Abstract

The theory of a superresolution experiment is developed. To achieve superresolution one must know in advance some properties of the objects, e.g., nonbirefringence, time independence, or wavelength independence. Assuming that the objects are nonbirefringent, it would be wasteful to use the two possible states of independent linear polarization of the light for simultaneously carrying the same information twice through the image-forming system. One can avoid this waste by inserting polarizers and certain double-refracting components into the system, so that the two states of polarization instead carry different information through the conventional image-forming system. The transfer function of such a superresolution system is derived for coherent and incoherent object illumination. It confirms qualitatively the results of previously reported experiments. A modification of the system is then proposed so that the one-dimensional restriction of the original concept is eliminated. The transfer function for the modified system is derived and numerical examples are presented. The modification imposes a further constraint on the class of allowed objects: the objects must be time-independent or only slowly time-varying.

© 1964 Optical Society of America

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References

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  1. E. Lau, Physik. Z. 38, 446 (1937). A. Blanc-Lapierre, M. Perrot, G. Peri, Compt. Rend. 236, 1540 (1953). P. Lacomme, Opt. Acta 1, 33 (1954). G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955). A. Lohmann, Opt. Acta 3, 97 (1956). D. Gabor, in Astronomical Optics, Z. Kopal, ed. (North-Holland, Amsterdam, 1956), p. 59. J. M. Cowley, A. F. Moody, Proc. Phys. Soc. (London) B70, 486, 497, 505 (1957) and Proc. Phys. Soc. (London) B76, 378 (1960). H. Wolter, Physica 24, 457 (1958). G. Nomarski, Abbilden und Sehen, H. Schober, R. Röhler, eds. (Akademische Buchdruckerei F. Straub, Munich, 1962), p. 172.
    [CrossRef]
  2. N. E. Lindenblad, U.S. Patent2,443,258 (1948). A. I. Kartashev, Opt. Spectry. 9, 204 (1960).
  3. M. Françon, Nuovo Cimento Suppl. 9, 283 (1952). W. Lukosz, Z. Naturforsch. 18a, 436 (1963). W. Lukosz, M. Marchand, Opt. Acta 10, 241 (1963). B. Morgenstern, D. P. Paris, “A Method for Exceeding the Cutoff Frequency of a Bandlimited Optical System”,IBM San Jose Tech. Rept. TR 02.281 (1963).
    [CrossRef]
  4. W. Gärtner, A. Lohmann, Z. Physik 174, 18 (1963).
    [CrossRef]
  5. P. M. Duffieux, G. Lansraux, Rev. Opt. 24, 65, 151, 215 (1945). M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 484.
  6. P. M. Duffieux, L’intégrale de Fourier et ses applications à l’optique (Oberthur, Rennes, 1946). H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408, 424 (1953).

1963 (1)

W. Gärtner, A. Lohmann, Z. Physik 174, 18 (1963).
[CrossRef]

1952 (1)

M. Françon, Nuovo Cimento Suppl. 9, 283 (1952). W. Lukosz, Z. Naturforsch. 18a, 436 (1963). W. Lukosz, M. Marchand, Opt. Acta 10, 241 (1963). B. Morgenstern, D. P. Paris, “A Method for Exceeding the Cutoff Frequency of a Bandlimited Optical System”,IBM San Jose Tech. Rept. TR 02.281 (1963).
[CrossRef]

1945 (1)

P. M. Duffieux, G. Lansraux, Rev. Opt. 24, 65, 151, 215 (1945). M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 484.

1937 (1)

E. Lau, Physik. Z. 38, 446 (1937). A. Blanc-Lapierre, M. Perrot, G. Peri, Compt. Rend. 236, 1540 (1953). P. Lacomme, Opt. Acta 1, 33 (1954). G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955). A. Lohmann, Opt. Acta 3, 97 (1956). D. Gabor, in Astronomical Optics, Z. Kopal, ed. (North-Holland, Amsterdam, 1956), p. 59. J. M. Cowley, A. F. Moody, Proc. Phys. Soc. (London) B70, 486, 497, 505 (1957) and Proc. Phys. Soc. (London) B76, 378 (1960). H. Wolter, Physica 24, 457 (1958). G. Nomarski, Abbilden und Sehen, H. Schober, R. Röhler, eds. (Akademische Buchdruckerei F. Straub, Munich, 1962), p. 172.
[CrossRef]

Duffieux, P. M.

P. M. Duffieux, G. Lansraux, Rev. Opt. 24, 65, 151, 215 (1945). M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 484.

P. M. Duffieux, L’intégrale de Fourier et ses applications à l’optique (Oberthur, Rennes, 1946). H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408, 424 (1953).

Françon, M.

