1120 J. Opt. Soc. Am. B/Vol. 16, No. 7 /July 1999 Johansen et al.

Influence of quadratic recombination on gratingrecording in photorefractive crystals

Per Michael Johansen and Henrik C. Pedersen

Optics and Fluid Dynamics Department, Risø National Laboratory, DK-4000 Roskilde, Denmark

Evgeny V. Podivilov

Institute of Automation and Electrometry, Russian Academy of Sciences,Koptyg Prospekt 1, 630090 Novosibirsk, Russia

Received December 7, 1998; revised manuscript received March 29, 1999

The influence of quadratic recombination is analyzed theoretically and numerically by formal derivation of thematerial wave equation including this effect. Analytically, it is shown how the quadratic recombination altersthe nonlinear properties of space-charge waves. The influence of higher-harmonic components induces both anonlinear frequency shift and a nonlinear shift in dissipation. Numerically, it is shown that 12 higher-harmonic components are necessary to cover the entire region of intensity contrast and that the quadratic re-combination effect gives significant corrections for small frequencies below the fundamental resonance fre-quency. © 1999 Optical Society of America [S0740-3224(99)01807-X]

OCIS codes: 090.7330, 160.5320, 190.4410, 190.5330.

1. INTRODUCTIONThe photorefractive response has been subject to a vastamount of investigation with the sole purpose of optimiz-ing the amplitude of the induced space-charge field andhence maximizing the energy transfer between incidentlaser beams. A fundamental problem in this optimiza-tion procedure is, however, that photorefractive materialsrespond linearly to an incident light intensity distributiononly within the range of a very low intensity contrast.Above this contrast value the material responds nonlin-early, and, as a consequence, analysis of the fundamentalresponse calls for taking into account various higher-harmonic components. An early attempt at includingthis effect, of paramount importance, was made by Ref-regier et al.,1 who tried to account for nonlinear effects byinferring an empirically based correction function.Later, a solution to the photorefractive band-transportmodel based on a nonperturbative analysis waspublished.2 With this method it was possible, by use ofFourier expansion, to obtain a solution for the first threespatial harmonics. In addition to the various analyticalattempts, a number of numerical simulations in the limitof large intensity contrast were published. In Ref. 3 itwas found that the higher-harmonic amplitudes growwith increasing intensity contrast. A detailed numericalanalysis accompanied with experimental verification wasalso made by Brost et al.4 The significant results fromthese analyses were that the presence of higher-harmonicgratings shifts the temporal resonance frequency of thefundamental grating and, in addition, alters the gratingformation time of the fundamental grating.

The problem with all the above-mentioned approachesis that the physics was not very clear. This problem con-cerning the physical insight was solved only by the pres-ence of the so-called space-charge wave theory5 (SCW),

0740-3224/99/071120-07$15.00 ©

covering the parametric wave interaction in photorefrac-tive crystals of the sillenite family. The SCW theory pro-vided a new way of describing and understanding the un-derlying dynamics of the space-charge field on the basis ofa single wave equation. From this wave equation funda-mental wave concepts such as characteristic frequency,dissipation, and wave quality factor can be deduced.Moreover, a clear identification of forced waves and para-metric waves, together with so-called eigennonlinearity,can also be made within the framework of this theory.The SCW theory was applied to the analyses of subhar-monic generation,5 parametric oscillation6 andamplification,7 transversal parametric oscillation,8 andac-field-induced subharmonics.9 Moreover, the SCWtheory was successfully applied to explain the existence ofthe so-called low-frequency pecularities10 that induce anadditional peak in the space-charge field amplitude at atemporal frequency lower than that of the fundamentalresonance. This effect is due to a nonlinear resonant in-teraction between the SCW’s and is highly dependent onthe intensity contrast. The analysis of this phenomenonwas performed without including the effect of subharmon-ics on the fundamental grating. An equivalent analysiswas made of the influence of higher harmonics on the fun-damental grating for the ac-field case.11 It was shownthat the effect of the higher harmonics was a reduction inthe fundamental space-charge field amplitude as com-pared with the linear fundamental amplitude calculatedwithout taking higher harmonics into account. It has tobe stressed that all calculations mentioned above wereperformed below the threshold of subharmonic genera-tion. The concept of nonlinear frequency shift introducedby Sturman and co-workers12 provided a basis for thephysical understanding of the influence of the higher har-monics in the dc-field running grating case. This expla-

