Abstract

We show how to explicitly determine the space-frequency window (phase-space window) for optical systems consisting of an arbitrary sequence of lenses and apertures separated by arbitrary lengths of free space. If the space-frequency support of a signal lies completely within this window, the signal passes without information loss. When it does not, the parts that lie within the window pass and the parts that lie outside of the window are blocked, a result that is valid to a good degree of approximation for many systems of practical interest. Also, the maximum number of degrees of freedom that can pass through the system is given by the area of its space-frequency window. These intuitive results provide insight and guidance into the behavior and design of systems involving multiple apertures and can help minimize information loss.

© 2013 Optical Society of America

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2012 (2)

2011 (2)

2010 (2)

2009 (2)

F. S. Oktem, and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett 16, 727–730 (2009).
[CrossRef]

Z. Zalevsky, V. Mico, and J. Garcia, “Nanophotonics for optical super resolution from an information theoretical perspective: a review,” J. Nanophoton. 3, 032502 (2009).
[CrossRef]

2008 (5)

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

J. Healy and J. Sheridan, “Bandwidth, compact support, apertures and the linear canonical transform in ABCD systems,” Proc. SPIE 6994, 69940W (2008).
[CrossRef]

M. Testorf and A. Lohmann, “Holography in phase space,” Appl. Opt. 47, A70–A77 (2008).
[CrossRef]

J. J. Healy, and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228–230 (2008).
[CrossRef]

A. Stern, “Uncertainty principles in linear canonical transform domains and some of their implications in optics,” J. Opt. Soc. Am. A 25, 647–652 (2008).
[CrossRef]

2007 (4)

2006 (4)

2005 (3)

2004 (2)

A. Stern, and B. Javidi, “Shannon number and information capacity of three-dimensional integral imaging,” J. Opt. Soc. Am. A 21, 1602–1612 (2004).
[CrossRef]

Z. Zalevsky, N. Shamir, and D. Mendlovic, “Geometrical superresolution in infrared sensor: experimental verification,” Opt. Eng. 43, 1401–1406 (2004).
[CrossRef]

2002 (1)

2000 (3)

1998 (1)

1997 (4)

D. Mendlovic, and A. Lohmann, “Space-bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14, 558–562 (1997).
[CrossRef]

D. Mendlovic, A. Lohmann, and Z. Zalevsky, “Space-bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
[CrossRef]

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

1996 (1)

1995 (1)

H. M. Ozaktas and O. Aytur, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

1994 (1)

1989 (1)

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

1985 (1)

1982 (1)

1981 (1)

1975 (1)

1974 (1)

1973 (1)

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

1971 (1)

1969 (1)

1955 (1)

Alieva, T.

Ark, S.

Asundi, A.

Aytur, O.

H. M. Ozaktas and O. Aytur, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

Barakat, R.

Barshan, B.

Bastiaans, M. J.

Bryanston-Cross, P.

Calvo, M.

Calvo, M. L.

Candan, C.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, M. A. Kutay, and C. Candan, “Fractional Fourier transform,” in Transforms and Applications Handbook, A. D. Poularikas, ed. (CRC, 2010), pp. 14-1–14-28.

Catrysse, P.

Claus, D.

Cohen, L.

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

L. Cohen, Integral Time-Frequency Analysis (Prentice-Hall, 1995).

Coskun, T.

di Francia, G. Toraldo

Dorsch, R. G.

Erden, M. F.

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

Ferreira, C.

Forbes, G.

Gabor, D.

D. Gabor, “Light and information,” in Progress In Optics, E. Wolf, ed. (Elsevier, 1961), Vol. I, Chap. 4, pp. 109–153.

Garcia, J.

Z. Zalevsky, V. Mico, and J. Garcia, “Nanophotonics for optical super resolution from an information theoretical perspective: a review,” J. Nanophoton. 3, 032502 (2009).
[CrossRef]

Gopinathan, U.

U. Gopinathan, G. Pedrini, B. Javidi, and W. Osten, “Lensless 3D digital holographic microscopic imaging at vacuum UV wavelength,” J. Disp. Technol. 6, 479–483 (2010).
[CrossRef]

Gori, F.

Guattari, G.

Guo, Z.

Healy, J.

J. Healy and J. Sheridan, “Bandwidth, compact support, apertures and the linear canonical transform in ABCD systems,” Proc. SPIE 6994, 69940W (2008).
[CrossRef]

Healy, J. J.

Hennelly, B.

