Abstract

The gyrator transform (GT) promises to be a useful tool in image processing, holography, beam characterization, mode transformation, and quantum information. We introduce what we believe to be the first flexible optical experimental setup that performs the GT for a wide range of transformation parameters. The feasibility of the proposed scheme is demonstrated on the gyrator transformation of Hermite–Gaussian modes. For certain parameters the output mode corresponds to the Laguerre–Gaussian one.

© 2007 Optical Society of America

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References

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  1. R. Simon and K. B. Wolf, "Structure of the set of paraxial optical systems," J. Opt. Soc. Am. A 17, 342-355 (2000).
    [CrossRef]
  2. K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).
  3. J. A. Rodrigo, T. Alieva, and M. L. Calvo, "Gyrator transform: properties and applications," Opt. Express 15, 2190-2203 (2007).
    [CrossRef] [PubMed]
  4. H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  5. J. A. Rodrigo, T. Alieva, and M. L. Calvo, "Gyrator transform for image processing," Opt. Commun. , in press, doi: 10.1016/j.optcom.2007.06.023.
  6. G. F. Calvo, "Wigner representation and geometric transformations of optical orbital angular momentum spatial modes," Opt. Lett. 30, 1207-1209 (2005).
    [CrossRef] [PubMed]
  7. T. Alieva and M. Bastiaans, "Orthonormal mode sets for the two-dimensional fractional Fourier transformation," Opt. Lett. 32, 1226-1228 (2007).
    [CrossRef] [PubMed]
  8. E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
    [CrossRef]
  9. J. A. Rodrigo, T. Alieva, and M. L. Calvo, "Optical system design for ortho-symplectic transformations in phase space," J. Opt. Soc. Am. A 23, 2494-2500 (2006).
    [CrossRef]
  10. J. Shamir, "Cylindrical lens described by operator algebra," Appl. Opt. 18, 4195-4202 (1979).
    [CrossRef] [PubMed]
  11. G. Nemes and A. E. Seigman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am. A 11, 2257-2264 (1994).
    [CrossRef]
  12. V.A.Soifer, ed., Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

2007 (2)

2006 (1)

2005 (1)

2004 (1)

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

2000 (1)

1994 (1)

1979 (1)

Abramochkin, E. G.

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

Alieva, T.

Alper Kutay, M.

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

Bastiaans, M.

Calvo, G. F.

Calvo, M. L.

Nemes, G.

Ozaktas, H. M.

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

Rodrigo, J. A.

Seigman, A. E.

Shamir, J.

Simon, R.

Volostnikov, V. G.

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

Wolf, K. B.

Zalevsky, Z.

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

Appl. Opt. (1)

J. Opt. A, Pure Appl. Opt. (1)

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 (3)

Opt. Commun. (1)

J. A. Rodrigo, T. Alieva, and M. L. Calvo, "Gyrator transform for image processing," Opt. Commun. , in press, doi: 10.1016/j.optcom.2007.06.023.

Opt. Express (1)

Opt. Lett. (2)

Other (3)

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

V.A.Soifer, ed., Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

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

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Figures (7)

Fig. 1
Fig. 1

(a) Setup scheme associated with the gyrator transform. The phase modulation functions (for the case z = 0.5 m and λ = 532 nm ) associated with each generalized lens when the setup performs the GT at angle α = 3 π 4 are shown below, where the gray levels indicate the phase distribution range [ π , π ] . (b) An assembled set of two cylindrical lenses that forms the generalized lenses L 1 and L 2 .

Fig. 2
Fig. 2

Operation curves ϕ 1 ( α ) for the generalized lenses (a) L 1 and (b) L 2 . The operation curve ϕ 2 ( α ) is derived from the relation ϕ 2 = ( ϕ 1 + π 2 ) .

Fig. 3
Fig. 3

Experimental optical system configuration. Two SLMs (Holoeye LCR-2500) are used for the input signal (HG mode) generation. BS is a beam splitter. A pinhole (PH) is placed at the Fourier plane of the 4 - f system. The GT at angle α is performed by three generalized lenses. The parameter α is changed by the proper rotation of the cylindrical lenses, which form every generalized lens. The output signal is registered using a CCD camera (Sony XCD-X710).

Fig. 4
Fig. 4

Hermite–Gaussian mode HG 7 , 4 obatined using the proposed hybrid hologram. The image (a) corresponds to HG 7 , 4 intensity distribution and (b) is an interference pattern, which reveals the HG 7 , 4 phase distribution.

Fig. 5
Fig. 5

Input HG modes HG 8 , 6 , HG 3 , 2 , and HG 3 , 3 are transformed by the GT at α = 3 π 4 into LG modes LG 6 , 2 + , LG 2 , 1 + , and LG 3 , 0 , respectively. The interferrograms for these LG modes with a plane wave are displayed in (a)–(c), correspondingly. Images (a) and (b) reveal a forklike structure, typical for the associated helicoidal phase distribution, which is not present for the case of (c) LG 3 , 0 .

Fig. 6
Fig. 6

Intermediate modes obtained by the GT of the input mode HG 5 , 3 (a) for the angles α = [ 3 π 4 , π ] , (b)–(f). The first and second rows correspond to the intensity and phase distributions (gray levels indicate the phase range [ π , π ] ), respectively, obtained by numerical simulations of the GT. The experimental intensity distributions are shown in the third row. The input mode ( α = 0 ) generated by the SLMs is also displayed in the third row, (a).

Fig. 7
Fig. 7

Intermediate modes obtained by the GT of the HG modes composition HG 5 , 2 + HG 2 , 5 (a) for the angles α = [ 3 π 4 , π ] , (b)–(e). The first and second rows correspond to the intensity and phase distributions (gray levels indicate the phase range [ π , π ] ), respectively, obtained by numerical simulations of the GT. The experimental intensity distributions are shown in the third row. The input mode ( α = 0 ) generated by the SLMs is also displayed in the third row, (a).

Equations (9)

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g o ( r o ) = R α [ g i ( r i ) ] ( r o ) = 1 sin α g 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 ) d x i d y i ,
g o ( x o , y o ) = exp ( i π x o 2 + y o 2 2 x o y o sin 2 φ λ f ) g i ( x o , y o ) ,
g 1 ( x 1 , y 1 ) = 1 i λ z exp ( i π x 1 2 + y 1 2 λ z ) g i ( x i , y i ) exp ( i 2 π x 1 x i + y 1 y i x i y i sin 2 φ 1 λ z ) d x i d y i .
g 2 ( x 2 , y 2 ) = 1 2 λ z sin 2 φ 2 exp ( i π x 2 2 + y 2 2 λ z ) g i ( x i , y i ) exp ( i π 2 x i y i sin 2 φ 1 ( x 2 y i + x i y 2 + x i y i + x 2 y 2 ) sin 2 φ 2 λ z ) d x i d y i .
g o ( x o , y o ) = 1 2 λ z sin 2 φ 2 g i ( x i , y i ) exp ( i 2 π ( x o y o + x i y i ) ( 2 sin 2 φ 1 sin 2 φ 2 1 ) ( x o y i + x i y o ) 2 λ z sin 2 φ 2 ) d x i d y i .
sin 2 φ 1 = cot ( α 2 ) ,
sin 2 φ 2 = ( sin α ) 2 .
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 ) ,
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 ) ,

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