1. Introduction

Plane waves are the most fundamental wave distributions, while structured waves which display spatial features essentially different from plane waves are naturally to appear in the wave equation with external potentials [1], such as atomic orbits [2,3], eigenstates of quantum dots [4–6], resonant modes in laser resonators [7–11], acoustic waves [12,13], Landau states in a magnetic field [14–18], etc. From the characteristics of the interference of multiple plane waves, it can be shown that the structured waves generally contain phase singularities [19]. Since the probability current density swirls around phase singularities, the term of vortex is used to represent the phase singularity [20]. The distribution of phase singularities is intimately associated with the orbital angular momentum density (OAMD). Although vortices appeared in many early studies of optical and quantum waves, the systematic exploration of phase singularities was first given in 1974 by Nye and Berry in ultrasonic fields [21] and by Hirschfelder et al. in quantum wave functions [22,23]. The quantum theory has been extensively exploited to study the distribution of phase singularities not only in quantum physics but also in different branches of physics [24–29]. Nowadays, the exploration of structured coherent waves with vortices becomes an emerging area of research.

For a spherical laser cavity, the eigenmodes are usually represented as Hermite-Gaussian (HG) modes with rectangular symmetry or Laguerre-Gaussian (LG) modes with circular symmetry. In addition to HG and LG modes, the generalized representation of eigenmodes can be expressed as the Hermite-Laguerre-Gaussian (HLG) modes by using the quantum SU(2) coupled oscillator model [30]. The SU(2) Lie algebra has been developed to investigate various quantum phenomena such as entanglement in coherent states [31], generation and evolution of vortex states [32,33], orbital magnetism in quantum dots [34], charge particles in external fields [35], and shell effects in nuclei and metallic clusters [36]. A systematic review for laser transverse modes characterized by the SU(2) coupled oscillator model can be found in Ref. [30]. In experimental aspects, the HLG modes can be generated by using a spherical laser cavity with astigmatism or by using an astigmatic mode converter (AMC) to transform HG modes.

Besides eigenmodes, the high-order transverse modes in the degenerate cavities often exhibit the so-called geometric modes with the intensities concentrated on the geometric rays. The domination of eigenmodes or geometric modes is mainly dependent on the transverse order N and the ratio Δf_Tx/Δf_Ty, where Δf_Tx and Δf_Ty are the transverse mode spacings in the two eigenaxes x and y, respectively. The laser cavity to be degenerate is determined by the condition of the ratio Δf_Tx/Δf_Ty to be the rational number p/q, where p and q are coprime integers. Experimental results revealed that the lasing transverse modes were usually dominated by the eigenmodes for N < p with N < q and dominated by the geometric modes for N > p with N > q. The geometric modes corresponding to HG eigenmodes are called the Lissajous geometric modes, the counterpart corresponding to LG eigenmodes are the trochoidal geometric modes. By nature, the geometric modes corresponding to HLG eigenmodes can be called the Lissajous-trochoidal geometric modes. In theoretical aspects, the quantum SU(2) coupled oscillator model has been used to derive the stationary coherent states that not only can display the wave intensities well localized on the classical trajectories but also can exhibit the continuous transformation between Lissajous and trochoid geometric curves [37,38]. Nevertheless, since the accidental degeneracies and the high-order Hermite polynomials are involved in computations, the spatial structures and phase singularities are rater difficult to evaluate. Recently, we have originally developed the integral method to accurately compute the stationary coherent states for manifesting the spatially phase singularities of the Lissajous geometric modes [39–45,30,46–48]. So far, the integral method has not been applied to the general case for the transition between the HLG eigenmode and counterpart geometric modes. It will be greatly useful to develop the generalized integral method from the quantum SU(2) coupled oscillator model for utterly understanding the laser transverse modes related to the transition between the HLG eigenmode and geometric modes.

