Download Measurement of The Nonlinear Refractive Index by Z

University of Ljubljana
Faculty of Mathematics and Physics
Department of Physics
Seminar Ib
Measurement of
The Nonlinear Refractive Index
by Z-scan Technique
Author: Eva Ule
Advisor: izred. prof. dr. Irena Drevenšek Olenik
Ljubljana, February 2015
Abstract
In this seminar the third-order nonlinear optical phenomena are described. The emphasis is on
the nonlinear refractive index n2 . We describe the Z-scan technique, which is a simple method to
determine n2 . At the end we show how Z-scan method can be used to examine protein concentration
in human blood.
Contents
1 Introduction
2
2 Nonlinear Optical Media
2.1 Nonlinear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
3
3 The Third-order Nonlinear Optics
3.1 Self-Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
4
4 Z-scan Method
4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Nonlinear Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
6
8
5 Use
5.1
5.2
5.3
5.4
of Z-Scan Technique to Measure Protein Concentration
Preparation of the Samples . . . . . . . . . . . . . . . . . . . .
Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conventional Colorimetric Method . . . . . . . . . . . . . . . .
Comparison of Results of Two Different Methods . . . . . . . .
6 Conclusion
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Introduction
Materials with large third-order optical nonlinearities have recently become the topic of a broad scientific
interest, mostly because of their possible application in high speed optical switching devices, which are
becoming more and more important and more commonly used. [1] The nonlinearities of the third order
are usually studied in centrosymmetric media, in which the second-order nonlinear susceptibility is zero.
2
Nonlinear Optical Media
In a linear dielectric medium there is a linear relation between the induced electric polarization and the
electric field
P = ε0 χE,
(1)
where ε0 is the permittivity of vacuum and χ is the dielectric susceptibility of the medium. In a nonlinear
dielectric medium the relation between P and E is nonlinear. This is illustrated in the figure 1.
Figure 1: The P-E relation: the dashed line represents the linear and the full line represents nonlinear
relation between P and E. [2]
The polarization P is a product of the individual dipole moment p, which is induced by the applied
electric field E, and the number density of dipole moments. The relation between p and E is linear when
E is small and becomes nonlinear as E acquires values from 105 to 108 V/m. External electric fields are
almost always smaller than the characteristic interatomic fields, even when laser beams are used, which
2
means that the nonlinearity is usually weak. So we can expand the function relating P to E in a Taylor’s
series about E = 0:
.
P = ε χE + ε χ(2) : EE + ε χ(3) ..EEE + ...
(2)
0
0
0
This is the basic representation for a nonlinear optical medium.
2.1
Nonlinear Wave Equation
A wave equation for the propagation of light in a nonlinear medium can be derived from Maxwell’s
equations
∂2P
1 ∂2E
(3)
∇2 E − 2 2 = µ0 2 ,
c0 ∂t
∂t
with E being the electric field and P the polarization. It is useful to write P as a sum of linear and
nonlinear parts
P = ε0 χE + PN L
(4)
and
.
PN L = ε0 χ(2) : EE + ε0 χ(3) ..EEE + ...
If we use the relations n2 = 1 + χ, c0 =
medium
√ 1
µ0 ε0
and c =
c0
n,
(5)
we can write a wave equation in nonlinear
1 ∂2E
∂ 2 PN L
= µ0
.
(6)
2
2
c ∂t
∂t2
This is the basic equation in the theory of nonlinear optics. Two approaches exist to approximately
solve this partial differential equation. The first one is the Born approximation and the second is the
coupled-wave theory.
∇2 E −
3
The Third-order Nonlinear Optics
In media with centrosymmetry (the properties of the medium are not altered by the transformation
r → −r), the second-order nonlinear susceptibility χ(2) is 0, so the third-order term becomes important
and equation (2) is
.
P = ε χE + 0 + ε χ(3) ..EEE
(7)
0
0
and where E is
e
(A exp [ikz − iωt] + A∗ exp [−ikz + iωt]) .
2
We combine equations (7) and (8) and we get
(1)
(3) 1
P = ε0 χ E + 3ε0 χ
[AA∗ ] E.
4
E=
(8)
(9)
2
Using hji = 12 ε0 c0 n(ω) |A| (j is the energy flux density) we rewrite the equation (9) in
3 (3)
hji
P = ε0 χ(1) + χef f
E = ε0 (εef f − 1)E.
2
ε0 c0 n(ω)
(10)
Using the definition of the refractive index
nef f =
√
εef f
(11)
we get the result
(3)
nef f
3 χef f hji
=n
e+ ·
=n
e + n2 hji .
