intrinsic and induced anisotropy of nonlinear absorption and refraction in zinc blende...

$: Intrinsic and induced anisotropy of nonlinear absorption and refraction in zinc blende semiconductors$
Schroeder et al. Vol. 12, No. 3 /March 1995/J. Opt. Soc. Am. B 401

Intrinsic and induced anisotropy of nonlinear absorptionand refraction in zinc blende semiconductors

W. Andreas Schroeder,* D. S. McCallum, D. R. Harken, Mark D. Dvorak, David R. Andersen, and Arthur L. Smirl

Photonics and Quantum Electronics Laboratory, Iowa Advanced Technology Laboratories,University of Iowa, Iowa City, Iowa 52242-1000

Brian S. Wherrett

Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK

Received July 11, 1994; revised manuscript received October 24, 1994

An ellipsometric picosecond pump–probe technique is used to measure the birefringence and dichroism inducedin the zinc blende semiconductors ZnSe, GaAs, and CdTe by an intense linearly polarized pump pulse at950 nm. We show that the induced birefringence and dichroism depend strongly on sample orientation(i.e., they are anisotropic). Furthermore, we demonstrate that by measuring the induced birefringence anddichroism for the pump polarized along the [100] and [110] crystal axes we can determine the sign andthe magnitude of the intrinsic anisotropy both in the bound-electronic nonlinear refractive index and inthe two-photon absorption coefficient; and, finally, we can extract the anisotropy in both the real and theimaginary parts of the product of the anisotropy parameter s and the diagonal element of x s3d, the third-ordersusceptibility tensor. The latter product is a measure of the intrinsic anisotropy of the material. We findthat the induced birefringence varies by roughly a factor of 2 with sample orientation in all three materials.The induced birefringence is, however, roughly an order of magnitude larger in size and opposite in sign forexcitation above half the band gap (in GaAs and CdTe) than it is for excitation below half the band gap (inZnSe). As expected, no dichroism is observed in ZnSe, but it is large in GaAs and CdTe, and, as with thebirefringence, the dichroism varies by roughly a factor of 2 with crystal orientation. In order to test theeffect of the measured anisotropy on device performance, we construct a simple on–off optical switch thatexploits the measured birefringence and dichroism. Figures of merit are defined, and switch performanceis investigated as a function of crystal orientation for excitation above and below the two-photon resonance(i.e., half the band gap). The figures of merit are shown to be extremely sensitive to crystal anisotropies forexcitation below half the band gap and less so for excitation above half the band gap.

1. INTRODUCTION

The performances of ultrafast nonlinear-optical de-vices such as waveguide switches,1 – 4 bistable etalons,5

and optical limiters6 – 10 are often determined by theproperties of the third-order susceptibility tensorx

s3dijkl of the nonlinear medium. For example, semi-

conductor nonlinear directional couplers1,4,11 rely onnonlinear refraction (i.e., the real part of x s3d) fortheir operation, and their performance is knownto be limited by two-photon absorption (i.e., by theimaginary part of x s3d). As a result, the measurementand theoretical understanding of the sign and magnitudeof the real and the imaginary parts of the third-ordersusceptibility tensor in semiconductors have receivedconsiderable attention in recent years.10,12 – 20 In par-ticular, use of the single-beam Z-scan technique21 – 24

has been very successful for determining the mag-nitudes and signs of both absorptive and refractiveoptical nonlinearities in a variety of semiconductormaterials. Recent work24 – 26 on zinc blende semi-conductors has indicated that the magnitudes of thenonlinear refractive indices n2 and the two-photonabsorption coefficients b in these materials depend oncrystal orientation as well as on the material itself.These anisotropies can be large [e.g., .40% in GaAs(Refs. 24–26)], and consequently, as we shall discuss

0740-3224/95/030401-15$06.00

here, some devices that depend on b and n2 for their op-eration will perform better for particular crystal cuts orgrowth directions.

Specifically, in our own recent studies25 we useda picosecond pump–probe technique to measure theanisotropy of the two-photon absorption coefficient asa function of crystal orientation and of pump and probepolarization. From these measurements we were able todetermine the magnitude and the sign of the anisotropyparameter s and to extract independently the magni-tude and the sign of the imaginary parts of all threeindependent x s3d tensor elements involved in two-photonabsorption for selected zinc blende semiconductors. Thistechnique, as applied, provided such information onlyfor the absorptive component of the nonlinearities andprovided no information on the refractive component.

Here we describe a complementary procedure that al-lows for the determination of the anisotropy both inthe instantaneous bound–electronic nonlinear refractiveindex (defined by n2) and in the nonlinear absorption co-efficient (defined by a2) and that can be used in the trans-parent region (i.e., below the two-photon resonance) andin the absorptive regime (i.e., above the two-photon reso-nance) of the material. The pump–probe technique thatwe use here is a novel extension of the Kerr ellipsome-try technique described by Pfeffer et al.,27 which, in turn,was based on earlier Kerr gate experiments.28,29 In this

1995 Optical Society of America

402 J. Opt. Soc. Am. B/Vol. 12, No. 3 /March 1995 Schroeder et al.

procedure an intense, linearly polarized pump pulse isused to induce a birefringence and a dichroism in theoptical properties of the material as experienced by aweaker probe pulse. Here, for our purposes, this bire-fringence is characterized by Dn2 n2

k 2 n2', where n2

k

and n2' are the nonlinear refractive indices measured by

probe light polarized parallel to and perpendicular to, re-spectively, the polarization of the pump pulse, and simi-larly, the dichroism is described by Da2 a2

k 2 a2',

where a2k and a2

' are the nonlinear absorption coeffi-cients measured by probe light polarized parallel to andperpendicular to, respectively, the polarization directionof the pump. We then measure this pump-induced bire-fringence and dichroism by determining the polarizationstate of the transmitted probe light by measuring itstransmission through a crossed analyzer, with and with-out an intervening quarter-wave plate in place.

It is well known that this optically induced anisotropyis observed even in intrinsically isotropic materials, suchas liquids and glasses.27 – 29 In such materials the princi-pal axes for the induced birefringence and dichroism aredetermined totally by the direction of polarization of theexciting radiation and are independent of sample orienta-tion, because both linear and third-order susceptibilitiesare intrinsically isotropic. By contrast, however, for zincblende materials the third-order susceptibility tensor isintrinsically anisotropic.24 – 26 Consequently, the magni-tudes of the induced anisotropies, Dn2 and Da2, shoulddepend on sample orientation. In this paper we mea-sure the anisotropies of the induced birefringence Dn2 andof the induced dichroism Da2 for selected sample orien-tations in ZnSe, where the photon energy is below thetwo-photon resonance, and in GaAs and CdTe, where thephoton energy is above the two-photon resonance. Wefind that Dn2 and Da2 vary by approximately a factorof 2 with sample orientation in these materials, indicat-ing that the intrinsic anisotropies in the induced birefrin-gence and dichroism are large.

In addition, we present macroscopic expressions for theanalyzer transmission, with and without the quarter-wave plate present, that include averaging over thepulse’s spatial and temporal profiles. These expressionsrelate the sample-orientation dependence of the inducedbirefringence Dn2 and of the induced dichroism Da2 tothe anisotropy of the nonlinear refractive index n2 andof the nonlinear absorption coefficient a2, respectively.We also present expressions that, in turn, relate eachof these quantities to the intrinsic anisotropy in the x s3d

tensor elements. Using these expressions, we extractthe magnitudes and signs of the anisotropies in n2 anda2, and we determine both the real and the imaginaryparts of the intrinsic anisotropy.

Finally, primarily to illustrate the effects of theanisotropies in n2 and a2 on device performance, wedemonstrate the operation of a simple on–off all-opticalpolarization rotation switch that exploits the inducedbirefringence and dichroism measured here. We demon-strate fast, efficient switching action for all three of thesemiconductor materials mentioned above. More impor-tantly, we demonstrate that the anisotropy in n2 anda2 measured here can cause a factor-of-5 variation inswitch performance as the sample orientation is changed.We define figures of merit for this switch, we compare

them with the figures of merit defined for several otherdevices, and we discuss the implications of anisotropyfor each.

