Spatial mixed binarymultiplication by photon echoes
D. Manganaris, P. Talagala, and M. K. Kim
We demonstrate mixed binary multiplication of two or three numbers represented as spatial bit patternsby using a backward stimulated photon echo in a low-temperature solid activated by rare-earth ions. Thephoton-echo output image is the convolution-correlation of input images in a holographic arrangement,which can be interpreted as multiplication in a mixed binary format. The computation is extremely fast(- 50 ns) and may be combined with temporal data processing for a versatile vector and matrix processor.
The backward stimulated photon echo (BSPE) hasattracted considerable attention recently for its poten-tial application in high-density high-speed opticalmemory and optical data processing. In particular,the storage and the phase conjugation of images byBSPE in solids have been demonstrated. 1,2 Long-termstorage of up to 12 h (Ref. 3) and storage of up to 248bits of temporal data4 have been demonstrated. Fur-thermore, the convolution and the cross-correlationproperties of photon echo for both temporal pulseshapes56 and the spatial beam profile7 have beendemonstrated. These experiments have exploited theholographic nature of the BSPE process, which canbe considered a time-delayed degenerate four-wavemixing of nonlinear optics. In a conventional hologra-phy, the object beam scattered from an object and theplane-wave write reference beam simultaneously im-pinge on the photographic plate and produce aninterference pattern, which is eventually recorded asa transmittance grating. The other plane-wave readreference beam diffracts from the transmittance grat-ing and gives rise to a phase-conjugate holographicimage. In a degenerate four-wave mixing experiment,the interference patterns are recorded as a refractive-index grating, which results from the nonlinear depen-dence of the index of refraction on the light intensity.The BSPE has the remarkable property that the lightbeams can interfere with each other even if they areseparated and nonoverlapping in time. This is possi-ble because the atomic dipole oscillation set up by theresonant laser pulses can remember and compare thephases of time-delayed laser pulses. This interferencepattern is recorded as a grating of the atomic quan-
The authors are with the Department of Physics and Astronomy,Wayne State University, Detroit, Michigan 48202.
tum states, which is usually referred to as the popula-tion grating. In BSPE the population grating hasboth spatial and temporal (or equivalently, spectral)structures, whereas the conventional and four-wave-mixing holograms involve spatial grating only.
One of the most important properties of a holo-graphic system is the convolution-correlation of spa-tial images,- 10 which is the basis of numerous opticaldata processing methods, including BSPE. In thispaper we demonstrate the multiplication of two orthree numbers represented as spatial bit patterns byusing BSPE in a low-temperature solid activated byrare-earth ions. The photon-echo output image is theconvolution-correlation of input images in a holo-graphic arrangement; this process can be interpretedas multiplication in a mixed binary format. Becauseof the additional temporal dimensionality of BSPE,the convolution and the correlation of temporallyprogrammed optical data is also possible. The multi-plication of temporal bit patterns by using BSPE hasbeen demonstrated by Babbitt and Mossberg.11 Thecomputation is extremely fast ( - 50 ns in our studybut subnanosecond speed is possible in principle), andit may be combined with temporal data processing fora versatile vector and matrix processor.
In a BSPE experiment, three laser pulses with anelectric field
9i = eF exp[i(k, r - ot)] (1)
are incident on the resonant two-level atoms of thesample crystal. Here Ei is the envelope of the ith pulseat t = 0, T, and T + T for i = 1, 2, and 3, respectively(see Fig. 1). The BSPE pulse emitted at t = 2
T + T is
9e = e, exp[i(ke r- t)]
o sin 0 sin 02 sin 03 exp[i(-k + k2 + k3) rexp(-iwt),I EE2E3(
Fig. 1. (a) Coordinates used in calculation of propagation of BSPEimages. (b) Temporal pulse sequence of BSPE.
where the area of a laser pulse of amplitude ei andlength ri is defined as
Oi = pe' (3)h
and p is the transition dipole moment. In the smallpulse area approximation, sin Oi = Oi and
Ee = AE1*e2E3 , (4)
where A is a constant. If k2 + k3 = 0, then ke = -k,and the backward echo is the phase conjugate of thefirst pulse.
