15 June 1998

Ž .Optics Communications 152 1998 207–214

Full length article

Giant enhancement of low frequency non-steady-state photo-EMFsignal in Bi SiO crystal under external DC bias12 20

S. Mansurova a,), S. Stepanov b, N. Korneev b, C. Dibon c

a UniÕersidad Autonoma de Estado de Morelos, FCQeI, AÕ. UniÕersidad 1001, Col. Chamilpa, CP 62210, CuernaÕaca, Morelos, Mexicob Instituto Nacional de Astrofisica Optica y Electronica, AP 51y216, Puebla 72000, Pue., Mexicoc Institut d’Optique Theorique et Appliquee, Bat 503, Centre Scientifique D’Orsay, Paris, France

Received 11 August 1997; revised 12 February 1998; accepted 12 February 1998

Abstract

We present experimental results which demonstrate that non-steady-state photo-EMF, induced by vibrating interferenceŽ 3 .fringes in Bi SiO photorefractive crystal can be dramatically by a factor of 10 in our experimental conditions12 20

enhanced in a resonance regime under application of an external DC field. We show that for increasing applied electric fieldthe resonance maximum moves from the dielectric relaxation frequency towards the low frequency end. Its magnitude growsas E2 for low values of the field, for high external field significant broadening of the resonance peak and linear growth of0

the signal resonance amplitude is observed. This effect and non-trivial dependence of the resonance signal amplitude on thespatial frequency of the pattern is explained by saturation of the space charge grating. Finally, the sensitivity of the biasedresonance adaptive photo-EMF detector is estimated. q 1998 Elsevier Science B.V. All rights reserved.

Keywords: Photorefractive media; Non-steady-state photo-EMF; Space charge electric field; Running grating; Resonance amplification

1. Introduction

Non-steady-state photoinduced electromotive force ef-Ž .fect photo-EMF effect reveals itself as an AC current

through a short-circuited photoconductive sample illumi-w xnated by a vibrating interference pattern 1 . It can be used

for high sensitivity adaptive detection of optical phasemodulated signals. In recent years different proposals forapplication of this technique in homodyne laser vibrometryw x w x2,3 , in fiber optical sensors 4 , for phase locking of

w xindependent lasers 5 have been reported. It is worthnoting here that unlike the other similar technique which

w xemploys a moving interference pattern 6 , the photo-EMFtechnique used does not impose strict conditions to theinterference fringe number in the interelectrode spacing

) E-mail: smansur@fcqi.uaem.mx

Ž lsNL, where l is the interelectrode spacing, L is the.interference fringe spacing, and N is an integer .

Most of these applications are related to the highfrequency region when the detected modulation frequencyis larger than the dielectric cut-off frequency equal to theinverse dielectric relaxation time t s´´ rs . In the lowdi 0 0

Žfrequency region i.e. for frequencies below the dielectric.cut-off frequency , the output signal is proportional to the

frequency of modulation and does not depend on theaverage light intensity. So, the lower the modulation fre-quency, the lower the output signal observed.

There is, however, a number of fields where the adap-Ž .tive detection of low frequency in the 1–100 Hz region

phase modulated optical signals is necessary. One impor-tant example is detection of slow periodic thermal changes

w xin experiments on detection of weak optical absorption 7 .Of course, one can use photodetectors with low photocon-ductivity and large dielectric relaxation time t . But thisdi

will result in a low photoresponse of detection and bring

0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0030-4018 98 00098-4

( )S. MansuroÕa et al.rOptics Communications 152 1998 207–214208

additional problems in matching the photodetector withpreamplifier.

Another problem of similar kind can arise in applica-tions of adaptive photodetectors based on semiconductorcrystals. Due to high photoconductivity the dielectric cut-off frequency lies in the MHz region in these materials.This naturally limits the number of important vibrometricapplications in the kHz frequency region.

