Abstract

Assuming a non-paraxial propagation operator, we study the propagation of an electromagnetic field with an arbitrary initial condition in a quadratic GRIN medium. We show analytically that at certain specific periodic distances, the propagated field is given by the fractional Fourier transform of a superposition of the initial field and of a reflected version of it. We also prove that for particular wavelengths, there is a revival and a splitting of the initial field. We apply this results, first to an initial field given by a Bessel function and show that it splits into two generalized Bessel functions, and second, to an Airy function. In both cases our results are compared with the numerical ones.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Revival and splitting of a Gaussian beam in gradient index media

V. Arrizon, F. Soto-Eguibar, A. Zuñiga-Segundo, and H. M. Moya-Cessa
J. Opt. Soc. Am. A 32(6) 1140-1145 (2015)

Analytical surrogate model for the aberrations of an arbitrary GRIN lens

John A. Easum, Sawyer D. Campbell, Jogender Nagar, and Douglas H. Werner
Opt. Express 24(16) 17805-17818 (2016)

Arbitrary GRIN component fabrication in optically driven diffusive photopolymers

Adam C. Urness, Ken Anderson, Chungfang Ye, William L. Wilson, and Robert R. McLeod
Opt. Express 23(1) 264-273 (2015)

References

  • View by:
  • |
  • |
  • |

  1. C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).
    [Crossref]
  2. E. Silvestre-Mora, P. Andres, and J. Ojeda-Castaneda, “Self-imaging in GRIN media,” Proc. SPIE 2730, Second Iberoamerican Meeting on Optics, 468 (February 5, 1996).
  3. M. T. Flores-Arias, C. Bao, M. V. Prez, and C. Gmez-Reino, “Talbot effect in a tapered gradient-index medium for nonuniform and uniform illumination,” J. Opt. Soc. Am. A 16, 2439–2446 (1999).
    [Crossref]
  4. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [Crossref]
  5. J. Ojeda-Castañeda and P. Szwaykowski, “Novel modes in α-power GRIN,” Proc. SPIE 1500, 246 (1991).
    [Crossref]
  6. H. M. Moya-Cessa, M. Fernández Guasti, V. M. Arrizon, and S. Chávez-Cerda, “Optical realization of quantum-mechanical invariants,” Opt. Lett. 34, 1459–1461 (2009).
    [Crossref] [PubMed]
  7. S. M. Chumakov and K. B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193, 51–53 (1994).
    [Crossref]
  8. A. Zuñiga-Segundo, B. M. Rodríguez-Lara, D. J. Fernández, and H. M. Moya-Cessa, “Jacobi photonic lattices and their SUSY partners,” Opt. Express 22, 987–994 (2014).
    [Crossref] [PubMed]
  9. G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
    [Crossref]
  10. H. Y. Fan and J. H. Chen, “On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations,” Front. Phys. 10, 1–6 (2015).
    [Crossref]
  11. F. Soto-Eguibar, V. Arrizon, A. Zuñiga-Segundo, and H. M. Moya-Cessa, “Optical realization of quantum Kerr medium dynamics,” Opt. Lett. 39, 6158–6161 (2014).
    [Crossref] [PubMed]
  12. V. Arrizon, F. Soto-Eguibar, A. Zuñiga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. Soc. of Am. A 32, 1140–1145 (2015).
    [Crossref]
  13. G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Il Nuovo Cimento B 105(3), 327–348 (1990).
    [Crossref]
  14. G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “A note on the theory of n-variable generalized Bessel functions,” Il Nuovo Cimento B 106(10), 1159–1166 (1991).
    [Crossref]
  15. G. Dattoli and A. Torre, “Note on discrete-like diffraction dynamics in free space: highlighting the variety of solving procedures,” J. Opt. Soc. of Am. B 31, 2214–2220 (2014).
    [Crossref]
  16. A. Perez-Leija, F. Soto-Eguibar, S. Chavez-Cerda, A. Szameit, H. M. Moya-Cessa, and D. N. Christodoulides, “Discrete-like diffraction dynamics in free space,” Opt. Express 21, 17951–17960 (2013).
    [Crossref] [PubMed]
  17. T. Eichelkraut, C. Vetter, A. Perez-Leija, H. M. Moya-Cessa, D. N. Christodoulides, and A. Szameit, “Coherent random walks in free space,” Optica 1, 268–271 (2014).
    [Crossref]
  18. A. Torre, “Propagating Airy wavelet-related patterns,” J. Opt. 17, 075604 (2015).
    [Crossref]
  19. M. A. Bandres and J. C. Gutíerrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Expr. 15, 16719–16728 (2007).
    [Crossref]
  20. A. Torre, “Gaussian modulated Ai- and Bi-based solutions of the 2D PWE: a comparison,” Appl. Phys. B99, 775–799 (2010).
    [Crossref]
  21. K. B. Wolf, Geometric Optics in Phase Space (Springer-Verlag, 2004).
  22. V. Namias, “The Fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Maths Applics. 25, 241–265 (1980).
    [Crossref]
  23. Pierre Pellat-Finet, Optique de Fourier. Théorie Métaxiale et Fractionnaire (Springer-Verlag, France2009).
    [Crossref]
  24. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).
  25. F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (NIST, 2010)

