- Email: [email protected]
Optical bistability and beam reshaping in nonlinear multilayered structures
1 November 1998
Optics Communications 156 Ž1998. 219–226
Full length article
Optical bistability and beam reshaping in nonlinear multilayered structures Xinghua Lu
a,)
, Yingxin Bai a , Shiqun Li a , Tianjie Chen
b
a
b
Department of Modern Applied Physics, Tsinghua UniÕersity, Beijing 10084, China Department of Physics, Southern Illinois UniÕersity at Carbondale, Carbondale, IL 62901-4401, USA Received 17 July 1997; revised 31 March 1998; accepted 11 May 1998
Abstract A new algorithm which takes into account the history of input intensity is presented to describe nonlinear characteristics such as optical bistability and multistability in nonlinear multilayered structures. The optical response along with the history of input intensity is demonstrated in a straightforward way with this algorithm. Beam reshaping due to different transmissivity in the transverse section is investigated when a laser beam passes through nonlinear multilayered structures. q 1998 Published by Elsevier Science B.V. All rights reserved.
1. Introduction Since the first observation of optical bistability in 1974 w1x there has been increasing interest in the study of the optical response of nonlinear materials because of its potential application for optical switching, optical memories, optical transistors, optical limiters and optical logic gates, etc. w2x. Optical bistability and multistability have been investigated in various nonlinear devices such as Fabry–Perot resonators w2x, thin films w3,4x and multilayered structures w5–11x. In recent years certain valuable applications with nonlinear one-dimensional structures, such as broad band optical limiters and optical logic operation devices, have been designed w12–15x. Optical response of two-dimensional nonlinear superlattices has also been investigated both theoretically and experimentally by Xu and Ming w16,17x. Based on energy integrals, an exact solution of the nonlinear equations was proposed by Chen and Mills w4–6x to investigate the optical response of a lossless nonlinear thin film, bilayers and superlattices. More recently, a transfer matrix formalism was proposed by Danckaert et al. w8x to describe the stationary optical response in nonlinear multilayered structures. This nonlinear transfer matrix formalism relies on the slowly varying envelope approximation ŽSVEA. and only treats those structures whose layer thickness is bigger than the wavelength in the material. Several other works based on transfer matrix formalism have also been reported recently w9–11x. In the case of multistability, there is more than one solution corresponding to a given input intensity, and it is impossible to derive the output intensity directly. Information about the history of input intensity is necessary to be taken into account to determine the exact solution, i.e., on which level the output intensity should be. In previous studies, the dependence of transmissivity on input intensity was studied by treating the output intensity as a parameter. In other words, the dependence of transmissivity on the output intensity, which is a mono-valued function, is derived in the first step for further investigation. Now there is a question: Is it possible to calculate the output intensity directly from the incident intensity if the history of the input intensity is known? An attempt was made by Zhao et al. w11x to calculate the direct transmissivity versus input
)
Corresponding author. E-mail: [email protected]
0030-4018r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 2 7 8 - 8
X. Lu et al.r Optics Communications 156 (1998) 219–226
220
Fig. 1. Schematic view of the multilayered structure.
intensity with a perturbation approach and iterating algorithm. However, their iterating algorithm does not converge and gives chaotic solutions when the input intensity reaches the bistability threshold, as will be discussed in detail in Section 2. In this paper, we present an improved algorithm based on convergent iteration to obtain the output intensity directly from the input intensity. Optical response such as bistability in nonlinear multilayered structures is simulated with this improved algorithm in Section 3. In Section 4, we will consider the reshaping of a laser beam injected into nonlinear multilayered structures. Different field intensity in transverse section will cause different transmissivity and result in an interesting transverse function for the output beam.
2. Model and theory We consider a plane-polarized electromagnetic wave with frequency v that propagates along the z direction and normal to the interfaces of the nonlinear multilayered structure shown in Fig. 1. Each layer exhibits an optical nonlinearity of Kerr form and is characterized by its linear refractive index nŽ0. j , nonlinear coefficient b j and thickness d j s z j y z jy1 . The nonlinear coefficient b j may be either positive or negative and the linear refractive index may be a complex number if there is absorption or gain in the medium. The full refractive index of the jth layer is written as < <2 n j Ž z . s nŽ0. j q bj E Ž z . ,
Ž1.
where EŽ z . is the electric field at point z in the structure. Because of the interference of forward and backward propagating waves, the electric field intensity varies on the scale of wavelength along the z direction and so does the refractive index in each layer. Instead of the energy integration in Refs. w4–6x, we use the thin layer approximation similar to that suggested by Agranovich et al. w9x to solve the nonlinear equations. If the thickness of each layer is much smaller than the incident wavelength, i.e. d j < l, the field intensity in each layer can be regarded as uniform and the refractive index can be approximately written as < <2 n j s nŽ0. j q bj E Ž z j . .
