n VanderLugt optical correlators and convolvers, aully complex multiplication of two Fourier-transformmages is performed by a nonlinear or modulationptical element. Higher speed operation is now pos-ible with current spatial light modulator �SLM� de-ices for image input. This has in turn enabled theigh-speed processing of more complex images andne-dimensional �1-D� data that has been rasterednto two-dimensional �2-D� images. We report themplementation of a new signal processing operationor VanderLugt correlators: the correlation of im-ges that have a spatial-frequency shift. For somepplications it is desirable to spatial-frequency shifthe information in one input image before the crossorrelation is performed. In the space domain thisorresponds to multiplying one image by a spatialarrier frequency �a local oscillator� to perform therequency shift before the correlation or the convolu-ion is obtained. In the frequency domain this cor-esponds to a lateral shift of one Fourier-transform
mage before the product is taken. We implementedhis operation in the frequency domain by using aotating mirror to shift the Fourier transform in ahotorefractive VanderLugt correlator.Photorefractive crystals have been used for both
orrelation and convolution in VanderLugt configu-ations. White and Yariv reported using Bi12SiO20BSO� to achieve these operations, but they did nottore the filter as a holographic image for later read-ut.1,2 Stepanov and Gural’nik reported storing thelter as a Fourier-transform hologram in lithium nio-ate, and they discussed the wavelength and accep-ance angle restrictions imposed by the Braggondition.3 BSO was also used as the active pho-orefractive material in joint transform correlators.4,5
he effect of misalignment �lateral shift� of theourier-plane filter has been treated as a disadvan-age of the VanderLugt filter,6 but we are not awaref any report of a beneficial use of such a shift.The advantage of using a photorefractive �PR� ma-
erial for the frequency-shift–correlation operation ishat the Fourier transform of one image can be storeds a hologram in the PR material and that multipleross correlations can be performed by reading outith images that have different spatial-frequency
hifts �all without reloading any image data into theLM input devices�. If, for example, the original
requency shift of the data was unknown, multipleross correlations can be performed with varying fre-uency shifts while searching for a correlation match.his configuration should allow high-speed process-
ng of multiple cross correlations of frequency-shiftedata during one data-load cycle of the input SLMs.In Section 2 we describe the correlator implemen-
ation from both mathematical and optical points ofiew, as well as the experimental test setup that usesSO as the PR material. The results of testing theorrelator are given in Section 3. Finally, the impli-ations of the tests and applications-related issuesre discussed in Section 4.
. Implementation of the Frequency-Shifted Correlator
he optical correlation operation is achieved in a two-tep maneuver. First, the 2-D Fourier transform, A,f an image a�x, y� or of a rastered time series inputignal, a�t�, is written as a volume, Fourier-transformologram into the PR BSO by use of a reference beam,. The reference beam is an unmodulated planeave at an angle �3° to the beam containing A.his is shown in Fig. 1�a�. The hologram exposure
rradiance is given by the following equation:
Exposure Irradiance � �R � A�2
� RA* � R*A � � A�2 � �R�2.(1)
s a result of this exposure, the BSO contains aefractive-index modulation that is proportional tohe exposure irradiance.
The second step is the readout that may be re-eated as required until the hologram becomesrased. A readout beam, B, is incident from theame direction as A �to produce a correlation at the
ig. 1. �a� Writing waves that store a hologram of the input, andb� readout waves that reconstruct the output of the photorefrac-ive BSO crystal in a VanderLugt optical correlator. EDC � dcias electric field.
utput�, as shown in Fig. 1�b�. This beam containshe 2-D Fourier transform of the second image b�x, y�r of a rastered, finite-length input signal, b�t�. Arime is added to denote a laterally shifted version of. The readout irradiance transmitted through theSO may be represented by
B��R � A�2 � B��RA* � R*A � � A�2 � �R�2�, (2)
hich is equal to
�RA*B�� � �R*AB�� � �B��R�2� � �B�� A�2�. (3)
he four terms in Eq. �3� correspond to the outputeams identified in Fig. 1�b�. The desired outputave beam 1 in Fig. 1 is represented by the first term
n Eq. �3� and satisfies the Bragg condition for dif-raction from the stored volume hologram. Beam 2,epresented by the second term, is not Bragg matchedut produces the convolution under the conditionsoted below. The third and fourth terms of Eq. �3�xit in a single direction, do not produce a usefulutput, and thus are blocked in the experimentaletup. We note that, if the readout beam were in-tead input along the reference beam direction, theiffracted beam that satisfied the Bragg conditionould produce the convolution of the two input sig-als rather than their correlation. This would ariserom the second term in Eq. �3�.
