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Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding
1 March 1999
Optics Communications 161 Ž1999. 141–148
Full length article
Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding Shouxian She ) , Sijiong Zhang Department of Physics, Northern Jiaotong UniÕersity, Beijing 100044, China Received 19 May 1998; revised 9 December 1998; accepted 15 December 1998
Abstract Formulas are presented for analyzing optical nonlinearity and bistability of TE modes in a periodic refractive index waveguide with nonlinear cladding, both by exact method and by the method of rms approximation. Numerical calculations for realistic periodic refractive index planar waveguide parameters show that the results of the two methods agree excellently when the number of layers of the periodic structure is sufficiently large so that the thin film approximation Žthe electric field is slowly varying over each period of the periodic structure. is valid. By using rms approximation, the dependence of the modal refractive index on the total power, bistability between core power and total power, and the influence of waveguide parameters on them are discussed. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear multilayer waveguide; Periodic structure; Multiple-quantum-well waveguide; Modal index; Optical power; Optical nonlinearity; Optical bistability
1. Introduction Recently, nonlinear multilayer waveguides with periodic refractive index structure including multiplequantum-well Žperiodic core. structures have attracted considerable attention because of their distinctive features: optical nonlinearity, very fast response and suitability for integration of device components into circuits w1–4x. In the past decade considerable attention and great effort has been focused on theoretical investigations w5–11x. The model studied in this paper is as follows Žas far as we know, investigation of this model has not been reported yet.. The core of the waveguide is a periodic refractive index structure consisting of wells and barriers of linear dielectrics, sandwiched between a nonlinear Kerr-type cladding and a linear substrate. Formulas and method of calculation for this model by the exact method and by the method of root mean square Žrms. approximation w12–15x are discussed in detail. Numerical calculations for realistic
)
Corresponding author. E-mail: [email protected]
periodic refractive index planar waveguide parameters show that the results of the two methods agree excellently when the number of layers of the periodic core is large enough so that the thin film approximation Ži.e., the electric field is slowly varying over each period of the periodic structure. is valid. ŽIn typical periodic structures as given below, the number of layers required ranges from 10 to 30.. It is emphasized that the method of rms approximation is simple and efficient, and saves a large amount of computations. 2. Theory. Calculation methods The cross-section of the periodic core optical waveguide studied in this paper is shown in Fig. 1. The core of the waveguide consists of M layers with refractive indices n1 and n 2 , thicknesses a1 and a 2 , respectively, spatial period L s a1 q a2 , and total thickness w s M L. The refractive index of the substrate is n 3 and the nonlinear refractive index of the cladding is n˜ 0 , for Kerr-type nonlinear cladding. For TE modes n˜ 20 s n 20 q a E 2 ,
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 0 0 4 - 8
a s c ´ 0 n20 n I
Ž1.
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
142
2.1. Exact analysis From the Helmholtz equations, the field functions are given as follows, EŽ x . s
(
2
a
q0 sech w k 0 q0 Ž x y x 0 . x
Ž x F 0. ,
E Ž x . s A j cos k 0 q1 w x y Ž j y 1 . L x 4 q B j sin k 0 q1 w x y Ž j y 1 . L x 4
Fig. 1. Schematic diagram of a nonlinear periodic core waveguide.
Ž N - n1 . ,
E Ž x . s A j cosh k 0 q1 w x y Ž j y 1 . L x 4 q B j sinh k 0 q1 w x y Ž j y 1 . L x 4 where n 0 is the linear refractive index, n I the nonlinear coefficient, c and ´ 0 the velocity of light and the dielectric constant in free space, respectively, and E the local electric field intensity of the guided optical wave, n1 ) n 0 ) n 2 G n 3. In Fig. 1, the x-axis is perpendicular to all interfaces of the waveguide, and the origin O lies at the interface between core and nonlinear cladding. Let N represent the modal index, N s brk 0 , where b is the propagation constant of the guided wave, k 0 s 2prl is the wave number in free space, and l is the wavelength in free space. For exact analysis and for rms approximation, we define
( q s (N
q1 s n12 y N 2 2
2
y n 22 ,
(
Ž n1 ) N . , q1 s N 2 y n2I
(
q3 s N 2 y n 23 ,
Ž nI - N . ,
(
For rms approximation, we need further to define two quantities as follows, qf s n2eff y N 2 Ž n eff ) N . , f
2
y n2eff Ž n eff - N .
Ž3.
where n eff is the equivalent refractive index of the periodic core. For TE waves, n eff is the rms of the core refractive index nŽ x . in one spatial period in the total thickness w, that is 1
n eff s
H L 0
L 2
1r2
n Ž x .d x
.
Ž4.
For the present case, the refractive indices of the well and barrier are n1 and n 2 , respectively. We define parameter R s a1rL,
(
n eff s Rn12 q Ž 1 y R . n22 .
Ž5.
In the limit R s 1, the periodic core waveguide reduces to a three-layer planar waveguide. And when a s 0, the periodic core waveguide reduces to a linear periodic core waveguide. Therefore, the methods in this paper are applicable to linear periodic core waveguides and to three-layer planar waveguides with nonlinear cladding.
j s 1,2, . . . , M ,
E Ž x . s C j cosh k 0 q2 w x y Ž j y 1 . L x y a1 4 q Dj sinh k 0 q2 w x y Ž j y 1 . L x y a1 4 ,
Ž j y 1. L F x F j L ,
j s 1,2, . . . , M ,
E Ž x . s CMq1exp w yk 0 q3 Ž x y w . x
Ž xGw.
Ž6.
where the constants x 0 , A j , Bj , C j , Dj Ž j s 1,2, . . . , M . and CMq1 to be solved can be derived through continuity conditions. Let the field intensity at the interface between core and cladding be E0 . Defining the parameter Õ s tanh Ž k 0 q0 x 0 .
Ž7.
we have E0 s
q0 s N 2 y n 20 .
Ž2.
( q s (N
Ž j y 1 . L F x F Ž j y 1 . L q a1 ,
Ž N ) n1 . ,
(
2
a
q0'1 y Õ 2 .
Ž8.
