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Title An efficient finite-element analysis of magnetooptic channel waveguides

Author(s) Koshiba, M.; Zhuang, X.P.

Citation Journal of Lightwave Technology, 11(9), 1453-1458https://doi.org/10.1109/50.241935

Issue Date 1993-09

Doc URL http://hdl.handle.net/2115/6039

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Type article

File Information JLT11_9.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

JOURNAL OF liGHTWAVE TECHNOlOGY. VOL 11 . NO. 9. SEPTEMBER 1993 1453

An Efficient Finite-Element Analysis of Magnetooptic Channel Waveguides

Masanori Koshiba, Senior Member, IEEE, and Xiu-Ping Zhuang

Abstract-A finite-element method based on the scalll l"-wave approximation is developed for the analysis of magnetooptic waveguides. A si mple and efficient itentive method is proposed for soh'ing a nonli near eigenvalue equation derived from the scalar fi nite-element approach. To show the validity and useful­ness of this method, examples are computed for magnetooptic rib-type and ridge-type waveguides. Subsequently, we discuss the waveguide structures which have larger nonreciprocal phase shift.

I. INTRODUCTION

A MAGNETOOPTIC waveguide is one of the key elements in nonreciprocal devices such as isolators and circulators.

Theoretical studies on the nonreciprocity of magnetooptic waveguides have mainly focused on planar (two-dimensional) waveguides (lH4] . lWo-dimensional waveguides can trap optical fi elds in the direction of the thickness (y direction), but allow the fie lds to spread in the horizontal direction (x direction). In order to faci litate the construction of integrated nonreciprocal devices, channel (three-dimensional) waveg­uides, which trap optical fields in both x and y directions, are more importanl. It is, in general, difficull 10 analyze three-dimensional waveguides with nonreciprocal properties, and approximate analytical methods, such as the Marcat ili method [5J and the effect ive index method have been used [6]-[8].

In Ihis paper, a new numerical solution method, which is more accurate and can be applied to various magnctooptic channel waveguides, is developed. This approach is based on the fi nite-element method and the scalar-wave approximation [9]- [J 1]. A simple and efficient iterative method is proposed for solving a nonlinear eigenvalue equation derived from the scalar fini te-element approach. The validity and usefulness of this method are confirmed by analyzing the magnetooptic rib­type and ridge-type wavegu ides. We also discuss possible ways to get larger nonreciprocal phase shift. Because the formula tion is based on the scalar-wave approximation, spurious solutions that are included in the vector fi nite-element method (12) do not appear.

Manuscripl r«eived AuguSI 14, 1992; revised February 5. 1993. This work was i»'rtially supported by Grant- ln·Aid for Scientific Researcll on Priority Area, Ultrafast and UlLra ·ParalJet Optoelectronics from tile Ministry of Education. Science and Culture of Japanese Government.

The aulho~ are .... illltlle Department of Elccl ronic Engineering, Hollaido University, Sapporo, 060 Japan.

IEEE Log Number 9210005.

11 . B ASIC EQUATIONS

With a time dependence of the form cxp (jwt) being im­plied, Maxwell 's equations are

V' x E = - jW/11JH

V' x H = jW(o[(r]E

\J · H = O

\J . ([,,·IE ) = 0

(1)

(2)

(3)

(4)

where W is the angu lar frequency, E and H are the electric and magnetic fi elds, respectively, (0 and ~o are the permitt ivity and permeabili ty of free space, respectively, and [frl is the relative permittiv ity tensor.

We consider magnetooptic channel waveguides as shown in Fig. 1, where all the materials are assumed to be lossless, light propagates along the z direction, and the dc magnetic field is applied in the x direction. The relative pcrmiltivity tensor of the magnetooplic material can be wrillen as

o n' , -;6

(5)

where 11.." 11. .. , and n~ are the refractive indexes in the x, y, and z directions, respectively, and {j represents the first-order magnetooptic effect, which causes nonreciprocal nature and is related to the Faraday rotation.

Rewriting (1}-{4) in component form, we have

8E, ·"E . H 8y + )JJ,., • II = -JWJ1.Q :z:

·"E aE, . H - JlJjJ ., - ax = -JWIJ,o ..

