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Romashko et al. Vol. 27, No. 2 /February 2010 /J. Opt. Soc. Am. B 311

Photorefractive vectorial wave mixingin different geometries

Roman V. Romashko,1,* Salvatore Di Girolamo,2 Yuri N. Kulchin,1 and Alexei A. Kamshilin2

1Optoelectronics Laboratory, Institute of Automation and Control Processes, Far Eastern Branch,Russian Academy of Sciences, 5 Radio Street, Vladivostok 690041, Russia

2Physics Department, University of Kuopio, P.O. Box 1627, FIN-70211 Kuopio, Finland*Corresponding author: romashko@dvo.ru

Received October 2, 2009; revised November 25, 2009; accepted December 14, 2009;posted December 14, 2009 (Doc. ID 118023); published January 22, 2010

We analyze vectorial wave mixing in a photorefractive crystal of cubic symmetry in different geometries ofbeam interactions—reflection, transmission, and orthogonal. It is shown that orthogonal geometry in contrastwith others supports an efficient phase demodulation of a depolarized object wave in linear mode without usingany polarization-filtering elements. As a result adaptive interferometers based on the orthogonal geometry canprovide a higher signal-to-noise ratio due to lower noise and lower optical losses. © 2010 Optical Society ofAmerica

OCIS codes: 090.2880, 120.1088.

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. INTRODUCTIONt is well known that an interferometer is a very sensitiveool which allows detecting ultra-small physical quanti-ies [1]. The dynamic nature of holograms recorded inhotorefractive materials makes possible such a detectionith adaptive interferometers not only in a laboratory butlso in a real-life environment (e.g., in space, in a vehicle,n a production line, etc.), which is typically quite un-table [2–5]. The application of dynamic holography to in-erferometric measuring systems makes them adaptive tolow temporal changes in environment and allows use ofnteracting waves with completely different wavefronts,hich can be also rather complex, e.g., speckled. At the

ame time the mixed waves must be properly polarized inrder to get efficient wave coupling in most of the knownechniques based on a photorefractive crystal (PRC).owever, the object wave is often depolarized in meaning

hat its polarization state is randomly varying in space. Aypical example is the light wave emerging from multi-ode optical fibers which are more preferable for design

f robust and flexible measurement systems. Therefore,n object wave must be properly polarized by transmit-ing it through a polarization filter in order to meetroper requirements of wave coupling. However, such altering as a rule causes significant optical losses (whichsually exceed 50%) and as a sequence leads to worseningf the interferometer’s sensitivity [5]. Moreover, the polar-zation filtering can become a reason for additional noisen an interferometer [6]. These two points, reduction inhe sensitivity and increase in the noise, can significantlyiminish the signal-to-noise ratio, and consequently, de-ection of ultra-small physical quantities may be impos-ible.

Recently we demonstrated that efficient phase demodu-ation can be achieved even when the speckled objectave with a randomly distributed polarization state isixed with the elliptically polarized reference wave in the

0740-3224/10/020311-7/$15.00 © 2

rthogonal geometry (OG) (i.e., when interacting wavesropagate in mutually orthogonal directions inside aRC) [7]. In this paper we present a detailed theoreticalnalysis of vectorial wave mixing (VWM) in cubic PRCs inhree basic geometries of wave coupling (transmission, re-ection, and orthogonal) on the basis of common ap-roach. The main purpose of the analysis is to explaineasons and conditions which provide linear phase de-odulation in these geometries including the realization

f polarization-insensitive phase demodulation in the OG.

. WAVE MIXING IN CUBICHOTOREFRACTIVE CRYSTAL. Theoretical Backgroundhotorefractive semiconductor crystals (CdTe, GaAs, InP,tc.) and crystals of the sillenite family (Bi12SiO20,i12TiO20, and Bi12GeO20) are very promising recordingaterials for exploitation in adaptive interferometers due

o their fast response in contrast to famous photorefrac-ive ferroelectrics (BaTiO3, LiNbO3, etc.) which possessigh electro-optic coefficients and support recording ofighly efficient holograms but have very slow response.oth photorefractive semiconductors and sillenites belong

o cubic systems of symmetry, which makes them opti-ally isotropic under nominal conditions. However, forma-ion of the space-charge field in these crystals under illu-ination by the interference pattern of interacting waves

eads to the induced non-uniform birefringence. The mainpecific feature of the photorefractive wave coupling in cu-ic crystals is the vectorial character of the interactionhich is determined by the small distance in k-space be-

ween two eigenwaves stipulated by light-induced aniso-ropy [8]. Consequently in these crystals, the energy andolarization exchange between light waves cannot gener-lly be held apart and, correspondingly, the VWM cannote reduced to the scalar one. Moreover, the wave coupling

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312 J. Opt. Soc. Am. B/Vol. 27, No. 2 /February 2010 Romashko et al.

s highly sensitive to the input wave polarizations, to therystal cut, to the applied electric field (including space-harge field), etc.

