Abstract

The optimization of an optical system is generally carried out by minimizing the wave-front aberrations, the x and y transverse aberrations, or a combination of both. In the last-named, most general case, the optimization implies the treatment of a large number of functions of quite a different nature. We propose to use, together with the Zernike polynomials that orthogonalize the wave-front aberrations, a new set of wave-front polynomials that orthogonalize the transverse aberrations. These polynomials turn out to be a simple linear combination of Zernike polynomials. The combination of these two sets of wave-front polynomials with proper weighting yields the possibility of optimizing the frequency response of both slightly and severely aberrated systems in a formally identical way. The advantage of the method is that one does not have to leave the domain of the wave-front aberration to characterize an optical system, even when severe aberrations are present. The polynomials that minimize the transverse aberrations yield optimum response at very low frequencies; other linear combinations of Zernike polynomials are shown to maximize the frequency response at relatively high spatial frequencies.

© 1987 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 465.
  2. H. H. Hopkins, “The use of diffraction-based criteria of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
    [CrossRef]
  3. A. M. Goodbody, Tables of Optical Frequency Response, Summer School on Applied Optics (Imperial College, London, 1959).
  4. R. W. Gostick, “OTF-based optimization criteria for automatic optical design,” Opt. Quantum Electron. 8, 31–37 (1974).
    [CrossRef]
  5. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).
  6. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1986), p. 258.
  7. J. Kross, “Beschreibung, Analyse und Bewertung der Bildfehler optischer Systeme durch interpolierende Darstellungen mit Hilfe von Zernike-Kreispolynomen,” Optik 29, 65–80 (1969).

1974 (1)

R. W. Gostick, “OTF-based optimization criteria for automatic optical design,” Opt. Quantum Electron. 8, 31–37 (1974).
[CrossRef]

1969 (1)

J. Kross, “Beschreibung, Analyse und Bewertung der Bildfehler optischer Systeme durch interpolierende Darstellungen mit Hilfe von Zernike-Kreispolynomen,” Optik 29, 65–80 (1969).

1966 (1)

H. H. Hopkins, “The use of diffraction-based criteria of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 465.

Goodbody, A. M.

A. M. Goodbody, Tables of Optical Frequency Response, Summer School on Applied Optics (Imperial College, London, 1959).

Gostick, R. W.

R. W. Gostick, “OTF-based optimization criteria for automatic optical design,” Opt. Quantum Electron. 8, 31–37 (1974).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The use of diffraction-based criteria of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
[CrossRef]

Kross, J.

J. Kross, “Beschreibung, Analyse und Bewertung der Bildfehler optischer Systeme durch interpolierende Darstellungen mit Hilfe von Zernike-Kreispolynomen,” Optik 29, 65–80 (1969).

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1986), p. 258.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 465.

Opt. Acta (1)

H. H. Hopkins, “The use of diffraction-based criteria of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
[CrossRef]

Opt. Quantum Electron. (1)

R. W. Gostick, “OTF-based optimization criteria for automatic optical design,” Opt. Quantum Electron. 8, 31–37 (1974).
[CrossRef]

Optik (1)

J. Kross, “Beschreibung, Analyse und Bewertung der Bildfehler optischer Systeme durch interpolierende Darstellungen mit Hilfe von Zernike-Kreispolynomen,” Optik 29, 65–80 (1969).

Other (4)

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 465.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1986), p. 258.

A. M. Goodbody, Tables of Optical Frequency Response, Summer School on Applied Optics (Imperial College, London, 1959).

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Figures (4)

Fig. 1
Fig. 1

Wave front of a wave converging to the image point P. The origin of the rectangular coordinate system is at O. The reference sphere is centered on the point P. A general point Q on the reference sphere is identified by its coordinates x and y. The radius of the reference sphere is R. The transverse aberrations are denoted by Δx and Δy.

Fig. 2
Fig. 2

(a) A radial section of a wave front affected by fourth-order spherical aberration. The unit of wave-front aberration is λ. The pupil coordinate ρ has been normalized and runs from 0 to 1. The wavefronts are represented as follows: solid line, 9ρ4 − 12ρ2; dotted line, 9ρ4 − 9ρ2; dashed line, 9ρ4 − 6ρ2; dashed–dotted line, 9ρ4 − 3ρ2. (b) The MTF curves corresponding to the wave fronts depicted in (a). The horizontal scale is in units NA/λ. The MTF has been calculated by the evaluation of the autocorrelation of the wave front (diffraction MTF).

