Modeling and simulation methodologyfor digital optical computing systems
Ahmed Louri and Jongwhoa Na
A modeling and simulation methodology for digital optical computing systems is introduced in thispaper. The methodology predicts maximum performance of a given optical computing architecture andevaluates its feasibility. As an application example, we apply this methodology to evaluate the feasibilityand performance of the optical content-addressable parallel processor proposed in Appl. Opt. 31, 3241(1992). The approach consists of two major phases. The first phase involves analytical studies of theeffects of design parameters such as cross talk, diffraction-limited beam spot diameter, and pitch onsystem performance parameters such as signal packing density and skew time. In the second phase, asimulation model and a simulator are introduced by the use of GLAD (General Laser Analysis and Design,an optical software package developed by Applied Optics Research) to evaluate the combined effects ofbit-error rate, bit rate, optical power efficiency, available source power, and signal contrast on theperformance parameters such as signal packing density, misalignment tolerance, and distance betweendevices. The methodology presented here investigates the model, not on a component-by-componentbasis, but as a whole, which produces a more realistic representation of the actual laboratoryprototype. The proposed methodology is intended to reduce the optical computing system design time aswell as the design risk associated with building a prototype system.
Key words: Digital optical computing, diffraction analysis, modeling and simulation, optical content-addressable parallel processor.
1. IntroductionIn recent years, several optical computing architec-tures and systems have been proposed.1 -6 Thesesystems are designed to exploit the advantages ofoptics such as noninterference between signals, inher-ent parallelism, and high spatial and temporal band-width. Although some of the proposed systems pre-sent results of laboratory prototypes and some reportresults based on first-order analysis, the systematicor automated modeling and simulation methodolo-gies have not yet been presented. Without the aid ofa general-purpose simulation model, the developmentperiods from an initial concept of an actual prototypehave been too long, and the accompanying costs havebeen too high. One study showed that the concep-tual design, engineering design, fabrication, and test-ing of an optical system typically takes 3-5 years.7
Moreover, problems arise when the target system
The authors are with the Department of Electrical and Com-puter Engineering, University of Arizona, Tucson, Arizona 85721.
Received 20 May 1993; revised manuscript received 20 Septem-ber 1993.
becomes so complex that there are simply too manyparameters to be considered. As the optical comput-ing and networking systems gain popularity, futuresystems will become increasingly complicated.Therefore it is necessary to have automated opticalsystem design and analysis tools.
In optical system research fields other than opticalcomputing, the importance of these automated toolsis already recognized. For example, an optical diskstorage system that utilizes a laser diode head and anoptical disk has been modeled and simulated.8 Anintegrated design tool called SCOPE (supercompactoptoelectronic simulator) has been proposed9 for mi-crowave optoelectronic systems that handle laserdiodes, light-emitting diodes, and photodetectors.In the optical interconnection network field, severalresearchers have reported the modeling and simula-tion study of optical interconnects.10 11 For opticalcomputing systems, up until now, there have been nomodeling and simulation tools for verifying the properfunctionality of an optical computing system as wellas its physical realizability. We should note thatsome efforts have been made to design computer-aided designs (CAD's) for optical computing sys-tems,12 13 but these efforts have been limited to onlythe functional aspects of systems.
In this paper we propose a modeling and simulationmethodology for digital optical computing systemsthat not only evaluates the feasibility of the systembut also tests its functionality and predicts its perfor-mance. As an application example, we apply theproposed methodology to the optical content-address-able parallel processor'4 (OCAPP). The approachconsists of two major phases. In the first phase,analytical studies are performed to investigate theeffects of design parameters such as cross talk (includ-ing power dissipation and noise), diffraction-limitedbeam spot diameter, and pitch on signal packingdensity, skew time (execution time), and systemvolume. In the second phase a simulation model anda simulator are introduced to evaluate the combinedeffects of bit-error rate (BER), bit rate (BR), opticalpower efficiency, and available source power on theperformance parameters such as maximum signalpacking density, misalignment tolerance, and maxi-mum distance between devices. The simulator isdesigned by the use of GLAD (General Laser Analysisand Design, an optical simulation software packagedeveloped by Applied Optics Researchl 5). GLAD per-mits detailed modeling of each system component ofsystems such as spatial light modulators (SLM's) inaddition to simulating the propagation of an opticalwave front passing through them. The proposedapproach enables a more complete evaluation of theconceptual design, which will eventually result infaster prototype development.
