Abstract

A superresolving optical system with a spatial resolution exceeding the classical limit is described. The gain in spatial bandwidth is obtained by reduction of the usable object field. The superresolving system is essentially a conventional system modified by the insertion of two masks (line or crossed gratings) into conjugate planes of object and image space. Its spread and transfer functions for coherent and incoherent illumination are derived theoretically. The experiments with different extended objects clearly show the expected increase of the spatial bandwidth; the experimental point images agree with the theoretical predictions.

© 1967 Optical Society of America

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  1. W. Lukosz, J. Opt. Soc. Am. 56, 1463 (1966).
    [Crossref]
  2. This result is valid for low apertures and small angles of deflection at the masks. The aberrations of the superresolving system, which in general appear if the aperture and deflection angles are large, have not yet been investigated.
  3. In a real microscopic arrangement there would be a high magnification between the object and the intermediate image plane, and between the planes of M and M′. So the lens behind the intermediate image would have to transmit low spatial frequencies only; the mask M′ would have to be correspondingly coarser than M.

1966 (1)

J. Opt. Soc. Am. (1)

Other (2)

This result is valid for low apertures and small angles of deflection at the masks. The aberrations of the superresolving system, which in general appear if the aperture and deflection angles are large, have not yet been investigated.

In a real microscopic arrangement there would be a high magnification between the object and the intermediate image plane, and between the planes of M and M′. So the lens behind the intermediate image would have to transmit low spatial frequencies only; the mask M′ would have to be correspondingly coarser than M.

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Figures (8)

Fig. 1
Fig. 1

Imaging of an object point P into its image point P′ by the superresolving system. OP, IP object and image plane; M, M′ masks; Dx aperture of the conventional system S; DxM extent of the mask necessary to obtain superresolution; Δx usable object field.

Fig. 2
Fig. 2

(a) Transmittances of the masks M and M′. (b) Cross-correlation function of M and M′ for Δx=0 (—) and Δx=dx/2 (— — —).

Fig. 3
Fig. 3

(a) Amplitude spread functions, (b) pupil functions’ and (c) “incoherent” transfer functions of (1) a conventional system with rectangular aperture, and (2,3) the corresponding superresolving system with the masks of Fig. 2, for Δx=0 and dx/2, respectively. In (1a) also the cross-correlation function M ˆ(x) is indicated for Δx=0 and dx/2.

Fig. 4
Fig. 4

Experimental set-up. L1 low-aperture lens whose spatial bandwidth is to be increased beyond the classical limit; L2 auxiliary lens to image the intermediate image IP of the object plane OP into the final image plane IP′; M, M′ line or square gratings with grating constant d; f focal length of both L1 and L2. a=1800 mm, f=500 mm, d=0.5 mm, D ≈ 5 mm, x=λ(af)/d ≈ 13 mm, f-number ≈ 360.

Fig. 5
Fig. 5

Resolution target imaged by (a) the conventional system (with rectangular aperture), and (b,c) the superresolving system, the masks being line and square gratings, respectively (Δx=0; Δxy=0).

Fig. 6
Fig. 6

Text imaged by (a) the conventional system (with rectangular aperture), (b) the superresolving system with square gratings as masks (Δxy=0).

Fig. 7
Fig. 7

Point images of (a) the conventional system with rectangular aperture, (b,c) the superresolving system with line gratings as masks for Δx=0 and Δx=dx/2, and (d,e) the superresolving system with crossed gratings as masks for Δxy=0 and Δxy=dx/2=dy/2.

Fig. 8
Fig. 8

(a,b,c) Point images analogous to those in Fig. 7 (a,b,c,) but for a circular and much smaller aperture. (In the central part of the Airy disk the cross-correlation function of the masks M and M′ is seen.)

Equations (42)