In this paper we propose an alternative approach tohigh sensitivity adaptive detection of low-frequency sig-nals which consists in the following. It is well known thata shifted space charge electric field grating can be reso-nantly amplified by using the moving grating technique in

w xphotorefractive crystals under external DC bias 8 . In thistechnique the moving interference pattern is usually ob-

Žtained by introducing a constant frequency shift this canbe done using, for example, a piezomirror driven by

w xperiodic sawtooth voltage 9 or an acousto-optical fre-w x .quency shifter 10 between two interfering plane waves .

This results in a phase shift increasing linearly with timeand the interference pattern in the bulk of the samplemoves with constant velocity. To reach the resonanceregime of holographic recording the velocity of the dis-placement of the interference fringes has to be equal to theresonance velocity of the running space charge electric

Ž w x.field grating i.e. ‘wave of recharging of traps’ 11 . Theimportant point here is that the higher the DC voltage

Žapplied, the lower the resonance velocity i.e. the reso-. w xnance frequency of the grating 8 .

Unlike this, in the photo-EMF configuration consideredŽ .here periodical sinusoidal phase modulation is introduced

Ž .in one beam. As a result we get a vibrating or oscillatinginterference pattern. Formally, it can be represented as asuperposition of a stationary grating and two other gratingsmoving with constant velocity in opposite directions. If thephase velocity of one of these moving interference gratingscoincides with the resonance velocity of the space chargeelectric field grating, resonance enhancement of the photo-EMF current occur. It can manifest itself as a resonancepeak which appears in the frequency transfer function. Asit follows from above, this peak shifts towards lowermodulation frequencies with increasing external voltagew x1 . This clearly can give an excellent possibility of effi-cient adaptive detection of phase modulated signals atfrequencies below the dielectric cut-off frequency.

General theoretical analysis of the photo-EMF effect inthe presence of DC bias for a monopolar photoconductor

w xwith long photocarrier lifetime was given in Ref. 12 . WeŽchoose this, more complicated, model which takes into

.account the photocarrier lifetime because it correspondsbetter to the Bi SiO samples under the experimental12 20

conditions we use below. Note, however, that similar basicresults concerning resonance amplification of the low-frequency photo-EMF signal can also be obtained using a

w xmore simple model developed in Ref. 1 ignoring thefinite carrier lifetime.

In the following we use the final expression for thephoto-EMF short-circuit current density amplitude ob-

w xtained in Ref. 12 :

m2Ds0Vj s4

=2i E yVt E q i EŽ .0 di 0 D

2 2 21yV tt q iV tqt 1qK L q i KLŽ .Ž .di di D 0

2i E yVt E y i EŽ .0 di 0 Dy .

2 2 21yV tt q iV tqt 1qK L y i KLŽ .Ž .di di D 0

1Ž .

Here m is the contrast of the interference fringes, D is theamplitude of phase modulation, s is the average photo-0

conductivity of the sample, V is the frequency of modula-tion, t s´´rs is the dielectric relaxation time, t is thedi 0

photocarrier lifetime, K is the spatial frequency of thegrating, L and L are the average diffusion and driftD 0

length of the photocarrier respectively, E sKk Tre isD B

the diffusion field, and E is the external DC field. As0

usual, m and D are assumed to be small.It is possible to show from this equation that under

external DC bias two resonance peaks with the characteris-tic frequencies

1X

V , 2Ž .rt KLdi 0

and

KL0YV , 3Ž .r

t

Ž .are observed Fig. 1 . Note that under increasing DCvoltage the first, more intensive, resonance with the ampli-tude

m2Ds E KL0 0 0Xj ,y 4Ž .r 2 22 1qK LŽ .D

and with the width

2 1qKLŽ .DDVs 5Ž .2

t KLŽ .di 0

shifts towards the low frequencies. Below we pay attentionfirst of all to the experimental investigation and discussionof this resonance maximum.

