wave particle duality considerations in optical computing
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Applied Optics Letters to the Editor
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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,
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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.
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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
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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.
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