Optik 121 (2010) 2172–2175

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Optik

0030-40

doi:10.1

n Corr

Shando

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journal homepage: www.elsevier.de/ijleo

Study on Z-scan characteristics of multilayer nonlinear media usingcoordinate transformation method

Yongliang Liu a,b,n, Weiping Zang b, Zhibo Liu b, Shengwen Qi a,b, Jianguo Tian b

a Key Laboratory of Biophysics in Universities of Shandong, Department of Physics, Dezhou University, Dezhou 253023, Chinab The Key Laboratory of Weak Light Nonlinear Photonics, Ministry of Education, Teda Applied Physics School, Nankai University, Tianjin 300457, China

a r t i c l e i n f o

Article history:

Received 22 March 2009

Accepted 4 September 2009

Keywords:

Numerical simulations

Multilayer

Coordinate transformation

Z-scan

26/$ - see front matter & 2009 Elsevier Gmb

016/j.ijleo.2009.11.002

esponding author at: Key Laboratory of B

ng, Department of Physics, Dezhou University

534 8985800.

ail address: liuyl593@yahoo.com.cn (Y. Liu).

a b s t r a c t

We present a method that combines coordinate transformation and Crank–Nicholson finite difference

scheme to simulate numerically Z-scan measurements of a multilayer nonlinear medium or a

complicated multilayer structure. The method would be useful to design and optimize optical limiters

and to determine the nonlinearities of cascading medium layers. The method has given a guideline for

determining the characteristics of multilayer medium.

& 2009 Elsevier GmbH. All rights reserved.

1. Introduction

Z-scan method has become a very popular tool [1] formeasurements of nonlinear index because of its simplicity andaccuracy. Passive optical limiters usually use tight focus geome-try, which is analogous to the geometry arrangement of Z-scanmeasurements, to focus the light into a nonlinear optical (NLO)element to reduce the limiting threshold. Unfortunately, NLOelements themselves always undergo laser-induced damages athigh inputs, which lowers the dynamic range (DR) of systems. Sothe tandem of nonlinear media was proposed [2,3] to improve DRof systems. On the other hand, inhomogeneous distributednonlinear media are sometimes met for Z-scan measurements,such as nanoparticles embedded in solid hosts or sol–gel [4]. Thismay form a structure of multiple nonlinear medium layers. Iteasily leads to an incorrect conclusion if we do not use a suitablemethod to analyze experimental data. We have proposed somemethods to treat this situation [5–7]. Some drawbacks aboutnumerical efficiency or accuracy have not been eliminated. Thecoordinate transform method has been shown to be a highlyefficient and accurate method for beam propagation in nonlinearmedium [8–9].

In this paper, we present a method that combines coordinatetransformation and Crank–Nicholson finite difference scheme to

H. All rights reserved.

iophysics in Universities of

, Dezhou 253023, China.

simulate numerically Z-scan measurements of a multilayernonlinear medium or a complicated multilayer structure.

For a demonstration, we numerically simulate a multilayermedium consisting of some linear medium layers and somenonlinear medium layers.

2. Methods

2.1. Coordinate transformation method

The coordinate transformation is well documented in Ref. [9].The paraxial wave equation in the Fresnel approximation can bewritten as

1

r

@

@rr@E

@r

� ��2ik

@E

@zþ k2

0wNLðr; z; tÞ� �

E¼ 0; ð1Þ

where wNL is the nonlinear susceptibility of the medium.For a nonlinear medium with Kerr effect and two-photon

absorption, real and imaginary parts of the nonlinear suscept-ibility are related to the Kerr coefficient and the two-photonabsorption as follows:

Re wNL r; tð Þ� �

¼ 2n0n2 Ej j2; ð2aÞ

Im wNL r; tð Þ� �

¼�n0b2

k0Ej j2; ð2bÞ

where n2 is Kerr coefficient, b2 is two-photon absorptioncoefficient, n0 is linear refractive index of medium and k0 is wavevector in vacuum.

Fig. 1. The relation between the equivalent medium length Lequ and the position of

medium x. The real medium length is L=1, 2, and 4, respectively. The unit of

medium length is Rayleigh length z0.

Y. Liu et al. / Optik 121 (2010) 2172–2175 2173

Assuming a TEM00 Gaussian beam traveling in the +z direction,we can write E as follows:

Eðr; z; tÞ ¼ E0ðtÞw0

wðzÞexp �

r2

w2ðzÞ�

ikr2

2RðzÞþ itan�1ðz=z0Þ

� ; ð3Þ

where w2ðzÞ ¼w20 1þz2=z2

0

�is the beam radius, RðzÞ ¼ z 1þz2

0=z2 �

is the radius of curvature of the wave-front at z, w0 is the radius ofbeam at the focus and z0 ¼ kw2

0=2 is the Rayleigh length of thebeam, k¼ 2p=l is the wave vector. E0ðtÞ denotes the electric fieldat the focus and contains the temporal envelope of the laser pulse.

