Analysis of the dual discrimination ability of the two-port photorefractive joint transform correlator.

Abstract

An all-optical joint transform correlator featuring two operative correlation planes(ports) with complementary performance is presented. We present the theory of operation, derive the input-output characteristics, and demonstrate computer simulations and experimental results. The two-port joint transform correlator is based on simultaneous use of two photorefractive wave-mixing architectures. The first port uses two-beam coupling, and the second port uses four-wave mixing. The performance of the two ports depends on an experimentally controlled beam intensity ratio and the photorefractive coupling coefficient. With appropriate selection of these parameters, the first port is capable of high discrimination, while simultaneously the second offers a low discrimination output. Our results show that the two-beam coupling port can achieve peak-to-noise and signal-to-noise ratio values better than the phase-only correlator, whereas the four-wave-mixing port performs similarly to the classical joint transform correlator. This leads to a potential application in which the correlator could be set up so that in one port a general class is detected (interclass) and, in the other, the specific item in a class is detected (intraclass).

Full text

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Analysis of the dual discrimination ability of the
two-port photorefractive joint transform correlator
GeorgeAsimellis, Mark Cronin-Golomb, Jehad Khoury,
Jonathan Kane, and Charles Woods
An all-optical joint transformcorrelator featuring twooperativecorrelation planes1ports2with complemen-
tary performanceis presented.Wepresent thetheory of operation, derivetheinput–output characteris-
tics, and demonstrate computer simulations and experimental results.
correlator is based on simultaneous use of two photorefractive wave-mixing architectures.
The first port uses two-beam coupling, and the second port uses four-wave mixing.
the two ports depends on an experimentally controlled beam intensity ratio and the photorefractive
coupling coefficient.With appropriate selection of these parameters, the first port is capable of high
discrimination, while simultaneously the second offers a low discrimination output.
that the two-beam coupling port can achieve peak-to-noise and signal-to-noise ratio values better than
the phase-only correlator, whereas the four-wave-mixing port performs similarly to the classical joint
transform correlator.This leads toa potential application in which thecorrelator could beset up sothat
in one port a general class is detected 1interclass2 and, in the other, the specific item in a class is detected
1intraclass2.
Key words:J oint transform correlator, photorefractive pattern recognition, discrimination ability.
r1995 Optical Society ofAmerica
The two-port joint transform
The performance of
Our results show
1.
Experimental demonstration and modeling of the
first two-port joint transform correlator 1TPJ TC2 was
introduced in Ref. 1. In this paper we develop the
device theory, analyze an interesting aspect of this
correlator, namely its dual discrimination ability, and
verify the conclusions and results with computer
simulation and experiment.
The TPJ TC is an all-optical photorefractive joint
transform correlator 1J TC2 with two output correla-
tion planes, called ports, that operate simultaneously
and have different discrimination characteristics.
Discrimination is defined as the ability to reject
1discriminate against2 any other signal not identical
to the reference signal. No optical pattern recogni-
tion scheme developed today satisfies together low
Introduction
and high discrimination requirements.
cal J TC is known for its low discrimination ability,2
whereas the binary J TC offers high discrimination.3
Tworecently proposedall-optical photorefractiveJ TC’s
have a tunable discrimination ability,4,5which de-
pends on a beam intensity ratio.
beam ratio, the discrimination can be either low or
high.The TPJ TC attempts to satisfy both low and
high discrimination requirements.
The model presented in this paper uses the physics
of photorefractive four-wave mixing and self-pumped
phase conjugation6,7together with principles of non-
linear signal processing.
forward pump, modified by the signal Fourier trans-
form, and a plane-wave reference beam.
viewed as an application of the two-beam-coupling
J TC proposedin Ref. 4.Thefour-wave-mixing geom-
etry employs the same forward pump and the phase
conjugate of the outgoing signal beam as the back-
ward pump beam.The Fourier transform of the
diffracted beam generates the second J TC port.
This port is an application of the four-wave-mixing
J TC.5
The performance of the two ports is shown to be
dependent on the photorefractive gain and the inten-
sity ratio of the signal-to-probe beams.
The classi-
Depending on the
The first port utilizes a
It can be
When the
G. Asimellis, M. Cronin-Golomb, and J . Khoury are with the
Electro-Optics Technology Center, Tufts University, Medford, Mas-
sachusetts 02115.J . Kane and C. Woods are with the Rome
Laboratories, Hanscom Air Force Base, Massachusetts 01731.
Khoury is alsoa visiting scientist at RomeLaboratories.
Received 10 April 1995; revised manuscript received 27 J uly
1995.
0003-6935@95@358154-13$06.00@0.
r1995 Optical Society ofAmerica.
J .
8154 APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 2