2. EXPERIMENTAL METHODOur experimental geometry and the nomenclature that weadopt are shown in Fig. 1. The optical source for theseexperiments is a synchronously amplified, Styryl 13 dyelaser system that produces 1-ps pulses at 950 nm at a10-Hz repetition rate.30,31 Each laser pulse is split intotwo parts to provide a pump pulse and a probe pulse.The pump pulse is s polarized (as indicated by theunit vector e in the inset of Fig. 1) and has a nearlycircularly symmetric Gaussian spatial profile with ameasured radius [half-width at the 1ye point of the ir-radiance (HWe21I)] of 500 mm at the sample. A delaystage in the probe path allows the nonlinear propertiesof the sample to be studied as a function of delay be-tween the pump and probe pulses. The delay stage alsoenables the temporal coincidence of the pump and probepulses to be determined to within 100 fs by the use ofbackground-free second-harmonic autocorrelation tech-niques. A half-wave plate and polarizer P1 are used torotate the probe polarization (indicated by p) to 45± withrespect to the pump polarization. The probe is thenrecombined with the pump so that they are spatially co-incident in the sample. We focus the probe pulse (usinglens L1) to a spot on the sample that is smaller thanthe pump (,100-mm HWe21I) to ensure that the probesamples an area that has experienced approximately uni-form pump irradiance. In all cases the pump-to-probepulse energy ratio is maintained at approximately 500:1so that the pump-to-probe fluence ratio always exceeds20:1. The angle between the pump and the probe beams(19±) is chosen to permit clear spatial separation of thetwo beams on transmission through the sample while en-suring that there will be negligible walk-off in the spatialoverlap of the two pulses as they propagate the length ofthe semiconductor crystals.

The combined effects of the pump-pulse-induced non-linear birefringence and dichroism cause the transmittedprobe to be elliptically polarized with the major axis of

Fig. 1. Experimental geometry for the Kerr ellipsometry mea-surement. P1 and P2 are, respectively, the polarizer and theanalyzer; L1 and L2 are lenses, QWP is a quarter-wave plate,and HWP is a half-wave plate. The excitation (pump) pulse,indicated by e, is vertically polarized, and the probe pulse, p, ispolarized at 45± with respect to it. The transmission axis of theanalyzer is indicated by a; d 0 corresponds to the transmissionaxis of the analyzer perpendicular to the transmission axis ofthe polarizer.


the ellipse rotated from direction p. This elliptical po-larization may be considered to be a combination of arotated linearly polarized component and a circularlypolarized component. In Section 3 we show analyti-cally that, if the sample orientation is chosen so thatthe principal axes for the pump-induced birefringenceand dichroism are parallel to and perpendicular to thepump field, then the probe polarization can be decom-posed into two components that are polarized along thesame directions and that propagate independently. Werestrict our measurements to such orientations. Forthese selected orientations the pump-induced anisotropyin the absorption coefficient sDa2 fi 0d causes a simplerotation of the probe polarization vector. By contrast,the pump-induced anisotropy in the nonlinear refractiveindex sDn2 fi 0d introduces a component of circular po-larization into the probe. Consequently, to measure in-dependently both the sign and the magnitude of bothDa2 and Dn2, it is necessary to measure separately thedirection and the degree of the rotation of the probe po-larization induced by the pump and to measure the senseand the amount of circular polarization induced in theprobe polarization by the pump.

We measure the degree of polarization rotation first byremoving the quarter-wave plate shown in Fig. 1 and byinitially aligning analyzer P2 to reject the incident probepolarization. That is, the transmission axis of the ana-lyzer (indicated by the unit vector a) is initially orientedalong direction r (i.e., d 0), orthogonal to the incidentprobe polarization, as illustrated in the inset of Fig. 1.The analyzer transmission is then measured as a func-tion of small positive and negative angles d for each pumpfluence.27 The use of lens L2 to collimate the transmit-ted probe beam ensures that the extinction ratio of thepolarizer pair P1 and P2 is ,1025 with the sample re-moved. Approximately the same ratio is measured witheach sample present but with the pump pulse blocked.The latter measurement confirms that none of thesamples exhibits any significant linear birefringence andthat the probe pulse is too weak and of the wrong in-cident polarization (with respect to the crystallographicaxes) to induce any birefringence or dichroism when act-ing alone. Consequently, in the absence of the pump,the polarization state of the probe is preserved, and theangular dependence of the probe irradiance transmittedby the analyzer can be written as

T sdd s1 2 Cdsin2 d 1 C , (1)

where C is the extinction ratio. For small angles T sddcan be approximated by the parabola d2 1 C. As we shallshow in Section 3, in the presence of the pump the trans-mission of the analyzer still exhibits a parabolic depen-dence on d for small angles and for small anisotropies.With the pump present we can determine the directionand the degree to which the probe polarization is rotatedby measuring the sign and the magnitude of the angleof minimum transmission, and, as we shall show, fromthat angle both the sign and the magnitude of Da2 canbe determined.

The magnitude and the sense of the circular polariza-tion introduced into the probe polarization by the pumpis determined by a second measurement, in which a

quarter-wave plate is inserted between the sample andthe analyzing polarizer, with one of its neutral axes ori-ented parallel to the incident probe polarization, as shownin Fig. 1. With the quarter-wave plate in position theanalyzer transmission is again measured for small posi-tive and negative angles. In this configuration thequarter-wave plate interchanges the roles of linear andcircular polarizations, and the sign and the magnitude ofthe angle of minimum transmission are determined bythe magnitude and the sense of the circular polarizationinduced in the probe polarization by the pump. As weshall show, from that angle the sign and the magnitudeof Dn2 can be obtained.

Therefore, by performing these two experiments, withand without the quarter-wave plate, we are able to de-termine the sign and the magnitude of the inducedanisotropies of both the nonlinear refractive index andthe nonlinear absorption coefficient. These anisotropies(i.e., Dn2 and Da2), however, depend on sample orien-tation. Consequently, by repeating both measurementsfor carefully selected crystal orientations, as discussed inSection 3, we are able to determine the strength of theintrinsic anisotropies in n2 and a2 that are associatedwith the intrinsic anisotropies in the real and imaginaryparts of the third-order susceptibility tensors.

In addition, we also measure the analyzer transmis-sion for a fixed analyzer orientation sd 0d as a func-tion of pump fluence and time delay, both with andwithout the quarter-wave plate in place and as a functionof sample orientation. The measurements as a func-tion of fluence and time confirm that the nonlinearitiesare third order and instantaneous, respectively. Themeasurements as a function of fluence and sample ori-entation demonstrate the dependence of switching actionon sample anisotropy and orientation. Measurementsare reported on three unintentionally doped zinc blendesemiconductors (ZnSe, GaAs, and CdTe). All experi-ments are conducted at 300 K. In all cases the short,1-ps pulse duration used in the investigations ensuresthat photo-induced free-carrier effects23,32 – 34 such asfree-carrier absorption and refraction can be neglected.This was confirmed by time-resolved differential trans-mission experiments.34

3. MACROSCOPIC DESCRIPTIONOF EXPERIMENTSIn this section we derive expressions for the transmissionof the analyzer, with and without the quarter-wave platepresent, in terms of the induced nonlinear dichroism Da2

and the induced nonlinear birefringence Dn2. More im-portantly, we relate these experimentally measured quan-tities to the macroscopic third-order susceptibilities of zincblende materials. These expressions clearly show thatDa2 and Dn2 depend on sample orientation and that adirect measure of the intrinsic anisotropy in x s3d can beextracted from measurements of Da2 and Dn2 for selectedsample orientations.

We begin by writing the Fourier component of the third-order polarization, Pi

s3dsvi, kid, associated with propaga-tion in the direction ki and at a frequency vi in terms ofthe component field amplitudes Ei referenced to the crys-


tallographic axes of the medium, as is commonly done35:

Pis3dsvi, kid

e0

4

Xjkl

xs3dijklsvj , vk, vldEj svj , kj dEksvk, kkd

3 Elsvl, kld , (2)

where e0 is the permittivity of free space, x s3d is thethird-order susceptibility tensor, and i, j , k, and l denotethe crystallographic axes of the medium. Photon energyand momentum conservation require the conditions vi vj 1 vk 1 vl and ki kj 1 kk 1 kl. For cubic crystals,i, j , k, and l denote the principal crystal directions [100],[010], and [001], which will be represented by the unitvectors x, y, and z, respectively.

