Abstract

A general theory of optical-aberration tests, encompassing many well-known practical tests, is presented, utilizing the formalism of spatial filtering, employing coherent illumination. The basic diffraction integrals of the Ronchi-grating test, Foucault knife-edge test, phase Foucault knife-edge test, and the Zernike phase-contrast test are derived. Typical situations have been analyzed and numerical results are shown and discussed.

© 1969 Optical Society of America

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References

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  1. L. C. Martin, Technical Optics, Vol. 2 (Pitman and Sons, Ltd., London, 1960), 2nd ed.
  2. I. Adachi, Atti Fond. Ronchi 15, 461, 550 (1960).
  3. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).
  4. M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley–Interscience, Inc., New York, 1965).
  5. In the integrations in this and succeeding sections, numerical constants have often been omitted, because the final normalization can be performed at the end of the analysis.
  6. G. Toraldo di Francia, in Optical Image Evaluation Symposium, Natl. Bur. Std. (U.S.) Circ. 526 (U. S. Gov’t Printing Office, Washington, D. C., 1954), p. 161.
  7. P. Erdös, J. Opt. Soc. Am. 49, 865 (1959).
    [CrossRef]
  8. H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, Oxford, 1950).
  9. Adachi has previously solved the case of an aberration-free defocused system and performed some low-frequency grating experiments, I. Adachi, Atti Fond. G. Ronchi,  18, 344 (1963).
  10. A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill Book Co., New York, 1962), p. 38.
  11. E. H. Linfoot, Recent Advances in Optics, 2 (Oxford University Press, Oxford, 1958), Ch. 2.
  12. H. Wolther in Handbuch der Physik 24, S. Flügge, Ed. (Springer–Verlag, Berlin, 1956), p. 258.
  13. E. H. Linfoot, Proc. Phys. Soc. (London) 58, 759 (1946).
    [CrossRef]
  14. S. C. Gascoigne, Monthly Not. Roy. Astron. Soc. 104, 326 (1945).Gascoigne treats the two-dimensional model. The author has rederived all of his results, using the theory of prolate spheroidal functions.
  15. F. Zernike, Physica 1, 689 (1934).
    [CrossRef]
  16. C. R. Burch, Monthly Not. Roy. Astron. Soc. 94, 384 (1934).

1963 (1)

Adachi has previously solved the case of an aberration-free defocused system and performed some low-frequency grating experiments, I. Adachi, Atti Fond. G. Ronchi,  18, 344 (1963).

1960 (1)

I. Adachi, Atti Fond. Ronchi 15, 461, 550 (1960).

1959 (1)

1946 (1)

E. H. Linfoot, Proc. Phys. Soc. (London) 58, 759 (1946).
[CrossRef]

1945 (1)

S. C. Gascoigne, Monthly Not. Roy. Astron. Soc. 104, 326 (1945).Gascoigne treats the two-dimensional model. The author has rederived all of his results, using the theory of prolate spheroidal functions.

1934 (2)

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

C. R. Burch, Monthly Not. Roy. Astron. Soc. 94, 384 (1934).

Adachi, I.

Adachi has previously solved the case of an aberration-free defocused system and performed some low-frequency grating experiments, I. Adachi, Atti Fond. G. Ronchi,  18, 344 (1963).

I. Adachi, Atti Fond. Ronchi 15, 461, 550 (1960).

Burch, C. R.

C. R. Burch, Monthly Not. Roy. Astron. Soc. 94, 384 (1934).

Erdös, P.

Gascoigne, S. C.

S. C. Gascoigne, Monthly Not. Roy. Astron. Soc. 104, 326 (1945).Gascoigne treats the two-dimensional model. The author has rederived all of his results, using the theory of prolate spheroidal functions.

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, Oxford, 1950).

Kay, I.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley–Interscience, Inc., New York, 1965).

Kline, M.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley–Interscience, Inc., New York, 1965).

Linfoot, E. H.

E. H. Linfoot, Proc. Phys. Soc. (London) 58, 759 (1946).
[CrossRef]

E. H. Linfoot, Recent Advances in Optics, 2 (Oxford University Press, Oxford, 1958), Ch. 2.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

Martin, L. C.

L. C. Martin, Technical Optics, Vol. 2 (Pitman and Sons, Ltd., London, 1960), 2nd ed.

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill Book Co., New York, 1962), p. 38.

