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Optik 121 (2010) 2172–2175
Contents lists available at ScienceDirect
Optik
0030-40
doi:10.1
n Corr
Shando
Fax: 86
E-m
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: [email protected] (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 andmultilayer 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).
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