wave particle duality considerations in optical computing

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Applied Optics Letters to the Editor Letters to the Editors should be addressed to the Editor, APPLIED OPTICS, Georgia Institute of Technology, Atlanta, GA 30332-0252. If authors will state in their covering communications whether they expect their institutions to pay the publication charge, publication time should be shortened (for those who do). Wave particle duality considerations in optical computing H. John Caulfield and Joseph Shamir University of Alabama in Huntsville, Center for Applied Optics, Huntsville, Alabama 35899. Received 3 October 1988. 0003-6935/89/122184-03$02.00/0. © 1989 Optical Society of America. The wave particle duality inherent in the propagation of light or particles can be exploited for energy efficient com- puting leading to energy requirement per calculation below kT. Although several reversible computers with similar characteristics were proposed in the past, only optical im- plementations can be made with the present technology. Present day digital computers are based on the propaga- tion of signals through a sequence of logic gates. The proper operation of each such logic gate requires a certain amount of energy with a thermodynamically determined lower bound 1 of kT, where k is the Boltzmann constant and T is the operating temperature of the device. Currently available computers operate far above this limit, practical values being around 10 4 kT. This state of the art did not change with the introduction of optical logic gates, which also need a similar amount of operating energy. Theoretically, the thermodynamic energy limit has been removed by the introduction of reversible logic gates 2 ; how- ever, even their optical implementation 3,4 needs quite an appreciable amount of switching energy, far more than kT. Fundamental investigations into reversible computing re- sulted in the concept of quantum computers 5-9 and an indi- cation that the thermodynamic limit may be relaxed so that it applies only to the detection or decision stage. In the quantum computer terminology one would speak about observable and unobservable variables. Most quan- tum mechanical restrictions apply only to the observables and come into play when a measurement or observation is performed. 7 From this point of view, we consider a quantum mechanical system prepared with certain initial conditions (the inputs), which is described by a wave function having many states that are inaccessible for an observation. To perform a measurement of an observable one may have to detect a particle that is localized in a detector of finite size at a given time. The event of detection eliminates the proba- bility for the detection of any other state. Although nature's computers (such as the brain) may ap- proach the performance of a quantum computer, technology is very far from implementing a similar artificial device. Nevertheless, several aspects of such computers may be real- ized by optical means. Simple examples of such systems are the coherent Fourier correlator 10 and the free space intercon- nection network. 11 In general, we may describe a wave particle (WP) comput- er by one or several coherent light sources that illuminate an optical system containing the input variables and a detector array that records the outputs. We assume that a detector can make a decision after recording m photons, where m must be fairly large to reduce the uncertainty due to statisti- cal fluctuations. With this assumption, the energy require- ment for a single decision is where h is Planck's constant and v is the frequency of the illuminating light. Focusing on a single source and single detector we may define an interconnectivity w, which is the number of wires that would have been used for making all the parallel elec- tronic interconnections that are implemented by diffraction and propagation of a coherent wave from the source to the detector. Since we deal with weighted interconnections that must be evaluated, we may deduce that the interconnectivity w is at least of the order of the number of calculations performed toward the derivation of each decision. Thus a conservative estimate for the energy requirement per calcu- lation for our WP computer is given by the relation To compare the energy performance of the WP computer to that of a digital computer we have to recall that in the latter each calculation is related to a decision (such as the output of a logic gate), and thus the energy for each calcula- tion is bounded by the thermodynamic limit 1 leading to a minimum energy requirement for the whole process given by In the last two equations we did not take into account the fact that presently available digital computers actually require ~10 4 kT of energy per operation and also assumed that the energy of a single electron is much below kT so that this quantum of energy contains a large number of electrons that take care of statistical fluctuations. To compare the energy performance of the two classes of computers we assume room temperature and visible light, yielding a photon energy For the whole computing process the WP needs an amount of energy given by [see Eq. (1)] Dividing Eq. (3) by Eq. (2) with the consideration of Eq. (5) and likewise Eq. (4) by Eq. (1), we obtain the two correspond- ing relations, 2184 APPLIED OPTICS / Vol. 28, No. 12 / 15 June 1989

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Page 1: Wave particle duality considerations in optical computing

Applied Optics Letters to the Editor

Letters to the Editors should be addressed to the Editor, APPLIED OPTICS, Georgia Institute of Technology, Atlanta, GA 30332-0252. If authors will state in their

covering communications whether they expect their institutions to pay the publication charge, publication time should be shortened (for those who do).