In this paper, we systematically overview the quantum wavefunctions and classical trajectories for the SU(2) coupled oscillator. We further develop the integral method based on the Fourier transformation to represent the stationary coherent state of the SU(2) coupled oscillator as an integral evolution of the time-dependent wave packet state over a complete period. The developed integral formula is then exploited to analyze the spatial structures of the stationary coherent states for manifesting the transition from low-order eigenmodes to geometric modes. The calculated results for the spatial distributions can not only reveal the quantum-classical connection but also display the plentiful variations of phase singularities and edge dislocations. More importantly, the overall review and the developed model can help readers employ the integral formula to reconstruct the structured optical waves related to HLG modes and Lissajous-trochoidal geometric modes that are experimentally transformed by using AMC for the HG modes and Lissajous geometric modes, respectively.

2. SU(2) quantum oscillator model

The quantum SU(2) states are related to the attractive oscillator system, whereas the quantum SU(1,1) states are associated with the repulsive oscillator system. Since the resonant modes of the laser cavity are analogous to the states of the confined system, the quantum SU(2) states can be analogously employed to characterize laser transverse modes generated from cavities with various astigmatism. The generalized Hamiltonian related to the SU(2) interaction for the coupled oscillator systems can be given by [33,37,38]

The classical counterpart of the system $\hat{H}$ in Eq. (1) can be expressed by using the dimensionless variables ${a_1} = {{({\tilde{x} + i\,{{\tilde{p}}_x}} )} / {\sqrt 2 }}$, $a_1^\dagger = {{({\tilde{x} - i\,{{\tilde{p}}_x}} )} / {\sqrt 2 }}$, ${a_2} = {{({\tilde{y} + i\,{{\tilde{p}}_y}} )} / {\sqrt 2 }}$, and $a_2^\dagger = {{({\tilde{y} - i\,{{\tilde{p}}_y}} )} / {\sqrt 2 }}$. The dimensionless variables $\tilde{x}$, $\tilde{y}$, ${\tilde{p}_x}$, and ${\tilde{p}_y}$ are given by $\tilde{x} = x\sqrt {\mu {\omega _o}/\hbar } $, $\tilde{y} = y\sqrt {\mu {\omega _o}/\hbar } $, ${\tilde{p}_x} = {p_x}\sqrt {1/\hbar \mu {\omega _o}} $, and ${\tilde{p}_y} = {p_y}\sqrt {1/\hbar \mu {\omega _o}} $, where μ is the mass of the particle, x and y are the position variables, and p_x and p_y are the momentum variables. In terms of $\tilde{x}$, $\tilde{y}$, ${\tilde{p}_x}$, and ${\tilde{p}_y}$, the classical Hamiltonian for $H$ in Eq. (1) is given by

From Eq. (6), the classical equation of motion for the Hamiltonian H can be derived as

Considering the frequency ratio ${\omega _1}/{\omega _2}$ to be a rational number $q/p$, the characteristic frequencies can be written as ${\omega _1} = q\omega$ and ${\omega _2} = p\omega$ with $\omega = 2{\omega _o}/(p + q)$ by using ${\omega _1} + {\omega _2} = 2{\omega _o}$ and ${\omega _1}/{\omega _2} = q/p$, where q and p are coprime integers,. Note that the integer q is always positive; however, the integer p can be either positive for ${\omega _o} > (\Omega /2)$ or negative for ${\omega _o} < (\Omega /2)$ . By using the relations of ${v_1} = (\tilde{x} + i\,{\tilde{p}_x})/\sqrt 2$ and ${v_2} = (\tilde{y} + i\,{\tilde{p}_y})/\sqrt 2$, the classical periodic orbit can be obtained from $\tilde{x} = \sqrt 2 Re ({v_1})$ and $\tilde{y} = \sqrt 2 Re ({v_2})$. Consequently, the parametric equations for the classical trajectory for ${\omega _1}/{\omega _2} = q/p$ can be found by using Eq. (8) and explicitly expressed as

Equation (9) indicates that the periodic orbits of the SU(2) coupled oscillators display a wide variety of famous curves that are determined by the values of α and β. For $\alpha = 0$, the orbits in Eq. (9) are Lissajous figures at an angle of β/2 with respect to the x-axis. For $\alpha = \pi /2$ and $\beta = \pi /2$, the orbits are hypotrochoidal or epitrochoidal curves, depending on whether the integer p is positive or negative [37]. Overall, the orbits in Eq. (9) can be called Lisssjous-trochoidal curves since these curves are associated with a continuous transformation between Lissajous figures and trochoidal curves by varying the values of the parameter α and β.

