tube. All the data are also plotted in Figure 4.1.
77
78
Substance
Measured optical rotation [nrad/µV]
Water
4.82 5.16 5.09 4.80 4.81 4.66 4.62 4.95 5.22 5.03 4.90 5.10
Ethanol
9.97 10.37 10.29
Methanol
8.32 8.31 7.57 7.74 7.75 7.73 8.10
n-Hexane
11.06 11.13 11.98 12.39 12.15
Cyclohexane
11.42 12.38 12.44 11.95 11.64 11.91 12.46 11.58 11.51
Hexene
10.30 10.27 10.03
Propanol
10.12 10.16 10.34 10.65 10.16 10.34
Isopropanol
10.01 10.14 10.07
Perfluorohexane
69.33 67.37 69.90 69.50
Mean Std%
4.93
3.94
10.21 2.07
7.93
3.86
11.74 5.19
11.92 3.49
10.20 1.45
10.30 1.93
10.07 0.65
69.03 1.64
Table 4.2: Original measured data of nuclear spin induced optical rotation. The data shown are rotation angles divided by the
traditional NMR signal, so they are proportional to NSOR constants.
F
H
19
1
Nucleus
Substance
Water(1 H)
Methanol(1 H)
Ethanol(1 H)
Propanol(1 H)
Isopropanol(1 H)
Hexene(1 H)
Hexane(1 H)
Cyclohexane(1 H)
Perfluorohexane(19 F )
VN [µrad/(M · cm)] Vd [µrad/(M · cm)] Vr [µrad/(M · cm)]
0.94±0.05
0.88
0.94
1.52±0.08
0.62
0.66
1.96±0.10
0.75
0.78
1.97±0.10
0.79
0.83
1.93±0.10
0.82
0.86
1.96±0.10
0.98
1.01
2.25±0.12
0.80
0.84
2.29±0.11
0.82
0.86
13.23±0.67
0.23
-
Table 4.3: Measured NSOR constant (VN ), Faraday rotation Verdet constant (Vd )
and a reference value of Verdet constant (Vr ) of 1 H (from water to cyclohexane) and
19
F (in perfluorohexane). NSOR constants are scaled to 1 M concentration of fully
polarized nuclei and Verdet constants are also expressed in units of µrad/M · cm using
the magnetic field generated by 1 M of polarized nuclei. The reference data of Verdet
constant are obtained from empirical dispersion relation based on the data of other
wavelengths, while the reference data at 405nm is not shown there,† Cf. International
Critical Tables of Numerical Data, Physics, Chemistry and Technology, Washburn,
1926-1933. Referenced data of perfluorohexane is not found yet.
From the data in Table 4.3 and Figure 4.1, for water, the Faraday rotation almost
accounts for all of the NSOR signal, in agreement with previous work in reference [25].
For other chemicals, the NSOR constants are enhanced relative to the scaled Verdet
constants by a factor greater than 2, indicating that hyperfine interactions play a large
role in NSOR. The data for water, methanol, and ethanol are generally consistent with
earlier first-principle theoretical calculations[34]. The enhancement of NSOR for 1 H
bound to carbon can be explained qualitatively by smaller electronegativity of carbon
compared with that of oxygen. The electronegativity using the Pauling scale of H,
C and O is 2.2, 2.55 and 3.44 respectively [65, 66]. For example, in water H2 O,
both 1 H are bonded with the oxygen nucleus, then the electrons are pulled to oxygen
nucleus due to the fact that the electronegativity of H is much smaller than O. Since
the short-distance hyperfine interaction induced optical rotation in NSOR is quite
sensitive to the distance between the electrons and nuclei, the NSOR enhancement
in water may be reduced so much that it could be negligible in measurement, which
79
3
Verdet constant
Ref Verdet constant
NSOR constant
2.5
µrad/(cm⋅M)
2
1.5
1
he
xa
an
C
yc
lo
ex
H
ne
e
e
en
n−
H
ex
no
l
Is
op
ro
pa
an
ol
l
op
no
Pr
Et
ha
no
ha
et
M
W
at
er
0
l
0.5
Figure 4.1: 1 H NSOR constants VN (cross points and error bars), scaled Faraday
rotation Verdet constants Vd (diamonds) from our measurements and scaled reference
Verdet constants (blue dots). All measurements of NSOR constants are shown for all
chemicals to indicate the degree of experimental scatter.
leads to that the Faraday rotation of water is almost equal to the NSOR signal.
On the other hand, the electronegativity of C is close to H, which results in greater
overlap of the electronic wave function with 1 H nuclei in CH2 and CH3 groups, giving a
larger hyperfine interaction than the OH group. First-principle calculations of optical
rotation constant of individual 1 H in CH2, CH3 and OH groups in ethanol, as shown in
Figure 4.2, agree with this explanation very well. Because the NSOR signal measured
under low magnetic fields is a averaged value of all 1 H nuclei in a molecule, as it is
not able to distinguish different 1 H through chemical shift at low fields, the molecules
with OH bonds exhibit a smaller NSOR constant than the molecules without OH
bond. This could further explain that NSOR of n-Hexane and Cyclohexane, which
is only composed of CH2 or CH3 groups, is larger than water, Methanol, Ethanol,
Propanol and Isopropanol. The data of Hexene is close to Propanol and Isopropanol,
80
1
Methyl group
Methylene group
Alcohol group
0.6
0.4
N
V [µrad/(M cm)]
0.8
0.2
0
400
600
800
1000
Wavelength [nm]
1200
1400
Figure 4.2: Calculated NSOR constant (VN ) of individual 1 H in different groups in
ethanol as a function of laser wavelength. These first-principle calculations are based
on DFT methods with the BHandHLYP/co-2 level for a single molecule, and the bulk
field effect (regular Faraday rotation in the macroscopic medium) is not included.
Experimentally equivalent nuclei are averaged. Reproduced from reference [34].
since the C=C bond have a larger electronegativity than C-C bond. The significant
difference between Methanol and Ethanol is probably due to the difference of the
fraction of CH3 , because Hydrogen in O-CH3 has smaller attraction force than that
in C-CH3. Of course, this empirical explanation based on classical electronegativity
theory could only provide a qualitative explanation. In Section 4.3, a quantitative
calculation based on first-principle calculations will be given,
4.2
Comparison of
19
F and 1H
Theory and calculations[25, 31, 34] predicted that NSOR in heavy nuclei are normally larger than light nuclei, since in heavier atoms the contact term enhancement
generally increases with the atomic number Z, which was also verified experimentally
in
129
Xe[25] and
19
F[29]. Therefore, molecules with heavier magnetic nuclei attract
81
more interests of applications of NSOR in chemistry. Due to the limitations in current low-field experiment setup, most of the other heavy nuclei are not applicable,
e.g. lithium(7 Li) and other magnetic metal ions solutions do not have enough spin
polarization density, condensed phosphoric (31 P) acid is too viscous for running, and
the natural abundances of 13 C, 15 N and 17 O are too small and their polarization relaxation time is quite short. The most applicable substance is Fluorine organic liquids,
since its gyromagnetic ratio allows high enough spin polarization density, and its coherence time in liquids is about 1∼10 seconds, which also allows the liquids flowing
through sample tube without serious polarization loss. The first experiment of 19 F in
perfluorohexane(C6 F14 ) and hexafluorobenzene(C6 F6 ) was explored in reference [29]
at high magnetic fields, but the signal-noise ratio (SNR) is relatively weak due to the
short optical path and other detection noise.
Here, we carry out the measurement of
19
F in perfluorohexane under low mag-
netic fields with help of multi-pass cavity and short wavelength (405nm), obtaining a
relatively higher SNR. In Figure 4.3, we compare the optical rotation spectrum of 1 H
in C6 H14 and
19
F in C6 F14 after 1000 seconds of integration, where the SNR is about
16 and 24 for 1 H in C6 H14 and
19
F in C6 F14 respectively. Since the spin polarization
ratio of 19 F in C6 F14 deduced from traditional NMR signal and spin density are much
smaller than 1 H in C6 H14 , the measured NSOR constants for 19 F is a factor of 6 larger
than in hydrocarbons, while the Faraday rotation Verdet constant is a factor of 3.5
smaller (see Table 4.3), consistent with earlier measurements of the Verdet constant
in fluorocarbons[29]. Therefore, the NSOR constant of
19
F in C6 F14 enhanced the
optical rotation due to the Faraday effect by a factor of about 57, which is partly
due to stronger hyperfine interaction in heavier atoms and partly due to high electronegativity of fluorine atoms. Other heavy nuclei, such as
205
Tl,
207
Pb and
133
Cs,
are expected to have much higher NSOR constants, which will be investigated in the
future.
82
Opti cal Rotati on ( nrad /mHz1/ 2 )
Opti cal Rotati on ( nrad /mHz1/ 2 )
40
100
30
20
10
0
0.4
0.5
Frequency (Hz)
0.6
80
60
40
20
0
0.4
0.5
0.6
Frequency (Hz)
Figure 4.3: Nuclear spin optical rotation spectrum of 1 H in C6 H14 and 19 F in C6 F14
after 1000 seconds of integration. Since B0 field is modulated on and off the resonance
at 0.5 Hz, the signal appears at this frequency. The SNR is about 16 and 24 for 1 H
in C6 H14 and 19 F in C6 F14 respectively.
4.3
Theoretical calculation
In this section, the theoretical calculations for NSOR constants is described, as qualitative agreement with measured data is obtained with first-principles calculations
using density-functional theory (DFT)[67] calibrated against ab initio coupled cluster with single and double operators (CCSD)[68] data. The following calculation is
done by Ikäläinen and Vaara according to the collaboration with our experiments.
As sketched in Section 2.3, the NSOR constant VN can be calculated through the
rotationally and ensemble-averaged antisymmetric polarizability [45, 69, 32, 34],
e3 ~ µ20
1X
`N,ν
1
γN
ετ ν Imhhr ; rτ , 3 iiω,0 ,
VN = − ωnA cIN
2
me 4π
6 τ ν
rN
(4.1)
where ω is the frequency, IN is the nuclear spin, γN is the gyromagnetic ratio, and ετ ν
is the Levi-Civita symbol. This expression is in terms of quadratic response theory
[70], involving time-dependent electric dipole interaction with the light beam taken
83
to second order, and the static hyperfine interaction
HNPSO =
X `iN
e~ µ0
γN IN ·
3
me 4π
riN
i
(4.2)
between the nuclear magnetic moment µN = γN ~IN and the electrons, involving the
electronic angular momentum about nucleus N , `N . For heavy-atom systems such as
liquid Xe, relativistic formulation should be employed [35]. In the present systems
that only include light nuclei (1 H, C and O) the non-relativistic form (Equation 4.1)
is sufficient.
Equation 4.1 does not include the long-range magnetic interactions between
molecules, as discussed in Reference [33]. For a long cylindrical sample the longrange field equals B =
1
3
µ0 M , resulting in an additional regular Faraday rotation.
Hence, a bulk inter -molecular correction VB given by
1
VB = nA µ0 µN V,
3
(4.3)
should be added to VN to be fully comparable to the experimental results.
In order to calculate the antisymmetric polarizability, optimized geometries of the
molecules were first obtained with the Gaussian software [71] at the B3LYP/aug-ccpVTZ level, while the Dalton program [72] was used to calculate NSOR constants at
405 nm. When running Dalton program, implementations of quadratic response functions were employed for the Hartree-Fock (HF), density-functional theory (DFT), and
coupled-cluster (CC) methods from reference [73, 74, 75] respectively. DFT was used
to obtain results of predictive quality for larger molecules. Therefore, its performance
was assessed through more accurate but also more time-consuming ab initio CC singles and doubles (CCSD) calculations for water, methanol, ethanol, propanol and
isopropanol. The DFT functionals BHandHLYP(50%), B3LYP (20%), and BLYP
(0%) were used, where the percentages denote the amount of exact HF exchange
84
admixture, which has been often seen to be the factor controlling DFT accuracy
for hyperfine properties [76, 34]. In addition, novel and compact sc. completenessoptimized (co) basis sets [77] were used to furnish near-basis-set limit results for VN ,
which requires an accurate description of the electronic structure both at the nuclear
sites and at the outskirts of the electron cloud. This is due to the involvement of
both the magnetic hyperfine and electric dipole operators. The efficiency of co sets
for magnetic properties has been verified in several studies [77, 78, 34, 35, 79]. The
co-2 set (10s7p3d primitive functions for C-O; 10s7p3d for H) was developed in Reference [78] for laser-induced
13
C shifts in hydrocarbons. The carbon exponents [78]
are used here also for oxygen. Co-0 (C-O: 12s10p4d1f , H: 8s8p5d) was generated
in Reference [79] for basis-set-converged NSOR for first-row main-group systems, as
calibrated by 1 H and
19
FSOR calculations for the FH molecule.
To calculate VB , the Faraday rotation Verdet constants were calculated at
the BHandHLYP/co-2 and B3LYP/co-2 levels for all molecules, as well as at the
CCSD/co-2 level for water, methanol, ethanol, propanol, and isopropanol. NSOR
constants computed with the co-2 basis were combined with VN obtained with co-0,
as the former property is not as sensitive to the basis-set quality as NSOR.
Figure 4.4 shows the calculated NSOR via various methods with two basis sets,
where all the data are weighted averages over all 1 H and
19
F nuclei of the molecules.
