Abstract

The wave-aberration function of systems with circular and square apertures can be expanded in terms of Zernike and Legendre polynomials. The polynomial terms form orthogonal sets; therefore each coefficient independently determined by an integral satisfies the principle of least squares. To evaluate the integral the pupil is divided into small areas where the wave-aberration function is approximated by the first three terms of a Taylor series expansion: the optical path difference and components of geometric aberration. In final form the coefficients are expressed by the sum of three bilinear terms by combining three matrices and six vectors. The former depend on the construction parameters and the latter on the ray pattern.

© 1992 Optical Society of America

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References

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  1. J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, (2), 85–88 (1964).
    [CrossRef]
  2. H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. London Section B 70, 1002–1005 (1957).
    [CrossRef]
  3. H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics (Macmillan, New York), 1964, Chap. 9, pp. 464–473.
  5. J. L. Rayces, H-C. Hsieh, “Lens design with Glatzel’s method and the Nijboer–Zernike coefficients,” J. Opt. Soc. Am. 60, 1557(A) (1970).
  6. J. L. Rayces, “Expansion of the aberration function in terms of Legendre polynomials,” J. Opt. Soc. Am. 62, 712(A) (1972).
  7. R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York), 1962, Sec. 17.6, p. 233.
  8. Ref. 7, Sec. 17.9, p. 237.
  9. W. B. King, “The approximation of vignetted pupil shape by an ellipse,” Appl. Opt. 7, 197–201 (1968).
    [CrossRef] [PubMed]
  10. W. B. King, “Dependence of the Strehl ratio on the magnitude of the variance of the wave aberration,” J. Opt. Soc. Am. 58, 655–661 (1968).
    [CrossRef]
  11. J. Kross, “Beschreibung, Analyse un Bewertung der Bildfehler optischer Systeme durch interpolierende Darstellungen mit Hilfe von Zernike Kreispolynomen,” Optik 1, 65–71 (1969).

1972 (1)

J. L. Rayces, “Expansion of the aberration function in terms of Legendre polynomials,” J. Opt. Soc. Am. 62, 712(A) (1972).

1970 (2)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

J. L. Rayces, H-C. Hsieh, “Lens design with Glatzel’s method and the Nijboer–Zernike coefficients,” J. Opt. Soc. Am. 60, 1557(A) (1970).

1969 (1)

J. Kross, “Beschreibung, Analyse un Bewertung der Bildfehler optischer Systeme durch interpolierende Darstellungen mit Hilfe von Zernike Kreispolynomen,” Optik 1, 65–71 (1969).

1968 (2)

1964 (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, (2), 85–88 (1964).
[CrossRef]

1957 (1)

H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. London Section B 70, 1002–1005 (1957).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York), 1964, Chap. 9, pp. 464–473.

Hamming, R. W.

R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York), 1962, Sec. 17.6, p. 233.

Hopkins, H. H.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. London Section B 70, 1002–1005 (1957).
[CrossRef]

Hsieh, H-C.

J. L. Rayces, H-C. Hsieh, “Lens design with Glatzel’s method and the Nijboer–Zernike coefficients,” J. Opt. Soc. Am. 60, 1557(A) (1970).

King, W. B.

Kross, J.

J. Kross, “Beschreibung, Analyse un Bewertung der Bildfehler optischer Systeme durch interpolierende Darstellungen mit Hilfe von Zernike Kreispolynomen,” Optik 1, 65–71 (1969).

Rayces, J. L.

J. L. Rayces, “Expansion of the aberration function in terms of Legendre polynomials,” J. Opt. Soc. Am. 62, 712(A) (1972).

J. L. Rayces, H-C. Hsieh, “Lens design with Glatzel’s method and the Nijboer–Zernike coefficients,” J. Opt. Soc. Am. 60, 1557(A) (1970).

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, (2), 85–88 (1964).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York), 1964, Chap. 9, pp. 464–473.

Yzuel, M. J.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

W. B. King, “Dependence of the Strehl ratio on the magnitude of the variance of the wave aberration,” J. Opt. Soc. Am. 58, 655–661 (1968).
[CrossRef]

J. L. Rayces, H-C. Hsieh, “Lens design with Glatzel’s method and the Nijboer–Zernike coefficients,” J. Opt. Soc. Am. 60, 1557(A) (1970).

J. L. Rayces, “Expansion of the aberration function in terms of Legendre polynomials,” J. Opt. Soc. Am. 62, 712(A) (1972).

Opt. Acta (2)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, (2), 85–88 (1964).
[CrossRef]

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Optik (1)

J. Kross, “Beschreibung, Analyse un Bewertung der Bildfehler optischer Systeme durch interpolierende Darstellungen mit Hilfe von Zernike Kreispolynomen,” Optik 1, 65–71 (1969).

