SPIE Proceedings [SPIE Information Technologies 2000 - Boston, MA (Sunday 5 November 2000)] Optical Devices for Fiber Communication II - Measurement of the electrostrictive constants

Measurement of the electrostrictive constants of silicaand their impact on poled silica devices

A. C. Liu, M. J. F. Digonnet, and G. S. KinoEdward L. Ginzton Laboratory,Stanford University, California

ABSTRACT

In an electro-optic poled-silica switch, modulation arises from both the electro-optic effect andelectrostriction. To investigate the magnitude of these two contributions, we measured the phase shiftinduced in a thin slab of unpoled silica subjected to a strong dc plus a small, low-frequency (0-19 MHz) acvoltage. The frequency responses for light polarized parallel and perpendicular to the applied field exhibit aconstant term due to electronic dc Kerr, and strong peaks due to electrostriction enhanced by mechanicalresonances of the slab. A theoretical model is presented that gives good quantitative agreement with theseobservations. From this comparison we infer the values of the dc Kerr and electrostrictive constants of silicafor each polarization. For the perpendicular polarization, electrostriction largely dominates in the frequencyrange under study. A potential electrostrictive modulator and the impact of electrostriction on thepolarization dependence of poled-silica devices are discussed.

Keywords: DC Kerr, electrostriction, third-order nonlinear effects, poled silica, polarization dependence,modulators, switches

1. INTRODUCTION

The large second-order nonlinearity observed in thermally poled silica has been attributed at least in

part to DC rectification of the third-order susceptibility χ(3) of silica by a permanent field Edc induced bypoling.[1-3] In projected guided-wave switches and modulators based on poled silica, an external signal atfrequency ω applied across the waveguide modulates the phase of an optical signal traveling in thewaveguide. This modulation is due not only to the induced second-order nonlinearity (the electro-opticeffect), but also to the intrinsic electrostriction of silica. Via electrostriction, the total applied field introducesa stress to the medium. This results in a strain, which induces, via the elasto-optic effect, an index change atω. Although it may be possible to reduce the magnitude of electrostriction in a practical poled phasemodulator by damping the acoustic resonances,[4,5] electrostriction in undamped fiber-size samples does notbecome negligible until extremely high frequencies (at least in the GHz range).[6-8] Since electrostrictioncan be strongly enhanced by resonance, and since these two effects have very different frequency andpolarization dependencies, in the dependencies on frequency and polarization of the modulation may berather complex. It is thus important to measure the absolute contributions of both effects across a wide rangeof frequency.

We recently reported such a measurement.[9] It consisted in recording with an interferometer thephase modulation induced in a slab of unpoled silica by a strong applied dc field mixed with a small ac fieldat ω. This measurement was carried out over the 0-20 MHz frequency range for optical signals polarizedeither parallel or perpendicular to the field. In this work we present a theoretical model that simulates thismeasurement by taking in account both the electronic and electrostrictive contributions of the Kerr effect,including their frequency and polarization dependencies. This measurement was inspired from Ref. [10],which used a similar technique in a fiber to separate the two contributions on the basis of their polarizationdependence. However, the authors took electrostriction to be polarization independent, which we believe isincorrect, as demonstrated below. The predictions of our model are in excellent quantitative agreement withour measured phase spectra. Comparison between experiment and theory enables us to infer the values of the

Optical Devices for Fiber Communication II, Michel J. F. Digonnet, Osman S. Gebizlioglu,Roger A. Greenwell, Dennis N. Horwitz, Dilip K. Paul, Editors, Proceedings of SPIEVol. 4216 (2001) © 2001 SPIE · 0277-786X/01/$15.00

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dc Kerr and electrostrictive constants of silica for both polarizations, and to make useful predictionsregarding the relative magnitudes of these two effects in a poled-silica phase modulator.