The rest of this paper is organized as follows:Section 2 presents a brief description of the OCAPPand GLAD. Section 3 proposed a modeling and simu-lation methodology for the OCAPP. Section 4 char-acterizes the performance of the diffraction-limitedOCAPP. Section 5 describes how GLAD is used tosimulate the OCAPP and the values of the param-eters considered. Section 6 summarizes the simula-tion work, and Section 7 concludes the paper.
2. Background
A. Modeling and Simulation
In this subsection we briefly discuss the underlyingconcepts of modeling and simulation. Modeling andsimulation of a system is a technique that acts as abridge between a conceptual design and a laboratoryprototype. In an abstract sense, modeling meanscollecting all possible information about a system.This information collection process may be accom-plished either by coding the structure and the behav-ior of the optical system by the use of conventionalprogramming languages such as C or PASCAL or by theuse of optical system analysis software packages suchas CODE V, OSLO, or GLAD. 16'17
Simulation may be defined as an experiment per-formed on a model. In computer system simulation,two aspects can be noted, namely the functional andthe physical aspects. The simulation of functional-ity means verifying the functions or algorithms thatare going to be performed on the system. It istypically done by using conventional programming
Fig. 1. Architecture of the OCAPP.
languages or simulation packages such as DEVS (Dis-crete EVent-based Simulation) or SIMSCRIPT.18 ,1 9
The simulation of the physical aspect of a computersystem means verifying the physical realizability ofthe conceptual design. This can be done by simulat-ing each component of the computer system by theuse of an optics simulation or a ray-tracing softwarepackage.
B. Optical Content-Addressable Parallel ProcessorIn Ref. 14, a parallel architecture called the OCAPPhas been introduced for the fast and efficient imple-mentation of symbolic computing tasks such as search-ing, sorting, information retrieval and database-knowledge-base processing. Figure 1 is a schematicdiagram of the OCAPP. The architecture is com-posed of a selection unit, a match/compare unit, aresponse unit, an output unit, and a control unit, thewords stored in the storage array. A detailed expla-nation and implementation of each unit of the OCAPPand the algorithms implemented on the OCAPP arepresented in Ref. 14.
This architecture is under construction in theOptical Computing and Parallel Processing Labora-tory at the University of Arizona. A laboratorysetup is shown in Fig. 2. The optical system iscomposed of three SLM's, a beam splitter, sphericaland cylindrical lens elements, spatial filtering assem-blies, mirrors, and two linear CCD (charge-coupleddevice) arrays. The OCAPP uses a collimated laser
beam as an input source and two linear CCD arrays asthe output detection unit. The operations of thematch/compare unit are performed by SLM1 andSLM2. SLM1 and SLM2 are used to hold two wordsor two bit slices to be matched or compared withrespect to each other, depending on the algorithmemployed. The selection unit is mapped into SLM3,which is used to enable/disable words and/or bit-slices of the two-dimensional (2-D) optical data arrayfrom SLM2. The response unit is omitted in thislayout because the first version of the OCAPP isconfigured as a relational database machine, whichdoes not use ordering between the matched words.
The optical layout for OCAPP architecture is fur-ther simplified into a simulation model, shown in Fig.3. This model is more suitable for the simulationstudy without loss of functionality of the originalOCAPP described in Ref. 14. The simulation modelof Fig. 3 constitutes the major optical path of thesystem of Fig. 2 that consumes most of the power.For clarity, the spatial filtering assembly and mirrorsare not considered in the simulation model as theycontribute little power loss in our application. Thismodified OCAPP model is studied with the simula-tion methodology described in Section 3.
C. General Laser Analysis and DesignThere are two types of commercial software packagefor the analysis and design of optical systems. Oneis a geometric code (such as CODE V16 or OSLO17 that isbased on ray-tracing optics, and the other is a physicaloptics code (such as GLAD) 20 that is based on diffrac-tion propagation of wave fronts. Although geomet-ric codes may be useful in analyzing the given systemto some extent, the physical optics code is able toprovide a more accurate and powerful tool by utilizingfast Fourier transforms.21 The physical optics codeprovides detailed beam intensity and phase profiles,whereas the geometric code is limited to providingsimple intensity profiles such as a constant or Gauss-ian profile.21 Moreover, the geometric code limitsthe diffraction propagation to strictly near field or farfield, whereas the physical code can handle any kindof diffraction propagation. For the above reasons,we chose to use GLAD for our purpose.