4 ε0 c0 n
e2
(12)
(This is called the optical Kerr effect because of its similarity to the electro-optic Kerr effect.) The
coefficient n2 has units cm2 /W and values from 10−16 to 10−14 in glasses, 10−14 to 10−7 in doped
glasses, 10−10 to 10−8 in organic materials and 10−10 to 10−2 in semiconductors. It depends on the
polarization state of the optical beam and it is sensitive to the operating wavelength. In most cases the
peak intensity of the laser beam is around 1 GW/cm2 .
3
3.1
Self-Focusing
An interesting effect that can be seen in the third-order nonlinear medium is self-focusing. If an intense
optical beam is transmitted through a thin sheet of nonlinear material exhibiting the optical Kerr effect,
the refractive index change maps the intensity pattern in the transverse plane. [2] For the beam with its
Figure 2: Self-focusing of the beam with Gaussian intensity profile [2]
highest intensity at the center, the maximum change of the refractive index is also at the center.
For a Gaussian beam with the intensity profile
2ρ2
I = I0 exp − 2
w
(13)
where ρ = (x2 + y 2 )1/2 is transversal coordinate and w is beam radius, the focal length of the effective
lens is given as:
w2
,
(14)
f=
2n2 I0 d
where d is the thickness of the sheet.
4
Z-scan Method
The nonlinear refractive index n2 can be measured by a Z-scan technique, which can simultaneously
measure nonlinear absorption and nonlinear refraction in solids, liquids and liquid solutions. It is a
single-beam technique that gives us both the sign and magnitude of refractive index nonlinearities. This
method is rapid, simple to perform and accurate, therefore it is often used. It is especially adequate for
determination of a nonlinear coefficient n2 for a particular wavelength.
This technique is used to study semiconductors, glasses, semiconductor-doped glasses, liquid crystals,
biological materials and every-day liquids e.g. tea. [3]
Figure 3: Schematic drawing of the Z-scan technique [4]
4
We use a Gaussian laser beam and then the transmittance of the beam through the experimental system
shown in figure 3 is measured. We have to change a position of the sample with respect to the focal
plane of a field lens, which is set at position z = 0.
Figure 4: The Z-scan measurement as represented in an online animation http://www.optics.unm.
edu/sbahae/z-scan.htm; we can see the change of the laser beam and the change of the transmittance
at the same time
The measurement starts far away from the focus (negative z), where the transmittance is relatively
constant (figure 4 a). Then the sample is moved towards the focus and then to the positive z (figure
4 b to h). If the material has a positive nonlinearity (n2 > 0), the T(z) graph has a valley first and
then a peak. For the sample with n2 < 0 the graph is exactly the opposite (first the peak and then the
5
valley) (figure 5). When self-focusing in the sample occurs, this tends to collimate the beam and causes
a beam narrowing at the aperture which results in an increase in the measured transmittance and when
self-defocusing occurs the beam broadens at the aperture and the transmittance decreases. The scan is
finished when the transmittance becomes linear again.
Figure 5: We can determine the sign of the n2 from a graph of transmittance T (z) [5]
Figure 6: Typical Z-scan curve; a) for n2 > 0 (barium floride) [6] and b) for n2 < 0 a lyotropic liquid
crystals [7]
There are not many materials with n2 > 0, the negative n2 is much more common. (This is attributed
to a thermal nonlinearity. [6])
4.1
Theory
As mentioned before, we use Gaussian beam with the magnitude of E
w0
r2
ikr2
E(r, z, t) = E0 (t)
exp − 2
−
· exp [−iφ(z, t)],
w(z)
w (z) 2R(z)
(15)
where w(z) is the radius of the beam at z, E0 is the electric field at the beam waist (z=0, r=0) and the
last term contains all the radially uniform phase variations. [8] If the sample length L is small enough,
so that there are no changes in the beam diameter within the sample, the medium is treated as thin.
This fact simplifies the problem and helps us a lot. We need the phase shift ∆φ
∆φ0 (t)
2r2
∆φ(z, r, t) =
exp − 2
.
(16)
1 + z 2 /z02
w (z)
The flux density of the incident beam has the Gaussian intensity profile so according to the equation (12)
even the change of the refractive index has the form of the Gaussian function in the transverse direction.
