Asymptotic Expansion of Multiple Integrals and the Method of Stationary Phase

An Approach to Ray Optics in Anisotropic Media

“…and M r«zl (15) With the above definitions, (7) and (8) can be rewritten as t t (18) where W = W -"W W t tz zt (19) and the equivalent transverse source transform I is given by « = * -W W • t t tz z z (20) The elements of the product matrix W W in (19) (21) Observe that W is a function of z in that • = e i ^O T z < 0 while for z > 0, e = e _ .…”

“…and L _ of (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24) are zero and hence m 1 mZ L > L ., L _ so that 6 in (11-26) is given by (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27). Thus, writing m ml me mn L = L = v : ' + y^ + z , since L is the same for all rays, and writing mm 6 as 6 , the stationary point or direct ray contributions to (18) In (22), the stationary point summation is over all points on the m 1^1 branch of the dispersion surface of the plasma medium at which v is in the direc--m tion of the displacement from the slot to the observation point.…”

“…Thus, for this combination of signs of th» Gaussian curvatures, and for values of cp such that cos . : lies between the roots of (C-ll), the values of L 2 m for which P P -P = 0 will be complex and consequently (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24) will have complex roots L , and L _. For real r and when cos ■ is not ml m2 between the roots of (C-ll), Q is the positive and P P-P will To derive the number of tangencies of the scattered ray to an actual caustic, as given in Table C which, as previously proved, is independent of a .…”

“…In studying the roots L , and L " of (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24) for the rays scattered ml m2 from a planar interface, those combinations of signs of the principal curvatures of the m and n branches of the dispersion surfaces for which the roots are real and those combinations for which the roots can b<» com-2 plex are first found. Since the Hessian P P P ~ is an invariant, it can be evaluated in the (a,T) coordinate system defined by the require- where the 1 and 2 subscripts refer to the partial derivatives of K {^,T\).…”

An Efficient Method for Computing Highly Oscillatory Physical Optics Integral

“…Traditional asymptotic expansion approximations [3,18,[38][39][40][41][42][43] are widely used in computational electromagnetics, optics, acoustics and geophysics areas. The asymptotic expansion approximation for the highly oscillatory PO integral [10,[17][18][19]] is a frequencyindependent approach.…”