A noncoherent optical vector-matrix multiplier using a linear LED source array and a linear P-I-N photodiode detector array has been combined with a 1-D adder in a feedback loop. The resultant iterative optical processor and its use in solving simultaneous linear equations are described. Operation on complex data is provided by a novel color-multiplexing system

Carlotto, Mark

Casasent, David

Psaltis, Demetri

English

Caltech Authors - Main

348 OPTICS LETTERS / Vol. 4, No. 11 / November 1979
Iterative color-multiplexed, electro-optical processor
Demetri Psaltis, David Casasent, and Mark Carlotto
Carnegie-Mellon University, Department of Electrical Engineering, Pittsburgh, Pennsylvania 15213
Received June 29, 1979
A noncoherent optical vector-matrix multiplier using a linear LED source array and a linear P-I-N photodiode de-
tector array has been combined with a 1-D adder in a feedback loop. The resultant iterative optical processor and
its use in solving simultaneous linear equations are described. Operation on complex data is provided by a novel
color-multiplexing system.
I. Introduction A =B-' C.
In an effort to increase the flexibility of optical systems
and to design optical processors that are more easily
fabricated, Cutrona' suggested a simple vector-matrix
multiplier in 1965. This system was capable of per-
forming any linear transformation and was thus not
limited to space-invariant operations. Since then,
many similar optical systems have been reported, in-
cluding extensive architecture studies,2 feedback sys-
tems, 3 systems using multiple 4 and single5 LED sources,
matrix-matrix multipliers,4 and discrete-Fourier-
transform systems6 .
In this Letter we combine much of this previous work
utilizing recent advances in LED's and detectors into
an iterative optical processor. The novel aspects of this
work are the iterative feedback concept and its appli-
cation to solving matrix equations (Section II) and a new
color-multiplexing method that enables the system to
handle complex data vectors and matrices (Section III).
A discussion of several applications of this system and
initial experimental results are included in Section
IV.
II. Iterative Optical Processor
The iterative optical processor that we have assembled
is shown schematically in Fig. 1. The Pl-P 2-- P3 portion
of the system is similar to one described earlier.6 A
linear N-element LED input array at Pi is spatially
modulated by the components An of the input vector
A. Plane PI is imaged vertically onto P2. The light
from each LED is spread horizontally across the corre-
sponding row at P2. Stored at P2 is the NXM matrix
B, where barn is the transmittance of element (n,m).
The mask at P2 is then focused vertically and imaged
horizontally onto a linear detector array at P3. The
output from the detector (m) is thus
Cm = Ei An nr,
n
Using the iterative algorithm
Ai+, = (I - B)Ai + C, (4)
we can find A when the i and i + 1 estimates Ai and
Ai,+ are equal, i.e., when the iterative equation in Eq.
(4) converges. In this case Eq. (4) reduces to C = BA,
and hence A is given by Eq. (3). To realize Eq. (4) in the
system of Fig. 1, we input an initial estimate Ai of A into
P1 , and as the P2 mask we use (I-B), where I is the
identity matrix. At P3 we obtain (I-B)Ai, and at the
output of the parallel adder we find Ai+1 given by Eq.(4). This value is then reinserted into the LED's at P1.
On the next iteration we obtain Ai+2 at the output of the
adder. These iterations continue until As = Ais+. At
this point, we have the solution A to Eq. (3).
III. Color Multiplexing
To accommodate complex data vectors and matrices,
we can decompose each complex input vector compo-
nent An into its projections on the 00, 1200, and 240°
axes (represented by 6, 1, and 2, respectively) in the
complex plane.7]8 Representing each barn similarly, we
can rewrite Eq. (2) as
C6 + Ci + C = (Bo + Bi + B§) (Ao + Ai + A2)
= (BoAo + BiAQ + B2AI)o
+ (BiAo + B2A2 + BOAO)i
+ (B2Ao + BoA2 + BiAO)2. (5)
(1)
which is the scalar form of the vector-matrix product
C = BA. (2)
The much more formidable problem is to determine
A given B and C, i.e., to solve
