Abstract

The general theory of optically compensated varifocal systems is applied to the case of a two-component system consisting of a single movable component placed behind a fixed component. This is the simplest case of a varifocal system and one that can be treated in all its details by elementary analytical methods. It is shown that for a given focal range four different systems are possible in all of which the focal length of the movable component has the same magnitude. Of these four systems only two are of practical interest. It is demonstrated that if the optimum conditions are realized the image deviation can be reduced to a sufficiently small value so that the two-component varifocal system can be of practical value in some specific applications, as for example in projection systems and viewfinders, as well as in telescopes of small magnification range.

© 1962 Optical Society of America

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References

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  1. L. Bergstein, J. Opt. Soc. Am. 48, 154 (1958).
    [Crossref]
  2. C. C. Allen, U. S. Patent696,788 (1902).
  3. H. J. Gramatzki, Brit. Patent449,434 (1953).
  4. H. J. Gramatzki, Probleme der Konstruktiven Optik (Akademie Verlag, Berlin, 1954).

1958 (1)

Allen, C. C.

C. C. Allen, U. S. Patent696,788 (1902).

Bergstein, L.

Gramatzki, H. J.

H. J. Gramatzki, Brit. Patent449,434 (1953).

H. J. Gramatzki, Probleme der Konstruktiven Optik (Akademie Verlag, Berlin, 1954).

J. Opt. Soc. Am. (1)

Other (3)

C. C. Allen, U. S. Patent696,788 (1902).

H. J. Gramatzki, Brit. Patent449,434 (1953).

H. J. Gramatzki, Probleme der Konstruktiven Optik (Akademie Verlag, Berlin, 1954).

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Figures (8)

Fig. 1
Fig. 1

The two-component optically compensated varifocal system.

Fig. 2
Fig. 2

The two-component optically compensated varifocal system and its Gaussian parameters, its over-all normalized focal length f(z) and image-plane deviation y(z) as functions of the (normalized) displacement z.

Fig. 3
Fig. 3

Possible two-component varifocal systems, the focal lengths f2 and f1 of their components, their maximum over-all focal length f(max), their final image distance l′(z) and focal range f(z). The systems shown in (Pa-1), (Pa-2) and (Pb) are P systems, the systems shown in (Na) and (Nb) are N systems. Only the systems shown in (Pa-1), (Pa-2), and (Na) are of practical interest.

Fig. 4
Fig. 4

The parameters of the optimum two-component varifocal systems as functions of the focal range τ. f1 is the normalized focal length of the movable component; f2 is the normalized focal length of the front component, and (f2)0=f2s21, is the focal length of the front component when the distance s21 between the rear principal plane of the front component and the front principal plane of the movable component in its extreme front position is zero; l1′(0) is the final image distance measured from the rear principal plane of the movable component in position z=0; fmax is the normalized maximum over-all focal length of the system; ( f max ) 0 = f max + ( ) 1 2 s 21, is the maximum over-all focal length for the case when s21=0; and ( f max ) 1 = ( f max ) 0 - ( ) 1 2, is the maximum over-all focal length when s21=1.0.

Fig. 5
Fig. 5

The maximum value y2(max) of the normalized image-plane deviation y(z) of two-component varifocal systems without full compensation at both ends of the operating range (and an optimized image deviation function) as a function of the focal range τ. The image-plane deviation y(z) as a function of the displacement z of the movable component is shown above the graph of y2(max).

Fig. 6
Fig. 6

The maximum value, YT(max)=[FT2(max)/Zm]×Θ(max), of the image-plane deviation of optimum two-component varifocal systems without full compensation at both ends of the operating range (and an optimized image deviation function) as a function of the focal range τ. [Θ(max)]0 is the value of Θ(max) for the case when s21=0 and [Θ(max)]1 is the value of Θ(max) for the case when s21=1.0; s21 is the distance between the rear principal plane of the front component and the front principal plane of the movable component in the position z=0. FT(max) is the maximum over-all focal length of the system and Zm is the maximum displacement of the movable component.

Fig. 7
Fig. 7

The two optimum two-component projection systems of example 1, their focal range f(z) and image-plane deviation y(z).

Fig. 8
Fig. 8

The varifocal viewfinder of example 2, its magnification range M(z) and image plane deviation δϕ(z) in diopters.

