Abstract

We present a real time all optical super resolution method for exceeding the diffraction limit of an imaging system which has a circular aperture. The resolution improvement is obtained using two fixed circular gratings which are placed in predetermined positions. The circular gratings generate synthetic circular duplications of the aperture, thus they are the proper choice for a circular aperture optical system. The method is applicable for both spatially coherent and incoherent illuminations, as well as for white light illumination. The resolution improvement is achieved by limiting the object field of view. The proposed method is presented analytically, demonstrated via numerical simulations, and validated by laboratory experiments.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  39. E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system using three fixed generalized gratings: experimental results,” J. Opt. Soc. Am. A 18(3), 514 (2001).
    [Crossref]
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    [Crossref] [PubMed]
  43. L. Granero, V. Micó, Z. Zalevsky, and J. García, “Superresolution imaging method using phase-shifting digital lensless Fourier holography,” Opt. Express 17(17), 15008–15022 (2009).
    [Crossref] [PubMed]
  44. L. Granero, Z. Zalevsky, and V. Micó, “Single-exposure two-dimensional superresolution in digital holography using a vertical cavity surface-emitting laser source array,” Opt. Lett. 36(7), 1149–1151 (2011).
    [Crossref] [PubMed]
  45. Z. Zalevsky, E. Gur, J. Garcia, V. Micó, and B. Javidi, “Superresolved and field-of-view extended digital holography with particle encoding,” Opt. Lett. 37(13), 2766–2768 (2012).
    [Crossref] [PubMed]
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    [Crossref]
  47. H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta: Int. J. Opt. 24(4), 505–515 (1977).
    [Crossref]
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    [Crossref]

2015 (1)

2014 (1)

K. Wicker and R. Heintzmann, “Resolving a misconception about structured illumination,” Nat. Photonics 8(5), 342–344 (2014).
[Crossref]

2012 (2)

2011 (1)

2010 (1)

2009 (3)

2008 (4)

2007 (1)

2006 (1)

2005 (2)

A. Zlotnik, Z. Zalevsky, and E. Marom, “Superresolution with nonorthogonal polarization coding,” Appl. Opt. 44(18), 3705–3715 (2005).
[Crossref] [PubMed]

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102(37), 13081–13086 (2005).
[Crossref] [PubMed]

2003 (1)

2002 (2)

A. Shemer, Z. Zalevsky, D. Mendlovic, N. Konforti, and E. Marom, “Time multiplexing superresolution based on interference grating projection,” Appl. Opt. 41(35), 7397–7404 (2002).
[Crossref] [PubMed]

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81(17), 3143–3145 (2002).
[Crossref]

2001 (1)

2000 (2)

E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system with two fixed generalized Damman gratings,” Appl. Opt. 39(29), 5318–5325 (2000).
[Crossref] [PubMed]

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

1999 (3)

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Optical system with improved resolving power,” Prog. Opt. 40, 271–341 (1999).

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Super resolution optical systems for objects with finite sizes,” Opt. Commun. 163(1-3), 79–85 (1999).
[Crossref]

A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P. Garcia Martinez, “Superresolving optical system with time multiplexing and computer decoding,” Appl. Opt. 38(35), 7245–7251 (1999).
[Crossref] [PubMed]

1998 (1)

1997 (4)

1996 (1)

1992 (1)

1986 (1)

1982 (1)

H. Bartelt and A. W. Lohmann, “Optical processing of one-dimensional signals,” Opt. Commun. 42(2), 87–91 (1982).
[Crossref]

1977 (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta: Int. J. Opt. 24(4), 505–515 (1977).
[Crossref]

1971 (1)

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3(5), 312–315 (1971).
[Crossref]

1969 (1)

1967 (1)

1966 (3)

1964 (1)

1963 (1)

W. Gartner and A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–23 (1963).

1960 (1)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spektrosk. 9, 204–206 (1960).

1952 (1)

M. Françon, “Amélioration de résolution d’optique,” Nouvo Climento 9, 283–290 (1952).

1896 (1)

L. Rayleigh, “XV. On the theory of optical images, with special reference to the microscope,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 42(255), 167–195 (1896).
[Crossref]

1873 (1)

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Für Mikroskopische Anat. 9(1), 413–418 (1873).
[Crossref]

Abbe, E.

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Für Mikroskopische Anat. 9(1), 413–418 (1873).
[Crossref]

Bachl, A.

Bartelt, H.

