Electromagnetic Characteristics of Transverse Acousto-Optic Waveguide Device in Integrated Optics

INVITED PAPER

Electromagnetic Characteristics of Transverse Acousto-Optic Waveguide Device in Integrated Optics

Yasumitsu MIYAZAKI^†a),Fellow

SUMMARY Among several optical devices in integrated optics, the fundamental characteristics of collinear optical switching devices have been studied about optical dielectric waveguides. Conventional waveguide- type acousto-optic (A-O) devices use collinear and longitudinal interac- tions with mode coupling based on the Bragg condition between opti- cal waves and surface acoustic waves (SAW). Collinear A-O devices of the waveguide-type show sufficient performance for wavelength-selective switching with narrow bandwidths. However, in these collinear A-O de- vices, interaction time is several microseconds for 10 mm waveguide de- vice length. In A-O devices of optical waveguides using transverse A-O in- teraction, where SAW propagates transversely to optical wave propagation direction, SAW propagation lengths needed for complete A-O interaction may become 10μm and interaction time may be several nanoseconds. In this paper, fundamental characteristics of the transverse A-O interaction are studied as an electromagnetic boundary value problem. Refractive indices in optical waveguides induced by A-O effects with SAW are shown by sine functions. Wave field characteristics in periodic structures for transverse di- rections are analyzed by analytic method of Hill’s equations for transverse spectral functions. Electromagnetic fields in regions with periodic struc- tures are discussed by the Mathieu functions and the perturbation method.

Dispersion characteristics of A-O eigen modes are studied for wavelengths of optical waves and SAW, with A-O coefficients.

key words: A-O waveguide, SAW, mode conversion, Mathieu function, per- turbation method

1. Introduction

Integrated optics and optical devices consisting of active and passive waveguides have been rapidly developed recently.

Among several optical devices, such as optical switches us- ing A-O effects, the fundamental characteristics and applica- tion processing systems of collinear optical switching device using A-O effects, have been studied about optical dielec- tric waveguides on LiNbO₃crystal substrates[1]–[3]. Con- ventional waveguide-type A-O devices use collinear inter- action with mode coupling based on the Bragg condition between optical waves and SAW both propagating in the same direction. Collinear A-O devices of the waveguide- type show sufficient performance for wavelength selective switching with narrow bandwidths in case of low switching speed[4]–[7]. However, since the SAW propagation speed is very slow, comparing with the optical wave speed, and in these collinear A-O devices, the interaction time is several microseconds for 10 mm waveguide device length. In A- O waveguide devices using the transverse A-O interaction,

Manuscript received April 3, 2015.

Manuscript revised August 2, 2015.

†The author is with Aichi Mathematical Technology Labora- tory, Souen, 5B, Nagoya-shi, 460–0024 Japan.

a) E-mail: [email protected] DOI: 10.1587/transele.E99.C.3

where SAW propagates transversely and at the right angle to the optical wave propagation direction, SAW propagation lengths needed for complete A-O interaction may become 10μm and interaction time may be several nanoseconds[8]–

[10]. Ultrahigh-speed switching can be accomplished in A- O waveguide devices with transverse interaction of optical modes and SAW.

In this paper, the fundamental characteristics of trans- verse A-O interaction are studied as an electromagnetic boundary value problem. Refractive indices in optical waveguides induced by A-O effects of SAW are shown by sine functions in the transverse direction and yield wave equations with functional coefficients. Wave field charac- teristics in inhomogeneous media of periodic structures for transverse directions given by A-O effects due to SAW are analyzed by analytic method of Hill’s equations for trans- verse spectral functions[11]. Electromagnetic fields in re- gions with periodic structures expressed by sine functions corresponding to SAW fields are shown by Mathieu func- tions with parameters and concerned with eigenvalues and a perturbation method.

By boundary conditions for electric and magnetic fields at boundaries between core and clad regions, eigen equa- tions for eigen modes in the transverse-type A-O wave- guides are derived. Dispersion characteristics of A-O eigen modes are shown by the perturbation method to Mathieu equation with A-O coefficients for wavelengths of optical waves and SAW. Switching and modulation properties due to propagation phase changes of the fundamental mode by A-O effects are shown. Based on these fundamental field characteristics of A-O waveguides, the mode coupling and switching in the transverse A-O waveguide devices con- sisting of coupled several optical waveguides controlled by SAW may be shown.

