Abstract

A new concept on differential movements for moving lens components is given. A unified varifocal differential equation is derived. The solvable region, solution exchange, quickly varifocal curve, etc., are discussed in detail. Some important types of zoom systems are discussed from the point of view of the varifocal differential equation.

© 1992 Optical Society of America

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References

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  1. Tao ChunKan, “The varifocal equation for a zoom system,” Kexue Tongbao, 22, 207–213 (1977).
  2. Tao ChunKan, Design of Zoom System, 1st ed. (Publishing House of Defence Industry, Beijing, July, 1988).
  3. G. H. Cooke, Japanese Patent48-6813 (6June1969).
  4. M. Herzberger, “Gaussian optics and Gaussian brackets,” J. Opt. Soc. Am. 33, 651–655 (1943).
    [CrossRef]
  5. L. Bergstein, “General theory of optically compensated varifocal systems,” J. Opt. Soc. Am. 48, 154–171 (1958).
    [CrossRef]
  6. R. Pegis, W. Peck, “First-order theory for linearly compensated zoom systems,” J. Opt. Soc. Am. 52, 905–911 (1962).
    [CrossRef]
  7. K. Tanaka, “Allgemeine Gaussche Theorie eines mechanisch Kompensierten Zoom-objectives,” Opt. Commun. 29, 138–140 (1979).
    [CrossRef]

1979 (1)

K. Tanaka, “Allgemeine Gaussche Theorie eines mechanisch Kompensierten Zoom-objectives,” Opt. Commun. 29, 138–140 (1979).
[CrossRef]

1977 (1)

Tao ChunKan, “The varifocal equation for a zoom system,” Kexue Tongbao, 22, 207–213 (1977).

1962 (1)

1958 (1)

1943 (1)

Bergstein, L.

ChunKan, Tao

Tao ChunKan, “The varifocal equation for a zoom system,” Kexue Tongbao, 22, 207–213 (1977).

Tao ChunKan, Design of Zoom System, 1st ed. (Publishing House of Defence Industry, Beijing, July, 1988).

Cooke, G. H.

G. H. Cooke, Japanese Patent48-6813 (6June1969).

Herzberger, M.

Peck, W.

Pegis, R.

Tanaka, K.

K. Tanaka, “Allgemeine Gaussche Theorie eines mechanisch Kompensierten Zoom-objectives,” Opt. Commun. 29, 138–140 (1979).
[CrossRef]

J. Opt. Soc. Am. (3)

Kexue Tongbao (1)

Tao ChunKan, “The varifocal equation for a zoom system,” Kexue Tongbao, 22, 207–213 (1977).

Opt. Commun. (1)

K. Tanaka, “Allgemeine Gaussche Theorie eines mechanisch Kompensierten Zoom-objectives,” Opt. Commun. 29, 138–140 (1979).
[CrossRef]

Other (2)

Tao ChunKan, Design of Zoom System, 1st ed. (Publishing House of Defence Industry, Beijing, July, 1988).

G. H. Cooke, Japanese Patent48-6813 (6June1969).

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Figures (12)

Fig. 1
Fig. 1

Schematic of the zoom system.

Fig. 2
Fig. 2

m31 and m32

Fig. 3
Fig. 3

Relation of b to m3.

Fig. 4
Fig. 4

Solvable region of m3

Fig. 5
Fig. 5

Solution exchange of m31 and m32.

Fig. 6
Fig. 6

Schematic of grad Γ.

Fig. 7
Fig. 7

Diagram of the system with a positive compensation component.

Fig. 8
Fig. 8

Solution exchange for the system with a positive compensation component.

Fig. 9
Fig. 9

Schematic diagram of a Varotal-30-type system.

Fig. 10
Fig. 10

Diagram of a Varotal-30-type system.

Fig. 11
Fig. 11

Δ1 and Δ2 in the condition of Eqs. (4.21).

Fig. 12
Fig. 12

System with three movable components.

Tables (2)

Tables Icon

Table I Relation of d ¯ 23 l to f3

Tables Icon

Table II Image Position Error

Equations (97)