1999 Optical Society of America

Johansen et al. Vol. 16, No. 7 /July 1999 /J. Opt. Soc. Am. B 1121

nation goes as follows: For an increasing intensity con-trast the higher-harmonic gratings start to be generated.The feedback from these gratings to the fundamental har-monic grating causes the eigenfrequency of this grating toshift to a larger value that is dependent on the fundamen-tal harmonic grating itself. This, in turn, means that,given that the fundamental grating is resonantly excited,the resonance frequency now shifts to a larger value, andthe growth of the fundamental grating stabilizes. Forthe ac-field case the physics is slightly different in thesense that, as the intensity contrast increases, the funda-mental grating amplitude grows and provides feedback tothe dissipation coefficient that increases and, in turn, re-duces the growth in fundamental amplitude. Again, thenonlinear correction acts as a stabilization mechanism.

One fundamental limitation in all of the above analysesis, however, that none of them includes the effect of qua-dratic recombination. This effect becomes important, es-pecially for high values of the intensity contrast. To in-clude and analyze, both theoretically and numerically,the influence of quadratic recombination is our purpose inthe present paper. The composition of the present paperis as follows. In Section 2 we formally derive the waveequation including quadratic recombination and discussanalytically the influence of higher-harmonic gratings ina simplified version of the wave equation. In Section 3 anumerical solution representing the entire spatial regionof fringe spacing is given, and a detailed analysis for vari-ous values of intensity contrast is performed. In Section4 the conclusions are drawn.

2. THEORYIn this section we outline the equation for the inducedspace-charge field in the limit of quadratic recombination.Our starting point is the set of band-transportequations13 in the simplified form:6

]ND1

]t5 NDsI 2 gRND

1n, (1)

]ND1

]t5 2m¹ • S nE 1

kBT

q¹n D , (2)

¹ • E 5q

e0es~ND

1 2 NA!, (3)

where the variables ND1, I, n, and E are the ionized do-

nor density, the light intensity distribution in the mate-rial, the density of electrons, and the total electric fieldconsisting of the space-charge field and the possible exter-nal applied field, respectively. The material parametersare the density of donors ND , the photoexcitation con-stant s, the recombination constant gR , the mobility m,the permittivity es , and the density of acceptors NA .The physical constants are the Boltzmann constant kB ,the absolute temperature T, the absolute value of theelectronic charge q, and the free-space permittivity e0 .

As stated above, Eqs. (1)–(3) are a simplified subset ofthe general set given in Ref. 13. This subset has, how-ever, been frequently used in the literature to account forthe formation of a space-charge field in photorefractive

sillenite crystals of BSO (Bi12SiO20), the crystal consid-ered in the present paper. The simplifications actuallymade are the following. First, the approximation ofweak thermal excitations has been invoked, i.e., b ! sI.This assumption is valid for practically all cw intensitydistributions used experimentally. Second, the genera-tion of ionized donor density is much smaller than thedensity of donors; i.e., we have assumed the approxima-tion ND

1 ! ND (in BSO ND1 > 1022 m23 and ND

> 1025 m23). Third, the adiabatic approximation]n/]t . 0 has been used. This approximation is validfor times much larger than the free-carrier recombinationtime tR (in BSO tR > 1027 s). Fourth, the charge contri-bution from the density of electrons in the conductionband is assumed to be small in comparison with the con-tribution from the ionized donors, i.e., n ! ND

1 2 NA (inBSO n > 5 3 1016 m23). A detailed discussion of theseapproximations, including the calculations of the indi-vidual parameters, can be found in Refs. 14–17.

The light pattern is considered to be of the form

I 5 I0 1 I1 , (4)

where I0 is the dc part of the light intensity and I1 is theac part. The ac part may be independent of time, i.e., astanding light interference pattern, or it may be periodicin time, originating from a running intensity pattern.

To solve the set of Eqs. (1)–(3) we apply an approachinspired by the form of the intensity pattern in Eq. (4), inwhich the variables are decomposed into dc and ac parts.Assuming this form of variation for the independent vari-ables ND

1, n, and E allows us to combine Eqs. (1)–(3) intothe following two equations:

n1

n05

I1

I02

e0es

qNAS ¹ • E1 1

1

v0¹ • E1D

1 1e0es

qNA¹ • E1

, (5)

21

mt¹ • E1 5 v0

qNA

e0esH ¹ • E1 1 FE0 • ¹ 1

kBT

q¹2

1 ~¹ • E1! 1 E1 • ¹G n1

n0J , (6)

where the parameter v0 is given by v0 5 sI0ND /NA , mtis the mobility lifetime product of free electrons, the life-time is given by t21 5 gRNA , and n0 5 tNDsI0 .