The second, high frequency resonance related to thefinite lifetime of the photocarriers runs into the opposite

Ž .direction with increasing external bias Fig. 1 . It corre-sponds to the characteristic time of the photocarrier drift

w xthrough one spatial period of the interference pattern 13 .This resonance was also observed experimentally in the

w xphoto-EMF configuration earlier 12 .The main purpose of this paper is to present original

experimental results which demonstrate the possibility of

( )S. MansuroÕa et al.rOptics Communications 152 1998 207–214 209

Ž .Fig. 1. Theoretical curves calculated from Eq. 1 for differentvalues of the external DC field. t ft and K s Ly1. E has thedi D 0

Ž . Ž . Ž . Ž . Ž .following values: a 0; b 3E ; c 6.25E ; d 12.5E ; eD D DŽ .25E ; f 50E .D D

efficient resonance amplification of the photo-EMF signalat low frequency of modulation under an external DC field

w xin Bi SiO . In agreement with the theory 1 the position12 20

and magnitude of this resonance peak depends strongly onthe external DC field and spatial frequency of the pattern.The effect of saturation of the recorded electric field

grating and its influence on the amplitude and width of theresonance peak are also discussed below. Finally, thesensitivity of the adaptive photodetector operating in theresonance regime is estimated.

2. Experimental samples and configuration

The experiments were performed using a Bi SiO12 20Ž . 3BSO sample of 4=1=10 mm size at the wavelength

Ž .of the doubled frequency cw Nd YAG laser ls532 nm .Ž 2.The front and back faces of the sample 4=10 mm were

polished to optical quality. The lateral faces of the sampleŽ 2.10=1 mm were coated with silver-paste electrodes.

For detection of the photo-EMF signal we used anexperimental arrangement which allowed us to obtain anautomatic scan of the output signal amplitude versus fre-quency of modulation, spatial frequency of the pattern, and

Ž . Ž .external DC voltage Fig. 2 . The step motor SM , elec-Ž . Ž .tro-optic modulator EOM , and high voltage supply HVS

were connected to the lock-in amplifier SRS-510 whichwas controlled by a PC. Our optical configuration con-sisted of a Mach–Zehnder interferometer with the electro-optic modulator placed in one arm. To change the spatial

Ž .period of the interference fringes one of the mirrors RMŽof the interferometer located in the input plane of a

.standard 4f scheme was rotated by a step motor. Our BSOsample biased by the high voltage supply was placed in the

Fig. 2. Experimental optical and electronic arrangement for automatic scan of non-steady-state photo-EMF effect. BSI – cubic beam splitter;M – mirror; RM – rotating mirror; BS2 – plate beam splitter; EOM – electro-optic modulator; L1 and L2 – lens with focal distance 100mm and 120 mm, respectively; R – load resistor; HVS – high voltage supply; SM – step motor; FG – function generator; PC – personalL

computer.

( )S. MansuroÕa et al.rOptics Communications 152 1998 207–214210

output image plane. The output signal UV from the loadŽ .resistor R s10 kV after the standard preamplifier LM-L

310 was detected by the lock-in amplifier.Because of the relatively large interelectrode spacing of

our sample we used additional uniform high intensityillumination of one half of the sample. This was done toobtain higher efficient electric field in the other part of thesample for the same external voltage applied. In addition,this enabled more uniform illumination within this ‘work-ing’ half of the sample. For this purpose we additionally

Ž .used a narrow 2 mm horizontal slit placed just in front ofthe sample.

