## Abstract

This letter describes a method for optically convolving a pair of two-dimensional functions. The method overcomes a sometimes serious limitation in more conventional filters such as Vander Lugt’s system.[1] Suppose that an optical system is to perform a linear operation on input data by multiplying the spectrum of the input by an appropriate attenuation and phase shift in the back focal plane of a converging lens using a filtering transparency. If the space-bandwidth products of both the input data and the desired impulse response are comparable with the space-bandwidth product of the optical system, the frequency-plane mask must be positioned to within a fraction of the diffraction limit of the input aperture. The alignment in the focal plane must be to within a fraction of the distance Δ, given by

where λ is the optical wavelength, *X* is the width of the aperture, and *f* is the focal length of the lens. For λ = 6 × 10^{−7}m, *f* = 1 m, and *X* = 0.035 m, Δ ≃ 4 × 10^{−5} m = 40 *μ*. The required accuracy is some fraction of this, one-tenth, for instance, or about 4 *μ*. A similar accuracy is required in the orthogonal direction.

In some applications (such as pattern recognition) the data may occupy only a small sector in the input aperture, and the filtering transparency positioning requirements will be proportionally less. However, if communication signals are being filtered, the input aperture typically will be completely filled, e.g., the input may be a TV-type raster intensity modulated by the signal and noise. In the latter case, the positioning becomes a serious problem, especially since there usually is no convenient reference point on the filtering transparency. Any positioning error means that the true value of the convolution is nowhere available at the output plane of the conventional system; thus, an error of more than a few microns may completely obliterate the desired output.

Consider the system in Fig. 1. A transparency with real transmittance *f*(*x*,*y*) is positioned below the optical axis with its center at the point (0, −*y*_{1}), and a second transparency with real transmittance *g*(−*x*, −*y*) is positioned above the optical axis with its center at the point (0,*y*_{2}). The two transparencies are illuminated by a collimated beam from the laser and Fourier transformed by the converging lens. Then, in the back focal plane, we have

where the asterisk denotes the complex conjugate.

If a piece of film is placed in the back focal plane, the exposure may be adjusted to yield a positive transparency with a transmittance that is proportional to the square of the light amplitude; in other words, the amplitude transmittance is proportional to

If this latter transparency is transformed by the system of Fig. 1, *F**(*f*_{1}*f*_{2})*G**(*f*_{1}*f*_{2}) exp[−*j*2*πf*_{2}(*y*_{1} + *y*_{2})] will be transformed to the convolution *f*(+*x*, +*y*)**g*(+*x*, +*y*), with the origin of the convolution at the point (0, −*y*_{1}, −*y*_{2}). This is the desired linear operation. The function *F*(*f*_{1}*f*_{2})*G*(*f*_{1}*f*_{2}) exp[*j*2*πf*_{2}(*y*_{1} + *y*_{2})] will be transformed to the convolution *f*(−*x*, −*y*)**g*(−*x*, −*y*) centered at the point (0,*y*_{1}+*y*_{2}), while the sum of the transforms of |*F*(*f*_{1},*f*_{2})|^{2}and |*G*(*f*_{1}*f*_{2})|^{2} will be located in between with their centers on the optical axis.

The distance between the bottom of the *g*(−*x*, −*y*) transparency (in Fig. 1) and the top of the *f*(*x*,*y*) transparency should be greater than max{*h** _{f}*,

*h*

*}, where*

_{g}*h*

*is the height of the*

_{f}*f*(

*x*,

*y*) transparency and

*h*

*is the height of the*

_{g}*g*(−

*x*, −

*y*) transparency. This spacing will prevent the transforms of |

*F*(

*f*

_{1}

*f*

_{2})|

^{2}and |

*G*(

*f*

_{1}

*f*

_{2})|

^{2}from overlapping

*f*(

*x*,

*y*)*

*g*(

*x*,

*y*).

When there is sufficient separation, the true value of the convolution for each argument is always available at a point with position fixed relative to the edges of *f*(*x*,*y*)**g*(*x*,*y*), and the disadvantage of the conventional filter mentioned earlier is thus overcome. However, it should be noted that once the filtering transparencies in the conventional filter have been generated, no further photographic steps are required. In our filter, no filtering transparency is generated, but a photographic step is necessary for each convolution.

The range of intensity in the back focal plane of the system of Fig. 1 may make it impossible to operate always on a portion of the H and D curve yielding square-law operation over significant areas of the transform plane. In such cases, the system of Fig. 2 is suggested. A reference beam (similar to the reference beam used to generate the filtering transparencies in the Vander Lugt filter) illuminates the back focal plane at an angle *θ* with respect to the optical axis. The sole purpose of this reference is to establish a bias point on the H and D curve in a region where the desired square-law operation can be obtained. If the information-bearing portion of the exposure is attenuated, the dynamic range of exposure variations about the bias point is correspondingly reduced, with the result that proper nonlinear behavior is assured over the entire transform plane. Let the total width from the top of the upper transparency to the bottom of the lower transparency be *L*, and let the focal length of the transforming lens in Fig. 2 be *f*. If the angle *θ* satisfies

aside from a bright spot on the optical axis, the extra terms generated by the presence of the biasing wave are deflected out of the region where the desired convolution appears.

This work was supported by the United States Air Force Avionics Laboratory.

## Figures

## References

**1. **A. Vander Lugt, IEEE Trans. **IT-10**, 139 (1964).