For the theoretical analysis of wave coupling in cubicrystals we used the theory of VWM, which was developedn [8] and applied to the analysis of adaptive VWM inter-erometers in [9]. We reduce our consideration to the casef the diffusion-dominated mode of wave coupling (wheno external electric field is applied to the crystal) and forptically non-active crystals. In addition we omit opticalosses caused by absorption in a crystal because they leadnly to diminishing of wave amplitudes and do not affecthe character of wave coupling. In these approximations,he interaction of two waves with vectorial amplitudes And B and wave vectors kA and kB can be described byhe simplified system of vectorial coupled-wave equations:

��

�lAA = − m�HB,

�

�lBB = m��HA, � �1�

here �=−�n03r41ED /�; n0 and r41 are the refractive index

nd electro-optic coefficient of the crystal, respectively; �s the wavelength; ED=2�kT / �e�� is an amplitude of theiffusion space-charge field; � is a space period of the in-erference pattern; k is the Boltzmann constant; T is thehermodynamic temperature; e is the elementary charge;

=AB� / ��A�2+ �B�2� is a complex function which can beefined as the half-contrast of the interference patternthe full contrast is m+m�). lA and lB are directions of thenteracting waves’ propagations, which are generally dif-erent. We consider three typical geometries of waves’ in-eractions which are shown in Fig. 1. In the transmission

ig. 1. (Color online) (a) Transmission, (b) reflection, and (c) oubic PRC.

eometry [Fig. 1(a)], the interacting waves propagate inhe crystal at a small angle to each other providing thatA� lB�x. In the reflection geometry [Fig. 1(b)], theseaves are counterpropagating so that lA�−lB�x while

he waves propagate in completely different directions inhe OG [Fig. 1(c)], lA�x , lB�y. Tensor H is a 2�2 cou-ling matrix whose components are obtained from theormalized tensor � of dielectric-permittivity changesaused by the grating’s electric field as the following8,10]:

Hij = �i���j�, i,j = s,p, ��� =r��K

r41�K�, �,�, = 1,2,3,

�2�

here r�� are components of the third-rank electro-opticensor, K are components of the grating’s wave vector=kA−kB which coincides with the space-charge field di-

ection; �s� and �p� are vectors which form the orthogonalasis for the wave’s vectorial amplitude. In collinear ge-metries (transmission and reflection), the interactingaves propagate along almost the same direction provid-

ng equality of the components of the coupling matrix Hesponsible for both interacting waves. In contrast, theseomponents are generally different for reference and ob-ect waves in the OG. However for the particular crystalrientation shown in Fig. 1(c), the interaction matrix isymmetrical. The components of the coupling matricesalculated by using Eq. (2) for all geometries depicted inig. 1 are summarized in Table 1. One can see that theoupling matrix has the same shape in all three geom-tries with zero components of the main diagonal. Thiseans that these geometries support pure anisotropic dif-

raction from a dynamic hologram recorded in the crystal.

nal geometries of TWM supporting anisotropic diffraction in a

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Romashko et al. Vol. 27, No. 2 /February 2010 /J. Opt. Soc. Am. B 313

Undepleted pump-beam approximation (in which themplitude B of the reference wave is constant) was usedor the analysis. In addition we assume that the half-ontrast m of the interference pattern is approximatelyhe same along the propagation coordinate lA,B. Thenrom the first line of Eq. (1) one can easily calculate thehange in the object wave amplitude at the output of therystal:

�A = − LAm�HB, �3�

here LA is the length of the crystal along the coordinateA. Consequently, the output amplitude of the object waves given by

A = A + �A = A − LAm�HB. �4�

Let us consider now the interaction of two waves in col-inear geometry (CG) and OG separately. Taking into ac-ount the transversal nature of electro-magnetic wavese represent their amplitudes by John’s vectors having

wo polarization components: one of them �s� is orthogo-al to the plane formed by wave vectors kA and kB, whilenother �p� belongs to the plane �kA ,kB�. We assume thatoth interacting waves are elliptically polarized so thatheir amplitudes are written as

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ap exp i�, B = bs

bp exp i , �5�

here � and are phase differences between orthogonalomponents of the object and reference waves, respec-ively.