When beam 1 emerges from the BSO crystal planend is imaged onto a camera, the detected image isepresented by
�RA*B��2, (4)
hich is the cross-power spectrum of a and b. Ifnstead we perform another 2-D Fourier transform oneam 1 before the camera, we can observe the mag-itude squared, cross correlation of the rastered im-ges a�x, y� and b�x, y�. Hereafter, this is referred tos the output plane for the processor.Frequency shifting of the input signal b is achieved
s shown in Fig. 2. Although the shift can be ap-lied along any arbitrary direction, the problem cane decomposed into orthogonal shifts along and per-endicular to the grating direction �plane of the inci-ent beams�. For images of rastered signals, onlyhe two directions parallel or perpendicular to the
ig. 2. Block diagram that illustrates the process of a 1-D fre-uency shift and the multiplication of two 2-D images in aanderLugt optical correlator �* denotes convolution and the starenotes correlation�.
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aster lines are of interest. Because shifting in thelane of the incident beams �perpendicular to therating k� is the most restricted owing to Braggatching, it will be treated as the exemplary case.ssuming that both time series are rastered first in
ows along x �and then to the next row increasing y byne step�, x corresponds to “fine time” and y to “coarseime.” After the 2-D Fourier transform to obtain��, ��, � �parallel to x� corresponds to “coarse fre-uency” and � to “fine frequency.” By shifting in thedirection �parallel to y�, we are performing small or
fine frequency” shifts on B. Larger frequency shiftsre possible by shifting in the � �parallel to x� direc-ion. When the shifted Fourier-transform B� is ap-lied to the BSO as the readout signal, the product of��, � � �� with the stored A��, �� is formed.hen beam 1 and the BSO crystal plane are imaged
nto a camera, the resulting image is the cross-powerpectrum
�RA*��, �� B��, � � ���2. (5)
f instead beam 1 were Fourier transformed onceore, the resulting output image would be the mag-
itude squared of the cross correlation between a andversion of b that was frequency shifted by �.
ecause the local oscillator is represented in thepace domain as exp� j2��y�, both positive and neg-tive spatial frequencies of B experience a uniformositive �or negative� frequency shift. From this, weee that, to implement the frequency shift-correlationperation in the space domain, requires a phase grat-ng whose period, 1��, must be varied. This wouldequire another phase-only SLM and would be cum-ersome to implement.
. Experimental Setup
he experimental implementation of the correlatorollows the VanderLugt configuration. Both theriting and the readout optics are illustrated in Fig.. Electromechanical shutters were used to selecthe correct illuminating beams and to shield the cam-ra during the hologram write step. For the exper-mental tests, both a and b were the same metallizedlass transparency described in Subsection 2.B.he laser source was a 50-mW single-longitudinal-ode, diode-pumped, doubled-cw Nd:YAG laser �� �
ig. 3. Optical setup of the photorefractive optical correlator.ollimated light is input from the power splitter and the beamxpanding telescopes described in the text.
