H. B. G. Casimir, "Reciprocity Theorems and Irreversible Processes," Proc. IEEE 51, 1570–1573 (1963).

A. Van der Lugt, "Operational Notation for the Analysis and Synthesis of Optical Data-Processing Systems," Proc. IEEE 54, 1055–1063 (1966).

The time-harmonic field quantities are assumed to vary according to e^{-ωt}. The scalar treatment of optical phenomena implies the neglection of possible interactions between the *x* and *y* components of the electromagnetic field vectors inside the optical system.

In mathematical terms ø satisfies the homogeneous Helmholtz equation in this region with a radiation condition at infinity.

If not stated otherwise, all integrations extend from -∞ to +∞. The double sign convention in (1) which is discussed further down, was suggested by M. J. Bastiaans (private communication).

Also "mixed" relations can be formulated where the excitation is described in the space domain and the response in the frequency domain (or vice versa).

To the best of our present knowledge all (finite) optical weighting functions are realizable. This fact strongly contrasts with the situation in time-domain filtering, where the principle of causality puts severe limitations on realizability.

At this stage, we take advantage of the double sign convention in (1). If the same signs had been chosen in (1) for either direction of transmission, Eq. (10a) would be transformed into *G*_{21}(R_{2}, R_{1})= G_{12}(-R_{2}, -R_{1}), according to (4). This asymmetry with respect to space and frequency description would prohibit us to develop the fundamental notion of duality in the further course of the paper.

*g*_{21}* denotes the complex conjugate of *g*_{21}.

In a more restricted sense, two systems are dual if the associated functions *H*_{21}(R) and *m*_{21}(r) have the same mathematical structure (this is the case, for instance, with lens and free space, see Sec. VI).

In this light the Fourier relationship (1) can be viewed as a unitary coordinate transformation.

Here "temporal" frequencies are meant and not "spatial" frequencies. Furthermore, *t* denotes the time variable.

While a spreadless system is completely specified by the function *m* (r) and the choice of the coordinate system r(*x*, *y*), we need for the specification of a shift-invariant system besides *H*(R) and the coordinate system an agreement about the signs in (1). Clearly, if the sign convention is altered, *H*(R) is transformed into *H*(-R).

In other words, the "powers" (=reciprocal focal distances) have to be added.

In all these cascades the overall system properties are independent of the order of arrangement.

Reciprocity of all systems yields the profit that all proofs need only be given for one direction of transmission.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 5 and 6.

E. O'Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 11.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, "Optical Data Processing and Filtering Systems," IRE Trans. Inf. Theory IT-6, 386–400 (1960).

E. L. O'Neill, "Spatial Filtering in Optics," IRE Trans. Inf. Theory IT-2, 56–65 (1956).