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74 J. Opt. Soc. Am. B / Vol. 16, No. 1 / January 1999 Tian et al. Persistent spectral hole burning in an organic material for temporal pattern recognition M. Tian, F. Grelet, I. Lorgeré, J.-P. Galaup, and J.-L. Le Gouët Laboratoire Aimé Cotton, bâtiment 505, 91405 Orsay Cedex, France Received March 31, 1998; revised manuscript received July 23, 1998 A broad-bandwidth persistent spectral hole-burning organic material is used as a fast optical processor that compares an input temporal-frequency profile with a recorded reference spectral shape. This patternrecognition procedure relies on a subpicosecond temporal cross-correlation process. The size of the phaseencoding spectral interval exceeds 1 THz. The storage material, the spectral encoder, and the interferometric detector are examined in detail for optimal pattern discrimination. Experimental temporal patternrecognition results are reported. © 1999 Optical Society of America [S0740-3224(99)02101-3] OCIS codes: 090.0090, 020.1670, 070.5010, 320.5390, 200.4740, 190.4380. 1. INTRODUCTION 1 Since Mossberg’s seminal work in 1982, it has been known that the photon-echo process opens a specific way to optical data storage and processing. Optical storage of data trains was demonstrated in various materials,2–5 and a rare-earth ion-doped crystal seems to be a good candidate for the practical use of an optical dynamic randomaccess memory.6 As for time-domain processing applications, several architectures were demonstrated in both organic7,8 and inorganic9–12 persistent spectral holeburning (PSHB) materials, but the cross correlation of a readout temporal pattern with an engraved one was effected only in ion-doped crystals,11,12 until our preliminary report.13 Indeed, the achievement of broadbandwidth temporal cross correlation in organic dyedoped amorphous materials leaves us faced with a triple challenge. First, in order to take full advantage of the broad spectral bandwidth of the material, one has to deal with subpicosecond time-scale temporal patterns that one must be able to shape and control. Second, one has to engrave a coherent spectral structure over a several terahertz interval within the absorption band of the photosensitive medium. Third, one has to detect the crosscorrelation signal with subpicosecond resolution. In Section 2 we outline the basics of temporal cross correlation in PSHB materials, insisting upon specific features such as the white-light storage of the spectral holograms and the interferometric detection of the diffracted signal. The experimental configuration is depicted in Section 3, and the critical adjustment of the spectral shaping device is detailed in Section 4. Experimental results are presented in Section 5. We conclude in Section 6. tering of an input image through a reference pattern recorded in a photosensitive medium. Storage and filtering take place in the Fourier-transform space. In the present work, correlation is effected between temporal shapes. The time-to-frequency Fourier transform of the reference pattern is recorded in a persistent spectral holeburning material. The reference image has to include phase and amplitude information. In order to record a field-amplitude distribution in a quadratic photosensitive material, one usually engraves the interference pattern of a known field and the shaped field to be saved. This is the principle of holography. We proceed in a slightly different way since we do not really save all the information contained in the image field. The laser beam is split into two parts. On one arm the spectral field amplitude Ẽ( n ) is shaped by a spectral phase factor m 1 ( n ) that changes it into the image field Ẽ( n )m 1 ( n ). The pattern m 1 ( n ) spans a D-wide frequency interval, in the terahertz range. The field on the other arm undergoes a delay T. The two fields propagate along wave vectors k1 and k2 , respectively. One makes them interfere in a low-optical-density PSHB slab. To lowest order in the burning dose, the fields engrave the space and frequency structure u Ẽ( n ) u 2 m 1* ( n )exp@22ipn T 1 i(k1 2 k2 ) • r# , provided the delay T is smaller than the inverse homogeneous width G 21 hom, and D is smaller than the absorption-band inhomogeneous width G in . This represents the reference pattern. The phase structure of m 1 ( n ) is saved, but that of Ẽ( n ) is lost. Let an input image, represented by its spectral amplitude Ẽ 8 ( n )m 2 ( n ), be directed to the recorded hologram along k1 . A signal is diffracted in direction k2 . Its spectrum, filtered by the hologram, reads as Ẽ 8 ( n )m 2 ( n ) 3 (1 2 iĤ) @ u Ẽ( n ) u 2 m 1* ( n )exp(22ipnT)#, where the Hilbert operator Ĥ is defined by 2. TEMPORAL PATTERN RECOGNITION This work is placed within the general framework of cross-correlation pattern recognition. This procedure can be traced back to the VanderLugt correlator, which operates in the space domain.14 It relies on the linear fil0740-3224/99/010074-09$15.00 Ĥ @ f ~ n !