Abstract

We study conical refraction in crystals where both diffraction and nonlinearity are present. We develop a new set of evolution equations. We find that nonlinearity induces a modulational instability when it is defocussing as well as focussing. We also examine the evolution of incident beams which contain analytic singularities, and in particular optical vortices, which do not feel the effect of conical refraction.

©2006 Optical Society of America

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References

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  1. W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. Royal Irish Acad. 171–144 (1837)
  2. H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Phil. Mag. 2112–120 and 207–210 (1833)
  3. M. Berry,“Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6289–300 (2004)
    [Crossref]
  4. M. Berry, M. R. Jeffrey, and L. Lunney, “Conical diffraction observations and theory,” Proc. Roy. Soc. A 4621629–1642 (2006)
    [Crossref]
  5. G. J. Roskes,“Some nonlinear multiwave multiphase interactions,” Stud. Appl. Math 55231–238 (1976)
  6. O. C. Wright, “Modulational instability in a defocussing coupled nonlinear Schrödinger system,” Physica D 821–10 (1995).
    [Crossref]
  7. J. Rothenberg, “Observation of buildup of modulational instability from wave breaking,” Opt. Lett. 1618–20 (1991).
    [Crossref] [PubMed]
  8. J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. 4651–654 (1987).
    [Crossref]
  9. J. Gutierrez-Vega and M. Banderes, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. 22289–295 (2005).
    [Crossref]
  10. A. Mamaev, M. Saffman, D. Anderson, and A. Zozulya, “Propagation of light beams in anisotropic nonlinear media,” Phys. Rev. A 54870–879 (1996).
    [Crossref] [PubMed]
  11. V. Lvov, Y. Lvov, A. C. Newell, and V. Zakharov,“Statistical description of acoustic turbulence,” Phys. Rev. E 56390–410 (1997).
    [Crossref]

2006 (1)

M. Berry, M. R. Jeffrey, and L. Lunney, “Conical diffraction observations and theory,” Proc. Roy. Soc. A 4621629–1642 (2006)
[Crossref]

2005 (1)

J. Gutierrez-Vega and M. Banderes, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. 22289–295 (2005).
[Crossref]

2004 (1)

M. Berry,“Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6289–300 (2004)
[Crossref]

1997 (1)

V. Lvov, Y. Lvov, A. C. Newell, and V. Zakharov,“Statistical description of acoustic turbulence,” Phys. Rev. E 56390–410 (1997).
[Crossref]

1996 (1)

A. Mamaev, M. Saffman, D. Anderson, and A. Zozulya, “Propagation of light beams in anisotropic nonlinear media,” Phys. Rev. A 54870–879 (1996).
[Crossref] [PubMed]

1995 (1)

O. C. Wright, “Modulational instability in a defocussing coupled nonlinear Schrödinger system,” Physica D 821–10 (1995).
[Crossref]

1991 (1)

1987 (1)

J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. 4651–654 (1987).
[Crossref]

1976 (1)

G. J. Roskes,“Some nonlinear multiwave multiphase interactions,” Stud. Appl. Math 55231–238 (1976)

1837 (1)

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. Royal Irish Acad. 171–144 (1837)

1833 (1)

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Phil. Mag. 2112–120 and 207–210 (1833)

Anderson, D.

A. Mamaev, M. Saffman, D. Anderson, and A. Zozulya, “Propagation of light beams in anisotropic nonlinear media,” Phys. Rev. A 54870–879 (1996).
[Crossref] [PubMed]

Banderes, M.

J. Gutierrez-Vega and M. Banderes, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. 22289–295 (2005).
[Crossref]

Berry, M.

M. Berry, M. R. Jeffrey, and L. Lunney, “Conical diffraction observations and theory,” Proc. Roy. Soc. A 4621629–1642 (2006)
[Crossref]

M. Berry,“Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6289–300 (2004)
[Crossref]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. 4651–654 (1987).
[Crossref]

Gutierrez-Vega, J.

J. Gutierrez-Vega and M. Banderes, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. 22289–295 (2005).
[Crossref]

Hamilton, W. R.

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. Royal Irish Acad. 171–144 (1837)

Jeffrey, M. R.

M. Berry, M. R. Jeffrey, and L. Lunney, “Conical diffraction observations and theory,” Proc. Roy. Soc. A 4621629–1642 (2006)
[Crossref]

Lloyd, H.

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Phil. Mag. 2112–120 and 207–210 (1833)

Lunney, L.

M. Berry, M. R. Jeffrey, and L. Lunney, “Conical diffraction observations and theory,” Proc. Roy. Soc. A 4621629–1642 (2006)
[Crossref]

Lvov, V.

V. Lvov, Y. Lvov, A. C. Newell, and V. Zakharov,“Statistical description of acoustic turbulence,” Phys. Rev. E 56390–410 (1997).
[Crossref]

Lvov, Y.

V. Lvov, Y. Lvov, A. C. Newell, and V. Zakharov,“Statistical description of acoustic turbulence,” Phys. Rev. E 56390–410 (1997).
[Crossref]

Mamaev, A.

A. Mamaev, M. Saffman, D. Anderson, and A. Zozulya, “Propagation of light beams in anisotropic nonlinear media,” Phys. Rev. A 54870–879 (1996).
[Crossref] [PubMed]

Newell, A. C.

