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Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications – Asymptotic solutions
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Microelectronics Reliability 48 (2008) 555–562 www.elsevier.com/locate/microrel
Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications – Asymptotic solutions Jianxin Zhu *, Zhihua Chen, Shuyuan Tang Department of Mathematics, Zhejiang University, Hangzhou 310027, China Received 22 July 2007; received in revised form 4 September 2007 Available online 7 November 2007
Abstract In two-dimensional optical waveguides with the varied refractive index, approximate analytic solutions of the leaky modes are derived by using an asymptotic analysis for both transverse electric (TE) and transverse magnetic (TM) modes. Numerical simulations show that the asymptotic solutions of the leaky modes are quite close to the exact ones. The results are useful in the eigenmode method, where the leaky modes appear if a perfectly matched layer (PML) is used to terminate the transverse directions of optical waveguides. 2007 Elsevier Ltd. All rights reserved.
1. Introduction For the on-chip optical interconnection proposed recently, one of the the major issues is that the refractive index of the waveguide prepared by microfabrication technique may vary in the microwaveguides [1]. When an unbounded waveguide is terminated by a perfectly matched layer (PML), leaky modes also appear [2,3]. The leaky modes play an important role because they can be used to partially represent the wave field related to the continuous spectrum of the radiation and evanescent modes [4–7]. With a PML, the waveguide supports a discrete sequence of modes [8–10], such as the propagating modes, the leaky modes and the Berenger modes [11]. Since the propagation constants of the PML modes are in general complex, it is not easy to find all PML modes in a given region of the complex plane. For the two-dimensional step-index waveguides, the propagation constants satisfy a nonlinear equation. Some numerical methods have been proposed [12–14] to solve this equation. If an initial guess for a propagation constant
*
Corresponding author. E-mail address: [email protected] (J. Zhu).
0026-2714/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2007.09.004
is available, the accurate solution can be easily obtained by using a nonlinear equation solver, such as Newton’s method. Rogier and De Zutter [8] found some approximate solutions of the PML modes for two-layer waveguides that are open in one side of the transverse direction. We developed some more approximate solutions of the PML leaky modes [6] for three-layer slab waveguides where the transverse variable is unbounded for both positive and negative directions. But when the refractive index n is varied with the transverse variable x, the nonlinear equation of the leaky modes can not be obtained. In order to get asymptotic solutions of the leaky modes, WKB method is used to create an approximate nonlinear equation. It turns out that the PML leaky modes and the original leaky modes have identical asymptotic behaviors [9]. Thus, our approximated solutions can be used as initial guesses for computing the PML leaky modes by an iterative method such as Multi-Rayleigh Quotient iteration [18]. Finally, the high-accuracy solutions of leaky modes can be obtained, and they are very helpful to have a better modelling of the microchip optical interconnects. Asymptotic solutions of the approximate equation are derived in Section 2. In Section 3, some examples are given to compare asymptotic solutions with exact ones.
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asymptotic solutions of TE and TM modes, respectively.
2. Asymptotic solutions For a two-dimensional waveguide with the varied refractive index, we let 8 > < n1 ðxÞ; x < 0; nðxÞ ¼ n2 ðxÞ; x > d; > : n0 ðxÞ; 0 < x < d: Denote n0(0) = n0(0+), n0(d) = n0(d), n1(0) = n1(0), n2(d) = n2(d+), n00 ð0Þ ¼ n00 ð0pþffiffiffiffiffiffi Þ; nffi 00 ðdÞ ¼ n00 ðd Þ; n01 ð0Þ ¼ þ 0 0 0 n1 ð0 Þ; n2 ðdÞ ¼ n2 ðd Þ; and i ¼ 1: Original problem is o 1 ou o 1 ou q þq þ j20 n2 ðxÞu ¼ 0; oz q oz ox q ox 1 < x < þ1;
ð1Þ
where
8 > < q1 ; x < 0; qðx; zÞ ¼ q0 ; 0 < x < d; > : q2 ; x > d:
Let u = /(x)ei(bz xt), where / is the mode profile, b is the propagation constant, and x is the angular frequency. For inhomogeneous medium, we can get the following nonlinear eigenvalue problem on the finite interval 0 < x < d. (I) TE mode:
(II) TM mode: d 1 d/ 2 n0 ðxÞ þ j20 n20 ðxÞ/ ¼ b2 /; dx n20 ðxÞ dx
ð4Þ ð5Þ ð6Þ
d/ ¼ dx d/ ¼ dx
x ¼ 0þ ;
where ^c1 and ^c2 are arbitrary constants. Using conditions (5) and (6), there is an approximation nonlinear equation of the eigenvalue b2, that is, n0 ðdÞn00 ðdÞj20 2c202
ic01 þ ic1
n0 ð0Þn00 ð0Þj20 2c201
n ðdÞn0 ðdÞj2 n ð0Þn0 ð0Þj2 ic02 ic2 0 2c02 0 ic01 ic1 þ 0 2c02 0 02 01 R d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 2i j0 n0 ðtÞb dt e 0 ; ð11Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c02 ¼ j20 n20 ðdÞ b2 and c01 ¼ j20 n20 ð0Þ b2 : Suppose |b| is large enough, then
n0 ðdÞn00 ðdÞj20 2c202
and
n0 ð0Þn00 ð0Þj20 2c201
are very close to zero. Eq. (11) becomes approximately R d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi c02 þ c2 c01 þ c1 2i j20 n20 ðtÞb2 dt e 0 : ð12Þ c02 c2 c01 c1 Denote d1 ¼ j20 n20 ð0Þ j20 n21 ð0Þ; d0 ¼ j20 n20 ð0Þ j20 n20 ðdÞ; d2 ¼ j20 n20 ð0Þ j20 n22 ðdÞ; d3 = d2 d0. We have R d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i 2c01 b2 b4 j20 n20 ðtÞb2 dt ¼ 1 þ þ þ ; s0 e2 0 1 c201 c401 ðd1 d3 Þ4 2 2 2 1 where b2 ¼ d1 þd82 þd0 ; b4p¼ 1 þ 5d2 þ d0 þ 2d2 d1 þ ffiffiffi 128 ð5d 4 kp kp 2d2 d0 þ 2d1 d0 Þ; and s0 ¼ 1 ¼ cosð 2 Þ þ i sinð 2 Þ for k = 0, 1, 2, and 3. So idc01 2c01 b2 b4 2 s0 e 1 þ 2 þ 4 þ ; 1 c01 c01 ðd1 d3 Þ4
that is, 0 < x < d; ð7Þ
n2 ð0Þ ic1 02 /; n1 ð0Þ 2 n ðdÞ ic2 02 /; n2 ðdÞ
Applying WKB method [15] to (4), we get its approximate solution as follows: R x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i j2 n2 ðtÞb2 dt e 0 00 /ðxÞ ^c1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 j20 n20 ðxÞ b2 R x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i j2 n2 ðtÞb2 dt e 0 00 þ ^c2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 0 < x < d; ð10Þ 4 j20 n20 ðxÞ b2
ic02 þ ic2 þ
For TE modes, we let qi = 1 (i = 0, 1, 2); for TM modes, let qi ¼ n2i ðxÞ; (i = 0, 1, 2). Outgoing condition is given in the positive and negative directions, that is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ou ¼ i j20 n21 ðxÞ b2 u; x < 0; ð2Þ ox qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ou ¼ i j20 n22 ðxÞ b2 u; x > d: ð3Þ ox
d2 / þ j20 n20 ðxÞ/ ¼ b2 /; 0 < x < d; dx2 d/ðxÞ ¼ ic1 /; x ¼ 0þ ; dx d/ðxÞ ¼ ic2 /; x ¼ d : dx
2.1. The derivation of TE modes
ð8Þ
x ¼ d ; ð9Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c1 ¼ j20 n21 ð0Þ b2 and c2 ¼ j20 n22 ðdÞ b2 : In the next two part, we will give the approximate
1 is0 d b2 b4 a0 c01 4 ðd1 d3 Þ a0 c0 e 1 þ 2 þ 4 þ ; 4 c01 c01 where a0 ¼ id : 2 Since b2 b4 a0 c01 1 þ 2 þ 4 þ a0 c01 e c01 c01 a a a a0 c01 þ 22 þ 33 þ 44 þ M a2 a3 a4 c c c 01 01 01 ¼ a0 c01 þ 2 þ 3 þ 4 þ e ¼ W eW ; c01 c01 c01
J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562 3 where a2 = b2, a3 ¼ 4di ðd2 þ d0 þ d1 Þ; a4 ¼ idb2a4 2a 0 a3 a2 a4 W ¼ a0 c01 þ c2 þ c3 þ c4 þ . 01 01 01 That is,
W eW
a22 2
; and
1 is0 d ðd1 d3 Þ4 : 4 1
It leads to [17] W Lambertðp; is40 d ðd1 d3 Þ4 Þ: We also rewrite the fourth root of 1 as iq1, then is0 = iq for q = 0, 1, 2, and 3. Let m = 4(1 + p) + q, we have 1 iq 4 ðd1 d3 Þ W W m ¼ Lambert p; 4 for p = 1, 2, . . .
