Abstract

In this work we formulate the main properties of the gyrator operation which produces a rotation in the twisting (position - spatial frequency) phase planes. This transform can be easily performed in paraxial optics that underlines its possible application for image processing, holography, beam characterization, mode conversion and quantum information.As an example, it is demonstrated the application of gyrator transform for the generation of a variety of stable modes.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons, NY, USA (2001).
  2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
    [CrossRef]
  3. E. G. Abramochkin and V. G. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991).
    [CrossRef]
  4. E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A.: Pure Appl. Opt. 6, S157- S161 (2004).
    [CrossRef]
  5. G. F. Calvo, "Wigner representation and geometric transformations of optical orbital angular momentum spatial modes," Opt. Lett. 30, 1207-1209 (2005).
    [CrossRef] [PubMed]
  6. R. Simon and K. B. Wolf, "Structure of the set of paraxial optical systems," J. Opt. Soc. Am. A 17, 342-355 (2000).
    [CrossRef]
  7. K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, Berlin, 2004).
  8. J. A. Rodrigo, T. Alieva, M. L. Calvo, "Optical system design for ortho-symplectic transformations in phase space," J. Opt. Soc. Am. A 23, 2494-2500 (2006).
    [CrossRef]
  9. T. Alieva, V. Lopez, F. Agullo Lopez, L. B. Almeida, "The fractional Fourier transform in optical propagation problems, " J. Mod. Opt. 41, 1037-1044 (1994).
    [CrossRef]
  10. M. Bastiaans and T. Alieva, "First-order optical systems with unimodular eigenvalues," J. Opt. Soc. Am. A 23, 1875-1883 (2006).
    [CrossRef]
  11. T. Alieva and M. Bastiaans, "Mode mapping in paraxial lossless optics," Opt. Lett. 30, 1461-1463 (2005).
    [CrossRef] [PubMed]
  12. M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions, Frankfurt am Main, Germany (1984).
  13. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products (Academic Press, NY, USA, 1996).
  14. V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs and J. Turunen, "Elliptic Laguerre- Gaussian beams," J. Opt. Soc. Am. A 23, 43-56, (2006).
    [CrossRef]

2006

2005

2004

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A.: Pure Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

2000

1994

T. Alieva, V. Lopez, F. Agullo Lopez, L. B. Almeida, "The fractional Fourier transform in optical propagation problems, " J. Mod. Opt. 41, 1037-1044 (1994).
[CrossRef]

1993

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

1991

E. G. Abramochkin and V. G. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991).
[CrossRef]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A.: Pure Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991).
[CrossRef]

Agullo Lopez, F.

T. Alieva, V. Lopez, F. Agullo Lopez, L. B. Almeida, "The fractional Fourier transform in optical propagation problems, " J. Mod. Opt. 41, 1037-1044 (1994).
[CrossRef]

Alieva, T.

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Almazov, A. A.

Almeida, L. B.

T. Alieva, V. Lopez, F. Agullo Lopez, L. B. Almeida, "The fractional Fourier transform in optical propagation problems, " J. Mod. Opt. 41, 1037-1044 (1994).
[CrossRef]

Bastiaans, M.

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Calvo, G. F.

Calvo, M. L.

Jefimovs, K.

Khonina, S. N.

Kotlyar, V. V.

Lopez, V.

T. Alieva, V. Lopez, F. Agullo Lopez, L. B. Almeida, "The fractional Fourier transform in optical propagation problems, " J. Mod. Opt. 41, 1037-1044 (1994).
[CrossRef]

Rodrigo, J. A.

Simon, R.

Soifer, V. A.

Turunen, J.

Van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A.: Pure Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Wolf, K. B.

J. Mod. Opt.

T. Alieva, V. Lopez, F. Agullo Lopez, L. B. Almeida, "The fractional Fourier transform in optical propagation problems, " J. Mod. Opt. 41, 1037-1044 (1994).
[CrossRef]

J. Opt. A.: Pure Appl. Opt.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A.: Pure Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991).
[CrossRef]

Opt. Lett.

Other

M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions, Frankfurt am Main, Germany (1984).

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products (Academic Press, NY, USA, 1996).

H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons, NY, USA (2001).

K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, Berlin, 2004).

Supplementary Material (2)

» Media 1: AVI (2549 KB)     
» Media 2: AVI (2914 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Graphical representation for the phase structure associated to the gyrator kernel for α = π/4, x o = y o = 0 (a) and 2x o = y o = 1 (b). These figures (a) and (b) correspond to the exponential argument of the kernel.

Fig. 2.
Fig. 2.

Intensity distributions corresponding to the GT of the circle function are displayed for different transformation angles α = 0 (a),7π/36 (b), π/4 (c), 11π/36 (d), and π/2 (e).Note that for α = π/2 the rotated Fourier transform is obtained.

Fig. 3.
Fig. 3.

Intensity (up row) and phase (low row) of the GT of HG 1,0 mode for different angles α. Figure (a) corresponds to transformation angle α = 0, π/4, π/2, 3π/4, π, 5π/4,3π/2,7π/4. (b) Intermediate sequence between angle α = 0 and α =π/4 is displayed. (2.5 MB) Movie: mode transformation for different angles α, where the input mode is HG 1,0. [Media 1]

Fig. 4.
Fig. 4.

