Abstract

Synthetic aperture with Fresnel elements (SAFE) is an incoherent holographic imaging system in which the complete hologram is a mosaic of several holograms captured from different points of view. In this paper we investigate a new scheme of SAFE which may be used as a basis for designing a new type of synthetic aperture telescopes. Laboratory in-door experiments may provide the proof of concept for such a new design.

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References

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  1. A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
    [CrossRef]
  2. P. R. Lawson, Selected Papers on Long Baseline Stellar Interferometry, (SPIE Press Book, 1997).
  3. R. Okayasu, M. Inoue, N. Kawaguchi, S. Kameno, K. Shibata, and Y. Asaki, “Space VLBI satellite HALCA and its engineering accomplishments,” Acta Astronaut. 50(5), 301–309 (2002).
    [CrossRef]
  4. M. E. Testorf and M. A. Fiddy, “Superresolution imaging-revisited,” Adv. Imaging Electron Phys. 163, 165–218 (2010).
    [CrossRef]
  5. S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging laser radar: laboratory demonstration and signal processing,” Appl. Opt. 44(35), 7621–7629 (2005).
    [CrossRef] [PubMed]
  6. V. Mico, Z. Zalevsky, P. García-Martínez, and J. García, “Synthetic aperture superresolution with multiple off-axis holograms,” J. Opt. Soc. Am. A 23(12), 3162–3170 (2006).
    [CrossRef]
  7. L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16(1), 161–169 (2008).
    [CrossRef] [PubMed]
  8. G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. 46(6), 993–1000 (2007).
    [CrossRef] [PubMed]
  9. B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18(2), 962–972 (2010).
    [CrossRef] [PubMed]
  10. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007).
    [CrossRef] [PubMed]
  11. J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express 15(5), 2244–2250 (2007).
    [CrossRef] [PubMed]
  12. J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
    [CrossRef]
  13. B. Katz, D. Wulich, and J. Rosen, “Optimal noise suppression in Fresnel incoherent correlation holography (FINCH) configured for maximum imaging resolution,” Appl. Opt. 49(30), 5757–5763 (2010).
    [CrossRef] [PubMed]
  14. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007) p. 7.
  15. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996) pp. 66−73.

2010

2008

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
[CrossRef]

L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16(1), 161–169 (2008).
[CrossRef] [PubMed]

2007

2006

2005

2002

R. Okayasu, M. Inoue, N. Kawaguchi, S. Kameno, K. Shibata, and Y. Asaki, “Space VLBI satellite HALCA and its engineering accomplishments,” Acta Astronaut. 50(5), 301–309 (2002).
[CrossRef]

1920

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
[CrossRef]

Asaki, Y.

R. Okayasu, M. Inoue, N. Kawaguchi, S. Kameno, K. Shibata, and Y. Asaki, “Space VLBI satellite HALCA and its engineering accomplishments,” Acta Astronaut. 50(5), 301–309 (2002).
[CrossRef]

Beck, S. M.

Brooker, G.

Buck, J. R.

Buell, W. F.

Dickinson, R. P.

Fiddy, M. A.

M. E. Testorf and M. A. Fiddy, “Superresolution imaging-revisited,” Adv. Imaging Electron Phys. 163, 165–218 (2010).
[CrossRef]

García, J.

García-Martínez, P.

Indebetouw, G.

Inoue, M.

R. Okayasu, M. Inoue, N. Kawaguchi, S. Kameno, K. Shibata, and Y. Asaki, “Space VLBI satellite HALCA and its engineering accomplishments,” Acta Astronaut. 50(5), 301–309 (2002).
[CrossRef]

Javidi, B.

Kameno, S.

R. Okayasu, M. Inoue, N. Kawaguchi, S. Kameno, K. Shibata, and Y. Asaki, “Space VLBI satellite HALCA and its engineering accomplishments,” Acta Astronaut. 50(5), 301–309 (2002).
[CrossRef]

Katz, B.

Kawaguchi, N.

R. Okayasu, M. Inoue, N. Kawaguchi, S. Kameno, K. Shibata, and Y. Asaki, “Space VLBI satellite HALCA and its engineering accomplishments,” Acta Astronaut. 50(5), 301–309 (2002).
[CrossRef]

Kozlowski, D. A.

Marechal, N. J.

Martínez-León, L.

Michelson, A. A.

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
[CrossRef]

Mico, V.

Okayasu, R.

R. Okayasu, M. Inoue, N. Kawaguchi, S. Kameno, K. Shibata, and Y. Asaki, “Space VLBI satellite HALCA and its engineering accomplishments,” Acta Astronaut. 50(5), 301–309 (2002).
[CrossRef]

Rosen, J.

Shibata, K.

R. Okayasu, M. Inoue, N. Kawaguchi, S. Kameno, K. Shibata, and Y. Asaki, “Space VLBI satellite HALCA and its engineering accomplishments,” Acta Astronaut. 50(5), 301–309 (2002).
[CrossRef]

Tada, Y.

Testorf, M. E.

M. E. Testorf and M. A. Fiddy, “Superresolution imaging-revisited,” Adv. Imaging Electron Phys. 163, 165–218 (2010).
[CrossRef]

Wright, T. J.

Wulich, D.

Zalevsky, Z.

Acta Astronaut.

R. Okayasu, M. Inoue, N. Kawaguchi, S. Kameno, K. Shibata, and Y. Asaki, “Space VLBI satellite HALCA and its engineering accomplishments,” Acta Astronaut. 50(5), 301–309 (2002).
[CrossRef]

Adv. Imaging Electron Phys.

M. E. Testorf and M. A. Fiddy, “Superresolution imaging-revisited,” Adv. Imaging Electron Phys. 163, 165–218 (2010).
[CrossRef]

Appl. Opt.

