Abstract

Equations for the third-order aberrations in the meridional plane of a symmetrical optical system in the presence of finite tilts and decentrations are derived, based on the stop-shift equations of H. H. Hopkins. To demonstrate the application of these formulas and to show the degree of agreement between these equations and an exact ray trace, a numerical example is included.

© 1964 Optical Society of America

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References

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  1. Many early references as to the first-order properties of tilted or decentered lenses may be found in the book by S. Czapski and O. Eppenstein, Grundzüge der Theorie der Optischen Instrumente (Johann Ambrosius Barth, Leipzig, Germany, 1924), especially on p. 65.
  2. A. Maréchal, Rev. Opt., 1–24 (1950). Many references are also found in the bibliography on p. 24.
  3. A. L. Ingalls and K. Pestrecov, J. Opt. Soc. Am. 38, 343–349 (1948).
    [Crossref] [PubMed]
  4. P. C. Foote and R. A. Woodson, J. Opt. Soc. Am. 39, 590–599 (1949).
  5. L. I. Epstein, J. Opt. Soc. Am. 39, 847–853 (1949).
    [Crossref]
  6. G. Wooters, J. Opt. Soc. Am. 40, 521–523 (1950).
    [Crossref]
  7. H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, New York, 1950), pp. 76–118.
  8. C. G. Wynne, Proc. Phys. Soc. (London) B65, 429–437 (1952).
  9. Reference 7, p. 87.
  10. Reference 8, p. 430.
  11. Reference 7, p. 104.
  12. Reference 8, p. 431.

1952 (1)

C. G. Wynne, Proc. Phys. Soc. (London) B65, 429–437 (1952).

1950 (2)

A. Maréchal, Rev. Opt., 1–24 (1950). Many references are also found in the bibliography on p. 24.

G. Wooters, J. Opt. Soc. Am. 40, 521–523 (1950).
[Crossref]

1949 (2)

P. C. Foote and R. A. Woodson, J. Opt. Soc. Am. 39, 590–599 (1949).

L. I. Epstein, J. Opt. Soc. Am. 39, 847–853 (1949).
[Crossref]

1948 (1)

Czapski, S.

Many early references as to the first-order properties of tilted or decentered lenses may be found in the book by S. Czapski and O. Eppenstein, Grundzüge der Theorie der Optischen Instrumente (Johann Ambrosius Barth, Leipzig, Germany, 1924), especially on p. 65.

Eppenstein, O.

Many early references as to the first-order properties of tilted or decentered lenses may be found in the book by S. Czapski and O. Eppenstein, Grundzüge der Theorie der Optischen Instrumente (Johann Ambrosius Barth, Leipzig, Germany, 1924), especially on p. 65.

Epstein, L. I.

Foote, P. C.

P. C. Foote and R. A. Woodson, J. Opt. Soc. Am. 39, 590–599 (1949).

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, New York, 1950), pp. 76–118.

Ingalls, A. L.

Maréchal, A.

A. Maréchal, Rev. Opt., 1–24 (1950). Many references are also found in the bibliography on p. 24.

Pestrecov, K.

Woodson, R. A.

P. C. Foote and R. A. Woodson, J. Opt. Soc. Am. 39, 590–599 (1949).

Wooters, G.

Wynne, C. G.

C. G. Wynne, Proc. Phys. Soc. (London) B65, 429–437 (1952).

J. Opt. Soc. Am. (4)

Proc. Phys. Soc. (London) (1)

C. G. Wynne, Proc. Phys. Soc. (London) B65, 429–437 (1952).

Rev. Opt. (1)

A. Maréchal, Rev. Opt., 1–24 (1950). Many references are also found in the bibliography on p. 24.

Other (6)

Many early references as to the first-order properties of tilted or decentered lenses may be found in the book by S. Czapski and O. Eppenstein, Grundzüge der Theorie der Optischen Instrumente (Johann Ambrosius Barth, Leipzig, Germany, 1924), especially on p. 65.

Reference 7, p. 87.

Reference 8, p. 430.

Reference 7, p. 104.

Reference 8, p. 431.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, New York, 1950), pp. 76–118.

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Figures (10)

Fig. 1
Fig. 1

Analysis of a lens suffering from decentration only.

Fig. 2
Fig. 2

Analysis of a lens as a result of tilting the stop.

Fig. 3
Fig. 3

Analysis of a lens whose object plane has been tilted.

Fig. 4
Fig. 4

The effect of tilting a lens about the entrance pupil. This is a “pure” tilt.

Fig. 5
Fig. 5

To aid analysis of a “pure” tilt, Fig. 4 is rotated through an angle θ.

Fig. 6
Fig. 6

The effect of a skewed lens. This is the general case in any optical system.

Fig. 7
Fig. 7

To aid analysis of a skewed lens, Fig. 6 is shown as a combination of a decentration and a “pure” tilt.

Fig. 8
Fig. 8

The formula for a triplet lens system to be evaluated in the presence of tilts and decentrations is shown.

Fig. 9
Fig. 9

The results of tracing a paraxial marginal ray and a paraxial principal ray through the triplet of Fig. 8 include third-order spherical aberration and coma coefficients. The paths of the rays are shown.

Fig. 10
Fig. 10

Lens B of the triplet of Fig. 8 is tipped and Lens C is decentered. An exact axial ray is traced through the resulting system.

