Abstract

We propose a simple approach for the phase space tomography reconstruction of the Wigner distribution of paraxial optical beams separable in Cartesian coordinates. It is based on the measurements of the antisymmetric fractional Fourier transform power spectra, which can be taken using a flexible optical setup consisting of four cylindrical lenses. The numerical simulations and the experimental results clearly demonstrate the feasibility of the proposed scheme.

© 2009 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 42, 1932-1937 (2003).
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    [CrossRef] [PubMed]
  10. T. Alieva, M. Bastiaans, and L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112-123 (2003).
    [CrossRef]
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    [CrossRef]
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  13. C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNully, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A 22, 1691-1699 (2005).
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    [CrossRef]
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    [CrossRef]
  16. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).
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  22. J. C. Wood and D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” IEEE Trans. Signal Process. 42, 3166-3177 (1994).
    [CrossRef]
  23. S. R. Deans, “Radon and Abel transforms,” in The Transforms and Applications Handbook, A.D.Poularikas, ed. (CRC Press, 1999), pp. 8.1-8.95.

2008

2007

A. V. Gitin, “Optical systems for measuring the Wigner function of a laser beam by the method of phase-spacial tomography,” Quantum Electron. 37, 85-91 (2007).
[CrossRef]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A 24, 3135-3139 (2007).
[CrossRef]

2006

2005

2004

2003

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 42, 1932-1937 (2003).

D. Dragoman, “Redundancy of phase-space distribution functions in complex field recovery problems,” Appl. Opt. 42, 1932-1937 (2003).
[CrossRef] [PubMed]

T. Alieva, M. Bastiaans, and L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112-123 (2003).
[CrossRef]

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

1997

1996

1995

1994

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

J. C. Wood and D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” IEEE Trans. Signal Process. 42, 3166-3177 (1994).
[CrossRef]

1993

1992

K. A. Nugent, “Wave field determination using 3-dimensional intensity information,” Phys. Rev. Lett. 68, 2261-2264 (1992).
[CrossRef] [PubMed]

1986

1983

1978

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237-246 (1972).

Alieva, T.

Andrés, P.

Barry, D. T.

J. C. Wood and D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” IEEE Trans. Signal Process. 42, 3166-3177 (1994).
[CrossRef]

Bastiaans, M.

T. Alieva, M. Bastiaans, and L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112-123 (2003).
[CrossRef]

Bastiaans, M. J.

Beck, M.

Brenner, K. H.

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

Calvo, M. L.

Clarke, L.

Deans, S. R.

S. R. Deans, “Radon and Abel transforms,” in The Transforms and Applications Handbook, A.D.Poularikas, ed. (CRC Press, 1999), pp. 8.1-8.95.

Dorsch, R. G.

Dragoman, D.

Fienup, J. R.

Furlan, W. D.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237-246 (1972).

Gitin, A. V.

A. V. Gitin, “Optical systems for measuring the Wigner function of a laser beam by the method of phase-spacial tomography,” Quantum Electron. 37, 85-91 (2007).
[CrossRef]

Gopinathan, U.

Granieri, S.

Kutay, M. A.

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

Liu, X.

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

Lohmann, A.

Mayer, M.

McAlister, D. F.

McNully, I.

Mendlovic, D.

Naughton, T. J.

Nugent, K. A.

Ozaktas, H. M.

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

Paterson, D.

Peele, A. G.

Raymer, M. G.

Roberts, A.

Rodrigo, J. A.

Saavedra, G.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237-246 (1972).

Semichaevsky, A.

Sheridan, J. T.

Situ, G.

Stankovic, L.

T. Alieva, M. Bastiaans, and L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112-123 (2003).
[CrossRef]

Tamura, S.

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 42, 1932-1937 (2003).

Teague, M. R.

Testorf, M.

Tran, C. Q.

Tu, J.

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 42, 1932-1937 (2003).

Wood, J. C.

J. C. Wood and D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” IEEE Trans. Signal Process. 42, 3166-3177 (1994).
[CrossRef]

Zalevsky, Z.

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842-844 (1996).
[CrossRef] [PubMed]

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

Appl. Opt.

IEEE Trans. Signal Process.

