Abstract

Optical systems with variable optical characteristics (zoom lenses) find broader applications in practice nowadays and methods for their design are constantly developed and improved. We describe a relatively simple method of the design of zoom lenses using the third-order aberration theory. It presents one of the possible approaches of obtaining the Seidel aberration coefficients of individual members of a zoom lens. The advantage of this method is that Seidel aberration coefficients of individual elements of a given optical system can be obtained simply by solving of a set of linear equations. By using these coefficients, one can determine residual aberrations of the optical system without detailed knowledge about the structure of its individual elements. Furthermore, we can determine construction parameters of the optical system, i.e., radii of curvature and thicknesses of individual elements of a given optical system. The proposed method makes it possible to determine which elements of the optical system can be designed as simple lenses and which elements must have a more complicated design, e.g., doublets or triplets.

© 2008 Optical Society of America

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References

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  1. B. Havelka, Geometrical Optics I, II (Czech Academy of Science, 1955).
  2. D. Argentieri, Ottica Industriale (Hoepli, 1942).
  3. H. H. Hopkins, Wave Theory of Aberrations (Oxford, 1950).
  4. G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).
  5. A. Cox, A System of Optical Design (Focal, 1964).
  6. P. Mouroulis and J. MacDonald, Geometrical Optics and Optical Design (Oxford, 1997).
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  10. H. Haferkorn, Bewertung Optisher Systeme (VEB Deutscher Verlag der Wissenschaften, 1986).
  11. W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).
  12. A. D. Clark, Zoom Lenses (Adam Hilger, 1973).
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    [CrossRef]
  15. G. Wooters and E. W. Silvertooth, “Optically compensated zoom lens,” J. Opt. Soc. Am. 55, 347-351 (1965).
    [CrossRef]
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  17. A. V. Grinkevich, “Version of an objective with variable focal length,” J. Opt. Technol. 73343-345 (2006).
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  19. G. H. Matter and E. T. Luszcz, “A family of optically compensated zoom lenses,” Appl. Opt. 9, 844-848 (1970).
    [CrossRef] [PubMed]
  20. K. Tanaka, “Erratum: Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type,” Appl. Opt. 21, 3805 (1982).
    [CrossRef] [PubMed]
  21. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 2: Generalization of Yamaji type V,” Appl. Opt. 21, 4045-4053 (1982).
    [CrossRef] [PubMed]
  22. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 3: Five-component type,” Appl. Opt. 22, 541-553 (1983).
    [CrossRef] [PubMed]
  23. H. Chretien, Calcul des Combinaisons Optiques (Masson, 1980).
  24. H. H. Hopkins and V. V. Rao, “The systematic design of two component objectives,” Opt. Acta 17, 497-514 (1970).
    [CrossRef]
  25. M. I. Khan, “Cemented triplets: a method for rapid design,” Opt. Acta 31, 873-883 (1984).
    [CrossRef]
  26. www.zemax.com
  27. www.sinopt.com

2006 (1)

2002 (1)

2001 (1)

1984 (1)

M. I. Khan, “Cemented triplets: a method for rapid design,” Opt. Acta 31, 873-883 (1984).
[CrossRef]

1983 (1)

1982 (3)

1970 (3)

1965 (1)

Argentieri, D.

D. Argentieri, Ottica Industriale (Hoepli, 1942).

Chretien, H.

H. Chretien, Calcul des Combinaisons Optiques (Masson, 1980).

Clark, A. D.

A. D. Clark, Zoom Lenses (Adam Hilger, 1973).

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part I (Oxford, 1929), Part II (Dover, 1960).

Cox, A.

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

Grinkevich, A. V.

Haferkorn, H.

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

Havelka, B.

B. Havelka, Geometrical Optics I, II (Czech Academy of Science, 1955).

Hopkins, H. H.

H. H. Hopkins and V. V. Rao, “The systematic design of two component objectives,” Opt. Acta 17, 497-514 (1970).
[CrossRef]

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

Khan, M. I.

M. I. Khan, “Cemented triplets: a method for rapid design,” Opt. Acta 31, 873-883 (1984).
[CrossRef]

Kienholz, D. F.

Luszcz, E. T.

MacDonald, J.

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

Matter, G. H.

Mikš, A.

Mouroulis, P.

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

Rao, V. V.

H. H. Hopkins and V. V. Rao, “The systematic design of two component objectives,” Opt. Acta 17, 497-514 (1970).
[CrossRef]

Silvertooth, E. W.