In most cases, the use of a basis set with higher quality leads to larger VN . The only
exception is water, where no systematic change is observed. Perfluorohexane shows a
difference of 10% between the two basis sets, while for the other molecules, the percentage ranges from 20–50%. In all cases other than perfluorohexane, the calculated
intramolecular NSOR in Equation 4.1 is smaller than the experimental result. After
adding the bulk correction VB , the agreement of BHandHLYP/co-0 and B3LYP/co-0
data with experiment is improved (see Figure 4.4), apart from the exaggerated B3LYP
results for perfluorohexane and water. However, the use of BHandHLYP/co-0 for wa85
Figure 4.4: VN for all molecules at the B3LYP/co-2 and B3LYP/co-0 levels of theory, as well as VN with the bulk (intramolecular) correction for B3LYP/co-0 and
BHandHLYP/co-0. CCSD/co-2 data is given for the smaller molecules. Experimental values with error limits are also shown. Results for perfluorohexane are divided
by a factor of six.
ter results in a good agreement with the measurements, both due to the reduced
intramolecular NSOR contribution as well as the more realistic V obtained at the
BHandHLYP level. For the larger molecules, the experimental values are reproduced
qualitatively. A detailed analysis would necessitate the incorporation of solvation and
intramolecular dynamics effects via molecular dynamics simulations [79]. In the case
of perfluorohexane, the known issues [76] of the present DFT functionals with the
hyperfine properties of
19
F may contribute to the observed overestimation.
Furthermore, the calculation for individual H in a group(Supporting information
in Reference [36]) reveals that the largest NSOR occur in the CH2 groups, while
the hydroxyl OH groups display distinctly smaller values than either the methyl or
86
87
Basis
co-2
co-0
co-2
co-0
co-0
co-2
co-0
co-0
co-2
co-0
co-2
co-0
co-2
Water
Methanol
Ethanol
Propanol Isopropanol
Hexene
Hexanea Cyclohexane Perfluorohexane
0.45
0.92
1.05
1.11
1.14
1.26
1.26
1.25
10.08
0.50
1.22
1.36
1.42
1.43
1.56
1.55
1.57
10.38
0.62
0.95
1.09
1.15
1.22
1.31
1.36
1.29
11.98
0.64
1.24
1.39
1.46
1.51
1.60
1.69
1.62
12.53
0.91
1.40
1.57
1.64
1.73
1.87
1.89
1.81
12.59
0.86
0.99
1.04
1.20
1.33
1.37
1.44
1.34
14.29
0.85
1.28
1.44
1.51
1.62
1.68
1.78
1.69
16.00
1.21
1.46
1.65
1.73
1.89
1.98
2.01
1.90
16.07
1.19
1.03
1.16
1.24
1.46
1.46
1.52
1.40
17.19
1.15
1.32
1.47
1.55
1.75
1.75
1.88
1.70
19.35
0.73
0.89
1.05
1.12
1.20
–
–
–
–
0.75
–b
–
–
–
–
–
–
–
1.03 c
1.05
1.24
1.32
1.43
–
–
–
–
0.94±0.05 1.52±0.08 1.96±0.10 1.97±0.10 1.93±0.10 1.96±0.10 2.25±0.12
2.29±0.11
13.23±0.67
Table 4.4: Calculated NSOR (VN ) [in µrad/(M · cm)] at 405 nm using the HF, DFT, and CCSD methods and the basis sets
co-0 and co-2. For B3LYP/co-0 and CCSD/co-2 levels of theory, bulk-corrected values are given. Experimental data is also
provided.
a
The most diffuse d-type function for hydrogen has been omitted in the co-0 basis used for hexane. b Not calculated due to
program limitations. c The CCSD/co-0 + bulk result equals 1.05 µrad/(M·cm).
CCSD + bulk
Exp.
CCSD
B3LYP + bulk
BLYP
BhandHLYP + bulk
B3LYP
BHandHLYP
Method
HF
methylene groups for all molecules. This supports the electronegativity argument for
the relatively small VN in water. In perfluorohexane, however, the CF3 -group feature
a larger VN than the CF2 group. The different methylene groups in propanol and
hexane, as well as the axial and equatorial hydrogens in cyclohexane give similar
results. Alteration in the magnitude of VN is observed for the CH2 groups in hexene.
The values for 1 H in the methyl groups are rather similar for all the molecules, with
isopropanol and hexane giving slightly larger NSOR than the other systems. The VN
appropriate to the hydroxyl groups differs between the molecules, with ethanol and
propanol giving very small signals. In hexene, the cis-type hydrogen shows a larger
rotation than the other protons situated next to the double bond.
In summary, we find that experimental NSOR signals do not scale with the Verdet
constant of the chemicals studied, but provide unique information about the nuclear
chemical environment. Qualitative agreement with measured data is obtained with
first-principles calculations using DFT calibrated against ab initio CCSD data. Hybrid DFT with 20% or 50% exact exchange is found to produce results closest to
experiment. The 1 H NSOR is able to clearly distinguish hydroxyl group from methyl
and methylene groups. Future application of these techniques to more complicated
molecules can provide unique new information about their conformation and electronic state.
88
Chapter 5
NSOR experiments under high
magnetic fields
In the NSOR experiment under low magnetic fields, all the signal are averaged values
of the same nuclei (e.g. 1 H) in a molecule, so it is impossible to distinguish the nuclei
in different groups in the same molecule. It is necessary to measure NSOR under
high magnetic fields, since it provides capability of chemical shifts and thereby it
is possible to measure the NSOR constants of the same type of nuclei in different
groups. Besides, under high fields, there are several aspects that could promote the
measurement of NSOR as follows.
(1) High magnetic fields not only exhibit chemical shifts among different functional groups, but also could reach a higher polarization ratio for the material, as the
net population ratio of polarized nuclear spin is approximately proportional to the
magnetic field B0 around room temperature,
ρ∆ =
~γN B0
2kB T
(5.1)
where γN is the nuclear gyromagnetic ratio, kB is the Boltzmann constant, and T is
the temperature. On the other hand, the NSOR signal is also proportional to the
89
spin polarization ratio. Therefore, the increment of magnetic fields could enhance the
NSOR signal linearly. Unlike the situation in low-field setup, in which the nuclear
spins are prepolarized in a external magnet and then adiabatically transferred to the
low field region resulting with serious polarization loss, the spin polarization under
high fields could be maintained without loss, since the sample is placed statically inside the magnet while B0 serves as the polarization field as well as the precessing field
simultaneously. Under a high magnetic field, normally 1∼10 Tesla in a superconducting magnet, the polarization could reach maximal efficiency, and the pulsed NMR
technique could be executed continuously with high repetition in aid of the long-time
integration of signals. The only concern under high magnetic field is the field homogeneity, because the sample for investigation always possesses some finite volume and
a small field inhomogeneity among this volume will deteriorate the coherence time,
leading to the signal linewidth broadening and then make the chemical shift less distinguishable. Current first-principle calculations could calculate the NSOR constants
for single nuclei in a specific functional group, and NSOR enhancement of each group
with chemical shift also attracts interests in the area of chemical analysis, hence we
aim to setup the apparatus for NSOR measurement under high fields.
(2) Second, performing high-field experiments makes it possible to measure the
NSOR of many heavy magnetic nuclei, while it is not unachievable under low fields.
As already stated in Section 4.2, except some organic liquids containing 1 H and
19
F
for investigation, most magnetic nuclei could not be measured successfully at low
fields, under which it requires long coherence time, high enough polarization and low
toxicity for bulk usage. Common magnetic nuclei, such as
207
13
C,
17
O,
14
N,
31
P,
29
Si,
Pb and so on, are not detectable regarding NSOR under low fields. For example,
although the relaxation time T1 of 13 C is about several seconds, the abundance of 13 C
is only 1.1% and its gyromagnetic ratio is also much smaller than 1 H, which leads to
very small spin polarization.
17
O has a rather short T1 ∼0.02s, so it could not keep
90
polarization for enough time for continuous-wave NMR measurement. Some solutions
of metal ion salts, e.g. 7 Li,
109
Ag,
207
Pb, could not reach high spin density, then the
spin polarization is not high enough for measurements beyond noise level. Although
31
P is also a good candidate for investigation, the liquid condensed H3 PO4 is too
viscous to flow quickly for continuous prepolarization in current low-field experiment
setup. Under high fields, most of the above issues are not a problem, as the sample do
not need to flow, the polarization is higher than under low fields, and pulsed NMR is
capable of measure the magnetic nuclei with short T1 and T2 . In summary, NSOR of
most magnetic nuclei are expected to be measured, as long as the signal could surpass
the noise after long time average.
(3) Under high fields, a multipass or optical cavity arrangement [80, 81], or a long
hollow fiber [82, 83, 84, 85], could be applied to not only reduce the sample volume
down to µL or even nL, but also overcome the space limitation in a superconducting
magnet. For example, if we implement a 10-meter-long single mode hollow fiber of
inner diameter 5µm, the sample volume is only of the order of V = πR2 l = 0.2µL,
while its sensitivity could be enhanced with 10-meter-long optical path, which is
estimated to be S/N∼1000 for 1 H, achieved by coiling this fiber inside a 10 Tesla
magnet and detecting with 1 Watt of laser at 400 nm wavelength for 1 second. Besides,
small sample volume usage in NMR is quite favorable, since sometimes it needs to use
some expensive, highly toxic chemicals or mass- and volume-limited samples [86, 87].
In our experiment, we implement the hollow fiber scheme to measure the NSOR
signal in a 2.7 T superconducting magnet. We tried to use several types of hollow
fiber, e.g. photonic crystal fibers, Teflon AF tubings, fused silica capillary tubings, to
realize long-length optical waveguide with high transmission. However, as far, we find
that the light suffers from depolarization and bending-induced linear birefringence in
liquid-core fibers, both of which attenuate the optical rotation signal significantly. In
addition, due to the possible bandwidth limitation of instruments and fibers, we have
91
not achieved the measurement of high-frequency optical rotation, including Faraday
rotation and NSOR, although the low-frequency measurement of Faraday rotation is
successful after using chiral liquids to suppress bending-induced birefringence.
5.1
Experiment setup at high fields
A high magnetic field that is larger than 1 Tesla is usually achieved by a cylindrical
superconducting magnet, in which the magnetic field is along the bore axis and the
bore area with homogenous field is quite narrow, typically having a diameter of 5∼20
cm. However, in the detection of NSOR, the direction of light propagation should be
perpendicular to the magnetic field direction, i.e. the light direction is perpendicular
to the axis of the magnet bore. Since the signal of NSOR is quite small, we need to
make the light propagating for long distance in the radial direction of the magnet bore.
There are two schemes for elongating optical path in the magnet, a optical cavity and
hollow fiber. A optical cavity, based on multipass or resonant optical cavity, could
guide light for long distance through many times of reflection in a narrow space, but
also bring in some optical loss when the light is reflected on the interface of mirrors
and liquids.
In the case of the hollow fiber, as long as the refractive index of the wall of fibers
is higher than the liquid, the light could be guided inside the hollow core of the fiber
through total internal reflection. In common fused silica capillary tubings, the index
of the tubing wall is 1.4696 (for 405 nm light), so it allows liquids that have a higher
index (>1.46) to propagate for long distance. If the fiber is made of Teflon AF or it is
a photonic crystal fiber, the effective refractive index could be smaller than 1.31, while
most transparent liquids, including water, have a refractive index > 1.33, then this
hollow fiber could be filled with most liquids to guide light. As far, benefiting from
recent material technologies, hollow fibers based on Teflon AF material and photonic
92
crystal fiber are applied widely in precision spectroscopy and trace analysis of liquids
[82, 83, 84, 85]. Here, we adopt the hollow fiber scheme for the detection of NSOR
in a high magnetic field, because the hollow fiber could provide a long optical path
with several meters and a flexible fiber could be coiled densely in a narrow space.
5.1.1
Experimental setup
The hollow fiber scheme is illustrated in Figure 5.1. The magnetic field B0 is uniformly distributed vertically in the space. The hollow fiber filled with liquids are
coiled circularly, as the light is guided through the fiber for a long optical path, then
the outgoing light at the end is collimated by a lens to be analyzed by a polarimetry. A double-circle RF coil (or toroid coil) creates a centrifugal RF field B1 on the
equatorial plane, so the RF field is always perpendicular to the optical fiber and the
magnetic field B0 along the coiled fiber everywhere. In the presence of B0 , the nuclear spins are initially polarized along the magnetic field. After applying a short π/2
pulse, the spins are excited to rotate to the equatorial plane, and then precessing at
the Larmor frequency according to the magnetic field B0 . The spin precession experiences a relaxation process, i.e. free induction decay in pulsed NMR, until all the spins
relax to the thermal equilibrium condition after some time. During this process, the
nuclear spin precession induces optical rotation (NSOR) at the same frequency, and
the optical rotation of the outgoing light beam could be measured by the polarimetry.
Such a single measurement cycle is not enough for NSOR detection, because the spin
precession time is limited by actual coherence time T∗2 , which is much shorter than
T1 (∼several seconds for 1 H), typically less than 100 ms due to decoherence arising
from the inhomogeneity of magnetic fields. So we need to repeat this process, applying π/2 pulses periodically (at a time interval > 2·T1 ), and accumulate the optical
rotation signal for many times. In addition, some decoupling pulses, e.g. CPMG
pulse, is used to elongate the actual coherence time T∗2 , then the spin precession
93
could induce the optical rotation for several seconds. In this scheme, we combine the
regular pulsed NMR technique and polarimetry analysis for the detection of NSOR.
B0
B1
M
B 0 =2.7T
RF coil
B1
B1
405nm
Laser
PD1
Lens
Lens
Connector 1
Connector 2
PBS
PD2
Figure 5.1: Design of liquid core hollow fibers and RF coil for the detection of NSOR
under high magnetic fields (the figure is not to scale). The magnetic field B0 is 2.7
T, corresponding to 115 MHz in 1 H NMR. Pulsed NMR technique is implemented to
measure NSOR signal.