Wave particle duality considerations in optical computing H. John Caulfield and Joseph Shamir

University of Alabama in Huntsville, Center for Applied Optics, Huntsville, Alabama 35899. Received 3 October 1988. 0003-6935/89/122184-03$02.00/0. © 1989 Optical Society of America.

The wave particle duality inherent in the propagation of light or particles can be exploited for energy efficient com­puting leading to energy requirement per calculation below kT. Although several reversible computers with similar characteristics were proposed in the past, only optical im­plementations can be made with the present technology.

Present day digital computers are based on the propaga­tion of signals through a sequence of logic gates. The proper operation of each such logic gate requires a certain amount of energy with a thermodynamically determined lower bound1

of kT, where k is the Boltzmann constant and T is the operating temperature of the device. Currently available computers operate far above this limit, practical values being around 104 kT. This state of the art did not change with the introduction of optical logic gates, which also need a similar amount of operating energy.

Theoretically, the thermodynamic energy limit has been removed by the introduction of reversible logic gates2; how­ever, even their optical implementation3,4 needs quite an appreciable amount of switching energy, far more than kT.

Fundamental investigations into reversible computing re­sulted in the concept of quantum computers5-9 and an indi­cation that the thermodynamic limit may be relaxed so that it applies only to the detection or decision stage.

In the quantum computer terminology one would speak about observable and unobservable variables. Most quan­tum mechanical restrictions apply only to the observables and come into play when a measurement or observation is performed.7 From this point of view, we consider a quantum mechanical system prepared with certain initial conditions (the inputs), which is described by a wave function having many states that are inaccessible for an observation. To perform a measurement of an observable one may have to detect a particle that is localized in a detector of finite size at a given time. The event of detection eliminates the proba­bility for the detection of any other state.

Although nature's computers (such as the brain) may ap­proach the performance of a quantum computer, technology is very far from implementing a similar artificial device. Nevertheless, several aspects of such computers may be real­ized by optical means. Simple examples of such systems are the coherent Fourier correlator10 and the free space intercon­nection network.11

In general, we may describe a wave particle (WP) comput­er by one or several coherent light sources that illuminate an

optical system containing the input variables and a detector array that records the outputs. We assume that a detector can make a decision after recording m photons, where m must be fairly large to reduce the uncertainty due to statisti­cal fluctuations. With this assumption, the energy require­ment for a single decision is

where h is Planck's constant and v is the frequency of the illuminating light.

Focusing on a single source and single detector we may define an interconnectivity w, which is the number of wires that would have been used for making all the parallel elec­tronic interconnections that are implemented by diffraction and propagation of a coherent wave from the source to the detector. Since we deal with weighted interconnections that must be evaluated, we may deduce that the interconnectivity w is at least of the order of the number of calculations performed toward the derivation of each decision. Thus a conservative estimate for the energy requirement per calcu­lation for our WP computer is given by the relation

To compare the energy performance of the WP computer to that of a digital computer we have to recall that in the latter each calculation is related to a decision (such as the output of a logic gate), and thus the energy for each calcula­tion is bounded by the thermodynamic limit1

leading to a minimum energy requirement for the whole process given by

In the last two equations we did not take into account the fact that presently available digital computers actually require ~10 4 kT of energy per operation and also assumed that the energy of a single electron is much below kT so that this quantum of energy contains a large number of electrons that take care of statistical fluctuations.

To compare the energy performance of the two classes of computers we assume room temperature and visible light, yielding a photon energy

For the whole computing process the WP needs an amount of energy given by [see Eq. (1)]

Dividing Eq. (3) by Eq. (2) with the consideration of Eq. (5) and likewise Eq. (4) by Eq. (1), we obtain the two correspond­ing relations,

2184 APPLIED OPTICS / Vol. 28, No. 12 / 15 June 1989

Page 2: Wave particle duality considerations in optical computing

Fig. 1. Coherent Fourier correlator: I, F, and D are the input, filter, and detector planes, respectively, each with N × N pixels. The Fourier transforming lenses are L situated at a focal distance

from each plane.

This relation indicates that in principle the WP computer has an energetic advantage over the digital computer for relatively large parallel interconnectivity (w > 100m). We should note, however, that the present state of the art puts this comparison in a much more favorable position for the WP computers since these already operate near their theo­retical limits while digital computers are still ~ 4 orders of magnitude away.

