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Transcript
JUST, Vol. IV, No. 1, 2016
Trent University
A study of reflection and transmission of
birefringent retarders
James Godfrey
Keywords
Optics — Computational Physics — Theoretical Physics
Champlain College
1. Interference in Rotating Waveplates
1.1 Research Goal
The objective of this project was to obtain a realistic theoretical prediction of the transmission curve for linearly-polarized
light incident normal on a [birefringent] retarder that behaves
as a Fabry-Perot etalon. The birefringent retarder considered
is a slab of birefringent crystal cut with the optic axis in the
face of the slab, and with parallel faces, and the waveplate has
no coating of any kind.
Results of this project hoped to possibly explain the results
of work done by a previous student, Nolan Woodley, for his
Physics project course in the 2013/14 academic year. In his
project, Woodley took many polarimeter scans which had a
roughly sinusoidal shape [as they should], but adjacent peaks
of different heights. These differing heights were completely
unexplained, and not predicted by theory.
1.2 Methodology
The projection of the normally-incident P-polarized (planepolarized) light’s E-vector onto the optic axis of the retarder
is proportional to cos(β ), where β is the angle between the
E-vector and the optic axis. The projection onto the other axis
of the retarder (the axis perpendicular to the optic axis in the
face of the waveplate) is proportional to sin(β ). The optic
axis is called the fast axis if its refractive index is lower than
the refractive index of the perpendicular axis, and it is called
the slow axis if the refractive index is higher. Without loss
of generality, it may be assumed that the optic axis is the fast
axis.
These axes each have associated characteristic refractive
indices, and thus P-polarized light oscillating along one axis
would be transmitted and reflected in different proportions, as
sufficiently described by the Fresnel equations, assuming the
air-retarder interfaces are completely lossless (no absorption).
The reflectance of a dielectric at normal incidence is given as:
R=
nt − ni
nt + ni
2
(1-1)
where nt is the refractive index of the transmitting medium
and ni is the refractive index of the incident medium.
The transmitted intensity of along each axis must be considered separately, since equation (1-1) dictates that the reflectance of the fast- and slow-axis components of the incident
light will be different. The reflectance along the fast axis (Ro )
and slow axis (Rs ) are may be expressed:
n f − ni
n f + ni
2
ns − ni
ns + ni
2
Ro =
Rs =
(1-2)
Assuming the sides of the etalon are essentially parallel, the retarder may now be treated as a low-finesse (lowreflectance, R << 1) Fabry-Perot etalon. Their coefficients of
finesse (Fo and Fs , respectively) will also be different:
Fo =
Fs =
4Ro
(1 − Ro )2
4Rs
(1-3)
(1 − Rs )2
The transmission of a Fabry-Perot etalon is typically given
by the Airy Function:
T=
1
1 + F · sin2
(1-4)
δ
2
where F is the etalon’s coefficient of finesse and δ is the round
trip phase shift of the wave; the accumulated phase associated
with traversing the cavity back and forth once.
The round trip phase shift of each component at normal
incidence is expressed:
δ = 2k · n · d
(1-5)
where k is the vacuum wavenumber of the incident light, n is
the index of refraction of the material through which the light
wave is propagating and d is the thickness of the etalon.
Since the etalon being considered is a waveplate (made
of birefringent material), it has two indices of refraction, and
A study of reflection and transmission of birefringent retarders — 2/5
is manufactured to have a specific length as to impart a specific desired phase shift upon propagating the length of the
waveplate. We shall define the waveplate order q as:
q≡
|δ o − δs | 1
·
2
2π
(1-6)
where δo and δs are the round trip phase shift for light waves
whose electric fields are oscillating in the direction of the fast
and slow axes, respectively.
The thickness of a waveplate d may then be given by the
following expression:
d=
q · λ0
.
|ns − n f |
Tnet
To =
Ts =
1
1 + Fo · sin2
2π·n f ·q
|ns −n f |
2π·ns ·q
|ns −n f |
1
1 + Fs · sin2
(1-8)
Considering that the fast- and slow-axis projections of
the P-polarized light’s electric field vector are proportional
to cos[β ] and sin[β ], respectively, intensity of the transmitted fast- and slow-axes components scales with cos2 [β ] and
sin2 [β ]. The net transmission of the waveplate may then be
simply expressed in terms of equations (1-8a) and (1-8b):
(1-7)
where λ 0 is the vacuum wavelength of the incident light.
