Berenger introduced the concept of a perfectly matched layer (PML) for reflectionless

absorption of electromagnetic waves, which can be employed as an alternative to the transparent boundary condition (TBC). The PML approach defines the truncation of the computation domain by layers (which absorb impinging plane waves) without any reflection, irrespective of their frequency and angle of incidence [15] – [18].

PML equations

We consider a 3-D optical waveguide surrounded by PML regions I, II, and III with thickness d as shown in Figure 2. Using the transversely scaled version of PML, Maxwell’s equations can be written as

Optical BPM - Equation 33-34

From Equation 33 and Equation 34 we get:

Optical BPM - Equation 35

with

Optical BPM - Equation 36

where

Optical BPM - Equation 37

Here E and H are the electric and magnetic field vectors, respectively, ω is the angular frequency, ε0 and μ0 are the permittivity and permeability of free space, respectively, n is the refractive index, and σe  and σm  are the electric and magnetic  conductivities of PML, respectively. The modified differential operator ∇’  used in Equation 33 and Equation 34 is defined as

Optical BPM - Equation 38

with

Optical BPM - Equation 39-40

where xˆ , yˆ , and zˆ are the unit vectors in the x , y , and z directions, respectively, and the values of sx are summarized in Table 1.

Optical BPM - Optical waveguide surrounded by PML

Figure 2: Optical waveguide surrounded by PML

Region

sx

sy

I

1

s

II

s

1

III

1

1

Table 1:   Values of sx and sy

The relation in Equation 37 is required to satisfy the PML impedance matching condition

Optical BPM - Equation 41

which means that wave impedance of PML medium exactly equals of the adjacent medium with refractive index n in the computation window, Optical BPM - Equation, regardless of the angle of propagation or frequency.

In the PML medium, we assume an m th-power of the electric conductivity as

Optical BPM - Equation 42

where ρ is the distance from the beginning of PML. Using the theoretical reflection coefficient R at the interface between the computational window and the PML medium

Optical BPM - Equation 43

the maximum conductivity σmax may be determined as

Optical BPM - Equation 44

where is the light velocity of free space.

The parameter s is written as

Optical BPM - Equation 45

where λ = 2πc ⁄ ω, d, x0, ρ are the free-space wavelength, the PML thickness, the position of the PML surface, and the theoretical reflection coefficient, respectively. Here, the PML is terminated with the perfect electric or magnetic conductor for TE or TM mode, respectively. Usually, a parabolic is assumed for the conductivity, =  2 .