By Lawrence M. Graves
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Extra resources for Calculus of Variations and its Applications: Proceedings of the Eighth Symposium in Applied Mathematics of the American Mathematical Society
Sample text
To make our theory quantitative, we associate a field u(s) with each ray. It is composed of an amplitude A (s) and a phase W(s) in the form (19) u(s) = A(s)e°l;*( . In (19), k = w/c is the propagation constant, determined by the angular frequency w of the field and the propagation velocity c in empty space. Equivalently, k = 21r/a, where X is the wavelength of the field in empty space. Thus our construction applies to a time-harmonic field. The time factor a-i" will be omitted. The total field at a point P is the sum of the fields (19) on all rays through P.
This is indeed the case, and is of particular importance in using our theory for the quantitative calculation of intensities. Away from discontinuity or boundary surfaces, all rays-both ordinary and diffracted-are determined by the usual laws of geometrical optics. These laws, plus the laws of reflection and refraction and the foregoing laws governing diffraction, completely determine all real rays. 32 JOSEPH B. KELLER As a consequence of the theory of diffraction just described, diffracted rays will exist, in addition to ordinary rays, in any medium which is bounded or in which the refractive index or any of its derivatives is discontinuous.
The value of s is the complex distance from (x,y) to the point of tangency, and (x,y) is the only real point on the ray. As an example, suppose that C is a circle of radius a. Let the polar coordi- A GEOMETRICAL THEORY OF DIFFRACTION 43 nates (a,o) denote a point on C and let (r,0) denote a point off C. Then (1) becomes r cos 0 = a cos 0 - s sin r sin 0 = a sin ¢ + s cos (2) Solving these equations for the point of tangency 0 and the distance s, we obtain = 0 ± cos-, (3) s= (4) { a r r2 - a2. From (3) we see that, for r ?