. AH aH, . 'E ' E - )VI-' ., - -- = ]wt"on .. v - w€ou z 8x

(6)

(7)

(8)

(9)

(10)

(11)

0733-8724/93$03.00 ~ 1993 tEEE

14~4 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL 11, NO.9, SEPTEMBER 1993

I- I/ . ! , ,

I ' 1/ • I c where (f is given by •

~ , • 1',1

, ~ , • , n,

(19)

~ , • , • Generally, a waveguide fo r the optical integrated circuit will , •

I • , I I

(.)

II.[

! • , (b)

• I I

• support the propagation of waves having two possible fi eld configurations, classified as the EZ and Ell modes [5), which are well approximated by the TE (Ell = 0, a leading function is E z) and TM (HII = 0, a leading function is Hz) modes, respectively. Substituting (17) and (18) into (9) and neglecting the terms of El" we obtain the following basic equation f9r

I • • I (

, ~ •

, ~

• n,

I!,) , r. II.[, I!,) ,

~ , • ~

I' II ' 1 • , • •

, , , ,

the E Z modes: 1

n~82Ez o'lE:s; /.I2 E ,"E - O --, a' + a 2 -,., z+ on:s; :s;-n" x y

(20)

where ko is the wavenumber of free space and is given by ko = 211" / A with A being the wavelength of free space.

, • I • I I I

• ' 1

Substituting (15) and (16) into (6) and neglecting the terms • of Hil I we obtain the fo llowing basic equation for the Ell

modes: , , (0) (d)

Fig. 1. Magnetooptic channel waveguide. (a) Magnetooptic rib waveguide. (b) Rib waveguide on a magnetooptic substrate. (e) Magnetooptic ridge waveguide with an isotropic loading layer. (d) Magnetooptic ridge waveguide with a magnetooptie loading layer.

8H..., 8HII . f.lH 0 - + -- - JV~ ,,= 8x oy (12)

,aE. a( 'E .') n z ox + 8y nil l' + Ju E"

- jvtJ( -joEl' + n~E,,) = 0 (13)

where tJ is the phase constant in the z direction and v is a directional parameter given by

{ + 1 for + z propagation v -

- - 1 for - z propagation. (14)

Assuming 0 « n~ (i = x , y, z) and 60/ ox ::::: 0, from (7). (8), and (IO}-{13) we obtain

E ~ _1_ [_Vfin; H. + • aH. n; ~ (aH. + aH,)] 11 Wfo (f (f 8y IItJq ox ox 8y

(15)

(16)

(17)

(18)

n~ 82 Hz + ~ (n~ 8Hz _ IItJ 6 Hz)

(J 8x2 8y (f 8y (1

f.l o 8Hz /.I2n~H k'H - 0 +"''''--8 -" - ..., + 0 z- . q y q (21)

The functiona ls are given by

F ~ J' { (n; aE; aE. + aE; aE. 10 n~ 8x ax oy 8y

+f32 E;Ez - k5n;E;Ez) dxdy (22)

for (20), and

F ~ J' { [n; aH; aH. + aH; (n~ aH. _ vfi S H.) In q ax ox 8y (f 8y (1'

-11/3 0 HO 8 Hz + /32 n~ HO Hz ~ k2 H* Hz] dxdy (23) (1z8y (1z O:s;

fo r (21). Here n is the cross section of the waveguide and asterisk denotes complex conjugate.

III . F INITE-ELEMENT APPROACH

Dividing the waveguide cross section n into a number of quadratic triangular elements [11], we expand the electric field Ez and the magnetic fie ld Hz in each element as

E. ~ {N)T{ E.). H. ~ {N)T {H.).

(24)

(25)

where {Ez} e and {Hz}" are the nodal electric and magnetic fie ld vectors for each element, respectively, {N} is the shape funct ion vector for the quadratic triangular element, and T denotes a transpose.