Ž2.
Assume a light wave of frequency v incident from the left in the region z - z 0 , transmitted on the right, where z ) z N . The electric field can be written in the form EŽ z,t . s EŽ z .expŽyi v t ., where
° E expŽ i k Ž z y z . . q E expŽ yi k Ž z y z . . , z - z , E Ž z . s~ F exp Ž i k Ž z y z . . q B exp Ž yi k Ž z y z . . , z - z - z , ¢ E exp Ž i k Ž zyz .. , z)z . I
j
0
j
0
jy1 T
R
0
j
j
Nq1
N
0
0
jy1
jy1
Ž3.
j
n
Here Fj and Bj are the amplitudes of the forward and backward propagating waves in the jth layer, while ES , E I , and E T are the amplitudes of the incident, reflected and transmitted waves, respectively, k j s 2p n jrl is the wave vector in the jth layer and l is the vacuum wavelength of the incident field. The boundary conditions that the tangential components of EŽ z . and its derivative in the z direction are continuous lead to the amplitude relations between the jth and Ž j q 1.th layers written in the matrix form as follows: 1 Fjy1 B jy1
2 s
1 2
ž ž
1q 1y
nj n jy1 nj n jy1
/ /
exp Ž yi k jy1 d jy1 . exp Ž qi k jy1 d jy1 .
1 2 1 2
ž ž
1y 1q
nj n jy1 nj n jy1
/ /
exp Ž yi k jy1 d jy1 . exp Ž qi k jy1 d jy1 .
Fj Bj
.
Ž4.
X. Lu et al.r Optics Communications 156 (1998) 219–226
221
We define the nonlinear transfer matrix as 1 N
Ms Ł
js1
2 1 2
ž ž
1q 1y
nj n jy1 nj n jy1
/ /
exp Ž yi k jy1 d jy1 . exp Ž qi k jy1 d jy1 .
1 2 1 2
ž ž
1y 1q
nj n jy1 nj n jy1
/ /
exp Ž yi k jy1 d jy1 . ,
Ž5.
exp Ž qi k jy1 d jy1 .
which satisfies EI E sM T . ER 0 The reflectivity and transmissivity can be written with the transfer matrix elements,
Fig. 2. Transmissivity calculated with the direct iterating algorithm with Ža. g s1, Žb. g s 0.05 and Žc. g s 0.01, respectively.
Ž6.
X. Lu et al.r Optics Communications 156 (1998) 219–226
222
Rs
IR II
s
< ER < 2 < EI < 2
s
M2,1 M1,1
2
,
Ts
IT II
s1yRs1y
M2,1 M1,1
2
,
Ž7.
where Mi , j are the elements of transfer matrix M. Since the nonlinear transfer matrix M depends on the field intensity in each layer, it is difficult to derive R and T for a given input intensity directly. Agranovich et al. w9x used an iterating algorithm to derive the dependence of transmissivity T on the output intensity, by calculating from the output side of the multilayered structures to the input side with Eq. Ž4.. Then the output intensity can be found by varying it as a parameter. Instead of this method, Zhao et al. w11x used another iterating algorithm to solve Eq. Ž6. for a given input intensity in a direct way. In their algorithm, the field intensity in each layer is taken to be zero first to calculate the zero-order nonlinear transfer matrix M Ž0.. Then the zero-order solution of electric fields E Ž0. Ž z . can be derived from Eqs. Ž6. and Ž4.. Substituting the electric fields into Eq. Ž5., we get the first-order nonlinear transfer matrix M Ž1. Ž< E Ž0. Ž z .< 2 .. The higher order nonlinear transfer matrix M Ž nq1. Ž< E Ž n. Ž z .< 2 . can be derived in sequence by repeating the above process. This algorithm can be expressed as follows M Ž0. ™ E Ž0. Ž z . ™ M Ž1. Ž < E Ž 1 . Ž z . < 2 . ™ E Ž1. Ž z . ™ PPP ™ E` Ž z . ,
Ž8.
