Abstract

Our work deals with the influence of light wavelength on values of wave aberration coefficients and image quality for an axial object point, and a technique for calculation of the dependence of aberration coefficients on the wavelength is proposed. Moreover, the formula for calculation of the monochromatic and polychromatic Strehl definition using chromatic aberration coefficients is derived and the tolerance limits are given. The proposed method of determination of chromatic aberration coefficients is shown for the case of the imaging of an axial object point by a rotationally symmetric optical system. Relations that enable calculation of chromatic aberration coefficients are derived. Finally, the proposed method is applied on an example of an achromatic doublet, and the results of calculations are compared with the values obtained using commercial optical design software (ZEMAX, OSLO).

© 2009 Optical Society of America

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References

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  1. A. Mikš, Applied Optics (Czech Technical University, 2009).
    [PubMed]
  2. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1968).
  3. A. Cox, A System of Optical Design (Focal Press, 1964).
  4. W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic Press, 1974).
  5. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
  6. P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University, 1997).
  7. H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).
  8. H. Haferkorn, Bewertung Optisher Systeme (VEB Deutscher Verlag der Wissenschaften, 1986).
  9. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).
  10. J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85-88 (1964).
    [CrossRef]
  11. G. W. Forbes, “Chromatic coordinates in aberration theory,” J. Opt. Soc. Am. A 1, 344-349 (1984).
    [CrossRef]
  12. G. W. Forbes, “Weighted truncation of power series and the computation of chromatic aberration coefficients,” J. Opt. Soc. Am. A 1, 350-355 (1984).
    [CrossRef]
  13. A. van den Bos, “Aberration and the Strehl ratio,” J. Opt. Soc. Am. A 17, 356-358 (2000).
    [CrossRef]
  14. G. Martial, “Strehl ratio and aberration balancing,” J. Opt. Soc. Am. A 8, 164-170 (1991).
    [CrossRef]
  15. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am. 72, 1258-1266 (1982).
    [CrossRef]
  16. V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73, 860-861(1983).
    [CrossRef]
  17. W. B. King, “Dependance of the Strehl ratio of the magnitude of the variance of the wave aberration,” J. Opt. Soc. Am. 58, 655-661 (1968).
    [CrossRef]
  18. K. Strehl, “Über luftschlieren und zonenfehler,” Z. Instrumentenkd. 22, 213-217 (1902).
  19. www.zemax.com.
  20. www.sinopt.com.

2000 (1)

1991 (1)

1984 (2)

1983 (1)

1982 (1)

1968 (1)

1964 (1)

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

1902 (1)

K. Strehl, “Über luftschlieren und zonenfehler,” Z. Instrumentenkd. 22, 213-217 (1902).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1968).

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).

Cox, A.

A. Cox, A System of Optical Design (Focal Press, 1964).

Forbes, G. W.

Haferkorn, H.

H. Haferkorn, Bewertung Optisher Systeme (VEB Deutscher Verlag der Wissenschaften, 1986).

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

Hopkins, H. H.

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

King, W. B.

Macdonald, J.

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University, 1997).

Mahajan, V. N.

Martial, G.

Mikš, A.

A. Mikš, Applied Optics (Czech Technical University, 2009).
[PubMed]

Mouroulis, P.

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University, 1997).

Rayces, J. L.

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

Strehl, K.

K. Strehl, “Über luftschlieren und zonenfehler,” Z. Instrumentenkd. 22, 213-217 (1902).

van den Bos, A.

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic Press, 1974).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1968).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

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

Z. Instrumentenkd. (1)

K. Strehl, “Über luftschlieren und zonenfehler,” Z. Instrumentenkd. 22, 213-217 (1902).

Other (11)

www.zemax.com.

www.sinopt.com.

A. Mikš, Applied Optics (Czech Technical University, 2009).
[PubMed]

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1968).

A. Cox, A System of Optical Design (Focal Press, 1964).

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic Press, 1974).

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University, 1997).

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

H. Haferkorn, Bewertung Optisher Systeme (VEB Deutscher Verlag der Wissenschaften, 1986).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).

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Figures (4)

Fig. 1
Fig. 1

Aberration of optical system.

Fig. 2
Fig. 2

Typical dependence of the wave aberration of an achromatic doublet on the wavelength ( r = 0.7 ) and comparison of wave aberration fitting using the chromatic aberration coefficients model with N = 3 , M = 2 (dashed line) and the chromatic aberration coefficients model with N = 3 , M = 4 .

Fig. 3
Fig. 3

Dependence of the Strehl definition of an achromatic doublet on the wavelength (parameter Λ). Comparison of SD calculated with the chromatic aberration coefficients model N = 3 , M = 4 using relation (13) and approximate relation (14) with values obtained from ZEMAX and OSLO.

Fig. 4
Fig. 4

Comparison of wave aberrations calculated using the chromatic aberration coefficients model with N = 3 , M = 4 (solid lines) and wave aberrations calculated using ZEMAX software (different markers) for five wavelengths.