In a CG, s-directions of both interacting waves are theame and p-directions approximately coincide so that thengular mismatch between them can be neglected as seenn Figs. 1(a) and 1(b). On the contrary in the OG, only-components of both waves are coinciding while-components are mutually orthogonal. This principal dif-erence determines a different character of wave couplingnd results in a different regime of phase demodulation.n the sections below we consider these two basic geom-tries.

. Collinear Geometry of VWMoth components (s and p) of interacting waves partici-ate in recording of the hologram in CG. Thus, the half-ontrast of the interference pattern is written as

m = �asbs + apbp exp�i�� − � �I0−1, �6�

here I0= �A�2+ �B�2 is the total light intensity. Now theolographic grating can be considered as a superpositionf two gratings mutually shifted by the phase of ��−�.

Table 1. Coupling Matrices in Different Geom-etries Which Support Anisotropic Diffraction

TransmissionGeometry

ReflectionGeometry

OrthogonalGeometry

H= �0 11 0 � H= �0 1

1 0 � H= � 0 1�2

1�2

0 �

ach polarization component of the mixed waves mayenerally diffract from any of these gratings and result inhe formation of an interference signal. Thus, in the gen-ral case, the interference signal should have four compo-ents.Hereafter we consider the case of fast transient phase

hift in the object wave, which occurs during a shorterime than the recording time of the hologram. Thus theologram can be treated as a frozen grating with respecto the fast shift of the interference pattern caused by thehase modulation. Therefore, the phase modulation � cane introduced into the object wave just by multiplying themplitude A in Eq. (4) with the phase factor of exp�i��hile m is kept constant. Consequently, the intensity of

he phase-modulated object wave mixed with the refer-nce wave in the PRC in the CG can be calculated usingqs. (4)–(6):

IA = IA + ���2IB + �I���, �7�

here IA and IB are the intensities of the waves A and B,espectively; �=�s+�p exp�i��−� , �s=�LAI0

−1asbs, andp=�LAI0

−1apbp. �I��� is the intensity change caused byhase modulation in the object wave, which is given by

�I��� = − 2�asbp�s cos�� − � + asbp�p cos�� − ��

+ apbs�s cos�� + �� + apbs�p cos�� + � . �8�

As pointed out above, the interference signal �I��� is aum of four terms which represent the diffraction of twoolarization components of the reference wave from twohase-shifted gratings and their subsequent interferenceith respective polarization components of the objectave. It is seen from Eq. (8) that if both interacting wavesre linearly polarized (i.e., if �=0 and =0), only the non-inear (quadratic) regime of phase demodulation takeslace: �I��� �1−�2 /2�, for ��1. However, the ellipticalolarization state of one (or both) interacting waves leadso the appearance of the modulated intensity componenthich is linearly proportional to �. Since the object wave

s generally considered as a wave which could be arbi-rarily polarized (including randomness of its phase dif-erence, �) let us select an elliptical polarization with � /2 for the reference wave. Then the object-wave inten-ity modulation can be found from Eq. (8) as

�I��� � − 2�LAI0−1�bsbp�as

2 − ap2�� + sin �asap�bs

2 − bp2��

+ cos �asap�bs2 + bp

2��1 − �2/2� . �9�

It is seen from Eq. (9) that linear phase demodulationecomes impossible if the object wave is linearly polarized�=0� at the angle of 45° to the s- (or p-) direction �as±ap�. On the contrary if the object wave is ellipticallyolarized ���0�, the modulated intensity always containsomponents linearly proportional to �. However, this isalid only if bs

2�bp2; otherwise the second term in Eq. (9)

isappears and we return to the same situation as withhe linearly polarized object wave. Note that in the case ofn elliptically polarized object wave the component of thentensity modulation, which is linearly proportional to �,ecomes more pronounced than the component propor-ional to �2. The latter completely disappears when �±� /2.

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314 J. Opt. Soc. Am. B/Vol. 27, No. 2 /February 2010 Romashko et al.

Let us consider now the case of a depolarized objectave, i.e., the wave with random spatial distribution of

he polarization state. Each part of such a wave contrib-tes its own component of the intensity modulation ac-ording to its specific polarization state. However, the to-al modulation of the object-wave intensity at the crystalutput is the sum of all contributions because a singlehotodetector is typically used in the adaptive interferom-ter. Considering that all polarization parameters of theepolarized wave are random and they possess all pos-ible values (for as ,ap� �−0,a , and for �� �−� ,� ), onean easily conclude from Eq. (9) that integrating of thehole field of the object wave on the detector will give

ero average for all the summands in the demodulationignal �I���.