Using the coordinate transformations [9]

Eðr; zÞ � E0ðtÞC r; zð ÞAð~r ; ~zÞ; ð4aÞ

C r; zð Þ ¼w0

wðzÞexp �

ikr2

2RðzÞþ itan�1ðzÞ

" #; ð4bÞ

d~z

dz¼

1

1þz2; ð4cÞ

where z¼ z=z0, r¼ r=w0, and ~r ¼ r=ð1þz2Þ. The phase of function

A ~r ; ~zð Þ changes very slowly with ~r . Rewriting Eq. (1) for thefunction A ~r ; ~zð Þ in these new coordinates, the following equationswill be obtained:

1~r@

@~r~r@A

@~r

� �þ4 1�~r2�

A�4i@A

@~z�p Að ÞA¼ 0; ð5aÞ

p Að Þ ¼4kDn tð Þ A

�� ��2n0

�2iDa tð Þ A�� ��2" #

z0; ð5bÞ

where Dn tð Þ ¼ n2I0 tð Þ is the change of refraction at focus for thenonlinear medium, Da tð Þ ¼ b2I0 tð Þ is the nonlinear absorptioncoefficient at focus for the nonlinear medium.

For the case of p=0, which corresponds to the beampropagation in vacuum, the function A ~r ; ~zð Þ has the followingform:

Að~r ; ~zÞ ¼ expð�~r2Þ: ð6Þ

It means that the form of the function A ~r ; ~zð Þ in vacuum keepsunchanged while the beam propagates in this coordinate system.Therefore, the use of the transformations in Eqs. (4) cansignificantly reduce the size of the data arrays and the computa-tional time, since fewer points can express the radial fielddistribution and the propagation distance is shrunken greatly.

The Z-scan experimental arrangement is the same as that inRef. [2]. If the front surface of the medium is located at z and themedium length is L, one can obtain the equivalent medium lengthLequ by Eq. (4c), which is defined as the medium lengthcorresponding to the distance between the front surface and theexit surface of medium in this new coordinate system:

Lequ ¼ tan�1ðzþLÞ�tan�1ðzÞ; ð7Þ

where z¼ z=z0, L¼ L=z0, and Lequ is the dimensionless quantity.It can be seen from Eq. (7) that the equivalent medium length

in new coordinate system is the function of the front surfaceposition of medium and the medium length. The equivalentmedium length as a function of the position of medium is shownin Fig. 1. We can see from Fig. 1 that the equivalent mediumlength Lequ is the function of the position of medium and mediumlength L. When the focal point of beam is located in the middle ofmedium, for a given length L, Lequ obtains its maximum. Lequ isslowly increased with the increase of medium length L and isquickly decreased with the increase of the distance between themiddle of medium and focal point of beam; if medium lengthL-1 and the focal point of beam is located in middle of medium,Lequ is equal to p. So the length of medium in new coordinates isshrunken greatly. Eqs. (1) and (5) are parabolic partial differential

equations and can be solved numerically by the Crank–Nicholsonfinite difference scheme [10].

2.2. Multilayer nonlinear medium

The multilayer nonlinear medium consists of a stack ofnonlinear medium and linear medium in our analysis. It can beseen that there is no explicit dependence on the medium positionin Eq. (5a). Therefore, the function Að~r ; ~zÞ, solved from Eq. (5a), isnot the explicit function of the medium position. The functionAð~r ; ~zÞ is only determined by the equivalent medium length that isthe function of medium position and medium length.

Suppose multilayer nonlinear medium has n layers, the lengthof ith layer isLi, the linear and nonlinear refraction index areni

