Applied Physics B

, Volume 121, Issue 3, pp 315–336 | Cite as

Fundamental laser modes in paraxial optics: from computer algebra and simulations to experimental observation

  • Christoph Koutschan
  • Erwin Suazo
  • Sergei K. Suslov


We study multi-parameter solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation which include oscillating laser beams in a parabolic waveguide, spiral light beams, and other important families of propagation-invariant laser modes in weakly varying media. A “smart” lens design and a similar effect of superfocusing of particle beams in a thin monocrystal film are also discussed. In the supplementary electronic material, we provide a computer algebra verification of the results presented here, and of some related mathematical tools that were stated without proofs in the literature. We also demonstrate how computer algebra can be used to derive some of the presented formulas automatically, which is highly desirable as the corresponding hand calculations are very tedious. In numerical simulations, some of the new solutions reveal quite exotic properties which deserve further investigation including an experimental observation.

Mathematics Subject Classification

Primary 35C05 35Q55 Secondary 68W30 81Q05 



This research was partially carried out during our participation in the Summer School on “Combinatorics, Geometry and Physics” at the Erwin Schrödinger International Institute for Mathematical Physics (ESI), University of Vienna, in June 2014. We wish to express our gratitude to Christian Krattenthaler for his hospitality. The first-named author was supported by the Austrian Science Fund (FWF): W1214, the second-named author by the Simons Foundation Grant #316295 and by the National Science Foundation Grant DMS-1440664, and the third-named author by the AFOSR Grant FA9550-11-1-0220. We are grateful to Eugeny G. Abramochkin, Sergey I. Kryuchkov, Vladimir I. Man'ko, and Peter Paule for valuable comments and to Miguel A. Bandres for kindly pointing out the reference [16] to our attention. Suggestions from the referees are much appreciated. Last but not least, we would like to thank Aleksei P. Kiselev for communicating the interesting articles [69, 70, 71].

Supplementary material

340_2015_6231_MOESM1_ESM.nb (9.9 mb)
Supplementary material 1 (nb 10104 KB)


  1. 1.
    D. Abdollahpour, S. Suntsov, D.G. Papazoglou, S. Tzortzakis, Spatiotemporal Airy light bullets in the linear and nonlinear regimes. Phys. Rev. Lett. 105(25), 253901 (2010)ADSCrossRefGoogle Scholar
  2. 2.
    E.G. Abramochkin, unpublished manuscript (in Russian) Google Scholar
  3. 3.
    E.G. Abramochkin, T. Alieva, J.A. Rodrigo, Solutions of paraxial equations and families of Gaussian beams, in Mathematical Optics: Classical, Quantum, and Computational Methods, ed. by V. Lakshmianrayanan, M.L. Calvo, T. Alieva (CRC Press, Boca Raton, 2013), pp. 143–192Google Scholar
  4. 4.
    E.G. Abramochkin, E. Razueva, Product of three Airy beams. Opt. Lett. 36(19), 3732–3734 (2011)ADSCrossRefGoogle Scholar
  5. 5.
    E.G. Abramochkin, V.G. Volostnikov, Two-dimensional phase problem: differential approach. Opt. Commun. 74(3), 139–143 (1989)ADSCrossRefGoogle Scholar
  6. 6.
    E.G. Abramochkin, V.G. Volostnikov, Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone. Opt. Commun. 74(3), 144–148 (1989)ADSCrossRefGoogle Scholar
  7. 7.
    E.G. Abramochkin, V.G. Volostnikov, Beam transformations and nontransformed beams. Opt. Commun. 83(1–2), 123–135 (1991)ADSCrossRefGoogle Scholar
  8. 8.
    E.G. Abramochkin, V.G. Volostnikov, Spiral light beams. Phys. Uspekhi 47(12), 1177–1203 (2004)ADSCrossRefGoogle Scholar
  9. 9.
    E.G. Abramochkin, V.G. Volostnikov, Modern Optics of Gaussian Beams (FizMatLit, Moscow, 2010). (in Russian)Google Scholar
  10. 10.
    G.P. Agrawal, A.K. Ghatak, C.L. Mehtav, Propagation of a partially coherent beam through selfoc fibers. Opt. Commun. 12(3), 333–337 (1974)ADSCrossRefGoogle Scholar
  11. 11.
