We consider a planar waveguide where x and z are the transverse and propagation directions, respectively, and there is no variation in the y direction ( ∂ ⁄ ∂y ≡ 0 ) .

Furthermore we consider that the planar optical waveguide with width W is surrounded by PML regions with thickness d as shown in Figure 3.

Optical BPM - Planar waveguide surrounded by PML

Figure 3: Planar waveguide surrounded by PML

With these assumptions and transversely-scaled version of PML, as defined in “Perfectly Matched Layer (PML)” on page 26, we get the following basic equation from Equation 35:

Optical BPM - Equation 46

with

Optical BPM - Equation 47- 48

where Ey  and Hy  are the y components of the electric and magnetic fields respectively, ω is the angular frequency, ε0  and μ0  are the permittivity and permeability of free space, respectively, n is the refractive index, k0  is the free-space wave number, and the parameter s is defined by Equation 45.

We separate the field Φ( x, z ) into two parts: the axially slowly varying envelop term of φ( x, z ) and the rapidly term of exp ( –jk0 nre f z ) . Here, nre f  is the reference index.

Then, Φ( x, y ) is expressed by

Optical BPM - Equation 49

By substituting Equation 49 into Equation 46, we obtain the following equation for slowly varying complex amplitude φ :

Optical BPM - Equation 50

The term ∂p ⁄ ∂is neglected for TM modes.

Finite Difference Approximant

To obtain the field solution at each cross section we discretise Equation 50 using Finite Differences scheme along x – direction [23] – [27].

Optical BPM - Finite Difference uniform mesh

Figure 4: Finite Difference uniform mesh

Formally, we have from Taylor expansion:

Optical BPM - Equation 51-52

Subtracting Equation 51 from Equation 51 and neglecting higher order terms:

Optical BPM - Equation 53

Thus, for TE modes we get

Optical BPM - Equation 54

Here we consider:

Optical BPM - Equation 55

Optical BPM - Equation 56

By substituting Equation 55 and Equation 56 into Equation 54, we get:

Optical BPM - Equation 57

Therefore, we can rewrite Equation 50 for TE modes as

Optical BPM - Equation 58

where

Optical BPM - Equation 59

Here,

Optical BPM - Equation 60-62

For TM modes we have:

Optical BPM - Equation 63

Applying FD scheme in the third term of Equation 63 we obtain:

Optical BPM - Equation 64

For small h one has:

Optical BPM - Equation 65-68

By substituting Equation 66, Equation 67, and Equation 68 into Equation 64, we get:

Optical BPM - Equation 69

Thus, using FD scheme into Equation 50 for TM modes, we get:

Optical BPM - Equation 70

with

Optical BPM - Equation 71-74