Ray Transfer Matrix Analysis - Resonator Stability

Resonator Stability

RTM analysis is particularly useful when modeling the behaviour of light in optical resonators, such as those used in lasers. At its simplest, an optical resonator consists of two identical facing mirrors of 100% reflectivity and radius of curvature R, separated by some distance d. For the purposes of ray tracing, this is equivalent to a series of identical thin lenses of focal length f=R/2, each separated from the next by length d. This construction is known as a lens equivalent duct or lens equivalent waveguide. The RTM of each section of the waveguide is, as above,

.

RTM analysis can now be used to determine the stability of the waveguide (and equivalently, the resonator). That is, it can be determined under what conditions light travelling down the waveguide will be periodically refocussed and stay within the waveguide. To do so, we can find all the "eigenrays" of the system: the input ray vector at each of the mentioned sections of the waveguide times a real or complex factor λ is equal to the ouput one. This gives:

.

which is an eigenvalue equation:

,

where I is the 2x2 identity matrix.

We proceed to calculate the eigenvalues of the transfer matrix:

,

leading to the characteristic equation

,

where

is the trace of the RTM, and

is the determinant of the RTM. After one common substitution we have:

,

where

is the stability parameter. The eigenvalues are the solutions of the characteristic equation. From the quadratic formula we find

Now, consider a ray after N passes through the system:

.

If the waveguide is stable, no ray should indifinetly move off the main axis, that is, λN must not grow without limit. Suppose, then both eigenvalues are real. Since, one of them has to be bigger than 1 (in absolute value), what implies that the ray, which corresponds the propagation of this eigenvector, would not converge. Therefore, ≤ 1, and the eigenvalues can be represented by complex numbers:

,

with the substitution g = cos(ϕ).

For let and be the eigenvectors with respect to the eigenvalues and respectively, which span all the vector space because they are orthogonal, the latter due to ≠ . The input vector can therefore be written as

,

for some constants and .

After N waveguide sectors, the output reads

,

what represents a periodic function.