THE METHOD OF STATIONARY PHASE 15 (i) has an amplitude equal to l/ \ times the amplitude of the primary wave where A = lir/k is the wave length, and (ii) its phase is one quarter of a period ahead of the primary wave. (This is a way of interpreting the factor /.) Fresnel made these two assumptions directly in his formulation of Huygens' principle, and this led many to regard his theory as being ad hoc, As we have seen, they are a consequence of the method of stationary phase and Helmholtz's formula. We still must discuss the question of when the critical points are non- degenerate. We shall treat points of type (a) the points of type (b) can be treated in an identical manner. Actually, the discussion is almost the same as in our treatment of emitted radiation: Let us define the "exponential map" E: 5 x R + ^ R 3 b y E{y,r) = y + rgradcpO). Then the critical points on S associated with a point P consist precisely of those y such that E(y,r) = P, where r = \\y - P\\. If grad p(y) is not tangent to S, then E is a diffeomorphism near (y, 0). Under this assumption we proceed as on page 9. (We can consider the situation on page 9 as the special case where grad cp is everywhere normal to S.) One shows that y is a degenerate critical point for P = E(y, r) if and only if (y, r) is a point at which the map E is singular. In this case we call P a focal point of the map E at y. As before, if P is not a focal point, then the index of the Hessian of p + r at y is the number of focal points on the ray segment from y to P (counted with multiplicity). We leave the details to the reader. Finally, we should observe that if grad tp is close to the normal to a surface S, then the focal points of the map E associated to p will be close to the corresponding focal points of the surface. In this way we get an explanation of the Gouy double mirror experiment mentioned at the beginning of the Chapter. REFERENCES, CHAPTER I 1. L. G. Gouy, C. R. Acad. Sci. Paris 110 (1890), 1251. 2. H. Poincare, Theorie mathematique de la lumiere, II, George Carre, Paris, 1892, pp. 168-174. 3. J. W. Milnor, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N. J., 1963. MR 29 #634. 4. P. D. Lax and R. S. Phillips, Scattering theory, Pure and Appl. Math., vol. 26, Academic Press, New York and London, 1967. MR 36 #530. 5. B. B. Baker and E. T. Copson, The mathematical theory of Huygens*principle, Clarendon Press, Oxford, 1953. 6. J. Larmor, Proc. London Math. Soc. (2) 19 (1919), 169-180. 7. M. Born and E. Wolf, Principles of optics, 4th ed., Pergamon Press, Oxford and New York, 1970. 8. R. S. Palais, Bull. Amer. Math. Soc. 75 (1969), 968-671. MR 40 #6593. 9. M. Golubitsky and V. Guillemin, Advances in Math. 15 (1975), 2.
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