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246   Chapter Eight


               infinitesimally thin bundle of rays B corresponding to small intervals
                  1 ,    2 around a central ray   1 ,   2 , and to z between z 0 and z 1 ,as
               shown in Fig. 8.3. By using Gauss’ theorem, this volume integral can
               be reduced to a surface integral, i.e.,


                                    2    3        2
                                                        a
                              ∇· A ∇	 d r =      A ∇	 · d  = 0      (8.28)
                                                  0
                                    0
                             B                 ∂B
               where ∂B refers to the outer surface of the bundle B, and d  is the
                                                                   a
               outward-pointing differential area element. It is easy to see that the
               only contributions to the surface integral come from the infinitesimal
               end faces of the bundle, since d  is perpendicular to the ray momen-
                                          a
               tum ∇	 at the sides of the bundle. Let the infinitesimally small area
               elements at both ends of the bundle be called  a 0 and  a 1 , respec-
               tively, so Eq. (8.28) can be written as
                                              2
                   2
                 A [X(z 0 ,  ),z 0 ]H(z 0 ,  )(− a 0 ) + A [X(z 1 ,  ),z 1 ]H(z 1 ,  )  a 1 = 0
                   0                          0
                                                                    (8.29)
               where the minus sign in the area element for the first term comes from
               the fact that ∇	 points into B at the beginning of the bundle segment
               and out of B at its end. In getting to this expression, we also used the
               fact that the z component of ∇	[X(z,  ),z] is simply H(z,  ). The in-
                                           2
                                                2
               tensity of the field is given by |A| ≈ A . The product of this intensity
                                                0
               and H (which is the refractive index times an obliquity factor) is pro-
               portional to the flux density traversing the area element. Therefore
                      y





                       Δa 0

                                    x

                    z 0


                                                               Δa 1

                                                   z 1  z


               FIGURE 8.3 Volume B, occupied by a segment of an infinitesimally thin
               bundle of rays.
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