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

H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. London Section B 70, 1002–1005 (1957).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics (Macmillan, New York), 1964, Chap. 9, pp. 464–473.

R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York), 1962, Sec. 17.6, p. 233.

Ref. 7, Sec. 17.9, p. 237.

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Figures (2)

Fig. 1
Fig. 1

Circular pupil divided into rings and sectors. The rays are traced through the geometrical center of each small area.

Fig. 2
Fig. 2

Square pupil divided into rows and columns. The rays are traced through the geometrical center of each small area.

Equations (37)

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W ( ρ , θ ) = A 00 + 1 2 n = 2 A n 0 R n 0 ( ρ ) + n = 1 m = 1 n R n m ( ρ ) ( A n m cos m θ + B n m sin m θ ) .
0 1 R n m ( ρ ) R n m ( ρ ) ρ d ρ = 1 2 ( n + 1 ) δ n n , 0 2 π cos m θ cos m θ d θ = π δ m m             ( m > 0 ) , 0 2 π sin m θ sin m θ d θ = π δ m m , 0 2 π sin m θ cos m θ d θ = 0.
Γ W ( ρ , θ ) R n 0 ( ρ ) ρ d ρ d θ = 1 2 A n 0 0 1 [ R n 0 ( ρ ) ] 2 ρ d ρ 0 2 π d θ = π 2 ( n + 1 ) A n 0 ,
Γ W ( ρ , θ ) R n m ( ρ ) cos m θ ρ d ρ d θ = A n m 0 1 [ R n m ( ρ ) ] 2 ρ d ρ 0 2 π cos 2 m θ d θ = π 2 ( n + 1 ) A n m , Γ W ( ρ , θ ) R n m ( ρ ) sin m θ ρ d ρ d θ = B n m 0 1 [ R n m ( ρ ) ] 2 ρ d ρ 0 2 π sin 2 m θ d θ = π 2 ( n + 1 ) B n m .
A n 0 = 2 ( n + 1 ) π Γ W ( ρ , θ ) R n 0 ( ρ ) ρ d ρ d θ , A n m = 2 ( n + 1 ) π Γ W ( ρ , θ ) R n m ( ρ ) cos m θ ρ d ρ d θ , B n m = 2 ( n + 1 ) π Γ W ( ρ , θ ) R n m ( ρ ) sin m θ ρ d ρ d θ .
ρ ¯ i = ½ ( ρ i - 1 + ρ i ) ,             θ ¯ j = ½ ( θ j - 1 + θ j ) .
ρ i = i I ,             θ j = 2 π T j ,
ρ i = i I ,             θ j = 2 π T j ,
W ¯ x = - N A x T A x ,             W ¯ y = - N A y T A y .
W ¯ ρ = sin θ ¯ W ¯ x + cos θ ¯ W ¯ y , W ¯ θ = ρ ¯ cos θ ¯ W ¯ x - ρ ¯ sin θ ¯ W ¯ y .
δ W ¯ ( ρ , θ ) = OPD ¯ + W ¯ ρ ( ρ - ρ ¯ ) + W ¯ θ ( θ - θ ¯ ) = U θ + V ρ + W ,
U = W ¯ θ ,             V = W ¯ ρ ,             W = OPD ¯ - V ρ ¯ - U θ ¯ .
A n 0 = 2 ( n + 1 ) π i j δ Γ ( U i j θ + V i j ρ + W i j ) R n 0 ( ρ ) ρ d ρ d θ , A n m = 2 ( n + 1 ) π i j δ Γ ( U i j θ + V i j ρ + W i j ) R n m ( ρ ) ρ d ρ cos m θ d θ , B n m = 2 ( n + 1 ) π i j δ Γ ( U i j θ + V i j ρ + W i j ) R n m ( ρ ) ρ d ρ sin m θ d θ .
C m ( θ ) = 2 π 0 θ d φ = 2 π θ , D m ( θ ) = 2 π 0 θ φ d φ = 2 2 π θ 2 ;
C m ( θ ) = 2 π 0 θ cos m φ d φ = 2 m π sin m θ , D m ( θ ) = 2 π 0 θ φ cos m φ d φ = 2 m 2 π ( cos m θ + m θ sin m θ - 1 ) ;
