Abstract

The fractional Fourier transform, which is a generalization of the classical Fourier transform, is introduced into an optical aperture synthesis (OAS) system by which imaging of an astronomical object can be achieved. We introduce fractional Fourier optical imaging and fractional Fourier-domain filtering (FFDF), and then present the schematic diagram of an OAS imaging system with FFDF. The modulation transfer function of an OAS system with FFDF is compared with that of an OAS system in the same condition. The result indicates that the OAS system with FFDF has larger practical cutoff frequency when the fill factor is smaller. Furthermore, the quality of imaging and restoration also demonstrates this conclusion.

© 2012 Optical Society of America

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  1. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  2. H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  3. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  4. Y. Z. Wang, Y. P. Huo, and Y. L. Du, “Recognition of images with small differences based on fractional correlation,” Chin. J. Lasers 1, 328–331 (2004).
  5. D. D. Dragoman and M. Dragoman, “Near and far optical beam characterization using fractional Fourier transform,” Opt. Commun. 141, 5–9 (1997).
    [CrossRef]
  6. Y. S. Bae, S. I. Jin, and H. S. Lee, “Crosstalk-noise of volume holographic memory using fractional Fourier transform,” Proc. SPIE 6352, 63523O (2006).
    [CrossRef]
  7. N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
    [CrossRef]
  8. H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  9. M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onaural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
    [CrossRef]
  10. L. Liu, Y. S. Jiang, H. Y. Wang, and Y. T. He, “Novel array configuration and its optimization for sparse aperture imaging systems,” Opt. Eng. 50, 053202 (2011).
    [CrossRef]
  11. Y. T. He, Y. S. Jiang, L. Liu, and C. W. Wang, “Passive interferometric array optimization based on redundant spacing calibration,” Opt. Express 17, 21598–21607 (2009).
    [CrossRef]
  12. A. H. Greenaway, “Self-calibrating dilute-aperture optics,” Proc. SPIE 1351, 738–748 (1990).
    [CrossRef]
  13. R. J. Eastwood, A. M. Johnson, and A. H. Greenaway, “Calculation and correction of piston phase aberration in synthesis imaging,” J. Opt. Soc. Am. A 26, 195–205 (2009).
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  14. D. Y. Wang, J. Han, H. C. Liu, S. Q. Tao, X. Y. Fu, and H. F. Guo, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46, 103201 (2007).
    [CrossRef]
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    [CrossRef]
  16. S. T. Liu, J. D. Xu, Y. Zhang, and C. F. Li, “Optical implementations of the fractional Fourier transform using lenses,” Acta Opt. Sin. 15, 1405–1408 (1995).
  17. F. J. Song, “Fractional Fourier transform and its optical implementation,” in Advanced Optical Information Processing, D. Zhai, ed. (Peking University, 1998), pp. 94–118.
  18. X. Yang, Q. Tan, and X. Wei, “Improved fast fractional-Fourier-transform algorithm,” J. Opt. Soc. Am. A 21, 1677–1681 (2004).
    [CrossRef]

2011

L. Liu, Y. S. Jiang, H. Y. Wang, and Y. T. He, “Novel array configuration and its optimization for sparse aperture imaging systems,” Opt. Eng. 50, 053202 (2011).
[CrossRef]

2009

2007

D. Y. Wang, J. Han, H. C. Liu, S. Q. Tao, X. Y. Fu, and H. F. Guo, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46, 103201 (2007).
[CrossRef]

2006

Y. S. Bae, S. I. Jin, and H. S. Lee, “Crosstalk-noise of volume holographic memory using fractional Fourier transform,” Proc. SPIE 6352, 63523O (2006).
[CrossRef]

2004

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[CrossRef]

Y. Z. Wang, Y. P. Huo, and Y. L. Du, “Recognition of images with small differences based on fractional correlation,” Chin. J. Lasers 1, 328–331 (2004).

