Mathematical Formulations
We consider a longitudinal varying 3-D optical waveguide, as shown in Figure 1, and the transverse cross-section of the optical waveguide surrounded by PML regions I, II, and III with thickness d, as shown in Figure 2, where x and y are the transverse directions and z is the propagation direction. Figure 1: Longitudinally varying 3-D optical…
Continuous Wave Equations for Anisotropic Media
Consider an anisotropic medium described by relative electric permittivity Therefore, the propagation of the electromagnetic field is described by the curl-curl equation where Φ , p , and q are shown in Table 6. Φ p q E 1 ε ˜ E ε–1 ˜ 1 Table 6: Definitions of Φ , p , and…
Full E-vector Formulation
E-Formulation: The vector wave equation for the electric fields On the transverse plane Equation (19) becomes: The solution of the wave equation Equation 31 can be separated as a slowly varying envelope and a fast-oscillating phase term: Using Gauss’ Law, we get: If the refractive index varies slowly along the propagation direction , which is valid…
Introduction
Modal analysis for optical waveguides is one of the most important area in modeling and simulation of guided wave photonic devices. The problem of the numerical boundary condition is expected to become much more accurate for the leaky modes, as the modal fields at the edges of the computation window are traveling waves and the…
Appendix I
E-Formulation The full vectorial wave equation is given by: Considering into Equation 1, we get: The term is null. We can separate Equation 2 into two equations, one for longitudinal terms and another for transversal terms: for longitudinal terms and for the transversal terms. By substituting Equation 3 into the first right hand term of Equation 4,…
Full H-vector Formulation
The FD-VBPM based on the E and H fields are equivalent and yield almost identical results [11]. Similar to the vector wave equation for the electric field, the equation for the magnetic field considering transversely scaled version of PML is: here . The double-curl Equation 1 involves three vector components of the magnetic field, while strictly…
H-Vectorial Modal Analysis for Anisotropic Waveguide
Assuming ∂ ⁄ ∂z = 0 and regarding n0 as an effective refractive index, Equation 23 is reduced to a basic equation for the guided-mode analysis of anisotropic optical waveguides. Hence, from (9) we get the following Helmholtz equation: One can cast the Helmholtz equation into the following matrix form: where and The differential operators are defined as:…
Appendix II
H-Formulation The full vectorial wave equation for H is given by: By substituting into Equation 1 we get: The term is null. We can separate Equation 2 into one for longitudinal terms and another one for transversal terms: for longitudinal terms, and By substituting into the first right hand term of Equation 4, we get: By substituting Equation…
H-Formulation
References
[1] [Berenger, 1994] J. P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys., No. 114, 1994, pp. 185-200. [2] [Teixeira, 1998] F. L. Teixeira and W. C. Chew, ” Systematic Derivation of Anisotropic PML Absorbing Media in Cylindrical and Spherical Coordinates”, IEEE Microwave and Guided Lett., vol. 8, No. 6,…
Introduction
Many kinds of optical fiber can be described, or in the case of graded index fibers, approximated by, a series of concentric layers of loss-less dielectric. When the index contrast in the structure is small, it is common to use the scalar wave equation to obtain the linearly-polarized modes (LP modes). However, when the index contrast…
Real-valued formulation
The time independent Maxwell curl equations with a positive time convention (ej ω t ) are and the divergence equations are The factor j does not suit the current purpose, so make the substitution where n0 = is the free space impedance, so that h and E have the same units. Then where . Eliminating the h gives the…
Debye Potential
For finding electromagnetic fields in a cylindrical geometry it is convenient to use cylindrical coordinates r, θ and z, and Debye potentials parallel to the axis of rotation, [2] It is supposed that the permittivity is constant in the region in which the above equations apply. The fibers are defined as a series of concentric layers of…
Separation of Variables
The potentials themselves are solutions of the scalar Helmholtz equation, and the particular solution is found by observing the boundary conditions imposed by physical considerations on E and h. The potentials are supposed to form modes, so a solution where the variables are separated is appropriate and a similar expression applies for the other Debye potential, ϕ…
Solution of the Linear System
The order of the components in the vector f is chosen to help factor the matrix in (9) into two matrices as Ff = Dc where c = [ A, B, C, D ]T , The advantage of this factoring is that it is easy to find the expressions for F–1 and D–1 . F–1 is found by…