M. Françon, Nuovo Cimento Suppl. 9, 283 (1952). W. Lukosz, Z. Naturforsch. 18a, 436 (1963). W. Lukosz, M. Marchand, Opt. Acta 10, 241 (1963). B. Morgenstern, D. P. Paris, “A Method for Exceeding the Cutoff Frequency of a Bandlimited Optical System”,IBM San Jose Tech. Rept. TR 02.281 (1963).
[CrossRef]

Gärtner, W.

W. Gärtner, A. Lohmann, Z. Physik 174, 18 (1963).
[CrossRef]

Lansraux, G.

P. M. Duffieux, G. Lansraux, Rev. Opt. 24, 65, 151, 215 (1945). M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 484.

Lau, E.

E. Lau, Physik. Z. 38, 446 (1937). A. Blanc-Lapierre, M. Perrot, G. Peri, Compt. Rend. 236, 1540 (1953). P. Lacomme, Opt. Acta 1, 33 (1954). G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955). A. Lohmann, Opt. Acta 3, 97 (1956). D. Gabor, in Astronomical Optics, Z. Kopal, ed. (North-Holland, Amsterdam, 1956), p. 59. J. M. Cowley, A. F. Moody, Proc. Phys. Soc. (London) B70, 486, 497, 505 (1957) and Proc. Phys. Soc. (London) B76, 378 (1960). H. Wolter, Physica 24, 457 (1958). G. Nomarski, Abbilden und Sehen, H. Schober, R. Röhler, eds. (Akademische Buchdruckerei F. Straub, Munich, 1962), p. 172.
[CrossRef]

Lindenblad, N. E.

N. E. Lindenblad, U.S. Patent2,443,258 (1948). A. I. Kartashev, Opt. Spectry. 9, 204 (1960).

Lohmann, A.

W. Gärtner, A. Lohmann, Z. Physik 174, 18 (1963).
[CrossRef]

Nuovo Cimento Suppl. (1)

M. Françon, Nuovo Cimento Suppl. 9, 283 (1952). W. Lukosz, Z. Naturforsch. 18a, 436 (1963). W. Lukosz, M. Marchand, Opt. Acta 10, 241 (1963). B. Morgenstern, D. P. Paris, “A Method for Exceeding the Cutoff Frequency of a Bandlimited Optical System”,IBM San Jose Tech. Rept. TR 02.281 (1963).
[CrossRef]

Physik. Z. (1)

E. Lau, Physik. Z. 38, 446 (1937). A. Blanc-Lapierre, M. Perrot, G. Peri, Compt. Rend. 236, 1540 (1953). P. Lacomme, Opt. Acta 1, 33 (1954). G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955). A. Lohmann, Opt. Acta 3, 97 (1956). D. Gabor, in Astronomical Optics, Z. Kopal, ed. (North-Holland, Amsterdam, 1956), p. 59. J. M. Cowley, A. F. Moody, Proc. Phys. Soc. (London) B70, 486, 497, 505 (1957) and Proc. Phys. Soc. (London) B76, 378 (1960). H. Wolter, Physica 24, 457 (1958). G. Nomarski, Abbilden und Sehen, H. Schober, R. Röhler, eds. (Akademische Buchdruckerei F. Straub, Munich, 1962), p. 172.
[CrossRef]

Rev. Opt. (1)

P. M. Duffieux, G. Lansraux, Rev. Opt. 24, 65, 151, 215 (1945). M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 484.

Z. Physik (1)

W. Gärtner, A. Lohmann, Z. Physik 174, 18 (1963).
[CrossRef]

Other (2)

N. E. Lindenblad, U.S. Patent2,443,258 (1948). A. I. Kartashev, Opt. Spectry. 9, 204 (1960).

P. M. Duffieux, L’intégrale de Fourier et ses applications à l’optique (Oberthur, Rennes, 1946). H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408, 424 (1953).

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Figures (8)

Fig. 1
Fig. 1

Principal arrangement of Gärtner and Lohmann’s experiment. L1, L2, and L3 are lenses with focal length f. POL and AN are polarizers. P1 and P2 are two identical birefringent prisms. SP, OP, FP and IP are source plane, object plane, Fraunhofer plane, and image plane, respectively.

Fig. 2
Fig. 2

Orientation of the coordinate systems in the four major planes of Fig. 1.

Fig. 3
Fig. 3

Effective pupil areas in the Fraunhofer plane if (a) Rochon or Sénarmont and (b) Wollaston prisms are used. RA/2 is the cutoff frequency for the coherent case.

Fig. 4
Fig. 4

(1) and (2) are the effective pupils for S > RA. For ϑ0ϑπ/2 only the single combinations (1,1) and (2, 2) will contribute to the total transfer function. Both contributions are identical and equal to D0. If −ϑ0 < ϑ < ϑ0 the additional combination (1, 2) will also contribute.