Therefore, in the CG of VWM via the hologram re-orded in the diffusion mode, the linear demodulation ofhe phase can be achieved only when the object wave has

predominant polarization state: linear (s- or-polarized) or elliptical (with a prevalent sense of rota-ion). It stipulates for necessity to use proper polarizationltering of the depolarized object wave before mixing itith the reference one to obtain linear phase demodula-

ion in the CG.

. Orthogonal Geometry of VWMn contrast to the CG, there are exclusively s-polarizationomponents of mixed waves which are interfering and re-ording the hologram in the OG, because p-componentsre mutually orthogonal. Therefore, the second term inq. (6) disappears and only one grating with the followingalf-contrast is recorded in the crystal:

m = asbsI0−1. �10�

onsequently, the intensity of the output object wave af-er mixing in PRS with the reference wave becomes theollowing:

IA = IA +�s

2

2IB + �I���. �11�

Considering again the case of an elliptically polarizedeference wave (in particular =� /2) and small phaseodulation ���1�, we can find the change in the object-ave intensity as the following:

�I��� = − �2�s�asbp� − sin �apbs� + cos �apbs�1 − �2/2� .

�12�

One can see that the first term in Eq. (12) is responsibleor linear phase-to-intensity transformation being non-ero for any polarization state of the object wave excepthen this wave is linearly polarized along the p-direction

i.e., when as=0). Note that in the OG no hologram is re-orded in the crystal when as=0 [see Eq. (10)], which re-ults in an impossibility for any phase demodulation forhe p-polarized object wave. The second term in Eq. (12)lso gives a linear contribution to the demodulation sig-al if the object wave is elliptically polarized ���0�, whilehe last term results in the quadratic response to thehase modulation. Considering again the depolarized ob-

ect wave, one can find that integration of the modulatedntensity �I�x ,y� over the whole optical field results inero second and third summands and non-zero first one.herefore, in contrast to the CG of VWM, here the linearegime of phase demodulation is achieved even for theepolarized (e.g., speckled) object wave without anyolarization filtering.

. EXPERIMENTAL RESULTShe first experimental observation of the linear demodu-

ation of the phase encrypted into the depolarized speck-ed wave without any polarization filtering was reportedn [7] where advantages of the OG were listed and dis-ussed. The high spatial frequency of the interference pat-ern formed in the OG provides high holographic effi-iency in the diffusion mode as it was shown in earlyublications [11,12] where the features of holographic re-ording in this geometry were discussed but application ofRC to phase-to-intensity transformation was not consid-red in that time. Here we present a comparative study ofWM in the photorefractive semiconductor CdTe:V pos-essing cubic symmetry in all three geometries shown inig. 1. The reference wave with the plane wavefront wasenerated by Nd:YAG laser (�=1064 nm; the coherenceength �100 m; output power 150 mW). The polarizationtate of the reference beam was set elliptical with the el-ipticity ratio bs /bp=1.44. The object wave was emergedrom the multimode fiber with a core diameter of 400 �m.aking into account a large number of guided modes

�33 700� supported in such a fiber, the object wave cane treated as completely depolarized due to a random spa-ial distribution of polarization states over the speckledavefront. The fiber was wrapped around a piezoelectric

ylinder. By applying an alternative voltage to the cylin-er the fiber length, and as consequence, the phase of thebject wave was harmonically modulated at the frequencyf 7.7 kHz with the amplitude of 0.15 rad.

The object wave was mixed with the reference wave inhe PRC CdTe:V measuring 7.0�6.0�6.6 mm3 alongxes [110], �110 , and [001], respectively. It is worth toote that the crystal was rotated around the axis [001] by5° in reflection geometry to ensure the required orienta-ion [Fig. 1(b)]. After passing the crystal, the whole objectave was collected into a conventional photodiode to de-

ect the power of the object wave. The response of the pho-odiode was measured by a digital oscilloscope. Oscillo-cope traces in different geometries of wave mixing arehown in Fig. 2. The performance of the adaptive interfer-meter in each of the three principal VWM geometriesas compared for two different polarization states of thebject wave: linearly polarized in the s-direction (which iset by means of the polarizer) and completely depolarizedwhich is just emerged from the fiber without any filter).races marked by M in Fig. 2 show the modulating volt-ge applied to the piezoelectric cylinder, which is directlyroportional to the phase modulation of the speckledave. Oscilloscope traces marked by D show responses of

he photodiode on the phase modulation of the objectave, which are proportional to the intensity modulationt the output of the crystal and were recorded by the os-

