Abstract

We present a new lensless incoherent holographic system operating in a synthetic aperture mode. Spatial resolution exceeding the Rayleigh limit of the system is obtained by tiling digitally several Fresnel holographic elements into a complete Fresnel hologram of the observed object. Each element is acquired by the limited-aperture system from different point of view. This method is demonstrated experimentally by combining three holographic elements recorded with white light illumination which is emitted from a binary grating.

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References

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  1. 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]
  2. 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]
  3. L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16(1), 161–169 (2008).
    [CrossRef] [PubMed]
  4. 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]
  5. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007).
    [CrossRef] [PubMed]
  6. J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express 15(5), 2244–2250 (2007).
    [CrossRef] [PubMed]
  7. J. Rosen and G. Brooker, “Non-Scanning Motionless Fluorescence Three-Dimensional Holographic Microscopy,” Nat. Photonics 2(3), 190–195 (2008).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 314−317.

2008 (2)

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 (3)

2006 (1)

2005 (1)

Beck, S. M.

Brooker, G.

Buck, J. R.

Buell, W. F.

Dickinson, R. P.

García, J.

García-Martínez, P.

Indebetouw, G.

Javidi, B.

Kozlowski, D. A.

Marechal, N. J.

Martínez-León, L.

Mico, V.

Rosen, J.

Tada, Y.

Wright, T. J.

Zalevsky, Z.

Supplementary Material (4)

» Media 1: AVI (1114 KB)     
» Media 2: AVI (1116 KB)     
» Media 3: AVI (1116 KB)     
» Media 4: AVI (1111 KB)     

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Figures (3)

Fig. 1
Fig. 1

Scheme of SAFE operating as synthetic aperture radar to achieve super-resolution. P indicates polarizer.

Fig. 2
Fig. 2

Results of SAFE for the first object with the RA, and with the SA. (a) is the phase distribution of the reflection masks displayed on the SLM at θ = 120° with the RA; (b)-(d) are the same as (a) using the SA; (e) is the magnitude of the final on-axis digital hologram with the RA, and (f) is the same as (e) with the SA; (g) is the phase of the final hologram with the actual aperture, and (h) is the phase with the synthetic aperture; (i) is the reconstruction of the hologram of the binary grating at the best focus distance for the RA and (j) is for the SA. Movies of the holographic reconstructions that correspond to Fig. 2(i) and Fig. 2(j), respectively, can be viewed by pressing the cursor on Media 1 and Media 2.

Fig. 3
Fig. 3

Results of SAFE for the second object with the RA and with the SA. (a) and (c) are, respectively, the magnitude and the phase of the final on-axis digital hologram with the RA; (b) and (d) are, respectively, the magnitude and the phase of the final on-axis digital hologram with the SA; (b) is assembled from the magnitudes of the three holographic elements shown in (e)-(g), and (d) is assembled from the phases of three holographic elements shown in (h)-(j); (k) is the reconstruction of the hologram of the binary grating at the best focus distance for the RA, and (l) is for the SA. Movies of the holographic reconstructions that correspond to Fig. 3(k) and Fig. 3(l), respectively, can be viewed by pressing the cursor on Media 3 and Media 4.

Equations (14)

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r e c t ( x α , y β ) { 1 ( | x | , | y | ) ( α / 2 , β / 2 ) 0 O t h e r w i s e .
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 ( r ¯ s ) Q [ 1 z s ] L [ r ¯ s z s ] ( C 2 Q [ 1 f 1 ] + C 3 Q [ 1 f 2 ] ) × r e c t ( x A x m A x , y A y n A y ) Q [ 1 z h ] | 2 ,
I h ( x o , y o ; x s , y s , z s ) = ( C 4 + C 5 ( r ¯ s ) Q [ 1 z r ] L [ r ¯ r z r ] + C 5 * ( r ¯ s ) Q [ 1 z r ] L [ r ¯ 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 = ( f 1 z s z h z s + f 1 z h ) ( f 2 z s z h z s + f 2 z h ) z s 2 ( f 1 f 2 ) = f 2 ( z s + z h ) ( f 1 z s z h z s + f 1 z h ) z s 2 r ¯ r = ( x r , y r ) = r ¯ 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 C C D ) } ,
| 2 λ z s D S L M | = | 2 λ z s D | < | 2 λ z r M T D C C D | = | 2 λ z r z s z h D | = | 2 λ D ( z s + z h ) ( f 1 z s z h z s + f 1 z h ) z h z s |
Δ min R A Δ min S A = 2 λ z r R A / M T D C C D R A 2 λ z r S A / M T D C C D S A = N ( | f 1 | z s + z h z s + | f 1 | z h ) N | f 1 | z s + z h z s + N | f 1 | z h
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
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 ( r ¯ s ) Q [ 1 z s ] L [ r ¯ s z s ] ( C 2 Q [ 1 f 1 ] + C 3 Q [ 1 f 2 ] ) × r e c t ( x A x m A x , y A y n A y ) Q [ 1 z h ] | 2 .
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 ( r ¯ s ) L [ r ¯ s z s ] ( C 2 Q [ f 1 z s f 1 z s ] + C 3 Q [ f 2 z s f 2 z s ] ) r e c t ( x A x m A x , y A y n A y ) Q [ 1 z h ] | 2 .
I h ( x o , y o ; x s , y s , z s ) = | C 1 ( r ¯ s ) L [ r ¯ s z s ] ( C 2 Q [ f 1 z s f 1 z s ] + C 3 Q [ f 2 z s f 2 z s ] ) Q [ 1 z h ] | 2 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 ) .
u ( x , y ; z h ) = ( C ( r ¯ s ) Q [ 1 z ] L [ A ¯ ] ) Q [ 1 z h ] = C ' ( r ¯ s ) Q [ 1 z + z h ] L [ A ¯ z z + z h ] .
I h ( x o , y o ; x s , y s , z s ) = | C ' ( r ¯ s ) L [ f 1 r ¯ s f 1 z s z + h f 1 z s f 1 z s ] Q [ 1 z + h f 1 z s f 1 z s ] + C ' ' ( r ¯ s ) L [ f 2 r ¯ s f 2 z s z + h f 2 z s f 2 z s ] Q [ 1 z + h f 2 z s f 2 z s ] | 2 × 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 ) .
= | C ' ( r ¯ s ) L [ f 1 r ¯ s z f h 1 z s z + h f 1 z s ] Q [ f 1 z s z f h 1 z s z + h f 1 z s ] + C ' ' ( r ¯ s ) L [ f 2 r ¯ s z f h 2 z s z + h f 2 z s ] Q [ f 2 z s z f h 2 z s z + h f 2 z s ] | 2 × 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 )

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