Abstract

It is shown how the results of a previous paper [ J. Opt. Soc. Am. 52, 20 ( 1962)] may be used to deduce computing formulas for the aberration produced by refraction at either a spherical or aspheric surface. The secondary and higher-order aberrations are seen to fall into two parts, one independent of, and the other dependent on, the aberrations of prior surfaces. The method is illustrated by deriving the computing formulas for the primary and secondary aberrations. The computing formulas are applied to a triplet, and the results are compared with the values of the ray aberration found by ray tracing.

© 1962 Optical Society of America

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References

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  1. E. F. Denby, J. Opt. Soc. Am. 52, 20 (1962).
    [Crossref]
  2. H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, Oxford, England, 1950).
  3. E. F. Denby, Ph.D. thesis, University of London, 1957.
  4. E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, England, 1955).
  5. A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, Oxford, England, 1929), Part 1.

1962 (1)

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, Oxford, England, 1929), Part 1.

Denby, E. F.

E. F. Denby, J. Opt. Soc. Am. 52, 20 (1962).
[Crossref]

E. F. Denby, Ph.D. thesis, University of London, 1957.

Hopkins, H. H.

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

Linfoot, E. H.

E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, England, 1955).

J. Opt. Soc. Am. (1)

Other (4)

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

E. F. Denby, Ph.D. thesis, University of London, 1957.

E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, England, 1955).

A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, Oxford, England, 1929), Part 1.

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Figures (2)

Fig. 1
Fig. 1

The aberration function nuδη plotted against D, the height at which the incident ray cuts the entrance pupil (0° field). The lightly dashed line indicates primary aberrations, the heavier dashed line, primary and secondary aberrations, and the solid line, trigonometrical tracing.

Fig. 2
Fig. 2

The aberration function nuδη plotted against D, the height at which the incident ray cuts the entrance pupil (23° field).

Tables (2)

Tables Icon

Table I Nomenclature for secondary aberration coefficients and their dependence on the field and aperture variables.

Tables Icon

Table II Contributions to the primary and secondary spherical aberration and zonal coma coefficients at each surface of a simple Cooke triplet.

Equations (16)