# 5 1 p © 1999 Optical Society of America R f~ n8! dn 8 . n 2 n8 (1) Tian et al. Vol. 16, No. 1 / January 1999 / J. Opt. Soc. Am. B The time-dependent signal that emerges from the PSHB plate reads as E s~ t ! 5 E Ẽ 8 ~ n ! m 2 ~ n ! exp~ 2i pn t ! 3 ~ 1 2 iĤ !@ u Ẽ ~ n ! u 2 m 1* ~ n ! exp~ 22i pn T !# dn . (2) When the laser source delivers Fourier-transform-limited short pulses, the power spectrum u Ẽ( n ) u 2 is constant over the spectral width of the temporal images, provided that the pulse duration is much shorter than D 21 . Then u Ẽ( n ) u 2 can be factored out of the Hilbert operator. However, one can proceed in quite the same way with a whitelight stochastic source. Indeed, engraving is usually an accumulation process that requires repeated interaction with low-energy writing pulses. Therefore one can replace the quantity u Ẽ( n ) u 2 by its statistical average S 5 ^ u Ẽ( n ) u 2 & . If the coherence time of the stochastic source is much smaller than D 21 , the spectrum is flat over the frequency aperture D of the shaping factors and it can be factored out of the Hilbert operator. In addition we assume that the temporal extension of the recorded image does not exceed the time delay T. Then the following approximation holds:15 Ĥ @ m 1* ~ n ! exp~ 22i pn T !# > im 1* ~ n ! exp~ 22i pn T ! , (3) and the signal reads as E s ~ t ! 5 2S E (4) E~ t ! 5 E dn Ẽ ~ n ! exp~ 2i pn t ! . (5) The fringe contrast C( t ) that is detected on a photodiode array may be expressed as UE U dn Ẽ S ~ n ! Ẽ 8 * ~ n ! exp~ 2i pn t ! , (6) which, according to Eq. (4), is transformed into C~ t ! } S UE duration and a 1-THz shaping window, the stochastic background is only 1% of the correlation-peak amplitude. For insight into the behavior of the interferometric signal, one can regard a stochastic pulse as a sequence of uncorrelated subpulses, the duration of which is given by the inverse width of the shaping window. Each subpulse gives rise to an elementary response that contributes to the interferometric signal through its interference with a delayed replica of its parent subpulse. Summing up all these contributions, the detector actually achieves a statistical averaging operation that tends to smooth out the individual response fluctuations. Since the power spectrum ^ u Ẽ 8 ( n ) u 2 & is constant over the shaping window, one finally obtains C ~ t 5 T! } UE U m 2 ~ n ! m 1* ~ n ! dn . (8) Except for the small stochastic background, the interferometric signal does differ from what would be obtained with the help of a femtosecond Fourier-transform-limited source. Let m 1 ( n ) and m 2 ( n ) belong to a family of functions m ( i ) ( n ) that satisfy the orthogonality condition: E m ~ i ! ~ n ! m ~ j ! * ~ n ! dn 5 d ij . (9) Ẽ 8 ~ n ! m 2 ~ n ! m 1* ~ n ! exp@ 2i pn ~ t 2 T !# dn . The signal is expected to take on the shape of a correlation peak when the input image shape factor m 2 ( n ) coincides with the recorded reference pattern m 1 ( n ). If the spectral width D stays in the terahertz range, the correlation signature flashes during a time interval D 21 , shorter than 1 ps. Interferometric detection is appropriate to capture this short-living burst.16 One builds the spatial interference fringes of the signal E s (t) and a delayed replica of the readout field E 8 (t 2 t ), where the spectral and temporal amplitudes are linked by the Fourier-transform equation: C~ t ! } 75 U u Ẽ 8 ~ n ! u 2 m 2 ~ n ! m 1* ~ n ! exp@ 2i pn ~ t 2 T !