2. SUMMARY OF EXPERIMENTAL RESULTS

This section summarizes the experimental results of the aforementioned experiment, described inmore ample details in Ref. [9]. In order to make simple comparisons with theory, the sample was chosen tobe a slab of unpoled high-purity silica (Infrasil 301) 25 mm on the side and with a thickness d=406 µm. Afilm of Cr/Au 3000 Å thick and 25 mm long was deposited on the slab's top and bottom surfaces to serve aselectrodes. The sample was attached, though not rigidly clamped to avoid acoustic damping, to a largealuminum ground electrode. A voltage V=Vdc+Vmcosωmt was applied between the electrodes, where Vdcwas a large dc voltage of the order of several hundred volts and Vm a small modulation voltage at frequencies

f=ωm/2π in the range of 0.5 to 19 MHz. The non-electroded faces of the slab were polished for optical

coupling. The phase shift ∆φ at ωm induced in the slab was measured by placing the sample in a fiber Mach-Zehnder interferometer stabilized against slow ambient temperature drifts. The probe signal was the beam ofa HeNe laser (λ=633 nm). It was linearly polarized, either parallel or perpendicular to the applied field, andtraveled mid-point between the electrodes. The absolute phase shift in the sample was calibrated against aLiNbO3 modulator. The minimum detectable phase shift was ~3 µrad.

The measured dependence of the phase shift on modulation frequency for each polarization is shownin Fig. 1.[9] The oscillations in the measured phase shift above 8 MHz are due to system noise and pick-up ofambient acoustic vibrations. As discussed earlier, ∆φ is the sum of a frequency-independent contribution dueto the electronic Kerr effect and a strongly resonant contribution due to electrostriction at f=7.34 MHz. Nearresonance electrostriction far exceeds the electronic contribution, and the electrostriction component for lightpolarized perpendicular to the applied field exceeds the parallel component by a ratio of ~2.3. Away fromresonance the induced index change is essentially independent of polarization.

3. MODEL OF LOW-FREQUENCY KERR MODULATION

This model aims to develop an analytical expression for the refractive index modulation ∆n inducedin a rectangular slab by a voltage V=Vdc+Vmcosωmt. This analysis is restricted to isotropic linear materialssuch as fused silica, in which the linear electro-optic and piezoelectric effects are forbidden. In this case theindex modulation is due to the sum of an electronic contribution ∆nKerr, which we assume at the outset is

independent of frequency, and a strongly resonant electrostrictive contribution ∆nEs(ωm). For an opticalsignal parallel and perpendicular to the applied field, the index modulations can be written as:

∆n// (ωm ) = ∆n//,Kerr + ∆n//,Es(ωm )

∆n⊥ (ωm ) = bk∆n//, Kerr + bes∆n//, Es(ωm)(1)

where bes and bk are frequency-independent coefficients that describe the polarization dependence of theelectrostrictive and quadratic Kerr effect, respectively.

In the low-frequency range of interest, the contribution of the electronic Kerr effect is proportional to

the third-order susceptibility χ(3)(-ω;ω,0,0):

Proc. SPIE Vol. 4216120


10-6

10-5

10-4

10-3

10-2

0 5 10 15 20

Parallel PolarizationPerpendicular Polarization

Phas

eS

hift

[rad

]

Modulation frequency [MHz]

Vm

= 15 V

Vdc

= 2220 V

Figure 1. Measured phase shift dependence on the modulation frequency in unpoled silicaslab for light polarized parallel and perpendicular to the applied field.

∆n//,Kerr =3

nχ1111

(3) (−ωm;ωm ,0,0)EdcEm

∆n⊥ ,Kerr =3

nχ1122

(3) (−ωm ;ωm ,0,0)Edc Em

(2)

where n is the refractive index of the material, Edc=Vdc/d is the applied dc field, Em=Vm/d is the applied ac

field, and χ(3)1111 and χ(3)1122 are the relevant third-order susceptibility tensor elements of the material.

For an isotropic material, these elements satisfy χ(3)1111=3χ(3)1122,[11] which means that the constant bkin Eq. 1 is equal to 1/3.