557
Let /(x) = n0(x)p(x), then Eq. (15) can be changed to n0xx 2n20x 2 2 2 ð18Þ 2 b p ¼ 0; pxx þ j0 n0 ðxÞ þ n0 n0 where n0xx is represented as n000 ðxÞ, and n0x is represented as n00 ðxÞ: Using c = ln n(x), Eqs. (15)–(17) turn into the following forms: pxx þ ½j20 n20 ðxÞ þ cxx c2x b2 p ¼ 0; 0 < x < d; ic n2 ð0Þ n0x p x ¼ 12 0 p; x ¼ 0þ ; n1 ð0Þ n0 ð0Þ ic2 n20 ðdÞ n0x px ¼ p; x ¼ d : n0 ðdÞ n22 ðdÞ
ð19Þ ð20Þ ð21Þ
Applying WKB method to Eqs. (19)–(21), we get
e
2i
R d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 0
j0 n0 ðxÞþcxx cx b dx
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ðdÞ n ðdÞ 1 0 0:25ðc202 þ cxx ðdÞ c2x ðdÞÞ yðdÞ þ i c202 þ cxx ðdÞ c2x ðdÞ þ n22 ðdÞ n0x0 ðdÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ð0Þ n ð0Þ 0 0:25ðc201 þ cxx ð0Þ c2x ð0ÞÞ1 yð0Þ þ i c201 þ cxx ð0Þ c2x ð0Þ n12 ð0Þ n0x0 ð0Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ð0Þ n ð0Þ 1 0 0:25ðc201 þ cxx ð0Þ c2x ð0ÞÞ yð0Þ þ i c201 þ cxx ð0Þ c2x ð0Þ þ n12 ð0Þ þ n0x0 ð0Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ðdÞ n ðdÞ ; 0 0:25ðc202 þ cxx ðdÞ c2x ðdÞÞ1 yðdÞ þ i c202 þ cxx ðdÞ c2x ðdÞ 2dn2 ðdÞ þ n0x0 ðdÞ 2
It appears that the case m = 0 (i.e., p = 1 and q = 0) does not correspond to an actual leaky mode, and it should be removed. Thus, we can obtain the following asymptotic formulas:
e
(A) Leading term approximation: 2iW c01 ; d
ð13Þ
(B) Fourth term approximation: c01
2i
R d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2
j0 n0 ðxÞþcxx cx b dx
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic2 n2 ðdÞ n ðdÞ 0 c202 þ cxx ðdÞ c2x ðdÞ þ n2 ðdÞ n0x0 ðdÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ð0Þ ð0Þ 0 i c201 þ cxx ð0Þ c2x ð0Þ n12 ð0Þ nn0x0 ð0Þ i
1
W a0 a2 a20 a3 a30 a4 2 3 4 ; a0 W W W
ð14Þ
where Im(c01) > 0 by choosing some integers p.
2.2. The derivation of TM modes
0 < x < d;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ð0Þ n ð0Þ 0 i c201 þ cxx ð0Þ c2x ð0Þ þ n12 ð0Þ þ n0x0 ð0Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ðdÞ ðdÞ 0 i c202 þ cxx ðdÞ c2x ðdÞ n22 ðdÞ þ nn0x0 ðdÞ 2
Remark. If n0(x) is a constant, then d0 = 0 and d3 = d2. The result is same as the one in [6].
Starting from d 1 d/ n20 ðxÞ þ j20 n20 ðxÞ/ ¼ b2 /; dx n20 ðxÞ dx d/ n2 ð0þÞ ¼ ic1 0 2 /; x ¼ 0þ ; dx n1 ð0Þ d/ n2 ðdÞ ¼ ic2 0 2 /; x ¼ d : dx n2 ðdÞ
where yðxÞ ¼ 2j20 n0 ðxÞn0 0ðxÞ þ cxxx 2cx cxx . When |b| is large enough, above equation is changed approximately as follows:
ð15Þ ð16Þ ð17Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c202 þ cxx ðdÞ c2x ðdÞ i n20 ðdÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i c202 þ cxx ðdÞ c2x ðdÞ
ðdÞ 2 þ n2icðdÞ nn0x3 ðdÞ 2
0
ðdÞ 2 n2icðdÞ þ nn0x3 ðdÞ 2 0 n20 ðdÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i c201 þ cxx ð0Þ c2x ð0Þ ð0Þ 1 þ n2icð0Þ þ nn0x3 ð0Þ 1 0 n20 ð0Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : i c201 þ cxx ð0Þ c2x ð0Þ ð0Þ 1 n2icð0Þ nn0x3 ð0Þ 1 0 n20 ð0Þ
Expanding the right-hand side of above equation as the inverse power series of c01, we have
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e
2i
J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562
c1 c2 c3 c4 c0 þ þ 2 þ 3 þ 4 þ c01 c01 c01 c01 e1 e2 e3 e4 e0 þ þ 2 þ 3 þ 4 þ c01 c01 c01 c01 1 ¼ c0 e0 þ ðc1 e0 þ c0 e1 Þ c01 1 þ 2 ðc0 e2 þ c1 e1 þ c2 e0 Þ c01 1 þ 3 ðc0 e3 þ c1 e2 þ c2 e1 þ c3 e0 Þ c01 1 þ 4 ðc0 e4 þ c1 e3 þ c2 e2 þ c3 e1 þ c4 e0 Þ þ c01 A1 A2 A3 A4 ¼ A0 þ þ þ þ þ ; c01 c201 c301 c401
R d pffiffiffiffiffiffiffiffiffiffiffi2ffi 0
uðxÞb dx
8 A0 > > > < A2 > A3 > > : A4
¼ c0 e0 ;
A1 ¼ c1 e0 þ c0 e1 ;
¼ c0 e2 þ c1 e1 þ c2 e0 ; ¼ c0 e3 þ c1 e2 þ c2 e1 þ c3 e0 ;
¼ c0 e4 þ c1 e3 þ c2 e2 þ c3 e1 þ c4 e0 and ^a2 ¼ d0 cxx ðdÞ þ c2x ðdÞ; ^b2 ¼ c2x ð0Þ cxx ð0Þ cx(d) = cx(d), cx(0) = cx(0+), cxx(d) = cxx(d) + cxx(0) = cxx(0 ). Further, ^
e2idðc01 þb2 Þ A0 þ
here and
A 1 A 2 A3 A4 þ þ þ : c01 c201 c301 c401
It leads to 2idðc01 þ ^b2 Þ 2ipm þ lnðA0 Þ þ where t1 ¼ AA10 and t2 ¼
2A2 A0 A21 A20
t1 t2 t3 t4 þ 2 þ 3 þ 4 ; c01 c01 c01 c01
:
where uðxÞ ¼ j20 n20 ðxÞ þ cxx ðxÞ c2x ðxÞ;
8 > > > e0 > > > > > > e2 > > > > < e3 > > > > > > e4 > > > > > > > :
n2 ðdÞþn2 ðdÞ
¼ n22 ðdÞn02 ðdÞ ; 2
0
2ic ðdÞn4 ðdÞ 2
2n6 ðdÞc2 ðdÞ
x 2 ¼ ðn2 ðdÞn 2 ðdÞÞ3 þ 2
¼ ¼
TE Example 2
x 2 c1 ¼ ðn2 ðdÞn 2 ðdÞÞ2 ; 0
ð^ a2 d2 Þn20 ðdÞn22 ðdÞ ðn20 ðdÞn22 ðdÞÞ2
0
5
;
icx ðdÞn22 ðdÞ½2d2 n40 ðdÞþ^ a2 n20 ðdÞn22 ðdÞþ^ a2 n42 ðdÞc2x ðdÞn42 ðdÞ ðn22 ðdÞn20 ðdÞÞ3 ð^ a2 d2 Þn20 ðdÞn22 ðdÞ½ð3^ a2 þd2 Þn22 ðdÞð^ a2 þ3d2 Þn20 ðdÞ 4ðn22 ðdÞn20 ðdÞÞ3
þ
c2x ðdÞn42 ðdÞð3d2 n20 ðdÞ^ a2 n22 ðdÞÞ 2ðn22 ðdÞn20 ðdÞÞ3
n2 ð0Þþn2 ð0Þ
;
0
1
2c2x ð0Þn61 ð0Þ
¼ ðn2 ð0Þn2 ð0ÞÞ3 þ 1
¼ ¼
0
3
2
2ic ð0Þn4 ð0Þ
x 1 ¼ n12 ð0Þn02 ð0Þ ; e1 ¼ ðn2 ð0Þn 2 ð0ÞÞ2 ; 1
4
; Im(β/κ0)
8 > > > c0 > > > > > > c2 > > > > < c3 > > > > > > c4 > > > > > > > :
0
ð^ b2 d1 Þn20 ð0Þn21 ð0Þ ðn21 ð0Þn20 ð0ÞÞ2
;
icx ð0Þn21 ð0Þ½^b2 n21 ð0Þn20 ð0Þ^ b2 n41 ð0Þþ2d1 n40 ð0Þþc2x ð0Þn41 ð0Þ 2 ðn1 ð0Þn20 ð0ÞÞ3
1
; 0.7
1
ð^ b2 d1 Þn20 ð0Þn21 ð0Þ½ð3^ b2 þd1 Þn21 ð0Þð3d1 þ^ b2 Þn20 ð0Þ
þ
1.6
1.9
2.2
Re(β/κ0)
4ðn21 ð0Þn20 ð0ÞÞ3 c2x ð0Þn41 ð0Þð3d1 n20 ð0Þ^ b2 n21 ð0ÞÞ 2ðn21 ð0Þn20 ð0ÞÞ3
1.3
Fig. 1. Comparison of the exact and asymptotic propagation constants of leaky modes for TE in Example 2.
