872 OPTICS LETTERS / Vol. 24, No. 13 / July 1, 1999

Mode-profile dependence of the electrostrictiveresponse in fibers

Eric L. Buckland*

112 Mariner’s Point Lane, Hickory, North Carolina 28601

Received February 5, 1999

The transverse optical intensity profile in fibers affects the efficiency of acoustic mode excitation by theoptical f ield and the subsequent response of the field to the excited acoustic modes. The magnitude of theelectrostrictive nonlinear coeff icient for a square-top intensity profile will exceed that of a Gaussian profile by afact of 2. For current fiber designs the range of values for n2str at zero frequency is expected to vary from 0.43to 0.71 3 10216 cm2 W21 based on mode profile alone. The relative contribution of electrostriction to the totalnonlinear response �electrostrictive 1 Kerr� in fibers increases proportionately as the mode profile f lattens. 1999 Optical Society of America

OCIS codes: 060.2270, 060.4370, 060.4510, 060.5530, 190.4370.

The effect of electrostriction on the intensity-dependent dynamics of light propagation in opticalfibers is currently an area of active research. Electro-striction, the process by which light excites acoustic vi-brations in matter, is the origin of stimulated Brillouinscattering in optical fibers, a process that limits theforward power transmission of narrow-bandwidth lightand couples counterpropagating waves.1 Whereasstimulated Brillouin scattering originates from thelongitudinal (axial) material response, the transverse(radial) electrostrictive response also inf luences lightpropagation. For example, transverse electrostric-tion is now accepted to be the origin of a long-rangesoliton interaction.2,3 Most recently, this transverseelectrostrictive response has been shown to cause asignif icant frequency-dependent nonlinear refractionin optical fibers, extending to modulation frequenciesof 1–2 GHz.4,5 Theory indicates a value of the electro-strictive nonlinear refractive index n2str of approxi-mately 0.57 3 10216 cm2 W21,6,7 although there hasbeen some discrepancy noted in measured values.

Without exception, the research to date has assumeda Gaussian optical mode. Although the fields of step-index (SI) fibers near cutoff are well approximatedby a Gaussian profile, this approximation is inade-quate for studying the electrostrictive response, asthis response depends strongly on the second trans-verse derivative of the intensity profile. Significanteffects that arise purely from the geometry of the opti-cal mode have been overlooked. The theory developedin previous studies is extended here, and the opticalfield is modeled as a Gaussian-like field for analysisof the geometry-dependent electrostrictive response.This model is more appropriate for describing bothsub-Gaussian intensity distributions (e.g., SI fibers farfrom cutoff and various dispersion-shifted fibers) andthe f lattened super-Gaussian profile of current large-effective-area fibers. The results show for what is be-lieved to be the first time that the magnitude of thephysical electrostrictive response is strongly dependenton the geometry of the optical mode, in stark contrastwith the Kerr nonlinearity. The relative contribution

0146-9592/99/130872-03$15.00/0

of electrostriction to the total nonlinear response de-pends on the mode geometry and is independent of thenonlinear effective area.

Electrostriction has the unique characteristic amongintensity-dependent processes in optical fibers of beingexplicitly dependent on the transverse optical intensityprofile. The acoustic wave equation describes thematerial density variation in the presence of an appliedelectromagnetic field,1

≠2r

≠t22 G0D�

≠r

≠t2 vs2D�r � 2

ge2

4pD�jEj2, (1)

where r is the time-dependent material density, vs �5.996 km�s is the speed of sound in the glass, ge �r0�≠e�≠r� is the electrostrictive coefficient, and jEj isthe slowly varying envelope of the propagating opticalfield. The density of the medium is r0, and e is the di-electric constant. In the present case the propagatingoptical signal excites the acoustic vibration, inducinga time-retarded perturbation to the refractive index.We can write the frequency-domain perturbation to therefractive index owing to electrostriction as6

Dn�V;a,p� � n2strP0B�V�Aeff

H �V;a,p� , (2)

where V is the optical modulation frequency in radi-ans, n2str � ge

2��4r0cn2vs2� is the zero-frequency (cw)electrostrictive nonlinear coefficient, P0 is the peakoptical power, Aeff is the effective nonlinear area ofthe optical mode. B�V� �

R`

2` dt exp�iVt�P �t��P0is the spectrum of the optical envelope, and H �V�is the electrostrictive frequency-response function.The refractive-index perturbation and the frequency-response function are parameterized here by variablesa and p that describe the transverse intensity profilef 2�r� � exp�2�r�a�p� of a Gaussian-like optical mode.

