Abstract

A new theorem on the ultimate limit of performance of optical systems is established: Not the bandwidth of the transferred spatial frequencies but only the number of degrees of freedom of the optical message transmitted by a given optical system is invariant. It is therefore possible (a) to extend the bandwidth by reducing the object area, (b) to extend the bandwidth in the x direction while proportionally reducing it in the y direction, so that the two-dimensional bandwidth is constant, and (c) to double the bandwidth when transmitting information about one state of polarization only.

To achieve this, the optical systems are modified by inserting two suitable masks (generally gratings) into optically conjugate planes of object and image space. The transfer and spread function of the modified systems are calculated for the case of coherent illumination.

© 1966 Optical Society of America

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References

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  1. M. von Laue, Ann. Physik 44, 1197 (1914).
  2. A. Lohmann, Opt. Acta 3, 97 (1956); W. Gärtner and A. Lohmann, Z. Physik 174, 18 (1963); A. W. Lohmann and D. P. Paris, Appl. Opt. 3, 1037 (1965).
    [Crossref]
  3. W. Lukosz, Z. Naturforsch. 18a, 436 (1963); W. Lukosz and M. Marchand, Opt. Acta. 10, 241 (1963).
    [Crossref]
  4. Die Lehre von der Bildenlstehung im Mikroskop von E. Abbe, bearbeitet und herausgegeben von O. Lummer und F. Reiche (Vieweg, Braunschweig, 1910).
  5. D. Gabor, Progress in Optics, Vol. I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), p. 109.
    [Crossref]
  6. A. Lohmann and H. Wegener, Z. Physik 143, 431 (1955).
    [Crossref]
  7. G. Toraldo di Francia, Rev. Opt. 28, 597 (1949).
  8. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York and London, 1959), pp. 381, 480.
  9. D. Gabor, in Astronomical Optics, Zdenek Kopal, Ed. (North-Holland Publishing Co., Amsterdam, 1956), p. 17.
  10. H. Gamo, in Progress in Optics, Vol. III, E. Wolf, Ed. North-Holland Publishing Co., Amsterdam, 1964), p. 202.
  11. D. M. MacKay, Inform. Control 1, 148 (1958).
    [Crossref]
  12. G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
    [Crossref]
  13. K. Miyamoto, J. Opt. Soc. Am. 50, 856 (1960; J. Opt. Soc. Am. 51, 910 (1961).
    [Crossref]
  14. L. Brillouin, Science and Information Theory (Academic Press Inc., New York, 1956), Ch. 8.
  15. If only one mask is inserted into object space a moiré pattern produced by object and mask appears in the image plane. The experiments of Blanc-Lapierre et al. [A. Blanc-Lapierre, M. Perot, and G. Peri, Compt. Rend. 236, 1540 (1953)] and Wolter [H. Wolter, Physica 24, 457 (1958); Physica 26, 75 (1960); Opt. Acta 7, 53 (1960)] {cf. also Herzberger [M. Herzberger, Optik 22, 645 (1965)]} and the similar electron-microscopic investigations of crystal lattices and their defects [J. W. Menter, Advan. Phys. 7, 299 (1958)] show that it is possible—if some a priori knowledge about the object exists—to obtain information about structures “unresolvable” by the same system used with uniform illumination. The reason is, that the moiré pattern contains the spatial difference and sum frequencies of the spatial frequencies of the object and the mask. But this appearance of spatial frequencies not existing in the object shows that the imaging is not space invariant. This makes the evaluation rather complicated [A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 55, 1007 (1965)].
    [Crossref]
  16. M. A. Grimm and A. W. Lohmann, J. Opt. Soc. Am. 55, 600A (1965); J. Opt. Soc. Am. 56, 1151 (1966).

