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Surface acoustic wave distribution and acoustooptic interactions in silica waveguide Bragg devices Chen Chen ∗ , Bangren Shi, Meng Zhao, Lijun Guo Physics Department, Changchun University of Science and Technology, 7089 Weixing Road, Changchun, 130022, PR China

a r t i c l e

i n f o

Article history: Received 3 December 2010 Accepted 21 May 2011

Keywords: Silica waveguide Surface acoustic wave Acoustooptic interaction

a b s t r a c t The efﬁciency of acoustooptic interaction in single-mode strip silica waveguide is analyzed theoretically for the ﬁrst time by determining the overlap integral between the optical and acoustic ﬁeld distributions. The results show that there is a good overlap of the optical and SAW ﬁelds in the low SAW frequency range. At high acoustic frequencies, the overlap integral decreases with increasing acoustic frequency. At 216 MHz, the maximum of 0.8544 for the overlap integral is obtained provided that the H/ equals 0.02. © 2011 Elsevier GmbH. All rights reserved.

1. Introduction

2. Calculation of the SAW ﬁeld distribution

Acoustooptic (AO) interactions have been used to perform a number of signal processing function including modulation [1], beam deﬂection [2], tunable ﬁltering [3], and spectrum analysis [4]. These functions are implemented in devices based on acoustooptic interactions in GaAs [5], LiNbo3 [6], quartz [7], etc. However, AO interactions in Si-based silica waveguides have considerable potential because of possibility to integrate directly laser diodes and detectors on the substrate and lower waveguide loss [8]. Hence a rigorous computation of the basis AO interaction in silica waveguides is necessary, but has not been reported so far. In this paper, we present a theoretical analysis of the AO interaction in silica waveguides by calculating the perturbed SAW distribution and the optical ﬁeld distribution in silica waveguides. The Bragg device being analyzed is showed in Fig. 1. It consists of an optical single-mode strip waveguide of width a and an interdigital transducer (IDT) exciting a SAW of beamwidth L equal to the ﬁnger overlap. The optical mode propagates in the x2-direction. The SAW propagating in the x1-direction creates a moving grating of periodic variation in the refractive index and hence the permittivity near the silica surface by the acoustooptic effect, on which in principle diffraction effects can occur. The device is realized on nonpiezoelectric material (silica), so the IDT needs a thin piezoelectric overlay of ZnO in order to excite the SAW.

The analysis of SAW’s has been given by many authors [9]. Here we follow the nomenclature of Campbell and Jones [10]. The conﬁguration being analyzed is illustrated in Fig. 2. The c-axis of the hexagonal ZnO crystal is collinear with the x3-axis. The equations of state are for free space (Region I) DIi = ε0 EIi BIi = 0 HIi

(1)

for ZnO (Region II) TIIij = Cijkl SIIkl − ekij EIIk DIIi = eikl SIIkl − εik EIIk BIIi = 0 HIIi

(2)

and for silica (Region III) TIIIij = cijkl sIIIkl DIIIi = εg EIIIi BIIIi = 0 HIIIi

(3)

where εg is the permittivity of the isotropic material (silica). The stress equations of motion are for ZnO II

∂TIIij ∂2 UIIi = ∂xj ∂t 2

(4)

and for silica ∗ Corresponding author. Tel.: +86 431 85583340; fax: +8613664437199. E-mail address: chen chen [email protected] (C. Chen). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.06.004

III

∂TIIIij ∂2 UIIIi = ∂xj ∂t 2

(5)

618

C. Chen et al. / Optik 123 (2012) 617–620

for free space

EIS EIT

c1 c3 c2

=0

for ZnO

(12)

⎤⎡

⎡

⎤

M1 ⎥ ⎢ M3 ⎥ ⎥ ⎢ G1 ⎥ ⎥⎢ ⎥ = 0 ⎥ ⎢ G3 ⎥ ⎦⎣ ⎦ M2 G2

⎢ UIIS ⎢ ⎢ ⎢ ⎣ UIIT

(13)

Fig. 1. SAW driven strip waveguide Bragg device.