;
Table 1 The relative error of asymptotic solution for TE case in Example 1 m
Exact result b
Solution b of (13)
Solution b of (14)
R.E. of (13)
R.E. of (14)
1 2 3 4
31.927284 + 20.670091i 34.471744 + 38.879799i 36.644954 + 56.098296i 38.484651 + 72.821842i
31.922255 + 20.700198i 34.476206 + 38.893398i 36.649356 + 56.105230i 38.488164 + 72.825836i
31.927813 + 20.670035i 34.471769 + 38.879711i 36.644946 + 56.098273i 38.484645 + 72.821836i
8.0254 · 104 2.7544 · 104 1.2257 · 104 6.4580 · 105
1.3986 · 105 1.7605 · 106 3.6342 · 107 1.0302 · 107
Table 2 The relative error of asymptotic solution for TM case in Example 1 m
Exact result b
Solution b of (23)
Solution b of (24)
R.E. of (23)
R.E. of (24)
1 2 3 4
18.928928 + 13.977974i 17.503529 + 29.461225i 16.819679 + 45.520900i 16.516545 + 61.536569i
19.108537 + 13.212853i 17.322568 + 29.150214i 16.681640 + 45.405300i 16.425496 + 61.484485i
18.952827 + 14.025118 17.513796 + 29.453669i 16.820165 + 45.517499i 16.516212 + 61.535407i
3.3400 · 102 1.0500 · 102 3.7102 · 103 1.6463 · 103
2.2463 · 103 3.7200 · 104 7.0794 · 105 1.8972 · 105
J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562
559
it1 ; and Im(c01) > 0 by where K m1 ¼ K m þ ^b2 2dK m choosing some integers m.
Finally, we obtain the following asymptotic formulas. (C) Leading term approximation: c01
3. Numerical results
pm i lnðA0 Þ ^ b2 ¼ K m ; d 2d
ð22Þ
(D) Two terms approximation: c01 K m þ ^ b2
it1 A1 i ¼ Km þ ^ b2 2dA0 K m 2dK m
ð23Þ
and (E) Three terms approximation: c01 K m1
it2 ið2A2 A0 A21 Þ ¼ K m1 ; 2 2dK m1 2dA20 K 2m1
ð24Þ
Example 1. Let n0(x) = 3.3, n1(x) = n2(x) = 3.17, d = 0.2 cm, k = 1.55 lm and j0 ¼ 2p k : Tables 1 and 2 show the effect of the asymptotic approximation for TE and TM cases, respectively. In next examples, since their exact solutions of the propagation constant b can not be obtained directly, they are substituted by well approximated ones, which are solved by some numerical method in a bounded waveguide with
TM Example 2
TM Example 4
10 7 9 6
7
Im(β/κ0)
Im(β/κ0)
8
6
5 4
5
3
4
2
3
1
2
0.5
0.6
0.7
0.8
0.2
0.4
Re(β/κ0)
0.6
0.8
Re(β/κ0)
Fig. 2. Comparison of the exact and asymptotic propagation constants of leaky modes for TM in Example 2.
Fig. 4. Comparison of the exact and asymptotic propagation constants of leaky modes for TM in Example 4.
TE Example 5
TE Example 3 5
3.5
4.5 3
Im(β/κ0)
Im(β/κ0)
4 2.5 2
3.5 3
1.5
2.5
1
2 1.5
0.5 0.6
0.8
1
1.2
1.4
1.6
1.8
Re(β/κ0) Fig. 3. Comparison of the exact and asymptotic propagation constants of leaky modes for TE in Example 3.
0.6
0.7
0.8
0.9
1
Re(β/κ0) Fig. 5. Comparison of the exact and asymptotic propagation constants of leaky modes for TE in Example 5.
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J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562
an appropriate PML. For TE cases, the exact solutions are marked by ‘‘+’’, and the asymptotic solutions from (13) and (14) are marked by ‘‘·’’, ‘‘’’, respectively (see Figs. 1, 3 and 5). For TM cases, the exact solutions are marked by ‘‘+’’, and the asymptotic solutions from (22)–(24) are marked by ‘‘·’’, ‘‘’’ and ‘‘Æ’’, respectively (see Figs. 2, 4 and 6).
TM Example 6 9 8
Im(β/κ0)
7 6
Example 2. Let n0(x) = 3.3(1 0.01((x 1.0)/2.5)2), n1(x) = n2(x) = 3.17, d = 2.0 cm, k = 1.55 lm and j0 ¼ 2p k : Tables 3 and 4 show the effect of the asymptotic approximation for TE and TM cases, respectively.
5 4 3 2 1 0.2
0.3
0.4
0.5
0.6
0.7
Example 3. Let n0 = 3.3(1 0.01((x 1.0)/2.5)2), n1 = 2.10, n2 = 3.17, d = 2.0 cm, k = 1.55 lm and j0 ¼ 2p : Table k 5 shows the effect of the asymptotic approximation for TE case.
Re(β/κ0) Fig. 6. Comparison of the exact and asymptotic propagation constants of leaky modes for TM in Example 6.
Example 4. Let n0 = 3.3(1 0.01((x 1.0)/2.5)2), n1 = 3.17, n2 = 1.0, d = 2.0 cm, k = 1.55 lm and j0 ¼ 2p : Table k 6 shows the effect of the asymptotic approximation for TM case.
