Abstract

An aberration function that is closely related to the transverse-ray aberration but which remains meaningful under all circumstances is described. As a preliminary to deriving computing formulas for the change in aberration with refraction, it is shown first how this function can be expanded in terms of refractive indices and quantities invariant with refraction, and second how these invariants can be expanded as functions of the object space coordinates of the incident ray. Refraction at both spherical and aspheric surfaces is considered and the expansions are taken as far as is necessary to calculate the primary and secondary aberrations, although they are capable of being taken further. The convergence of the expansions is discussed.

© 1962 Optical Society of America

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References

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  1. J. Focke, Jenaer Jahrbuch 1951 (Kommissionsverlag Gustav Fischer Jena, 1951).
  2. H. A. Buchdahl, Optical Aberration Coefficients (Oxford University Press, New York, 1954).
  3. A. E. Conrady, Applied Optics and Optical Design, Part I (Oxford University Press, New York, 1929).
  4. M. T. Zolli, Atti Fondazione Giorgio Ronchi 10, 267 (1955).

1955 (1)

M. T. Zolli, Atti Fondazione Giorgio Ronchi 10, 267 (1955).

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Oxford University Press, New York, 1954).

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part I (Oxford University Press, New York, 1929).

Focke, J.

J. Focke, Jenaer Jahrbuch 1951 (Kommissionsverlag Gustav Fischer Jena, 1951).

Zolli, M. T.

M. T. Zolli, Atti Fondazione Giorgio Ronchi 10, 267 (1955).

Atti Fondazione Giorgio Ronchi (1)

M. T. Zolli, Atti Fondazione Giorgio Ronchi 10, 267 (1955).

Other (3)

J. Focke, Jenaer Jahrbuch 1951 (Kommissionsverlag Gustav Fischer Jena, 1951).

H. A. Buchdahl, Optical Aberration Coefficients (Oxford University Press, New York, 1954).

A. E. Conrady, Applied Optics and Optical Design, Part I (Oxford University Press, New York, 1929).

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Figures (1)

Fig. 1
Fig. 1

The boundary ∑ between the two media of refractive indices n and n′ is tangent to YOX. The entrance pupil plane is yōx, and ηoξ represents the Gaussian object plane. The incident ray ABC intersects ∑ at A, the entrance pupil at B, and the object plane at C. The normal to ∑ at A is AR, and RP is normal to AR.

Equations (55)

