Abstract

An optical vortex (phase singularity) with a high topological strength resides on the axis of a high-order light beam. The breakup of this vortex under elliptic perturbation into a straight row of unit-strength vortices is described. This behavior is studied in helical Ince–Gauss beams and astigmatic, generalized Hermite–Laguerre–Gauss beams, which are perturbations of Laguerre–Gauss beams. Approximations of these beams are derived for small perturbations, in which a neighborhood of the axis can be approximated by a polynomial in the complex plane: a Chebyshev polynomial for Ince–Gauss beams, and a Hermite polynomial for astigmatic beams.

© 2006 Optical Society of America

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References

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  1. L.Allen, S.M.Barnett, and M.J.Padgett, eds., Optical Angular Momentum (Institute of Physics Publishing, 2003).
    [CrossRef]
  2. L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  3. D. McGloin and K. Dholakia, Contemp. Phys. 46, 15 (2005).
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  4. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999).
  5. M. A. Bandres and J. C. Gutiérrez-Vega, J. Opt. Soc. Am. A 21, 873 (2004).
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  8. E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, S157 (2004).
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  9. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 241, 237 (2004).
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  10. F. S. Roux, Opt. Commun. 223, 31 (2003).
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  23. V. Bargmann, Rev. Mod. Phys. 34, 829 (1962).
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  24. J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison-Wesley, 1994).

2006 (1)

2005 (1)

D. McGloin and K. Dholakia, Contemp. Phys. 46, 15 (2005).
[CrossRef]

2004 (4)

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 241, 237 (2004).
[CrossRef]

M. V. Berry and M. R. Dennis, J. Opt. A, Pure Appl. Opt. 6, S178 (2004).
[CrossRef]

M. A. Bandres and J. C. Gutiérrez-Vega, J. Opt. Soc. Am. A 21, 873 (2004).
[CrossRef]

2003 (2)

F. S. Roux, Opt. Commun. 223, 31 (2003).
[CrossRef]

L.Allen, S.M.Barnett, and M.J.Padgett, eds., Optical Angular Momentum (Institute of Physics Publishing, 2003).
[CrossRef]

2001 (1)

M. R. Dennis, ''Topological singularities in wave fields,'' Ph.D. dissertation (Bristol University, 2001), Chap. 2.

2000 (2)

1999 (1)

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999).

1996 (1)

1995 (1)

I. Freund, Phys. Rev. E 52, 2348 (1995).
[CrossRef]

1994 (2)

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison-Wesley, 1994).

1992 (2)

S. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef] [PubMed]

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

1988 (1)

J. F. Nye, J. V. Hajnal, and J. H. Hannay, Proc. R. Soc. London Ser. A 417, 7 (1988).
[CrossRef]

1975 (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., J. Math. Phys. 16, 512 (1975).
[CrossRef]

1965 (1)

M.Abramowitz and I.Stegun, eds., Handbook of Mathematical Functions (Dover, 1965), Chap. 22.

1964 (1)

F. M. Arscott, Periodic Differential Equations (Pergamon, 1964), pp. 451-456.

1962 (1)

V. Bargmann, Rev. Mod. Phys. 34, 829 (1962).
[CrossRef]

1953 (1)

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, 1953), Vol. 1.

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

Allen, L.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Aravind, P. K.

S. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef] [PubMed]

Arscott, F. M.

F. M. Arscott, Periodic Differential Equations (Pergamon, 1964), pp. 451-456.

Bandres, M. A.

Bargmann, V.

V. Bargmann, Rev. Mod. Phys. 34, 829 (1962).
[CrossRef]

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Bekshaev, A. Y.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 241, 237 (2004).
[CrossRef]

Bently, J. B.

Berry, M. V.

M. V. Berry and M. R. Dennis, J. Opt. A, Pure Appl. Opt. 6, S178 (2004).
[CrossRef]

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., J. Math. Phys. 16, 512 (1975).
[CrossRef]

Chávez-Cerda, S.

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, 1953), Vol. 1.

Courtial, J.

A. O'Neil and J. Courtial, Opt. Commun. 181, 35 (2000).
[CrossRef]

Danakas, S.

S. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef] [PubMed]

Davis, J. A.

Dennis, M. R.

M. V. Berry and M. R. Dennis, J. Opt. A, Pure Appl. Opt. 6, S178 (2004).
[CrossRef]

M. R. Dennis, ''Topological singularities in wave fields,'' Ph.D. dissertation (Bristol University, 2001), Chap. 2.

Dholakia, K.

D. McGloin and K. Dholakia, Contemp. Phys. 46, 15 (2005).
[CrossRef]

Freund, I.

I. Freund, Phys. Rev. E 52, 2348 (1995).
[CrossRef]

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

Gutiérrez-Vega, J. C.

Hajnal, J. V.

J. F. Nye, J. V. Hajnal, and J. H. Hannay, Proc. R. Soc. London Ser. A 417, 7 (1988).
[CrossRef]

Hannay, J. H.