Ž`. Ž
and is supposed to lead to a convergent solution E z .. However, it is correct only when the input intensity is weaker than the threshold of bistability or multistability. In the case of bistability or multistability, the iteration leads to divergent solutions. Fig. 2a presents the transmissivity as a function of the iterating order with such an iterating algorithm, which gives a divergent solution. In order to overcome the divergence, we find it effective to replace < E Ž j. Ž z .< 2 with the weighted averages of < E Ž j. Ž z .< 2 and < E Ž jy1. Ž z .< 2 , which has the form of < EXŽ j. Ž z .< 2 s g < E Ž j. Ž z .< 2 q Ž1 y g .< E Ž jy1. Ž z .< 2 , in the Ž j q 1.th nonlinear transfer matrix M Ž jq1., where g , satisfying 0 - g F 1, is a parameter used to control the weight of < E Ž j. Ž z .< 2. Because < EXŽ j. Ž z .< 2 is
Fig. 3. Bistability characteristics of nonlinear multilayered structure Ž40 layers. calculated with Ža. the direct iterating algorithm and Žb. the roundabout algorithm.
X. Lu et al.r Optics Communications 156 (1998) 219–226
223
between < E Ž j. Ž z .< 2 and < E Ž jy1. Ž z .< 2 , if the series < E Ž j. Ž z .< 2 is convergent, the series < EXŽ j. Ž z .< 2 is also convergent and has the same limit. So, the limit of the nonlinear transfer matrix M Ž`. is the solution of Eq. Ž6. EI E s M Ž`. T . ER 0
Ž9.
We have discovered in our studies that this improved algorithm gives convergent solutions with proper selection of the weight g . Generally speaking, the solutions converge with sufficiently small g and too small weight g may require a large iterating order. In practice, at first we choose a larger value of g , e.g. g s 1, then if the solution does not converge, a smaller weight g is chosen. Fig. 2b and 2c show two convergent series of transmissivity calculated with g equal to 0.05 and 0.01, respectively. However, such a convergent algorithm leads to only one solution for any given input intensity and is insufficient for the case of multistability which requires multi-solutions. Why is this so? And how can we find other Žor proper. solutions? This can be explained from the nature of bistability and multistability, which indicates that the output of a nonlinear system depends not only on the amplitude but also on the history of the input intensity. There is no way to derive the corresponding solution without including the history of the input intensity in the calculation. We find that it is the zero-order nonlinear transfer matrix M Ž0. that contains the information of the history of input intensity. In the above algorithm, the electric field in each layer is taken to be zero, which means that the input intensity increases from zero and the solution only corresponds to the lower output level. If the input intensity decreases from a higher value above the upper threshold, we need to take the solution of the transfer matrix M for higher input intensity as the zero-order transfer matrix M Ž0. in iteration. Then the algorithm leads to another solution Župper output level.. Thus, an effective iterating algorithm which takes into account the history of the input intensity has been established to solve the optical response in nonlinear multilayered structures. For a given history of the input intensity, we first separate the time domain into some discrete parts. The zero-order nonlinear transfer matrix M Ž0. for each time point is given by the nonlinear transfer matrix solved at the previous time point. ŽThe zero-order nonlinear transfer matrix at the first time point is established by taking the electric fields in each layer to be zero.. In the next section, we perform some numerical simulations.
3. Numerical calculation and discussion Consider a nonlinear multilayered structure with alternating refractive index n H equal to 2.3 and n L equal to 1.48. The refractive indexes in region z - z 0 and z ) z N are 1.0 and 1.52, respectively. Although the absorption and gain can also be taken into account in simulation, we do not include them here because the main task of this paper is just to propose a new algorithm. The thickness of the low refractive index layer is 630r4n L nm and 630r4n H nm for the high refractive index.
Fig. 4. Multistability characteristic of nonlinear multilayered structure Ž120 layers. calculated with the direct iterating algorithm.
224
X. Lu et al.r Optics Communications 156 (1998) 219–226
Fig. 5. Output intensity profile in the transverse section for different central input intensity corresponding to points Ža. A, Žb. B and Žc. C in Fig. 3a, respectively. The beam width is much bigger than the incident wavelength.