We adopt a 400 MHz (1 H) NMR superconducting magnet (Oxford instruments),
but the magnetic field B0 is downgraded to 2.7 Tesla, corresponding to the frequency
of 115 MHz in 1 H NMR. Although a higher magnetic field gives a higher spin polarization, the bandwidth of optical detectors (photodiode amplifier) is limited and
the noise level will be much larger than shot-noise level at high frequency, while the
magnetic field only increase the nuclear spin polarization linearly. We managed to
make a high bandwidth photodiode amplifier with near thermal-noise and shot-noise
level, as the bandwidth goes up to 140MHz. Therefore, we balance the magnitude
of the magnetic field and the bandwidth & noise level of the optical detector, to
94
approach a maximal signal-to-noise ratio in measurements practically. In addition,
unlike 5mm sample tubes used in the regular NMR spectroscopy, the diameter of
the densely coiled fiber bundle is as large as 1.0-1.5 inch typically, then the magnetic
field homogeneity is rather poor, which gives an actual T∗2 less than 1ms. Therefore,
the shim coils of the superconducting magnet are optimized to give a homogenous
magnetic field in a ring area with 1.0-1.5 inch diameter. As a best result, we could
achieve T∗2 ∼5ms for the 1 H in water, while it is about 0.1 s in the same magnet for a
5 mm NMR sample tube.
The double-circle RF coil is connected with two trimmer capacitors C1 and C2 ,
as shown in Figure 5.2, in order to match the impedance of the RF amplifier. The
’Parallel-Series’ configuration of C1 and C2 is tuned to ensure the circuit is resonant at
the Larmor frequency. In addition, the equivalent impedance of the circuits is tuned
to be 50Ω, which matches the impedance of RF amplifier and cables, maximizing the
RF power on the coil. When C2 C1 , we can get the resonance frequency ω0 and
impedance Z0 of the circuit,
r
C1 + C2
L0 C1 C2
,
LC1
Z0 =
R(C1 + C2 )2
ω0 =
(5.2)
where L0 is the inductance of the coil. For a 1-inch-diameter double-circle coil as
shown in Figure 5.2, we tuned the trimmer C1 (NMA-HV series from Voltronics Corporation) around 4pF and C2 around 320 pF, achieving a 115 MHz resonant circuit
with minor RF reflection, which means a good impedance matching, by measuring
the reflected RF power by a directional coupler and wattmeter (from Bird Electronics
Corporation).
The above double-circle coil could create a rather uniform RF pulse across the
fiber bundle, but it could not be used to regular Faraday rotation due to the fact that
the magnetic field is perpendicular to the light propagating direction, while we need to
95
RF
400
Transmission
line
C2
NMR
2.7T
C1
L, R
NMR RF Coil
and Tank Circuit
B0
Superconducting
Magnet
Figure 5.2: The NMR RF coil and tank circuit inside a 2.7 Tesla superconducting
magnet. (left) Diagram of the RF circuit configuration inside the magnet. (right)
Picture of the coil and tank circuit in this experiment. The coiled fiber around
a (white) Teflon rod is placed between the two circles of the coil. The magnetic
field induce by RF pulse is centrifugal and perpendicular to the B0 . The aluminum
container is used for RF shielding.
measure the high frequency Faraday rotation for signal bandwidth testing and other
experimental apparatus inspections. Therefore, we implement a toroid coil, as shown
in Figure 5.3, to enable the Faraday rotation measurements. In this configuration,
the magnetic field is circularly distributed and roughly parallel to the light propagating direction in the fiber. The toroid coil has a higher inductance and the field
homogeneity is a bit worse than the double-circle coil. We only use 12 turns of toroid,
because more turns of coil result with a higher inductance, while a higher inductance
should be used with smaller capacitances of C1 and C2 for impedance matching and
capacitors are hardly manufactured or tuned less than 0.5 pF in practice. For this
coil, we use C1 =5.0 pF and C2 =100 pF to get resonance condition (115 MHz) and
impedance matching.
96
Teflon body
Coil
(a) A toroid coil.
Coil
(b) Cross section of the fiber-coil geometry
Figure 5.3: Toroid coil for NSOR and Faraday rotation measurement. (a) Picture
of the toroid coil (12 turns). A 10-meter-long hollow fiber is densely coiled in the
groove layer in a Teflon body (OD=1.5 inch), (b) Cross section of the left picture.
The hollow fiber is indicated in small circles (black) filled with liquids (green).
The hollow fibers used in this experiment consists of fused silica or Teflon AF,
and most of them have an outer diameter (OD) of 180µm, 250µm or 360µm. For
all of these fiber, a liquid-optical interface is manufactured for guiding light and
liquid through the hole of hollow fibers, while most material used in this system
have a high degree of chemical resistance. Shown here in Figure 5.4 is schematic
drawing of a enclosed connector combining a fiber connector and liquid port for this
purpose, while both ends of the fiber are linked to such a connector. The main body
of the connector and most threaded adapters are made of Polyether ether ketone
(PEEK), which is chemically resistant to most organic liquids, including all chemicals
for investigation in this thesis. A 0.75 mm hole is drilled along the axis of the connector
body thoroughly, then the fiber is inserted into the connector body until it reach a
window (made of MgF2 or Fused silica) at the right end, while it is also clamped and
sealed at the left end of the connector by PEEK adapters, sleeves and micro tubing
fittings (from LabSmith and IDEX Health&Science). A glass capillary tubing, whose
outer diameter (OD) is a little smaller than 0.75 mm and the inner diameter is a little
larger than the outer diameter of the fiber, is used to keep the fiber straight inside the
97
connector. With aid of a threaded liquid port on the top of the connector, the liquids
is introduced into the connector body and then the hollow fiber through a PEEK
tubing with ID=150µm and OD=360µm. For a long hollow fiber, the liquid friction
is rather high, we use a high pressure syringe to push the liquid into the PEEK tubings
and hollow fibers. Sometimes we even use a high-pressure (50∼300 psi) Helium gas
cylinder to apply the force consistently, then some gas bubbles remained in corners
of the connector are minimized under high pressure and will not affect light guiding
at the interface of the hollow fiber and the window. Also a inline PEEK microfilter
with 1µm frit is interconnected with the PEEK tubing, thus filtering out small dusts
and other unsolvable particles before the liquid entering the fiber connector. The
liquid flow is shown in blue in Figure 5.4, and the liquid inside the system could be
purged out by blowing high-pressure inert gas from the PEEK tubing before filling
the hollow fiber with other kinds of liquids. Figure 5.5 shows the picture of this
liquid-light interface, where the OD diameter of the connector body is 0.5 inch.
The incident laser is collimated into the hollow fiber by using a fused silica lens
with anti-reflection coating, according to the Gaussian beam waist formula, D1 =
4λf /(3πD0 ), where, D0 (mm) the input beam diameter , f (mm) the focal length
of the lens, λ (µm) the wavelength and D1 (mm) the output beam diameter at the
focal plane. As shown in Figure 5.6, the beginning of the fiber is placed at the
focal plane of the lens, then the collimated light beam is quasi-parallelly injected
into the fiber core that is filled with liquid, while the window in the fiber connector
isolates the liquid from air. Despite the above collimation issue of laser beam, the
more important light polarization issue should also be considered. As the theory of
mode matching and polarization birefringence is described in Section 5.2, we need to
appropriately choose the collimating lens and the size of the fiber core, then the light
beam could be well matched with the fiber core, which means the size of fundamental
mode of light beam is almost equal to the fundamental mode of the hollow fiber as
98
Figure 5.4: Schematic drawing of the fiber connector and liquid port. The hollow
fiber is a flexible capillary tubing or a photonic crystal fiber. All the adapters, ports
and connector body are made of commercial polyether ether ketone (PEEK), as well
as the liquid filling tubing. At the right end of the connector, a window (MgF2 or
Fused silica) is placed between the fiber end and a PEEK cap, as a Teflon O-ring
creates a seal at the interface. A glass capillary tubing, whose inner diameter is a
little larger than the outer diameter of the fiber, is used to keep the fiber straight
inside the connector, while it is almost as long as the hole inside the connector and
its right end is near the window. The filled liquid is shown in blue.
a waveguide when the incident angle of light beam is as small as possible. Only at
mode match condition, the linear polarization of light beam is maintained optimally,
since the fundamental mode of light beam keep polarization to maximal extent when
propagating inside a fiber as a waveguide, while higher-order modes hardly keep their
polarization inside bent fiber, which is similar to the depolarization phenomenon in
regular multimode fibers. For example, for a laser beam of diameter 1 mm and the
fiber core with 50µm, the lens should have a focal length 5 cm for mode matching.
In addition, we also find that the bending-induced birefringence (BIB) occurs in the
liquid core hollow fiber in a coiled geometry, and this effect deteriorates the linear
polarization of light beams, which leads to a small fraction of linear polarization for
analysis in polarimetry. To suppress this effect, we manage to use chiral liquids to
99
Figure 5.5: Picture of the fiber connector and adapter. The diameter of the connector
body is 0.5 inch. The hollow fiber is a flexible fused silica capillary tubing (Polymicro)
coated with polyimide, OD∼360µm, ID∼50-150µm.
reduce the bending-induced birefringence, because the chiral liquids inherently rotates
the polarization plane through circular birefringence but still maintain the linearity of
polarization, and chiral liquids has a much higher rotation capability than BIB, which
helps suppress the depolarization and circular polarization effects induced by BIB.
The theory of mode matching and bending-induced birefringence is briefly described
in Section 5.2. Once the light beam emerges from the other window at the end of
the fiber, the light beam is in divergent profile, so another lens is used to collimate
the divergent beam, making the light beam as a collimated outgoing beam for the
subsequent polarization analysis by means of a polarimetry.
5.1.2
Three types of hollow fibers
In 1970s, when the research of solid glass core fiber was just launched for optical
communication [88, 89, 90], the liquid core hollow fiber was also investigated for
the same application, since it has the least degree of imperfection as the liquid core
is homogeneous and free of mechanical stress [91, 92, 93]. Although, due to the
thermal and mechanical instability, the liquid core hollow fiber can not make a success
of commercial optical communication, while the solid glass core fiber could be well
100
Window
Collimating lens
Liquid
Fiber connector body
Fiber cladding
Laser beam
D0
core
diameter (ID)
Beam
diameter
f
Focal length
Liquid core
D1
Collimated
beam diameter
Figure 5.6: Collimating the laser beam into the liquid core of the hollow fiber.
The collimated beam diameter D1 at the left interface of the fiber is equal to
D1 = 4λf /(3πD0 ), which should be smaller than the fiber core diameter (ID). The zigzag total-internal-reflection pattern illustrates the light propagation path, although
in small core fiber, the fiber performs as a waveguide for the light beam. The laser
beam has a beam diameter 1 mm typically. The fiber core diameter (ID) ranges from
50µm to 150µm. The focal length is about 1cm∼10cm, which is chosen for both of
spot size matching and mode matching, depending on the laser beam diameter and
fiber core diameter. The liquid is shown in blue. The core index n1 is larger than the
cladding index n2 .
manufactured with ultralow imperfection and high stability, the liquid core hollow
fiber still play an important role in some scientific areas, i.e. chemical analysis,
spectroscopy and sensors [94, 95, 82, 83], since the hollow fiber could elongate optical
path for some liquid chemicals instead of optical cavity. As far as now, there are
three types of liquid core hollow fibers in use, fused silica capillary tubings, Teflon
AF tubings and photonic-crystal fibers, as introduced in the following.
In the early years, only fused silica capillary tubings are available for making liquid
core hollow fiber [93]. When using fused silica capillary tubing, the liquid for filling
should have a higher refractive index than the fused silica (n∼1.46), therefore only
some chemical liquids, e.g. aromatic and halocarbon compounds, could meet this
requirement, while most other common chemicals and water-based solutions have a
lower index than fused silica. This issue severely limits the utility of fused silica
capillary tubings in chemistry.
101
One way to extend the usability of fused silica capillary tubing is coat the inner
wall of the capillary tubing with some low-refractive-index material. Teflon AF 2400,
a clear amorphous perfluoropolymer having a refractive index of 1.29, is a good choice
for this type of coating [96, 97], because its ultralow refractive index is even lower than
water (n=1.33). Nowadays, the commercial Teflon AF tubing is also available, fully
consisting of pure Teflon AF 2400, although its wall is thicker than fused silica tubing.
The Teflon AF could be used as liquid core hollow fiber, allowing filling most of the
chemical liquids of a refractive index exceeding 1.29, so it becomes a practical tool
for Raman, fluorescence and absorption spectroscopy for liquid and water solution
samples [98, 83]. We also try to use the Teflon AF tubing (purchased from Biogeneral
Inc.) as liquid core hollow fiber, but find that the light scattering on the inner wall
surface is considerable, probably due to the poor surface quality of Teflon tubing in
manufacturing as far, so the optical loss is serious when the length of tubing exceeds
2 m for the liquid sample water and Toluene, . Since a long optical path about 10m
is required to measure NSOR signal at high magnetic field, the Teflon tubing with
current manufacturing technology is not usable for our experiment.