X. Yang, Q. Tan, and X. Wei, “Improved fast fractional-Fourier-transform algorithm,” J. Opt. Soc. Am. A 21, 1677–1681 (2004).
[CrossRef]

1997

D. D. Dragoman and M. Dragoman, “Near and far optical beam characterization using fractional Fourier transform,” Opt. Commun. 141, 5–9 (1997).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onaural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

1995

S. T. Liu, J. D. Xu, Y. Zhang, and C. F. Li, “Optical implementations of the fractional Fourier transform using lenses,” Acta Opt. Sin. 15, 1405–1408 (1995).

1994

1993

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

1990

A. H. Greenaway, “Self-calibrating dilute-aperture optics,” Proc. SPIE 1351, 738–748 (1990).
[CrossRef]

1980

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Arikan, O.

M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onaural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

Bae, Y. S.

Y. S. Bae, S. I. Jin, and H. S. Lee, “Crosstalk-noise of volume holographic memory using fractional Fourier transform,” Proc. SPIE 6352, 63523O (2006).
[CrossRef]

Barshan, B.

Dragoman, D. D.

D. D. Dragoman and M. Dragoman, “Near and far optical beam characterization using fractional Fourier transform,” Opt. Commun. 141, 5–9 (1997).
[CrossRef]

Dragoman, M.

D. D. Dragoman and M. Dragoman, “Near and far optical beam characterization using fractional Fourier transform,” Opt. Commun. 141, 5–9 (1997).
[CrossRef]

Du, Y. L.

Y. Z. Wang, Y. P. Huo, and Y. L. Du, “Recognition of images with small differences based on fractional correlation,” Chin. J. Lasers 1, 328–331 (2004).

Eastwood, R. J.

Fu, X. Y.

D. Y. Wang, J. Han, H. C. Liu, S. Q. Tao, X. Y. Fu, and H. F. Guo, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46, 103201 (2007).
[CrossRef]

Greenaway, A. H.

Guo, H. F.

D. Y. Wang, J. Han, H. C. Liu, S. Q. Tao, X. Y. Fu, and H. F. Guo, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46, 103201 (2007).
[CrossRef]

Han, J.

D. Y. Wang, J. Han, H. C. Liu, S. Q. Tao, X. Y. Fu, and H. F. Guo, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46, 103201 (2007).
[CrossRef]

He, Y. T.

L. Liu, Y. S. Jiang, H. Y. Wang, and Y. T. He, “Novel array configuration and its optimization for sparse aperture imaging systems,” Opt. Eng. 50, 053202 (2011).
[CrossRef]

Y. T. He, Y. S. Jiang, L. Liu, and C. W. Wang, “Passive interferometric array optimization based on redundant spacing calibration,” Opt. Express 17, 21598–21607 (2009).
[CrossRef]

Huo, Y. P.

Y. Z. Wang, Y. P. Huo, and Y. L. Du, “Recognition of images with small differences based on fractional correlation,” Chin. J. Lasers 1, 328–331 (2004).

Jiang, Y. S.

L. Liu, Y. S. Jiang, H. Y. Wang, and Y. T. He, “Novel array configuration and its optimization for sparse aperture imaging systems,” Opt. Eng. 50, 053202 (2011).
[CrossRef]

Y. T. He, Y. S. Jiang, L. Liu, and C. W. Wang, “Passive interferometric array optimization based on redundant spacing calibration,” Opt. Express 17, 21598–21607 (2009).
[CrossRef]

Jin, S. I.

Y. S. Bae, S. I. Jin, and H. S. Lee, “Crosstalk-noise of volume holographic memory using fractional Fourier transform,” Proc. SPIE 6352, 63523O (2006).
[CrossRef]

Johnson, A. M.

Joseph, J.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[CrossRef]

Kutay, M. A.

M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onaural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

Lee, H. S.

Y. S. Bae, S. I. Jin, and H. S. Lee, “Crosstalk-noise of volume holographic memory using fractional Fourier transform,” Proc. SPIE 6352, 63523O (2006).
[CrossRef]

Li, C. F.

S. T. Liu, J. D. Xu, Y. Zhang, and C. F. Li, “Optical implementations of the fractional Fourier transform using lenses,” Acta Opt. Sin. 15, 1405–1408 (1995).

Liu, H. C.

D. Y. Wang, J. Han, H. C. Liu, S. Q. Tao, X. Y. Fu, and H. F. Guo, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46, 103201 (2007).
[CrossRef]

Liu, L.