and for silica

In addition to the equations of motion, Maxwell’s equations must be satisﬁed everywhere. They are for free space

∇ × ∇ × EI = −0 ε0

∂2 EI ∂t 2

(6)

for ZnO ∂2 DII ∂t 2

(7)

and for silica

∇ × ∇ × EII = −0 εg

∂2 EIII ∂t 2

(8)

We consider surface waves propagating in the x1 direction with exponentially varying amplitudes in the x3 direction, and no amplitude variations in the x2 direction. The assumed partial wave solutions for (4)–(8) are written for free space ω c exp Vs i

for ZnO UIIi = Mi exp EIIi =

UIIIi = Ai exp EIIIi =

ω

Vs

ω

˛x3 exp iω

Vs

ω Vs

ˇx3 exp iω

ω

ω C exp Vs i

Vs

ω G exp Vs i

and for silica

UIIIS UIIIT EIIIS

EIIIT

∇ × ∇ × EII = −0

EIi =

x1

x1 Vs

ˇx3 exp iω

x3 exp iω

ω Vs

Vs

−t

−t

Vs

Vs

x3 exp iω

(9)

C1 C3 C2

=0

(14)

=0

(15)

For simplicity, Rayleigh mode is considered here. The total electric ﬁelds and mechanical displacements are given in the various regions by appropriate linear combinations of the partial wave solutions. They are subject to the following boundary conditions: the electric ﬁeld in the propagation direction and the electric displacement normal to the interfaces are continuous at both interfaces; T13 and T33 vanish at the free surface and are continuous along with displacements U1 and U3 at the interface between Regions II and III; i.e., at x3 = 0,

⎧ ⎪ ⎨ EI1 = EII1

DI3 = DII3

⎪ ⎩ TII13 = 0

(16)

TII33 = 0

x1

x1

A1 A3 A2

−t

−t

x1 Vs

and at x3 = H (10)

TII13 = TIII13

TII33 = TIII33 ⎪ ⎪ ⎪ ⎪ ⎩ UII1 = UIII1

−t

⎧ EII1 = EIII1 ⎪ ⎪ ⎪ ⎪ ⎨ DII3 = DIII3

UII3 = UIII3

(11)

The phase velocity of the surface wave is Vs . The unknown decay constants ˛, ˇ, and are found in terms of the surface wave phase velocity by utilizing the equations of motion. By the successive substitution of the assumed partial wave solutions into the equations of state and equations of motion, we can obtain characteristic equations with the following form

Fig. 2. Layered media structure of ZnO and silica.

(17)

Due to the algebraic intricacy, the problems can only be solved numerically by assigning a velocity in the characteristic equation and then solving (12), (15) for the decay constants. These decay constants together with the pre-assigned velocity are substituted into the determinant of the coefﬁcients of the unknown partial wave amplitudes to see whether the boundary conditions are also satisﬁed. After the velocity and the decay constants are found, the displacement and the electromagnetic ﬁeld in each region can be found. The material constants for ZnO and SiO2 are taken from Refs. [11,12]. The mechanical displacements for ZnO ﬁlm on silica are plotted as a function of the normalized depth (x3/) from the free surface which are shown in Figs. 3–5 for three values of H/, where H is the ZnO thickness and is the SAW wavelength. It can be seen that when the normalized thickness H/ of ZnO is very small, the wave has displacements approaching that of the bulk silica value (Fig. 3). For a very thick layer, i.e., several wavelengths, the wave propagates along the free surface of the ZnO layer and has the characteristics of a Rayleigh wave in bulk ZnO (Fig. 5). By repeatedly calculating the mechanical displacements with different values H/, we ﬁnd that considering the major contribution of U3 to the SAW, when H/ is less than or equal to 0.05, the

C. Chen et al. / Optik 123 (2012) 617–620

619

Fig. 6. Cross section of a strip silica waveguide.

Fig. 3. Mechanical displacements of Rayleigh mode for H/ = 0.02 (H: ZnO thickness; : SAW wavelength).

Fig. 7. Electric ﬁeld of the guided mode.