# . (7) Let the input image be carried by a t L -long stochastic pulse with a coherence time much smaller than D 21 . Then C( t ) differs only by a term of order 1/(D t L ) 1/2 from a statistical average expression, where u Ẽ 8 ( n ) u 2 is replaced by the expectation value ^ u Ẽ 8 ( n ) u 2 & . With a 10-ns pulse Then the correlation signal C( t ) cancels at t 5 T, except when the input image coincides with the recorded reference pattern. The stored reference pattern has been expressed to lowest order in the burning dose, in an optically thin sample. Description can be extended to optically thick slabs, where larger diffraction efficiency values are expected. With a simple argument one can determine the validity range of the linear dose dependence of the engraving. It relies on the linear character of the hole-burning process itself. As long as the incident dose is completely absorbed by the medium, the depletion of the activemolecule population is proportional to the total incident energy. Thus the optical-density reduction is a linear function of the dose. In other words, total bleaching near the input side of the plate does not affect the linear behavior of the optical density as long as the dose is completely absorbed. As the accumulated dose increases, bleaching progresses, and an important fraction of available energy ultimately reaches the exit surface. A part of the dose passes through and does not modify the medium. Then the optical density no longer behaves as a linear function of the dose. Since the hologram-diffracted signal combines contributions from all the photosensitive volume, it is expressed in terms of the optical density, which is actually the relevant quantity to consider. In order to build the functions m ( i ) ( n ), one can divide the shaping spectral window into N even elementary intervals of width d. A phase factor e n ( i ) 5 61 is ascribed to each one of these slices so that the biphase pattern m ( i ) ( n ) reads as 76 J. Opt. Soc. Am. B / Vol. 16, No. 1 / January 1999 ~i! m ~n! 5 1 AD N21 ( e ~ni ! P n50 S D n d 2n , Tian et al. (10) where P(x) 5 1, when u x u , 1/2, and vanishes otherwise. The pattern orthogonality is then conditioned by the following rule: N21 ( e ~ni ! e ~n j ! 5 N d ij . (11) n50 Several procedures can be used to build a set of N phasefactor arrays that satisfy this condition. For instance, the Hadamard codes are defined as the rows of matrices generated by the recursion formula:17 H 2 ~ p11 ! 5 F @ H 2~ p!# @ H 2~ p!# @ H 2~ p!# 2@ H 2 ~ p ! # G , (12) Fig. 1. which starts from the 2 3 2 seed matrix, H2 5 F 1 1 1 21 G . (13) In the pth-order matrix, N 5 2 p rows of N elements obey the orthogonality condition. The row number is used to label the phase-factor array. Therefore all the phase factors are set equal to 11 in m ( 0 ) ( n ), which is uniform over the D-wide shaping interval. Function m ( 0 ) ( n ) is named the gate function. The product of two functions still belongs to the same function set. A function multiplied by the gate function equals the function itself. The square of a function equals the gate function. 3. EXPERIMENTAL CONFIGURATION The setup is presented on Fig. 1. The light pulses are 10 ns long. They are provided by a low-repetition-rate (15Hz) broadband laser. In this demonstration experiment the reference phase factor m 1 ( n ) and the input image m 2 ( n ) are built by the same spectral shaper. In the recording step the shutters Sh1 and Sh2 are open and closed, respectively. The shaped pulse, propagating along k1 , interferes with the initial pulse that propagates along k2 . They engrave the reference hologram in the liquid-helium-cooled PSHB sample. In the patternrecognition step the shutter state is reversed. The spectral shaper imprints the phase factor m 2 ( n ) on the light pulse that is directed to the sample along k1 . Fringes built by the diffracted signal and the interferometric reference beam are detected on a CCD linear array. The time intervals T and t are adjusted with the help of the optical delay lines DL1 and DL 2, respectively. The PSHB sample is a 0.5-mm-thick plate of octaethylporphyrin-doped polystyrene. The lifetime of the PSHB engraving is practically limited to a few hours, as imposed by the cryostat helium-holding time. Repeated illumination by the input pulses also contributes to the erasure of the engraved reference image, which is not fixed. The light pulses are spectrally shaped with the help of a two-grating device that was originally proposed by Froehly18 and was further experimented by Weiner.19 The frequency components of the pulse are spatially dis- Experimental setup. persed by the first diffraction grating, and they are focused by a lens on a linear array of liquid-crystal phase modulators (LCM’s). After phase filtering, the different spectral components are collimated by a second lens and collected into a single