1965 (1)

M. A. Grimm and A. W. Lohmann, J. Opt. Soc. Am. 55, 600A (1965); J. Opt. Soc. Am. 56, 1151 (1966).

1963 (1)

W. Lukosz, Z. Naturforsch. 18a, 436 (1963); W. Lukosz and M. Marchand, Opt. Acta. 10, 241 (1963).
[Crossref]

1960 (1)

1958 (1)

D. M. MacKay, Inform. Control 1, 148 (1958).
[Crossref]

1956 (1)

A. Lohmann, Opt. Acta 3, 97 (1956); W. Gärtner and A. Lohmann, Z. Physik 174, 18 (1963); A. W. Lohmann and D. P. Paris, Appl. Opt. 3, 1037 (1965).
[Crossref]

1955 (2)

A. Lohmann and H. Wegener, Z. Physik 143, 431 (1955).
[Crossref]

G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
[Crossref]

1953 (1)

If only one mask is inserted into object space a moiré pattern produced by object and mask appears in the image plane. The experiments of Blanc-Lapierre et al. [A. Blanc-Lapierre, M. Perot, and G. Peri, Compt. Rend. 236, 1540 (1953)] and Wolter [H. Wolter, Physica 24, 457 (1958); Physica 26, 75 (1960); Opt. Acta 7, 53 (1960)] {cf. also Herzberger [M. Herzberger, Optik 22, 645 (1965)]} and the similar electron-microscopic investigations of crystal lattices and their defects [J. W. Menter, Advan. Phys. 7, 299 (1958)] show that it is possible—if some a priori knowledge about the object exists—to obtain information about structures “unresolvable” by the same system used with uniform illumination. The reason is, that the moiré pattern contains the spatial difference and sum frequencies of the spatial frequencies of the object and the mask. But this appearance of spatial frequencies not existing in the object shows that the imaging is not space invariant. This makes the evaluation rather complicated [A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 55, 1007 (1965)].
[Crossref]

1949 (1)

G. Toraldo di Francia, Rev. Opt. 28, 597 (1949).

1914 (1)

M. von Laue, Ann. Physik 44, 1197 (1914).

Blanc-Lapierre, A.

If only one mask is inserted into object space a moiré pattern produced by object and mask appears in the image plane. The experiments of Blanc-Lapierre et al. [A. Blanc-Lapierre, M. Perot, and G. Peri, Compt. Rend. 236, 1540 (1953)] and Wolter [H. Wolter, Physica 24, 457 (1958); Physica 26, 75 (1960); Opt. Acta 7, 53 (1960)] {cf. also Herzberger [M. Herzberger, Optik 22, 645 (1965)]} and the similar electron-microscopic investigations of crystal lattices and their defects [J. W. Menter, Advan. Phys. 7, 299 (1958)] show that it is possible—if some a priori knowledge about the object exists—to obtain information about structures “unresolvable” by the same system used with uniform illumination. The reason is, that the moiré pattern contains the spatial difference and sum frequencies of the spatial frequencies of the object and the mask. But this appearance of spatial frequencies not existing in the object shows that the imaging is not space invariant. This makes the evaluation rather complicated [A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 55, 1007 (1965)].
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York and London, 1959), pp. 381, 480.

Brillouin, L.

L. Brillouin, Science and Information Theory (Academic Press Inc., New York, 1956), Ch. 8.

Gabor, D.

D. Gabor, in Astronomical Optics, Zdenek Kopal, Ed. (North-Holland Publishing Co., Amsterdam, 1956), p. 17.

D. Gabor, Progress in Optics, Vol. I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), p. 109.
[Crossref]

Gamo, H.

H. Gamo, in Progress in Optics, Vol. III, E. Wolf, Ed. North-Holland Publishing Co., Amsterdam, 1964), p. 202.

Grimm, M. A.

M. A. Grimm and A. W. Lohmann, J. Opt. Soc. Am. 55, 600A (1965); J. Opt. Soc. Am. 56, 1151 (1966).

Lohmann, A.