For given waveguide parameters and a given value of modal indices, the field distribution, the powers in each region and the total power can be deduced as follows. Applying continuity conditions at the interfaces, recurrent formulas for A j , Bj , C j , Dj Ž j s 1,2, . . . , M . and CMq1 can be deduced. For N - n1 A1 s
(
2
a
q0'1 y Õ 2 ,
B1 s
q0 q1
ÕA1 ,
C1 s A1cos Ž k 0 q1 a1 . q B1 sin Ž k 0 q1 a1 . , q1 D 1 s w yA1sin Ž k 0 q1 a1 . q B1cos Ž k 0 q1 a1 . x , q2 A j s C jy1cosh Ž k 0 q2 a 2 . q Djy1 sinh Ž k 0 q2 a2 . , q0 Bj s C jy1 sinh Ž k 0 q2 a2 . q Djy1cosh Ž k 0 q2 a 2 . , q1 C j s A j cos Ž k 0 q1 a1 . q Bj sin Ž k 0 q1 a1 . , q1 Dj s yA j sin Ž k 0 q1 a1 . q Bj cos Ž k 0 q1 a1 . q2
Ž j s 2,3, . . . , M . CMq1 s
q2 q3
w CM sinh Ž k 0 q2 a2 . q DM cosh Ž k 0 q2 a2 . x . Ž 9a .
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
For N ) n1 A1 s
(
2
and, for both N - n1 and N ) n1
q0'1 y Õ 2 ,
a
q0
B1 s
q1
DM
ÕA1 ,
D1 s
A j s C jy1cosh Ž k 0 q2 a 2 . q Djy1 sinh Ž k 0 q2 a2 . , q0 Bj s C jy1 sinh Ž k 0 q2 a2 . q Djy1cosh Ž k 0 q2 a 2 . , q1
Õs
q1
yA j sin Ž k 0 q1 a1 . q Bj cos Ž k 0 q1 a1 .
q2
q1 B1
q2 q3
Ps
w CM sinh Ž k 0 q2 a2 . q DM cosh Ž k 0 q2 a2 . x . Ž 9b.
From formulas Ž9a. or Ž9b., we get the following. For N - n1 A1 s
(
2
q0'1 y Õ 2 ,
a
q 2 Dj Bj
s
Aj
q1 C j 1y
q 2 Dj q1 C j
q2 A j
s
C jy1
1y
q1
P3 s
y tanh Ž k 0 q2 a2 .
q2 A j
2
tanh Ž k 0 q2 a2 .
a
q 2 Dj Bj
s
Aj
q1 C j 1y
q1 C j
q1 B j Djy1 C jy1
s
q2 A j 1y
B1 s
q0 q1
2
d x s P0 q Pcore q P3
0
Hy` E
N
w
2 cm0
H0
N
q`
2 cm0
Hy`
2
d xs
N
4 q0
4 cm0 k 0 a
E 2d x s
E2d xs
M
I1 s
Ž 13.
N 4 cm0 k 0
ž
Ž1 y Õ . ,
I1
q
q1
N
2 CMq1
4 cm0 k 0
q3
I2 q2
/
,
,
Ž 14a.
Ý Ij1 ,
M
I2 s
Ý Ij2 , js1
I j1 s Ž A2j q Bj2 . Ž k 0 q1 a1 . q Ž A2j y Bj2 . sin Ž k 0 q1 a1 . cos Ž k 0 q1 a1 .
ÕA1 ,
q 2 A j B j sin2 Ž k 0 q1 a1 .
y tanh Ž k 0 q1 a1 .
q 2 Dj
N 2 cm0
js1
q0'1 y Õ 2 ,
Hy` E
where
Ž j s 1,2, . . . , M . .
For N ) n1
(
Ž 12.
where m 0 is the permittivity in free space and c m 0 s 377 V. P0 , Pcore and P3 are the powers in the nonlinear cladding, core and substrate, respectively;
, tan Ž k 0 q1 a1 .
q1 B j
Ž N - n1 .
q 0 A1
q`
2 cm0
Pcore s
Ž 10a.
A1 s
N
P0 s
ÕA1 ,
q tan Ž k 0 q1 a1 .
q1 B j Djy1
q0
B1 s
Ž 11 .
q1 B1
Ž N ) n1 . , Õ s
q 0 A1
Ž j s 2,3, . . . , M . CMq1 s
.
and finally E0 and all constants A j , B j , C j , Dj Ž j s 1,2, . . . , M . and CMq1 are determined by Eqs. Ž8. and Ž9a. or Eq. Ž9b.. Total power P and core power Pcore of the waveguide are calculated by the following formulas according to the function of field distribution,
C j s A j cos Ž k 0 q1 a1 . q Bj sin Ž k 0 q1 a1 . , Dj s
1 q Ž q3rq2 . tanh Ž k 0 q2 a2 .
For given N, from expressions Ž11., Ž10a. and Ž10b., DMrCM , BjrA j , DjrC j can be obtained and the parameter Õ is determined by
yA1sin Ž k 0 q1 a1 . q B1cos Ž k 0 q1 a1 . ,
q2
q3rq2 q tanh Ž k 0 q2 a2 .
sy
CM
C1 s A1cos Ž k 0 q1 a1 . q B1sin Ž k 0 q1 a1 . , q1
143
Ž N - n1 . ,
I j1 s Ž A2j y Bj2 . Ž k 0 q1 a1 . , q Ž A2j q Bj2 . sinh Ž k 0 q1 a1 . cosh Ž k 0 q1 a1 .
tanh Ž k 0 q1 a1 .
q 2 A j B j sinh2 Ž k 0 q1 a1 . y tanh Ž k 0 q2 a2 .
q1 B j q2 A j
Ž j s 1,2, . . . , M . tanh Ž k 0 q2 a2 .
Ž N ) n1 . ,
I j2 s Ž C j2 y Dj2 . Ž k 0 q2 a2 . q Ž C j2 q Dj2 . sinh Ž k 0 q2 a 2 . cosh Ž k 0 q2 a2 .
Ž 10b.
q 2C j Dj sinh2 Ž k 0 q2 a 2 . .
Ž 14b.
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
144
2.2. RMS approximation
For n eff - N
It has been shown by perturbation theory w8x and by transfer matrix technique w12,13x that, if the electric field is slowly varying over each period of a periodic structure Žthe electric field is not necessarily slowly varying over the entire region considered., then the thin film approximation Žfor each period. is valid and the periodic refractive index core can be treated as a uniform core of refractive index w13x n eff s wŽ1rL.H0L n 2 Ž x .d x x1r2 as given by formula Ž4.. It is rational to assert that the approximation is valid provided that k 0 < N 2 y n 2eff < L < 1, where n eff is the rms value of the refractive index in the spatial period L. It can be seen that the above condition can be written as
(
(
M 4 2p < N 2 y n 2eff <
w
l
,
(
2
a
q0 sech w k 0 q0 Ž x y x 0 . x
Ž x F 0. ,
E Ž x . s A cos Ž k 0 qf x . q B sin Ž k 0 qf x .
Ž N - n eff . ,
E Ž x . s A cosh Ž k 0 qf x . q B sinh Ž k 0 qf x .
Ž 15.