32 nm�. The ratio of the signal and the referenceeams was determined by a rotating half-wave platend a polarizing beam splitter located in front of theeam expanding telescopes.The Fourier-transforming lenses L1–L3 were all f �
00-mm focal length, 50-mm diameter achromaticoublets. The image and Fourier-transform planesn Fig. 3 are all separated by f from the lenses and areabeled with an I or an F to denote an image or aourier-transform plane. A galvanometer-controlledirror �galvo-mirror� was placed in the first image
lane of the input transparency to change the tilt ofhe beam and thus to shift its position in the Fourier-ransform plane located in the BSO crystal. Follow-ng the BSO crystal, L4, an f � 150-mm focal length,0-mm diameter achromatic doublet, could be ad-usted to provide either an image of the BSO �a cross-ower spectrum of the inputs� at the CCD camera orFourier transform of the BSO �a spatial image of the
orrelation plane� at the camera. The overall opticalpatial-frequency response extended to beyond 30ine pairs per millimeter �lp�mm�, and the 10-mmperture of the BSO placed an absolute limit at 50p�mm for this configuration, as determined by thepatial-frequency test object described below.A personal computer running LabVIEW software
ontrolled the rotation of the galvo-mirror, the shut-er sequencing and exposure durations, and the videorame acquisition. In preliminary tests the 90% riseime of the hologram diffraction efficiency was foundo be �70 ms for the power levels used. Thus, toxpose a hologram, both the reference and the signaleam shutters were opened for 100 ms with thealvo-mirror at its zero position. For readout onlyhe shutter in the transparency beam was openedith the galvo-mirror set to a new position to give a
requency shift. This shutter was open for 66 ms �2ideo frames� to obtain at least one complete videorame from the sequence of frames that was acquired.he room lights were extinguished during testingecause they were found to cause excessive erasure ofhe hologram.
. Test Data for the Correlator
o test the correlator operation, a metallized glassransparency was placed in the front focal plane of lens1. This transparency was the Edmund Variablerequency Resolution Target with 5–120 lp�mm re-ions in steps of 5 lp�mm. Each region was 1 mmide and was separated by approximately 0.3 mm
rom the next. An image of this target after it trav-led through the system with the BSO removed ishown in Fig. 4. Only the low-frequency region wassed, and an appropriate mask was used to transmitnly two frequencies at a time �i.e., 5 and 10, 10 and5, or 15 and 20 lp�mm�. This simplification al-owed us to model the spatial-frequency content of thewo regions as square-wave signals with fundamen-al frequencies f1 and f2. Using the orthogonality ofhe Fourier components, these results can be ex-ended to the general case of more complicated im-ges. Using this essentially 1-D object also reduced
he frequency shifting analysis to one dimension.he object spatial-frequency components and thehift direction were both in the horizontal plane. Asxplained in Section 2 this shift direction is the mostestricted of all possible directions owing to Braggatching and is being treated as the exemplary case.
n no way does this choice of input data indicate aimitation in the model or in the performance of theR correlator.During operation, the maximum optical power was
pplied to the reference beam and the minimum tohe signal beam. This resulted in 8 mW�cm2 in theeference beam over the 10 mm 10 mm BSO aper-ure. Although the dc spot of the signal beam con-ained only approximately 8 �W at the BSO, thisepresented an irradiance of approximately 0.8
�cm2. Normally, neutral density filters were in-erted in this beam �for both write and readout� withptical density �OD� in the range of 1–3 �10–30-dBttenuation�. For the case of 20-dB attenuation, thec spot and the reference wave were about equal inrradiance, resulting in good contrast in the output.
The two BSO crystals used in the experimentsere 10 10 5 mm long and 10 10 2.5 mm
ong. They were oriented identically with propaga-ion in the �110� direction, electrodes to apply anlectric field in the �1�10� direction, and the third faceriented as �001�. The applied dc voltage was typi-ally 8 kV. Excluding the autocorrelation imagehat used the 5-mm-long crystal �Fig. 5; shown at theeginning of Section 3� all of the other data wereollected with the 2.5-mm-long crystal. The inter-ering write beams, as well as the dc electric field,ere in the horizontal plane �along with the Fourier-
ransform shift direction and the spatial frequenciesf the test object�. All beams were polarized verti-ally.
ig. 4. Input image used in the correlator tests. An appropriateask was used so that only two of these spatial frequencies weresed at a time for the actual tests.
We should point out that the longitudinal positionf the BSO crystal was found to be very critical. Toll in a symmetric correlation image, and to give aearly symmetric power spectrum, the BSO z posi-ion had to be carefully adjusted. Some lack of sym-etry between the positive and the negative spatial
requencies was still visible in the data.