The quantum eigenstates of the coupled oscillator $\hat{H}$ in Eq. (1) can be analytically derived with the SU(2) algebra via defining a new pair of operators

In terms of $b_1^\dagger$ and $b_2^\dagger$, the coupled oscillator $\hat{H}$ can be untangled as

Consequently, the quantum eigenstates of the Hamiltonian $\hat{H}$ can be given by

Substituting Eq. (10) into Eq. (12) and using the fact ${|{0,0} \rangle _{\hat{H}}} = {|{0,0} \rangle _{{{\hat{H}}_o}}}$, the eigenstates ${|{{n_1},{n_2}} \rangle _{\hat{H}}}$ can be explicitly expressed as a linear combination of the eigenstates of ${\hat{H}_o}$:

3. Quantum stationary coherent states manifesting classical periodic orbits

The quantum stationary coherent states manifesting the classical periodic orbits can be derived by using the Schrödinger coherent state that is obtained from the generating function for the Hermite polynomials given by [52]

Note that the generating function is valid for all complex values of τ. Setting $\tau = u/\sqrt 2$ and multiplying the term ${e^{ - (|u{|^2} + {{\tilde{x}}^2})/2}}$ on the both sides of Eq. (18), after some rearrangement and in terms of ${\psi _n}({\tilde{x}} )$, the Schrödinger coherent state can be given by

From Eqs. (19) and (20), the time-dependent Gaussian wave packet state linking to the trajectory $\tilde{x} = \sqrt 2 Re ({v_1})$ and $\tilde{y} = \sqrt 2 Re ({v_2})$ can be expressed as the product of two Schrödinger coherent states:

In terms of $a_1^\dagger$ and $a_2^\dagger$, the wave packet state $g(\tilde{x},{v_1})g(\tilde{y},\;{v_2})$ can be written as

Consequently, Eq. (12) can be used to express the wave packet state $g(\tilde{x},{v_1})g(\tilde{y},\;{v_2})$ as the coherent superposition of the quantum eigenstates $\psi _{{n_{\,1}},{n_{\,2}}}^{(\hat{H})}(\tilde{x},\tilde{y})$:

From Eqs. (8) and (9), the classical trajectory can be found to only rely on the relative phase between ϕ₁ and ϕ₂. Therefore, it is concisely convenient to express the parameters u₁ and u₂ as

Substituting Eq. (25) into Eq. (24), the summation terms in the wave packet state $g(\tilde{x},{v_1})g(\tilde{y},\;{v_2})$ can be rearranged as

From ${\omega _1} = q\omega$, ${\omega _2} = p\omega$ and Eq. (13), the eigenvalues can be found to mainly depend on the factor $q{n_1} + p{n_2}$. For a given $({N_1},{N_2})$, the eigenstates $\psi _{{n_1},{n_2}}^{(\hat{H})}(\tilde{x},\tilde{y})$ with $({n_1},{n_2})$ meeting $q{n_1} + p{n_2} = q{N_1} + p{N_2}$ can be therefore confirmed to be degenerate. As a consequence, the time-dependent wave packet $g(\tilde{x},{v_1})g(\tilde{y},\;{v_2})$ in Eq. (27) can be expressed as the superposition of the stationary coherent states with the eigenvalues to be related to $q{n_1} + p{n_2}$.

