Abstract

It is shown that polynomials related to the desired magnification and shift of focus of a zoom system may be used to derive a continued fraction related to a family of lenses having the desired properties. This is used to develop the method for first-order design of linearly compensated zoom systems outlined by William Peck at the 1961 Spring Meeting of the Optical Society of America. It is also shown that certain types of designs are apt to be very difficult to achieve with optical compensation, yet readily achieved with linear compensation. Numerical calculations illustrating the design procedure are carried out.

© 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. F. G. Back and H. Lowen, J. Opt. Soc. Am. 48, 149 (1958).
    [CrossRef]
  3. R. Kingslake, J. Soc. Motion Picture Television Engrs. 69, 534 (1960).
  4. It is clear, and is further demonstrated in Sec. 5, that optically compensated systems represent but a very limited subset of the class of all linearly compensated zoom systems. From this alone, it may be contended that linearly compensated systems enjoy a greater range of applicability than optically compensated systems.
  5. M. Herzberger, J. Opt. Soc. Am. 33, 651 (1943).
    [CrossRef]
  6. M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958), pp. 457–462.
  7. L. Bergstein, reference 1.
  8. M. Herzberger, reference 6.
  9. This convention is the reverse of that ordinarily applied but is somewhat more convenient in optical problems.
  10. H. S. Wall, Continued Fractions (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1948), Chap. IX, p. 16.
  11. Generally it is desirable that these properties be part of the specifications of a zoom lens which means that the coefficients of Pn may not be varied over a large range. The exact coefficients may be decided upon (if any variation is allowed) by systematic variation of the coefficients.
  12. L. Bergstein, reference 1, pp. 161–163.

1960 (1)

R. Kingslake, J. Soc. Motion Picture Television Engrs. 69, 534 (1960).

1958 (2)

1943 (1)

Back, F. G.

Bergstein, L.

L. Bergstein, J. Opt. Soc. Am. 48, 154 (1958).
[CrossRef]

L. Bergstein, reference 1, pp. 161–163.

L. Bergstein, reference 1.

Herzberger, M.

M. Herzberger, J. Opt. Soc. Am. 33, 651 (1943).
[CrossRef]

M. Herzberger, reference 6.

M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958), pp. 457–462.

Kingslake, R.

R. Kingslake, J. Soc. Motion Picture Television Engrs. 69, 534 (1960).

Lowen, H.

Wall, H. S.

H. S. Wall, Continued Fractions (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1948), Chap. IX, p. 16.

J. Opt. Soc. Am. (3)

J. Soc. Motion Picture Television Engrs. (1)

R. Kingslake, J. Soc. Motion Picture Television Engrs. 69, 534 (1960).

Other (8)

It is clear, and is further demonstrated in Sec. 5, that optically compensated systems represent but a very limited subset of the class of all linearly compensated zoom systems. From this alone, it may be contended that linearly compensated systems enjoy a greater range of applicability than optically compensated systems.

M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958), pp. 457–462.

L. Bergstein, reference 1.

M. Herzberger, reference 6.

This convention is the reverse of that ordinarily applied but is somewhat more convenient in optical problems.

H. S. Wall, Continued Fractions (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1948), Chap. IX, p. 16.

Generally it is desirable that these properties be part of the specifications of a zoom lens which means that the coefficients of Pn may not be varied over a large range. The exact coefficients may be decided upon (if any variation is allowed) by systematic variation of the coefficients.

L. Bergstein, reference 1, pp. 161–163.

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Figures (1)

Fig. 1
Fig. 1

Nomenclature and sign convention (paraxial optics).

Tables (3)

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Table I Origins and terminal points for the paraxial quantities associated with the (general) jth lens.

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Table II Tabular form for the computation of linearly compensated zoom systems.

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Table III Focal lengths (f’s) of lenses and separations (D’s) between lenses for a three lens linearly compensated zoom system. The parameter u has the range 0 ≤ u ≤ 1. Three phases in the development of the lens are shown. They are initial, modification 1, and modification 2 (final).