Substituting (24) into (22) and using the variational princi­ple, we obtain the following eigenvalue equation for the EZ modes:

[K]{E.) - n;.IM ]{E.) ~ {OJ (26)

ZHUANO AND KOSHIBA: AN EFFICIENT FINITE-ELEMENT ANALYSIS

where {Ex} is the global electric field vector, {O} is a null vector, neff = 131 ko is the effective refractive index, and the submatrices of [K] and [M] are given by

[K) ~ I: j" r [n;{N}{N }T _ n; 8{N} 8{N}T J~ n z Ox Ox

o

8{N} 8{N}T] ~~ ~!:l-= !:l-= uxuy

uy uy (27)

[M) ~ ;;: f1 {N}{N}T dXdy. (28)

Here x = kox, Y = koy, and the summation L~ extends over all different elements.

Similarly, substituting (25) into (23), and using the varia­tional principle, we obtain the following eigenvalue equation for the Ell modes:

[K(n •• )j{H} - n:. [Mj{H.} ~ {O} (29)

with

[K(n •• ») ~ I:j" r [{N}{N}T _ n; 8{N } 8{N}T J~ 0" Ox Ox o

_ n; 8{N} a{N}T 0" au ay

+ vn •• 6 ({N) 8{N}T + 8{N} {N}T)] a fry fry

. axdfi (30)

[M ) ~ I: 11 ~ {N}{N}T dXdy. (31) o

IV. METHOD OF NUMERICAL CALCULATION

We define the normalized nonreciprocal phase shift ¢ as follows [7], [8] :

(32)

where I is a waveguide length, and the retardations ¢/ and ¢", respectively, for the +z and -z propagations arc given by

2,1 <P/ = T(n/x - n/lI ) (33)

2,1 ¢b = T(nbx - nby). (34)

Here nIx , n/ ll are the effective refractive indexes of the fundamental get (Efd, Ell {Efd modes in the case of +z propagation, respectively, and nbx, nby are the effective refractive indexes of the Eft, Eft modes in the case of ~z propagation, respectively.

Substituting (33) and (34) into (32) and noting that the effective refractive indexes of the Eft mode propagating in the

1455

+z and - z directions are the same (n/ x = nbx), we obtain

2, ¢ = -;x{nbll ~ N/y ). (35)

Although (26) for the EX modes is a linear generalized eigenvalue equation, (29) for the EY modes is a nonlinear generalized eigenvalue equation. Hence, we use the following iterative scheme.

(i) Specify..\, nx, ny, nz, and 0 as input data and calculate the coefficient matrix 1M]. (ii) Assign initial value to n~ff in an arbitrary way. A convenient way to choose this value is 10 use that for the 0 = 0 case; we adopt this way in the present paper. (iii) Calculate the nonlinear coefficient matrix [K{n~fd] . (iv) To obtain a new value of neff, solve the eigenvalue equation (29). (v) Iterate procedures (iii) and (iv) until the solution (eigenvalue, neff) converges within the desired criterion.

Fig. 2 shows the flowchart of the iterative process, where ..6. is the value for judging the convergence. In this calculation we set ..6. = 10- 10 , and the convergent solution is obtained within four or five iterations.

V. NUMERICAL RESULTS AND DISCUSSION

We consider the magnetooptie channel waveguides as shown in Fig. 1, where the wavelength ..\ is 1.152/'Lm and the refractive indexes of a substrate and a top layer are n~ = 1.95 and nc = 1.0, respectively. The refractive index no: = n1l = n z ~ n and the off-diagonal component of the relative permittivity tensor 0 of magnetooptic materials are given in Table I [2], [7], [8], (13]. For simplicity, we assume the artificial boundary walls x = ±X /2, y = - Y~, and y = t + Yc (magnetooptic rib waveguides) or y = t+h+Yc (magnetooptic ridge waveguides) far from the core region.

A. Magnerooptic Rib Waveguides

We consider the magnetooptic rib waveguides as shown in Fig. l(a) and (b), where the rib width Wand rib height dare 311m and 12 nm, respectively, X = 20.6/'Lm, Y& = 1.9 11m, and Yc = 0.5 11m. A magnetooptic material is used as a guided layer [7]. [8] and a substrate [4), [14] in Fig. l(a) and (b), respectively.

