Abstract

One can achieve high-resolution (superresolution) imaging, beyond the classical limit, by exploiting certain degrees of freedom such as time and polarization for the object under consideration. We present an implementation, based on polarization coding, that requires insertion of a single mask into the object plane followed by postprocessing of the detected signal. We describe the procedure and provide experimental evidence for the implementation of the proposed technique.

© 2005 Optical Society of America

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References

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  1. D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev, Z. Zalevsky, “One-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 24, 2353–2359 (1997).
    [CrossRef]
  2. A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. (USSR) 9, 204–206 (1960).
  3. E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, I. Kiryus-chev, “Superresolution optical system with two fixed generalized Damman gratings,” Appl. Opt. 39, 5318–5325 (2000).
    [CrossRef]
  4. J. Solomon, Z. Zalevsky, D. Mendlovic, “Superresolution by code division multiplexing,” Appl. Opt. 42, 1451–1462 (2003).
    [CrossRef] [PubMed]
  5. W. Gärtner, A. Lohmann, “Ein experiment zur überschrei-tung der Abbeschen Auflösungsgrenze,” Z. Phys. 174, 18–23 (1963).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 126–142.
  7. F. Xu, J. E. Ford, Y. Fainman, “Polarization-selective computer-generated holograms: design, fabrication and applications,” Appl. Opt. 34, 256–265 (1995).
    [CrossRef] [PubMed]
  8. N. Nieuborg, A. Kirk, B. Morlion, H. Thienpont, I. Veretennicoff, “Polarization-selective diffractive optical elements with an index-matching gap material,” Appl. Opt. 36, 4681–4685 (1997).
    [CrossRef] [PubMed]
  9. J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Damman gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
    [CrossRef]

2003 (1)

2000 (1)

1997 (2)

N. Nieuborg, A. Kirk, B. Morlion, H. Thienpont, I. Veretennicoff, “Polarization-selective diffractive optical elements with an index-matching gap material,” Appl. Opt. 36, 4681–4685 (1997).
[CrossRef] [PubMed]

D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev, Z. Zalevsky, “One-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 24, 2353–2359 (1997).
[CrossRef]

1995 (1)

1989 (1)

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Damman gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

1963 (1)

W. Gärtner, A. Lohmann, “Ein experiment zur überschrei-tung der Abbeschen Auflösungsgrenze,” Z. Phys. 174, 18–23 (1963).
[CrossRef]

1960 (1)

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

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Damman gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Fainman, Y.

Ford, J. E.

Gärtner, W.

W. Gärtner, A. Lohmann, “Ein experiment zur überschrei-tung der Abbeschen Auflösungsgrenze,” Z. Phys. 174, 18–23 (1963).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 126–142.

Jahns, J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Damman gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Kartashev, A. I.

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

Kirk, A.

Kiryuschev, I.

D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev, Z. Zalevsky, “One-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 24, 2353–2359 (1997).
[CrossRef]

Kiryus-chev, I.

Konforti, N.

E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, I. Kiryus-chev, “Superresolution optical system with two fixed generalized Damman gratings,” Appl. Opt. 39, 5318–5325 (2000).
[CrossRef]

D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev, Z. Zalevsky, “One-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 24, 2353–2359 (1997).
[CrossRef]

Lohmann, A.

W. Gärtner, A. Lohmann, “Ein experiment zur überschrei-tung der Abbeschen Auflösungsgrenze,” Z. Phys. 174, 18–23 (1963).
[CrossRef]

Lohmann, A. W.

D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev, Z. Zalevsky, “One-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 24, 2353–2359 (1997).
[CrossRef]

Mendlovic, D.

Morlion, B.

Nieuborg, N.

Prise, M. E.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Damman gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Sabo, E.

Solomon, J.

Streibl, N.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Damman gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Thienpont, H.

Veretennicoff, I.

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Damman gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Xu, F.

Zalevsky, Z.