In the pump–probe experiments described in Section 2,however, we wish to detect a field of arbitrary polar-ization generated by pulses of fixed polarization as thecrystal orientation is varied. Consequently we find itconvenient to write our expressions in the frame of refer-ence defined by the incident and detected radiation ratherthan by the crystallographic axes. We introduced suchnotation previously,25 and we will follow and extend ithere. Given incident fields Eb, Ec, and Ed with frequen-cies vb, vc, and vd and with polarization unit vectors b,c, and d, we can rewrite Eq. (2) to give one of the Fouriercomponents of the polarization at a frequency va in thedirection a as

P s3da sva, kad

e0

4

Xbcd

Xijkl

xs3dijklsvb, vc, vddai

pbj ckdl

3 Ebsvb, kbdEcsvc, kcdEdsvd, kdd , (3)

where aip ap ? i, bj b ? j , ck c ? k, and dl d ? l

are the direction cosines for the projections of Pap, Eb,

Ec, and Ed onto the directions i, j , k, and l, respectively,and where we have taken the unit vectors and directioncosines to be complex to allow for the possibility of circularor elliptical polarization. The summation over i and thedirection cosine ai

p are included to take into accountthe contribution of all P

pi along the ap direction. The

summation over b, c and d must include all permutationsof the three fields for which the frequencies and wavevectors satisfy the conditions va vb 1 vc 1 vd andka kb 1 kc 1 kd in order to yield the total polarizationin the direction a with a frequency va and a propagationvector ka.

Moreover, recognizing that the only nonzero third-ordersusceptibility tensor elements for materials that haveisotropic or cubic 432, m3m, or 4 3m symmetry are of theform xiiii, xiijj , xijji, and xijij , where i, j x, y, or z, wecan, in turn, write Eq. (3) in the simpler form

P s3da sva, kad

e0

4

Xbcd

xeff svb, vc, vddEbsvb, kbd

3 Ecsvc, kcdEdsvd kdd , (4)

where

xeff svb, vc, vdd xxxyy svb, vc, vddsap ? bdsc ? dd

1 xxyyxsvb, vc, vddsap ? ddsb ? cd

1 xxyxy svb, vc, vddsap ? cdsb ? dd

1 Ssvb, vc, vddPi

aipbicidi , (5)

and the parameter S is defined as

Ssvb, vc, vdd xxxxxsvb, vc, vdd 2 f xxxyy svb, vc, vdd

1 xxyxy svb, vc, vdd 1 xxyyxsvb, vc, vddg

s1svb, vc, vddRef xxxxxsvb, vc, vddg

1 is2svb, vc, vddImf xxxxxsvb, vc, vddg

(6)

and is taken as a measure of the intrinsic anisotropyof the material, because for an isotropic material thediagonal element is equal to the sum of the three non-diagonal elements. The real quantities s1 and s2 corre-spond to the conventional anisotropy parameters for thereal and the imaginary parts of the susceptibility, respec-tively, as defined in the literature.24,25,36 We define Sin this way in order to allow for the possibility that theanisotropy parameter s may be different for the real andthe imaginary parts of the susceptibility and may be dif-ferent for different processes (i.e., different combinationsof vb, vc, and vd). We emphasize that the dot prod-ucts refer only to the relative polarizations of the elec-tromagnetic fields and not to the crystallographic axesand that the dependence on crystal orientation is totallycontained in the summation

Pai

pbicidi. Equation (6)is valid for all resonant and nonresonant, degenerateand nondegenerate instantaneous processes in cubic andisotropic materials.

Here we restrict our discussions to degeneratepump–probe experiments with linearly polarized light,and we find it convenient to decompose the probe intocomponents parallel Ep

k and perpendicular Ep' to the

pump:

Epsv, kpd Epksv, kpde 1 Ep

'sv, kpdq , (7)

where kp indicates the probe propagation direction, e isthe unit vector in the polarization direction of the pump(or exciting) pulse, and q is a unit vector perpendicular tothat direction, as shown in Fig. 1. Then, by performinga little algebra, one can apply the formalism indicatedby Eqs. (4)–(6) to obtain expressions for the nonlinearpolarizations Pe and Pq that are polarized parallel andperpendicular, respectively, to the pump and that areassociated with propagation in the probe direction at afrequency v:

Pesv, kpd e0

4fxk

T Eepsv, kedEesv, kedEp

ksv, kpd

1 xCEepsv, kedEesv, kedEp

'sv, kpdg , (8)

Pqsv, kpd e0

4fx'

T Eepsv, kedEesv, kedEp

'sv, kpd

1 xCEepsv, kedEesv, kedEp

ksv, kpdg , (9)

where the total effective susceptibilities are given by

xkT 2A 2 2D

√1 2

Pi

ei4

!, x'

T 2B 1 2DPi

ei2qi

2,

xC 2DPi

ei3qi (10)


and where the coefficients A, B, and D indicate the fol-lowing combinations of susceptibility tensor elements:

A xxxxxs2v, v, vd 1 xxxxxsv, 2v, vd

1 xxxxxsv, v, 2vd , (11)

B xxyyxs2v, v, vd 1 xxxyy sv, 2v, vd

1 xxyxy sv, v, 2vd , (12)

D Ss2v, v, vd 1 Ssv, 2v, vd 1 Ssv, v, 2vd . (13)

Notice that D is the sum of the anisotropy param-eters for the two-photon, single-photon, and nonreso-nant processes and, therefore, is an indicator of theoverall anisotropy. In writing Eqs. (8)–(13) we haveinvoked symmetry, which dictates that xijjis2v, v, vd xijij s2v, v, vd, xiijj sv, 2v, vd xijjisv, 2v, vd, andxiijj sv, v, 2vd xijij sv, v, 2vd.37 Also notice that theperpendicular and the parallel components of the probefield are coupled through the nonlinear polarization bythe cross susceptibility xC . We can decouple the probefield components by choosing crystal orientations suchthat xC 2D

Pei

3qi 0. For any given propagationdirection there are eight pump-polarization directions forwhich this condition is satisfied, with the exception thatif k is parallel to the [111] direction then xC is zero forall polarization directions. We shall restrict ourselves tosuch cases. Also notice that the effective susceptibilitiesx

kT and x

'T are induced susceptibilities. That is, they

reflect the magnitude of the absorptive and refractivechanges induced by the pump in the parallel and theperpendicular components of the probe.

We can now evaluate the individual effects of both thereal (refractive) and the imaginary (absorptive) parts ofx

kT and x

'T on the parallel and the perpendicular com-

ponents of the probe by expressing each component ofthe probe in terms of an amplitude and a phase, Ep

k Ak exp sifkd and Ep

' A' exp sif'd, and by substitut-ing Eqs. (7)–(9) into the reduced wave equation to obtain

≠Ak

≠z 2

v

4n2c2e0Im x

kT IeA

k 212

a2kIeA

k , (14)

≠fk

≠z

v

4n2c2e0Re x

kT Ie

2p

l0n2

kIe (15)

for the parallel components of the amplitude and thephase and

≠A'

≠z 2

v

4n2c2e0Im x'

T IeA' 2

12

a2'IeA

' , (16)

≠f'

≠z 2

v

4n2c2e0Re x'

T Ie 2p

l0n2

'Ie (17)

for the perpendicular components. Here z is taken as thecoordinate along the probe propagation direction withthe origin (z 0) at the front sample surface and l0 isthe vacuum wavelength.

The decoupled equations (14)–(17) can be readily solvedto yield expressions for the parallel and perpendicu-lar components of the probe field following transmis-sion through the sample. The irradiance transmitted bythe analyzer, in the absence of the quarter-wave plate,

can subsequently be found by projection of the resultingtotal probe field onto the transmission direction for theanalyzer [i.e., forming the dot product between the to-tal probe field and the unit vector a cosspy4 1 dde 2

sinspy4 1 ddq]. The resulting probe irradiance trans-mitted by the analyzer as a function of angle a is givento second order in the products d, Dn2, and Da2 by theexpression

I sn0ly4d Ips0, r, tds1 2 R1ds1 2 R2d

3

("d 1

Da2s1 2 R1dIes0, r, tdL4

#2

1

"pDn2s1 2 R1dIes0, r, tdL

l0

#2), (18)

where L is the sample thickness, R1 and R2 are the frontand the back surface reflection coefficients, respectively,and Ipsz 0, r, td and Iesz 0, r, td are the incidentpump and probe irradiances, respectively, as a function oftransverse coordinate r and retarded time t. The experi-mentally extracted quantities Dn2 and Da2 are defined interms of the parallel and the perpendicular coefficients as

Dn2 n2k 2 n2

' 1

4n2ce0ResDxd , (19)

Da2 a2k 2 a2

' v

2n2c2e0ImsDxd , (20)

where Dx is the difference in the total induced suscepti-bilities for components of the probe polarized parallel andperpendicular to the pump:

Dx xkT 2 x'

T

2

"A 2 B 2 D

√1 2

Pi

ei4 1

Pi

ei2qi

2

!#. (21)

In obtaining Eq. (18) we have neglected linear absorptionand have assumed that the pump is undepleted by nonlin-ear absorption on propagation through the sample. Theformer expedient is well justified for all samples to bestudied here, as the photon energy is less than the bandgap and the linear absorption at 950 nm was less than 7%.We ensured the latter by conducting all investigations in aregime in which two-photon absorption (if present) causedless than 10% depletion of the pump.