For our first example we consider a coherent Fourier corre­lator with an input plane composed of N X N pixels and a filter plane having the same number of pixels (see Fig. 1). The input function diffracts the illuminating light so that each pixel in the input plane is connected to each pixel of the filter plane as indicated by the two pixels (1) and (2) in the figure. Since each of the N2 input pixels are connected to each of the N2 filter pixels where a multiplication is per­formed, interconnectivity in the first part of the system is w' = N4. In the second half of the system, each of the N2 pixels in the filter diffracts light toward each detector adding N2

interconnections. Thus the interconnectivity required by each detector toward the derivation of a decision is w = N4 + N2.

As an application example of this correlator we attempt to distinguish between two objects, A and B, by observing the output of a single detector element positioned at the origin of the detector plane D. We prepare a spatial filter matched to object A and illuminate the system by a constant flux of photons originating in a coherent laser. After calibration we determine a time interval δt and consider object B to be identified if the number of counts in the detector m during that time interval is m < M, where M is a large number, say 1000, and consider object A to be identified if m > M. This experiment means that we are able to make a decision by detecting m photons, each of which performed the w calcula­tions given above. To derive the same decision by using a conventional digital computer we are forced to make an intermediary decision (with the help of a logic gate), at least once, for each of the w calculations before proceeding to the output plane.

In the above example our WP computer exploited the characteristics of an analog computer based on wave propa­gation with the particle aspects of light manifested only at the decision stage. The performance of this WP computer may be compared with the digital computer that operates exclusively on particles at every stage of the computation satisfying relations (3) and (4). Substitution of the present value of w into relations (2) and (4) with (5) yields

retaining relations (1) and (3) in their original form. Assuming an array of N = 1000 and taking m to be of the

order of M = 1000 we may neglect the N2 term compared with the AT4 term and obtain

to be compared with

The above considerations may lead to the incorrect im­pression that photonic processors based on the wave particle duality have unlimited capacity with the finite expenditure of energy. This impression is naturally false. In our experi­ment we arrived at a single decision by detecting m photons. Thus in principle we extracted just a single bit of information with the appropriate expenditure of energy. To arrive at a similar single bit decision using a conventional digital com­puter, we need full precision computations for all the w intermediary calculations. Assuming binary weights for the interconnections, the result of such a calculation is an output with a value which may vary through a dynamic range of w that is not needed for a binary decision. However, if we wish to determine the exact value of this output we must have at least w decision levels at the detector for the digital computer as well as for the WP computer. The implication for the latter is that instead of m photons we must detect wm pho­tons giving up the energetic advantage. The difference be­tween the two computing processes is that with the digital computer we always have to perform a full precision compu­tation while with the WP computer the degree of precision is our free choice, and we do not have to waste unnecessary computation power.

In addition to the above reservations several limitations, some of which originate from fundamental principles,13-14

must be observed. Furthermore, not all analog computers are WP computers, and not all WP computers actually pos­sess the energy advantage. In the following we discuss some of the limitations based on free-space optical interconnec­tion networks (or a matrix-matrix multipler) introduced in Ref. 11 and further analyzed in Refs. 12 and 13. The basic configuration of an interconnection network is shown in Fig. 2, where the two lenses, L1 and L2, image a hologram array H onto the detector plane D. Each of the N × N holograms in the array directs a weighted amount of light to each of the N × N pixels of an input matrix presented on the SLM sand­wiched between the two lenses. The process is completed after integration onto the detector array D. Since the illumi­nating beam R is a coherent laser beam, in principle, each photon performs the complete process and its particle aspect is manifested only at the detector plane. Since plane D is the

Fig. 2. Architecture for a coherent holographic interconnection network: R, coherent reference beam; H, hologram array; SLM,

input spatial light modulator; L, lenses; D, detector array.