The representation for the stationary coherent states can be obtained by using the Fourier transform. The kernel of the Fourier transform is given by

Applying Eq. (28) to Eq. (27), the stationary coherent state for a given $({N_1},{N_2})$ can be derived from

For a given $({N_1},{N_2})$, the group for the degenerate eigenstates $\psi _{{n_1},{n_2}}^{(\hat{H})}(\tilde{x},\tilde{y})$ can be expressed as ${n_1} = {N_1} - pK$ and ${n_2} = {N_2} + qK$, where K is an arbitrary integer. The stationary coherent state $\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )$ in Eq. (29) can therefore be derived as

Equation (30) reveals that the total number of the degenerate eigenstates $\psi _{{N_1} - pK,{N_2} + qK}^{(\hat{H})}(\tilde{x},\tilde{y})$ for a given $({N_1},{N_2},\phi )$ in the stationary coherent state $\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )$ is ${N_t} = [{N_1}/p] + [{N_2}/q] + 1$. For ${N_t} > > 1$, the spatial distributions ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ can be confirmed to be well localized on the classical orbits. The summation expression in Eq. (30) can be associated with the representation of the SU(2) coherent state that has been used to analyze quantum wave functions corresponding to classical orbits [37,38]. Furthermore, Eq. (30) also reveals how the stationary coherent state to be generated from the integral evolution of the time-dependent wave packet state. The integral evolution was originally proposed for numerically analyzing the nonintegrable system [53,54]. We are the first group to develop the integral evolution for dealing with the 2D and higher-dimensional coupled oscillator systems. More importantly, we first exploited the SU(2) coupled oscillator model to manifest the laser transverse modes covering the HLG and geometric modes [39–45,30,46–48].

4. Results and discussion

First of all, we demonstrate the case with $\alpha = 0$ and $\beta = 0$ that corresponds to the 2D anisotropic harmonic oscillator. Figure 1 shows the calculated patterns of the stationary coherent states $\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )$ in Eq. (30) with the parameters of $(q,p) = (5,4)$, and $\phi = 0$ for different orders of $({N_1},{N_2}) = (Mp,Mq)$ with the factor M = 1, 2, 3, 8, and 10. As discussed above, the physical meaning of the parameter M can be related to the total number ${N_t}$ of eigenmodes participating in the coherent state $\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )$ as ${N_t} = 2M + 1$. On the other hand, the case of $(q,p) = (5,4)$ is used here just for illustrative example. Consequently, the factor M is used here to display the transition of the wave patterns ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ from low-level to high-level states. The wave pattern for M = 1 can be seen to be similar to the HG eigenmodes of the 2D anisotropic harmonic oscillator. To be more specific, when the indices $({N_1},{N_2})$ are small and satisfy ${N_1} < p$ and ${N_2} < q$, the stationary coherent states $\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )$ represent pure single eigenstates and their wave patterns are independent of the phase factor ϕ since no degenerate states are in the low-level energies. With increasing the factor M, it can be seen that more and more degenerate eigenstates participate in the stationary coherent states $\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )$ to make the wave patterns gradually change from the widespread distributions of HG eigenstates to the concerned structures corresponding to classical orbits. As expected, the wave patterns are localized on the Lissajous figures that are the classical periodic orbits for the cases with $\alpha = 0$ and $\beta = 0$.

 

Fig. 1. Calculated patterns of the stationary coherent states ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ in Eq. (30) with the parameters of $(q,p) = (5,4)$, $\alpha = 0$, $\beta = 0$, and $\phi = 0$ for different orders of $({N_1},{N_2}) = (Mp,Mq)$ with the factor M = 1, 2, 3, 8, and 10.

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Next, we discuss the case with $\alpha = \pi /2$ and $\beta = \pi /2$ that corresponds to the problem of a charged particle moving in a 2D isotropic harmonic oscillator potential in the xy plane and subjected to a constant external magnetic field along the z axis. Figure 2 shows the calculated results of the stationary coherent states $\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )$ with the parameters of $(q,p) = (8,1)$, and $\phi = 0$ for different orders of $({N_1},{N_2}) = (Mp,Mq)$ with the factor M = 1, 2, 3, 8, and 10. The wave pattern for M = 1 can be seen to be similar to the LG eigenmodes in the 2D isotropic harmonic oscillator. As shown in Fig. 1 for the case with $\alpha = 0$ and $\beta = 0$, the stationary coherent state $\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )$ gradually transits from the extensively spatial characteristics of LG eigenstates to the localized structures with intensities concentrated on classical orbits with increasing the index order $({N_1},{N_2})$. The wave patterns are localized on the trochoidal curves that are the classical periodic trajectories for the cases with $\alpha = \pi /2$ and $\beta = \pi /2$.