The nonlinear coefficients are 10y10 ŽcmrV. 2 and y10y10 ŽcmrV. 2 for high and low refractive index, respectively. There is a total of 40 layers in the structure. The incident wavelength is 550 nm and input intensity increases from zero to < E I < 2 s 9 = 10 8 ŽVrcm. 2 , then decreases to zero. The input intensity changes at a constant rate and the time domain is divided into 100 parts. We suppose that the input intensity changes so slowly that we can take the stationary solution at each time point. To meet the thin film approximation mentioned in Section 2, we subdivide each layer into 20 sub-layers, so the
Fig. 6. Output phase in the transverse section, corresponding to Fig. 5c.
X. Lu et al.r Optics Communications 156 (1998) 219–226
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actual layer number in our calculation is 800. We have verified the numerical accuracy by increasing the number of sub-layers and ensuring the results are insensitive to the changes. Fig. 3a shows the output intensity versus the history of input intensity Žhysteresis curve. calculated with the new algorithm developed in Section 2. It can be seen that the output beam amplitude depends not only on the amplitude but also on the history of the input intensity. The output jumps from lower level to upper level when the input intensity increases to the upper threshold and from upper level to lower level when the input intensity decreases to the lower threshold. This hysteresis curve shows no difference from that derived by other methods, for example, the roundabout algorithm used by Agranovich et al. w9x. Fig. 3b presents the output intensity versus input intensity derived with their roundabout algorithm. There are three solutions in Fig. 3b, one of which is unstable and has no meaning, while the new algorithm presents only two stable solutions and has excluded the unstable one automatically. Since the nonlinear effect is enhanced by the reflection at each boundary, we suggest that the nonlinear characteristics will change with the number of layers in the structure. Fig. 4 shows a similar hysteresis curve when the number of layers increases to 120. One can see that the threshold falls and multistability occurs. The dashed lines indicate the possible paths of the curve.
4. Beam reshaping through nonlinear multilayered structures Assume a laser beam with Gaussian section function E Ž r . s E0 exp Ž yr 2r2 d 2 . ,
Ž 10.
incident on a nonlinear multilayered structure, where E0 is the central amplitude and d is the incident beam width. The different beam intensity causes different transmissivity in the transverse section, so the output intensity differs from the Gaussian form. The exact solution of the output beam remains a rather difficult task, because the uneven refractive index in the transverse section causes diffusion in the nonlinear layers. However, when the beam width is much longer than the wavelength, the diffusion is comparatively very weak. So we can neglect it and make the approximation that the transmissivity is determined directly by the local input intensity in the transverse section. We now suppose the multilayered structure has a bistability characteristic as shown in Fig. 3. Fig. 5 shows the output intensity in the transverse section at different time points A, B and C in Fig. 3a. If the central intensity of input is above the threshold, the output intensity profile in the transverse section is a quasi-step function with even amplitude in the central part. When the input intensity decreases from position B to C, the width and central intensity of the output remain stable. There are different input intensity profiles but very similar output intensity profiles in Fig. 5b and 5c and the same input intensity profiles but rather different output intensity profiles in Fig. 5a and 5c. We also present in Fig. 6 the output phase in the transverse section, corresponding to Fig. 5c. Similar to the beam intensity, the output phase displays a quasi-step function. The phase in the central part is somewhat higher than that in the outer part. This shows the possibility of self-focusing, and the output beam may converge after some distance. The nonlinear multilayered structure in this way acts like a focusing lens.
5. Summary In summary, we have developed an algorithm with rapid convergence to investigate the optical response in nonlinear multilayered structures. Optical bistability, multistability and beam reshaping are demonstrated in a direct way. New characteristics in beam reshaping will be helpful for the design of novel applications. Further investigations in such interesting topics will be of great value.
Acknowledgements This research was supported by the National Natural Science Foundation of China.
References w1x S.L. McCall, H.M. Gibbs, G.G. Churchill, T.N.C. Venkatesan, Bull. Am. Phys. Soc. 20 Ž1975. 636. w2x H.M. Gibbs, Optical Bistability: Controlling Light with Light, Academic Press, New York, 1985.