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φ = φ 1 + φ 2 - d φ 1 φ 2 ,
i d L i = 0 ,
L = f ( 2 - 1 / m - m ) ,
L min = 4 f .
d = ( 1 - m 2 2 ) d q .
m 3 2 ( 1 - m 2 2 ) d q ,
( 1 - m 3 2 ) d Δ .
m 3 2 ( 1 - m 2 2 ) d q + ( 1 - m 3 2 ) d Δ = 0 ,
( 1 - m 2 2 ) d q + 1 - m 3 2 m 3 2 d Δ = 0
d q = f 2 m 2 2 d m 2 ,
d Δ = f 3 d m 3 .
1 - m 2 2 m 2 2 f 2 d m 2 + 1 - m 3 2 m 3 2 f 3 d m 3 = 0.
d L = 1 - m 2 m 2 f d m ,
U ( m 2 , m 3 ) = f 2 ( 1 / m 2 + m 2 ) + f 3 ( 1 / m 3 + m 3 ) = C .
m 2 = m 2 l ,             m 3 = m 3 l ;
f 2 ( 1 / m 2 l + m 2 l ) + f 3 ( 1 / m 3 l + m 3 l ) = C ,
f 2 ( 1 / m 2 - 1 / m 2 l + m 2 - m 2 l ) + f 3 ( 1 / m 3 - 1 / m 3 l + m 3 - m 3 l ) = 0.
m 3 2 - b m 3 + 1 = 0 ,
b = - f 2 f 3 ( 1 m 2 - 1 m 2 l + m 2 - m 2 l ) + ( 1 m 3 l + m 3 l ) .
m 31 = b + ( b 2 - 4 ) 1 / 2 2 , m 31 = b - ( b 2 - 4 ) 1 / 2 2 ,
q = f 2 ( 1 / m 2 - 1 / m 2 l ) ,
m 2 = 1 1 / m 2 l + q / f 2
Δ 1 = f 3 ( m 31 - m 3 l ) , Δ 2 = f 3 ( m 32 - m 3 l ) .
m 31 > 0 ,             m 32 > 0.
m 31 > 1 ,             0 < m 32 < 1.
m 31 < 1 ,             m 32 < 0.
m 31 < 1 ,             m 32 > 1.
b = 1 / m 3 + m 3 .
b min = 2.
b < 2.
b 2             or             b - 2.
m 3 l = f 3 f 2 ( 1 - m 2 l ) - d 23 l + f 3 ,
d m 3 d m 2 = 1 - m 2 2 m 2 2 f 2 / 1 - m 3 2 m 3 2 f 3 .
m 2 = ± 1.
m 2 = - 1 ,             m 3 = - 1.
Γ = m 2 l m 3 l m 2 m 3
( grad Γ ) p = ( M 2 ) p i ¯ i + ( M 3 ) p j ¯ = ( Γ m 2 ) i ¯ + ( Γ m 3 ) p j ¯ = ( - 1 ) ( Γ ) p ( m 2 ) p i ¯ + ( - 1 ) ( Γ ) p ( m 3 ) p j ¯ ,
tan α = ( M 3 ) p ( M 2 ) p = ( m 2 ) p ( m 3 ) p .
1 - m 2 2 m 2 2 f 2 d m 2 + 1 - m 3 2 m 3 2 f 3 d m 3 = 0.
m 2 = 1 1 / m 2 l + q / f 2 .
b = - f 2 f 3 ( 1 m 2 - 1 m 2 l + m 2 - m 2 l ) + ( 1 m 3 l + m 3 l ) .
m 31 = b + ( b 2 - 4 ) 1 / 2 2 , m 32 = b - ( b 2 - 4 ) 1 / 2 2 .
Δ 1 = f 3 ( m 31 - m 3 l ) , Δ 2 = f 3 ( m 32 - m 3 l ) .
Γ 1 = m 2 l m 3 l m 2 m 31 , Γ 2 = m 2 l m 3 l m 2 m 32 .
m 31 = m 32 = - 1.
b = - f 2 f 3 ( 1 m 2 - 1 m 2 l + m 2 - m 2 l ) + ( 1 m 3 l + m 3 l ) ,
m 3 l = f 3 f 2 ( 1 - m 2 l ) - d 23 l + f 3 .
m 2 = - 1 ,             b = - 2.
- 2 = 1 f 3 ( - 2 - 1 m 2 l - m 2 l ) + ( 1 m 3 l + m 3 l ) .
A u = 1 m 3 l + m 3 l = 1 f 3 ( 2 + 1 m 2 l + m 2 l ) - 2 ;
m 3 l 2 - A u m 3 l + 1 = 0.
m 3 l = A u ± ( A u 2 - 4 ) 1 / 2 2 ,
