Abstract

A new family of paraxial optical beams has been recently introduced in the literature [Opt. Express 21, 17951 (2013)] [CrossRef]  , having the significant feature of exhibiting discrete-like diffraction patterns reminiscent of those observed in periodic evanescently coupled waveguide lattices. In this connection, we wish to highlight the symmetry properties of the paraxial wave equation, properties that, in our opinion, are still not exploited at their full potentiality, and the effectiveness of differential equation-solving procedures based on the generating-function method.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Perez-Leija, F. Soto-Eguibar, S. Chavez-Cerda, A. Szameit, H. Moya-Cessa, and D. N. Christodoulides, “Discrete-like diffraction dynamics in free space,” Opt. Express 21, 17951–17960 (2013).
    [Crossref]
  2. G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cimento B 105, 327–348 (1990).
    [Crossref]
  3. G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cimento B 106, 21–51 (1991).
    [Crossref]
  4. G. Dattoli and A. Torre, Theory and Applications of the Generalized Bessel Functions (Aracne, 1996).
  5. L. S. Brown and T. W. B. Kibble, “Interaction of intense laser beams with electrons,” Phys. Rev. 133, A705–A719 (1964).
    [Crossref]
  6. H. R. Reiss, “Foundations of strong-field physics,” in Lectures on Ultrafast Intense Laser Science, K. Yamanouchi, ed. (Springer-Verlag, 2010)., pp. 41–84.
  7. V. I. Usachenko and S.-I. Chu, “Strong-field ionization of laser-irradiated light homonuclear diatomic molecules: a generalized strong-field approximation-linear combination of atomic orbitals model,” Phys. Rev. A 71, 063410 (2005).
    [Crossref]
  8. A. Jaron-Becker, “Molecular dynamics in strong laser fields,” IEEE J. Sel. Top. Quantum Electron. 18, 105–112 (2012).
    [Crossref]
  9. C. S. E. van Ditzhuijzen, A. Tauschinsky, and H. B. van Linden van den Heuvell, “Observation of Stueckelberg oscillations in dipole-dipole interactions,” Phys. Rev. A 80, 063407 (2009).
    [Crossref]
  10. A. Jaron-Becker, A. Becker, and F. H. M. Faisal, “Dependence of strong-field photoelectron angular distributions on molecular orientation,” J. Phys. B 36, L375–L380 (2003).
    [Crossref]
  11. G. Dattoli, A. Renieri, and A. Torre, Lectures in Free-Electron Laser Theory and Related Topics (World Scientific, 1995).
  12. V. Strakhovenko, X. Artru, R. Chehab, and M. Chevallier, “Generation of circularly polarized photons for a linear collider polarized positron source,” Nucl. Instrum. Methods Phys. Res. A 547, 320–333 (2005).
    [Crossref]
  13. F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33, 2689–2691 (2008).
    [Crossref]
  14. A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77, 043804 (2008).
    [Crossref]
  15. E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt+Uxx = 0 and iUt+Uxx − c/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).
    [Crossref]
  16. C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U = 0,” J. Math. Phys. 16, 499–511 (1975).
    [Crossref]
  17. W. Miller, Symmetry and Separation of Variables (Addison-Wesley, 1977).
  18. A. Torre, “Linear and quadratic exponential modulation of the solutions of the paraxial wave equation,” J. Opt. 12, 035701 (2010).
    [Crossref]
  19. A. Torre, “Paraxial wave equation: Lie-algebra based approach,” in Mathematical Optics: Classical, Quantum, and Computational Methods, V. Lakshminarayanan, M. L. Calvo, and T. Alieva, eds. (CRC Press, 2013), Chap. 10, pp. 341–417.
  20. D. Babusci, G. Dattoli, and M. Del Franco, “Lectures on mathematical methods for physics,” (ENEA, 2010).
  21. G. Dattoli, A. Torre, and M. Carpanese, “The Hermite-Bessel functions: a new point of view on the theory of the generalized Bessel functions,” Radiat. Phys. Chem. 51, 221–228 (1998).
    [Crossref]
  22. M. P. Appell, “Sur l’équation ∂2z/∂x2 ∂z/∂y = 0 et la théorie de la chaleur,” J. Math. Pures Appl. 8, 187–216 (1892).
  23. D. V. Widder, The Heat Equation (Academic, 1975).
  24. H. Leutwiler, “On the Appell transformation,” in Potential Theory, J. Kràl, J. Lukws, I. Netuka, and J. Vesely, eds. (Plenum, 1988), pp. 215–222.
  25. K. Shimomura, “The determination of caloric morphisms on Euclidean domains,” Nagoya Math. J. 158, 133–166 (2000).
  26. M. Brzezina, “Appell type transformation for the Kolmogorov operator,” Mathematische Nachrichten 169, 59–67 (1994).
    [Crossref]
  27. A. Torre, “The Appell transformation for the paraxial wave equation,” J. Opt. 13, 015701 (2011).
    [Crossref]
  28. A. Torre, “Appell transformation and canonical transforms,” SIGMA 7, 072 (2011), http://www.emis.de/journals/SIGMA/S4.html .
  29. A. Torre, “Appell transformation and symmetry transformations for the paraxial wave equation,” J. Opt. 13, 075710 (2011).
    [Crossref]
  30. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [Crossref]
  31. A. E. Siegman, Lasers (University Science, 1986).
  32. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  33. T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, 1976).
  34. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966).
  35. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 7th ed. (Academic, 2007) #3.462.2.
  36. P. Appell and J. Kampé de Fériet, Fonctions Hypergeométriques and Hypersphériques. Polynomes d’Hermite (Gauthier-Villars, 1926).
  37. A. A. Sukhorukov, A. S. Solntev, and J. S. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
    [Crossref]