Equations (5) and (6) are exact within the initial as-sumptions. Hence they are valid for all regions of the pa-rameters. Usually, when the above equations are com-bined to yield a single equation for the space-charge field,the denominator in Eq. (5) is set equal to unity, which cor-responds to the assumption that the effect of nonlinear re-combination is small,6 i.e., neglecting the product of theterms n1ND1

1 in Eq. (1).In the following we assume that E1 ! Eq(k), where

Eq(k) is the saturation field14 given by Eq(k)5 qNA /ke0es . This assumption indicates that the sec-ond term in the denominator of Eq. (5) is very small, i.e.,(e0es /qNA)¹ • E1 ! 1. We will test in Section 3 that

1122 J. Opt. Soc. Am. B/Vol. 16, No. 7 /July 1999 Johansen et al.

this assumption holds true by plotting it for typicalparameters15 for the photorefractive material BSO.Having adopted this assumption means that we can ex-pand the denominator, retaining only the first-order term,and insert this term into Eq. (6) to obtain a single equa-tion for the space-charge field, which can be cast in thefollowing form:

vector of magnitude K 5 2p/L, where L is the fringespacing, and V is the running grating frequency.

This prescribed intensity distribution allows us to con-sider a one-dimensional situation. But, as we are deal-ing with a nonlinear wave equation and do not know thespatial structure of the space-charge field, we introduce aFourier representation in the form

S E0 • ¹ 1kBT

q¹2 2

1

mtD¹ • E1 2 S zI0 2 v0

kBT

q¹2 2 v0E0 • ¹ D¹ • E1

5 zE0 • ¹I1 1 zkBT

q¹2I1 1 z¹ • ~I1E1! 2 ¹ • ~E1¹ • E1! 2 v0¹ • ~E1¹ • E1! 2

v0

I0S E0 • ¹ 1

kBT

q¹2D ~I1¹ • E1!

1v0

zI0S E0 • ¹ 1

kBT

q¹2D $@v0~¹ • E1! 1 ~¹ • E1!#¹ • E1%, (7)

where z 5 sqND /e0es . Note that the operators in thefirst parenthesis of the last singly underlined term act onall the terms included in the braces.

Equation (7) is the so-called material wave equation,and by comparing it with the wave equation obtained by asimilar method6 but without the quadratic recombinationterms included, we note the following: The left-handside, which determines the characteristics of the eigen-waves is the same as in Ref. 6. The first two terms on theright-hand side, constituting the linear driving terms, arealso the same and originate from the intensity distribu-tion incident on the crystal. The three doubly underlinedterms on the right-hand side are quadratic nonlinearterms equivalent to those obtained earlier in Ref. 6. Thefirst of these is a parametric term, since the ac intensitydistribution oscillates harmonically. Moreover, the threedoubly underlined terms represent the nonlinearity of theproblem in the case in which only linear recombination istaken into account. The singly underlined terms are thenonlinear contributions originating from the quadratic re-combination term, i.e., from the product n1ND1

1 in Eq.(1). The first singly underlined term is a parametricterm, whereas the second is a quadratic nonlinear termresponsible, together with the second and third doublyunderlined terms, for the self-nonlinearity, i.e., the inter-action of the space-charge wave with itself.

This material wave equation can be applied to theanalysis of situations in which the externally applied elec-tric field is time dependent and the intensity distributionis a standing light interference pattern. For the case ofan ac square-wave field applied to a photorefractive sille-nite crystal of BSO the threshold for excitation of spatialsubharmonics by the inclusion of higher-harmoniccomponents9 has been analyzed recently. Equation (7)can also be applied to the analysis of the situation, whichis considered in the present paper, in which the externallyapplied electric field is static and the intensity pattern ismoving harmonically (as an intensity wave), according to

I 5 I0 1 I1 5 I0 1 I0m cos~Kx 2 Vt !, (8)

where m is the intensity contrast, K is the grating wave

E1~x, t ! 5 (k

E0ek~t !exp~ikx 2 iVkt ! 1 c.c., (9)

where ek is the time-dependent dimensionless amplitude,c.c. denotes the complex conjugate quantity, and the sum-mation is to be performed over all physical values of k.Since the space-charge field is real, the relation e2k5 ek* must be valid. Now, by inserting Eqs. (8) and (9)into Eq. (7) and isolating the terms proportional toexp(ikx), we can rewrite the wave equation in the follow-ing form:

]ek

]t1 i~vk 2 Vk 2 igk!ek

5m2

Ak~dk,K 1 dk,2K! 1m2

~Bk,Kek2K 1 Bk,2Kek1K!