beam by a second grating. Both lenses have the same focal distance f 5 25 cm, and the two gratings (Jobin–Yvon 520 12 090, 40 mm 3 60 mm, blazed at 620 nm) have the same parameter a 5 2400 grooves/mm. The grating grooves are set in the vertical direction so that the plane of incidence is horizontal. In an ideal configuration, each lens is located at equal distance f from the LCM and the closest grating. Therefore the lenses form a 21 magnification telescope and the system is symmetric with respect to the plane where the LCM is positioned. Maximum diffraction efficiency on the gratings is obtained when the light polarization is oriented in the plane of incidence. It has to be made consistent with the LCM action. The horizontally polarized field that emerges from the grating G1 undergoes a 90° rotation provided by a half-wave plate inserted before the liquid-crystal cell. This device acts itself as a half-wave plate that makes polarization return to the horizontal direction. A polarizer inserted before the second grating G2 selects this polarization component. Mechanical stresses on the glass windows of the LCM cell induce a small birefringence. We compensate it by inserting a properly oriented quarter-wave plate. The LCM delimits a 1.1-THz-wide window that is parted into 64 independent pixels. We used the device for building a family of 32 Hadamard orthogonal codes, from the combination of 32 elementary intervals. Each one spreads over two pixels, and its width d equals 35 GHz. The delay T is set equal to d 21 , so that Eq. (3) is valid. The shaping device requires careful adjustment. The critical parameters are examined in Section 4. 4. ADJUSTMENT OF THE SPECTRO-TEMPORAL SHAPING DEVICE A. Spectral Resolution It is commonly admitted that the finite size of the incident light beam limits the spectral resolution of the shaping Tian et al. Vol. 16, No. 1 / January 1999 / J. Opt. Soc. Am. B device.19 We show that the pattern-recognition operation in our experiment circumvents this diffraction effect. On the first grating the incident field is represented by its amplitude E i (x, n ) that depends on frequency n and on transverse coordinate x (see Fig. 2). Let this function be decoupled into the following product: E i ~ x, n ! 5 E ~ x ! J~ n ! . E F ~ j , n ! 5 Ẽ F G j 2 b~ n 2 n0!f J~ n ! , a l 0f (15) where Ẽ( j ) 5 * dxE(x)exp(22ipxj) and where b5 ]u ]n U n0 occurs at the end of the pattern-recognition process. The detected fringe contrast can be expressed as C~ t 5 T ! } l 02 a 52 c cos u 0 E U S DU UE S D S UE U dX Ẽ 3 (14) We assume that E i (x, n ) is essentially a diaphragmlimited plane wave. Through the grating G1 and the lens L1 the different spectral components are focused in the plane F where the light modulator is positioned. Let the geometric image of the central component n 0 be selected as the origin of position coordinate j. The spectralcomponent images are affected by diffraction. Provided its aperture is larger than the transverse size of the beam, the lens L1 operates as a plain Fourier transformer. Therefore in focal plane F the field distribution reads as 77 } X gl 0 f m2 2 D U aX aX 1 n m 1* 1 n dn gbf gbf m 2 ~ n ! m 1* ~ n ! dn . (17) Each point on the sample is affected by the same spectral shift during the storage and readout steps. Therefore the finite incident-beam size does not limit the spectral resolution of pattern recognition.20 This conclusion relies on the assumption that the sample and the modulator are optically conjugate. This position is more critical as the incident beam diameter becomes smaller. Despite its symmetric look, the shaping device actually works in an asymmetric way, as illustrated on Fig. 3. In the first section of the device the incident-beam diameter determines the diffraction spot size on the modulator, but this is the optics aperture in the second section that im- stands for the angular-dispersion coefficient and where the astigmatism coefficient a5 ]u ]w U n0 cos w 0 52 cos u 0 connects the diffraction angle u with the incidence angle w. The two transverse dimensions of the beam are affected in different ways. In the plane of incidence the beam is resized by the factor a 21 . The vertical dimension is unchanged. In the focal plane the field is shaped by the liquid-crystal modulator that is represented by the transmission factor m @ j /( b f ) # , as a function of the horizontal coordinate j. The sample located in plane P and the shaping plane F are set in conjugate positions through the lenses L2 and Lg. One assumes that the diffraction effect caused by the finite aperture D of these optics can be neglected. Practically this means that l 0 f/D is smaller than the width d of the modulator pixels. Let the geometric image of the central spectral component be located at position X 5 0 on the sample. Then the geometric image of point j at frequency n is located at X 5 2( g/ a )( j 2 n b f ), where g 5 2f L g /f is the magnification factor from F to P. Therefore the field distribution in the sample reads as E P ~ X, n ! 