Equations (92)

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f k F k Z m ,             d k , k - 1 D k , k - 1 Z m ,             s k , k - 1 S k , k - 1 Z m ,
z = ( Z / Z m ) .
ψ 1 ( 2 , 1 ) b 1 = d 21 .
f ( z ) = - f 1 f 2 / ( z + b 1 ) .
x 1 ( z ) = f 1 2 / ( z + b 1 ) ,
y ( z ) = ( z 2 + c 1 z + c 2 ) / ( z + b 1 ) ,
c 1 = b 1 - ( x - y 0 ) , c 2 = y 0 b 1 ;
x x 1 ( 0 ) = f 1 2 / b 1 .
γ 1 = z 1 + z 2 , γ 2 = z 1 z 2 ,
y ( z ) = ( z - z 1 ) ( z - z 2 ) z + b 1 = z 2 - γ 1 z + γ 2 z + b 1 .
b 1 - ( x - y 0 ) = - γ 1 ,
y 0 = ( γ 2 / b 1 ) .
y ( 0 ) = y ( 1 ) ,
γ 2 = ( 1 - γ 1 ) b .
y ( 1 ) y 0 = ( γ 2 / b 1 ) = 1 - γ 1 ,
x - b 1 = 1.0.
F max F min = f max f min .
r F ( 0 ) F ( 1 ) = f ( 0 ) f ( 1 ) ,
τ f ( 0 ) - f ( 1 ) f ( 0 ) + f ( 1 ) = r - 1 r + 1 = ± - 1 + 1 ,
( r - 1 ) b 1 = 1.0.
d 21 = b 1 = 1 r - 1 = 1 - τ 2 τ ,
x = b 1 + 1 = r r - 1 = 1 + τ 2 τ ,
f 1 = ± ( r ) 1 2 r - 1 = ± ( 1 - τ 2 ) 1 2 2 τ .
f 1 = ( ) 1 2 / ( - 1 ) .
f 2 = s 21 - d 21 - f 1 ,
f 2 = 1 1 ( r ) 1 2 + s 21 .
f max = ( ) 1 2 f 2 ,
l 1 ( z ) = - ( d 21 + f 1 + z ) = - [ ( r ) 1 2 ( r ) 1 2 1 - ( 1 - z ) ] .
l 1 ( z ) = x ( z ) + f 1 = [ ( r ) 1 2 ( r ) 1 2 1 - z ] + y ( z , 0 ) = ( r ) 1 2 ( r ) 1 2 1 - r z 1 + ( r - 1 ) z ,
l 0 i ( z ) = l 1 ( z ) - l 1 ( z ) = ( r ) 1 2 ± 1 ( r ) 1 2 1 + y ( z , 0 ) .
- l 1 ( z ˆ ) = l 1 ( z ˆ ) = ( 1 ± 1 ) ( r ) 1 2 r - 1 .
1 2 l 0 i ( z ˆ ) = - l 1 ( z ˆ ) = l 1 ( z ˆ ) = 2 ( r ) 1 2 / ( r - 1 ) = 2 f 1 .
1 2 l 0 i ( z ˆ ) = - l 1 ( z ˆ ) = l 1 ( z ˆ ) = 0.
f ( z ) = ( r ) 1 2 f 2 1 + ( r - 1 ) z ,
f P ( max ) = f P ( 0 ) = ( ) 1 2 ( ) 1 2 - 1 { 1 - [ ( ) 1 2 - 1 ] s 21 } ,
f N ( max ) = f N ( 1 ) = - ( ) 1 2 - 1 [ 1 + ( ) 1 2 - 1 ( ) 1 2 s 21 ] ,
y ( z , 0 ) y ( z ) - y 0 = - ( r - 1 ) z ( 1 - z ) 1 + ( r - 1 ) z .
z ˆ z ˆ 12 = 1 ( r ) 1 2 + 1 = 1 2 [ 1 - ( r ) 1 2 - 1 ( r ) 1 2 + 1 ] 1 2 ( 1 - g ) ,
g ( r ) 1 2 - 1 ( r ) 1 2 + 1             and             g = ( ) 1 2 - 1 ( ) 1 2 + 1 .
y ( z ˆ ) - y 0 = - ( r 1 2 - 1 ) / ( r 1 2 + 1 ) = - g .
y ( 0 ) = y ( z ˆ ) = y ( 1 ) = y ( max ) = 1 2 g = 1 2 ( 1 2 - 1 ) / ( 1 2 + 1 ) ,
z 1 , 2 = 1 4 [ 2 - g ( 2 - g 2 ) 1 2 ] .
y ( z ˆ ) = y * ( max ) = g = ( 1 2 - 1 ) / ( 1 2 + 1 ) .
Y T ( z ) = F T 2 ( max ) Z m [ y ( z ) f 2 ( max ) ] = F T 2 ( max ) Z m Θ ( z ) ,
Y T ( max ) F T 2 ( max ) / Z m Θ ( max ) = [ 1 2 - 1 2 ( 1 2 + 1 ) ] 1 f 2 2 .
Θ P ( max ) = ( 1 2 - 1 ) 3 2 ( 1 2 + 1 ) [ 1 1 - ( 1 2 - 1 ) s 21 ] 2 ,
Θ N ( max ) = ( 1 2 - 1 ) 3 2 2 ( 1 2 + 1 ) [ 1 1 + ( 1 2 - 1 ) s 21 / 1 2 ] 2 ,