1 (k8

Ck,k8~v0 2 iVk8!ek2k8ek8 , (10)

where dk,K is the Kronecker delta. In arriving at Eq. (10)we have applied the so-called slowly varying envelope ap-proximation (SVEA) to neglect the time derivative ap-pearing in the nonlinear terms, i.e., u]ek8 /]tu ! uVk8ek8u.Moreover, the condition of temporal synchronism hasbeen applied, i.e., Vk 5 Vk2k8 1 Vk8 .

From the linear undriven part of Eq. (10) we recognizevk as the characteristic frequency of an eigenwave and gkas the damping (dissipation) coefficient of this wave. Thecoefficients are given by

vk 5 v0

E0@Eq~k ! 2 EM~k !#

@ED~k ! 1 EM~k !#2 1 E02 ,

gk 5 v0

@Eq~k ! 1 ED~k !#@ED~k ! 1 EM~k !# 1 E02

@ED~k ! 1 EM~k !#2 1 E02 ,

(11)

where the characteristic photorefractive fields are givenby ED(k) 5 kBTk/q, Eq(k) 5 qNA /ke0es , and EM(k)5 1/kmt. The other coefficients appearing in Eq. (10)

Johansen et al. Vol. 16, No. 7 /July 1999 /J. Opt. Soc. Am. B 1123

can be explained as follows. The magnitude of Ak deter-mines the force generating the fundamental grating, i.e.,the grating with k 5 K. The coefficient Bk,K is respon-sible for the nonlinear coupling between the intensitywave and the space-charge field, i.e., the parametric non-linearity, and the coefficient Ck,k8 is the magnitude of theself-nonlinearity, i.e., the coupling of the space-chargefield to itself. These coefficients are given by

Ak 5 2iv0

Eq~k !

E0

ED~k ! 2 iE0

ED~k ! 1 EM~k ! 2 iE0,

Bk,k8 5 iv0

Eq~k8!

E0Ck,k8 ,

Ck,k8 5 iE0

Eq~k8!

Eq~k ! 1 @ED~k ! 2 iE0#k 2 k8

k

ED~k ! 1 EM~k ! 2 iE0. (12)

Equation (10) is a quite general form of the wave equa-tion, and, as such, it also covers the situation of a square-wave electric field externally applied to a photorefractivecrystal together with a standing light interference pat-tern. To analyze this situation, one has to makeVk , Vk8 5 0 and replace E0 with E0p(t), where p(t) isthe time-dependent function accounting for the temporalvariation of the electric field. This means that the waveequation changes from a differential equation with con-stant coefficients to one with time-dependent coefficients.When these substitutions are inserted into Eqs. (10)–(12),they reduce to the equations obtained in Ref. 9.

Equations (10)–(12) will constitute the basis for our nu-merical calculations of the feedback from higher harmon-ics on the fundamental photorefractive space-charge fieldperformed in Section 3. Here, however, we proceed ana-lytically by adopting approximations, which is valid insome practical cases, i.e., uED(k) 1 EM(k)u ! E0! Eq(k). In the limit of these approximations the qual-ity factor defined by Qk 5 uvk /gku is much larger thanunity, which is very important for the generation of space-charge waves.5,12 In this limit Eq. (10) simplifies to

]ek

]t1 i~vk 2 Vk 2 igk!ek

5 2im2

vk~dk,K 1 dk,2K 1 ek2K 1 ek1K!

1 i(k8

k8

kVk8ek2k8ek8 , (13)

where we have inserted the simplified coefficients

vk 5 v0

Eq~k !

