# Laser Transverse Modes with Ray-Wave Duality: A Review
## 2 Ideal Spherical Cavity  
### Eigenfrequency Spectrum  
The round-trip propagation matrix for a stable spherical resonator is generalized as  
\[\mathbf{M}=\left[\begin{array}{cc}A&B\\ C&D\end{array}\right] \tag{1}\]with \(\det(M)=1\). Two eigenvalues of the matrix \(\mathbf{M}\) can be found to be  
\[\lambda_{\pm}=(A+D)/2\pm\sqrt{\left[(A+D)/2\right]^{2}-1} \tag{2}\]  
Conveniently, two eigenvalues are expressed as \(\lambda_{\pm}=e^{\pm i\theta}\), where  
\[\cos\theta=(A+D)/2 \tag{3}\]  
From [PERSON]'s theorem, \(\mathbf{M}^{Q}\) with a positive integer \(Q\) can be shown as [57]  
\[\mathbf{M}^{Q}=\left[\begin{array}{cc}\frac{\sin[(Q+1)\theta]-D\sin(Q\theta )}{\sin\theta}&\frac{B\sin(Q\theta)}{\sin\theta}\\ \frac{C\sin(Q\theta)}{\sin\theta}&\frac{D\sin(Q\theta)-\sin(Q-1)\theta]}{\sin \theta}\end{array}\right] \tag{4}\]  
If the parameter \(\theta/2\pi\) is equal to a rational number \(P/Q\) with \(P\) and \(Q\) to be coprime positive integers, Equation (4) can be used to confirm that \(\mathbf{M}^{Q}=\mathbf{I}\), where \(\mathbf{I}\) is the unit matrix. The feature of \(\mathbf{M}^{Q}=\mathbf{I}\) means that an optical ray inside the cavity is back to the initial position after the \(Q\) round trips. The fraction \(\theta/2\pi\) represents the ratio of the phase variations between the transverse and longitudinal parts. Therefore, the fraction \(P/Q\) is just the mode-spacing ratio \(\Delta f_{T}/\Delta f_{L}\), where \(\Delta f_{T}\) and \(\Delta f_{L}\) are the frequency spacings for the transverse and longitudinal modes, respectively. Note that the value of \(Q\) is infinite for the ratio \(\Delta f_{T}/\Delta f_{L}\) to be an irrational number. From Equation (3), the mode-spacing ratio can be explicitly given by  
\[\frac{\Delta f_{T}}{\Delta f_{L}}=\frac{1}{2\pi}\cos^{-1}\left(\frac{A+D}{2}\right) \tag{5}\]  
Considering a concave-flat cavity with a mirror with radius of curvature \(R_{c}\) at \(z=-L_{c}\) and a flat output coupler at \(z=0\), the round-trip propagation matrix starting from \(z=0\) is  
\[\left[\begin{array}{cc}A&B\\ C&D\end{array}\right]=\left[\begin{array}{cc}1-\frac{2L_{c}}{R_{c}}&2\Big{(} 1-\frac{L_{c}}{R_{c}}\Big{)}\\ -\frac{2L_{c}}{R_{c}}&1-\frac{2L_{c}}{R_{c}}\end{array}\right] \tag{6}\]  
Using Equations (5) and (6), the frequency ratio \(\Delta f_{T}/\Delta f_{L}\) can be obtained as  
\[\frac{\Delta f_{T}}{\Delta f_{L}}=\frac{1}{\pi}\sin^{-1}\left(\sqrt{\frac{L_{ c}}{R_{c}}}\right) \tag{7}\]  
Consequently, the eigenfrequencies of the cavity can be expressed as  
\[f(\mathbf{n};L_{c})=\left(n_{1}+n_{2}+1\right)\ \Delta f_{T}+n_{3}\ \Delta f_{L} \tag{8}\]  
where \(\mathbf{n}=(n_{1},n_{2},n_{3})\), the integers \(n_{1}\) and \(n_{2}\) are the transverse mode indices, and the integer \(n_{3}\) is the longitudinal mode index. Let \(\mathbf{n}_{c}\) be the index for the central mode, the relative eigenfrequency distribution can be expressed as  
\[\begin{array}{ll}F(\delta\mathbf{n};L_{c})&=\frac{f(\mathbf{n}_{c}+\delta \mathbf{n};L_{c})-f(\mathbf{n}_{c};L_{c})}{\Delta f_{L}}\\ &=(\delta n_{1}+\delta n_{2})(\Delta f_{T}/\Delta f_{L})+\delta n_{3}S\end{array} \tag{9}\]  
where \(\delta\mathbf{n}=\mathbf{n}-\mathbf{n}_{c}=(\delta n_{1},\delta n_{2},\delta n _{3})\). The connection between geometric modes and eigenfrequency degeneracies can be manifested from the relative spectrum \(F(\delta\mathbf{n};L_{c})\). Figure 1 shows the spectrum \(F(\delta\mathbf{n};L_{c})\) as a function of \(L_{c}/R_{c}\) for \(|\delta n_{i}|\leq 10\) with \(i=1,2,3\). Numerous degeneracies and gaps with \(\Delta f_{T}/\Delta f_{L}=P/Q\) can be seen to appear at the effective cavity lengths of \(L_{c}=L_{P/Q}=R_{c}\sin^{2}(\pi P/Q)\).