Abstract

Objects which vary strongly as a function of x but only slowly in y occupy a cigar-shaped area in the spatial-frequency domain. Such an object spectrum is badly matched to the frequency-transfer domain of a lens, which is usually circular. By means of spatial modulation, the object spectrum can be adapted to the transfer domain of the lens. In this way, the one-dimensional resolution limit or bandwidth of the lens can be overcome, as shown by experiment and theory.

© 1966 Optical Society of America

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References

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  1. M. Françon, Nuovo Cimento Suppl. 9, 283 (1952);W. Lukosz, Z. Naturforsch. 18a, 436 (1963);W. Lukosz and M. Marchand, Opt. Acta 10, 241 (1963);B. Morgenstern and D. P. Paris, J. Opt. Soc. Am. 54, 1282 (1964);A. Lohmann and D. P. Paris, J. Opt. Soc. Am. 54, 579 (1964).
    [Crossref]
  2. A. Lohmann, Opt. Acta 3, 97, (1957);W. Gartner and A. Lohmann, Z. Physik 174, 18, (1963);A. Lohmann and D. P. Paris, Appl. Opt. 3, 1037 (1964).
    [Crossref]
  3. A. J. Downes, Brit. Patent129747 (1918);N. E. Lindenblad, U. S. Patent2,443,258 (1948);A. I. Kartashev, Opt. Spectry. 9, 204 (1960);J. D. Armitage, A. Lohmann, and D. P. Paris, Japan. J. Appl. Phys. Suppl. 1 4, 273 (1965);H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1073, 1078, (1964).
    [Crossref]
  4. A. Lohmann and D. P. Paris, J. Opt. Soc. Am. 55, 1007 (1965).
    [Crossref]
  5. M. Herzberger, Optik 22, 645 (1965).

1965 (2)

1957 (1)

A. Lohmann, Opt. Acta 3, 97, (1957);W. Gartner and A. Lohmann, Z. Physik 174, 18, (1963);A. Lohmann and D. P. Paris, Appl. Opt. 3, 1037 (1964).
[Crossref]

1952 (1)

M. Françon, Nuovo Cimento Suppl. 9, 283 (1952);W. Lukosz, Z. Naturforsch. 18a, 436 (1963);W. Lukosz and M. Marchand, Opt. Acta 10, 241 (1963);B. Morgenstern and D. P. Paris, J. Opt. Soc. Am. 54, 1282 (1964);A. Lohmann and D. P. Paris, J. Opt. Soc. Am. 54, 579 (1964).
[Crossref]

Downes, A. J.

A. J. Downes, Brit. Patent129747 (1918);N. E. Lindenblad, U. S. Patent2,443,258 (1948);A. I. Kartashev, Opt. Spectry. 9, 204 (1960);J. D. Armitage, A. Lohmann, and D. P. Paris, Japan. J. Appl. Phys. Suppl. 1 4, 273 (1965);H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1073, 1078, (1964).
[Crossref]

Françon, M.

M. Françon, Nuovo Cimento Suppl. 9, 283 (1952);W. Lukosz, Z. Naturforsch. 18a, 436 (1963);W. Lukosz and M. Marchand, Opt. Acta 10, 241 (1963);B. Morgenstern and D. P. Paris, J. Opt. Soc. Am. 54, 1282 (1964);A. Lohmann and D. P. Paris, J. Opt. Soc. Am. 54, 579 (1964).
[Crossref]

Herzberger, M.

M. Herzberger, Optik 22, 645 (1965).

Lohmann, A.

A. Lohmann and D. P. Paris, J. Opt. Soc. Am. 55, 1007 (1965).
[Crossref]

A. Lohmann, Opt. Acta 3, 97, (1957);W. Gartner and A. Lohmann, Z. Physik 174, 18, (1963);A. Lohmann and D. P. Paris, Appl. Opt. 3, 1037 (1964).
[Crossref]

Paris, D. P.

J. Opt. Soc. Am. (1)

Nuovo Cimento Suppl. (1)

M. Françon, Nuovo Cimento Suppl. 9, 283 (1952);W. Lukosz, Z. Naturforsch. 18a, 436 (1963);W. Lukosz and M. Marchand, Opt. Acta 10, 241 (1963);B. Morgenstern and D. P. Paris, J. Opt. Soc. Am. 54, 1282 (1964);A. Lohmann and D. P. Paris, J. Opt. Soc. Am. 54, 579 (1964).
[Crossref]

Opt. Acta (1)

A. Lohmann, Opt. Acta 3, 97, (1957);W. Gartner and A. Lohmann, Z. Physik 174, 18, (1963);A. Lohmann and D. P. Paris, Appl. Opt. 3, 1037 (1964).
[Crossref]

Optik (1)

M. Herzberger, Optik 22, 645 (1965).

Other (1)

A. J. Downes, Brit. Patent129747 (1918);N. E. Lindenblad, U. S. Patent2,443,258 (1948);A. I. Kartashev, Opt. Spectry. 9, 204 (1960);J. D. Armitage, A. Lohmann, and D. P. Paris, Japan. J. Appl. Phys. Suppl. 1 4, 273 (1965);H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1073, 1078, (1964).
[Crossref]

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Figures (7)

F. 1
F. 1

Essentially one-dimensional object. (a) Spatial-frequency spectrum; circle indicates transfer domain of a lens, (b) x-dependent object with finite y-extension.

F. 2
F. 2

Setup of superresolution experiment. O: object, in contact with mask M1, which is slightly rotated around optical axis. The moiré fringes from O and M1 are imaged by lens L into plane MO, where the second mask M2 is inserted. Cylinder lens C smears out y-structures and yields image I.