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M ( x , y ) = j , l = 0 , ± 1 , m j , l exp 2 π i ( j x / d x + l y / d y ) ,
M ( x , y ) = j , l = 0 , ± 1 , m j , l exp 2 π i ( j x / d x + l y / d y ) ,
M ˆ ( x , y ) = 1 d x d y 0 d x 0 d y M ( x ¯ , y ¯ ) M ( x ¯ + x , y ¯ + y ) d x ¯ d y ¯
M ˆ ( x , y ) = j , l = 0 , ± 1 , m ˆ j , l exp 2 π i ( j x / d x + l y / d y ) ,
m ˆ j , l = m j , l m - j , - l .
k x = k sin α x ,             k y = k sin α y ,
F ˆ ( x , y ) = F ( x , y ) M ˆ ( x , y ) .
f ˆ ( k x , k y ) = j , l = 0 , ± 1 , m ˆ j , l f ( k x - 2 π j / d x , k y - 2 π l / d y ) .
d ¯ x = d x a / a M ,             d ¯ y = d y a / a M
or             ( α x ) j = j λ d x a M a = j λ / d ¯ x , k ( α x ) j = j 2 π / d ¯ x .
λ / d ¯ x = 2 α x ,
2 π / d ¯ x = 2 k x ,
Δ x = λ ( a - a M ) / d x ,             Δ y = λ ( a - a M ) / d y .
D x M = s x [ ( a - a M ) / a ] D = n x d x ,
n x = Δ x 2 s x α x / λ = Δ x 2 s x k x / 2 π ;
Ĝ ( x , y ) = F ˆ ( x , y ) 2 ,
Ĝ ( x , y ) = G ( x , y ) M ˆ ( x , y ) 2 ,
G ( x , y ) = F ( x , y ) 2
ĝ ( k x , k y ) = - + Ĝ ( x , y ) exp [ - i ( k x x + k y y ) ] d x d y , Ĝ ( x , y ) = 1 ( 2 π ) 2 - + ĝ ( k x , k y ) exp [ i ( k x x + k y y ) ] d k x d k y .
ĝ ( k x , k y ) = 1 ( 2 π ) 2 - + f ˆ * ( k ¯ x , k ¯ y ) f ˆ ( k ¯ x + k x , k ¯ y + k y ) d k ¯ x d k ¯ y .
ĝ ( k x , k y ) = ĝ * ( - k x , - k y ) .
ĝ ( k x , k y ) = j , l = 0 , ± 1 , m ˆ j , l g ( k x - j 2 π / d ¯ x , k y - l 2 π / d ¯ y ) ,
M ˆ ( x , y ) 2 = j , l = 0 , ± 1 , m ˆˆ j , l exp 2 π i ( j x / d x + l y / d y ) .
m ˆˆ j , l = s , t = 0 , ± 1 , m ˆ s , t * m ˆ s + j , t + l .
M ( x ) = { 1 0             for             x d x / 4 d x / 4 < x < d x / 2 M ( x ) = M ( x + m d x )             if             m = 0 , ± 1 , ,
M ( x ) = M ( x - Δ x ) .
M ( x ) = 1 2 { 1 + ( 4 / π ) [ cos 2 π x / d x - 1 3 cos 6 π x / d x + ] }
M ˆ ( x ) = 1 d x 0 d x M ( x ¯ ) M ( x ¯ + x ) d x ¯ = 1 2 ( 1 - 2 x - Δ x / d x )             for             x - Δ x d x / 2 M ˆ ( x ) = M ˆ ( x + m d x )             if             m = 0 , ± 1 ,
M ˆ ( x ) = j = 0 , ± 1 , m ˆ j exp ( 2 π i j x / d x )
m ˆ 0 = 1 4 , m ˆ j = 1 π 2 j 2 exp ( - 2 π i j Δ x / d x ) for j = ± 1 , ± 3 , m ˆ j = 0             for             j = ± 2 , ± 4 , ,
M ˆ ( x ) = 1 4 { 1 + ( 8 / π 2 ) [ cos 2 π ( x - Δ x ) / d x + 1 9 cos 6 π ( x - Δ x ) / d x + ] .
M ˆ ( x ) 2 = j = 0 , ± 1 , m ˆˆ j exp ( 2 π i j x / d x )
m ˆˆ 0 = 1 12 m ˆˆ j = 1 2 π 2 j 2 exp ( - 2 π i j Δ x / d x )             for             j = ± 1 , ± 2 , ,
M ˆ ( x ) 2 = 1 12 { 1 + ( 12 / π 2 ) [ cos 2 π ( x - Δ x ) / d x + 1 4 cos 4 π ( x - M x ) / d x + ] } .
f ˆ ( k x , k y ) = 1 4 { f ( k x , k y ) + ( 4 / π 2 ) [ exp ( - 2 π i Δ x / d x ) f ( k x - 2 π / d ¯ x , k y ) + exp ( 2 π i Δ x / d x ) f ( k x + 2 π / d ¯ x , k y ) + 1 9 exp ( - 6 π i Δ x / d x ) f ( k x - 6 π / d ¯ x , k y ) + 1 9 exp ( 6 π i Δ x / d x ) f ( k x + 6 π / d ¯ x , k y ) + ] } .
ĝ ( k x , k y ) = 1 12 { g ( k x , k y ) + ( 6 / π 2 ) [ exp ( - 2 π i Δ x / d x ) g ( k x - 2 π / d ¯ x , k y ) + exp ( 2 π i Δ x / d x ) g ( k x + 2 π / d ¯ x , k y ) + 1 4 exp ( - 4 π i Δ x / d x ) g ( k x - 4 π / d ¯ x , k y ) + 1 4 exp ( 2 π i Δ x / d x ) g ( k x + 4 π / d ¯ x , k y ) + ] } .
f ( k x ) = { 1 0             for             k x k x k x > k x
F ( x ) ( sin k x x ) / k x x .
M ( x , y ) = M ( x ) M ( y )
M ( x , y ) = M ( x ) M ( y ) ,
M ˆ ( x , y ) = M ˆ ( x ) M ˆ ( y ) ,
m ˆ j , l = m ˆ j m ˆ l ,