Abstract

We report on discrete, nondiffracting, paraxial beams with a Bessel spatial envelope in 1D periodic structures of dielectric media. Anisotropy of the envelope profile is demonstrated to behave in the same manner as extraordinary waves in uniaxial crystals.

© 2008 Optical Society of America

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References

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  1. A. Barthelemy, S. Maneuf, and C. Froehly, "Soliton propagation and self-confinement of laser-beams by Kerr optical non-linearity," Opt. Commun. 55, 201-206 (1985).
    [CrossRef]
  2. D. Faccio, A. Averchi, A. Couairon, A. Dubietis, R. Piskarskas, A. Matijosius, F. Bragheri, M. A. Porras, A. Piskarskas, and P. D. Trapani, "Competition between phase-matching and stationarity in Kerr-driven optical pulse filamentation," Phys. Rev. E 74, 047603 (2006).
    [CrossRef]
  3. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003).
    [CrossRef] [PubMed]
  4. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003).
    [CrossRef]
  5. J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  6. J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  7. G. Indebetouw, "Nondiffracting optical fields: some remarks on their analysis and synthesis," J. Opt. Soc. Am. A 6, 150-152 (1989).
    [CrossRef]
  8. J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
    [CrossRef]
  9. M. A. Porras, G. Valiulis, and P. D. Trapani, "Unified description of Bessel X waves with cone dispersion and tilted pulses," Phys. Rev. E 68, 016613 (2003).
    [CrossRef]
  10. S. Longhi, K. Janner, and P. Laporta, "Propagating pulsed Bessel beams in periodic media," J. Opt. B: Quantum Semiclassical Opt. 6, 477-481 (2004).
    [CrossRef]
  11. S. Longhi and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235,123 (2004).
    [CrossRef]
  12. S. Longhi, "Localized and nonspreading spatiotemporal Wannier wave packets in photonic crystals," Phys. Rev. E 71, 016603 (2005).
    [CrossRef]
  13. O. Manela, M. Segev, and D. N. Christodoulides, "Nondiffracting beams in periodic media," Opt. Lett. 30, 2611-2613 (2005).
    [CrossRef] [PubMed]
  14. K. Staliunas and R. Herrero, "Nondiffractive propagation of light in photonic crystals," Phys. Rev. E 73, 016601 (2006).
    [CrossRef]
  15. J. Hudock, N. K. Efremidis, and D. N. Christodoulides, "Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays," Opt. Lett. 29, 268-270 (2004).
    [CrossRef] [PubMed]
  16. A. Ciattoni and C. Palma, "Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis," Opt. Commun. 224, 175-183 (2003).
    [CrossRef]
  17. J. A. Fleck and M. D. Feit, "Beam propagation in uniaxial anisotropic media," J. Opt. Soc. Am. 73, 920-926 (1983).
    [CrossRef]
  18. A. Ciattoni, B. Crosignani, and P. D. Porto, "Vectorial theory of propagation in uniaxially anisotropic media," J. Opt. Soc. Am. A 18, 1656-1661 (2001).
    [CrossRef]
  19. A. Ciattoni and C. Palma, "Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond," J. Opt. Soc. Am. A 20, 2163-2170 (2003).
    [CrossRef]
  20. J. Sipe and H. Winful, "Nonlinear Schrodinger solitons in a periodic structure," Opt. Lett. 13, 132-134 (1988).
    [CrossRef] [PubMed]
  21. C. M. de Sterke and J. E. Sipe, "Envelope-function approach for the electrodynamics of nonlinear periodic structures," Phys. Rev. A 38, 5149-5165 (1988).
    [CrossRef] [PubMed]
  22. C. M. de Sterke and J. E. Sipe, "Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures," Phys. Rev. A 39, 5163-5178 (1989).
    [CrossRef] [PubMed]
  23. R. deL. Kronig and W. G. Penney, "Quantum mechanics of electrons in crystal lattices," Proc. R. Soc. London, Ser. A 130, 499-513 (1931).
    [CrossRef]
  24. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85, 1863-1866 (2000).
    [CrossRef] [PubMed]

2006

D. Faccio, A. Averchi, A. Couairon, A. Dubietis, R. Piskarskas, A. Matijosius, F. Bragheri, M. A. Porras, A. Piskarskas, and P. D. Trapani, "Competition between phase-matching and stationarity in Kerr-driven optical pulse filamentation," Phys. Rev. E 74, 047603 (2006).
[CrossRef]