2. Optical Functional Waveguide with A-O Effects and Collinear Interaction

Interactions between SAW and optical waves are expressed by refractive index tensors. For A-O effects, physical rela- tions among electric fieldEk, electric displacementDi, parti- cle displacementuj, stress tensorTi jand strain tensorSk jare expressed using physical coefficients of elastic coefficients C_{i jkl}^E , piezoelectric coefficientseki j, dielectric constantsεi j, A-O coefficientspi jkl. Refractive index changesΔnildue to strains of SAW are[7]

Copyright c2016 The Institute of Electronics, Information and Communication Engineers

Δnil=−1 2

k,j

n²_ik nil

⎛⎜⎜⎜⎜⎜

⎝

α, β

pk jαβS_αβ

⎞⎟⎟⎟⎟⎟

⎠n²_jl. (1) For the one-dimensional case,

Δn=−n³

2 pS. (2)

Refractive index changes induced by the acoustic longitu- dinal wave with angular frequencyΩ, wave numberK and amplitudeS₀, propagating in thezdirection in homogeneous media are given, for thex1 =x,x2 =yandx3 =zcompo- nents, as

Δn1=−1

2n³p12S3, Δn3=−1

2n³p11S3, (3) whereS₃ =S₀sin(Ωt−Kz) andS₁=S₂=0.

For displacement vector ua and electric frequency of SAWω, using the double inner product of dyadics and ten- sors,

Δε(r, ω)=p:s(r, ω), s=bua(ri,z, ω)e^−jK(ω)z,

wherebis the material constant of elastic properties.

Figure 1 shows the collinear and longitudinal optical waveguide of two-dimensional case with SAW propagating in thezdirection. For A-O devices such as optical modu- lators, switches, couplers and separators using A-O effects, one or two A-O waveguides are combined.

General characteristics of active optical waveguide with A-O effects are shown by the following Fourier compo- nents of dielectric constants in the core and clad waveguide regions for several electric frequencies of SAWωi.

Δε¯=

i

Δε¯⁽ⁱ⁾(r,t), Δε=

i

ε⁽ⁱ⁾(r, ω), (4) where

Δε= ^∞

−∞Δε(r,¯ t)e^−jωtdt, (5) Δε¯= 1

2π

∞

−∞Δεe^jωtdω.

The optical electric fields in the waveguides are E(r, ω)= ^∞

−∞

E(r,¯ t)e^−jωtdt, (6)

∇×∇×E(r, ω)−μεω²E(r, ω)= μ

⎡⎢⎢⎢⎢⎢

⎣ (−ω²)Δε(ω)E(ω−ω)dω+ (−2ω)(ω− ω)Δε(ω)E(ω−ω)dω+

(−(ω−ω)²)Δε(ω)E(ω−ω)dω

⎤⎥⎥⎥⎥⎥

⎦. (8) Optical frequencies are higher than SAW frequencies,ω= ωiωand then right hand term is

μ

⎡⎢⎢⎢⎢⎢

⎣ (−ω²)Δε(ω)E(ω−ω)dω

⎤⎥⎥⎥⎥⎥

⎦. (9)

In active optical waveguide devices with A-O effects consisting of several optical waveguides (i), using eigen modesφ_αand eigenvaluesβαin a normal waveguide with- out A-O effects, we consider Green’s dyadics

Γi j(r,r)=

α

φ_α(r, ω)φ_α(r, ω)

M_α(ω) e^{−jβα(ω)|z−z|} (10) whereM_α(ω) is the normalization factor of eigen modes,

φ_α(rt, ω)φ_α(rt, ω)d²rt=δαα.

Electric fieldsEiare generated by field source at the input.

