Abstract

The application of optical transfer theory to the process of image formation requires that the image-forming system be linear and space invariant. In a space-invariant system, the point image retains its shape while the point source explores the object plane. The purpose of this paper is to investigate image-forming systems which are linear but space variant. Such systems may exceed performance limitations which are inherent in linear space-invariant systems. A method for experimentally determining space variance is devised. The degree of space invariance is defined and evaluated for several examples of space-variant systems.

© 1965 Optical Society of America

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References

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  1. R. V. Pole (Thomas J. Watson Research Center, Yorktown Heights, New York) proposed the term “space invariance” instead of the formerly used “stationarity.” The term “nonisoplanatism” which has been used in this context, refers traditionally to the space variance caused by aberrations only.
  2. P. Lacomme, Opt. Acta 1, 33 (1954).
    [Crossref]
  3. H. Wolter, Physica 24, 457 (1958).
    [Crossref]
  4. J. W. Menter, Advan. Phys. 7, 299 (1958).
    [Crossref]
  5. E. Lau, Physik. Z. 38, 446 (1937);A. Lohmann and D. Paris, Optik 22, 226 (1965).
  6. A. Lohmann, Hausmitteilungen J. Schneider und Co. 14, 52 (1962).
  7. Integrals without limits of integration are here and henceforth to be taken from −∞ to +∞.
  8. H. Wolter, Arch. Elek. Übertragung 13, 393 (1959);Opt. Acta 7, 53 (1960).
  9. Unfortunately, this term implies that “invariance” can have a variable degree, while really something is either invariant or it is not. However, so far we were unable to find a more appropriate term.
  10. P. B. Fellgett and E. H. Linfoot, Phil. Trans. Roy. Soc. London A247, 369 (1955).

1962 (1)

A. Lohmann, Hausmitteilungen J. Schneider und Co. 14, 52 (1962).

1959 (1)

H. Wolter, Arch. Elek. Übertragung 13, 393 (1959);Opt. Acta 7, 53 (1960).

1958 (2)

H. Wolter, Physica 24, 457 (1958).
[Crossref]

J. W. Menter, Advan. Phys. 7, 299 (1958).
[Crossref]

1955 (1)

P. B. Fellgett and E. H. Linfoot, Phil. Trans. Roy. Soc. London A247, 369 (1955).

1954 (1)

P. Lacomme, Opt. Acta 1, 33 (1954).
[Crossref]

1937 (1)

E. Lau, Physik. Z. 38, 446 (1937);A. Lohmann and D. Paris, Optik 22, 226 (1965).

Fellgett, P. B.

P. B. Fellgett and E. H. Linfoot, Phil. Trans. Roy. Soc. London A247, 369 (1955).

Lacomme, P.

P. Lacomme, Opt. Acta 1, 33 (1954).
[Crossref]

Lau, E.

E. Lau, Physik. Z. 38, 446 (1937);A. Lohmann and D. Paris, Optik 22, 226 (1965).

Linfoot, E. H.

P. B. Fellgett and E. H. Linfoot, Phil. Trans. Roy. Soc. London A247, 369 (1955).

Lohmann, A.

A. Lohmann, Hausmitteilungen J. Schneider und Co. 14, 52 (1962).

Menter, J. W.

J. W. Menter, Advan. Phys. 7, 299 (1958).
[Crossref]

Pole, R. V.

R. V. Pole (Thomas J. Watson Research Center, Yorktown Heights, New York) proposed the term “space invariance” instead of the formerly used “stationarity.” The term “nonisoplanatism” which has been used in this context, refers traditionally to the space variance caused by aberrations only.

Wolter, H.

H. Wolter, Arch. Elek. Übertragung 13, 393 (1959);Opt. Acta 7, 53 (1960).

H. Wolter, Physica 24, 457 (1958).
[Crossref]

Advan. Phys. (1)

J. W. Menter, Advan. Phys. 7, 299 (1958).
[Crossref]

Arch. Elek. Übertragung (1)

H. Wolter, Arch. Elek. Übertragung 13, 393 (1959);Opt. Acta 7, 53 (1960).

Hausmitteilungen J. Schneider und Co. (1)

A. Lohmann, Hausmitteilungen J. Schneider und Co. 14, 52 (1962).

Opt. Acta (1)

P. Lacomme, Opt. Acta 1, 33 (1954).
[Crossref]

Phil. Trans. Roy. Soc. London (1)

P. B. Fellgett and E. H. Linfoot, Phil. Trans. Roy. Soc. London A247, 369 (1955).

Physica (1)

H. Wolter, Physica 24, 457 (1958).
[Crossref]

Physik. Z. (1)

E. Lau, Physik. Z. 38, 446 (1937);A. Lohmann and D. Paris, Optik 22, 226 (1965).

Other (3)

Integrals without limits of integration are here and henceforth to be taken from −∞ to +∞.

Unfortunately, this term implies that “invariance” can have a variable degree, while really something is either invariant or it is not. However, so far we were unable to find a more appropriate term.

R. V. Pole (Thomas J. Watson Research Center, Yorktown Heights, New York) proposed the term “space invariance” instead of the formerly used “stationarity.” The term “nonisoplanatism” which has been used in this context, refers traditionally to the space variance caused by aberrations only.

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Figures (2)

F. 1
F. 1

Space-variant image formation. Line-spread function A(xx′; x′) of the space-variant image-forming system SVS changes its shape, while the line source δ(xx′) explores the object plane. O: object plane, I: image plane.

F. 2
F. 2

Measurement of C(ξ,x′,x″). Two slits at x′ and x″ in the object plane O of the space-variant image-forming system SVS are coherently illuminated. A double slit (width ξ) travels across the image plane I. The contrast of the interference fringes in plane F serves as a measure for the cross-correlation function C. P., polarizer; A, analyzer; HW, half-wave plates; f, focal length of lens L.

