Inverse Diffraction Problem Research Articles

Solvability of certain inverse diffraction problems

Solvability of certain inverse diffraction problems

The Solution of Some Inverse Diffraction Problems in Terms of the Diffraction Coefficients

The form of the recently developed series representation of the axially symmetrical diffraction fields enables us to establish the pupil function from the series coefficients γk (u). A new interpretation of the Strehl intensity ratio is given. It represents the ratio of the energy coming to the image focus and that coming to the other foci of the filter. Further, by the use of the three-term approximation, the transmissivity of the filter giving the desired two-point resolution and attaining the maximum value of the Strehl criterion is found and compared with the exact solution proposed by Luneberg. Finally, it is shown that the Luneberg filter transmissivity represents a common approximative solution of the apodization problems that utilize the integral criteria of apodization.

Inverse wave propagation in an inhomogeneous waveguide

A solution is given for the problem of inverse propagation of a scalar wave in inhomogeneous rectangular two-dimensional waveguide. The sound speed is assumed to vary in depth and inverse propagation means the calculation of the field at range x1 in terms of the field at range x2 where x2≳x1. The method is analogous to that used by Wolf, Shewell, and Lalor for the inverse diffraction problem in a homogeneous half-space. It is found that the field at x1 can be expressed in terms of two integrals over the field at x2. The kernel of the first integral is bounded and expresses physically the result at x1 of the waves at x2 reversing their direction of propagation and decay, i.e., they now propagate and decay in the direction of x1. A reciprocity relation for this term is possible. The kernel of the second integral is singular and expresses the mathematical fact of the residual effect of the evanescent waves at x1 as they reverse their direction at x2 and now grow exponentially. Consequences of the neglect of this singular term are discussed.

The inverse diffraction problem for a reactance plane

The inverse diffraction problem for a reactance plane

Physical optics inverse diffraction

A general method for solving the inverse diffraction problem is presented. It is based on an identity of Bojarski which states that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\gamma(x)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Gamma(p)</tex> are a Fourier transform pair. Here <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\gamma(x)</tex> is the characteristic function of the target ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\gamma=1</tex> inside the target, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\gamma = 0</tex> outside), <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p = (2\omega/c)J,\omega</tex> is the frequency, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</tex> is a unit vector specifying the aspect, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Gamma(P)</tex> can be obtained by measurement of the backscattered electromagnetic far field at frequency <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega = (c/2)|P|</tex> and aspect <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J=|p|^{-1}p</tex> . If data is obtained in any subset <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</tex> of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> space, the method yields partial or complete information about the target geometry. It is used to rederive earlier results very simply and to obtain a significant new solution, in which the target geometry is completely determined using frequencies only in a practical frequency band and aspects in a narrow cone.

A new approach to the inverse diffraction problem

In recent studies of the inverse diffraction problem, a solution was sought in the form of a linear integral transform. In this paper a formal solution is obtained in terms of a differential rather than an integral operator. A useful representation for the differential operator, which is valid for fields whose spatial frequency spectrum is band-limited to a circle whose radius is equal to the wave number of the field, is also given.

Inverse Wave Propagator

In a recent publication, Wolf and Shewell gave a formal solution to the inverse diffraction problem, i.e., to finding the field distribution in the plane z = 0 from the knowledge of the field in an arbitrary plane z = z1 &amp;gt; 0 in the half-space into which the field is propagated. The solution involved the use of a singular kernel. In the present paper the inverse diffraction problem is treated in a rigorous manner. Our method makes use of the representation of the field as an angular spectrum of plane waves and demonstrates the usefulness of this type of representation. It is shown that by the use of a suitable truncation procedure one may avoid the use of a singular kernel or the generalized function theory.

Inverse Diffraction and a New Reciprocity Theorem*

A solution is obtained to the inverse diffraction problem for a monochromatic scalar wave field propagated into the half space z&gt;0. It is shown how to determine the field distribution throughout the region 0≤z&lt;z1 from the knowledge of the field distribution in the plane z=z1&gt;0. The solution takes a particularly simple form when the spatial-frequency spectrum of the distribution in the plane z=z1 (or in any other plane z=const&gt;0) is bandlimited to a circle whose radius is equal to the wavenumber of the field. In this case, the solution to the inverse diffraction problem may be expressed in a form strictly similar to that for the direct-propagation problem (exterior boundary-value problem), given by Rayleigh’s diffraction formula of the first kind. A comparison of these two solutions leads to the formulation of a new reciprocity theorem, valid for a wide class of wave fields.