A. Lohmann, Opt. Acta 3, 97 (1956); W. Gärtner and A. Lohmann, Z. Physik 174, 18 (1963); A. W. Lohmann and D. P. Paris, Appl. Opt. 3, 1037 (1965).
[Crossref]

A. Lohmann and H. Wegener, Z. Physik 143, 431 (1955).
[Crossref]

Lohmann, A. W.

M. A. Grimm and A. W. Lohmann, J. Opt. Soc. Am. 55, 600A (1965); J. Opt. Soc. Am. 56, 1151 (1966).

Lukosz, W.

W. Lukosz, Z. Naturforsch. 18a, 436 (1963); W. Lukosz and M. Marchand, Opt. Acta. 10, 241 (1963).
[Crossref]

MacKay, D. M.

D. M. MacKay, Inform. Control 1, 148 (1958).
[Crossref]

Miyamoto, K.

Peri, G.

If only one mask is inserted into object space a moiré pattern produced by object and mask appears in the image plane. The experiments of Blanc-Lapierre et al. [A. Blanc-Lapierre, M. Perot, and G. Peri, Compt. Rend. 236, 1540 (1953)] and Wolter [H. Wolter, Physica 24, 457 (1958); Physica 26, 75 (1960); Opt. Acta 7, 53 (1960)] {cf. also Herzberger [M. Herzberger, Optik 22, 645 (1965)]} and the similar electron-microscopic investigations of crystal lattices and their defects [J. W. Menter, Advan. Phys. 7, 299 (1958)] show that it is possible—if some a priori knowledge about the object exists—to obtain information about structures “unresolvable” by the same system used with uniform illumination. The reason is, that the moiré pattern contains the spatial difference and sum frequencies of the spatial frequencies of the object and the mask. But this appearance of spatial frequencies not existing in the object shows that the imaging is not space invariant. This makes the evaluation rather complicated [A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 55, 1007 (1965)].
[Crossref]

Perot, M.

If only one mask is inserted into object space a moiré pattern produced by object and mask appears in the image plane. The experiments of Blanc-Lapierre et al. [A. Blanc-Lapierre, M. Perot, and G. Peri, Compt. Rend. 236, 1540 (1953)] and Wolter [H. Wolter, Physica 24, 457 (1958); Physica 26, 75 (1960); Opt. Acta 7, 53 (1960)] {cf. also Herzberger [M. Herzberger, Optik 22, 645 (1965)]} and the similar electron-microscopic investigations of crystal lattices and their defects [J. W. Menter, Advan. Phys. 7, 299 (1958)] show that it is possible—if some a priori knowledge about the object exists—to obtain information about structures “unresolvable” by the same system used with uniform illumination. The reason is, that the moiré pattern contains the spatial difference and sum frequencies of the spatial frequencies of the object and the mask. But this appearance of spatial frequencies not existing in the object shows that the imaging is not space invariant. This makes the evaluation rather complicated [A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 55, 1007 (1965)].
[Crossref]

Toraldo di Francia, G.

G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
[Crossref]

G. Toraldo di Francia, Rev. Opt. 28, 597 (1949).

von Laue, M.

M. von Laue, Ann. Physik 44, 1197 (1914).

Wegener, H.

A. Lohmann and H. Wegener, Z. Physik 143, 431 (1955).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York and London, 1959), pp. 381, 480.

Ann. Physik (1)

M. von Laue, Ann. Physik 44, 1197 (1914).

Compt. Rend. (1)

If only one mask is inserted into object space a moiré pattern produced by object and mask appears in the image plane. The experiments of Blanc-Lapierre et al. [A. Blanc-Lapierre, M. Perot, and G. Peri, Compt. Rend. 236, 1540 (1953)] and Wolter [H. Wolter, Physica 24, 457 (1958); Physica 26, 75 (1960); Opt. Acta 7, 53 (1960)] {cf. also Herzberger [M. Herzberger, Optik 22, 645 (1965)]} and the similar electron-microscopic investigations of crystal lattices and their defects [J. W. Menter, Advan. Phys. 7, 299 (1958)] show that it is possible—if some a priori knowledge about the object exists—to obtain information about structures “unresolvable” by the same system used with uniform illumination. The reason is, that the moiré pattern contains the spatial difference and sum frequencies of the spatial frequencies of the object and the mask. But this appearance of spatial frequencies not existing in the object shows that the imaging is not space invariant. This makes the evaluation rather complicated [A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 55, 1007 (1965)].
[Crossref]