    S.A. Akhmanov, Y.E. Dyakov, A.S. Chirkin, An Introduction to Statistical Radiophysics and Optics (Nauka, Moscow, 1981). (in Russian)Google Scholar
  12. 12.
    S.A. Akhmanov, S.Y. Nikitin, Physical Optics (Clarendon Press, Oxford, 1997)Google Scholar
  13. 13.
    C. Ament, P. Polynkin, J.V. Moloney, Supercontinuum generation with femtosecond self-healing Airy pulses. Phys. Rev. Lett. 107(24), 243901 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    M.A. Bandres, Accelerating beams. Opt. Lett. 34(24), 3791–3793 (2009)ADSCrossRefGoogle Scholar
  15. 15.
    M.A. Bandres, J.C. Gultiérrez-Vega, Airy-Gauss beams and their transformation by paraxial optical systems. Opt. Express 15(25), 16719–16728 (2007)ADSCrossRefGoogle Scholar
  16. 16.
    M.A. Bandres, M. Guizar-Sicairos, Paraxial group. Opt. Lett. 34(1), 13–15 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    M.A. Bandres, I. Kaminer, M. Mills, B.M. Rodriguez-Lara, E. Greenfield, M. Segev, D.N. Christodoulides, Accelerating optical beams. Opt. Photonic News 24(6), 30–37 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    R. Bekenstein, M. Segev, Self-accelerating optical beams in highly nonlocal nonlinear media. Opt. Express 19(24), 23706–23715 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    P.A. Bélanger, Packetlike solutions of the homogeneous-wave equation. J. Opt. Soc Am. A 1(7), 723–724 (1984)ADSCrossRefGoogle Scholar
  20. 20.
    M.V. Berry, N.L. Balazs, Nonspreading wave packets. Am. J. Phys. 47(2), 264–267 (1979)ADSCrossRefGoogle Scholar
  21. 21.
    I.M. Besieris, A.M. Shaarawi, R.W. Ziolkowski, Nondispersive accelerating wave packets. Am. J. Phys. 62(6), 519–521 (1994)ADSCrossRefGoogle Scholar
  22. 22.
    I.M. Besieris, A.M. Shaarawi, A note on an accelerating finite energy Airy beam. Opt. Lett. 32(16), 2447–2449 (2007)ADSCrossRefGoogle Scholar
  23. 23.
    I.M. Besieris, A.M. Shaarawi, Accelerating Airy beams with non-parabolic trajectories. Opt. Commun. 331, 235–238 (2014)ADSCrossRefGoogle Scholar
  24. 24.
    I. Bialynicki-Birula, Photon as a quantum particle. Acta Phys. Pol. B 37(3), 935–946 (2006)ADSGoogle Scholar
  25. 25.
    I. Bialynicki-Birula, Z. Bialynicka-Birula, Canonical separation of angular momentum of light into its orbital and spin parts. J. Opt. 13(6), 064014 (2011)ADSCrossRefGoogle Scholar
  26. 26.
    C.P. Boyer, R.T. Sharp, P. Winternitz, Symmetry breaking interactions for the time dependent Schrödinger equation. J. Math. Phys. 17(8), 1439–1451 (1976)MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 27.
    M. Born, E. Wolf, Principles of Optics, 7th edn. (Pergamon Press, Oxford, 1999)CrossRefGoogle Scholar
  28. 28.
    C. Brosseau, Polarization and coherence optics: historical perspective, status, and future directions, in Progress in Optics, ed. by E. Wolf (Elsevier, Amsterdam, 2009), pp. 149–208Google Scholar
  29. 29.
    B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. PhD thesis, University of Innsbruck, Austria, 1965Google Scholar
  30. 30.
    R.-P. Chen, C.-F. Yin, X.-X. Chu, H. Wang, Effect of Kerr nonlinearity on an Airy beam. Phys. Rev. A 82, 043832 (2010)ADSCrossRefGoogle Scholar
  31. 31.
    R.-P. Chen, H.-P. Zheng, C.-Q. Dai, Wigner distribution function of an Airy beam. J. Opt. Soc. Am. A 28(6), 1307–1311 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    A. Chong, W.H. Renninger, D.N. Christodoulides, F.W. Wise, Airy-Bessel wave packets as versatile linear light bullets. Nat. Photonics 4(2), 103–106 (2010)ADSCrossRefGoogle Scholar
  33. 33.