Appl. Opt. (5)

Opt. Eng. (1)

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Damman gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

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

Z. Phys. (1)

W. Gärtner, A. Lohmann, “Ein experiment zur überschrei-tung der Abbeschen Auflösungsgrenze,” Z. Phys. 174, 18–23 (1963).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 126–142.

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Figures (12)

Fig. 1
Fig. 1

1-D band-limited imaging system. The polarization-sensitive spatial filter in the object plane creates differently polarized replicas in the optical Fourier plane. The aperture in this plane blocks frequencies greater than a certain limit.

Fig. 2
Fig. 2

(a) Spectra of grating fs(x) with diffraction order relative intensities S0, S1, S2, all with polarization s; (b) the spectra of grating fp(x) with diffraction order relative intensities P0, P1, P2, all with polarization p; (c) the various diffraction orders of the synthetic grating polarized with the resultant polarizations Pol0, Pol1, and Pol2.

Fig. 3
Fig. 3

(a) FT of the spatial filter for N = 2. Order 0 is carried as light polarized in Pol0 linear polarization, orders ±1 by Pol1, and orders ±2 by Pol2. (b) FT of some real input field. (c) FT of the input field multiplied by the polarization-sensitive spatial filter. Two solid vertical lines signify the spatial frequency limits of the imaging system.

Fig. 4
Fig. 4

Example for N = 2 and linear polarization angles ϕ0 = 45°, ϕ1 = 0°, and φ2 = 90°. (a) FT of the output field with the input field given by Fig. 2(b). The other signals are t 0 in (b), t 1 in (c), t 2 in (d), r 0 in (e), r 1 in (f), and r 2 in (g).

Fig. 5
Fig. 5

Set s0, …, sN−1: (a), (b) s0 and s1; (c) the reconstructed spectrum.

Fig. 6
Fig. 6

Experimental setup: First is a collimating configuration with two lenses, L1 and L2, and a pinhole; next, polarizer P at angle 35° that provides p and s illumination. A Mach–Zehnder interferometer with a polarization beam splitter (PBS), a regular beam splitter (BS), two mirrors (M1 and M2), and two masks (MaskS and MaskP; Ronchi gratings) creates the effective polarization-sensitive element, which is imaged to the object plane where the object grating is positioned. A camera, preceded by a polarizer, located at plane (u, v) captures the Fourier transform of the object plane.

Fig. 7
Fig. 7

(a) Top view image of the sinusoidal input object 2-D FT; (b) central cross section of (a) versus normalized frequency ν/νc.

Fig. 8
Fig. 8

Spectrum of the cosine object imaged by a tested imaging system without superresolution. Object frequencies ±νob are blocked.

Fig. 9
Fig. 9

(a)–(c) The three central cross-sections of the captured images t 0, t 1, and t 2 versus normalized frequency ν/νc. (d)–(f) The three central cross sections of computed images r 0, r 1, and r 2 versus normalized frequency ν/νc.

Fig. 10
Fig. 10

Simulation results: (a)–(c) The three central cross sections of t 0, t 1, and t 2 shown versus normalized frequency ν/νc. (d)–(f) The three central cross-sections of r 0, r 1, and r 2 plotted versus normalized frequency ν/νc.

Fig. 11
Fig. 11

(a), (b) Reconstructed image of the input sinusoidal object FT and its central cross section, respectively, versus normalized frequency ν/νc.

Fig. 12
Fig. 12

Experimental setup: First is a collimating configuration with two lenses, L1 and L2, and a pinhole; next, polarizer P at angle 35° that provides p and s illumination. A Mach–Zehnder interferometer with a polarization beam splitter (PBS), a regular beam splitter (BS), two mirrors (M1 and M2), and two masks (MaskS and MaskP; Ronchi gratings) creates the effective polarization-sensitive element, which imaged at the input plane. The object mask (Ronchi grating) is positioned remotely to produce a proper scale relatively to the effective polarization-sensitive element. A camera, preceded by a polarizer, located at plane (u, v) captures the Fourier transform of the object.

