Step-profile waveguides

The periodic shape of the ray path in Fig. 1-4 is a consequence of the translational invariance of the waveguide, and gives rise to a ray invariant which is constant along the path and specifies the ray direction at any position in the cross-section. For the step-profile waveguide, invariance is expressed by Eq. (1-4), so we set... 

Fig. 3-1 Pulse propagation is described in terms of a superposition of bound rays, each ray propagating distance z along (a) a step-profile waveguide or (b) a graded-profile waveguide.

The clad power-law profiles for planar waveguides are defined by Eq. (1-59) and illustrated in Fig. 1-10. Here we determine which of these profiles, i.e. the value of q, gives minimal pulse spreading. Unlike the step-profile waveguide, it is not immediately obvious from the geometry of the sinusoidal-like ray paths... 

Fig. 6-1 The electric field intensity IE decreases exponentially beyond (a) the interface of a step-profile waveguide and (b) the turning-point caustic of a graded-profile waveguide.

Step-profile waveguides and the scalar wave equation... 

This equation is sometimes referred to as the scalar wave equation. However, in this book we describe a field component as a solution of the scalar wave equation only if (i) it satisfies Eq. (11-45) for all values of x and y, including the interface, and consequently (ii) is continuous and has continuous first derivatives everywhere - see Section 33-1. This is not the case for any cartesian component of on a step-profile waveguide of arbitrary cross-sectional shape, since the Vj In n terms are nonzero at the interface. Thus, to solve for Cgj everywhere, we impose the boundary conditions of Maxwell s equations on the solutions of Eq. (11-45) derived in each homogeneous region. The component h j is derived similarly. Alternatively we can solve Eq. (11-44) with V lnn terms retained. The transverse components then follow from Eq. (11-43). 

The step-profile waveguide has a core of uniform refractive index n, surrounded by a cladding of uniform refractive index n, which is assumed unbounded. Thus the only variation in profile is a step, or jump, discontinuity at the core-cladding interface in Fig. 11-1 (a). This profile has exact analytical solutions for the modal fields on planar waveguides, circularly symmetric fibers and elliptical fibers. 

The simplest example of a step-profile waveguide is the symmetric planar waveguide of Fig. 12-1 with profile... 

The derivation of the modal fields, given above, is based on the longitudinal components, for which the longitudinal components of the vector wave equation decouple on step-profile waveguides, as explained in Section 11-15. However, for the special case of the planar waveguide, the TE and TM mode fields can also be derived starting with g, and hy, respectively, as we show later in Sections 12-14 and 12-15. 

The construction of ray paths within the core of the step-profile waveguides of Chapters 1 and 2 is based on straight-line trajectories, which are solutions of the ray-path equation of Eq. (1-18) in a uniform medium. When the core is graded, the cartesian component equations of the ray-path equation follow directly, as in Eqs. (1-19) and (2-49). Here we derive the corresponding component equations in directions defined by the cylindrical polar coordinates (r, 0, z) of Fig. 2-1, for application to fibers with graded profiles n(r) in Chapter 2, and, by simple generalization, to slowly varying fibers with profiles n(r, z) in Chapter 5. 

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