Slyusarev, G. G.

G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).

Tanaka, K.

Wather, A.

Welford, W. T.

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

Wooters, G.

Yamaji, K.

K. Yamaji, Progress in Optics (North-Holland, 1967), Vol. VI.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Technol. (1)

Opt. Acta (2)

H. H. Hopkins and V. V. Rao, “The systematic design of two component objectives,” Opt. Acta 17, 497-514 (1970).
[CrossRef]

M. I. Khan, “Cemented triplets: a method for rapid design,” Opt. Acta 31, 873-883 (1984).
[CrossRef]

Other (15)

www.zemax.com

www.sinopt.com

H. Chretien, Calcul des Combinaisons Optiques (Masson, 1980).

A. E. Conrady, Applied Optics and Optical Design, Part I (Oxford, 1929), Part II (Dover, 1960).

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

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

A. D. Clark, Zoom Lenses (Adam Hilger, 1973).

K. Yamaji, Progress in Optics (North-Holland, 1967), Vol. VI.

B. Havelka, Geometrical Optics I, II (Czech Academy of Science, 1955).

D. Argentieri, Ottica Industriale (Hoepli, 1942).

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

G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).

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

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

A. Mikš, Applied Optics (Czech Technical U. Press, 2000).

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Figures (4)

Fig. 1
Fig. 1

Two-element optical system.

Fig. 2
Fig. 2

Spot diagrams of two-element zoom lens for y = 0 mm and S I = S I I = S I I I = 0 .

Fig. 3
Fig. 3

Spot diagrams of three-element zoom lens for y = 0 mm and S I = S I I = S I I I = 0 .

Fig. 4
Fig. 4

Spot diagrams of three-element zoom lens for y = 14 mm and S I = S I I = S I I I = 0 .

Tables (4)

Tables Icon

Table 1 Parameters of Cemented Doublet

Tables Icon

Table 2 Seidel Coefficients S I and S I I of the Doublet

Tables Icon

Table 3 Parameters of the Optical System ( e = 0 , f 1 = 50 mm , f 2 = 50 mm )

Tables Icon

Table 4 Parameters of Three-Element Zoom Lens

Equations (79)

