## Abstract

We theoretically analyzed the relationship between quantum Green’s functions of two-dimensional harmonic oscillators and radial-order Laguerre–Gaussian laser modes of spherical resonators. By using a nearly hemispherical resonator and a tight focusing in the end-pumped solid-state laser, we successfully generated various laser transverse modes analogous to quantum Green’s functions. We further experimentally and numerically verified that the transverse order associated with quantum Green’s functions is noticeably raised with increasing the pump power induced by the thermal effect. More importantly, the high lasing efficiency and the salient structure enable the present laser source to be used in exploring the light–matter interaction.

© 2017 Chinese Laser Press

## 1. INTRODUCTION

The study of optical pattern formation has been one of the most active fields of research for decades, since there are some interesting similarities in behavior between optical and hydrodynamics systems [1–6]. Due to the isomorphism between the paraxial wave equation and the Schrödinger equation [7–9], the transverse modes of laser spherical cavities can be described with the eigenfunctions of two-dimensional (2D) quantum harmonic oscillators that can be analytically expressed as Hermite–Gaussian (HG) functions with Cartesian symmetry and Laguerre–Gaussian (LG) functions with circular symmetry. The ${\mathrm{HG}}_{n,m}$ modes [10,11] and ${\mathrm{LG}}_{p,l}$ modes [12–14] have been directly generated in end-pumped solid-state lasers with the selective pumping [15], where $n$ and $m$ are the order indices in the $x$ and $y$ directions, respectively; $p$ and $l$ are the radial and azimuthal indices, respectively. The Ince–Gaussian modes, another form of eigenfunctions to the paraxial wave equation, have been recently introduced [16] and also experimentally observed in stable resonators [17–19]. There are some intriguing features in the spatial structures of high-order transverse modes, such as low divergence of Bessel beams [20], orbital angular momentum of helical beams [21], free acceleration of Airy beams [22], and evolution of a light pulse in a nonlinear laser cavity [23]. Moreover, several interesting issues for generating optical vortices have been explored, such as the formation of vortex lattices in transverse-mode-locked processing [24,25], the phenomenon of Berezinskii–Kosterlitz–Thouless phase transition [26–28], and twisted speckle patterns [29]. These characteristics enable high-order transverse structures to be exploited in numerous applications, including optical tweezers and microscopy [30], particle trapping [31], and quantum information [32].

Recently, the connection between the pattern formation of quantum Green’s functions and classical periodic orbits for the 2D harmonic oscillator has been numerically analyzed by plotting the classical trajectories starting from the initial position $({\tilde{x}}_{s},{\tilde{y}}_{s})$ [33]. As seen in Fig. 1, two cases for $({\tilde{x}}_{s},{\tilde{y}}_{s})=(0,0)$ and $({\tilde{x}}_{s},{\tilde{y}}_{s})=(0,2.6)$ display the correspondence between quantum Green’s functions and classical periodic-orbit bundles. Three different total numbers of classical trajectories of 6, 12, and 1000 are plotted in Fig. 1 to reveal the formation of periodic-orbit bundles. It certainly provides important insights into mesoscopic physics to manifest the connection between the spatial distributions of quantum Green’s functions and the classical periodic orbits. According to Hamilton’s opticomechanical theory [34,35], modern laser resonators have been widely used to analogously explore the formation of quantum coherent waves in the mesoscopic regime [36–41]. Although a numerical analysis of quantum Green’s functions has been reported, so far there is no experimental observation to demonstrate that the lasing transverse patterns can be related to quantum Green’s functions for the 2D harmonic oscillator.

In this work, we analyze the transverse pattern formation in the spherical cavity with tight excitation by using the inhomogeneous Schrödinger equation for the 2D quantum harmonic oscillator. It is verified that the point-excited resonant modes, corresponding to the quantum Green’s function, are exactly the radial-order ${\mathrm{LG}}_{p,0}$ modes for the source at the origin. We further develop an end-pumped solid-state laser with a nearly hemispherical resonator to imitate the point-like source and to generate the resonant modes. It is confirmed that the tightly excited resonant modes can be efficiently obtained from low to very high orders just by increasing the pump power due to the thermal effect. It is believed that the tightly excited resonant modes can be used not only to develop novel applications in optical manipulation but also to explore the spatial patterns of quantum Green’s functions in an analogous way.

