Maxwell's equations are
ε ∂t E = rot H,
∂t H = - rot E.
Assuming that all partial derivatives of the fields with respect to z
are equal to zero, we get two independant modes:
the transverse magnetic (TM) mode with Ez ,
Hx , Hy field components;
the transverse electric (TE) mode with Hz ,
Ex , Ey field components.
For the TM mode Maxwell's equations are reduced to
ε ∂t Ez =
∂x Hy - ∂y Hx ,
∂t Hx =
- ∂y Ez ,
∂t Hy =
∂x Ez .
Discretization of the simulation domain for the TM polarisation
is shown below.
(Ez )i-1, j+1
—
(Hy )i-½, j+1
—
(Ez )i, j+1
—
(Hy )i+½, j+1
—
(Ez )i+1, j+1
|
(Hx )i-1, j+½
|
|
(Hx )i, j+½
|
|
(Hx )i+1, j+½
|
(Ez )i-1, j
—
(Hy )i-½, j
—
(Ez )i, j
—
(Hy )i+½, j
—
(Ez )i+1, j
|
(Hx )i-1, j-½
|
|
(Hx )i, j-½
|
|
(Hx )i+1, j-½
|
(Ez )i-1, j-1
—
(Hy )i-½, j-1
—
(Ez )i, j-1
—
(Hy )i+½, j-1
—
(Ez )i+1, j-1
Three field components (Hx )i, j+½ ,
(Hy )i+½, j ,
(Ez )i, j are placed in the same Float32 texel
(Hx, Hy, Ez, 0).
Therefore the finite-difference expressions for field components are
(Hy )x,y = (Hy )x,y +
[(Ez )x+d,y - (Ez )x,y ]dt/dx,
(Hx )x,y = (Hx )x,y +
[(Ez )x,y - (Ez )x,y+d ]dt/dx,
(Ez )x,y = (Ez )x,y +
[(Hx )x,y-d - (Hx )x,y +
(Hy )x,y - (Hy )x-d,y ]dt/dx.Electromagnetic wave simulation updated 7 Apr 2014