Modal field distribution EŽ x ., total power P and core power Pcore corresponding to modal refractive index N s brk 0 can be determined using continuity conditions at the interfaces. We define parameters Õ and E0 similar to expressions Ž7. and Ž8., where < Õ < F 1. Obviously, when Õ tends to 1, the cladding is a linear dielectrics, and Õ - 0 Ž x 0 - 0. corresponds to the cases where the maximum of E shifts into the core nonlinear cladding. For such cases, the positions of the maximum of the field intensity are located at x 0 Ž x 0 - 0.. By continuity of EŽ x . and EX Ž x ., we get the following. For n eff ) N 2
a
q0'1 y Õ 2 ,
q0'1 y Õ 2 ,
q0
Bs
qf
ÕA,
Ž 16b.
The eigenvalue equations are as follows. For n eff ) N Õs
qf qf tan Ž k 0 qf w . y q3
Ž 17a.
q0 qf q q3tan Ž k 0 qf w .
or tan Ž k 0 qf w . s
Õsy
qf Ž q3 y Õq0 . qf2 y Õq0 q3
.
Ž 17b.
Bs
q0 qf
ÕA,
C s A cos Ž k 0 qf w . q B sin Ž k 0 qf w .
qf qf tanh Ž k 0 qf w . q q3
Ž 18a.
q0 qf q q3tanh Ž k 0 qf w .
or qf Ž q3 q Õq0 .
tanh Ž k 0 qf w . s y
qf2 q Õq0 q3
.
Ž 18b.
For given parameters n˜ 0 , n1, n 2 , n 3 , R, core thickness w and given value of modal index N, the corresponding parameter Õ Ž< Õ < F 1. can be solved through eigenvalue Eqs. Ž17a. and Ž17b. or Ž18a. and Ž18b., and E0 and the field functions of guided waves can be determined by using expressions Ž16a. or Ž16b., the total power P, core power Pcore and substrate power P3 of the waveguide are given by the following formulas, P0 s
E Ž x . s C exp w yk 0 q3 Ž x y w . x Ž x G w . .
(
2
a
C s A cosh Ž k 0 qf w . q B sinh Ž k 0 qf w . .
Ž N ) n eff .
Ž w G x G 0. ,
As
(
For n eff - N
where M the number of periods and w the core thickness, l is the optical wavelength. Calculations for realistic periodic structures Žsee below. show that the above condition can be satisfied for number of periods ranging from 10 to 30 in such typical cases and thus the rms approximation is valid in these cases, saving large amount of computations. The method of analysis of propagation characteristics is as follows. From the nonlinear Helmholtz equation and using the rms approximation, the field function is given by EŽ x . s
As
N
Pcore s
P3 s
0
2 cm0
Hy` E
N
w
2 cm0
H0
N
q`
2 cm0
2
d xs
N
E 2d x s
Hy` E
2
q0
cm0 k 0 a
d xs
N 4 cm0 k 0 N
Ž1 y Õ . ,
I, C2
4 c m 0 k 0 q3
,
Ž 19.
where Is
1 qf
Ž A2 q B 2 . k 0 qf w
q Ž A2 y B 2 . sin Ž k 0 qf w . cos Ž k 0 qf w . q2 AB sin2 Ž k 0 qf w . Is
1 qf
Ž n eff ) N . ,
Ž 20a.
Ž A2 y B 2 . k 0 qf w
q Ž A2 q B 2 . sinh Ž k 0 qf w . cosh Ž k 0 qf w .
Ž 16a.
q2AB sinh2 Ž k 0 qf w .
Ž n eff - N . .
Ž 20b.
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
145
Therefore, for any given value of N, the curves of N–P and corresponding Pcore –P curve can be calculated. Furthermore, for given parameters of the waveguide, we can get the curves of nonlinearity and bistability, the threshold power for optical bistability, the influence of the waveguide parameters on the characteristics of guided waves, and curves of modal field distribution for all cases.
3. Examples and numerical analysis. Discussion Numerical calculations have been processed for a nonlinear periodic planar waveguide with realistic parameters. For one of the realistic models Žmodel A., the free-space wavelength l s 0.515 mm ŽArq laser., the cladding is a
Fig. 3. Field distribution of a periodic core structure with M s10, 20, 30 periods, Rs 0.8 and 0.9 for modal refractive index N s 3.45 Ž l s 0.83 mm, w s1.5 mm, n 0 s 3.42, n I s10y9 m2 rW, n1 s 3.465, n 2 s 3.315, n 3 s n 2 .. Solid line: exact; dotted line: rms approximation.
Fig. 2. Field distribution of a periodic core structure with M s10, 20, 30 periods, Rs 0.7 and 0.9 for modal refractive index N s 1.566 Ž l s 0.515 mm, w s1.4 mm, n 0 s1.55, n I s10y9 m2 rW, n1 s1.57, n 2 s1.52, n 3 s n 2 .. Solid line: exact; dotted line: rms approximation.
NBBA liquid crystal Ža Kerr-type nonlinear medium. with very strong nonlinearity, for which n 0 s 1.55 and n I s 10y9 m2rW; the material of the well is Coring 7059 glass, its linear refractive index is n1 s 1.57, that of the barriers is soda lime glass and its refractive index is n 2 s 1.52, the refractive index of the substrate is n 3 s n 2 s 1.52. The above values of the material parameters were initially used by Stegeman et al. w14x in their calculation, and subsequently cited by some other researchers w15,16x in their theoretical investigations of nonlinear optical waveguides. For another realistic model Žmodel B., we take the following typical values Žrefer to Refs. w5–8x.: free-space wavelength l s 0.83 mm, MQW core is made of layers,
146
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
the refractive index of the well and barrier fall in the range typical of Al xGa 1yx As materials, i.e., n g w3.315, 3.465x, and we take n1 s 3.465, n 2 s 3.315, core width w s 1.5 mm Žfrom Ref. w8x. and for the nonlinear Kerr-type cladding, n 0 s 3.42, n I s 2 = 10y9 m2rW, for the linear substrate, n 3 s 3.42. Calculations are processed for both model A and model B to verify the validity of rms approximation in characteristics analysis. We now compare the calculation results by the two methods, so as to verify the validity of the rms approximation when the number of layers in a spatial period is sufficiently large. For waveguide model A, Fig. 2a and b, show the curves of field distribution calculated by the exact method Žsolid lines. and by rms approximation Ždotted lines., corresponding respectively to R s 0.7 and R s 0.9, M s 10, 20, 30, with parameters n1 s 1.57, n 2 s 1.52, n 3 s n 2 , n I s 10y9 m2rW, wavelength l s 0.515 mm and core width w s 1.4 mm, and modal index N s 1.566. It is seen ŽFig. 2a. that for R s 0.7, the agreement of the two methods is good for M s 20 and is excellent for M 0 30, and ŽFig. 2b. that for R s 0.9, the agreement is good for M s 10 and is excellent for M 0 20. Now the thin film approximation is valid for M 4 2p N 2 y n2eff Ž wrl., we have, for R s 0.7, n2eff s 2.4185, N 2 s Ž1.566. 2 s 2.4524, so the approximation is valid for M 4 3.4, for R s 0.9, n2eff s 2.4495, N 2 s Ž1.566. 2 s 2.4524, so that the approximation is valid for M 4 0.92. It is seen that the rms approximations are valid for both cases. For waveguide model B, Fig. 3a and b show the curves of field distribution calculated by the two methods corresponding to R s 0.8 and R s 0.9, M s 10, 20, 30, with parameters n1 s 3.465, n 2 s 3.315, n 3 s n 2 , n 0 s 3.42,
(
Fig. 4. N – P curve for a periodic core structure with M s 5 periods, Rs 0.7 Ž l s 0.515 mm, w s1.4 mm, n 0 s1.55, n I s 10y9 m2 rW, n1 s1.57, n 2 s1.52, n 3 s n 2 .. Solid line: exact; dotted line: rms approximation.