K. Staliunas and R. Herrero, "Nondiffractive propagation of light in photonic crystals," Phys. Rev. E 73, 016601 (2006).
[CrossRef]

2005

S. Longhi, "Localized and nonspreading spatiotemporal Wannier wave packets in photonic crystals," Phys. Rev. E 71, 016603 (2005).
[CrossRef]

O. Manela, M. Segev, and D. N. Christodoulides, "Nondiffracting beams in periodic media," Opt. Lett. 30, 2611-2613 (2005).
[CrossRef] [PubMed]

2004

S. Longhi, K. Janner, and P. Laporta, "Propagating pulsed Bessel beams in periodic media," J. Opt. B: Quantum Semiclassical Opt. 6, 477-481 (2004).
[CrossRef]

S. Longhi and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235,123 (2004).
[CrossRef]

J. Hudock, N. K. Efremidis, and D. N. Christodoulides, "Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays," Opt. Lett. 29, 268-270 (2004).
[CrossRef] [PubMed]

2003

A. Ciattoni and C. Palma, "Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis," Opt. Commun. 224, 175-183 (2003).
[CrossRef]

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003).
[CrossRef]

M. A. Porras, G. Valiulis, and P. D. Trapani, "Unified description of Bessel X waves with cone dispersion and tilted pulses," Phys. Rev. E 68, 016613 (2003).
[CrossRef]

A. Ciattoni and C. Palma, "Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond," J. Opt. Soc. Am. A 20, 2163-2170 (2003).
[CrossRef]

2001

2000

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

1996

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

1989

G. Indebetouw, "Nondiffracting optical fields: some remarks on their analysis and synthesis," J. Opt. Soc. Am. A 6, 150-152 (1989).
[CrossRef]

C. M. de Sterke and J. E. Sipe, "Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures," Phys. Rev. A 39, 5163-5178 (1989).
[CrossRef] [PubMed]

1988

J. Sipe and H. Winful, "Nonlinear Schrodinger solitons in a periodic structure," Opt. Lett. 13, 132-134 (1988).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, "Envelope-function approach for the electrodynamics of nonlinear periodic structures," Phys. Rev. A 38, 5149-5165 (1988).
[CrossRef] [PubMed]

1987

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

1985

A. Barthelemy, S. Maneuf, and C. Froehly, "Soliton propagation and self-confinement of laser-beams by Kerr optical non-linearity," Opt. Commun. 55, 201-206 (1985).
[CrossRef]

1983

1931

R. deL. Kronig and W. G. Penney, "Quantum mechanics of electrons in crystal lattices," Proc. R. Soc. London, Ser. A 130, 499-513 (1931).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt.

S. Longhi, K. Janner, and P. Laporta, "Propagating pulsed Bessel beams in periodic media," J. Opt. B: Quantum Semiclassical Opt. 6, 477-481 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature (London)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003).
[CrossRef]

Opt. Commun.

A. Barthelemy, S. Maneuf, and C. Froehly, "Soliton propagation and self-confinement of laser-beams by Kerr optical non-linearity," Opt. Commun. 55, 201-206 (1985).
[CrossRef]

A. Ciattoni and C. Palma, "Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis," Opt. Commun. 224, 175-183 (2003).
[CrossRef]

Opt. Lett.

Phys. Rev. A

C. M. de Sterke and J. E. Sipe, "Envelope-function approach for the electrodynamics of nonlinear periodic structures," Phys. Rev. A 38, 5149-5165 (1988).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, "Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures," Phys. Rev. A 39, 5163-5178 (1989).
[CrossRef] [PubMed]

Phys. Rev. B

S. Longhi and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235,123 (2004).
[CrossRef]

Phys. Rev. E

S. Longhi, "Localized and nonspreading spatiotemporal Wannier wave packets in photonic crystals," Phys. Rev. E 71, 016603 (2005).
[CrossRef]

K. Staliunas and R. Herrero, "Nondiffractive propagation of light in photonic crystals," Phys. Rev. E 73, 016601 (2006).
[CrossRef]

D. Faccio, A. Averchi, A. Couairon, A. Dubietis, R. Piskarskas, A. Matijosius, F. Bragheri, M. A. Porras, A. Piskarskas, and P. D. Trapani, "Competition between phase-matching and stationarity in Kerr-driven optical pulse filamentation," Phys. Rev. E 74, 047603 (2006).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

M. A. Porras, G. Valiulis, and P. D. Trapani, "Unified description of Bessel X waves with cone dispersion and tilted pulses," Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A

R. deL. Kronig and W. G. Penney, "Quantum mechanics of electrons in crystal lattices," Proc. R. Soc. London, Ser. A 130, 499-513 (1931).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

(a) Dispersion equation at vacuum wavelength λ 0 = 1.55 μ m for a layered medium made of dielectric ( n 2 = 1.46 ) slabs of width b = 1.7 μ m immersed in air ( a = 0.6 μ m ) . (b) Isofrequency curves (only first quadrant) for equidistant values γ of the axial PM belonging to the first band. Bandgaps are plotted in yellow and PMs units are μ m 1 .

