Greatness in Simplicity: a Review of Laser Transverse Modes of Spherical Resonators

The SU(2) transformation in the Jordan-Schwinger map is exploited to derive the wave function for the Hermite-Laguerre-Gaussian (HLG) modes which are the generalized transverse modes of spherical laser resonators. The image is numerical patterns for the high-order HLG mode.

The quantum harmonic oscillator is an indispensable example for understanding the quantum classical correspondence, quantifying the concepts of radiation fields and quantum optics. The eigenmodes of two-dimensional (2D) quantum harmonic oscillators can be resolved into Hermite-Gaussian (HG) modes with rectangular symmetry or Laguerre-Gaussian (LG) modes with circular symmetry. Since the paraxial wave equation of the spherical laser cavity is the same as the Schrödinger equation of the two-dimensional harmonic oscillator, the HG and LG eigenmodes play important roles in the exploration of laser transverse modes. With the advent of end-pumped configurations, high-order HG and LG modes can be efficiently generated in diode-pumped solid-state lasers. The Ince-Gaussian (IG) mode is another eigenfunction form of the paraxial wave equation, which has recently been introduced, and it has also been experimentally observed in a stable cavity.

The general form that characterizes scalar monochromatic light waves is a complex-valued function, which can be expressed as real and imaginary parts. The intersections of the nodal lines of the real and imaginary parts are all isolated zeros, the so-called phase singularities. Since singularities are surrounded by cyclic phases, they are usually related to the formation of optical vortices. Therefore, the terms phase singularity and optical vortex are often used interchangeably. Optical vortex beams carrying orbital angular momentum (OAM) can be used in various fields, including optical tweezers, trapping and guiding cold atoms, radio communications, and quantum information processing. Several techniques have been proposed to generate optical vortex beams. One approach with an astigmatic mode converter (AMC) has been widely used to generate the optical vortex beams which are associated with the so-called Hermite-Laguerre-Gaussian (HLG) beam. The HLG beam is a continuous evolution between the HG and LG beams, which can be achieved continuously by rotating the AMC around the optical axis by an angle.

On the other hand, the ellipse is a classic periodic orbit in a 2D isotropic harmonic oscillator. The SU(2) coherent state has been used to prove the quantum wave function localized on an elliptical orbit. It is generally found that quantum wave functions localized on classical periodic orbits are related to striking quantum phenomena, such as the conductance fluctuations of mesoscopic semiconductor billiard balls, the oscillation of light separation cross-sections, and the shell effect in metals in the cluster. In laser physics, diode-pumped solid-state lasers have been experimentally used to observe the elliptical mode corresponding to the SU(2) coherent state. Mathematically, the SU(2) ellipse mode is the superposition of the degenerate HG mode. Recently it was verified that the ellipse mode and the HG mode can form a quantum Fourier transform pair. According to the representation of the quantum coherent state, the elliptical mode as the integral of the two-dimensional Gaussian wave packet on the elliptical orbit was creatively derived, without involving the HG mode. The derived integral formula can be widely used to calculate not only the elliptical mode but also the HLG mode in an ultra-efficient way.

Recently, a research group led by Prof. Yongfu Chen from Taiwan Chiao Tung University published a review in Chinese Optics Letters Volume 18, No. 9, 2020 (Yongfu Chen, et al. Laser transverse modes of spherical resonators: A review). In this review article, Schwinger's SU(2) transformation is used to systematically outline the wave representation of the HLG transverse mode of a spherical laser resonator. The SU(2) representation of the HLG mode means that the HG mode and the LG mode are analytically connected through the rotation transformation on the Poincaré sphere. On another topic, the wave representation of elliptic modes by extending the one-dimensional (1D) Schrödinger coherent state to the 2D coherent state is reviewed. The integral formula of the ellipse module is obtained based on the integral of the Gaussian wave packet on the ellipse trajectory. Quantum Fourier transform is further reviewed to decompose the HG mode into coherent superpositions of elliptical modes corresponding to a bunch of elliptical orbits. It is also outlined that the decomposition of the HG mode can be extended to represent the HLG mode as a coherent superposition of the elliptical modes corresponding to the elliptical orbital beam under the SU(2) transformation. The overwhelming advantage of expressing the HLG mode based on the ellipse mode is the direct manifestation of the wave-ray (quantum classical) connection without involving the special functions of Hermite and Laguerre polynomials. Finally, it is outlined how to use the wave representation of the HLG mode to characterize the propagation and evolution of the vortex structure of the HG beam converted from a single-angle AMC to an arbitrary-angle. Since in-depth studies on laser modes with orbital angular momentum have been conducted in the past few years, this analysis in the review undoubtedly provides not only teaching insights for quantum physics, but also theoretical formulae for generating optical vortices.

