High-precision Berenger modes of dual-layer micro-waveguides terminated with a perfectly matched layer for on-chip optical interconnections

Microelectronics Reliability 53 (2013) 1164–1167

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Research note

High-precision Berenger modes of dual-layer micro-waveguides terminated with a perfectly matched layer for on-chip optical interconnections Jianxin Zhu a,⇑, Zhaochen Zhu b a b

Department of Mathematics, Zhejiang University, Hangzhou 310027, China School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China

a r t i c l e

i n f o

Article history: Received 22 August 2012 Received in revised form 25 February 2013 Accepted 26 April 2013 Available online 20 May 2013

a b s t r a c t In this paper, some high-precision Berenger mode solutions of dual-layer micro-waveguides terminated by a perfectly matched layer are derived. Numerical examples show that these asymptotic solutions are very close to their exact ones. They can act as initial guesses for solving the optical dispersion equation by Newton’s method. This method is useful and simplifies the micro on-chip optical waveguide design. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction To reduce the power dissipation of signal transmission and to increase the bandwidth of very large scale of integrated circuits, optical interconnects has been proposed to be an attractive technological option for future microelectronics [1,2]. A micro-waveguide based on oxynitride materials is usually regarded as a dual-layer step-index and unbounded region. In practical numerical computation, the unbounded waveguide can be considered as an approximately terminated bounded region by a perfectly matched layer (PML) [3]. In this structure, three kinds of modes, such as propagating modes, leaky modes and Berenger modes, appear [3–7]. The behavior of Berenger modes mainly depends on the characteristics of the PML. Furthermore, leaky modes and Berenger modes play an important role since they can be used to partially represent the optical wave related to the continuous spectrum of the radiation and evanescent modes [8]. To design a high-performance device, we require the information of high precision distribution of these modes. There were many research works on the optical characteristics and applications having been further reported [9–14]. However, it is difficult to find out the solutions using existing numerical solvers [15,16]. Therefore, propagating modes, leaky modes were also studied in Refs. [17,6,16], respectively. High precision modes were also given in these studies. Although some approximated Berenger modes are also given [6], their precisions are too low to estimate accurately the optical wave field of the micro-waveguide for microelectronic on-chip integration applications. In this paper, high-precision Berenger modes are derived for two-layer micro-waveguides terminated by a PML. Numerical sim-

ulations for both TE and TM cases show that our results on Berenger modes are much better than the ones obtained by previously [6]. 2. Mathematical model of optical wave propagation For a dual-layer planar waveguide shown in Fig. 1, let the refractive index satisfy n(x) = n0 as 0 < x < d and n(x) = n1 as x > d. We start from the governing equation (Helmholtz equation ) of optical wave propagation as follows:

q

    @ 1 @u @ 1 @u þq þ j20 n2 ðxÞu ¼ 0 for x > 0; @z q @z @x q @x

x!þ1

E-mail address: [email protected] (J. Zhu). 0026-2714/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.microrel.2013.04.008



1 for TE case ; j ¼ 2p is called the vacuum n2 ðxÞ for TM case 0 k0 wavenumber, and k0 = 1.55 lm. Since the domain of the problem in (1) is an open region, it cannot be solved by normal numerical methods. Thus, for an open waveguide, we terminate it by a finite boundary x = D, and put a perfectly matched layer (PML) between x = H and x = D to eliminate the wave reflection [2]. These treatments make the problem (1) be approximated by: where q ¼

8 u þ uxx þ j20 n20 u ¼ 0; > > > zz n o > > @ 1 @u > þ j20 n21 u ¼ 0; < uzz þ 1þi1rðxÞ @x 1þirðxÞ @x > uð0; zÞ ¼ 0 ðfor TEÞ; > > > > > : ujx¼d ¼ ujx¼dþ ;

in which ⇑ Corresponding author.

ð1Þ

lim u ¼ 0;

rðxÞ ¼



@u j @x x¼0

0 < x < d; d 6 x < D;

¼ 0 ðfor TMÞ; uðD; zÞ ¼ 0;  1 @u 1 q @x  ¼ q x¼d



@u ; @x  x¼dþ

ð2Þ Ct 3 1þt 2

0;

xH where t ¼ DH ;

H6x6D . d
We discuss the characteristic problems of both TE and TM cases of Eq. (2) in Sections 3 and 4, respectively.