2013 (2)

A. Perez-Leija, F. Soto-Eguibar, S. Chavez-Cerda, A. Szameit, H. Moya-Cessa, and D. N. Christodoulides, “Discrete-like diffraction dynamics in free space,” Opt. Express 21, 17951–17960 (2013).
[Crossref]

A. A. Sukhorukov, A. S. Solntev, and J. S. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
[Crossref]

2012 (1)

A. Jaron-Becker, “Molecular dynamics in strong laser fields,” IEEE J. Sel. Top. Quantum Electron. 18, 105–112 (2012).
[Crossref]

2011 (3)

A. Torre, “The Appell transformation for the paraxial wave equation,” J. Opt. 13, 015701 (2011).
[Crossref]

A. Torre, “Appell transformation and canonical transforms,” SIGMA 7, 072 (2011), http://www.emis.de/journals/SIGMA/S4.html .

A. Torre, “Appell transformation and symmetry transformations for the paraxial wave equation,” J. Opt. 13, 075710 (2011).
[Crossref]

2010 (1)

A. Torre, “Linear and quadratic exponential modulation of the solutions of the paraxial wave equation,” J. Opt. 12, 035701 (2010).
[Crossref]

2009 (1)

C. S. E. van Ditzhuijzen, A. Tauschinsky, and H. B. van Linden van den Heuvell, “Observation of Stueckelberg oscillations in dipole-dipole interactions,” Phys. Rev. A 80, 063407 (2009).
[Crossref]

2008 (2)

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33, 2689–2691 (2008).
[Crossref]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77, 043804 (2008).
[Crossref]

2005 (2)

V. I. Usachenko and S.-I. Chu, “Strong-field ionization of laser-irradiated light homonuclear diatomic molecules: a generalized strong-field approximation-linear combination of atomic orbitals model,” Phys. Rev. A 71, 063410 (2005).
[Crossref]

V. Strakhovenko, X. Artru, R. Chehab, and M. Chevallier, “Generation of circularly polarized photons for a linear collider polarized positron source,” Nucl. Instrum. Methods Phys. Res. A 547, 320–333 (2005).
[Crossref]

2003 (1)

A. Jaron-Becker, A. Becker, and F. H. M. Faisal, “Dependence of strong-field photoelectron angular distributions on molecular orientation,” J. Phys. B 36, L375–L380 (2003).
[Crossref]

2000 (1)

K. Shimomura, “The determination of caloric morphisms on Euclidean domains,” Nagoya Math. J. 158, 133–166 (2000).

1998 (1)

G. Dattoli, A. Torre, and M. Carpanese, “The Hermite-Bessel functions: a new point of view on the theory of the generalized Bessel functions,” Radiat. Phys. Chem. 51, 221–228 (1998).
[Crossref]

1994 (1)

M. Brzezina, “Appell type transformation for the Kolmogorov operator,” Mathematische Nachrichten 169, 59–67 (1994).
[Crossref]

1991 (1)

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cimento B 106, 21–51 (1991).
[Crossref]

1990 (1)