15 June 1989 / Vol. 28, No. 12 / APPLIED OPTICS 2185

Page 3: Wave particle duality considerations in optical computing

Fig. 3. Architecture for partially incoherent interconnection net­work. LAD, laser diode array; H, hologram array; D, detector array.

conjugate of plane H, there is a one to one correspondence between the holograms in the array and detector pixels. Each hologram produces a fan-out of N2 followed by a similar fan-in; thus, although there are N4 parallel interconnections, each detector makes a decision by detecting photons, each of which performed only w = N2 calculations. Compared to the Fourier correlator the energy benefit is substantially re­duced, but it still exists for large arrays. Returning to our previous numbers (N = 1000, m = 1000), we obtain for this case

The coherent wave nature is necessary for the energy benefits discussed here as indicated by the modified archi­tecture of Fig. 3, where the coherent illumination was re­placed by a laser diode array (LDA), and the imaging config­uration is replaced by free space diffraction. From the interconnection point of view13 this architecture operates in a similar way to that of Fig. 2 when the input vector is introduced as an intensity modulation of the lasers in the array. However, since the lasers are mutually incoherent, at least one photon is required from each laser to be incident on each detector, thus reducing the interconnectivity to w = 1. With this architecture one, therefore, cannot expect any energy advantage over a digital computer, although the total number of actual interconnections is still N4. For large N this is impractical by any other means where physical leads must be installed. The WP advantage can be regained by replacing the LDA with a coherently illuminated SLM.

It should be noted that the same laser diode array replac­ing the uniform illumination in the architecture of Fig. 2 will not change the results obtained for that architecture due to the limited spatial coherence requirements when the imag­ing condition is met.

In addition to spatial coherence, temporal coherence must be also considered. Observation of Fig. 2 shows that the distance from one hologram to the various pixels of the SLM depends on position, resulting in a time skew.14 For a coher­ent processor this means that the coherence time of the source must exceed this time skew, leading to a limitation on the size of the array that can be processed. Also, a longer coherence time means a larger uncertainty in the detection time of each photon. This, of course, is not a limitation, since the photon flux may be increased for compensation. In principle, this photon flux has no fundamental limitations since photons are bosons, in contrast to electrons that must obey the Pauli exclusion principle for fermions.

In conclusion: We have shown with the help of specific examples that a combination of analog computation based on wave propagation with digital decision based on particle detection may have a substantial energy benefit over digital (particle) computers. The intrinsic penalty implied by this

2186 APPLIED OPTICS / Vol. 28, No. 12 / 15 June 1989

benefit is a reduction in accuracy. Since for many applica­tions full range digital accuracies are not required, this novel approach for computing may prove very useful. A more rigorous derivation of the fundamental characteristics of a WP computer, as well as its noise performance, is under study.

Joseph Shamir also works in the Electrical Engineering Department of the Technion.

References 1. R. Landauer, "Irreversibility and Heat Generation in the Com­

puting Process," IBM J. Res. 5, 183-000 (1961). 2. E. Fredkin and T. Toffoli, "Conservative Logic," Int. J. Theor.

Phys. 21, 219-253 (1982). 3. J. Shamir, H. J. Caulfield, W. Miceli, and R. J. Seymor, "Optical

Computing and the Fredkin Gate," Appl. Opt. 25, 1604-1607 (1986).

4. R. Cuykendall, "Three-Port Reversible Logic," Appl. Opt. 27, 1772-1779 (1988).

5. R. P. Feynman, "Quantum Mechanical Computers," Opt. News 11, No. 2, 11-20 (1985); Found. Phys. 16, 507-531 (1986).

6. C. H. Bennett and R. Landauer, "The Fundamental Physical Limits of Computation," Sci. Am. 253, No. 1, 48-56 (1985).

7. D. Deutsch, "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer," Proc. R. Soc. London Ser. A 400, 97-117 (1985).

8. A. Peres, "Reversible Logic and Quantum Computers," Phys. Rev. A 32, 3266-3276 (1985).

9. R. Landauer, "Computation and Physics: Wheeler's Meaning Circuit?," Found. Phys. 16, 551-564 (1986).

10. A. VanderLugt, "Signal Detection by Complex Spatial Filter­ing," IEEE Trans. Inf. Theory IT-10, 139-000 (1964).

11. H. J. Caulfield, "Parallel N4 Weighted Optical Interconnec­tions," Appl. Opt. 26, 4039-4040 (1987).

12. J. Shamir, H. J. Caulfield, and M. M. Mirsalehi, "Improved Architectures for Massive Holographic Interconnection Networks," Proc. Soc. Photo-Opt. Instrum. Eng. 963, 283-287 (1988).

13. J. Shamir, H. J. Caulfield, and R. B. Johnson, "Massive Holo­graphic Interconnections and Their Limitations," Appl. Opt. 28, 311-324 (1989).

14. J. Shamir, "Fundamental Speed Limitations on Parallel Proces­sing," Appl. Opt. 26, 1567 (1987).