Fig. 3
Fig. 3

(a) A radial section of a wavefront affected by sixth-order spherical aberration. The unit of wave-front aberration is λ. The pupil coordinate ρ has been normalized and runs from 0 to 1. The wave fronts are represented as follows: solid line, 20 ρ 6 - 36 ρ 4 + 18 ρ 2 [ R 6 0 ( ρ ) - R 4 0 ( ρ ) ]; dotted line, 20 ρ 6 - 30 ρ 4 + 12 ρ 2 R 6 0 ( ρ ); dashed line, 20 ρ 6 - 24 ρ 4 + 8 ρ 2 [ R 6 0 ( ρ ) + R 4 0 ( ρ ) + R 2 0 ( ρ ) ]; dashed–dotted line, 20 ρ 6 - 18 ρ 4 + 4 ρ 2 [ R 6 0 ( ρ ) + 2 R 4 0 ( ρ ) + 2 R 2 0 ( ρ ) ]. (b) The MTF curves corresponding to the wave fronts depicted in (a).

Fig. 4
Fig. 4

(a) A radial section of a wave front affected by eighth-order spherical aberration. The unit of wave-front aberration is λ. The wave fronts are represented as follows: solid line, 70 ρ 8 - 160 ρ 6 + 120 ρ 4 - 32 ρ 2 [ R 8 0 ( ρ ) - R 6 0 ( ρ ) ]; dotted line, 70 ρ 8 - 140 ρ 6 + 90 ρ 4 - 20 ρ 2 R 8 0 ( ρ ); dashed line, 70 ρ 8 - 120 ρ 6 + 66 ρ 4 - 12 ρ 2 [ R 8 0 ( ρ ) + R 6 0 ( ρ ) + R 4 0 ( ρ ) + R 2 0 ( ρ ) ]; dashed–dotted line, 70 ρ 8 - 100 ρ 6 + 42 ρ 4 - 4 ρ 2 [ R 8 0 ( ρ ) + 2 R 6 0 ( ρ ) + 2 R 4 0 ( ρ ) + 2 R 2 0 ( ρ ) ]. (b) The MTF curves corresponding to the wave fronts depicted in (a).

Equations (57)