Inform. Control (1)

D. M. MacKay, Inform. Control 1, 148 (1958).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Acta (1)

A. Lohmann, Opt. Acta 3, 97 (1956); W. Gärtner and A. Lohmann, Z. Physik 174, 18 (1963); A. W. Lohmann and D. P. Paris, Appl. Opt. 3, 1037 (1965).
[Crossref]

Rev. Opt. (1)

G. Toraldo di Francia, Rev. Opt. 28, 597 (1949).

Z. Naturforsch. (1)

W. Lukosz, Z. Naturforsch. 18a, 436 (1963); W. Lukosz and M. Marchand, Opt. Acta. 10, 241 (1963).
[Crossref]

Z. Physik (1)

A. Lohmann and H. Wegener, Z. Physik 143, 431 (1955).
[Crossref]

Other (6)

L. Brillouin, Science and Information Theory (Academic Press Inc., New York, 1956), Ch. 8.

Die Lehre von der Bildenlstehung im Mikroskop von E. Abbe, bearbeitet und herausgegeben von O. Lummer und F. Reiche (Vieweg, Braunschweig, 1910).

D. Gabor, Progress in Optics, Vol. I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), p. 109.
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York and London, 1959), pp. 381, 480.

D. Gabor, in Astronomical Optics, Zdenek Kopal, Ed. (North-Holland Publishing Co., Amsterdam, 1956), p. 17.

H. Gamo, in Progress in Optics, Vol. III, E. Wolf, Ed. North-Holland Publishing Co., Amsterdam, 1964), p. 202.

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Figures (6)

Fig. 1
Fig. 1

Telescopic system (of unit magnification) used with plane waves as a spatial-frequency filter. S light source; C condenser; OP, IP object and image plane; L1, L2 lenses of equal focal length; FP Fraunhofer plane with filtering diaphragm or aperture stop.

Fig. 2
Fig. 2

Arbitrary optical system used with spherical waves as a spatial-frequency filter. S light source; C condenser; OP, IP object and image plane; EP, EP entrance and exit pupil.

Fig. 3
Fig. 3

Spatial frequencies transferred by an optical system with rectangular aperture for (a) central and (b) oblique illumination.

Fig. 4
Fig. 4

The essential feature common to all systems with a resolving power exceeding the classical limit is the insertion of two masks M and M into optically conjugate planes of object and image space. Each of the above masks is assumed to be a grating producing only two diffraction orders. OP, IP object and image plane.

Fig. 5
Fig. 5

Principle of optical arrangement increasing the resolution for a reduced field of view. OP object plane; 5 optical system with aperture stop; IP image plane. The masks M and M—in optically conjugate planes of object and image space—are gratings, which in this figure are assumed to produce only two diffraction orders.

Fig. 6
Fig. 6

Principle of an optical arrangement increasing resolution in the kx direction by sacrificing resolution in the ky direction. M, M masks (gratings); OP, IP object and image plane, L (band limited) lens. Below, second imaging stage IPIP, in which all spatial frequencies | k y | > k ¯ y are filtered out by the system S.