The index modulation due to electrostriction stems from the applied field, which induces a strain andthus an index change through the elasto-optic effect. In an isotropic material such as unpoled silica, the onlynon-zero elasto-optic tensor elements (in contracted index notation) are p11 and p12.[12] Furthermore, for arectangular slab with a thickness much smaller than the other two dimensions, the transverse straincomponent S11 is negligible. Consequently, the electrostrictive index modulations are given by:

∆n//,Es ≈1

2n3p11S33(ωm)

∆n⊥ ,Es ≈1

2n 3p12S33(ωm )

(3)

where S33(ωm) is the longitudinal strain component. Equation (3) shows that for this slab geometry, thepolarization ratio bes for electrostriction in Eq. 1 is simply bes≈p12/p11. For silica, p11=0.121 andp12=0.27,[13] so that bes≈2.23. This result states that electrostriction induces a stronger index change for theperpendicular polarization than for the parallel polarization, by a factor of ~2.23. The same calculation

Proc. SPIE Vol. 4216 121


repeated without the approximation S11=0 made here yields an exact value for bes of 2.15,[14] which lendscredence to our approach. This result differs from the claim made in Ref. [10] that in a fiber geometry with asimilar electrode configuration, the electrostrictive index change is independent of polarization.

The frequency dependence S33(ωm) can be determined from the equation of motion of a particle inthe material, which leads to the standard wave equation for acoustic propagation in a rectangular slab.[15]The solution of this equation that satisfies the boundary conditions of the rectangular slab (zero longitudinalstress at the electroded surfaces of the slab) yields the dependence of the strain S33(ωm) along the direction zof the applied field:

S33(ωm) =Γ33

c33 + jωmη33

cos[β(z − d 2)]

cos β d 2EdcEm (4)

where Γ33 is the electrostrictive coefficient in contracted index notation, c33 is the material elastic

coefficient, η33 is the viscosity of the material, which describes the acoustic loss, and β is the acoustic wave

propagation constant. For a material with a specific density ρ0, it is given by:

β 2 =ωm

2 ρ0

c33 + jωmη33

(5)

By inserting the solution for the strain (Eq. 4) into Eq. 3, one obtains the spatial, polarization, andfrequency dependence of the index modulation due to electrostriction. These expressions, together with Eq.2, are then inserted in Eq. 1 to obtain the total field-induced index modulation:

∆n// (ωm ) = [3

nχ1111

(3) (−ωm;ωm ,0,0) +1

2

n3 p11Γ33

c33 + jωmη33

cos[β(z − d 2)]

cosβ d 2]Edc Em (6a)

∆n⊥ (ωm ) = [3

nχ1122

(3) (−ωm;ωm ,0,0) +1

2

n3 p12Γ33

c33 + jωmη 33

cos[β(z − d 2)]

cosβ d 2]Edc Em (6b)

For each polarization, the index change is made of a dc term (first term) that depends on polarizationbut is nearly independent of frequency, and a modulated term (second term) with an amplitude that dependson polarization. The electrostrictive term exhibits a resonance at the theoretical frequency f0=v/2d, where

v=(c33/ρ0)1/2 is the longitudinal wave acoustic velocity.

4. INTERPRETATION OF EXPERIMENTAL RESULTS

Figure 2a reproduces the measured phase shift spectrum for the parallel polarization of Fig. 1 as wellas a the best theoretical fit to this spectrum plotted from Eq. 6a. Figure 2b does the same for theperpendicular polarization, with a best fit was plotted from Eq. 6b. These fits were calculated for the

parameter values of silica, namely ρ0=2.2x103 Kg/m3, c33=7.81x1010 Kg/m2/s,[15] and z=d/2=2.03x10-4 m

(since the laser beam passed through the middle of the slab). The three remaining parameters, χ(3)1111, Γ33,

and η33, have unknown values and were used as fitting parameters. To first order, χ(3)1111 and Γ33 control

the magnitude of the offset in the spectrum; Γ33 and η33 control the magnitude of the resonance; and η33controls the linewidth of the resonance. In Figs. 2a and 2b, the



10-7

10-6

10-5

10-4

10-3

10-2

0 5 10 15 20

Phase Shift

Theory

Electrostriction

KerrPh

ase

Shi

ft[m

rad]

Modulation Frequency [MHz]

(a)

10-6

10-5

10-4

10-3

10-2

0 5 10 15 20

Phase Shift

Theory

Electrostriction

Kerr

Phas

eS

hift

[rad

]