In order to relate this analysis to experiment, becausethe photodetectors monitor the pulse energies we must in-tegrate Eq. (18) over the temporal and spatial profiles ofour pulses; however, because the probe spot size is muchsmaller than the pump, the spatial integral over the pumpcan be neglected. In addition, we assume that the pulsesare temporally coincident; that is, we experimentally setthe time delay between the two pulses to be zero. Con-sequently, by integrating over the temporal profile of thepump and the probe and over the spatial profile of theprobe, we obtain an expression for the normalized trans-mission without the quarter-wave plate (defined as theprobe energy transmitted through the analyzer dividedby the probe pulse energy incident upon the sample):


T sn0ly4d s1 2 R1ds1 2 R2d

3

0B@0B@0B@

8<:d 123

"s1 2 R1dDa2Ie0L

4

#9=;2

18

15

8<:"

ps1 2 R1dDn2Ie0Ll0

#2

116

"s1 2 R1dDa2Ie0L

4

#29=;

1CA1CA1CA , (22)

where Ie0 is the peak-on-axis pump irradiance incidentupon the sample and where we have assumed a sech2stytdpulse shape for both pump and probe pulses.

Equation (22) quantitatively verifies and emphasizesthe qualitative statements made in Section 2 to describethe technique. Notice that in the absence of the pump,i.e., in the absence of any induced nonlinear birefrin-gence sDn2Ie0 0d or of induced nonlinear dichroism(Da2Ie0 0), the transmission of the analyzer has aparabolic dependence on analyzer angle d, for small d,and that the minimum transmission occurs at d 0 andis expected to be zero, apart from a small offset caused bythe leakage of the polarizer–analyzer pair. By compari-son, in the presence of the pump the angle dependenceof the transmission is still parabolic, although the mini-mum is not at d 0, and there is an offset from zerotransmission. The angle of minimum analyzer transmis-sion is determined by Da2, the difference in the nonlin-ear absorption coefficients experienced by the parallel andperpendicular components of the probe. This confirmsthat, when one component is attenuated more than theother, the net effect of Da2 is to produce a simple rota-tion of the probe polarization on propagation through thesample. From Eq. (18) it is evident that the degree ofthis rotation varies in time as the irradiance Ie variesthrough the pulse. Averaging this time-varying polariza-tion rotation over the temporal profile of the pulses pro-duces a background transmission, proportional to sDa2d2,that contributes to the offset. Notice that the offset as-sociated with sDa2d2 was not present in Eq. (18) beforethe temporal averaging, and we emphasize that it willnot be present in steady-state experiments. In contrastto the polarization rotation produced by Da2, the nonlin-ear birefringence Dn2 induces a degree of circular polar-ization into the probe pulse. As indicated by Eq. (22),this circularly polarized component contributes a sec-ond background term, which is proportional to sDn2d2, tothe probe pulse transmission. Consequently, with thisgeometry both the sign and the magnitude of Da2 can bedetermined from the position of the minimum transmis-sion relative to d 0; however, only the magnitude of Dn2

can be determined, and that only by indirect means oncethe value for Da2 is known.

The sign and the magnitude of Dn2 can be determineddirectly, however, if the quarter-wave plate is insertedbetween the sample and the analyzer with one neutralaxis parallel to the incident probe polarization. In thissecond experimental geometry the effects of the nonlin-ear birefringence and anisotropic nonlinear absorption areeffectively exchanged. Indeed, repeating the above cal-culation with the slow axis of the quarter-wave plate ori-ented parallel to the probe polarization vector results in

the following expression for the probe energy transmittedthrough the quarter-wave plate and the analyzer:

T sly4d s1 2 R1ds1 2 R2d

3

0B@0B@0B@

8<:d 223

"ps1 2 R1dDn2Ie0L

l0

#9=;2

18

15

8<:"

s1 2 R1dDa2Ie0Ll0

#2

116

"ps1 2 R1dDn2Ie0L

l0

#29=;

1CA1CA1CA . (23)

From this expression it is clear that one can find both thesign and the magnitude of Dn2 by determining the valueof d at which the minimum of the parabola occurs.

Thus, by measuring the analyzer transmission for smallangles, with and without the quarter-wave plate present,we can directly determine the magnitudes and the signsof both Dn2 and Da2. From Eqs. (19) and (20) we canreadily see that these two quantities are related to theindividual third-order susceptibility tensor elements andto the orientation of the sample by Eq. (21). We empha-size that the dependences of Dn2 and Da2 on the orien-tation of the sample are completely determined by thereal and the imaginary parts, respectively, of the lastterm in Eq. (21), that is, by 2Ds1 2

Pi ei

4 1P

i ei2qi

2d.Both the real and the imaginary parts of D can be de-termined by measurement of Dn2 and Da2 for the twosample orientations shown in Fig. 2. For each of thesegeometries the direction of propagation is along a prin-cipal crystal axis (taken as z). For the first orientationthe pump is polarized along the [100] direction (i.e., e xand q 2y), and for the second, along the [110] direction[i.e., e sx 1 ydy

p2, q sx 2 ydy

p2], as shown in Fig. 2.

[We emphasize that xC 0 for each of these orientations,and therefore Eqs. (8) and (9) are decoupled in each case.]We can then extract the real and the imaginary parts ofD from the differences in Dn2 and Da2 for the two orien-tations, using the following two expressions:

Dn2f100g 2 Dn2f110g 1

4n2ce0Res2Dd

1

2n2ce0hs1s2v, v, vdRef xxxxxs2v, v, vdg

1 s1sv, 2v, vdRef xxxxxsv, 2v, vdg

1 s1sv, v, 2vdRef xxxxxsv, v, 2vdgj , (24)

Fig. 2. Orientation of the excitation pulse polarization withrespect to the crystallographic axes. The polarization of theexcitation pulse remains vertical while the sample is rotated suchthat the excitation polarization lies along (a) the [100] axis and(b) the [110] axis.


Da2f100g 2 Da2f110g v

2n2c2e0Ims2Dd

v

n2c2e0hs2s2v, v, vdImf xxxxxs2v, v, vdg

1 s2sv, 2v, vdImf xxxxxsv, 2v, vdg

1 s2sv, v, 2vdImf xxxxxsv, v, 2vdgj . (25)

In fact, the third-order nonlinear birefringence Dn2 forany arbitrary incident pump polarization e that decouplesEqs. (8) and (9) (and recall that there are at least eightsuch polarization directions for each propagation direc-tion) can be written in terms of Dn2f100g and Dn2f110g:

Dn2feg Dn2f110g 1 sDn2f100g 2 Dn2f110gd

3

"Pi

sei4 2 ei

2qi2d

#. (26)

A similar expression can be written for the orientationdependence of nonlinear dichroism:

Da2feg Da2f110g 1 sDa2f100g 2 Da2f110gd

3

"Pi

sei4 2 ei

2qi2d

#. (27)

The measurement of Dn2 and Da2 (and the conse-quent determination of the real and the imaginary partsof D) for these two orientations not only determinesthe intrinsic anisotropy in the induced birefringence andthe induced dichroism but also completely specifies theintrinsic anisotropy in the third-order nonlinear refrac-tion n2 and the nonlinear absorption a2. We can empha-size the latter point by calculating n2 and a2 for a singlepulse with linear polarization e propagating through asample of arbitrary orientation. The procedure for thiscalculation parallels the steps outlined in Eqs. (7)–(17)for the pump–probe experiment, except that in this casethere is only a single pulse. The resulting expressionsfor the single-pulse n1 and a2,

n2 1

4n2ce0Ref xT g

14n2ce0

Re

264A 2 D

0B@1 2X

i

ei4

1CA375 ,

(28)

a2 v

2n2c2e0Imf xT g

v

2n2c2e0

3 Im

264A 2 D

0B@1 2X

iei

4

1CA375 , (29)

are the same as those contained in Eqs. (19) and (20),respectively, except that the total susceptibility xT for asingle pulse is half the x

kT experienced by the parallel

probe in the presence of the pump.25,39,40 Finally, it isstraightforward to show explicitly that the orientationdependence (anisotropy) of n2 is completely specified bymeasuring Dn2 for these two orientations by rewritingEq. (28) as

n2feg n2f100g 2 1/2sDn2f100g 2 Dn2f110gd

√1 2

Pi

ei4

!.