Table 3 The relative error of asymptotic solution for TE case in Example 2 m
Exact result b
Solution b of (13)
Solution b of (14)
R.E. of (13)
R.E. of (14)
6 7 8 9 10 11 12 13 14 15
8.405613 + 2.240053i 6.516805 + 3.574983i 4.784198 + 5.83967i 3.858838 + 8.482219i 3.442450 + 10.940137i 3.237798 + 13.191868i 3.129862 + 15.295930i 3.072181 + 17.295962i 3.043025 + 19.220440i 3.031326 + 21.088201i
8.350168 + 2.331169i 6.465522 + 3.708539i 4.779191 + 6.002421i 3.888505 + 8.633277i 3.484863 + 11.079736i 3.286326 + 13.326365i 3.182811 + 15.429381i 3.129131 + 17.430837i 3.104078 + 19.358344i 3.096554 + 21.230264i
8.343727 + 2.2947754i 6.450073 + 3.674752i 4.756127 + 5.980562i 3.868389 + 8.622147i 3.469156 + 11.073391i 3.273898 + 13.32227 i 3.172724 + 15.426509i 3.120756 + 17.428706i 3.096997 + 19.356698i 3.090476 + 21.228954i
0.012261 0.019247 0.0215686 0.016520 0.012721 0.010526 0.009196 0.008334 0.007750 0.0073373
0.00949 0.016148 0.019029 0.015051 0.011849 0.00996 0.008802 0.008047 0.007531 0.007166
Table 4 The relative error of asymptotic solution for TM case in Example 2 m
Exact result b
Solution b of (23)
Solution b of (24)
R.E. of (23)
R.E. of (24)
10 11 12 13 14 15 16 17 18
3.209989 + 8.822371i 2.713684 + 11.238592i 2.431673 + 13.465754i 2.252737 + 15.555120i 2.130175 + 17.546192i 2.041584 + 19.465271i 1.974959 + 21.329954i 1.923269 + 23.152433i 1.882185 + 24.941440i
2.961755 + 8.632321i 2.526157 + 11.132919i 2.289611 + 13.399737i 2.142810 + 15.510877i 2.043540 + 17.515464i 1.972366 + 19.443774i 1.919132 + 21.315314i 1.878023 + 23.143266i 1.845470 + 24.936892i
3.228186 + 8.75776133i 2.730608 + 11.2009311i 2.450432 + 13.4421507i 2.273112 + 15.5397933i 2.151677 + 17.5363799i 2.063823 + 19.4595443i 1.997666 + 21.327581i 1.946302 + 23.153041i 1.905451 + 24.944833i
0.033301 0.018618 0.011448 0.007539 0.005201 0.003703 0.002694 0.001987 0.001479
0.007149 0.003571 0.002203 0.001622 0.001337 0.001173 0.001065 0.000991 0.000940
Table 5 The relative error of asymptotic solution for TE case in Example 3 m
Exact result b
Solution b of (13)
Solution b of (14)
R.E. of (13)
R.E. of (14)
8 9 10 11 12 13
3.720599 + 5.3860768i 2.903792 + 8.3614252i 2.617303 + 10.934894i 2.498615 + 13.234072i 2.448795 + 15.362522i 2.432718 + 17.377481i
3.854492 + 5.563151i 3.015709 + 8.456972i 2.701728 + 11.001253i 2.565028 + 13.288107i 2.503310 + 15.410629i 2.479054 + 17.422706i
3.709002 + 5.443356i 2.903358 + 8.407232i 2.619634 + 10.974873i 2.502269 + 13.271602i 2.453443 + 15.399253i 2.438266 + 17.414362i
0.033912 0.016625 0.009550 0.006357 0.004673 0.003690
0.008927 0.005175 0.003561 0.002799 0.002379 0.002125
J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562
561
Table 6 The relative error of asymptotic solution for TM case in Example 4 m
Exact result b
Solution b of (23)
Solution b of (24)
R.E. of (23)
R.E. of (24)
9 10 11 12 13 14
2.666497 + 5.4977708i 1.830561 + 8.5207237i 1.502239 + 11.089345i 1.330306 + 13.379734i 1.225011 + 15.500628i 1.154245 + 17.509487i
2.367969 + 5.134397i 1.611944 + 8.380557i 1.3493853 + 11.012340i 1.2161435 + 13.329667i 1.1355542 + 15.465301i 1.0817335 + 17.483593i
2.887592 + 5.427700i 1.967417 + 8.474227i 1.603880 + 11.058126i 1.411039 + 13.357143i 1.291413 + 15.483693i 1.210118 + 17.496759i
0.076964 0.029797 0.015294 0.009271 0.006185 0.004387
0.037957 0.016584 0.009501 0.006235 0.004407 0.003265
Table 7 The relative error of asymptotic solution for TE case in Example 5 m
Exact result b
Solution b of (13)
Solution b of (14)
R.E. of (13)
R.E. of (14)
8 9 10 11 12 13 14
2.871394 + 8.343677i 2.614595 + 10.938301i 2.473316 + 13.233021i 2.442923 + 15.357283i 2.415184 + 17.382631i 2.423352 + 19.305651i 2.437238 + 21.193212i
2.997865 + 8.453837i 2.68599 + 10.999956i 2.55062 + 13.287519i 2.489778 + 15.410375i 2.466124 + 17.422629i 2.463085 + 19.356486i 2.472254 + 21.232342i
2.884520 + 8.404006i 2.603262 + 10.973558i 2.487409 + 13.271006i 2.439559 + 15.398993i 2.425055 + 17.414280i 2.428744 + 19.350093i 2.443035 + 21.227289i
0.019007 0.008388 0.007026 0.004554 0.003691 0.003316 0.002461
0.00699 0.003292 0.00300 0.002690 0.001889 0.002300 0.001601
Table 8 The relative error of asymptotic solution for TM case in Example 6 m
Exact result b
Solution b of (23)
Solution b of (24)
R.E. of (23)
R.E. of (24)
9 10 11 12 13 14
2.660849 + 5.4865224i 1.808461 + 8.4977042i 1.487248 + 11.094186i 1.317280 + 13.369376i 1.209966 + 15.502616i 1.143919 + 17.503316i
2.343885 + 5.125013i 1.593123 + 8.377984i 1.3333623 + 10.111698i 1.2016354 + 13.328997i 1.1219826 + 15.464866i 1.0687936 + 17.483289i
2.86505382 + 5.416793i 1.94832151 + 8.470549i 1.587446 + 11.056353i 1.396160 + 13.356098i 1.277524 + 15.483004i 1.196907 + 17.496271i
0.078846 0.028358 0.015620 0.009117 0.006157 0.004432
0.03538 0.016398 0.009568 0.005954 0.004523 0.003047
Example 5. Let n0 = 3.3(1 0.01((x 1.0)/2.5)2), n1 = 2.10(1 0.01((x 1.0)/2.5)2), n2 = 3.17(1 0.01((x 1.0)/ 2.5)2), d = 2.0 cm, k = 1.55 lm, and j0 ¼ 2p : Table 7 shows k the effect of the asymptotic approximation for TE case. Example 6. Let n0 = 3.3(1 0.01((x 1.0)/2.5)2), n1 = 1.0(1 0.01((x 1.0)/2.5)2, n2 = 3.17(1 0.01((x 1.0)/ 2.5)2, d = 2.0 cm, k = 1.55 lm and j0 ¼ 2p : Table 8 k shows the effect of the asymptotic approximation for TM case. 4. Conclusions In order to compute the leaky modes in the optical waveguide where the refractive index is varied with the transverse variable, we derive some asymptotic analytic solutions. Numerical results illustrate that these solutions are very close to the exact ones as the norms of the modes become larger. Our results are useful in the eigenmode expansion method [16] when PMLs are used. In this method, it is necessary to compute the perturbed leaky modes and the Berenger modes. Our asymptotic analytic solutions can reduce the effort of computing these modes.
Acknowledgements The work described in this paper was supported by the Natural Science Foundation of China (Project No. 10571162) and the Natural Science of Zhejiang Province (Project No. Y605181). References [1] Wong H, Filip V, Wong CK, Chung PS. Silicon integrated photonic begins to revolution. Microelectron Reliab 2007;47:1–10. [2] Berenger JP. A perfectly matched layer for the absorption of electromagnetic-wave. J Comput Phys 1994;114(2):185–200. [3] Chew WC, Weedon WH. A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates. Microwave Opt Technol Lett 1994;7(13):599–604. [4] Snyder AW, Love JD. Optical waveguide theory. London, UK: Chapman & Hall; 1983. [5] Marcuse D. Theory of dielectric optical waveguides. 2nd ed. Boston, MA: Academic; 1991. [6] Zhu J, Lu YY. Leaky modes of slab waveguides – asymptotic solutions. IEEE J Lightwave Technol 2006;24(3):1619–23. [7] Vassallo C. Optical waveguide concepts. Amsterdam, The Netherlands: Elsevier; 1991. [8] Rogier H, De Zutter D. Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer. IEEE Trans Microwave Theory Tech 2001;49(4):712–5.
