Buchhave, Preben

Kukhtarev, Nickolai

Lyuksyutov, Sergei

An analytical solution of the photoconductive material equations for dynamic holographic gratings of arbitrary contrast has been obtained. A method of measuring high-contrast correlation functions has been suggested and tested experimentally. Good agreement with the analytical expression for the current at high contrast and experiment is found. An interesting effect of the holographic transversal photocurrent is predicted and a number of high-contrast-related topics are discussed

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Optical and electric properties of dynamic holographic gratings with arbitrary contrastKukhtarev, Nickolai; Buchhave, Preben; Lyuksyutov, SergeiPublished in:Physical Review A (Atomic, Molecular and Optical Physics)Link to article, DOI:10.1103/PhysRevA.55.3133Publication date:1997Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Kukhtarev, N., Buchhave, P., & Lyuksyutov, S. (1997). Optical and electric properties of dynamic holographicgratings with arbitrary contrast. Physical Review A (Atomic, Molecular and Optical Physics), 55(4), 3133-3136.DOI: 10.1103/PhysRevA.55.3133Optical and electric properties of dynamic holographic gratings with arbitrary contrastN. V. Kukhtarev,* P. Buchhave, and S. F. Lyuksyutov†Department of Physics, Technical University of Denmark, DK2800 Lyngby, Denmark~Received 29 May 1996!An analytical solution of the photoconductive material equations for dynamic holographic gratings of arbi-trary contrast has been obtained. A method of measuring high-contrast correlation functions has been suggestedand tested experimentally. Good agreement with the analytical expression for the current at high contrast andexperiment is found. An interesting effect of the holographic transversal photocurrent is predicted and anumber of high-contrast-related topics are discussed. @S1050-2947~97!01704-6#PACS number~s!: 42.40.Eq, 42.65.2kDynamic photoexcited gratings ~PG’s! are widely usedfor characterization of materials, for signal processing andfor information storage @1#. Recently, a surge of interest inPG’s was inspired by the discovery of highly efficient mate-rials for grating recording, e.g., photorefractive polymers @2#and photorefractive liquid crystals @3#, and by the observa-tion of photoconductivity in rare-earth–doped laser crystals@4#.Effects due to the nonlinearity of the material equationsare strongly enhanced with increasing contrast of the lightinterference pattern. Early solutions of the material equationsof the photogeneration-diffusion-drift-trapping ~PDDT!model, also called the band transport model @5#, were validonly for gratings of small contrast ~linearized solutions!. Ex-tensions of the linear solution have been obtained includingquadratic @6# and cubic @7# expansion in terms of the contrastm , and numerical solutions have also been obtained @8#.However, these approaches suffered from analytical andcomputational complexity. The latest attempt @9# at the solu-tion of the PDDT model gives results valid only for thediffusion-dominated case, and does not include a descriptionof photoconductive gratings. A ‘‘high-contrast correction’’was introduced in the description of ferroelectrics @10,11#and also for semiconductors @10,12,13#, and experimental re-sults of the harmonic content of the space-charge field hasbeen reported in @14#. The electric current along the gratingvector is proportional to ~12m2!1/2 in the small-contrast ap-proximation. In the case of high contrast, when m!1, thecurrent according to this expression must vanish in contra-diction to experimental results. To account for the experi-mental results empirical ‘‘artificial’’ correction functions forthe amplitude of holographic grating as a function of themodulation of the light interference pattern have been sug-gested @6,15# in an attempt to describe the high-contrast case.In this paper we present a simple analytical solution for thePDDT model, giving the spatial distribution of photoconduc-tivity, space-charge field, and ionized donors. The solution isbased on a simple but profound observation from numericalsimulations of the full set of PDDT equations @17# regardingthe form of these variables as a function of the spatial coor-dinate in direction of the grating vector: Whereas the space-charge field and the ionized donor density have a highlynonlinear relation to the ‘‘driving force,’’ the light intensitypattern impressed