It follows that the phase shift is also the Gaussian function of the transverse coordinate with respect to
the centre of the beam. The most important term for us is ∆φ0 (t), which is defined
∆φ0 (t) = kn2 I0 (t)Lef f ,
6
(17)
where I0 is the irradiance at the focus z = 0 and Lef f = (1 − exp[−αL])/α is the effective propagation
length inside the sample with the sample length L and the linear absorption coefficient α. At the exit
of the sample there is the complex electric field Ee which contains the nonlinear phase distortion. We
use the Gaussian decomposition (GD) method, in which the complex electric field at the exit plane of
the sample is decomposed into a summation of Gaussian beams through a Taylor series expansion of the
nonlinear phase term. [8] When all these beams come to the aperture plane they are again united in one
single beam. We get the electric field at the aperture Ea (r, t) and it is function of ∆φ0 . If we spatially
integrate |Ea (r, t)|2 up to the aperture radius ra we get the transmitted power PT (∆φ0 (t)). Now we can
calculate the normalized Z-scan transmittance T (z) which can be seen in the figure 4
R∞
PT (∆φ0 (t))dt
(18)
T (z) = −∞R ∞
S −∞ Pi (t)dt
where Pi (t) is the instantaneous input power within the sample and S is the aperture linear transmittance
S = 1 − exp(−2ra2 /wa2 ) (wa is the beam radius at the aperture and ra is the aperture radius). S is an
important parameter because a large aperture reduces the variations in T (z).1
Usually, the most important quantity is ∆Tp−v , which is the difference between the highest (peak) and
the lowest (valley) value of transmittance Tp − Tv . Based on a numerical fitting, there is a relation
Figure 7: ∆Tp−v [9]
between Tp−v and ∆φ0 (which is a function of n2 )
∆Tp−v ' 0.406(1 − S)0.25 |∆φ0 |
(19)
for |∆φ0 | ≤ π. Using the equation (17) we can calculate the nonlinear refractive index n2
n2 =
∆φ0
.
kI0 Lef f
(20)
1 The size of the aperture is signified by its transmittance S in the linear regime, i.e. when the sample has been placed
far away from the focus. In most reported experiments, 0.1 < S < 0.5 has been used for determining nonlinear refraction.
Obviously, the S=1 case corresponds to collecting all the transmitted light and therefore is insensitive to any nonlinear
beam distortion due to nonlinear refraction. Such a scheme is referred to as an open aperture Z-scan. [5]
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4.2
Nonlinear Absorption
Z-scan method is also used to determine the coefficient of nonlinear absorption β. The whole absorption
is defined
α = α0 + βI,
(21)
where α0 is the linear absorption coefficient and I is the intensity of the beam.
Figure 8: Typical T (z) graph for ZnSe with open aperture (λ = 532 nm) and the nonlinear absorption
β = 5.8 cm/GW [8]
Our final graph is represented in the picture 9. As we can see, the peak and the valley are asymmetric
and this is the sign of the nonlinear absorption.
Figure 9: Typical Z-scan graph (closed aperture) for the material with the nonlinear absorption (β 6= 0)
for n2 > 0 and n2 < 0
It is quite remarkable that we can determine the sign of n2 and the presence of the nonlinear absorption
only from the form of the transmittance graph.
5
Use of Z-Scan Technique to Measure Protein Concentration
in Blood
The proteins which are made of amino acids, are very important compounds in our body. They make
new cells, maintain muscles, support the immune system, etc. The two major groups of proteins in the
human blood are albumin and globulin. Albumin is a protein, which is important for tissue growth, but
its main function is to regulate the colloidal osmotic pressure2 of blood [10]. It is mostly produced in
the liver. Globulins are produced either by the immune system or by the liver. We know alpha, beta
and gamma types of them and they have various roles in our body.
2 Colloid osmotic pressure is a form of osmotic pressure exerted by proteins; notably albumin, in a blood vessel’s plasma
(blood/liquid) that usually tends to pull water into the circulatory system. [11]
8
If the level of total protein in the blood is low, this can be an indicator of the liver or kidney disorder.
Thus the values of these substances in the blood are important and usually measured by colorimetric
method. It is because of the nonlinear optical properties of total protein and albumin that we can use
Z-scan technique to determine their values.
5.1
Preparation of the Samples
For sample preparation the serum was used and small amount of proper reagent was added. Then the
solution was incubated for specified time at 37 ◦ C. In both cases (for total protein and for albumin) some
chemical reactions happened and the formed color was proportional to the concentration.