Equations (23)

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u image ( x ) = u object ( x ) h ( x ) ,
[ t 0 ( n ) t N ( n ) ] = Pm [ r 0 ( n ) r N ( n ) ] , { Pm } i , j = | cos ( ψ i φ j ) | ,
( 0 ) s 0 = r 0 , ( 0 w ) s w 0 = SWAP { s 0 } , ( 1 ) s 1 = r 1 s w 0 , ( 1 w ) s w 1 = SWAP { s 1 } , ( N 1 ) s N 1 = r N 1 s w N 2 .
Signal recovered = APART 0 N c { s 0 } + APART 1 N c { s 1 } + + APART ( N 1 ) N c { s N 1 } ,
amp p / amp s = tan ( φ ) .
g ( x ) = n = 0 M ( 1 ) n rect [ x ( x n + 1 + x n ) / 2 x n + 1 x n ] , 0 x 0.5 .
g ( x ) = m = A m exp ( 2 π m i x ) ,
A m = 2 0 0.5 g ( x ) cos ( 2 π m x ) d x .
A 0 = 4 n = 1 M ( 1 ) n + 1 x n + ( 1 ) M + 1 , A m = 2 m π n = 1 M ( 1 ) n + 1 sin ( 2 π m x n ) , m 0 .
U ( u , v ) = a exp [ j π λ z ( u 2 + v 2 ) ] element ( p , q ) × object ( p , q ) exp [ j 2 π λ z ( u p + v q ) ] d p d q .
U M ( u , v ) = a exp [ j π λ z 2 ( u 2 + v 2 ) ] × element M ( η , ξ ) object ( s η , s ξ ) × exp [ j 2 π λ z 2 ( u η + v ξ ) ] d η d ξ ,
element M ( η , ξ ) = n = N N a n exp ( j π n 2 λ d z 1 Λ 2 z 2 ) × exp ( j 2 π Λ n η ) ,
element ( η , ξ ) = n = N N a n exp ( j 2 π Λ n η ) ,
exp ( j π n 2 λ d z 1 Λ 2 z 2 )
U ( u , v ) = a exp [ j π λ z 2 ( u 2 + v 2 ) ] element ( η , ξ ) × object ( s η , s ξ ) exp [ j 2 π λ z 2 ( u η + v ξ ) ] d η d ξ ,
exp [ j π λ z 2 ( u 2 + v 2 ) ] .
element ( p ) = n = N N a n exp ( j 2 π Λ n p ) ,
U 1 ( p ) = exp ( j π λ z 1 p 2 ) exp ( j 2 π Λ p ) .
U 2 ( η ) = exp ( j π λ d η 2 ) exp ( j 2 n π Λ p ) × exp ( j π λ z 1 p 2 ) exp ( j π λ d p 2 ) × exp ( j 2 π λ d p η ) d p = C exp ( j π n 2 λ d z 1 Λ 2 z 2 ) exp ( j π λ z 2 η 2 ) × exp ( j 2 n π z 1 Λ z 2 η ) ,
U 3 ( x ) = exp ( j π λ z 2 x 2 ) U 2 ( η ) object ( η ) × exp ( j π λ z 2 η 2 ) exp ( j 2 π λ z 2 η x ) d η = C exp ( j π n 2 λ d z 1 λ 2 z 2 ) exp ( j π λ z 2 x 2 ) object ( η ) × exp ( j 2 π z 1 Λ z 2 η ) exp ( 2 π λ z 2 η x ) d η = A exp ( j π n 2 λ d z 1 Λ 2 z 2 ) exp ( j π λ z 2 x 2 ) × object ( s η ) exp ( j 2 π Λ η ) exp ( j 2 π λ z 1 η x ) d η ,
U M ( u , v ) = a exp [ j π λ z 2 ( u 2 + v 2 ) ] × element M ( η , ξ ) object ( s η , s ξ ) × exp [ j 2 π λ z 2 ( u η + v ξ ) ] d η d ξ ,
element M ( η , ξ ) = n = N N a n exp ( j π n 2 λ d z 1 Λ 2 z 2 ) × exp ( j 2 π Λ n η ) .
exp ( j π n 2 λ d z 1 Λ 2 z 2 )

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