From the Green’s formula, we have

Etotl= Γi j μω²Δεj(ω)Ej(ω−ω)dωdv−

S0 Si

n·[(∇×Etotl)×Γi j+Etotl×(∇×

Γi j)]dSdω. (11) The electric fields in active A-O waveguides can be ex- panded as the following field expansion series with ampli- tudesa_αiby normal modesφ_αi

Ei(rt, ω)=

α

a_αi(z, ω)φ_αi(rt, ω)e^{−jβαi(ω)z}. (12) Using integral operatorH=

dω, from Eqs. (11) and (12), in A-O waveguides, mode coupling equations for mode am- plitudesa_αi by mode couplings and perturbations due to A- O effects are derived as

da

dz =HAa, a=(a1,a2, · · · ·an)^t, (13)

Fig. 2 Collinear A-O waveguide coupler

Aij=(−jζij)e^jΔkijz,

where the mode coupling coefficientsAij,ζijare

−jζ_αα(z, ω, ω^,, ω−ω)= 1

M_α (μω²)bφαi(rt, ω)p(rt,z) :ua(rt,z, ω)×

φαi(r_t,, ω−ω)d²r_t, (14) and the phase differences concerned with the Bragg relations Δkijare

Δki j(ω, ω,z)=βi(ω)−βj(ω−ω)−K(ω,z). (15) Using normalized mode amplitudes b⁽ⁱ⁾s and normalized mode coupling coefficients css, the mode coupling equa- tions are derived as,

db⁽ⁱ⁾₀

dz =−jβ⁽ⁱ⁾₀ b⁽ⁱ⁾₀ −jc⁽ⁱ⁾₀₁b⁽ⁱ⁾₁ −j

s

c⁽ⁱ_0s^,j)b⁽ⁱs^,j), db⁽ⁱ⁾₁

dz =−jβ⁽ⁱ⁾₁ b⁽ⁱ⁾₁ −jc⁽ⁱ⁾₁₀b⁽ⁱ⁾₀ −j

sc⁽ⁱ_1s^,j)b⁽ⁱs^,j), (16) db⁽ⁱs^,j)

dz =−jβ⁽ⁱs^,j)b⁽ⁱs^,j)−jc_0s^(i,j)b⁽ⁱ⁾₀ −jc⁽ⁱ_1s^,j)b⁽ⁱ⁾₁.

A collinear A-O waveguide coupler consisting of two waveguides with longitudinal SAW is shown in Fig. 2. Here, dominant mode amplitude b⁽ⁱ⁾₀ is given by Eq. (17), when b⁽ⁱ_s^,j) =0, andc⁽ⁱ⁾₀₁ = c⁽ⁱ⁾₁₀ = cfor the fundamental modes of b⁽ⁱ⁾₀ =andb⁽ⁱ⁾₁ in two waveguides.

b⁽ⁱ⁾₀ ² = |c|²

|c|²+|Δβ|² sin²

|c|²+|Δβ|²z

, (17)

where the phase relations of optical waves in two wave- guides and SAW are

Δβ=K−Δβopt, Δβopt =β⁽ⁱ⁾₀ −β⁽ⁱ⁾₁ . (18) The mode amplitudesb⁽ⁱ⁾₀ show sharp wave filtering and sep- arator characteristics controlled by SAW.

3. Structure of Transverse A-O Waveguide Device In the transverse A-O waveguide shown in Fig. 3, optical

Fig. 3 Transverse A-O waveguide device

waves propagate in the longitudinal z direction and SAW propagates in the transverse xdirection, where in theydi- rection the waveguide has uniform characteristics. Region I is the core part with refractive indexn1 and widthd, and regions II and III are the clad parts with refractive indices n2andn3. Transverse plane of the waveguide is thexycross section. StrainS(x,t) induced by SAW with velocityΩ/K and phase factorφis given by

S(x,t)=S₀sin(Ωt−K x−φ). (19) Refractive index changes in the homogeneous media of re- gions iwith refractive indicesni, using elasto-optic coeffi- cientspi, are

Δni=−1

2n³_ipiS(x,t). (20)

When the dielectric constants of core and clad re- gions in the transverse A-O waveguide are ˜ε1, ˜ε2 and ˜ε3, wave numbers of optical waves in the transversexdirection change to βx±K by SAW with the wave numbers K and the propagation constantsβz in the longitudinalzdirection are controlled to

β²−(βx±K)², whereβ=ω√ με˜is the wave number of media without SAW. These characteristics of wave spectrum and dispersions are shown in Fig. 4.