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[9] Rogier H, De Zutter D. Berenger and leaky modes in optical fibers terminated with a perfectly matched layer. J Lightwave Technol 2002;20(7):1141–8. [10] Rogier H, Knockaert L, De Zutter D. Fast calculation of the propagation constants of leaky and Berenger modes of planar and circular dielectric waveguides terminated by a perfectly matched layer. Microwave Opt Technol Lett 2003;37(3): 167–71. [11] Derudder H, Olyslager F, De Zutter D. An efficient series expansion for the 2-D Green’s function of a microstrip substrate using perfectly matched layers. IEEE Microwave Guided Wave Lett 1999;9(12):505–7. [12] Ye RZL, Yevick D. Noniterative calculation of complex propagation constants in planar waveguides. J Opt Soc Am A, Opt Image Sci 2001;18(11):2819–22.
[13] Gaal SB, Hoekstra HJWM, Lambeck PV. Determining PML modes in 2-D stratified media. J Lightwave Technol 2003;21(1):293–8. [14] Knockaert L, Rogier H, De Zutter D. An FFT-based signal identification approach for obtaining the propagation constants of the leaky modes in layered media. AEU, Int J Electron Commun 2005;59(4):230–8. [15] Jensen FB, Kuperman WA, Porter MB, Schmidt H. Computational ocean acoustics. American Institute of Physics; 1994. [16] Derudder H, De Zutter D, Olyslager F. Analysis of waveguide discontinuities using perfectly matched layers. Electron Lett 1998;34(22):2138–40. [17] Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE. On the Lambert W function. Adv Comput Math 1996;5(4):329–59. [18] Zhu J, Zou Q. Eigenvalue computation in slab waveguides with some perfectly matched layers. Proc SPIE 2005;5279:172–80.
Microelectronics Reliability 48 (2008) 555–562 www.elsevier.com/locate/microrel
Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications – Asymptotic solutions Jianxin Zhu *, Zhihua Chen, Shuyuan Tang Department of Mathematics, Zhejiang University, Hangzhou 310027, China Received 22 July 2007; received in revised form 4 September 2007 Available online 7 November 2007
Abstract In two-dimensional optical waveguides with the varied refractive index, approximate analytic solutions of the leaky modes are derived by using an asymptotic analysis for both transverse electric (TE) and transverse magnetic (TM) modes. Numerical simulations show that the asymptotic solutions of the leaky modes are quite close to the exact ones. The results are useful in the eigenmode method, where the leaky modes appear if a perfectly matched layer (PML) is used to terminate the transverse directions of optical waveguides. 2007 Elsevier Ltd. All rights reserved.
1. Introduction For the on-chip optical interconnection proposed recently, one of the the major issues is that the refractive index of the waveguide prepared by microfabrication technique may vary in the microwaveguides [1]. When an unbounded waveguide is terminated by a perfectly matched layer (PML), leaky modes also appear [2,3]. The leaky modes play an important role because they can be used to partially represent the wave field related to the continuous spectrum of the radiation and evanescent modes [4–7]. With a PML, the waveguide supports a discrete sequence of modes [8–10], such as the propagating modes, the leaky modes and the Berenger modes [11]. Since the propagation constants of the PML modes are in general complex, it is not easy to find all PML modes in a given region of the complex plane. For the two-dimensional step-index waveguides, the propagation constants satisfy a nonlinear equation. Some numerical methods have been proposed [12–14] to solve this equation. If an initial guess for a propagation constant
*
Corresponding author. E-mail address: [email protected] (J. Zhu).
0026-2714/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2007.09.004
is available, the accurate solution can be easily obtained by using a nonlinear equation solver, such as Newton’s method. Rogier and De Zutter [8] found some approximate solutions of the PML modes for two-layer waveguides that are open in one side of the transverse direction. We developed some more approximate solutions of the PML leaky modes [6] for three-layer slab waveguides where the transverse variable is unbounded for both positive and negative directions. But when the refractive index n is varied with the transverse variable x, the nonlinear equation of the leaky modes can not be obtained. In order to get asymptotic solutions of the leaky modes, WKB method is used to create an approximate nonlinear equation. It turns out that the PML leaky modes and the original leaky modes have identical asymptotic behaviors [9]. Thus, our approximated solutions can be used as initial guesses for computing the PML leaky modes by an iterative method such as Multi-Rayleigh Quotient iteration [18]. Finally, the high-accuracy solutions of leaky modes can be obtained, and they are very helpful to have a better modelling of the microchip optical interconnects. Asymptotic solutions of the approximate equation are derived in Section 2. In Section 3, some examples are given to compare asymptotic solutions with exact ones.
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J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562
asymptotic solutions of TE and TM modes, respectively.
2. Asymptotic solutions For a two-dimensional waveguide with the varied refractive index, we let 8 > < n1 ðxÞ; x < 0; nðxÞ ¼ n2 ðxÞ; x > d; > : n0 ðxÞ; 0 < x < d: Denote n0(0) = n0(0+), n0(d) = n0(d), n1(0) = n1(0), n2(d) = n2(d+), n00 ð0Þ ¼ n00 ð0pþffiffiffiffiffiffi Þ; nffi 00 ðdÞ ¼ n00 ðd Þ; n01 ð0Þ ¼ þ 0 0 0 n1 ð0 Þ; n2 ðdÞ ¼ n2 ðd Þ; and i ¼ 1: Original problem is o 1 ou o 1 ou q þq þ j20 n2 ðxÞu ¼ 0; oz q oz ox q ox 1 < x < þ1;
ð1Þ
where
8 > < q1 ; x < 0; qðx; zÞ ¼ q0 ; 0 < x < d; > : q2 ; x > d:
Let u = /(x)ei(bz xt), where / is the mode profile, b is the propagation constant, and x is the angular frequency. For inhomogeneous medium, we can get the following nonlinear eigenvalue problem on the finite interval 0 < x < d. (I) TE mode:
(II) TM mode: d 1 d/ 2 n0 ðxÞ þ j20 n20 ðxÞ/ ¼ b2 /; dx n20 ðxÞ dx
ð4Þ ð5Þ ð6Þ
d/ ¼ dx d/ ¼ dx
x ¼ 0þ ;
where ^c1 and ^c2 are arbitrary constants. Using conditions (5) and (6), there is an approximation nonlinear equation of the eigenvalue b2, that is, n0 ðdÞn00 ðdÞj20 2c202
ic01 þ ic1
n0 ð0Þn00 ð0Þj20 2c201
n ðdÞn0 ðdÞj2 n ð0Þn0 ð0Þj2 ic02 ic2 0 2c02 0 ic01 ic1 þ 0 2c02 0 02 01 R d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 2i j0 n0 ðtÞb dt e 0 ; ð11Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c02 ¼ j20 n20 ðdÞ b2 and c01 ¼ j20 n20 ð0Þ b2 : Suppose |b| is large enough, then
n0 ðdÞn00 ðdÞj20 2c202
and
n0 ð0Þn00 ð0Þj20 2c201
are very close to zero. Eq. (11) becomes approximately R d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi c02 þ c2 c01 þ c1 2i j20 n20 ðtÞb2 dt e 0 : ð12Þ c02 c2 c01 c1 Denote d1 ¼ j20 n20 ð0Þ j20 n21 ð0Þ; d0 ¼ j20 n20 ð0Þ j20 n20 ðdÞ; d2 ¼ j20 n20 ð0Þ j20 n22 ðdÞ; d3 = d2 d0. We have R d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i 2c01 b2 b4 j20 n20 ðtÞb2 dt ¼ 1 þ þ þ ; s0 e2 0 1 c201 c401 ðd1 d3 Þ4 2 2 2 1 where b2 ¼ d1 þd82 þd0 ; b4p¼ 1 þ 5d2 þ d0 þ 2d2 d1 þ ffiffiffi 128 ð5d 4 kp kp 2d2 d0 þ 2d1 d0 Þ; and s0 ¼ 1 ¼ cosð 2 Þ þ i sinð 2 Þ for k = 0, 1, 2, and 3. So idc01 2c01 b2 b4 2 s0 e 1 þ 2 þ 4 þ ; 1 c01 c01 ðd1 d3 Þ4
that is, 0 < x < d; ð7Þ
n2 ð0Þ ic1 02 /; n1 ð0Þ 2 n ðdÞ ic2 02 /; n2 ðdÞ
Applying WKB method [15] to (4), we get its approximate solution as follows: R x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i j2 n2 ðtÞb2 dt e 0 00 /ðxÞ ^c1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 j20 n20 ðxÞ b2 R x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i j2 n2 ðtÞb2 dt e 0 00 þ ^c2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 0 < x < d; ð10Þ 4 j20 n20 ðxÞ b2
ic02 þ ic2 þ
For TE modes, we let qi = 1 (i = 0, 1, 2); for TM modes, let qi ¼ n2i ðxÞ; (i = 0, 1, 2). Outgoing condition is given in the positive and negative directions, that is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ou ¼ i j20 n21 ðxÞ b2 u; x < 0; ð2Þ ox qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ou ¼ i j20 n22 ðxÞ b2 u; x > d: ð3Þ ox
d2 / þ j20 n20 ðxÞ/ ¼ b2 /; 0 < x < d; dx2 d/ðxÞ ¼ ic1 /; x ¼ 0þ ; dx d/ðxÞ ¼ ic2 /; x ¼ d : dx
2.1. The derivation of TE modes
ð8Þ
x ¼ d ; ð9Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c1 ¼ j20 n21 ð0Þ b2 and c2 ¼ j20 n22 ðdÞ b2 : In the next two part, we will give the approximate
1 is0 d b2 b4 a0 c01 4 ðd1 d3 Þ a0 c0 e 1 þ 2 þ 4 þ ; 4 c01 c01 where a0 ¼ id : 2 Since b2 b4 a0 c01 1 þ 2 þ 4 þ a0 c01 e c01 c01 a a a a0 c01 þ 22 þ 33 þ 44 þ M a2 a3 a4 c c c 01 01 01 ¼ a0 c01 þ 2 þ 3 þ 4 þ e ¼ W eW ; c01 c01 c01
J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562 3 where a2 = b2, a3 ¼ 4di ðd2 þ d0 þ d1 Þ; a4 ¼ idb2a4 2a 0 a3 a2 a4 W ¼ a0 c01 þ c2 þ c3 þ c4 þ . 01 01 01 That is,
W eW
a22 2
; and
1 is0 d ðd1 d3 Þ4 : 4 1
It leads to [17] W Lambertðp; is40 d ðd1 d3 Þ4 Þ: We also rewrite the fourth root of 1 as iq1, then is0 = iq for q = 0, 1, 2, and 3. Let m = 4(1 + p) + q, we have 1 iq 4 ðd1 d3 Þ W W m ¼ Lambert p; 4 for p = 1, 2, . . .