    R. Cordero-Soto, R.M. Lopez, E. Suazo, S.K. Suslov, Propagator of a charged particle with a spin in uniform magnetic and perpendicular electric fields. Lett. Math. Phys. 84(2–3), 159–178 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  34. 34.
    R. Cordero-Soto, S.K. Suslov, Time reversal for modified oscillators. Theor. Math. Phys. 162(3), 286–316 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    R.T. Couto, Green’s functions for the wave, Helmholtz and Poisson equations in a two-dimensional boundless domain. Revista Brasileira de Ensino de Física 35(1), 1304 (2013)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Y.A. Danilov, G.I. Kuznetsov, Y.A. Smorodinsky, On the symmetry of classical and wave equations. Sov. J. Nucl. Phys. 32(6), 801–804 (1980)Google Scholar
  37. 37.
    J.A. Davis, M.J. Mitry, M.A. Bandres, D.M. Cottrell, Observation of accelerating parabolic beams. Opt. Express 16(17), 12866–12871 (2008)ADSCrossRefGoogle Scholar
  38. 38.
    J.A. Davis, M.J. Mitry, M.A. Bandres, I. Ruiz, K.P. McAuley, D.M. Cottrell, Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns. Appl. Opt. 48(17), 3170–3176 (2009)ADSCrossRefGoogle Scholar
  39. 39.
    Y.N. Demkov, Channeling, superfocusing, and nuclear reactions. Phys. Atomic Nucl. 72(5), 779–785 (2009)ADSCrossRefGoogle Scholar
  40. 40.
    Y.N. Demkov, J.D. Meyer, A sub-atomic microscope, superfocusing in channeling and close encounter atomic and nuclear reactions. Eur. Phys. J. B 42, 361–365 (2004)ADSCrossRefGoogle Scholar
  41. 41.
    D.M. Deng, Propagation of Airy-Gaussian beams in a quadratic-index medium. Eur. Phys. J. D 65, 553–556 (2010)ADSCrossRefGoogle Scholar
  42. 42.
    D.M.R. Dennis, J.B. Götte, R.P. King, M.A. Morgan, M.A. Alonso, Paraxial and nonparaxial polynomial beams and the analytic approach to propagation. Opt. Lett. 36(22), 4452–4454 (2011)ADSCrossRefGoogle Scholar
  43. 43.
    A.S. Desyatnikov, D. Buccoliero, M.R. Dennis, Y.S. Kivshar, Suppression of collapse for spiraling elliptic solutions. Phys. Rev. Lett. 104, 053902 (2010)ADSCrossRefGoogle Scholar
  44. 44.
    I.H. Deutsch, J.C. Garrison, Paraxial quantum propagation. Phys. Rev. A 43(5), 2498–2513 (1991)ADSCrossRefGoogle Scholar
  45. 45.
    V.V. Dodonov, V.I. Man’ko, Invariants and correlated states of nonstationary quantum systems, in Invariants and the Evolution of Nonstationary Quantum Systems, Proceedings of Lebedev Physics Institute, vol. 183, pp. 71–181, Nauka, Moscow, 1987 (in Russian); English translation published by Nova Science, Commack, New York, 1989, pp. 103–261Google Scholar
  46. 46.
    J. Durnin, Exact solutions for nondiffracting beams I. The scalar theory. J. Opt. Soc. Am. A 4(4), 651–654 (1987)ADSCrossRefGoogle Scholar
  47. 47.
    J. Durnin, J.J. Miceli Jr, J.H. Eberly, Diffraction-free beams. Phys. Rev. Lett. 58(15), 1499–1501 (1987)ADSCrossRefGoogle Scholar
  48. 48.
    G. Eichmann, Quasi-geometric optics of media with inhomogeneous index of refraction. J. Opt. Soc. Am. 61(2), 161–168 (1971)ADSCrossRefGoogle Scholar
  49. 49.
    V. Fock, Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld. Zs. für Phys. 47, 446–448 (1928). translated to English: A Comment on Quantization of the Harmonic Oscillator in a Magnetic Field, in Selected Works: Quantum Mechanics and Quantum Field Theory, ed. by L.D. Faddeev, L.A. Khalfin, I.V. Komarov (Chapman & Hall/CRC, Boca Raton, London, New York, Washington, DC, 2004), pp. 29–31Google Scholar
  50. 50.