226 w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x
X. Lu et al.r Optics Communications 156 (1998) 219–226 R. Reinisch, G. Vitrant, Phys. Rev. B 39 Ž1989. 5775. W. Chen, D.L. Mills, Phys. Rev. B 35 Ž1986. 524. W. Chen, D.L. Mills, Phys. Rev. B 36 Ž1987. 6269. W. Chen, D.L. Mills, Phys. Rev. Lett. 58 Ž1987. 160. J. He, M. Cada, IEEE J. Quantum Electron. 27 Ž1991. 1182. J. Danckaert et al., Phys. Rev. B 44 Ž1991. 8214. V.M. Agranovich, S.A. Kiselev, D.L. Mills, Phys. Rev. B 44 Ž1991. 10917. S.D. Gupta, D.S. Ray, Phys. Rev. B 38 Ž1988. 3628. Y. Zhao, D. Huang, C. Wu, R. Shen, J. Non. Opt. Phys. Mat. 4 Ž1995. 1. K.M. Yoo, R.R. Alfano, Optics Lett. 16 Ž1991. 1823. M. Scaora, J.P. Dowling, C.M. Bowden, M.J. Bloemer, Phys. Rev. Lett. 73 Ž1994. 1368. M. Cada et al., Appl. Phys. Lett. 60 Ž1992. 404. P. Tran, Optics Lett. 21 Ž1996. 1138. B. Xu, N.B. Ming, Phys. Rev. Lett. 71 Ž1993. 1003. B. Xu, N.B. Ming, Phys. Rev. Lett. 71 Ž1993. 3959.
Optics Communications 156 Ž1998. 219–226
Full length article
Optical bistability and beam reshaping in nonlinear multilayered structures Xinghua Lu
a,)
, Yingxin Bai a , Shiqun Li a , Tianjie Chen
b
a
b
Department of Modern Applied Physics, Tsinghua UniÕersity, Beijing 10084, China Department of Physics, Southern Illinois UniÕersity at Carbondale, Carbondale, IL 62901-4401, USA Received 17 July 1997; revised 31 March 1998; accepted 11 May 1998
Abstract A new algorithm which takes into account the history of input intensity is presented to describe nonlinear characteristics such as optical bistability and multistability in nonlinear multilayered structures. The optical response along with the history of input intensity is demonstrated in a straightforward way with this algorithm. Beam reshaping due to different transmissivity in the transverse section is investigated when a laser beam passes through nonlinear multilayered structures. q 1998 Published by Elsevier Science B.V. All rights reserved.
1. Introduction Since the first observation of optical bistability in 1974 w1x there has been increasing interest in the study of the optical response of nonlinear materials because of its potential application for optical switching, optical memories, optical transistors, optical limiters and optical logic gates, etc. w2x. Optical bistability and multistability have been investigated in various nonlinear devices such as Fabry–Perot resonators w2x, thin films w3,4x and multilayered structures w5–11x. In recent years certain valuable applications with nonlinear one-dimensional structures, such as broad band optical limiters and optical logic operation devices, have been designed w12–15x. Optical response of two-dimensional nonlinear superlattices has also been investigated both theoretically and experimentally by Xu and Ming w16,17x. Based on energy integrals, an exact solution of the nonlinear equations was proposed by Chen and Mills w4–6x to investigate the optical response of a lossless nonlinear thin film, bilayers and superlattices. More recently, a transfer matrix formalism was proposed by Danckaert et al. w8x to describe the stationary optical response in nonlinear multilayered structures. This nonlinear transfer matrix formalism relies on the slowly varying envelope approximation ŽSVEA. and only treats those structures whose layer thickness is bigger than the wavelength in the material. Several other works based on transfer matrix formalism have also been reported recently w9–11x. In the case of multistability, there is more than one solution corresponding to a given input intensity, and it is impossible to derive the output intensity directly. Information about the history of input intensity is necessary to be taken into account to determine the exact solution, i.e., on which level the output intensity should be. In previous studies, the dependence of transmissivity on input intensity was studied by treating the output intensity as a parameter. In other words, the dependence of transmissivity on the output intensity, which is a mono-valued function, is derived in the first step for further investigation. Now there is a question: Is it possible to calculate the output intensity directly from the incident intensity if the history of the input intensity is known? An attempt was made by Zhao et al. w11x to calculate the direct transmissivity versus input
)
Corresponding author. E-mail: [email protected]
0030-4018r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 2 7 8 - 8
X. Lu et al.r Optics Communications 156 (1998) 219–226
220
Fig. 1. Schematic view of the multilayered structure.
intensity with a perturbation approach and iterating algorithm. However, their iterating algorithm does not converge and gives chaotic solutions when the input intensity reaches the bistability threshold, as will be discussed in detail in Section 2. In this paper, we present an improved algorithm based on convergent iteration to obtain the output intensity directly from the input intensity. Optical response such as bistability in nonlinear multilayered structures is simulated with this improved algorithm in Section 3. In Section 4, we will consider the reshaping of a laser beam injected into nonlinear multilayered structures. Different field intensity in transverse section will cause different transmissivity and result in an interesting transverse function for the output beam.