Figure 5 shows the refractive index distributions due to A-O effects with SAW and dynamical characteristics of sinusoidal SAW responses in the core and clad regions of the transverse A-O waveguide.

Fig. 4 Mode spectrum in A-O waveguide

Fig. 5 Strain and refractive index distribution due to SAW λoptical wavelength

Λwavelength of SAW, Tsperiod of SAW

4. Electromagnetic Field in Transverse A-O Wave- guide

Field and wave equations in the core (I) and clad (II), (III) regions of transverse A-O waveguides controlled by SAW are shown by changes of refractive indices and dielectric constants in each region as

ε˜i=εi+ Δεi, ε˜i=n˜²_iε0, Δεi= Δn²iε0,

˜

n1=n1+ Δn1, n˜2(=n˜3)=n2+ Δn2, (21) Δεi=−Δεissin(Ωt−K x−φ), (22)

∂t ∂t

From Eq. (21), in the region of A-O effects, the electric field is given as

∇²−με˜∂²

∂t²

E=−μΔεissin(Ωt−K x−φ)∂²E

∂t²

− ∇(∇ln ˜εE). (25) The magnetic field is given by, when the angular frequency of SAWΩis smaller than the optical frequencyω,

H= j

μω∇ ×E, ωΩ. (26)

Here, the electromagnetic fieldsE⁽ⁱ⁾(x,z,t) in the core and clad regions, (I) and (II), for the uniform field in the y-direction are shown as, if the propagation factor of thez direction ise^−jβzzand the time factor ise^jωt,

E⁽ⁱ⁾(x,z,t)=E⁽ⁱ⁾_t (x)e^{−jβzz+jωt}, (27) where

Ey(x,z,t)=Et(x)e^jωt−jβzz, Hx(x,z,t)=Ht(x)e^jωt−jβzz.

The transverse electric fields satisfy, using the frequency re- lation ofωΩ

∂²

∂x²+μω²(εi+Δεissin(K x+φ−Ωt))−β²z

E⁽ⁱ⁾_t (x)=0, (28) when the phase relation isφ−Ωt=ϕ(t),

∂²

∂x² +μω²(εi+ Δεissin(K x+ϕ))−β²z

E⁽ⁱ⁾_t (x)=0.

(29) In Eq. (29), we define the parameters of SAW and waveguidesW,η,ai,qi, following as, forypolarization and E⁽ⁱ⁾_t =E⁽ⁱ⁾y iy, defining new coordinatex,

K x+ϕ−π

2 =2η=K x, ai=W²

μω²εi−β²z

= 4

μω²εi−β²z

K² ,

−2qi=W²μω²Δεis=4μω²Δεis

K² , (30)

β²i =μω²ε˜i=n˜²_iε0.

Fig. 6 Space-time interaction of optical modes and SAW

Here, for wavelengthΛof SAW, we defineKandW, as W=2

K=Λ

π, K= 2 W=2π

Λ and d dx=K

2 d dη= 1

W d dη Equation (29) of transverse electric fieldsE⁽ⁱ⁾_t (x) is reduced to the following Mathieu equation with parameter q and eigenvaluesaforηcoordinate variable

d²

dη² +(ai−2qicos 2η)

E⁽ⁱ⁾_t (η)=0. (31) In the transverse A-O waveguide, the propagation character- istics of optical waves and SAW are shown, for distancex, and timet, in Fig. 6.

The propagation velocity of SAW V = Ω

K is smaller than those of optical modes v=1/√εeqμ, and optical waves pass through the transverse cross section of SAWxy plane with short-time interactions and high-speed responses.

Whereεeqis equivalent dielectric constant of optical modes, βz

β = √εeq.

