Abstract

Phase-space optics is used to relate the problem of designing diffractive optical elements for any first-order optical system to the corresponding design problem in the Fraunhofer diffraction regime. This, in particular, provides a novel approach for the fractional Fourier transform domain. For fractional Fourier transforms of arbitrary order, the diffractive element is determined as the optimum design computed for a generic Fourier transform system, scaled and modulated with a parabolic lens function. The phase-space description also identifies critical system parameters that limit the performance and applicability of this method. Numerical simulations of paraxial wave propagation are used to validate the method.

© 2006 Optical Society of America

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References

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2004 (2)

O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef]

M. G. Ertosun, H. Ath, M. Ozaktas, and B. Barshan, "Complex signal recovery from fractional Fourier transform intensities: order and noise dependence," Opt. Commun. 244, 61-70 (2004).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

1997 (1)

1996 (2)

D. F. V. James and G. S. Agarwal, "The generalized Fresnel transformation and its application to optics," Opt. Commun. 126, 207-212 (1996).
[CrossRef]

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

1994 (1)

1993 (1)

1991 (1)

1988 (1)

1980 (1)

J. R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-305 (1980).

1971 (1)

H. Dammann and K. Görtler, "High-efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Agarwal, G. S.

D. F. V. James and G. S. Agarwal, "The generalized Fresnel transformation and its application to optics," Opt. Commun. 126, 207-212 (1996).
[CrossRef]

Arrizón, V.

Ath, H.

M. G. Ertosun, H. Ath, M. Ozaktas, and B. Barshan, "Complex signal recovery from fractional Fourier transform intensities: order and noise dependence," Opt. Commun. 244, 61-70 (2004).
[CrossRef]

Barshan, B.

M. G. Ertosun, H. Ath, M. Ozaktas, and B. Barshan, "Complex signal recovery from fractional Fourier transform intensities: order and noise dependence," Opt. Commun. 244, 61-70 (2004).
[CrossRef]

Bastiaans, M.

M. Bastiaans, "Application of the Wigner distribution function in optics," in The Wigner Distribution--Theory and Applications in Signal Processing, W. Mecklenbräuker and F. Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.

Bernhardt, M.

Bryngdahl, O.

Dammann, H.

H. Dammann and K. Görtler, "High-efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Dong, B.-Z.

Dorsch, R. G.

Dragoman, D.

D. Dragoman, "The Wigner distribution function in optics and optoelectronics," in Progress in Optics XXXVII, E. Wolf, ed. (Elsevier, 1997), pp. 1-56.
[CrossRef]

Ertosun, M. G.

M. G. Ertosun, H. Ath, M. Ozaktas, and B. Barshan, "Complex signal recovery from fractional Fourier transform intensities: order and noise dependence," Opt. Commun. 244, 61-70 (2004).
[CrossRef]

Fienup, J. R.

J. R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-305 (1980).

Görtler, K.

H. Dammann and K. Görtler, "High-efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Gu, B.-Y.

Herzig, H.-P.

O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef]

Jahns, J.

James, D. F. V.

D. F. V. James and G. S. Agarwal, "The generalized Fresnel transformation and its application to optics," Opt. Commun. 126, 207-212 (1996).
[CrossRef]

Kettunen, V.

O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001), Chap. 3.4.4, pp. 104-107.

Lohmann, A. W.

Mait, J. N.

J. N. Mait, "Fourier array generators," in Micro-optics, H. P. Herzig, ed. (Taylor & Francis, 1997), pp. 293-323.

Mendlovic, D.

Ojeda-Castañeda, J.

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001), Chap. 3.4.4, pp. 104-107.

Ozaktas, M.

M. G. Ertosun, H. Ath, M. Ozaktas, and B. Barshan, "Complex signal recovery from fractional Fourier transform intensities: order and noise dependence," Opt. Commun. 244, 61-70 (2004).
[CrossRef]

Ripoll, O.

O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 1.4 and 3.2, pp. 26-37 and 92-100.

Sinzinger, S.