beamratiois much smaller than theexponential gain,
both ports behave similarly to the classical J TC,
whereas at very large beam ratios both ports exceed
the performance of the phase-only correlator in terms
of peak intensity, peak-to-noise ratio 1PNR2, and
signal-to-noiseratio1SNR2.
pare with the gain, the two ports have different
characteristics.In particular, the two-beam-cou-
pling port has high discrimination, and the second
port has low discrimination.
preferable when an image has to be selected from
among others of the same class, but tends to reject a
correct object if the object is somewhat rotated or
scaled.In such cases a low discrimination scheme
works better.
At beam ratios that com-
High discrimination is
2.
The advent of charged-coupled-device 1CCD2 arrays
and spatial light modulators 1SLM’s2 made possible
real-time optical correlation systems based on the
J TC.8–11
Yet these innovations did not change the
fundamental concept of the classical J TC, namely the
use of quadratic processing in a square-law receiver,
be it a photographic film or an intensity-responding
camera.The quadratic processing of the spectral
amplitudeisequivalent tothat oftheclassical matched
filter.It is well known that the matched filter is
characterized by poor performance.12
A variety of nonlinear operations have been pro-
posed toimprovetheperformanceof theJ TC in terms
of correlation peak intensity, noise robustness, dis-
crimination ability, SNR, andPNR.
mention the binarization and other fractional power
laws studied by J avidi.13
evaluate the binary J TC thresholding level are cur-
rently under study.14
Other J TC variations include
fringe adjustment15and image preprocessing.16
thesemodifications arerealizableby serial operations
that include a digital processing unit and additional
CCD camerasandSLM’s.
mance improvements, however, have come at the
expense of the advantages that the optical parallel
processing is best known for, namely processing speed
and operational simplicity.
The holographic properties of photorefractive crys-
tals can be the nucleus of a fast, all-optical J TC
architecture. In particular, a photorefractive crystal
placed at theFourier-transform planecan providethe
appropriatespatial frequency mixing neededfor corre-
lation becausetheinterferencegrating can bewritten
on the intensity-responding real-time holographic
crystal.The use of photorefractives in an all-optical
J TC instead of an electronically assisted J TC offers
thefollowing advantages:
Background
For example, we
Advanced techniques to
All
Theunquestionableperfor-
x
A single photorefractive crystal can be substi-
tuted for a CCD camera, a digital processing unit, and
a SLM display.Only basic optic elements are neces-
sary for J TC operation, and processing speed is no
longer compromised.This results in a compact, fast,
low-energy budget and low-cost design.
x
The photorefractive crystal can achieve much
larger resolution compared with the current SLM
displays and CCD cameras.
x
The inherent photorefractive nonlinearities of
the various beam-coupling energy-transfer mecha-
nisms provide a wide variety of spatial frequency
mixing tools.
Photorefractives have already been introduced by a
number of researchers in linear correlation filter
architectures.17–19
The nonlinear aspects of photore-
fractivemixing havebeen considered tobesideeffects
and, until recently, they have not been actively uti-
lized and investigated.The two recently proposed
photorefractive nonlinear J TC’s actively use these
nonlinearities to achieve significant performance im-
provementsovertheprevious
schemes.4,5
These performance improvements can
be assessed by many metrics such as peak intensity,
PNR, SNR, and discrimination ability.
A setup of thetwo-port correlator is shown in Fig. 1.
Interference by coherent beams 1 1signal beam2 and 4
1probe beam2 establishes a grating inside the photore-
fractive crystal.The readout beam is a plane wave,
and the signal beam carries the joint Fourier trans-
form of the scene 1s2 and reference 1r2 signals, which
are placed alongside the input plane.
crystal, the probe beam is selectively amplified or
deamplified by the joint power spectrum.
quent Fourier transformation of this beam yields
correlation plane 1, which is henceforth called port 1.
Correlation plane 2 is obtained as follows:
pumped phase-conjugate mirror is placed in the prop-
agation path of beam 1, so that a replica of the
outgoing signal beam reenters the crystal.
pumped phase conjugator provides self-alignment
and aberration corrections, sothat the entering beam
is Bragg matched with the grating.
theexisting grating insidethecrystal generates beam
3, which lies along the propagation path of the
reference beam.A beam splitter is used to separate
photorefractive
On exiting the
Subse-
a self-
The self-
Diffraction from
Fig. 1.Experimental arrangement.
10 December 1995 @ Vol. 34, No. 35 @ APPLIED OPTICS8155