L. Liu, Y. S. Jiang, H. Y. Wang, and Y. T. He, “Novel array configuration and its optimization for sparse aperture imaging systems,” Opt. Eng. 50, 053202 (2011).
[CrossRef]

Y. T. He, Y. S. Jiang, L. Liu, and C. W. Wang, “Passive interferometric array optimization based on redundant spacing calibration,” Opt. Express 17, 21598–21607 (2009).
[CrossRef]

Liu, S. T.

S. T. Liu, J. D. Xu, Y. Zhang, and C. F. Li, “Optical implementations of the fractional Fourier transform using lenses,” Acta Opt. Sin. 15, 1405–1408 (1995).

Lohmann, A. W.

Mendlovic, D.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Nishchal, N. K.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[CrossRef]

Onaural, L.

M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onaural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

Onural, L.

Ozaktas, H. M.

M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onaural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

Qian, L.

X. F. Zhu, F. Wu, Q. Y. Wu, and L. Qian, “Image restoration for sparse aperture systems based on wavelet-Wiener algorithm,” Proc. SPIE 7513, 75131B (2009).
[CrossRef]

Singh, K.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[CrossRef]

Song, F. J.

F. J. Song, “Fractional Fourier transform and its optical implementation,” in Advanced Optical Information Processing, D. Zhai, ed. (Peking University, 1998), pp. 94–118.

Tan, Q.

Tao, S. Q.

D. Y. Wang, J. Han, H. C. Liu, S. Q. Tao, X. Y. Fu, and H. F. Guo, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46, 103201 (2007).
[CrossRef]

Wang, C. W.

Wang, D. Y.

D. Y. Wang, J. Han, H. C. Liu, S. Q. Tao, X. Y. Fu, and H. F. Guo, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46, 103201 (2007).
[CrossRef]

Wang, H. Y.

L. Liu, Y. S. Jiang, H. Y. Wang, and Y. T. He, “Novel array configuration and its optimization for sparse aperture imaging systems,” Opt. Eng. 50, 053202 (2011).
[CrossRef]

Wang, Y. Z.

Y. Z. Wang, Y. P. Huo, and Y. L. Du, “Recognition of images with small differences based on fractional correlation,” Chin. J. Lasers 1, 328–331 (2004).

Wei, X.

Wu, F.

X. F. Zhu, F. Wu, Q. Y. Wu, and L. Qian, “Image restoration for sparse aperture systems based on wavelet-Wiener algorithm,” Proc. SPIE 7513, 75131B (2009).
[CrossRef]

Wu, Q. Y.

X. F. Zhu, F. Wu, Q. Y. Wu, and L. Qian, “Image restoration for sparse aperture systems based on wavelet-Wiener algorithm,” Proc. SPIE 7513, 75131B (2009).
[CrossRef]

Xu, J. D.

S. T. Liu, J. D. Xu, Y. Zhang, and C. F. Li, “Optical implementations of the fractional Fourier transform using lenses,” Acta Opt. Sin. 15, 1405–1408 (1995).

Yang, X.

Zhang, Y.

S. T. Liu, J. D. Xu, Y. Zhang, and C. F. Li, “Optical implementations of the fractional Fourier transform using lenses,” Acta Opt. Sin. 15, 1405–1408 (1995).

Zhu, X. F.

X. F. Zhu, F. Wu, Q. Y. Wu, and L. Qian, “Image restoration for sparse aperture systems based on wavelet-Wiener algorithm,” Proc. SPIE 7513, 75131B (2009).
[CrossRef]

Acta Opt. Sin.

S. T. Liu, J. D. Xu, Y. Zhang, and C. F. Li, “Optical implementations of the fractional Fourier transform using lenses,” Acta Opt. Sin. 15, 1405–1408 (1995).

Chin. J. Lasers

Y. Z. Wang, Y. P. Huo, and Y. L. Du, “Recognition of images with small differences based on fractional correlation,” Chin. J. Lasers 1, 328–331 (2004).

IEEE Trans. Signal Process.