The electric fields in the core (I) and clad (II), (III) re- gions, forypolarization,E⁽ⁱ⁾_t (x)=iyEy⁽ⁱ⁾(x), and thezcom- ponent of magnetic fieldsHz⁽ⁱ⁾ are, using Mathieu functions and coefficientsC⁽ⁱ⁾_ν ,S⁽ⁱ⁾_ν

E_y⁽ⁱ⁾(η)=

ν=0

C_ν⁽ⁱ⁾ce_ν(η,qi)+S⁽ⁱ⁾_ν se_ν(η,qi) , H⁽ⁱ⁾_z (η)= 1

jωμ 1 W

ν=0

C⁽ⁱ⁾_ν ce_ν(η,qi)+S⁽ⁱ⁾_ν se_ν(η,qi) . (32) Here, for smallq parameter cases, we have Mathieu func- tions based on sine and cosine functions as

ai=ν²+^∞

r=0

αrq^r_i, α2= 1

2(ν²−1), (33) ce_ν(η,q)=cosνz−1

4q

cos(ν+2)η

(ν+1) −cos(ν−2)η (ν−1)

+O(q²), se_ν(η,q)=sinνz−1

4q

sin(ν+2)η

(ν+1) −sin(ν−2)η (ν−1)

+O(q²).

From the continuity conditions of electric and magnetic

fields at boundaries between core and clad, x = ±(d/2), when refractive indices of clad regions II and III, ˜n2 =

˜

n₃, using coefficientsC⁽¹⁾_ν ,S⁽¹⁾_ν ,C⁽²⁾_ν , S⁽²⁾_ν and defining, for boundaries ofηcoordinates,

η_±= K 2

±d 2

+1 2

ϕ−π 2

,

we have the eigen equation and derive the dispersion char- acteristics of eigen modes giving propagation constants.

As a simple case, we next consider the transverse A-O effects in the core region, without A-O effects in the clad regions. In case of Δε2 = Δε3 = 0, and ˜ε1 = ε1+ Δε1, ε1 > ε2 =ε3,Δε1 =−Δε1ssin(Ωt−K x−φ), where only the core region has A-O effects, we discuss simple case of mode conversions due to A-O effects in the core region and no A- O effects in the clad region. Forϕ=π/2, the TE symmetric (even) mode for y polarization, the electric and magnetic fieldsE⁽¹⁾y ,E⁽²⁾y ,Hz⁽¹⁾,Hz⁽²⁾, are, using Mathieu functions and C_ν⁽¹⁾,C⁽²⁾_ν in the core and the clad regions,

E⁽¹⁾_y =C⁽¹⁾_ν ce_ν(η,q1)e^−jβzz, H_z⁽¹⁾=− j

ωμ 1

WC⁽¹⁾_ν ce_ν(η,q1)e^−jβzz, |x| ≤ d

2, (34) E⁽²⁾_y =C⁽²⁾_ν e^−jβzze^{±α(2)x x}, H⁽²⁾_z = ±α⁽²⁾x

ωμ jC_ν⁽²⁾e^−jβzze^{±α(2)x x},

|x| ≥ d 2 Here, in Eq. (33), we definea_i=a₁,αr =αr1,q_i=q₁, and for spectrum parameters

−α⁽ⁱ⁾x

2=β²i −β²z =β⁽ⁱ⁾x

2, α⁽²⁾x =α⁽³⁾x , and coefficientsC⁽²⁾=C⁽³⁾.

The boundary condition of the electric and magnetic fieldsEyandHzat the boundary interface at,x=±d

2 are C⁽¹⁾_ν ce_ν

η^(d),q1

=C⁽²⁾_ν e^−αxd2, 1

WC⁽¹⁾_ν ce_ν η^(d),q1

=αxC⁽²⁾_ν e^−αxd2,

−c e_ν(η,q₁)

η=+^K₄d = +c e_ν(η,q₁)

η=−^K₄d, (35)

where η^(d) = K

4d and αx=α⁽²⁾x .

From the field continuity conditions of Eq. (35), we have mode dispersion relations of TE symmetric modes as

C⁽¹⁾_ν

C⁽²⁾_ν = e^−αxd2 ce_ν

η^(d),q1

= αxe^−αxd2 1

Wce_ν

η^(d),q₁, (36) c e_ν

η^(d),q1

c e_ν

η^(d),q₁ =−αxW, where

Fig. 7 Eigen characteristicsaandbof TE modes for Mathieu functions cenandsen

β²z−α²x=β²₂, αxW = β²₁−β²₂

W²−1, β²z+

1 W

2

=β²₁.

The propagation constantsβ^(ν)z of the optical guidedν modes and dispersion characteristics derived by solutions of eigen equation of Eq. (36) for Mathieu functions will be con- cretely discussed in a subsequent paper to be published in the future. The eigen characteristics of Mathieu functions yield parameter relations ofqanda,bconcerned with the TE symmetric modes as shown in Fig. 7.