2. Model and theory We consider a plane-polarized electromagnetic wave with frequency v that propagates along the z direction and normal to the interfaces of the nonlinear multilayered structure shown in Fig. 1. Each layer exhibits an optical nonlinearity of Kerr form and is characterized by its linear refractive index nŽ0. j , nonlinear coefficient b j and thickness d j s z j y z jy1 . The nonlinear coefficient b j may be either positive or negative and the linear refractive index may be a complex number if there is absorption or gain in the medium. The full refractive index of the jth layer is written as < <2 n j Ž z . s nŽ0. j q bj E Ž z . ,
Ž1.
where EŽ z . is the electric field at point z in the structure. Because of the interference of forward and backward propagating waves, the electric field intensity varies on the scale of wavelength along the z direction and so does the refractive index in each layer. Instead of the energy integration in Refs. w4–6x, we use the thin layer approximation similar to that suggested by Agranovich et al. w9x to solve the nonlinear equations. If the thickness of each layer is much smaller than the incident wavelength, i.e. d j < l, the field intensity in each layer can be regarded as uniform and the refractive index can be approximately written as < <2 n j s nŽ0. j q bj E Ž z j . .
Ž2.
Assume a light wave of frequency v incident from the left in the region z - z 0 , transmitted on the right, where z ) z N . The electric field can be written in the form EŽ z,t . s EŽ z .expŽyi v t ., where
° E expŽ i k Ž z y z . . q E expŽ yi k Ž z y z . . , z - z , E Ž z . s~ F exp Ž i k Ž z y z . . q B exp Ž yi k Ž z y z . . , z - z - z , ¢ E exp Ž i k Ž zyz .. , z)z . I
j
0
j
0
jy1 T
R
0
j
j
Nq1
N
0
0
jy1
jy1
Ž3.
j
n
Here Fj and Bj are the amplitudes of the forward and backward propagating waves in the jth layer, while ES , E I , and E T are the amplitudes of the incident, reflected and transmitted waves, respectively, k j s 2p n jrl is the wave vector in the jth layer and l is the vacuum wavelength of the incident field. The boundary conditions that the tangential components of EŽ z . and its derivative in the z direction are continuous lead to the amplitude relations between the jth and Ž j q 1.th layers written in the matrix form as follows: 1 Fjy1 B jy1
2 s
1 2
ž ž
1q 1y
nj n jy1 nj n jy1
/ /
exp Ž yi k jy1 d jy1 . exp Ž qi k jy1 d jy1 .
1 2 1 2
ž ž
1y 1q
nj n jy1 nj n jy1
/ /
exp Ž yi k jy1 d jy1 . exp Ž qi k jy1 d jy1 .
Fj Bj
.
Ž4.
X. Lu et al.r Optics Communications 156 (1998) 219–226
221
We define the nonlinear transfer matrix as 1 N
Ms Ł
js1
2 1 2
ž ž
1q 1y
nj n jy1 nj n jy1
/ /
exp Ž yi k jy1 d jy1 . exp Ž qi k jy1 d jy1 .
1 2 1 2
ž ž
1y 1q
nj n jy1 nj n jy1
/ /
exp Ž yi k jy1 d jy1 . ,
Ž5.
exp Ž qi k jy1 d jy1 .
which satisfies EI E sM T . ER 0 The reflectivity and transmissivity can be written with the transfer matrix elements,
Fig. 2. Transmissivity calculated with the direct iterating algorithm with Ža. g s1, Žb. g s 0.05 and Žc. g s 0.01, respectively.
Ž6.
X. Lu et al.r Optics Communications 156 (1998) 219–226
222
Rs
IR II
s
< ER < 2 < EI < 2
s
M2,1 M1,1
2
,
Ts
IT II
s1yRs1y
M2,1 M1,1
2
,
Ž7.