3. Experimental results

3.1. Frequency transfer function and spatial frequency( )dependence without external Õoltage E s00

As it is shown in Fig. 3 the frequency transfer functionŽi.e. the dependence of the photo-EMF signal on the

.modulation frequency without external voltage demon-Ž .strates the typical behavior predicted by Eq. 1 . The signal

grows linearly until some characteristic cut-off frequencyV

X which is higher for higher average light intensity. ThenŽthere is a frequency independent part in our experiments it

.is clearly seen for low light intensities only followed byan inverse linear decay beginning at some other cut-offfrequency V

Y. Note that this second characteristic fre-quency does not depend on the average light intensity, so

Žwe can attribute it to the photocarrier lifetime or gener-.ally, to the photoconductivity relaxation time . On the

ŽFig. 3. Frequency transfer function without external voltage E s0.0 . BSO crystal at ls532 nm for spatial period of the interfer-

ence pattern L s 2p rK f 2 mm, m s 0.9 and I f 1602 Ž . Ž . Ž .mWrmm . ` I ; e I r3; I I r20.0 0 0

Fig. 4. Spatial frequency scan of photo-EMF signal. BSO crystalat ls532 nm, for modulation frequency V r2ps1 kHz, V

X)

V ) VY , mf0.9 and I f0.8 mWrmm2.0

other hand, the first characteristic frequency, which de-pends linearly on the average intensity of the illumination,can be attributed to the dielectric relaxation time.

The spatial frequency scan of the photo-EMF signalŽ . XFig. 4 for the intermediate modulation frequency V )V

)VY also exhibits a typical behavior with an approxi-

mately linear growth, a maximum and a linear decay as1rK after the maximum. From the figure we can deter-mine the diffusion length L of the photocarriers whichD

Fig. 5. Frequency transfer function for different values of theexternal DC field. BSO crystal at ls532 nm, 1rLf900linrmm, ms0.6 and I f3.2 mWrmm2. The values of E are0 0

Ž . Ž . Ž . Ž .the following: B 0; ` 50 Vrcm; ^ 500 Vrcm; e 1000Ž . Ž .Vrcm; = 2000 Vrcm; I 4000 Vrcm.

( )S. MansuroÕa et al.rOptics Communications 152 1998 207–214 211

corresponds to the point of the maximum in this curve.This value was estimated as 1.4 mm which gives a mt

product equal to 1.02=10y6 cm2rV, corresponding wellw xto the literature data on BSO 12 . It is worth noting that,

in fact, in our experiments the position of the maximum inthe spatial frequency dependence was slightly influencedby the average light intensity.

3.2. Frequency transfer function under the external fieldE and the field dependence0

V Ž .Fig. 5 shows the J V dependences for differentvalues of the external DC electric field E . From Fig. 3 we0

could see that without external voltage for the light inten-

Ž .Fig. 6. a Resonance maximum position versus external electricŽ .field applied. b Resonance amplitude versus external electric

field applied.

Fig. 7. Frequency transfer function for different spatial frequen-cies. BSO crystal at ls532 nm, 1rL f900 linrmm, ms0.6,max

I f3.2 mWrmm2; E s4 kVrcm. The values of L are the0 0Ž . Ž . Ž .following: ` L ; ^ 0.3L ; I 3L .max max max

sity I s3 mWrmm2 two characteristic frequencies prac-0

tically overlap. After application of the external voltage theŽ .resonance maximum although broad enough appears in

Ž .the low frequency 10 Hz–1 kHz part of the curve. Theresonance frequency depends on the external electric field

Ž .as 1rE Fig. 6a . In turn, the resonance amplitude grows0

as E2 at least for low values of the external field and then0Ž .this dependence becomes linear Fig. 6b . Note that for

ŽE s4 kVrcm the resonance signal amplitude at Vr2p0. 3s10 Hz is higher by a factor 2.5=10 than the signal at

the same frequency without external field.In the high frequency region the displacement of the

Ž .second higher characteristic cut-off frequency proportion-ally to the applied electric field can be traced. At highvalues of the external field growth of the signal also occursin this region.

Frequency transfer functions for different spatial fre-quencies and fixed external DC voltage are presented inFig. 7. We can see that the position of the resonancemaximum is inversely proportional to the spatial frequencyof the grating. Our experimental data show also that theresonance amplitude is higher for the lower spatial fre-quencies.

