Abstract

We investigate the counterpropagation of paraxial nondiffracting optical beams through a medium hosting a bulk reflection grating in the quasi-Bragg matching condition. The impact of the relative magnitude of the Bragg detuning parameter and the grating depth on the plane wave dispersion relation allows us to identify three distinct regimes where counterpropagation and interaction of nondiffracting beams show qualitatively different features, encompassing longitudinally invariant, periodic or exponential intensity profiles. In one of the identified regimes the dispersion relation is not monotonic and the consequent “longitudinal degeneracy” allows the investigation of new class of nondiffracting beams characterized by a double spectral ring profile.

© 2007 Optical Society of America

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References

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  1. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941)
  2. J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  3. D. McGloin, V. Garcs-Chvez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003).
    [CrossRef] [PubMed]
  4. J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, "Optical dipole traps and atomic waveguides based on Bessel light beams," Phys. Rev. A 63, 063602 (2001).
    [CrossRef]
  5. J. C. Gutirrez-Vega, M. D. Iturbe-Castillo, and S. Chvez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1492-1494 (2000).
  6. M. A. Bandres, J. C. Gutirrez-Vega, and S. Chvez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
    [CrossRef] [PubMed]
  7. R. Piestun and J. Shamir, "Generalized propagation-invariant wave fields," J. Opt. Soc. Am. A 15, 3039-3044 (1998).
    [CrossRef]
  8. J. Lu and J. F. Greenleaf, "Ultrasonic nondiffracting transducer for medical imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438-447 (1990).
    [CrossRef] [PubMed]
  9. A. Ciattoni, C. Conti, and P. Di Porto, "Vector electomagnetic X waves," Phys. Rev. E 69, 036608 (2004).
    [CrossRef]
  10. J. N. Brittingham, "Focus waves modes in homogeneous Maxwell’s equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
    [CrossRef]
  11. A. Ciattoni, and C. Palma, "Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis," Opt. Commun. 224, 175-183 (2003).
    [CrossRef]
  12. O. Manela, M. Segev, and D. N. Christodoulides, "Nondiffracting beams in periodic media," Opt. Lett. 30, 2611- 2613 (2005).
    [CrossRef] [PubMed]
  13. A. Yariv, and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  14. A. Ciattoni, C. Rizza, E. DelRe and E. Palange, "Counterpropagating spatial Kerr soliton in reflection gratings," Opt. Lett. 31, 1507-1509 (2006)
    [CrossRef] [PubMed]

2006 (1)

2005 (1)

2004 (2)

2003 (2)

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

D. McGloin, V. Garcs-Chvez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003).
[CrossRef] [PubMed]

2001 (1)

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, "Optical dipole traps and atomic waveguides based on Bessel light beams," Phys. Rev. A 63, 063602 (2001).
[CrossRef]

2000 (1)

J. C. Gutirrez-Vega, M. D. Iturbe-Castillo, and S. Chvez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1492-1494 (2000).

1998 (1)

1990 (1)

J. Lu and J. F. Greenleaf, "Ultrasonic nondiffracting transducer for medical imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438-447 (1990).
[CrossRef] [PubMed]

1987 (1)

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

1983 (1)

J. N. Brittingham, "Focus waves modes in homogeneous Maxwell’s equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
[CrossRef]

Arlt, J.

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, "Optical dipole traps and atomic waveguides based on Bessel light beams," Phys. Rev. A 63, 063602 (2001).
[CrossRef]

Bandres, M. A.

Brittingham, J. N.

J. N. Brittingham, "Focus waves modes in homogeneous Maxwell’s equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
[CrossRef]

Christodoulides, D. N.

Chvez-Cerda, S.

M. A. Bandres, J. C. Gutirrez-Vega, and S. Chvez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
[CrossRef] [PubMed]

J. C. Gutirrez-Vega, M. D. Iturbe-Castillo, and S. Chvez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1492-1494 (2000).

Ciattoni, A.

A. Ciattoni, C. Rizza, E. DelRe and E. Palange, "Counterpropagating spatial Kerr soliton in reflection gratings," Opt. Lett. 31, 1507-1509 (2006)
[CrossRef] [PubMed]

A. Ciattoni, C. Conti, and P. Di Porto, "Vector electomagnetic X waves," Phys. Rev. E 69, 036608 (2004).
[CrossRef]

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

Conti, C.

A. Ciattoni, C. Conti, and P. Di Porto, "Vector electomagnetic X waves," Phys. Rev. E 69, 036608 (2004).
[CrossRef]

DelRe, E.

Dholakia, K.

D. McGloin, V. Garcs-Chvez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003).
[CrossRef] [PubMed]

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, "Optical dipole traps and atomic waveguides based on Bessel light beams," Phys. Rev. A 63, 063602 (2001).
[CrossRef]

Di Porto, P.

A. Ciattoni, C. Conti, and P. Di Porto, "Vector electomagnetic X waves," Phys. Rev. E 69, 036608 (2004).
[CrossRef]

Durnin, J.

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

Eberly, J. H.

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

Garcs-Chvez, V.

Greenleaf, J. F.

J. Lu and J. F. Greenleaf, "Ultrasonic nondiffracting transducer for medical imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438-447 (1990).
[CrossRef] [PubMed]

Gutirrez-Vega, J. C.