For a precisely Gaussian �p � 2� intensity distribu-tion the response function can be described as a sumover acoustic modes n, such that2,8

H �V� � AXn�1

BnCn

V2 2 Vn2 2 2iGnV

. (3)

1999 Optical Society of America

July 1, 1999 / Vol. 24, No. 13 / OPTICS LETTERS 873

Bn �R2p

0

RR0 fn�r�D�f2�r�rdrdf and Cn �

R2p

0

RR0 3

fn�r�f2�r�rdrdf are overlap integrals representing theexcitation of acoustic waves by the optical wave as apump and the response to the acoustic waves of theoptical wave as a probe, respectively. This function isnormalized through A such that the cw value of therefractive-index perturbation, n2strH�0�, matches the-ory and empirical results. This normalization hidesthe dependence of the response on the optical intensityprofile.

The response function can be expressed alternativelyby use of the infinite cladding model of the acousticresponse. The frequency-response function in thismodel is given by

H �V;a,p� �vs2

4p2

Z `

0

b�q�c�q�V2 2 vs2q2 2 2iGV

qdq . (4)

This follows directly from Eq. (4) in Ref. 6, recast tomaintain notational uniformity with Eq. (3). The cor-responding overlap integrals are b�q� � 2q2

R2p

0

R`

0 3

J0�qr�f2�r�rdrdf and c�q� � �p�a2G�2�p��R2p

0

R`

0 3J0�qr�f2�r�rdrdf, where q is the acoustic wave vectorand the acoustic modes form a continuum of the formJ0�qr�, where J0 is the zeroth-order Bessel function ofthe first kind. Here, G�2�p� is the exponential gammafunction, and the prefactor to the integral arises fromthe normalization of the optical intensity distribution.These two models converge when G�fR . 1, where G

is the acoustic damping rate and fR � vs�2R is theresonant frequency of the acoustic echoes.

The continuum model of Eq. (4) is useful for exam-ining the material and the geometric dependences ofthe electrostrictive response function. First, the fre-quency extent of the response varies in proportion toa characteristic frequency, fa � vs�a, that representsthe inverse of the transit time for the acoustic wave totraverse the optical core. The damping coefficient Gaffects the envelope of the frequency response weaklybut is a primary factor in determining the visibilityof resonances in the discrete-mode model. Other ma-terial properties do not inf luence H �V� but certainlydetermine the magnitude of n2str, as indicated above.

The geometry of the optical mode, however, has astriking effect on the magnitude of H �V�. As shownin Fig. 1, as the mode profile is f lattened (p is in-creased), the efficiency of excitation of acoustic modesby the optical intensity and the subsequent responseof the optical field to the acoustic modes are in-creased. The material response is directly affected bythe optical mode profile; this is not a result of anychange in the effective mode area. The magnitudeof the profile-dependent response can be evaluatedanalytically at zero frequency by use of Eq. (4) for thecases p � 1, 2, `. These exact values for H �0; a, p ��1, 2, `�� are plotted in Fig. 2, along with numericalresults across this range of parameter p. The nu-merical results are identical for both the continuumand the discrete models. In the limiting case of thesquare-wave profile �p � `�, the value of the effec-tive nonlinear coefficient n2str�effective� � n2strH �0� isequivalent to the theoretical prediction for the plane-wave model,1 and this value is twice that reported

previously for the Gaussian profile.9 This limit canbe realized with a SI multimode fiber. The practi-cal range for current single-mode fibers is from ap-proximately p � 1.5 for SI fibers far from cutoff andzero-dispersion (dispersion-shifted) fibers to p � 2.9for recent large-effective-area fibers, as deduced fromrecent literature.10 Even in this limited range, thevalue of n2strH �0� is expected to vary from 0.43 to0.71 3 10216 cm2 W21 based on the mode-profile ef-fect alone, assuming a Gaussian-mode value of 0.57 310216 cm2 W21. The Kerr nonlinearity does not havethis profile dependence; the strength of the electrostric-tive nonlinearity relative to the Kerr nonlinearity asa function of p is expressed by the right-hand axisof Fig. 2.

As the mode-profile shape is modified, the effectivenonlinear area can be modified as well (equiva-lently for both the electrostrictive and the Kerrresponses). One can evaluate the combined effect ofthe modified response H �V; a, p� and the effective

Fig. 1. (top) Acousto-optic overlap integrals 2b�q� and(bottom) c�q�, calculated with the continuum-mode model.