Equations (45)

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δ ( x x ) A ( x x ; x ) , u ( x ) δ ( x x ) u ( x ) A ( x x ; x ) , u ( x ) = u ( x ) δ ( x x ) d x u ( x ) A ( x x ; x ) d x ,
υ ( x ) = u ( x ) A ( x x ; x ) d x .
A ( x x ; x ) A ( x x ) .
C ( ξ , x , x ) = A ( x + ξ / 2 x ; x ) A * ( x ξ / 2 x ; x ) d x = C * ( ξ , x , x ) ,
A ( x 0 + ξ / 2 x ; x ) e 2 π i R ( x 0 + ξ / 2 ) ,
y = λ f R .
A ( x 0 ξ / 2 x ; x ) e 2 π i R ( x 0 ξ / 2 ) .
I ( R ; ξ , x , x ) = C ( 0 , x , x ) + C ( 0 , x , x ) + 2 Re [ C ( ξ , x , x ) e 2 π i R ξ ] .
I ( R ; ξ , x , x ) = C ( 0 , x , x ) + C ( 0 , x , x ) + 2 | C ( ξ , x , x ) | cos [ 2 π R ξ + α ( ξ , x , x ) ] .
Γ ( x x ) = ( source ) S ( R ) e 2 π i R ( x x ) d R ,
Φ OB ( ξ ) = u 1 ( x + ξ / 2 ) u 2 * ( x ξ / 2 ) d x .
Φ OB ( ξ ) = ũ 1 ( R ) ũ 2 * ( R ) e 2 π i R ξ d R ,
ũ ( R ) = u ( x ) e 2 π iRx d x .
Φ IM ( ξ ) = υ 1 ( x + ξ / 2 ) υ 2 * ( x ξ / 2 ) d x = υ 1 ( x ) υ 2 * ( x ) A ( x + ξ / 2 x ; x ) × A * ( x ξ / 2 x ; x ) dxd x d x .
Φ IM ( ξ ) = u 1 ( x ) u 2 ( x ) C ( ξ , x , x ) d x d x .
A ( x x ; x ) = { A p ( x x ) for x > 0 A p ( x x ) for x < 0 ,
A p ( x + ξ / 2 x ) A n * ( x ξ / 2 x ) d x
u 1 ( x ) = 0 for x 0 u 2 ( x ) = 0 for x 0 .
υ 1 ( x ) = u 1 ( x ) A p ( x x ) d x υ 2 ( x ) = u 2 ( x ) A n ( x x ) d x ,
Φ IM ( ξ ) = u 1 ( x ) u 2 * ( x ) × [ A p ( x + ξ / 2 x ) A n * ( x ξ / 2 x ) d x ] d x d x = u 1 ( x ) u 2 * ( x ) C p n ( ξ x + x ) d x d x .
Φ IM ( ξ ) = Φ OB ( η ) C p n ( ξ η ) d η .
Φ IM ( R ) = Φ OB ( R ) C p n ( R ) .
Φ OB = ũ 1 ũ 2 * , Φ IM = υ 1 υ 2 * , C p n = à p à n * .
σ ( x , x ) = C ( x x , x , x ) / [ C ( 0 , x , x ) C ( 0 , x , x ) ] 1 2 .
| σ ( x , x ) | 1 .
à ( R ; x ) = A ( x ; x ) e 2 π iRx d x ,
σ ( x , x ) = Ã ( R ; x ) Ã * ( R ; x ) d R [ | Ã ( R ; x ) | 2 d R | Ã ( R ; x ) | 2 d R ] 1 2 = σ * ( x , x ) .
I I = H σ ( x , x ) d x d x / H d x d x .
σ ( x , x ) = { 1 if either x , x > 0 or x , x < 0 σ p n if x > 0 , x < 0 σ p n * if x < 0 , x > 0 ,
σ p n = A p ( x ) A n * ( x ) d x [ | A p ( x ) | 2 d x | A n ( x ) | 2 d x ] 1 2 .
= ( 1 + Re σ p n ) / 2 .
A ( x x ; x ) = A 0 ( x x ) M ( x ) ,
σ ( x , x ) = e i [ φ ( x ) φ ( x ) ] ,
= | lim h 1 h h / 2 h / 2 e i φ ( x ) d x | 2 .
A ( x x ; x ) = A 0 [ x x d ( x ) ] ,
à 0 ( R ) = A 0 ( x ) e 2 π iRx d x
σ ( x , x ) = | Ã 0 ( R ) | 2 e 2 π i R [ d ( x ) d ( x ) ] d R | Ã 0 ( R ) | 2 d R .
p ( ρ ; x ) = | p ( ρ ; x ) | e i k W ( ρ ; x ) ,
p ( ρ ; x ) = { e i k W ( ρ ; x ) for | ρ | b 0 for | ρ | > b .
σ ( x , x ) = 1 2 b b b e i k [ W ( ρ ; x ) W ( ρ ; x ) ] d ρ .
| σ ( x , x ) 1 | ,
| 1 2 b b b { e i k [ W ( ρ ; x ) W ( ρ ; x ) ] 1 } d ρ | .
W ( ρ ; x + Δ x ) W ( ρ ; x ) = n = 1 [ δ n W ( ρ ; x ) δ x n ] x = x ( Δ x ) n n ! .
| m = 1 n = 1 ( i k ) m m ! n ! ( Δ x ) n W m n av | ,
W m n av = 1 2 b b b [ δ n W ( ρ ; x ) δ x n ] x = x m d ρ .