Page 3

the diffracted beam, and a transform lens produces
output correlation plane 2, which is henceforth called
port 2.
A common element in the photorefractive J TC’s
discussed in Refs. 4 and 5 and theTPJ TC is that their
input–output curve in the spatial frequency plane is
similar to that of the limiter-quadratic detector20
1LQD2 but with a saturative soft clipping instead of
hard clipping. Such nonlinearities amplify only the
weak portions of the signal 1in the quadratic region2
and deamplify the intense portions 1in the limiting
region2.The LQD is a proven robust alternative to
the classical quadratic detector 1QD2 for the problem
of a signal embedded in non-Gaussian noise.
theoretical investigation21it has been shown that the
relative efficiency of the LQD with respect to the QD
is independent of the noise variance 1s22 if the limiter
value is proportional to s.
selection of the limiter value, the LQD has consider-
ably more false-alarm stability and better perfor-
mancethan theQD for non-Gaussian noise.
A purely quadratic J TC produces broad and dim
correlation peaks.It is typically a scheme with low
discrimination and is preferable when the inputs
have a relative scale and rotation difference or the
noise has Gaussian characteristics.
purely limiting 1alsoknown as hard clipping2 produces
thin and intense peaks when the signals are identical
and no peak if they are different.
preferable when maximum discrimination is desired,
which reduces, for example, the false-alarm probabil-
ity.
In the input–output curves of the photorefractive
J TC’s thesignal spectral intensity appears multiplied
by the signal-to-probe beam ratio.
parameter is controlled by the experimental settings
and substantially affects the crystal response to the
input signal intensity.When this parameter is small,
so that most of the modified spectral signal intensity
is below the limiter level, the crystal response is
predominantly quadratic, whereas for a large param-
eter the crystal response is mainly limiting.
tion from the one to the other mode is possible by
simply increasing the beam ratio.
Section 3, the transition occurs at different values of
beam ratio for the two ports of the TPJ TC.
thebasis of thedifferent performanceof thetwoports.
In Section 3 we also present the expressions that
describe the operational modes, analyze the effect of
these modes on the relative performance of the two
ports, and discuss the conditions under which one
port may operate near the limiting region while the
other port operates near thequadratic region.
In a
With an appropriate
A J TC that is
A limiting J TC is
The value of this
Transi-
As is shown in
This is
3.
In deriving the TPJ TC input-output characteristics
we consider the transmission grating geometry, with
the notation of Fig. 1.We use the method of grating
integral, which is fully presented elsewhere,22,23with
Theory
the following assumptions:
same wavelength l, all wave vectors are in the plane
of the paper, all waves are polarized perpendicular to
this plane, and the grating wave vector is parallel to
the crystal optical axis.The absorption coefficients
are assumed to be small and are omitted.
intensity I0and input intensity ratiom are defined as
all beams are of the
The total
I05o
i51
4
Ai2,m 5
A11022
A41022.
112
The coupled-wave equations can be written as
1boldfaced characters indicatecomplex quantities2
d
dzA11z2 5 2
1
2GC1z2A41z2,
d
dzA41z2 5
1
2G*C*1z2A11z2,
d
dzA21z2 5
1
2GC1z2A31z2,
d
dzA31z2 5 2
1
2G*C*1z2A21z2,
122
where G is the amplitude complex coupling constant
and C1z2 is thegrating interferencefunction23:
C1z2 5
A11z2A4*1z2 1 A21z2A3*1z2
I0
.
132
Of special interest is thegrating action integral, u1z2
which is defined as
u1z2 5e
0
z1
2g0C1z20dz,
142
where g is the real part of G.
externally superimposed electric fields on the crystal
1diffusion limit2, the coupling constant is real and no
phase coupling occurs between the mutually interact-
ing beams. If L is the crystal thickness along the
direction of propagation and U 5 u1L2, the solutions of
Eqs. 122 can be expressed in terms of wave amplitudes
only as
In the absence of
A11z2 5 A1102cos3u1z24 2 A4102sin3u1z24,
A41z2 5 A4102cos3u1z24 1 A1102sin3u1z24,
A21z25A21L2cos3u1z22U41A31L2sin3u1z22U4,
A31z25A31L2cos3u1z22U42A21L2sin3u1z22U4.
15a2
15b2
15c2
15d2
The wave amplitudes as expressed above offer a
clean representation of the energy-transfer mecha-
nisms andthegrating diffraction efficiency.
more, the validity of these solutions is not confined
for thin interaction lengths only or for small coupling
constants. To evaluate the value of the grating
action integral, we consider the boundary conditions,
Further-
8156APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 4

which are
A1102 5 A10,A4102 5 A40,A21L2 5 A2L,A31L2 5 0.
162
Because A31L2 5 0, the grating can be considered as
exclusively written by beams 1 and 4 if A2is small
compared with A1 and A4.
mixing approximation.In such a case the following
known solutions for A11z2 and A41z2 from the two-wave-
mixing theory,7
This is the two-wave-
A11z2 5 A103
1 1 m21
1 1 m21exp1gz24
1@2
,
A41z2 5 A403
1 1 m
1 1 m exp12gz24
1@2,
172
must satisfy thegeneral solutions of Eqs. 15a2 and 15b2.
By incorporating the conditions 162 and expressions 172
into Eq. 142, we obtain the solution for the grating
action integral:
u1gz, m2 5 tan213
exp1
gz
222 1
1
Œmexp1
Œm 1
gz
224
.
182
The grating diffraction efficiency h is defined by the
portion of beam A2at z 5 L that is deflected in the
direction of beam 4 at z 5 0.
easy toseefrom Eq. 15d2 that
Because A31L2 5 0, it is
h 5
I30
I2L
5 sin23u1gL, m24.
192
Beam 1 illuminates a transparency that contains
the two input signals, whose transmissivity is de-
noted by r1x, y 1 y02 and s1x, y 2 y02, and is subse-
quently Fourier transformed by a lens with focal
length f1.As a result, beam A10incident upon the
crystal 1signal beam2 is modulated by the sum of the
individual complex Fourier transforms of the scene
and thereferencesignals, R 1nx, ny2 1 S1nx, ny2.
and ny indicate the spatial frequency coordinates.
If A˜10is the signal beam amplitude before the input
transparency, then
Herenx
A105 A˜10
0R 1nx, ny2 1 S1nx, ny20
lf1
.
1102
Themodified input intensity ratiois written as
m 5
A˜102
A402
0R 1nx, ny2 1 S1nx, ny202
1lf122
5 meffE1nx, ny2,
1112
where meffis the effective beam ratio at the Fourier-
transform plane and E1nx, ny2 is the normalized signal
intensity spectrum.
follows:
We define meffand E1nx, ny2 as
meff5
A˜102
A402
E02
1lf122,
1122
E1nx, ny2 5
0R 1nx, ny2 1 S1nx, ny202
E02
,
1132
where E0is the integrated transmissivity of the input
signal r 1 s. Thesedefinitions aresuch that both the
effective beam ratio and the normalized signal inten-
sity are dimensionless. In addition, any changes in
thesignal beam intensity by alterations in thearea of
the input signals are accounted for:
A˜102E02can beviewedas theintegratedintensity after
passage of beam 1 through the transparency.
effective beam ratio 1simply called beam ratio2 is
controlled by the experimental settings and the nor-
malized signal intensity 1alternatively referred to as
energy2 depends on theinput characteristics.
By the substitution of Eqs. 1112 and 182 into Eqs. 192,
15a2, and 15b2, the grating diffraction efficiency h and
thewaveamplitudes A1Land A2Ltaketheform
the product
The
h 5 sin25
tan213
exp1
gL
222 1
1
1meffE 21@2exp1
1meffE 21@21
gL
2246
,
1142
A1L5 A4053meffE11 2 h241@22 1h21@26,
1152
A4L5 A405311 2 h241@21 3meffEh41@26.
1162
The phase-conjugate mirror creates a replica of
beam A1L.Under certain conditions, including one
in which thephase-conjugatereflectivity b is indepen-
dent of the intensity spatial fluctuations, one can
express A2Las A2L5 bA1L.
sions for A2Land A3LintoEq. 15d2 weobtain
Substituting the expres-
A305 2bA4053meffE11 2 h241@22 1h21@261h21@2.
1172
Correlation planes 1 and 2 are produced after
inverse Fourier transformation of beams A4Land A30,
respectively.By setting A4L5 A40f11nx, ny2 and A305
2bA40f21nx, ny2, we derive the input–output functions
f11nx, ny2 and f21nx, ny2 for thetwoports:
f11nx, ny2 5 31 2 h1nx, ny241@21 3meffE1nx, ny2h1nx, ny241@2,
1182
f21nx, ny2 5 5meffE1nx, ny231 2 h1nx, ny24h1nx, ny261@2
2 h1nx, ny2.
1192
4.
The two input–output operators f11nx, ny2 and f21nx, ny2
describe nonlinear processing of the joint intensity
spectrum.The characteristics of these nonlineari-
Analysis
10 December 1995 @ Vol. 34, No. 35 @ APPLIED OPTICS8157