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O o = l ,             O ō = l ¯ .
η = Y + ( l - Z ) F / G .
η = Y + ( l - Z ) F / G .
Y h 0 Δ ( n / l ) + h 0 Δ ( n F / G ) - Z h 0 Δ ( n F / l G ) = Δ ( n η h 0 / l ) , = Δ ( n u η )
Δ ( n u η ) = Δ ( n u δ η ) ,
δ η k = δ η k + 1 , u k = u k + 1 , n k = n k + 1 ,
k = 1 m Δ ( n k u k δ η k ) = n m u m δ η m - n 1 u 1 δ η 1 .
n h 0 tan - n h 0 tan 0 - n u δ η ,
[ Δ ( n k u k δ η k ) ] / n m u m .
( R - Z ) M / r + N Y / r ,             - ( R - Z ) L / r - N X / r ,             - L Y / r + M X / r .
r E = [ N Y + ( R - Z ) M ] sin I - X cos I r F = - [ N X + ( R - Z ) L ] sin I - Y cos I r G = [ M X - L Y ] sin I - ( Z - R ) cos I .
cos I = ( 1 - sin 2 I ) 1 2 = 1 - 1 2 sin 2 I - 1 8 sin 4 I + O ( sin 6 I ) , 1 / G = ( r / R ) [ 1 - 1 2 sin 2 I - 1 8 sin 4 I - ( Z / R ) cos I - ( L Y - M X ) ( sin I ) / R ] - 1 + O ( sin 6 I ) .
0 1 2 sin 2 I + 1 8 sin 4 I + ( Z / R ) cos I + ( L Y - M X ) ( sin I ) / R < 1 ,
n u η 0 / h 0 = n Y / l - n ( 1 - Z / l ) F / G , = n ( Y / l - Y / R ) + n ( Y / l - Y / R ) Z / R + n ( Y / l - Y / R ) Z 2 / R 2 - n ( L + N X / R ) sin I - n ( sin I ) ( L Y - M X ) Y / R 2 - n L ( sin I ) Z / R + n ( L Y - M X ) ( sin I ) Y Z / R 3 }
- 1 2 n L sin 3 I - n L ( L Y - M X ) ( sin 2 I ) / R + n L ( sin I ) ( Z / l - Z / R )
- n ( L Y - M X ) 2 ( sin 2 I ) Y / R 3 - n N X ( L Y - M X ) ( sin 2 I ) / R 2 + n N X ( Z / l - Z / R ) ( sin I ) / R + n ( L Y - M X ) ( sin I ) ( Z / l - Z / R ) Y / R 2 - 1 2 n N X ( sin 3 I ) / R - n L ( L Y - M X ) 2 ( sin 3 I ) / R 2 - 1 2 n ( L Y - M X ) ( sin 3 I ) Y / R 2 + n L ( L Y - M X ) ( sin 2 I ) ( Z / l - Z / R ) / R + 1 2 L n ( sin 3 I ) Z / R + 1 2 n L ( sin 3 I ) ( Z / l - Z / R ) - n L ( L Y - M X ) ( sin 4 I ) / R - 3 8 n L sin 5 I . }
n ( l - ρ ) / l ρ = n ( l - ρ ) / l ρ ,
S R = h ( d z / d h ) - 1 .
Z = 1 2 h 2 / ρ + 1 8 h 4 ( 1 + E 1 ) / ρ 3 + 1 1 6 h 6 ( 1 + E 2 ) / ρ 5 + O ( h 8 ) ,
d z / d h = h / ρ + 1 2 h 3 ( 1 + E 1 ) / ρ 3 + 3 8 h 5 ( 1 + E 2 ) / ρ 5 + O ( h 7 ) .
R = O S + S R = ρ - 1 2 h 2 E 1 / ρ - 1 8 h 4 ( 3 E 2 - 4 E 1 - 2 E 1 2 ) / ρ 3 + O ( h 6 ) ,
1 / R = [ 1 + 1 2 h 2 E 1 / ρ 2 + 1 8 h 4 ( 3 E 2 - 4 E 1 ) / ρ 4 ] / ρ + O ( h 6 ) .
r = ( h 2 + S R 2 ) 1 2
r = ρ - 1 2 h 2 E 1 / ρ + O ( h 4 ) .
( 1 / l - 1 / R ) = ( 1 / l - 1 / ρ ) - 1 2 h 2 E 1 / ρ 3 - 1 8 h 4 ( 3 E 2 - 4 E 1 ) / ρ 5 + O ( h 6 ) ,
( l - R ) / r = ( l - ρ ) / ρ + 1 2 h 2 l E 1 / ρ 3 + O ( h 4 ) ,
( l - R ) 2 / r 2 = ( l - ρ ) 2 / ρ 2 + E 1 l ( l - ρ ) h 2 / ρ 4 + O ( h 4 ) .
( l ¯ - R ) / r = ( l ¯ - ρ ) / ρ + 1 2 h 2 l ¯ E 1 / ρ 3 + O ( h 4 )
( l ¯ - R ) 2 / r 2 = ( l ¯ - ρ ) 2 + E 1 l ¯ ( l ¯ - ρ ) h 2 / ρ 4 + O ( h 4 ) .
( l - R ) ( l ¯ - R ) / r 2 = ( l - ρ ) ( l ¯ - ρ ) / ρ 2 + 1 2 E 1 h 2 [ l ( l ¯ - ρ ) + l ¯ ( l - ρ ) ] / ρ 4 + O ( h 4 ) .
L x + M y + N ( l ¯ - R ) = 0
L ξ + M η + N ( l - R ) = 0.
L 2 + M 2 + N 2 = 1 ,
L = [ - y ( l - R ) + η ( l ¯ - R ) ] N / ( ξ y - η x ) , M = [ x ( l - R ) - ξ ( l ¯ - R ) ] N / ( ξ y - η x ) , N = ± [ ξ y - η x ] [ α ( l - R ) 2 - 2 β ( l - R ) ( l ¯ - R ) + γ ( l ¯ - R ) 2 + ( ξ y - η x ) 2 ] - 1 2 , }