    V.A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon Press, London, 1965)Google Scholar
  51. 51.
    S. Fürhapter, A. Jesacher, S. Bernet, M. Ritsch-Marte, Spiral interferometry. Opt. Lett. 30(15), 1953–1955 (2005)ADSCrossRefGoogle Scholar
  52. 52.
    L. Gagnon, P. Winternitz, Lie symmetries of a generalised non-linear Schrödinger equation: II. Exact solutions. J. Phys. A Math. Gen. 22, 469–497 (1989)MathSciNetADSzbMATHCrossRefGoogle Scholar
  53. 53.
    J.A. Giannini, R.I. Joseph, The role of the second Painlevé transcendent in nonlinear optics. Phys. Lett. A 141(8), 417–419 (1989)MathSciNetADSCrossRefGoogle Scholar
  54. 54.
    G. Gibson, J. Courtial, M.J. Padgett, M. Vasnetsov, V. Pas’ko, S.M. Barnett, S. Franke-Arnold, Free-space information transfer using light beams carrying orbital angular momentum. Opt. Express 12(22), 5448–5456 (2004)ADSCrossRefGoogle Scholar
  55. 55.
    A.M. Goncharenko, Gaussian Beams of Light (Nauka & Tekhnika, Minsk, 1977). (in Russian)Google Scholar
  56. 56.
    F. Gori, G. Guattari, C. Padovani, Bessel–Gauss beams. Opt. Commun. 64(6), 491–495 (1987)ADSCrossRefGoogle Scholar
  57. 57.
    D.M. Greenberg, Comment on “Nonspreading wave packets”. Am. J. Phys. 48(3), 256 (1980)ADSCrossRefGoogle Scholar
  58. 58.
    Y. Gu, G. Gbur, Scintillation of Airy beam arrays in atmospheric turbulence. Opt. Lett. 35, 3456–3458 (2010)ADSCrossRefGoogle Scholar
  59. 59.
    Y. Gu, Statistics of optical vortex wander on propagation through atmospheric turbulence. J. Opt. Soc. Am. A 30(4), 708–716 (2013)ADSCrossRefGoogle Scholar
  60. 60.
    G. Gbur, T.D. Visser, Coherence vortices in partially coherent beams. Opt. Commun. 222, 117–125 (2003)ADSCrossRefGoogle Scholar
  61. 61.
    A.V. Gurevich, Nonlinear Phenomena in the Ionosphere (Springer, Berlin, 1978)CrossRefGoogle Scholar
  62. 62.
    J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, T. Omatsu, Optical-vortex laser ablation. Opt. Express 18(3), 2144–2151 (2010)ADSCrossRefGoogle Scholar
  63. 63.
    M.R. Hatzvi, Y.Y. Schechner, Three-dimensional optical transfer of rotating beams. Opt. Lett. 37, 32074 (2012)CrossRefGoogle Scholar
  64. 64.
    H.A. Haus, J.L. Pan, Photon spin and the paraxial wave equation. Am. J. Phys. 61(9), 818–821 (1993)ADSCrossRefGoogle Scholar
  65. 65.
    J.D. Jackson, Classical Electrodynamics, 2nd edn. (Wiley, New York, 1975)zbMATHGoogle Scholar
  66. 66.
    I. Kaminer, M. Segev, D.N. Christodoulides, Self-accelerating self-trapped optical beams. Phys. Rev. Lett. 106, 213903 (2011)ADSCrossRefGoogle Scholar
  67. 67.
    J. Kasparian, J.-P. Wolf, Laser beams take a curve. Science 324, 194–195 (2009)ADSCrossRefGoogle Scholar
  68. 68.
    C. Kharif, E. Pelinovsky, A. Slunyaev, Rogue Waves in the Ocean (Springer, Berlin, 2009)zbMATHGoogle Scholar
  69. 69.
    A.P. Kiselev, Localized light waves: paraxial and exact solutions of the wave equation (a review). Opt. Spectrosc. 102(4), 603–622 (2007)ADSCrossRefGoogle Scholar
  70. 70.
    A.P. Kiselev, M.V. Perel, Highly localized solutions of the wave equation. J. Math. Phys. 41(4), 1934–1955 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  71. 71.
    A.P. Kiselev, A.B. Plachenov, P. Chamorro-Posada, Nonparaxial wave beams and packets with general astigmatism. Phys. Rev. A 85(4), 043835 (2012)ADSCrossRefGoogle Scholar
  72. 72.