Some resonance maximum can also be observed in theexternal field dependence of the photo-EMF signal formodulation frequency lower than the dielectric relaxation

X Ž .frequency V Fig. 8 . It can be seen from this curve thatwith increasing electric field the photo-EMF signal de-

Žcreases. But after some minimum which is more pro-.nounced for lower modulation frequencies it grows

Žquadratically, passes through some resonance see the.curves obtained for 90 and 300 Hz and after this begins to

decay again. On the other hand, for modulation frequency

( )S. MansuroÕa et al.rOptics Communications 152 1998 207–214212

Fig. 8. External field dependence of photo-EMF signal for differ-ent frequencies of modulation. BSO crystal at ls532 nm, 1rL

f400 linrmm, ms0.6, I s3.2 mWrmm2. The values of0Ž . Ž . Ž . Ž .V r2p are the following: ) 10 Hz; ^ 30 Hz; B 90 Hz; l

Ž .300 Hz; ` 1000 Hz.

Ž Xhigher than the dielectric relaxation frequency V r2p,1.kHz the electric field dependence after passing through

the minimum reaches some saturation level.

4. Discussion

It is clearly seen that the experimental curves obtainedŽ .for zero external field correspond to the basic Eq. 1 quite

well. Indeed, without the external bias the photo-EMFdemonstrates behavior typical for the monopolar photocon-ductor with long photocarrier lifetime. This is evidenced,first of all, by the presence of two characteristic points inthe frequency transfer function and their typical depen-dence on the average light intensity.

Appearance of the resonance maximum at frequencieslower than V

X and its evolution for low external electricŽfield in particular, the shift of its position as 1rE and0

2.growth of its amplitude as E also agree well with Eqs.0Ž . Ž .1 – 3 . This can be easily seen from comparison of the

Ž . Ž .theoretical curves Fig. 1 calculated from Eq. 1 forŽ y1.conditions close to our experiment t ft , KfL withdi D

the experimental dependences shown in Fig. 5.However, for higher applied external fields quantitative

agreement of the theoretical data and experimental resultsis not so good. Here linear growth of the resonanceamplitude with increasing DC field is observed, instead of

Ž .a quadratic one predicted by Eq. 1 . In practice this resultsin a lower signal gain compared with the theoreticalestimate. The width of the resonance maximum in theexperimental curves also differs at least by one order ofmagnitude from the theoretical estimate. We believe that

Ž .both of these disagreements can be at least partly at-tributed to a non-uniform broadening of the resonancepeak due to a spatial inhomogeneity of the light intensityand the external field in the crystal interelectrode spacing.This inhomogeneity can also be the result of non-ohmic

Žcontacts as well see, for example, results obtained forw x.GaAs in Ref. 14 . Assumption of non-ohmic contacts

needs, however, additional detailed experimental investiga-tions in this particular BSO sample.

Another reason of this disagreement could be a satura-tion of the signal observed for high values of the externalapplied field. Indeed, the standard theory of moving grat-

w xing formation 8 predicts that in a resonance regime theamplitude of the electric field grating grows quadratically

Žwith the external field as E syimE KL r2 m<1,sc 0 0.KL <1 . On the other band, this amplitude cannot clearlyD

be higher than the externally applied DC electric field E .0

This means that when the mKL r2 product approaches 10

the amplitude of the recorded electric field grating beginsto saturate. Now its amplitude is limited not by the en-hancement mechanism due to the large photocarrier drift

w xlength, but by the value of the external field itself 15 . Inour experimental conditions the initial value m was nearlyequal to 0.6 so this saturation can start for KL f3.0

ŽProbably this fact also explains the relatively low com-.paring with the theoretical estimate value of the resonance

gain and significant broadening of the resonance peak forhigh applied voltage. Indeed, Fig. 9 demonstrates that forlow values of the contrast, the resonance maximum ismore pronounced than that for mf1. This means that inorder to obtain a sharp resonance peak it is necessary to

Fig. 9. Resonance peak in frequency transfer function of photo-EMF for different values of the effective contrast of the interfer-ence fringes. BSO crystal at ls532 nm, 1rLf900 linrmm,I f3 mWrmm2; E s4 kVrcm. The values of m are the0 0

Ž . Ž . Ž .following: e 1; I 0.6; ^ 0.2.