Intensity (up row) and phase (low row) for different angles of the GT of HG 1,0 affected by scaling factors sx = s = s -1 y : s = 1/2 (a, b, c) and s = 2 (d, e ,f), respectively.

Fig. 5.
Fig. 5.

Intensity (up row) and phase (low row) of the GT (for the angle α) of HG 1,0 mode shifted by vt = (1mm,0) (a, b, c) and vt = (1mm,-1mm) (d, e, f).

Fig. 6.
Fig. 6.

Intensity (up row) and phase (low row) distributions for GT of the HG 3,0+HG 0,3 input mode are displayed for different angles α = 0 (input mode),π/8, π/5, 2π/9, π/4 (LG + 0,3+LG - 0,3mode). (2.8 MB) Movie: mode transformation for different angles α, where the input mode is HG 3,0 + HG 0,3. [Media 2]

Tables (1)

Tables Icon

Table 1. Selected functions and their gyrator transforms

Equations (57)

Equations on this page are rendered with MathJax. Learn more.

f o ( r o ) = R α [ f i ( r i ) ] ( r o ) = f i ( x i , y i ) K α ( x i , y i , x o , y o ) dx i dy i
= 1 sin α f i ( x i , y i ) exp ( i 2 π ( x o y o + x i y i ) cos α ( x i y o + x o y i ) sin α ) dx i dy i ,
r o q o = X Y Y X r i q i = T ( α ) r i q i ,
X = cos α 0 0 cos α , Y = 0 sin α sin α 0 ,
R α [ f i ( x i , y i ) ] ( x o , y o ) = R α [ f i ( x i , y i ) ] ( x o , y o ) ,
R α [ f i ( r i ) ] ( r o ) ( R α [ g i ( r e ) ] ( r o ) ) * d r o = f i ( r i ) exp ( i 2 π ( x o y o + x i y i ) cos α ( x i y o + x o y i ) sin α ) d r i
× 1 sin 2 α g i * ( r e ) exp ( i 2 π ( x o y o + x e y e ) cos α + ( x e y o + x o y e ) sin α ) d r e d r o
= 1 sin 2 α f i ( r i ) g i * ( r e ) exp ( i 2 π ( x i y i x e y e ) cos α sin α )
× exp ( i 2 π ( x i y e ) y o + x o ( y i y e ) sin α ) d r o d r i d r e
= f i ( r i ) g i * ( r e ) exp ( i 2 π ( x i y i x e y e ) cos α sin α ) δ ( r i r e ) d r i d r e
= f i ( r i ) g i ( r i ) * d r i .
R α [ f i ( r i v ) ] ( r o ) = exp ( ( v x v y sin 2 α 2 r o t v ˜ sin α ) ) R α [ f i ( r i ) ] ( r o v cos α )
R α [ f i ( r i ) exp ( i 2 π k t r i ) ] ( r o ) = exp ( ( k x k y sin 2 α 2 k t r o cos α ) ) R α [ f i ( r i ) ] ( r o + k ˜ sin α ) ,
R α [ f i ( Sr i ) ] ( r o ) = σ β cos β σ α cos α exp ( i 2 π x o y o ( 1 ( cos β cos α ) 2 ) cot α ) R β [ f i ( r i ) ] ( cos β cos α Sr o ) ,
S = s x 0 0 s y and cot β = cot α s x s y .
R α [ f i ( x i s , y i s 1 ) ( r o ) = R α [ f i ( r i ) ] ( x o s , y o s 1 ) ,
R α [ f i ( x i s , y i s 1 ) ] ( r o ) = R π α [ f i ( r i ) ] ( x o s , s 1 y o ) ,
R α [ f i ( x i s , y i s 1 ) ] ( r o ) = R α [ f i ( r i ) ] ( x o s , s 1 y o ) .
R α [ f i ( x i , y i ) ] ( r o ) = R α [ f i ( r i ) ] ( x o , y o ) .
HG m , n ( r ; w ) = 2 1 2 H m ( 2 π x w ) H n ( 2 π y w ) 2 m m ! w 2 n n ! w exp ( π w 2 r 2 ) ,
K ± π 4 ( x i , y i , x o , y o ) = 2 exp ( ± i 2 π [ x o y o + x i y i 2 ( x i y o + x o y i ) ] ) .
LG p , l ± ( r ; w ) = w 1 min ( m , n ) ! max ( m , n ) ! ( 2 π ( x w ± i y w ) ) l L p l ( 2 π w 2 r 2 ) exp ( π w 2 r 2 ) ,
f i ( r i ) = n , m a n , m exp ( i 2 π ( x i k x n + y i k y m ) ) .
f o ( r o ) = R α [ f i ( r i ) ] ( r o ) = n , m a n , m R α [ exp ( i 2 π ( x i k x n + y i k y m ) ) ] ( r o ) .