0andni2, the distance between focus point and the entrance face

of first layer is z. We can obtain the length of ith layer in vacuumas

Livac ¼

Li

ni0

: ð8Þ

The coordinate of entrance face of ith layer is

di ¼ zþXi�1

k ¼ 1

Lk

nk0

: ð9Þ

From Eq.(7), we can obtain the equivalent medium length Liequ

of ith layer as

Liequ ¼ tan�1 diþLi

vac

� �tan�1 dið Þ: ð10Þ

3. Numerical simulation

To simulate numerically the beam propagation in a nonlinearmedium, two beam-propagation method (BPM) algorithms areusually used. They are 1D Fast Fourier Transform (FFT) basedspectral algorithm [11–13] and second-order accuracy Crank–Nicholson finite difference scheme [10]. The first algorithmutilizes the expansion of propagation operator into the Taylorseries and thus can ultimately reach any desired order ofaccuracy. But FFT-BPM algorithm has the following disadvantagesdue to the nature of the FFT [14]: (1) it requires a longcomputation time; (2) the propagation step size has to be small;(3) the number of sampling points must be a power of 2, andso on. The second algorithm is unitary and unconditionallysteady. The numerical efficiency in this case is better than thatof FFT-BPM [15]. Hence, we apply coordinate transformationmethod and Crank–Nicholson finite difference scheme to the

Fig. 2. Close-aperture Z-scan curves of multilayer medium. The number of layer of

multilayer medium, Ln is 3, 5, and 7, respectively.

Fig. 3. Close-aperture Z-scan curves of multilayer medium. The length of linear

layer of multilayer medium Ll is 0.5, 1, and 1.5, respectively.

Y. Liu et al. / Optik 121 (2010) 2172–21752174

numerical simulation. Below we consider Z-scan of multilayerstructure consisting of linear and nonlinear mediums. Thecalculation procedure can be summarized as follows:

1.

At the entrance face of first medium, the beam propagatesthrough the first medium, and electric field amplitude andphase at the exit face of the medium can be obtained by Eqs.(5). This procedure can be dealt with by Crank–Nicholson finitedifference scheme.

2.

After going through the first linear or nonlinear medium, thebeam continues to propagate through the next medium. Thisprocedure can be dealt with by Crank–Nicholson finitedifference scheme also. If this is not last medium repeat step2, otherwise go to step 3.

3.

Fig. 4. Closed-aperture Z-scan curves for a uniform nonlinear medium and

multilayer medium, the number of layer is 7.

Finally the beam propagates through vacuum to the far fieldaperture. This procedure can be treated by Fresnel diffractionintegral. Of course, we can also use coordinate transformationand Crank–Nicholson finite difference scheme to obtain theelectric field at the far field aperture, but it takes much longercomputation time than Fresnel diffraction integral since thedistance from the second thin medium to far field aperture istoo far.

Therefore, numerical simulation can deal with any complicatedmultilayer structure.

4. Result and discussion

As examples of application of method proposed in this paper,we numerically simulate some Z-scan of multilayer structure.

The first example is a multilayer medium consisting of linearand nonlinear medium alternately. The length of each layer invacuumLi

vac is z0. Let the first layer be nonlinear medium and thenumber of layer be 3, 5, and 7. The change in refractive index is setto Dn¼ n2I0 ¼ 1:5� 10�5. Fig. 2 gives a close-aperture Z-scancurve for the number of layer, Ln is 3, 5, and 7.

Results show that the coarse structure of plot contain thewhole length of multilayer, the distance between the peak andthe valley of normalized transmittance approximately equal tothe whole length of multilayer, as a uniform thick medium case.The detailed structure of plot contains the information of thestructure of multilayer. The detailed structure of plot here refersto the ripples between the peak and the valley of normalizedtransmittance.

The ripples between the peak and the valley of normalizedtransmittance change with the length of linear layer betweennonlinear layers. Fig. 3 gives a close-aperture Z-scan curve for the

length of linear layer, Ll is 0.5, 1, and 1.5z0, respectively. Resultsshow that the characteristic of ripples changes with the length oflinear layer. As the length of linear layer increases, the distancebetween ripples and the amplitudes of oscillation of ripplesincrease. So if the length of linear layer is short enough, theoscillation of ripples is weak enough. Fig. 4 gives the multilayermedium and uniform nonlinear medium, they have the samelength. The change in refractive index in multilayer is settoDnmm ¼ n2I0 ¼ 1:5� 10�5and that in uniform medium is settoDnum ¼ 3Dnmm=4. The length of linear layer is 0.4z0 and thenumber of layer is seven. These two samples have an almost sameplot of normalized transmittance. In other words, if we examinethe nonlinear refraction of multilayer medium, and simulate theexperimental data using uniform medium, in Fig. 4, the error ofresult will reach 25%.

So in order to get the real characteristic of medium, we mustconfirm the structure of the sample measured first. The sameproblem appears open z-scan experiments. Fig. 5 shows openZ-scan curves. The parameters are the same as in Fig. 2, exceptthat the change in refractive index in multilayer is replaced by thenonlinear absorption coefficientDa¼ b2I0 ¼ 1cm�1.

