Editing Ray transfer matrix analysis (section)

== Resonator stability ==
RTM analysis is particularly useful when modeling the behavior of light in [[optical resonator]]s, 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]] {{mvar|R}}, separated by some distance {{mvar|d}}. For the purposes of ray tracing, this is equivalent to a series of identical thin lenses of focal length {{math|1= ''f'' = ''R''/2}}, each separated from the next by length {{mvar|d}}. This construction is known as a ''lens equivalent duct'' or ''lens equivalent [[waveguide]]''. The {{abbr|RTM}} of each section of the waveguide is, as above,
<math display="block">\mathbf{M} =\mathbf{L}\mathbf{S} = \begin{pmatrix} 1 & d \\ \frac{-1}{f} & 1-\frac{d}{f} \end{pmatrix} .</math>

{{abbr|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 traveling down the waveguide will be periodically refocused 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 {{mvar|λ}} is equal to the output one. This gives:
<math display="block"> \mathbf{M} \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix} = \begin{bmatrix}x_2 \\ \theta_2\end{bmatrix} = \lambda \begin{bmatrix} x_1 \\ \theta_1 \end{bmatrix} . </math>
which is an [[eigenvalue]] equation:
<math display="block"> \left[ \mathbf{M} - \lambda\mathbf{I} \right] \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix} = 0 , </math>
where <math display="inline">\mathbf{I} = \left[\begin{smallmatrix} 1&0 \\ 0&1 \end{smallmatrix}\right]</math>  is the {{val|2|×|2}} [[identity matrix]].

We proceed to calculate the eigenvalues of the transfer matrix:
<math display="block">\det \left[  \mathbf{M} - \lambda\mathbf{I} \right] = 0 , </math>
leading to the [[Characteristic polynomial#Characteristic equation|characteristic equation]]
<math display="block"> \lambda^2 - \operatorname{tr}(\mathbf{M}) \lambda + \det( \mathbf{M}) = 0 , </math>
where
<math display="block"> \operatorname{tr} ( \mathbf{M} ) = A + D = 2 - \frac{d}{f} </math>
is the [[trace (linear algebra)|trace]] of the {{abbr|RTM}}, and
<math display="block">\det(\mathbf{M}) = AD - BC  = 1 </math>
is the [[determinant]] of the {{abbr|RTM}}.  After one common substitution we have:
<math display="block"> \lambda^2 - 2g \lambda + 1 = 0 , </math>
where
<math display="block"> g \overset{\mathrm{def}}{{}={}}  \frac{ \operatorname{tr}(\mathbf{M}) }{ 2 } = 1 - \frac{ d }{ 2 f } </math>
is the ''stability parameter''. The eigenvalues are the solutions of the characteristic equation. From the [[Quadratic equation#Quadratic formula and its derivation|quadratic formula]] we find
<math display="block"> \lambda_{\pm}  =  g \pm \sqrt{g^2 - 1} . </math>

Now, consider a ray after {{mvar|N}} passes through the system:
<math display="block"> \begin{bmatrix}x_N \\ \theta_N \end{bmatrix} = \lambda^N \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix}. </math>

If the waveguide is stable, no ray should stray arbitrarily far from the main axis, that is, {{mvar|λ{{sup|N}}}} must not grow without limit. Suppose {{nowrap|<math> g^2 > 1</math>.}} Then both eigenvalues are real. Since {{nowrap|<math> \lambda_+ \lambda_- = 1</math>,}} one of them has to be bigger than 1 (in absolute value), which implies that the ray which corresponds to this eigenvector would not converge. Therefore, in a stable waveguide, {{nowrap|<math> g^2 \leq 1</math>,}} and the eigenvalues can be represented by complex numbers:
<math display="block"> \lambda_{\pm} = g \pm i \sqrt{1 - g^2} = \cos(\phi) \pm i \sin(\phi) =  e^{\pm i \phi} , </math>
with the substitution {{math|1=''g'' = cos(''ϕ'')}}.

For <math> g^2 < 1 </math> let <math> r_+ </math> and <math> r_- </math> be the eigenvectors with respect to the eigenvalues <math> \lambda_+ </math> and <math> \lambda_- </math> respectively, which span all the vector space because they are orthogonal, the latter due to {{nowrap|<math>\lambda_+ \neq \lambda_-</math>.}} The input vector can therefore be written as
<math display="block"> c_+ r_+ + c_- r_- , </math>
for some constants <math> c_+ </math> and {{nowrap|<math> c_- </math>.}}

After {{mvar|N}} waveguide sectors, the output reads
<math display="block"> \mathbf{M}^N (c_+ r_+ + c_- r_-) = \lambda_+^N c_+ r_+ + \lambda_-^N c_- r_- = e^{i N \phi} c_+ r_+ + e^{- i N \phi} c_- r_- , </math>
which represents a periodic function.