Fig. 5. N – P curve for a periodic core structure with M s 5 periods, Rs 0.9 Žwaveguide parameters are the same as in Fig. 4.. Solid line: exact; dotted line: rms approximation.
n I s 2 = 10y9 m2rW, wavelength l s 0.83 mm and core width w s 1.5 mm, and modal index N s 3.45. It is seen ŽFig. 3a. that for R s 0.8, the agreement is good for M s 20, excellent for M 0 30, and that ŽFig. 3b. for R s 0.9, the agreement is excellent. Here, for R s 0.8, n 2eff s 11.803, N 2 s Ž3.45. 2 s 11.90, the thin film approximation is valid for M 4 3.6, for R s 0.9, n2eff s 11.90, N 2 s 11.90, the thin film approximation is valid for M 4 1. It is seen that the rms approximation is valid for both cases also. Figs. 4–9 show the numerical results for waveguide model A Ž l s 0.515 mm.. Figs. 4 and 5 show the N–P curves calculated by the exact method and rms approximation corresponding to w s 1.4 mm and R s 0.7 and 0.9, respectively. The figures show that the results of the two methods agree well for
Fig. 6. Dependence of modal refractive index N on total power P Žwaveguide parameters are the same as in Fig. 4..
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
147
Fig. 7. Pcore – P curves Ž w s1.4 mm, Rs 0.7, 0.8, 0.9, 1.0. Žwaveguide parameters are the same as in Fig. 4..
Fig. 9. Dependence of modal refractive index N on total power P for core width w s1.2, 1.4, 1.6 mm Ž Rs 0.9. Ž l s 0.515 mm, refractive indices are the same as in Fig. 4..
M s 5. For M s 10, the N–P curves given by the two methods cannot be distinguished in the figures. Fig. 6 shows the curves of the dependence of modal index N on total power P for TE 0 mode Žcore thickness w s 1.4 mm, R s 0.7, 0.8, 0.9 and 1.0.. It is seen that the TE 0 mode exhibits stronger nonlinearity as power P becomes large and optical bistability appears at higher power when R is greater than 0.8. Fig. 7 shows the curves of the dependence of core power Pcore on total power P for the TE 0 mode Žcore thickness w s 1.4 mm, R s 0.7, 0.8, 0.9 and 1.0.. The figure shows that optical bistability between core power and total power is apparent when R G 0.9. The greater the R, the stronger the optical bistability. From Figs. 6 and 5, it is seen that optical bistability begins to appear approxi-
mately in the neighborhood of N s n eff . Furthermore, the threshold power of appearance for optical bistability in a nonlinear periodic core structure is lower than that for a nonlinear three-layer planar waveguide. Fig. 8 shows the curves of field distribution in the cladding, core and substrate of the waveguide for core thickness w s 1.4 mm, R s 0.9 and N s 1.560, 1.563, 1.566, 1.569, respectively. The curves in the figure show that the maximum value of the field begins to shift from the core region into the nonlinear cladding when N increases from N s 1.563 to N s 1.565 Žcorresponding to equivalent refractive index n eff .. When N further increases from N s 1.565 to 1.572, the maximum value of the field shifts further to the left and the maximum value of field gradually increases. That means that the TE 0 mode transforms from optical guided waves to optical surface waves. Fig. 9 shows N–P curves for core thickness w s 1.2, 1.4, 1.6 mm when R s 0.9. It is seen that optical bistability becomes stronger when the thickness w is increased.
4. Conclusions
Fig. 8. Field distribution for core width w s1.4 mm, Rs 0.9 and N s1.56, 1.563, 1.566, 1.569 Žwaveguide parameters are the same as in Fig. 4..
First, the waveguide parameters Žcore thickness w, ratio R s a1rL, and indices of refraction. have important effect on optical nonlinearity for TE 0 mode. For fixed values of w and R, the features of modal fields are independent of the number of layers M of periodic core provided M is large enough. Second, when the core thickness w and ratio R are small, optical nonlinearity appears at lower power, but optical bistability does not appear. When the core thickness w and ratio R are large, TE 0 mode exhibits strong optical nonlinearity and optical stability, and when the core thickness w and ratio R are increased, these effects be-
148
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
come stronger. The threshold power of optical bistability for the nonlinear periodic core waveguide is lower than that for the corresponding nonlinear three-layer waveguide. Finally, when modal refractive index N increases from values less than the rms equivalent refractive index n eff to values larger than n eff , TE 0 mode transforms from guided wave to surface wave and the turning point lies in the neighborhood of n eff .
w8x
w9x
w10x
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waveguides, IEEE J. Quantum Electron. QE-28 Ž7. Ž1992. 1678–1688. G.M. Alman, L.A. Malter, H. Shen, M. Dutta, Refractive index approximation from linear perturbation theory for planar MQW waveguides, IEEE J. Quantum Electron. QE-28 Ž3. Ž1992. 650–657. A.P. Zhao, S.R. Cvetkovic, Numerical modeling of nonlinear TE waves in multiple-quantum-well waveguides, IEEE Photon. Technol. Lett. 4 Ž6. Ž1992. 623–626. S.R. Cvetkovic, A.P. Zhao, Finite element formalism for linear and nonlinear guided waves in multiple-quantum-well waveguides, J. Opt. Soc. Am. B 10 Ž8. Ž1993. 1401–1407. S. Selleri, M. Zoboli, Stability analysis of nonlinear TE polarized waves in multiple-quantum-well waveguides, IEEE J. Quantum Electron. QE-31 Ž10. Ž1995. 1785–1789. J.M. Skinner, R. Shall, B.L. Weiss, Modal propagation within MQW waveguides, IEEE J. Quantum Electron. QE-25 Ž1. Ž1989. 6–11. Z.Q. Cao, New formula for equivalent refractive index of MQW waveguides, Acta Optica Sinica 11 Ž6. Ž1991. 553– 557. G.I. Stegeman, C.T. Seaton, J. Chiwell, S.D. Smith, Nonlinear waves guided by thin films, Appl. Phys. Lett. 44 Ž1984. 830–832. C.T. Seaton, J.D. Valera, R.L. Shoemaker, G.I. Stegeman, J.T. Chilwell, S.D. Smith, Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media, IEEE J. Quantum Electron. QE-21 Ž7. Ž1985. 774– 783. K. Hayata, M. Nagai, M. Koshiba, Finite-element formalism for nonlinear layer-guided waves, IEEE Trans. Microwave Theory Tech. 36 Ž7. Ž1988. 1207–1215.