    M. Kline, I.W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience Publishers, New York, 1965)zbMATHGoogle Scholar
  73. 73.
    H. Kogelnik, On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation. Appl. Opt. 4(12), 1562–1569 (1965)ADSCrossRefGoogle Scholar
  74. 74.
    H. Kogelnik, T. Li, Laser beams and resonators. Appl. Opt. 5(10), 1550–1567 (1966)ADSCrossRefGoogle Scholar
  75. 75.
    C. Koutschan, Advanced applications of the holonomic systems approach. PhD thesis, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria, 2009Google Scholar
  76. 76.
    C. Koutschan, HolonomicFunctions (user’s guide), RISC Report Series, Johannes Kepler University, Linz, Austria, 2010;
  77. 77.
    C. Koutschan, Creative telescoping for holonomic functions, in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, ed. by C. Schneider, J. Blümlein Springer Series “Texts and Monographs in Symbolic Computation” (Springer, Wien, 2013), pp. 171–194Google Scholar
  78. 78.
  79. 79.
    C. Koutschan, P. Paule, S.K. Suslov, Relativistic Coulomb Integrals and Zeilbergers Holonomic Systems Approach II, in AADIOS 2012—Algebraic and Algorithmic Aspects of Differential and Integral Operators Session, ed. by M. Barkatou et al., Lecture Notes in Computer Sciences, vol. 8372 (Springer, 2014) pp. 135–145Google Scholar
  80. 80.
    C. Koutschan, E. Suazo, S.K. Suslov, Mathematica notebook MultiParameterModes.nb, supplementary electronic material to the article Fundamental Laser Modes in Paraxial Optics: From Computer Algebra and Simulations to Experimental Observation,, 2015
  81. 81.
    C. Krattenthaler, S.I. Kryuchkov, A. Mahalov, S.K. Suslov, On the problem of electromagnetic-field quantization. Int. J. Theor. Phys. 52(12), 4445–4460 (2013); see also arXiv:1301.7328v2 [math-ph] 9 Apr 2013
  82. 82.
    Y.A. Kravtsov, Y.I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, Berlin, 1990)CrossRefGoogle Scholar
  83. 83.
    S.G. Krivoshlykov, I.N. Sissakian, Optical beam and pulse propagation in inhomogeneous media. Application to multiple parabolic-index waveguides. Opt. Quantum Electron. 12, 463–475 (1980)ADSCrossRefGoogle Scholar
  84. 84.
    S.G. Krivoshlykov, N.I. Petrov, I.N. Sissakian, Spacial coherence of optical beams in longitudinally inhomogeneous media with quadratic refractive index profiles. Sov. J. Quantum Electron. 15(3), 330–338 (1985)ADSCrossRefGoogle Scholar
  85. 85.
    S.G. Krivoshlykov, N.I. Petrov, I.N. Sissakian, Correlated coherent states and propagation of arbitrary Gaussian beams in longitudinally inhomogeneous quadratic media exhibiting absorption or amplification. Sov. J. Quantum Electron. 16(7), 933–941 (1986)ADSCrossRefGoogle Scholar
  86. 86.
    S.G. Krivoshlykov, E.G. Sauter, Transformation of paraxial beams in arbitrary multimode parabolic-index fiber tapers by using a quantum-theoretical approach. Appl. Opt. 31(7), 2017–2024 (1992)ADSCrossRefGoogle Scholar
  87. 87.
    S.I. Kryuchkov, N. Lanfear, S.K. Suslov, The Pauli–Lubański vector, complex electrodynamics, and photon helicity. Phys. Scr. 90(7), 074065 (2015)ADSCrossRefGoogle Scholar
  88. 88.
    S.I. Kryuchkov, S.K. Suslov, J.M. Vega-Guzmán, The minimum-uncertainty squeezed states for atoms and photons in a cavity. J. Phys. B Atomic Mol. Opt. Phys. 46, 104007 (2013). (IOP Select and Highlight of 2013)ADSCrossRefGoogle Scholar
  89. 89.
    E.A. Kuznetsov, S.K. Turitsyn, Talanov transformations in self-focusing problems and instability of stationary waveguides. Phys. Lett. A 112(6–7), 273–275 (1985)ADSCrossRefGoogle Scholar
  90. 90.
