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 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.
References
[1] K. Staliunas, V. J. Sanchez-Morcillo. Transverse Patterns in Nonlinear Optical Resonators(2003).
[2] M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss. Transverse laser patterns. I. Phase singularity crystals. Phys. Rev. A, 43, 5090-5113(1991).
[3] D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, P. Glorieux. Two-dimensional optical lattices in a CO2 laser. Phys. Rev. A, 46, 5955-5958(1992).
[4] E. Louvergneaux, D. Hennequin, D. Dangoisse, P. Glorieux. Transverse mode competition in a CO2 laser. Phys. Rev. A, 53, 4435-4438(1996).
[5] K. Staliunas, G. Slekys, C. O. Weiss. Nonlinear pattern formation in active optical systems: shocks, domains of tilted waves, and cross-roll patterns. Phys. Rev. Lett., 79, 2658-2661(1997).
[6] V. B. Taranenko, K. Staliunas, C. O. Weiss. Pattern formation and localized structures in degenerate optical parametric mixing. Phys. Rev. Lett., 81, 2236-2239(1998).
[7] D. Dragoman, M. Dragoman. Quantum-Classical Analogies(2004).
[8] V. Bužek, T. Quang. Generalized coherent state for bosonic realization of SU(2) Lie algebra. J. Opt. Soc. Am. B, 6, 2447-2449(1989).
[9] G. S. Agarwal, J. Banerji. Entanglement by linear SU(2) transformations: generation and evolution of quantum vortex states. J. Phys. A, 39, 11503-11519(2006).
[10] Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, S. C. Wang. Generation of Hermite–Gaussian modes in fiber-coupled laser-diode end-pumped lasers. IEEE J. Quantum Electron., 33, 1025-1031(1997).
[11] H. Laabs, B. Ozygus. Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping. Opt. Laser Technol., 28, 213-214(1996).
[12] Y. F. Chen, Y. P. Lan. Dynamics of the Laguerre Gaussian TEM0,l* mode in a solid-state laser. Phys. Rev. A, 63, 063807(2001).
[13] C. J. Flood, G. Giuliani, H. M. van Driel. Preferential operation of an end-pumped Nd:YAG laser in high-order Laguerre–Gauss modes. Opt. Lett., 15, 215-217(1990).
[14] S. Ngcobo, K. Aït-Ameur, N. Passilly, A. Hasnaoui, A. Forbes. Exciting higher-order radial Laguerre–Gaussian modes in a diode-pumped solid-state laser resonator. Appl. Opt., 52, 2093-2101(2013).
[15] A. Hu, J. Lei, P. Chen, Y. Wang, S. Li. Numerical investigation on the generation of high-order Laguerre–Gaussian beams in end-pumped solid-state lasers by introducing loss control. Appl. Opt., 53, 7845-7853(2014).
[16] M. A. Bandres, J. C. Gutiérrez-Vega. Ince–Gaussian modes of the paraxial wave equation and stable resonators. J. Opt. Soc. Am. A, 21, 873-880(2004).
[17] U. T. Schwarz, M. A. Bandres, J. C. Gutiérrez-Vega. Observation of Ince–Gaussian modes in stable resonators. Opt. Lett., 29, 1870-1872(2004).
[18] T. Ohtomo, K. Kamikariya, K. Otsuka, S. Chu. Single-frequency Ince–Gaussian mode operations of laser-diode-pumped microchip solid-state lasers. Opt. Express, 15, 10705-10717(2007).
[19] N. Barré, M. Romanelli, M. Brunel. Role of cavity degeneracy for high-order mode excitation in end-pumped solid-state lasers. Opt. Lett., 39, 1022-1025(2014).
[20] J. Durnin, J. J. Miceli, J. H. Eberly. Diffraction-free beams. Phys. Rev. Lett., 58, 1499-1501(1987).
[21] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman. Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes. Phys. Rev. A, 45, 8185-8189(1992).
[22] G. A. Siviloglou, J. Broky, A. Dogariu, D. N. Christodoulides. Observation of accelerating Airy beams. Phys. Rev. Lett., 99, 213901(2007).
[23] A. Yu Okulov, A. N. Oraevsky. Space-temporal behavior of a light pulse propagating in a nonlinear nondispersive medium. J. Opt. Soc. Am. B, 3, 741-746(1986).
[24] Y. F. Chen, Y. P. Lan. Transverse pattern formation of optical vortices in a microchip laser with a large Fresnel number. Phys. Rev. A, 65, 013802(2001).
