Programmable optical interconnections by multilevel syntheticacousto-optic holograms
Eero Tervonen, Ari T. Friberg, and Jan Westerholm
Department of Technical Physics, Helsinki University of Technology, SF-02150 Espoo, Finland
Jari Turunen and Mohammad R. Taghizadeh
Department of Physics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK
Received February 22, 1991
Reconfigurable high-fidelity, high-efficiency weighted optical interconnection patterns are demonstrated forthe first time to our knowledge with the aid of optimized multilevel phase gratings achieved electronically asphase-modulated sinusoidal refractive-index variations propagating in an acousto-optic Bragg cell.
Optical interconnections based on gratings with anoptimized distribution of power among the dif-fraction orders play an important role in many elec-tro-optical and all-optical processing and computingarchitectures. Such components generate one-dimensional (ID) or two-dimensional (2D) arrays ofequal-intensity spots,' but they may also providemore complicated interconnections, which are re-configurable in real time if the grating profile canbe controlled, as, e.g., in photorefractive materials2
and in liquid-crystal 3 and magneto-optic 4 spatiallight modulators. Acousto-optic devices have re-cently also been used to demonstrate a reconfig-urable array generator5'6 and a photonic switch.7Advantages of this approach include high light effi-ciency (as much as 851%) and short switching time,limited to 1-10 As by the sound-wave transit acrossthe deflector aperture.
A straightforward method of generating an appro-priate electronic signal for the acousto-optic deviceis to fan in the signals from several fixed-frequencyoscillators.7 Unwanted intermodulation effects ap-pear, however, and the diffraction efficiency intothe desired beams is not optimized.8 Alternatively,an efficient acousto-optic phase-only grating profilecan be constructed using the methods of syntheticholography; this is a novel approach that was firstapplied in Ref. 5 to control the spatial coherenceproperties of beamlike wave fields. In the presentLetter we extend this method to generate fully arbi-trary, programmable, space-invariant, acousto-opticinterconnects with high diffraction efficiency andlow reconstruction noise. Binary and four-levelphase-modulated sinusoidal gratings are demon-strated that reconstruct both regular and arbitraryiD spot patterns.
Figure 1 shows an acousto-optic deflector drivenby a phase-modulated cosinusoidal signal Re{exp[i+(t) - ifl~t]} of a carrier frequency fl,. When ap-plied to the transducer T, the signal generates aphase grating propagating at the speed of sound V inthe material. A coherent light beam of wavelength
A is incident upon the device at the first Bragg angleOB = A/2Ac, where A, = 27rV/fl,. The index modu-lation and cell thickness are presumed such that themaximum Bragg efficiency is obtained. We also as-sume that the modulating signal is periodic, with aperiod A = 2rV/fl >> A,. The volume phase grat-ing then acts as a hybrid hologram5 : the high-frequency carrier deflects the incident beam by anangle 2 0B (also adding a frequency shift of flc,), whilethe low-frequency modulating signal generates aphase-delay profile of the form O(x) in the Bragg-diffracted wave front. This hybrid scheme hasseveral distinct advantages over the usual normal-incidence Raman-Nath diffraction (see Sec. V ofRef. 5).
Assuming a uniform frequency response and aquasi-monochromatic Gaussian incident beamUo(xco) = [So(wo)]"2 exp(-x2 /wO2 ) with spot size w0,we obtain the equation [cf. Eq. (33) of Ref. 5]
for the cross-spectral density9 in the array plane(the back focal plane of a lens of focal length F).Here Gn are the harmonics amplitudes of the low-frequency modulation grating, i.e. (T = 2v/Q),
= T
Gn = T` f exp[io(t) - i2vrnt1T]dt, (2)
WF = FA/7wo is the e-2 beam half-width of a singlediffraction order in the Fourier plane, and u0 =FA/A is the separation between the centers of twoadjacent diffraction orders. If uo > 2 WF, the beamsare spatially well separated and essentially mutuallyuncorrelated (as seen by evaluating the optical in-tensity and the complex degree of spectral coher-ence9 ). The quantity C = uo/2wF is the compressionratio of the device, and hence for an array generatorC 2 1. The small frequency shifts nfl between thebeams are irrelevant in most applications. We note
Fig. 1. Schematic of the experimental setup. AOD,acousto-optic deflector; OB, Bragg angle; T, transducer; L,lens; D, detector in the Fourier plane; C, fixed-frequencyoscillator; S, electronic grating synthesizer; M, phasemodulator and amplifier.