Another candidate for the liquid core hollow fiber is the Photonic-crystal fiber
(PCF), which is a type of microstructured optical fibers consisting of a periodic array
of microscopic air holes along the entire fiber length [99, 100, 101]. With an ability
to confine light in hollow cores that is not possible in conventional optical fiber, PCF
have many applications in fiber-optic communications, fiber lasers, highly sensitive
gas sensors, and other scientific areas. As shown in Figure 5.7(a), a typical photonic crystal fiber has a bundle of small cladding holes and relatively large central
hole, allowing specific spectrum of light propagating through the central hole without
scattering outside from cladding holes, because PCF could confine light by band gap
effects instead of total-internal-reflection effects in conventional optical fibers. Benefiting from this special structure, a new type of liquid core hollow fiber is available,
102
if the central hole is filled with liquids while the cladding holes are not [84, 85], the
average index of air-silica cladding hole is sufficiently low (as low as air), and thereby
the filling of liquid into the core turns the photonic bandgap guidance into total internal reflection guidance. The tricky part here is to achieve selective filling of liquids
in different types of holes in PCF. People have developed several efficient methods to
seal the ends of cladding holes but still leave the central hole open, such as fusing the
cladding holes at the fiber ending with a conventional fiber fusion splicer [102], gluing
the cladding hole at ends with UV curable glue by utilizing the dependence of filling
speed on the size of the air holes [103, 104], closing the cladding holes when manufacturing PCF [102], and micromachining the cladding holes by using high power
femtosecond IR laser [105]. For the capability in our lab, we use a fusion splicer and
a Bunsen burner to quickly close the cladding holes, as well as use UV curable glue to
seal the cladding holes. As shown in Figure 5.7(c), we adopt the three-step method as
described in Reference [103] to selectively seal the cladding holes of a PCF (HC-800-1,
from NKT Photonics) with UV curable glue (NOA 71, from Norland Products). We
succeed to fill water in this selectively sealed PCF and guide 405 nm laser in a 1meter-long PCF with optical transmission efficiency of 10%. However, since the core
(9.5µm) of this PCF is relatively small and it is hard to align the fiber to be straight
at the beginning, the output light beam is almost completely depolarized arising from
mode conversion during propagation. Furthermore, the PCF fiber is very fragile, expensive, and the small central hole suffers the jam issue because of small dusts from
connector or liquids, which cause some difficulties in practical operations of a long
PCF exceeding 1 m, therefore we have not managed to use PCF fiber as a practical tool to measure NSOR in this experiment as far, although liquid selectively-filled
PCF could guide light for almost all transparent liquids even with ultralow refractive
index.
103
(a)
(b)
(c)
Figure 5.7: Photonic crystal fiber (PCF). (a) SEM imaging of the PCF. (Model:
HC-800-1. Purchased from NKT Photonics. Picture is adapted from Thorlabs) (b)
Optical microscopy imaging of the original PCF. Some light is guided into the fiber
from the other end for illumination. There are two type of holes, the central hole
(OD=9.5µm), and many cladding holes aside (OD=2.3µm). (c) The cladding hole is
sealed with UV cure epoxy (from Norland Products), while the central hole is still
open, only allowing liquids flowing through the central hole.
Figure 5.8: Flexible capillary tubings made of fused silica (TSH series, from Polymicro
Technologies). (left). ID=40µm, OD=180µm; (right). ID=250µm, OD=365µm. All
the tubings are coated with thin-walled polyimide.
For practical considerations, most hollow fibers used in this experiment is fused
silica capillary tubings, due to its low-cost usage, various choices of inner and outer
diameters and its flexibility as well as sturdiness. As shown in Figure 5.8, the flexible
fused silica capillary tubings used in our experiment mostly come from Polymicro
Technologies (TSH series), which provides high quality capillary tubings at low cost.
We make various choices of inner diameter (ID) and outer diameters (OD) of tubings,
104
and find that the capillary tubings of OD=360µm and ID=50µm∼150µm have a best
performance of filling liquids and guiding light, as these thick but flexible tubings
could be aligned very straightly at the beginning for accepting incoming collimated
laser beams, thereby reducing mode conversion when the light is introduced into the
liquid core hollow fiber. Of course, the usage of capillary tubings limits the choice
of liquids, as the refractive index of liquids should exceed fused silica, therefore we
only use several high index liquids, e.g. Toluene, Xylene and Limonene, for the
investigation of light guiding and polarization in liquid core hollow fibers.
5.1.3
Pulse sequence
Due to random fluctuations of the local magnetic field, the transverse relaxation time
T2 (i.e. dephasing time) of 1 H nuclei is about 1∼10 s in liquid samples at room
temperature. However, if the magnetic field is inhomogeneous across the sample
volume, especially when the size of sample is relatively large, the actual transverse
relaxation time T∗2 is reduced much more. The relaxation process of nuclear spin
magnetization is according to
∗
M = M0 · e−t/T2 ,
(5.3)
and the linewidth of signal peak at frequency domain broadens as ∆f 1 =1/(π·T2 ) ,
2
where the linewidth ∆f 1 is the full width at half height of a peak in the spectrum.
2
In the superconducting magnet used in this experiment, the magnet core is designed
for standard 5 mm NMR sample tubes, so the magnetic field is rather inhomogenous
as the outer diameter of coiled fiber ring is 1.5 inch, although we have optimized the
shim coils for maximal field homogeneity.
The free induction decay (FID) signal of 1 H in liquid Toluene after applying a
simple π/2 (90◦ ) pulse is shown in the range of 0 ms < t < 3 ms in Figure 5.11,
105
and its spectrum is shown in Figure 5.9, which is measured by a homemade NMR
spectrometer (For details, see Thesis [106]). Because there are two types of hydrogen
in Benzyl and CH3 groups, two peaks are shown in the spectrum. The linewidth
∆f 1 could be estimated from the full width at half height of a peak, which is about
2
300 Hz as shown in Figure 5.9, so the actual dephasing time T∗2 ∼1ms, which is much
shorter than T2 of sample in a small 5mm tube. Since the longitude relaxation
time T1 , originating from recovery of the longitude component of the nuclear spin
magnetization through thermal equilibrium relaxation, is usually several seconds for
1
H at room temperature, then in a simple π/2 pulse scheme, the effective time for
measurement is dominated by T∗2 , which is as low as 1 ms, leading to a low signal-tonoise ratio (SNR) in measurement. Therefore, some decoupling pulse sequence should
be applied to elongate the coherence time.
NMR Amplitude [mV/Hz]
0.3
0.2
0.1
0.0
-1000
-1500
-2000
-2500
Detuning [Hz]
-3000
-3500
-4000
Figure 5.9: The conventional NMR spectrum of Toluene, measured with a OD=1.5
inch toroid coil (see Figure 5.3). The x-axis is the relative frequency shift from
115.325049 MHz. There is a chemical shift (∼5ppm) between Benzyl (left peak) and
CH3 (right peak) groups. The full width at half height ∆f 1 is estimated as 300Hz,
2
which is equivalent to the actual dephasing time T∗2 =1 ms.
In order to improve SNR, we implement Carr-Purcell-Meiboom-Gill (CPMG)
pulse sequence [107, 108] to sustain the nuclear-spin coherence for longer periods
of time. As illustrated in Figure 5.10, in CPMG pulse sequence, a initial π/2 pulse
106
creates transverse magnetization, then after time τ /2 when the induction signal is
decayed sufficiently, the spin ensemble is out of phase due to the magnetic field inhomogeneity, a subsequent inversion pulse (πy ) reverses the spin direction, which also
inverses the phase difference, thereby resulting in a complete refocusing of spin magnetization after some time of evolution when the spin ensemble have the same phase
again; subsequent inversion pulses are repeatedly applied at time interval τ , in order
to refocus the spin ensemble for long time. Figure 5.11 shows the induction signal
of 1 H in Toluene when applying the first three inversion pulses in CPMG sequence,
where the spin magnetization could be refocused perfectly at time interval of 6 ms.
Carr-Purcell-Meiboom-Gill (CPMG) sequence
Figure 5.10: CPMG pulse sequence. π/2x is the 90◦ pulse applied on the x-axis on
the transverse x − y plane, and πy (inversion pulse) is the 180◦ pulse applied on the
y-axis on the same plane. The time interval between π/2x and the first πy pulse is
τ /2, while the subsequent πy pulses are separated by τ .
Inversion pulses in CPMG sequence could be applied for many times, then the
nuclear-spin coherence could be sustained for long time. Therefore, the effective time
in each cycle (T1 ) is elongated for larger SNR. As shown in Figure 5.12, inversion
pulses are repeated by 800 times in 5 s, the NMR induction signal decays slowly. The
envelop profile of the whole curve signify the ideal relaxation time after removing
environmental decoherence by periodic inversion pulses, and it shows that the decoherence time T2 is about 2 s, which is much larger than the actual T∗2 (∼1 ms) under
inhomogenous magnetic fields.
107
0.20
0.15
0.10
Signal (V)
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.25V
0
5
10
Time (ms)
15
20ms
Figure 5.11: The NMR induction signal after applying a CPMG pulse. A πx /2
pulse creates transverse magnetization, then the signal decays quickly in 3ms due to
inhomogeneous dephasing, in which some spin precess a bit faster while some other
spins are slow. The first 180◦ inversion pulse (πy ) applied at t=3ms flip the spin
direction, then the fast and slow spin are reversed, resulting in a complete refocusing
at time 6ms when all the spins have the same phase. Subsequently, periodic πy pulses
are applied at interval τ =6ms, thereby creating a periodic refocusing for up to several
seconds. Here, the pulse width is 9µs and 18µs of the π/2 and π pulse respectively.
5.1.4
High bandwidth photodiode amplifier
The resonant frequency for 1 H nuclei is 115 MHz at 2.7 Tesla, and the optical rotation frequency of NSOR is also equal to this value. In order to measure such a
high-frequency optical signal, the photodiode amplifier for detection should have a
high bandwidth larger than 115 MHz, as well as low noise level. At high frequency,
electronic circuits always have unavoidable parasitic capacitance, which exists between the parts of electronic components and wires because of their proximity to
each other. The parasitic capacitance, as well as the inherent terminal capacitance
in the photodiode, dominates the response of circuits at high frequency, sets the
circuit bandwidth and induces unexpected noise gain, while at low frequency these
capacitance are negligible.
108
Figure 5.12: The long-time performance of CPMG pulse sequence. After applying
inversion pulses (πy ) after 800 repetitions in 5 s, the NMR induction signal decays
slowly. The envelop profile of of the whole curve signify the ideal relaxation time after
removing environmental decoherence by periodic inversion pulses, and it shows that
the decoherence time T2 is about 2s, which is much larger than the actual T∗2 (∼1ms)
under inhomogenous magnetic field. Some sharp peaks that are significantly larger
than relaxation signal are the RF bursts when applying high-voltage RF pulses.
As shown in Figure 5.13, the photodiode amplifier circuit at high frequency is
the same as at low frequency, as shown in Section 3.1.5. However, the terminal capacitance Ci in the photodiode should be considered, as the photodiode could be
modeled as a current source ip bypassed with its terminal capacitance Ci . The feedback capacitance Cf compensates phase in circuit by counteracting with Ci , as well as
control noise level and bandwidth. At high frequency, the parasitic capacitance, e.g.
induced by leads of operational amplifier chip and pads on circuits, usually provides
this feedback capacitance, as this parasitic Cf is around 0.2 pF∼0.5 pF depending
on the manufacturing of the circuit. Apart from Ci and Cf , the input capacitance of
operational amplifier (op amp), such as common mode input capacitance Ccm and
differential mode input capacitance Cdm , also affect the behavior of circuit. The
109
Figure 5.13: The model of photodiode and the circuit for photodiode amplifier. The
photodiode is modeled as a current source ip bypassed with its terminal capacitance
Ci .
bandwidth of photodiode amplifier circuit at high frequency is generally determined
as [109]
s
f−3dB =
fGBW
,
2πRf (Cf + Ci + Cin )
(5.4)
where the bandwidth f
here is defined as -3dB bandwidth, f
is the gain
-3dB
GBW
bandwidth product of op amp, Rf is the feedback resistance, and Cin is the input
capacitance of circuits, including the input capacitance of op amp and other parasitic
capacitance in circuits besides Cf . Apparently, a small feedback resistance Rf , small
capacitance including Ci , Cin and parasitic capacitance Cf , and a high gain bandwidth product (GBW) of op amp is required to achieve a high-bandwidth circuit.
We choose some high-bandwidth and small-input-capacitance electronic components to manufacture the photodiode amplifier circuit. The photodiode used in this
circuits is a Si PIN photodiode (Model: S5973-02, Hamamatsu Photonics K.K.),
which has a high-speed response 1 GHz, with a spectrum response range from ultraviolet to near infrared, especially optimized for violet light. The photodiode (S5973-2)
has a terminal capacitance 2.5 pF, but could be suppressed below 0.5 pF after apply-
110
ing a 10 V reverse voltage on the photodiode. The op amp is ADA4817-1 (Analog
Devices, Inc.) with a high gain bandwidth product (GBW) ∼ 410 MHz, and its
common mode input capacitance Ccm =1.3 pF, differential mode input capacitance
Cdm =0.1 pF, so the input capacitance Cin is approximated as 1.4 pF.
In addition, in order to get a small parasitic capacitance, the whole circuit is
designed based on small surface mount electronic components with compact configuration, as shown in Figure 5.14. When the circuit is configured with Rf =510Ω and
without extra compensation capacitance, the measured noise level (no light input) is
√
√
about 6 nV/ Hz, near the calculated thermal noise level 3 nV/ Hz from the feed√
p
back resistor Rf (n = πkB Rf T). The shot noise level is 1.6 nV/ Hz, assuming
0.1 mW laser power is received by one photodiode. The bandwidth f-3dB is about
140 MHz, a bit smaller than the theoretical estimation 233 MHz (see Equation 5.4),
providing Cf + Ci + Cin ∼2.4 pF, partly due to the bandwidth limitation arising from
measurement instruments. In summary, the bandwidth and noise level of this photodiode amplifier circuit is capable of measuring high-frequency optical rotation signals
at 115 MHz.
Figure 5.14: Picture of the high bandwidth photodiode amplifier. (a) The compact
photodiode amplifier of one channel. Photodiode (Model: S5973-02, Hamamatsu
Photonics K.K.) is placed at the center. The operational amplifier is ADA48171 (Analog Devices, Inc.). (b) Two channel photodiode amplifier with differential
output. Circuits are enclosed in the blue aluminum box.
111
5.2
Polarization in liquid core hollow fibers
Maintaining the polarization linearity of light beam is crucial for the detection of
optical rotation, as the polarimetry is only capable of analyze rotation angle of the
linear polarized part of light. If the light beam is partially depolarized or circularly polarized, the detection efficiency will be reduced, leading to a smaller SNR.