IF Al -A iA1 1 , THEN CBOA
Fig. 1. Schematic diagram of an iterative optical pro-
cessor.
0146-9592/79/110348-03$0.50/0 0 1979, Optical Society of America
November 1979 / Vol. 4, No. 11 / OPTICS LETTERS 349
GRATINGMATRIX
MASK
Fig. 2. Schematic diagram of a color-multiplexed optical
processor.
From Eq. (5), we see that by properly arranging the lo-
cation of the components of B and by properly com-
bining the various outputs at P3, complex vector-matrix
multiplication is possible.
This technique increases the size of the LED array
required at Pi by a factor of 3 and the space-bandwidth
product (SBWP) of the P2 mask by a factor of 9. To
avoid this, color multiplexing can be utilized. The
schematic diagram of such a system is shown in Fig. 2.
Three input LED arrays with nonoverlapping spectra
are modulated in parallel at P1 with the three complex
components A6, Ai, and A2 of A. All three linear LED
arrays simultaneously illuminate the mask B at P2 as
before. To separate the three components Co, Ci, and
C2 of C at P3 , a grating is inserted between P2 and P3.
This separates the three colors vertically at P3, where
the three components of C are separately detected on
three linear photodiode arrays. Summation of the
proper photodiode outputs at P3 yields the correct
complex vector-matrix product. An iterative version
of this color-multiplexed system follows directly from
Fig. 2. The advantages of such a system are obviously
a reduction by three in the 1-D SBWP of the input PI
array and a reduction by three in the SBWP of the P2
mask (only horizontal spatial multiplexing in the mask
elements is needed). The nonuniqueness of the com-
plex number representation in Eq. (5) can be avoided
by subtracting the smallest component from the rest.
IV. Experimental Confirmation
The inherent parallelism afforded by the system's ar-
chitecture allows us to increase N and hence the SBWP
without affecting the system's speed (or vice versa).
Corrections for static and dynamic LED source errors
as well as P2 mask errors are also directly included in
such an iterative feedback system. The system that we
are currently fabricating uses N = 20 input LED's
(present LED and laser-diode technology can easily
extend this to N = 100). To reduce support electronics,
input and output multiplexing is used. To reduce the
effects of nonlinear LED source response and to in-
crease system accuracy, pulse-width modulation of both
the LED output and the P2 mask is employed. This
increases the cycle time of the system to 1 lisec. Al-
though 10-nsec cycle times are possible with amplitude
modulation and without electronic multiplexing, 1-gsec
cycle times and the 8-bit accuracy that we expect to
obtain are still admirable performance characteris-
tics.
The absolute maximum size of the input LED array
is ultimately determined by the diffraction limit of the
optics used in the system. To avoid this limitation on
our system, we are considering several designs that will
eliminate the need for optical elements (lenses). One
such approach is the use of fiber-optic bundles that
would transfer the light from the LED's to the mask and
from the mask to the detector array. A second ap-
proach involves integrating the source, mask, and de-
tector in a single device by the use of long rectangular
source and detector elements.
The original proposed use for this system was as an
adaptive phased-array radar signal processor.9 To
demonstrate the use of this iterative processor for this
application, we considered the simple scenario of a
single noise source and a two-element receiver array
with receiver noise also present. For this application,
the optimum weight vector A to be applied to the array
is related to the covariance matrix B of the noise field
and the steering vector C by
C* = BA. (6)
To solve Eq. (6) using the system of Fig. 1, we com-
puted B for the case under study and fabricated a 6 X
6 element pulse-width modulated mask (I-B) at P2. A
prototype system with 8 input LED's and a 20-element
P-I-N detector array was used in these initial experi-
ments. An initial estimate Ai for A was fed to 6 of the
8 input, LED's. The vector-matrix product (I-B)A 1