Page 5

ties are subject to the values of the experiment-
controlled beam ratio meff and the photorefractive
gain factor gL.The nature of these characteristics
and the way that they can be used for performance
optimization areinvestigated in this analysis.
There are twofundamental aspects of this process-
ing: the mixing of the spectra, which is responsible
for the correlation properties of the J TC, and the
selectiveamplification of thejoint intensity spectrum,
which is responsible for the different characteristics
of the two ports. The crystal responds to the inten-
sity of the signal, not to the amplitude.
accountablefor themixing of thespectra, and wenote
that both input–output operators are functions of the
signal energy E1nx, ny2.Thecrystal responseis differ-
ent for different signal energy values, as is demon-
strated by the energy dependence of the diffraction
efficiency h1nx, ny2, as shown in Eq. 1142.
denceresults in theselectiveamplification.
It is well understood that the diffraction efficiency
is negligible for any of the limits meffE 9 exp1gL2 and
meffE : exp1gL2.Wecan examinethebehavior of the
input–output response curves of the two ports at
these limits by using series expansion and keeping
only the first-order terms.
thefollowing twocases of approximations:
This is
This depen-
By doing so we arrive at
1a2
Approximations at small beam ratio values:
A beam ratio is small when, for most of the spatial
frequencies 1nx, ny2, the condition meffE 1nx, ny2 9
exp1gL2 is true. Such low beam ratio values are
obtainedwith a strong readout beam,A4.
tion efficiency and the input–output operators are
expressed as
Thediffrac-
h1gL, meffE 2
mE9exp1gL2
5 meffE exp12
gL
224 sinh21
gL
42
1202
f11nx, ny2
mE9exp1gL2
5 1 1 meffE exp12
3
4gL22 sinh1
gL
42,
1212
f21nx, ny2
mE9exp1gL2
5 meffE exp12
3
4gL22 sinh1
gL
42,
1222
Both input–output operators are quadratic with
respect tothe signal amplitude 1linear with the signal
energy2. Usually the low spatial frequencies, which
correspond to the dc autocorrelation terms, are very
intense.Thusat small beamratiostheTPJ TC ampli-
fies the low frequencies.This is typical of the classi-
cal J TC quadratic receiver, which has a very low
discrimination ability.
1b2
Approximations at largebeam ratios:
ratiois largewhen, for most of thespatial frequencies
1nx, ny2, the condition meffE1nx, ny2 : exp1gL2 holds.
This is accomplished with a strong signal beam, A1.
Thediffraction efficiency and theinput–output opera-
A beam
tors are
h1gL, meffE 2
mE:exp1gL2
5
exp1
gL
224 sinh21
meffE
gL
42
,
1232
f11nx, ny2
mE:exp1gL2
5 exp1
gL
222
1
meffEexp1
gL
224 sinh21
gL
42,
1242
f21nx, ny2
mE:exp1gL2
5 exp1
gL
222 1 2
1
meffEexp1gL2
3 4 sinh21
gL
42.
1252
In this region both operators have an inverse
relation with the input intensity.
high frequencies, which correspond to the cross-
correlation grating, that are of low intensity.
fore at large beam ratios the TPJ TC is amplifying the
high frequencies of the spectra.
meffE = ` the operators are purely limiting, and their
output is independent of the signal amplitude.
J TC with amplitude removal is equivalent to the
phase-extraction correlator, which is characterized by
a very high discrimination ability.
Typically it is the
There-
At the limit of
A
Figure 2 displays two sets of the input–output
operators f11nx, ny2 of the two-beam-coupling port 1solid
curves2 and f21nx, ny2 of the four-wave-mixing port
1dashed curves2 versus the product meffE.
11 has been added tof2sothat at the low beam ratio
limit all the curves converge to11.2
curves reach a plateau when meffE : exp1gL2, and
they are linear with the signal energy when meffE 9
exp1gL2.
The curves are different at intermediate beam ra-
tios.For such beam ratios, the product meffE1nx, ny2
for small E satisfies the quadratic approximation
1A bias of
As expected, all
Fig. 2.
port 1solid curves2 and the four-wave-mixing port 1dashed curves2
for gain ranging from 26 to16.
TPJ TC input-output curves for the two-beam-coupling
8158APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 6