Fig. 5
Fig. 5

Transfer function D(R, ϑ; S) of the Gärtner-Lohmann experiment for incoherently illuminated objects using a diffraction limited lens. RA is the cutoff frequency of the lens. For small angles α it follows that S = α/λ, where α is the angle between the directions of propagation of the two plane waves emerging from each of the prisms.

Fig. 6
Fig. 6

Frequency vector R and time-dependent shift vector S(t) in the Fraunhofer plane. S rotates uniformly.

Fig. 7
Fig. 7

Transfer function D(R;S) of the modified Gärtner-Lohmann experiment for incoherently illuminated objects using a diffraction limited lens. RA is the cutoff frequency of the lens.

Fig. 8
Fig. 8

Proposed experimental arrangement for resolution increase in a microscope with high numerical aperture. (SP = source plane; OP = object plane; MO1, MO2 = microobjectives; POL, AN = polarizers; P1, P2 = Wollaston prisms; EP = eyepiece.)

Equations (33)

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R = r λ f .
e i k · r ,
k · k = ( 2 π λ ) 2 .
k = k z z 0 2 π R s ,
k z = 2 π 1 λ 2 R s · R s .
e i [ k z z 0 + 2 π ( ε R s ) ] · r ,
k z = 2 π 1 λ 2 ( ε R s ) · ( ε R s ) .
e 2 π i ( R s ) · r ,
u ( r ) e 2 π i ( R s ) · r ε .
u ˜ ( R + R s ε ) ,
p L ( R ) = p L ( λ f R ) .
u ˜ ( R + R s ε ) p L ( R ) .
u ˜ ( R + R s ε ) p L ( R ) = u ˜ ( R + R s ε ) p L ( R ) δ ( R R ) d A ,
δ ( R R ) e 2 π i ( R ε ) · r .
υ ( r , ε ) = u ˜ ( R + R s ε ) p L ( R ) e 2 π i ( R ε ) · r d A .
υ ( r , ε ) = e 2 π i R s · r u ˜ ( R ) p L ( R R s + ε ) e 2 π i R · r d A .
υ ( r ) = υ ( r , ε ) + υ ( r , ε ) = e 2 π i R s · r u ˜ ( R ) [ p L ( R R s + ε ) + p L ( R R s + ε ) ] e 2 π i R · r d A .
υ ( r ) = u ˜ ( R ) [ p L ( R + ε ) + p L ( R + ε ) ] e 2 π i R · r d A .
D coh ( R ) = p L ( R + ε ) + p L ( R + ε ) .
( a ) Rochon prism ε = 0 ε = α λ R x ( 0 ) , ( b ) Sénarmont prism ε = α λ R x ( 0 ) ε = 0 , ( c ) Wollaston prism ε = α 2 λ R x ( 0 ) ε = α 2 λ R x ( 0 ) ,
p L ( R + ε ) + p L ( R + ε ) .
D ( R ) = 1 C p ( R + R 2 ) p * ( R R 2 ) d A ,
D incoh ( R ) = 1 C [ p L ( R + ε + R 2 ) p L * ( R + ε R 2 ) + p L ( R + ε + R 2 ) p L * ( R + ε R 2 ) + p L ( R + ε + R 2 ) p L * ( R + ε R 2 ) + p L ( R + ε + R 2 ) p L * ( R + ε R 2 ) ] d A ,
ε = 1 2 ( ε + S ) S = ε ε , ε = 1 2 ( ε S ) ε = ε + ε .
D incoh ( R ) = 1 C [ D L ( R ) + D L ( R + S ) + D L ( R S ) 2 ] ,
D 0 ( R ) = { 2 π [ cos 1 R R A R R A 1 ( R R A ) 2 ] for | R | R A 0 for | R | R A .
D incoh ( R , ϑ ; S ) = 1 C [ D 0 ( R ) + D 0 ( R 2 + S 2 + 2 R S cos ϑ ) + D 0 ( R 2 + S 2 2 R S cos ϑ ) 2 ] ,
D ( R , ϑ ; S ) = D ( R , π ϑ ; S ) = D ( R , π + ϑ ; S ) = D ( R , ϑ ; S ) .
sin ϑ 0 = R A / S .
α Sén . = α Ro . = 1 2 α Woll .
D ( R ; S ) = 1 π 0 π D incoh ( R ; S ) d φ = 1 C { D L ( R ) + 1 2 π 0 π [ D L ( R + S ) + D L ( R S ) d φ ] } .
D ( R , θ; S ) = 1 C { D 0 ( R ) + 1 4 π 0 2 π [ D 0 ( R 2 + S 2 + 2 R S cos ( θ φ ) ) + D 0 ( R 2 + S 2 2 R S cos ( θ φ ) ) ] d φ } ,
D ( R ; S ) = 1 C { D 0 ( R ) + 1 π 0 π / 2 [ D 0 ( R 2 + S 2 + 2 R S cos φ ) + D 0 ( R 2 + S 2 2 R S cos φ ) ] d φ } .

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