F. 3
F. 3

Explanation in spatial-frequency domain. (a) object, (b) mask M1, (c) effective object consisting of O and M1, (d) transfer domain of lens L, (e) passed through lens, (f) after interacting with mask M2, (g) y-smearing by νy low-pass, (h) final image.

F. 4
F. 4

Experimental setup. A: Aperture as in Fig. 3(d). F: Filter for y-smearing as in Fig. 3(g).

F. 5
F. 5

Experimental result. The object frequency increases slowly from the left to the right. Only the lower-right part of the field is covered with masks M1 and M2.

F. 6
F. 6

Visualization of condition tg φΔy=Nd/cosφ.

F. 7
F. 7

Optical transfer function (incoherent). TL for aberration-free lens with square aperture. T for over-all process.

Equations (31)

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ũ ( ν x , ν y ) = u ( x , y ) e 2 π i ( x ν x + y ν y ) dxdy
u ( x , y ) = u ( x ) rect ( y / Δ y ) ; rect ( α ) = { 1 if | α | 1 2 0 if | α | > 1 2 .
u ( x , y ) = 1 + cos ( 2 π ν x ) .
M ( x , y ) = 1 + cos { 2 π ν 0 ( x cos φ + y sin φ ) } .
u E ( x , y ) = u ( x , y ) M ( x , y ) = u + M 1 + cos ( 2 π ν x ) cos { 2 π ν 0 ( x cos φ + y sin φ ) } .
1 2 cos { 2 π [ x ( ν + ν 0 cos φ ) + y ν 0 sin φ ] } + 1 2 cos { 2 π [ x ( ν ν 0 cos φ ) y ν 0 sin φ ] } .
ν + 2 = ( ν + ν 0 cos φ ) 2 + ν 0 2 sin 2 φ = ν 2 + ν 0 2 + 2 ν ν 0 cos φ ; ν 2 = ν 2 + ν 0 2 2 ν ν 0 cos φ .
1 + 1 2 cos { 2 π [ x ( ν ν 0 cos φ ) y ν 0 sin φ ] }
1 + 1 2 cos { 2 π [ x ( ν ν 0 cos φ ) y ν 0 sin φ ] } 1 + cos { 2 π ν 0 ( x cos φ + y sin φ ) } = 1 + 1 2 cos { 2 π [ ] } + cos { } + 1 2 cos { 2 π [ ] } + cos { } .
1 4 cos ( 2 π ν x ) + 1 4 cos { 2 π [ x ( ν 2 ν 0 cos φ ) 2 y ν 0 sin φ ] } .
u ι ( x , y ) = 1 + 1 4 cos ( 2 π ν x ) .
u ( x , y ) = u ( x ) rect ( y / Δ y ) .
u ( x ) = ũ ( ν x ) w ( x ν x ) d ( ν x ) ; w ( α ) = exp ( 2 π i α ) rect ( y / Δ y ) = Δ y sinc( π ν y Δ y ) w ( y ν y ) d ν y ; sinc ( α ) = sin α / α M 1 ( x ) = A n w ( n ν 0 x ) .
M 1 ( x cos φ + y sin φ ) .
u E ( x , y ) = A n ũ ( ν x ) sinc ( π ν y Δ y ) w { } d ν x d ν y w { } = w { x ( ν x + n ν 0 cos φ ) + y ( ν y + n ν 0 sin φ ) } .
w { } w { } T L ( ν x + n ν 0 cos φ , ν y + n ν 0 sin φ ) .
u 2 ( x , y ) = A n ũ sinc T L ( ) w { } d ν x d ν y .
M 2 ( x cos φ + y sin φ ) = B m w { m ν 0 ( x cos φ + y sin φ ) } .
υ ( x , y ) = u 2 ( x , y ) M 2 ( x , y ) = A n B m ũ sin T L w { } d ν x d ν y w { } = w { x [ ν x + ( n + m ) ν 0 cos φ ] + y [ ν y + ( n + m ) ν 0 sin φ ] } .
sinc { π Δ y [ μ y ( n + m ) ν 0 sin φ ] } w ( y μ y ) d μ y .
sinc { π Δ y [ μ y ( n + m ) ν 0 sin φ ] } δ ( μ y ) d μ y = sinc { π Δ y ( n + m ) ν 0 sin φ } .
ν 0 Δ y sin φ = N ( integer ) .
υ ( x ) = A n B n ũ ( ν x ) T L w ( x ν x ) d ν x T L = T L ( ν x + n ν 0 cos φ , n ν 0 sin φ ) .
u ( x ) = ũ ( ν x ) w ( x ν x ) d ν x ,
T ( ν x ) = A n B n T L ( ν x + n ν 0 cos φ , n ν 0 sin φ ) A n B n T L ( n ν 0 cos φ , n ν 0 sin φ ) .
ν 0 Δ y sin φ = N ( integer )
Δ y t g φ = N d / cos φ ( d = 1 / ν 0 ) .
A n = B n = 1 / n 0 sinc ( π n / n 0 ) . ( A n / A 0 ) 2 = sinc 2 ( π n / n 0 ) = C n T L ( ν x , ν y ) = [ 1 | ν x | / ν A ] [ 1 | ν y | / ν A ] ; zero if | ν x | , | ν y | > ν A T ( ν x ) = C n T L ( ν x + n ν A , n ν A t g φ ) .
{ Δ ν y } L n 0 { Δ ν y } 0 n 0 { Δ ν y } I
{ Δ ν x } I n 0 { Δ ν x } L .
{ Δ ν x Δ ν y } IMG { Δ ν x Δ ν y } LENS .