3,exp1gL24, whereas for large E it satisfies the limit-
ing condition 3.exp1gL24.
sponse has mixed characteristics, namely quadratic
for the weak points of the intensity spectrum and
saturating 1limiting2 for the intense points, with a
smooth transition for the intermediate intensities.
Such a soft-clipping detector has an improved perfor-
mance compared with both the hard-clipping detec-
tors andtheQD’s.This is attributedtothereduction
of thefrequency-planecontrast by selectiveamplifica-
tion of the weak parts of the signal.
plane in a J TC is dominated by the intense on-axis dc
term; yet the important information lays at the
interference grating written by the cross-correlation
term, which is normally weak.
tively amplified by a soft-clipping detector, while the
intense dc term is relatively deamplified.
QD amplifies thewholespectrum, and predominantly
the dc terms, whereas a purely hard-clipping detector
does not amplify the weak portion of the signal.
Photorefractive selective amplification at the spatial
frequency plane has been proposed for contrast ma-
nipulation,24image enhancement,25and noise reduc-
tion.26
For both ports of the TPJ TC, the limiter level,
wherethetransition from thequadratictothethresh-
olding region occurs, depends on the gain gL.
beam ratio at which transition occurs is approxi-
mately exp1gL2, but it is different for the two ports.
We note from Fig. 2 that for positive gain the
two-beam-coupling port saturates earlier than the
four-wave-mixing port.At negative gain, the four-
wave-mixing port saturates first.
sition is smooth and not abrupt, we attempt to
estimate the meffE value at the limiter level by the
point of inflection of the input–output curve, which
can be found analytically.
values for the point of inflection of the two input–
output curves.
We note that 1i2 the meffE values at the points of
inflection are approximately related to a geometrical
succession, whose step is different for the two ports;
1ii2 both ports require larger signal intensities for
increasing photorefractive gain to saturate, and 1iii2
the separation of the two curves is emphasized for
largeand positivephotorefractivegains.
Figure 3 displays the TPJ TC input-output curves
for a gain of 18 to show in detail the different
responses of the two ports.
sponds to the two-beam-coupling port, and the four-
wave-mixing port is indicated by the dashed curve.
Therefore the output re-
The frequency
This grating is selec-
A purely
The
Because the tran-
Table 1 presents the meffE
The solid curve corre-
At very large beam ratios 3meff. 1044, both curves
have a limiting response, which is identical tothat of
theinversefilter and is noted by thehorizontal dotted
linein thefigure. At small beam ratios, theresponse
is linear in thesignal energy for thefour-wave-mixing
port.The similarity with the matched-filter re-
sponse curve 1a line with a slope of 12 is clearly
identified. There is a relatively broad range of beam
ratiovalues for which thetworesponsecurves display
different characteristics, namely the two-beam-cou-
pling port has a response close to that of the phase-
only correlator 1a line with a slope of 0.52, whereas the
four-wave-mixing port performs like the classical
matched filter.
Similar separation occurs for negative photorefrac-
tivegain values, which areachieved simply by revers-
ing the crystal orientation.
values necessary for separation are of the order of
1024, which means that thesignal beam has tobevery
weak.With a very weak beam, the self-pumped
conjugator does not operate, and the very weak beam
limit is impractical from the experimental point of
view.
We summarize our theoretical predictions regard-
ing the relative performance of the two ports with
positivephotorefractivegain in Table2.
However, the beam ratio
5.
The objective of this analysis was to examine the
performance of the TPJ TC by comparison of the
Computer Simulation
Table 1.Point of Inflection for Various Photorefractive Gains
Photorefractive
Gain gL
meffE at the Point of Inflection
Port 1 Port 2
8
6
4
2
0.2
1085.7
203.2
28.7
8782.5
1040.1
120.4
11.74.8
1.2 1.3
Table 2.TPJTC Performance at Different Operating Points
Beam RatioPort 1Port 2
meffE 9 exp1gL2
Quadratic—low dis-
crimination
Nearly saturating—
high discrimina-
tion
Saturating—very
high discrimina-
tion
Quadratic—low dis-
crimination
Nearly qua-
dratic—low dis-
crimination
Saturating—high
discrimination
meffE < exp1gL2
meffE : exp1gL2
Fig. 3.
curve2 and the four-wave-mixing port 1dashed curve2 for a gain of
8. The slopes of three known filters 1classical matched, phase
only, and inverse2 aredrawn for comparativeanalysis.
Input–output curves of the two-beam-coupling port 1solid
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Page 7