(30)

Similarly, Eq. (29) can be written as

a2feg a2f100g 2 1/2sDa2f100g 2 Da2f110gd

√1 2

Pi

ei4

!.

(31)

Equations (26), (27), (30), and (31), together withEqs. (22) and (23), constitute our fundamental results.A number of figures of merit have been defined for vari-ous optical devices that utilize instantaneous third-orderoptical nonlinearities.2,41 – 43 As we discuss in more detailin the following sections, each figure of merit is definedin terms of the above parameters (n2, a2, Dn2, and Da2),either individually or in combination. Consequently, op-timizing the device performance requires optimization ofthe crystal orientation as well as of the material system.

4. INDUCED NONLINEARBIREFRINGENCE BELOW HALF THEBAND GAP: ZINC SELENIDEAt the laser wavelength of 950 nm the photon en-ergy (1.31 eV) is less than half the band gap of ZnSe(2.68 eV).44 At this wavelength the two-photon absorp-tion coefficient b should be zero for all sample orienta-tions. Consequently we would expect to see only inducednonlinear birefringence but no induced nonlinear dichro-ism in this material. These expectations are confirmedby the data shown in Fig. 3, which were obtained with a2-mm-thick, undoped ZnSe crystal.

Figure 3 shows the results of measuring the probetransmission T sdd as a function of the analyzer angle d

with and without the quarter-wave plate for the same twosample orientations shown in Fig. 2. The results for thepump pulse polarized along the [100] direction are shownin Fig. 3(a), and those for the pump polarized along the[110] direction in Fig. 3(b). The data taken without the

Fig. 3. Probe transmission as a function of analyzer angle forthe pump pulse polarized along (a) the [100] axis and (b) the [110]axis of the ZnSe sample. The filled (open) symbols indicate datataken without (with) the quarter-wave plate. The peak-on-axispump irradiance was (a) 3 3 1013 Wym2 and (b) 1 3 1013 Wym2.


quarter-wave plate are indicated by the filled dots, andthose taken with the quarter-wave plate in place are in-dicated by the open dots.

Even before the data are quantitatively analyzed, sig-nificant information can be deduced from the qualita-tive features shown in Fig. 3. From the discussionsof Section 3 [see Eq. (22)] we know that, without thequarter-wave plate, the magnitude of the angle of mini-mum transmission is proportional to the induced dichro-ism Da2 and the direction indicates the sign (a negativeangle indicates a positive Da2). Inspection of the datataken without the quarter-wave plate between the sampleand the analyzer (filled dots) clearly reveals that, withinexperimental error, the minimum of the parabola for eachorientation is located at d 0, indicating that there is noinduced dichroism sDa2 0d in this material at this wave-length, as expected. Similarly [see Eq. (23)], with thequarter-wave plate in place the shift in the minimum po-sition is proportional to the nonlinear birefringence Dn2,and a shift to positive angles indicates a positive Dn2.Although the minimum of the transmission parabola isat a positive angle for each pump polarization, it is at alarger angle for the [110] orientation than for the [100]orientation, despite the fact that the pump irradiance issignificantly less in the former case. Consequently wecan conclude that Dn2 for ZnSe at this wavelength isnonzero and positive and that it is larger when the pumppolarization is along [110] than when it is along [100].This difference in Dn2 for the two orientations indicatesa significant intrinsic anisotropy in the real part of x s3d.

The solid curves shown in Fig. 3 are the result ofusing Eqs. (22) and (23) to fit the data without and withthe quarter-wave plate, respectively, with Da2 and Dn2 asfree parameters. The parameters extracted from thesefits are Dn2f100g 0.14 3 1024 cm2yGW, Dn2f110g 0.34 3 1024 cm2yGW, and Da2f100g Da2f110g 0.These values (which are listed in the column headings ofTable 1) can be substituted into Eq. (26) to yield a particu-larly simple expression for the induced birefringence inZnSe for any arbitrary orientation that decouples Eqs. (8)and (9):

Dn2feg

"0.34 2 0.20

Pi

sei4 2 ei

2qi2d

#3 1024 cm2yGW.

(32)

This dependence of the induced birefringence on sampleorientation will be critically important for devices that de-pend on a rotation of the incident polarization for their op-eration. In Section 6 we will illustrate this dependenceon sample orientation for a simple polarization rotationswitch, the on–off ratio of which varies with the square ofDn2. Consequently the more-than-factor-of-2 variationin Dn2 evident in Eq. (32) will result in a more-than-factor-of-4 variation in device performance. Of course,

the specific dependence on Dn2 varies from device to de-vice, depending on the details of construction and on theunderlying principles of operation.

The measurement of the induced birefringence for thetwo orientations shown in Fig. 2 also completely deter-mines the magnitude and the sign of the intrinsic varia-tion of n2 with sample orientation, which we can write as

n2feg n2f100g 1 s0.1 3 1024 cm2yGWd

√1 2

Pi

ei4

!

n2f100g 2 s1s2v, v, vdn2f100g

√1 2

Pi

ei4

!(33)

by using Eqs. (30) and (28) and by assuming that thenonlinear index of refraction is dominated by the realpart of the two-photon susceptibility, x s3ds2v, v, vd.The assumption is reasonable because, at the wave-length used here, the photon energy is ,48% of theband gap, and consequently the nearest resonance is thatof two-photon absorption. In this spectral region n2 isdominated by, and enhanced by, its proximity to the two-photon resonance.16,45 The technique does not, however,provide an independent measure of n2f100g. The en-hancement of n2 just below half the band gap, togetherwith the absence of any two-photon absorption and lowbackground linear absorption, makes this an attractivespectral region in which to operate. The expected highfigures of merit in this region were the primary reasonsfor including this material in the current studies.

This intrinsic anisotropy in n2 will clearly affect theoperation of devices that depend on the nonlinear phasefor their operation. Examples of such semiconductordevices are directional couplers,46,47 X junctions,48 andMach–Zehnder49 and bistable Fabry–Perot50 switches.A figure of merit has been defined in an effort to iden-tify materials that are suitable for such applications.41

This figure of merit is given by jDnjyal, where Dn isthe nonlinear index change and a includes both linearand nonlinear absorption. When such devices are oper-ated at a photon energy of less than half the band gapthis figure of merit simplifies to n2Iya0l, where I is theirradiance and a0 is the below-band-gap linear absorp-tion coefficient. Of course, the precise threshold valueof the figure of merit necessary for successful operationvaries from device to device, but clearly optimizationdepends on sample orientation as well as on choice ofmaterial system.

The connection between the anisotropy in the observedinduced birefringence and that in the third-order suscep-tibility of ZnSe is contained in Eq. (24). Under the cir-cumstances discussed here, Eq. (24) can be written to agood approximation as

RefDg 2n2ce0sDn2f100g 2 Dn2f110gd

s1s2v, v, vdRef xxxxxs2v, v, vdg , (34)

Table 1. Measured Induced Birefringence Dn2Dn2Dn2 and Dichroism DbDbDb of the Zinc Blende SemiconductorsZnSe, GaAs, and CdTe for the Exciting Pulse Polarized along the [100] and [110]Crystal Axes

Semiconductor Dn2f110g s1024 cm2yGWd Dn2f100g s1024 cm2yGWd Dbf110g scmyGWd Dbf100g scmyGWd

ZnSe 0.34(60.09) 0.14(60.04) – –GaAs 25.0(61.3) 23.3(60.9) 32(68) 12(63)CdTe 22.4(60.6) 21.4(60.4) 20(65) 9(62)


Table 2. Measured Values of Real and ImaginaryComponents of D [Eqs. (24) and (25)]

for the Zinc Blende Semiconductors ZnSe, GaAs,and CdTe

Semiconductor ResDd s10215 cm2yV2d ImsDd s10215 cm2yV2d

ZnSe 20.72 6 0.36 –GaAs 11 6 6 210 6 5CdTe 4.3 6 2.2 23.6 6 1.8

Fig. 4. Transmission of probe pulse through the [110]-orientedZnSe sample as a function of delay between the pump and theprobe pulses for analyzer angle d 0.

and a value of 27.2 3 10216 cm2yV2 can be extracted forthe product s1s2v, v, vdRef xxxxxs2v, v, vdg, as shownin Table 2.

