Abstract

Based on the ray transformation matrix formalism, we propose a simple method for generation of paraxial beams performing anisotropic rotation in the phase space during their propagation through isotropic optical systems. The widely discussed spiral beams are the particular case of these beams. The propagation of these beams through the symmetric fractional Fourier transformer is demonstrated by numerical simulations.

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References

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  1. E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
    [CrossRef]
  2. E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125(4-6), 302–323 (1996).
    [CrossRef]
  3. E. Abramochkin and V. Volostnikov, “Spiral light beams,” Phys. Usp. 47(12), 1177–1203 (2004).
    [CrossRef]
  4. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Centrifugal transformation of the transverse structure of freely propagating paraxial light beams,” Opt. Lett. 31(6), 694–696 (2006).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  14. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. 33(17), 1603–1629 (2000).
    [CrossRef]
  15. E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004).
    [CrossRef]
  16. T. Alieva and A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).

2007

2006

2005

2004

E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral light beams,” Phys. Usp. 47(12), 1177–1203 (2004).
[CrossRef]

2000

R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17(2), 342–355 (2000).
[CrossRef]

A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. 33(17), 1603–1629 (2000).
[CrossRef]

1999

T. Alieva and A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).

1996

E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125(4-6), 302–323 (1996).
[CrossRef]

1993

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
[CrossRef]

1970

Abramochkin, E.

E. Abramochkin and V. Volostnikov, “Spiral light beams,” Phys. Usp. 47(12), 1177–1203 (2004).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125(4-6), 302–323 (1996).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
[CrossRef]

Alieva, T.

Barbé, A.

T. Alieva and A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).

Bastiaans, M. J.

Bekshaev, A.

Bekshaev, A. Y.

Calvo, G. F.

Calvo, M. L.

Collins, S. A.

Rodrigo, J. A.

Simon, R.

Soskin, M.

Soskin, M. S.

Vasnetsov, M. V.

Volostnikov, V.

E. Abramochkin and V. Volostnikov, “Spiral light beams,” Phys. Usp. 47(12), 1177–1203 (2004).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125(4-6), 302–323 (1996).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
[CrossRef]

Wolf, K. B.

Wünsche, A.

A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. 33(17), 1603–1629 (2000).
[CrossRef]

J. Mod. Opt.

T. Alieva and A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).

J. Opt. A, Pure Appl. Opt.

E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. Math. Gen.

A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. 33(17), 1603–1629 (2000).
[CrossRef]

Opt. Commun.

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125(4-6), 302–323 (1996).
[CrossRef]

Opt. Lett.

Phys. Usp.

E. Abramochkin and V. Volostnikov, “Spiral light beams,” Phys. Usp. 47(12), 1177–1203 (2004).
[CrossRef]

Other

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Supplementary Material (1)

» Media 1: AVI (2654 KB)     

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Figures (1)

Fig. 1
Fig. 1

Intensities (top row) and phases (bottom row) of the beam Ψ 2 U g ( α ) ( r , 1 ) , given by Eq. (12), for various values of the angle α: (a) α = π / 4 (a spiral beam), (b) α = π / 8 , (c) α = 0 (combination of HG modes). (Media 1) Intensity and phase evolution of the beam | R U f ( φ , φ ) [ Ψ 2 U g ( α ) ( r , 1 ) ] | 2 for φ [ 0 , 2 π / 3 ] .

Equations (16)

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( r o p o ) = [ A B C D ] ( r i p i ) = T ( r i p i ) .
T = T L T S T O = [ I 0 G I ] [ S 0 0 S 1 ] [ X Y Y X ] ,
U f ( γ x , γ y ) = [ exp ( i γ x ) 0 0 exp ( i γ y ) ] , U r ( θ ) = [ cos θ sin θ sin θ cos θ ] , U g ( α ) = [ cos α i sin α i sin α cos α ] ,
| R T a r [ Ψ ( r i ) ] ( r o ) | 2 = | R T f ( φ , φ ) [ Ψ ( r i ) ] ( r o ) | 2 .
H m , n U 0 ( r ) = ( 1 ) m + n exp ( x 2 + y 2 ) 2 m + n 1 / 2 ( π m ! n ! ) 1 / 2 ( U 11 x + U 12 y ) m ( U 21 x + U 22 y ) n exp ( 2 x 2 2 y 2 ) ,
H m , n U g ( 0 ) ( r ) = H m , n I ( r ) = i n ( π 2 m + n 1 m ! n ! ) 1 / 2 exp ( x 2 y 2 ) H m ( 2 x ) H n ( 2 y ) , H m , n U g ( ± π / 4 ) ( r ) = ( ± i ) n ( 1 ) min ( π / 2 ) 1 / 2 min ! max ! exp ( r 2 ± i ( m n ) ψ ) ( 2 r ) | m n | L min | m n | ( 2 r 2 ) ,
R U f ( φ , φ ) [ Ψ ( r ) ] = exp ( i ϕ ) R U a r ( γ , U 0 ) [ Ψ ( r ) ] ,
R U a r ( γ , U 0 ) R U f ( φ , φ ) [ Ψ ( r ) ] = exp ( i ϕ ) Ψ ( r ) .
R U a r ( γ , U 0 ) R U f ( φ , φ ) [ H m , n U 0 ( r ) ] = exp ( i ϕ ) H m , n U 0 ( r ) ,
Ψ U 0 ( r , v ) = m , n c m n H m , n U 0 ( r )
m ( 1 v ) + n ( 1 + v ) = const ,
R U a r ( v φ , U 0 ) R U f ( φ , φ ) [ Ψ U 0 ( r , v ) ] = exp [ i ϕ ( v , φ ) ] Ψ U 0 ( r , v ) .
Ψ n U 0 ( r , 1 ) = m = 0 c m n H m , n U 0 ( r ) ,
f ( r ) = m = 0 n = 0 f m n H m , n U 0 ( r ) = n = 0 Ψ n U 0 ( r , 1 ) ,
m 1 ( 1 v ) + n 1 ( 1 + v ) = m 2 ( 1 v ) + n 2 ( 1 + v ) v = ( m 2 + n 2 ) ( m 1 + n 1 ) ( m 2 n 2 ) ( m 1 n 1 ) = k l
Ψ 2 U g ( α ) ( r , 1 ) = n = 0 ( 3 n + 2 ) ! ( 3 n ) ! 3 3 n H 3 n + 2 , 2 U g ( α ) ( r ) .

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