E0,

gk 5v0

E02 $Eq~k !@ED~k ! 1 EM~k !# 1 E0

2%,

Ak 5 Bk,K 5 2ivk , Ck,k8 5 2k8

k. (14)

For a sufficiently small value of the intensity contrast mthe photorefractive response is linear, in which case thesteady-state linear solution to Eq. (13) for the fundamen-tal component of the space-charge field is easily foundwhen the nonlinear terms are neglected,

eK,L 5m

2

vK

V 2 vK 1 igK. (15)

For a slight increase in the intensity contrast above thatof the pure linear response but below the value of contrastat which subharmonics start to be generated, the higher-spatial-harmonic components are known to play an im-portant role. Assuming this limit on the contrast and in-cluding for simplicity only the second-harmoniccomponent in the space-charge field, we can obtain thesteady-state amplitude equations from Eq. (13) in theform

@~vK 1 dvK 2 V! 2 i~ gK 1 dgK!#eK 5 2m

2vK ,

e2K 5V

vK 2 4V 2 2ig2KeK

2, (16)

which is valid close to the resonance V 5 vK . The rela-tions V2K 5 2V and v2K 5 vK/2 have also been inserted.The nonlinear frequency shift dvK and the nonlinear shiftin dissipation dgK are given by

dvK 5 5V24V 2 vK

~vK 2 4V!2 1 4g2K2 ueKu2,

dgK 5 10V2g2K

~vK 2 4V!2 1 4g2K2 ueKu2. (17)

It is generally the case for nonlinear systems that thenonlinear shift in dissipation is smaller than the nonlin-ear frequency shift, i.e., dgk ! dvk .

A solution for ueKu2 is easily found from Eqs. (16) and(17), and the result is shown graphically in Fig. 1, wherethe modulus square of the space-charge field is plottedversus the parameter d 5 1 2 vK /V, which is the detun-ing away from the linear fundamental resonance; i.e., thefundamental space-charge field is resonantly excited atd 5 0. The curves are shown for an applied field of E05 7 3 105 V/m, a fringe spacing of L 5 20 mm, andthree different values of the contrast m. The solid curveis plotted for m 5 0.01, and the curve has its peak at thefundamental resonance. This indicates that only thefundamental harmonic component is present in the am-plitude. For m 5 0.06, as depicted by the dotted curve,we see that the amplitude is slightly shifted away fromthe linear fundamental resonance, shown in the figure bythe amount dvK /V, owing to the presence of a nonlinearfrequency shift. This indicates that the second-harmoniccomponent is now present in the space-charge field ampli-tude. This effect becomes more pronounced in thedashed curve shown for m 5 0.1. Here the influence ofthe second-harmonic component is even more clearlypresent, since the resonance is shifted further away, d> 0.12, from the linear fundamental resonance. Again,this is due to the feedback from the nonlinear frequencyshift. From these curves it is clearly seen that the effect

1124 J. Opt. Soc. Am. B/Vol. 16, No. 7 /July 1999 Johansen et al.

of the nonlinear feedback from the second-harmonic 2Kgrating to the fundamental harmonic K grating is notonly a shift in the resonance frequency but also a chargein the shape and width of the curves as well as in the am-plitudes.

The problem of recombination has been treated in dif-ferent ways in the photorefractive literature.14 The termcovering the so-called linear saturation process, i.e., theterm n0ND1

1 in Eq. (1), is often neglected. Neglectingthis term changes only the linear properties of a space-charge wave, since it can be shown that it corresponds toneglecting the field EM(k) in the denominator of the ex-pression for the characteristic frequency in Eq. (11) and,at the same time, in the denominator of the dissipationcoefficient, neglecting ED(k) in the first square bracketand the last term E0

2. In Fig. 2 we have analyzed theinfluence of this effect on the fundamental space-chargefield. The curves are displayed for the same parameters

Fig. 1. Modulus of the space-charge (s-c) field divided by m ver-sus d for three different values of intensity contrast. The non-linear frequency shift is indicated in the curves by dvK /V.

Fig. 2. Modulus of the space-charge field versus d for an inten-sity contrast of m 5 0.01. The dashed curve is depicted withoutthe effect of linear saturation, whereas the solid curve includesthis effect.

as in Fig. 1 at an intensity contrast of m 5 0.01. Thedashed curve is plotted for the case in which the linearsaturation is neglected, and the solid curve includes thiseffect. It is seen that the saturation effect not only limitsthe amplitude of the space-charge field but also changesthe shape of the curve, which is, of course, evident, sinceboth the characteristic frequency and the damping coeffi-cient are modified.

3. NUMERICAL SOLUTIONHere we solve numerically Eq. (10)–(12), covering the en-tire spatial region, and the aim is to investigate qualita-tively the influence of quadratic recombination on thefundamental space-charge field amplitude. From vari-ous numerical experiments we found that the inclusion of12 harmonics was sufficient to cover the entire range ofintensity contrast. But before discussing the specific re-sults from this simulation, let us briefly recapture the in-fluence of the various forms of recombination. As statedin Section 2, the linear saturation process influences onlythe linear wave properties through the characteristic fre-quency and the dissipation coefficient. The quadratic re-combination, it turns out, influences only the nonlinearwave properties. The influence of this effect appears bymeans of the singly underlined terms in the wave equa-tion in Eq. (7), and, consequently, it leads to the term@ED(k) 2 iE0#(k 2 k8)/k, appearing in the denominatorof the eigennonlinearity coefficient Ck,k8 in Eq. (12).