The parameters a and b are represented, for the Mathieu functionscenandsen, as eigen valuesain Eq. (33).

SinceΔεandq are very small, the perturbation method to the Mathieu equations is useful in A-O waveguides.

5. Eigen Mode Characteristics by Perturbation Method to Mathieu Equation

The propagation constants and the eigen mode fields of A-O waveguides are derived by the perturbation method to so- lutions of the Mathieu equations, Eqs. (29) and (31), with A-O effects based on field expansions by the normal eigen modes in slab waveguides satisfying the boundary condition without A-O effects.

The wave fields controlled by A-O effects, E⁽ⁱ⁾y (x) in Eq. (29) are expanded by the eigen mode functionsΦm = Ψm(x)e^−jβΦ(m)z in the normal waveguides without A-O ef- fects of, ˜εi=εi,Δεi=0 as follows.

E_y⁽ⁱ⁾(x)=

m

am(βz)Ψm(x). (37) From Eq. (29), using integration in the xcross section, we have the following linear equation for amplitudesam

m

am(β)ΨmΨm

β²z−β²_Φ

−ω²μ( ˜εi−εi) dx=0.

(38) Defining the mode perturbation and mode conversion fac- tors with the mode normalizations as,

D_{2s−1,2t−1} =D_2s,2t=β²_Φ (s=t)

=0 (st).

we have the following linear equation β²I−D

+ζ

a=0. (40)

From Eq. (40), the eigen characteristic equation of the eigen modes in the A-O waveguide, using componentsHpγof ma- trixD−ζ, by the perturbation method for smallΔε, we have eigen vectorsα_γ and eigen vectorsβ²_γ

α_γ =a_γ−

pγ

Hpγ

β²−Hpp

ap+· · ·, β²_γ=H_γγ+

p

H_γpHpγ

β²−Hpp

+· · ·. (41) The propagation constants of theγth mode in the A-O waveguide are

β_γ=β²_Φ−K_γγ. (42)

Here, the media perturbation term due to A-O effects is ε˜1−ε1= Δε= Δεssin(K x+ϕ) (43) For the fundamental even TE₀mode of amplitudeC0as

Ψ(0)=C₀cosβ1xx,

the perturbation factors concerned with A-O effects are given by

K00=−ω²μΔεs

1 d

d/2

−d/2

sin(K x+ϕ) cos(β1xx) cos(β_1xx)dx

=−ω²μΔεsC00, (44) where the mode coupling and perturbation factorsC00are

C00= 1 d

d/2

−d/2

sin(K x+ϕ) cos(β1xx) cos(β_1xx)dx. (45) When thexcomponent of wave spectrum isβ1x =β_1x, we have

C₀₀=1 2sinϕ

⎡⎢⎢⎢⎢⎢

⎢⎢⎢⎢⎢

⎢⎢⎢⎢⎣

sin dK

2

dK 2

+1 2

w=±1

sin

(K+2wβ1x)d 2

(K+2wβ1x)d 2

+δ 2

⎤⎥⎥⎥⎥⎥

⎥⎥⎥⎥⎥

⎥⎥⎥⎥⎦

(46)

Fig. 8 Dynamic phase properties of controlling SAW ϕ(t)=φ−Ωt Λ =2π

K Ts=2π Ω

Fig. 9 Mode conversion factors due to A-O effects of SAW ξ=aK,a(K±2β1x)

where,K=2β1xforw=1,δ=1 andK2β1xforw=±1, δ=0.

From Eqs. (42) and (44), the propagation constants of perturbed eigen modes due to A-O effects in the A-O wave- guides are given by, for the fundamental dominant mode,

β₍₀₎² =β²₍₀₎−K₀₀=β²₍₀₎+ω²μΔεsC₀₀. (47) These evaluation yield phase characteristics of the propaga- tion modes. The transverse field distributions of perturbed eigen modes in A-O waveguides are shown as,

α(0)a(0),

Ψ₍₀₎ Ψ(0) (48)

for the fundamental dominant mode, in the first approxima- tion, whenΔεdue to A-O effects is small.