(b)

Figure 2. Theoretical fit to the measured phase shift spectrum for signal polarized(a) parallel, and (b) perpendicular to the applied field.

dotted and dashed curves are the respective contribution of the electronic and electrostrictive components,and the solid curves are the sum of the two. For both polarizations the agreement between theory andexperiment is very good, except in the region where the response is of the order of 3 µV and is dominated bynoise, which lends credence to our interpretation of the inherent physical mechanisms and of their frequency

and polarization dependencies. In particular, it confirms the near frequency independence of χ(3) over thefrequency range of interest, and the value of bes=2.23 for the polarization ratio of electrostriction. For silica

the longitudinal velocity is v=(c33/ρ0)1/2=5.96 Km/s, so that the theoretical resonance frequency isf0=v/2d=7.339 MHz. This value is in excellent agreement with the measured value of 7.34 MHz. The



measured 3-dB bandwidth of the resonance peak, ∆f=0.02 MHz, predicts a resonant enhancement ofQ=f0/∆f=367. It agrees well with the observed amplitude increase of 350 at resonance.

The values of the fitting parameters are χ(3)1111(-ωm;ωm,0,0)=1.9x10-22 m2/V2, Γ33=3.44x10-11

N/V2, and η33=7.08 Kg/m/s. Therefore χ(3)1122(-ωm;ωm,0,0)=χ(3)1111(-ωm;ωm,0,0)/3=0.63x10-22

m2/V2. The third-order susceptibility around dc, χ(3)1111(-ωm;ωm,0,0), is close to the range of values (1.7-

2.1x10-22 m2/V2) reported for the susceptibility of silica fibers at optical frequencies, χ(3)1111(-ωm;ωm,-

ωm,ωm).[10,16] It suggests that in this material, the dispersion of χ(3) between dc and optical frequency issmall. Also, the 1:3 polarization ratio still holds around dc, as expected from symmetry considerations. Thefitted value of η33 is greater than reported in the literature by about three orders of magnitude,[17] whichmay be due to electrical loss in the system. We noticed that the slab tapers slightly (by an angle of ~6 mrad)in the direction of propagation, which broadens the resonance. When this effect was included in the model,

the fitted values of χ(3)1111 and Γ33 did not change appreciably, but η33 was lowered by a factor of 2. Our

confidence in the inferred value of η33 is consequently rather low, but it has no impact on the validity of the

other constants we infer. As an independent check, the inferred value of the electrostriction constant Γ33 is

reasonably close to other measured values, namely 4.25x10-11 N/V2 (calculated from [18]) and 3.8x10-11

N/V2 (calculated from [19]). However, it differs markedly from the theoretical value of 0.97x10-11 N/V2

reported in [20]), which suggests that the latter may be strongly underestimated.

5. IMPLICATIONS FOR POLED SILICA MODULATORS

Electrostriction can lead to extremely large phase shifts at a relatively low modulation voltage. In apoled silica sample, in which the dc field is not supplied externally but built-in during poling, we expect thatthis feature can produce useful, albeit narrowband, phase modulators. The main constraint is that the sample(fiber or waveguide) must not be too rigidly clamped. As an example, we show the calculated electronic andelectrostrictive contributions to r33 (Fig. 3a) and r13 (Fig. 3b) for a poled silica sample with the sameacoustic loss as measured in our slab but with a built-in field close to the dielectric breakdown field of silica(Edc=800 V/µm). These curves were calculated from Eqs. 6 rewritten in terms of electro-optic coefficients.(In other words, they are the fits of Figs. 2 rescaled for a dc field of 800 V/µm and replotted as electro-opticcoefficients). They show that at resonance, the contributions to the electro-optic coefficient r33 are 0.2 pm/Vand 12 pm/V for the electronic Kerr effect and electrostriction, respectively. The corresponding figures forr13 are 0.07 pm and 28 pm/V. The electro-optic coefficient due to electrostriction in our sample is thereforeover ten times larger than the electro-optic coefficients reported for thermally poled silica, which aretypically under 0.5 pm/V. This shows that in an unclamped sample, electrostriction may be used to fabricatenarrowband, low-voltage phase modulators in the MHz range and higher.

