1388 J. Opt. Soc. Am. B/Vol. 11, No. 8/August 1994

Coherent and incoherent pump-probe specular inverseFaraday effect in media with instantaneous nonlinearity

Yu. P. Svirko

General Physics Institute, 38, Vavilov Street, Moscow 117333, Russia

N. I. Zheludev

Department of Physics, University of Southampton, Southampton S017 1BI, England

Received November 8, 1993; revised manuscript received March 28, 1994

A wave theory of the specular inverse Faraday effect (SIFE) that reveals the existence of coherent andincoherent terms of the phenomenon is presented. Recovery of various cubic nonlinear tensor componentsfrom the SIFE observation in transparent and opaque materials is examined.

INTRODUCTION

A number of pump-probe, nonlinear, polarization-sensitive reflectivity techniques have been experimen-tally developed for measurement of optical nonlinearitiesin solids, the first example being the pioneering work ofAuston and Shank,1 in which transient reflectivity nearthe Brewster angle was applied in a study of the dynamicsof the refractive index of optically created electron-holeplasma in germanium. Transient dynamic ellipsometrywas later used in the monitoring of silicon melting by apulsed excimer laser.2 The optical Kerr effect in a con-figuration termed on reflection was used for the study ofexcitons in ZnSe.3 Dynamics of the nonlinear nonlocaloptical response of dense free electron-hole plasma4 andsubpicosecond time-resolved phonon oscillation have beeninvestigated in GaAs by specular polarization-sensitivemethods.5

In the present paper we concentrate on the novel phe-nomenon of the specular inverse Faraday effect (SIFE).This is the reflection version of the conventional trans-mission inverse Faraday effect, a classical electrodynamiceffect that in the simplest case consists of the mag-netization of a medium in the presence of a rotatingelectric field. In a sense the classical effect is oppositeto the rotation of the polarization plane in a magneticfield and is therefore called the inverse Faraday effect.6r0

The induced magnetization may be detected directly withinstruments sensitive to magnetic momentum, such as su-perconducting quantum-interference magnetometers."" 2

The result of optical stimulation can also be investigatedwith a second weak probe beam through detection of thepump-induced polarization-plane rotation or by measur-ing the pump-induced differential absorption for left- andright-hand-side circularly polarized probe beams. Thismethod has been used widely in laser spectroscopy.' 3 '19

In the SIFE scheme the polarization state of the probewave reflected from the medium interface, rather than ofthe transmitted wave, is measured. In the most commonconfiguration a weak linearly polarized light wave acts asa probe, and the pump wave, used for stimulation of the

medium surface, is circularly polarized (see Fig. 1). Thenonlinear polarization change of the reflected probe beamis a figure of merit of the effect. During the past fewyears this specular technique has attracted extensive ex-perimental attention in the spectroscopy of solids. It wasfirst demonstrated by Kuwata and used for the measure-ment of the strong resonant excitonic optical nonlinearityof CuCl and HgI2.20 Recent progress in time-resolvedpolarization measurement techniques led to the experi-mental discovery that SIFE has coherent terms that de-pend on the relative phase between the probe and thepump waves and incoherent terms that do not.21 Co-herent and incoherent SIFE was observed and used formeasurements of nonresonant optical nonlinearities insemiconductor crystals of GaAs and InSb (Ref. 21) and su-perconducting crystals of YBa2Cu307 8 *22 Strong light-induced polarization-plane rotation was recently observedin the diluted magnetic semiconductor CdMnTe.2 3 TheSIFE measuring technique developed in Refs. 20-24 per-mits reliable measurements of optical nonlinearities indifferent regimes as the optical energy and intensity arevaried, providing information on resonant and nonreso-nant susceptibilities in opaque and transparent materials.Use of the SIFE in magnetic studies is also promising,since the microscopic aspects of the induced magnetismand its subsequent relaxation might be measured withthe ultimate time resolution.

