Abstract

The angular-spectrum representation of wave fields is shown to yield scalar Helmholtz-equation solutions (in a half-space) that tend in the L2 limit of the mean to prescribed boundary values on an infinite plane boundary for all square-integrable boundary values. This result is employed to obtain several important properties of source-free wave fields, i.e., wave fields (satisfying square-integrable boundary values) that contain only homogeneous plane waves in their angular-spectrum representations: (1) the function describing a source-free wave field can be extended into the region behind the plane boundary to give a bounded continuous solution of the scalar Helmholtz equation in all space; (2) the function describing a wave field can be extended to the whole space of three complex variables as an entire function satisfying a certain inequality if, and only if, the wave field is source free; (3) the two-dimensional autocorrelation function of the wave field on planes parallel to the boundary plane is independent of the distance of the plane from the boundary if, and only if, the field is source free; and (4) a boundary value exists that produces a three-dimensional, pseudoscopic, real image of a wave field if, and only if, the wave field is source free. A series-mode expansion for source-free fields in terms of the boundary value is derived and is shown to be absolutely and uniformly convergent. The series is transformed into a two-dimensional Taylor series (with coefficients determined in terms of the boundary value) and another series that displays explicitly the contribution to the wave field due to each partial derivative of boundary value. The series are valid representations of wave fields in a broad class that is the natural extension of the source-free class of fields when the restriction that the boundary value be square integrable is removed. The series are used to derive two angular-spectrum representations for wave fields in the extended class and to discuss the properties of those wave fields.

© 1969 Optical Society of America

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