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cimento B 105, 327–348 (1990).
[Crossref]

1975 (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U = 0,” J. Math. Phys. 16, 499–511 (1975).
[Crossref]

1974 (1)

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt+Uxx = 0 and iUt+Uxx − c/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).
[Crossref]

1970 (1)

1964 (1)

L. S. Brown and T. W. B. Kibble, “Interaction of intense laser beams with electrons,” Phys. Rev. 133, A705–A719 (1964).
[Crossref]

1892 (1)

M. P. Appell, “Sur l’équation ∂2z/∂x2 ∂z/∂y = 0 et la théorie de la chaleur,” J. Math. Pures Appl. 8, 187–216 (1892).

Appell, M. P.

M. P. Appell, “Sur l’équation ∂2z/∂x2 ∂z/∂y = 0 et la théorie de la chaleur,” J. Math. Pures Appl. 8, 187–216 (1892).

Appell, P.

P. Appell and J. Kampé de Fériet, Fonctions Hypergeométriques and Hypersphériques. Polynomes d’Hermite (Gauthier-Villars, 1926).

Artru, X.

V. Strakhovenko, X. Artru, R. Chehab, and M. Chevallier, “Generation of circularly polarized photons for a linear collider polarized positron source,” Nucl. Instrum. Methods Phys. Res. A 547, 320–333 (2005).
[Crossref]

Babusci, D.

D. Babusci, G. Dattoli, and M. Del Franco, “Lectures on mathematical methods for physics,” (ENEA, 2010).

Becker, A.

A. Jaron-Becker, A. Becker, and F. H. M. Faisal, “Dependence of strong-field photoelectron angular distributions on molecular orientation,” J. Phys. B 36, L375–L380 (2003).
[Crossref]

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U = 0,” J. Math. Phys. 16, 499–511 (1975).
[Crossref]

Brown, L. S.

L. S. Brown and T. W. B. Kibble, “Interaction of intense laser beams with electrons,” Phys. Rev. 133, A705–A719 (1964).
[Crossref]

Brzezina, M.

M. Brzezina, “Appell type transformation for the Kolmogorov operator,” Mathematische Nachrichten 169, 59–67 (1994).
[Crossref]

Carpanese, M.

G. Dattoli, A. Torre, and M. Carpanese, “The Hermite-Bessel functions: a new point of view on the theory of the generalized Bessel functions,” Radiat. Phys. Chem. 51, 221–228 (1998).
[Crossref]

Chavez-Cerda, S.

Chehab, R.

V. Strakhovenko, X. Artru, R. Chehab, and M. Chevallier, “Generation of circularly polarized photons for a linear collider polarized positron source,” Nucl. Instrum. Methods Phys. Res. A 547, 320–333 (2005).
[Crossref]

Chevallier, M.

V. Strakhovenko, X. Artru, R. Chehab, and M. Chevallier, “Generation of circularly polarized photons for a linear collider polarized positron source,” Nucl. Instrum. Methods Phys. Res. A 547, 320–333 (2005).
[Crossref]

Christodoulides, D. N.

Chu, S.-I.

V. I. Usachenko and S.-I. Chu, “Strong-field ionization of laser-irradiated light homonuclear diatomic molecules: a generalized strong-field approximation-linear combination of atomic orbitals model,” Phys. Rev. A 71, 063410 (2005).
[Crossref]

Collins, S. A.

Dattoli, G.

G. Dattoli, A. Torre, and M. Carpanese, “The Hermite-Bessel functions: a new point of view on the theory of the generalized Bessel functions,” Radiat. Phys. Chem. 51, 221–228 (1998).
[Crossref]

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cimento B 106, 21–51 (1991).
[Crossref]

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cimento B 105, 327–348 (1990).
[Crossref]

G. Dattoli and A. Torre, Theory and Applications of the Generalized Bessel Functions (Aracne, 1996).

G. Dattoli, A. Renieri, and A. Torre, Lectures in Free-Electron Laser Theory and Related Topics (World Scientific, 1995).

D. Babusci, G. Dattoli, and M. Del Franco, “Lectures on mathematical methods for physics,” (ENEA, 2010).

Del Franco, M.

D. Babusci, G. Dattoli, and M. Del Franco, “Lectures on mathematical methods for physics,” (ENEA, 2010).

Dreisow, F.

Faisal, F. H. M.