H. Bartelt and A. W. Lohmann, “Optical processing of one-dimensional signals,” Opt. Commun. 42(2), 87–91 (1982).
[Crossref]

Bo, F.

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81(17), 3143–3145 (2002).
[Crossref]

Chi, K. R.

K. R. Chi, “Microscopy: ever-increasing resolution,” Nature 462(7273), 675–678 (2009).
[Crossref] [PubMed]

Chowdhury, S.

Cojoc, D.

Cox, I. J.

Dammann, H.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta: Int. J. Opt. 24(4), 505–515 (1977).
[Crossref]

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3(5), 312–315 (1971).
[Crossref]

De Nicola, S.

Dhalla, A.-H.

Di Francia, G. T.

Dorsch, R. G.

Farkas, D.

Ferraro, P.

Ferreira, C.

Finizio, A.

Françon, M.

M. Françon, “Amélioration de résolution d’optique,” Nouvo Climento 9, 283–290 (1952).

Garcia, J.

García, J.

Garcia Martinez, P.

García-Martínez, P.

Gartner, W.

W. Gartner and A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–23 (1963).

Görtler, K.

H. Dammann and K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3(5), 312–315 (1971).
[Crossref]

Granero, L.

Grilli, S.

Grimm, M. A.

Gur, A.

Gur, E.

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102(37), 13081–13086 (2005).
[Crossref] [PubMed]

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

Heintzmann, R.

K. Wicker and R. Heintzmann, “Resolving a misconception about structured illumination,” Nat. Photonics 8(5), 342–344 (2014).
[Crossref]

Ilovitsh, A.

Ilovitsh, T.

Izatt, J.

Javidi, B.

Jia, J.

Kartashev, A. I.

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spektrosk. 9, 204–206 (1960).

Kiryuschev, I.

Klotz, E.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta: Int. J. Opt. 24(4), 505–515 (1977).
[Crossref]

Konforti, N.

Leith, E. N.

Levanon, N.

Limon, O.

Liu, C.

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81(17), 3143–3145 (2002).
[Crossref]

Liu, H.

Liu, L.

Liu, Z.

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81(17), 3143–3145 (2002).
[Crossref]

Lohmann, A. W.

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Optical system with improved resolving power,” Prog. Opt. 40, 271–341 (1999).

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Super resolution optical systems for objects with finite sizes,” Opt. Commun. 163(1-3), 79–85 (1999).
[Crossref]

D. Mendlovic, D. Farkas, Z. Zalevsky, and A. W. Lohmann, “High-frequency enhancement by an optical system for superresolution of temporally restricted objects,” Opt. Lett. 23(10), 801–803 (1998).
[Crossref] [PubMed]

D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, and D. Farkas, “Two-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36(26), 6687 (1997).

D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, and D. Farkas, “Two-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36(26), 6687–6691 (1997).
[Crossref] [PubMed]

D. Mendlovic and A. W. Lohmann, “Space-bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14(3), 558–562 (1997).
[Crossref]

D. Mendlovic, A. W. Lohmann, and Z. Zalevsky, “Space–bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14(3), 563 (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(3), 470 (1996).
[Crossref]

H. Bartelt and A. W. Lohmann, “Optical processing of one-dimensional signals,” Opt. Commun. 42(2), 87–91 (1982).
[Crossref]

M. A. Grimm and A. W. Lohmann, “Superresolution image for one-dimensional objects,” J. Opt. Soc. Am. 56(9), 1151–1156 (1966).
[Crossref]

A. W. Lohmann and D. P. Paris, “Superresolution for nonbirefringent objects,” Appl. Opt. 3(9), 1037–1043 (1964).
[Crossref]

W. Gartner and A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–23 (1963).

Lukosz, W.

Marom, E.

Mendlovic, D.

A. Shemer, Z. Zalevsky, D. Mendlovic, N. Konforti, and E. Marom, “Time multiplexing superresolution based on interference grating projection,” Appl. Opt. 41(35), 7397–7404 (2002).
[Crossref] [PubMed]

E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system using three fixed generalized gratings: experimental results,” J. Opt. Soc. Am. A 18(3), 514 (2001).
[Crossref]

E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system with two fixed generalized Damman gratings,” Appl. Opt. 39(29), 5318–5325 (2000).
[Crossref] [PubMed]

A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P. Garcia Martinez, “Superresolving optical system with time multiplexing and computer decoding,” Appl. Opt. 38(35), 7245–7251 (1999).
[Crossref] [PubMed]

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Optical system with improved resolving power,” Prog. Opt. 40, 271–341 (1999).