5 Ẽ S DS D X aX 1 n J~ n ! . m gl 0 f gbf Fig. 2. Spectral shaping device. The spectral components of the incident beam are dispersed by grating G1 and focused in plane F by lens L1. They are collimated by lens L2 and collected by grating G2 into a single beam that is focused on the memory element by the lens system Lg . Coordinates x, j, and X are orthogonal to the optical axis. The angles of incidence and of diffraction on G1 are represented by w 0 and u 0 , respectively. (16) The spectral distribution at each position on the sample faithfully reflects the transmission profile of the spatial modulator. However, a a X/( g b f ) frequency shift affects the spectrum at each position X so that the different sample points are illuminated by different spectra. As a consequence, the space-integrated spectrum on the sample is actually blunted by the diffraction profile Ẽ (X/gl 0 f ). In our experiment, space integration only Fig. 3. Asymmetric operation of the two-grating device. The diffraction spot size on the LCM is determined by the incident beam diameter, but, as for the imaging of the LCD on the sample, the diffraction limit depends on the output optics aperture. 78 J. Opt. Soc. Am. B / Vol. 16, No. 1 / January 1999 Tian et al. poses the diffraction limit of the spectral resolution. Optics aberrations have to be considered in full-aperture operation. Proper orientation of the lenses reduces the spherical aberration that is expected to dominate under infinite conjugate ratio conditions. As for doublet L2, its less steeply curved surface should face the modulator. We shall see that this orientation conflicts with the elimination of group-velocity dispersion. B. Group-Velocity Dispersion First, the device must not modify the spectro-temporal properties of the incident light in the absence of the LC modulator. The group-velocity dispersion vanishes when the second grating coincides with the image of the first one through the two lenses. Let u denote the distance from exact conjugation along the optical axis of the lens pair. Then the spectral component n undergoes a phase shift with respect to the central component n 0 . The phase shift is expanded in powers of n 2 n 0 . The second-order term contributes to the group-velocity dispersion and reads as21 F ~ 2 !~ n ! 5 p H F u l 0a l 0 cos~ u 0 ! G JS 2 n 2 n0 n0 D 2 , (18) where u 0 represents the diffraction angle on the first grating. At u 0 5 70° and l 0 5 620 nm, a phase shift of p radians is observed at 2 THz from the central frequency when the second-grating position is 2 mm away from exact conjugation. The intergrating distance has thus to be adjusted with a 1023 precision. To achieve this precision, we monitor the group-velocity dispersion with the help of the interferometric detector. Indeed, the output-field spectral amplitude can be expressed as Ẽ( n )exp@iF(n)#, where Ẽ( n ) stands for the input field and where F(n) represents the dispersion phase shift. One makes the two fields interfere on the photodiode array where the fringe contrast reads as C~ t ! } UE U dn u Ẽ ~ n ! u exp@ 2i pn t 1 iF ~ n !# . 2 C. Astigmatism The distance between lenses L1 and L2 has to be set equal to 2 f. Then they combine into a 21 magnification telescope, and the astigmatism effects of the two gratings compensate each other. However, special care must be taken when the incident field differs from a plane wave. An incident spherical wave would exhibit two focusing positions after lens L1. It would converge in F' and F i along vertical and horizontal directions, respectively. Let F' be located at distance h, ahead of the L1 focal plane. Then F i stays at distance h / a 2 from this plane. When a polychromatic beam is directed to the shaping device, the different spectral components draw a misleading horizontal focal line at position F' . When set at the beam waist, the sample is actually conjugate with a linear vertical object positioned at F' . However, the horizontal linear modulator array has to be set at F i in order to be imaged on the sample. The distance h ( a 22 2 1) between F i and F' increases dramatically when one enlarges the diffraction angle u 0 to improve the spectral dispersion. 