In practice, almost all fibers, including single mode fibers, are birefringent, because
of fiber imperfection, scattering, external stress, electromagnetic fields and bending
[110, 111, 112, 113]. In a single-mode solid core fiber, the light still propagates in
near degenerate modes, HEx11 and HEy11 , with orthogonal polarization; so when the
fiber is bended or stressed, the fiber become locally anisotropic and causes a fast- and
slow -axis with an index difference (birefringence) for the pair of degenerated modes,
resulting in partial depolarization of light beam finally [114, 112]. In a multimode
solid core fiber, apart from the anisotropy issue for the degenerated polarized modes
for each mode, different modes have different guiding properties inside fiber, resulting in phase shifts between different modes, usually called modal dispersion, so light
beams propagating multimode fibers are depolarized normally.
Liquid core hollow fibers are actually step-index fibers. As far as we have tested,
the liquid core fibers filled with Toluene and Limonene are running in multi-modes.
Although the bending does not induce anisotropy of the liquid core, the cladding
still suffers from bending-induced birefringence, which performs like a linear retarder,
attenuates the polarization linearity of light, and reduces the SNR in experiments of
optical rotation. Besides, the mode conversion and fiber twist also leads to birefringence and depolarization. We try to understand these issues and implemented some
methods, like optimal mode matching and linear-birefringence suppression via chiral
liquids, in order to maintain the polarization linearity of light in the multimode liquid
core hollow fibers.
112
In this section, the theory of light guiding inside a multimode fiber will be reviewed
first. Then the issue of mode matching when the light beam is launching into the fiber
core, the bending-induced birefringence in liquid core fiber, and some issues about
bandwidth of fiber will be described sequentially.
5.2.1
Multimode liquid core fiber
When the high-index liquid is filled in a fused silica capillary tubing, this liquid core
fiber could be considered as a step-index fiber,
n(r) = n1 0 < r < a core,
(5.5)
= n2 r > a cladding,
as illustrated in Figure 5.15, where a is the core radius. Since the refractive index in
liquid core is close to the cladding normally, n1 ≈ n2 (n2 =1.46958 for fused silica),
it allows using weakly guiding approximation [115, 116], under which the modes
are assumed to be nearly transverse and can have an arbitrary state of polarization
consisting of two independent sets of modes in x- or y−polarization. In the following,
the analysis of step-index fiber follows the work originally by Gloge [115], as well as
other works [117, 116, 118] and some special conditions for the light guiding in the
liquid core hollow fiber in our experiments. In the weakly guiding approximation, the
transverse component of the electric field (Ex or Ey ) satisfies
∇2 Ψ = 0 µ0 n2
∂ 2Ψ
.
∂t2
(5.6)
For an ideally circular and straight fiber, n only depends on the radial coordinate
r, we can assume the wavefunction with the form
Ψ(r, φ, z, t) = R(r)Φ(φ)ei(ωt−βz)
113
(5.7)
a
Incident Waves
Core
z
Cladding
Air
z=0
Figure 5.15: The model of liquid core hollow fiber. The liquid core has index n1 ,
while cladding has index n2 . When (n1 − n2 )/n1 is small, the fiber could be modeled
as a weakly guiding fiber.
where β is the propagation constant and ω is the angular frequency of light, which
leads to a equation in cylindrical coordinates r and φ,
r2
R
d2 R 1 dR
+
dr2
r dr
1 d2 Φ
+ r2 n2 (r)k02 − β 2 = −
= +l2
2
Φ dφ
(5.8)
where k0 = ω/c = 2π/λ is the wave number in free space. Therefore, Φ should in the
form of cos(lφ) or sin(lφ), and Φ(φ) is periodic due to the circular symmetry of fiber,
i.e. Φ(φ + 2π) = Φ(φ), the number l should be quantized as
l = 0, 1, 2, ...,
(5.9)
where for each discrete l, there even allows several guided modes, which is known
as the guided modes of the system, designated as (Linear Polarized) LPlm modes
(m=1,2,3,...).
For simplicity, it is convenient to define the normalized frequency V , the normalized propagation constant b and numerical aperture NA
q
q
2π
2
2
V = k0 a n1 − n2 =
a n21 − n22 ,
λ
114
(5.10)
b=
β 2 /k02 − n22
n2e − n22
=
,
n21 − n22
n21 − n22
NA =
(5.11)
q
n21 − n22
(5.12)
where, ne = β/k0 (k0 = 2π/λ) is called effective index. Only for k02 n22 < β 2 <
k02 n21 (i.e. 0 < b < 1), the fields R(r) are oscillatory in the core and decay in the
cladding[119, 116], which are called guided modes in a fiber.
From Equation 5.8, the radial part of the equation follows as a standard Bessel
equation,
r
2
d2 R 1 dR
r2
2
2
+
+ V (1 − b) 2 − l R = 0;
dr2
r dr
a
2
2
d R 1 dR
2
2
2 r
+
r
− V b 2 + l R = 0;
dr2
r dr
a
0<r<a
(5.13)
r>a
Therefore, the solutions are given by [119, 116]
√
1 − b · ar )
√
for 0 < r < a,
Rco (r) = A
Jl (V 1 − b)
√
Kl (V b · ar )
√
for r > a.
Rcl (r) = A
Kl (V b)
Jl (V
(5.14)
Which has already incorporated the boundary condition ψco (a− ) = ψcl (a+ ) = A
(A is a constant).
Furthermore, considering the other boundary condition,
0
ψco
(r) |a− = ψcl0 (r) |a+ , we can obtain
√
√
√ Kl+1 (V b)
√
Jl+1 (V 1 − b)
√
√
Fl (b) = V 1 − b
−V b
= 0,
Jl (V 1 − b)
Kl (V b)
(5.15)
where Jn (x) and Kn (x) are the Bessel J-type function and modified Bessel K-type
function respectively[120]. This equation is called the characteristic equation for the
weakly guiding step-in fiber, and it limits the choice of b for given fiber parameter V
115
and discrete number l. Usually, there are several roots (b1 , ..., bm ) of Equation 5.15,
which are designated as blm , corresponding to the guiding mode LPlm . For example,
when V = 10, l=0, there are three choices for b, as shown in Figure 5.16, thus, there
are three guiding modes LP01 , LP02 , LP03 .
100
80
60
40
20
0
0
0.2
0.4
b
0.6
0.8
1
Figure 5.16: The graph of |Fl (b)| (see Equation 5.15) at V = 10 and l=0 . There are
three roots of b in the range of [0, 1], b01 , b02 , b03 , i.e. m = 1, 2, 3. These roots of b
corresponds to guiding modes LP01 , LP02 , LP03 . The largest b is taken as the first
solution and set m to 1.
From the definition in Equation 5.10, the normalized waveguide number V is
parameter of the fiber, which is determined by the wavelength λ, core diameter a,
core index n1 and cladding index n2 . When V is changed in terms of these parameters
for different fibers, the value of blm also varies for a fixed mode LPlm , which could
only be numerically calculated from Equation 5.15. . Figure 5.17 shows the b − V
characteristics for each guiding mode LPlm . Only when there is a root b between
[0,1], it allows a guiding mode LPlm . From these numerical results, there is cutoff V
value for each mode, e.g. when V < 2.4, it only allows one mode LP01 ; when V <
116
3.83, it then allows two modes LP01 and LP11 . The cutoff frequency Vl is determined
when b → 0 in Equation 5.15, which is equivalent to the condition [116]
J1 (V ) = 0 for l = 0,
(5.16)
Jl−1 (V ) = 0 for l ≥ 1 (Vc 6= 0).
Therefore, the cutoff frequency V is 2.4048 for LP11 , 3.8317 for LP21 , and so on.
When V < 2.4048, it is called single-mode condition, under which only one mode LP01 ,
called fundamental mode, could be guided through the fiber core (Although the cutoff
frequency of LP01 is 0, if V is too small, 0 < V < 1.5, the light power will reside more
in the cladding region than the core region, then the fiber is vulnerable to external
disturbance, bending and contamination, making it impossible to work in practice).
The fundamental mode supports two degenerate and orthogonal polarization modes,
while higher modes supports ≥4 degenerate polarization modes.
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
Figure 5.17: b − V characteristics of weakly guiding fibers. The b value is the root of
Equation 5.15, and it depends on parameter V for each mode. Only when there is a
root b between [0,1], it allows a guiding mode LPlm . There is cutoff V value for each
mode.
117
Bringing in the angular function Φ(φ) in Equation 5.8, based on ψ(r, φ)=R(r)Φ(φ),
we can obtain the transverse distribution of an x−polarized electrical field component
in a LPlm mode,
√
Jl (V 1 − blm · ar )
√
ψco (r, φ) = A
cos(lφ) for 0 < r < a,
Jl (V 1 − blm )
√
Kl (V blm · ar )
√
ψcl (r, φ) = A
cos(lφ)
for r > a,
Kl (V blm )
(5.17)
where blm is the normalized propagation constant for its corresponding mode and A is
a positive constant, while for y−polarized component, the angular function becomes
sin(lφ). From these expressions, as to the mode designation for linear polarization
mode LPlm , the azimuthal mode number l relates to the angular variation of the transverse electric field. If l is zero for LP0m , the transverse electric field is independent of
φ, which means it has circular symmetry. The radial mode number m corresponds to
the mth root in Equation 5.15, resulting in m−nodes of axial distribution of LPlm .
Some axial distribution of ψ(r, φ) is plotted in Figure 5.18.
LP01
LP02
LP11
LP12
LP21
LP22
LP31
LP41
Figure 5.18: Calculated axial intensity distribution in the fiber core (r ≤ a) of LPlm
modes guided by a step-index fiber with v=7.1. The colorful contour maps represent
the intensity of the square of electrical field amplitude |e|2 ; Red: high intensity; Blue:
low intensity. LP01 mode is very close to a Gaussian beam profile. A Gaussian beam
launching into the fiber core axial-symmetrically only excites LP0m modes, which
have circular symmetry as the incident Gaussian beam.
118
In traditional mode designations in the study of metallic waveguides in microwave
technology, there are rotationally symmetric TE (transverse electric) and TM (transverse magnetic) modes, and φ-dependent fields as hybrid modes HE and EH. For
a step-index fiber, only TE0m , TM0m , HElm and EHlm could be guided [115, 117].
The traditional designation is equivalent to LPlm , but the index is slight different,
LPlm ↔HEl+1,m , EHl+1,m for l 6= 1, and LP1m ↔HE2,m , TE0m , TM0m . The fundamental mode LP01 is equivalent to HE11 . In each LPlm mode for l ≥ 1, it contains several
modes, e.g. LP11 contains TE01 , TM01 and HE21 . Besides HE1m modes, other modes
are not linearly polarized, although some combinations of them could form a linearly
polarized mode.
The fundamental mode HE11 , i.e. LP01 , plays an important role in high-bandwidth
fiber communication, as well as the polarized light transportation in single-mode and
multimode fibers. In the core region, the transverse electric and magnetic fields of
HE11 mode are approximately linearly polarized, and there are two mutually orthogonal polarized degenerate mode with x− and y−polarization. In addition, the
transverse field in HE11 mode resides more in the central area of the core, as shown as
LP01 in Figure 5.18. Other higher modes HE1m have more energy distributed in the
cladding, making it more vulnerable to external disturbance, bending. In addition,
the mode conversion from HElm to higher modes could introduce phase shift and depolarization, as the propagation constant blm , and thereby the effective index ne (see
Equation 5.22), is slightly different for each modes. In practice, it is favorable to use
a single-mode fiber, or operating a multimode fiber as a single-mode one with some
methods. If a Gaussian beam is collimated axial-symmetrically into a circular fiber,
it only excites HE1m (m=1, 2, . . ., n) modes, i.e. circular-symmetry modes [119],
thus we mainly consider the HE1m modes in the following.
As to a capillary tubing with a core diameter 50 µm filled with Toluene, n1 =1.526
(Toluene) and n2 =1.4696 (Fused silica), for guiding 405 nm laser, the normalized
119
frequency V =159, allowing about 50 HE1m modes (if the light is not optimally coupled
into fiber because of some displacement or angular asymmetry, it allows more than
10000 modes including all HElm and TE0m modes). The cutoff core diameter for
single mode guiding of liquid Toluene core fibers is 0.76 µm, which is beyond the
capability of current commercial hollow fiber manufacturing technology. In the case
for Limonene (n = 1.484), the results are similar: in a 50µm core fiber, V =80, it allows
25 HE1m modes, and the single-mode cutoff diameter is 1.5 µm. For a capillary tubing
with a smallest core diameter 2µm that we can get commercially (TSP002, Polymicro
Technologies), it allows 2 modes, but it also suffers more from liquid imperfection,
e.g. dust, thermal fluctuation and thermal lensing, which leads to complicate mode
conversions inside fibers and thereby depolarization.
5.2.2
Mode matching
When a light beam is coupled into the core of a multimode fiber, it could excite many
modes of this fiber. Here, ‘excite’ means the transverse electrical field component
of the light beam could be matched with or decomposed into the distribution of
some modes in the fiber. When a Gaussian beam is coupled axial-symmetrically into
a circular fiber, it only excites HE1m modes. Furthermore, if the diameter of the
laser beam, focal length of the lens and core diameter is appropriately configured,
only the fundamental mode HE11 could be significantly excited, which is called mode
matching [121]. Under this condition, the launched fundamental mode will propagate
through a long multimode (liquid or solid core) fiber, as long as the fiber is free from
considerable bending, distortion and any other external stress, then the single mode
operation in multimode fibers could be achieved perfectly, even for high-bandwidth
optical communications with tens of gigahertz [92, 122]. Any deviations from this
optimal coupling will give rise to mode conversions at the beginning of the fiber,
resulting in a sharply decreasing polarization linearity [121].
