December 15, 1991 / Vol. 16, No. 24 / OPTICS LETTERS

Self-organizing photorefractive frequency demultiplexer

Mark Saffman, Claus Benkert, and Dana Z. Anderson

Department of Physics and Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colorado 80309-0440

Received July 8, 1991

We demonstrate a self-organizing photorefractive circuit that demultiplexes a beam that has two signals im-posed on separate optical carrier frequencies into two beams, each containing one of the signals on its carrier.Unlike conventional demultiplexing techniques, this method requires little a priori knowledge about the carrierfrequencies. The signal channels must be spatially uncorrelated, and their frequency separation must be morethan the photorefractive response bandwidth (hertz to kilohertz). The optical circuit uses no local oscillator,and the photorefractive response bandwidth places no upper bound on the channel bandwidth. Experimentalresults for demultiplexing a beam that has two signals, with a BaTiO3 circuit, show contrast ratios of betterthan 40:1 at the outputs.

Consider the task of demultiplexing a number ofsignals carried on a multimode fiber, and assumethat we have no a priori information about the car-rier frequencies other than that they are different.How can the channels at the output of the fiber beseparated? One might use a grating, but then thecarriers would need to be well separated in wave-length, by at least an angstrom or so, and we do notknow ahead of time that they are. One might useinterferometric techniques, but these require addi-tional frequency scan and locking electronics thatmust also avoid possible degeneracies in resonanceconditions. To make things worse, neither approachcan handle the high spatial content of the specklepatterns in a straightforward manner.

Our a priori information about the signals islimited to the fact that they have different frequen-cies, and, in the case of a highly multimode fiber,that the individual speckle patterns of the signalswill be uncorrelated. Using this information alone,and drawing on some elementary concepts fromneural networks, we have constructed a photorefrac-tive optical circuit that self-organizes to demultiplexincoming optical signals according to their temporaland spatial differences.

An optical circuit that demultiplexes two signalchannels is shown schematically in Fig. 1. Theself-organizing behavior' arises from competitiveinteractions in multimode photorefractive ring reso-nators.2' The combination of the competitive in-teraction between resonators in the gain mediumand a local interaction in each resonator, which werefer to as reflexive gain, leads to a steady-state op-erating condition in which each resonator chooses tooscillate at one of the pump frequencies. A portionof the oscillating energy is then coupled out of theresonators to give the demultiplexed outputs. Notethat the optical circuit is purely signal driven:there is no local oscillator, and there is no localpower source.

Arguments based on steady-state energy transferin photorefractive two-beam coupling will clarify theoperation of the circuit. We will, for simplicity, con-

sider a two-resonator circuit demultiplexing a two-signal input beam. To demultiplex N frequencieswe would include N resonators, each sharing thesame pump volume, and each with a reflexive photo-refractive interaction. The resonators supportmany transverse modes, and to simplify the analysiswe assume that the oscillating-mode superpositionis always on resonance, so pump-detuning effectsneed not be considered. When the circuit is abovethreshold the two rings compete for energy in theshared gain medium. Without additional interac-tions the rings are at most neutrally coupled,4 andany mixture of the incident frequencies can oscillatein each ring.

The key to achieving the demultiplexing behavioris a nonlinearity that raises the cavity losses whentwo signals try to oscillate in the same ring. Thereare two distinct cases to be considered. In the firstcase the two signals may attempt to oscillate withidentical spatial-mode superpositions. This is ahigh-loss configuration since the oscillating signalsare now spatially matched, and each one will scatteroff the grating owing to the interference of the otheroscillating signal and its pump. Thus one spatial-mode superposition oscillating at two frequenciesis disfavored. In the second case, where the twosignals choose orthogonal spatial-mode superposi-tions, the desired nonlinearity is provided by thereflexive gain interaction. A portion of the oscillat-ing energy is taken out of each ring and then coupledback into the same ring by photorefractive two-beamcoupling, as is detailed in Fig. 2. The standardequations for photorefractive two-beam coupling5'6can be used to derive an expression for the steady-state transmission T of the resonator beam throughthe reflexive gain interaction. If only one signal os-cillates in the resonator beam we find that T =1/[1 + r exp(-GR)] in the absence of reflective andabsorptive losses. The pump-to-signal ratio inci-dent upon the photorefractive medium is given byr = 6/(1 - 8), where 8 is the intensity reflectioncoefficient of the beam splitter, exp(GR) is the small-signal gain in the photorefractive medium, and

0146-9592/91/241993-03$5.00/0 0 1991 Optical Society of America

1993

1994 OPTICS LETTERS / Vol. 16, No. 24 / December 15, 1991

Ren .IReflexive

Energy hn u P&ttoefravtie E-egyTr-afer Pump T Cranfer

omp." OuP.t"Pert Port

Wol Uk2

Fig. 1. Schematic diagram of the self-organizing circuit.

resonator

EnergyTransfer

Fig. 2. Reflexive gain interaction. A fraction 8 of theincident intensity is removed from the resonator beam andthen coupled in again by photorefractive two-wave mixing.