A. Jaron-Becker, A. Becker, and F. H. M. Faisal, “Dependence of strong-field photoelectron angular distributions on molecular orientation,” J. Phys. B 36, L375–L380 (2003).
[Crossref]

Giannessi, L.

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cimento B 105, 327–348 (1990).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 7th ed. (Academic, 2007) #3.462.2.

Heinrich, M.

Jaron-Becker, A.

A. Jaron-Becker, “Molecular dynamics in strong laser fields,” IEEE J. Sel. Top. Quantum Electron. 18, 105–112 (2012).
[Crossref]

A. Jaron-Becker, A. Becker, and F. H. M. Faisal, “Dependence of strong-field photoelectron angular distributions on molecular orientation,” J. Phys. B 36, L375–L380 (2003).
[Crossref]

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U = 0,” J. Math. Phys. 16, 499–511 (1975).
[Crossref]

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt+Uxx = 0 and iUt+Uxx − c/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).
[Crossref]

Kampé de Fériet, J.

P. Appell and J. Kampé de Fériet, Fonctions Hypergeométriques and Hypersphériques. Polynomes d’Hermite (Gauthier-Villars, 1926).

Kato, T.

T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, 1976).

Kibble, T. W. B.

L. S. Brown and T. W. B. Kibble, “Interaction of intense laser beams with electrons,” Phys. Rev. 133, A705–A719 (1964).
[Crossref]

Lederer, F.

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77, 043804 (2008).
[Crossref]

Leutwiler, H.

H. Leutwiler, “On the Appell transformation,” in Potential Theory, J. Kràl, J. Lukws, I. Netuka, and J. Vesely, eds. (Plenum, 1988), pp. 215–222.

Lorenzutta, S.

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cimento B 106, 21–51 (1991).
[Crossref]

Magnus, W.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966).

Maino, G.

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cimento B 106, 21–51 (1991).
[Crossref]

Mezi, L.

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cimento B 105, 327–348 (1990).
[Crossref]

Miller, W.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U = 0,” J. Math. Phys. 16, 499–511 (1975).
[Crossref]

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt+Uxx = 0 and iUt+Uxx − c/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).
[Crossref]

W. Miller, Symmetry and Separation of Variables (Addison-Wesley, 1977).

Moya-Cessa, H.

Nolte, S.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33, 2689–2691 (2008).
[Crossref]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77, 043804 (2008).
[Crossref]

Oberhettinger, F.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966).

Perez-Leija, A.

Pertsch, T.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33, 2689–2691 (2008).
[Crossref]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77, 043804 (2008).
[Crossref]

Reiss, H. R.

H. R. Reiss, “Foundations of strong-field physics,” in Lectures on Ultrafast Intense Laser Science, K. Yamanouchi, ed. (Springer-Verlag, 2010)., pp. 41–84.

Renieri, A.

G. Dattoli, A. Renieri, and A. Torre, Lectures in Free-Electron Laser Theory and Related Topics (World Scientific, 1995).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 7th ed. (Academic, 2007) #3.462.2.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Shimomura, K.

K. Shimomura, “The determination of caloric morphisms on Euclidean domains,” Nagoya Math. J. 158, 133–166 (2000).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Sipe, J. S.

A. A. Sukhorukov, A. S. Solntev, and J. S. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
[Crossref]

Solntev, A. S.

A. A. Sukhorukov, A. S. Solntev, and J. S. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
[Crossref]

Soni, R. P.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966).

Soto-Eguibar, F.

Strakhovenko, V.

V. Strakhovenko, X. Artru, R. Chehab, and M. Chevallier, “Generation of circularly polarized photons for a linear collider polarized positron source,” Nucl. Instrum. Methods Phys. Res. A 547, 320–333 (2005).
[Crossref]

Sukhorukov, A. A.

A. A. Sukhorukov, A. S. Solntev, and J. S. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
[Crossref]

Szameit, A.

Tauschinsky, A.

C. S. E. van Ditzhuijzen, A. Tauschinsky, and H. B. van Linden van den Heuvell, “Observation of Stueckelberg oscillations in dipole-dipole interactions,” Phys. Rev. A 80, 063407 (2009).
[Crossref]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Torre, A.

A. Torre, “The Appell transformation for the paraxial wave equation,” J. Opt. 13, 015701 (2011).
[Crossref]

A. Torre, “Appell transformation and canonical transforms,” SIGMA 7, 072 (2011), http://www.emis.de/journals/SIGMA/S4.html .