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J. Zhu, Z. Zhu / Microelectronics Reliability 53 (2013) 1164–1167

Berenger Modes for TM Case

0

25

n=n 0 d

20

0

Im (β/κ )

n=n1 H

PML

D

15

10

Fig. 1. A dual-layer waveguide terminated by a PML.

5

0

Berenger Modes for TE Case

0

5

10

Fig. 3. Comparisons of the exact and approximate Berenger modes for TM case.

Im (β/κ 0 )

20

where Im(W1) > 0, and

15

8 d1 b  dÞ; a2 ¼ d1 ; > a ¼ ið D a3 ¼  4a ; > 4 > < 2

d1 1 2 2 2  d ; d ¼ j a4 ¼ 4a 1 2 0 n0  n1 ; 16 1     > > p ffiffiffiffiffi > b  dÞ d1 ; m P 0 except LambertW 0; i  ð D b  dÞpffiffiffiffiffi : W 1 ¼ LambertW m;  i  ð D d1 : 2 2

10

5

0

In fact, formula (D1) is same as the one given in Ref. [6].

0

5

10

4. Berenger modes for TM case

15

Re (β/κ 0)

Consider the characteristic problem of Eq. (2) for TM case:

Fig. 2. Comparisons of the exact and approximate Berenger modes for TE case.

3. Berenger modes for TE case

1þirðxÞ dx

8 d2 /ðxÞ 2 2 > 2 þ j0 n0 /ðxÞ ¼ k/ðxÞ; > > > dx   > > 1 < 1 d  d/ þ j2 n2 /ðxÞ ¼ k/ðxÞ; 1þirðxÞ dx

1þirðxÞ

0 < x < d; d < x < D;

0 1

dx

> > /ð0Þ ¼ 0; /ðDÞ ¼ 0; > >  > >  : /ðxÞjx¼d ¼ /ðxÞjdþ ; d/ðxÞ dx 

x¼d

  ¼ d/ðxÞ dx 

x¼dþ

1

ð3Þ

;

1

ð5Þ

W 1 aa2 a2 a3 a3 a4  2 3  4 a W1 W1 W1

lim

x!dþ

d/ðxÞ ; dx

ð7Þ

1

Let je2ic0 d j  1 or 6. That is, n2



1 þ n02 T 1 1

n2

ð8Þ

:

1  n02 T 1 1

8   bS 20 > 2 2 > > ðLTÞ : kð0Þ ; if omitting O c12 ; > m  j0 n1 þ b > 1 4ð D dÞ2 > > >  2 , >   > < 2 ð1Þ 2 2 1 1 b b b km  j0 n1 þ S 0 þ S 1 2 2 ð0Þ if omitting O c4 ; ½4ð D  dÞ ; j0 n1 km 1 > > > > " #2 , > > h i   > > 2 ð2Þ 2 2 1 b b b  dÞ ; if omitting O 16 ; > 4ð D þb S 2 2 2 1 ð1Þ 2 > : ðTTÞ : km  j0 n1 þ S 0 þ S 1 j2 n2 kð1Þ c1 m ðj n k Þ 0 1

 2 W1  ; a

ð3Þ km  j20 n21 

¼

1 n21

By the derivation [18], we obtain the following formulae for the TM case:

Similar to the derivation in [18] by an inverse power series, we obtain the following formulae:

j

1 lim d/ðxÞ n20 x!d dx

    n2 n2 b 1 þ n02 T 1  1  n02 T 1 e2ic1 ð D dÞ 1 1 ¼    : n2 n2 1  n02 T 1  1 þ n02 T 1 e2ic1 ðbD dÞ 1

ð4Þ

1  T1 b  e2ic1 ð D dÞ : 1 þ T1

2 2 0 n1

e2ic0 d

e

Let je2ic0 d j  1 or 6. That is,

kð0Þ m

d < x < D;

0 1

dx

b are same as the ones defined in Here the notations c0, c1, T1 and D Section 3. Also, it leads to the following nonlinear equation for the eigenvalue:

2ic1 ðb D dÞ

b ðT 1  1Þ þ ðT 1 þ 1Þe2ic1 ð D dÞ ¼ : ðT þ 1Þ þ ðT  1Þe2ic1 ðbD dÞ

1þirðxÞ

0 < x < d;