Equations (58)

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Δ U + k 2 U = 0 ,
A ( x , y ) = + A ( x ¯ , y ¯ ) F A ( x x ¯ , y y ¯ ) d x ¯ d y ¯ .
A ( x , y ) = 1 ( 2 π ) 2 + a ( k x , k y ) × exp [ i ( k x x + k y y ) ] d k x d k y .
a ( k x , k y ) = + A ( x , y ) exp [ i ( k x x + k y y ) ] d x d y .
a ( k x , k y ) = a ( k x , k y ) f a ( k x , k y ) .
U k ( x , y , z ) = const exp [ i ( k x x + k y y + k z z ) ]
U ( x , y , z ) = 1 ( 2 π ) 2 + u ( k x , k y ) × exp [ i ( k x x + k y y + k z z ) ] d k x d k y
k z = k sin α x ; k y = k sin α y .
f a ( k x , k y ) = f ( k x + k x s , k y + k y s ) .
F A ( x , y ) = F ( x , y ) exp [ i ( k x s x + k y s y ) ] ,
f ( k x , k y ) = { 1 for | k x | k x , | k y | k y 0 elsewhere
k x k x s k x k x k x s ; k y k y s k y k y k y s .
k x = 2 π n x / L x , k y = 2 π n y / L y ; n x , n y = 0 , ± 1 , .
N = 2 · N t · ( 1 + L x k x / π ) ( 1 + L y k y / π ) .
N 2 · N t · L x L y · k x k y / π 2 = 2 · N t · S · W ,
Δ x = 2 z 0 λ / d .
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 ) ] .
U M ( x , y , z = z 0 ) = U ( x , y , z = z 0 ) M ( x , y ) .
U ( M ) ( x , y , z ) = 1 ( 2 π ) 2 + u ( M ) ( k x , k y ) × exp i ( k x x + k y y + k z z ) d k x d k y ,
k z = + [ k 2 k x 2 k y 2 ] 1 2 ,
u M ( k x k y ) = j , l = 0 , ± 1 , m j , l u ( k x 2 π j / d x , k y 2 π l / d y ) × exp i z 0 { [ k 2 ( k x 2 π j / d x ) 2 ( k y 2 π l / d y ) 2 ] 1 2 ( k 2 k x 2 k y 2 ) 1 2 } .
U M ( x , y , z = 0 ) = j , l = 0 , ± 1 , m j , l U ( x Δ x j , y Δ y l ) × exp i [ ( x Δ x j / 2 ) ( k x s ) j + ( y Δ y l / 2 ) ( k y s ) l ]
u M ( k x , k y ) = u M ( k x , k y ) f ( k x , k y ) .
u ˆ ( k x , k y ) = j , l = 0 , ± 1 , m j , l u M ( k x 2 π j / d x , k y 2 π l / d y ) × exp i z 0 { [ k 2 ( k x 2 π j / d x ) 2 ( k y 2 π l / d y ) 2 ] 1 2 ( k 2 k x 2 k y 2 ) 1 2 } .
u ˆ ( k x , k y ) = p , q = 0 , ± 1 ,... f p , q ( k x , k y ) × u ( k x 2 π p / d x , k y 2 π q / d y ) · exp i z 0 { [ k 2 ( k x 2 π p / d x ) 2 ( k y 2 π q / d y ) 2 ] 1 2 ( k 2 k x 2 k y 2 ) 1 2 } ,
f p , q ( k x , k y ) = j , i = 0 , ± 1 ,... m j , i m p j , q l × f ( k x 2 π j / d x , k y 2 π l / d y ) .
u ˆ ( k x , k y ) = u ( k x , k y ) f ˆ ( k x , k y ) ,
f ˆ ( k x , k y ) = f p = 0 , q = 0 ( k x , k y ) = j , l = 0 , ± 1 ,... m i j × f ( k x 2 π j / d x , k y 2 π l / d y ) ,