m 3 l 1 = A u + ( A u 2 - 4 ) 1 / 2 2 m 3 l 2 = A u - ( A u 2 - 4 ) 1 / 2 2
d 23 l = m 2 l - 1 + f 3 - f 3 m 3 l 2 .
m ¯ 2 l = - 1 ,             m ¯ 3 l = - 1.
d ¯ 23 l = m ¯ 2 l - 1 + f 3 - f 3 m ¯ 3 l ,
d ¯ 23 l = 2 f 3 - 2.
m ¯ 2 l = - 1 ,             d ¯ 23 l = 0.8 ,             f 2 = - 1 ,             f 3 = 1.4 ,
m 4 2 m 3 2 ( 1 - m 2 2 ) d q 2 .
m 4 2 ( 1 - m 3 2 ) d Δ ,
( 1 - m 4 2 ) d q 4 .
m 4 2 m 3 2 ( 1 - m 2 2 ) d q 2 + m 4 2 ( 1 - m 3 2 ) d Δ + ( 1 - m 4 2 ) d q 4 = 0 ,
d q = f 2 m 2 2 d m 2 ,
d q 4 = f 4 d m 4 ,
d Δ = ( 1 - m 2 2 ) d q 2 + f 2 m 3 2 d m 3 ,
1 - m 2 2 m 2 2 f 2 d m 2 + 1 - m 3 2 m 3 2 f 3 d m 3 + 1 - m 4 2 m 4 2 f 4 d m 4 = 0.
i = 2 N 1 - m i 2 m i 2 f i d m i = 0
U ( m 2 , m 3 , m 4 ) = f 2 ( 1 / m 2 + m 2 ) + f 3 ( 1 / m 3 + m 3 ) + f 4 ( 1 / m 4 + m 4 ) = C .
m 2 = m 2 l ,             m 3 = m 3 l ,             m 4 = m 4 l ,
f 2 ( 1 / m 2 - 1 / m 2 l + m 2 - m 2 l ) + f 3 ( 1 / m 3 - 1 / m 3 l + m 3 - m 3 l ) + f 4 ( 1 / m 4 - 1 / m 4 l + m 4 - m 4 l ) = 0.
m 3 2 - b m 3 + 1 = 0.
b = - f 2 f 3 ( 1 m 2 - 1 m 2 l + m 2 - m 2 l ) - f 4 f 3 ( 1 m 4 - 1 m 4 l + m 4 - m 4 l ) + ( 1 m 3 l + m 3 l ) ,
m 31 = b + ( b 2 - 4 ) 1 / 2 2 , m 32 = b - ( b 2 - 4 ) 1 / 2 2 .
m 2 = 1 1 / m 2 l - q 2 / f 2 .
m 4 = m 4 l + q 2 / f 4 .
Δ = f 3 ( 1 / m 3 - 1 / m 3 l ) - q 2 + f 2 ( m 2 - m 2 l ) .
d m 4 d m 2 = f 2 f 4 1 m 2 2 .
d m 3 d m 2 = - 1 - m 4 2 m 4 2 f 2 m 2 2 - 1 - m 2 2 m 2 2 f 2 1 - m 3 2 m 3 2 f 3 = 0.
m 2 2 m 4 2 = 1.
d m 3 d m 2 = - m 3 2 f 3 ( 1 m 2 2 - 1 ) f 2 ,
d m 3 d m 2 = ( - 1 - m 4 2 m 4 2 f 2 m 2 2 - 1 - m 2 2 m 2 2 f 2 ) / 1 - m 3 2 m 3 2 f 3 ,
m 3 2 = 1 - m 4 2 ( m 2 2 m 4 2 - m 4 2 ) .
m 2 2 m 4 2 = 1 , m 3 2 = 1 ,
m 2 m 4 = 1 , m 3 = - 1 ,
Δ 1 = Δ 2 ,
d Δ 1 d m 2 = d Δ 2 d m 2 0
m 3 2 m 4 2 d q 2 + ( 1 - m 3 2 ) m 4 2 d q 3 + ( 1 - m 4 2 ) d q 4 = 0.
m 3 2 d q 2 + ( 1 + m 3 2 ) d q 3 + 1 - m 4 2 m 4 2 d q 4 = 0.
d q 4 = f 4 d m 4 ,
d q 3 = f 3 m 3 2 d m 3 + d q 2 .
d q 2 + i = 3 4 1 - m i 2 m i 2 f i d m i = 0.
q 4 = f 4 ( m 4 l - m 4 ) ,
m 4 = m 4 l + q 4 / f 4 .
m 3 2 - b m 3 + 1 = 0 ,
b = f 4 f 3 ( 1 m 4 - 1 m 4 l + m 4 - m 4 l ) + ( 1 m 3 l + m 3 l ) - q 2 f 3 ,
m 31 = b + ( b 2 - 4 ) 1 / 2 2 , m 32 = b - ( b 2 - 4 ) 1 / 2 2 .
q 3 = f 3 ( 1 / m 3 - 1 / m 3 l ) + q 2 .

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