on the crystal, the free-carrier distributioncontains a negligible amount of higher harmonics. This factallows us to find a simple analytical solution for the free-carrier distribution for an arbitrarily high index of modula-tion of the impressed light distribution. This in turn leads toa simple analytical expression for the high-contrast correc-tion ~also called the reduction factor!. Since the other vari-ables may be expressed directly in terms of the carrier dis-tribution, we can now find simple Fourier expansions for thespace-charge field and the ion density. The ability to expressthe spatial variation of the physical quantities in simple ana-lytical equations for arbitrary contrast has great significancefor the understanding of the photorefractive process and forthe prediction of experiments and applications, and, since thePDDT model has applications or equivalents in other photo-induced processes, we expect that some of these results mayhave importance in other fields as well.We consider illumination with a sinusoidal light intensitygrating:I5I011m cos~Kx !, ~1!where the contrast m can have any value between 0 and 1. I0is the total intensity of incident light, K is the grating vector,and x is the spatial coordinate along the grating normal. ThePDDT model @5,16# provides the following relations for thetotal current J , the space-charge field Esc , and the photocar-rier concentration n assuming linear recombination and neg-ligible depletion of photoactive impurity centers:J5em^n21&21E0 , ~2!E5ED~ lnn !81E0~n^n21&!21, ~3!l~ lnn !91j~n21!8^n21&21512N1NA~4!where N15(NA/n)G is the density of charged ions,ED5(D/m)K is the diffusion field, E0 is the external appliedelectric field along the grating vector, m is the mobility ofcarriers, D is the diffusion constant, NA is the concentrationof acceptors, l5ED/EQ , and j5E0/EQ . The saturation field*Permanent address: Physics Department, Alabama A&M Uni-versity, Normal, AL 35762.†Permanent address: Institute of Physics, National Academy ofSciences, 252650 Kiev, Ukraine.PHYSICAL REVIEW A APRIL 1997VOLUME 55, NUMBER 4551050-2947/97/55~4!/3133~4!/$10.00 3133 © 1997 The American Physical SocietyEQ equals NA/(«0«K), where «0 is the free-space permittiv-ity, and « is the static dielectric constant. The derivative istaken with respect to u5Kx . The generation function G isrelated to the intensity pattern ~1! as G5G0~11m¯ cosu!,where G05(bd1b i1sI0)(N2NA)/(gNA), bd is the rate ofphotocarriers generation in the dark, bi is the generation rateunder incoherent light related to scattering and reflection in-side the material, m¯5m/11(bd1b i)/sI0 is the contrastreduced due to dark conductivity and the influence of inco-herent scattering, and ^ & indicates spatial averaging.Looking for periodic solution of Eq. ~4! in the formn5n0(p~apcospu1bpsinpu !, ~5!with a051, and assuming that high harmonics are negligible,we obtain the fundamental harmonic solution after averagingEq. ~4! and calculating coefficients for cosu and sinu . Aver-aging Eq. ~4! means that we select a solution that satisfiesconditions of neutrality both integrally over one period andlocally for the fundamental harmonic. As a result we find n0 ,a1 , and b1 , or equivalently n0 , a , and w, where a15a cosw,b152a sinw, and tanw52j/~11l!:a5M11~Md!2 @12dA11M 2~d221 !# , ~6!n0G05Ma~11l! . ~7!Here d5l/~11l! and M5m¯ (11l)/A(11l)21j2. Thephysical meaning of w is a phase shift between the lightpattern and the carrier grating, which is clear if we rewritethe expression for the fundamental harmonics of n in thefollowing form:n15n0~11a1cosu1b1sinu !5n011a cos~u1w!.~8!For the evaluation of the higher harmonics, we rearrangeEq. ~5! in the following form:n5n011a cos~u1w!1 f , ~9!and for f , which is simply the sum of higher harmonics, weuse an iteration approach based on placing linear terms inEq. ~4! on the left-hand side and nonlinear terms on theright-hand side. Neglecting f in all nonlinear terms, we ob-tain the following equation:lS n1n01 f D 92S n1n01 f D5 l~n18!2n0n11n18jA12a2n12Gn0.~10!The procedure for calculating the higher harmonics ana-lytically is relatively simple. Below we give the expressionfor the second harmonic:n25n0a2cos@2~u1w!#1b2sin@2~u1w!