M. A. Bandres, J. C. Gutirrez-Vega, and S. Chvez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
[CrossRef] [PubMed]

J. C. Gutirrez-Vega, M. D. Iturbe-Castillo, and S. Chvez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1492-1494 (2000).

Iturbe-Castillo, M. D.

J. C. Gutirrez-Vega, M. D. Iturbe-Castillo, and S. Chvez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1492-1494 (2000).

Lu, J.

J. Lu and J. F. Greenleaf, "Ultrasonic nondiffracting transducer for medical imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438-447 (1990).
[CrossRef] [PubMed]

Manela, O.

McGloin, D.

Miceli, J. J.

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

Palange, E.

Palma, C.

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

Piestun, R.

Rizza, C.

Segev, M.

Shamir, J.

Soneson, J.

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, "Optical dipole traps and atomic waveguides based on Bessel light beams," Phys. Rev. A 63, 063602 (2001).
[CrossRef]

Wright, E. M.

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, "Optical dipole traps and atomic waveguides based on Bessel light beams," Phys. Rev. A 63, 063602 (2001).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. Lu and J. F. Greenleaf, "Ultrasonic nondiffracting transducer for medical imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438-447 (1990).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

J. N. Brittingham, "Focus waves modes in homogeneous Maxwell’s equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
[CrossRef]

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

Opt. Commun. (1)

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. (5)

Phys. Rev. A (1)

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, "Optical dipole traps and atomic waveguides based on Bessel light beams," Phys. Rev. A 63, 063602 (2001).
[CrossRef]

Phys. Rev. E (1)

A. Ciattoni, C. Conti, and P. Di Porto, "Vector electomagnetic X waves," Phys. Rev. E 69, 036608 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (2)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941)

A. Yariv, and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Supplementary Material (1)

» Media 1: AVI (3154 KB)     

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Figures (2)

Fig. 1.
Fig. 1.

Plot of the plane wave dispersion relation β 2 = (κ 2 + χ)2 - 1 in the three different regimes: (a) χ > 1, (b) |χ| < 1 and (c) χ < -1. We have used black and red lines to report pure real (homogeneous plane waves) and pure imaginary (inhomogeneous plane waves) β, respectively. Note in (c) the longitudinal degeneracy occurring for β(κ 1) = β(κ 2). The labels reported along the κ and β axes are relevant points characterizing the dispersion curves.

Fig. 2.
Fig. 2.

Example of counterpropagating nondiffracting beams (movie: counterprop.avi, 3154 KB) of Eqs.(1) for v + (n) = δ n,1 v + (1) and v - (n) = δ n,-1 v - (-1) (with v - (-1) = 2v + (+1)) where the periodic regime has been selected with a longitudinal period 2π/β = 3Λ/4. [Media 1]

Equations (14)

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E ( r ) = { exp [ ik ( 1 + δ ) z + i ϕ 2 ] A + ( r ) + exp [ ik ( 1 + δ ) z i ϕ 2 ] A ( r ) } e ̂ x ,
i A + ζ + ( 2 ξ 2 + 2 η 2 ) A + χ A + + A = 0 ,
i A ζ + ( 2 ξ 2 + 2 η 2 ) A χ A + A + = 0 ,
β 2 = ( κ 2 + χ ) 2 1
A + ( ρ , ζ ) = d 2 κ exp ( i κ ρ ) [ T ( ζ , κ ) V + ( κ ) + R ( ζ , κ ) V ( κ ) ] ,
A ( ρ , ζ ) = d 2 κ exp ( i κ ρ ) [ R ( Λ ζ , κ ) V + ( κ ) + T ( Λ ζ , κ ) V ( κ ) ] ,
T ( ζ , κ ) = β cos [ β ( Λ ζ ) ] + i ( κ 2 + χ ) sin [ β ( Λ ζ ) ] β cos ( β Λ ) + i ( κ 2 + χ ) sin ( β Λ ) ,
R ( ζ , κ ) = i sin ( β ζ ) β cos ( β Λ ) + i ( κ 2 + χ ) sin ( β Λ ) ,
A + ( ρ , ζ ) = T ( ζ , κ 0 ) A + ( ρ , 0 ) + R ( ζ , κ 0 ) A ( ρ , Λ ) ,
A ( ρ , ζ ) = R ( Λ ζ , κ 0 ) A + ( ρ , 0 ) + T ( Λ ζ , κ 0 ) A ( ρ , Λ ) ,
A + ( ρ , ζ ) = exp ( i β 0 ζ ) [ v 1 κ 1 2 + χ β 0 J 0 ( κ 1 ρ ) + v 2 κ 2 2 + χ β 0 J 0 ( κ 2 ρ ) ] ,
A ( ρ , ζ ) = exp ( i β 0 ζ ) [ v 1 κ 1 2 + χ β 0 J 0 ( κ 1 ρ ) + v 2 κ 2 2 + χ + β 0 J 0 ( κ 2 ρ ) ] ,
A + ( ρ , ζ ) = exp ( i χ 2 1 ζ ) [ v 1 χ χ 2 1 + v 2 χ χ 2 1 J 0 ( 2 χ ρ ) ] ,
A ( ρ , ζ ) = exp ( i χ 2 1 ζ ) [ v 1 χ + χ 2 1 + v 2 χ + χ 2 1 J 0 ( 2 χ ρ ) ] ,

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