Iliescu, D.

Javidi, B.

Koc, A.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Koç, A.

Kutay, M. A.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef]

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

H. M. Ozaktas, M. A. Kutay, and C. Candan, “Fractional Fourier transform,” in Transforms and Applications Handbook, A. D. Poularikas, ed. (CRC, 2010), pp. 14-1–14-28.

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

Lindberg, J.

J. Lindberg, “Mathematical concepts of optical superresolution,” J. Opt. 14, 083001 (2012).
[CrossRef]

Lohmann, A.

Lohmann, A. W.

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

A. W. Lohmann, “The space-bandwidth product, applied to spatial filtering and holography,” Research paper RJ-438 (IBM San Jose Research Laboratory, 1967).

A. W. Lohmann, Optical Information Processing, Lecture notes (Optik+Info, 1986).

Manko, V.

Maycock, J.

McDonald, J.

McElhinney, C.

Mendlovic, D.

Miao, J.

Mico, V.

Z. Zalevsky, V. Mico, and J. Garcia, “Nanophotonics for optical super resolution from an information theoretical perspective: a review,” J. Nanophoton. 3, 032502 (2009).
[CrossRef]

Miller, D. A. B.

Naughton, T.

Newsam, G.

Oktem, F. S.

F. S. Oktem, and H. M. Ozaktas, “Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space-bandwidth product,” J. Opt. Soc. Am. A 27, 1885–1895 (2010).
[CrossRef]

F. S. Oktem, and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett 16, 727–730 (2009).
[CrossRef]

F. S. Oktem, “Signal representation and recovery under partial information, redundancy, and generalized finite extent constraints,” Master’s thesis (Bilkent University, 2009).

Onural, L.

Osten, W.

U. Gopinathan, G. Pedrini, B. Javidi, and W. Osten, “Lensless 3D digital holographic microscopic imaging at vacuum UV wavelength,” J. Disp. Technol. 6, 479–483 (2010).
[CrossRef]

Ozaktas, H.

Ozaktas, H. M.

F. S. Oktem, and H. M. Ozaktas, “Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space-bandwidth product,” J. Opt. Soc. Am. A 27, 1885–1895 (2010).
[CrossRef]

F. S. Oktem, and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett 16, 727–730 (2009).
[CrossRef]

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef]

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

H. M. Ozaktas and O. Aytur, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[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, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

H. M. Ozaktas, M. A. Kutay, and C. Candan, “Fractional Fourier transform,” in Transforms and Applications Handbook, A. D. Poularikas, ed. (CRC, 2010), pp. 14-1–14-28.

Paolucci, S.

Pedrini, G.

U. Gopinathan, G. Pedrini, B. Javidi, and W. Osten, “Lensless 3D digital holographic microscopic imaging at vacuum UV wavelength,” J. Disp. Technol. 6, 479–483 (2010).
[CrossRef]

Peng, X.

Pierri, R.

Piestun, R.

Rodrigo, J.

Rodrigo, J. A.

Ronchi, L.

Sari, I.

Shamir, N.

Z. Zalevsky, N. Shamir, and D. Mendlovic, “Geometrical superresolution in infrared sensor: experimental verification,” Opt. Eng. 43, 1401–1406 (2004).
[CrossRef]

Sheridan, J.

J. Healy and J. Sheridan, “Bandwidth, compact support, apertures and the linear canonical transform in ABCD systems,” Proc. SPIE 6994, 69940W (2008).
[CrossRef]

B. Hennelly and J. Sheridan, “Optical encryption and the space bandwidth product,” Opt. Commun. 247, 291–305 (2005).
[CrossRef]

Sheridan, J. T.

Simon, R.

Solimene, R.

Starikov, A.

Stern, A.

Testorf, M.

Walker, B. H.

B. H. Walker, Optical Design for Visual Systems (SPIE, 2000).

Wandell, B.

Wolf, K.

Wolf, K. B.

K. B. Wolf, “Construction and properties of canonical transforms,” in Integral Transforms in Science and Engineering (Plenum, 1979), Chap. 9.

Xu, L.

Zalevsky, Z.