2015 (3)

H. Y. Fan and J. H. Chen, “On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations,” Front. Phys. 10, 1–6 (2015).
[Crossref]

V. Arrizon, F. Soto-Eguibar, A. Zuñiga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. Soc. of Am. A 32, 1140–1145 (2015).
[Crossref]

A. Torre, “Propagating Airy wavelet-related patterns,” J. Opt. 17, 075604 (2015).
[Crossref]

2014 (4)

2013 (1)

2010 (1)

A. Torre, “Gaussian modulated Ai- and Bi-based solutions of the 2D PWE: a comparison,” Appl. Phys. B99, 775–799 (2010).
[Crossref]

2009 (1)

2007 (1)

M. A. Bandres and J. C. Gutíerrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Expr. 15, 16719–16728 (2007).
[Crossref]

1999 (1)

1994 (2)

S. M. Chumakov and K. B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193, 51–53 (1994).
[Crossref]

G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[Crossref]

1993 (1)

1991 (2)

J. Ojeda-Castañeda and P. Szwaykowski, “Novel modes in α-power GRIN,” Proc. SPIE 1500, 246 (1991).
[Crossref]

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “A note on the theory of n-variable generalized Bessel functions,” Il Nuovo Cimento B 106(10), 1159–1166 (1991).
[Crossref]

1990 (1)

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Il Nuovo Cimento B 105(3), 327–348 (1990).
[Crossref]

1980 (1)

V. Namias, “The Fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Maths Applics. 25, 241–265 (1980).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

Agarwal, G. S.

G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[Crossref]

Andres, P.

E. Silvestre-Mora, P. Andres, and J. Ojeda-Castaneda, “Self-imaging in GRIN media,” Proc. SPIE 2730, Second Iberoamerican Meeting on Optics, 468 (February 5, 1996).

Arrizon, V.

V. Arrizon, F. Soto-Eguibar, A. Zuñiga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. Soc. of Am. A 32, 1140–1145 (2015).
[Crossref]

F. Soto-Eguibar, V. Arrizon, A. Zuñiga-Segundo, and H. M. Moya-Cessa, “Optical realization of quantum Kerr medium dynamics,” Opt. Lett. 39, 6158–6161 (2014).
[Crossref] [PubMed]

Arrizon, V. M.

Bandres, M. A.

M. A. Bandres and J. C. Gutíerrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Expr. 15, 16719–16728 (2007).
[Crossref]

Bao, C.

Boisvert, R. F.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (NIST, 2010)

Chavez-Cerda, S.

Chávez-Cerda, S.

Chen, J. H.

H. Y. Fan and J. H. Chen, “On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations,” Front. Phys. 10, 1–6 (2015).
[Crossref]

Christodoulides, D. N.

Chumakov, S. M.

S. M. Chumakov and K. B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193, 51–53 (1994).
[Crossref]

Clark, C. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (NIST, 2010)

Dattoli, G.

G. Dattoli and A. Torre, “Note on discrete-like diffraction dynamics in free space: highlighting the variety of solving procedures,” J. Opt. Soc. of Am. B 31, 2214–2220 (2014).
[Crossref]

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “A note on the theory of n-variable generalized Bessel functions,” Il Nuovo Cimento B 106(10), 1159–1166 (1991).
[Crossref]

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Il Nuovo Cimento B 105(3), 327–348 (1990).
[Crossref]

Eichelkraut, T.

