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
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 modes [10,11] and modes [12–14] have been directly generated in end-pumped solid-state lasers with the selective pumping , where and are the order indices in the and directions, respectively; and are the radial and azimuthal indices, respectively. The Ince–Gaussian modes, another form of eigenfunctions to the paraxial wave equation, have been recently introduced  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 , orbital angular momentum of helical beams , free acceleration of Airy beams , and evolution of a light pulse in a nonlinear laser cavity . 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 . These characteristics enable high-order transverse structures to be exploited in numerous applications, including optical tweezers and microscopy , particle trapping , and quantum information .
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 . As seen in Fig. 1, two cases for and 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 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 3), the resonant function can be derived as 4) is explicitly the quantum Green’s function for the 2D harmonic oscillator. For the resonant condition, the zeros of the denominator in Eq. (4) determine the predominantly contributed eigenmodes. For , the function in Eq. (4) can be expressed as a superposition of the degenerate eigenmodes with : 43] 5) and using Eq. (6), after some algebra, the resonant function in Eq. (5) can be derived as 8) can be used to show that . As a consequence, cannot exist for odd . In contrast, substituting with into Eq. (8) for even , it can be obtained that . Substituting this result into Eq. (7), the resonant function can be simplified as 43] 44]. Furthermore, the mode with can be related to the zero-order Bessel beam by the asymptotic formula 
Considering the transverse distribution of the pump source to deal with a realistically end-pumped configuration in solid-state lasers, the inhomogeneous wave equation for deriving the resonant modes is given by2. The numerical calculations for the location of the excitation source at the origin (i.e., ), , and are depicted in Figs. 2(a)–2(c), respectively. It can be seem that the coefficient is approximately a constant for the pump-to-mode size ratio within the value of 0.3. Here the critical criterion of the pump-to-mode size ratio is validated that the laser transverse modes with a tight pumping can be analogous to quantum Green’s functions of 2D harmonic oscillators with a point source. In the following, we experimentally demonstrate that the tight excitation can be genuinely realized by using an end-pumped solid-state laser with a nearly hemispherical cavity.
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. % crystal with a length of 2 mm and an aperture of . The large aperture of the gain medium was very critical for generating extremely high-order transverse modes. The crystal was coated on both end surfaces to be antireflective at lasing wavelength ( 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 () and high-transmittance coating () 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 was approximately 25 μm.
To achieve the comparable effect of tight pumping, the pump radius should be considerably smaller than the cavity mode radius in the gain medium. Since the pump radius is limited by the brightness of the pump source, enlarging is the critical criterion. Here a nearly hemispherical resonator was exploited to obtain a large cavity mode radius for satisfying the criterion of . 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 was designed to be nearby 19.95 mm. Under the circumstance , the beam waist at the output coupler can be calculated by using . With and , the beam waist is approximately 18 μm. With , the mode radius in the gain medium can be calculated by using , where and is the distance between the laser crystal and the output coupler. In the experiment, is approximately 7.0 mm with which can be found to be approximately 126 μm. Consequently, the ratio 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 in the experiment is comparably tight enough that the pump radius should be smaller than 38 μm for the cavity mode radius to mimic the quantum Green’s function with a point source. For the present hemispherical configuration, the ratio is nearly equal to 1/2, where and are the transverse and longitudinal mode spacings, respectively.
In the end-pumping scheme, the location of the excitation source can be precisely controlled by the manual translation stages, where and represent the distances away from the center of the optical axis in the and directions, respectively. The position is related to the theoretical parameter by and . To compare with the resonant function in Eq. (7), the displacement was fixed to be zero and was adjustable to correspond to the theoretical parameter . Figure 4 shows experimental results for the output power and the far-field patterns of lasing modes obtained by varying the pump power for the source at the origin, i.e., . 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 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 , as shown in the bottom of Fig. 4. Based on experimental observation, the relationship between the effective order index and the pump strength can be empirically expressed as . Since is exactly the mode with , the maximum radial order for the mode can be found to be up to in the present experiment. As discussed in Eq. (11), the high-radial-order mode can behave as the zero-order Bessel beam.
Quite recently, Dong and collaborators have developed several passively -switched microchip lasers to generate various high-order transverse modes by means of adjusting the longitudinally focal position , the incident angle [47,48], or the polarization angle  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:5 indicates the calculation result for the coefficient as a function of the transverse order of eigenmodes with various pump positions in the direction. It is clearly identified that the influence of the thermal effect is relatively more significant on the fundamental mode for the pump source at the origin represented by the red solid curve in Fig. 5. On the other hand, the fundamental mode with the maximum value of the overlap integral is corresponding to the minimum threshold power. For the near-threshold power, the fair thermal lensing causes the fundamental mode with the lowest threshold breaking into oscillation at first. Considering the pump power far above the threshold value, the thermal effect of the fundamental mode rapidly increases, which gives rise to an unstable fundamental mode and is replaced by higher-order modes. With the excitation source away from the origin, the maximum value of the overlap integral will be gradually shifted to higher-order transverse modes represented by the blue dashed curve for and the green dashed–dotted curve for , as shown in Fig. 5. Here the transverse order of lasing modes can be symmetrically enlarged with increasing the pump intensity for the pump size much smaller than the cavity mode size by considering the thermal effect of different transverse order modes.
Even though the far-field pattern of the lasing mode agrees well with the spatial distribution of a single mode , 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 . The focal length of the cylindrical lenses was and the distance was precisely adjusted to be for the operation of the 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 mode. In other words, the lasing mode is not a pure 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 with . For the lasing mode in Fig. 4(d), the mode components are fitted as with . The low-order LG components in the main mode come from the frequency locking with different longitudinal orders. Since the mode-spacing ratio 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 for the excitation source at , i.e., . 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 . It can be clearly seen that all the experimental lasing modes agree very well with the spatial patterns shown in the bottom of Fig. 7 from low to high orders. Figure 8 shows experimental results obtained at . Once again, all the experimental lasing modes are consistent with the theoretical distributions shown in the bottom of Fig. 8.
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 mode that can be asymptotic to the zero-order Bessel beam in the limit . 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.
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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