Q B R = Θ ,             Q B = s ¯ ,             and             Q C = s .
s - s ¯ = [ ( l - l ¯ ) 2 + ( ξ - x ) 2 + ( η - y ) 2 ] 1 2 .
s ¯ = B R cos Θ - ( - B C 2 - B R 2 + R C 2 ) / 2 B C .
s ¯ = [ ( l - l ¯ ) ( l ¯ - R ) - α + β ] [ ( l - l ¯ ) 2 + ( ξ - x ) 2 + ( η - y ) 2 ] - 1 2
s = [ ( l - l ¯ ) ( l - R ) + γ - β ] [ ( l - l ¯ ) 2 + ( ξ - x ) 2 + ( η - y ) 2 ] - 1 2 .
r 2 sin 2 I = R B 2 - s ¯ 2 = [ α ( l - R ) 2 - 2 β ( l - R ) ( l ¯ - R ) + γ ( l ¯ - R ) 2 + ( ξ y - η x ) 2 ] [ s - s ¯ ] - 2 .
sin 2 I = [ α ( l - ρ ) 2 - 2 β ( l - ρ ) ( l ¯ - ρ ) + γ ( l ¯ - ρ ) 2 ] [ ρ 2 ( l - l ¯ ) 2 ] - 1 + h 2 E 1 [ α l ( l - ρ ) - β { l ¯ ( l - ρ ) + l ( l ¯ - ρ ) } + γ l ¯ ( l ¯ - ρ ) ] × [ ρ 4 ( l - l ¯ ) 2 ] - 1 - [ α 2 ( l - ρ ) 2 - 2 α β ( l - ρ ) × ( l - ρ + l ¯ - ρ ) + 2 α γ ( l - ρ ) ( l ¯ - ρ ) ] [ ρ 2 ( l - l ¯ ) 4 ] - 1 - [ β 2 ( l - ρ + l ¯ - ρ ) 2 - 2 β γ ( l ¯ - ρ ) ( l - ρ + l ¯ - ρ ) + γ 2 ( l ¯ - ρ ) 2 ] [ ρ 2 ( l - l ¯ ) 4 ] - 1 + O ( α 3 ) .
Y = ( η l ¯ - y l ) / ( l ¯ - l ) + O ( y 3 ) X = ( ξ l ¯ - x l ) / ( l ¯ - l ) + O ( x 3 ) .
h 2 = X 2 + Y 2 = ( α l 2 - 2 β l l ¯ + γ l ¯ 2 ) ( l - l ¯ ) - 2 + O ( α 2 )
E 1 { α l ( l - ρ ) - β [ l ¯ ( l - ρ ) + l ( l ¯ - ρ ) ] + γ l ¯ ( l ¯ - ρ ) } × { α l 2 - 2 β l l ¯ + γ l ¯ 2 } { ρ 4 ( l - l ¯ ) 4 } - 1 .
L sin I = [ - y ( l - R ) + η ( l ¯ - R ) ] [ ± r ( s - s ¯ ) ] - 1 , M sin I = [ x ( l - R ) - ξ ( l ¯ - R ) ] [ ± r ( s - s ¯ ) ] - 1 , N sin I = [ ξ y - η x ] [ ± r ( s - s ¯ ) ] - 1 .
L sin I = [ - y ( l - ρ ) + η ( l - ρ ) ] [ 1 - 1 2 ( α - 2 β + γ ) ( l - l ¯ ) - 2 ] × [ ρ ( l - l ¯ ) ] - 1 - 1 2 E 1 [ l y - l ¯ η ] [ α l 2 - 2 β l l ¯ + γ l ¯ 2 ] × [ ρ 3 ( l - l ¯ ) 3 ] - 1 + O ( y 5 ) .
M sin I = [ x ( l - ρ ) - ξ ( l - ρ ) ] [ 1 - 1 2 ( α - 2 β + γ ) ( l - l ¯ ) - 2 ] × [ ρ ( l - l ¯ ) ] - 1 + 1 2 E 1 [ l x - l ¯ ξ ] [ α l 2 - 2 β l l ¯ + γ l ¯ 2 ] × [ ρ 3 ( l - l ¯ ) 3 ] - 1 + O ( y 5 ) ,
N sin I = [ ξ y - η x ] [ ρ ( l - l ¯ ) ] - 1 + O ( y 4 ) .
Y = ( y s - η s ¯ ) ( s - s ¯ ) - 1 - ( η - y ) r ( cos I ) ( s - s ¯ ) - 1 .
Y = ( y l - η l ¯ ) ( l - l ¯ ) - 1 - 1 2 [ y - η ] [ α l 2 - 2 β l l ¯ + γ l ¯ 2 ] × [ ρ ( l - l ¯ ) 3 ] - 1 + O ( y 5 ) .
X = ( x l - ξ l ¯ ) ( l - l ¯ ) - 1 - 1 2 [ x - ξ ] [ α l 2 - 2 β l l ¯ + γ l ¯ 2 ] × [ ρ ( l - l ¯ ) 3 ] - 1 + O ( y 5 ) .
h 2 = { α l 2 - 2 β l l ¯ + γ l ¯ 2 } { 1 - [ α l - β ( l + l ¯ ) + γ l ¯ ] [ ρ ( l - l ¯ ) 2 ] - 1 } × { l - l ¯ } - 2 + O ( Y 6 ) ,
h 4 = ( α l 2 - 2 β l l ¯ + γ l ¯ 2 ) 2 ( l - l ¯ ) - 4 + O ( y 6 ) .
Z = 1 2 h 2 / ρ + 1 8 h 4 ( 1 + E 1 ) / ρ 3 + O ( h 6 ) ,
Z = 1 2 [ α l 2 - 2 β l l ¯ + γ l ¯ 2 ] [ ρ ( l - l ¯ ) 2 ] - 1 + 1 8 E 1 [ α l 2 - 2 β l l ¯ + γ l ¯ 2 ] 2 × [ ρ 3 ( l - l ¯ ) 4 ] - 1 + 1 8 [ α 2 l 3 ( l - 4 ρ ) + 4 α β l 2 ( 3 l ¯ + l ρ - l l ¯ ) + 2 α γ l l ¯ ( - 2 l ρ - 2 l ¯ ρ + l l ¯ ) ] [ ρ 3 ( l - l ¯ ) 4 ] - 1 + 1 8 [ 4 β 2 l l ¯ ( - 2 l ρ - 2 l ¯ ρ + l l ¯ ) + 4 β γ l ¯ 2 ( 3 l ρ + l ¯ ρ - l l ¯ ) + γ 2 l ¯ 3 ( l ¯ - 4 ρ ) ] [ ρ 3 ( l - l ¯ ) 4 ] - 1 + O ( y 6 ) .