A1A -- I \J . l 1-DSLM3 Cylindrical detector
Lens arrayFig. 3. Simplified model of the OCAPP.
3. Modeling and Simulation of the OpticalContent-Addressable Parallel ProcessorIn this section, a two-phase modeling and simulationmethodology for digital optical computing systems isproposed, and the simulation results are explained.The main objectives of the methodology are findingmaximum values of performance parameters of agiven optical computing system as well as providing alaboratory prototype model for fast prototype develop-ment. Performance parameters considered here in-clude signal packing density, misalignment toler-ances, distance between components, power efficiency,and skew time.22 23 Maximum values can be foundby manipulating cross talk, BER, BR, and opticalpower efficiency of a given system. During the firstphase, a theoretical analysis of the system is per-formed. During the second phase, GLAD is used for adetailed simulation and evaluation of the system.In what follows, we describe each phase in detail.
A. Phase 1: Diffraction Analysis of the OpticalContent-Addressable Parallel ProcessorIn the first phase, a preliminary analysis is performedto narrow down the range of values of parametersused in the simulation. The analysis provides upperbounds of performance parameters such as diffraction-limited signal packing density, skew time, and crosstalk. Knowing these bounds would enable one toavoid unnecessary simulation experiments and tohave a better understanding of the overall simulationwork. Another point to note is that some param-eters identified at the preliminary analysis phase canbe used in the second phase. For example, skewtime, which is estimated in the first phase, is used inthe calculation of the BR that is used in evaluatingthe required optical input power. A summary ofparameters and definitions used in this paper is givenin Table 1.
In the first phase, the pitch, signal power, noisepower, cross talk, and diffraction-limited beam spotdiameter are used as design parameters that can bemanipulated whereas the signal packing density isused as a performance parameter. First, the diffrac-tion-limited signal packing density is calculated byobtaining the minimum pitch for a given system crosstalk. The cross talk is expressed in terms of thepitch. This is possible because the cross talk is theratio of the noise power to the signal power, and thenoise power can be expressed in terms of pitch. Thenoise power is obtained by integrating the outputintensity distribution over the neighboring detectorapertures. The neighboring detector aperture canbe expressed in terms of the diffraction-limited beamspot diameter and pitch. The skew time is obtainedby calculating the difference between the maximumand the minimum optical path lengths. The skewtime is then used to estimate the cycle time andmaximum bit rate of the OCAPP. Finally, the vol-ume of the OCAPP and optical power dissipation-limited signal packing density are calculated. In
a Length of a pixel of a SLMdD Diffraction-limited beam spot diametern Number of pixels per row (or column)p Pitch: center-to-center distance between two
adjacent pixelsp Signal packing density: number of pixels per
1 cm2
,qt System optical power efficiencyd Distance between SLM'sf Cylindrical lens focal lengthX Optical wavelengthI Length of the SLML Length of the systemN System fanoutNf Fresnel numberTskew Skew time: propagation time difference from
input to output among the various opticalpaths
V Volume of the systemBER Bit error rateBR Bit rateX Cross talk: ratio of noise power to signal powerPi. Required optical power per beamPsi gnal Collected optical power at the designated detector
elementPnoise Collected optical power at the detectors other
than the designated detector elementQ Ratio of the rms signal voltage to the total rms
cross-talk voltager Ratio of current to the detector in the OFF state
over the ON stateUP Power dissipation density (in watts per square
centimeter)
maximum number of pixels in the optical data plane.In order to determine the maximum signal packingdensity PM, the individual pixels must be packed astightly as possible. Therefore PM is obtained byfinding the minimum pitch pm of the 2-D array.
The pitch p can be related to the cross talk Xcalculation because the cross-talk calculation re-quires the evaluation of the collected noise powerPnoise, which uses p as an integration parameter. Inother words, to calculate Pnoise, the intensity distribu-tion must be integrated over the neighboring detectoraperture, which has a diameter of dD, and separatedfrom the designated detector aperture by multiples ofp. 2 4
,25 Therefore, by setting X to some value, we can
calculate Pm of the array. Once pm is known, we candirectly calculate PM and the maximum number ofpixels in the array.
