University of LjubljanaFaculty of Mathematics and Physics

Department of Physics

Seminar Ib

Measurement ofThe Nonlinear Refractive Index

by Z-scan Technique

Author: Eva Ule

Advisor: izred. prof. dr. Irena Drevensek Olenik

Ljubljana, February 2015

Abstract

In this seminar the third-order nonlinear optical phenomena are described. The emphasis is onthe nonlinear refractive index n2. We describe the Z-scan technique, which is a simple method todetermine n2. At the end we show how Z-scan method can be used to examine protein concentrationin human blood.

Contents

1 Introduction 2

2 Nonlinear Optical Media 22.1 Nonlinear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 The Third-order Nonlinear Optics 33.1 Self-Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Z-scan Method 44.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Nonlinear Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Use of Z-Scan Technique to Measure Protein Concentration in Blood 85.1 Preparation of the Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.3 Conventional Colorimetric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 Comparison of Results of Two Different Methods . . . . . . . . . . . . . . . . . . . . . . . 11

6 Conclusion 11

1 Introduction

Materials with large third-order optical nonlinearities have recently become the topic of a broad scientificinterest, mostly because of their possible application in high speed optical switching devices, which arebecoming more and more important and more commonly used. [1] The nonlinearities of the third orderare 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 theelectric field

P = ε0χE, (1)

where ε0 is the permittivity of vacuum and χ is the dielectric susceptibility of the medium. In a nonlineardielectric 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 nonlinearrelation between P and E. [2]

The polarization P is a product of the individual dipole moment p, which is induced by the appliedelectric field E, and the number density of dipole moments. The relation between p and E is linear whenE is small and becomes nonlinear as E acquires values from 105 to 108 V/m. External electric fields arealmost always smaller than the characteristic interatomic fields, even when laser beams are used, which

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means that the nonlinearity is usually weak. So we can expand the function relating P to E in a Taylor’sseries about E = 0:

P = ε0χE + ε0χ(2) : EE + ε0χ

(3)...EEE + ... (2)

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’sequations

∇2E− 1

c20

∂2E

∂t2= µ0

∂2P

∂t2, (3)

with E being the electric field and P the polarization. It is useful to write P as a sum of linear andnonlinear parts

P = ε0χE + PNL (4)

and

PNL = ε0χ(2) : EE + ε0χ

(3)...EEE + ... (5)

If we use the relations n2 = 1 + χ, c0 = 1√µ0ε0

and c = c0n , we can write a wave equation in nonlinear

medium

∇2E− 1

c2∂2E

∂t2= µ0

∂2PNL∂t2

. (6)

This is the basic equation in the theory of nonlinear optics. Two approaches exist to approximatelysolve this partial differential equation. The first one is the Born approximation and the second is thecoupled-wave theory.

3 The Third-order Nonlinear Optics

In media with centrosymmetry (the properties of the medium are not altered by the transformationr→ −r), the second-order nonlinear susceptibility χ(2) is 0, so the third-order term becomes importantand equation (2) is

P = ε0χE + 0 + ε0χ(3)

...EEE (7)

and where E isE =

e

2(A exp [ikz − iωt] +A∗ exp [−ikz + iωt]) . (8)

We combine equations (7) and (8) and we get

P = ε0χ(1)E + 3ε0

(χ(3) 1

4

)[AA∗]E. (9)

Using 〈j〉 = 12ε0c0n(ω) |A|2 (j is the energy flux density) we rewrite the equation (9) in

P = ε0

(χ(1) +

3

2χ(3)eff

〈j〉ε0c0n(ω)

)E = ε0(εeff − 1)E. (10)

Using the definition of the refractive index

neff =√εeff (11)

we get the result

neff = n+3

4·χ(3)eff 〈j〉ε0c0n2

= n+ n2 〈j〉 . (12)

(This is called the optical Kerr effect because of its similarity to the electro-optic Kerr effect.) Thecoefficient n2 has units cm2/W and values from 10−16 to 10−14 in glasses, 10−14 to 10−7 in dopedglasses, 10−10 to 10−8 in organic materials and 10−10 to 10−2 in semiconductors. It depends on thepolarization state of the optical beam and it is sensitive to the operating wavelength. In most cases thepeak intensity of the laser beam is around 1 GW/cm2.

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3.1 Self-Focusing

An interesting effect that can be seen in the third-order nonlinear medium is self-focusing. If an intenseoptical 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

I = I0 exp

[−2ρ2

w2

](13)

where ρ = (x2 + y2)1/2 is transversal coordinate and w is beam radius, the focal length of the effectivelens is given as:

f =w2

2n2I0d, (14)

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 simultaneouslymeasure nonlinear absorption and nonlinear refraction in solids, liquids and liquid solutions. It is asingle-beam technique that gives us both the sign and magnitude of refractive index nonlinearities. Thismethod is rapid, simple to perform and accurate, therefore it is often used. It is especially adequate fordetermination 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]

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We use a Gaussian laser beam and then the transmittance of the beam through the experimental systemshown in figure 3 is measured. We have to change a position of the sample with respect to the focalplane 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 transmittanceat the same time