f o ( r o ) = exp ( i 2 π x o y o tan α ) sin α n , m a n , m R α exp ( i 2 π nm k x k y tan α ) exp ( i 2 π n k x x o + m k y y o cos α ) .
f o ( r o ) = exp ( i 2 π x o y o tan α l ) sin α l f i ( 1 cos α l r o ) ,
f o ( r o ) = R α [ f i ( r i ) ] ( r o ) = 1 sin α f i ( r i ) exp ( ( r i t y 1 Xr i 2 r i t Y 1 r o + r o t XY 1 r o ) ) d r i .
f o ( r o ) = 1 sin α f i ( r i v ) exp ( ( r i t Y 1 Xr i 2 r i t Y 1 r o + r o t XY 1 r o ) ) d r i
= 1 sin α f i ( u ) exp ( ( ( u + v ) t Y 1 X ( u + v ) 2 ( u + v ) t Y 1 r o + r o t XY 1 r o ) ) d u
= 1 sin α f i ( u ) exp ( iπϕ ) d u .
exp ( iπϕ ) = exp ( ( v t YXv + r o t [ ( X Y 1 X ( Y 1 ) t Y ] v ) ) ×
exp ( ( r i t Y 1 Xr i 2 r i t Y 1 ( r o Xv ) + ( r o Xv ) t XY 1 ( r o Xv ) ) ) ,
v t Y 1 Xu = u t ( Y 1 X ) t v = u t X t ( Y 1 ) t v ,
XY t = YX t ,
X t X + Y t Y = I .
f o ( r o ) = R α [ f i ( r i v ) ] ( r o )
= exp ( ( v t YXv + r o t [ ( XY 1 X ( Y 1 ) t Y ] v ) ) R α [ f i ( r i ) ] ( r o Xv )
= exp ( ( v x v y sin 2 α 2 r o v ˜ sin α ) ) R α [ f i ( r i ) ] ( r o v cos α ) ,
f o ( r o ) = R α [ f i ( Sr i ) ] ( r o )
= exp ( i 2 π x o y o cot α ) s x s y sin α f i ( x i y i ) exp ( i 2 π ( x i y i cot α s x s y 1 sin α ( x i y o s x + x o y i s y ) ) ) d x i d y i .
f o ( r o ) = exp ( i 2 π x o y o cot α ) s x s y sin α f i ( x i , y i ) exp ( i 2 π ( x i y i cot β 1 sin α ( x i y o s x + x o y i s y ) ) ) d x i dy i
= exp ( i 2 π x o y o cot α ) sin β s x s y sin α exp ( i 2 π cot β y o x o sin 2 β s x s y sin 2 α ) R β [ f i ( r i ) ] x o sin β s y sin α y o sin β s x sin α
= σ β cos β σ α cos α exp ( i 2 π x o y o cot α ( 1 ( cos β cos α ) 2 ) ) R β [ f i ( r i ) ] ( cos β cos α sr i ) ,
f o ( r o ) = R α [ exp ( i 2 π cx i y i ) ( r o ) ]
= exp ( i 2 π x o y o cot α ) sin α exp ( i 2 π ( x i y i ( c + cot α ) 1 sin α ( x i y o + x o y i ) ) ) d x i d y i
= exp ( i 2 π x o y o cot α ) sin α exp ( i 2 π x o y i sin α ) d y i exp ( i 2 π x i ( y i ( c + cot α ) y o sin α ) ) d x i
= exp ( i 2 π x o y o cot α ) sin α exp ( i 2 π x o y i sin α ) δ ( y i ( c + cot α ) y o sin α ) d y i
= exp ( i 2 π x o y o cot α ) sin α exp ( i 2 π x o y o ( c + cot α ) sin 2 α ) ,
δ ( v ) = exp ( i 2 πvx ) dx .
R α [ exp ( i 2 π cx i y i ) ( r o ) ] = 1 sin α exp ( i 2 π c cot α 1 c + cot α x o y o ) .
f 0 x o y o = 1 sin α exp ( i 2 π x 0 y 0 cot α ) g o x o y o ,
g o x o y o = exp ( γ r i 2 ) exp ( i 2 π ( x i y i cot α 1 sin α ( x i y o + x o y i ) ) ) dx i dy i
= exp ( x i 2 γ ) exp ( i 2 π x i y o sin α ) dx i exp ( y i 2 γ ) exp ( i 2 π ( x i cot α x 0 sin α ) y i ) dy i
= π γ exp ( x i 2 γ ) exp ( π 2 γ ( x i cot α x o sin α ) 2 ) exp ( i 2 π x i y o sin α ) dx i .
exp ( μx 2 + βx ) dx = π μ exp ( β 2 4 μ ) ,
g o x o y o = π 2 γ 2 d exp ( π 2 γd sin 2 α ( x o 2 + y o 2 ) ) exp ( i 2 π π 2 2 sin 2 α x o y o cot α ) ,
f o ( r o ) = exp ( i 2 π ( 1 1 cos 2 α + ( γ π ) 2 sin 2 α ) x o y o cot α ) cos 2 α + ( γ π ) 2 sin 2 α exp ( γ r 0 2 cos 2 α + ( γ π ) 2 sin 2 α ) .

Metrics