ð6Þ

where k is an eigenvalue, and /(x) is the corresponding eigenfunction. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ Dþ Denote c0 ¼ j20 n20  k; c1 ¼ j20 n21  k; T 1 ¼ cc1 and D RD 0 i d rðsÞds. It leads to the following nonlinear equation for the eigenvalue:

e

8 2 d /ðxÞ > > þ j20 n20 /ðxÞ ¼ k/ðxÞ; > > dx2 >   > > 1 < 1 d  d/ þ j2 n2 /ðxÞ ¼ k/ðxÞ; > > lim /ðxÞ ¼ limþ /ðxÞ; > > > x!d x!d > > : 0 / ð0Þ ¼ 0; /ðDÞ ¼ 0:

Consider the characteristic problem of Eq. (2) for TE case:

2ic0 d

15

Re (β/κ 0)

25

ðD1Þ

!2 ;

ðD2Þ

0 1

m

where

8 n2 n2 b > S 0 ¼ ln n02 þn12 þ ð2m þ 1Þpi; > > 0 1 > > < j2 n20 n21 b S 1 ¼  n02 þn 2 ; 1 0 > > > 4 2

> j n n2 > :b S 2 ¼ 14 20 0 21 2 n40  3n41 ; and m P 0: ðn1 þn0 Þ

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J. Zhu, Z. Zhu / Microelectronics Reliability 53 (2013) 1164–1167

Table 1 Comparisons of relative errors of Berenger modes for TE case. m 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8

bm 5.418013367 + 12.38601825i 4.016623836 + 3.742221989i 10.60866744 + 25.40407804i 7.747435534 + 19.27860106i 16.91132683 + 37.49805387i 13.73440243 + 31.48694901i 23.39773729 + 49.35908040i 20.13605908 + 43.44871050i 29.99946148 + 61.09933391i 26.68751086 + 55.24028441i 36.67414957 + 72.76878978i 33.32937953 + 66.94103001i 43.39929210 + 84.39193618i 40.03140189 + 78.58512566i 50.16141393 + 95.98254180i 46.77635647 + 90.19068917i

ð0Þ bm

4.973265185 + 12.55321280i 3.264678888 + 4.871980264i 10.65847526 + 25.39810598i 7.689893163 + 19.14393292i 16.92222731 + 37.50212538i 13.75212981 + 31.49501066i 23.40404248 + 49.36149138i 20.14427431 + 43.45169894i 30.00348997 + 61.10098155i 26.69248815 + 55.24225952i 36.67694204 + 72.76998854i 33.33270595 + 66.94242625i 43.40133969 + 84.39284838i 40.03377895 + 78.58616638i 50.16297860 + 95.98325966i 46.77813841 + 90.19149543i

bð3Þ m

RE0m

R E3m 1

4.875000341 + 12.52545985i 2.919120436 + 4.642459230i 10.63339738 + 25.39042844i 7.645623391 + 19.13140505i 16.91102414 + 37.49819640i 13.73602289 + 31.48969937i 23.39773883 + 49.35906756i 20.13603952 + 43.44866167i 29.99946102 + 61.09932928i 26.68751027 + 55.24027759i 36.67414938 + 72.76878754i 33.32937925 + 66.94102684i 43.39929202 + 84.39193497i 40.03140177 + 78.58512403i 50.16141389 + 95.98254108i 46.77635641 + 90.19068825i

0.351 10 0.247 100 0.182 102 0.705 102 0.283 103 0.567 103 0.124 103 0.183 103 0.640 104 0.873 104 0.373 104 0.482 104 0.236 104 0.294 104 0.159 104 0.193 104

0.415 101 0.259 100 0.103 102 0.861 102 0.813 105 0.930 104 0.237 106 0.110 105 0.684 107 0.112 106 0.276 107 0.426 107 0.128 107 0.185 107 0.666 108 0.907 108

ð2Þ bm

RE0m

RE2m

Table 2 Comparisons of relative errors of Berenger modes for TM case. m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

bm 4.952610002 + 1.010470789i 4.756669069 + 7.237545340i 8.249078822 + 13.88310833i 11.63864020 + 19.88019652i 15.07649409 + 25.73942311i 18.53510394 + 31.54107252i 22.00563793 + 37.31220459i 25.48372356 + 43.06531316i 28.96688617 + 48.80689566i 32.45362593 + 54.54066235i 35.94298076 + 60.26888607i 39.43430514 + 65.99303669i 42.92714979 + 71.71410681i 46.42119239 + 77.43279085i 49.91619569 + 83.14958918i 53.41198122 + 88.86487148i

bð0Þ m 4.874025120 + 0.569126189i 5.198575506 + 7.574155175i 8.323534447 + 13.92763454i 11.67945949 + 19.90082849i 15.10094996 + 25.75242893i 18.55133092 + 31.54990925i 22.01719472 + 37.31860167i 25.49237057 + 43.07015997i 28.97359844 + 48.81069522i 32.45898696 + 54.54372116i 35.94736098 + 60.27140160i 39.43795100 + 65.99514192i 42.93023158 + 71.71589457i 46.42383156 + 77.43432794i 49.91848118 + 83.15092488i 53.41397961 + 88.86604293i