Fan, H. Y.

H. Y. Fan and J. H. Chen, “On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations,” Front. Phys. 10, 1–6 (2015).
[Crossref]

Fernández, D. J.

Fernández Guasti, M.

Flores-Arias, M. T.

Giannessi, L.

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Il Nuovo Cimento B 105(3), 327–348 (1990).
[Crossref]

Gmez-Reino, C.

Gomez-Reino, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).
[Crossref]

Gutíerrez-Vega, J. C.

M. A. Bandres and J. C. Gutíerrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Expr. 15, 16719–16728 (2007).
[Crossref]

Lorenzutta, S.

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “A note on the theory of n-variable generalized Bessel functions,” Il Nuovo Cimento B 106(10), 1159–1166 (1991).
[Crossref]

Lozier, D. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (NIST, 2010)

Maino, G.

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “A note on the theory of n-variable generalized Bessel functions,” Il Nuovo Cimento B 106(10), 1159–1166 (1991).
[Crossref]

Mendlovic, D.

Mezi, L.

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Il Nuovo Cimento B 105(3), 327–348 (1990).
[Crossref]

Moya-Cessa, H. M.

Namias, V.

V. Namias, “The Fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Maths Applics. 25, 241–265 (1980).
[Crossref]

Ojeda-Castaneda, J.

E. Silvestre-Mora, P. Andres, and J. Ojeda-Castaneda, “Self-imaging in GRIN media,” Proc. SPIE 2730, Second Iberoamerican Meeting on Optics, 468 (February 5, 1996).

Ojeda-Castañeda, J.

J. Ojeda-Castañeda and P. Szwaykowski, “Novel modes in α-power GRIN,” Proc. SPIE 1500, 246 (1991).
[Crossref]

Olver, F. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (NIST, 2010)

Ozaktas, H. M.

Pellat-Finet, Pierre

Pierre Pellat-Finet, Optique de Fourier. Théorie Métaxiale et Fractionnaire (Springer-Verlag, France2009).
[Crossref]

Perez, M. V.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).
[Crossref]

Perez-Leija, A.

Prez, M. V.

Rodríguez-Lara, B. M.

Silvestre-Mora, E.

E. Silvestre-Mora, P. Andres, and J. Ojeda-Castaneda, “Self-imaging in GRIN media,” Proc. SPIE 2730, Second Iberoamerican Meeting on Optics, 468 (February 5, 1996).

Simon, R.

G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[Crossref]

Soto-Eguibar, F.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

Szameit, A.

Szwaykowski, P.

J. Ojeda-Castañeda and P. Szwaykowski, “Novel modes in α-power GRIN,” Proc. SPIE 1500, 246 (1991).
[Crossref]

Torre, A.

A. Torre, “Propagating Airy wavelet-related patterns,” J. Opt. 17, 075604 (2015).
[Crossref]

G. Dattoli and A. Torre, “Note on discrete-like diffraction dynamics in free space: highlighting the variety of solving procedures,” J. Opt. Soc. of Am. B 31, 2214–2220 (2014).
[Crossref]

A. Torre, “Gaussian modulated Ai- and Bi-based solutions of the 2D PWE: a comparison,” Appl. Phys. B99, 775–799 (2010).
[Crossref]

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “A note on the theory of n-variable generalized Bessel functions,” Il Nuovo Cimento B 106(10), 1159–1166 (1991).
[Crossref]

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Il Nuovo Cimento B 105(3), 327–348 (1990).
[Crossref]

Vetter, C.

Wolf, K. B.

S. M. Chumakov and K. B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193, 51–53 (1994).
[Crossref]

K. B. Wolf, Geometric Optics in Phase Space (Springer-Verlag, 2004).

Zuñiga-Segundo, A.