Fig. 2. Magnitude of the electrostrictive response H �0; p�and the strength of electrostriction relative to the Kerrresponse. The open circles are the exact calculations.The solid curve represents numerical results. The dashedlines indicate the regions of primary interest for currentfiber designs.

874 OPTICS LETTERS / Vol. 24, No. 13 / July 1, 1999

Fig. 3. Temporal variation in the refractive index owing toelectrostriction driven by an optical impulse. The mode-field radius is vff � 3.5 mm. The ordinate is scaled withn2strH �0; 2� � 0.57 3 10216 cm2 W21.

area Aeff �a; p� by considering fibers designed bymodification of the shape of the mode profile sothat equivalent spot sizes but different effectiveareas are achieved. The far-field spot size11 can becalculated from the near-field profile by use of the rela-tion12 vff

2 � 2R`

0 f2�r�rdr�

R`

0 �≠f �r��≠r�2rdr to obtainvff

2 � 8G�2�p�p22a2 for the p-parameterized distribu-tion. The effective nonlinear area,13 Aeff � �

R2p

0

R`

0 3

f 2�r�rdrdf�2�R2p

0

R`

0 f4�r�rdrdf, can be evaluated in

terms of the spot size and the p parameterization,yielding

Aeff � pvff22�2/p22�p . (5)

The combined effect of a f lattened p . 2 mode pro-file on fibers is demonstrated in Fig. 3. In this fig-ure the electrostrictive impulse response functions forfibers designed with a fixed spot size vff � 3.5 mmare plotted for increasing values of p. Three elementsinf luence the electrostrictive impulse responses shownin Fig. 3: (a) n2strH �0� increases with p, increasingthe physical response; (b) Aeff increases with p, de-creasing the optical intensity and thus the aggregateresponse; and (c) the intensity radius a increases withp, increasing the effective electrostrictive time con-stant, ta � 1�fa. The resultant magnitude of the elec-trostrictive response is thus seen to decrease as theprofile f lattens (with the spot size held constant),and the electrostrictive response will integrate over alonger time scale. Furthermore, the reduced magni-

tude of the first echo (inset of Fig. 3) implies reducedelectrostriction-induced timing jitter.

In conclusion, the profile of the optical mode ex-plicitly modifies the eff iciency of the acousto-opticinteraction in fibers. The magnitude of the elec-trostrictive nonlinearity is expected to vary by almosta factor of 2 for current optical fiber designs based onthe mode geometry alone. This effect is not presentfor the Kerr nonlinearity, so the relative contributionof electrostriction to the total nonlinear response is anequally strong function of the optical mode profile.

The author thanks R. W. Boyd and A. Mellonifor helpful discussions; e-mail address, buckland@abts.net.

*Present address, Corning Incorporated, Corning,New York 14831.

References

1. R. W. Boyd, Nonlinear Optics (Academic, Boston,Mass., 1992).

2. E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, andA. M. Prokhorov, Appl. Phys. B 54, 175 (1992).

3. E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, andA. N. Starodumov, Opt. Lett. 15, 314 (1990).

4. E. L. Buckland and R. W. Boyd, Opt. Lett. 22, 676(1997).

5. A. Melloni, M. Martinelli, and A. Fellegara, FiberIntegr. Opt. 18, (1998).

6. E. L. Buckland and R. W. Boyd, Opt. Lett. 21, 119(1996).

7. A. Melloni, M. Frasca, A. Garavaglia, A. Tonini, and M.Martinelli, Opt. Lett. 23, 691 (1998).

8. A. Fellagara and S. Wabnitz, Opt. Lett. 23, 1357 (1998).9. Note that n2str as reported in Eq. 3(a) of Ref. 6 implic-

itly contains a factor of 1�2 that follows from the Gauss-ian optical mode. General agreement between thetheoretical predictions in Refs. 6 and 7 occurs in spiteof this factor-of-2 discrepancy in n2str . This agreementis due to an offsetting discrepancy in the value of gethat is yet to be resolved.

10. V. L. da Silva, Y. Liu, A. J. Antos, G. E. Berkey,and M. A. Newhouse, in Optical Fiber CommunicationConference, Vol. 2 of 1996 OSA Technical Digest Series(Optical Society of America, Washington, D.C., 1996),p. 202.

11. ‘‘Mode field diameter, variable aperture in the farfield,’’ ANSI/TIA/EIA-455-167 (American NationalStandards Institute, New York, 1998).

12. E.-G. Neumann, ed., Single-Mode Fibers (Springer-Verlag, Berlin, 1988).

13. G. P. Agrawal, Nonlinear Fiber Optics (Academic, SanDiego, Calif., 1989).