correlation peaks with known J TC schemes, namely
theclassical J TC andthephase-only J TC.
plane was an array of 512 3 1024 pixels, in which the
scene and the reference signals, combined in an array
of 128 3 256 pixels, were centered on the optical axis.
We used the following sets of scene and reference:
1a2 a set of two identical large disks, and 1b2 a set of a
largeand small disks. Thelargeand thesmall disks
had radii of 28 and 14 pixels, respectively, and their
center-to-center separation was 128 pixels.
Thenonlinear functions f11nx, ny2 and f21nx, ny2 in Eqs.
1182 and 1192 were applied in the spatial frequency
plane with the values for the beam ratio1meff2 of 1021,
102, and 104, typical values for small, intermediate,
andlargebeamratios, respectively.
tive gain 1gL2 was 18.The fast Fourier transforma-
tions werecorrected sothat theenergy was preserved
in all planes.
In the output plane only the 160 3 160 pixel region
area around the first-order correlation peak was
selected for display. Figure 4 shows the computer
simulation results with the classical J TC and the
phase-only J TC for the similar 1top row2 and the
dissimilar 1bottom row2 disks.
the frequency-plane processing was quadratic in the
spectral amplitude, whereas for the phase-only27J TC
the absolute value of the spectral amplitude was
considered.The z-axis ratio between Figs. 41b2, 41c2,
and 41e2, 41f2 is 5:1 because of the relative decrease in
peak intensity.This axis convention is held for all
Theinput
Thephotorefrac-
For the classical J TC,
cases when similar-disk and dissimilar-disk correla-
tion peak intensities arecompared.
Figure5 displays theTPJ TC correlation peak inten-
sity plots with the similar disks for different beam
ratios.Figure6 repeats this for thedissimilar disks.
SNR and PNR measurements for the corresponding
amplitude arrays surrounding the correlation peaks
areshown in Table3 for thesimilar disks and in Table
4 for the dissimilar disks.
PNR definitions as described in Ref. 28.
As expected, the shape of the peaks and the values
of intensity, SNR, and PNR depend on theoperational
point, which is controlled by the beam ratio.
beam ratio values 3Figs. 51a2, 51b2, 61a2, 61b24 give both
ports characteristics of the classical J TC.
the similar shapes of the correlation peaks between
both ports of the TPJ TC with meff 5 0.1 and the
classical J TC.These peaks are typically broad and
not intense. Such a correlator sometimes fails to
detect objects in the presence of nonoverlapping non-
Gaussian noise, but performs well with clean inputs.
Its low discrimination is somewhat tolerable to rota-
tion and scaling of the twoinputs, as indicated by the
sizablecorrelation peak with thedissimilar disks.
Large beam ratio values produce peaks with a
much smaller full-width at half-maximum 1FWHM2,
as well as larger PNR and intensity values than the
correlation peak of the classical J TC or the phase-
only J TC.Both ports outperform both the classical
We used the SNR and
Small
We note
Fig. 4.
classical J TC correlation peak intensity, 1c2 the phase-only J TC correlation peak intensity, 1d2 the input plane with dissimilar disks, 1e2 the
classical J TC correlation peak intensity, 1f2 thephase-only J TC correlation peak intensity.
Computer simulations with similar 1top row2 and dissimilar 1bottom row2 disks:
1a2 the input plane with similar disks, 1b2 the
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Page 8

and the phase-only J TC’s:
port shown in Fig. 51e2, with beam ratiomeff5 104, has
PNR and SNR values three times larger than the
corresponding peak produced by the phase-only J TC,
as shown in Fig. 41c2.
Such a correlator successfully detects an object in
the presence of nonoverlapping non-Gaussian noise
and performs excellently with clean inputs.
characterized by a very high discrimination ability,
the peak from the first
It is
but is very sensitive to input rotation and scaling.
This is demonstrated by the fact that the output with
the dissimilar disks is seriously degraded for both
ports.
A disadvantage in operating well past the transi-
tion point is the increased presence of higher har-
monic terms. The severity of the nonlinearity is
such that most of the signal is hard clipped.
output of a hard-clipping nonlinear processor can be
The
Fig. 5.
and the four-wave-mixing port 2 1bottom row2.
1a2 Port 1, beam ratioat 0.1; 1b2 port 2, beam ratioat 0.1; 1c2 port 1, beam ratioat 102; 1d2 port 2, beam ratioat 102; 1e2 port 1, beam ratioat 104;
1f2 port 2, beam ratioat 104.
TPJ TC computer simulations with thesimilar disks:intensity plots of correlation peaks of thetwo-beam-coupling port 1 1toprow2
The same columns correspond to the same beam ratio.The photorefractive gain was 8.
Fig. 6.
row2 and thefour-wave-mixing port 2 1bottom row2.
8.
1a2 Port 1, beam ratioat 0.1; 1b2 port 2, beam ratioat 0.1; 1c2 port 1, beam ratioat 102; 1d2 port 2, beam ratioat 102; 1e2 port 1, beam ratioat
104; 1f2 port 2, beam ratioat 104.
TPJ TC computer simulations with the dissimilar disks:intensity plots of correlation peaks of the two-beam-coupling port 1 1top
Thesamecolumns correspond tothesamebeam ratio.Thephotorefractivegain was
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Page 9