Substituting equation (1.7) into (1.5) for each component, and
then substituting that and equation (1.3) into (1.4), we get the

1
=
1 + Fs · sin2
2
transmission functions the fast- and slow-axis components:
Tnet = cos2 (β ) · To + sin2 (β ) · Ts
1
1
= (Ts + To ) + (Ts − To )cos(2β )
2
2
(1-9)
Substituting equation (1-8) into (1-9) yields:
!!−1
!!−1 
2π
·
n
·
q
2π · ns · q
f
+
+ 1 + Fo · sin2 ns − n f ns − n f 
!!−1
!!−1 
2π
·
n
·
q
1
2π
·
n
·
q
f
s
 cos (2β ) (1-10)
1 + Fs · sin2 − 1 + Fo · sin2 2
ns − n f ns − n f Equations (1-9) and (1-10) were implemented in Mathematica, and used to generate plots of net transmittance Tnet
against waveplate rotation angle β . The parameters chosen
were those of calcite for yellow incident light (λ0 = 589 nm);
n f = 1.4864 and ns = 1.6584. Calcite was chosen since it has
a fairly high birefringence (|n f − ns | > 0.1), unlike other commonly used materials such as quartz, so that any transmission
properties related to the birefringence would be accentuated.
1.3 Notable Findings and Possible Next Steps
When modelling a waveplate as a [lossless] birefringent FabryPerot etalon, the transmission curve (as a function of waveplate rotation angle, β ) is a sinusoid of period π (see Fig. 1).
The β -dependence, and particularly the fact that the transmission curve is π-periodic in β , arises from reflection symmetry
of the electric field vector’s projections on the fast and slow
axes of the waveplate; the components along each axis are
of equal magnitude whether it forms an angle β with the fast
axis to the left or right of a particular axis.
Though originally only quarter-wave plates were of interest, retarders of arbitrary thickness were shown to possess a
similar transmission curve for [almost] all thicknesses. The
amplitude of the sinusoid varies extremely with waveplate order q, and thus waveplate thickness. The maximum change in
transmittance ∆T is twice the coefficient of the cos[2β ] term
in equations (1-9) and (1-10). The trough-to-peak amplitude
of the transmission curve is given as:
2
∆T = 1 + Fo · sin 2π ·
q
·nf
|ns − n f |
−1
− 1 + Fs · sin2 2π ·
q
· ns
|ns − n f |
−1
(1-11)
where:
• n f and ns are the characteristic indices of refraction of
the waveplate’s slow and fast axes, respectively
• ni is the index of refraction of the incident medium, Fo is
the fast axis’ coefficient of finesse, Fo ≡ (n f –ni )2 /(n f +
ni )2
• Fs is the slow axis’ coefficient of finesse, Fs ≡ (ns –ni )2 /(ns +
ni )2 ,
• k is the vacuum wavenumber of the incident light,
• λ is the vacuum wavelength of the incident light
A study of reflection and transmission of birefringent retarders — 3/5
Figure 1. Two plots of simulated transmission data for two different thicknesses of calcite waveplate rotated through a full cycle of β . The figure
on the left is a waveplate with q = 0.25; the two orthogonal components of the incident P-polarized light entering in-phase emerge with a phase
difference of π/2(λ /4). Similarly, the second plot corresponds to a calcite plate of appropriate thickness such that the imparted phase difference
is 9π/2.
• q is the waveplate mode number (0.25, 1.25, 2.25, ...
correspond to quarter-wave plates; 0.5, 1.5, 2.5, ... correspond to half-wave plates; 1, 2, 3, ... correspond to
full-wave plates).
the experimentally obtained data.
1.4 Relevant Literature
• Optics, 4th Ed. by E. Hecht
• Introduction to Optics, 3rd Ed. by F. L. Pedrotti, L. S.
Pedrotti, and L. M. Pedrotti
2. Transmission and Reflection from
Birefringent Crystals
Figure 2. A plot of equation (1-11) over the range 0 ≥ q ≥ 1. Negative
values of ∆T correspond two transmission curves that start at a
minimum, such as the first plot in Fig. 1-1. ∆T can be seen to range
between roughly -0.15 and 0.22, and it seems to have a periodic
envelope.