Z. Zalevsky, V. Mico, and J. Garcia, “Nanophotonics for optical super resolution from an information theoretical perspective: a review,” J. Nanophoton. 3, 032502 (2009).
[CrossRef]

Z. Zalevsky, N. Shamir, and D. Mendlovic, “Geometrical superresolution in infrared sensor: experimental verification,” Opt. Eng. 43, 1401–1406 (2004).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and A. Lohmann, “Understanding superresolution in wigner space,” J. Opt. Soc. Am. A 17, 2422–2430 (2000).
[CrossRef]

K. Wolf, D. Mendlovic, and Z. Zalevsky, “Generalized Wigner function for the analysis of superresolution systems,” Appl. Opt. 37, 4374–4379 (1998).
[CrossRef]

D. Mendlovic, A. Lohmann, and Z. Zalevsky, “Space-bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

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

Appl. Opt. (4)

IEEE Signal Process. Lett (1)

F. S. Oktem, and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett 16, 727–730 (2009).
[CrossRef]

IEEE Trans. Signal Process. (1)

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

J. Disp. Technol. (1)

U. Gopinathan, G. Pedrini, B. Javidi, and W. Osten, “Lensless 3D digital holographic microscopic imaging at vacuum UV wavelength,” J. Disp. Technol. 6, 479–483 (2010).
[CrossRef]

J. Nanophoton. (1)

Z. Zalevsky, V. Mico, and J. Garcia, “Nanophotonics for optical super resolution from an information theoretical perspective: a review,” J. Nanophoton. 3, 032502 (2009).
[CrossRef]

J. Opt. (1)

J. Lindberg, “Mathematical concepts of optical superresolution,” J. Opt. 14, 083001 (2012).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (15)

G. Forbes, V. Manko, H. Ozaktas, R. Simon, and K. Wolf, “Wigner distributions and phase space in optics,” J. Opt. Soc. Am. A 17, 2274 (2000).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and A. Lohmann, “Understanding superresolution in wigner space,” J. Opt. Soc. Am. A 17, 2422–2430 (2000).
[CrossRef]

P. Catrysse, and B. Wandell, “Optical efficiency of image sensor pixels,” J. Opt. Soc. Am. A 19, 1610–1620 (2002).
[CrossRef]

A. Stern, and B. Javidi, “Shannon number and information capacity of three-dimensional integral imaging,” J. Opt. Soc. Am. A 21, 1602–1612 (2004).
[CrossRef]

R. Piestun and D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A 17, 892–902 (2000).
[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]

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Figures (7)

Fig. 1.
Fig. 1.

Illustration of a space-frequency corridor.

Fig. 2.
Fig. 2.

Space-frequency window of a system with two apertures.

Fig. 3.
Fig. 3.

Space-frequency window of a system with four apertures.

Fig. 4.
Fig. 4.

(a) Apertured optical system with input plane at z=0 and output plane at z=2 m. The horizontal axis is in meters. The lens focal lengths fj in meters and the aperture sizes Δj in centimeters are given right above them. (b) and (c) Evolution of a(z) and M(z) as functions of z. λ=0.5μm and s=0.3mm [1,37].

Fig. 5.
Fig. 5.

Space-frequency window of the system at the input plane in the dimensionless (a) and dimensional (b) spaces.

Fig. 6.
Fig. 6.

Space-frequency window of the system at the output plane in the dimensionless (a) and dimensional (b) spaces.

Fig. 7.
Fig. 7.

(a) Signal support is wholly contained within the system window so there is no loss of information. (b) Part of the signal support lying within the system window will pass, and the parts lying outside will be blocked.

Equations (13)

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fT(x)(CTf)(x)CT(x,x)f(x)dx,CT(x,x)1Beiπ/4eiπ(DBx221Bxx+ABx2)
fa(x)(Faf)(x)Ka(x,x)f(x)dx,Ka(x,x)Aϕseiπ(cotϕs2x22cscϕs2xx+cotϕs2x2),Aϕ=1icotϕ,ϕ=aπ/2
Fa=[cos(aπ/2)s2sin(aπ/2)sin(aπ/2)s2cos(aπ/2)],
T=[ABCD]=[10qs21][M001M][cosϕs2sinϕsinϕs2cosϕ].
fT(x)=exp(iπqs2x2)1Mfa(xM).
a={2πarctan(1s2BA),ifA02πarctan(1s2BA)+2,ifA<0,
M=A2+(B/s2)2,
q={s2CA1s2B/AA2+(B/s2)2,ifA0s2DB,ifA=0.
Wf(x,σx)=f(x+x/2)f*(xx/2)ei2πσxxdx.
WfT(x,σx)=Wf(DxBσx,Cx+Aσx).
ΔxTjΔj,
ΔxajΔj/Mj,
Δxaj/sΔj/Mjs,

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