V. Lvov, Y. Lvov, A. C. Newell, and V. Zakharov,“Statistical description of acoustic turbulence,” Phys. Rev. E 56390–410 (1997).
[Crossref]

Roskes, G. J.

G. J. Roskes,“Some nonlinear multiwave multiphase interactions,” Stud. Appl. Math 55231–238 (1976)

Rothenberg, J.

Saffman, M.

A. Mamaev, M. Saffman, D. Anderson, and A. Zozulya, “Propagation of light beams in anisotropic nonlinear media,” Phys. Rev. A 54870–879 (1996).
[Crossref] [PubMed]

Wright, O. C.

O. C. Wright, “Modulational instability in a defocussing coupled nonlinear Schrödinger system,” Physica D 821–10 (1995).
[Crossref]

Zakharov, V.

V. Lvov, Y. Lvov, A. C. Newell, and V. Zakharov,“Statistical description of acoustic turbulence,” Phys. Rev. E 56390–410 (1997).
[Crossref]

Zozulya, A.

A. Mamaev, M. Saffman, D. Anderson, and A. Zozulya, “Propagation of light beams in anisotropic nonlinear media,” Phys. Rev. A 54870–879 (1996).
[Crossref] [PubMed]

J. Opt. A (1)

M. Berry,“Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6289–300 (2004)
[Crossref]

J. Opt. Soc. Am. (2)

J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. 4651–654 (1987).
[Crossref]

J. Gutierrez-Vega and M. Banderes, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. 22289–295 (2005).
[Crossref]

Opt. Lett. (1)

Phil. Mag. (1)

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Phil. Mag. 2112–120 and 207–210 (1833)

Phys. Rev. A (1)

A. Mamaev, M. Saffman, D. Anderson, and A. Zozulya, “Propagation of light beams in anisotropic nonlinear media,” Phys. Rev. A 54870–879 (1996).
[Crossref] [PubMed]

Phys. Rev. E (1)

V. Lvov, Y. Lvov, A. C. Newell, and V. Zakharov,“Statistical description of acoustic turbulence,” Phys. Rev. E 56390–410 (1997).
[Crossref]

Physica D (1)

O. C. Wright, “Modulational instability in a defocussing coupled nonlinear Schrödinger system,” Physica D 821–10 (1995).
[Crossref]

Proc. Roy. Soc. A (1)

M. Berry, M. R. Jeffrey, and L. Lunney, “Conical diffraction observations and theory,” Proc. Roy. Soc. A 4621629–1642 (2006)
[Crossref]

Stud. Appl. Math (1)

G. J. Roskes,“Some nonlinear multiwave multiphase interactions,” Stud. Appl. Math 55231–238 (1976)

Trans. Royal Irish Acad. (1)

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. Royal Irish Acad. 171–144 (1837)

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Figures (3)

Fig. 1.
Fig. 1. Evolution of amplitude of the most unstable mode, semi-log scale. The x’s are for a perturbation with k = 11 and τI 0 = 2, the o’s are for k = 9 and τI 0 = -2. The lines are for reference, and show the predicted growth rates. Parameter are β = 0.1, p = 1
Fig. 2.
Fig. 2. Total intensity at z = 1 for (a) Gaussian with τ = 0, (b) with τ = 1, (c) modulated Gaussian (F(ζ) = ζ) with τ = 0, and (d) with τ = 1.
Fig. 3.
Fig. 3. Total intensity graphed radially at z=0 (black), 1 (blue), and 2 (red) cases as in Fig. 2 The peak for (b) at z = 2, r ≈ .2 goes off scale to ≈ 6.8

Equations (9)

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Ω = ( ω 2 c 2 n x 2 k y 2 k z 2 k x k y k x k z k x k y ω 2 c 2 n y 2 k x 2 k z 2 k y k z k x k z k y k z ω 2 c 2 n z 2 k x 2 k y 2 ) .
E = ( ( 0 1 0 ) B + n 2 ( 1 n x 2 cos θ 0 1 n z 2 sin θ ) A ) exp ( inω c ( x sin θ + z cos θ ) iωt ) + ( cc ) + E 1
B z n c B t = p B x + p A y + 2 B + i τ 1 A 2 + γ B 2 1 + δ ( A 2 + B 2 ) B
A z n c A t = p A x + p B y + 2 A + i τ 2 A 2 + γ B 2 1 + δ ( A 2 + B 2 ) A .
σ 4 + 2 σ 2 k 2 ( β 2 k 2 + p 2 βτI 0 ) + k 4 ( β 2 k 2 p 2 ) ( β 2 k 2 p 2 2 βτI 0 ) = 0
F + z = p ( r + 1 r i r α ) F p 2 F e F F + ( 2 r 2 + 1 r r + 1 r 2 2 α 2 ) F +
+ 2 F e F ( r + i r α ) F + + 2 FF * ( F + 2 + F 2 ) F +
F z = p ( r + i r α ) F + + ( 2 r 2 + 1 r r + 1 r 2 2 α 2 + 2 i r α 1 r 2 ) F
+ 2 F e F ( r + i r α 1 r ) F + 2 FF * ( F + 2 + F 2 ) F

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