This theoretical groundwork is at the stage where it could
be experimentally tested. A simple [but somewhat expensive]
experimental setup could use an uncoated known-order (q)
calcite waveplate aligned normal with a highly P-polarized
collimated light-source of known power and a calibrated photodiode or power meter. The retarder should be mounted on a
rotating-mount controlled by a step motor, and rotated through
a full cycle of β while stopping to take power readings with
the photodiode every 3.6o (or some other similarly small rotation angle). Beginning the rotation of the fast axis aligned
with the fast axis of the calcite retarder, and scanning in small
steps of β , one should be able to obtain a data set that, when
normalized to the incident power and plotted, is comparable
to those in Figure 1-1. Knowing the indices of refraction of
the calcite waveplate for the wavelength of the incident light
and the waveplate order q, a prediction of the intensity scan
can be made using equation (1-10), and can be compared to
2.1 Research Goal
The goal of this project was to determine how Brewster’s
angle of a birefringent waveplate (in a waveplate the optic
axis is necessarily in the face of the optic) changes with the
orientation of the waveplate. The light considered was not
necessarily polarized in the plane of incidence, but might as
well have been, as the TM-mode is solely responsible for the
Brewster’s angle phenomenon. It should be noted that the Eray and O-ray necessarily feel a different index of refraction,
and thus should have different Brewster’s angles, but the same
angle of reflection.
2.2 Methodology
The focus of this project was definitely determining Brewster’s
angle of the E-ray, θ pe , as Brewster’s angle of the O-ray, θ po , is
easily determined using the conventional formula:
no
o
θ p = arctan
(2-1)
ni
where no is the characteristic ordinary refractive index of
the birefringent material and ni is the refractive index of the
incident medium.
However, determining θ pe is not so simple, as the boundary
conditions at the air-crystal interface give rise to a different
set of Fresnel equations than those of the O-ray. This is due
to the index of refraction of the E-ray, neo , actually being a
function of the transmitted angle θeo , unlike no . The version of
A study of reflection and transmission of birefringent retarders — 4/5
Snel’s Law of Refraction1 that must be satisfied by the E-ray
is given by:
ni sin(θi ) = neo (θeo ) sin(θeo )
1
2
1
1
1
2
2 o
o
+
−
cos (φ ) sin (θe )
neo (θe ) =
2
2
2
(ne )
(no )
(ne )
(2-2)
where no and ne are the characteristic ordinary and extraordinary refractive indices of the birefringent material and φ is
the waveplate rotation angle; the angle formed between the
plane of incidence and the optic axis of the waveplate (this
angle is analogous to β from section 1).
The symbol φ was used instead of β to simplify programming; Yang’s article used φ throughout, and thus it was easier
to compare the equations in my program to those in the article. In his article, Yang describes the E-ray’s transmitted
field, reflected field, and incident field. From these the TE(denoted ⊥) and TM-mode (denoted k) amplitude reflection
coefficients (Re⊥ and Rek , respectively) and amplitude transmission coefficients (T⊥e and Tke , respectively) may be obtained.
For each mode, each reflection and transmission coefficient is
simply expressed as the ratio of reflected or transmitted field
amplitude over the incident field amplitude. Finding θ pe at a
given φ is now simply a matter of finding the θi -root of Rek .
2.3 Notable Findings and Suggested Next Steps
Using the conventional TM-mode amplitude reflection coefficient for the O-ray, and the modified TM-mode reflection
coefficient derived from Yang’s equations, it is now possible
to determine the two Brewster angles θ po and θ pe . Figure 2-1
demonstrates that the theory in Yang’s article agrees well with
previously-established theory; the plot shown below agrees
with Figure 23-3 in Introduction to Optics, 3rd Ed. by Pedrotti.
The program used to generate Figure 2-1 could be modified to generate a plot of Rok and Rek at a constant θi while
the φ ranges from 0o to 90o , and analyzed to determine the
estimated total intensity of the reflected ray at every φ . It may
be beneficial to use an uncoated waveplate made of a material
with a high birefringence (such as calcite) so that θ po and θ pe
are noticeably different.
A fairly simple experimental setup could be constructed to
obtain data to compare to the theoretical prediction mentioned
above. A suggested setup would include a waveplate on a
rotating mount (preferably one controlled by a step motor), a
source of P-polarized light (of known power/intensity) such
as a diode laser, and a calibrated photodiode or power meter.
The P-polarized light should be incident on the waveplate at
an oblique angle, and polarized in the plane of incidence. If
necessary, use a linear polarizer, although this may facilitate
some measurement of the intensity of the incident light after it
passes through the polarizer. The photodiode or power meter
1 yes,
‘Snel’ is actually the correct spelling
Figure 3. A plot of the amplitude coefficients against θi for a quartz
waveplate (no = 1.543, ne = 1.552). Dotted lines represent the
amplitude reflection coefficients and solid thick lines represent
amplitude transmission coefficients. The blue line is the height on the
graph at which an interface coefficient has the value of 0. The black,
dotted line (TM-mode reflection coefficient) can be seen vary such
that [1 ≥ (Re|| + T||e ) ≥ −1] as θi ranges for 0o to 90o , and TE-mode
coefficients behave such that Re⊥ + T⊥e = 1 for all θi . The point at which
Rek crosses through 0 on the vertical axis corresponds to θ pe ≈ 57.5o
for φ = 20o .
should also be set up in the plane of incidence, normal to
and in the path of the reflected beam. Rotating the waveplate
should change the intensity of the reflected beam without
changing its path. Since the incident light is polarized in
the plane of incidence, the intensity of the reflected beam is
a sum of the reflected intensities of the E- and O-ray TMmode components. By rotating the waveplate in small φ increments, taking power measurements of the reflected beam,
and normalizing the power measurement to the incident power,
an experimental data set comparable to the plot mentioned in
the above paragraph could be taken.