-10 -5 0 5 10
Fig. 2. Experimental result: six-beam equal-intensitypattern with suppressed even orders generated by abinary-phase acousto-optic grating.
also that any spot fluctuations and cross talk due tograting motion are minimal, since the illuminatedregion of the deflector always contains several peri-ods of the modulating signal.
The desired array {JGnj 2} within the predeter-mined signal window n = N,, .. ., N 2 is obtained bydesigning the grating structure using, e.g., nonlin-ear optimization techniquesl'0" with the generalgrating-profile shape dictated by the available fabri-cation methods. These shapes include the so-calledDammann grating,'2 which constitutes simply aperiodic train of optimized binary pulses andproduces an inversion-symmetric pattern. Regulararrays with an even number of beams are gener-ated by grating profiles of the type +(x + A/2) =+(x) + 7r. This profile implies that all even diffrac-tion orders (including the zeroth) vanish, i.e., aso-called even-orders-missing (EOM) grating is gen-erated. Consider a six-beam binary-phase acousto-optic EOM grating with a diffraction efficiency of84.6% and a negligible reconstruction error, calcu-lated using the method of Ref. 10. The experimen-tal 6-beam array with C = 2 and a homogeneity of5% is presented in Fig. 2; in this instance the grat-ing profile was made by a pulse train generator anda mixer (Mini-Circuits Model ZAY-3) as in Ref. 5.In general, however, binary acousto-optic EOMprofiles are readily produced electronically: if anodd number of optimized transition points togglethe output signal between two possible levels thatcorrespond to the phases 0 and 7r, the resulting grat-ing structure (an even number of transition points)is automatically of the desired type.
To generate arbitrary (non-inversion-symmetric)programmable arrays, grating profiles with more.than two phase levels are needed. Continuous pro-files would be ideal in terms of diffraction effi-ciency, but linear phase modulators with sufficient
speed and accuracy are not available. On the otherhand, modulators with K equally spaced phase levelscan be constructed by combining mixers. The holo-gram profile should also be digitally addressable,i.e., consist of L cells of equal width, each having aconstant phase delay. Electronic signals for hybridholograms of this type can be generated as illus-trated schematically in Fig. 3. A preoptimizedhologram structure is selected from the memoryand copied into the ring buffer, in which it is cycli-cally repeated. The speed of the ring buffer is acritical factor; fast buffers can be achieved by usingshift registers or one-of-many data selectors. Thecompression ratio C may be varied simply by adjust-ing the clock frequency.
The design of fully reconfigurable array gen-erators with L cells, K discrete phase levels, anda maximum fan out of N beams consists of twostages. In the first stage the basic solution withequal-intensity beams is obtained. This combin-atorial optimization problem can be solved bythe method of simulated annealing.'" The uniformsolution is characterized by its diffraction effi-ciency 7m = InIGn0
2 and reconstruction error PU =maxnI1 - NIGnI2/'171, where n = N1,...,N 2 andN = N2 - N, + 1. Our numerical calculationswith small values of K ('20) have shown thatreasonable efficiencies (iq> > 80%) and reconstruc-tion errors (Pu < 10%) are obtained if the numberKL of the degrees of freedom exceeds the num-ber N of diffraction orders roughly by a factor ofG 10-40. A precise, universally valid figure isdifficult to give owing to the highly nonlinear na-ture of the problem.
The second stage of the design consists of thecalculation of the hologram profiles for the non-uniform-intensity arrays, using the same values ofK and L. The maximum intensity for any beamshould be the same as the one obtained in stage one.The goal distribution for an arbitrary member ofthese arrays then is {I-uIn/N}, with In E [0, 1] defin-ing the relative beam weights. Obviously, for theequal-intensity array, In = 1 for all n E N,,...,N 2, and hence the goal efficiency in the nonuniformarray must be 71a = 77u nI0 /N. These efficienciesare normally readily achieved with optimizationby simulated annealing. Moreover, the reconstruc-tion errors pa= maxnJIn- NIGn 2/i ul are close to
l Control | I Clock I FixedI unit I I generator I oscillator
- - - - + -- --
___ __ __ __ I I … … … I
I II | Power
|amplifier|
Output signal to the transducer
Fig. 3. Electronics block diagram for synthesizing multi-level acousto-optic holograms. The blocks in part II re-quire fast electronic components, whereas those in part Ican be implemented with a personal computer.