The magnitude of the nonreciprocal phase shift I¢I as a function of LaGa:YIG film thickness is shown in Fig. 3(a) by the solid line. The results obtained agree approximately with the experimental results [7]. The results of the planar wave­guide (d = 0) arc also shown in Fig. 3(a), by the dotted line. For larger values of t, the rib waveguide structure considered here is almost like the planar structure, and therefore the value of I¢[ for the three-dimensional waveguide approaches that for the two-dimensional waveguide. For smaller values of t, on the other hand, the results of the two-dimensional waveguide deviate from those of the three-dimensional waveguide. The value of I¢I as a function of YIG, Bi:YIG, or Bi:GdIG film thickness is shown in Fig. 3(b). The value of [¢[ becomes larger with an increase of the Faraday rotation coefficient 0. Chen and Kumarswami [3] have reported that the guided layer thickness to give the maximum nonreciprocal phase

1456

i = i + 1

"

Material LaGa:YIG

YIG Bi:YIO

6i:GdlG

Ce:YIO

start

O=-O

solve eq.(29)

nL = n."

i =- I

v=-I v =-1

calculate calculate (K(n .1/ = n;,o)] [K("'JI = ,,;,0)]

solve eq.(29) solve eq.(29)

i=i+l

n, = ne/I ", = n.//

" n;+! _ ";1 < /l nHI-n, i</l

,~ ,~

n/r =- " ;+1 n lr = n;+,

4> = ¥-(n lr - n/,)

stop

Fig. 2. Calculation of the E Y modes.

TABLE J MAGNETOOPTLC MATEllIAL CONSTANTS

0 , 2, 18 3.2 x 10 - 4

2.18 3.4 x 10 - 4

2.25 - 8.9 x 10- 4

2.40 -4.3 x 10 - 3

2.23 0.019

shift for lanar magnetooptic waveguides is approximated by kot n2 - n; '" 2.2. Using this relation, the values of t become OAl/~m, 0.361J,m, and 0.29 11m for YIG, Bi:YIG, and Bi :GdIG, respectively. These values approximately coincide with those in Fig. 3(b).

JO URNAL OF LIGHTWAVE TECHNOLOGY. VO L. I I, NO. 9, SEPTEMBER 1993

"',---------, • E~pe ri me nt

• - 0.5

, 0 '" ,. 0

I ( I'm)

(0)

" Bi:Gd!G

-• - "

o 0.4 0.' t (I'm)

(b)

Fig. 3. Nonreciprocal phase shift of a magnctooptic rib waveguide as a function of guided layer thickness. (a) LaOa:YIO film. (b) YIO, Bi:YIG, or Bi:GdIG film.

The magnitude of the nonreciprocal phase shift of a rib waveguide on a YIG substrate as a funct ion of guided layer thickness is shown in Fig. 4(a). The difference between the refractive indexes of a YlG fi lm and an isot(()pic substrate in Fig. 3(b) is 0.23 . The same index difference is obtained for n, = 2.41 in Fig. 4(a). Comparing these two cases, we can find that the nonreciprocal phase shift becomes larger for the structure using a magnetooptic material as a substrate. Furthermore, the nonreciprocal phase shift becomes larger with an increase of the refractive index difference between a guided layer and a substrate. For a rib waveguide with a guided layer of nf = 3.44 on a YIG, Bi:YIG, or Bi:GdIG substrate, the nonreciprocal phase shift as a function of guided layer thickness is shown in Fig. 4(b). A larger value of 11>1 is obtained for a Bi:GdIG substrate.

B. Magllerooptic Ridge Waveguides

We consider the magnetooptic ridge waveguides as shown in Fig. J(c) and (d), where W = 31'm, d = 20 nm, X = 20.6/~m, Y3 = 1.7 ~m, and Yc = O.Sp.m. The guided layer thickness t is optimized so as to give the maximum nonreciprocal phase shift in case of a rib waveguide. Isotropic and magnetooptic materials are used as a high refractive index loading layer

ZHUANG AND KOSHIBA: AN EFFICIENT FINITE-ELEMENT ANALYSIS

'.0 n. _3. ((

' .0

n._2. 83

] I nl , • 2.0 , -

~. I • ,. 0

no"2. 2(

~~-0

0 o . • o. • ... t (11m)