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W ( ρ , ϕ ) = n m R n m ( ρ ) [ a n m cos ( m ϕ ) + a ¯ n m sin ( m ϕ ) ] ,
Variance ( W ) = ( W - W ¯ ) 2 = n = 1 ( a 2 n 0 ) 2 n + 1 + m = 1 i = 0 ( a 2 i + m m ) 2 2 ( 2 i + m + 1 ) .
Δ x = R W δ x λ ,
Δ y = R δ W δ y λ ,
Δ x ( ρ , ϕ ) = cos ϕ δ W δ ρ - sin ϕ ρ δ W δ ϕ ,
Δ y ( ρ , ϕ ) = sin ϕ δ W δ ρ - cos ϕ ρ δ W δ ϕ ,
Δ x ( ρ , ϕ ) = m = 0 n = m a n m 2 ( { d d ρ R n m ( ρ ) ] + m R n m ( ρ ) ρ } × cos ( m - 1 ) ϕ + { d d ρ [ R n m ( ρ ) ] - m R n m ( ρ ) ρ } × cos ( m + 1 ) ϕ ) ,
Δ y ( ρ , ϕ ) = m = 0 n = m a n m 2 ( - { d d ρ R n m ( ρ ) ] + m R n m ( ρ ) ρ } × sin ( m - 1 ) ϕ + { d d ρ [ R n m ( ρ ) ] - m R n m ( ρ ) ρ } × sin ( m + 1 ) ϕ ) .
Δ 2 = ( Δ x ) 2 + ( Δ y ) 2
W ( ρ , ϕ ) = i = 0 n - m 2 α 2 i + m m R 2 i + m m ( ρ ) cos ( m ϕ ) .
Δ x ( ρ , ϕ ) = [ f ( n , m , ρ ) cos ( m - 1 ) ϕ + g ( n , m , ρ ) cos ( m + 1 ) ϕ ] ,
Δ y ( ρ , ϕ ) = [ - f ( n , m , ρ ) sin ( m - 1 ) ϕ + g ( n , m , ρ ) sin ( m + 1 ) ϕ ] ,
f ( n , m , ρ ) = 1 2 ( d d ρ + m ρ ) i = 0 n - m 2 α 2 i + m m R 2 i + m m ( ρ ) ,
g ( n , m , ρ ) = 1 2 ( d d ρ - m ρ ) i = 0 n - m 2 α 2 i + m m R 2 i + m m ( ρ ) .
1 2 ( d d ρ + m ρ ) R n m ( ρ ) = k = 0 n - m 2 ( n - 2 k ) R n - 1 - 2 k m - 1 ( ρ ) .
1 2 ( d d ρ - m ρ ) R n m ( ρ ) = k = 0 n - m 2 - 1 ( n - 2 k ) R n - 1 - 2 k m + 1 ( ρ ) .
f ( n , m , ρ ) = i = 0 n - m 2 k = 0 i α 2 i + m m ( 2 i + m - 2 k ) R 2 i - 2 k + m - 1 m - 1 ( ρ ) ,
g ( n , m , ρ ) = i = 0 n - m 2 k = 0 i - 1 α 2 i + m m ( 2 i + m - 2 k ) R 2 i - 2 k + m - 1 m + 1 ( ρ ) .
i = 0 i max k = 0 i f ( i , k , ρ ) = l = 0 i max i = l i max f ( i , i - l , ρ ) ,
f ( n , m , ρ ) = l = 0 n - m 2 ( 2 l + m ) R 2 l + ( m - 1 ) m - 1 ( ρ ) ( i = l n - m 2 α 2 i + m m ) ,
g ( n , m , ρ ) = l = 1 n - m 2 ( 2 l + m ) R 2 l + ( m - 1 ) m + 1 ( ρ ) ( i = l n - m 2 α 2 i + m m ) .
Δ 2 = Δ x 2 + Δ y 2 = f 2 cos 2 ( m - 1 ) ϕ + g 2 cos 2 ( m + 1 ) ϕ + f 2 sin 2 ( m - 1 ) ϕ + g 2 sin 2 ( m + 1 ) ϕ + 2 f g [ cos ( m - 1 ) ϕ cos ( m + 1 ) ϕ - sin ( m - 1 ) ϕ sin ( m + 1 ) ϕ ]
Δ 2 = f 2 + g 2 .
R n 1 m 1 ( ρ ) R n 2 m 2 ( ρ ) unit circle
Δ 2 = l = 0 n - m 2 ( 2 l + m ) 2 ( 2 l + m ) ( i = l n - m 2 α 2 i + m m ) 2 + l = 1 n - m 2 ( 2 l + m ) 2 ( 2 l + m ) ( i = l n - m 2 α 2 i + m m ) 2 ,
Δ 2 = m ( i = 0 n - m 2 α 2 i + m m ) 2 + l = 1 n - m 2 2 ( 2 l + m ) × ( i = 1 n - m 2 α 2 i + m m ) 2 .
α n - 2 m = - α n m , α n - 2 i m = 0             for i = 2 , , n - m 2 ,
W ( ρ , ϕ ) = α n m [ R n m ( ρ ) - R n - 2 m ( ρ ) ] cos ( m ϕ ) .
Δ 2 = 0 + 2 ( n - m + m ) ( α n - m + m m ) 2 ,
Δ 2 = 2 n ( α n m ) 2 .
Δ 2 = 4 n ( α n m ) 2 m = 0 , = 2 n ( α n m ) 2 m 0.