To calculate X, we calculate the field distribution atthe output plane u2(x, y) of a pixel located at thecenter of the input plane (SLM1 of Fig. 3) for a giveninput field distribution ul(x, y). As we have a colli-mated laser beam as a source, u1 (x, y) can be approxi-mated as a normally incident unit amplitude planewave. Assuming a square aperture for the SLMpixel, the field distribution immediately after thesquare pixel of dimension a is given by
ul(x,y) = rect(x/a,y/a) = rect(x/a)rect(y/a). (3)
Because the rect function is separable and the powerof the lens exists only along the y axis, the outputdistribution at they axis will be a Fraunhofer diffrac-tion pattern that can be expressed as
exp(jkz)exp(jkx 2/2z) a ayU2(Y) = j- a sct - -what follows, the parameters in Phase 1 are calcu-
lated based on the architecture shown in Fig. 3.
1. Diffraction-Limited Beam Spot DiameterThe diffraction-limited beam spot diameter dD for agiven system configuration is calculated here. dDwill be used below for the optical signal power calcula-tion. In order to check the extreme case, we assumethat the SLM's, beam splitter, and cylindrical lens arein contact. In the case of square input aperture, dDis given by
dD X (1)a
where a is the length of a pixel of the SLM, f is thefocal length of the cylindrical lens, and X is the laserwavelength (refer to Fig. 3). For simplicity, weassume that the lengths of the pixel on the SLM andthat of a detector have the same value, which is a.Then dD becomes
dD = a = Xf. (2)
2. Diffraction-Limited Signal Packing DensityThe signal packing density p is one of the mostimportant performance parameters as it limits the
(4)
The output field distribution u2(x) along the x axiswill be a Fresnel diffraction pattern as there is nofocal power in the x direction in the cylindrical lens:
exp(jkz) Ca/2
U12(X) - jXz J/2 exp[ - (X1 - x)2ldxl (5)
Now we check the Fresnel number Nf, which isdefined to be a2 /Xf, to study u2(x). For the followingestimation, we assume that we have = 633 nm andf = 0.1 m. For the given and f, with Eq. (2), abecomes 356 Rim. With the above data, Nf becomes- 2. This number implies that the diffraction pat-
tern of u2(x) will be neither a geometric projection ofaperture function nor the Fraunhofer diffractionpattern. Figure 4 shows the intensity distributionof the diffraction pattern of u2(x), which we calculatedby solving Fresnel integrals at the cylindrical lensfocal plane. Next we calculate X between channels.Figure 5 shows the geometry used in the signal andnoise power calculation. The parameter X can bedefined as
Fig. 4. Intensity distribution of the Fresnel diffraction pattern ofa square aperture of a SLM. The dimension of apixel is 356 tom x356 pm.
where Pjgnal is the power collected over the centerpixel (pixel A of Fig. 5) of the detector (assuming pixelA is the intended destination). The signal powercollected at pixel A is
IdD/2 dD/2
Psignal= J J I(x,y)dxdy. (7)-dD/2 -dD/2
On the other hand, Pnoise is the power collected bythe neighboring detector elements around the in-tended detector element. For simplicity, if we in-clude only two neighboring detector elements (pixel Band pixel C of Fig. 5) in our calculation, Pnoise is givenby
I Pnoise 2PnlX (8)
where P is the power obtained from the closestneighboring pixel and can be calculated as
(dD/2 rp+dD/2
Pnl = J I(x,y)dxdy,-dD/2 p-dD/2
-6.00
m -8.00
Y -10.00
i -12.00
U -14.00
-16.00400 500 600 700
Pitch (m)Fig. 6. Cross talk for various pitches of the SLM array in thediffraction-limited OCAPP. Diffraction-limited beam spot diam-eter is set to 356 pm.