Now we use the Fresnel diffraction theory12 toderive the spatial profile of the BSPE. To simplify thenotation, we consider a one-dimensional profile. Refer-ring to Fig. 1, we see that the input image mask Ei (xi)is placed a distance zi in front of the input lens of focallength /i, which in turn is at a distance si from thesample crystal. Experimentally, we can select thesedistances for beams 1, 3, and e individually by separat-ing the beams with beam splitters before insertinglenses and object masks. Then the input field arrivingat the crystal is
ee(xOx) = AEl*(xo)E2 (Xo)E3(Xo), (8)
-m which propagates a distance se to the lens of focallength fe (normally se = si and fe = A). The echooutput field at Ze from the lens fe is then given by
Ee{(xe) = Aal*%-a30te eXp(iyeXe2 )
x)([heXe]{exp[i(-13 + P2 + 133 + pe )XO2
]5-* [hixo]IEl(xl)exp(iyixi2 )1
x 5[h 2 xo]IE2(x2 )exp(iy2 x22)1 F[h3 xoI {E3(x3 )exp(i-y3x3
2)}}. (9)
In the confocal arrangement where si = zi = fi (i = 1,2, 3, e), we have
Ee -c E1 ® E2 E 3 , (10)
where 0 and denote correlation and convolution,respectively. Equation (9) can be used to simulate thepropagation of the photon-echo pulse profile givenany input pulse profiles. For example, Fig. 2 shows asimple phase conjugation of the photon-echo profilewhen pulse 1 has a step function or a Gaussian profileand pulses 2 and 3 have a plane-wave profile. Theinput beams are not shown in the figure. The figureshows plots of the echo intensity (arbitrary scale, notlabeled) across beam profile x at various distances zfrom the exit lens fe = A. The confocal arrangementis used with fi = = si = 500 mm (i = 1, 2, 3). Notethe phase-conjugate reconstruction of the image iscomplete at Ze = 500 mm. Equation (9) also showsthat the position of exact phase conjugation or corre-lation-convolution is given by the relation
i- 1 + 32 + 3 + 13e = 0. (11)
The phase-conjugate image is scaled by the factors hiwhen the arrangement is not exactly confocal. We
ei(Xo) ti exp(i ix02 ) -[hixo] Ei (Xi)eXp(iYX,2 )
where
I(-ikqi \112i= \2srzjs j exp[ik(zi + si)/2],
ksi - qi kzi - qi kqi2 S,2 hi 2 Zi2 zisi
qi zi si T
and the Fourier transform is defined as
[K]{f (X) = f dxf(x)exp(-i-x).
(5)
()
Fig. 2. Propagation of phase-conjugate photon echo calculated(6) from Eq. (9), with square and Gaussian profiles in pulse 1 and a
plane-wave profile in pulses 2 and 3. Here x is the distance acrossthe beam profile and z is the distance from the exit lens. The echointensity has an arbitrary scale (not labeled). The confocal arrange-ment is assumed with /i = zi = si = 500 mm (i = 1, 2, 3, e). Noticethat the reconstruction of the phase-conjugate image is complete at
may note that Eq. (9) and relation (10) apply to alldegenerate four-wave-mixing geometry, including con-ventional holography and BSPE.
The experiment is performed on the 3PO-3H4 transi-tion ( = 477.8 nm) of Pr3+ ions doped 0.1 at. % in aLaF3 single crystal, immersed in a liquid heliumcryostat. The laser pulses are provided by a YAG-pumped pulsed dye laser, whose output is divided intothree parts with beam splitters and delayed to a fewtens of nanoseconds with optical delay lines. Eachinput pulse typically has energy of a few tens ofmicrojoules and a beam diameter of approximately 3mm. Often it was necessary to use a telescope and aspatial filter to expand and to clean up the beamprofile. The multimode beam structure of the pulseddye laser was the limiting factor in the quality of thephoton-echo images. We used a confocal arrangementwith 50-cm lenses (i.e., f = si = zi = 50 cm). Thephoton-echo images are captured with a charge injec-tion-device-camera and a video tape recorder. ThePockels cell shutter arrangement is used to preventdirect or scattered input lasers from saturating thecamera. The input images that represent binarynumbers are made of vertical holes of 0.5-mm widthcut out of a black tape on a slide glass.
The experimental results of photon-echo convolu-tion-correlation are shown in Figs. 3 and 4. In Fig.3(a) a simple phase conjugation of the input image ofthree vertical slits is shown. In Fig. 3(b) the inputimages represent binary numbers 11 (= 3
dec ), 11,and 1. The output image is interpreted as a mixedbinary number 121MB = 1 x 22 + 2 x 21 + 1 x 20 =9dec, which is the product of the input numbers.Similarly, Fig. 3(c) shows the mixed binary multiplica-tion 11 11 11 = 331MB = 2 7 de,* More examples ofmixed binary multiplication of three-digit binarynumbers are shown in Fig. 4, along with curves thatrepresent the electric-field profile calculated from Eq.9, if we assume Gaussian input profiles. These exam-ples show that the convolution-correlation of photon-echo images can be interpreted as the mixed binary
#1 #2 #3
(a) N111 (7) x 1 x 1
(b)
11 (3) x 11 (3) x 1
11 (3) x 11 (3) x 11 (3)
echo
= 111 (7)
= 121 (9)
= 1331 (27)
Fig. 3. Mixed binary multiplication by BSPE. (a) 11 x 1 x 1 = 11,(b) 11 x 11 x 1 = 121, (c) 11 x 11 x 11 = 1331.