Astrophys. J.

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Photonics

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Other

P. R. Lawson, Selected Papers on Long Baseline Stellar Interferometry, (SPIE Press Book, 1997).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007) p. 7.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996) pp. 66−73.

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Figures (7)

Fig. 1
Fig. 1

Scheme of SAFE operating as synthetic aperture radar to achieve super-resolution. P, polarizer; BPF, bandpass filter; SLM, spatial light modulator; and CCD, charged coupled device.

Fig. 2
Fig. 2

Possible configurations of recording holograms in the case of point-like object: (a) for FINCH, where f d > z h . In this configuration a hologram can be recorded, but, as indicated in the text, this hologram is suboptimal. (b) for FINCH, where f d < z h . In this configuration a hologram cannot be recorded because there is no interference between the plane and the spherical waves arriving from the same part of the SLM. (c) for T-SAFE, where f d < z h . In this configuration the recorded hologram is optimal. The red and green areas indicate the spherical and plane waves, respectively. The rectangles in (a) and (b) symbolize the diffractive element of constant phase, where the lens symbol in all of the figures stands for the quadratic phase element, both the constant phase and quadratic phase elements are displayed on the same SLM.

Fig. 3
Fig. 3

Proposed design of T-SAFE which is based on spherical (black line) and flat (red line) mirrors rather than on SLMs. Four interfering steps, needed to obtain the synthetic aperture hologram, are shown. PM stands for phase modulator.

Fig. 4
Fig. 4

Experimental setup. The two uncorrelated object points are created by two HeNe lasers and imaged by the T-SAFE.

Fig. 5
Fig. 5

Experimental results of a regular FINCH obtained for the complete and narrow apertures: (a) is one of the three masks displayed on the SLM for the complete wide aperture; (b) the corresponding recorded hologram; (c) the computed absolute and (d) phase of the complex-valued hologram of 1 mm gap between the source points; (e)-(h) the same as (a)-(d) but for the narrow aperture; (i) the best in-focus reconstructed plane in case of the complete aperture, for a gap between the source points of 1 mm; (j)-(l) the same as (i) but in case of narrow apertures for a gap between the source points of 1 mm, 1.5 mm and 2 mm, respectively.

Fig. 6
Fig. 6

Experimental results obtained for the case of synthetic aperture: (a)-(d) present four out of twelve masks displayed on the SLM during the recording process of the synthetic aperture hologram, and their corresponding recorded holograms are shown in (e-h); (i)-(l) computed magnitude and (m)-(p) phase of the complex holograms; (q) magnitude and (r) phase of the computed synthetic aperture hologram; (s) best in-focus reconstructed plane for the synthetic aperture for the gap between the points of 1 mm.

Fig. 7
Fig. 7

Experimental results obtained for the case of synthetic aperture for several different distances between the two point objects: (a)-(c), (d)-(f), (g)-(i) and (j)-(l), show the magnitude, phase and best in-focus reconstructed planes for gaps between the points - 2 mm, 1.5 mm, 1 mm and 0.75 mm, respectively.

Equations (14)

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P m n ( x , y ) = ( C 2 Q [ 1 / f d ] + C 3 ) r e c t [ ( x A x m ) / A x , ( y A y n ) / A y ] .
r e c t ( x α , y β ) { 1 ( | x | , | y | ) ( α / 2 , β / 2 ) 0     Otherwise .
I h ( x o , y o ; x s , y s , z s ) = n = 1 N 2 N 1 2 m = 1 M 2 M 1 2 | C 1 ( x s , y s ) Q [ 1 z s ] L [ x s z s , y s z s ] P m n ( x , y ) Q [ 1 z h ] | 2 ,
I h ( x o , y o ; x s , y s , z s ) = ( C 4 + C 5 ( x s , y s ) Q [ 1 z r ] L [ x r , y r z r ] + C 5 * ( x s , y s ) Q [ 1 z r ]         × L [ x r , y r z r ] ) n = 1 N 2 N 1 2 m = 1 M 2 M 1 2 r e c t ( x o A x m A x , y o A y n A y ) ,
z r = ± ( z s + z h ) ( f d z s z h z s + f d z h ) z s 2 z s ± ( f d z h ) , x r = x s z h z s , y r = y s z h z s ,
Δ min = max { λ / N A i n , λ / ( M T N A o u t ) } = max { 2 λ z s / D S L M , 2 λ | z r | / ( M T D H ) } ,
z s D S L M | z r | M T D H .
D H = D S L M | f d z h | f d .
f d z h .
P m n ( x , y ) = C 2 Q [ 1 f d ] r e c t [ x A x ( m + 1 / 2 ) A x , y A y ( n + 1 / 2 ) A y ] + C 3 r e c t [ x + A x ' ( m + 1 / 2 ) A x , y + A y ' ( n + 1 / 2 ) A y ] .
A x ' A x = A y ' A y = z h f d f d .
H ( x o , y o ) = I s ( x s , y s , z s ) I h ( x o , y o ; x s , y s , z s ) d x s d y s d z s .
H ( x o , y o ) = C 5 ( x s 1 , y s 1 ) Q [ 1 z r ] L [ ( x r 1 , y r 1 ) z r ] + C 5 ( x s 2 , y s 2 ) Q [ 1 z r ] L [ ( x r 2 , y r 2 ) z r ] = 2 C 5 Q [ 1 z r ] L [ ( x r 1 + x r 2 ) , ( y r 1 + y r 2 ) 2 z r ] cos { π λ z r [ ( x r 2 x r 1 ) x o + ( y r 2 y r 1 ) y o ] } ,
x r k = x s k z h z s , y r k = y s k z h z s , k = 1 , 2.

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