Optics Communications 161 Ž1999. 141–148
Full length article
Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding Shouxian She ) , Sijiong Zhang Department of Physics, Northern Jiaotong UniÕersity, Beijing 100044, China Received 19 May 1998; revised 9 December 1998; accepted 15 December 1998
Abstract Formulas are presented for analyzing optical nonlinearity and bistability of TE modes in a periodic refractive index waveguide with nonlinear cladding, both by exact method and by the method of rms approximation. Numerical calculations for realistic periodic refractive index planar waveguide parameters show that the results of the two methods agree excellently when the number of layers of the periodic structure is sufficiently large so that the thin film approximation Žthe electric field is slowly varying over each period of the periodic structure. is valid. By using rms approximation, the dependence of the modal refractive index on the total power, bistability between core power and total power, and the influence of waveguide parameters on them are discussed. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear multilayer waveguide; Periodic structure; Multiple-quantum-well waveguide; Modal index; Optical power; Optical nonlinearity; Optical bistability
1. Introduction Recently, nonlinear multilayer waveguides with periodic refractive index structure including multiplequantum-well Žperiodic core. structures have attracted considerable attention because of their distinctive features: optical nonlinearity, very fast response and suitability for integration of device components into circuits w1–4x. In the past decade considerable attention and great effort has been focused on theoretical investigations w5–11x. The model studied in this paper is as follows Žas far as we know, investigation of this model has not been reported yet.. The core of the waveguide is a periodic refractive index structure consisting of wells and barriers of linear dielectrics, sandwiched between a nonlinear Kerr-type cladding and a linear substrate. Formulas and method of calculation for this model by the exact method and by the method of root mean square Žrms. approximation w12–15x are discussed in detail. Numerical calculations for realistic
)
Corresponding author. E-mail: [email protected]
periodic refractive index planar waveguide parameters show that the results of the two methods agree excellently when the number of layers of the periodic core is large enough so that the thin film approximation Ži.e., the electric field is slowly varying over each period of the periodic structure. is valid. ŽIn typical periodic structures as given below, the number of layers required ranges from 10 to 30.. It is emphasized that the method of rms approximation is simple and efficient, and saves a large amount of computations. 2. Theory. Calculation methods The cross-section of the periodic core optical waveguide studied in this paper is shown in Fig. 1. The core of the waveguide consists of M layers with refractive indices n1 and n 2 , thicknesses a1 and a 2 , respectively, spatial period L s a1 q a2 , and total thickness w s M L. The refractive index of the substrate is n 3 and the nonlinear refractive index of the cladding is n˜ 0 , for Kerr-type nonlinear cladding. For TE modes n˜ 20 s n 20 q a E 2 ,
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 0 0 4 - 8
a s c ´ 0 n20 n I
Ž1.
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
142
2.1. Exact analysis From the Helmholtz equations, the field functions are given as follows, EŽ x . s
(
2
a
q0 sech w k 0 q0 Ž x y x 0 . x
Ž x F 0. ,
E Ž x . s A j cos k 0 q1 w x y Ž j y 1 . L x 4 q B j sin k 0 q1 w x y Ž j y 1 . L x 4
Fig. 1. Schematic diagram of a nonlinear periodic core waveguide.
Ž N - n1 . ,
E Ž x . s A j cosh k 0 q1 w x y Ž j y 1 . L x 4 q B j sinh k 0 q1 w x y Ž j y 1 . L x 4 where n 0 is the linear refractive index, n I the nonlinear coefficient, c and ´ 0 the velocity of light and the dielectric constant in free space, respectively, and E the local electric field intensity of the guided optical wave, n1 ) n 0 ) n 2 G n 3. In Fig. 1, the x-axis is perpendicular to all interfaces of the waveguide, and the origin O lies at the interface between core and nonlinear cladding. Let N represent the modal index, N s brk 0 , where b is the propagation constant of the guided wave, k 0 s 2prl is the wave number in free space, and l is the wavelength in free space. For exact analysis and for rms approximation, we define
( q s (N
q1 s n12 y N 2 2
2
y n 22 ,
(
Ž n1 ) N . , q1 s N 2 y n2I
(
q3 s N 2 y n 23 ,
Ž nI - N . ,
(
For rms approximation, we need further to define two quantities as follows, qf s n2eff y N 2 Ž n eff ) N . , f
2
y n2eff Ž n eff - N .
Ž3.
where n eff is the equivalent refractive index of the periodic core. For TE waves, n eff is the rms of the core refractive index nŽ x . in one spatial period in the total thickness w, that is 1
n eff s
H L 0
L 2
1r2
n Ž x .d x
.
Ž4.
For the present case, the refractive indices of the well and barrier are n1 and n 2 , respectively. We define parameter R s a1rL,
(
n eff s Rn12 q Ž 1 y R . n22 .
Ž5.
In the limit R s 1, the periodic core waveguide reduces to a three-layer planar waveguide. And when a s 0, the periodic core waveguide reduces to a linear periodic core waveguide. Therefore, the methods in this paper are applicable to linear periodic core waveguides and to three-layer planar waveguides with nonlinear cladding.
j s 1,2, . . . , M ,
E Ž x . s C j cosh k 0 q2 w x y Ž j y 1 . L x y a1 4 q Dj sinh k 0 q2 w x y Ž j y 1 . L x y a1 4 ,
Ž j y 1. L F x F j L ,
j s 1,2, . . . , M ,
E Ž x . s CMq1exp w yk 0 q3 Ž x y w . x
Ž xGw.
Ž6.
where the constants x 0 , A j , Bj , C j , Dj Ž j s 1,2, . . . , M . and CMq1 to be solved can be derived through continuity conditions. Let the field intensity at the interface between core and cladding be E0 . Defining the parameter Õ s tanh Ž k 0 q0 x 0 .