A. Torre, “Appell transformation and symmetry transformations for the paraxial wave equation,” J. Opt. 13, 075710 (2011).
[Crossref]

A. Torre, “Linear and quadratic exponential modulation of the solutions of the paraxial wave equation,” J. Opt. 12, 035701 (2010).
[Crossref]

G. Dattoli, A. Torre, and M. Carpanese, “The Hermite-Bessel functions: a new point of view on the theory of the generalized Bessel functions,” Radiat. Phys. Chem. 51, 221–228 (1998).
[Crossref]

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cimento B 106, 21–51 (1991).
[Crossref]

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cimento B 105, 327–348 (1990).
[Crossref]

G. Dattoli and A. Torre, Theory and Applications of the Generalized Bessel Functions (Aracne, 1996).

G. Dattoli, A. Renieri, and A. Torre, Lectures in Free-Electron Laser Theory and Related Topics (World Scientific, 1995).

A. Torre, “Paraxial wave equation: Lie-algebra based approach,” in Mathematical Optics: Classical, Quantum, and Computational Methods, V. Lakshminarayanan, M. L. Calvo, and T. Alieva, eds. (CRC Press, 2013), Chap. 10, pp. 341–417.

Tünnermann, A.

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77, 043804 (2008).
[Crossref]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33, 2689–2691 (2008).
[Crossref]

Usachenko, V. I.

V. I. Usachenko and S.-I. Chu, “Strong-field ionization of laser-irradiated light homonuclear diatomic molecules: a generalized strong-field approximation-linear combination of atomic orbitals model,” Phys. Rev. A 71, 063410 (2005).
[Crossref]

van Ditzhuijzen, C. S. E.

C. S. E. van Ditzhuijzen, A. Tauschinsky, and H. B. van Linden van den Heuvell, “Observation of Stueckelberg oscillations in dipole-dipole interactions,” Phys. Rev. A 80, 063407 (2009).
[Crossref]

van Linden van den Heuvell, H. B.

C. S. E. van Ditzhuijzen, A. Tauschinsky, and H. B. van Linden van den Heuvell, “Observation of Stueckelberg oscillations in dipole-dipole interactions,” Phys. Rev. A 80, 063407 (2009).
[Crossref]

Widder, D. V.

D. V. Widder, The Heat Equation (Academic, 1975).

IEEE J. Sel. Top. Quantum Electron. (1)

A. Jaron-Becker, “Molecular dynamics in strong laser fields,” IEEE J. Sel. Top. Quantum Electron. 18, 105–112 (2012).
[Crossref]

J. Math. Phys. (2)

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt+Uxx = 0 and iUt+Uxx − c/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).
[Crossref]

C. P. Boyer, E. G. Kalnins, and W. Miller, “Lie theory and separation of variables. 6. The equation iUt+Δ2U = 0,” J. Math. Phys. 16, 499–511 (1975).
[Crossref]

J. Math. Pures Appl. (1)

M. P. Appell, “Sur l’équation ∂2z/∂x2 ∂z/∂y = 0 et la théorie de la chaleur,” J. Math. Pures Appl. 8, 187–216 (1892).

J. Opt. (3)

A. Torre, “The Appell transformation for the paraxial wave equation,” J. Opt. 13, 015701 (2011).
[Crossref]

A. Torre, “Appell transformation and symmetry transformations for the paraxial wave equation,” J. Opt. 13, 075710 (2011).
[Crossref]

A. Torre, “Linear and quadratic exponential modulation of the solutions of the paraxial wave equation,” J. Opt. 12, 035701 (2010).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. B (1)

A. Jaron-Becker, A. Becker, and F. H. M. Faisal, “Dependence of strong-field photoelectron angular distributions on molecular orientation,” J. Phys. B 36, L375–L380 (2003).
[Crossref]

Mathematische Nachrichten (1)

M. Brzezina, “Appell type transformation for the Kolmogorov operator,” Mathematische Nachrichten 169, 59–67 (1994).
[Crossref]

Nagoya Math. J. (1)

K. Shimomura, “The determination of caloric morphisms on Euclidean domains,” Nagoya Math. J. 158, 133–166 (2000).

Nucl. Instrum. Methods Phys. Res. A (1)

V. Strakhovenko, X. Artru, R. Chehab, and M. Chevallier, “Generation of circularly polarized photons for a linear collider polarized positron source,” Nucl. Instrum. Methods Phys. Res. A 547, 320–333 (2005).
[Crossref]