120
When a Gaussian beam is launched axial-symmetrically into the core of a circular
fiber, it only excites HE1m modes, and the excitation efficiency is given [119, 121] by

P1m
 1
Z
U
K
(W
)
2a
m
0
m
 J0 (Um R) e−R2 /(w1 /a)2 RdR
=2 
V K1 (Wm ) w1
J0 (Um )
0
2
Z+∞
K0 (Wm R) −R2 /(w1 /a)2
+
e
RdR
K0 (Wm )
(5.18)
01
where w1 is the radius of collimated Gaussian beam at the surface of the fiber, a is
the radius of fiber core , V is the normalized frequency of the fiber (See definition
in Equation 5.10), and Jn (x) and Kn (x) are Bessel J-type and modified Bessel Ktype function. The parameter Um is numerically determined by J0 (U ) = 0, which is
p
2 , both of which are
actually the cutoff frequency of mode HE2m and Wm = V 2 − Um
m-dependent constants. The parameter Um is listed in Table 5.1. When the liquid of
Toluene is filled in the fiber core of diameter 2a = 50µm, V = 159. The calculated
excitation efficiency in terms of w1 /a is plotted in Figure 5.19, where the excitation
profiles for the first 6 modes are shown. For a fixed light wavelength λ and fiber
core radius a, the radius w1 of Gaussian beam collimated on the fiber core could be
controlled by the collimating lens and the Gaussian beam spot size, since
w1 =
4λf
3πw0
(5.19)
where f is the focal length and w0 is the radius of the Gaussian bean when it incidences
on the collimating lens. From Figure 5.19, we can see that a maximal excitation efficiency of HE11 is about 95% around w1 /a=0.68. The mode matching profile is similar
for Limonene core fiber (2a=50µm, V =80), i.e. the optimal mode matching condition
is met around w1 /a∼0.68, as well as for other high V value [121]. From Equation
5.18, the relative ratio of mode excitation at mode m1 and m2 almost depends on
the ratio of K0 (Wm1 )K1 (Wm2 )/(K1 (Wm1 )K0 (Wm2 )), but numerical calculations show
121
that K0 (Wm )/K1 (Wm ) is insensitive to V , but almost equal to 1 for the lowest modes
m <10, which are the main excited modes in normal condition 0.2 < w1 /a < 1.2.
Therefore, the mode matching condition w1 /a∼0.68 is robust for various liquid core
fibers. Under the condition of mode matching, the optimal beam diameter (2w1 ) is
34µm for a liquid core fiber of diameter 50µm. For a typical beam diameter 1mm, to
meet the mode matching condition, the best choice of lens is f ≈5 cm.
Excitation efficiency [%]
100
80
60
40
20
0
0.2
0.4
0.6
0.8
1
1.2
Figure 5.19: Mode excitation efficiency versus w1 /a. w1 is Gaussian beam radius, and
a is the fiber core radius. V =159. The incident Gaussian beam is supposed to axialsymmetrically launch into the fiber core. Other high modes, e.g. HE17 , HE18 , etc.,
are not shown here, since they all have small excitation efficiency around w1 /a=0.7,
while we are more concerned with the fundamental mode HE11 . A maximal excitation
efficiency of HE11 is about 95% around w1 /a=0.68. Under this condition, the optimal
beam diameter 2w1 is 34 µm for a liquid core fiber of diameter 50 µm. The relative
mode excitation efficiency is insensitive to the change of V .
5.2.3
Bending induced birefringence
For an ideally manufactured and straight fiber without external stress, electromagnetic fields or imperfections, the excited modes from the beginning of the fiber will
122
m
1
2
3
4
5
6
Um
2.4048
5.5201
8.6537
11.792
14.931
18.071
Table 5.1: The cutoff frequency Um for mode HE2m . Um are the roots of J0 (Um ) = 0.
keep propagating as described by the Equation 5.8 within the weakly guiding approximation, until it exits from the end of the fiber. In practice, if a fiber is longer
than several meters, it is inevitably bent in operation. In our experiments for the
measurement of optical rotation, a 10-m-long fiber needs to be curled for many loops.
However, the bending of a fiber will introduce birefringence, as the each linearly
polarized guiding mode (HE1m ) has two degenerate modes, which also applies for
the fundamental mode HE11 in all fibers, no matter it is single-mode or multimode
[123, 124, 114]. In addition, since the bending of fiber violates the ideal circular symmetry of a step-index fiber, it also induces mode conversion, which leads to optical
loss and depolarization, as well as birefringence. In this section, the bending induced
birefringence for an individual mode will be studied, while the bending induced mode
conversion will be introduced in the next section.
For the guiding modes HE1m , since each mode has two degenerate and orthogonal
modes, the x− and y− polarized modes, their propagation constants βx and βy are not
equal to each other, if the fiber’s circular and axial symmetry is violated by bending,
as shown in Figure 5.20, the fiber is linearly birefringent. The difference βx − βy is
known as the linear birefringence. If βy > βx , the phase velocity of x−polarized mode
is fast than that of the y−polarized mode, then the x axis is called fast axis of the
birefringent fiber. If βy < βx , the y axis is the fast axis. The linear birefringence is
defined as
123
Bl =
βx − βy
= nx − ny ,
k
(5.20)
where nx and ny is the effective index for x− and y−polarized modes in the fiber. For
a fiber is under distortion, in a magnetic fields, or the fiber core is composed of chiral
material, it also induced circular birefringence, as a fiber also supports two circularly
polarized modes that propagate with two propagation constants, βR and βL .
Figure 5.20: A bent liquid core fiber with radius R. For fused silica cladding, the
plane of curvature is the fast plane.
For a liquid core fiber bent locally in a circular arc of radius R shown in Figure
5.20, the inner and outer boundaries of the cladding arc is different, because the arc
length is either elongated or compressed by bending. Strictly speaking, the tension
exerted on the inner cladding and outer cladding is slight different, here for qualitative
analysis, the detailed computation is not included. Due to the deformation caused
by bending, the index of cladding will also be changed due to the a mechanical stress
effect [114]. From the mechanical analysis and numerical calculation in Reference
[114, 117], it is found that the plane of curvature is the fast plane.
Recall from Equation 5.10 and 5.11,
q
V = k0 a n21 − n22
124
(5.21a)
b=
n2e − n22
n21 − n22
(5.21b)
In first order approximation, ne ∼ n1 ∼ n2 , then from Equation 5.21b, we can
obtain
ne ≈ n2 + b(n1 − n2 )
(5.22)
Therefore, in the liquid core fiber, let alone the thermal fluctuation or some other
effects on the liquid core, normally only the cladding index n2 will be changed in a
bent fiber. Now suppose there is a small perturbation, ∆n2 , exerted on the cladding
index n2 , then the small change of the effective index ne according to ∆n2 is
∆ne = (1 − b)∆n2 + ∆b · (n1 − n2 )
(5.23)
Actually, for fixed l and m, b is a function of V , determined numerically, so
db ∂V
db −V · n2
∂b
=
=
∂n2
dV ∂n2
dV n21 − n22
(5.24)
Apply the first approximation again, n1 + n2 ∼ 2n1 , it follows that
∂b
V db
(n1 − n2 ) = −
∂n2
2 dV
(5.25)
Finally, the variation of ne in terms of the perturbation of n2 is concluded as
∆ne = (1 − b)∆n2 −
V db
∆n2
2 dV
V db
= (1 − b −
)∆n2
2 dV
(5.26)
In a bent liquid core fiber, the parameter b is still the same for x− and y−polarized
modes in the same HE1m mode, while n2 is usually different since the fiber is
anisotropic after bending. If the fiber is bent such that the effective index ne for the
125
two degenerate modes, x− and y−polarization modes, are different, especially when
one of polarization modes is on the plane of curvature while the other one is normal
to, as shown in Figure 5.20, then the linear birefringence could be given as
Bl ≈
V db
1−b−
2 dV
∆n2(x) − ∆n2(y) (5.27)
The linear birefringence acts like a linear phase retarder, delay the phase of the
polarized mode in slow-axis, resulting in a elliptic polarization. Since the fiber is
curled by many loops, it induces periodical changing of birefringence, where the beat
length LB could be defined as
LB ≈
1−b−
V db
2 dV
λ
∆n2(x) − ∆n2(y) .
(5.28)
When the light inside the bent fiber propagates for every length LB , the polarization
state restores to the original state, therefore an initial linearly polarized light becomes
elliptically polarized (0<δ<90◦ ), circularly polarized when a 90◦ linear phase shift
is achieved as happens in a quarter-wave (λ/4) plate, then go back to elliptically
polarized (90◦ <δ<180◦ ), and becomes linearly polarized again when the phase shift
reaches 180◦ . In the remaining half circle, the phase shift δ goes from 180◦ to 360◦ ,
running over elliptic, circular and linear polarizations again. The periodic polarization
transformation is illustrated in Figure 5.21. Another period begins, when the light
propagates in the subsequent beat length. For single-mode fiber, with appropriate
bending, the emerging light beam from the exit of fiber could be still linearly polarized,
providing the phase shifts between the degenerate modes is times of 180◦ . In a
multimode fiber, it is more complicated, the beat length LB for each mode is not the
same, since blm is different for each modes.
Besides the linear birefringence, in the presence of distortion, external magnetic
field or the liquid in the core is chiral, all of which induce optical rotations via circular
126
Fast axis
Slow axis
gth
n
e
l
t
Bea
Figure 5.21: Periodic phase shift in a linear birefringent fiber. In one beam length,
the phase shift δ is changing from 0 to 360◦ .
birefringence, there will be a superposition of the circular birefringence and bending
induced linear birefringence effects, and the fiber behaves as linear phase retarder and
polarization rotator simultaneously. For an infinitesimal piece of fiber with length ∆L,
suppose the linear birefringence induces a linear phase shift δ and an optical rotation
α, from Jones calculus and previous results in the case of coexistence of two types of
birefringence [125],



 Ex 


Ey
l1 +∆l


 A −B   Ex 
=


B A∗
Ey
(5.29)
l1
where
φ
φ
A = cos
+ i · cos χ sin
2
2
(5.30a)
φ
B = sin χ sin
2
(5.30b)
127
2
φ 2
δ
=
+ α2
2
2
tan χ =
α
δ/2
(5.30c)
(5.30d)
When there is no circular birefringence, from Equation 5.30c
φ=δ
(5.31)
which means the fiber is purely linear birefringent, performing as a pure linear phase
retarder.
When the linear phase shift is much larger than the optical rotation angle α, then
φ |αδ ∼ δ +
2α2
+ O [α] 4
δ
(5.32)
which means that the linear phase shift δ dominates the whole phase shift, while
the small circular birefringence rotation angle α is suppressed into second order, and
even dividing a larger δ. This effect seriously deteriorates the measurement of optical
rotation, for example, the actual Faraday rotation will be reduced much more than a
half.
But when there is a circular birefringence rotation angle α that is much larger
than a linear birefringence phase shift δ, from Equation 5.30c,
φ |δα ∼ 2α +
δ2
+ O [δ] 4
4α
(5.33)
which means the dependence of φ on δ is reduced to 2nd order, as well as dividing
a large α, ∆φ ∼ δ 2 /(4α). Therefore, when β δ, the linear birefringence phase
shift δ could be neglected. Therefore, here comes a simple method to suppress linear
birefringence in a bent fiber: filling the fiber core with chiral liquids, which have
128
an larger rotation angle than the linear birefringence phase shift. This methods is
applied in the Limonene filled fiber in our experiment.
Since the linear phase shift is due to the birefringence of the fast- and slow-axis
when light is guided in a bent fiber, there are some methods devoted to compensate
the phase shifts between these two axis. One way is curling the fiber into two loops
at right angles to each other and forming a right-angle figure 8 configuration [126], as
shown in Figure 5.22(a). In this right-angle 8 configuration, the fast- and slow-plane
is interchanged, when two linear polarized modes enter into the two orthogonal loops
alternatively, thereby the effects of fast- and slow-plane are applied on the x− and
y−polarized modes equally after the light exits from these double circles. The other
method is implementing a Faraday rotator mirror (also called ortho-conjugation reflector) [127, 128, 129], which could coupled the light back in the fiber with modes
conjugation, as shown in Figure 5.22(b). For the light emerging from a linearly birefringent fiber, the Faraday rotator consisting of a permanent magnet is configuration
that it could rotate the light polarization plane with 45◦ for each pass, then it induced
a 90◦ rotation after the light is reflected from a conjugated mirror, which could collimate the light beam back into the fiber. Finally, when the light is coupled back into
the fiber, the mode propagated in fast axis is now propagating in the slow axis, therefore the total phase shift is equalized for both modes. Of course, these two methods
are only effective when the circular birefringence arising from chiral liquid or Faraday rotation is stronger than the linear birefringence arising from the fiber, otherwise
the circular birefringence is suppressed by the linear birefringence. Also, the fiber
should be operated to guide fundamental mode mainly via optimal mode matching
as far as possible, since using neither of these two methods could not compensate the
intermodal phase shift.
129
Figure 5.22: Right-angle 8-configuration and Faraday rotator mirror. Both of these
two methods are used to compensate the phase shift between fast- and slow-axis of
the guiding light beam.
5.2.4
Mode conversion and bandwidth of multimode fiber
The mode matching described in Section 5.2.2 only involves the mode excitation
when the light beam is coupled into the fiber. When the light is guided inside the
fiber for some length, there will inevitably induced some mode conversion [130], or
named as mode coupling, which transfer one mode into other modes, leading to power
and polarization state redistribution. The mode conversion could occurs when there
is scattering, external field turbulence [131], distortion [132, 122] and especially the
bending of fiber [133, 122, 134, 135]. When the fiber is coiled in many loops densely,
the distortion that is induced when curling the fiber induces strong mode conversion,
and thereby depolarization and pulse dispersion. This case occurs in our experiments,
as we need to coil a 10-m-long fiber with a small radius < 2cm, then the mechanical
tension and distortion exerted on the inner layers of fiber is considerable. Therefore,
the fiber distortion is one of the main problems in our experiments, as well as the
dust issue in liquid core fibers. Of course these two issues could be solved partially
when the fiber is curled uniformly and cleanly handled with extra microfilter. Apart
from linear birefringence, the inevitable bending of fiber also have several unfavorable
effects on the light propagation, as well as in the measurement of optical rotation,
which will be covered in several aspects as follows.