# ~11!wherea252l~12A12a2!A12a2~4l11 !a2 ,and b252(j/l)a2. Analysis of the condition a@b2, b2shows that it is valid for the whole range of parameters. Thispoint justifies the validity of the solution for the fundamentalharmonic in form ~6! for small as well as high contrast m .Equation ~9! with n0 , a , and w given by Eqs. ~6! and ~7!provides the desired analytical solution for the carrier con-centration for arbitrary value of the light intensity modula-tion, m , without using the small modulation assumption oran assumed functional form for the carrier distribution. Wecan obtain a simple expression for the case of small contrast,M 2~d221!!1, by using the general solution ~6!. We finda.mRF11 l2 ~mR !2G , ~12!whereR5H S 11 bd1b isI0DA~11l!21j2J 21is the reduction factor first introduced in @16#, and recentlyinvestigated by using cross talk between two slightly differ-ent gratings in photorefractive BSO @18#. Note that the firstnonlinear term in the expression for the contrast m is cubic,which reflects the fact that a(m) is a nonsymmetric functionin solution ~6!. For high contrast when d2@~12M 2!~12d2!,from Eq. ~6! we obtaina5M ~12d2!~11M 2!211~Md!2 . ~13!Having obtained the solution for n , the spatial distributionof the space-charge field E(x) and the concentration of ion-ized donors N1 can also be found from the basic equations~2!–~4!. The electric current is an explicit function of thecontrast,J5emn0E0A12a2, ~14!where a can be close to 1, and significantly change the cur-rent at high values of the contrast m . Thus current measure-ments provide an interesting tool for verifying the presenttheory and for sensing the condition of the carrier grating,especially at high contrast. As an example, knowledge of theanalytical dependence of a on the contrast m gives the de-gree of coherence or the correlation function of the interact-ing beams. Thus we may use relation ~14! in the case of aBSO crystal for determination of the important Debyescreening length @1,4#, and also for the study of the influenceof scattering and other sources of incoherence, which mayreduce the modulation of the carriers.For the electric field and its spatial Fourier harmonics,from Eqs. ~3! and ~5! we obtain3134 55N. V. KUKHTAREV, P. BUCHHAVE, AND S. F. LYUKSYUTOVE5E0A12a22EDa sin~u1w!11a cos~u1w!5 12 (p51,2,.. .~~E02iED!A ~p !ei~u1w!p1c.c.!, ~15!whereA5A12a221a5A11M 2~d221 !21M ~11d! ~16!is the amplitude of the fundamental harmonic, which forsmall modulation m takes the formA52M2~11l! H 11 M24 ~12d2!J . ~17!For N1 we obtain the following expression:N15NAGn5a~11l!NA~11m¯ cosu !M @11a cos~u1w!# .The amplitude of the E-field grating @Eqs. ~17! and ~18!# isthe starting point for the calculation of beam coupling anddiffraction efficiency in photorefractive materials. From thegeneral expression Eq. ~14! we can obtain the results, pre-sented in @10#, for the special case of E050~j50! andl[(kD/kQ)2!1.The experimental configuration is sketched in Fig. 1 @18#.Two plane-wave beams O1 and O2 from a single-mode Ar-ion laser polarized in the plane of incidence and separated byan angle Du are incident on a sample of photorefractiveBSO. The ~110! surface is perpendicular to the wave vectorK of the grating. The fringe spacing was chosen in the range0.5–900 mm. The external electric voltage E055 kV is ap-plied along the grating wave vector. We measure the currentbetween the negative connection to the sample and ground.A phase modulation is applied to beam O2 with a piezoelec-tric mirror driven by a sine voltage, resulting in a harmonicmotion of the optical interference pattern inside the crystal.By this means the photorefractive grating can be switched onand off without changing the total intensity of the incidentlight. According to formula ~15!, the current must have itsmaximum value when there is no modulation of the lightincident on the sample, i.e., when the contrast a equals zero.We then have J5Jmax5emn0E0 . Modulation of the incidentlight results in modulation of the free-carrier distribution,leading to a decrease of the photoconductivity and of themeasured current. Thus by measuring the current with andwithout modulation of the incident light we can find the ratioJ/Jmax5A12a2, which gives us the value of the modulationof the carrier distribution a , and, in general, informationabout the reduction factor R , because a5mR . For m51, thevalue of a is simply equal to R , where R itself depends onthe modulation of light m .The experimental dependence of the contrast a of the car-rier grating on the wave number k of the interference fringesis shown in Fig. 1. The wavelength for the experiment wasselected to be 488 nm. The theoretical dependence ~solidcurve 1! is calculated from Eq. ~6! using the following pa-rameters valid for BSO: Ld50.77 mm, m¯50.8, and E055kV/cm. Attempting to create a good fit to the measuredpoints using the existing theory by choosing other values forFIG. 