2.4 Relevant Literature
• Introduction to Optics, 3rd Ed. by F. L. Pedrotti, L. S.
Pedrotti, and L. M. Pedrotti
• W.Q. Zhang (2000): New phase shift formulas and
stability of waveplate in oblique incident beam
• T. Yang (2006): An improved description of Jones vectors of the electric fields of incident and refracted rays
in a birefringent plate
Please note that Yang’s article contained several errors:
• in equation (14), the third component in the numerator should be positive (although it did not affect any
calculated results)
e should be squared under the square
• in equation (21), Dtk
root
• in equations (24c) and (24d), remove/scratch out the
first closing round bracket [)] in both, or add an opening
e is not
round bracket [(] between each cos and θe o; At⊥
divided by neo cos(θi )
A study of reflection and transmission of birefringent retarders — 5/5
3. Serendipitous Learning
A beautiful part of research is that, in attempting to answer a
single question, you end up answering an incredible number of
questions on the way to answering the first one. This section
is dedicated to some of the wonderful unexpected [practical]
learning that during the summer.
When doing polarimetry, it is important to consider the
type of photodiode being used; between different silicon photodiodes the results of a rotating waveplate scan were vastly
different (i.e. different percentage change, and completely different characteristic shape of intensity profile). Perhaps some
PDs have different sensitivities to certain polarizations of
light; the TSL251 had large percentage changes (≈ 5 − 12%)
in its scans, and had [roughly] sinusoidal intensity profile of
period π/2, whereas the OPT101A had a non-periodic shape
and small percentage changes (≈ 0.5 − 2%). The TSL251
also seemed to hit its local minima when circularly-polarized
light was incident (at β =45o , 135o , 225o , 315o ), and maxima
when linearly-polarized light was incident (at β = 0o , 90o ,
180o , 270o , 360o ).
Another fun fact, the output power of a diode laser operating at near room temperature can be very sensitive to changes
in the temperature of the gain medium; a 1o C increase in
the temperature of the gain medium increased the laser’s output power as much 2 percent. Although somewhat puzzling,
it seemed to be easier to keep the diode laser at a constant
temperature at a temperature of about 17.0o C, as opposed to
about 18.6o C, which was the initial set point of the laser’s
cooling device. Intuitively, one might think it would be easier
to maintain a temperature closer to room temperature.
When taking any measurement with a photodiode, devise
an experiment where you never have to move the photodiode; it will make your life significantly less difficult 100% of
the time. In an experiment where the photodiode is moved
manually, it is an extremely onerous task to keep all of the
components of the apparatus aligned for every intensity measurement.
Finally, arguably the most important thing that was learned
this summer: Wolfram Mathematica is a powerful, versatile,
and simply fantastic program, and everybody should use it for
everything. Perhaps the previous statement is an exaggeration,
but it really is a fantastic program. From rendering beautiful high-resolution videos of time-evolving electric fields
from arrays of dipole oscillators to preparing simple plots,
Mathematica is a simple and effective choice of programming
language. It’s got all of the numerical tools of Matlab, all of
the analytical solving techniques of Maple, and even has builtin connectivity with Wolfram Alpha’s massive database. (And
no, the research in this report was not sponsored by Wolfram
in any way, although Mathematica was an indispensable tool
in all research mentioned in this report.)
(a)
(b)
Figure 4. (a) A plot of transmitted intensity (arbitrary units) against
waveplate rotation angle β ; intensity measurements taken with
OPT101A photodiode. The difference between the maximum and
minimum reading taken is 1.10%. The 3 coloured sets are 3 individual
scans, and the black set is the mean of the 3 individual scans. The
scans seems to be non-sinusoidal in shape. (b) A plot of transmitted
intensity (arbitrary units) against waveplate rotation angle β ; intensity
measurements taken with TSL251 photodiode. The difference
between the maximum and minimum reading taken is 5.39%, much
larger than the OPT101A scan. Scans seem to roughly follow a
sinusoidal shape of period π/2. The shapes of the two plots are
apparently fundamentally different.