130
In most cases, as long as the fiber imperfection is not considerable and distortion is
controlled well when curling a long fiber, the bending is the main source contributing
to the mode conversion [92, 136, 122], and the bending induced mode conversion (or
called mode coupling) has been confirmed experimentally and explained theoretically
[136, 130, 137, 138, 139, 140, 135]. In the region that the fiber is bent, the geometry
condition differs from the circular and axial symmetry in a straight fiber, so the mode
structures and field distributions in this region are also distinct from the straight
region. Some analysis based on perturbation theory could explain most phenomenon
about mode conversion in the curved region [134, 135, 137], and it is concluded as
each HE1m modes excited by the axially incidence Gaussian beam only couples to the
HE2m , HE2,m−1 , TE0,m and TE0,m−1 [134]; In particular, the fundamental mode HE11
only couples to the HE21 and TE01 , which means the mode only strongly couples
with its adjacent modes in terms of l and m; In addition, it is also shown that the
polarization direction of all modes except the HE1m modes changes [137], and field
distributions are also deformed in the plane of curvature [138, 137, 141]. The bending
of fiber, not only leads to linear birefringence for any linearly polarized mode in all
fibers as described in Section 5.2.3, but also induces optical loss, pulse dispersion and
depolarization, because the fundamental mode could be coupled into higher modes
in a multimode fiber.
As described in Section 5.2.1, higher modes distribute more and more power in
the cladding. The mode conversion into higher modes will leave the light waveguide
vulnerable to external perturbation. Furthermore, high order modes will transform
into radiation modes in some possibility in bending area, resulting in optical loss.
As to the large core multimode fibers used in our experiments, a =25µm, V =80 for
Limonene and 159 for Toluene, the optical leaking via cladding is negligible, as well
as the PCF fiber (HC-800-1), but the power attenuation is due to liquid imperfection
and dust. As we have tested, the strong optical loss in Teflon AF tubing is mainly
131
due to scattering even when the tubing is slightly bent, as the inner wall quality of
the Teflon AF tubing is not very good.
The mode conversion from the fundamental mode HE11 to higher modes induces
birefringence and depolarization. If the fundamental modes is converted into HE1m
modes through first order and higher order conversion, the propagation constant blm
is different for them, which leads to some difference between their phase velocity
and group velocity. Although the phase shift in two degenerate modes in each HE1m
modes (name as intramodal birefringence) could be compensated in some configuration as shown in Figure 5.22, the intermodal phase shift is not easy to handle.
As we explored, the Toluene or Xylene core fiber appears significant intramodal and
intermodal linear birefringence, which surpasses the Faraday rotation. Only when
we use Limonene, the optical rotation could be observed repeatably, since the chiral
induced optical rotation is much larger than the intermodal and intramodal linear
birefringence. Apart from birefringence, the depolarization is an inevitable problem
throughout all of our measurements about liquid core fibers. The depolarization is
induced because the HE1m modes is coupled into non-polarized modes, e.g. HE11
could be significantly converted into TE01 , which is actually a scalar mode without
polarization; For l ≥ 2, most HElm modes, e.g. HE21 , are also non-polarized [117].
Since the coil diameter for fiber in our experiment is < 2 cm, then mode conversion
into non-polarized modes is considerable. As we measured, the maximal polarization
ratio we could get is about 60% for a densely coiled liquid core fiber, while it could
reach >90% for a relatively short and non-curved one.
As stated above, after mode conversion in a multimode fiber, many modes could
be excited in the same fiber, the phase velocity νph for mode LPlm (HEl+1,m ) is,
νph =
c
c
c
ω
=
≈
≈ (1 + ∆1 − blm ∆1 )
β
ne
n2 + blm (n1 + n2 )
n1
132
(5.34)
where ∆1 = (n1 − n2 )/n1 . The phase speed is between c/n1 and c/n2 , because
0 < blm < 1. Similarly, for a fiber length L, the group velocity is given [117] as
νgr =
dβ
=
dω
c
d(kne )
dk
=
d
(kn1 )
dk
c
− ∆1 n1 1 −
d
dV
(V blm )
(5.35)
Because the propagation constant blm depends on the mode number l and m, then the
phase velocity νph and group velocity νgr differs among different modes, which results
in intermodal dispersion for long distance light propagation. The effect of intermodal
dispersion introduces pulse distortion and broadening. The intermodal phase shift
between two modes LPl1 m1 and LPl2 m2 is
θl2 m2 − θl1 m1 =
ωL
(n1 − n2 ) (bl1 m1 − bl2 m2 )
c
(5.36)
For a ω=115 MHz, L=10 m, n1 =1.484 and n2 =1.4696, we can find the maximal phase
shift between two different modes is 20◦ . In addition, the intramodal phase shift, e.g.
for the fundamental mode HE11 , is
|θx − θy | =
V db
ωL
· (1 − b −
)∆n2 ,
c
2 dV
(5.37)
which results in a rotation angle ∼1.1◦ (∆n2 is adopted from the calculation in [142],
when acl =180µm, bending radius R =1.9 cm). Although for HE11 mode, the phase
shift is pretty small, it is increasing fast for higher modes. Therefore, if the fiber
is bent and twisted so that the fundamental mode is converted into higher modes,
larger phase shift will be induced. When the phase shift is considerable comparing
with the signal phase period 2π (360◦ ), the high frequency signal will be attenuated
according to the interference between different modes, thus limiting the bandwidth of
the signal. The difference of intramodal phase velocity and group velocity also exists
for a single-mode fiber, because of the propagation constant shift on the fast- and
133
slow-axis on a anisotropic fiber. Both the intermodal and intramodal dispersion limit
the bandwidth of the light signal for transmission.
Axial ray
impulse
input
Extreme meridional ray
Dispersive
output
Figure 5.23: The axial ray and extreme meridional ray. The distance difference
induces the phase time delay, thereby the phase delay, at the output. As a result, an
impulse input with gives a dispersive output with finite time width δt.
Here we show a simple model for the bandwidth of multimode fiber (intermodal
dispersion) based on the classical analysis of geometric optics theory. As shown in
Figure 5.23, for a multimode fiber, when the light beam is coupled into the fiber,
there are many choices incident angle as long as the the first reflection angle of the
light beam is less than the critical angle θc , where the critical angle is defined as
sin θc = n2 /n1 under the total internal reflection condition. In this figure, it shows
the axial ray (blue), as well as the extreme meridional ray (red) (reflection angle =
θc ). The axial ray has a shortest distance during propagation inside the fiber, while
the extreme meridional ray has a longest distance, which results in a time delay for
them. The difference of distance shown here is equivalent to the difference in index
in the analysis of wave optics in the previous context. The time delay could be easily
calculated as
Tmax =
Ln1
c·cos θ
Tmin =
=
Ln21
cn2
Ln1
c
δt = Tmax − Tmin =
(5.38)
Ln21
cn2
−
Ln1
c
where L is the length of fiber, c is the light speed, and cos θ = sin θc . Because
n1 − n2 n1 or n2 , the expression of δt could be further approximated as
134
Ln21
δt =
cn2
n1 − n2
n1
≈
L(n21 − n22 )
L(N A)2
=
,
2cn1
2cn1
(5.39)
where N A is the numerical aperture. Now assume an impulse input (i.e. all light
components have the same phase) of light is injected into the fiber, the output beam
will possess time width δt due to the time delay of different components of light. The
output pulse with time width δt is thereby the impulse response of the fiber,
1
δt
, |t| <
δt
2,
δt
h(t) =0, |t| >
2
h(t) =
(5.40)
where the amplitude is 1/δt for normalization. The fourier transform H(f ) of the
impulse response h(t) determines the frequency bandwidth of the impulse response
as a output beam.
H(f ) = F[h(t)] = sinc(f δt ), (sincx ≡
sin πx
)
πx
(5.41)
as plotted in Figure 5.24. The first zero of H(f ) at 1/δt is defined as the essential
bandwidth of fiber. In other words, from the geometric optics, the bandwidth of a
multimode fiber is
BW =
1
2cn1
=
,
δt
L(N A)2
which is a good estimation for a multimode fiber.
135
(5.42)
1
Bandwidth
0
0
Figure 5.24: The frequency spectrum of impulse response h(t):H(f ) = sincf δt . The
essential bandwidth of fiber is defined as the first zero at 1/δt.
For a 10-m-long hollow fiber filled with Limonene, the bandwidth is about 2.0
GHz. This calculation is rough estimation, while a more accurate calculation is based
on time-dependent power flow equation [143, 144, 145], but the bandwidth is still
beyond 150 MHz. Since the optical rotation in high-field NSOR has a high frequency,
e.g. 115 MHz for the 2.7 T magnet, the bandwidth of rotation signal should also be
concerned. It is reported that an attenuation of 7 dB/km was achieved over kilometer
lengths of liquid-core multimode optical fiber with up to hundreds of MHz bandwidth
[92, 146]. However, in our experiment of the Limonene core fiber, when performing
a high-frequency 115 MHz Faraday rotation measurement, we almost can only get
<1% signal as expected from theoretical calculation, while in the low frequency (1-20
KHz) we can get a good agreement between measured value and theoretical value.
It is possible that the bandwidth is limited by other instruments or some unknown
effects arising from densely bent fiber. As far, for the Limonene core fiber, we have
not achieved a high-bandwidth light guiding.
5.3
Experiments with hollow fibers
The liquid core hollow fiber used in this section is the fused silica capillary tubing filled
with liquids, so only liquids with high index (n>1.47), for example, Toluene, Xylene
136
and Limonene, could guide light throughout the fiber. As far as we attempted, thin
hollow fibers with OD≤150 µm tend to be bent in the beginning, resulting in a poor
mode matching and high degree of mode conversion in the beginning part of the fiber;
For the hollow fibers with ID < 10 µm, it is vulnerable to liquid imperfection, such
as dust, although we have implemented microfilter (frit size ∼ 1-2 µm). Capillary
tubings with relative thick diameter (OD=360 µm) and large core (ID=50∼150 µm)
is favorable for the easy of practical operations in liquid filling and optical alignment
to approach optimal mode matching.
As described in Section 5.2.1, when a ID=50 µm capillary tubing is filled with
Toluene and Limonene, it becomes a multimode step-index fiber, with high normalized frequency V =159 and V = 80 respectively. In order to mainly excite the
fundamental mode in the multimode fiber, a mode matching condition w1 /a = 0.68
should be applied in terms of the choice of lens and incident beam diameter, as well
as injecting the Gaussian beam axially symmetrical to the fiber core. In practice,
the beam alignment is not perfect and the bending of fiber induces birefringence and
depolarization, as discussed in Section 5.2.3 and 5.2.4, we could not obtain a perfect
linearly polarized output light beam emerging from the fiber.
In this section, the transmission increment is described as a practical issue discovered in Toluene and Limonene core fiber. In Subsection 5.3.2, the depolarization for
all liquid core fibers and large linear birefringence in Toluene core fiber is discussed.
In the final subsection, we found the choice of chiral liquids (Limonene) suppresses the
linear birefringence, and a Faraday rotation test under low frequency is performed,
while this test under high frequency is not successful, mainly due to the bandwidth
limitation arising from intramodal and intermodal dispersion.
137
5.3.1
Transmission increase
The first step in manufacturing a liquid core fiber is filling pure liquid into a hollow
fiber. In our experiment, we filled Toluene and Limonene in fused silica capillary
tubings. Since the index of Toluene and Limonene is higher than fused silica, light
could be guided in this step-index fiber. When the light is coupled into these fibers,
we found that it could not reach a maximal transmission in short time, especially for
a long liquid core fiber. Although it is not related to the physical insights in optical
rotation experiments, technically speaking, it obstructs all the liquid-core fiber based
measurements and attenuates the signal-to-noise ratio if the light transmission is low.
1.2
5m fiber
Light intensity [mW]
1
0.8
0.6
0.4
0.2
0
0
5
10
Time [hour]
15
20
Figure 5.25: Transmission increase in 5-m-long hollow fiber. The liquid filled in the
fiber core is Limonene. The ID of fiber is 50 µm. The input power is 12.5 mW.
Figure 5.25 shows a typical slow transmission increase in a 5m-long hollow fiber
filled with Limonene. Once the liquid is filled in the fiber and all the openings are
closed up, the liquid become static inside the fiber core. Soon after the light is
launched into fiber, some transmission is immediately set up in seconds. As shown in
Figure 5.25, about 0.17 mW power could be detected in the output. However, as time
138
goes on, the transmission slowly increase, and finally reach a maximal value after 20
hours. If the transmission is expressed in a exponential form I0 (1 − e−t/T ), the time
constant is in Figure 5.25 is estimated as 4 hours. For short fiber, the time constant
is much short, e.g. for a 50cm-long fiber, time constant is around 10 minutes. As the
length increases, the time constant grow rapidly. In addition, for different liquids, the
time constant for Toluene is less than Limonene significantly. As the fiber connector
and fiber cleaving is checked carefully, it is confirmed that the slow transmission
increment arises from the liquid core fiber itself. The slow transmission increase is
possibly due to small bubbles induced by the gas dissolved in liquids, thermal effects,
e.g. thermal lensing or impurities in liquids.