(0)

00 Bl :GdIG

30

! , • " • , -•

"

(b)

Fig. 4. Nonreciprocal phase shm of I rib waveguide on iI magnetooptic substrate as a funaioll of guided layer thickness. <a) YIG substrate. (b) YIG, Bi :YIG, or Bi;GdIG substrate.

in Fig. l(c) and (d), respectively. For a ridge waveguide with a LaGa:YIG fi lm of t = 0.44f.lm [7], the magnitude of the nonreciprocal phase shift as a function of isotropic loading layer thickness is shown in Fig. 5(a) by the solid line, where n, = 2.60. The results obtained agree approximately with the experimental results [7]. The results of the planar waveguide (d = 0) Brc also shown in Fig. 5(a), by the dotted line. Since the rib height d is extremely small compared with the guided layer thickness, the ridge waveguide structure considered here is almost like the planar structures. Fig. 5(b) shows the magnitude of the nonreciprocal phase shift of a ridge waveguide with a Bi :GdIG fi lm of t = O.281lm as a function of isotropic loading layer thickness, where n J = 2.60 or 3.44. The value of 11>1 becomes larger with an increase of the refractive index n J of a loading layer. While the nonreciprocal phase shift becomes larger, the change of 11>1 against the loading layer thickness in the vicin ity of its maximum value is very abrupt. That is to say, this structure has a small tolerance in the loading layer thickness control.

The magnitude of the nonreciprocal phase shift of a ridge waveguide with a Bi:GdIG film of t = O.281lm as a function of magnetooplic loading layer thickness is also shown in Fig. 5(b), where a Ce:YIG fi lm is used as a loading layer. We

' .0

•

'.0

fi , • • , -• "0

0. 0 0

90

" ~ 0 , • ,

• -30

00

bp.~ !I ... n t

•

o. , o .• o. • o . • o .• h (U"')

(0)

n. " 3. (4

0.2 0.. 0. 6 0. 8

I C II II)

(b)

J451

Fig. S. Nonreciprocal phase shift of a magnetooptic ridge waveguide as ft function of loading layer thickness. <I) LaGa:YIG guided layer of f = 0.44/lm. (b) Bi:GdIG guided layer of t = 0.28I'm.

can find that when using a magnetooptic material as a loading layer, the value of 14>1 becomes larger and the change of ItPl against the loading layer thickness becomes gentler.

VI. CONCLUSIONS

A scalar finite-e lement method was developed for the anal­ysis of magnetooptic channel waveguides. In this approach, the nonphysical spurious solutioflS do not appear. To show the validity of this method, computed results were compared with the earlier experimental results. Also, the structures with larger nonreciprocal phase shift were investigated in detail.

The nonreciprocal phase shift is larger for an isotropic guided layer on a magnetooptic substrate than for a magne­tooptic guided layer on an isotropic substrate. The larger the difference between the refractive indexes of a guided layer and a substrate becomes, the larger the nonreciprocal phase shift becomes. In the structure with an additional loading layer, the nonreciprocal phase shift is larger and its change against the loading layer thickness becomes gentler when using a magnetooptic material as a loading layer than when using an isotropic material as a loading layer.

Although a scalar finite-element approach is very con­venient, its formulation is approximate in a strict sense.

1458

We are now working on a vector finite-element method for magnetooptic channel wavegu ides.

REFERENCES

IIJ T. Mizumoto and Y. Naito, ~Nonreciprocal propagation characteristics of YIG th in film," JEEE Trans. Microwaw: Theory T~da., vol. MTf-30, pp. 922-925, June 1982.

12] T. Miwmoto, K. Oochi, T. Harada, and Y. Naito, gMeasurement of optical nonreciprocal phase shift in a Bi-substituted Gd3 Fe&0 12 film and appl ication to waveguide-type optical circulator," J. Lightwaw: T~cJuwI., vol. IT-4, pp. 347-352, Mlr. 1986.

[3J C.-L Chen and A. Kumarswami, "Nonreciprocal TM-mode thin film phlsc shifters," Appl. Opt., vol. 25, pp. 3664--3670, Oct. 1986.