Appl. Phys. (1)

A. Torre, “Gaussian modulated Ai- and Bi-based solutions of the 2D PWE: a comparison,” Appl. Phys. B99, 775–799 (2010).
[Crossref]

Front. Phys. (1)

H. Y. Fan and J. H. Chen, “On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations,” Front. Phys. 10, 1–6 (2015).
[Crossref]

Il Nuovo Cimento B (2)

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Il Nuovo Cimento B 105(3), 327–348 (1990).
[Crossref]

G. Dattoli, A. Torre, S. Lorenzutta, and G. Maino, “A note on the theory of n-variable generalized Bessel functions,” Il Nuovo Cimento B 106(10), 1159–1166 (1991).
[Crossref]

J. Inst. Maths Applics. (1)

V. Namias, “The Fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Maths Applics. 25, 241–265 (1980).
[Crossref]

J. Opt. (1)

A. Torre, “Propagating Airy wavelet-related patterns,” J. Opt. 17, 075604 (2015).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. of Am. A (1)

V. Arrizon, F. Soto-Eguibar, A. Zuñiga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. Soc. of Am. A 32, 1140–1145 (2015).
[Crossref]

J. Opt. Soc. of Am. B (1)

G. Dattoli and A. Torre, “Note on discrete-like diffraction dynamics in free space: highlighting the variety of solving procedures,” J. Opt. Soc. of Am. B 31, 2214–2220 (2014).
[Crossref]

Opt. Commun. (1)

G. S. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[Crossref]

Opt. Expr. (1)

M. A. Bandres and J. C. Gutíerrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Expr. 15, 16719–16728 (2007).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Optica (1)

Phys. Lett. A (1)

S. M. Chumakov and K. B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193, 51–53 (1994).
[Crossref]

Proc. SPIE (1)

J. Ojeda-Castañeda and P. Szwaykowski, “Novel modes in α-power GRIN,” Proc. SPIE 1500, 246 (1991).
[Crossref]

Other (6)

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics (Springer-Verlag, 2002).
[Crossref]

E. Silvestre-Mora, P. Andres, and J. Ojeda-Castaneda, “Self-imaging in GRIN media,” Proc. SPIE 2730, Second Iberoamerican Meeting on Optics, 468 (February 5, 1996).

K. B. Wolf, Geometric Optics in Phase Space (Springer-Verlag, 2004).

Pierre Pellat-Finet, Optique de Fourier. Théorie Métaxiale et Fractionnaire (Springer-Verlag, France2009).
[Crossref]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (NIST, 2010)

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 (6)

Fig. 1
Fig. 1 The square of the propagated electric field at the first critical distance when the initial condition is J0 (x + 3). The quadratic GRIN media parameters are n0 = 1.5 and g = 10 mm−1. The beam has k = 1099.7, which corresponds to a wave number κ = 8062.27 m−1. a). In the Z direction the 3D graphic goes from zc − 200 to zc + 200, where zc = 4.18 × 109, or Zc = 3.80 × 108 m. b). The black continuous line is the graphic of expression (19) and the red dotted line is the numerical solution Eq. (49).
Fig. 2
Fig. 2 The square of the propagated electric field at the first critical distance when the initial condition is J3 (x + 3). The quadratic GRIN media parameters are n0 = 1.5 and g = 10 mm−1. The beam has k = 1099.7, which corresponds to a wave number κ = 8062.27 m−1. a). In the Z direction the 3D graphic goes from zc − 200 to zc + 200, where zc = 4.18 × 109, or Zc = 3.80 × 108 m. b). The black continuous line is the graphic of expression (19) and the red dotted line is the numerical solution Eq. (49).
Fig. 3
Fig. 3 The square of the propagated electric field at the first critical distance when the initial condition is J0 (x + 3). The quadratic GRIN media has n0 = 1.5 and g = 10mm−1. We have taken l = 1, m = 8 × 105 and thus kc = 1264.91 (κ= 10666.6 m−1). a). In the Z–direction the graphics goes from zc − 1000 to zc + 1000. b). The black continuous line is the graphic of expression (20) and the red dotted line is the numerical solution, Eq (49).
Fig. 4
Fig. 4 The square of the propagated electric field at the first critical distance when the initial condition is J3 (x + 3). The quadratic GRIN media has n0 = 1.5 and g = 10mm−1. We have taken l = 1, m = 8 × 105 and thus kc = 1264.91 (κ = 10666.6 m−1). a). In the Z–direction the graphics goes from zc − 1000 to zc + 1000. b). The black continuous line is the graphic of expression (20) and the red dotted line is the numerical solution Eq. (49).
Fig. 5
Fig. 5 The square of the electric field at the critical distance when the initial condition is Ai(x). The quadratic GRIN media parameters are n0 = 1.5 and g = 10 mm−1, and the wave number is k = 1000.5 (κ = 6673.33 m−1). a). Behavior of the propagation around the critical distance zc. The graphics shows from zc − 200 to zc + 200. b). The black continuous line is the graphic of expression (21) and the red dotted line is the numerical solution.
Fig. 6
Fig. 6 The square of the propagated electric field at the critical distance when the initial condition is Ai(x). The quadratic GRIN media has n0 = 1.5 and g = 10 mm−1. We have taken l = 1, m = 4 × 105 and then the wave number is k = 894.43 (κ = 5333.37 m−1). a). Behavior of the propagation around the critical distance zc. The graphics shows from zc − 200 to zc + 200. b). The black continuous line is the graphic of expression (22) and the red dotted line is the numerical solution.