In order to explain possible reason for this issue, we used degassed Limonene
processed through liquid nitrogen degasification technique, as well as use helium gas
to pump the liquid into fibers as helium normally has a small solubility in liquids, but
it did not help speed up the transmission increment. Since the slow transmission and
equilibrium looks like a thermal equilibrium, we guess there is some thermal effects
when the light is propagating in the liquid medium. A typical thermal effect is the
thermal lensing, the liquid region is hotter on the beam axis, compared with the outer
regions, causing some transverse gradient of the refractive index. The thermal lensing
occurred in fibers will change the propagating angle of guiding light and cause optical
leakage from the cladding when the angle is larger than critical angle. But in the long
term, when the liquid in the whole fiber is heat up sufficiently and achieve some point
of equilibrium, the optical loss will be much smaller, then a maximal transmission is
reached. This explanation is consistent with the transmission loss when the liquid
become flowing, as shown in Figure 5.26, and then after stopping the liquid flow, the
equilibrium well set up again after long time.
Another possible reason is liquid impurities arising from external contamination or
photochemical reactions under laser illumination, which might have some absorption
139
1
0.9
5m fiber
Release flow valve
Light intensity [mW]
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
Time [s]
400
500
600
Figure 5.26: Transmission drop when the liquid is flowing. Tested in 5 m, ID=150
µm fiber filled with Limonene. The orange line indicates the time when the liquid
valve is released.
of 405 nm laser and it finally becomes saturated after long time exposure under light.
Figure 5.27 shows a comparison of a fresh fiber and a reused fiber. Here, fresh means
it is the first time to use the fiber, while a reused fiber refer to liquid core fibers
that have been illuminated with light and reflowed with liquids. It seems that, for
a 50cm-long fiber, the time constant for a fresh fiber is very short, while it is much
longer in a reused fiber. Therefore, some external contamination or photochemical
reactions introduces some impurities into the fiber and they have some affinity to
the inner surface of the hollow fiber. As far, we have not completely explain the
slow transmission increase, the only method to overcome this problem is just wait for
long time until the light transmission reaches maximum, because we found that the
equilibrium transmission state could be maintained stably for more than one day.
5.3.2
Depolarization and linear birefringence
In this thesis, the key component for the high-field NSOR measurement is a long but
densely coiled liquid core fiber. Due to the bending of fiber, the linear birefringence
and depolarization inevitably occurs, as discussed theoretically in Section 5.2.3 and
140
1.8
1.2
1.6
Fresh fiber
1.2
Light intensity [mW]
Light intensity [mW]
1.4
1
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
0
0
0.2
0
Reused fiber
1
100
200
Time [s]
300
400
0.2
0
1000
2000
Time [s]
3000
Figure 5.27: The comparison of transmission in a fresh fiber and a reused fiber. Tested
in a 50 cm, ID=150 µm fiber filled with Limonene.
5.2.4. The output light beam emerging from the liquid core fiber is partially depolarized, partially circular polarized and partially linearly polarized. Here we define
polarization linearity as the ratio of linear polarization of light.
The measurement of polarization linearity is performed in an apparatus illustrated
in Figure 5.28. In this scheme, a linearly polarized laser beam is injected into a
densely coiled liquid core fiber, and the output light beam exiting from the other end
of the fiber is analyzed by a rotating half wave plate (λ/2 waveplate for 405 nm) and
a polarimetry. The depolarized and circular polarized parts of the light will not be
affected by the half wave pate, but only the linear polarized part of the light is rotated
by the half wave plate. Once the half waveplate rotates about an angle θ, the linear
polarization plane of the light is rotated by an angle 2θ. Thus when the half waveplate
is continuously rotating with aid of a stepper motor, the linear polarization plane of
the output beam is thereby rotating periodically, then the light intensity received by
two photodiode after the polarization beam splitter also oscillates.
141
4000
Rotating
λ/2 Waveplate
Linear
polarizer
polarizing
beam splitter
Photodiode
Laser
Liquid Core fiber
Photodiode
Figure 5.28: Measurement of polarization linearity of the light emerging from the
liquid core fiber. The polarization is rotated by a λ/2 waveplate. Powered by a
stepper motor, the λ/2 waveplate is continuously rotating at low frequency.
The oscillating light intensity of two channels of photodiodes is plotted in Figure
5.29. The waveplate is driven to rotate at 1.4 Hz, then the oscillating frequency is
2.8 Hz. As we can see from this figure, the two sum of the two channels keeps as a
constant value 1.3 mW, because the whole intensity from the two channels is conserved
no matter how the waveplate is rotating or polarization beam splitter is placed. If the
light beam is completely polarized, the peak-to-peak amplitude is equal to the sum.
As shown in this figure, the peak-to-peak amplitude of each channel is about about 0.4
mW, which actually means only this part of light could be modulated by the half wave
plate, in other words, this part of light is linearly polarized. Therefore, the fraction of
linear polarization, polarization linearity, is about 30%. Thereby, if there is a small
optical rotation signal for measurement, the denominator in calculation formula for
rotation angle in Equation 3.4 should be take the effective value as 0.4 mW×2 = 0.8
mW, instead of 1.3 mW×2 = 2.6 mW. The signal-to-noise ratio is attenuated, because
the ratio of linear polarization is reduced for the same transmission.
142
Figure 5.29: Output light intensity from two channels after periodically modulated
by a rotating λ/2 waveplate. It is measured in a 4-m-long fiber filled with Toluene.
The modulation frequency of the rotating λ/2 waveplate is 1.4 Hz.
Besides the depolarization that could be measured in the above method, another
effect, the linear birefringence, also plays an important role in the polarization linearity. From the discussion in Section 5.2.3 as shown in Figure 5.21, due to the
intramodal or intermodal linear phase shift, the polarization part of the light is periodically changing. This effect keeps the polarization, but if it is in presence of a
external field induced optical rotation, it could suppress the optical rotation when
the optical rotation angle is much smaller than the linear phase shift. We observe
this effect in the Toluene filled liquid core fiber. When measuring the polarization
linearity, we can always get a considerable value in the final output beam by properly
bend or twist the fiber at some local region, which also means the beat length is short.
However, once we perform the measurement of Faraday rotation, we can only get a
small rotation angle which is just 1%-20% of theoretical value, depending on the fiber
coil diameter and loop number. If the 8-m-long fiber is coiled densely with a diameter of 2 inch, the measure rotation angle is negligible. It confirms that the Faraday
143
rotation induced circular birefringence is almost suppressed by linear birefringence
arising from the bent fiber, because the Faraday rotation angle is normally smaller
than 1◦ for liquid in this configuration even with a strong current of several amperes.
Since the NSOR angle is even smaller than Faraday rotation, due to the low nuclear
spin polarization, then it is impossible to detect NSOR signal for Toluene in a liquid
core fiber. One way to suppress the linear birefringence effect is to use chiral liquids
instead of Toluene, which will be covered in next section.
5.3.3
Chiral liquids
Based on the argument in Section 5.2.3, if a optical rotation arising from circular
birefringence prevails the linear phase shift arising from the linear birefringence of
the fiber, then the bending induced linear birefringence in a curved fiber could be
suppressed. Filling the fiber core with chiral liquids is a good method, since lots
of chiral liquids have a large optical rotation, while other methods inducing circular
birefringence have some drawbacks; for example, the twist of fiber is not easy to
control in this experiment, and the electromagnetic field induced optical rotation (e.g.
Faraday rotation) has a too small rotation angle, which is usually smaller than linear
phase shift. The chiral liquid Limonene ((R)-(+) Limonene or (L)-(-)-Limonene)
is a good choice for measurement, because it has a large specific rotation constant
◦
−1
3 −1
[α]20
D ≈120 (dm [g/cm ] )) and a large enough index n = 1.484 (405 nm).
First we perform the measurement of Faraday rotation of Limonene filled hollow
fibers at low frequency (f =1 kHz∼20 kHz), which could be realized easily based on
the instruments used in the NSOR experiment at low fields. The apparatus takes the
similar form as shown in Figure 3.23, but the sample cell is replaced by the densely
coiled fiber of 8 meters. As shown in Figure 5.30, the 8m-long fiber is wound densely
around a Teflon rod with a diameter of 2 inch; a thick copper wire passes through
144
the center of the fiber coil with axial symmetry, in which a strong current oscillating
at kHz frequency is running.
R
I
Figure 5.30: The measurement of Faraday rotation in a long coiled fiber at low frequency. The fiber is wound around a Teflon rod with diameter of 2 inch. The straight
thick copper wire passes through the center of the fiber coil with axial symmetry.
In this configuration, the optical rotation induced by the Limonene ((D)-(+)◦
−1
3 −1
Limonene, Sigma Aldrich, >97%, [α]20
D ≈120 (dm [g/cm ] )) in each loop is about
161◦ , which is a very large angle, while the Faraday rotation angle in this configuration is typically smaller than 1◦ . Here, the dispersion of two circular polarized
components because of the slight index difference arising from circular birefringence
is just 0.0016◦ for a 10-m-long fiber, which could be neglected in terms of the effects
on bandwidth. The incident 13.8 mW laser beam at 405 nm gives a output light
beam with intensity of 0.372 mW, so the light coupling efficiency is about 3% in this
8-m-long and densely coiled fiber. The effective polarization of the output light beam
is about 33%, according to the measurement as shown in Figure 5.28 and Figure
5.29. The AC current on the wire creates a circular magnetic field parallel to the
fiber although the fiber coil, and the magnitude of the field is determined by
µ0 I
,
πD
145
B=
(5.43)
where, I is the current, D is the diameter of the fiber coil. When I=1 A, the
magnetic field is 0.079 G. First, the Verdet constant Vd of Limonene is measured
as 10.42µrad/(G · cm) at 405 nm in a cylindrical cell as shown in Figure 5.30.
The calculated Faraday optical rotation angle in this coiled-fiber configuration is
θ = Vd LB =10.42µrad/(G · cm) × 8m×0.079G=0.658mrad, while the measured optical rotation angle is 0.708 mrad (i.e. Vd =11.21 µrad/(G · cm)), about 7% larger
than the theoretical value, which is tolerant due to the measurement error in current
source, fiber length, light intensity and polarization ratio. We also perform the measurement in a toroid coil as shown in Figure 5.3, which is configured for high field
magnetic field. The measured rotation constant is about 3.3µrad/(G · cm), only 33%
of the Verdet constant, which may be because of the inhomogeneous field distribution
of the toroid coil, as well as unexpected fiber distortion when the fiber is curled more
densely and confined by the copper coil. When changing the current frequency from
1 kHz to 20 kHz, this value almost keeps the same with about 2% tolerance, which
means the bandwidth of this fiber is at least larger than 20 kHz.
The high-frequency Faraday rotation of Limonene inside hollow fibers is measured
with a toroid coil and a 115 MHz RF current source via RF power amplifier, then
the current could goes up to several amperes. However, we only get about less than
1% rotation signal comparing with theoretical calculation. Therefore, the NSOR
experiment could not be carried out, because the optical rotation is even much smaller
than Faraday rotation, and the signal is too small to detect due to the low signalto-noise ratio. The only explanation for this unexpected small result is that the
bandwidth is limited by some instruments or unknown effects in densely bent fiber.
The way to improve the bandwidth is achieving a nearly single-mode liquid-core fiber,
as well as reducing dust and impurities in liquids and avoiding fiber distortion when
curling the long fiber.
146
Chapter 6
Conclusions
In this work, we explore potential applications of Nuclear Spin induced Optical Rotation in the area of chemistry. Based on the continuous-wave spin-lock technique
under a low magnetic field (5 Gauss), we perform precision measurements of NSOR
signal for several pure organic liquids that contain the nucleus 1 H, and find that it is
able to clearly distinguish some of these chemicals by means of NSOR at low fields.
With aid of a multipass cavity, the optical path is elongated to about 3 meters, which
improves the sensitivity of NSOR, as the optical rotation signal is proportional to the
length of optical path. In addition, a 405 nm laser is implemented to get a larger optical rotation angle, as well as to obtain lowest absorptions of light in those chemicals.
Combining these two techniques, we succeed to achieve a large Signal-to-Noise ratio of
NSOR detection, thereby demonstrate the capability of distinguishing 1 H in different
chemicals through NSOR. In addition, we also manage to measure the NSOR signal
of
19
F at low fields, which has a much larger optical rotation constant than 1 H. All
of our experimental results agree with quantitative analysis based on first-principle
quantum chemistry calculations.
In order to extend applications of NSOR to measure signals of the same type
of nuclei inside a molecule via chemical shifts, we explore an experimental scheme
147
based on liquid-core hollow fibers to approach the measurement of NSOR under high
magnetic fields, as well as set up a new type of nanoliter NMR. By filling liquids inside
a long hollow fiber and coiling it densely, it is possible to detect NSOR signals with
a long optical path, as well as higher spin polarizations and distinguishable chemical
shifts, inside a narrow core of a superconducting magnet. According to our attempts,
because of the robustness and low-cost of fused silica capillary tubings, filling highindex liquids in those tubings is an appropriate choice of liquid-core fiber in practice.
By coiling a long fiber densely for many loops around a small rod, we find that it
is possible to measure optical rotation signals inside a narrow-bore superconducting
magnet. However, those liquid-core capillary tubings perform like multimode stepindex fibers, and thereby exhibit linear birefringence and depolarization, significantly
reducing the light polarization for the measurement of optical rotation. Based on
the theory of birefringence in fibers, we succeed to suppress the linear birefringence
by filling chiral liquids in hollow fibers, and approach near single-mode operation by
means of launching light beam into the fiber core under the mode match condition.
As far as we have explored, NSOR signal in bent liquid-core fiber has not been
obtained at high frequency. The best solution to realized the high-field NSOR is
to implement a single-mode polarization maintaining liquid core fiber, which might
become possible in near future due to the recent advances in photonic crystal fiber
[99, 100, 147, 148, 149]. Our work on the liquid-core fiber provides the basis for future
fiber-based NSOR experiments under high magnetic fields.
148
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