Equations (50)

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[ ] = 1 , [ a 1 ] = a 1 , [ a 1 , a 2 , , a n ] = a n [ a 1 , a 2 , , a n 1 ] + [ a 1 , a 2 , , a n 2 ] .
[ a 1 , a 2 ] = a 2 [ a 1 ] + [ ] = a 1 a 2 + 1 , [ a 1 , a 2 , a 3 ] = a 3 [ a 1 , a 2 ] + [ a 1 ] = a 1 a 2 a 3 + a 1 + a 3 , [ 2 ] = 2 , [ 2 , 7 ] = 7 [ 2 ] + [ ] = 14 + 1 = 15 , [ 2 , 7 , 9 ] = 9 [ 2 , 7 ] + [ 2 ] = 9 · 15 + 2 = 137 , [ 2 , 7 , 9 , 8 ] = 8 [ 2 , 7 , 9 ] + [ 2 , 7 ] = 8 · 137 + 15 = 1111 , [ a 1 + b 1 u ] = a 1 + b 1 u , [ a 1 + b 1 u , c 1 ] = c 1 [ a 1 + b 1 u ] + [ ] = a 1 c 1 + b 1 c 1 u + 1 , [ a 1 + b 1 u , c 1 , a 2 + b 2 u ] = ( a 2 + b 2 u ) ( a 1 c 1 + c 1 b 1 u + 1 ) + a 1 + b 1 u = b 1 b 2 c 1 u 2 + ( b 1 c 1 a 2 + b 2 a 1 c 1 + b 2 + b 1 ) u + a 1 a 2 c 1 + a 1 + a 2 .
[ a 1 , a 2 , a 3 , , a n ] [ a 2 , a 3 , , a n ] = a 1 + 1 a 2 + 1 a 3 + 1 + 1 a n .
F = a n + b n a n 1 + b n 1 + b 1 a 0 + b 0 ,
F = a n + Q b n Q a n 1 + Q b n 1 a n 2 + b n 2 + b 1 a 0 + b 0 ,
P n + 1 = c n + 1 u n + 1 + c n u n + c n 1 u n 1 + + c 1 u + c 0 ,
P n = d n u n + d n 1 u n 1 + + d 1 u + d 0 ,
P n + 1 P n = R n u + S n + 1 R n 1 u + S n 1 + 1 R n 2 u + S n 2 + 1 + 1 R 0 u + S 0 .
P 3 P 2 = 2 u 3 u 2 + 3 u 5 u 2 + u 1 = 2 u + 3 u 2 + 5 u 5 u 2 + u 1 = 2 u 3 + 8 u 8 u 2 + u 1 = 2 u 3 + 1 u 2 + u 1 8 u 8 = 2 u 3 + 1 ( 1 / 8 ) u + 2 u 1 8 u 8 = 2 u 3 + 1 ( 1 / 8 ) u + ( 1 / 4 ) + 1 8 u 8 .
j = 0 n D j
x j x j = f j 2 ( 1 j n ) ,
( 1 / L j ) ( 1 / L j ) = 1 / f j = K j ( 1 j n ) ,
L j / L j = h j / h j = m j ( 1 j n ) ,
h 0 / h j = M j = m 1 m 2 m 2 m j = M j 1 m j ( 1 j n ) ,
X j = x j x j + 1 = D j f j f j + 1 ( 1 j n 1 ) ,
X 0 = x 1 = D 0 f 1 ,
X n = D n f n ,
L n + 1 + X n = x n = D n f n + L n + 1 = L n f n ,
m j = L j K j + 1 ( 1 j n ) ,
D j = + L j + 1 L j ( 1 j n ) ,
D 0 = L 1 .
[ D 0 , K 1 , D 1 , K 2 , , D n 1 , K n ] = M n ,
[ D 0 , K 1 , D 1 , K 2 , , D n 1 , K n , D n ] = M n L n + 1 ,
[ D 0 , K 1 ] = 1 K 1 D 0 = 1 + K 1 L 1 = m 1 = M 1 .