涡旋光束高阶横模的量子式简化

高阶Hermite-Laguerre-Gaussian (HLG)模式的数值图形

量子谐波振荡器是理解量子经典对应关系、量化辐射场和量子光学概念必不可少的工具。二维量子谐波振荡器的本征模可以分解为矩形对称的Hermite-Gaussian（HG）模式或圆形对称的Laguerre-Gaussian（LG）模式。由于球形激光腔的傍轴波动方程与二维谐波振荡器的薛定谔方程相同，因此HG和LG本征模在探索激光横模中起着重要作用。

随着端面泵浦固体激光的出现，二极管泵浦固体激光器中可以高效的生成高阶HG和LG模式。最近提出的因斯-高斯（IG）模式是傍轴波动方程的另一本征函数形式，并且也已在稳定腔中得到实验观察。

表征标量单色光波的一般形式是复值函数，可以表示为实部和虚部。实部和虚部节线的交点都是孤立的零，即所谓的相位奇点。由于奇点被循环相位包围，它们通常与光学涡旋的形成有关，因此相位奇点和光学涡旋经常互换使用。携带轨道角动量（OAM）的旋涡光束可用于各种领域，包括光镊、捕获和引导冷原子、无线电通信和量子信息处理。

几种产生光学涡旋光束的技术已被相继提出，其中借助像散模式转换器（AMC）的方法已被广泛用于产生与“厄米-拉盖尔-高斯（HLG）光束”相关的光学涡旋光束，而HLG光束是HG和LG光束之间的连续演变，可以通过将AMC绕光轴旋转一定角度来连续实现。

另一方面，二维各向同性谐波振荡器中的经典周期轨道呈椭圆形，而 SU(2)相干态常被用来证明量子波函数位于椭圆轨道上。通常经典周期轨道上的量子波函数与经典的量子现象有关，例如介观半导体台球的电导涨落、光剥离截面的震荡特性以及金属团簇中的壳效应。

在激光物理学中，二极管泵浦固态激光器可用于实验性的观察与SU(2)相干态对应的椭圆模式。在数学上，SU(2)椭圆模式是简并HG模式的叠加。最近，椭圆模式和HG模式已被验证可以形成量子傅里叶变换对。借助量子相干态，可以创造性地绕开HG模式，成功推导出椭圆模式作为椭圆轨道上二维高斯波包的积分。该积分公式不仅可以广泛且高效地计算椭圆模式，还可用于HLG模式的计算。

台湾交通大学陈永富教授领导的研究团队在Chinese Optics Letters 2020年第18卷第9期发表的综述中（Yongfu Chen, et al. Laser transverse modes of spherical resonators: A review），使用Schwinger的SU(2)变换系统地概述了球形激光谐振器中HLG横模的波表示。 HLG模式的SU(2)表示意味着HG模式和LG模式可以通过庞加莱球上的旋转变换进行解析连接。

另一方面，综述中通过将一维薛定谔相干态扩展到二维相干态来回顾椭圆模的波表示。椭圆模的积分公式可以通过椭圆轨迹上高斯波包的积分获得，并通过进一步使用量子傅里叶变换将HG模式分解为与一串椭圆轨道相对应的椭圆模式的相干叠加。

该综述还概述了HG模式的分解，通过对应椭圆轨道光束在SU（2）的转换，可以将HLG模式表示扩展为椭圆模式的相干叠加。基于椭圆模式表示的HLG模式的巨大优势在于波射线（量子经典）连接的直接体现，而无需涉及Hermite和Laguerre多项式的特殊应用。

最后，该综述概述了如何使用HLG模式的波表示来表征从单角度AMC转换为任意角度的HG光束涡结构的传播和演化。

由于过去几年研究人员已经对具有轨道角动量的激光模式进行了深入研究，因此本篇综述不仅为量子物理学提供了教学参考，还为光学涡旋的产生提供了理论公式。