# OptiBPM Tutorials

## Full H-vector Formulation

The FD-VBPM based on the E and H fields are equivalent and yield almost identical results . 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…

## References

           [Berenger, 1994] J. P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys., No. 114, 1994, pp. 185-200.            [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,   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…