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Romashko et al. Vol. 27, No. 2 /February 2010 /J. Opt. Soc. Am. B 315

illoscope simultaneously with M-traces. As one can see inhe upper row of the oscilloscope traces in Fig. 2, all threeWM geometries support linear phase-to-intensity trans-

ormation when the object wave is linearly polarized,hich is confirmed by observing the same frequency for

he phase and intensity modulations. However, the inten-ity modulation disappears for the depolarized objectave in both transmission and reflection geometries [see

he bottom row in Figs. 2(a) and 2(b)]. Nevertheless, theG of VWM supports efficient phase demodulation in the

inear mode with high signal-to-noise ratio both for theinearly polarized and completely depolarized object wavesee Fig. 2(c)]. These experimental results confirm theheoretical conclusions given in the section above.

. DISCUSSIONt is well known that the highest sensitivity of an inter-erometer is achieved in the linear regime of the phase-to-ntensity transformation. This regime occurs when a non-iffracted object beam and diffracted reference beam havehe phase difference of ±� /2 rad (the quadrature condi-ion). This phase difference particularly depends on theechanism of the hologram formation. As known [2,13],

he quadrature condition is directly achieved in the driftode of hologram recording when the object and refer-

nce waves have the same linear and parallel polarizationtates and when a strong electric dc-field is applied to theRC. However, applying a strong external field to theRC may be unacceptable in many practical applications.n the diffusion-dominated mechanism (without externaleld), the quadrature condition is satisfied by either (i)ppropriate installation of phase retarders in mixedaves and subsequent polarization filtering or (ii) em-loying the VWM technique in the geometry supportingn anisotropy diffraction [5]. It is worth to note that in thease of VWM (in commonly used collinear geometries) thebject and reference waves should have linear and ellipti-al polarization states, respectively [9]. Thus an appropri-te polarization filtering of the object wave prior or afterRC is strictly required in any adaptive interferometryechniques based on photorefractive wave mixing. Alter-atively, the quadrature condition can be achieved by in-roducing into the reference wave an external step-likehase shift by means of an optical modulator [14,15] but

ig. 2. Oscilloscope traces of modulation voltage �M� and pho-odetector current �D� obtained in three geometries of VWM: (a)ransmission, (b) reflection, and (c) orthogonal. Upper row: thebject wave is linearly polarized; bottom row: the object wave isepolarized.

uch a technique is more complicated and it is character-zed by degradation in the time of the demodulated signal.

However, polarization filtering of the nominally depo-arized object beam leads to the appearance of the specificolarization noise [6] if the object beam has originallypeckled intensity distribution as it occurs, for instance,or the light emerged from a multimode fiber. This noise,hich relates with the random and unstable polarization

tate of the light of the dynamic speckle pattern, is inevi-able and limits the sensitivity in any arrangement of thedaptive interferometer with PRC. The origin of thisoise is the stochastic distribution of the polarizationtates among different speckles at the output of the fiber6]. Any external impact on the fiber (including that onehich leads to phase modulation caused by the measur-nd) affects the propagation of spatial modes through theber, thus, changing both the size and polarization statef each individual speckle. However, there should be noodulation of the light power measured by the photodi-

de if all the emerged light is directly collected into thehotodiode because the total power of all speckles doesot depend on the external impact. By installing the po-

arizer into the object wave, we select only one projectionrom various polarization states. The total power of theserojections is now affected by any external impact on theber. Consequently, the photodiode will register the inten-ity modulation caused by the measurand even withoutixing with the reference wave if the distribution of the

olarization states among speckles depends on the mea-urand (as it occurs when the measurand modulatestrains of the fiber). However, the working point (aroundhich the averaged polarization component does oscil-

ate) is unstable due to slow variations in the temperaturer pressure, which results in stochastic changes in bothhe amplitude and frequency of the registered modula-ion. After mixing the object and reference beams in PRC,he intensity modulation of the polarization-filtered objecteam is added to �I��� leading to an instability of theight-power-modulation amplitude, which we interpret ashe noise of the output signal. In spite of the fact that theharacteristic time of these instabilities is much slowertens of seconds) than the response time of the PRC, theyannot be compensated for by the dynamic hologram.

