Modes in Slab waveguides Consider a plane wave polarized in the y direction and propagating in z direction in an unbounded dielectric medium in the Cartesian coordinates. The vector wave equations(42. 6) lead to the scalar equations d-e 0 (42.9a) d-H a2H=0 (42.9b) The solutions are E= Aej(t-Kz (42.10a) EA (42.10b) n where a is a constant and n= whe is the intrinsic impedance of the medium Because the film is bounded by the upper and lower layers, the rays follow the zigzag paths as shown in Fig. 42.3. The upward and downward traveling waves interfere to create a standing wave pattern within the film the fields transverse to the z axis, which have even and odd symmetry about the x axis, are given, respectively, by E,=A cos(hy)e/t-pa) (42.11a) Ey= A sin(hy) (42.11b) where Band h are the components of K parallel to and normal to the z axis, respectively. The fields in the upper and lower layers are evanescent fields decaying rapidly with attenuation factors a, and a2, respectively, and are b E= A,e (42.12b) Only waves with raypaths for which the total phase change for a complete(up and down) zigzag path is an egral multiple of 2T undergo constructive interference, resulting in guided modes. Waves with raypaths not satisfying this mode condition interfere destructively and die out rapidly. In terms of a raypath with an angle of incidence 8, =0 in Fig. 42.3, the mode conditions [Haus, 1984] for fields transverse to the z axis and with ven and odd symmetry about the x axis are given, respectively, by tan sin-8-n2 (42.13a) sin e (42.13b) e 2000 by CRC Press LLC© 2000 by CRC Press LLC Modes in Slab Waveguides Consider a plane wave polarized in the y direction and propagating in z direction in an unbounded dielectric medium in the Cartesian coordinates. The vector wave equations (42.6) lead to the scalar equations: (42.9a) (42.9b) The solutions are Ey = Aej(wt – kz) (42.10a) (42.10b) where A is a constant and h = is the intrinsic impedance of the medium. Because the film is bounded by the upper and lower layers, the rays follow the zigzag paths as shown in Fig. 42.3. The upward and downward traveling waves interfere to create a standing wave pattern. Within the film, the fields transverse to the z axis, which have even and odd symmetry about the x axis, are given, respectively, by Ey = A cos(hy)ej(wt – bz) (42.11a) Ey = A sin(hy)ej(wt – bz) (42.11b) where band h are the components of k parallel to and normal to the z axis, respectively. The fields in the upper and lower layers are evanescent fields decaying rapidly with attenuation factors a3 and a2 , respectively, and are given by (42.12a) (42.12b) Only waves with raypaths for which the total phase change for a complete (up and down) zigzag path is an integral multiple of 2p undergo constructive interference, resulting in guided modes. Waves with raypaths not satisfying this mode condition interfere destructively and die out rapidly. In terms of a raypath with an angle of incidence qi = q in Fig. 42.3, the mode conditions [Haus, 1984] for fields transverse to the z axis and with even and odd symmetry about the x axis are given, respectively, by (42.13a) (42.13b) ¶ ¶ ¶ 2 2 0 E z E y - = y ¶ ¶ ¶ 2 2 0 H z H x - = x H E A e x y jt z = - = - h h ( ) w k me E Ae e y jt z y d = - - Ê Ë Á ˆ ¯ ˜ - 3 3 2 a ( ) w b E Ae e y jt z y d = - + Ê Ë Á ˆ ¯ ˜ - 2 2 2 a ( ) w b tan cos sin hd n n n 2 1 1 1 2 2 2 2 Ê 1 2 Ë Á ˆ ¯ ˜ = - [ ] q q / tan cos sin hd n n n 2 2 1 1 1 2 2 2 2 1 2 - Ê Ë Á ˆ ¯ ˜ = - [ ] p q q /