2. ~a! Orientation of tilted interference pattern for forma-tion anisotropic current in the y direction. ~b! Dependence of theanisotropic current j' on angle w. Open squares: measurement;solid curve: theory.FIG. 1. Black circles correspond to experimental points for ap-plied voltage 5 kV and m51. Solid curves calculated from Eq. ~6!,curve 1 is for Ld50.77 mm, m¯50.8; curve 2 for Ld50.77 mm,m¯50.7; and curve 3 for Ld50.3 mm, m¯50.8.55 3135OPTICAL AND ELECTRIC PROPERTIES OF DYNAMIC . . .m¯ ~m¯50.7, curve 2!, and for Ld ~Ld50.30 mm, curve 3! failby giving a different functional form. The discrepancy forsmall values of k between experiment and theory may beexplained by the growing contribution from multiple reflec-tions ~Fabry-Pe´rot modes! for small incident angles. Thisleads to an increase in incoherent scattering, decrease of con-trast and increase of holographic current ~in accordance withthe experiment!.This experiment allows a direct measurement of the con-trast a of the carrier distribution in the sample without usingthe photorefractive effect. This is a matter of importance,because the redistribution of carriers in a photoconductivesample is determined mainly by the recombination time andnot on the Maxwell relaxation time, which is important in thephotorefractive effect. Thus current measurements as de-scribed above may, in the case of photoconductive samples,lead to fast correlators.The present solution may, due to its simple analyticalform, throw light on a great number of interesting problems.As examples we may mention that we find a slowdown ofthe grating build-up time with increasing contrast. Such aslowdown was reported recently in a numerical study @19#.Further we may mention the study of beam-coupling effects.Using the expressions for n0 , a , and w we can obtain a so-lution for the intensities of the coupled waves in both pho-toconductive and photorefractive samples. In the case ofphotoconductive samples, we find that the direction of en-ergy transfer depends on the sign of the external electric fieldE0 . Sign sensitivity of the energy transfer may be useful insome applications such as visualization of near-contact elec-tric field distribution and of charged defect clusters in semi-conductors. These findings will be reported in a separatepublication.This solution described above, which was derived for ex-ternal field E0 along the grating vector K , has been general-ized for a tilted grating, where E0 and K form angle w, Fig.2~a!. For a tilted geometry we predict appearance of a trans-verse current, which can be detected by an additional pair ofelectrodes. Detailed calculations of this effect willbe presented in @20#. Here we give a formula for the depen-dence of the transverse current j' on angle w: j'5E0~s'2si!sin 2w/2, where s'5s iA12a2; s'5emn . This effectcan be very useful for processing of both optical and electri-cal signals. It resembles the Hall effect, but does not requirea magnetic field. Preliminary experiments in sillenites showgood agreement with the predicted dependence of j' on theintersection angle w @Fig. 2~b!#.In conclusion, we solved the material equations of thePDDT model for periodic modulation of photoexcitationwith arbitrarily high contrast. This solution allows the ex-perimental determination of Debye screening length and thereduction factor related to incoherence factors in BSOsamples by a very simple method based on measurement ofsteady-state current. This high-contrast solution can also beused for the interpretation of the measured correlation func-tion of interacting beams with high degree of coherence.S.L. acknowledges the support from Danish Technical-Scientific Research Council Grant No. 9400102. N.V.K. ac-knowledges the support from U.S. Army SSDC under Con-tract No. DASG 60-95-2-00.@1# M. P. Petrov, S. I. Stepanov, and A. N. Khomenko, in Photo-refractive Crystals in Coherent Optical Systems ~Springer,New York, 1991!.@2# S. Ducharme, J. C. 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Optical and electric properties of dynamic holographic gratings with arbitrary contrast

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Optical and electric properties of dynamic holographic gratings with arbitrary contrast

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