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1 s 1 s = 1 f = φ ,
φ = ( n 1 ) ( 1 r 1 r ) ,
m = s s = 1 1 + s φ .
X = r + r r r .
Y = s + s s s .
r = 2 ( n 1 ) φ ( X + 1 ) , r = 2 ( n 1 ) φ ( X 1 ) .
Y = s + s s s = m + 1 m 1 = 1 2 s φ = 1 2 s φ .
S I = i = 1 K h i 4 M i ,
S I I = i = 1 K h i 3 h ¯ i M i + i = 1 K h i 2 N i ,
S I I I = i = 1 K h i 2 h ¯ i 2 M i + 2 i = 1 K h i h ¯ i N i + i = 1 K φ i ,
S I V = i = 1 K φ i n i ,
S V = i = 1 K h i h ¯ i 3 M i + 3 i = 1 K h ¯ i 2 N i + i = 1 K h ¯ i h i ( 3 + 1 n i ) φ i ,
M i = φ i 3 ( A i X i 2 + B i X i Y i + C i Y i 2 + D i ) , N i = φ i 2 ( E i X i + F i Y i ) , A i = n i + 2 4 n i ( n i 1 ) 2 , B i = n i + 1 n i ( n i 1 ) , C i = 3 n i + 2 4 n i ,
D i = n i 2 4 ( n i 1 ) 2 , E i = B i / 2 , F i = 2 n i + 1 2 n i , φ i = ( n i 1 ) ( 1 r i 1 r i ) = 1 s i 1 s i , X i = r i + r i r i r i , Y i = s i + s i s i s i = m i + 1 m i 1 = 1 2 s i φ i = 1 2 s i φ i , Y i + 1 = h i φ i h i + 1 φ i + 1 ( Y i 1 ) 1.
r i = 2 ( n i 1 ) φ i ( X i + 1 ) , r i = 2 ( n i 1 ) φ i ( X i 1 ) .
δ y = y P ( y P 2 + x P 2 ) 2 ( s 1 s ¯ 1 ) 3 u 1 3 u K S I + y 1 ( 3 y P 2 + x P 2 ) 2 ( s 1 s ¯ 1 ) 3 u 1 2 u K u ¯ 1 S I I y 1 2 y P 2 ( s 1 s ¯ 1 ) 3 u 1 u K u ¯ 1 2 ( 3 S I I I + I 2 S I V ) + y 1 3 2 ( s 1 s ¯ 1 ) 3 u K u ¯ 1 3 S V , δ x = x P ( y P 2 + x P 2 ) 2 ( s 1 s ¯ 1 ) 3 u 1 3 u K S I + 2 y 1 y P x P 2 ( s 1 s ¯ 1 ) 3 u 1 2 u K u ¯ 1 S I I y 1 2 x P 2 ( s 1 s ¯ 1 ) 3 u 1 u K u ¯ 1 2 ( S I I I + I 2 S I V ) ,
I = h 1 h ¯ 1 ( 1 s 1 1 s ¯ 1 ) = u 1 h ¯ 1 u ¯ 1 h 1 .
h ¯ 1 = s 1 s ¯ 1 s ¯ 1 s 1 .
h ¯ j = h j ( h ¯ 1 + i = 2 j d i 1 h i 1 h i ) ,
M ¯ i = M i ( φ i = 1 , m i = 1 ) , N ¯ i = N i ( φ i = 1 , m i = 1 ) , M ¯ i = A i X i 2 + D i , N ¯ i = E i X i .
δ s = 2 H 2 M ¯ ,
δ m = H 2 ( M ¯ + N ¯ / 2 ) ,
M ¯ i = 1 4 [ n i + 2 n i ( n i 1 ) 2 X i 2 + n i 2 ( n i 1 ) 2 ] ,
N ¯ i = 1 2 [ n i + 1 n i ( n i 1 ) X i ] .
M ¯ i = D i + A i E i 2 N ¯ i 2 ,
D i = n i 2 4 ( n i 1 ) 2 , A i E i 2 = n i ( n i + 2 ) ( n i + 1 ) 2 .
n = 1.5 A / E 2 = 0.84 , n = 2.0 A / E 2 = 0.89.
M ¯ i = D i + 0.86 N ¯ i 2 .
M i = φ i 3 ( M ¯ i + 2 N ¯ i Y i + 1.06 Y i 2 ) ,
N i = φ i 2 ( N ¯ i + 1.31 Y i ) ,
M ¯ i = f i 3 M i 2 f i 2 N i Y i + 1.56 Y i 2 ,
N ¯ i = f i 2 N i 1.31 Y i .
D = M ¯ 0.86 N ¯ 2 .
n = D D 0.5 .
r = 2 ( n 1 ) φ ( X + 1 ) , r = 2 ( n 1 ) φ ( X 1 ) .
1 s = φ 2 ( Y 1 ) , 1 s = φ 2 ( Y + 1 ) .
S I j = i = 1 K h j i 4 φ i 3 M ¯ i + 2 i = 1 K h j i 4 φ i 3 Y j i N ¯ i + 1.06 i = 1 K h j i 4 φ i 3 Y j i 2 ,
S I I j = i = 1 K h j i 3 h ¯ j i φ i 3 M ¯ i + i = 1 K h j i 2 φ i 2 ( 2 h j i h ¯ j i φ i Y j i + 1 ) N ¯ i + i = 1 K h j i 2 φ i 2 Y j i ( 1.06 h j i h ¯ j i φ i Y j i + 1.31 ) ,
S I I I j = i = 1 K h j i 2 h ¯ j i 2 φ i 3 M ¯ i + 2 i = 1 K h j i h ¯ j i φ i 2 ( h j i h ¯ j i φ i Y j i + 1 ) N ¯ i + i = 1 K h j i h ¯ j i φ i 2 Y j i ( 1.06 h j i h ¯ j i φ i Y j i + 2.62 ) + i = 1 K φ i ,