Figure 10: UV-Vis spectra of standard a) total protein and b) albumin with reagent with marked
wavelengths of the lasers used for Z-scan measurement [12]
5.2
Measurements
Nd:YAG laser (532 nm) is usually used to study total protein and He-Ne laser (633 nm) for albumin
only. Figures 11 and 12 show results of a typical study. The intensities of the laser beams used were
I0 =7.824 kW/cm2 (Nd:YAG) and I0 =1.758 kW/cm2 (He-Ne laser).
Figure 11: T(z) graph for blood with standard concentration of total protein
9
Figure 12: T(z) graph for blood with standard concentration of albumin
Normally the level of total protein is in the range from 6 to 8.3 g/dl and of albumin is from 3.2 to 5
g/dl and this range is considered in the figures 11 and 12. If we compare the figures 11 and 12 with the
figure 5 we can see that n2 of total protein and albumin is negative. However, we are interested only in
absolute values, therefore we ignore the sign.
Figure 13: Linear variation of Tp−v with concentration of total protein and albumin [12]
Figure 14: Linear variation of n2 with concentration of total protein and albumin [12]
From this data we can calculate n2 . The values are presented in figure 14. From these results it follows
that measurements of n2 at two different wavelengths can provide information on concentration of both
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types of the proteins in the blood sample. The volume of the sample needed for these measurements is
relatively small (µl).
5.3
Conventional Colorimetric Method
This is a method to determine the concentration of a chemical compound (or chemical element) in a
solution (e.g. in the blood) with the aid of a color reagent. Once the reagent is added, the solution is put
into the colorimeter (a device used to test the concentration of a solution by measuring its absorbance
at specific wavelength of light [13]). It is widely used in medical laboratories for determination of blood
sugar, proteins, etc.
5.4
Comparison of Results of Two Different Methods
Colorimetric method
6.33
6.83
6.50
7.83
7.33
Z-scan method
6.22
6.90
6.54
7.79
7.26
Table 1: Concentration of total protein (g/dl)
Colorimetric method
3.42
3.85
3.68
4.20
4.02
Z-scan method
3.49
3.78
3.75
4.13
4.08
Table 2: Concentration of albumin (g/dl)
Five different samples of blood collected from five volunteers were studied. All the samples were
examined with both methods: colorimetric and Z-scan method. As we can see in the tables 1 and 2, the
results are in a good agreement for both protein and albumin. The samples were also separately tested
for another protein (globulin) and those values matched as well.
Because of these promising results, there are suggestions to use Z-scan method also for determination of
the values of LDL-cholesterol, glucose, total cholesterol, triglycerides etc. [12]
6
Conclusion
Even though the third order nonlinear effects are quite weak (n2 has values from 10−16 to 10−8 cm2 /W)
they are very important in experiments with pulsed laser beams. We can determine the third order
nonlinear refractive index n2 using Z-scan method, which is fast, simple and cheap and thus widely
applied. In this seminar we concentrate only on examinations of blood, but Z-scan method is used in
many different processes where the value of n2 is needed.
References
[1] M. Yin, H.P. Li, S.H. Tang, W. Ji, Appl. Phys. B 70, 587-591 (2000).
[2] Bahaa E. A. Saleh, Malvin Carl Teich, Fundamentals of Photonics (John Wiley, New York, 1991).
[3] S. M. Mian, B. Taheri, and J. P. Wicksted, J. Opt. Soc. Am. B 13 (1995).
[4] http://simphotek.com/bckg/images/z-scan.2.png (October 23 2014).
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[5] M. G. Kuzyk and C. W. Dirk, Characterization Techniques and Tabulations for Organic Nonlinear
Materials (Marcel Dekker, 1998).
[6] M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, Opt. Lett. 14, 955-957 (1989).
[7] S.L. Gomez, F.L.S. Cuppo, and A.M. Figueiredo Neto, Braz. J. Phys. 33, 813 (2003).
[8] M. Sheik-bahae, A. A. Said, T.H. Wei, D. J. Hagan, E. W. Van Stryland, IEEE J. Quant. Electron.
QE-26, 760 (1990).
[9] S. L. Gomez, F. L. S. Cuppo, A. M. Figueiredo Neto, T. Kosa, M. Muramatsu, and R. J. Horowicz,
Phys. Rev. E 59, 3059 (1999).
[10] http://en.wikipedia.org/wiki/Albumin (July 30 2014).
[11] http://en.wikipedia.org/wiki/Oncotic_pressure (October 23 2014).
[12] A. N. Dhinaa, P. K. Palanisamy, J. Biomedical Science and Engineering 3, 285-290 (2010).
[13] http://en.wikipedia.org/wiki/Colorimetry_(chemical_method) (October 23 2014).
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