It is worth noting that the configuration of two-waveixing characterized by a linear detection of phase shifts

ncrypted in a depolarized object wave was also proposedy Delaye et al. [16]. In that configuration, no polarizeras placed before the PRC. However, the detection of the

ntensity modulated beam was arranged by placing a 45°-olarizing beam-splitter after the PRC in order to dividehe output beam and to send its components with orthogo-al polarizations toward two identical photodetectorshose electric signals were electronically processed after-ard by the differential amplifier. However, the differen-

ial amplifier does not compensate the above described po-arization noise caused by the polarizing elementnstalled after the PRC, but even doubles it due to deduct-ng the intensities of orthogonal polarization componentsf the object wave. Moreover, the application of a dc elec-ric field is required in this configuration to achieve linearetection of phase transients.In contrast, the OG of wave mixing in PRC supports

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316 J. Opt. Soc. Am. B/Vol. 27, No. 2 /February 2010 Romashko et al.

inear phase-to-intensity transformation without any con-entional polarization element. The absence of the polar-zation elements leads to diminishing of the polarizationoise. In the ideal case when all the light that emerged

rom the multimode fiber is completely collected into ahotodiode, any redistribution of the size and polarizationtate among the speckles should not lead to the modula-ion of the photodiode’s current if the responsivity of itsctive area is spatially uniform. However, the dynamicologram in the OG is recorded only by the vertical polar-

zation component of the speckled wave. The instability ofhis component results in the changes in the diffractionfficiency of the dynamic hologram thus affecting the am-litude of the useful signal [see Eq. (12) and suppose aseing unstable]. These signal instabilities are natural forny adaptive interferometers with PRC and have a differ-nt nature from those observed in the scheme with theonventional polarizer where they are caused by slowariations of the average phase of the object wave. In con-rast to the polarizer, the dynamic hologram adapts slowariations of the average phase between the recordingeams, which makes the photodiode’s signal more stable.n other words, by replacing the conventional polarizinglement by dynamic polarization-sensitive holographicrating we diminish the polarization noise because of thetabilization of the working point of the adaptive interfer-meter. Experimental verification of the polarizationoise diminishing was demonstrated in [7] using a photo-efractive CdTe:V crystal.

Sensitivity of an adaptive interferometer to smallhase excursions of the object wave is one of the key pa-ameters for such a system [5]. It can be estimated by theatio of the modulated light power received by the photo-eceiver to the magnitude of the average received power17]. Let us compare the sensitivity of different configura-ions of VWM-interferometers. Comparing the interfer-meter in the OG with that in the reflection geometry,here are three reasons for a smaller sensitivity of theormer configuration in the case of the depolarized objectave: (i) the components of the coupling matrix H are re-uced by the factor of �2 (see Table 1); (ii) the space-harge field ED is reduced by the factor of �2 due to theespective reduction in the spatial frequency �−1; (iii) theodulation depth of the light power at the photo-receiver

s two times smaller which results in the sensitivity re-uction by the factor of �2 because of the absence of theolarizer. In total, the sensitivity of the interferometer inhe reflection geometry is 23/2 times better than that inhe OG. However in some crystals, increasing of thepace-charge field in the reflection geometry may bemaller than �2 because of the trap’s saturation.

However, the sensitivity of the interferometer in theG is much higher than that in the transmission geom-try mainly because of significant increasing of the diffu-ion space-charge field at much higher spatial frequen-ies. Therefore, we can conclude that the adaptiventerferometer based on VWM in the OG is more advan-ageous than that in the CG. Configuration in the OG isery simple and it allows us to use depolarized light ashe object wave (including the speckled one). Polarizationltering neither in front nor behind PRC is required to ob-

ain appropriate wave coupling and realize the linear re-ime of phase demodulation. No external electric field isecessary to apply to a PRC as well.

. CONCLUSIONn this paper we analyzed the VWM in PRCs of cubic sym-etry in three basic geometries—reflection, transmission,

nd orthogonal ones. Conditions which determine the lin-ar regime of phase demodulation at a diffusion dynamicologram were found. We have shown that OG supportsn efficient phase-to-intensity transformation not only for

properly polarized object wave (which is strictly re-uired in collinear geometries), but also when it is com-letely depolarized.

CKNOWLEDGMENTShe authors acknowledge financial support from the Rus-ian Foundation for Basic Research (RFBR) (grant 07-02-105), Ministry of Science and Education of Russian Fed-ration, and from the Academy of Finland (projects18487 and 133559).

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