W = n m b n m B n m ( ρ ) cos ( m ϕ ) ,
B n m ( ρ ) = R n m ( ρ ) - R n - 2 m ( ρ ) .
Δ 2 = n = 1 4 n ( b 2 n 0 ) 2 + m = 1 [ m ( b m m ) 2 + i = 1 2 ( 2 i + m ) ( b 2 i + m m ) 2 ] ,
OTF ( f x ) = 1 S p pupil exp ( 2 π j f x R δ W δ x ) d x d y ,
OTF ( s x ) = 1 π x 2 + y 2 1 exp ( 2 π j s x Δ x ) d x d y ,
OTF ( s x ) 2 = 1 - 4 π 2 s x 2 ( Δ ) 2 + 4 π 2 s x 2 ( Δ ) 2 ,
MTF ( s x ) = [ OTF ( s x ) 2 ] 1 / 2 = 1 - 2 π 2 s x 2 [ ( Δ ) 2 - ( Δ ) 2 ] = 1 - 2 π 2 s x 2 Δ 2 ,
Δ rms 0.10 s x ( λ / NA ) .
R 4 0 ( ρ ) - R 2 0 ( ρ ) = 6 ρ 4 - 8 ρ 2 + 2             [ B 4 0 ( ρ ) ] , R 4 0 ( ρ ) = 6 ρ 4 - 6 ρ 2 + 1 , R 4 0 ( ρ ) + R 2 0 ( ρ ) = 6 ρ 4 - 4 ρ 2 ,
W ( ρ ) = R 6 0 ( ρ ) + γ ( R 4 0 + R 2 0 ) .
1 2 ( d d ρ + m ρ ) R n m ( ρ ) = s = 0 n - m 2 ( n - 2 s ) R n - 1 - 2 s m - 1 ( ρ ) .
R n m ( ρ ) = k = 0 n - m 2 ( - 1 ) k ( n - k ) ! k ! ( n + m 2 - k ) ! ( n - m 2 - k ) ! ρ n - 2 k
1 2 q = 0 n - m 2 ( - 1 ) q ( n - q ) ! ( n - 2 q + m ) q ! ( n + m 2 ) ! ( n - m 2 - q ) ! ρ n - 2 q - 1 = s = 0 n - m 2 ( n - 2 s ) p = s n - m 2 ( - 1 ) p - s ( n - 1 - s - p ) ! ( p - s ) ! ( n + m 2 - 1 - p ) ! ( n - m 2 p ) ! ρ n - 2 p - 1 .
k = 0 q ( - 1 ) k ( n - 2 k ) q ! ( n - 1 - q - k ) ! ( q - k ) ! ( n - q ) ! 1
1 2 [ d d ρ - m ρ ] R n m ( ρ ) = q = 0 n - m 2 ( - 1 ) q 2 ( n - q ) ! ( n - 2 q - m ) q ! ( n + m 2 - q ) ! ( n - m 2 - q ) ! ρ n - 2 q - 1 .
g ( n , m , ρ ) = q = 0 n - m 2 - 1 k = 0 q ( - 1 ) q + k 2 ( n - 1 - q - k ) ! ( n - 2 k ) ( n - 2 q - m ) ( q - k ) ! ( n + m 2 - q ) ! ( n - m 2 - q ) ! ρ n - 2 q - 1 ,
k = 0 n - m 2 - 1 ( n - 2 k ) q = k n - m 2 - 1 ( - 1 ) q - k ( n - n - q - k ) ! ( q - k ) ! ( n + m 2 - q ) ! ( n - m 2 - q - 1 ) ! ρ n - 2 q - 1 .
k = 0 n - m 2 - 1 ( n - 2 k ) R n - 1 - 2 k m + 1 ( ρ ) ,
W ( ρ , ϕ ) = m = 0 n = m b n m B n m ( ρ ) cos ( m ϕ ) ,
B n m ( ρ ) = R n m ( ρ ) - R n - 2 m ( ρ )             ( n m + 2 ) ,
B n m ( ρ ) = R n m ( ρ )             ( n = m ) ,
W ( ρ , ϕ ) = m = 0 n = m ( b n m - b n + 2 m ) R n m ( ρ ) cos ( m ϕ ) .
Δ 2 = l = 1 4 l [ i = l ( b 2 i 0 - b 2 i + 2 0 ) ] 2 + m = 1 { m [ i = 0 ( b 2 i + m m - b 2 i + m + 2 m ) ] 2 + l = 1 2 ( 2 l + m ) [ i = l ( b 2 i + m m - b 2 i + m + 2 m ) ] 2 } .
Δ 2 = l = 1 4 l ( b 2 l 0 ) 2 + m = 1 [ m ( b m m ) 2 + l = 1 2 ( 2 l + m ) ( b 2 l + m m ) 2 ]
Δ x ( ρ , ϕ ) = n = 2 2 n b n 0 R n - 1 1 ( ρ ) cos ( ϕ ) + m = 1 n = m n b n m { R n - 1 m - 1 ( ρ ) cos ( m - 1 ) ϕ + R n - 1 m + 1 ( ρ ) cos ( m + 1 ) ϕ } ,
Δ x ( ρ , ϕ ) = n = 2 2 n b n 0 R n - 1 1 ( ρ ) cos ( ϕ ) + m = 1 n = m n b n m { R n - 1 m - 1 ( ρ ) cos ( m - 1 ) ϕ + R n - 1 m + 1 ( ρ ) cos ( m + 1 ) ϕ } ,

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