The measurement starts far away from the focus (negative z), where the transmittance is relativelyconstant (figure 4 a). Then the sample is moved towards the focus and then to the positive z (figure4 b to h). If the material has a positive nonlinearity (n2 > 0), the T(z) graph has a valley first andthen a peak. For the sample with n2 < 0 the graph is exactly the opposite (first the peak and then the

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valley) (figure 5). When self-focusing in the sample occurs, this tends to collimate the beam and causesa beam narrowing at the aperture which results in an increase in the measured transmittance and whenself-defocusing occurs the beam broadens at the aperture and the transmittance decreases. The scan isfinished 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 liquidcrystals [7]

There are not many materials with n2 > 0, the negative n2 is much more common. (This is attributedto a thermal nonlinearity. [6])

4.1 Theory

As mentioned before, we use Gaussian beam with the magnitude of E

E(r, z, t) = E0(t)w0

w(z)exp

[− r2

w2(z)− ikr2

2R(z)

]· exp [−iφ(z, t)], (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 thelast 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

∆φ(z, r, t) =∆φ0(t)

1 + z2/z20exp

[− 2r2

w2(z)

]. (16)

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 tothe centre of the beam. The most important term for us is ∆φ0(t), which is defined

∆φ0(t) = kn2I0(t)Leff , (17)

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where I0 is the irradiance at the focus z = 0 and Leff = (1− exp[−αL])/α is the effective propagationlength inside the sample with the sample length L and the linear absorption coefficient α. At the exitof the sample there is the complex electric field Ee which contains the nonlinear phase distortion. Weuse the Gaussian decomposition (GD) method, in which the complex electric field at the exit plane ofthe sample is decomposed into a summation of Gaussian beams through a Taylor series expansion of thenonlinear phase term. [8] When all these beams come to the aperture plane they are again united in onesingle beam. We get the electric field at the aperture Ea(r, t) and it is function of ∆φ0. If we spatiallyintegrate |Ea(r, t)|2 up to the aperture radius ra we get the transmitted power PT (∆φ0(t)). Now we cancalculate the normalized Z-scan transmittance T (z) which can be seen in the figure 4

T (z) =

∫∞−∞ PT (∆φ0(t))dt

S∫∞−∞ Pi(t)dt

(18)

where Pi(t) is the instantaneous input power within the sample and S is the aperture linear transmittanceS = 1 − exp(−2r2a/w

2a) (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) andthe 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

kI0Leff. (20)

1The size of the aperture is signified by its transmittance S in the linear regime, i.e. when the sample has been placedfar 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 nonlinearbeam 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 absorptionis 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 absorptionβ = 5.8 cm/GW [8]

Our final graph is represented in the picture 9. As we can see, the peak and the valley are asymmetricand 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 absorptiononly from the form of the transmittance graph.

5 Use of Z-Scan Technique to Measure Protein Concentrationin Blood

The proteins which are made of amino acids, are very important compounds in our body. They makenew cells, maintain muscles, support the immune system, etc. The two major groups of proteins in thehuman blood are albumin and globulin. Albumin is a protein, which is important for tissue growth, butits main function is to regulate the colloidal osmotic pressure2 of blood [10]. It is mostly produced inthe liver. Globulins are produced either by the immune system or by the liver. We know alpha, betaand gamma types of them and they have various roles in our body.

2Colloid 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]

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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 colorimetricmethod. It is because of the nonlinear optical properties of total protein and albumin that we can useZ-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 thesolution was incubated for specified time at 37 ◦C. In both cases (for total protein and for albumin) somechemical 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 markedwavelengths 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 albuminonly. Figures 11 and 12 show results of a typical study. The intensities of the laser beams used wereI0=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

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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 5g/dl and this range is considered in the figures 11 and 12. If we compare the figures 11 and 12 with thefigure 5 we can see that n2 of total protein and albumin is negative. However, we are interested only inabsolute 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 followsthat 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 isrelatively small (µl).

5.3 Conventional Colorimetric Method

This is a method to determine the concentration of a chemical compound (or chemical element) in asolution (e.g. in the blood) with the aid of a color reagent. Once the reagent is added, the solution is putinto the colorimeter (a device used to test the concentration of a solution by measuring its absorbanceat specific wavelength of light [13]). It is widely used in medical laboratories for determination of bloodsugar, proteins, etc.

5.4 Comparison of Results of Two Different Methods

Colorimetric method Z-scan method

6.33 6.226.83 6.906.50 6.547.83 7.797.33 7.26

Table 1: Concentration of total protein (g/dl)

Colorimetric method Z-scan method

3.42 3.493.85 3.783.68 3.754.20 4.134.02 4.08

Table 2: Concentration of albumin (g/dl)

Five different samples of blood collected from five volunteers were studied. All the samples wereexamined with both methods: colorimetric and Z-scan method. As we can see in the tables 1 and 2, theresults are in a good agreement for both protein and albumin. The samples were also separately testedfor 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 ofthe 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 ordernonlinear refractive index n2 using Z-scan method, which is fast, simple and cheap and thus widelyapplied. In this seminar we concentrate only on examinations of blood, but Z-scan method is used inmany 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 NonlinearMaterials (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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