557
Let /(x) = n0(x)p(x), then Eq. (15) can be changed to n0xx 2n20x 2 2 2 ð18Þ 2 b p ¼ 0; pxx þ j0 n0 ðxÞ þ n0 n0 where n0xx is represented as n000 ðxÞ, and n0x is represented as n00 ðxÞ: Using c = ln n(x), Eqs. (15)–(17) turn into the following forms: pxx þ ½j20 n20 ðxÞ þ cxx c2x b2 p ¼ 0; 0 < x < d; ic n2 ð0Þ n0x p x ¼ 12 0 p; x ¼ 0þ ; n1 ð0Þ n0 ð0Þ ic2 n20 ðdÞ n0x px ¼ p; x ¼ d : n0 ðdÞ n22 ðdÞ
ð19Þ ð20Þ ð21Þ
Applying WKB method to Eqs. (19)–(21), we get
e
2i
R d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 0
j0 n0 ðxÞþcxx cx b dx
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ðdÞ n ðdÞ 1 0 0:25ðc202 þ cxx ðdÞ c2x ðdÞÞ yðdÞ þ i c202 þ cxx ðdÞ c2x ðdÞ þ n22 ðdÞ n0x0 ðdÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ð0Þ n ð0Þ 0 0:25ðc201 þ cxx ð0Þ c2x ð0ÞÞ1 yð0Þ þ i c201 þ cxx ð0Þ c2x ð0Þ n12 ð0Þ n0x0 ð0Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ð0Þ n ð0Þ 1 0 0:25ðc201 þ cxx ð0Þ c2x ð0ÞÞ yð0Þ þ i c201 þ cxx ð0Þ c2x ð0Þ þ n12 ð0Þ þ n0x0 ð0Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ðdÞ n ðdÞ ; 0 0:25ðc202 þ cxx ðdÞ c2x ðdÞÞ1 yðdÞ þ i c202 þ cxx ðdÞ c2x ðdÞ 2dn2 ðdÞ þ n0x0 ðdÞ 2
It appears that the case m = 0 (i.e., p = 1 and q = 0) does not correspond to an actual leaky mode, and it should be removed. Thus, we can obtain the following asymptotic formulas:
e
(A) Leading term approximation: 2iW c01 ; d
ð13Þ
(B) Fourth term approximation: c01
2i
R d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2
j0 n0 ðxÞþcxx cx b dx
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic2 n2 ðdÞ n ðdÞ 0 c202 þ cxx ðdÞ c2x ðdÞ þ n2 ðdÞ n0x0 ðdÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ð0Þ ð0Þ 0 i c201 þ cxx ð0Þ c2x ð0Þ n12 ð0Þ nn0x0 ð0Þ i
1
W a0 a2 a20 a3 a30 a4 2 3 4 ; a0 W W W
ð14Þ
where Im(c01) > 0 by choosing some integers p.
2.2. The derivation of TM modes
0 < x < d;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ð0Þ n ð0Þ 0 i c201 þ cxx ð0Þ c2x ð0Þ þ n12 ð0Þ þ n0x0 ð0Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ic n2 ðdÞ ðdÞ 0 i c202 þ cxx ðdÞ c2x ðdÞ n22 ðdÞ þ nn0x0 ðdÞ 2
Remark. If n0(x) is a constant, then d0 = 0 and d3 = d2. The result is same as the one in [6].
Starting from d 1 d/ n20 ðxÞ þ j20 n20 ðxÞ/ ¼ b2 /; dx n20 ðxÞ dx d/ n2 ð0þÞ ¼ ic1 0 2 /; x ¼ 0þ ; dx n1 ð0Þ d/ n2 ðdÞ ¼ ic2 0 2 /; x ¼ d : dx n2 ðdÞ
where yðxÞ ¼ 2j20 n0 ðxÞn0 0ðxÞ þ cxxx 2cx cxx . When |b| is large enough, above equation is changed approximately as follows:
ð15Þ ð16Þ ð17Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c202 þ cxx ðdÞ c2x ðdÞ i n20 ðdÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i c202 þ cxx ðdÞ c2x ðdÞ
ðdÞ 2 þ n2icðdÞ nn0x3 ðdÞ 2
0
ðdÞ 2 n2icðdÞ þ nn0x3 ðdÞ 2 0 n20 ðdÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i c201 þ cxx ð0Þ c2x ð0Þ ð0Þ 1 þ n2icð0Þ þ nn0x3 ð0Þ 1 0 n20 ð0Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : i c201 þ cxx ð0Þ c2x ð0Þ ð0Þ 1 n2icð0Þ nn0x3 ð0Þ 1 0 n20 ð0Þ
Expanding the right-hand side of above equation as the inverse power series of c01, we have
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e
2i
J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562
c1 c2 c3 c4 c0 þ þ 2 þ 3 þ 4 þ c01 c01 c01 c01 e1 e2 e3 e4 e0 þ þ 2 þ 3 þ 4 þ c01 c01 c01 c01 1 ¼ c0 e0 þ ðc1 e0 þ c0 e1 Þ c01 1 þ 2 ðc0 e2 þ c1 e1 þ c2 e0 Þ c01 1 þ 3 ðc0 e3 þ c1 e2 þ c2 e1 þ c3 e0 Þ c01 1 þ 4 ðc0 e4 þ c1 e3 þ c2 e2 þ c3 e1 þ c4 e0 Þ þ c01 A1 A2 A3 A4 ¼ A0 þ þ þ þ þ ; c01 c201 c301 c401
R d pffiffiffiffiffiffiffiffiffiffiffi2ffi 0
uðxÞb dx
8 A0 > > > < A2 > A3 > > : A4
¼ c0 e0 ;
A1 ¼ c1 e0 þ c0 e1 ;
¼ c0 e2 þ c1 e1 þ c2 e0 ; ¼ c0 e3 þ c1 e2 þ c2 e1 þ c3 e0 ;
¼ c0 e4 þ c1 e3 þ c2 e2 þ c3 e1 þ c4 e0 and ^a2 ¼ d0 cxx ðdÞ þ c2x ðdÞ; ^b2 ¼ c2x ð0Þ cxx ð0Þ cx(d) = cx(d), cx(0) = cx(0+), cxx(d) = cxx(d) + cxx(0) = cxx(0 ). Further, ^
e2idðc01 þb2 Þ A0 þ
here and
A 1 A 2 A3 A4 þ þ þ : c01 c201 c301 c401
It leads to 2idðc01 þ ^b2 Þ 2ipm þ lnðA0 Þ þ where t1 ¼ AA10 and t2 ¼
2A2 A0 A21 A20
t1 t2 t3 t4 þ 2 þ 3 þ 4 ; c01 c01 c01 c01
:
where uðxÞ ¼ j20 n20 ðxÞ þ cxx ðxÞ c2x ðxÞ;
8 > > > e0 > > > > > > e2 > > > > < e3 > > > > > > e4 > > > > > > > :
n2 ðdÞþn2 ðdÞ
¼ n22 ðdÞn02 ðdÞ ; 2
0
2ic ðdÞn4 ðdÞ 2
2n6 ðdÞc2 ðdÞ
x 2 ¼ ðn2 ðdÞn 2 ðdÞÞ3 þ 2
¼ ¼
TE Example 2
x 2 c1 ¼ ðn2 ðdÞn 2 ðdÞÞ2 ; 0
ð^ a2 d2 Þn20 ðdÞn22 ðdÞ ðn20 ðdÞn22 ðdÞÞ2
0
5
;
icx ðdÞn22 ðdÞ½2d2 n40 ðdÞþ^ a2 n20 ðdÞn22 ðdÞþ^ a2 n42 ðdÞc2x ðdÞn42 ðdÞ ðn22 ðdÞn20 ðdÞÞ3 ð^ a2 d2 Þn20 ðdÞn22 ðdÞ½ð3^ a2 þd2 Þn22 ðdÞð^ a2 þ3d2 Þn20 ðdÞ 4ðn22 ðdÞn20 ðdÞÞ3