Ž7.
we have E0 s
q0 s N 2 y n 20 .
Ž2.
( q s (N
Ž j y 1 . L F x F Ž j y 1 . L q a1 ,
Ž N ) n1 . ,
(
2
a
q0'1 y Õ 2 .
Ž8.
For given waveguide parameters and a given value of modal indices, the field distribution, the powers in each region and the total power can be deduced as follows. Applying continuity conditions at the interfaces, recurrent formulas for A j , Bj , C j , Dj Ž j s 1,2, . . . , M . and CMq1 can be deduced. For N - n1 A1 s
(
2
a
q0'1 y Õ 2 ,
B1 s
q0 q1
ÕA1 ,
C1 s A1cos Ž k 0 q1 a1 . q B1 sin Ž k 0 q1 a1 . , q1 D 1 s w yA1sin Ž k 0 q1 a1 . q B1cos Ž k 0 q1 a1 . x , q2 A j s C jy1cosh Ž k 0 q2 a 2 . q Djy1 sinh Ž k 0 q2 a2 . , q0 Bj s C jy1 sinh Ž k 0 q2 a2 . q Djy1cosh Ž k 0 q2 a 2 . , q1 C j s A j cos Ž k 0 q1 a1 . q Bj sin Ž k 0 q1 a1 . , q1 Dj s yA j sin Ž k 0 q1 a1 . q Bj cos Ž k 0 q1 a1 . q2
Ž j s 2,3, . . . , M . CMq1 s
q2 q3
w CM sinh Ž k 0 q2 a2 . q DM cosh Ž k 0 q2 a2 . x . Ž 9a .
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
For N ) n1 A1 s
(
2
and, for both N - n1 and N ) n1
q0'1 y Õ 2 ,
a
q0
B1 s
q1
DM
ÕA1 ,
D1 s
A j s C jy1cosh Ž k 0 q2 a 2 . q Djy1 sinh Ž k 0 q2 a2 . , q0 Bj s C jy1 sinh Ž k 0 q2 a2 . q Djy1cosh Ž k 0 q2 a 2 . , q1
Õs
q1
yA j sin Ž k 0 q1 a1 . q Bj cos Ž k 0 q1 a1 .
q2
q1 B1
q2 q3
Ps
w CM sinh Ž k 0 q2 a2 . q DM cosh Ž k 0 q2 a2 . x . Ž 9b.
From formulas Ž9a. or Ž9b., we get the following. For N - n1 A1 s
(
2
q0'1 y Õ 2 ,
a
q 2 Dj Bj
s
Aj
q1 C j 1y
q 2 Dj q1 C j
q2 A j
s
C jy1
1y
q1
P3 s
y tanh Ž k 0 q2 a2 .
q2 A j
2
tanh Ž k 0 q2 a2 .
a
q 2 Dj Bj
s
Aj
q1 C j 1y
q1 C j
q1 B j Djy1 C jy1
s
q2 A j 1y
B1 s
q0 q1
2
d x s P0 q Pcore q P3
0
Hy` E
N
w
2 cm0
H0
N
q`
2 cm0
Hy`
2
d xs
N
4 q0
4 cm0 k 0 a
E 2d x s
E2d xs
M
I1 s
Ž 13.
N 4 cm0 k 0
ž
Ž1 y Õ . ,
I1
q
q1
N
2 CMq1
4 cm0 k 0
q3
I2 q2
/
,
,
Ž 14a.
Ý Ij1 ,
M
I2 s
Ý Ij2 , js1
I j1 s Ž A2j q Bj2 . Ž k 0 q1 a1 . q Ž A2j y Bj2 . sin Ž k 0 q1 a1 . cos Ž k 0 q1 a1 .
ÕA1 ,
q 2 A j B j sin2 Ž k 0 q1 a1 .
y tanh Ž k 0 q1 a1 .
q 2 Dj
N 2 cm0
js1
q0'1 y Õ 2 ,
Hy` E
where
Ž j s 1,2, . . . , M . .
For N ) n1
(
Ž 12.
where m 0 is the permittivity in free space and c m 0 s 377 V. P0 , Pcore and P3 are the powers in the nonlinear cladding, core and substrate, respectively;
, tan Ž k 0 q1 a1 .
q1 B j
Ž N - n1 .
q 0 A1
q`
2 cm0
Pcore s
Ž 10a.
A1 s
N
P0 s
ÕA1 ,
q tan Ž k 0 q1 a1 .
q1 B j Djy1
q0
B1 s
Ž 11 .
q1 B1
Ž N ) n1 . , Õ s
q 0 A1
Ž j s 2,3, . . . , M . CMq1 s
.
and finally E0 and all constants A j , B j , C j , Dj Ž j s 1,2, . . . , M . and CMq1 are determined by Eqs. Ž8. and Ž9a. or Eq. Ž9b.. Total power P and core power Pcore of the waveguide are calculated by the following formulas according to the function of field distribution,
C j s A j cos Ž k 0 q1 a1 . q Bj sin Ž k 0 q1 a1 . , Dj s
1 q Ž q3rq2 . tanh Ž k 0 q2 a2 .
For given N, from expressions Ž11., Ž10a. and Ž10b., DMrCM , BjrA j , DjrC j can be obtained and the parameter Õ is determined by
yA1sin Ž k 0 q1 a1 . q B1cos Ž k 0 q1 a1 . ,
q2
q3rq2 q tanh Ž k 0 q2 a2 .
sy
CM
C1 s A1cos Ž k 0 q1 a1 . q B1sin Ž k 0 q1 a1 . , q1
143
Ž N - n1 . ,
I j1 s Ž A2j y Bj2 . Ž k 0 q1 a1 . , q Ž A2j q Bj2 . sinh Ž k 0 q1 a1 . cosh Ž k 0 q1 a1 .
tanh Ž k 0 q1 a1 .
q 2 A j B j sinh2 Ž k 0 q1 a1 . y tanh Ž k 0 q2 a2 .
q1 B j q2 A j
Ž j s 1,2, . . . , M . tanh Ž k 0 q2 a2 .
Ž N ) n1 . ,
I j2 s Ž C j2 y Dj2 . Ž k 0 q2 a2 . q Ž C j2 q Dj2 . sinh Ž k 0 q2 a 2 . cosh Ž k 0 q2 a2 .
Ž 10b.
q 2C j Dj sinh2 Ž k 0 q2 a 2 . .
Ž 14b.