Figure 3b also shows that in an unclamped sample, even away from resonance the electro-opticcoefficient r13 is largely electrostrictive. A practical measurement of the electro-optic coefficient is thereforelikely to produce an r13 value considerably larger than expected from an independent measurement of d13(which is unaffected by electrostriction). One must thus be careful in interpreting such measurements. Incontrast, r33 has a dominantly electronic origin (see Fig. 3a), so that for this polarization the ambiguity is notpresent.

As the modulation frequency is increased, the induced phase shift goes through severalelectrostriction resonances, and the polarization dependence of the phase shift is expected to change. This isillustrated in Fig. 4, which plots the ratio of electro-optic coefficients R=r13/r33, again for the



0.01

0.1

1

10

100

0 5 10 15 20

Ele

ctro

-opt

icco

effi

cien

tr33

[pm

/V]


Electrostriction

Kerr

(a)

0.01

0.1

1

10

100

0 5 10 15 20

Ele

ctro

-opt

icco

effi

cien

tr13

[pm

/V]


Electrostriction

Kerr

(b)

Figure 3. Predicted frequency dependencies of the electro-optic coefficient(a) r33, and (b) r13 of poled silica assuming a built-in field of 800 V/µm.

same loss as in our experimental device. At low frequency, R oscillates as the phase shift goes throughsuccessive harmonics of the resonance. At every frequency for which R=1, the overall modulation amplitudeis independent of polarization. A consequence of this property is that a device can be polarizationindependent by operating at one of these particular frequencies. Another consequence is that a measurementof the ratio of R, which has been invoked in the past to identify the symmetry of poled silica, does not

necessarily lead to 1/3, because of electrostriction. The acoustic loss increases rapidly with frequency (as f2),and at high enough frequency electrostriction becomes negligible. The modulation is then due entirely to theelectronic Kerr effect and R goes to 1/3, as shown in Fig. 4.



0

0.5

1

1.5

2

2.5

3

3.5

0 1 108 2 108 3 108 4 108 5 108

Rat

ior 33

/r13

Frequency (Hz)

Limit with no electrostriction = 3

η33

=7.08 Kg/m/s

Figure 4. Predicted frequency dependence of the electro-optic coefficient ratio r33/r13 forpoled silica.

It is interesting to study the effect of the acoustic loss, and thus the effectiveness of acousticdamping, on the phase modulation produced by such a device. This can be done simply by increasing the lossη33 in Eq. 6. Figure 5 shows the evolution of the index change spectrum (normalized to the applied field) forthe perpendicular polarization as the acoustic loss is increased in 10-fold increments starting from themeasured loss of our experimental device. As the acoustic loss is increased, the electrostriction peaks becomeweaker and eventually vanish. For extremely high loss, electrostriction is only present at very lowfrequencies. For example, for a loss 1000 times higher than our device's loss

10-13

10-12

10-11

10-10

0 2 107 4 107 6 107 8 107 1 108

Loss=7.079 Kg/m/sLoss=70.79 Kg/m/sLoss=707.9 Kg/m/sLoss=7079 Kg/m/s

Inde

xch

ange

per

unit

appl

ied

fiel

d

Frequency (Hz)

Perpendicular polarization

Figure 5. Evolution of the electrostriction peaks with increasing acoustic loss.



electrostriction is present only below ~5 MHz. Note that this level of loss is extremely high: it corresponds toabout 13 dB/µm at 10 MHz. Furthermore, even in the presence of such tremendous losses, below ~5 MHzelectrostriction still exceeds the electronic component. The same qualitative comment applies for the parallelpolarization. This theory therefore shows that (1) a sample must be extremely well damped beforeelectrostriction is fully gone, and (2) unlike stated in some references, the level of acoustic attenuationrequired to fully remove electrostriction may be difficult to attain, at least at some frequencies.