Finally, we confirmed that the nonlinearities measuredhere in ZnSe were indeed instantaneous third-order op-tical Kerr nonlinearities by measuring the transmissionof the analyzer as a function of time delay between thepump and the probe pulses, as shown in Fig. 4. For thesemeasurements the quarter-wave plate was removed, theanalyzer angle d was set to zero, the pump polarizationwas arranged along the [110] axis, and the pump flu-ence was adjusted to 1.1 mJycm2. For this arrangementone can readily show from Eq. (18) that, for an instanta-neous third-order nonlinearity, the analyzer transmissionwill be proportional to the correlation of the pulse enve-lope with the square of that envelope. The location ofthe peak and the temporal width of the signal in Fig. 4are consistent with those expected for such a process.The small asymmetry with respect to zero delay in Fig. 4(which is not present in the second-harmonic intensityautocorrelation) is attributed to the slight asymmetry ofthe pulses generated by the amplification process in theStyryl 13 laser system.51

5. INDUCED DICHROISM ANDBIREFRINGENCE ABOVE HALFTHE BAND GAP

A. Gallium ArsenideUnlike ZnSe, GaAs, which has a band gap of 1.42 eV atroom temperature,44 is a strong two-photon absorber at950 nm.7,10,17,19,23 Consequently, in contrast to ZnSe,we would expect GaAs to exhibit induced dichroism(with Da2 ; Db) as well as induced birefringence.

The optically induced and intrinsic anisotropies of a0.47-mm-thick GaAs crystal were measured with a pro-cedure similar to that described for ZnSe. The resultsare shown in Fig. 5 for a pump fluence of 0.29 mJycm2

and for the same two orientations with and without thequarter-wave plate. Again the solid curves in Fig. 5are the result of fitting the data, using Eqs. (22) and(23), with Dn2 and Db as free parameters. The val-ues extracted from these fits are Dbf100g 12 cmyGW,Dbf110g 32 cmyGW, Dn2f100g 23.3 3 1024 cm2yGW,and Dn2f110g 25.0 3 1024 cm2yGW and are listed inTable 1 for convenience. We emphasize that this tech-nique allows the induced and the intrinsic anisotropyin the birefringence and the dichroism to be extractedsimultaneously from a single set of measurements and,consequently, provides significant information about theanisotropy in both the nonlinear refractive index n2 andthe two-photon absorption coefficient b, as we discuss inthe following paragraphs.

The influence of a nonzero two-photon absorption isevident in Fig. 5 in the shifts of the minima of the parabo-las to negative angles when the quarter-wave plate is notpresent. The values extracted from the fits for Db forthe two orientations can be used in Eq. (27) to yield anexpression for the dichroism for any arbitrary sample ori-entation for which xC 0:

Dbfeg 32 2 20

"Pi

sei4 2 ei

2qi2d

#cmyGW. (35)

Notice that the intrinsic anisotropy in the two-photon ab-sorption coefficient causes the induced dichroism to varyby more than a factor of 2.5 (from 12 to 32 cmyGW) as thesample is rotated. This measured dichroism can thenbe used in Eq. (31) together with Eq. (29) to describethe orientation dependence of the two-photon absorptioncoefficient:

Fig. 5. Probe transmission as a function of analyzer angle forthe pump pulse polarized along (a) the [100] axis and (b) the [110]axis of the GaAs sample. The filled (open) symbols indicate datataken without (with) the quarter-wave plate. The peak on-axispump irradiance was 2.5 3 1012 Wym2 in both (a) and (b).


bfeg bf100g 1 10

√1 2

Pi

ei4

!cmyGW

bf100g 2 s2s2v, v, vdbf100g

√1 2

Pi

ei4

!. (36)

Equation (36) confirms the intrinsic anisotropy in b thatwe have measured in separate studies, using a comple-mentary technique.25 Those studies provide a value of,215 cmyGW for the product of bf100g (19 cmyGW)and s2s2v, v, vd s20.76d, compared to the 10 mea-sured here. Although the former value is somewhatlarger than the value obtained here, it is well within therange of uncertainties given in Table 1 for the presenttechnique, and the value obtained here is well withinthe errors quoted in our previous paper.25 If we usebf100g 19 cmyGW in Eq. (36), we readily see that b

ranges from 19 to 27 cmyGW between the [100] and the[111] orientations, compared to the 19 to 30 cmyGW re-ported previously.25

Recognizing that the imaginary part of the suscepti-bility will be dominated by the two-photon resonance,we can rewrite Eq. (25) to obtain an expression that di-rectly relates the measured dichroism to the product ofthe anisotropy parameter and the diagonal element of thesusceptibility tensor:

ImsDd n2c2e0

vsDbf100g 2 Dbf110gd

s2s2v, v, vdImf xxxxxs2v, v, vdg . (37)

Substituting the measured values for Dbf100g andDbf110g into this expression, we find that s2s2v, v, vdImf xxxxxs2v, v, vdg 210 3 10215 cm2yV2. Thisvalue is in good agreement with the value of 214 3

10215 cm2yV2 that we obtain by multiplying the valuesof s2s2v, v, vd 20.76 and Imf xxxxxs2v, v, vdg 19 3 10215 cm2yV2 that we obtained in separatemeasurements25 and with the almost identical valueobtained from Z-scan measurements hs2s2v, v, vd 20.74, Imf xxxxxs2v, v, vdg 19 3 10215 cm2yV2j.24

(Notice in comparing Refs. 24 and 25 that there is afactor-of-3 difference in the definition of x.) Moreover,the measurements of the anisotropy in the two-photonabsorption presented here are consistent with the pre-dictions fs2s2v, v, vd 20.72g of a simple scaling lawbased on k ? p theory25 and, in turn, are in good, butnot precise, agreement with the more detailed k ? pcalculations of Hutchings and Wherrett,26 who calcu-late that s 21.0 for "vyEg 0.92. It should benoted, however, that the latter calculation used bandstructure parameters for GaAs near 0 K, whereas ourmeasurements were performed at 300 K.

The present ellipsometric pump–probe technique pro-vides information on the anisotropy of the nonlinear re-fractive index n2 and on the anisotropy of the real part ofthe susceptibility that could not be extracted by our previ-ous technique.25 Unlike for ZnSe, where the photon en-ergy was less than half the band gap and Dn2 was positive,Dn2 is negative for both orientations of the GaAs crystal,where the photon energy is above half the band gap and,in fact, near the single-photon resonance. These twonegative values for the induced birefringence can be used

in Eqs. (26) and (28) to write general expressions for thedichroism:

Dn2feg

"25.0 1 1.7

Pi

sei4 2 ei

2qi2d

#3 1024 cm2yGW,

(38)

and for the nonlinear index:

n2feg n2f100g 2 s0.9 3 1024 cm2yGWd

√1 2

Pi

ei4

!. (39)

Note that n2 is expected to be negative for allorientations.16 The change in sign of n2 as the photonenergy is increased from less than half of the band gap(in ZnSe) to slightly below the band gap (in GaAs) isconsistent with the description of dependence of n2 on"vyEg in Ref. 16.

The anisotropy in the real part of the susceptibilityin the region near, but below, the band edge is not sosimple as that of the imaginary part. In this spectralregion the imaginary part is dominated by the two-photonresonant term in the susceptibility. By comparison, thereal part consists of important contributions from the two-photon resonant term and the quadratic Stark term, withlesser contributions from the Raman and linear Starkeffects.16 We reflect these complications by keeping allthree degenerate susceptibilities when we rewrite Eq. (24)for the case of GaAs:

ResDd 2n2ce0sDn2f100g 2 Dn2f110gd

s1s2v, v, vdRef xxxxxs2v, v, vdg

1 s1sv, 2v, vdRef xxxxxsv, 2v, vdg

1 s1sv, v, 2vdRef xxxxxsv, v, 2vdg . (40)

From our measurements it is not possible to determinethe contribution of each process to the anisotropy; how-ever, by using the measured values for Dn2 in this expres-sion we obtain ResDd 11 3 10215 cm2yV2 for the overallanisotropy parameter.