( )S. MansuroÕa et al.rOptics Communications 152 1998 207–214 213

have illumination as uniform as possible and to use lowinitial contrasts of the recorded interference pattern. On theother hand, the photo-EMF signal is proportional to m2

and, for this reason, we can lose in the absolute value ofthe output signal.

The experimentally obtained transfer function presentedin Fig. 7 for different spatial frequencies also exhibits

Ž .unusual features. Indeed, Eq. 4 predicts that the reso-nance amplitude is maximal for the spatial frequencywhich corresponds to the maximum in the absence ofexternal voltage. However, our experimental data showthat the resonance amplitude is higher for lower spatialfrequencies. This fact requires probably a more detailedanalysis, but we think that this disagreement can also beassociated with the effect of the grating amplitude satura-tion discussed above.

The dependence of the signal on the external fieldshows, however, a good qualitative agreement with the

Ž .predictions of the theory. One can see Fig. 8 that we canachieve a resonance regime of detection for lower frequen-cies of modulation by applying higher external voltage.One has to remember that in a real experimental situationwe are obviously restricted by the value of the DC field forwhich the electrical breakdowns are observed. Note also,that the minima observed in these curves correspond to theeffect of the photo-EMF signal sign change predicted in

w xRef. 1 for frequencies much higher than the dielectricrelaxation frequency. However, unlike this simplifiedmodel, we have no zero signal here probably because of aclose vicinity of the dielectric relaxation frequency and thesecond characteristic frequency associated with the finitelifetime of the carriers.

One can see that application of an external DC fieldŽleads to significant 200 times in our experimental condi-

.tions enhancement of the signal compared with the maxi-mum value of the signal without voltage. This enhance-ment is observed for frequencies one order of magnitudebelow the dielectric relaxation frequency which can bepromising for some special applications of adaptive pho-todetectors mentioned in the introduction. However, inpractice we are interested mainly in maximal signal-to-noise ratio. To estimate this important parameter below we

w xuse the approach developed in Ref. 16 .Without going into details note that under a DC exter-

nal field the generation-recombination noise of the biasedsample

1r22mt E02J s 4e g V D f 6Ž .g r r 0 ž /Lx

is well above the thermal noise which is due to its photo-conductivity. Here e is the electron charge, g is the0

average volume carrier generation rate, V is the volume ofthe sample, L is the interelectrode space, and D f is thex

bandwidth in which the detection is performed. The maxi-

Ž .mal signal value at resonance is determined by Eq. 4 . Forthe optimal case KL s1 we get:D

S 1 U02s Dm . 7Ž .N 8 k TRD f' B

At first glance, here we have the possibility to increasethe signal-to-noise ratio by simply increasing the externalvoltage. In fact, one should remember that this situationŽi.e. the quadratic dependence of the output signal on the

.voltage applied is valid for the non-saturation regimeonly. This regime is observed, however, only up to U s0

Ž . Ž .2 L r mKmt . Substituting this limit value in Eq. 7 wex

obtain that:

S 1 gVs m .(ž /N 4 D fmax

For the optimal power of the reference beam P GP thisr s

relation transforms to

S 1 g Vss . 8Ž .(ž /N 2 D fmax

Here g is the average photocarrier generation rate due tos

the signal beam with the power P only. This estimate fors

the signal-to-noise ratio observed under grating saturationis only 4 times lower than the fundamental limit deter-mined by the generation-recombination noise in conven-tional photoconductive photodetectors. After reaching satu-ration the following increase of the biasing voltage doesnot improve the signal-to-noise ratio.