S I V j = i = 1 K φ i n i ,
S V j = i = 1 K h j i h ¯ j i 3 φ i 3 M ¯ i + i = 1 K h ¯ j i 2 φ i 2 ( 2 h j i h ¯ j i φ i Y j i + 3 ) N ¯ i + i = 1 K h ¯ j i 2 φ i 2 Y j i ( 1.06 h j i h ¯ j i φ i Y j i + 3.93 ) + i = 1 K h ¯ j i h j i ( 3 + 1 n i ) φ i ,
x i = M ¯ i ,
x i + K = N ¯ i ,
a j i = h j i 4 φ i 3 ,
a j , i + K = 2 h j i 4 φ i 3 Y j i ,
a j , 2 K + 1 = i = 1 K 1.06 h j i 4 φ i 3 Y j i 2 ,
a j + L , i = h j i 3 h ¯ j i φ i 3 ,
a j + L , i + K = h j i 2 φ i 2 ( 2 h j i h ¯ j i φ i Y j i + 1 ) ,
a j + L , 2 K + 1 = i = 1 K h j i 2 φ i 2 Y j i ( 1.06 h j i h ¯ j i φ i Y j i + 1.31 )
a j + 2 L , i = h j i 2 h ¯ j i 2 φ i 3 ,
a j + 2 L , i + K = 2 h j i h ¯ j i φ i 2 ( h j i h ¯ j i φ i Y j i + 1 ) ,
a j + 2 L , 2 K + 1 = i = 1 K h j i h ¯ j i φ i 2 Y j i ( 1.06 h j i h ¯ j i φ i Y j i + 2.62 ) + φ i ,
a j + 3 L , i = h j i h ¯ j i 3 φ i 3 ,
a j + 3 L , i + K = h ¯ j i 2 φ i 2 ( 2 h j i h ¯ j i φ i Y j i + 3 ) ,
a j + 3 L , 2 K + 1 = i = 1 K h ¯ j i 2 φ i 2 Y j i ( 1.06 h j i h ¯ j i φ i Y j i + 3.93 ) + 3.6 h ¯ j i h j i φ i ,
b j = S I j a j , 2 K + 1 , b j + L = S I I j a j + L , 2 K + 1 , b j + 2 L = S I I I j a j + 2 L , 2 K + 1 , b j + 3 L = S V j a j + 3 L , 2 K + 1 ,
Gx = b ,
G = ( a p q ) , b = ( b 1 , b 2 , , b 4 L ) T , x = ( M ¯ 1 , M ¯ 2 , , M ¯ K , N ¯ 1 , N ¯ 2 , , N ¯ K ) T ,
S target = ( S I 1 , S I 2 , , S I L , S I I 1 , S I I 2 , , S I I L , S I I I 1 , S I I I 2 , , S I I I L , S V 1 , S V 2 , , S V L ) T ,
x = G ¯ 1 b .
S res = S calc S target = Gx b ,
C I = i = 1 K h i 2 φ i ν i P i λ ,
C I I = i = 1 K h i h ¯ i φ i ν i P i λ ,
ν = n d 1 n F n C , P = λ n F n λ n F n C ,
i = 1 2 φ i = φ , i = 1 2 φ i ν i = 0.
i = 1 2 φ i = φ , i = 1 2 φ i ν i = 0 , i = 1 2 φ i ν i P i d = 0.
i = 1 3 φ i = φ , i = 1 3 φ i ν i = 0 , i = 1 3 φ i ν i P i d = 0 , i = 1 3 φ i ν i P i g = 0.
φ = φ 1 + φ 2 d φ 1 φ 2 , f = f 1 f 2 f 1 + f 2 d ,
s 1 = f ( 1 m 1 + d f 2 ) , s 2 = f ( 1 m d f 1 ) ,
d 2 e d + ( f 1 + f 2 ) e + f 1 f 2 ( m 1 ) 2 / m = 0.
d = 1 2 [ e ± e 2 4 ( f 1 + f 2 ) e 4 f 1 f 2 ( m 1 ) 2 / m ] .
d = 1 2 [ e ± e 2 4 ( f 1 + f 2 ) e 4 f 1 f 2 ( m 1 ) 2 / m ] ,
f = f 1 f 2 f 1 + f 2 d , s 1 = f ( 1 m 1 + d f 2 ) , s 2 = f ( 1 m d f 1 ) ,
h 1 = 1 , h 2 = s 2 s 1 m , h ¯ 1 = s 1 s ¯ 1 s ¯ 1 s 1 = s 1 ( p 1 + s 1 ) p 1 , h ¯ 2 = h 2 h ¯ 1 + d ,
Y 1 = 1 2 f 1 s 1 , Y 2 = 1 2 f 2 s 2 , φ 1 = 1 / f 1 , φ 2 = 1 / f 2 .
f 1 = 50 mm , f 2 = 50 mm , e = 0 , s ¯ 1 = 0 , m min = 0.333 , m max = 1 , K = 2 , L = 9.
S I = S I I = S I I I = 0.
f 3 = s 3 / ( 1 m 3 ) , f 2 = 1 / ( 1 / f 1 + 1 / f 3 ) ,
d 1 = f 1 + f 2 m 3 f 1 f 2 / f , d 2 = f ( 1 d 1 / f 1 s 3 / f ) / m 3 ,

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