However, theoretical treatment of the SIFE has notbeen properly developed. The only theoretical paperknown to us on the specular pump-probe polarizationphenomena that partially addressed the SIFE effectis Ref. 25. However, the approach used there ignoresthe coherent contribution to the effect, which has ledto the evisceration of the important physical proper-ties of the phenomenon and limits its use in interpre-tation of the existing experiments. In this paper wepresent a consistent phenomenological theory of the SIFEin media with instantaneous nonlinearity and for the firsttime to our knowledge consider the following aspects: (i)the treatment of the SIFE as a degenerate four-wavemixing phenomenon, disclosing the coherent and inco-

0740-3224/94/081388-06$06.00 ©1994 Optical Society of America

Y. P. Svirko and N. L Zheludev

Vol. 11, No. 8/August 1994/J. Opt. Soc. Am. B 1389

A. 1

Probe Pump

Fig. 1. Experimental arrangements for observation of the SIFE.

herent contributions to the SIFE; (ii) the investigationof the role of the real and the imaginary parts of opticalnonlinearities; (iii) the manifestations of the effect as in-duced probe polarization-azimuth rotation and alterationof probe-wave ellipticity; (iv) the procedure for the re-covery of various tensor components of nonlinear opticalsusceptibilities from the SIFE data. Our main objec-tive in this paper is to disclose the full nonlinear opticalpicture of the effect and the full potential of SIFE spec-troscopy, specifically for opaque materials, for which theapplication of transmission spectroscopy is impossible.

SIFE AS A NONLINEAR OPTICALINTERACTIONIn the most convenient SIFE configuration the probe isdirected normally to the crystal surface, and the pumpdirection makes a small angle with respect to the probe.The pump-induced alteration of the polarization stateof the probe reflected from the investigated surface isdetected and used in spectroscopic interpretations. Interms of nonlinear optics one can easily see the anal-ogy between the SIFE and a degenerate four-wave mixingprocess. The only peculiarity of the SIFE process is thatthe nonlinear interaction actually takes part in the thinlayer of the crystal that is responsible for the reflection.In the case of strongly absorbing materials the depth of this skin layer may be estimated as the smaller of thewavelength or the pump-probe penetration depth.

Below we conduct a phonomenological analysis of theSIFE, considering the instantaneous optical response andpresuming that the pump and the probe waves have nar-row spectra centered at frequency w. The concept of opti-cal susceptibilities may be used here in a straightforwardway, and the following material equation for the vector ofthe electric displacement strength, D, may be employed:

where A exp(ikz) is the pump and a exp(ikz) is the probewave, k = cn/c is the modulus of the wave vector, and thewaves are assumed to propagate along the z direction of aCartesian coordinate frame, where the x and they axes liein the plane of the reflecting surface of the crystal. Thecollinear propagation of the pump and the probe ensuresthe necessary phase-matching conditions for the coherentpart of the pump-probe interaction, however, since thenonlinear interaction length I is very small in a specularnonlinear process. The angle 0 between the pump andthe probe may reach several degrees, while the phase-mismatch parameter kR 7r02nlA-' remains a fractionof unity, and the treatment below is still completely ap-plicable when some angle is introduced to separatethe probe and the pump spatially. It is also suitable forthe practical situation in which the pump and the probewaves, being initially collinear, are focused on the targetby the same lens that makes their wave fronts parallel inthe focal plane.21 -2 4 However, in the general case of non-normal incidence of the probe and the pump waves thesituation is much more complicated. For example, thephase-conjugate and the mirror-reflected waves may notpropagate in the same direction. Moreover, becauseof weak phase-matching requirements arising from theshort interaction length there may be multiple-order scat-tered waves spread over different angles from the normalto the surface. (For a detailed explanation of the sim-plest case of a normally incident pump wave with theprobe wave propagating at some angle, see Ref. 26.)