was formed at P3. After addition of C*, we obtain the
next iteration Ai+, for A. This process is continued
until Ai = Ai+,. The P3 outputs at selected iteration
steps are shown in Fig. 3. As seen, the A outputs in-
crease at each iteration closer to the final value. After
seven iterations, all six A-component outputs are within
10% of their final values and within several per cent of
(a)
PD)
(C)
Fig. 3. Output-plane patterns at selected iterations.
350 OPTICS LETTERS / Vol. 4, No. 11 / November 1979
magnitude of the eigenvalues of B are all less than 1
(this can be ensured by proper normalization of B).
The feedback process can be stopped when Ai and Ai+,
are sufficiently close.
v+g9+0
(b)
V
- _
(c)
Fig. 4. Color-multiplexed complex vector-matrix multipli-
cation. (a) Input, (b) mask output, (c) system output.
(Printed in black and white: v, violet; g, green; o, orange).
the theoretical values calculated by computer simula-
tion.
The results of a color-multiplexed complex vector-
matrix multiplication are shown in Fig. 4. For this
experiment an arc-lamp source with three spectral lines
(violet XI, green X2, and orange X3) was used as the
source. A two-element input vector with components
Xai = (1,0), a,\2 = (1,1), and aC\3 = (0,1) was used as the
2 X 3 input [see Fig. 4(a)]. The output from a 2 X 6
mask B is shown in Fig. 4(b). Since XA reads out row 1,
X2 rows 1 and 2, and X3 row 3, turquoise- (violet plus
green) and yellow- (green plus orange) colored light
outputs result from P2. The 3 rows (6 elements per
row) in the P3 output shown in Fig. 4(c) are violet (top
row of P2 mask), green (both rows), and orange (bottom
row), and they represent the 3 X 6 = 18 possible cross
products of Eq. (5). In practice, one detector array is
used per row or color.
The iterative process is stable and will converge to the
one correct solution for all initial choices of Ai if the
V. Summary
Noncoherent optical-matrix multipliers have been
found to be most useful when reconfigured as iterative
systems to solve matrix equations. The use of color
multiplexing allows complex vector-matrix multipli-
cations to be realized and provides an added dimension
to the optical processor. For example, use of color-
multiplexing enables us to encode a 2-D pattern or many
1-D signals in the input plane for image or multichannel
signal processing.
The support of Rome Air Development Center
(contract F-30602-78-C-0272) for this work is most
appreciated.
References
1. L. Cutrona, in Optical and Electro-Optical Information
Processing, J. Tippett et al., eds. (MIT Press, Cambridge,
Massachusetts, 1965), pp. 97-98.
2. A. Edison and M. Nobel, Final Report, Contract
AF19(628-4199), General Electric Company (November
1966).
3. D. Mengent et al., U.S. Patent 3,525,856 (October 6,
1966).
4. W. Schneider and W. Fink, Opt. Acta 22, 879 (1975).
5. M. Monahan, in Digest of International Computing
Conference, Washington, D:C. (Institute of Electrical and
Electronics Engineers, New York; 1975), pp. 25-33.
6. J. W. Goodman, A. R. Dias, and L. M. Woody, Opt. Lett.
2,1 (1978).
7. C. B. Burckhardt, Appl. Opt. 9, 1949 (1970).
8. J. W. Goodman and L. M. Woody, Appl. Opt. 16, 2611
(1977).
9. D. Psaltis, D. Casasent, and M. Carlotto, Proc. Soc.
Photo-Opt. Instrum. Eng. 180 (to be published, 1979).
V
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Iterative color-multiplexed, electro-optical processor

A noncoherent optical vector-matrix multiplier using a linear LED source array and a linear P-I-N photodiode de-tector array has been combined with a 1-D adder in a feedback loop. The resultant iterative optical processor and its use in solving simultaneous linear equations are described. Operation on complex data is provided by a novel color-multiplexing system. I. Introduction A =B- &apos; C. In an effort to increase the flexibility of optical systems and to design optical processors that are more easily fabricated, Cutrona &apos; suggested a simple vector-matrix multiplier in 1965. This system was capable of per

Iterative color-multiplexed, electro-optical processor

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