The observed growth of the signal can be, nevertheless,useful when we are limited by the noise of the amplifier.Note also that the requirements on matching the photode-tector with the preamplifier are not so strict in this case.This occurs because, as mentioned earlier, the thermalnoise is well below the generation-recombination noise inthe biased mode of operation of the photo-EMF photode-tector. In turn, the sensitivity of the configuration to theamplitude noise of the laser is increased here due to DCcurrent flowing through the photodetector.

5. Conclusion

Summarizing, we report original experimental resultsŽ 3on giant resonance enhancement ;10 in our experimen-

.tal conditions of the non-steady-state photo-EMF signalfor low modulation frequencies in Bi SiO crystal under12 20

an external DC field. We show that the amplitude ofŽresonance, its position below the dielectric relaxation

.frequency of the illuminated sample , and the width de-pend strongly on the external DC field applied, spatialfrequency of the grating, illumination conditions, and thecontrast of the interference pattern. Influence of saturationof the resonantly recorded electric field grating on the

( )S. MansuroÕa et al.rOptics Communications 152 1998 207–214214

amplitude and width of the resonance maximum is demon-strated. Finally, it is shown that the resonance regime ofoperation of biased adaptive photo-EMF detectors in somecases can increase sensitivity for detection of low-frequencyphase modulated optical signals.

Acknowledgements

This research work was sponsored by the ConsejoŽ .Nacional de Ciencia y Technologia CONACyT , Mexico,

in the framework of the research project 0354P-E.

References

w x1 M.P. Petrov, I.A. Sokolov, S.I. Stepanov, G.S. Trofimov, J.Ž .Appl. Phys. 68 1990 2216.

w x Ž .2 N. Korneev, S. Stepanov, Optics Comm. 115 1995 35.w x3 P.V. Mitchell, S.W. McCahon, M.B. Klein, T.R. O’Meara,

G.J. Dunning, D.M. Pepper, in: Technical Digest of TopicalMeeting on Photorefractive Materials, Effects and Devices,Optical Society of America, Washington, DC, 1995, pp.432–434.

w x4 N. Korneev, M. Rendon, S. Stepanov, in: Technical Digest ofTopical Meeting on Photorefractive Materials, Effects andDevices, Optical Society of America, Washington, DC, 1995,pp. 496–499.

w x Ž .5 F. Davidson, C.-C. Wang, S. Trivedi, Optics Lett. 19 1994774.

w x6 Th. Merlet, D. Dolfi, J.P. Huignard, IEEE J. Quantum Elec-Ž .tron. 32 1996 778.

w x7 P. Rodriguez, N. Korneev, S. Stepanov, in: Technical Digestof CLEO’97 Conference, 1997, OSA Technical Digest Se-ries, p. 45.

w x8 S.I. Stepanov, V.V. Kulikov, M.P. Petrov, Optics Comm. 44Ž .1982 19.

w x9 Ph. Refreiger, L. Solymar, H. Rajbenbach, J.P. Huignard, J.Ž .Appl. Phys. 58 1985 45.

w x10 U. Haken, M. Hundhausen, L. Ley, Appl. Phys. Lett. 63Ž .1993 3066.

w x11 R.F. Kazarinov, R.A. Suris, B.I. Fuks, Sov. Phys. Semicond.Ž .7 1973 102.

w x Ž .12 I. Sokolov, S. Stepanov, J. Opt. Soc. Am. B 8 1993 1483.w x13 J.P. Partanen, J.M.C. Jonathan, R.W. Hellwarth, Appl. Phys.

Ž .Lett. 57 1990 2404.w x14 H. Rajbenbach, J.M. Verdiell, J.P. Huignard, Appl. Phys.

Ž .Lett. 53 1998 541.w x15 Ph. Refreiger, L. Solymar, H. Rajbenbach, J.P. Huignard,

Ž .Electron. Lett. 20 1984 656.w x Ž .16 S.I. Stepanov, Appl. Optics 33 1994 915.