5. EXPERIMENTAL RESULTS A. Testing the Spectral Shaper and the Interferometric Detector We test the encoding and detection systems in the same way as we monitor the group-velocity dispersion. One makes the initial field Ẽ( n ) and the spectrally shaped field Ẽ m ( n ) interfere on the photodiode array, where the fringe contrast reads as C~ t ! } UE U dn Ẽ * ~ n ! Ẽ m ~ n ! exp~ 2i pn t ! . A transmission factor m @ j /( b f ) # is encoded by the LCM. When the setup is correctly adjusted, the resulting shaped amplitude reads as Ẽ m ~ n ! 5 Ẽ ~ n ! m ~ n ! . (19) At optimum distance, F ( 2 ) ( n ) cancels and C( t ) is narrower. The spherical aberration of the lenses may also contribute to the group-velocity dispersion. This effect is reduced with the help of properly oriented doublets. The less steeply curved surface should face the nearest grating, where the focal point is located. This is the opposite of the condition that optimizes spectral resolution. As a matter of fact, the spherical aberration induces negligible dispersion in the present experiment, where the useful bandwidth of the two-grating device is only 1.1 THz. Therefore we select the L2 orientation that improves spectral resolution. It should be noticed that the spectral shaper need not be corrected for group-velocity dispersion in the patternrecognition experiment. Indeed, the correlation signal only depends on the phase difference between the recorded pattern and the readout field. Velocity dispersion has to be eliminated for the side experiment that was designed to check the capabilities of the shaper and of the interferometric detector. (20) (21) The laser spectral intensity is assumed to be uniform over the shaping window. Substitution of Eq. (21) into Eq. (20) leads to C~ t ! } UE U dn m ~ n ! exp~ 2i pn t ! , (22) which is nothing but the Fourier transform of the encoding function. Figure 4 represents the spectral structures of several biphase encoding functions, their temporal profiles calculated by Fourier transform and the optical delay variations of the measured fringe contrast. The curve number refers to the row position in the Hadamard matrix, which labels the encoding functions. The good agreement between theoretical and experimental profiles demonstrates the correct operation of the pulse shaper and of the interferometric detector. The profiles displayed in the upper boxes of Fig. 4 correspond to the uniform phase factor m ( 0 ) ( n ) 5 1, with maximum fringe contrast at zero delay. All the other encoding functions m ( k ) ( n ) (k Þ 0) are orthogonal to m ( 0 ) ( n ), in the sense defined by Eq. (9). In our experiment it is crucial that this orthogonality property be correctly transferred to the Tian et al. Vol. 16, No. 1 / January 1999 / J. Opt. Soc. Am. B Fig. 4. Experiment side: contrast of the interference fringes of the laser beam with its time-delayed, spectrally shaped replica. Theory side: frequency-to-time Fourier transform of the spectral shapes that are drawn in insets. Function numbers refer to their position in a 32-code Hadamard set. 79 Then the broadband source is converted into a tunable probe by insertion of a 0.08-THz-wide moving slit in the two-grating device. The intensity transmitted or diffracted by the exposed material is recorded as a function of the slit position. The resulting spectra are displayed in Figs. 5(c) and 5(d). With a dose of ;2.7 mJ/THz on an exposed area of ;0.25 mm2, the burning depth approaches 1/3 of the initial optical density. The measured transmission results from an average over the space- and frequency-engraved fringes. However, even at positions where the brightest illumination was received, the final optical density remains large enough to comply with the linear-dosedependence condition discussed in Section 1. It should be noticed that larger dose burning is also affected by the specific properties of persistent spectral hole burning in octaethylporphyrin. In this material the chemical structure of the photoproducts does not differ from that of the educts. Therefore the burnt molecules do not leave the absorption band and ultimately hamper the bleaching