Fig. 4. Mechanical displacements of Rayleigh mode for H/ = 1.2 (H: ZnO thickness; : SAW wavelength).

maximum of U3 is seen to occur in silica layer similar to Fig. 3, at 0.056 × depth from the free surface; when H/ is in the range from 0.06 to 0.1, the maximum of U3 appears in the interface between ZnO and silica; When H/ is greater than or equal to 0.2, the maximum of U3 is seen to occur in ZnO layer similar to Fig. 5, at 0.11 × depth from the free surface. Based on the above analysis, we would expect to obtain strong AO interaction in silica providing H/ is less than or equal to 0.05. In the following analysis, we select H/ = 0.02 to discuss. 3. Optical ﬁeld distribution Figs. 6 and 7 show the cross section of a single-mode strip waveguide and the electric ﬁeld distribution of the guided mode. The strip waveguide (width a and height b) has refractive index

n1, the substrate index n2 and the cover index n3. Single-mode transmission can be achieved choosing appropriate waveguide dimensions and proper values of refractive indices [13]. The optical ﬁeld for TE mode has been given by [14]:

Um (x3) =

⎧ Aexp(−ıx3) 0 ≤ x3 < ∞ ⎪ ⎪ ⎪ ⎨ A ⎪ ⎪ ⎪ ⎩

cos 3

cos(x3 + 3 )

− b ≤ x3 < 0

A cos(b − 3 )exp[(x3 + b)] cos 3

(18) − ∞ ≤ x3 < −b

where , ı and are the propagation constants in the ﬁlm, cover and substrate regions respectively and b is the waveguide depth. 4. Guided wave acoustooptic interaction The acoustooptic diffraction efﬁciency of an optical guided wave perfectly phase matched to the SAW for the isotropic case, with incident and diffracted optical modes being the same with respect to polarization, is given by the well-known formula [15]

= sin

2

n2eff L 2 cosB

|Bmax || |

(19)

where neff is the effective modal refractive index, is the free space optical wavelength, is the overlap integral, L is the interaction length or acoustic aperture and B is the Bragg angle. The AO diffraction efﬁciency strongly depends on the overlap between the optical and acoustic ﬁelds. which depends solely on the waveguide parameters and the acoustic frequency is given by

Fig. 5. Mechanical displacements of Rayleigh mode for H/ = 2.8 (H: ZnO thickness; : SAW wavelength).

|Um |2 Uaij dx3 | | = |Um |2 dx3

(20)

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C. Chen et al. / Optik 123 (2012) 617–620

Fig. 10. Overlap integral as a function of acoustic frequency; H/ = 0.02. Fig. 8. Strains of the SAW in silica; SAW frequency = 250 MHz; H/ = 0.02.

where Uaij =

Bij Bijmax

S Bij = pEijkl Skl + rijk Ek

i, j, k, l = 1, 2, 3

where Bij is the change in the optical indicatrix created by SAW, Um is the optical mode proﬁle of the waveguide, pEijkl is the strain S optic tensor at constant E, Skl are the strain components of SAW, rijk is the electrooptic tensor at constant S and Ek are the electric ﬁeld components of SAW. With the SAW propagating along x1 and the TE polarized light along x2, we have

B1 = p11 S1 + p13 S3 where 1 Sij = 2

∂Uj ∂Ui + ∂xi ∂xj

(21)

(22)

Substituting (11) into (22), the strain ﬁelds can be calculated in silica and are shown in Fig. 8. The values of p11 and p13 for silica are 0.121 and 0.27, respectively. Substituting the p11 and p13 of silica into (21) and combining the known distributions of strains from Fig. 8, a plot of (21), as shown in Fig. 9, can be obtained. Since the silica is nonpiezoelectric, the contribution to Bij due to the acoustooptic effect is dominant. At x3 = 2.288 m, the maximum of (21) is obtained. Substituting the known ﬁeld distributions (18) and (21) into (20), the overlap integral can be evaluated. Plot of (20) versus frequency is shown in Fig. 10 for silica waveguide. There is a good overlap of the optical and SAW ﬁelds in the low SAW frequency