Figures 8 and 9 show the dynamic phase properties of controlling SAW and the mode conversion factors due to the A-O effects of SAW, discussed in Eq. (47).

The transverse spectrum factors β1x in the normal waveguides are analytically given by V values, from the eigen equation of the fundamental even TE mode,

tanβ1x

d 2 =α2x

β1x. Near cutoff, definingζas

V =

n²₁−n²₂β0

d

2, ζ =V−mπ,

Fig. 10 Transverse spectrumβ1xof dominant TE mode in normal slab waveguide

dot line: approximation, solid line: exact

we have approximately the spectrum factors as follows, α2x= 1

d

4V²+1−1 , β1x=

√2 d

4V²+1−11/2

, βz=

(n2β0)²− 2 d²

1+2V²−

4V²+1 .

and the transverse spectrum factorsβ1xare shown in Fig. 10.

The phase characteristicsΦof the fundamental guided even TE₀ mode perturbed by the A-O effects of SAW are given by, for propagation lengthz,

Φ=(βz+ Δβz)z, ΔΦ= Δβzz, (49) whereΔΦis the phase modulation due to the A-O effects of SAW in the A-O waveguide.

From Eq. (47),ΔΦis evaluated, using refractive index perturbation Δns and refractive index of the core in wave- guide n1, by perturbation factorC00 of Eq. (46), when the effective wavelength of the fundamental TE mode in the nor- mal waveguide without the A-O effects isλeff =2π/βz, as follows

ΔΦ=βz

Δns

2n1

C₀₀z

= π λeff

Δns

n₁ C00z (50)

For example, when the waveguide parameters aren₁=2.27, n2=2.20, d=3 μm, λ=1 μm, and V=3.4, for Δns = 10⁻³, C₀₀ = 3/4, and z = 1 mm, we have ΔΦ = 2.35 (radian). This typical example provides excellent technical data for accomplishment of high-speed signal switching and processing.

6. Transverse A-O Wave Separator

Optical transverse A-O couplers and wave separators con- sisting of parallel waveguides with transversely propagat- ing SAW shown in Fig. 11, can be discussed, based on the eigen mode characteristics of the transverse A-O waveguide.

When one waveguide has the A-O effects, the propagation

Fig. 11 Transverse A-O wave separator

constants are controlled by SAW for wave separation and wave filtering given in Eq. (17).

7. Conclusions

Transverse A-O waveguides with SAW have high efficient and high-speed switching characteristics. Electromagnetic field characteristics in the transverse A-O waveguides with refractive indices transversely controlled by SAW are stud- ied by eigen function expansion methods for Hill’s equa- tions. Field characteristics are discussed by the Mathieu function expansion method and the perturbation method for the core and clad regions with the A-O effects due to SAW.

Eigen equations and eigen modes are shown, using field ex- pressions of Mathieu functions, by the field boundary condi- tion and the perturbation method based on the eigen modes in the normal waveguides without the A-O effects. The dis- persion characteristics for wave numbers and wavelengths of optical waves and SAW are shown. Transverse A-O wave couplers and wave separators are discussed.

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Yasumitsu Miyazaki received his B.E, M.E., and D.Eng. degrees in Electronic Engi- neering from Nagoya University in 1963, 1965, and 1969. He was appointed as an assistant pro- fessor and an associate professor at Nagoya Uni- versity in 1972 and 1976. From 1981 to 2003, he was a professor in the Department of Infor- mation and Computer Sciences, Toyohashi Uni- versity of Technology. From 2003 to 2011, he was a professor in the Department of Electronics and Information Engineering, Aichi University of Technology and since 2011, he has been an emeritus professor. He has engaged in research in the field of electromagnetic waves including waves in millimeter waveguides, optical fibers, integrated optics, electromagnetic scattering, and diffraction. He also studies biological phenomena of elec- tromagnetic fields and optical neural computing. From 1973 to 1975, he was with the Institute of High Frequency-Technics, Technical University of Braunschweig (Germany) and engaged in research on electromagnetic fields for optical communications. In 1996, he was a guest professor in IHFT at the Technical University of Berlin. He received the Yonezawa Memorial Paper award in 1970. He is a life member of IEEE and IEE Japan.