GR = FRIR, with FR the photorefractive coupling con-stant and IR the interaction length. In the steadystate the reflexive interaction simply contributes tothe passive resonator losses.7 When two signals,each with orthogonal transverse-mode superposi-tions and different carrier frequencies, oscillate inone ring, two orthogonal gratings build up in thereflexive gain crystal. The effective steady-statecoupling strength between two waves that have thesame frequency when several mutually incoherentbeams are present can be written as GReff =GR(Iint/I), where I is the total intensity and Iint is theportion of the intensity that writes the photorefrac-tive grating. Therefore, when two frequencies arepresent, the effective coupling for each mode super-position is reduced from the single-frequency case.T is correspondingly reduced, hence multifrequencyoscillation is again disfavored.

The reflexive gain interaction also serves to en-sure a high contrast ratio between the signal inten-sities in each ring. Even when the input signals arespatially uncorrelated, fanning gratings in the gainmedium will couple both input signals to both reso-nator rings, which tends to reduce the oscillatingcontrast ratio. Nonetheless, a high steady-statecontrast ratio can be achieved because the reflexivegain interaction actively enhances the contrast ratio,counteracting the contrast ratio reduction in thegain medium.

The contrast ratio enhancement in the reflexivegain interaction can be calculated as follows. Letthe oscillating beam consist of two fields, having dif-ferent frequencies and occupying orthogonal modesuperpositions, with intensities I, and I2. Thecontrast ratio at the input to the reflexive gain in-teraction is defined as q = I/I2. The effectivesteady-state coupling strengths for the two signalsare GRleff = GRq/(1 + q) and GR2,off = GR1/(1 + q),and the contrast ratio at the output of the reflexivegain interaction is then given by

+ rexp(-GR1 +) (1)

1+ rexp(-GRl q)

The contrast ratio enhancement follows, because forq > 1, q > q, and for q < 1, q < q.

We have attempted to provide some heuristic ar-guments for the observed behavior of the demulti-plexer. It is also useful to know the values ofcircuit parameters that are necessary to ensure thedesired behavior. In order to demultiplex N signalsthere must be sufficient gain for the resonators tobe above threshold. The oscillation threshold maybe derived from small-signal gain considerationsanalogous to those for a single-crystal photorefrac-tive resonator.8 For an N-ring circuit pumped bya beam having N signals of equal intensity wemust have

(1 -L) 1x(PN >1 21 + r exp(-GR/N) exp(Gp/N) > 1, (2)

where L is the passive cavity loss. The conditionsthat ensure stability of the demultiplexing state tosmall perturbations, and the instability of all othersteady-state conditions, have yet to be determined.

We have experimentally demonstrated the circuitdescribed above as shown in Fig. 3. The outputfrom an Ar-ion laser operating at 514.5 nm is splitinto two and then frequency shifted with acousto-optic cells to generate two beams with a frequencyseparation of 280 MHz. The beams are thencoupled into a 1.5-m-long step-index, 100-/Lm-core,multimode fiber. By arranging each beam to have adifferent angle of incidence on the fiber end face, weobtain uncorrelated output speckle patterns, evenwith a relatively short fiber.9 The output of thefiber is then used to pump the photorefractive reso-nators. By modulating the acousto-optic cells withlow-frequency square waves, the carrier content in

100-gmmulti,,wdef

Dzet 4

Moodulator Y em_

- ,,,odl=2.7/:1Hz a@ C 3x,,K~d217.8 8Hz Bar~sr[ R~ea~l.7 kffea = etg+ IJ4OMHz\ vL (2=sgo- 140 MHz s~A

Pump ~ ~ iru

+ | J BalaO~t uEodl= 2.7 kHz

100-0 v,-J7.8kHztublimodefiber

Fig. 3. Experimental demonstration of a self-organizingfrequency demultiplexer. The circuit configuration isequivalent to that shown in Fig. 1. Each resonator ringhas a length of -2 m, of which 1 m is 100-jum-core multi-mode fiber. The two resonator rings are positioned ontop of each other with the modes crossing in the pumpcrystal and occupying separate volumes in the reflexivegain crystal. The pump crystal is 450 -cut BaTiO3, and thereflexive gain crystal is 00-cut BaTiO3. The resonatorparameters are exp(Gp) = 1700, exp(GR) = 230, 1 - L =0.01, and a = 0.96.