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Super resolution optical systems for objects with finite sizes,” Opt. Commun. 163(1-3), 79–85 (1999).
[Crossref]

D. Mendlovic, D. Farkas, Z. Zalevsky, and A. W. Lohmann, “High-frequency enhancement by an optical system for superresolution of temporally restricted objects,” Opt. Lett. 23(10), 801–803 (1998).
[Crossref] [PubMed]

D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, and D. Farkas, “Two-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36(26), 6687 (1997).

D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, and D. Farkas, “Two-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36(26), 6687–6691 (1997).
[Crossref] [PubMed]

D. Mendlovic, A. W. Lohmann, and Z. Zalevsky, “Space–bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14(3), 563 (1997).
[Crossref]

D. Mendlovic and A. W. Lohmann, “Space-bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14(3), 558–562 (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(3), 470 (1996).
[Crossref]

Merola, F.

Mico, V.

Micó, V.

Paris, D. P.

Paturzo, M.

Preter, E.

Rayleigh, L.

L. Rayleigh, “XV. On the theory of optical images, with special reference to the microscope,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 42(255), 167–195 (1896).
[Crossref]

Sabo, E.

Shemer, A.

Sheppard, C. J. R.

Sun, P. C.

Sylman, D.

Wang, Y.

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81(17), 3143–3145 (2002).
[Crossref]

Wicker, K.

K. Wicker and R. Heintzmann, “Resolving a misconception about structured illumination,” Nat. Photonics 8(5), 342–344 (2014).
[Crossref]

Yuan, C.

Zalevsky, Z.

A. Ilovitsh, T. Ilovitsh, E. Preter, N. Levanon, and Z. Zalevsky, “Super-resolution using Barker-based array projected via spatial light modulator,” Opt. Lett. 40(8), 1802–1805 (2015).
[Crossref] [PubMed]

Z. Zalevsky, E. Gur, J. Garcia, V. Micó, and B. Javidi, “Superresolved and field-of-view extended digital holography with particle encoding,” Opt. Lett. 37(13), 2766–2768 (2012).
[Crossref] [PubMed]

L. Granero, Z. Zalevsky, and V. Micó, “Single-exposure two-dimensional superresolution in digital holography using a vertical cavity surface-emitting laser source array,” Opt. Lett. 36(7), 1149–1151 (2011).
[Crossref] [PubMed]

D. Sylman, V. Micó, J. García, and Z. Zalevsky, “Random angular coding for superresolved imaging,” Appl. Opt. 49(26), 4874–4882 (2010).
[Crossref] [PubMed]

L. Granero, V. Micó, Z. Zalevsky, and J. García, “Superresolution imaging method using phase-shifting digital lensless Fourier holography,” Opt. Express 17(17), 15008–15022 (2009).
[Crossref] [PubMed]

J. García, V. Micó, D. Cojoc, and Z. Zalevsky, “Full field of view super-resolution imaging based on two static gratings and white light illumination,” Appl. Opt. 47(17), 3080–3087 (2008).
[Crossref] [PubMed]

V. Mico, O. Limon, A. Gur, Z. Zalevsky, and J. García, “Transverse resolution improvement using rotating-grating time-multiplexing approach,” J. Opt. Soc. Am. A 25(5), 1115–1129 (2008).
[Crossref] [PubMed]

Z. Zalevsky, J. García, and V. Micó, “Transversal superresolution with noncontact axial movement of periodic structures,” J. Opt. Soc. Am. A 24(10), 3220–3225 (2007).
[Crossref] [PubMed]

Z. Zalevsky, P. García-Martínez, and J. García, “Superresolution using gray level coding,” Opt. Express 14(12), 5178–5182 (2006).
[Crossref] [PubMed]

A. Zlotnik, Z. Zalevsky, and E. Marom, “Superresolution with nonorthogonal polarization coding,” Appl. Opt. 44(18), 3705–3715 (2005).
[Crossref] [PubMed]

A. Shemer, Z. Zalevsky, D. Mendlovic, N. Konforti, and E. Marom, “Time multiplexing superresolution based on interference grating projection,” Appl. Opt. 41(35), 7397–7404 (2002).
[Crossref] [PubMed]

E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system using three fixed generalized gratings: experimental results,” J. Opt. Soc. Am. A 18(3), 514 (2001).
[Crossref]

E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system with two fixed generalized Damman gratings,” Appl. Opt. 39(29), 5318–5325 (2000).
[Crossref] [PubMed]

A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P. Garcia Martinez, “Superresolving optical system with time multiplexing and computer decoding,” Appl. Opt. 38(35), 7245–7251 (1999).
[Crossref] [PubMed]

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Optical system with improved resolving power,” Prog. Opt. 40, 271–341 (1999).