 

Fig. 2. Calculated results of the stationary coherent states ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ with the parameters of $(q,p) = (8,1)$, $\alpha = \pi /2$, $\beta = \pi /2$, and $\phi = 0$ for different orders of $({N_1},{N_2}) = (Mp,Mq)$ with the factor M = 1, 2, 3, 8, and 10.

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This result indicates that in addition to the order $({N_1},{N_2})$, the mean orbital angular momentum $\left\langle {{L_z}} \right\rangle$ depends on the coupling parameters α and β. The mean orbital angular momentum $\left\langle {{L_z}} \right\rangle$ can reach maximum for the cases with $\alpha = \pi /2$ and $\beta = \pi /2$, as discussed in Fig. 2. On the other hand, the mean orbital angular momentum $\left\langle {{L_z}} \right\rangle$ is zero for the cases with $\alpha = 0$ or $\beta = 0$.

Figure 3 shows the variation of the wave patterns ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ with the parameter β for a fixed $\alpha = \pi /2$ for the low-level states without degeneracy, where $(q,p) = (9,8)$ and $({N_1},{N_2}) = (2,3)$, $\phi = 0$. The overall feature of the wave patterns is clear to change gradually from rectangular HG mode to circular LG mode with increasing β from 0 to π/2. The second row of Fig. 3 shows the numerical results for the phase structures calculated by $\Theta (\tilde{x},\tilde{y}) = {\tan ^{ - 1}}\{ {\mathop{\rm Im}\nolimits} [\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )]/Re [\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )]\}$. It can be seen that a 2D array of isolated singularities appears when the parameter β increases from zero. The isolated singularities finally evolve into a single vortex when the parameter β is up to $\pi /2$. Figure 4 shows the dependence of the wave patterns ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ on the parameter β for a fixed $\alpha = \pi /2$ for the high-level states with accidental degeneracies, where the values of other parameters used in the calculation are $(q,p) = (4,1)$ and $({N_1},{N_2}) = (6,16)$, and $\phi ={-} \pi$. The localized structures of the wave patterns ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ can be seen to transform continuously from Lissajous to trochoidal orbits with increasing β from 0 to π /2. At the second row of Fig. 4, numerical results for the phase structures $\Theta (\tilde{x},\tilde{y})$ are shown to manifest the formation and variation of isolated singularities.

 

Fig. 3. First row: variation of the wave patterns ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ with the parameter β for a fixed $\alpha = \pi /2$ for the low-level states without degeneracy, where $(q,p) = (9,8)$ and $({N_1},{N_2}) = (2,3)$, and $\phi = 0$. Second row: numerical results for the phase structures.

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Fig. 4. First row: dependence of the wave patterns ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ on the parameter β for a fixed $\alpha = \pi /2$ for the high-level states with accidental degeneracies, where the values of other parameters used in the calculation are $(q,p) = (4,1)$ and $({N_1},{N_2}) = (6,16)$, and $\phi ={-} \pi$. Second row: numerical results for the phase structures.