expressed in an infinite series of harmonic terms,
whose weight is such that they can no longer be
neglected.As a result of light diffraction into the
higher harmonics the intensity at both ports is re-
duced.
Intermediate beam ratio values 3<exp1gL24 offer a
wide range of beam ratio values at which the two
ports of the TPJ TC operate in a different mode.
two-wave-mixing port operates closer to the flat sec-
tion, whereas thesecondport is closer tothequadratic.
In the case of the similar disks, the first port output
compared with that of the second port has a thinner
and more intense peak.
inputs, the first port output is drastically degraded,
whereas the second port output is less affected and
turns out to offer a brighter spot.
when Figs. 51c2 and 51d2 are compared with Figs. 61c2
and 61d2.
Thus the TPJ TC at intermediate beam ratios offers
a dual performance pattern, with port 2 fitted for the
low discrimination interclass identification and port 1
suitable for high discrimination intraclass identifica-
tion.At this range the performance of port 1 com-
pares in shape with that of the phase-only J TC, and
port 2 compares with the classical J TC.
compare Figs. 51c2 and 61c2 with Figs. 41c2 and 41f2 and
thePNR and theSNR values in Tables 3 and 4.
We conducted comparative simulations with the
median-thresholded14J TC and the TPJ TC with
meff5 102and gL 5 8. The numerical results for the
PNR and the SNR and the shape of the correlation
peaks were similar to that of the phase-only J TC.
This result was expected, given the similar perfor-
manceof thebinary J TC and thephase-only J TC with
similar inputs.
The
In the case of dissimilar
This is indicated
One can
6.
The experimental setup is shown in Fig. 1.
coherent illumination was provided by an argon laser
Experimental
The
at l 5 0.514 µm, split intotwobeams 11 and42 toallow
for different beam intensities.
on a barium titanate crystal with an angle of 20° and
a beam waist of 4 mm in diameter.
the reference signals were placed in the path of beam
1 andpassedthrough lens L11f15 16 cm2 sothat their
Fourier transforms appeared at the barium titanate
crystal. Exiting beam 1 was phase conjugated by a
second barium titanate crystal, placed in a self-
pumped configuration.This phase-conjugate beam
was used as backward pump beam 2.
Port 1, the two-wave-mixing output, was recovered
when beam 4 was allowed to propagate through
Fourier-transform lens L21f25 13.5 cm2.
four-wave-mixing output, was recoveredwhen a beam
splitter was placed in the path of beam 4 so that
diffracted phase-conjugate beam 3 was reflected off of
the beam splitter and passed through Fourier-
transform lens L31f35 13.5 cm2.
There were two different types of input pairs that
weexamined, namely a set of similar objects and a set
of dissimilar objects. Physically the objects were
disks cut intoa piece of sheet metal with a separation
of 1.5 mm. In the case in which the disks were
similar their radii were 0.3875 mm, versus that in
which they weredissimilar, wheredisk 1 had a radius
of 0.3875 mm and disk 2 had a radius of 0.175 mm.
The same analogies at the input planes have been
applied in thecomputer simulation.
We have covered two different beam intensity ra-
tios, one small and one large.
positiveand negativephotorefractivegains by revers-
ing the orientation of the crystal C axis.
positive gain, beam 1 pumps beam 4, and with a
negativegain, beam 4 pumps beam 1.
The beam intensities were 1i2 at the small beam
ratio, I1 5 2.51 3 104µW@mm2and I4 5 1.885 3
104µW@mm2; and 1ii2 at thelargebeam ratio, I15 2.51
3 104µW@mm2and I45 4.12 3 102µW@mm2.
intensity for beam 1 was measured after the passage
through the metal sheet.
definition given in Eq. 1112, we calculated that the
effective beam ratio for the first case 1small beam
ratio2 was meff5 197 and for the second 1large beam
ratio2 meff5 9007.Thesmall beam ratiosatisfies the
condition meff< exp1gL2 while the large beam ratio
satisfies meff: exp1gL2.
The experimental results are shown for the similar
images in Fig. 7 and for the dissimilar images in Fig.
8.On the left-hand side of the figures is the two-
wave-mixing output 1port 12, and on the right-hand
side of the figures is the four-wave-mixing output
1port 22.In both figures, cases A–D correspond tothe
large beam ratio, while cases E–H correspond to the
small beamratio.ThecasesA, B andG, H arefor the
positive gains, and C, D and E, F are for the negative
gains.
At the large beam ratio at either port and for both
positiveand negativegains, thecorrelator operates in
the saturation region, as we can see from Fig. 3.
The beams impinged
The scene and
Port 2, the
We applied both
With a
The
Using these data and our
As
Table 3. SNR and PNR for the TPJTC and the Classical and the
Phase-Only JTC’s with Similar Disks
J TCSNRPNR
Two-Port
meff5 0.1
meff5 102
meff5 104
Classical
Phase-only
Port 1
6.69
17.71
58.36
Port 2
6.05
7.64
29.48
Port 1
4.72
17.01
54.26
Port 2
3.87
5.82
25.12
6.64
17.12
3.84
18.92
Table 4. SNR and PNR for TPJTC and the Classical and the Phase-only
JTC’s with Dissimilar Disks
J TCSNRPNR
Two-Port
meff5 0.1
meff5 102
meff5 104
Classical
Phase-only
Port 1
9.16
6.16
8.07
Port 2
8.72
8.35
6.56
Port 1
3.09
4.49
6.88
Port 2
2.92
3.21
4.28
8.72
7.13
2.92
5.33
8162APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 10