WAVE PROPAGATION IN THENONLINEAR MEDIUM

To approach the nonlinear reflection problem, we firstshould consider the wave-propagation problem in the non-linear medium behind the interface, since we need toknow the general solution of the wave problem beforeconsideration of boundary conditions. We consider theintensity of the probe to be much smaller than the pumpintensity, and we neglect the effect of the probe on thepump. In such a case the magnitude of the probe wave inthe nonlinear medium is governed by the following equa-tion:

da 2 ikdz = (aijaj + flijaj*), (3)

which may be readily deduced from the wave equation bythe slowly varying envelope approximation method. InEq. (3)

aij = - ij + (c) 2iDi = eijEj + X(3) EjEEm.-

Choosing the material equation in this form, we ignenergy losses associated with the generation of optiharmonics on the second-order nonlinearities. Howe-under cubic nonlinearities X(3)1 we assume the effectvalues, incorporating contributions from the direct the cascade nonlinearities through implied second-orterms in Eq. (1). Here the total electric field inmedium is

E = A exp(ikz) + a exp(ikz) + c.c.,

+ 2( k )XVjmAiAm*,(1)

ore;calrer,

fij = ( ) X (3AlAm-

(4)

(5)

;ive Until the present stage we have not made any assumptionLnd concerning the symmetry of the nonlinear medium. Onder the basis of Eq. (3) the wave problem may be solved forthe arbitrary initial pump and probe polarizations and crystal

symmetry. However, hereafter for the sake of simplicityand physical clarity we consider only isotropic media.

(2) These, however, are the most important practical cases.

Y. P. Svirko and N. I. Zheludev

Aiv

1390 J. Opt. Soc. Am. B/Vol. 11, No. 8/August 1994

If the reflection takes place along an isotropic medium oran isotropic direction, we have

e= eyy = n2(1 + iK), k = &)n/c, (6)

while the nonlinearity tensor Xijlm(, two, t, - &)) has onlytwo independent components2 7 and may be presented inthe following form:

Xijlm = X1(3ij'lm + 65il 5jm + ime5 l) + X25im-5jl,

Xxx)xx = 3X + X2,

(3) =xxyy Xi

Xyyx= X2 + X1. (7)

Here we have extracted Xi, which is a totally symmetri-cal (with respect to permutation of the indices) part of thecubic susceptibility tensor. This representation, Eqs. (7),is the most general form of the fourth-rank tensor in anisotropic medium. It does not imply any reservationsconcerning the existence of optical resonances, dispersion,or dissipation. The component Xi is the only term ofthe cubic nonlinearity allowed in both absorbing and ab-sorptionless media for which the frequency dependence ofthe optical response may be ignored. On the other hand,

X2 = Xxyyx - Xxxyy vanishes if frequency dispersion is neg-ligible; i.e., X2 is essential only in absorbing materials.Now, from Eqs. (6) and (7), the coefficients aij and .ij inEqs. (4) and (5) are

(3) (3)=2 _ Xxxxx A 2 + 2 2Mx-n2 X n Y

(3)

n2

a 2 A yyxA*Al (9)cy=2[ XXYY AxAy* + xy I X*y, 9

IA3 2 2 X(3)Oay = iK + 2 XxxtX2+2xy A7~~2 n2

= 2 !xy AxAy* + X2 Ax*Ay, (10);S _ XYXY A2 n X A 2

2 2

yy- 2Ax 2 ~ Y./3 A 2

(3)

,By= 2 2xy AxA (1n

Until now no assumptions concerning the polarization ofthe pump and the probe have been made; i.e., the resultsabove could be used for analysis of various pump-probe

polarization combinations. They may be used, for ex-ample, not only for the specular Faraday effect but alsothe reflectivity Kerr effect, where the pump and the probeare initially linearly polarized at 450 with respect to eachother as in Ref. 3.