Nuovo Cimento B (2)

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cimento B 105, 327–348 (1990).
[Crossref]

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cimento B 106, 21–51 (1991).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

L. S. Brown and T. W. B. Kibble, “Interaction of intense laser beams with electrons,” Phys. Rev. 133, A705–A719 (1964).
[Crossref]

Phys. Rev. A (4)

V. I. Usachenko and S.-I. Chu, “Strong-field ionization of laser-irradiated light homonuclear diatomic molecules: a generalized strong-field approximation-linear combination of atomic orbitals model,” Phys. Rev. A 71, 063410 (2005).
[Crossref]

C. S. E. van Ditzhuijzen, A. Tauschinsky, and H. B. van Linden van den Heuvell, “Observation of Stueckelberg oscillations in dipole-dipole interactions,” Phys. Rev. A 80, 063407 (2009).
[Crossref]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77, 043804 (2008).
[Crossref]

A. A. Sukhorukov, A. S. Solntev, and J. S. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
[Crossref]

Radiat. Phys. Chem. (1)

G. Dattoli, A. Torre, and M. Carpanese, “The Hermite-Bessel functions: a new point of view on the theory of the generalized Bessel functions,” Radiat. Phys. Chem. 51, 221–228 (1998).
[Crossref]

SIGMA (1)

A. Torre, “Appell transformation and canonical transforms,” SIGMA 7, 072 (2011), http://www.emis.de/journals/SIGMA/S4.html .

Other (14)

D. V. Widder, The Heat Equation (Academic, 1975).

H. Leutwiler, “On the Appell transformation,” in Potential Theory, J. Kràl, J. Lukws, I. Netuka, and J. Vesely, eds. (Plenum, 1988), pp. 215–222.

A. E. Siegman, Lasers (University Science, 1986).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, 1976).

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 7th ed. (Academic, 2007) #3.462.2.

P. Appell and J. Kampé de Fériet, Fonctions Hypergeométriques and Hypersphériques. Polynomes d’Hermite (Gauthier-Villars, 1926).

G. Dattoli, A. Renieri, and A. Torre, Lectures in Free-Electron Laser Theory and Related Topics (World Scientific, 1995).

A. Torre, “Paraxial wave equation: Lie-algebra based approach,” in Mathematical Optics: Classical, Quantum, and Computational Methods, V. Lakshminarayanan, M. L. Calvo, and T. Alieva, eds. (CRC Press, 2013), Chap. 10, pp. 341–417.

D. Babusci, G. Dattoli, and M. Del Franco, “Lectures on mathematical methods for physics,” (ENEA, 2010).

W. Miller, Symmetry and Separation of Variables (Addison-Wesley, 1977).

H. R. Reiss, “Foundations of strong-field physics,” in Lectures on Ultrafast Intense Laser Science, K. Yamanouchi, ed. (Springer-Verlag, 2010)., pp. 41–84.

G. Dattoli and A. Torre, Theory and Applications of the Generalized Bessel Functions (Aracne, 1996).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1.
Fig. 1.

(x,z) counterplots (to the left) and x-profiles at different z (to the right) of the squared amplitude |u(x/w0,z/zc)|2 for n=0, n=3, and n=6. The parameters are as in [1], namely, w0=10μm, λ=633nm, thus yielding zc=0.993mm, and α=1 and σ2=103. Specifically, the x-profiles are plotted for z=0.5cm (solid line), z=1cm (dotted line), and z=3 cm (dashed line).

Fig. 2.
Fig. 2.

x-profiles of the input functions, u0(x/w0), for n=0, n=3, and n=6. The parameters are as in [1], namely, w0=10μm, λ=633nm, thus yielding zc=0.993mm, and α=1 and σ2=103.