Fig. 7.
gain; C, port 1, largebeam ratio, negativegain; D, port 2, largebeam ratio, negativegain; E, port 1, small beam ratio, negativegain; F, port
2, small beam ratio, negativegain; G, port 1, small beam ratio, positivegain; H, port 2, small beam ratio, positivegain.
TPJ TC experimental results with similar disks:A, port 1, large beam ratio, positive gain; B, port 2, large beam ratio, positive
10 December 1995 @ Vol. 34, No. 35 @ APPLIED OPTICS8163

Page 11

a result, the dissimilar disks 1Figs. 8A and 8B2 did not
display a distinct correlation peak but only a pair of
rings, while the similar disks 1Figs. 7A and 7B2
showed distinct, intense and sharp correlation peaks,
with the rings having a much smaller intensity than
the peaks.
the computer simulation results, in which one should
compare Figs. 51e2 and 51f2 with 7A and 7B and Figs.
61e2 and 61f2 with 8A and 8B.
lies in the fact that we do not actually observe full
These results are in full agreement with
The notable difference
Fig. 8.
gain; C, port 1, largebeam ratio, negativegain; D, port 2, largebeam ratio, negativegain; E, port 1, small beam ratio, negativegain; F, port
2, small beam ratio, negativegain; G, port 1, small beam ratio, positivegain; H, port 2, small beam ratio, positivegain.
TPJ TC experimental results with dissimilar disks:A, port 1, large beam ratio, positive gain; B, port 2, large beam ratio, positive
8164 APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 12

rings but rather crescents.
discrepancy can beunderstood if it is realized that the
resolution of the crystal has been assumed to be
isotropic and that the recording process is highly
nonlinear.
At the small beam ratiothe performance of the two
ports changes significantly on change of the beam
ratioand between the twoports.
Fig. 2, at a gain of 16 and beam ratio100, two-wave-
mixing port 1 is closetothelimiting region, whileport
2 is at the quadratic region.
port 1 at the dissimilar disks 1Fig. 8G2 did not display
any correlation peak, and we observed only the dc
spot.The absence of a correlation peak is indicated
by the computer simulation result shown in Fig. 61c2.
At the same time, port 2 showed very broad peaks,
which also agree with the simulation 3compare Fig.
8H with Fig. 61d24.This clearly demonstrates that
port 1 has high discrimination and port 2 has low
discrimination. The results with the similar disks
1Figs. 7G and 7H2 showed distinct, intense and sharp
correlation peaks, with the rings having a much
smaller intensity than the peaks for port 1 only, while
port 2 had correlation peaks similar to the cone.
Theseresults arein full agreement with thecomputer
simulation results, for which one should compare
Figs. 51e2 and 51f2 with 7G and 7H.
At a negative photorefractive gain, the small beam
ratio corresponds to the limiting region.
expect tosee results similar tothe ones obtained with
thelargebeam ratio.This is demonstrated in Fig. 7,
where the shapes of Figs. 7E and 7F are similar to
thoseof Figs. 7C and 7D.
The explanation of this
As we can see from
The results show that
Thus we
7.
A new all-optical TPJ TC based on four-wave mixing
in photorefractive media has been presented and
analyzed.It is, to our knowledge, the first J TC that
permits two simultaneously performing ports that
feature complementary discrimination ability, en-
abling both intraclass and interclass discrimination.
TheTPJ TC architecturemaintains parallel process-
ing at all planes, and bypasses the need for a serial
digital processor and additional CCD’s and SLM’s at
the spatial frequency plane.
with a soft-clipping quadratic-limiter nonlinearity.
The characteristics of this nonlinearity can be con-
trolled by the ratio between the incident intensities
parameter and the photorefractive gain 1gL2.
analysis has found that when the intensity ratio is
much smaller than exp1gL2 both ports behave like the
QD, whereas at intensity ratios much larger than
exp1gL2 both ports have a thresholding behavior.
the quadratic region both PNR and SNR values
indicate a similar performance to the classical J TC,
whereas in the limiting region the performance of
both ports in terms of PNR and SNR exceed that of
thephase-only J TC.
With approximately equal intensities of the signal
and the readout beams and positive photorefractive
gain we achieved limiting behavior in the two-wave-
Conclusions
The TPJ TC operates
Our
In
mixing port and quadratic behavior in the four-wave-
mixing port. At these settings the discrimination of
the first port was very high.
able when false warnings are to be eliminated.
second port had a classical J TC performance, with
broad but dim correlation peaks.
native scheme is preferable when the inputs have a
relativescaleand rotation difference.
This scheme is prefer-
The
This less discrimi-
References and Notes
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New York, 19932, Chap. 4.
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27. The phase-only filter, introduced by J . L. Horner and P. D.
Gianino in ‘‘Phase-only matched filtering,’’ Appl. Opt. 23,
812–816 119842, has as output fromthefilter planetheabsolute
value of the reference spectral amplitude.
between thephase-only filter andthephase-only J TC is that in
theJ TC thespectral amplitudeis madeof thejoint spectrumof
thesceneand thereference.
The difference
28. Our definition for the SNR can be found in J . L. Horner and P.
D. Gianino, ‘‘Signal-dependent phase distortion in optical
correlators,’’ Appl. Opt. 26, 2484–2487 119872.
tion for the PNR was originally called peak-to-correlation
energy, and the PNR we used is equivalent to the peak-to-
correlation energy introduced in Eq. 182 of J . L. Horner,
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Our defini-
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