T. Alieva, M. Bastiaans, and L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112-123 (2003).
[CrossRef]

J. C. Wood and D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” IEEE Trans. Signal Process. 42, 3166-3177 (1994).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19-30 (2003).
[CrossRef]

Opt. Lett.

Optik

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237-246 (1972).

Phys. Rev. E

J. Tu and S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 42, 1932-1937 (2003).

Phys. Rev. Lett.

K. A. Nugent, “Wave field determination using 3-dimensional intensity information,” Phys. Rev. Lett. 68, 2261-2264 (1992).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137-1140 (1994).
[CrossRef] [PubMed]

Quantum Electron.

A. V. Gitin, “Optical systems for measuring the Wigner function of a laser beam by the method of phase-spacial tomography,” Quantum Electron. 37, 85-91 (2007).
[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).

S. R. Deans, “Radon and Abel transforms,” in The Transforms and Applications Handbook, A.D.Poularikas, ed. (CRC Press, 1999), pp. 8.1-8.95.

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Figures (7)

Fig. 1
Fig. 1

Scheme of the optical system for the antisymmetric FRFT power spectra measurements. The parameter α is varied by rotating the cylindrical lens while the free-space intervals z are fixed. The FRFT power spectra are registered by a CCD camera (WXGA, pixel size 4.6 μ m ).

Fig. 2
Fig. 2

(a) Intensity distribution and (b) phase of the H 43 ( x , y ) field. In our case a curvature R = 2 m and a scale w = 0.75 mm have been used.

Fig. 3
Fig. 3

WD of the H 4 ( x ) mode reconstructed from the simulated (a) N = 45 and (b) N = 180 projections. (c) Theoretically calculated WD .

Fig. 4
Fig. 4

WD of the H 3 ( y ) mode reconstructed from the simulated (a) N = 45 and (b) N = 180 projections. (c) Theoretically calculated WD.

Fig. 5
Fig. 5

Radon–Wigner maps for N = 45 projections obtained experimentally for (a) H 4 ( x ) and (b) H 3 ( y ) and by simulation for the same modes, (c) and (d), respectively.

Fig. 6
Fig. 6

WD of the H 4 ( x ) mode reconstructed from N = 45 (a) experimental and (b) simulated FRFT power spectra.

Fig. 7
Fig. 7

WD of the H 3 ( y ) mode reconstructed from N = 45 (a) experimental and (b) simulated FRFT power spectra.

Equations (14)

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F α x , α y ( r o ) = f ( r i ) K α x , α y ( r i , r o ) d r i ,
K α q ( q i , q o ) = ( i s sin α q ) 1 2 exp [ i π ( q i 2 + q o 2 ) cos α q 2 q i q o s sin α q ]
S f α x , α y ( r o ) = f ( r i ) K α x , α y ( r i , r o ) d r i 2 .
S f α x , α y ( r o ) = Γ ( r i , r i ) K α x , α y ( r i , r o ) [ K α x , α y ( r i , r o ) ] * d r i d r i ,
S f α x , α y ( r o ) = Γ x ( x i , x i ) K α x ( x i , x o ) K α x ( x i , x o ) d x i d x i Γ y ( y i , y i ) K α y ( y i , y o ) K α y ( y i , y o ) d y i d y i = S f x α x ( x o ) S f y α y ( y o ) .
S f y α ( y o ) d y o = S f y β ( y o ) d y o = A y ,
S f α x , α y ( r o ) d y o = A y S f x α x ( x o )
S f α x , α y ( r ) d y S f α x , α y ( r ) d x = S f α x , α y ( r ) S f α x , α y ( r ) d x d y .
W f ( r , p ) = C Γ ( r + r 2 , r r 2 ) exp ( i 2 π p r ) d r ,
S f 0 , 0 ( r ) = C Γ ( r , r ) = W f ( r , p ) d p ,
W f x ( x , p x ) W f y ( y , p y ) d y d p y = A y W f x ( x , p x ) .
S f q α q ( q ) = W f q ( q cos α q s p q sin α q , s 1 q sin α q + p q cos α q ) d p q ,
H 43 ( x , y ) = H 4 ( x ) H 3 ( y ) ,
H n ( q ) = h ( 2 n n ! ) 1 2 H n ( 2 π q w ) exp [ π ( q w ) 2 ] exp [ i π q 2 λ R ] ,

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