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

(2iζ+2ξ2)v(ξ,ζ)=0,v(ξ,0)=v0(ξ).
v(ξ,ζ)=12πiζei(ξξ)22ζv0(ξ)dξ,
v(ξ,ζ)=S^u(ξ,ζ)=Ωa+cζei12c(a+cζ)(cξ+ε)2u(ξ+αζ+βa+cζ,b+dζa+cζ),S^WSL(2,C),
v(ξ,ζ)=S^(abcd)SL(2,C)u(ξ,ζ)=1a+cζeic2(a+cζ)ξ2u(ξa+cζ,b+dζa+cζ),
v0(ξ)=Jn(αξ),
Jn(x)=12πππeixsinφinφdφ,
v(ξ,ζ)=eiα2ζ412πππeiαξsinφ+iα2ζ4cos(2φ)inφdφ.
J(m)n(x,y;τ)=l=τlJnml(x)Jl(y),
J(m)n(x,y;eiθ)=12πππeixsinφ+iysin(mφ+θ)inφdφ,
v(ξ,ζ)=eiα24ζ(2)Jn(αξ,α24ζ;i).
u0(ξ)=eξ22σ2Jn(αξ)
eξ22σ2G(σ)=(10iσ21)SL(2,C).
u(ξ,ζ)=S^GSL(2,C)v(ξ,ζ)=1Q(σ,ζ)eiα24ζQ(σ,ζ)eξ22σ21Q(σ,ζ)J(2)n(αξQ(σ,ζ),α24ζQ(σ,ζ);i),
Q(σ,ζ)=1+iσ2ζ,
w(ξ,ζ)=1iζeiξ22ζu(ξζ,1ζ),
v(ξ,ζ)zzc1iζeiξ22ζv˜0(ξζ),
v˜0(ξ)=12π+eiξξv0(ξ)dξ.
w(ξ,ζ)=Σ(σ,ζ)eα24Σ(σ,ζ)eξ22Σ(σ,ζ)J(2)n(iαξΣ(σ,ζ),iα24Σ(σ,ζ);i),
Σ(σ,ζ)=σ21+iσ2ζ.
I(m)n(x,y;τ)=l=τlInml(x)Il(y),
I(2)n(x,y;ieiθ)=(i)n2πππexsinφ+ysin(2φ+θ)inφdφ,
I(2)n(x,y;1)=(i)n2πππexsinφ+ycos(2φ)inφdφ=(i)n(2)Jn(ix,iy;i).
w(ξ,ζ)=(i)nΣ(σ,ζ)eα24Σ(σ,ζ)eξ22Σ(σ,ζ)I(2)n(±αξΣ(σ,ζ),α24Σ(σ,ζ);1).
w(ξ,0)=F^[eξ22σ2Jn(αξ)](ξ)=σeα2σ24eξ2σ22I(2)n(αξσ2,α2σ24;1),
Xn(a,b)=eax2+bxxndx,
n=0tnn!Xn(a,b)=eax2+bxetxdx,
n=0tnn!Xn(a,b)=πaeb24aeb2at+14at2
Hn(x,y)=n!m=0[n/2]xn2mymm!(n2m)!,
n=0tnn!Hn(x,y)=ext+yt2.
Xn(a,b)=πaeb24aHn(b2a,14a).
In(a,b)=eax2+bxJn(x)dx,
n=0tnn!In(a,b)=eax2+bxn=0tnn!Jn(x)dx.
n=0tnn!Jn(αx)=ex2(t1t),
n=0tnn!In(a,b)=πaeb24aeb4a(t1t)+116a(t1t)2,
Jn(x)=m=0(1)mm!(n+m)!(x2)n+2mJHn(x,y)=m=0(1)mm!(n+m)!Hn+2m(x,y)2n+2m.
n=0tnn!JHn(x,y)=ex2(t1t)+y4(t1t)2,
In(a,b)=πaeb24aJHn(b2a,14a).
(τ2ξ2)ψ(ξ,τ)=0,ψ(ξ,0)=ψ0(ξ).
v(ξ,ζ)=14πτe(ξξ)24τψ0(ξ)dξ,
ψ0(ξ)=u0(ξ)=eξ22σ2Jn(αξ),
ψ(ξ,τ)=1R(ξ,τ)eξ22σ21R(ξ,τ)JHn(2αξR(ξ,τ),α2τR(ξ,τ)),
R(ξ,τ)=1+2τσ2.
hn(x,t)=n!m=0[n/2]tmm!(n2m)!xn2m=Hn(x,t).
u(ξ,ζ)=1Q(σ,ζ)eξ22σ21Q(σ,ζ)JHn(2αξQ(σ,ζ),iα2ζ2Q(σ,ζ)),
JHn(x,y)=ey2l=+Jn2l(x)Il(y2),
JHn(x,y)=ey2J(2)n(x,iy2;i),
(iy+22x2+1)J(2)n(x,y;i)=0,J(2)n(x,0;i)=Jn(x);(y2x2)JHn(x,y)=0,J(2)n(x,0)=Jn(x),

Metrics