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The stationary coherent states in the SU(2) coupled oscillator can be used to analyze the structured optical waves that can be experimentally generated from laser transverse modes with astigmatic transformation. Abramochkin and Volostnikov [55] originally used an AMC formed by a matched pair cylindrical lenses to generate the so-called HLG beams. The HLG beam, a continuous evolution between Hermite-Gaussian and Laguerre-Gaussian beams, can successively be realized by rotating the AMC about the optical axis by an angle η from 0 to π/4. The HLG modes display plentiful evolutions of point dislocations and edge dislocations. The AMC was later used to generate the light fields with nonzero orbital angular momentum [56]. Here we demonstrate that the stationary coherent state ${\Psi _{{N_{\,1}},{N_{\,2}}}}(\tilde{x},\tilde{y})$ with a fixed $\alpha = \pi /2$ for the indices $({N_1},{N_2})$ satisfying ${N_1} \le p$ and ${N_2} \le q$ can be used to analyze the HLG modes which are generated by using an AMC to transform the Hermite-Gaussian mode. Figure 5 shows the case with $({N_1},{N_2}) = (15,5)$ for demonstration, where the experimental patterns are obtained by using a diode-pumped solid-state laser with off-axis pumping [57,58] to generate the Hermite-Gaussian mode that is subsequently converted by an AMC with different angle η. The theoretical patterns ${|{{\Psi _{{N_{\,1}},{N_{\,2}}}}(\tilde{x},\tilde{y})} |^2}$ are calculated with $q = 7$ and $p = 16$ for different β. The one-to-one correspondence can be seen to be given by $\beta \textrm{ = 2}\eta$. As mentioned above, the theoretical wave patterns are independent of the phase factor ϕ because they are pure eigenstates without degeneracy. The third row of Fig. 5 shows numerical results for the phase structures. The formation and variation of isolated singularities can be clearly seen to vary from rectangular to circular distributions.

 

Fig. 5. First row: experimental transverse patterns obtained by using a diode-pumped solid-state laser with an external AMC with different angle η. Second row: theoretical patterns ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ calculated with $({N_1},{N_2}) = (15,5)$, $q = 7$ and $p = 16$ for different β = 2η. Third row: numerical results for the phase structures.

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In addition to HG modes, it has been experimentally observed that the tiny symmetry breaking can lead to the three-dimensional (3D) high-order laser modes with the transverse patterns to be well concentrated on the 2D Lissajous figures. The transverse wave functions of these laser modes can be described by the high-level stationary coherent state ${\Psi _{{N_{\,1}},{N_{\,2}}}}(\tilde{x},\tilde{y})$ with $\alpha = \pi /2$ and $\beta = 0$. As the generation of HLG modes by using an AMC, the transverse Lissajous pattern can be continuously transformed to the trochoidal pattern by passing through an AMC with the angle η changing from 0 to π /4. The one-to-one correspondence between the transformed laser patterns and the stationary coherent state ${\Psi _{{N_{\,1}},{N_{\,2}}}}(\tilde{x},\tilde{y})$ can be obtained with $\beta \textrm{ = 2}\eta$. Figure 6 shows the case with $(q,p) = (4,1)$ for demonstration, where the theoretical patterns $|{{\Psi _{{N_{\,1}},{N_{\,2}}}}(\tilde{x},\tilde{y})} |$ are calculated with $({N_1},{N_2}) = (30,80)$, and $\phi ={-} \pi$. Numerical results for the phase structures are also shown at the third row of Fig. 6. Once again, the theoretical patterns are in excellent agreement with the experimental results.

 

Fig. 6. First row: experimental transverse patterns obtained by using a diode-pumped solid-state laser with an external AMC with different angle η. Second row: theoretical patterns ${|{\Psi _{{N_1},{N_2}}^{(q,p)}(\tilde{x},\tilde{y},\phi )} |^2}$ calculated with $(q,p) = (4,1)$, $({N_1},{N_2}) = (30,80)$, and $\phi ={-} \pi$ for different β = 2η. Third row: numerical results for the phase structures.

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5. Conclusion

In summary, we have systematically reviewed the SU(2) coupled oscillator model to deliver the quantum eigenstates and stationary coherent states for characterizing laser transverse modes in an analogous way based on the quantum-classical connection. We further overviewed the integral formula for the representation of the stationary coherent states derived from the evolution of the time-dependent wave packet state. Several examples for the stationary coherent states are illustratively presented to display the spatial distributions and phase singularities for the quantum-classical transition. It is believed that the generalized integral method derived from the quantum SU(2) coupled oscillator model can provide a comprehensive understanding for the laser transverse modes related to the transition between the HLG eigenmode and geometric modes. Finally, it is worthwhile to mention that the quantum SU(2) states are related to the attractive oscillator system, whereas the quantum SU(1,1) states are associated with the repulsive oscillator system. Since the transverse modes of the laser cavity are analogous to the states of the confined system, all experimentally observed laser modes so far are linked to the SU(2) states instead of SU(1,1) states.