December 15, 1991 / Vol. 16, No. 24 / OPTICS LETTERS 1995

Ring

I

1.t2 b5

Ring2

It 7

:Z I-~~~~~

I I I I I - I I

_ /_I I _ _ _ _tI, ls

Fig. 4. Transient behavior of the circuit. The curvesrepresent the envelopes of the square waves driving theacousto-optic modulators. Signal I, on carrier a), oscil-lates in resonator ring 1, and signal I2 on carrier 0 2 oscil-lates in resonator ring 2.

.O

4 b

Ring1

Ring2

TrM n i n 11iA 11 M ii 11111 0 )101 2I ( 111

U EC U Ej L X U ULI

Fig. 5. Outputs of the two resonator rings. The signalcarrier frequencies are separated by 280 MHz.

each ring can be monitored by detecting the outputintensities with low-speed photodiodes.

When the input signals are presented to the cir-cuit it takes a few seconds for the self-organizationprocess to reach steady state. The mapping of theinput signals into resonator modes is initially noisedriven, so which signal will oscillate in which ring israndom. We observe the dynamical evolution by de-tecting the envelopes of the carrier modulations, asshown in Fig. 4. With a total pump intensity of10 mW, focused to a 0.58-mm-diameter spot in thepump crystal, the circuit takes approximately 10 s toreach the high-contrast-ratio demultiplexing stateshown in Fig. 5. This operating state is then stablefor as long as the pump beam is present. The totalsteady-state oscillating intensity in the two ringsreached 0.4% of the pump intensity. Measurementswith a spectrum analyzer show an output contrastratio of better than 40:1 in each ring. This is con-sistent with our previous holographic measurementsof the orthogonality of speckle patterns transmittedover a short length of fiber,9 where we found best-case cross-talk levels of 50:1. In order for thecircuit to demultiplex the input signals, their inten-sities must be matched to within approximately

10%, otherwise the stronger signal will oscillate inboth rings. We are currently investigating alterna-tive designs that may relax this constraint.

This circuit has potential application to opticalcommunications systems. We showed recently thatthe spatial-mode superposition can be used as amultiplex parameter to transmit multiple communi-cation channels on a single multimode fiber.9 Inthat case it was necessary to record, and periodi-cally refresh, the holograms that demultiplexed thechannels. The circuit described here demultiplexesspatially distinct channels while continuously adapt-ing to slow variations in the spatial-mode super-position owing to drifts in the carrier frequency orperturbations to the fiber. The circuit can also beused with channels that have been transmitted on asingle-mode fiber. Taking the output of the singlemode fiber and sending it through a sufficientlylong length of multimode fiber, before pumping theself-organizing circuit, will impress a distinct spatialmode on each channel.'0

In the parlance of neural networks this circuit de-multiplexes through a self-organizing process. Thecircuit learns to associate distinct input signals withdifferent resonator modes. The essential charac-teristics of the input signals that allow them to beseparated are their spatial and temporal orthogonal-ity. We have used acousto-optic modulators to gen-erate well-defined input signals. In principle anytype of transduction mechanism that provides dis-tinct input frequencies with spatially distinct repre-sentations could be used with this circuit.

We are grateful for the support of the U.S. AirForce Office of Scientific Research through contract90-0198. Mark Saffman acknowledges supportprovided by a U.S. Air Force Office of ScientificResearch laboratory graduate fellowship.

References

1. T. Kohonen, Self-Organization and AssociativeMemory, 2nd ed. (Springer-Verlag, Berlin, 1989).

2. D. Z. Anderson, C. Benkert, B. Chorbajian, and A.Hermanns, Opt. Lett. 16, 250 (1991).

3. C. Benkert and D. Z. Anderson, Phys. Rev. A 44, 4633(1991).

4. D. Z. Anderson and R. Saxena, J. Opt. Soc. Am. B 4,164 (1987).

5. N. V Kukhtarev, V B. Markov, S. G. Odulov, M. S.Soskin, and V L. Vinetskii, Ferroelectrics 22, 949(1979).

6. N. V Kukhtarev, V B. Markov, S. G. Odulov, M. S.Soskin, and V L. Vinetskii, Ferroelectrics 22, 961(1979).

7. Interestingly, when the pump consists of a single fre-quency the influence of the reflexive gain interactionon the dynamical evolution of the circuit leads to aflip-flop behavior between the rings, equivalent tothat reported in Ref. 2.

8. J. 0. White, M. Cronin-Golomb, B. Fischer, and A.Yariv, Appl. Phys. Lett. 40, 450 (1982).

9. M. Saffman and D. Z. Anderson, Opt. Lett. 16, 300(1991).

10. M. Koga and T. Matsumoto, IEEE Photon. Technol.Lett. 2, 487 (1990).