Periodic Port Mode Fields
Periodic, Diffraction Order, and Orthogonal Polarization Port features all use plane-wave electric mode fields of the form
,
where Em, km, and r are the amplitude, the wave vector, and the position vector, respectively. Here, m is the mode index. Since this field represents a plane wave, the amplitude must be orthogonal to the wave vector,
.
As plane-wave mode fields are assumed, the material properties in the domain adjacent to the port boundary must be homogeneous and isotropic.
As discussed in Additional Variables for Periodic Structure Calculations, the wave vector for the plane wave for mode m is defined by
,
where k is the wave number for the domain adjacent to the port boundary, α1m and α2m are the elevation and azimuth angles for mode m, a1 and a2 are the reference axes, and n is the port boundary normal.
A linearly polarized plane wave can have one of two polarization states — either TE or TM polarization (also called s and p polarizations). For TE polarization, the electric field is polarized in a direction orthogonal to the plane of incidence, spanned by the wave vector and the port boundary normal. Thus, the unit polarization vector for TE polarization can be written (the notation s and p will be used in the following equations for TE and TM polarization, respectively)
.
For TM polarization, the electric field must be orthogonal both to TE polarization and the wave vector. Thus, the TM unit polarization vector can be written as
.
From the definitions of the two polarization vectors, it is clear that the cross product of the TM and TE polarization vectors gives a vector in the direction of the wave vector.
For right-handed circularly polarized (RHCP) light, if you align the right hand thumb with the wave vector, the remaining fingers should point in the rotation direction when time changes, but the position is fixed. Thus, defining the RHCP polarization vector as
,
will give the following temporal evolution of the polarization vector
.
As seen from this expression, at t = 0, polarization is along the TE direction. One quarter of a period later, the polarization is along the negative TM direction. This means that the rotation direction is from (positive) TM to (positive) TE direction. So, the definition for the RHCP polarization vector above satisfies the RHCP definition.
For left-handed circularly polarized light (LHCP), the polarization vector is defined by
.
For the Periodic Port, when Polarization is set to User defined, the amplitude E0 is provided by the user (we set m = 0 here, as the Periodic Port represents the lowest diffraction order). The amplitude E0 can be expressed in terms of the two linear polarization vectors,
.
The amplitude for the Orthogonal Polarization Port is orthogonal to the amplitude of the Periodic Port. That is,
,
where Eorths and Eorthp are the expansion coefficients for the orthogonal polarization mode field Eorth and (E0)* means conjugation of the amplitude E0. To satisfy the equation above, Eorth can be defined as
.
For Diffraction Order ports, the amplitude for out-of-plane modes represents TE polarization and the amplitude for in-plane modes represents TM polarization.
The mode fields described above are the unnormalized mode fields. The normalized mode fields are scaled to produce a mode power that equals the specified (port) input power.