From Fermat's principle, light takes the shortestpath between two points. As the OCAPP has a 3-Dstructure, there are inherent path-length differencesbetween pixels of the input and the output opticaldata planes. This path-length difference generates aclock skew problem that can affect the accuracy aswell as the operating speed of the optical computingsystem. This problem will be aggravated in systemsin which the output signals are designed to be fedback to the input stage. Therefore, to calculate theoperating speed of the OCAPP and avoid the aboveproblems, we must identify the skew time of thesystem and the longest signal path to satisfy thesynchronization requirement.2 6
In Fig. 3, the three SLM's perform imaging opera-tions. Assuming that the length of the OCAPP is L(from SLM1 to the detectors) and that is the lengthof an SLM, the time taken to travel the shortest pathof the system, if the switching time of the SLM's isignored, is given by
(9)Tmin = L/c. (11)
wherep represents the pitch between pixels. For Pn1calculation, I(x) (the Fresnel diffraction pattern) isintegrated over the integration interval p ± dD/2along the x axis at the cylindrical lens focal plane.
Figure 6 shows the calculated cross talk for variouspitches when dD is fixed at 356 [im, = 633 nm, andf = 0.1 m. Once the pitch is found, as shown in Fig.7, the signal packing density can be estimated withthe following relation25 :
p = 1/p 2 .
On the other hand, thelongest path is given by
T =
time taken to travel the
(L - f) + [(1/2)2 + f2]1/
2
C
Therefore the skew time Tkew is
T~kew = Tm. - Tmin =
(10)
[(1/2)2 + f2 ]1/2- f
C
t KI I I
I IX
._
r-
.,
uzp
Pixel CFig. 5. Model of the detector aperture used for estimating thecross talk.
800700600500400300 400 450 500 550 600
Pitch (m)
Fig. 7. Diffraction-limited estimation of signal packing density ofthe SLM versus the pitch.
It should be noted that 1 = np, where n is thenumber of pixels per row (or column) of the SLM, andp is the pitch. Substituting I = np into Eq. (13)yields
Tskew = 22c
(14)
It can be seen that the skew time grows linearlywith the number of pixels per dimension.
3. System VolumeThe volume of an optical system affects the ease ofpackaging as well as the feasibility of the system.As SLM's are connected by imaging, the length of theOCAPP L from Fig. 3 is given by
L = 2d + 2f, (15)
where d is the distance between two SLM's and f isthe focal length of the cylindrical lens. The systemvolume v is given by
Table 2. Summary of Parameters Studied in the Analysis Phase
Design Assumed CalculatedParameter Value Parameter Value
Cylindrical lens 10 cm Diffraction- 356 .mfocal length f limited spot
diameter dD
Length of SLM 1.6 cm Minimum pitch 410 pLmI Pm
System cross -10 dB Diffraction- 594 pixels/cm 2
talk X limited sig-nal packingdensity p
Wavelength X 633 nm Power dissipa- 333 pixels/cm 2
tion density-limited p
Power dissipa- 1 W/cm 2 Skew time 26.67 pstion density TskewJ1p Light propaga- 1.360 ns
tion timeTprop
Volume v 25.6 cm 3
u = L12= 2(d + f)12 = 2(d + f)(np)2. (16)
It can be seen that the volume is proportional to thesquare of the number of pixels per dimension.
4. Power Dissipation-Limited Signal PackingDensityNow we consider the effect of power dissipationdensity on the signal packing density. It is knownthat the maximum intensity of the beam is limited bythe maximum real power dissipation density up,which has a typical value of 1 W/cm2.25 27 Then themaximum allowable heat dissipation per input beam,Pcrit, iS
pPerit (17)
As shown by Ref. 25, it is p that limits signalpacking density more severely than diffraction effects.Even for low threshold lasers currently available, athreshold current of 1 mA is required for minimaloperation. Assuming that laser operation requires- 3 mW per beam, then, for oup = 1 W/cm 2, p becomes
333 pixels/cm2. For an SLM of 2 cm x 2 cm activearea, the maximum number of pixels available on theSLM becomes 1332 pixels. Table 2 summarizes theresults obtained from the analysis phase.
B. Phase 2: Modeling and Simulation of the OpticalContent-Addressable Parallel Processor by the Use of GLAD
The second phase consists of modeling and simulat-ing OCAPP. The main objective here is to provide arealistic evaluation of the system by providing thecombined effects of design parameters on performance.Specifically, we would like to determine the maxi-mum signal packing density, maximum misalign-ment tolerance, and maximum distance betweendevices for a given operating BR, BER, and optical
power efficiency. As the GLAD model can provide arealistic representation of the model and simulatediffraction propagation of wave fronts by using 220observation points of the model, it is expected togenerate the closest data to the prototype being built.