E m ( θ ) = 2 π 0 θ sin m φ d φ = - 2 m π ( cos m θ - 1 ) , F m ( θ ) = 2 π 0 θ φ sin m φ d φ = 2 m 2 π ( sin m θ - m θ cos m θ + m θ ) , S n m ( ρ ) = ( n + 1 ) 0 ρ R n m ( σ ) d σ = ( n + 1 ) s = 0 1 / 2 ( n - m ) K ( n m , s ) n - 2 s + 2 ρ n - 2 s + 2 , T n m ( ρ ) = ( n + 1 ) 0 ρ R n m ( σ ) σ 2 d σ = ( n + 1 ) s = 0 1 / 2 ( n - m ) K ( n m , s ) n - 2 s + 3 ρ n - 2 s + 3 ,
K ( n m , s ) = ( - 1 ) s ( n - s ) ! s ! [ ½ ( n + m ) - s ] ! [ ½ ( n - m ) - s ] ! .
Δ C j ( m ) = C m ( θ j ) - C m ( θ j - 1 ) , Δ D j ( m ) = D m ( θ j ) - D m ( θ j - 1 ) , Δ E j ( m ) = E m ( θ j ) - E m ( θ j - 1 ) , Δ F j ( m ) = F m ( θ j ) - F m ( θ j - 1 ) , Δ S j ( n , m ) = S n m ( ρ i ) - S n m ( ρ i - 1 ) , Δ T i ( n , m ) = T n m ( ρ i ) - T n m ( ρ i - 1 ) .
A n m = i j [ U i j Δ S i ( n m ) Δ D j ( m ) + V i j Δ T i ( n m ) Δ C j ( m ) + W i j Δ S i ( n m ) Δ C j ( m ) ] , B n m = i j [ U i j Δ S i ( n m ) Δ F j ( m ) + V i j Δ T i ( n m ) Δ E j ( m ) + W i j Δ S i ( n m ) Δ E j ( m ) ] ,
W ( x , y ) = n m A n m P n ( x ) P m ( y ) .
Γ P n ( x ) P m ( y ) P n ( x ) P m ( y ) d x d y = 4 δ n n δ m m ( 2 n + 1 ) ( 2 m + 1 ) ,
Γ W ( x , y ) P n ( x ) P m ( y ) d x d y = 4 A n m ( 2 n + 1 ) ( 2 m + 1 ) ;
A n m = ¼ ( 2 n + 1 ) ( 2 m + 1 ) Γ W ( x , y ) P n ( x ) P m ( y ) d x d y .
y ¯ = ½ ( y i - 1 + y i ) ,             x ¯ = ½ ( x j - 1 + x j ) .
y i = i I ,             x j = j J .
δ W ¯ ( x , y ) = OPD ¯ + W ¯ x ( x - x ¯ ) + W ¯ y ( y - y ¯ ) = U y + V x + W ,
U - W ¯ y ,             V = W ¯ x ,             W = OPD ¯ - V x ¯ - U y ¯ .
A n m = ¼ ( 2 n + 1 ) ( 2 m + 1 ) i j Γ × ( U y + V x + W ) P n ( x ) P m ( y ) d x d y .
G n ( x ) = 0 x P n ( t ) d t = s = 0 1 / 2 ( n + 1 ) L ( n , s ) n - 2 s + 1 x n - 2 s + 1 , H n ( x ) = 0 x P n ( t ) d t = s = 0 1 / 2 ( n + 1 ) L ( n , s ) n - 2 s + 2 x n - 2 s + 2 ,
L ( n , s ) = ( - 1 ) 2 n ( 2 n - 2 s ) ! ( n - s ) ! ( n - 2 s ) 1 s ! .
Δ G j ( n ) = G n ( x j ) - G n ( x j - 1 ) , Δ G i ( m ) = G m ( y i ) - G m ( y i - 1 ) , Δ H j ( n ) = H n ( x j ) - H n ( x j - 1 ) , Δ H i ( m ) = H m ( y i ) - H m ( y i - 1 ) .
A n m = i j [ U i j Δ H i ( m ) Δ G j ( n ) + V i j Δ G i ( m ) Δ H j ( n ) + W i j Δ G i ( m ) Δ G j ( n ) ] .
W λ ( x , y ) = n m Q n m a n m P n ( x ) P m ( y ) ,
a n m = A n m λ Q n m ,             Q n m = [ ( 2 n + 1 ) ( 2 m + 1 ) ] 1 / 2 .
Var ( W λ ) = n m a n m 2             ( n + m 2 ) ,
m = 0 a 00 , m = 1 a 20 , a 11 , b 11 , m = 2 a 40 , a 31 , b 31 , a 22 , b 22 , m = 3 a 60 , a 51 , b 51 , a 42 , b 42 , a 33 , b 33 , m = 4 a 80 , a 71 , b 71 , a 62 , b 62 , a 53 , b 53 , a 44 , b 44 .
i ( P ) 1 - 4 π 2 Var ( W λ ) .

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