Jul 1, 1978 - David M. Pepper, John AuYeung, Dan Fekete, and Amnon Yariv. California ... ly dependent optical fields in the Fourier-transform plane of a lens system. The use of a ..... See, for example, A. Yariv, Quantum Electronics (Wiley,.
July 1978 / Vol. 3, No. 1 / OPTICS LETTERS
7
Spatial convolution and correlation of optical fields via degenerate four-wave mixing David M. Pepper, John AuYeung, Dan Fekete, and Amnon Yariv California Institute of Technology, Pasadena, California 91125 Received March 13, 1978
A nonlinear optical technique is described that performs, essentially instantaneously, the functions of spatial correlation and convolution of spatially encoded waves. These real-time operations are accomplished by mixing spatially dependent optical fields in the Fourier-transform plane of a lens system. The use of a degenerate four-wave mix-
ing scheme eliminates (in the Fresnel approximation) phase-matching restrictions and (optical) frequency-scaling factors. Spatial bandwidth-gain considerations and numerical examples, as well as applications to nonlinear microscopy, are presented.
In recent years, coherent and incoherent image pro-
E4 =-2 A4 (XYZ)
in a variety of applica-
cessing has been demonstrated
tions, including pattern recognition, guidance systems, and other data-processing techniques. Present methods used to generate convolution and correlation operations of spatially encoded optical images include digital
processing and Van der Lugt-type holograms.1' 2
It has been proposed by one of the authors (A. Yariv)
that nonlinear optical techniques can be used to perform real-time holographic functions.3' 4 The specific application of nonlinear three-wave mixing to perform
the operation of convolution and correlation has been proposed by Eremeeva et al. 5 The proposed scheme was demonstrated in the case of simple (luminous spot) images. These schemes, as well as that proposed in
Refs. 3 and 4, suffered from restrictions on the spatial bandwidth of the information, which were due to the need for phase matching. Second, the use of multiple wavelengths introduces spatial scaling that may be objectionable. In the analysis that follows we propose the use of
"time-reversed"
propagation,
63
-
Cwt)] +
whereA1,4(XYO) -U 1 ,4(XY)and A 2(XY,4f)
c.c.,
(1)
U2 (XY).
Fields E1 and E2 are essentially counterpropagating, with E4 being parallel to these fields and separated either spatially [e.g.,shifted by (x 3,y)] or via orthogonal polarizations. The ui contain the input information to be convolved or correlated, which can, for example, be
in the form of phase and/or amplitude transparencies. The ui are assumed to be illuminated by plane waves, all of the same frequency, co.
After propagating through lens LI, A1 has the following form (in the Fresnel approximation), Al(xy; f < z • 3f) = exp(iknA) exp(ikz) iXf
X
Y julx',y')
(x'2 +
exp[yf (2-
(2)
where 7ja}= & is the Fourier transform of a, and
exp(iknA) exp[- i (X2 +
four-wave mixing for real-time correlation and convo-
lution operations. The process of four-wave mixing, which has recently been applied to the problem of
exp[i(hz
y2)J
(3)
is the transmission function of a thin lens.2 The argu-
7 is shown to be free of
the phase-matching problem and to require a single frequency. Consider the nature of the field produced as a result of the simultaneous mixing of three optical fields, all of
radian frequency co,incident upon a thin medium possessing a third-order nonlinear optical susceptibility, 3
), centered at the common focal plane of two identical lenses (or mirrors) of focal length f. The geometry is illustrated in Fig. 1. Each field is specified XNL
spatially at the front focal plane of its respective lens with the followingamplitudes:
El= 2
2
-IA
2
Al(x,y,z) exp[i(kz- cot)]+ c.c., Fig. 1. 2 (X,Y,Z)
exp[-i(kz + cot)]+ c.c., 0146-9592/78/O700-0007$0.50/O
Convolution/correlation
geometry. All input optical
fields are at frequency a. BS is a beam splitter necessaryto view the desired output, E3 , which is evaluated at a plane located a distance f from the lens L1 . © 1978, Optical Society of America
8
OPTICS LETTERS / Vol. 3, No. 1 / July 1978
ments of the Fourier transform are [x = x/Xf and fy =
where f, and fy are the spatial frequencies in the x and y directions, respectively; in Eq. (3), A corresponds to the thickness of the lens. Similarly, the complex amplitude of E 2 has the form (after propagating through lens L2), y/Xf,
A2(XY; f
`
Z