1. Modeling the Optical Content-AddressableParallel Processor by the Use of GLAD
GLAD employs a modular-building-block approach tomodel each component in sequence as the beampropagates through the system.15 The building-block approach permits a beam train of any configura-tion to be modeled by assembling blocks in the correctorder. To design a simulation model for OCAPP, wemust first model its components. Among the compo-nents of OCAPP, as shown in Fig. 3, the SLM is themost complicated component in the system. Tomodel an SLM, we generate a prototype mask of aspecific number of pixels, pixel size, pixel pitch, andphysical dimension. Simulations are performed forsystems that contain SLM's of varying signal packingdensities. In order to maintain consistency amongthese various OCAPP models, the aggregate SLMdimension is held at a constant 1.6 cm x 1.6 cm size.For example, one of the models was a pixelated SLMconsisting of 8 pixels x 8 pixels in a matrix configura-tion. This 8 x 8 SLM model determined the aggre-gate 1.6 cm x 1.6 cm dimension as the pixel pitch was0.2 cm (i.e., the pixel size and the interpixel gap areboth 0.1 cm). Then, for each specific bit pattern ofthe optical data plane of the SLM, the desired targetpattern is overlaid on the prototype mask pattern.GLAD contains many commands to model componentssuch as mirrors, lenses, apertures, etc. An initialfield distribution for the beam by the use of geometricdata such as the beam center, coordinates, waist size,and location can be defined with a command likeGAUSSIAN. Once the optical configuration and theinitial optical beam distribution are available, the
PROP (PROPagation) command is used to simulatediffraction propagation.
2. Simulation of the Optical Content-AddressableParallel Processor by the Use of GLAD
a. Signal Packing Density. The simulation algo-rithm is illustrated in Fig. 8. Part-(A) of Fig. 8describes the procedure for the maximum signalpacking density PM. The maximum signal packingdensity is obtained by simulating the model to obtainoptical signal and noise power and calculating therequired optical input power Pin. Once Pin is calcu-lated, we compare it with the available optical sourcepower. If the calculated Pin with a given signalpacking density p is greater (less) than the availableoptical power, the model with a decreased (increased)p is prepared for the next simulation experiment.
In the following calculations, we set BER = 10-7.23The BER can be represented as27
BER = /2 , exp( 2) (18)
where Q is the ratio of the rms signal voltage to thetotal rms cross-talk voltage. For a given BER =10-17, Q = 8.5. For this given Q, the required opticalinput power Pin can be calculated as2 8
Pin=(1r) Q A iNA2)/2 (19)( -r) Xe 'l
P w.i{LA ......................................................................................................
Model with specified misalignment tolerance &Part - (C) I maximum signal packing density ..' idmaxlaiiimu opnnsai ngx 1,
Model with specified distance between devices &misalignment tolerance & maximum signal acking densi
Fig. 8. Simulation algorithm for the OCAPP.
where r is the ratio of current to the detector in thelow illumination state relative to the high illumina-tion state, N is the system fan-out, nt is the product ofthe quantum efficiency of the detector and the effi-ciency of the optical system, and (iNA
2 )1/2 is the rmscurrent noise generated by the detector and preampli-fier circuit.
Finally, to calculate Pin, we should determine theparameter r. As r represents the ratio of currents atthe high illumination state to low illumination state,we obtain it by comparing the power incident uponthe detector aperture at high and low illumination.The power for the two states is obtained by simulat-ing the OCAPP model with a given SLM pixel pattern.To obtain power at the high illumination state, thedesired pixel of each SLM is made transparent whileothers are set to opaque. Similarly, to obtain thepower at the low illumination state, we set the pixelsat the same column to opaque and make all the otherpixels transparent. The whole column is cleared toavoid the effect of the cylindrical lens in the OCAPP.The factor (iNA 2
), which is expressed in terms of theBR, is calculated based on the data presented in Ref.10, and N is set to 1 because of the one-to-one imagingbetween SLM's in the OCAPP. 't is set to -0.051by considering a 50% ON-state power transmissionefficiency for an ON-state pixel of the SLM, 50% powerdivision at the beam splitter, and 4% reflection lossper surface (5 optical surfaces).