    N. Lanfear, R.M. López, S.K. Suslov, Exact wave functions for generalized harmonic oscillators. J. Russ. Laser Res. 32(4), 352–361 (2011)CrossRefGoogle Scholar
  91. 91.
    M. Lax, W.H. Louisell, W.B. McKnight, From Maxwell to paraxial wave optics. Phys. Rev. A 11(4), 1365–1370 (1975)ADSCrossRefGoogle Scholar
  92. 92.
    A. Lotti et al., Stationary nonlinear Airy beams. Phys. Rev. A 84, 021807(R) (2011)ADSCrossRefGoogle Scholar
  93. 93.
    D. Leibfried, R. Blatt, C. Monroe, D. Wineland, Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75(1), 281–324 (2003)ADSCrossRefGoogle Scholar
  94. 94.
    R.M. López, S.K. Suslov, J.M. Vega-Guzmán, Reconstracting the Schrödinger groups. Phys. Scr. 87(3), 038118 (2013)CrossRefGoogle Scholar
  95. 95.
    R.M. López, S.K. Suslov, J.M. Vega-Guzmán, On a hidden symmetry of quantum harmonic oscillators. J. Differ. Equ. Appl. 19(4), 543–554 (2013)zbMATHCrossRefGoogle Scholar
  96. 96.
    R.K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964)Google Scholar
  97. 97.
    A. Mahalov, E. Suazo, S.K. Suslov, Spiral laser beams in inhomogeneous media. Opt. Lett. 38(15), 2763–2766 (2013)ADSCrossRefGoogle Scholar
  98. 98.
    A. Mahalov, S.K. Suslov, An “Airy gun”: self-accelerating solutions of the time-dependent Schrödinger equation in vacuum. Phys. Lett. A 377, 33–38 (2012)ADSCrossRefGoogle Scholar
  99. 99.
    A. Mahalov, S.K. Suslov, Wigner function approach to oscillating solutions of the 1D-quintic nonlinear Schrödinger equation. J. Nonlinear Opt. Phys. Mater. 22(2), 1350013 (2013)ADSCrossRefGoogle Scholar
  100. 100.
    A. Mahalov, S.K. Suslov, Solution of paraxial wave equation for inhomogeneous media in linear and quadratic approximation. Proc. Am. Math. Soc. 143(2), 595–610 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Entanglement of the orbital angular momentum states of photons. Nature 412(19), 313–316 (2001)ADSCrossRefGoogle Scholar
  102. 102.
    M.E. Marhic, Oscillating Hermite-Gaussian wave functions of the harmonic oscillator. Lett. Nuovo Cim. 22(8), 376–378 (1978)MathSciNetCrossRefGoogle Scholar
  103. 103.
    M.A.M. Marte, S. Stenholm, Paraxial light and atom optics: the Schrödinger equation and beyond. Phys. Rev. A 56(4), 2940–2953 (1997)ADSCrossRefGoogle Scholar
  104. 104.
    M. Meiler, R. Cordero-Soto, S.K. Suslov, Solution of the Cauchy problem for a time-dependent Schödinger equation. J. Math. Phys. 49, 072102 (2008)MathSciNetADSCrossRefGoogle Scholar
  105. 105.
    W. Miller Jr, Symmetry and Separation of Variables, Encyclopedia of Mathematics and Its Applications, vol. 4 (Addison-Wesley Publishing Company, Reading, 1977)Google Scholar
  106. 106.
    G. Molina-Terriza, J.P. Torres, L. Torner, Twisted photons. Nat. Phys. 3(5), 305–310 (2007)CrossRefGoogle Scholar
  107. 107.
    U. Niederer, The maximal kinematical invariance group of the free Schrödinger equations. Helv. Phys. Acta 45, 802–810 (1972)MathSciNetGoogle Scholar
  108. 108.
    U. Niederer, The maximal kinematical invariance group of the harmonic oscillator. Helv. Phys. Acta 46, 191–200 (1973)Google Scholar
  109. 109.
    A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics (Birkhäuser, Basel, 1988)zbMATHCrossRefGoogle Scholar
  110. 110.
    A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, Berlin, 1991)zbMATHCrossRefGoogle Scholar
  111. 111.
    B. Øksendal, Stochastic Differential Equations (Springer, Berlin, 2000)Google Scholar
  112. 112.