m ˆ j , l = m j , l m j , l .
F ˆ ( x , y ) = F ( x , y ) M ˆ ( x , y ) ,
M ˆ ( x , y ) = 1 d x d y 0 d y 0 d x M ( x ¯ , y ¯ ) M ( x ¯ + x , y ¯ + y ) d x d y = j , l = 0 , ± 1 , · · · m ˆ j , l exp 2 π i ( j x / d x + l y / d y ) .
U ˆ p , q ( x , y ) = + U ( x ¯ Δ x p , y ¯ Δ y q ) F p , q ( x x ¯ , y y ¯ ) × exp 2 π i [ ( x ¯ Δ x p / 2 ) p / d x + ( y ¯ Δ y q / 2 ) q / d y ] d x ¯ d y ¯ ,
F p , q ( x , y ) = F ( x , y ) M p , q ( x , y ) ,
M p , q ( x , y ) = 1 d x d y 0 d y 0 d x M ( x ¯ , y ¯ ) M ( x ¯ + x , y ¯ + y ) × exp [ 2 π i ( p x ¯ / d x + q y ¯ / d y ) ] d x ¯ d y ¯ .
M ( x , y ) = cos 2 π x / d ; M ( x , y ) = cos 2 π ( x Δ ) / d .
M ˆ ( x , y ) = 1 2 cos [ 2 π ( x Δ ) / d ] .
f ˆ ( k x , k y ) = 1 4 exp ( 2 π i Δ / d ) f ( k x 2 π / d , k y ) + 1 4 exp ( 2 π i Δ / d ) f ( k x + 2 π / d , k y )
f ˆ ( k x , k y ) = 1 4 [ f ( k x 2 π / d , k y ) + f ( k x + 2 π / d , k y ) ] .
x ˜ = x cos α + y sin α .
M ( x , y ) M ( x ˜ ) = j = 0 , ± 1 , m j exp ( 2 π i j x ˜ / d )
M ( x , y ) M ( x ˜ ) = j = 0 , ± 1 , m j exp ( 2 π i j x ˜ / d ) .
u ( k x , k y ) = 0 for | k y | > k ¯ y .
U M ( x , y , z = z 0 ) = U ( x , y , z = z 0 ) M ( x , y )
u M ( k x , k y ) = j = 0 , ± 1 , m j u ( k x 2 π j cos α / d , k y 2 π j sin α / d ) × exp i z 0 { [ k 2 ( k x 2 π j cos α / d ) 2 ( k y 2 π j sin α / d ) 2 ] 1 2 ( k 2 k x 2 k y 2 ) 1 2 } .
tan α = k y / N k x .
u m ( k x , k y ) = u M ( k x , k y ) f ( k x , k y ) .
U ¯ ( x , y , z = z 0 ) = U m ( x , y , z = z 0 ) M ( x , y )
u ¯ ( k x , k y ) = p = 0 , ± 1 , f p ( k x , k y ) × u ( k x 2 π p cos α d , k y 2 π p sin α d ) ,
f p ( k x , k q ) = j = 0 , ± 1 , m j m p j × f ( k x 2 π j cos α d , k y 2 π j sin α d ) .
g ( k y ) = { 1 0 for | k y | k y > k y ,
u ˆ ( k x , k y ) = u ˆ ( k x , k y ) g ( k y ) = u ( k x , k y ) f ˆ ( k x , k y ) ,
f ˆ ( k x , k y ) = g ( k y ) j = 0 , ± 1 , m ˆ j × f ( k x 2 π j cos α d , k y 2 π j sin α d ) , with m ˆ j m j m j .
F ˆ ( x , y ) = + [ F ( x , y ¯ ) M ˆ ( x , y ) ] G ( y y ¯ ) d y ¯ ,
M ˆ ( x , y ) M ˆ ( x ˜ ) = 1 d 0 d M ( x ¯ ) M ( x ¯ + x ˜ ) d x = j = 0 , ± 1 , m ˆ j exp ( 2 π i j x ¯ / d )
G ( y ) = ( k ¯ y / π ) sin k ¯ y y / k ¯ y y
f ˆ ( k x , k y ) = 1 4 exp ( 2 π i Δ / d ) f ( k x 2 π / d , k y ) + 1 4 exp ( 2 π i Δ / d ) f ( k x + 2 π / d , k y )
F ˆ ( x , y ) = F ( x , y ) M ˆ ( x , y ) ,