Equations (49)

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

[ 2 X 2 + 2 Z 2 + κ 2 n 2 ( X ) ] E ( X , Z ) = 0 ,
n 2 ( X ) = n 0 2 ( 1 g 2 X 2 ) ,
( 2 z 2 + 2 x 2 + k 2 x 2 ) E ( x , z ) = 0 ,
2 E ( x , z ) z 2 = ( k 2 p ^ 2 x 2 ) E ( x , z ) ,
E ( x , z ) = exp [ i z k 2 p ^ 2 x 2 ] E ( x , 0 ) = exp [ i z k 1 p ^ 2 + x 2 k 2 ] E ( x , 0 ) ,
E ( x , z ) = exp [ i z k 1 2 n ^ + 1 k 2 ] E ( x , 0 ) ,
E ( x , z ) = exp [ i z k ( 1 1 2 k 2 ) ] exp ( i z k n ^ ) E ( x , 0 ) .
E ( x , z ) = exp [ i z ( k 1 2 k ) ] 𝔉 z k { E ( x , 0 ) } .
E ( x , z ) = exp ( i γ 1 k 3 z ) exp ( i γ 2 k 3 z n ^ ) exp [ i 1 2 k 3 z n ^ 2 ] E ( x , 0 ) ,
γ 1 k 4 k 2 2 1 8 , γ 2 k 2 + 1 2 .
φ m ( x ) = ( 1 π ) 1 / 4 1 2 m m ! exp ( 1 2 x 2 ) H m ( x ) ,
E ( x , z ) = exp ( i γ 1 k 3 z ) exp ( i γ 2 k 3 z n ^ ) j = 0 c j exp [ i 1 2 k 3 z j 2 ] φ j ( x ) .
E ( x , z = l π k 3 ) = exp ( i l γ 1 π ) exp ( i l γ 2 π n ^ ) [ j = 0 c 2 j φ 2 j ( x ) + i l j = 0 c 2 j + 1 φ 2 j + 1 ( x ) ] .
2 j = 0 c 2 j φ 2 j ( x ) = j = 0 c j φ j ( x ) + ( 1 ) j j = 0 c j φ j ( x ) = E ( x , 0 ) + E ( x , 0 )
2 j = 0 c 2 j + 1 φ 2 j + 1 ( x ) = j = 0 c j φ j ( x ) ( 1 ) j j = 0 c j φ j ( x ) = E ( x , 0 ) E ( x , 0 ) ;
E ( x , z = l π k 3 ) = exp ( i l γ 1 π ) [ 1 + i l 2 exp ( i l γ 2 π n ^ ) E ( x , 0 ) + 1 i l 2 exp ( i l γ 2 π n ^ ) E ( x , 0 ) ] .
E ( x , z = l π k 3 ) = exp ( i l γ 1 π ) [ 𝔉 l γ 2 π { 1 + i l 2 E ( x , 0 ) + 1 i l 2 E ( x , 0 ) } ] .
E ( x , z = l π k c 3 ) = exp ( i l γ c π ) [ 1 + i l 2 E ( x , 0 ) + 1 i l 2 E ( x , 0 ) ] ,
E ( x , z = l π k 3 ) = exp { i [ 1 ( γ 1 + 1 2 γ 2 ) l π + 1 2 ( x 2 + b 2 2 ) tan ( l π γ 2 ) ] } sec ( l π γ 2 ) × { 1 + i l 2 J ν ( 2 ) [ x b sec ( l π γ 2 ) + a , b 2 4 tan ( l π γ 2 ) ; i ] + 1 i l 2 J ν ( 2 ) [ x b sec ( l π γ 2 ) + a , b 2 4 tan ( l π γ 2 ) ; i ] } .
E ( x , z = l π k c 3 ) = exp ( i l γ c π ) [ 1 + i l 2 J ν ( b x + a ) + 1 i l 2 J ν ( b x + a ) ] .