    A.Y. Okulov, Angular momentum of photons and phase conjugation. J. Phys. B Atomic Mol. Opt. Phys. 41, 101001 (2008)ADSCrossRefGoogle Scholar
  113. 113.
    F.W.J. Olver, Airy and related functions, in NIST Handbook of Mathematical Functions, ed. by F.W.J. Olwer, D.M. Lozier, et al. (Cambridge University Press, Cambridge, 2010). see also:
  114. 114.
    X. Pang, G. Gbur, T.D. Visser, The Gouy phase of Airy beams. Opt. Lett. 36(13), 2492–2494 (2011)ADSCrossRefGoogle Scholar
  115. 115.
    R. Piestun, Y.Y. Schechner, J. Shamir, Propagation-invariant wave fields with finite energy. J. Opt. Soc. Am. A 17(2), 294–303 (2000)ADSCrossRefGoogle Scholar
  116. 116.
    P. Polynkin, M. Kolesik, J.V. Moloney, G.A. Siviloglou, D.N. Christodoulides, Curved plasma channel generation using ultraintense Airy beams. Science 324, 229–232 (2009)ADSCrossRefGoogle Scholar
  117. 117.
    S.A. Ponomarenko, G.P. Agrawal, Do solitonlike self-similar waves exist in nonlinear optical media? Phys. Rev. Lett. 97, 013901 (2006)ADSCrossRefGoogle Scholar
  118. 118.
    R. Pratesi, L. Ronchi, Generalized Gaussian beams in free space. J. Opt. Soc. Am. 67(9), 1274–1276 (1977)ADSCrossRefGoogle Scholar
  119. 119.
    A. Rudnick, D.M. Marom, Airy-soliton interactions in Kerr media. Opt. Express 19(25), 25570–25582 (2011)ADSCrossRefGoogle Scholar
  120. 120.
    S.M. Rytov, Y.A. Kravtsov, V.I. Tatarskii, Principles of Statistical Radiophysics: Wave Propagation Through Random Media (Springer, Berlin, 1989)zbMATHCrossRefGoogle Scholar
  121. 121.
    Y.Y. Schechner, R. Piestun, J. Shamir, Wave propagation with rotating intensity distributions. Phys. Rev. E 54(1), R50–R53 (1996)ADSCrossRefGoogle Scholar
  122. 122.
    U.T. Schwarz, M.A. Banderes, J.C. Gutiérrez-Vega, Observation of Ince–Gaussian modes in stable resonators. Opt. Lett. 29(16), 1870–1872 (2004)ADSCrossRefGoogle Scholar
  123. 123.
    A.E. Siegman, Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions. J. Opt. Soc. Am. 63(9), 1093–1094 (1973)MathSciNetADSCrossRefGoogle Scholar
  124. 124.
    A.E. Siegman, Lasers (Univ. Sci. Books, Mill Valey, 1986)Google Scholar
  125. 125.
    G.A. Siviloglou, D.N. Christodoulides, Accelerating finite energy Airy beams. Opt. Lett. 32(2), 979–981 (2007)ADSCrossRefGoogle Scholar
  126. 126.
    G.A. Siviloglou, J. Broky, A. Dogariu, D.N. Christodoulides, Observation of accelerating Airy beams. Phys. Rev. Lett. 99, 213901 (2007)ADSCrossRefGoogle Scholar
  127. 127.
    R. Smith, Giant waves. J. Fluid Mech. 77(3), 417–431 (1976)MathSciNetADSzbMATHCrossRefGoogle Scholar
  128. 128.
    W.J. Smith, Modern Optical Engineering: The Design of Optical Systems, 3rd edn. (McGraw-Hill, New York, 2000)Google Scholar
  129. 129.
    A. Sommerfeld, Partial Differential Equations in Physics (Academic Press, New York, 1949)zbMATHGoogle Scholar
  130. 130.
    P. Sprange, B. Hafizi, Comment on nondiffracting beams. Phys. Rev. Lett. 66(6), 837 (1991)ADSCrossRefGoogle Scholar
  131. 131.
    S. Steinberg, Applications of the Lie algebraic formulas of Baker, partial differential equations. J. Differ. Equ. 26, 404–434 (1977)ADSzbMATHCrossRefGoogle Scholar
  132. 132.
