time domain Modal model
General Formula
In the time domain modal model, a freely propagating field, E(x,y,z,t), in a vacuum is expressed using a set of Hermite-Gaussian functions in the following way.
Various field properties are defined in See Summary of field properties. The base of a Hermite-Gaussian function can be characterized by the waist size, w0, and the waist position. In the expression about, the origine of the z coordinate is chosen at the waist position, and the direction of the field propagation is chose to be the positive direction.
Because the Hermite-Gaussian funtions, See , provides a complete set in the (x,y) space and the field expression, See , is a solution of the paraxial approximation of the Maxwell's equation, one can choose any Hermite-Gaussian base, and once chosen, the field is fully described by the coeffients amn and z.
Propagation
An initial condition can be specified by a set of amplitudes, amn, of a chosen set of Hermite-Gauissian functions at a given time t at a given location z. Then the field propagation can be calculated by See , i.e., when an initial condition is given and the amplitudes of all Hermite-Gaussian modes are calculated, the propagation of fields, or field amplitudes at location z, can be calculated by multipliying the phase due to the normal wave oscillation, k (z-z), and the guoy phase change, (m+n+1)(η00(z)-η00(z)).
The simulation of e2e traces the change of coefficients amn. When a field moves, See is applied to update the coefficients and, when there is any interaction, the coefficient is updated using the matrix operation explained in the next subsection.
Operation
A operation on a field, like changing the propagation direction or dispalcing the waist position, can be represented by a mode decomposition matrix, 
, which converts one set of coeffieints amn to another.
In these equations, 
means that the value of z is evaluated at z after the operation is applied. An example is an operation to rotate the field direction discussed in the following section.
One important point to note is that the mode decomposition matrix is calculated at a given location. The matrix relates a set of amplitudes calculated at a position to another set of amplitudes at the same position. Then the propagator defined in See is used to calculate the field at different locations.
One explicit example will be in order. When the waist position of a field is shifted by Δz, the new field can be expressed by using the original field by shifting the z coordinate. A more general discussion is given below, but the following is an explicit calculation up to the first order effect.
basic operations
Rotation Operator
Intuitive meaning of Reflection operator metrix
Simple case reflection copuling from TEM00 model to TEM01 and TEM10 mode.
The meaning of the reflection matrix -i 2 θ is as follows. In the following calculation, the waist position is assumed to be at the reflection point, and z << z0.
and 
. Expression valid when 
Reflection by a mirror with non-matching cutvature
Basic
This is a special case of the Hermite-Gaussian base change. The reflection by a mirror is given by See and See .
L is the location of the mirror (distance from the waist of the field), and Rm and R(L) are the curvatures of the mirror and the incoming field on the mirror. If the mirror curvature matches with that of the beam, z'=-L and z0' = z0, i.e., the beam is just reflected back.
The curvature of the reflected field, Rref, is given by See . When the mirror curvature matches with that of the incoming field, Rref=-R(L), i.e., expanding (shrinking) field changes to shrinking (expanding) field.
Mode decomposition matrix of the reflection by curvature mismatched mirror
This is the Vinet's formula, except the sign of "i" due to the different convention of exp(-ikz).
As is derived in See curvature mismatch matrix, the curvature mismatch matrix can be written in the following way.
null test
Because the same phenomeca can be calculated using different set of Hermite-Gaussians, there are several cases which can be used to validate calculating by comparing results using different set of Hermite-Gaussians. In this section, two cases are studied. One is a field in a flat-curved FP cavity with the end mirror tilted, and the other is a field whose waist position is slightly shifted from that of an resonant mode of a FP cavity.
HarmitE Gaussian functions
Hm(x) is the Hermite polynomial of order m. The following relations are used repeatedly in the calculations which follow:
Eqn. See is the orthonormality condition; eqns. See and See are recursion relations to be used to derive Hermite polynomials of any order, beginning with H0(x) = 1.
steady state multimode fields in a fabry-peroit cavity
Steady state fields in a Fabry-Peroit cavity can be calculated by solving the following consistency equation See .
In this equation, Ain and Acav are arrays of coefficients, amn in See , of incoming and intra cavity fields.
tj and rj are amplitude transmittance and reflectance of mirror j. PRij is a propagator from mirror i to mirror j, corresponding to See , and MRj is a mode decomposition matrix disucssed in See basic operations.
When the field is propagating back from mirror 2 to mirror 1, the z axis direction is revsered, and the numerical value of Δη21 is the same as Δη12. Hereafter, Δη is used to denote the change of gouy phase from mirror 1 to mirror 2, Δη12.
When the Hermite-Gaussian base is the eigenstate of the FP cavity, there is no coupling among modes on reflection, i.e., MRi is diagonal. When the FP cavity is slightly off from this idealistic case, mode mixing occurs when the field is reflected by mirrors, i.e., MRi is not diagonal.
A mode in the cavity, Acav,mn, induced by a specific mode of the incoming field, Ain,0, can be rewriten in the following form. (The mode of the incoming field can be any. Here sufix 0 is used to clarify that Ain,0 refers to an amplitude of a certain mode.)