Fig. 9. SAW induced indicatrix change for TE polarized light; SAW frequency = 250 MHz; H/ = 0.02.

range. At high acoustic frequencies, the overlap integral decreases with increasing acoustic frequency. Because of the minimum of |B1 | near the silica surface (see Fig. 9), the overlap integral is always less than 0.9 for TE polarization. At 216 MHz, the maximum of the overlap integral is obtained, which is 0.8544. 5. Conclusion The AO interaction in silica waveguide is theoretically investigated using the calculation of SAW and optical ﬁeld distribution for the ﬁrst time. There is a good overlap of the optical and SAW ﬁelds in the low SAW frequency range. At high acoustic frequencies, the overlap integral decreases with increasing acoustic frequency. At 216 MHz, the maximum of 0.8544 for the overlap integral is obtained provided that H/ equals 0.02. By the use of the SAW distribution and the calculation of the AO interaction, presented in this analysis, sophisticated design of guided SAW and AO devices based on silica waveguide can be achieved. References [1] C. Tsai, Guided-wave acoustooptic Bragg modulators for wide-band integrated optic communications and signal processing, IEEE Trans. Circuits Syst. 26 (1979) 1072–1098. [2] D.V. Semenov, E. Nippolainen, A.A. Kamshilin, Scanning ultra fast distance sensor based on acousto-optic deﬂection, in: Northern Optics Conference Proceedings, Bergen, Norway, 2006, pp. 17–22. [3] N.A. Riza, F.N. Ghauri, Compact tunable microwave ﬁlter using retroreﬂective acousto-optic ﬁltering and delay controls, Appl. Opt. 47 (2007) 1032– 1039. [4] N.V. Masalsky, Waveguide acoustooptic devices for the real-time spectral analysis of broadband optical signals, Laser Phys. 16 (2006) 1352–1355. [5] M.M. de Lima, M. Beck, R. Hey, Compact Mach–Zehnder acousto-optic modulator, Appl. Phys. Lett. 89 (2006) 121104-1–1121104-3. [6] N. Goto, Y. Miyazaki, Design of tapered SAW waveguide for wavelengthselective optical switches using weighted acoustooptic interaction, Electr. Eng. Jpn. 154 (2006) 36–46. [7] L.N. Magdich, Y.V. Pisarevskii, N.N. Semenovskii, Certain features of the effect of diffraction on the acoustooptic interaction in an elastically anisotropic medium, J. Commun. Technol. Electron. 53 (2008) 1442–1446. [8] T. Kominato, Y. Ohmori, H. Okazaki, M. Yasu, Very low-loss GeO2 -doped silica waveguides fabricated by ﬂame hydrolysis deposition method, Electron. Lett. 26 (1990) 327–329. [9] R.M. White, Surface elastic waves, Proc. IEEE 58 (1970) 1238–1276. [10] J.J. Campbell, W.R. Jones, A method for estimating optimal crystal cuts and propagation directions for excitation of piezoelectric surface waves, IEEE Trans. Sonics Ultrason. 15 (1968) 209–217. [11] M.B. S Dühring, O. Sigmund, Improving the acousto-optical interaction in a Mach–Zehnder interferometer, J. Appl. Phys. 105 (2009) 1083529–1083539. [12] L.L. Brizoual, F. Sarry, F. Moreira, O. Elmazria, FEM modelling of surface acoustic wave in diamond layered structure, Phys. Stat. Sol (A) 203 (2006) 3179–3184. [13] H. Ou, Different index contrast silica-on-silicon waveguides by PECVD, Electron. Lett. 39 (2003) 212–213. [14] A. Yariv, Coupled-mode theory for guided-wave optics, IEEE J. Quant. Electron. 9 (1973) 919–933. [15] C.S. Tsai, M.A. Alhaider, L.T. Nguyen, B. Kim, Wide-band guided-wave acoustooptic Bragg diffraction and devices using multiple tilted surface acoustic waves, Proc. IEEE 64 (1976) 318–328.

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