Clearly, the figure of merit jDnjyal defined above forphase-sensitive devices will be different for operation inthis spectral region from that for operation below halfthe band gap, where two-photon absorption is negligible.In fact, this figure of merit becomes n2y2bl for opera-tion above half the band gap.42,43 There are two obvi-ous differences between this figure of merit and the onediscussed in connection with our ZnSe results. First,notice that this one does not depend linearly on irra-diance. Therefore one cannot improve performance bysimply increasing the operating intensity. Second, no-tice that n2y2bl does not depend so strongly on sampleorientation. While both n2 and b depend strongly on ori-entation, each is larger for the pump polarized along the[110] direction, and each increases by roughly the sameamount. Thus, in this case, the anisotropies in the nu-merator and the denominator tend partially to cancel.Consequently we see that the importance of the intrin-sic anisotropy of zinc blende materials to device operationcan depend on several factors, including the spectral re-gion chosen for operation and the processes that deter-mine device performance.


Fig. 6. Probe transmission as a function of analyzer angle forthe pump pulse polarized along (a) the [100] axis and (b) the [110]axis of the CdTe sample. The filled (open) symbols indicate datataken without (with) the quarter-wave plate. The peak on-axispump irradiance was 2.1 3 1012 Wym2 in both (a) and (b).

As with ZnSe, we verified that the observed signal inGaAs was due to an instantaneous third-order nonlin-earity by confirming that it had a prompt temporal re-sponse and that the probe transmission was proportionalto the square of the pump irradiance at low fluences(,,0.4 mJycm2 in this case). At higher fluences beamdepletion associated with the two-photon absorption mod-erated the dependence of the signal on pump irradiance.

B. Cadmium TellurideThe third-order nonlinear optical properties of CdTe fora photon energy of 1.31 eV (Ref. 44) are expected to bevery similar to those of GaAs, because the two zinc blendematerials have similar band structures with similar band-gap energies. The band gap for CdTe is 1.49 eV. Con-sequently, as with GaAs, we are exciting CdTe well abovethe two-photon resonance and relatively near the single-photon resonance. Similar results to those shown forGaAs in Fig. 5 are shown in Fig. 6 for a 2-mm-thick CdTecrystal. The values of Dn2 and Db extracted from the fitsshown by the solid curves in Fig. 6 are listed in Table 1.

It is clear from these results that the third-order non-linear optical properties of CdTe are indeed very simi-lar to those of GaAs. Both Dn2 and Db are anisotropicwith larger values obtained for the pump polarized alongthe [110] crystal axis, and they can be written [by usingEqs. (26) and (27), respectively] in the form

Dn2feg

"22.4 1 1.0

Pi

sei4 2 ei

2qi2d

#3 1024 cm2yGW,

(41)

Dbfeg 20 2 11

"Pi

sei4 2 ei

2qi2d

#cmyGW. (42)

Notice that the birefringence and the dichroism in CdTeare slightly smaller than those in GaAs. The valuesgiven in Table 1 for Dn2 and Db can also be used together

with Eqs. (30) and (31) to write expressions for n2 and b

for CdTe that are similar to Eqs. (36) and (39), respec-tively, for GaAs:

n2feg n2f100g 2 s0.5 3 1024d

√1 2

Pi

ei4

!cm2yGW, (43)

bfeg bf100g 1 5.5

√1 2

Pi

ei4

!cmyGW

bf100g 2 s2s2v, v, vdbf100g

√1 2

Pi

ei4

!. (44)

As with GaAs, bf100g for CdTe is known from ourprevious measurements25 to have a value of bf100g 14 cmyGW and a value of s2s2v, v, vd 20.46, yield-ing a s2s2v, v, vdbf100g product of ,26 cmyGW,which is in excellent quantitative agreement with the25.5 cmyGW measured here.

The imaginary part of the intrinsic anisotropy pa-rameter D is expected to be dominated by the two-photon-resonant term, and a value of ss2v, v, vdImf xxxxxs2v, v, vdg 23.6 3 10215 cm2yV2 can be cal-culated by use of Eq. (37). This number is in excellentagreement with the value of 24.1 3 10215 cm2yV2 thatone obtains by multiplying ss2v, v, vd 20.46 andImf xxxxxs2v, v, vdg 0.9 3 10214 cm2yV2 extracted fromRef. 25. By comparison, the real part of D in the spectralregion near the band gap, as with GaAs, is determinedby contributions from several processes, mainly the two-photon resonant and the quadratic Stark terms.16 UsingEq. (40), we find that RefDg 4.3 3 10215 cm2yV2.

6. EFFECT OF ANISOTROPY ON APOLARIZATION ROTATION SWITCHA simple way to demonstrate the potential effect of the in-trinsic anisotropy of zinc blende materials on device per-formance is to operate the apparatus shown in Fig. 1 asan all-optical on–off switch. To do this, we remove thequarter-wave plate and we set the analyzer to reject theincident polarization (d 0) in order to obtain the maxi-mum on–off ratio. We operated the switch in this geom-etry, using each of the three samples described above asthe nonlinear material. Representative measurementsof the on–off ratio and the switch transmission as a func-tion of pump irradiance are shown in Fig. 7. For the pur-poses of comparison, the GaAs data have been scaled tothe same length (2 mm) as those of the other two crys-tals. The original data for GaAs were obtained at fourtimes higher irradiance than indicated by the points onthe graph. For our purposes here, we define the on–offratio as the ratio of the switch transmission with thepump present to the transmission when the pump is ab-sent. Using this definition, one can easily show fromEqs. (1) and (22) that the on–off (or contrast) ratio R forthis geometry is approximately given by

R >8

15C

3

8<:"

ps1 2 R1dDn2Ie0Ll0

#2

1

"s1 2 R1dDbIe0L

4

#29=; ,

(45)


Fig. 7. Transmission of the polarization rotation switch as afunction of peak on-axis pump pulse irradiance for ZnSe (filledcircles), CdTe (filled squares), and GaAs (filled triangles) withthe pump pulse polarized along the [110] axis and for ZnSe (opencircles) with the pump polarized along the [100] axis. The ZnSeand CdTe crystals are each 2 mm long, and the GaAs crystal is0.5 mm long. For the purposes of comparison, the GaAs datahave been scaled to a 2-mm crystal length. The original datafor GaAs were obtained at four times higher irradiance thanindicated by the points on the graph. The solid lines are fits toexpression (45) for the values of Dn2 and Db quoted in Table 1.There is no depletion evident in the ZnSe data, whereas depletionassociated with two-photon absorption is apparent in the GaAsand CdTe data.

where, as before, we have assumed that the pump andthe signal (probe) are undepleted (i.e., bL small), and werecall that the extinction ratio C for the crossed polar-izer–analyzer pair is approximately 1025.

We first consider the results shown in Fig. 7 for ZnSe.Notice that a probe transmission of up to 20% and an ex-tremely large contrast ratio R of approximately 20,000:1were readily obtained for ZnSe at the maximum availablelaser pulse intensity. For ZnSe excited below half theband gap, Db 0. In this regime expression (45) pre-dicts that the on–off ratio (and the transmission) shouldincrease with the square of the irradiance. The agree-ment with the latter tendency is indicated by the solidlines, which each have a gradient of 2, in Fig. 7. Expres-sion (45) also predicts that the contrast ratio should in-crease with the square of the induced birefringence Dn2.As we have shown here, this induced birefringence shoulddepend strongly on sample orientation, and indeed it does.As is shown in Fig. 7, for a given fluence the on–off ra-tio for the [110] orientation is roughly a factor of 5 largerthan that for the [100] orientation, consistent with themore than twofold increase in the induced birefringencereported in Section 4 for the former orientation.