where Mi , j are the elements of transfer matrix M. Since the nonlinear transfer matrix M depends on the field intensity in each layer, it is difficult to derive R and T for a given input intensity directly. Agranovich et al. w9x used an iterating algorithm to derive the dependence of transmissivity T on the output intensity, by calculating from the output side of the multilayered structures to the input side with Eq. Ž4.. Then the output intensity can be found by varying it as a parameter. Instead of this method, Zhao et al. w11x used another iterating algorithm to solve Eq. Ž6. for a given input intensity in a direct way. In their algorithm, the field intensity in each layer is taken to be zero first to calculate the zero-order nonlinear transfer matrix M Ž0.. Then the zero-order solution of electric fields E Ž0. Ž z . can be derived from Eqs. Ž6. and Ž4.. Substituting the electric fields into Eq. Ž5., we get the first-order nonlinear transfer matrix M Ž1. Ž< E Ž0. Ž z .< 2 .. The higher order nonlinear transfer matrix M Ž nq1. Ž< E Ž n. Ž z .< 2 . can be derived in sequence by repeating the above process. This algorithm can be expressed as follows M Ž0. ™ E Ž0. Ž z . ™ M Ž1. Ž < E Ž 1 . Ž z . < 2 . ™ E Ž1. Ž z . ™ PPP ™ E` Ž z . ,
Ž8.
Ž`. Ž
and is supposed to lead to a convergent solution E z .. However, it is correct only when the input intensity is weaker than the threshold of bistability or multistability. In the case of bistability or multistability, the iteration leads to divergent solutions. Fig. 2a presents the transmissivity as a function of the iterating order with such an iterating algorithm, which gives a divergent solution. In order to overcome the divergence, we find it effective to replace < E Ž j. Ž z .< 2 with the weighted averages of < E Ž j. Ž z .< 2 and < E Ž jy1. Ž z .< 2 , which has the form of < EXŽ j. Ž z .< 2 s g < E Ž j. Ž z .< 2 q Ž1 y g .< E Ž jy1. Ž z .< 2 , in the Ž j q 1.th nonlinear transfer matrix M Ž jq1., where g , satisfying 0 - g F 1, is a parameter used to control the weight of < E Ž j. Ž z .< 2. Because < EXŽ j. Ž z .< 2 is
Fig. 3. Bistability characteristics of nonlinear multilayered structure Ž40 layers. calculated with Ža. the direct iterating algorithm and Žb. the roundabout algorithm.
X. Lu et al.r Optics Communications 156 (1998) 219–226
223
between < E Ž j. Ž z .< 2 and < E Ž jy1. Ž z .< 2 , if the series < E Ž j. Ž z .< 2 is convergent, the series < EXŽ j. Ž z .< 2 is also convergent and has the same limit. So, the limit of the nonlinear transfer matrix M Ž`. is the solution of Eq. Ž6. EI E s M Ž`. T . ER 0
Ž9.
We have discovered in our studies that this improved algorithm gives convergent solutions with proper selection of the weight g . Generally speaking, the solutions converge with sufficiently small g and too small weight g may require a large iterating order. In practice, at first we choose a larger value of g , e.g. g s 1, then if the solution does not converge, a smaller weight g is chosen. Fig. 2b and 2c show two convergent series of transmissivity calculated with g equal to 0.05 and 0.01, respectively. However, such a convergent algorithm leads to only one solution for any given input intensity and is insufficient for the case of multistability which requires multi-solutions. Why is this so? And how can we find other Žor proper. solutions? This can be explained from the nature of bistability and multistability, which indicates that the output of a nonlinear system depends not only on the amplitude but also on the history of the input intensity. There is no way to derive the corresponding solution without including the history of the input intensity in the calculation. We find that it is the zero-order nonlinear transfer matrix M Ž0. that contains the information of the history of input intensity. In the above algorithm, the electric field in each layer is taken to be zero, which means that the input intensity increases from zero and the solution only corresponds to the lower output level. If the input intensity decreases from a higher value above the upper threshold, we need to take the solution of the transfer matrix M for higher input intensity as the zero-order transfer matrix M Ž0. in iteration. Then the algorithm leads to another solution Župper output level.. Thus, an effective iterating algorithm which takes into account the history of the input intensity has been established to solve the optical response in nonlinear multilayered structures. For a given history of the input intensity, we first separate the time domain into some discrete parts. The zero-order nonlinear transfer matrix M Ž0. for each time point is given by the nonlinear transfer matrix solved at the previous time point. ŽThe zero-order nonlinear transfer matrix at the first time point is established by taking the electric fields in each layer to be zero.. In the next section, we perform some numerical simulations.