. Test Results
sample experimental autocorrelation of an inputmage containing 10 and 15 lp�mm spatial-frequencyomponents is shown in Fig. 5, along with a MATLABimulation of the expected image. This image wasbtained by positioning lens L4 so that a Fourierransform of the light exiting the BSO crystal fell onhe CCD camera. It shows the expected strong fea-ures at 0.2 mm, where 10 lp�mm is shifted 2 periodsnd the 15 line pairs�mm is shifted 3 periods.eaker features are at integer shifts of only 1 of theseperiods. A diffuse area at �1.25 mm that is visible
n both the experimental and the simulation imagesepresents the spacing between the adjacent regionsf 10 and 15 lp�mm. Autocorrelation images ob-ained at higher laser irradiance developed distor-ions and artifacts owing to saturation of the writtenologram. Those obtained at lower intensitieshowed features identical to those seen in Fig. 5 butith smaller signal-to-noise ratios that made it moreifficult to see the weak features. After looking atimilar images both with and without shifting in theourier plane, it became evident that little was to be
ig. 5. Sample autocorrelation image �with zero frequency shift�or an input image containing 10 and 15 lp�mm frequency compo-ents. Top half, experimental image; bottom half, MATLAB sim-lation based on the 10 and 15 lp�mm portions of Fig. 4. The BSOrystal was 5 mm long with 16-dB attenuation of both the writend the read beams. The experimental image has been contrastnhanced and brightness adjusted to make the weaker featuresore visible.
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earned about the frequency-shift performance of theorrelator by examination of the correlation imagesven though the correlation had all the componentsnd features expected based on the input images.A more useful approach for examining the
requency-shifting performance was to record theross-power spectrum of the inputs. This waschieved by placing L4 and the camera so that anity-magnification image of the BSO plane was
ormed on the camera. First, a study of the auto-ower spectrum allowed us to observe both the posi-ions of the frequency components of interest andheir power levels. The power levels at the BSOere adjusted by the placement of neutral densitylters in the transparency path. Thus both therite image and the frequency-shifted readout imageere attenuated by the same amount. A 2-D image
f the auto-power spectrum is shown in Fig. 6�a�, andplot of a section through its frequency components
s shown in Fig. 6�b�. In this case, the input con-ained 15 and 20 lp�mm frequency components, andhe cross sections consisted of an average of six rowsf adjacent pixels. In Fig. 6�b� both pairs of positivend negative fundamental frequencies, as well as thec, are visible. Third-harmonic components of thequare-wave inputs were sometimes visible. For
ig. 6. �a� Auto-power spectrum of images containing frequencorizontal section through this auto-power spectrum, and �c� a h
mages with a frequency shift of 5 lp�mm. Horizontal �spatial frnd �c� are the digital numbers of the digitized video signal ampli
his reason inputs containing 5 lp�mm were not used,nd inputs containing 10 lp�mm were seldom used.or higher input frequencies, the third harmonic fellell outside the optical system response bandwidth.To test the frequency-shift performance, the galvo-irror was first visually adjusted so that the fre-
uency spots appeared to shift onto each other as theirror was dithered between the two positions.his value of voltage was close to a shift of 5 lp�mm.hen a sequence of cross-power spectra with smallhanges in galvo-mirror voltage �0.001-V steps� wereecorded to determine the voltage corresponding to ahift of exactly 5 lp�mm. A plot of a section throughhe image for a shift of exactly 5 lp�mm is shown inig. 6�c�. The �15 and 20 lp�mm peaks remainisible because they represent the product of the Fou-ier transform shifted 5 lp�mm to the right and mul-iplied by the original transform. The �20 and 15p�mm peaks have disappeared because there is noignal in one of the multiplied images. The large dceak leaves some residual output, especially at 5p�mm, where the shifted dc peak multiplies noisend low-frequency Fourier components of the originalmage.
