urface acoustic wave distribution and acoustooptic interactions in silicaaveguide Bragg devices
hen Chen ∗, Bangren Shi, Meng Zhao, Lijun Guohysics Department, Changchun University of Science and Technology, 7089 Weixing Road, Changchun, 130022, PR China
r t i c l e i n f o
rticle history:eceived 3 December 2010
a b s t r a c t
The efficiency of acoustooptic interaction in single-mode strip silica waveguide is analyzed theoreticallyfor the first time by determining the overlap integral between the optical and acoustic field distributions.
ccepted 21 May 2011
eywords:ilica waveguideurface acoustic wave
The results show that there is a good overlap of the optical and SAW fields in the low SAW frequencyrange. At high acoustic frequencies, the overlap integral decreases with increasing acoustic frequency. At216 MHz, the maximum of 0.8544 for the overlap integral is obtained provided that the H/� equals 0.02.
Acoustooptic (AO) interactions have been used to perform aumber of signal processing function including modulation [1],eam deflection [2], tunable filtering [3], and spectrum analysis [4].hese functions are implemented in devices based on acoustoop-ic interactions in GaAs [5], LiNbo3 [6], quartz [7], etc. However,O interactions in Si-based silica waveguides have considerableotential because of possibility to integrate directly laser diodesnd detectors on the substrate and lower waveguide loss [8]. Hence
rigorous computation of the basis AO interaction in silica waveg-ides is necessary, but has not been reported so far. In this paper, weresent a theoretical analysis of the AO interaction in silica waveg-ides by calculating the perturbed SAW distribution and the opticaleld distribution in silica waveguides.
The Bragg device being analyzed is showed in Fig. 1. It consists ofn optical single-mode strip waveguide of width a and an interdig-tal transducer (IDT) exciting a SAW of beamwidth L equal to thenger overlap. The optical mode propagates in the x2-direction.he SAW propagating in the x1-direction creates a moving gratingf periodic variation in the refractive index and hence the permit-ivity near the silica surface by the acoustooptic effect, on which inrinciple diffraction effects can occur. The device is realized on non-iezoelectric material (silica), so the IDT needs a thin piezoelectricverlay of ZnO in order to excite the SAW.
The analysis of SAW’s has been given by many authors [9]. Herewe follow the nomenclature of Campbell and Jones [10]. The con-figuration being analyzed is illustrated in Fig. 2. The c-axis of thehexagonal ZnO crystal is collinear with the x3-axis.
The equations of state are for free space (Region I)
In addition to the equations of motion, Maxwell’s equationsust be satisfied everywhere. They are for free space
× ∇ × EI = −�0ε0∂2EI
∂t2(6)
for ZnO
× ∇ × EII = −�0∂2DII
∂t2(7)
nd for silica
× ∇ × EII = −�0εg∂2EIII
∂t2(8)
We consider surface waves propagating in the x1 directionith exponentially varying amplitudes in the x3 direction, and no
mplitude variations in the x2 direction. The assumed partial waveolutions for (4)–(8) are written for free space
Ii = ω
Vsciexp
(ω
Vs˛x3
)exp
[iω
(x1Vs
− t)]
(9)
or ZnO
UIIi = Miexp(
ω
Vsˇx3
)exp
[iω
(x1Vs
− t)]
EIIi = ω
VsGiexp
(ω
Vsˇx3
)exp
[iω
(x1Vs
− t)] (10)
nd for silica
UIIIi = Aiexp(
ω
Vs�x3
)exp
[iω
(x1Vs
− t)]
EIIIi = ω
VsCiexp
(ω
Vsx3
)exp
[iω
(x1Vs
− t)] (11)
The phase velocity of the surface wave is Vs. The unknown decayonstants ˛, ˇ, � and are found in terms of the surface wave phaseelocity by utilizing the equations of motion. By the successive
ubstitution of the assumed partial wave solutions into the equa-ions of state and equations of motion, we can obtain characteristicquations with the following form
Fig. 2. Layered media structure of ZnO and silica.
(2012) 617– 620
for free space[EIS
EIT
] [c1c3c2
]= 0 (12)
for ZnO⎡⎢⎢⎢⎢⎣
UIIS
UIIT
⎤⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎣
M1M3G1G3M2G2
⎤⎥⎥⎥⎥⎦ = 0 (13)
and for silica[UIIIS
UIIIT
] [A1A3A2
]= 0 (14)
[EIIIS
EIIIT
] [C1C3C2
]= 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 thepartial wave solutions. They are subject to the following bound-ary conditions: the electric field in the propagation direction andthe electric displacement normal to the interfaces are continuousat both interfaces; T13 and T33 vanish at the free surface and arecontinuous along with displacements U1 and U3 at the interfacebetween Regions II and III; i.e., at x3 = 0,⎧⎪⎨⎪⎩
Due to the algebraic intricacy, the problems can only be solvednumerically by assigning a velocity in the characteristic equationand then solving (12), (15) for the decay constants. These decay con-stants together with the pre-assigned velocity are substituted intothe determinant of the coefficients of the unknown partial waveamplitudes to see whether the boundary conditions are also sat-isfied. After the velocity and the decay constants are found, thedisplacement and the electromagnetic field in each region can befound. The material constants for ZnO and SiO2 are taken from Refs.[11,12].