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
144
2.2. RMS approximation
For n eff - N
It has been shown by perturbation theory w8x and by transfer matrix technique w12,13x that, if the electric field is slowly varying over each period of a periodic structure Žthe electric field is not necessarily slowly varying over the entire region considered., then the thin film approximation Žfor each period. is valid and the periodic refractive index core can be treated as a uniform core of refractive index w13x n eff s wŽ1rL.H0L n 2 Ž x .d x x1r2 as given by formula Ž4.. It is rational to assert that the approximation is valid provided that k 0 < N 2 y n 2eff < L < 1, where n eff is the rms value of the refractive index in the spatial period L. It can be seen that the above condition can be written as
(
(
M 4 2p < N 2 y n 2eff <
w
l
,
(
2
a
q0 sech w k 0 q0 Ž x y x 0 . x
Ž x F 0. ,
E Ž x . s A cos Ž k 0 qf x . q B sin Ž k 0 qf x .
Ž N - n eff . ,
E Ž x . s A cosh Ž k 0 qf x . q B sinh Ž k 0 qf x .
Ž 15.
Modal field distribution EŽ x ., total power P and core power Pcore corresponding to modal refractive index N s brk 0 can be determined using continuity conditions at the interfaces. We define parameters Õ and E0 similar to expressions Ž7. and Ž8., where < Õ < F 1. Obviously, when Õ tends to 1, the cladding is a linear dielectrics, and Õ - 0 Ž x 0 - 0. corresponds to the cases where the maximum of E shifts into the core nonlinear cladding. For such cases, the positions of the maximum of the field intensity are located at x 0 Ž x 0 - 0.. By continuity of EŽ x . and EX Ž x ., we get the following. For n eff ) N 2
a
q0'1 y Õ 2 ,
q0'1 y Õ 2 ,
q0
Bs
qf
ÕA,
Ž 16b.
The eigenvalue equations are as follows. For n eff ) N Õs
qf qf tan Ž k 0 qf w . y q3
Ž 17a.
q0 qf q q3tan Ž k 0 qf w .
or tan Ž k 0 qf w . s
Õsy
qf Ž q3 y Õq0 . qf2 y Õq0 q3
.
Ž 17b.
Bs
q0 qf
ÕA,
C s A cos Ž k 0 qf w . q B sin Ž k 0 qf w .
qf qf tanh Ž k 0 qf w . q q3
Ž 18a.
q0 qf q q3tanh Ž k 0 qf w .
or qf Ž q3 q Õq0 .
tanh Ž k 0 qf w . s y
qf2 q Õq0 q3
.
Ž 18b.
For given parameters n˜ 0 , n1, n 2 , n 3 , R, core thickness w and given value of modal index N, the corresponding parameter Õ Ž< Õ < F 1. can be solved through eigenvalue Eqs. Ž17a. and Ž17b. or Ž18a. and Ž18b., and E0 and the field functions of guided waves can be determined by using expressions Ž16a. or Ž16b., the total power P, core power Pcore and substrate power P3 of the waveguide are given by the following formulas, P0 s
E Ž x . s C exp w yk 0 q3 Ž x y w . x Ž x G w . .
(
2
a
C s A cosh Ž k 0 qf w . q B sinh Ž k 0 qf w . .
Ž N ) n eff .
Ž w G x G 0. ,
As
(
For n eff - N
where M the number of periods and w the core thickness, l is the optical wavelength. Calculations for realistic periodic structures Žsee below. show that the above condition can be satisfied for number of periods ranging from 10 to 30 in such typical cases and thus the rms approximation is valid in these cases, saving large amount of computations. The method of analysis of propagation characteristics is as follows. From the nonlinear Helmholtz equation and using the rms approximation, the field function is given by EŽ x . s
As
N
Pcore s
P3 s
0
2 cm0
Hy` E
N
w
2 cm0
H0
N
q`
2 cm0
2
d xs
N
E 2d x s
Hy` E
2
q0
cm0 k 0 a
d xs
N 4 cm0 k 0 N
Ž1 y Õ . ,
I, C2
4 c m 0 k 0 q3
,
Ž 19.
where Is
1 qf
Ž A2 q B 2 . k 0 qf w
q Ž A2 y B 2 . sin Ž k 0 qf w . cos Ž k 0 qf w . q2 AB sin2 Ž k 0 qf w . Is
1 qf
Ž n eff ) N . ,
Ž 20a.
Ž A2 y B 2 . k 0 qf w
q Ž A2 q B 2 . sinh Ž k 0 qf w . cosh Ž k 0 qf w .
Ž 16a.
q2AB sinh2 Ž k 0 qf w .
Ž n eff - N . .
Ž 20b.
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
145
Therefore, for any given value of N, the curves of N–P and corresponding Pcore –P curve can be calculated. Furthermore, for given parameters of the waveguide, we can get the curves of nonlinearity and bistability, the threshold power for optical bistability, the influence of the waveguide parameters on the characteristics of guided waves, and curves of modal field distribution for all cases.
3. Examples and numerical analysis. Discussion Numerical calculations have been processed for a nonlinear periodic planar waveguide with realistic parameters. For one of the realistic models Žmodel A., the free-space wavelength l s 0.515 mm ŽArq laser., the cladding is a
Fig. 3. Field distribution of a periodic core structure with M s10, 20, 30 periods, Rs 0.8 and 0.9 for modal refractive index N s 3.45 Ž l s 0.83 mm, w s1.5 mm, n 0 s 3.42, n I s10y9 m2 rW, n1 s 3.465, n 2 s 3.315, n 3 s n 2 .. Solid line: exact; dotted line: rms approximation.
Fig. 2. Field distribution of a periodic core structure with M s10, 20, 30 periods, Rs 0.7 and 0.9 for modal refractive index N s 1.566 Ž l s 0.515 mm, w s1.4 mm, n 0 s1.55, n I s10y9 m2 rW, n1 s1.57, n 2 s1.52, n 3 s n 2 .. Solid line: exact; dotted line: rms approximation.