Once Pin is available, the number of pixels allowedper SLM can be obtained by comparing the requiredoptical input power with the available source power.As shown in Part-(A) of Fig. 8, if the calculated poweris less than (or greater than) the available power, amodel with an increased (or decreased) number ofpixels on the SLM plane is simulated. The maxi-mum number of pixels is determined when the re-quired optical input power is less than or equal to theavailable source power. The available optical powermust be less than the actual power as there are othersources of power losses such as component misalign-ment and aberrations.
Figure 9 shows the optical power collected at thedetector plane for models with various numbers of
4)>
0
0
AS
,
- e
16
i4
12
10
8
64
2A
0 100
Number of
200 300 400
pixels per SLMFig. 9. Simulated detected optical signal power versus number ofpixels of the optical data plane in the OCAPP.
10 March 1994 / Vol. 33, No, 8 / APPLIED OPTICS 1555
Optical Pswer (ROP) to Availsble ROP AOP tial Powm(A ROP < AOP
. .1 ROP ~~~~AOP...................................... ................. . O P... O
---. - I Ideal model with maximum signal packing density I
F d inax n n n. . ..sa..igum ent tolerance...............
Apply initial mnisalignment
Decrease Determine output intensity distribution Increasemisalignment. calculate contrast ratio, misalignmenLcalculate re utred ot tical wer.
P AROP AOP....................Compare ROP
ROP >AOP o .~Al
density) to an initial value]
pixels. The x axis of the graph represents the num-ber of pixels of an SLM. The results of Fig. 9 areused to calculate Pi,, as shown in Fig. 10. Thesimulation was started with the 4 x 4 model, whichrequired 1.09-mW (68.19 W x 42) optical sourcepower. As a 5-mW He-Ne laser was selected as thesource, the 8 x 8 model was simulated as the nextstep. The 8 x 8 model simulation requires 4.37 mW(68.23 W x 82), which is still smaller than our sourcepower. Next a 10 x 10 model was tested and foundto require 7.33 mW (73.29 VLW x 102), which exceedsthe 5-mW requirement. Therefore the 8 x 8 modelwas selected for the simulation experiments. As areference, a 20 x 20 model requires 57.04 mW(142.67 pW x 162).
b. Misalignment Tolerance. Once the maximumsignal packing density for a given model becomesavailable, then the maximum misalignment toleranceis found by applying part-(B) of the procedure de-scribed in Fig. 8. To find the effects of misalign-ments on the optical collected signal and noise power,each individual misalignment is applied to each com-ponent of the model so that its effect on the collectedoptical power and required optical input power can bedetermined.
The procedure starts with the minimum resolvablemisalignment. The unit of simulation is set to 50ulm. The unit of simulation means the distancebetween two sample data points used in the diffrac-tion calculations, which are an adjacent pair amongthe 220 data points. The minimum misalignmentthat can be applied becomes 50 pum.
Following part-(B) of Fig. 8, we extensively simu-late the 8 x 8 model for misalignment tolerance.Figures 11 and 12 show the simulation result ofrequired optical power. The two graphs show thatup to 500 m (half of a pixel width in the 8 x 8 model),the effect of applied lateral and longitudinal misalign-ments is not severe. However, as the amount ofmisalignment increases, the misalignment applied atSLM1 dominates the misalignment tolerance in boththe x and the y directions. In the case of lateralmisalignment, the maximum misalignment tolerance
0.
3
'-
._
.)
4)
4)
140
130
120
110
100
90
80
70
., 0
404-4
Q :3
. 50.001 1 1 _1 1 1
45.00 50.00 55.00 60.00 65.00 70.00 75.00
Misalignment (jam)
Fig. 11. Required optical power versus lateralapplied.
ISLMl115LM2
misalignments
becomes 700 gim. This value is obtained by compar-ing the line designated by SLM1 with the allowedbeam power per pixel (5 mW/64 pixels = 78 ,uW).Also, for the longitudinal misalignment, the misalign-ment tolerance of 610 plm can be obtained from thegraph.
c. Distance between Components. In optical com-puting systems, the mounting devices for opticalcomponents are important in that we must align tensor hundreds of individual beams with several devices.Also the beam broadens as it propagates because ofthe beam-spreading effect. Therefore, to enhancethe signal integrity of the model, the distances be-tween components must be minimized to the extentthat mounting devices permit. From the systemoptimization perspective, we need to find the allow-able range for d. This is calculated as shown inpart-(C) of Fig. 8.