J. F. Nye, J. V. Hajnal, and J. H. Hannay, Proc. R. Soc. London Ser. A 417, 7 (1988).
[CrossRef]

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, 1953), Vol. 1.

Iturbe-Castillo, M. D.

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., J. Math. Phys. 16, 512 (1975).
[CrossRef]

McGloin, D.

D. McGloin and K. Dholakia, Contemp. Phys. 46, 15 (2005).
[CrossRef]

Miller, W.

C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., J. Math. Phys. 16, 512 (1975).
[CrossRef]

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999).

J. F. Nye, J. V. Hajnal, and J. H. Hannay, Proc. R. Soc. London Ser. A 417, 7 (1988).
[CrossRef]

O'Neil, A.

A. O'Neil and J. Courtial, Opt. Commun. 181, 35 (2000).
[CrossRef]

Roux, F. S.

F. S. Roux, Opt. Commun. 223, 31 (2003).
[CrossRef]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison-Wesley, 1994).

Schechner, Y. Y.

Shamir, J.

Shvartsman, N.

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

Soskin, M. S.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 241, 237 (2004).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Vasnetsov, M. V.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 241, 237 (2004).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

Woerdman, J. P.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Contemp. Phys. (1)

D. McGloin and K. Dholakia, Contemp. Phys. 46, 15 (2005).
[CrossRef]

J. Math. Phys. (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., J. Math. Phys. 16, 512 (1975).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

M. V. Berry and M. R. Dennis, J. Opt. A, Pure Appl. Opt. 6, S178 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A, Pure Appl. Opt. 6, S157 (2004).
[CrossRef]

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

Opt. Commun. (3)

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 241, 237 (2004).
[CrossRef]

F. S. Roux, Opt. Commun. 223, 31 (2003).
[CrossRef]

A. O'Neil and J. Courtial, Opt. Commun. 181, 35 (2000).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (3)

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

S. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (1)

I. Freund, Phys. Rev. E 52, 2348 (1995).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. F. Nye, J. V. Hajnal, and J. H. Hannay, Proc. R. Soc. London Ser. A 417, 7 (1988).
[CrossRef]

Rev. Mod. Phys. (1)

V. Bargmann, Rev. Mod. Phys. 34, 829 (1962).
[CrossRef]

Other (7)

J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison-Wesley, 1994).

M. R. Dennis, ''Topological singularities in wave fields,'' Ph.D. dissertation (Bristol University, 2001), Chap. 2.

F. M. Arscott, Periodic Differential Equations (Pergamon, 1964), pp. 451-456.

M.Abramowitz and I.Stegun, eds., Handbook of Mathematical Functions (Dover, 1965), Chap. 22.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, 1953), Vol. 1.

L.Allen, S.M.Barnett, and M.J.Padgett, eds., Optical Angular Momentum (Institute of Physics Publishing, 2003).
[CrossRef]

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999).

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Figures (1)

Fig. 1
Fig. 1

Intensity, real (solid) and imaginary (dashed) zero contours for beams in the waist plane: (a) LG beam ψ LG , 4 , 0 ; (b) cylindrically perturbed ψ LG , 4 , 0 + 0.2 ψ LG , 0 , 0 ; (c) helical IG beam ψ IG , 4 , 4 ; (d) HLG beam ψ HLG , 4 , 0 π 12 . Panel (a) is the unperturbed beam corresponding to the others. In the elliptically perturbed cases (c) and (d), the order 4 vortex has unfolded into a row of four strength 1 vortices, interspersed with saddles with the same phase i.

Equations (7)

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ψ LG , l , p = [ 2 l + 1 p ! ] 1 2 R l exp ( i l ϕ ) [ π ( l + p ) ! ] 1 2 w 0 l + 1 exp ( R 2 w 0 2 ) L p l ( 2 R 2 w 0 2 ) ,
ψ IG , N , l e = [ A C N l e C N l ( i u , e ) C N l ( v , e ) + A S N l e S N l ( i u , e ) S N l ( v , e ) ] exp ( R 2 w 0 2 ) ,
ψ IG , N , l e A [ cos ( l v ) cosh ( l u ) + i sin ( l v ) sin ( l u ) ] A cosh [ l ( u + i v ) ] = A T l [ ( x + i y ) f 0 ] ,
ψ HLG , l , p α = m = N 2 N 2 d l 2 , m N 2 ( 2 α ) ( 1 ) m m ψ LG , 2 m , N 2 m ,
ψ LG , l , p [ ( l + p ) ! ] 1 2 ( x + i sign ( l ) y ) l ( π p ! ) 1 2 l ! w 0 l + 1
d m , m j ( 2 α ) [ ( j + m ) ! ( j m ) ! ( j m ) ! ( j + m ) ! ] 1 2 ( 1 ) m m α m m ( m m ) ! ,
ψ HLG , l , p α [ ( l + p ) ! π p ! ] 1 2 α l 2 w 0 l ! H l ( x + i y 2 w 0 α ) ,

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