Further detailed analysis of the overlap region ofwo frequency components was performed. The re-
15 and 20 lp�mm �negative image for enhanced visibility�, �b�ntal section through a cross-power spectrum for the same inputncy� scale is in camera pixel number, and the vertical scales in �b�s. The BSO is 2.5 mm long.
ion around the overlap of the �15 and shifted �20p�mm peaks was examined. A subset of the cross-ower spectra in this region for closely spaced shiftsre plotted in Fig. 7 as the galvo-mirror voltage wascanned through the exact overlap value of �0.151 V.hese curves illustrate the expected rise and fall of
he product peak as one component was shifted pasthe other. This behavior was also simulated by usef the �15 and �20 lp�mm peaks from the auto-ower spectrum �see Fig. 6�b��. The square root ofhe profiles of these two peaks were numericallyhifted and multiplied. Like the experimental data,he values of the simulated shift were adjusted forresentation in volts. These simulated curves are
ig. 7. Horizontal cross sections through the �15 lp�mm cross-ower spectrum peak, with small variations in the frequency shiftgalvo-mirror voltage� about the 5 lp�mm shift ��0.151 V�.
ig. 8. Simulation of the cross sections shown in Fig. 7, derivedrom a numerical shift and multiplication of the �20 and �15p�mm peaks measured in Fig. 6�b�.
hown in Fig. 8, and they show a strong resemblanceo the experimental data. Using the two differentonideal profiles of the �15 and the �20 lp�mm auto-ower spectra peaks was necessary in the simulationo reproduce the slight irregularities and asymmetryf the experimental cross-power spectra peaks.To test the dynamic range and saturation behavior,
n input with spatial frequencies of �10 and �15p�mm was used. The beam with the transparencyboth the write and read beams� was successivelyttenuated in increments of OD � 0.2 �2-dB steps�.or each attenuation value, the crosspower spectrumas recorded with a Fourier-transform shift of 5
p�mm. For this measurement to produce meaning-ul data, the CCD camera had to either have a linearesponse, or be calibrated so that the data could beinearized.
Linearization of the camera was accomplished byeasuring its responsivity curve with the write–
eadout beam after it passed through the BSO. Theourier-transform plane was imaged onto the cam-ra. This technique had the additional advantage ofroviding a reference against which the diffractionfficiency of the BSO could be estimated. The CCDamera was a highly sensitive, inexpensive black-nd-white 1�3 in. format security camera �Defenderecurity Model 82-6095� with a 6.25 6.25 �m pixelize and a sensitivity of 0.2 lx. The camera wasperated with both the back light compensation andhe auto-electronic shutter switches in the off posi-ion. A National Instruments IMAQ PCI-1407rame grabber digitized the camera output. For cal-bration, the transparency beam was attenuated in-dB steps, and the power spectrum �containing allrequencies from 5 to 20 lp�mm� was recorded. Thelot of the camera output in the form of log�digitalumber� versus attenuation �in OD or decibels� forach spatial-frequency component was uniformlyoncave downward. That is, there was severe com-ression for higher digital numbers ��120� and ex-ansion of the response for low digital numbers�30�. Although the center region approximated aamma �slope� of 1, it was curved over the entireegion. This response curve was fit to a cubic poly-omial and was used to correct the data obtainedrom the dynamic range measurement. After lin-arization, the camera saturation level of the mea-ured data was reset to 0 dB.The cross-power spectra dynamic range data after
inearization are plotted in Fig. 9. For the outputower range below approximately �4 dB, the dataave a slope of 2, as expected from unsaturated writend readout processes. Above this output powerevel, up to the camera saturation level, the slope ispproximately 1, indicating that the write processas saturated and that only the variation in readout
ntensity was reflected in the output power. For thetrong �10 and 15 lp�mm components that wereultiplied by the shifted spectrum, the diffraction
fficiency was found to range from 10�3 to nearly0�2 at write saturation. At the low-power end ofhe dynamic range, the performance was limited only
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. Discussion
number of performance issues arise in the evalua-ion of the results of these experiments. Some ofhem include the dynamic range and the saturationf the written hologram in the BSO, the Bragg match-ng restrictions on the field of view of the second inputmage, and the erasure of the stored hologram byepeated readouts. We explore these issues below.