Fig. 4. Experimental result: binary interconnectionpattern with goal distribution {In} = {0,1,1,0,0,1,0} fordiffraction orders n = -3,..., +3.
-8 -4 0 4 8
Fig. 5. Experimental result: three-level interconnec-tion pattern with goal distribution {In} = {1, 0.5, 0.5, 1,0, 0, 1, 1} for diffraction orders n = -4,..., +3.
Pu. We note that this definition of reconstruc-tion error emphasizes the absolute rather than arelative error.'0 "'
The maximum number of resolvable spots de-pends on the Bragg bandwidth AflB, the size of thedeflector aperture (which determines WF), andthe speed of sound V in the material. The sweeprange in the Fourier plane is Ax = FAAflB/27rV, andwith a compression ratio C the maximum number ofspots is thus Nmax = Ax/2wFC = FAA fB/47rCVwF.For a nontruncated Gaussian beam, Nm. = AflBWO/4CV. Whether this maximum number of beamscan be achieved in practice depends on the speed ofthe electronic grating generator. We assume thatthe number of phase levels is fixed and that the pixelfrequency is flp. Then the fundamental frequencyof the grating is fl = Qp/L, and the grating period isA = 2irVL/flp. Recalling from the optimizationthat KL > GN and requiring a compression ratio C,we have for a nontruncated Gaussian beam Nmax' =KflPwo/4GCV. Hence the electronics is able toexploit the full performance of the deflector ifNmax' > Nmax, i.e., if fp > GAMB/K.
In the experiments a quadriphase modulatorPM-110 by Anzac, driven by ordinary LS TTLcircuits, was used. The ring buffer was fabri-cated with 74LS166A shift registers from TexasInstruments, and the highest reliable pixel rate wasQpl27r = 25 MHz. The TeO2 deflector (MatsushitaEFL-D250R; AfIB/21T = 36 MHz, V = 660 m/s,clear aperture of 5 mm) was illuminated with aGaussian beam of w0 = 2.5 mm. Taking into ac-count the truncation by the deflector aperture(which results in WF 1.3FA/'7rwo), we haveNmax'C C 15. This is a rather modest value com-pared with NmaxC = 165. However, apart from anincrease in K, the value of Nmax'C could be improvedby reducing G (and sacrificing efficiency) or by em-ploying fast ECL devices in which pixel rates ex-ceeding 100 MHz are standard.
In Figs. 4 and 5 we present experimental resultsof binary and three-level reconfigurable one-to-many interconnection patterns. The acousto-optichologram that produced Fig. 4 consists of 32 cellswith four discrete phase levels. The theoreticaldiffraction efficiency of the optimized solution is-a = 66%o, while the reconstruction error is pal = 5%o,which matches the measured value. The compres-sion ratio is approximately 2.5, which correspondsto a grating frequency of fl/21T = 20.8/32 MHz =650 kHz. The optimized four-level phase hologramwith 64 equally wide cells producing the weightedinterconnection pattern of Fig. 5 has a theoreticaldiffraction efficiency of q,, = 74% and a recon-struction error of Pa = 8%. The measured pa isapproximately 10%, and the beam compression ratiois C = 1.3, which corresponds to 0/2vw = 22.6/64MHz = 350 kHz. The observed diffraction effi-ciencies were in all experiments close to the theo-retical values multiplied by the Bragg efficiency,which was approximately 60%.
In conclusion, we have demonstrated the feasibil-ity of synthetic acousto-optic multilevel phase holo-grams in the generation of reconfigurable iD spotarrays. Practical limitations concerning the com-pression ratio, the number of output beams, thehologram design, and the speed of electronics havebeen assessed. For increased parallelism, 2D inter-connects are required. Partially reconfigurable 2Dpatterns can be achieved by crossing two acousto-optic deflectors or a fixed array generator and anacousto-optic hologram. Fully programmable inter-connects could be produced with multichanneldevices as illustrated, e.g., in Fig. 14 of Ref. 5.
We thank H. Ichikawa and A. Dunlop for usefulcooperation. Financial support from the Academyof Finland, the Science and Engineering ResearchCouncil (UK), the Emil Aaltonen Foundation, andthe Jenny and Antti Wihuri Foundation is gratefullyacknowledged.
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