NBBA liquid crystal Ža Kerr-type nonlinear medium. with very strong nonlinearity, for which n 0 s 1.55 and n I s 10y9 m2rW; the material of the well is Coring 7059 glass, its linear refractive index is n1 s 1.57, that of the barriers is soda lime glass and its refractive index is n 2 s 1.52, the refractive index of the substrate is n 3 s n 2 s 1.52. The above values of the material parameters were initially used by Stegeman et al. w14x in their calculation, and subsequently cited by some other researchers w15,16x in their theoretical investigations of nonlinear optical waveguides. For another realistic model Žmodel B., we take the following typical values Žrefer to Refs. w5–8x.: free-space wavelength l s 0.83 mm, MQW core is made of layers,
146
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
the refractive index of the well and barrier fall in the range typical of Al xGa 1yx As materials, i.e., n g w3.315, 3.465x, and we take n1 s 3.465, n 2 s 3.315, core width w s 1.5 mm Žfrom Ref. w8x. and for the nonlinear Kerr-type cladding, n 0 s 3.42, n I s 2 = 10y9 m2rW, for the linear substrate, n 3 s 3.42. Calculations are processed for both model A and model B to verify the validity of rms approximation in characteristics analysis. We now compare the calculation results by the two methods, so as to verify the validity of the rms approximation when the number of layers in a spatial period is sufficiently large. For waveguide model A, Fig. 2a and b, show the curves of field distribution calculated by the exact method Žsolid lines. and by rms approximation Ždotted lines., corresponding respectively to R s 0.7 and R s 0.9, M s 10, 20, 30, with parameters n1 s 1.57, n 2 s 1.52, n 3 s n 2 , n I s 10y9 m2rW, wavelength l s 0.515 mm and core width w s 1.4 mm, and modal index N s 1.566. It is seen ŽFig. 2a. that for R s 0.7, the agreement of the two methods is good for M s 20 and is excellent for M 0 30, and ŽFig. 2b. that for R s 0.9, the agreement is good for M s 10 and is excellent for M 0 20. Now the thin film approximation is valid for M 4 2p N 2 y n2eff Ž wrl., we have, for R s 0.7, n2eff s 2.4185, N 2 s Ž1.566. 2 s 2.4524, so the approximation is valid for M 4 3.4, for R s 0.9, n2eff s 2.4495, N 2 s Ž1.566. 2 s 2.4524, so that the approximation is valid for M 4 0.92. It is seen that the rms approximations are valid for both cases. For waveguide model B, Fig. 3a and b show the curves of field distribution calculated by the two methods corresponding to R s 0.8 and R s 0.9, M s 10, 20, 30, with parameters n1 s 3.465, n 2 s 3.315, n 3 s n 2 , n 0 s 3.42,
(
Fig. 4. N – P curve for a periodic core structure with M s 5 periods, Rs 0.7 Ž l s 0.515 mm, w s1.4 mm, n 0 s1.55, n I s 10y9 m2 rW, n1 s1.57, n 2 s1.52, n 3 s n 2 .. Solid line: exact; dotted line: rms approximation.
Fig. 5. N – P curve for a periodic core structure with M s 5 periods, Rs 0.9 Žwaveguide parameters are the same as in Fig. 4.. Solid line: exact; dotted line: rms approximation.
n I s 2 = 10y9 m2rW, wavelength l s 0.83 mm and core width w s 1.5 mm, and modal index N s 3.45. It is seen ŽFig. 3a. that for R s 0.8, the agreement is good for M s 20, excellent for M 0 30, and that ŽFig. 3b. for R s 0.9, the agreement is excellent. Here, for R s 0.8, n 2eff s 11.803, N 2 s Ž3.45. 2 s 11.90, the thin film approximation is valid for M 4 3.6, for R s 0.9, n2eff s 11.90, N 2 s 11.90, the thin film approximation is valid for M 4 1. It is seen that the rms approximation is valid for both cases also. Figs. 4–9 show the numerical results for waveguide model A Ž l s 0.515 mm.. Figs. 4 and 5 show the N–P curves calculated by the exact method and rms approximation corresponding to w s 1.4 mm and R s 0.7 and 0.9, respectively. The figures show that the results of the two methods agree well for
Fig. 6. Dependence of modal refractive index N on total power P Žwaveguide parameters are the same as in Fig. 4..
S. She, S. Zhangr Optics Communications 161 (1999) 141–148
147
Fig. 7. Pcore – P curves Ž w s1.4 mm, Rs 0.7, 0.8, 0.9, 1.0. Žwaveguide parameters are the same as in Fig. 4..
Fig. 9. Dependence of modal refractive index N on total power P for core width w s1.2, 1.4, 1.6 mm Ž Rs 0.9. Ž l s 0.515 mm, refractive indices are the same as in Fig. 4..
M s 5. For M s 10, the N–P curves given by the two methods cannot be distinguished in the figures. Fig. 6 shows the curves of the dependence of modal index N on total power P for TE 0 mode Žcore thickness w s 1.4 mm, R s 0.7, 0.8, 0.9 and 1.0.. It is seen that the TE 0 mode exhibits stronger nonlinearity as power P becomes large and optical bistability appears at higher power when R is greater than 0.8. Fig. 7 shows the curves of the dependence of core power Pcore on total power P for the TE 0 mode Žcore thickness w s 1.4 mm, R s 0.7, 0.8, 0.9 and 1.0.. The figure shows that optical bistability between core power and total power is apparent when R G 0.9. The greater the R, the stronger the optical bistability. From Figs. 6 and 5, it is seen that optical bistability begins to appear approxi-
mately in the neighborhood of N s n eff . Furthermore, the threshold power of appearance for optical bistability in a nonlinear periodic core structure is lower than that for a nonlinear three-layer planar waveguide. Fig. 8 shows the curves of field distribution in the cladding, core and substrate of the waveguide for core thickness w s 1.4 mm, R s 0.9 and N s 1.560, 1.563, 1.566, 1.569, respectively. The curves in the figure show that the maximum value of the field begins to shift from the core region into the nonlinear cladding when N increases from N s 1.563 to N s 1.565 Žcorresponding to equivalent refractive index n eff .. When N further increases from N s 1.565 to 1.572, the maximum value of the field shifts further to the left and the maximum value of field gradually increases. That means that the TE 0 mode transforms from optical guided waves to optical surface waves. Fig. 9 shows N–P curves for core thickness w s 1.2, 1.4, 1.6 mm when R s 0.9. It is seen that optical bistability becomes stronger when the thickness w is increased.
4. Conclusions
Fig. 8. Field distribution for core width w s1.4 mm, Rs 0.9 and N s1.56, 1.563, 1.566, 1.569 Žwaveguide parameters are the same as in Fig. 4..
First, the waveguide parameters Žcore thickness w, ratio R s a1rL, and indices of refraction. have important effect on optical nonlinearity for TE 0 mode. For fixed values of w and R, the features of modal fields are independent of the number of layers M of periodic core provided M is large enough. Second, when the core thickness w and ratio R are small, optical nonlinearity appears at lower power, but optical bistability does not appear. When the core thickness w and ratio R are large, TE 0 mode exhibits strong optical nonlinearity and optical stability, and when the core thickness w and ratio R are increased, these effects be-
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S. She, S. Zhangr Optics Communications 161 (1999) 141–148
come stronger. The threshold power of optical bistability for the nonlinear periodic core waveguide is lower than that for the corresponding nonlinear three-layer waveguide. Finally, when modal refractive index N increases from values less than the rms equivalent refractive index n eff to values larger than n eff , TE 0 mode transforms from guided wave to surface wave and the turning point lies in the neighborhood of n eff .
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