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Super resolution optical systems for objects with finite sizes,” Opt. Commun. 163(1-3), 79–85 (1999).
[Crossref]

D. Mendlovic, D. Farkas, Z. Zalevsky, and A. W. Lohmann, “High-frequency enhancement by an optical system for superresolution of temporally restricted objects,” Opt. Lett. 23(10), 801–803 (1998).
[Crossref] [PubMed]

D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, and D. Farkas, “Two-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36(26), 6687 (1997).

D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, and D. Farkas, “Two-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36(26), 6687–6691 (1997).
[Crossref] [PubMed]

D. Mendlovic, A. W. Lohmann, and Z. Zalevsky, “Space–bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14(3), 563 (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(3), 470 (1996).
[Crossref]

Zhai, H.

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Appl. Opt. (10)

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D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, and D. Farkas, “Two-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36(26), 6687 (1997).

D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, and D. Farkas, “Two-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36(26), 6687–6691 (1997).
[Crossref] [PubMed]

A. Shemer, Z. Zalevsky, D. Mendlovic, N. Konforti, and E. Marom, “Time multiplexing superresolution based on interference grating projection,” Appl. Opt. 41(35), 7397–7404 (2002).
[Crossref] [PubMed]

P. C. Sun and E. N. Leith, “Superresolution by spatial-temporal encoding methods,” Appl. Opt. 31(23), 4857–4862 (1992).
[Crossref] [PubMed]

A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P. Garcia Martinez, “Superresolving optical system with time multiplexing and computer decoding,” Appl. Opt. 38(35), 7245–7251 (1999).
[Crossref] [PubMed]

E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system with two fixed generalized Damman gratings,” Appl. Opt. 39(29), 5318–5325 (2000).
[Crossref] [PubMed]

A. Zlotnik, Z. Zalevsky, and E. Marom, “Superresolution with nonorthogonal polarization coding,” Appl. Opt. 44(18), 3705–3715 (2005).
[Crossref] [PubMed]

J. García, V. Micó, D. Cojoc, and Z. Zalevsky, “Full field of view super-resolution imaging based on two static gratings and white light illumination,” Appl. Opt. 47(17), 3080–3087 (2008).
[Crossref] [PubMed]

D. Sylman, V. Micó, J. García, and Z. Zalevsky, “Random angular coding for superresolved imaging,” Appl. Opt. 49(26), 4874–4882 (2010).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81(17), 3143–3145 (2002).
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K. Wicker and R. Heintzmann, “Resolving a misconception about structured illumination,” Nat. Photonics 8(5), 342–344 (2014).
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K. R. Chi, “Microscopy: ever-increasing resolution,” Nature 462(7273), 675–678 (2009).
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[Crossref]

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Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Optical system with improved resolving power,” Prog. Opt. 40, 271–341 (1999).

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Figures (9)