þ
c2x ðdÞn42 ðdÞð3d2 n20 ðdÞ^ a2 n22 ðdÞÞ 2ðn22 ðdÞn20 ðdÞÞ3
n2 ð0Þþn2 ð0Þ
;
0
1
2c2x ð0Þn61 ð0Þ
¼ ðn2 ð0Þn2 ð0ÞÞ3 þ 1
¼ ¼
0
3
2
2ic ð0Þn4 ð0Þ
x 1 ¼ n12 ð0Þn02 ð0Þ ; e1 ¼ ðn2 ð0Þn 2 ð0ÞÞ2 ; 1
4
; Im(β/κ0)
8 > > > c0 > > > > > > c2 > > > > < c3 > > > > > > c4 > > > > > > > :
0
ð^ b2 d1 Þn20 ð0Þn21 ð0Þ ðn21 ð0Þn20 ð0ÞÞ2
;
icx ð0Þn21 ð0Þ½^b2 n21 ð0Þn20 ð0Þ^ b2 n41 ð0Þþ2d1 n40 ð0Þþc2x ð0Þn41 ð0Þ 2 ðn1 ð0Þn20 ð0ÞÞ3
1
; 0.7
1
ð^ b2 d1 Þn20 ð0Þn21 ð0Þ½ð3^ b2 þd1 Þn21 ð0Þð3d1 þ^ b2 Þn20 ð0Þ
þ
1.6
1.9
2.2
Re(β/κ0)
4ðn21 ð0Þn20 ð0ÞÞ3 c2x ð0Þn41 ð0Þð3d1 n20 ð0Þ^ b2 n21 ð0ÞÞ 2ðn21 ð0Þn20 ð0ÞÞ3
1.3
Fig. 1. Comparison of the exact and asymptotic propagation constants of leaky modes for TE in Example 2.
;
Table 1 The relative error of asymptotic solution for TE case in Example 1 m
Exact result b
Solution b of (13)
Solution b of (14)
R.E. of (13)
R.E. of (14)
1 2 3 4
31.927284 + 20.670091i 34.471744 + 38.879799i 36.644954 + 56.098296i 38.484651 + 72.821842i
31.922255 + 20.700198i 34.476206 + 38.893398i 36.649356 + 56.105230i 38.488164 + 72.825836i
31.927813 + 20.670035i 34.471769 + 38.879711i 36.644946 + 56.098273i 38.484645 + 72.821836i
8.0254 · 104 2.7544 · 104 1.2257 · 104 6.4580 · 105
1.3986 · 105 1.7605 · 106 3.6342 · 107 1.0302 · 107
Table 2 The relative error of asymptotic solution for TM case in Example 1 m
Exact result b
Solution b of (23)
Solution b of (24)
R.E. of (23)
R.E. of (24)
1 2 3 4
18.928928 + 13.977974i 17.503529 + 29.461225i 16.819679 + 45.520900i 16.516545 + 61.536569i
19.108537 + 13.212853i 17.322568 + 29.150214i 16.681640 + 45.405300i 16.425496 + 61.484485i
18.952827 + 14.025118 17.513796 + 29.453669i 16.820165 + 45.517499i 16.516212 + 61.535407i
3.3400 · 102 1.0500 · 102 3.7102 · 103 1.6463 · 103
2.2463 · 103 3.7200 · 104 7.0794 · 105 1.8972 · 105
J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562
559
it1 ; and Im(c01) > 0 by where K m1 ¼ K m þ ^b2 2dK m choosing some integers m.
Finally, we obtain the following asymptotic formulas. (C) Leading term approximation: c01
3. Numerical results
pm i lnðA0 Þ ^ b2 ¼ K m ; d 2d
ð22Þ
(D) Two terms approximation: c01 K m þ ^ b2
it1 A1 i ¼ Km þ ^ b2 2dA0 K m 2dK m
ð23Þ
and (E) Three terms approximation: c01 K m1
it2 ið2A2 A0 A21 Þ ¼ K m1 ; 2 2dK m1 2dA20 K 2m1
ð24Þ
Example 1. Let n0(x) = 3.3, n1(x) = n2(x) = 3.17, d = 0.2 cm, k = 1.55 lm and j0 ¼ 2p k : Tables 1 and 2 show the effect of the asymptotic approximation for TE and TM cases, respectively. In next examples, since their exact solutions of the propagation constant b can not be obtained directly, they are substituted by well approximated ones, which are solved by some numerical method in a bounded waveguide with
TM Example 2
TM Example 4
10 7 9 6
7
Im(β/κ0)
Im(β/κ0)
8
6
5 4
5
3
4
2
3
1
2
0.5
0.6
0.7
0.8
0.2
0.4
Re(β/κ0)
0.6
0.8
Re(β/κ0)
Fig. 2. Comparison of the exact and asymptotic propagation constants of leaky modes for TM in Example 2.
Fig. 4. Comparison of the exact and asymptotic propagation constants of leaky modes for TM in Example 4.
TE Example 5
TE Example 3 5
3.5
4.5 3
Im(β/κ0)
Im(β/κ0)
4 2.5 2
3.5 3
1.5
2.5
1
2 1.5
0.5 0.6
0.8
1
1.2
1.4
1.6
1.8
Re(β/κ0) Fig. 3. Comparison of the exact and asymptotic propagation constants of leaky modes for TE in Example 3.
0.6
0.7
0.8
0.9
1
Re(β/κ0) Fig. 5. Comparison of the exact and asymptotic propagation constants of leaky modes for TE in Example 5.
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J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562
an appropriate PML. For TE cases, the exact solutions are marked by ‘‘+’’, and the asymptotic solutions from (13) and (14) are marked by ‘‘·’’, ‘‘’’, respectively (see Figs. 1, 3 and 5). For TM cases, the exact solutions are marked by ‘‘+’’, and the asymptotic solutions from (22)–(24) are marked by ‘‘·’’, ‘‘’’ and ‘‘Æ’’, respectively (see Figs. 2, 4 and 6).
TM Example 6 9 8
Im(β/κ0)
7 6
Example 2. Let n0(x) = 3.3(1 0.01((x 1.0)/2.5)2), n1(x) = n2(x) = 3.17, d = 2.0 cm, k = 1.55 lm and j0 ¼ 2p k : Tables 3 and 4 show the effect of the asymptotic approximation for TE and TM cases, respectively.
5 4 3 2 1 0.2
0.3
0.4
0.5
0.6
0.7
Example 3. Let n0 = 3.3(1 0.01((x 1.0)/2.5)2), n1 = 2.10, n2 = 3.17, d = 2.0 cm, k = 1.55 lm and j0 ¼ 2p : Table k 5 shows the effect of the asymptotic approximation for TE case.