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( n u η ) / h 0 = 1 2 ( n 3 / n 2 ) L sin 3 I - ( n 2 / n ) ( L Y - M X ) L ( sin 2 I ) / R + ( n 2 / n ) L ( sin I ) ( Z / l - Z / R ) + n Y ( 1 / l - 1 / R ) + 0 ( sin 5 I ) .
[ y 0 ( l - ρ ) - η 0 ( l ¯ - ρ ) 2 l ρ 3 ( l - l ¯ ) 3 ] [ n 2 n ] [ α 0 l t ( l - ρ ) - 2 β 0 l t ( l ¯ - ρ ) + γ 0 [ l t ( l ¯ - ρ ) 2 + ρ 2 ( l - l ¯ ) 2 ] [ l - ρ ] - 1 ] - E 1 [ y 0 l - η 0 l ¯ 2 ρ 3 ( l - l ¯ ) 3 ] n [ α 0 l 2 - 2 β 0 l l ¯ + γ 0 l ¯ 2 ] + 0 ( y 0 5 ) ,
( n u η ) h 0 = [ 1 2 y α ] [ m 3 ( l - l ¯ ) 3 ] { [ l - ρ ρ ] 2 n 2 [ - l ρ + ( l - ρ ρ ) ( n n ) ] [ 1 n ] - E 1 l 3 n ρ 3 } - [ 1 2 η α + y ϐ ] [ m 2 m ¯ ( l - l ¯ ) 3 ] { [ l - ρ ρ ] [ l ¯ - ρ ρ ] n 2 [ - l ρ + ( l - ρ ρ ) ( n n ) ] [ 1 n ] - E 1 l 2 l ¯ n ρ 3 } + [ 1 2 y γ + η ϐ ] [ m m ¯ 2 ( l - l ¯ ) 3 ] { [ l ¯ - ρ ρ ] 2 n 2 [ - l ρ + ( l - ρ ρ ) ( n n ) ] [ 1 n ] - E 1 l l ¯ 2 n ρ 3 } + [ 1 2 y γ ] [ m m ¯ 2 l ( l - l ¯ ) ] [ n 2 ρ n ] - [ 1 2 η γ ] [ m ¯ 3 ( l - l ¯ ) 3 ] { [ l ¯ - ρ ρ ] 3 [ ρ l - ρ ] n 2 [ - l ρ + ( l - ρ ρ ) ( n n ) ] [ 1 n ] + [ l ¯ - ρ ρ ] [ ρ l - ρ ] [ ( l - l ¯ ) 2 l ] - E 1 l ¯ 3 n ρ 3 } + 0 ( y 5 ) .
Δ ( n u δ η ) = Δ ( n u η ) = S 1 y ( x 2 + y 2 ) + S 2 [ η ( x 2 + y 2 ) + 2 y ( ξ x + η y ) ] + ( S 3 + S 4 ) y ( ξ 2 + η 2 ) + 2 S 3 η ( ξ x + η y ) + S 5 η ( ξ 2 + η 2 ) + 0 ( y 5 )
Δ ( n u δ ξ ) = S 1 x ( x 2 + y 2 ) + S 2 [ ξ ( x 2 + y 2 ) + 2 x ( ξ x + η y ) ] + ( S 3 + S 4 ) x ( ξ 2 + η 2 ) + 2 S 3 ξ ( ξ x + η y ) + S 5 ξ ( ξ 2 + η 2 ) + 0 ( y 5 ) .
Δ ( n ū δ x ) = S ¯ 1 ξ ( ξ 2 + η 2 ) + S ¯ 2 [ x ( ξ 2 + η 2 ) + 2 ξ ( ξ x + η y ) ] + ( S ¯ 3 + S ¯ 4 ) ξ ( x 2 + y 2 ) + 2 S ¯ 3 x ( ξ x + η y ) + S ¯ 5 x ( x 2 + y 2 ) + 0 ( y 5 ) ,
Δ ( n ū δ y ) = S ¯ 1 η ( ξ 2 + η 2 ) + S ¯ 2 [ y ( ξ 2 + η 2 ) + 2 η ( ξ x + η y ) ] + ( S ¯ 3 + S ¯ 4 ) η ( x 2 + y 2 ) + 2 S ¯ 3 x ( ξ x + η y ) + S 5 x ( x 2 + y 2 ) + 0 ( y 5 ) ,
- 2 S ¯ 1 = B 2 h ¯ 0 Δ ( ū / n ) + ( E 1 / ρ 3 ) h ¯ 0 4 Δ ( n ) , etc .
S ¯ 2 = S 5 - 1 2 H Δ ( ū 2 ) , S ¯ 3 = S 3 - 1 2 H Δ ( u ū ) , S ¯ 4 = S 4 , S ¯ 5 = S 2 - 1 2 H Δ ( u 2 ) .
( n u η ) / h 0 = [ y ( l - ρ ) - η ( l ¯ - ρ ) 2 l ρ 3 ( l - l ¯ ) 3 ] [ n 2 n ] [ α l ( l - ρ ) t - 2 β l ( l ¯ - ρ ) t + γ l ( l ¯ - ρ ) 2 ( l - ρ ) - 1 t + γ ρ 2 ( l - l ¯ ) 2 ( l - ρ ) - 1 ]
+ [ y ( l - ρ ) - η ( l ¯ - ρ ) 8 l ρ 3 ( l - l ¯ ) 5 ] [ n 2 n ] [ α 2 L + α β M + β 2 N + α γ O + β γ P + γ 2 Q ] + 0 ( y 7 ) ,
S 0 z 1 y ( x 2 + y 2 ) 2 + S 1 z 2 η ( x 2 + y 2 ) 2 + S 0 z 2 y ( x 2 + y 2 ) ( ξ x + η y ) + S 1 6 η ( x 2 + y 2 ) ( ξ x + η y ) + S 0 6 y ( ξ x + η y ) 2 + S 0 1 y ( x 2 + y 2 ) ( ξ 2 + η 2 ) + S 1 2 η ( x 2 + y 2 ) ( ξ 2 + η 2 ) + S 0 2 y ( ξ x + η y ) ( ξ 2 + η 2 ) + S 1 7 η ( ξ x + η y ) 2 + S 1 3 η ( ξ x + η y ) ( ξ 2 + η 2 ) + S 0 4 y ( ξ 2 + η 2 ) 2 + S 0 5 η ( ξ 2 + η 2 ) 2 ,
( S 1 / m 3 ) y α + ( S 2 / m 2 m ¯ ) η α + ( 2 S 2 / m 2 m ¯ ) y β + [ ( S 3 + S 4 ) / m m ¯ 2 ] y γ + 2 ( S 3 / m m ¯ 2 ) η β + ( S 5 / m ¯ 3 ) η γ ,
δ x = [ ξ γ S ¯ 1 + ( x γ + 2 ξ ϐ ) S ¯ 2 + ξ α ( S ¯ 3 + S ¯ 4 ) + 2 x ϐ S ¯ 3 + x α S ¯ 5 ] [ n ū ] - 1 ,
δ ξ = [ x α S 1 + ( ξ α + 2 x ϐ ) S 2 + x γ ( S 3 + S 4 ) + 2 ξ ϐ S 3 + ξ γ S 5 ] [ n u ] - 1 ,
- δ η / η = O . S . C . = 1 - ( l m - l ¯ m ) u m sin U 1 / ( L m - l ¯ m ) u 1 sin U m ,