Fig. 2
Fig. 2

Field strength E x 2 in a transverse plane of localized TE waves with a phase matching at R 0 = ( 0 , 0 ) and propagation constant: (a) γ = 5.750 μ m 1 ( ν = 0.145 μ m 1 ) and (b) γ = 5.748 μ m 1 ( ν = 0.210 μ m 1 ) .

Fig. 3
Fig. 3

Behavior of the periodic function given in Eq. (25) in the first band. (a) Squared modulus of F k for different values of k y ; solid curve for k y = 0 , dashed curve for k y = 0.366 μ m 1 , and blue dotted curve for k y = π L ; and (b) relative error ε versus k y (inset: ε for k y 0.4 μ m 1 ).

Fig. 4
Fig. 4

Bessel envelope function J 0 2 ( ν R ) superposed upon the field intensity E x 2 , the latter evaluated numerically along the y axis for γ = 5.748 μ m 1 .

Equations (30)

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2 E + k 0 2 n 2 ( y ) E = 0 ,
E 0 F k ( y ) exp ( i k r ) ,
H F k = k y 2 F k .
H = y 2 + 2 i k y y ( k y 2 k 0 2 n 2 ) .
E x ( R , z ) = E 0 ( q ) F k ( y ) exp ( i k z z ) exp ( i q R ) d 2 q ,
E 0 ( k x , k y ) = a ( k x , k y ) δ [ γ 2 k z 2 ( k x , k y ) ] ,
h ( k y , k y 2 ) A k y 2 + B ( k y 2 k m 2 ) ,
k x 2 + η 2 k y 2 + k z 2 = k m 2 .
( k x ν ) 2 + ( k y η ν ) 2 = 1 ,
E 0 ( k x , k y ) = exp ( i C ) δ ( k x 2 + η 2 k y 2 ν 2 ) ,
H F k m = [ 2 i k y y ( k y 2 k m 2 ) ] F k m .
2 k y y F k m k y 2 F k m ,
2 k m k y k x 2 + k z 2 .
E x ( R , z ) = exp ( i γ z ) F k m ( y ) W ( R ) ,
W ( R ) = δ ( k x 2 + η 2 k y 2 ν 2 ) exp [ i ( k x x + k y y ) ] d k x d k y .
( k x , k y ) = ( q cos θ , q η sin θ ) ,
( x , y ) = ( R cos ϕ , R η 1 sin ϕ ) ,
W ( R ) = η 0 2 π 0 δ ( q 2 ν 2 ) exp [ i q R cos ( θ ϕ ) ] q d q d θ .
W ( R ) = 2 1 η 0 2 π exp [ i ν R cos ( θ ϕ ) ] d θ = π η J 0 ( ν R ) ,
x 2 + η 2 y 2 = 0.59 π 2 ν 2 ,
γ = ( k m 2 ν 2 ) 1 2 k m ν 2 2 k m
α = ( k y 2 k 0 2 n 1 2 ) 1 2 .
β = ( k 0 2 n 2 2 k y 2 ) 1 2 .
G = cosh ( α a ) cos ( β b ) + α 2 β 2 2 α β sinh ( α a ) sin ( β b ) .
F k ( y ) = exp ( i k y y ) { β sinh [ α ( a y ) ] + exp ( i k y L ) × [ β cos ( β b ) sinh ( α y ) + α cosh ( α y ) sin ( β b ) ] } ,
F k ( y ) = exp ( i k y y ) { β cos ( β y ) sinh ( α a ) α cosh ( α a ) sin ( β y ) + α exp ( i k y L ) sin [ β ( b + y ) ] } ,
cos ( k y L ) 1 k y 2 L 2 2 ,
G 1 + G m ( k y 2 k m 2 ) ,
η = ( 2 G m L 2 ) 1 2 .
F k = 0 L F k ( y ) 2 d y .

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