where w0 is the beam radius,
p0 is the focal plane on the
z-axis,
Ebg0 is the background electric field amplitude and the spot radius for different positions along the propagation axis is given by
Note that the time-harmonic ansatz in COMSOL is
ejωt and with this convention, the beam above propagates in the +
z direction. The equations are modified accordingly for beams propagating along the other coordinate axes.
where the beam is assumed to be propagating in the z direction, the focal plane is spanned by the
x and
y coordinates,
e is the unit magnitude transverse polarization in the focal plane,
l and
m denote the indices for the wave vectors, the index
n accounts for the two polarizations per wave vector
klm,
almn is the amplitude,
un(
klm) is the unit magnitude polarization, and
r is the position vector.
Multiplying with the conjugate of the exponential factor above and the polarization factor un(
klm) and applying a surface integral over the entire focal plane allows us to extract the amplitudes as
where kt,lm is the magnitude of the transverse wave vector component.