## 2. RELATIONSHIP BETWEEN QUANTUM GREEN’S FUNCTIONS AND RADIAL-ORDER LG MODES

Using the 2D isotropic harmonic oscillator to model the transverse part of the spherical cavity, the transverse eigenmodes can be given by [42]

Considering the transverse distribution of the pump source $F(\mathbf{r};{\mathbf{r}}_{s})$ to deal with a realistically end-pumped configuration in solid-state lasers, the inhomogeneous wave equation for deriving the resonant modes ${\mathrm{\Psi}}_{v}(\mathbf{r};{\mathbf{r}}_{s})$ is given by

## 3. EXPERIMENTAL RESULTS AND DISCUSSION

The experimental setup was a concave-flat laser cavity with selectively longitudinal pumping, as shown in Fig. 3. The gain medium was an a-cut 2.0 at. % $\mathrm{Nd}\text{:}{\mathrm{YVO}}_{4}$ crystal with a length of 2 mm and an aperture of $10\text{\hspace{0.17em}}\mathrm{mm}\times 10\text{\hspace{0.17em}}\mathrm{mm}$. The large aperture of the gain medium was very critical for generating extremely high-order transverse modes. The $\mathrm{Nd}\text{:}{\mathrm{YVO}}_{4}$ crystal was coated on both end surfaces to be antireflective at lasing wavelength ($R<0.2\%$ at 1064 nm). The laser crystal was wrapped with indium foil and mounted in a water-cooled copper holder to ensure stable laser output. The front mirror was a 20 mm radius-of-curvature concave mirror with antireflection coating at pumping wavelength (808 nm) on the entrance face and with high-reflectance coating at lasing wavelength ($>99.8\%$) and high-transmittance coating ($T>95\%$) at pumping wavelength on the second surface. The output coupler was a flat mirror with transmission as low as 0.8% at lasing wavelength. The pump source was a 3.0 W 808 nm fiber-coupled laser diode with a core diameter of 100 μm and a numerical aperture of 0.16. A lens with a 25 mm focal length was used to focus the pump beam into the laser crystal. The pump radius ${w}_{p}$ was approximately 25 μm.

To achieve the comparable effect of tight pumping, the pump radius ${w}_{p}$ should be considerably smaller than the cavity mode radius ${w}_{c}$ in the gain medium. Since the pump radius ${w}_{p}$ is limited by the brightness of the pump source, enlarging ${w}_{c}$ is the critical criterion. Here a nearly hemispherical resonator was exploited to obtain a large cavity mode radius for satisfying the criterion of ${w}_{p}\ll {w}_{c}$. For a 20 mm radius-of-curvature concave mirror, it was experimentally found that the tight excitation could be effectively realized when the optical cavity length $L$ was designed to be nearby 19.95 mm. Under the circumstance $L=19.95\text{\hspace{0.17em}}\mathrm{mm}$, the beam waist at the output coupler can be calculated by using ${w}_{o}=\sqrt{\lambda \sqrt{L(R-L)}/\pi}$ [45]. With $R=20\text{\hspace{0.17em}}\mathrm{mm}$ and $\lambda =1064\text{\hspace{0.17em}}\mathrm{nm}$, the beam waist ${w}_{o}$ is approximately 18 μm. With ${w}_{o}=18\text{\hspace{0.17em}}\mathrm{\mu m}$, the mode radius in the gain medium can be calculated by using ${w}_{c}={w}_{o}\sqrt{1+{(\mathrm{\Delta}z/{z}_{R})}^{2}}$, where ${z}_{R}=\pi {w}_{o}^{2}/\lambda $ and $\mathrm{\Delta}z$ is the distance between the laser crystal and the output coupler. In the experiment, $\mathrm{\Delta}z$ is approximately 7.0 mm with which ${w}_{c}$ can be found to be approximately 126 μm. Consequently, the ratio ${w}_{p}/{w}_{c}$ can be found to be approximately 0.2 smaller than the value of critical criterion 0.3 verified in Fig. 2. In other words, the pump radius ${w}_{p}\approx 25\text{\hspace{0.17em}}\mathrm{\mu m}$ in the experiment is comparably tight enough that the pump radius ${w}_{p}$ should be smaller than 38 μm for the cavity mode radius ${w}_{c}\approx 126\text{\hspace{0.17em}}\mathrm{\mu m}$ to mimic the quantum Green’s function with a point source. For the present hemispherical configuration, the ratio ${f}_{T}/{f}_{L}$ is nearly equal to 1/2, where ${f}_{T}$ and ${f}_{L}$ are the transverse and longitudinal mode spacings, respectively.