For the SIFE, however, the pump wave is circularlypolarized, i.e., Ax = A+/21/2, Ay = iA+/2" 2 . In this casethe right and the left circular components of the probe

wave a, = (a. ± ia,)/2 1 2 are governed by the followingequations:

da+ = ik (aa+ + pa+ )X

da- ik (k a-), (12)

where

a = iK + X1 A+A+ ,

n

2(2xi + X2) AA*A- = iK + 2+X2 A A *n

The general solution of this set of equations is

a+ = b+ exp( 2z) + b+* exp( 2*)

ikIL-z)a- = bexp 2)

where we introduce the following new variables:

,+ = i Im a + [(Re a)2 - 112]112,

= - + aA+* + a+ -

Rea +Re a)2 - 112 1 2

(13)

(14)

(15)

BOUNDARY PROBLEM

Now, since we know the general solution for a propagatingwave in a nonlinear medium, we are in a position to con-sider the boundary problem and study the polarization-sensitive nonlinear reflection itself. To find themagnitude of the reflected wave, we should take intoaccount the continuity of the tangential componentsof the electric- and the magnetic-field strengths onthe surface, that is, E±l(z = 0) = E± 2 (z = 0) andB~l(z = 0) = B±2(z = 0). Here the indices 1 and 2,respectively, denote the vacuum and the crystal (seeFig. 1). The circular components of the probe-waveelectric-field strength in a vacuum are equal to

E+(z) = a±j exp(iwt + ikz)

+ ar exp(-iwt - ikz) + c.c., (16)

where the indices i and r denote the magnitudes of theincident and the reflected probe waves. Now by use ofthe general solutions (14) and the Maxwell equation curlE = -c'lB/at the continuity conditions may be pre-sented in terms of the magnitudes of the incident, thereflected, and the transmitted light waves in the follow-ing forms:

E+l(z = 0) = E+2 (Z = 0) V ai+ + ar+ = bt+ + 6bt+

B+l(z = 0) = B+2(Z = 0) V as+ - ar+

= n bt+(1 + A ) + bt+*(1-

E 1J(z = 0) = E- 2(Z = 0) V a + ar- = bt-,

B-(z = 0) = B 2 (z = 0) V as - ar- = nbt-(1+ -

(17)

Y. P. Svirko and N. I. Zheludev

Vol. 11, No. 8/August 1994/J. Opt. Soc. Am. B 1391

From Eqs. (17) the magnitudes of the reflected wave maybe obtained in a straightforward way:

1- nar+ = + i+

1+n

(1 +)2(i Im + + 1 _ 1612 Re t+)

x a+ + (1 + n)2 1 1612 Re(,.+)ai+±X

1-n n1 + n -I + n)2rai (18)

If we now take into consideration the following relations,which are readily obtainable from Eqs. (15),

in 2 * - 12a,=(1+ n)2 (XiA+ 2 j 2 AIa) (24)

This orthogonal component arises as the result of boththe reflected probe azimuth rotation and the induced el-lipticity of the reflected light. The polarization azimuthrotation 8a itself is given by

8 = 1 tai(- S 2r 2 \ Sir )

-A+12

n(l - n2) I[X2 - Xi exp(2iok)], (25)

while the induced degree of ellipticity 8c7 is given by

877 = 2 sin( S)1 + 2 = Re a1 - If12 Re ,+

- _2 _ R 31-lfi2 ReMt+ (19) _ A+ 12

n(l - n2) Re[X2 - xi exp(2iik)],

then the solution for the reflected-wave magnitudes maybe presented in the more compact form

I-n n)*ar+1= + ai+ _ (aa+ + ai+*),

1 + n (1 /3aj+*)

where S are the Stokes vector components of thereflected probe wave: So = arxarx* + aryary*, S =

arxarx * - aryary*, S2 = 2 Re(arxary*), S3 = 2 Im(arxarY*).In Eqs. (25) and (26) we have introduced

1-n nar= +n ai- - - - ai.- (20)

Equations (20) can actually be used for calculations of thedifference between left and right reflection coefficients,which is measured in what could be called nonlinear dif-ferential reflection specular spectroscopy.

For the remainder of this paper we specifically considerthe case of a linearly polarized incident probe wave alongthe x direction,

as+ = ai- = a/, (21)

where in terms of the magnitudes of the incident wavethe Cartesian coordinates of the reflected probe,

ar+ + ar-arx= AS X

ar+ - ar-ar,,= 2

0 = arg(A+ai*), (27)

which is the phase difference between the pump and theprobe waves.