Toraldo di Francia, G.

G. Toraldo di Francia, in Optical Image Evaluation Symposium, Natl. Bur. Std. (U.S.) Circ. 526 (U. S. Gov’t Printing Office, Washington, D. C., 1954), p. 161.

Wolther, H.

H. Wolther in Handbuch der Physik 24, S. Flügge, Ed. (Springer–Verlag, Berlin, 1956), p. 258.

Zernike, F.

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

Atti Fond. G. Ronchi (1)

Adachi has previously solved the case of an aberration-free defocused system and performed some low-frequency grating experiments, I. Adachi, Atti Fond. G. Ronchi,  18, 344 (1963).

Atti Fond. Ronchi (1)

I. Adachi, Atti Fond. Ronchi 15, 461, 550 (1960).

J. Opt. Soc. Am. (1)

Monthly Not. Roy. Astron. Soc. (2)

S. C. Gascoigne, Monthly Not. Roy. Astron. Soc. 104, 326 (1945).Gascoigne treats the two-dimensional model. The author has rederived all of his results, using the theory of prolate spheroidal functions.

C. R. Burch, Monthly Not. Roy. Astron. Soc. 94, 384 (1934).

Physica (1)

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

Proc. Phys. Soc. (London) (1)

E. H. Linfoot, Proc. Phys. Soc. (London) 58, 759 (1946).
[CrossRef]

Other (9)

L. C. Martin, Technical Optics, Vol. 2 (Pitman and Sons, Ltd., London, 1960), 2nd ed.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, Oxford, 1950).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley–Interscience, Inc., New York, 1965).

In the integrations in this and succeeding sections, numerical constants have often been omitted, because the final normalization can be performed at the end of the analysis.

G. Toraldo di Francia, in Optical Image Evaluation Symposium, Natl. Bur. Std. (U.S.) Circ. 526 (U. S. Gov’t Printing Office, Washington, D. C., 1954), p. 161.

A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill Book Co., New York, 1962), p. 38.

E. H. Linfoot, Recent Advances in Optics, 2 (Oxford University Press, Oxford, 1958), Ch. 2.

H. Wolther in Handbuch der Physik 24, S. Flügge, Ed. (Springer–Verlag, Berlin, 1956), p. 258.

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Figures (12)

F. 1
F. 1

Schematic of testing arrangement. The light source and auxiliary equipment is to the left of plane A which is the exit pupil of the lens under test, plane B is the modulating plane, plane C is the receiving plane (image of the exit pupil).

F. 2
F. 2

Schematic of Ronchi test, illustrating the influence of higher harmonics.

F. 3
F. 3

Fringe system caused by a lens having third-order spherical aberration C040= 1.5λ as viewed in the paraxial receiving plane. The Ronchi grating has a frequency, β = π/3.

F. 4
F. 4

Fringe system caused by a lens having optimum balanced third-order spherical aberration (C040 = − C020 = 1.5λ). The Ronchi grating has a frequency β=π/3.

F. 5
F. 5

Distribution of illuminance caused by a lens having third-order coma C131 = 0.125 λ as viewed in the paraxial receiving plane. The Foucault knife edge is perpendicular to the p axis; x0 = 1, y0 = 0.

F. 6
F. 6

Distribution of illuminance caused by a lens having third-order coma C131 = 0.125 λ as viewed in the paraxial receiving plane. The Foucault knife edge is set at 45° to the p axis; x0 = 0.707, y0 = 0.707.

F. 7
F. 7

Distribution of illuminance caused by a lens having third-order coma C131 = 0.125 λ as viewed in the paraxial receiving plane. The Foucault knife edge is parallel to the p axis; x0 = 0, y0 = 1.

F. 8
F. 8

Distribution of: illuminance caused by an aberration-free lens, as viewed in the paraxial receiving plane. The phase-retarding knife edge is perpendicular to the p axis and the amount of phase retardation is δ = 0.25 λ.

F. 9
F. 9

Distribution of illuminance caused by an aberration-free lens as viewed in the paraxial receiving plane. The phase-retarding knife edge is perpendicular to the p axis and the amount of phase retarding is δ = 0.50λ.