process. The model that we developed22 shows that this does not occur as long as the hole depth does not exceed 1/3 of the initial optical density. Linear dose dependence also requires that one avoid transient depletion of the ground state during the laser pulse. About 90% of the excited molecules are trapped in the metastable triplet state during several tens of milliseconds. Therefore at the end of a laser shot the groundstate population is reduced by factor exp@2(Bw/c)#, where B stands for the stimulated-emission Einstein coefficient and w represents the spatial and spectral density of burning energy per shot. From available data on free-base porphyrin,23 one obtains (B/c) 21 5 0.35 mJ/THz/cm2/ pulse spectra and correctly detected by the correlator. According to the definition of m ( 0 ) ( n ), Eq. (22) can be rewritten as C~ t ! } UE U dn m * ~ n ! m ~ 0 ! ~ n ! exp~ 22i pn t ! , (23) which means that the othogonality of m ( k ) ( n ) (k Þ 0) and m ( 0 ) ( n ) is revealed at t 5 0. This is properly verified on the experimental data of Fig. 4. B. Hole Depth and Spectral Response The reference pattern is engraved in the photosensitive material as a space- and frequency-dependent bleaching. In order to specify the exposure conditions, we measure the depth of the hole that is burnt in the absorption profile. We also check the spectral response of the material. To this end the experimental setup undergoes a temporary modification. First, the laser bandwidth is enlarged to ;10 THz (see upper box on Fig. 5), and the central flat region of the spectrum is selected with the help of a 1.1-THz-wide window that replaces the LCM in the two-grating device. The shaped beam is split into two angled arms in order to engrave a space and frequency hologram within the photosensitive material. The delay between the two writing pulses is adjusted to T > 70 ps. Thus the period of the spectral interference fringes they engrave is ;15 GHz. Fig. 5. (a) Laser spectrum. (b) Optical density of the sample before engraving. (c) Optical density after engraving within a 1.1 T-Hz window, with an average laser energy of 0.7 mJ/THz/ shot for 6 min. (d) Diffracted-energy spectrum. 80 J. Opt. Soc. Am. B / Vol. 16, No. 1 / January 1999 shot. Experimental spectra were recorded with incident energy density per shot w 0 5 0.2 mJ/THz/cm2/shot. Comparing the initial and the final spectral distributions of optical density in Fig. 5, one notices that some bleaching is effected outside the burning window, on its red side. It corresponds to molecules that had been burnt through the phonon sideband of the homogeneous line. This effect contributes to optical-density reduction, but not to the space and the frequency hologram buildup. Indeed, the period of the recorded spectral fringes has to be larger that the homogeneous width. On the contrary, the 15-GHz fringe period is much smaller than the 0.5THz typical width of the phonon sideband. Finally, the spectral distribution of the holographic signal, drawn in Fig. 5(d), is uniform, as required for the pattern-recognition experiment. C. Pattern Recognition The experimental setup recovers the configuration illustrated in Fig. 1. One is faced with contradictory requirements. On the one hand, the laser spectrum must be uniform over the storage window. On the other hand, the detection becomes more noisy as the interferometric reference bandwidth exceeds the width of the shaping window. As a tentative trade-off, we equalize the laser spectrum with the help of a Lyot filter. The spectrum is made nearly flat within the 1.1-THz-wide LCM window and is ;3 THz wide at half-maximum intensity. The optics are readjusted in such a way that the beam area on the sample is ;4 times smaller than in Subsection 5.B. The reference and the object beam intensities are balanced so that they evenly contribute to the spatial and the spectral energy density on the sample. Dose calibration is effected by monitoring spectral hole depth. A dose of 0.63 mJ/THz is needed to achieve a 0.5 reduction of the optical density, which initially equals 1.5. This is consistent with previous data. The holographic storage dose is set equal to this value, which represents the linearregime upper limit. The burning window is centered at the top of the absorption band. The reference pattern No 16 is stored in the PSHB material. The burning dose is provided by a 2-min pulse sequence, with a repetition rate of 15 pulses per second. The average pulse