The mechanical displacements for ZnO film on silica are plottedas a function of the normalized depth (x3/�) from the free surfacewhich are shown in Figs. 3–5 for three values of H/�, where H isthe ZnO thickness and � is the SAW wavelength. It can be seen thatwhen the normalized thickness H/� of ZnO is very small, the wavehas displacements approaching that of the bulk silica value (Fig. 3).For a very thick layer, i.e., several wavelengths, the wave propagatesalong 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 withdifferent values H/�, we find that considering the major contribu-tion 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. 3. Mechanical displacements of Rayleigh mode for H/� = 0.02 (H: ZnO thick-ness; �: SAW wavelength).
F�
m0fbt0wHH
3
wT
F�
Fig. 6. Cross section of a strip silica waveguide.
ig. 4. Mechanical displacements of Rayleigh mode for H/� = 1.2 (H: ZnO thickness;: SAW wavelength).
aximum of U3 is seen to occur in silica layer similar to Fig. 3, at.056 × � depth from the free surface; when H/� is in the rangerom 0.06 to 0.1, the maximum of U3 appears in the interfaceetween ZnO and silica; When H/� is greater than or equal to 0.2,he maximum of U3 is seen to occur in ZnO layer similar to Fig. 5, at.11 × � depth from the free surface. Based on the above analysis,e would expect to obtain strong AO interaction in silica providing/� is less than or equal to 0.05. In the following analysis, we select/� = 0.02 to discuss.
. Optical field distribution
Figs. 6 and 7 show the cross section of a single-mode stripaveguide and the electric field distribution of the guided mode.
he strip waveguide (width a and height b) has refractive index
ig. 5. Mechanical displacements of Rayleigh mode for H/� = 2.8 (H: ZnO thickness;: SAW wavelength).
Fig. 7. Electric field of the guided mode.
n1, the substrate index n2 and the cover index n3. Single-modetransmission can be achieved choosing appropriate waveguidedimensions 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
cos3cos(�x3 + 3) − b ≤ x3 < 0
A
cos3cos(�b − 3)exp[�(x3 + b)] − ∞ ≤ x3 < −b
(18)
where �, ı and � are the propagation constants in the film, coverand substrate regions respectively and b is the waveguide depth.
4. Guided wave acoustooptic interaction
The acoustooptic diffraction efficiency of an optical guided waveperfectly phase matched to the SAW for the isotropic case, withincident and diffracted optical modes being the same with respectto polarization, is given by the well-known formula [15]
� = sin2
( n2
effL
2� cos�B|�Bmax||� |
)(19)
where neff is the effective modal refractive index, � is the free spaceoptical wavelength, � is the overlap integral, L is the interactionlength or acoustic aperture and �B is the Bragg angle. The AO diffrac-tion efficiency strongly depends on the overlap � between theoptical and acoustic fields. � which depends solely on the waveg-uide parameters and the acoustic frequency is given by
|� | =∣∣∣∣∫
|Um|2Uaijdx3∫|Um|2dx3
∣∣∣∣ (20)
620 C. Chen et al. / Optik 123 (2012) 617– 620
w
�
wU
oic
a
�
S
sasfta(
(qo
Fq
Fig. 8. Strains of the SAW in silica; SAW frequency = 250 MHz; H/� = 0.02.
here Uaij = �Bij
�Bijmax
Bij = pEijklSkl + rS
ijkEk i, j, k, l = 1, 2, 3
here �Bij is the change in the optical indicatrix created by SAW,m is the optical mode profile of the waveguide, pE
ijklis the strain
ptic tensor at constant E, Skl are the strain components of SAW, rSijk
s the electrooptic tensor at constant S and Ek are the electric fieldomponents of SAW.
With the SAW propagating along x1 and the TE polarized lightlong x2, we have
B1 = p11S1 + p13S3 (21)
where
ij = 12
(∂Ui
∂xj+ ∂Uj
∂xi
)(22)
Substituting (11) into (22), the strain fields can be calculated inilica and are shown in Fig. 8. The values of p11 and p13 for silicare 0.121 and 0.27, respectively. Substituting the p11 and p13 ofilica into (21) and combining the known distributions of strainsrom Fig. 8, a plot of (21), as shown in Fig. 9, can be obtained. Sincehe silica is nonpiezoelectric, the contribution to �Bij due to thecoustooptic effect is dominant. At x3 = 2.288 �m, the maximum of21) is obtained.
Substituting the known field distributions (18) and (21) into20), the overlap integral can be evaluated. Plot of (20) versus fre-uency is shown in Fig. 10 for silica waveguide. There is a goodverlap of the optical and SAW fields in the low SAW frequency
ig. 9. SAW induced indicatrix change for TE polarized light; SAW fre-uency = 250 MHz; H/� = 0.02.
[
[
[
[
[
[
Fig. 10. Overlap integral as a function of acoustic frequency; H/� = 0.02.
range. At high acoustic frequencies, the overlap integral decreaseswith increasing acoustic frequency. Because of the minimum of|�B1| near the silica surface (see Fig. 9), the overlap integral isalways less than 0.9 for TE polarization. At 216 MHz, the maximumof the overlap integral is obtained, which is 0.8544.
5. Conclusion
The AO interaction in silica waveguide is theoretically investi-gated using the calculation of SAW and optical field distributionfor the first time. There is a good overlap of the optical and SAWfields 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 isobtained provided that H/� equals 0.02. By the use of the SAWdistribution and the calculation of the AO interaction, presented inthis analysis, sophisticated design of guided SAW and AO devicesbased on silica waveguide can be achieved.
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