5. Conclusion

In summary, we present a method that combines coordinatetransformation and Crank–Nicholson finite difference scheme tosimulate numerically Z-scan measurements of a multilayernonlinear medium or a complicated multilayer structure. Themethod would be useful to design and optimize optical limitersand to determine the nonlinearities of cascading medium layers.

Fig. 5. Open Z-scan curves of multilayer medium. The number of layer of

multilayer medium, Ln is 3, 5, and 7, respectively. The parameters are the same as

in Fig. 2, except that the nonlinear refraction is replaced by nonlinear absorption.

Y. Liu et al. / Optik 121 (2010) 2172–2175 2175

As an example, we numerically simulate the Z-scan for close-aperture and open-aperture. The results show that in order todetermine the characteristics of nonlinear multilayer medium, thestructure of medium must be determined first.

Acknowledgments

This research is supported by project 60025512 supported bythe National Natural Science Foundation of China, the NaturalScience Foundation of Shandong Province (Grant Y2006A01), akey project of the Ministry of Education (Grant 00026), theScience and Technology Foundation of Education Department ofShandong (Grant J08LI53), the Foundation for University KeyTeachers of the Ministry of Education, and the Fok Ying TungEducation Foundation (Grant 71008).

References

[1] M. Sheik-Bahae, A.A. Said, T.H. Wei, D.J. Hagan, E.W. Van Stryland, Sensitivemeasurement of optical nonlinearities using a single beam, IEEE. J. QuantumElectron. 26 (1990) 760–769.

[2] P.A. Miles, Bottleneck optical limiters: the optimal use of excited-stateabsorbers, Appl. Opt. 33 (1994) 6965–6979.

[3] T. Xia, D.J. Hagan, A. Dogariu, A.A. Said, E.W. Van Stryland, Optimization ofoptical limiting devices based on excited-state absorption, Appl. Opt. 36(1997) 4110–4122.

[4] M. Kyoung, M. Lee, Nonlinear absorption and refractive index measurementsof silver nanorods by the Z-scan technique, Opt. Commun. 171 (1999)145–148.

[5] W.P. Zang, J.G. Tian, Z.B. Liu, W.Y. Zhou, X.J. Yang, C.P. Zhang, G..Y. Zhang, Z-scan characteristics of cascading nonlinear media, Chin. Phys. Lett. 20 (2003)1067–1069.

[6] W.P. Zang, J.G. Tian, Z.B. Liu, W.Y. Zhou, C.P. Zhang, G.Y. Zhang, Study on Z-scan characteristics of cascaded nonlinear media, Appl. Phys. B 77 (2003)529–533.

[7] W.P. Zang, J.G. Tian, Z.B. Liu, W.Y. Zhou, C. Mei, F. Song, C.P. Zhang, Analysis onZ-scan characteristics of cascade structure, Acta Phys. Sin. 53 (2004)1820–1825.

[8] W.P. Zang, J.G. Tian, Z.B. Liu, W.Y. Zhou, F. Song, C.P. Zhang, Coordinatetransformation for fast simulation of Z-scan measurements, Chin. Phys. Lett.21 (2004) 662–665.

[9] W.P. Zang, J.G. Tian, Z.B. Liu, W.Y. Zhou, F. Song, C.P. Zhang, The coordinatetransformation and coordinate mapping for numerical simulation of Z-scanmeasurements, Opt. Commun. 244 (2005) 71–77.

[10] C.F. Gerald, P.O. Wheatley, in: Appl. Numer. Anal., Addison Wesley, 1984Chap. 8.

[11] J.A. Fleck, J.R. Morris, M.D. Feit, Time-dependent propagation of high energylaser beams through the atmosphere, Appl. Phys. 10 (1976) 129–160.

[12] M.D. Feit, J.A. Fleck, Simple spectral method for solving propagation problemsin cylindrical geometry with fast Fourier transforms, Opt. Lett. 14 (1989)662–664.

[13] S. Hughes, J.M. Burzer, G. Spruce, B.S. Wherrett, Fast fourier transformtechniques for efficient simulation of Z-scan measurements, J. Opt. Soc. Am. B12 (1995) 1888–1893.

[14] K. Kawano, T. Kitoh, in: Introduction to Optical Waveguide Analysis, Wiley,New York, 2001 Chap.5.

[15] R. Scarmozzino, R.M. Osgood, Comparison of finite-difference and Fourier-transform solutions of the parabolic wave equation with emphasis onintegrated-optics applications, J. Opt. Soc. Am. A 8 (1991) 724–731.