The reflected probe-wave polarization variation, i.e.,the alterations of the angle of polarization azimuth rota-tion 6ar and ellipticity 8'gsr in terms of optical susceptibil-ities and incoming pump-beam intensity It± as measuredoutside the crystal are

{ ar | _ 327rIt±| 27r J cn(n - 1)(n + 1)3

x { Rj{-X2 + XI[cos(2tk) + i sin(20)]}. (28)

Here ± indicates, for the solutions corresponding to theright and the left circularly polarized pump,

(22) CIA- 12 (1 - n2 )

32ir(1 + n2 ) (29)

are given by the following equations:

ar = 2(1 + n)2 (a - -)ai + 3ai*],

arx - [(a + (-)a + 3ai*].1 +n 2(1 +n)2

(23)

SPECULAR POLARIZATIONEFFECTS IN SIFE

One can see that ary, i.e., the component of the reflectedprobe orthogonal to the initial probe polarization, appearsas the result of the presence of the pump. In terms ofnonlinear optical susceptibilities and amplitudes of thepump wave,

The first term on the right in Eq. (28) does not dependon the relative phase between the pump and the probeand does not require them to be mutually coherent tocontribute to the SIFE; this is the incoherent contribu-tion. The last two terms, on the other hand, depend onthe relative phase 0 between pump and probe and are re-sponsible for the coherent contribution to the SIFE. Wenow consider the two different cases:

(a) In a transparent medium, for which frequency dis-persion may be neglected and correspondingly X2 = 0, thephase-independent term in Eq. (3) disappears and onlythe coherent SIFE remains. The value and the sign ofthe net effect on the reflected probe-beam polarization ro-tation oscillates with the phase retardation 0 between thepump and the probe. If the pump and the probe are mu-tually incoherent or suffer from strong phase fluctuations,

(26)

Y. P. Svirko and N. I. Zheludev

1392 J. Opt. Soc. Am. B/Vol. 11, No. 8/August 1994

the time-averaged value of the induced rotation will tendto zero, and no SIFE observations are possible.

(b) In absorbing materials, for which dispersion is im-portant, X2 0, and the phase-independent terms inEq. (28) lead to the incoherent SIFE. In general bothcoherent and incoherent contributions are present. Thephase-independent term does not disappear even if thepump and the probe are mutually incoherent.

Equation (28) implies that all variations of the po-larization state of the reflected probe wave are con-trolled by four material parameters, namely, Re XI, Re X2,

Im V1, and Im X2, and that the complex-conjugate compo-nents of the nonlinear susceptibilities are responsible fordifferent optical effects, e.g., the induced polarizationazimuth rotation and ellipticity, and hence could be mea-sured simultaneously. Consequently SIFE spectroscopycould give direct information on both real and imaginaryparts of both independent cubic nonlinearity tensor com-ponents in an isotropic transparent or completely opaquemedium. The coherent contribution may be recoveredfrom the total effect by observation of the dependence ofthe polarization rotation on 0.

OBSERVATION OF SIFE WITHPULSE LASERS

With instantaneous optical nonlinearity in the pump-probe SIFE configuration, when optical pulses are used,the induced probe-beam polarization alteration may be ob-served only if the probe and the pump beams overlap onthe sample. Since recovery of the incoherent and the co-herent contributions to the SIFE requires measurement ofthe reflected probe-beam polarization alteration as a func-tion of the pump-probe relative phase 0, in practice re-covery depends on the measurements of cross-correlationfunctions of the observable effect and the pump pulseas the optical delay (i.e., r = 0eiw) varies. Because theexperimentally observable value of the polarization ef-fect in most techniques (see Refs. 20-24, for example) isgiven by the convolution of the instantaneous value of thepolarization-plane rotation of the probe wave and the in-tensity of the probe beam, this cross-correlation functionhas the following form:

K(r) = dtI(t)6a(t, ).