F. 10
F. 10

Distribution of illuminance caused by a lens having third-order coma C131 = 0.125 λ as viewed in the paraxial receiving plane. The phase-retarding knife edge is set at 45° to the p axis and the amount of phase retardation is δ = 0.50 λ.

F. 11
F. 11

Distribution of illuminance due to a lens having third-order astigmatism C131 = 0.0625 λ in the receiving plane C2 =0.0625 λ. The phase-retarding knife-edge is set at 45° to the p axis and the amount of phase retardation is δ = 0.25 λ.

F. 12
F. 12

Distribution of illuminance caused by a lens having third-order astigmatism C131 = 0.0625 λ in the receiving plane C2 = 0.0625 λ. The phase-retarding knife edge is set at 45° to the p axis and the amount of phase retardation is δ = 0.50 λ.

Equations (35)

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a ( x , y ) = aperture A ( p , q ) e i ( x p + y q ) d p d q ,
A ( p , q ) = { A 0 ( p , q ) e i k W ( p , q ) ( p , q aperture ) 0 ( p , q aperture ) .
p p / P m , q q / q m ,
υ p x p m , υ q y q m .
a ( υ p , υ q ) = aperture A ( p , q ) exp ( i υ p p + i υ q q ) d p d q .
à ( p 1 , q 1 ) = a ( υ p , υ q ) × exp ( i υ p p 1 i υ q q 1 ) d υ p d υ q .
à ( p 1 , q 1 ) = a ( υ p , υ q ) M ( υ p , υ q ) × exp ( i υ p p 1 i υ q q 1 ) d υ p d υ q .
à ( p 1 , q 1 ) = aperture A ( p , q ) d p d q M ( υ p , υ q ) × exp [ i ( p p 1 ) υ p + i ( q q 1 ) υ q ] d υ p d υ q .
M ( υ p ) = n = B n exp ( i n β υ p ) , ( β π L ) ,
à ( p 1 , q 1 ) = n = B n aperture A ( p , q ) d p d q × exp [ i υ p ( p p 1 + n β ) ] d υ p × exp [ i υ q ( q q 1 ) ] d υ q ,
= n = B n aperture A ( p p 1 + n β ) δ ( q q 1 ) d p d q .
b b f ( x ) δ ( x z ) d x = { f ( z ) | z | < b , 0 | z | > b ,
à ( p 1 , q 1 ) = { n = B n A n ( p 1 n β , q 1 ) ( p 1 n β ) 2 + q 1 2 1 , 0 ( p 1 n β ) 2 + q 1 2 > 1 .
W ( x 0 , y 0 , p , q ) = C 020 ( p 2 + q 2 ) + C 040 ( p 2 + q 2 ) 2 + C 311 ( x 0 2 + y 0 2 ) ( x 0 p + y 0 q ) + C 222 ( x 0 p + y 0 q ) 2 + C 220 ( x 0 2 + y 0 2 ) ( p 2 + q 2 ) + C 131 ( x 0 p + y 0 q ) ( p 2 + q 2 ) ,
W 020 = 1 2 ( N . A . ) 2 δ z δ z / 8 F 2 ,
M ( υ p ) = { 0 υ p < 0 , 1 υ p > 0 .
à ( p 1 , q 1 ) = aperture A ( p , d ) d p d q exp [ i ( q q 1 ) υ q ] d υ q × exp [ i ( p p 1 ) υ q ] d υ q , = aperture A ( p , d ) δ ( q q 1 ) d p d q × 0 exp [ i ( p p 1 ) υ p ] d υ p .
0 exp [ i ( p p 1 ) υ p ] d υ p = π δ ( p p 1 ) + 1 i ( p p 1 ) .
