# Laser Transverse Modes with Ray-Wave Duality: A Review
## 1 Introduction  
In paraxial approximation, the wave equation for spherical resonators was verified to be analogous to the Schrodinger equation for two-dimensional (2D) harmonic oscillators [1]. This analogy has been fruitfully employed to visualize the quantum wave functions from generating various laser transverse modes in spherical cavities [2]. The eigenmodes of the spherical cavities are analytically found to be Hermite-Gaussian (HG) modes in rectangular coordinates and Laguerre-Gaussian (LG) modes in polar coordinates [3]. The development of diode-pumped solid-state lasers enables us to experimentally generate both high-order HG and LG modes with the selectively focused excitation [4; 5; 6; 7; 8; 9; 10; 11]. Besides the HG and LG eigenmodes, the so-called geometric modes with ray-wave duality can be systematically generated in the degenerate cavities [12; 13; 14; 15; 16]. The observations of eigenmodes or geometric modes are confirmed to be dependent on the transverse order \(N\) and the frequency ratio \(\Delta f_{T}/\Delta f_{L}\), where \(\Delta f_{L}\) is the longitudinal mode spacing and \(\Delta f_{T}\) is the transverse mode spacing. The laser resonator that satisfies the condition of \(\Delta f_{T}/\Delta f_{L}=P/Q\) is called the degenerate cavity, where \(P\) and \(Q\) are coprime positive integers. The optical ray tracing in a degenerate cavity with \(\Delta f_{T}/\Delta f_{L}=P/Q\) can be found to be back to the initial point after the \(Q\) round tracings. To be brief, a periodic orbit comprises the ray tracing with the \(Q\) round trips in a degenerate cavity with \(\Delta f_{T}/\Delta f_{L}=P/Q\). Experimental observations displayed that if the transverse order \(N\) was less than the number \(Q\), the observed modes were often dominated by the HG or LG eigenmodes [4; 5]. However, the lasing modes were found to be governed by the geometric modes under the circumstance of \(N>>Q\)[13; 14; 15; 16]. One representation for the geometric modes was theoretically derived from the inhomogeneous wave equation [14]. The other representation proposed for expressing the geometric modes was the Fox-Li method by using an initial field matching the gain distribution [17]. Nevertheless, both methods cannot simultaneously represent the eigenmodes and geometrical modes. Recently, we have successfully exploited the formulism of quantum harmonic oscillators to develop an analytical theory to unify the representation for the eigenmodes and geometric modes [18]. The marriage between the resonant modes and the eigenfrequency distributions is of interest for exploring novel mesoscopic optics [19; 20; 21; 22]. Due to the analogy betweenquantum mechanics and optics, large-area vertical-cavity surface emitting lasers have been developed to explore the subtle relationship between level degeneracies and periodic orbits in mesoscopic quantum billiards [23; 24; 25; 26; 27]. Furthermore, two dimensional (2D) photonic systems analogous to quantum Hall [28] and quantum spin Hall [29] systems have been proposed and experimentally demonstrated in optical regime. Quantum level degeneracies have been found to link the conductance fluctuation to the prevalence of classical periodic orbits [30]. Analogous to mesoscopic quantum phenomena, the emergence of geometric modes has been evidenced to be related to the degeneracies of eigenfrequencies in spherical cavities [31; 32; 33; 34]. Laser geometric modes have been paid much attention because of various applications including ray-wave connection, light-matter interaction, optical communication, optical microscopy, and material processing [13; 14; 17; 35; 36; 37; 38]. One of the most striking geometric modes is the transverse intensities exhibiting the Lissajous figures [31; 32; 33; 34; 39; 40; 41]. In 1964, [PERSON] et al. [42] proved that under foldback conditions, an incident laser beam can be reflected forth and back between two spherical mirrors to exhibit an elliptical pattern.