It is clear that for a cw laser (r - -o) an expressionsimilar to the initial formula (28) may be obtained fromEq. (33). One can also see from Eq. (33) that the relativeproportions of the coherent and the incoherent terms ofthe SIFE may be deduced by comparison of the oscillatingand the smooth parts of the cross-correlation function, andeven with a pulsed pump and probe the oscillating natureof the coherent contribution in K(r) is not destroyed (seeFig. 2).

The SIFE is a small effect. However, with cw mode-locked lasers it may be accurately observed and the co-herent and the incoherent components separated. A fewexamples can give some idea about the scale of the ef-fect for nonresonant nonlinearities. In GaAs the in-duced polarization rotation constant, i.e., induced probepolarization rotation in radians per one unit of pumppulse intensity, was experimentally found to be 2 x10-11 rad cm2 W-', and in InSb, 9 X 10-1' rad cm2 W-1at 532 nm, giving rise to an induced rotation of -101 radat a pump intensity of 5 MW cm-2 in GaAs andInSb.2 ' In a YBa2Cu3 07-_ superconductor this con-stant was 2.2 X 10-11 rad cm2 W-1,22 while in dilutedsemimagnetic Cd06 MnO4Te it reaches the value of2 X 10-10 rad cm2 W-1.2 3 In SIFE the short nonlinearinteraction distance of the skin layer is in fact an ad-vantage. We believe that the strict requirements for thecollinearity of the pump and the probe, which are imposedby the phase-matching conditions of the coherent inverseFaraday effect, explain why the coherent effect has notbeen discovered yet in transmission. Another traditionalproblem of beam self-focusing that complicates the cor-rect evaluation of optical susceptibilities because of themodification of the beam profile in transmission experi-ments does not appear to be important at all in the SIFE.However, the most clear advantage of the SIFE as a spec-troscopic method appears in transient measurements inwhich submicrometer nonlinear interaction lengths en-sure minimal possible pump and probe pulse distortion bymaterial dispersion. Recent SIFE results obtained witha mode-locked titanium:sapphire laser in GaAs carrierspin-relaxation measurements prove this point.24

K(T)(30)

For the simplest case of pump and probe having bell-likepulse shapes,

Ip = .r exp - - ]

IT [P (TP)]W 2~~~

41 = exp -IIV1/T Ti T

we can get the following result:

K(r) = (2r)" 2rpn(n2 _ 1 I 11-X2 + yl exp(2iw

X exp( 2)|

(31)

(32)

Al_rli Fig. 2. Observable value of the SIFE K(r) as a function of

pump-probe delay r (relative units). The half-duration of thepump and the probe pulses is rp. The oscillating contribution

(33) corresponds to the coherent term, whereas the darker bell-shapedcurve represents the incoherent term of the SIFE.

ncohet 1IoI u4t = i /incoherent contribution

Y. P. Svirko and N. I. Zheludev

Vol. 11, No. 8/August 1994/J. Opt. Soc. Am. B 1393

CONCLUSIONS

We have developed the wave theory of the SIFE effectin the case of instantaneous nonlinearity and treated theeffect as a four-wave mixing phenomenon, revealing thecoherent and the incoherent contributions to the effect.We have shown that measurements of the alterations ofthe polarization state of the reflected probe wave in thepresence of the pump in isotropic media give informationabout the real and the imaginary parts of all nonzerocomponents of the cubic nonlinearity tensor, providing asimple and sensitive method for spectroscopy of opaqueand transparent materials.

Finally, the results obtained here nicely reflect the tworemarkable correspondence principles of polarization op-tics, which we formulate here in the following manner:

* A polarization phenomenon in transmitted and re-flected light is determined by the complex-conjugatedparts of the same optical susceptibility.

* Small changes in ellipticity and polarization azi-muth of a linearly polarized wave are determined bythe complex-conjugated parts of the same optical sus-ceptibility.

ACKNOWLEDGMENTS

This work was partially supported by the Royal Soci-ety, London, and the Science and Engineering ResearchCouncil.

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Y. P. Svirko and N. I. Zheludev