energy on the sample amounts to 0.44 mJ/THz/shot. The 32 test functions are successively imprinted on the input beam. The result is given in Fig. 5(a). An intense correlation peak is detected when the input pattern coincides with the reference one. False correlation is also detected for some input functions. The true-to-false signal ratio is larger than 15 for any input. The most important false correlation occurs for input phase factors Nos 0 and 24 in Fig. 6(a). The spectral structure of the diffracted field results from the product of the reference and the input spectral shapes. As explained in Section 2, the square of a shape function equals the gate function (No 0), and the product of any two different shape functions is another function in the same family. When the input pattern is the gate function, the diffracted-field spectrum should be given by the product of functions No 0 and 16, which coincides with the shape factor No 16. In the same way, the emission spectrum generated by input function No 24 should be described by Tian et al. function No 8, which results from the product of functions Nos 16 and 24. The signal at the output of the interferometric detector is nothing but the frequency-to-time Fourier transform of the emitted-field spectral amplitude. At zero delay it coincides with the integral of the spectral amplitude over the frequency-domain storage window. Among the 32 orthogonal shape factors, functions Nos 8 and 16 exhibit the narrowest Fourier-transform structures at zero time delay (see Fig. 4). When the diffracted field is spectrally shaped by these functions, the correlation signal is very sensitive to the source uniformity and to the adjustment of the delay line. The same false-correlation feature is observed for other reference patterns. For instance, the set of crosscorrelation signals in Fig. 6(b) have been obtained after storage of code No 12 in the PSHB material. Then the largest false correlation is observed when the shape No 28 is imprinted on the input pulse. The product of functions Nos 12 and 28 again gives the function No 16. With reference pattern No 12 stored in the PSHB material, and keeping the input function No 28, we record the interferometric detector as a function of the time delay. The resulting profile is displayed in Fig. 7(a). The amplitude at t 5 0 corresponds to the false correlation with input No 28. In the same way, one records the timedelay dependence of the signal with input function No 12 [see Fig. 7(b)]. Fig. 6. Experimental cross correlation between a recorded reference pattern and 32 input codes. The interferometric signal is detected at fixed delay t 5 T. Three different shapes (Nos 16, 12, and 1 in the 32-code Hadamard set) are successively stored in the PSHB material. Tian et al. Vol. 16, No. 1 / January 1999 / J. Opt. Soc. Am. B 81 This work includes elements from two Ph.D. dissertations defended in 1997 at Université Paris XI by François Grelet and Mingzhen Tian. REFERENCES 1. 2. Fig. 7. (a), (b) Interferometric detection of the diffracted field, as a function of the time delay ( t -T). Function No 12 is stored as the reference pattern. Readout is performed by input functions No 28 (a) and No 12 (b). (c), (d) Interferometric detection of the laser field shaped by the codes No 16 (c) and No 0 (d). The optical setup and the storage material may both contribute to the generation of the false-correlation signals. In order to elucidate the respective part they play, we compare the diffracted field data, diagrammed in Figs. 7(a) and 7(b), with the direct interferometric detection of functions Nos 16 and 0, as reported in Subsection 5.A. The latter recordings do not involve any contribution from the material. The relevant profiles are copied in Figs. 7(c) and 7(d). In Fig. 7(c) the interferometric signal at t 5 0 for function No 16 does not vanish. Therefore the spectral shaper, the laser spectrum, or the detector still have some defects. The signal ratio at zero delay between code No 0 [Fig. 7(d)] and code No 16 [Fig. 7(c)] is ;16. This is consistent with the ratio of the correlation signals, as observed in Figs. 7(a) and 7(b). Therefore in the linear regime the material does not seem to contribute to the false signal. The residual distortion is at least partly attributed to the nonuniformity and uncontrollable instability of both spatial and spectral distributions of the laser beam. 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