Re(β/κ0) Fig. 6. Comparison of the exact and asymptotic propagation constants of leaky modes for TM in Example 6.
Example 4. Let n0 = 3.3(1 0.01((x 1.0)/2.5)2), n1 = 3.17, n2 = 1.0, d = 2.0 cm, k = 1.55 lm and j0 ¼ 2p : Table k 6 shows the effect of the asymptotic approximation for TM case.
Table 3 The relative error of asymptotic solution for TE case in Example 2 m
Exact result b
Solution b of (13)
Solution b of (14)
R.E. of (13)
R.E. of (14)
6 7 8 9 10 11 12 13 14 15
8.405613 + 2.240053i 6.516805 + 3.574983i 4.784198 + 5.83967i 3.858838 + 8.482219i 3.442450 + 10.940137i 3.237798 + 13.191868i 3.129862 + 15.295930i 3.072181 + 17.295962i 3.043025 + 19.220440i 3.031326 + 21.088201i
8.350168 + 2.331169i 6.465522 + 3.708539i 4.779191 + 6.002421i 3.888505 + 8.633277i 3.484863 + 11.079736i 3.286326 + 13.326365i 3.182811 + 15.429381i 3.129131 + 17.430837i 3.104078 + 19.358344i 3.096554 + 21.230264i
8.343727 + 2.2947754i 6.450073 + 3.674752i 4.756127 + 5.980562i 3.868389 + 8.622147i 3.469156 + 11.073391i 3.273898 + 13.32227 i 3.172724 + 15.426509i 3.120756 + 17.428706i 3.096997 + 19.356698i 3.090476 + 21.228954i
0.012261 0.019247 0.0215686 0.016520 0.012721 0.010526 0.009196 0.008334 0.007750 0.0073373
0.00949 0.016148 0.019029 0.015051 0.011849 0.00996 0.008802 0.008047 0.007531 0.007166
Table 4 The relative error of asymptotic solution for TM case in Example 2 m
Exact result b
Solution b of (23)
Solution b of (24)
R.E. of (23)
R.E. of (24)
10 11 12 13 14 15 16 17 18
3.209989 + 8.822371i 2.713684 + 11.238592i 2.431673 + 13.465754i 2.252737 + 15.555120i 2.130175 + 17.546192i 2.041584 + 19.465271i 1.974959 + 21.329954i 1.923269 + 23.152433i 1.882185 + 24.941440i
2.961755 + 8.632321i 2.526157 + 11.132919i 2.289611 + 13.399737i 2.142810 + 15.510877i 2.043540 + 17.515464i 1.972366 + 19.443774i 1.919132 + 21.315314i 1.878023 + 23.143266i 1.845470 + 24.936892i
3.228186 + 8.75776133i 2.730608 + 11.2009311i 2.450432 + 13.4421507i 2.273112 + 15.5397933i 2.151677 + 17.5363799i 2.063823 + 19.4595443i 1.997666 + 21.327581i 1.946302 + 23.153041i 1.905451 + 24.944833i
0.033301 0.018618 0.011448 0.007539 0.005201 0.003703 0.002694 0.001987 0.001479
0.007149 0.003571 0.002203 0.001622 0.001337 0.001173 0.001065 0.000991 0.000940
Table 5 The relative error of asymptotic solution for TE case in Example 3 m
Exact result b
Solution b of (13)
Solution b of (14)
R.E. of (13)
R.E. of (14)
8 9 10 11 12 13
3.720599 + 5.3860768i 2.903792 + 8.3614252i 2.617303 + 10.934894i 2.498615 + 13.234072i 2.448795 + 15.362522i 2.432718 + 17.377481i
3.854492 + 5.563151i 3.015709 + 8.456972i 2.701728 + 11.001253i 2.565028 + 13.288107i 2.503310 + 15.410629i 2.479054 + 17.422706i
3.709002 + 5.443356i 2.903358 + 8.407232i 2.619634 + 10.974873i 2.502269 + 13.271602i 2.453443 + 15.399253i 2.438266 + 17.414362i
0.033912 0.016625 0.009550 0.006357 0.004673 0.003690
0.008927 0.005175 0.003561 0.002799 0.002379 0.002125
J. Zhu et al. / Microelectronics Reliability 48 (2008) 555–562
561
Table 6 The relative error of asymptotic solution for TM case in Example 4 m
Exact result b
Solution b of (23)
Solution b of (24)
R.E. of (23)
R.E. of (24)
9 10 11 12 13 14
2.666497 + 5.4977708i 1.830561 + 8.5207237i 1.502239 + 11.089345i 1.330306 + 13.379734i 1.225011 + 15.500628i 1.154245 + 17.509487i
2.367969 + 5.134397i 1.611944 + 8.380557i 1.3493853 + 11.012340i 1.2161435 + 13.329667i 1.1355542 + 15.465301i 1.0817335 + 17.483593i
2.887592 + 5.427700i 1.967417 + 8.474227i 1.603880 + 11.058126i 1.411039 + 13.357143i 1.291413 + 15.483693i 1.210118 + 17.496759i
0.076964 0.029797 0.015294 0.009271 0.006185 0.004387
0.037957 0.016584 0.009501 0.006235 0.004407 0.003265
Table 7 The relative error of asymptotic solution for TE case in Example 5 m
Exact result b
Solution b of (13)
Solution b of (14)
R.E. of (13)
R.E. of (14)
8 9 10 11 12 13 14
2.871394 + 8.343677i 2.614595 + 10.938301i 2.473316 + 13.233021i 2.442923 + 15.357283i 2.415184 + 17.382631i 2.423352 + 19.305651i 2.437238 + 21.193212i
2.997865 + 8.453837i 2.68599 + 10.999956i 2.55062 + 13.287519i 2.489778 + 15.410375i 2.466124 + 17.422629i 2.463085 + 19.356486i 2.472254 + 21.232342i
2.884520 + 8.404006i 2.603262 + 10.973558i 2.487409 + 13.271006i 2.439559 + 15.398993i 2.425055 + 17.414280i 2.428744 + 19.350093i 2.443035 + 21.227289i
0.019007 0.008388 0.007026 0.004554 0.003691 0.003316 0.002461
0.00699 0.003292 0.00300 0.002690 0.001889 0.002300 0.001601
Table 8 The relative error of asymptotic solution for TM case in Example 6 m
Exact result b
Solution b of (23)
Solution b of (24)
R.E. of (23)
R.E. of (24)
9 10 11 12 13 14
2.660849 + 5.4865224i 1.808461 + 8.4977042i 1.487248 + 11.094186i 1.317280 + 13.369376i 1.209966 + 15.502616i 1.143919 + 17.503316i
2.343885 + 5.125013i 1.593123 + 8.377984i 1.3333623 + 10.111698i 1.2016354 + 13.328997i 1.1219826 + 15.464866i 1.0687936 + 17.483289i
2.86505382 + 5.416793i 1.94832151 + 8.470549i 1.587446 + 11.056353i 1.396160 + 13.356098i 1.277524 + 15.483004i 1.196907 + 17.496271i
0.078846 0.028358 0.015620 0.009117 0.006157 0.004432
0.03538 0.016398 0.009568 0.005954 0.004523 0.003047
Example 5. Let n0 = 3.3(1 0.01((x 1.0)/2.5)2), n1 = 2.10(1 0.01((x 1.0)/2.5)2), n2 = 3.17(1 0.01((x 1.0)/ 2.5)2), d = 2.0 cm, k = 1.55 lm, and j0 ¼ 2p : Table 7 shows k the effect of the asymptotic approximation for TE case. Example 6. Let n0 = 3.3(1 0.01((x 1.0)/2.5)2), n1 = 1.0(1 0.01((x 1.0)/2.5)2, n2 = 3.17(1 0.01((x 1.0)/ 2.5)2, d = 2.0 cm, k = 1.55 lm and j0 ¼ 2p : Table 8 k shows the effect of the asymptotic approximation for TM case. 4. Conclusions In order to compute the leaky modes in the optical waveguide where the refractive index is varied with the transverse variable, we derive some asymptotic analytic solutions. Numerical results illustrate that these solutions are very close to the exact ones as the norms of the modes become larger. Our results are useful in the eigenmode expansion method [16] when PMLs are used. In this method, it is necessary to compute the perturbed leaky modes and the Berenger modes. Our asymptotic analytic solutions can reduce the effort of computing these modes.
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