A useful figure of merit M for this particular devicecan be defined as the ratio of the on–off ratio R achievedto the energy (per unit area) deposited in the sampleduring the switching. For ZnSe at this wavelength, be-cause Db 0 the fluence required for operation of theswitch is given by a0LIe0t, where t is the pulse width.In this regime the figure of merit M can be found fromexpression (45) to vary as

M ø8fps1 2 R1dDn2Ie0Lg2

15Cl02a0LIe0t

. (46)

From this expression it is clear that M , like R, will in-

crease quadratically with the birefringence Dn2. It alsoappears that M will increase linearly with crystal lengthwithout limit; however, this is simply an artifact of themodel, which assumes no beam depletion and is true onlyfor L ,, a0

21. The figure of merit in this spectral regionalso increases linearly with irradiance. Consequently, inthe sense of improving the on–off ratio per photon ab-sorbed, one can improve the device performance simply byincreasing the irradiance. Of course, the improvement isthe result of having only linear absorption present in thisspectral range. The price paid for this anomaly is thatDn2 is an order of magnitude smaller than it is in GaAsand CdTe, where Db fi 0.

Unlike in ZnSe, both the induced dichroism and theinduced birefringence contribute to the switching actionin GaAs and CdTe. That is, the Dn2 and Db terms inexpression (45) are both nonzero and of comparable mag-nitude. Moreover, as can be seen from Table 1, the in-duced nonlinearities are roughly an order of magnitudelarger near the band edge in GaAs than they are belowhalf the band gap in ZnSe. For that reason the irradi-ance that one needs to achieve a given contrast ratio ismore than a factor of 60 lower for GaAs than for ZnSe (forthe same length crystal), as shown in Fig. 7. By com-parison, Dn2 and Db are smaller in CdTe than in GaAs.Consequently the irradiance that one needs to achieve agiven contrast ratio is roughly a factor of 7 times largerin CdTe than in GaAs, as shown in Fig. 7.

For excitation above the two-photon resonance and forundepleted pump and probe pulses the energy per unitarea absorbed by the sample can be estimated by bIe0

2Lt

and the corresponding figure of merit given by

M >

8

8<:"

ps1 2 R1dDn2Ie0Ll0

#2

1

"s1 2 R1dDbIe0L

4

#29=;

15CbIe02Lt

.

(47)

There are notable differences between expressions (47)and (46). First, expression (47) is independent of theincident intensity. Second, it is much less sensitiveto sample orientation, because anisotropic parametersappear in both the numerator and the denominator.Even this expression contains some anisotropy, how-ever, as anisotropic parameters enter quadratically in thenumerator and linearly in the denominator. Finally, itis apparent from expression (47) that two-photon absorp-tion will eventually degrade and limit device performanceas the pump and probe pulses are depleted. The onset ofthis behavior is apparent in Fig. 7, where the transmis-sions of CdTe and GaAs are quadratic at low intensity butshow a less than quadratic dependence at larger pumpintensities.

The device geometry used here was chosen primarilybecause it provided a convenient means of demonstratingthe potential influence of intrinsic anisotropies in zincblende nonlinearities on device performance. Never-theless, this simple polarization rotation switch hassome interesting features when compared with otherdevices that utilize instantaneous optical Kerr non-linearities, for example, a waveguide nonlinear di-rectional coupler that was reported recently in the


literature.46,47 The latter device was 6 mm long, wasfabricated from AlGaAs, and operated at less thanhalf the band gap with 350-fs pulses. It can be seenthat the polarization rotation switch achieves a muchlarger contrast ratio (20,000:1 versus a maximum of,10:1) at a significantly lower intensity (4 3 1013 versus4 3 1014 Wym2) in a significantly shorter (2-mm) piece ofmaterial, but at the cost of throughput. The through-put of the directional coupler is ,75%. The through-put of the polarization rotation switch depends on yourpoint of view. The pump is undepleted and thereforealmost completely transmitted. The probe transmissionthrough the polarizer is ,20% for the stated irradiancein the on state (pump present) and nearly completelyrejected in the off state (no pump). Nevertheless, evenin the off state the rejected probe light is not absorbedbut is merely deflected. How much of this energy canbe reused depends on the system architecture. Also, incontrast to the figures of merit just discussed for the po-larization rotation switch, the relevant figures of meritdefined for the waveguide coupler are n2Iya0l (belowhalf the band gap) and n2y2bl (above half the band gap).The contrasting figures of merit for the two devices andfor these two spectral regions illustrate that the defini-tions of such quantities depend on operating wavelength,dominant active process, device geometry, and intendedapplication.

7. CONCLUSIONWe have shown that standard picosecond pump–probeellipsometric techniques can be readily adapted to mea-sure the intrinsic anisotropies in the instantaneous third-order bound-electronic nonlinear optical response of zincblende semiconductors. We directly measured the bire-fringence, characterized by Dn2, and the dichroism, char-acterized by Da2, that were induced by an intense 1-pspump pulse at 950 nm that was polarized first along the[100] and then along the [110] crystallographic axes ofthe zinc blende semiconductors ZnSe, GaAs, and CdTe.We then used the values for Dn2 and Da2 measured forthese two specific orientations to determine the sign andthe magnitude of the intrinsic anisotropy in the nonlinearrefractive index n2 and the two-photon absorption coeffi-cient b. Finally, we used Dn2f100g and Dn2f110g to de-termine the product of the anisotropy parameter and thereal part of the total diagonal tensor element xs3d

xxxx, andwe used Da2f100g and Da2f110g to obtain the product ofthe anisotropy parameter and the imaginary part of thediagonal tensor element x s3d

xxxx.For ZnSe the excitation was near, but less than, half the

band gap energy. In this spectral regime we observedno dichroism because there is no measurable nonlinearabsorption. The birefringence, however, showed a largeanisotropy varying by roughly a factor of 2 between thetwo orientations used here. By comparison, in GaAs andCdTe the photon energy was above the two-photon reso-nance. In fact, it was reasonably near a single-photonresonance (i.e., near the band gap). In this spectral re-gion the birefringence reversed sign and was an order ofmagnitude larger than that measured below half the bandgap (i.e., compared to ZnSe). The measured dichroism

was also large and varied by roughly a factor of 2 withcrystal orientation.

To investigate and illustrate the possible effect of crys-tal orientation on the performance of a device, we usedthe induced birefringence and dichroism measured in eachof these semiconductors to construct an all-optical on–offswitch. We defined figures of merit and investigated theeffects of crystal orientation on device performance forthe two spectral regimes studied here (namely, below halfthe band gap and near the band gap). For the purelyrefractive anisotropy observed in ZnSe we obtained ahigh probe throughput of more than 20% and an ex-tremely high contrast ratio (more than 20,000:1) at anintensity similar to that required to produce switchingaction in a nonlinear waveguide coupler. In this spec-tral regime the figures of merit and device performancewere strongly dependent on orientation and varied by fac-tors of 5. For GaAs and CdTe, where both absorptiveand refractive anisotropies contribute, higher contrast ra-tios were obtained at lower pump intensities, but theswitch throughput was ultimately limited by two-photonabsorption. The figures of merit for operation above halfthe band gap were shown to be much less sensitive tocrystal orientation.

ACKNOWLEDGMENTThis study was supported in part by the U.S. Office ofNaval Research.

*Present address, Department of Physics, University ofIllinois at Chicago, Chicago, Illinois 60607-7059.

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37. In the notation used in this paper xijkls2v, v, vd specifi-cally refers to those terms in the nonlinear susceptibilitythat resonate whenever 2v is equal to a transition frequency(two-photon-resonant terms). Similarly, xijklsv, 2v, vdrefers to one-photon-resonant terms and xijklsv, v, 2vd tononresonant terms. An alternative notation also appearsin the literature (e.g., in Ref. 35) in which one guarantees in-trinsic permutation symmetry by defining the susceptibilityas a permutation over all frequency/polarization pairings.The relationship

xs3dijkls2v; 2v, v, vd 1/3fxijkls2v, v, vd

1 xijklsv, 2v, vd 1 xijklsv, v, 2vdg

allows one to make the connection between the notation ofRef. 35 (on the left-hand side) and that used in this study.The frequency ordering on the left-hand side has no physicalsignificance. Accounting for this relationship, then

xkT ; 6x s3d

xxxxs2v; 2v, v, vd ,

x'T ; 6x s3d

xyyxs2v; 2v, v, vd .

These results are a special case of the relationship

xeffective je ? pj2x s3dxyyxs2v; 2v, v, vd

1 x s3dxyxy s2v; 2v, v, vd

1 je ? ppj2x s3dxyyxs2v; 2v, v, vd

for two-beam experiments in the isotropic limit.38

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intrinsic and induced anisotropy of nonlinear absorption and refraction in zinc blende...

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