à ( p 1 , q 1 ) = π aperture A ( p , q ) δ ( q q 1 ) δ ( p p 1 ) d p d q + 1 i A ( p , q ) δ ( q q 1 ) ( p p 1 ) d p d q .
à ( p 1 , q 1 ) = { π A ( p 1 , q 1 ) + 1 i A ( p , q 1 ) ( p q 1 ) d p , ( p 1 , q 1 aperture ) , 1 i A ( p , q 1 ) ( p q 1 ) d p , ( p 1 , q 1 aperture ) .
B B A ( p , q 1 ) ( p q 1 ) d p = A ( p , q 1 ) ( p q 1 ) d p + A ( p , q 1 ) ( p q 1 ) d p .
M ( υ p ) = { 1 υ p < 0 , e i k δ υ p > 0 ,
à ( p 1 , q 1 ) = aperture A ( p , q ) d p d q exp [ i ( q q 1 ) υ q ] d υ q × M ( υ q ) exp [ i ( p p 1 ) υ p ] d υ p = aperture A ( p , d ) δ ( q q 1 ) d p d q M ( υ p ) × exp [ i ( p p 1 ) υ p ] d υ p .
M ( υ p ) exp [ i ( p p 1 ) υ p ] d υ p = 0 exp [ i ( p p 1 ) υ p ] d υ p + e i k δ 0 exp [ i ( p p 1 ) υ p ] d υ p = 0 exp [ i ( p p 1 ) υ p ] d υ p + e i k δ 0 exp [ i ( p p 1 ) υ p ] d υ p = π δ ( p p 1 ) ( e i k δ + 1 ) + [ i ( p p 1 ) ] 1 ( e i k δ 1 ) .
à ( p 1 , q 1 ) = { π ( e i k δ + 1 ) A ( p 1 , q 1 ) i ( e i k δ 1 ) A ( p , q 1 ) ( p q 1 ) d p , i ( e i k δ 1 ) A ( p , q 1 ) ( p p 1 ) d p .
M ( υ p , υ q ) = M ( υ ) = { 0 υ < υ 0 , 1 υ > υ 0 ,
à ( p 1 q 1 ) = aperture A ( p , q ) exp [ i ( υ p p + υ q q ) ] d p d q × M ( υ ) exp [ i ( υ p p 1 + υ q q 1 ) ] d υ p d υ q .
υ p = υ cos ϕ , p p 1 = ρ cos θ , υ q = υ sin ϕ , q q 1 = ρ sin θ .
à ( p 1 , q 1 ) = aperture A ( p , q ) d p d q υ 0 0 2 π e i ρ υ cos ( ϕ θ ) d ϕ υ d υ , = aperture A ( p , q ) d p d q υ 0 J 0 ( υ ρ ) υ d υ .
υ 0 J 0 ( υ ρ ) υ d υ = 0 J 0 ( υ ρ ) υ d υ 0 υ 0 J 0 ( υ ρ ) υ d υ , = δ ( ρ ) ρ υ 0 J 1 ( υ 0 ρ ) ρ .
à ( p 1 , q 1 ) = aperture A ( p , q ) δ { [ ( p p 1 ) 2 + ( q q 1 ) 2 ] 1 2 } [ ( p p 1 ) 2 + ( q q 1 ) 2 ] 1 2 d p d q υ 0 aperture A ( p , q ) J 1 { υ 0 [ ( p p 1 ) 2 + ( q q 1 ) 2 ] 1 2 } [ ( p p 1 ) 2 + ( q q 1 ) 2 ] 1 2 d p d q .
à ( p 1 , q 1 ) = A ( p 1 , q 1 ) υ 0 aperture A ( p , q ) × J 1 { υ 0 [ ( p p 1 ) + ( q q 1 ) 2 ] 1 / 2 } { ( p p 1 ) 2 + ( q q 1 ) 2 } 1 2 d p d q .
M ( υ p , υ q ) = M ( υ ) = { e i k δ 0 < υ < υ 0 1 υ > υ 0 .
M ( υ ) exp { i [ ( p p 1 ) υ p + ( q q 1 ) υ q ] } d υ p d υ q = e i k δ 0 υ 0 0 2 π e i ρ υ cos ( ϕ θ ) d ϕ υ d υ + υ 0 0 2 π e i ρ υ cos ( ϕ θ ) d ϕ υ d υ , = e i k δ 0 υ 0 J 0 ( υ p ) υ d υ + υ 0 J 0 ( ρ υ ) υ d υ , = υ 0 J 1 ( υ 0 ρ ) ρ ( e i k δ 1 ) + δ ( ρ ) ρ .
à ( p 1 , q 1 ) = A ( p 1 , q 1 ) + υ 0 ( e i k δ 1 ) aperture A ( p , q ) × J 1 { υ 0 [ ( p p 1 ) 2 + ( q q 1 ) 2 ] 1 2 } { ( p p 1 ) 2 + ( q q 1 ) 2 } 1 2 d p d q .