Y T N ( max ) Y T P ( max ) = Θ N ( max ) Θ P ( max ) = 1 [ 1 - ( - 1 ) s 21 ( 1 2 ) + ( 1 2 - 1 ) s 21 ] 2 .
M = - F 0 B / F E .
F 0 = 1 2 f 2 M ( max ) Z m .
( S 10 ) P = [ 1 2 1 2 - 1 + 1 2 f 2 M ( max ) ] Z m = { ( 1 2 1 2 - 1 ) × [ M ( max ) - 1 M ( max ) ] + 1 2 s 21 M ( max ) } Z m ,
( S 10 ) N = [ - 1 1 2 - 1 + 1 2 f 2 M ( max ) ] Z m = { ( 1 1 2 - 1 ) × [ M ( max ) - 1 ] + 1 2 s 21 M ( max ) } Z m ,
δ ϕ ( z ) = - Z m y ( z ) F 0 2 = - [ M 2 ( max ) Z m ] [ y ( z ) f 2 ( max ) ] = - M 2 ( max ) Z m Θ ( z ) ,
δ ϕ ( max ) = M 2 ( max ) Z m Θ ( max ) ,
( r ) N = 1 / = 1 / 1.5             and             τ = - 0.2.
y ( max ) = 0.05051 ,
Y T ( max ) F T 2 ( max ) / Z m = Θ ( max ) = 0.001134 ( 1 + 0.18350 s 21 ) 2 .
Y T ( max ) = 0.00102 F T 2 ( max ) Z m .
Z m = 20.0 mm ,
Y T ( max ) = ( 0.00102 ) ( 45.0 ) mm = 0.046 mm .
F 1 = - ( 2.4495 ) ( 20.0 ) mm = - 48.990 mm , S 21 = ( 0.3 ) ( 20.0 ) mm = 6.000 mm ,
F 2 = + ( 5.4495 + 0.3 ) ( 20.0 ) mm = + 114.990 mm .
L 1 ( z ) = - [ 5.4495 - 1.5 ( 1 - z ) 1 + 0.5 ( 1 - z ) ] ( 20.0 ) mm ,
F ( max ) = - ( 7.0417 ) ( 20.0 ) mm = - 140.833 mm .
F T ( max ) = 30.0 mm .
W = - 140.833 / 30.0 = - 4.6944.
S 10 = ( 1.5 ) ( 20.0 ) mm = 30.0 mm .
F 0 = ( 118.990 / 5.6944 ) mm = + 20.896 mm ,
L 0 = ( 118.990 / 4.6944 ) mm = + 25.347 mm .
S = ( 1.8 ) ( 20.0 ) mm = 36.0 mm ,
L = ( 36.0 + 25.35 ) mm = 61.35 mm .
( r ) P = = 1.5             and             τ = + 0.2 ,
Θ ( max ) = 0.001701 ( 1 - 0.22475 s 21 ) 2 .
Y T ( max ) = 0.00196 F T 2 ( max ) Z m = 0.088 mm .
F 1 = + ( 2.4495 ) ( 20.0 ) mm = + 48.990 mm , S 21 = ( 0.3 ) ( 20.0 ) = 6.000 mm ,
F 2 = - ( 4.4495 - 0.3 ) ( 20.0 ) mm = - 82.990 mm .
L 1 ( z ) = ( 5.4495 - 1.5 z 1 + 0.5 z ) 20.0 mm ,
F ( max ) = + 5.0821 ( 20.0 ) mm = + 101.641 mm .
W = ( 101.641 / 30.0 ) = 3.3880.
F 0 = ( 78.990 / 2.3880 ) mm = 33.077 mm ,
L 0 = ( 78.990 / 3.3880 ) mm = 23.314 mm .
L = ( 36.0 + 23.314 ) mm = 59.314 mm .
F 2 2 Z m 1 2 - 1 2 ( 1 2 + 1 ) [ M 2 ( max ) δ ϕ ( max ) ] ,
= 4.0
M ( max ) = 2.0
F 2 2 Z m > 1 6 δ ϕ ( max ) .
δ ϕ ( max ) = 0.5 diopter .
F 2 2 / Z m 333.3 mm .
Z m = 30.0 mm
F 2 = + ( 3.333 ) ( 30.0 ) mm = + 100.00 mm .
F 1 = - ( 0.667 ) ( 30.0 ) mm = - 20.00 mm , S 21 = ( 1.333 ) ( 30.0 ) mm = 40.00 mm , F 0 = + ( 3.333 ) ( 30.0 ) mm = + 100.00 mm , S 10 = ( 2.333 ) ( 30.0 ) mm = 70.00 mm ,
F 1 = + ( 0.667 ) ( 30.0 ) mm = + 20.00 mm , S 21 = ( 4.333 ) ( 30.0 ) mm = 130.00 mm , F 0 = + ( 3.333 ) ( 30.0 ) mm = + 100.00 mm , S 10 = ( 5.333 ) ( 30.0 ) mm = 160.00 mm ,