Fig. 1
Fig. 1 An illustration of the 4F optical system for the proposed SR approach, where two additional gratings are placed before the object and the image planes.
Fig. 2
Fig. 2 Circular synthetic aperture example for spatially coherent illumination. (a) CTF of the original 4F system, with circular cutoff frequency of D/2λF. (b) Synthetic aperture generated simultaneously for all angles by the circular gratings with a basic frequency of twice the cutoff frequency. (c) Final synthetic aperture, with circular cutoff frequency of 3D/2λF.
Fig. 3
Fig. 3 Circular synthetic aperture example for spatially incoherent illumination. (a) Incoherent transfer function of the original 4F system, with circular cutoff frequency of D/λF. (b) Synthetic aperture generated simultaneously for all angles by the circular gratings with a basic frequency υ0 which equals to the cutoff frequency. (c) Final synthetic aperture, with circular cutoff frequency of 2D/λF.
Fig. 4
Fig. 4 Circular synthetic aperture example for white light illumination. (a) OTF of the original 4F system for RGB wavelengths, with circular cutoff frequency of D/λR,G,BF. (b) Synthetic aperture generated simultaneously for all angles by the circular gratings with a basic frequency υ0 which equals to the cutoff frequency of the red wavelength. (c) Summation for each wavelength individually. (d) Final synthetic aperture, with circular cutoff frequency of D(1/λR + 1/λB)/F.
Fig. 5
Fig. 5 SR numerical simulation results for spatially coherent monochromatic illumination. (a) HR reference image. (b) LR image. (c) SR image, presented in Log scale. (d-f) x4 digital zooms of (a-c) respectively, presented in normal intensity scale.
Fig. 6
Fig. 6 SR numerical simulation results for white light illumination. (a) HR reference image. (b) LR image. (c) SR image, presented in Log scale. (d-f) x4 digital zoom of (a-c) respectively, presented in normal intensity scale.
Fig. 7
Fig. 7 (a) The experimental setup: A red laser, expanded × 10, provides illumination in the 4F system. Two circular gratings are placed into the setup, the 1st grating is between the beam expander and the object, and the 2nd grating is between the final lens and the camera. (b) The central part of the printed 22lp/mm circular grating.
Fig. 8
Fig. 8 SR experimental results for spatially coherent monochromatic illumination. (a) HR reference image. (b) LR image. (c) SR image, presented in Log scale. (d-f) x4 digital zooms of (a-c) respectively, presented in normal intensity scale.
Fig. 9
Fig. 9 SR experimental results for white light illumination. (a) HR reference image. (b) LR image. (c) SR image, presented in Log scale. (d-f) x4 digital zooms of (a-c) respectively, presented in normal intensity scale.

Equations (26)