6. CONCLUSIONS

We measured the frequency and polarization dependence of the electrostrictive and electroniccontributions of the Kerr effect at low frequency (0-20 MHz) in an unpoled silica slab. The electroniccontribution is found to be essentially independent of frequency, while the electrostrictive contributionexhibits strong peaks due to mechanical resonances in the slab. A theoretical model is presented that gives

good quantitative agreement with observations. The silica constants inferred from this work are χ(3)1111(-

ωm;ωm,0,0)=1.94x10-22 m2/V2 for the dc electronic Kerr coefficient, and Γ33=3.44x10-11 N/V2 for theelectrostrictive constant. The dc electronic Kerr coefficient is essentially the same as the optical Kerrcoefficient, which suggests little to no dispersion. For light polarized perpendicular to the applied field,electrostriction is observed to largely exceed the electronic component over the entire frequency range understudy. The polarization dependence of the electrostriction coefficient is found to be 2.35, This st udy showsthat electrostriction is expected to be present in electro-optic devices based on poled silica unless eitheracoustic waves are extremely well damped or the device is operated at very high frequencies (GHz range).

7. REFERENCES

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3. A. Le Calvez, E. Freysz, and A. Ducasse, "A model for second harmonic generation in poled glasses,"Eur. Phys. J. D}, vol. 1, no. 2, 223-226, 1998.

4. M. C. Farries, and A. J. Rogers, "Temperature dependence of the Kerr effect in a silica optical fibre,"Electron. Lett., vol. 19, no. 21, 890-891, 1983.

5. X. C. Long, R. A. Myers, and S. R. J. Brueck, "Measurement of the linear electro-optic coefficient inpoled amorphous silica," Opt. Lett., vol. 19, no. 22, 1819-1821, 1994.

6. E. L. Buckland, and R. W. Boyd, "Measurement of the frequency response of the electrostrictivenonlinearity in optical fibers," Opt. Lett., vol. 22, no. 10, 676-678, 1997.

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8. P. J. Hardman, P. D. Townsend, A. J. Poustie, and K. J. Blow, "Experimental investigation of resonantenhancement of the acoustic interaction of optical pulses in an optical fiber," Opt. Lett., vol. 21, no. 6,393-395, 1996.

9. A. C. Liu, M. J. F. Digonnet, and G. S. Kino, "The DC Kerr Coefficient of Silica: Theory andExperiment," Proc. of the SPIE - The International Society for Optical Engineering, vol. 3542, 102-107,1998.

10. A. Melloni, M. Frasca, A. Garavaglia, A. Tonini, and M. Martinelli, "Direct measurement ofelectrostriction in optical fibers," Opt. Lett., vol. 23, no. 9, 691-693, 1998.

11. P. N. Butcher, and D. Cotter, The Elements of Nonlinear Optics, vol. 9 of Cambridge Studies in ModernOptics, Cambridge: Cambridge University Press, 1990.

12. J. F. Nye, Physical properties of crystals. Clarendon Press, Second edition, 1985.



13. R. W. Dixon, "Photoelastic properties of selected materials and their relevance for applications toacoustic light modulators and scanners," J. Appl. Phys., vol. 38, no. 13, 5149-5153, 1967.

14. M. Paillette, "Kerr effect and electrostriction in lead silicate glasses." Journal de Physique, vol. 37, no.7-8, 855-864, 1976.

15. G. S. Kino, Acoustic waves : devices, imaging, and analog signal processing. Englewood Cliffs, N. J.:Prentice-Hall, 1987.

16. T. Kato, Y. Suetsugu, M. Takagi, E. Sasaoka, and M. Nishimura, "Measurement of the nonlinearrefractive index in optical fiber by the cross-phase-modulation method with depolarized pump light,"Opt. Lett., vol. 20, no. 9, 988-990, 1995.

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20. M. Khoshnevisan, and P. Yeh, "Relationship between nonlinear electrostrictive Kerr effects andacousto-optics," Proc. of the SPIE - The International Society for Optical Engineering, vol. 739, 82-86,1987.



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SPIE Proceedings [SPIE Information Technologies 2000 - Boston, MA (Sunday 5 November 2000)] Optical Devices for Fiber Communication II - Measurement of the electrostrictive constants