E ( x , z = l π k 3 ) = sec ( l π γ 2 ) exp { i [ 1 2 l π γ 2 + 1 2 ( x 2 b 2 a ) tan ( l π γ 2 ) + b 6 12 tan 3 ( l π γ 2 ) ] } { 1 + i l 2 exp [ i b 3 2 x tan ( l π γ 2 ) sec ( l π γ 2 ) ] Ai [ b x sec ( l π γ 2 ) + a b 4 4 tan 2 ( l π γ 2 ) ] + 1 i l 2 exp [ i b 3 2 x tan ( l π γ 2 ) sec ( l π γ 2 ) ] Ai [ b x sec ( l π γ 2 ) + a b 4 4 tan 2 ( l π γ 2 ) ] } .
E ( x , z = l π k c 3 ) = exp ( i l γ c π ) [ 1 + i l 2 Ai ( x + 1 ) + 1 i l 2 Ai ( x + 1 ) ] .
J n ( b x + a ) = 1 2 π π π d τ exp { i [ n τ ( b x + a ) sin ( τ ) ] } .
𝔉 α { J n ( b x + a ) } = e i α n ^ 1 2 π π π d τ exp { i [ n τ ( b x + a ) sin ( τ ) ] } = 1 2 π π π d τ exp ( i n τ ) exp ( i α n ^ ) exp [ i ( b x + a ) sin ( τ ) ] .
exp [ i ζ ( p ^ 2 + x 2 ) ] = exp [ i f ( ζ ) x 2 ] exp [ i g ( ζ ) ( x p ^ + p ^ x ) ] exp [ i f ( ζ ) p ^ 2 ] ,
𝔉 α { J n ( b x + a ) } = exp ( i α 2 ) exp [ i f ( α 2 ) x 2 ] 2 π π π d τ exp ( i n τ ) exp [ i g ( α 2 ) ( x p ^ + p ^ x ) ] exp [ i f ( α 2 ) p ^ 2 ] exp [ i ( b x + a ) sin ( τ ) ] .
exp [ i f ( α 2 ) p ^ 2 ] exp [ i ( b x + a ) sin ( τ ) ] = exp [ i f ( α 2 ) b 2 sin 2 ( τ ) ] exp [ i ( b x + a ) sin ( τ ) ]
𝔉 α { J n ( b x + a ) } = exp ( i α 2 ) exp [ i f ( α 2 ) x 2 ] 2 π π π d τ exp ( i n τ ) exp [ i f ( α 2 ) b 2 sin 2 ( τ ) ] exp [ i g ( α 2 ) ( x p ^ + p ^ x ) ] exp [ i ( b x + a ) sin ( τ ) ] .
exp [ i g ( α 2 ) ( x p ^ + p ^ x ) ] exp [ i ( b x + a ) sin ( τ ) ] = = exp [ g ( α 2 ) ] exp { i ( exp [ 2 g ( α 2 ) ] b x + a ) sin ( τ ) } ,
𝔉 α { J n ( b x + a ) } = exp ( i α 2 ) exp [ i f ( α 2 ) x 2 ] exp [ g ( α 2 ) ] × 1 2 π π π d τ exp ( i n τ ) exp [ i f ( α 2 ) b 2 sin 2 ( τ ) ] exp { i ( exp [ 2 g ( α 2 ) ] b x + a ) sin ( τ ) } .
𝔉 α { J n ( b x + a ) } = exp [ g ( α 2 ) ] exp [ i α 2 + i f ( α 2 ) x 2 + i 2 b 2 f ( α 2 ) ] 1 2 π π π d τ exp ( i n τ ) exp [ i 2 f ( α 2 ) b 2 cos ( 2 τ ) ] exp { i ( exp [ 2 g ( α 2 ) ] b x + a ) sin ( τ ) } .
J n ( m ) ( x , y ; c ) = l = c l J n m l ( x ) J l ( y ) ,
J n ( m ) ( x , y ; e i θ ) = 1 2 π d φ exp [ i x sin φ + i y sin ( m φ + θ ) i n φ ] ,