M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onaural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

J. Inst. Math. Appl.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. D. Dragoman and M. Dragoman, “Near and far optical beam characterization using fractional Fourier transform,” Opt. Commun. 141, 5–9 (1997).
[CrossRef]

Opt. Eng.

L. Liu, Y. S. Jiang, H. Y. Wang, and Y. T. He, “Novel array configuration and its optimization for sparse aperture imaging systems,” Opt. Eng. 50, 053202 (2011).
[CrossRef]

D. Y. Wang, J. Han, H. C. Liu, S. Q. Tao, X. Y. Fu, and H. F. Guo, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46, 103201 (2007).
[CrossRef]

Opt. Express

Proc. SPIE

A. H. Greenaway, “Self-calibrating dilute-aperture optics,” Proc. SPIE 1351, 738–748 (1990).
[CrossRef]

X. F. Zhu, F. Wu, Q. Y. Wu, and L. Qian, “Image restoration for sparse aperture systems based on wavelet-Wiener algorithm,” Proc. SPIE 7513, 75131B (2009).
[CrossRef]

Y. S. Bae, S. I. Jin, and H. S. Lee, “Crosstalk-noise of volume holographic memory using fractional Fourier transform,” Proc. SPIE 6352, 63523O (2006).
[CrossRef]

Other

F. J. Song, “Fractional Fourier transform and its optical implementation,” in Advanced Optical Information Processing, D. Zhai, ed. (Peking University, 1998), pp. 94–118.

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Figures (10)

Fig. 1.
Fig. 1.

Lohmann I single-lens system.

Fig. 2.
Fig. 2.

Fractional Fourier optical imaging system.

Fig. 3.
Fig. 3.

Fractional Fourier optical imaging system with FFDF.

Fig. 4.
Fig. 4.

Schematic diagram of the OAS imaging system with FFDF.

Fig. 5.
Fig. 5.

Array configurations. (a) UC-6 array; (b) Golay-6 array.

Fig. 6.
Fig. 6.

MTF curves of six kinds of filter radius, b/a=0.3, 0.5, 0.7, 0.9, 1.1, and 1.3, with F=0.1 of the UC-6 array along the u axis (left) and the v axis (right).

Fig. 7.
Fig. 7.

MTF curves of six kinds of filter diameter, b/a=0.3, 0.5, 0.7, 0.9, 1.1, and 1.3, with F=0.1 of the Golay-6 array along the u axis (left) and the v axis (right).

Fig. 8.
Fig. 8.

MTF (top view) with different F=0.1, 0.2, 0.3, and 0.4 of the UC-6 array.

Fig. 9.
Fig. 9.

Image simulation and restoration of the UC-6 array based on two kinds of optical system (F=0.1). (a) object; (b) 0.9 order system (b/a=1); (c) OAS system.

Fig. 10.
Fig. 10.

Image simulation and restoration of the Golay-6 array based on two kinds of optical system (F=0.1). (d) 0.9 order OAS system (b/a=1); (e) OAS system.

Equations (13)

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

z=f1tan(θ/2),
Fp[f(s)]=f(s)exp[iπcotθλf1(s2+x2)]exp[i2πcscθλf1sx]ds,
Fα{Fβ[f(x,y)]}=Fα+β{f(x,y)},
{d1=f1tan(p1π/4)d2=f2tan(p2π/4),
{f1=fsin(p1π/2)f2=fsin(p2π/2).
{d1=2f[sin(p1π/4)]2d2=2f[sin(p2π/4)]2.
u(x1,y1)=Fp1{f(x0,y0)},
g(x2,y2)=Fp2{u(x1,y1)l(x1,y1)},
Ii(x4,y4)=Ig(x4,y4)PSF(x4,y4),
PSF(x4,y4)=|F{Fp2{Fp1{p(x1,y1)}l(x2,y2)}}|2,
MTF{x4,y4}=|F{PSF(x4,y4)}|.
Fp[f(s)]=exp[iπtan(θ2)x2]{[f(s)exp(iπtan(θ2)s2)][exp(iπcsc(θ)s2)]}.
Fp[f(s)]=exp[iπtan(θ2)x2]F1{F{f(s)exp[iπtan(θ2)]s2}F[exp(iπs2cscθ)]}.

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