3. Numerical calculation and discussion Consider a nonlinear multilayered structure with alternating refractive index n H equal to 2.3 and n L equal to 1.48. The refractive indexes in region z - z 0 and z ) z N are 1.0 and 1.52, respectively. Although the absorption and gain can also be taken into account in simulation, we do not include them here because the main task of this paper is just to propose a new algorithm. The thickness of the low refractive index layer is 630r4n L nm and 630r4n H nm for the high refractive index.
Fig. 4. Multistability characteristic of nonlinear multilayered structure Ž120 layers. calculated with the direct iterating algorithm.
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Fig. 5. Output intensity profile in the transverse section for different central input intensity corresponding to points Ža. A, Žb. B and Žc. C in Fig. 3a, respectively. The beam width is much bigger than the incident wavelength.
The nonlinear coefficients are 10y10 ŽcmrV. 2 and y10y10 ŽcmrV. 2 for high and low refractive index, respectively. There is a total of 40 layers in the structure. The incident wavelength is 550 nm and input intensity increases from zero to < E I < 2 s 9 = 10 8 ŽVrcm. 2 , then decreases to zero. The input intensity changes at a constant rate and the time domain is divided into 100 parts. We suppose that the input intensity changes so slowly that we can take the stationary solution at each time point. To meet the thin film approximation mentioned in Section 2, we subdivide each layer into 20 sub-layers, so the
Fig. 6. Output phase in the transverse section, corresponding to Fig. 5c.
X. Lu et al.r Optics Communications 156 (1998) 219–226
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actual layer number in our calculation is 800. We have verified the numerical accuracy by increasing the number of sub-layers and ensuring the results are insensitive to the changes. Fig. 3a shows the output intensity versus the history of input intensity Žhysteresis curve. calculated with the new algorithm developed in Section 2. It can be seen that the output beam amplitude depends not only on the amplitude but also on the history of the input intensity. The output jumps from lower level to upper level when the input intensity increases to the upper threshold and from upper level to lower level when the input intensity decreases to the lower threshold. This hysteresis curve shows no difference from that derived by other methods, for example, the roundabout algorithm used by Agranovich et al. w9x. Fig. 3b presents the output intensity versus input intensity derived with their roundabout algorithm. There are three solutions in Fig. 3b, one of which is unstable and has no meaning, while the new algorithm presents only two stable solutions and has excluded the unstable one automatically. Since the nonlinear effect is enhanced by the reflection at each boundary, we suggest that the nonlinear characteristics will change with the number of layers in the structure. Fig. 4 shows a similar hysteresis curve when the number of layers increases to 120. One can see that the threshold falls and multistability occurs. The dashed lines indicate the possible paths of the curve.
4. Beam reshaping through nonlinear multilayered structures Assume a laser beam with Gaussian section function E Ž r . s E0 exp Ž yr 2r2 d 2 . ,
Ž 10.
incident on a nonlinear multilayered structure, where E0 is the central amplitude and d is the incident beam width. The different beam intensity causes different transmissivity in the transverse section, so the output intensity differs from the Gaussian form. The exact solution of the output beam remains a rather difficult task, because the uneven refractive index in the transverse section causes diffusion in the nonlinear layers. However, when the beam width is much longer than the wavelength, the diffusion is comparatively very weak. So we can neglect it and make the approximation that the transmissivity is determined directly by the local input intensity in the transverse section. We now suppose the multilayered structure has a bistability characteristic as shown in Fig. 3. Fig. 5 shows the output intensity in the transverse section at different time points A, B and C in Fig. 3a. If the central intensity of input is above the threshold, the output intensity profile in the transverse section is a quasi-step function with even amplitude in the central part. When the input intensity decreases from position B to C, the width and central intensity of the output remain stable. There are different input intensity profiles but very similar output intensity profiles in Fig. 5b and 5c and the same input intensity profiles but rather different output intensity profiles in Fig. 5a and 5c. We also present in Fig. 6 the output phase in the transverse section, corresponding to Fig. 5c. Similar to the beam intensity, the output phase displays a quasi-step function. The phase in the central part is somewhat higher than that in the outer part. This shows the possibility of self-focusing, and the output beam may converge after some distance. The nonlinear multilayered structure in this way acts like a focusing lens.
5. Summary In summary, we have developed an algorithm with rapid convergence to investigate the optical response in nonlinear multilayered structures. Optical bistability, multistability and beam reshaping are demonstrated in a direct way. New characteristics in beam reshaping will be helpful for the design of novel applications. Further investigations in such interesting topics will be of great value.
Acknowledgements This research was supported by the National Natural Science Foundation of China.
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