𝔉 α { J n ( b x + a ) } = exp [ g ( α 2 ) i α 2 + i f ( α 2 ) ( x 2 + b 2 2 ) ] × J n ( 2 ) [ e 2 g ( α 2 ) b x + a , b 2 2 f ( α 2 ) ; i ] .
𝔉 α { J n ( b x + a ) } = sec α exp { i 2 [ α + ( x 2 + b 2 2 ) tan α ] } × J n 2 ( b x sec α + a , b 2 4 tan α ; i ) .
Ai ( b x + a ) = 1 2 π d τ exp [ i τ ( b x + a ) ] exp ( i τ 3 3 ) .
𝔉 α { Ai ( b x + a ) } = e i α n ^ 1 2 π d τ exp ( i τ b x ) exp ( i τ a ) exp ( i τ 3 3 ) = 1 2 π d τ exp ( i τ 3 3 ) exp ( i τ a ) e i α n ^ exp ( i τ b x ) .
exp [ i ζ ( p ^ 2 + x 2 ) ] = exp [ i f ( ζ ) x 2 ] exp [ i g ( ζ ) ( x p ^ + p ^ x ) ] exp [ i f ( ζ ) p ^ 2 ] ,
𝔉 α { Ai ( b x + a ) } = exp ( i α 2 ) exp [ i f ( α 2 ) x 2 ] 2 π d τ exp ( i τ 3 3 ) exp [ i g ( α 2 ) ( x p ^ + p x ) ] × exp ( i τ a ) exp [ i f ( α 2 ) p ^ 2 ] exp ( i τ b x ) .
exp [ i f ( α 2 ) p ^ 2 ] exp ( i τ b x ) = exp [ i f ( α 2 ) b 2 τ 2 ] exp ( i τ b x ) ,
𝔉 α { Ai ( b x + a ) } = exp ( i α 2 ) exp [ i f ( α 2 ) x 2 ] 2 π d τ exp ( i τ 3 3 ) exp [ i f ( α 2 ) b 2 τ 2 ] × exp ( i τ a ) exp [ i g ( α 2 ) ( x p ^ + p ^ x ) ] exp ( i b x τ ) .
exp [ i g ( α 2 ) ( x p ^ + p ^ x ) ] exp ( i b x τ ) = exp [ g ( α 2 ) ] exp { i exp [ 2 g ( α 2 ) ] b x τ } ,
𝔉 α { Ai ( b x + a ) } = exp ( i α 2 ) exp [ i f ( α 2 ) x 2 ] exp [ g ( α 2 ) ] × 1 2 π d τ exp ( i τ 3 3 ) exp ( i τ a ) exp [ i f ( α 2 ) b 2 τ 2 ] exp { i exp [ 2 g ( α 2 ) ] b x τ } .
𝔉 α { Ai ( b x + a ) } = exp { i α 2 + i f ( α 2 ) x 2 + i 2 3 b 6 f 3 ( α 2 ) i b 3 f ( α 2 ) exp [ 2 g ( α 2 ) ] x g ( α 2 ) i b 2 f ( α 2 ) a } × 1 2 π d τ exp ( i τ 3 3 ) exp ( i τ a ) exp ( i τ { exp [ 2 g ( α 2 ) ] b x b 4 f 2 ( α 2 ) } ) .
𝔉 α { Ai ( b x + a ) } = sec α exp { i [ α 2 + 1 2 tan α ( x 2 b 2 a x b 3 sec α + b 6 6 tan 2 α ) ] } × Ai ( x b sec α + a b 4 4 tan 2 α ) .
E ( x , 0 ) = j = 0 c j φ j ( x ) ,
c j = E ( x , 0 ) φ j ( x ) d x .
E ( x , z ) = j = 0 c j exp [ i z k 2 1 2 j ] φ j ( x ) .
E ( x , z ) = j = 0 N max c j exp [ i z k 2 1 2 j ] φ j ( x ) .

Metrics