. Saturation
e examined hologram saturation by using both theuto-power and the cross-power spectra. Initialuto-power spectra measurements were used to sethe optical power levels used in the experiments thatollowed. The saturation data for the cross-powerpectra were shown in Fig. 9. The performance ofhe correlator was found to be linear below the write-aturation level down to the detection limit imposedy the CCD camera used. For approximately 8 mW�m2 in each beam and for a 100 ms exposure time, theritten hologram was found to saturate. At thatoint the diffraction efficiency of the Fourier compo-ents that were multiplied in the BSO was at a max-
mum of 0.5% to 1%.Spurious output at frequencies for which thereere no elements to multiply were more than 20 dBelow the components that correctly multiplied, ex-ept for the case of the dc components. They yieldednonzero output by the multiplication of the noise or
he weak low-frequency �5 lp�mm� components of thenshifted spectrum. In most applications only fre-
ig. 9. Cross-power spectrum linearized signal power versuswrite and readout� attenuation at each spatial frequency by use ofmages with 10 and 15 lp�mm components and a shift of 5 lp�
m. Vertical scale is normalized to a camera saturation of 0 dB.his frequency shift should generate strong outputs at �10 and15 lp�mm.
uencies to either side of dc are of interest; thus thesec-related components could be effectively ignored.lternatively, a low-pass spatial filter could remove
he dc components before multiplication.
. Bragg Matching
uring the development of this correlator, we ob-erved the restriction known to accompany Braggismatch. This resulted in a 1-D limit on the width
f the correlation images. In our configuration withhe 5-mm-long BSO crystal, the maximum width ofhe autocorrelation was observed to be �3 mm. Usef the simple formula for object size versus crystalength given by White and Yariv2 and a PR crystalength of 5 mm yields a 2.9-mm width for the Bragg
atched correlation image. A more detailed analy-is of the Bragg limitation on the field of view wasiven by both Yu and Yin7 and by Sun et al.8
. Erasure
t is attractive to consider the possibility of writingne holographic filter and then performing successiveultiplication �cross-correlation� operations with
hifted versions of the second Fourier transform.e tested this case by performing repeated cross-
ower spectra with the same frequency-shifted Fou-ier transform. Thus a strong readout spot wasepeatedly diffracting from the same written spot forach readout cycle. This test was conducted with0-dB attenuation of the write and read beams, i.e.,ust at the write saturation level. For a sequence of0 repeated reads, no measurable deterioration of theross-power spectral components was observed.rasure does not appear to be a problem at the high-st practical write–read levels for at least 20 readycles.
If the read beam power could be increased indepen-ently of the write beam power, it would be possibleo diffract more power into the readout beam. How-ver, for the test conditions reported, the readouteam exposure energy was near that used for therite and reference beams �at a strong spatial-
requency spot�. Recall that the write exposure was00 ms, that the read exposure was 66 ms, and thatt saturation the write and read beam intensitiesere exactly equal to each other and approximatelyqual to the reference beam intensity. Thus there isittle opportunity to increase the read beam exposurenergy without reaching the levels used to write orrase the hologram.
. Conclusions
e have constructed an optical correlator based onhe VanderLugt configuration that was capable oferforming the correlation of one input image with apatial-frequency shifted second image. This oper-tion required a fully complex multiplication of twoourier-transform images, one having been stored asvolume hologram in a PR crystal, and the other
patial-frequency shifted before readout. The fre-uency shift was achieved by use of a galvanometerirror placed at an image plane of the input trans-
arency. This horizontally shifted its Fourier trans-orm in the PR BSO crystal. Both the small- and thearge-scale features of the multiplication processere verified and were found to match theoreticalredictions. Very-fine frequency shifts were possi-le with the galvanometer mirror. Linear write–ead operation was verified for optical power levelselow the saturation level for the written hologram.urthermore, multiple cross correlations �as many as0 successive readouts� of the same stored hologramere demonstrated without measurable hologram
rasure.
This research was supported by Innovative Signalnalysis, Richardson, Texas.
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