[4J K. Matsubara and H. Yajima, "Analysis ofY-branching optical circulator using magnetooptic medium as a substrate," J . Lightwaw: TechtwL, vol. 9, pp. 106 1- 1067, Sept. 1991.

[5] E. A. J. Marcatili , ~Dielectric rectangular waveguide and directional coupler for integrated optics," Bell SySI. Tech. J., vol. 48, pp. 2071- 2102, Sept. 1969.

[6] M. Tateda and T. Kimura, "Analysis of rectangular waveguide is01ator," J. Ligh/wave Technol., vol. LT-I, pp. 214-223, Mar. 1983.

[7] H. Inuzuka, Y. Okamura, and S. Yamamoto, "Nonreciprocal phase characteristics of single-mode magneto-optic rib waveguides," Trans. IEeE Japan, vol. nl-C, pp. 702-708, May 1988 <in Japanese).

[8] H. Inuzuka, Y. Okamura, and S. Yamamoto, "Magnetooptic non­reciprocal phase shift in (YLa) 3(FeGah Ot2 single-mode channel waveguides," J. Appl. PhyJ., vol. 64, no. 3, pp. 1575-15n, Aug. 1988.

(9] M. Koshiba, K. Hayata, and M. Suzuki, "Approximate scalar finite­element analysis of anisotropic optical w.veguides,~ Electron. Lefl., vol. 18, pp. 411-413, May 1982.

(10] - -, ~On Iccuracy of approximate scalar finite-ele ment analysis of di· electric optical waveguides, ~ Trans.IECEJapan, vol. E66, pp. 157- 158, Feb. 1983.

(IIJ --, "Approximate scalar finite-element analysis of anisotropic optical waveguides with off.-diagonal elements in a permittivity tensor,H IEEE Troll$. Microwa~ Theory Tech., vol. MTT-32, pp. 587-593, June 1984.

[12] B. M. A. Rahman, F. A. Fernandez, and J. B. Davies. "Review of finite element methods for microwave and optical waveguides," Proc. IEEE, vol. 79, pp. 1442-1448, Oct. 1991.

(13) M. Gomi, K. Satoh, and M. Abe, "Giant Faraday rOUltion of Ce­substituted YIQ films epiUlx ially grown by RF spunering,M Japan. J. App/. Phys., vol. 27, pp. L1 5J6.-L1 538, Aug. 1988.

JOURNAL OF LIGHTWAVE TECH NOLOGY, VOL.. It, NO.9, SEPTEMBER t993

(14] Y. Miyazaki, M. Mori, and K. Akao, "Nonreciprocal and loss properties of waveguide type optical i$Ola tol using magneloopt ic effects in garnet crystals," IOOC '77 Tech. Dig., P·1 (rokyo), July 1977.

Masanori Koshlba (SM'84) was born in Sapporo, Japan, on November 23, 1948. He received the B.S., M.S., and Ph.D. degrees in electronic engineering from Hokkaido University, Sapporo, Japan, in 1911, 1913, and 1976, respectively.

In 1976, he joined the Department of Elec­tronic Engineering, Kitami Institute of Technology, Kitami, Ja!}&n. From 1979 to 1987, he was an Associa te Professor of Electronic Engineering at Hokkaido University, and in 1987 he became a Professor there. He haS 'been engaged in research

on lightwave technology, surface acoustic waves, magnetostatic waves, microwave fi eld theory, and applications of fin ite-element and boundary­clement methods to field problems.

Dr. Koshiba is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), the Ins tilUle of Television Engineers of Japan, the Institute of Electrical Engineels of Japan, the Japan Socie.ty for Simulation Technology, and the Japan Society for Computational Methods in Engineering. In 1987, he WI$ awarded the 1986 Paper Award by the IEICE.

Xlu-Plng Zhuang Wll$ born in Anhui Province, China, on July I, 1963. She received the B.S. degree in communication engineering from Guilin College of Electronic Engineering, Guilin, Otina, in 1986. She is presently studying toward the M.S. degree in electronic engineering at Hokhido University, Sapporo, Japan.

Ms. Zhuang is a member of the Institute of Elec­tronics, Information and Communication Engineers of Japan.