The results of the numerical simulations are shown inFig. 3, where the fundamental space-charge field ampli-tude is shown versus normalized frequency, V/vK , forvarious values of the intensity contrast and both with andwithout the effect of quadratic recombination. The linearfundamental resonance is reached for V/vK > 1, andfrom the lowest curve (shown for m 5 0.1) it is seen thatthe nonlinear frequency shift is relatively small, as is theinfluence of quadratic recombination. The quadratic re-

Fig. 3. Amplitude of the fundamental space-charge field versusnormalized frequency V/vK for four different values of the inten-sity contrast both with and without the quadratic recombinationeffect included.

Johansen et al. Vol. 16, No. 7 /July 1999 /J. Opt. Soc. Am. B 1125

Fig. 4. Numerical plot of the term (e0es /qNA)¹ • E1 versus normalized distance x/L for four different values of the normalized fre-quency V/vK .

combination has also only a limited influence on the nextcurve, which is plotted for m 5 0.2. The nonlinear fre-quency shift is, however, more significant for this curve,and a small additional peak at V/vK > 0.25 is present.When the intensity contrast is altered to m 5 0.6, a rela-tively large frequency shift is seen, and the influence fromquadratic recombination is starting to play a significantrole. Moreover, additional peaks are now clearly seen forlow values of the normalized frequency, i.e., V/vK> 0.1, 0.4. These peaks have been reported before10 andoriginate from the so-called low-frequency peculiaritiescaused by the influence of higher-harmonic componentson the fundamental amplitude. This tendency is evenclearer in the top curve drawn for m 5 1. Here the non-linear frequency shift is large, approximately 1.5, and theinfluence of quadratic recombination significantlysmoothes out the peaks of the curve for frequencies lowerthan V/vK > 2.5.

In the last curves displayed in Fig. 4 we have tested theassumption (e0es /qNA)¹ • E1 ! 1 applied to make anexpansion and, subsequently, arrive at the closed-formmaterial wave equation in Eq. (7). The curves are plot-ted in direct space versus normalized distance x/L for dif-ferent values of the normalized frequency V/vK . Fromthe curves it is seen that the approximation is clearly ful-filled for higher values of V/vK . For smaller values, i.e.,V/vK 5 0.05, however, the assumption is critical, indicat-ing that it may be necessary to include higher-order termsin the expansion. This is in clear agreement with Fig. 3,showing that the influence of quadratic recombination is

crucial for small values of normalized frequency wherethe low-frequency peaks are also present.

All curves in this section are plotted for E0 5 73 105 V/m and L 5 20 mm.

4. CONCLUSIONSWe have, for what is the first time to our knowledge, dealtanalytically with the effect of quadratic recombination.This is achieved by derivation of a material wave equa-tion that includes this effect. It is shown that quadraticrecombination alters the nonlinear wave properties by theeigennonlinearity. The introduction of the approxima-tion of a large quality factor allowed us to obtain analyti-cal expressions for the nonlinear frequency shift and thenonlinear correction to the dissipation coefficient, whichmade it possible to show graphically the shift away fromthe linear fundamental resonance for increasing intensitycontrast. In the same approximation we showed that thelinear saturation process influences only linear waveproperties, and hence it changes the shape and amplitudeof the linear space-charge field. It was concluded that 12higher-harmonic components are necessary to cover nu-merically the entire region of intensity contrast. More-over, the quadratic recombination effect highly influencesthe fundamental space-charge wave amplitude at fre-quencies below the resonance frequency. Again, the non-linear frequency shift was clearly determined, and thepeaks at lower frequencies, the so-called low-frequencypeculiarities, were clearly present. Finally, when the ini-

1126 J. Opt. Soc. Am. B/Vol. 16, No. 7 /July 1999 Johansen et al.

tial assumption made for obtaining the material waveequation was tested, it was found that for low values ofthe normalized frequency this assumption may breakdown; this outcome calls for the the inclusion of higher-order terms in the expansion. For larger values of thenormalized frequency, however, the first-order expansionmade here gives an adequate description of the influenceof quadratic recombination.

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