The reflection of an electromagnetic plane wave by an infinite set of plates. I

Asymptotic behaviour of multiple scattering on an infinite number of parallel half-planes

“…We restrict ourselves to the case of the Dirichlet boundary conditions on the demi-plans treated in [7]. From the results of this work one can write down explicit expressions for reflection and transmission coefficients in the form of infinite products.…”

“…[7] are analyzed in semiclassical limit of large momentum. The most interesting (and difficult) case of small incident angle is considered.…”

A brief historical perspective of the Wiener–Hopf technique

“…The integral equation could then be solved, as described above, by applying Fourier Transforms and using the Wiener-Hopf technique. Although, somewhat cumbersome, the method provided a direct route for solving problems such as the famous Sommerfeld half-plane problem [9], and practical applications of and extensions to the technique began to appear frequently in the literature (see, for example, the now classic papers by Copson [10], Carlson and Heins [11,12] and Levine and Schwinger [13]). It was not, however, until Jones [14] demonstrated that the method could be considerably simplified by applying transforms directly to the boundary-value problem that the versatility and power of the method began to be realised.…”

Physical Optics of Metal Plate Media Part 1, Theoretical Considerations

“…It is well-known that there is no propagation within the medium for A > 2a and that for i < 2a the ratio of the phase velocity in free space to the phase velocity in the medium is 2 0 | I ~~~~~~~~~~~~~~~~~n (2) This quantity is called the index of refraction of the medium. Let k be the propagation i constant in free space; k = 217/X, and let K = nk be the corresponding quantity in the medium IEquation (2) is equivalent to I2 k 2 ss 2 + ().…”

“…If a new coordinate system is introduced with its origin 0' at (0, 0, 1), then a, exp (-jk 1 z cos 9 ) and as exp (jk 2 z cos 02) will be replaced by al' exp (-jkiz' cos 9) and as' exp (jk 2 In order that the preceding results be valid, it is not necessary that there should be a single plane or point of discontinuity. It is sufficient, that in some part of the space, say, for large negative z, there should be two plane waves progressing in opposite directions (or in directions making angles 9, and iT -91 with the z-axis), and that in another part of the space there should be a similar pair of waves.…”