Figure 13 shows the required optical power of a 4 x4, 8 x 8, 10 x 10, and 16 x 16 OCAPP for variousvalues of d. The figure shows that for an 8 x 8OCAPP, the change in the required optical powerover various distances is almost indistinguishable.Therefore we can conclude that the 8 x 8 OCAPP isalmost independent of the change in the distancebetween SLM's. However, as Fig. 13 shows, therequired optical input power for larger OCAPP (16 x16) increases exponentially with respect to the dis-tance parameter.
4. DiscussionTable 3 summarizes the results obtained from thesimulation phase. From the analysis of Phase 1 (the
0
40
0 100 200 300 400
Number of pixels per SLM
Fig. 10. Calculated required optical power Pin versus the numberof pixels of the optical data plane in the OCAPP.
90.00
80.00
70.00
60.0050.00
I I 71 _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~--E----------- --- ----------- -----
Fig. 13. Required power versus distance between SLM's.
results of which are summarized in Table 2), the
theoretical upper bound of the maximum signalpacking density is set by the heat removal capability.In addition to showing this heat removal factor, Table3 shows that the input source power is also animportant limiting factor in maximizing the signalpacking density. To maximize the signal packingdensity, two directions may be pursued: first, properdevice cooling techniques should be studied to in-crease the heat removal capacity, and second, opticalsources such as surface-emitting laser diode arraysshould be employed to deliver more power to thesystem.2 9
For misalignment, we found that the tolerancedepends on the direction and the location in thesystem where misalignment occurred. For example,in the case of the 8 x 8 model simulation, misalign-ments occurring at SLM1 are crucial, and misalign-ments occurring at SLM2 and SLM3 are tolerable aslong as the value of misalignment is less than theSLM pixel diameter. For SLM1, the lateral misalign-ment tolerance is found to be 700 [lm, whereas thelongitudinal misalignment tolerance is 610 pm. Asthe signal packing density increases (i.e., the pixel size
Table 3. Summary of Parameters Studied in the Simulation Phase
Wavelength X 633 nm Longitudinal mis- 610 plmalignment toler-ancea
System fanout N 1 Distance between 20 cmSLM'sa
Optical power 0.051efficiency Ift
BER 10-17
BR 0.75 Gbits/sCylindrical lens 10 cm
focal length fLength of SLM I 1.6 cm
aThis value is obtained from the 8 x 8 model simulation.
decreases), the misalignment tolerance requirementwill generally become more stringent.
For the distance between SLM's, the simulation of4 x 4 and 8 x 8 OCAPP models showed that thedistance between components is rather insensitive tothe signal packing density. However, for larger sig-nal packing density, (e.g., 10 x 10 or 16 x 16 models),the required optical power increases rapidly as thedistance increases.
5. Conclusions
A modeling and simulation methodology is proposedto evaluate the performance as well as the feasibilityof digital optical computing systems. As a particularexample, the OCAPP is modeled and simulated.The proposed methodology integrates various systemdesign parameters such as BER, BR, and opticalpower efficiency to determine maximum performanceparameters such as maximum signal packing density,misalignment tolerance, and distance between compo-nents. In the analysis phase of the methodology, adiffraction-limited OCAPP model is examined to deter-mine upper-bound values of design and performanceparameters. Then in the simulation phase, a GLADmodel is created and the model's performance pa-rameters are investigated by extensive simulations.Following the proposed simulation methodology, themaximum signal packing density, misalignment toler-ances, and the maximum distance between compo-nents are identified. The proposed methodology isintended to reduce optical computing systems' designtime as well as the design risk associated with build-ing the prototype system. The overall cost will alsobe reduced significantly because modeling and simula-tion permits design errors to be corrected beforeexpensive and time-consuming prototype construc-tion.
The authors thank G. N. Lawrence of the OpticalSciences Center, University of Arizona, for assistancein and permission for use of GLAD. This research wassponsored by National Science Foundation grantMIP-9113688.
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