Stern, M.

M. Stern, "Binary optics fabrication," in Micro-optics, H. P. Herzig, ed. (Taylor & Francis, 1997), pp. 53-85.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 1.4 and 3.2, pp. 26-37 and 92-100.

Testorf, M.

Wyrowski, F.

Yang, G.-Z.

Zalevsky, Z.

Zhang, Y.

Appl. Opt. (2)

J. Opt. Soc. Am. A (6)

Opt. Commun. (3)

D. F. V. James and G. S. Agarwal, "The generalized Fresnel transformation and its application to optics," Opt. Commun. 126, 207-212 (1996).
[CrossRef]

H. Dammann and K. Görtler, "High-efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

M. G. Ertosun, H. Ath, M. Ozaktas, and B. Barshan, "Complex signal recovery from fractional Fourier transform intensities: order and noise dependence," Opt. Commun. 244, 61-70 (2004).
[CrossRef]

Opt. Eng. (2)

J. R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-305 (1980).

O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef]

Opt. Lett. (1)

Other (6)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001), Chap. 3.4.4, pp. 104-107.

M. Stern, "Binary optics fabrication," in Micro-optics, H. P. Herzig, ed. (Taylor & Francis, 1997), pp. 53-85.

J. N. Mait, "Fourier array generators," in Micro-optics, H. P. Herzig, ed. (Taylor & Francis, 1997), pp. 293-323.

M. Bastiaans, "Application of the Wigner distribution function in optics," in The Wigner Distribution--Theory and Applications in Signal Processing, W. Mecklenbräuker and F. Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.

D. Dragoman, "The Wigner distribution function in optics and optoelectronics," in Progress in Optics XXXVII, E. Wolf, ed. (Elsevier, 1997), pp. 1-56.
[CrossRef]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 1.4 and 3.2, pp. 26-37 and 92-100.

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Figures (6)

Fig. 1
Fig. 1

Setup for substituting arbitrary first-order optical system.

Fig. 2
Fig. 2

FRFT systems.

Fig. 3
Fig. 3

Equivalence of optical systems in phase space: (a) representation of the input signal, (b) FRFT of the input signal, (c) input signal modulated with lens function, (d) Fourier transform and scaling operation of the signal W Lens .

Fig. 4
Fig. 4

Phase-space diagram of the semianalytical design procedure: (a) representation of the numerically optimized Fourier transform design, (b) WDF of the signal in the Fraunhofer diffraction plane, (c) modulation of W opt ( x , ν ) with a quadratic phase function, (d) FRFT of W DOE ( x , ν ) .

Fig. 5
Fig. 5

Numerical evaluation of the design method. (a) Discrete Fourier transform of the phase-only design computed by the IFTA; (b)–(d) simulated intensity output of the optical system in Fig. 2 for different orders of the FRFT. The size of the output pattern is set to be the same for all orders P.

Fig. 6
Fig. 6

Simulated intensity output of the FRFT system for different orders P. The size of the DOE is chosen to be same for all orders P.

Equations (8)

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W ( x , ν ) = u ( x + x / 2 ) u * ( x x / 2 ) exp ( i 2 πν x ) d x ,
| u ( x ) | 2 = W ( x , ν ) .
( A B C D ) = ( 1 0 f 0 f 2 1 ) ( 0 f F f 0 f 0 f F 0 ) ( 1 0 f 0 f 1 1 ) = ( f F f 1 f F f 0 f F f 0 f 1 f 2 f 0 f F f F f 2 ) ,
ϕ = P π / 2.
Δ η L = Δ x ( P = 1 ) f 0 f 1 sin ( ϕ ) = N DOE δ x 0 cos ( ϕ ) ,
Δη ( P ) = λ f 0 δ x 0 sin ( ϕ ) .
λ f 0 δ x 0 N DOE ( 1 α ) > sin ( ϕ ) cos ( ϕ ) 1 2 .
λ f 0 δ x 0 N DOE ( 1 α ) > tan 1 ( ϕ ) .

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