    E. Suazo, S.K. Suslov, Soliton-like solutions for nonlinear Schrödinger equation with variable quadratic Hamiltonians. J. Russ. Laser Res. 33(1), 63–82 (2012)CrossRefGoogle Scholar
  133. 133.
    S.K. Suslov, On integrability of nonautonomous nonlinear Schrödinger equations. Proc. Am. Math. Soc. 140(9), 3067–3082 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  134. 134.
    M. Tajiri, Similarity reduction of the one and two dimensional nonlinear Schrödinger equations. J. Phys. Soc. Jpn. 52(2), 1908–1917 (1983)MathSciNetADSCrossRefGoogle Scholar
  135. 135.
    V.I. Talanov, Focusing of light in cubic media. JETP Lett. 11, 199–201 (1970)ADSGoogle Scholar
  136. 136.
    W. Tang, A. Mahalov, Stochastic Lagrangian dynamics for charged flows in the E−F regions of ionosphere. Phys. Plasmas 20, 032305 (2013)ADSCrossRefGoogle Scholar
  137. 137.
    T. Tao, A pseudoconformal compactification of the nonlinear Schrödinger equation and applications. N. Y. J. Math. 15, 265–282 (2009)zbMATHGoogle Scholar
  138. 138.
    A.V. Timofeev, Geometrical optics and the diffraction phenomenon. Phys. Uspekhi 48(6), 609–613 (2005)ADSCrossRefGoogle Scholar
  139. 139.
    A. Torre, Gaussian modulated Ai- and Bi-based solutions of the 2D PWE: a comparison. Appl. Phys. B 99, 775–799 (2010)ADSCrossRefGoogle Scholar
  140. 140.
    A. Torre, Paraxial equation, Lie-algebra-based approach, in Mathematical Optics: Classical, Quantum, and Computational Methods, ed. by V. Lakshmianrayanan, M.L. Calvo, T. Alieva (CRC Press, Boca Raton, 2013), pp. 341–417Google Scholar
  141. 141.
    J. Turunen, A.T. Friberg, Propagation-invariant optical fields, in Progress in Optics, ed. by E. Wolf (Elsevier, Amsterdam, 2009), pp. 1–88Google Scholar
  142. 142.
    K. Unnikrishnan, A.R.P. Rau, Uniqueness of the Airy packet in quantum mechanics. Am. J. Phys. 64(8), 1034–1035 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  143. 143.
    L.A. Vainshtein, Electromagnetic Waves, 2nd edn. (Radio i Svyaz’, Moscow, 1988). (in Russian)Google Scholar
  144. 144.
    M.B. Vinogradova, O.V. Rudenko, A.P. Sukhorukov, Theory of Waves, 2nd edn. (Nauka, Moscow, 1990). (in Russian)Google Scholar
  145. 145.
    V.S. Vladimirov, Equations of Mathematical Physics (Marcel Dekker, New York, 1971)Google Scholar
  146. 146.
    S.N. Vlasov, V.I. Talanov, The parabolic equation in the theory of wave propagation. Radiophys. Quantum Electron. 38(1–2), 1–12 (1995)MathSciNetADSGoogle Scholar
  147. 147.
    A. Walther, The Ray and Wave Theory of Lenses (Cambridge University Press, Cambridge, 1995)CrossRefGoogle Scholar
  148. 148.
    E.M. Wright, J.C. Garrison, Path-integral derivation of the complex ABCD Huygens integral. J. Opt. Soc. Am. A 4(9), 1751–1755 (1987)MathSciNetADSCrossRefGoogle Scholar
  149. 149.
    A. Wünsche, Coherence vortices in partially coherent beams. J. Opt. Soc. Am. A 6(9), 1320–1329 (1989)ADSCrossRefGoogle Scholar
  150. 150.
    A.M. Yao, M.J. Padgett, Orbital angular momentum: origins, behavior and applications. Adv. Opt. Photonics 3(2), 161–204 (2011)CrossRefGoogle Scholar
  151. 151.
    A. Yariv, Quantum Electronics (Wiley, New York, 1988)Google Scholar
  152. 152.
    A. Yariv, P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th edn. (Oxford University Press, Oxford, 2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Christoph Koutschan
    • 1
  • Erwin Suazo
    • 2
  • Sergei K. Suslov
    • 3
  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  2. 2.School of Mathematical and Statistical SciencesUniversity of Texas of Rio Grande ValleyEdinburgUSA
  3. 3.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

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