By Emil Wolf (ed.)

ISBN-10: 0444501045

ISBN-13: 9780444501042

Provides 5 articles at the following themes of analysis in optics: advanced rays; homodyne detection and quantum-state reconstruction; scattering of sunshine in eikonal approximation; the orbital angular momentum of sunshine; and the optical Kerr impact and quantum optics in fibres.

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**Example text**

5b) The resulting field outside the caustic therefore takes the form u = A I eik@l + A2eik@ For points inside the caustic, however, a1,2, z1,2, and a a1,2= g, f i Arccosh-, T ~ =,T ~ i m , r v1,2 = ag, f i (a Arccosh! 7) m), where Arccosh(s) = In (s + fi) , and the amplitudes A1,2 are given by A~ = ceini4(a2 - r 2 -1/4 ) , = ce-'3~'4 (a2 - y 2 -1/4 ) . 8) Contours of phase, that is, of v', correspond to radial lines, whereas the amplitude contours are concentric circles. 29 COMPLEX RAYS IN PHYSICAL PROBLEMS I , § 41 The complex rays in this case are given by x 1 , ~= x0(a1,2)+ T P ~ ~ u ) , + ~pyO(a1,2), ~ 1 .

Shin and Felsen [1978] studied multiple reflection of Gaussian beams, and Kudou and Yokota [199I] analyzed reflection and transmission of Hermite-Gaussian beams incident on a curved dielectric layer. Norris [19861 considered the propagation of a Gaussian beam through a spherical interface. Analytic continuation of the interface into complex space is common in this work. 4. GAUSSIAN BEAMS AND COMPLEX RAYS 49 DIFFRACTION OF GAUSSIAN BEAMS The representation of a Gaussian beam as a spherical or cylindrical wave from a complex source gives an effective method for the solution of a wide class of diffraction problems.

Ik(z - ZO) + ~ The paraxial approximation converts 2(2 - zo) 44 THEORY AND APPLlCATIONS OF COMPLEX RAYS With zo = ikw; Gaussian beam: u(x, z ) = = [I, 55 i a , R is complex and eq. 3) The beam’s waist falls in the plane z = 0 where the field is proportional to exp(-x2/2w;), and a is its Rayleigh range. 5) This use of complex sources was introduced repeatedly in the late 1960’s. Kravtsov [1967a,b] showed that the 2D Gaussian beam of eq. 3) has its focus at ( x , z ) = (0,ikw;). Similarly, Deschamps [1967, 1968, 19711 and Keller and Streifer [1971] identified eqs.

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