In fact, the formula (LT) is same as the one given in Ref. [6]. 5. Numerical simulations Suppose the first layer a silicon nitride with refractive index n0 = 2.1 and the second layer is silicon oxide with refractive index n1 = 1.45 [2], d = 0.4 lm, D = 0.8 lm, H = 0.7 lm, and C = 16. Due to qffiffiffiffiffiffiffi pffiffiffi kð0Þ the propagation constant b ¼ k, we denote bð0Þ m ¼ m ; qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi ð3Þ kð2Þ kð3Þ bð2Þ m ¼ m , and bm ¼ m . The exact propagation constant is   ð2Þ  denoted as b⁄. And RE0m ¼ jbm  bð0Þ m j=jbm j; RE2m ¼ jbm  bm j=jbm j,  and RE3m ¼ jbm  bð3Þ m j=jbm j are represented as the relative errors ð2Þ ð3Þ of bð0Þ m ; bm and bm , respectively. In Figs. 2 and 3, the exact solutions b⁄ are all marked by ‘‘+’’ for both TE and TM cases. In Fig. 2, approximate solutions obtained by formulae (D1) and (D2) for TE case are marked by ‘‘h’’ and ‘‘ ’’, respectively. In Fig. 3, approximate solutions obtained by the formulae (LT) and (TT) for TM case are also marked by ‘‘h’’ and ‘‘ ’’, respectively. As shown in Figs. 2 and 3, and Tables 1 and 2, formulae (D2) and (TT) require a high order approximation for the eigenvalues when they are large. The numerical results conform with the asymptotic analysis of Berenger modes.

6. Conclusions For the Berenger modes in a dual-layer optical micro-waveguide terminated by a PML, we have derived some high-precision

1.368310151 + 0.055953517i 4.951136145 + 7.461324725i 8.241192858 + 13.88738009i 11.63853686 + 19.87976495i 15.07651495 + 25.73942715i 18.53510228 + 31.54107467i 22.00563737 + 37.31220482i 25.48372336 + 43.06531326i 28.96688608 + 48.80689571i 32.45362588 + 54.54066237i 35.94298073 + 60.26888609i 39.43430512 + 65.99303670i 42.92714978 + 71.71410681i 46.42119238 + 77.43279085i 49.91619569 + 83.14958919i 53.41198122 + 88.86487148i

1

0.887 10 0.641 101 0.537 102 0.199 102 0.929 103 0.505 103 0.305 103 0.198 103 0.136 103 0.973 104 0.720 104 0.548 104 0.426 104 0.338 104 0.273 104 0.223 104

0.314 100 0.342 101 0.555 103 0.193 104 0.712 106 0.742 107 0.140 107 0.447 108 0.181 108 0.849 109 0.514 109 0.291 109 0.120 109 0.111 109 0.103 109 0.000 100

asymptotic solutions. The derivation follows a systematic approach that makes use of inverse power series of c1, assuming that the norm of c1 is large. The precision of the asymptotic formulas increases when the norm of c1 is increased. Numerical results also illustrate that these solutions are very close to the exact ones as the norms of the modes turn larger. Furthermore, our results are better than the ones solved by the formulae reported in [6]. Therefore, our results are very useful to compute accurately the optical wave field of the microelectronic devices by the eigenmode expansion method [19] when a PML is used. At last, the design of microelectronic devices will be improved and their reliability will be promoted. Acknowledgements This research was supported by the Natural Science Foundation of China (NSFC) under the Grant (No. 11071217) and the Key Project of the Major Research Plan of NSFC (No. 91130004). References [1] Wong H, Filip V, Wong CK, Chung PS. Silicon integrated photonic begins to revolution. Microelectron Reliab 2007;47(1):1–10. [2] Wong CK, Wong H, Chan M, Chow YT, Chan HP. Silicon oxynitride integrated waveguide for on-chip optical interconnects applications. Microelectron Reliab 2008;48(2):212–8. [3] Berenger JP. A perfectly matched layer for the absorption of electromagneticwave. J Comput Phys 1994;114(2):185–200. [4] 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.

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