In the end-pumping scheme, the location of the excitation source $(\mathrm{\Delta}x,\mathrm{\Delta}y)$ can be precisely controlled by the manual translation stages, where $\mathrm{\Delta}x$ and $\mathrm{\Delta}y$ represent the distances away from the center of the optical axis in the $x$ and $y$ directions, respectively. The position $(\mathrm{\Delta}x,\mathrm{\Delta}y)$ is related to the theoretical parameter $({\tilde{x}}_{s},{\tilde{y}}_{s})$ by ${\tilde{x}}_{s}=\sqrt{2}\mathrm{\Delta}x/{w}_{c}$ and ${\tilde{y}}_{s}=\sqrt{2}\mathrm{\Delta}y/{w}_{c}$. To compare with the resonant function ${G}_{N}(\tilde{x},\tilde{y};{d}_{s})$ in Eq. (7), the displacement $\mathrm{\Delta}x$ was fixed to be zero and $\mathrm{\Delta}y$ was adjustable to correspond to the theoretical parameter ${d}_{s}$. Figure 4 shows experimental results for the output power and the far-field patterns of lasing modes obtained by varying the pump power ${P}_{\text{in}}$ for the source at the origin, i.e., ${d}_{s}=0$. The far-field patterns were measured by projecting the output beam on a paper screen at a distance of 50 cm away from the laser cavity and using a digital camera to capture the scattered light. It can be seen that the threshold pump power ${P}_{\mathrm{th}}$ is as low as 50 mW and the output power is up to 0.36 W at a pump power of 2.4 W. It is worthwhile to mention that the order of the lasing mode gradually increases with increasing the pump power. It is clear that all the lasing modes are in good agreement with the spatial patterns $|{G}_{N}(\tilde{x},\tilde{y};0)|$, as shown in the bottom of Fig. 4. Based on experimental observation, the relationship between the effective order index $N$ and the pump strength can be empirically expressed as ${P}_{\text{in}}/{P}_{\mathrm{th}}\approx 2N+1$. Since ${G}_{N}(\tilde{x},\tilde{y};0)$ is exactly the ${\mathrm{LG}}_{p,0}$ mode with $N=2p$, the maximum radial order for the ${\mathrm{LG}}_{p,0}$ mode can be found to be up to $p=15$ in the present experiment. As discussed in Eq. (11), the high-radial-order ${\mathrm{LG}}_{p,0}$ mode can behave as the zero-order Bessel beam.

Quite recently, Dong and collaborators have developed several passively $Q$-switched $\mathrm{Nd}\text{:}{\mathrm{YVO}}_{4}$ microchip lasers to generate various high-order transverse modes by means of adjusting the longitudinally focal position [46], the incident angle [47,48], or the polarization angle [49] of the pump beam. Since the cavity mode sizes in microchip lasers strongly depend on the pump intensity, the transverse orders of lasing modes are asymmetrically expanded within the pump area with increasing the pump power for considering the asymmetrical distribution of the inversion population inside the gain medium [47–49]. For the present hemispherical configuration with the critical condition of stable cavities, there exists a critical pump power at which the thermal effect will easily cause the laser cavity to be unstable. The thermal effect leads to a lensing behavior in the gain medium due to the beam profile of a fiber-coupled laser diode approximately described as a Gaussian distribution. Therefore, the thermal effect can be estimated by the overlap integral between the intensity distribution of eigenmodes and the thermal lensing profile:

Even though the far-field pattern of the lasing mode agrees well with the spatial distribution of a single mode ${G}_{N}(\tilde{x},\tilde{y};0)$, it is important to explore whether there are other different mode components in the individual lasing mode. To analyze the mode components, a cylindrical-lens mode converter outside the laser resonator was used to transform the lasing mode from the LG basis to the HG basis [50]. The focal length of the cylindrical lenses was $f=25\text{\hspace{0.17em}}\mathrm{mm}$ and the distance was precisely adjusted to be $\sqrt{2}f$ for the operation of the $\pi /2$ converter. Figure 6 shows the transformed patterns for the lasing modes in Figs. 4(c) and 4(d). It can be seen that the transformed pattern is not a pure single ${\mathrm{HG}}_{p,p}$ mode. In other words, the lasing mode is not a pure ${\mathrm{LG}}_{p,0}$ mode but includes some low-order modes. The mode components of the lasing mode are deduced by using the numerical fitting to reconstruct the transformed patterns, as shown in the right-hand side of Fig. 6. For the lasing mode in Fig. 4(c), the mode components are fitted as ${G}_{2p}(\tilde{x},\tilde{y};0)+[(1/4)\sum _{k=0}^{2}{G}_{2k}(\tilde{x},\tilde{y};0)]$ with $p=4$. For the lasing mode in Fig. 4(d), the mode components are fitted as ${G}_{2p}(\tilde{x},\tilde{y};0)+[(1/8)\sum _{k=0}^{2}{G}_{2k}(\tilde{x},\tilde{y};0)]$ with $p=7$. The low-order LG components in the main ${\mathrm{LG}}_{p,0}$ mode come from the frequency locking with different longitudinal orders. Since the mode-spacing ratio ${f}_{T}/{f}_{L}$ is nearly equal to 1/2, different transverse modes can be frequency locked with the help of different longitudinal orders. On the other hand, the transformed patterns reveal that the lasing modes are mainly dominated by the standing waves.

Figure 7 shows experimental results for the output power and the lasing modes obtained by varying the pump power ${P}_{\text{in}}$ for the excitation source at $({\tilde{x}}_{s},{\tilde{y}}_{s})=(0,1)$, i.e., ${d}_{s}=1$. The output power can reach 0.33 W at a pump power of 2.4 W. The overall efficiency is slightly lower than the result obtained for the source at $({\tilde{x}}_{s},{\tilde{y}}_{s})=(0,0)$. It can be clearly seen that all the experimental lasing modes agree very well with the spatial patterns $|{G}_{N}(\tilde{x},\tilde{y};1)|$ shown in the bottom of Fig. 7 from low to high orders. Figure 8 shows experimental results obtained at $({\tilde{x}}_{s},{\tilde{y}}_{s})=(0,2.6)$. Once again, all the experimental lasing modes are consistent with the theoretical distributions $|{G}_{N}(\tilde{x},\tilde{y};2.6)|$ shown in the bottom of Fig. 8.

## 4. CONCLUSIONS

In summary, we have theoretically explored the pattern formation of quantum Green’s functions with the point excitation. The point-excited resonant mode has been verified to be exactly the radial-order ${\mathrm{LG}}_{p,0}$ mode that can be asymptotic to the zero-order Bessel beam in the limit $p\to \infty $. In the experiment, an end-pumped solid-state laser with a nearly hemispherical resonator was employed to generate the tightly excited resonant modes from low to very high orders in an efficient way. It is believed that the present finding not only creates an important innovation to generate the structured beams for laser applications but also provides a remarkable method to visualize the quantum–classical correspondence.

## Funding

Ministry of Science and Technology, Taiwan (MOST) (106-2628-M-009-001); Japan Society for the Promotion of Science (JSPS) (JP 15H03571, JP 15K13373, JP 16H06507).

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