[25] A. Yu Okulov. Vortex–antivortex wavefunction of a degenerate quantum gas. Laser Phys., 19, 1796-1803(2009).
[26] V. L. Berezinskii. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group II. Quantum systems. Sov. Phys. J. Exp. Theor. Phys., 34, 610-616(1972).
[27] J. M. Kosterlitz, D. J. Thouless. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C, 6, 1181-1203(1973).
[28] Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, J. Dalibard. Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas. Nature, 441, 1118-1121(2006).
[29] A. Yu Okulov. Twisted speckle entities inside wavefront reversal mirrors. Phys. Rev. A, 80, 013837(2009).
[30] M. Woerdemann, C. Alpmann, M. Esseling, C. Denz. Advanced optical trapping by complex beam shaping. Laser Photon. Rev., 7, 839-854(2013).
[31] V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, Y. S. Kivshar. Giant optical manipulation. Phys. Rev. Lett., 105, 118103(2010).
[32] W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, A. Zeilinger. Quantum orbital angular momentum of elliptically symmetric light. Phys. Rev. A, 87, 033806(2013).
[33] Y. F. Chen, J. C. Tung, P. H. Tuan, Y. T. Yu, H. C. Liang, K. F. Huang. Characterizing classical periodic orbits from quantum Green’s functions in two-dimensional integrable systems: harmonic oscillators and quantum billiards. Phys. Rev. E, 95, 012217(2017).
[34] E. Schrödinger. Collected Papers on Wave Mechanics(1982).
[35] P. Holland. The Quantum Theory of Motion(1993).
[36] Y. F. Chen, Y. P. Lan, K. F. Huang. Observation of quantum-classical correspondence from high-order transverse patterns. Phys. Rev. A, 68, 043803(2003).
[37] Y. F. Chen, T. H. Lu, K. W. Su, K. F. Huang. Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces. Phys. Rev. Lett., 96, 213902(2006).
[38] J. Courtois, A. Mohamed, D. Romanini. Degenerate astigmatic cavities. Phys. Rev. A, 88, 043844(2013).
[39] J. C. Tung, P. H. Tuan, H. C. Liang, K. F. Huang, Y. F. Chen. Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves. Phys. Rev. A, 94, 023811(2016).
[40] K. F. Huang, Y. F. Chen, H. C. Lai, Y. P. Lan. Observation of the wave function of a quantum billiard from the transverse patterns of vertical cavity surface emitting lasers. Phys. Rev. Lett., 89, 224102(2002).
[41] Y. T. Yu, P. H. Tuan, P. Y. Chiang, H. C. Liang, K. F. Huang, Y. F. Chen. Wave pattern and weak localization of chaotic versus scarred modes in stadium-shaped surface-emitting lasers. Phys. Rev. E, 84, 056201(2011).
[42] A. Messiah. Quantum Mechanics(1966).
[43] N. N. Lebedev. Special Functions & Their Applications(1972).
[44] A. Hasnaoui, A. Bencheikh, K. Aït-Ameur. Tailored TEMp0 beams for large size 3-D laser prototyping. Opt. Lasers Eng., 49, 248-251(2011).
[45] N. Hodgson, H. Weber. Laser Resonators and Beam Propagation(2005).
[46] J. Dong, S. C. Bai, S. H. Liu, K. I. Ueda, A. A. Kaminskii. A high repetition rate passively Q-switched microchip laser for controllable transverse laser modes. J. Opt., 18, 055205(2016).
[47] J. Dong, Y. He, S. C. Bai, K. I. Ueda, A. A. Kaminskii. A Cr4+:YAG passively Q-switched Nd:YVO4 microchip laser for controllable high-order Hermite–Gaussian modes. Laser Phys., 26, 095004(2016).
[48] J. Dong, Y. He, X. Zhou, S. Bai. Highly efficient, versatile, self-Q-switched, high-repetition-rate microchip laser generating Ince–Gaussian modes for optical trapping. Quantum Electron., 46, 218-222(2016).
[49] H. S. He, M. M. Zhang, J. Dong, K. I. Ueda. Linearly polarized pumped passively Q-switched Nd:YVO4 microchip laser for Ince–Gaussian laser modes with controllable orientations. J. Opt., 18, 055205(2016).
[50] M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman. Astigmatic laser mode converters and transfer of orbital angular momentum. Opt. Commun., 96, 123-132(1993).