[ D 0 , K 1 , D 1 ] = M 1 D 1 D 0 = M 1 L 1 + M 1 L 2 + L 1 = L 1 + M 1 L 2 + L 1 = M 1 L 2 .
[ D 0 , K 1 , , K j , D j , K j + 1 ] = K j + 1 M j L j + M j = M j ( K j + 1 L j + 1 + 1 ) = M j m j + 1 = M j + 1 .
[ D 0 , K 1 , , K j + 1 , D j + 1 ] = D j + 1 M j + 1 + M j L j + 1 = M j + 1 ( L j + 2 L j + 2 ) + M j L j + 1 = M j + 1 L j + 2 M j ( m j + 1 L j + 1 ) + M j L j + 1 = M j + l L j + 2 M j L j + 1 + M j L j + 1 = M j + 1 L j + 2 .
M n L n + 1 / M n = L n + 1 .
x n = ( f n ) 2 X n 1 + ( f n 1 ) 2 X n 2 + ( f n 2 ) 2 + ( f j + 1 ) 2 ( 1 ) n + 1 j X j + ( f j ) 2 ( 1 ) n X 1 + ( f 1 ) 2 ( 1 ) n + 1 X 0 .
x 1 = f 1 2 / x 1 = f 1 2 / X 0 = f 1 2 / X 0 .
x j + 1 = ( f j + 1 ) 2 / x j + 1 = ( f j + 1 ) 2 / ( x j X j ) = ( f j + 1 ) 2 / ( X j x j ) .
[ D 0 , K 1 , , K n , D n ] [ D 0 , K 1 , , K n ] = X n + ( f n ) 2 X n 1 + ( f n 1 ) 2 X n 2 + ( f n 2 ) 1 \ 2 ( 1 ) n X 1 + ( f 1 ) 2 ( 1 ) n + 1 X 0 .
D j = d j + p j u ,
X j = d j + p j u f j f j + 1 or d j = X j p j u + f j + f j + 1 ,
X 0 = d 0 + p 0 u f 1 or d 0 = X 0 p 0 u + f 1 ,
X n = d n + p n u f n or d n = X n p n u + f n .
M n L n + 1 = [ D 0 , K 1 , , K n , D n ] = P n + 1 , M n = [ D 0 , K 1 , , K n ] = P n ,
c n + 1 = p 0 j = 1 n ( p j K j ) .
c n + 1 = c n + 1 ( c n + 1 / c n + 1 ) 2 K 1 = c n + 1 ( c n + 1 / c n + 1 ) 2 ( c n + 1 / c n + 1 ) = c n + 1 .
( c n + 1 / j = 0 n R j ) 2 ,
P 4 = M 3 L 4 = u 4 2 u 3 + 1.25 u 2 0.25 u + 0.0078125 , P 3 = M 3 = 2 u 3 + u + 1.
L 4 = P 4 / P 3 = 05 u 1 + 1 2.666667 u 0.888889 + 1 0.511848 u 0.832865 + 1 0.566477 u + 0.732891 .
( c n + 1 / c n + 1 ) 2 = 1 / ( 0.5 · 2.666667 · 0.511848 · 0.566477 ) 2 = 6.690753.
L 4 = 05 u 1 + 1 2.666667 u 0.888889 + 1 0.511848 u 0.832865 + 6.690753 3.790158 u + 4.903593 .
j = 0 3 p j
j = 0 3 p j = 0.5 + 2.666667 + 0.511848 3.790158 = 1.111644.
L 4 = 0.5 u 1 + 1.416866 3.778309 u 1.259436 + 1.41.6866 0511848 u 0.832865 + 6.690753 3.790158 u + 4.903593 ,
j = 0 3 p j = 0.000001.
j = 0 3 p j = 0
j = 0 3 p j = 0