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E o b j ( x , y , z = 0 ) = E ˜ 0 ( u , v ) e 2 π i ( x u + y v ) d u d v
E 0 ( x , y , z 0 ) = E ˜ 0 ( u , v ) e 2 π i [ x u + y v z 0 λ 1 λ 2 ( u 2 + v 2 )   ] d u d v
g ( r ) = m A m e 2 π i r m υ 0 = m A m e 2 π i r [ cos 2 ( θ ) + sin 2 ( θ ) ] m υ 0 = m A m e 2 π i [ x cos ( θ ) + y sin ( θ ) ] m υ 0
E 0 ( x , y , z 0 + ) = m A m E ˜ 0 ( u , v ) e 2 π i { x [ u + cos ( θ ) m υ 0 ] + y [ v + sin ( θ ) m υ 0 ] z 0 λ 1 λ 2 ( u 2 + v 2 )   } d u d v
E 0 ( x , y , z = 0 ) = m A m E ˜ 0 ( u , v ) e 2 π i { x [ u + cos ( θ ) m υ 0 ] + y [ v + sin ( θ ) m υ 0 ] z 0 λ ϕ 1 } d u d v
ϕ 1 = 1 λ 2 ( u 2 + v 2 )   1 λ 2 { [ u + cos ( θ ) m υ 0 ] 2 + [ v + sin ( θ ) m υ 0 ] 2 }  
E 0 ( μ , η , z = 2 F + ) = m A m E ˜ 0 [ μ λ f cos ( θ ) m υ 0 , η λ f sin ( θ ) m υ 0 ] e 2 π i ( z 0 λ ϕ 2 ) c i r c ( μ 2 + η 2 Δ ρ / 2 )
ϕ 2 = 1 λ 2 { [ μ λ F cos ( θ ) m υ 0 ] 2 + [ η λ F sin ( θ ) m υ 0 ] 2 }   1 λ 2 [ ( μ λ F ) 2 + ( η λ F ) 2 ]   λ [ μ cos ( θ ) m υ 0 + η sin ( θ ) m υ 0 ] F λ 2 m 2 υ 0 2 2
E 0 ( x , y , z = 4 F ) = m A m E ˜ 0 [ μ λ F cos ( θ ) m υ 0 , η λ F sin ( θ ) m υ 0 ] e 2 π i ( x μ λ F + y η λ F z 0 λ ϕ 2 ) c i r c ( μ 2 + η 2 Δ ρ / 2 ) d μ d η
E 0 ( x , y , z = [ 4 F z 0 ] ) = m A m E ˜ 0 [ μ λ F cos ( θ ) m υ 0 , η λ F sin ( θ ) m υ 0 ] e 2 π i ( x μ λ F + y η λ F z 0 λ ϕ 2 z 0 λ ϕ 3 ) c i r c ( μ 2 + η 2 Δ ρ / 2 ) d μ d η
ϕ 3 = 1 λ 2 [ ( μ λ F ) 2 + ( η λ F ) 2 ]   1 μ 2 + η 2 2 F 2
E 0 ( x , y , z = [ 4 F z 0 ] + ) = m n A m A n E ˜ 0 [ μ λ F cos ( θ ) m υ 0 , η λ F sin ( θ ) m υ 0 ] e 2 π i { x [ μ λ F + cos ( θ ) n υ 0 ] + y [ η λ F + sin ( θ ) n υ 0 ] z 0 λ ϕ 2 z 0 λ ϕ 3 } c i r c ( μ 2 + η 2 Δ ρ / 2 ) d μ d η
E 0 ( x , y , z = 4 F ) = m n A m A n E ˜ 0 [ μ λ F cos ( θ ) m υ 0 , η λ F sin ( θ ) m υ 0 ] e 2 π i { x [ μ λ F + cos ( θ ) n υ 0 ] + y [ η λ F + sin ( θ ) n υ 0 ] z 0 λ ϕ 2 z 1 λ ϕ 3 + z 1 λ ϕ 4 } c i r c ( μ 2 + η 2 Δ ρ / 2 ) d μ d η
ϕ 4 = 1 λ 2 { [ μ λ F + cos ( θ ) n υ 0 ] 2 + [ η λ F + sin ( θ ) n υ 0 ] 2 }   1 λ 2 2 { [ μ λ F + cos ( θ ) n υ 0 ] 2 + [ η λ F + sin ( θ ) n υ 0 ] 2 }
E 0 ( x , y , z = 4 F ) = m n A m A n E ˜ 0 ( u ^ , v ^ ) e 2 π i { x [ u ^ + cos ( θ ) ( m + n ) υ 0 ] + y [ v ^ + sin ( θ ) ( m + n ) υ 0 ] ϕ t o t } c i r c { [ u ^ + cos ( θ ) m υ 0 ] 2 + [ v ^ + sin ( θ ) m υ 0 ] 2 Δ ρ / 2 λ F } d u ^ d v ^
ϕ t o t = z 0 λ ϕ 2 + z 1 λ ϕ 3 z 1 λ ϕ 4
E 0 ( x , y , z = 4 F ) = n = m A m A n E ˜ 0 ( u ^ , v ^ ) c i r c { [ u ^ + cos ( θ ) m υ 0 ] 2 + [ v ^ + sin ( θ ) m υ 0 ] 2 Δ ρ / 2 λ F } e 2 π i ( x u ^ + y v ^ ) d u ^ d v ^
E 0 ( x , y , z = 4 F ) = E ˜ 0 ( u ^ , v ^ ) c i r c ( u ^ 2 + v ^ 2 Δ ρ / 2 λ F ) e 2 π i ( x u ^ + y v ^ ) d u ^ d v ^ + E ˜ 0 ( u ^ , v ^ ) ( c i r c { u ^ 2 + v ^ 2 3 Δ ρ / 2 λ F } c i r c { u ^ 2 + v ^ 2 Δ ρ / 2 λ F } ) e 2 π i ( x u ^ + y v ^ ) d u ^ d v ^ = E ˜ 0 ( u ^ , v ^ ) c i r c ( u ^ 2 + v ^ 2 3 Δ ρ / 2 λ F ) e 2 π i ( x u ^ + y v ^ ) d u ^ d v ^
r m , n = λ z 0 υ 0 ( m + n )
d max = λ z 0 υ 0
P ( x , y , z = 4 F ) = n = m A m A n c i r c { [ u ^ + cos ( θ ) m υ 0 ] 2 + [ v ^ + sin ( θ ) m υ 0 ] 2 Δ ρ / 2 λ F } e 2 π i ( x u ^ + y v ^ ) d u ^ d v ^
S ( x , y , z = 4 F ) = | P ( x , y , z = 4 F ) | 2
S x 0 , y 0 = S ( x , y ) δ ( x x 0 , y y 0 )
I o b j ( x , y , z = 0 ) = I o b j ( x 0 , y 0 , z = 0 ) δ ( x x 0 , y y 0 ) d x 0 d y 0
I i m g ( x , y , z = 4 F ) = I o b j ( x 0 , y 0 , z = 0 ) S ( x x 0 , y y 0 , z = 4 F ) d x 0 d y 0
O T F ( u , v ) = S ˜ ( u , v ) = P ˜ ( μ + u 2 , η + v 2 ) P ˜ * ( μ u 2 , η v 2 ) d μ d η | P ˜ ( μ , η ) | 2 d μ d η

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