#1 #2 #3 echo
AAA AAY AA JN LL(a)
111 (7) x 11 (3) 11 (3) = 13431 (63)
(b) / A / A. A;\AAM A,
101 (5) x 101 (5) X 11 (3) = 112211 (75)
Fig. 4. Mixed binary multiplication by BSE. (a) 101 x 101 x 11 =112211, (b) 111 x 11 x 11 = 13431. The calculated profiles of theBSPE electric field are also shown.
multiplication of binary numbers represented as one-dimensional bit patterns. Note that when interpret-ing the photon-echo mixed binary patterns, we needto take into account the transverse coherence of thelaser beam. That is, if the entire beam profile iscoherent, then the detected intensity of a digit isproportional to the square of its mixed binary value.For example, in Fig. 4(a) the intensity pattern of themixed binary number 1-3-4-3-1 is actually 1-9-16-9-1.We observe that this is qualitatively true in ourexperiments. Thus the mixed binary representationrequires discrimination of intensity levels. At present,the quality of photon-echo images is limited by themultimode structure of the pulsed dye laser, evenwith spatial filtering. Because of the large variation ofintensity across the input laser, the intensity ofindividual bits on a pattern is not uniform. Thismakes it difficult to obtain a quantitative measure-ment of bit intensity ratios in the output pattern. Inorder to improve the image quality and to multiply amore complex set of bit patterns, it will be necessaryto use a single-mode laser, perhaps the pulsed amplifi-cation of a cw ring dye laser.
The experimental results demonstrate that theBSPE can be used in principle to multiply two orthree numbers that are represented as one-dimen-sional bit patterns. It would also be possible toarrange multiplication of two-dimensional patternsand to use that to multiply vectors or matrices. Thereare many possible input pattern arrangements andways to interpret the output patterns, and in princi-ple most of the Fourier image-processing techniquesof conventional holography can be used here. TheBSPE process is extremely fast in the nanosecondregime, but what is the most significant finding inBSPE image processing is that the Fourier transformprocess applies to the temporal as well as the spatialdata. The spatial processing results of this paper can
also be accomplished by using other cw four-wave-mixing techniques. But the true potential of BSPE asan efficient parallel processor may be realized whenthe spatial and the temporal processing are combinedin one system. Now that the spatial and the tempo-ral" processing have been demonstrated separately,the next logical step would be combining the twoprocesses. The method may prove useful in a systemwhere large amounts of optical data are to be pro-cessed in real time, such as for robotic vision andsatellite survey.
In conclusion, we have demonstrated mixed binarymultiplication of spatial bit patterns by using back-ward stimulated photon echo in a low-temperaturesolid activated by rare-earth ions. At present theoutput image quality is limited by the laser profile,but the method can be extended to multiplication oftwo-dimensional bit patterns and temporal bit pat-terns for a versatile system of vector or matrixprocessor.
References1. M. K. Kim and R. Kachru, "Long-term image storage and
phase conjugation by a backward-stimulated echo inPr3 +:LaF3 ," J. Opt. Soc. Am. B 4, 305-308 (1987).
2. M. K. Kim and R. Kachru, "Storage and phase conjugation of
multiple images using backward-stimulated echoes inPr 3
+:LaF 3 ," Opt. Lett. 12, 593-595 (1987).3. M. K. Kim and R. Kachru, "Multiple-bit long-term data
storage by backward stimulated echo in Eu3 +:YA103," Opt.Lett. 14, 423-425 (1989).
4. M. Mitsunaga and N. Uesugi, "248-bit optical data storage inEu3 +:YA103 by accumulated photon echo," Opt. Lett. 15,
195-197 (1990).5. V. A. Zuikov, V. V. Samartsev, and R. G. Usmanov, "Correla-
tion of the shape of light echo signals with the shape of theexcitation pulses," Sov. Phys. JETP Lett. 32, 270-274 (1980).
6. Y. S. Bai, W. R. Babbitt, N. W. Carlson, and T. W. Mossberg,"Real-time optical waveform convolver/cross correlator," Appl.Phys. Lett. 45, 714-716 (1984).
7. E. Y. Xu, S. Krll, D. L. Heustis, R. Kachru, and M. K. Kim,"Nanosecond image processing using stimulated photonechoes," Opt. Lett. 15, 562-564 (1990).
8. J. C. AuYeung, in Optical Phase Conjugation, R. A. Fisher, ed.(Academic, New York, 1986), Chap. 9.
9. D. M. Pepper, J. AuYeung, D. Fekete, and A. Yariv, "Spatialconvolution and correlation of optical fields via degeneratefour-wave mixing," Opt. Lett. 3, 7-9 (1978).
10. J. 0. White and A. Yariv, "Real-time image processing viafour-wave mixing in a photorefractive medium," Appl. Phys.Lett. 37, 5-7 (1980).
11. W. R. Babbitt and T. W. Mossberg, "Mixed binary multiplica-tion of optical signals by convolution in an inhomogeneouslybroadened absorber," Appl. Opt. 25, 962-965 (1986).
12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,New York, 1968).