Tables (6)

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Table 1 Parameters of Achromatic Doublet f = 100 mm , Pupil Diameter D = 20 mm

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Table 2 Longitudinal Spherical Aberration Values Δ s ( q i , Λ i ) and Calculated Chromatic Aberration Coefficients W n m a

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Table 3 Coefficients C p and D p

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Table 4 Monochromatic Strehl Definition Calculated from Relations (13, 14) and Values Obtained from ZEMAX and OSLO

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Table 5 Dependence of the Calculated Polychromatic Strehl Ratio on the Number of Wavelengths a

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Table 6 Dependence of the Calculated Polychromatic Strehl Ratio on the Number of Wavelengths a

Equations (37)

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Δ x = X x 0 = R n · W x R , Δ y = Y y 0 = R n · W y R ,
F = 1 2 n sin σ k ,
Δ x = 2 F W w , Δ y = 2 F W v .
Δ y = Δ s tan σ = Δ s y R R + Δ s Δ s y R R ( 1 Δ s / R ) Δ s 2 n F v ,
Δ s = 4 F 2 v W v .
W = W 1 ( v 2 + w 2 ) + W 2 ( v 2 + w 2 ) 2 + W 3 ( v 2 + w 2 ) 3 + = n = 1 N W n ( v 2 + w 2 ) n = n = 1 N W n q n ,
W n = W n 0 + W n 1 Λ + W n 2 Λ 2 + = m = 0 M W n m Λ m ,
W ( q , Λ ) = m = 0 M ( n = 1 N W n m q n ) Λ m .
Δ s ( q , Λ ) 8 F 2 = m = 0 M ( n = 1 N n ( W n m ) q n 1 ) Λ m = m = 0 M A m ( q ) Λ m .
Λ = A + B λ .
A = λ max + λ min λ max λ min , B = 2 λ max λ min .
ω = Δ λ 1 + 5 2 Δ λ ,
Hw = s ,
w j = W n m , s i = 1 8 F 2 Δ s ( q i , Λ i ) ,
h i j ( q i , Λ i ) = n q i n 1 Λ i m ,
I ( λ ) = U ( λ ) U * ( λ ) ,
U ( λ ) = 1 S S e i k W ( r , φ , λ ) d S ,
e i k W 1 + i k W k 2 2 W 2 + .
U = 1 + i k S S W d S k 2 2 S S W 2 d S , U * = 1 i k S S W d S k 2 2 S S W 2 d S ,
I ( λ ) = U ( λ ) U * ( λ ) = 1 k 2 S S W 2 ( λ ) d S + ( k S S W ( λ ) d S ) 2 + ( k 2 2 S S W 2 ( λ ) d S ) 2 .
I ( λ ) = 1 k 2 ( W 2 ( λ ) ¯ W ( λ ) ¯ 2 ) = 1 k 2 E 0 ( λ ) ,
E 0 ( λ ) = W 2 ( λ ) ¯ W ( λ ) ¯ 2
W ( λ ) ¯ = 1 S S W ( λ ) d S , W 2 ( λ ) ¯ = 1 S S W 2 ( λ ) d S .
I 0.8.
W 2 ¯ W ¯ 2 λ 2 20 π 2 λ 2 197 .
E 0 ( λ ) = i = 1 N j = 1 N K i j W i ( λ ) W j ( λ ) = i = 1 N j = 1 j i N E i j ( λ ) ,
K i j = K j i = [ 1 i + j + 1 1 ( i + 1 ) ( j + 1 ) ] , i j E i j = 2 K i j W i ( λ ) W j ( λ ) i = j E i i = K i i W i ( λ ) W i ( λ ) .
I polychrom = λ min λ max P ( λ ) I ( λ ) λ 2 d λ λ min λ max P ( λ ) λ 2 d λ = 1 1 P ( Λ ) I ( Λ ) ( Λ A ) 2 d Λ 1 1 P ( λ ) ( Λ A ) 2 d Λ ,
I polychrom = 1 1 P ( Λ ) I ( Λ ) ( Λ A ) 2 d Λ 1 1 P ( λ ) ( Λ A ) 2 d Λ = 1 1 P ( Λ ) [ 1 k 2 E 0 ( Λ ) ] ( Λ A ) 2 d Λ 1 1 P ( λ ) ( Λ A ) 2 d Λ = 1 4 π 2 B 2 1 1 P ( Λ ) E 0 ( Λ ) ( Λ A ) 4 d Λ 1 1 P ( λ ) ( Λ A ) 2 d Λ ,
E 0 ( Λ ) = i = 1 N j = 1 j i N E i j ( Λ ) = p = 0 2 M C p Λ p .
C p = m = q n = p m p q i = 1 N j = 1 N K i j ( W i m ) ( W j n ) ,
I polychrom = 1 p = 0 2 M C p D p ,
D p = α 1 1 Λ p P ( Λ ) ( Λ A ) 4 d Λ , α = 4 π 2 B 2 1 1 P ( Λ ) ( Λ A ) 2 d Λ .
I polychrom = 0.9217 ,
I polychrom = 0.9138.
I = a b f ( x ) d x = j = 1 n G j f ( x j ) .
a b x q d x = ( b q + 1 a q + 1 ) / ( q + 1 ) = j = 1 n G j x j q , q = 0 , 1 , 2 , , m .

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