Tables (2)

Tables Icon

Table I Third order SII, versus exact SII, centered system.

Tables Icon

Table II Third-order SII versus exact SII, tilted and decentered system.

Equations (15)

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S I * = S I , S II * = [ ( h + e ) / h 0 ] S II , S III * = [ ( h + e ) / h 0 ] 2 S III , S IV * = [ ( h + e ) / h 0 ] 2 S IV , S V * = [ ( h + e ) / h 0 ] 3 S V , S VI * = [ ( h + e ) / h 0 ] 4 S VI .
S ¯ II * = S V * - ( h + e ) ( n u ) { [ ( h + e ) / d ] 2 - [ ( h + e ) / d ] 2 } , S ¯ III * = S III * - ( h + e ) ( n u ) { u [ ( h + e ) / d ] - u [ ( h + e ) / d ] } , S ¯ V * = S II * - ( h + e ) ( n u ) [ u 2 - u 2 ] .
δ S I = 0 , δ S II = ( Q δ E ) S I , δ S III = 2 ( Q δ E ) S II + ( Q δ E ) 2 S I , δ S IV = 0 , δ S V = ( Q δ E ) ( S IV + 3 S III ) + 3 ( Q δ E ) 2 S II + ( Q δ E ) 3 S I , δ S VI = ( Q δ E ) ( 3 S ¯ II + S V ) + ( Q δ E ) 2 ( 3 S ¯ III + 3 S III + 2 S IV ) + ( Q δ E ) 3 ( S ¯ IV + 3 S II ) + ( Q δ E ) 4 S I .
( Q δ E ) = δ B ¯ / B = ( ū i - ū ) / u , ( Q δ E ) = ( h + e d - h d ) / u , ( Q δ E ) = e / u d .
S I ** = S I , S II ** = [ ( h + e ) / h 0 ] S II + ( e / u d ) S I , S III ** = ( h + e h 0 ) 2 S III + 2 ( e u d ) ( h + e h 0 ) S II + ( e u d ) 2 S I , S IV ** = [ ( h + e ) / h 0 ] 2 S IV , S V ** = ( h + e h 0 ) 3 S V + ( e u d ) × [ ( h + e h 0 ) 2 S IV + 3 ( h + e h 0 ) 2 S III ] + 3 ( e u d ) 2 ( h + e h 0 ) S II + ( e u d ) 3 S I , S VI ** = ( h + e h 0 ) 4 S VI + ( e u d ) × { 4 ( h + e h 0 ) 3 S V - 3 ( h + e ) ( n u ) × [ ( h + e d ) 2 - ( h + e d ) 2 ] } + ( e u d ) 2 × { 6 ( h + e h 0 ) 2 S III + 2 ( h + e h 0 ) 2 S IV - 3 ( h + e ) × ( n u ) [ u ( h + e d ) - u ( h + e d ) ] } + ( e u d ) 3 × { 4 ( h + e h 0 ) S II - ( h + e ) ( n u ) [ u 2 - u 2 ] } + ( e u d ) 4 S I .
x = - h θ , u = y / d , u 1 = y / ( d + h θ ) ,
u 1 = y / ( d - x ) .
Q ¯ δ Ē = ( u 1 - u ) / ū , Q ¯ δ Ē = [ y / ( d - x ) - y / d ] / ū .
S I * = S I , S II * = [ ( h + d θ ) / h 0 ] S II , S III * = [ ( h + d θ ) / h 0 ] 2 S III , S IV * = [ ( h + d θ ) / h 0 ] 2 S IV , S V * = [ ( h + d θ ) / h 0 ] 3 S V , S VI * = [ ( h + d θ ) / h 0 ] 4 S VI .
S I * = S I , S II * = [ ( h + θ D + θ d ) / h 0 ] S II + ( θ D / u d ) S I , S III * = ( h + θ D + θ d h 0 ) 2 S III + 2 ( θ D u d ) ( h + θ D + θ d h 0 ) S II + ( θ D u d ) 2 S I , S IV * = [ ( h + θ D + θ d ) / h 0 ] 2 S IV , S V * = ( h + θ D + θ d h 0 ) 3 S V + ( θ D u d ) [ ( h + θ D + θ d h 0 ) 2 S IV + 3 ( h + θ D + θ d h 0 ) 2 S III ] + 3 ( θ D u d ) ( h + θ D + θ d h 0 ) S II + ( θ D u d ) 3 S I , S VI * = ( h + θ D + θ d h 0 ) 4 S VI + ( θ D u d ) × { 4 ( h + θ D + θ d h 0 ) 3 S V - 3 ( h + θ D + θ d ) ( n u ) × [ ( ( h + θ d ) + θ D d ) 2 - ( h + θ D + θ d d ) 2 ] } + ( θ D u d ) 2 { 6 ( h + θ D + θ d h 0 ) 2 S III + 2 ( h + θ D + θ d h 0 ) 2 S IV - 3 ( h + θ D + θ d ) ( n u ) × [ u ( ( h + θ d ) + θ D d ) - u ( h + θ D + θ d d ) ] } + ( θ D u d ) 3 [ 4 ( h + θ D + θ d h 0 ) S II - ( h + θ D + θ d ) ( n u ) ( u 2 - u 2 ) ] + ( θ D u d ) 4 S I .
d = y u - y ¯ ū = 0.58572828 - 0.24788263 - 0.00050135 0.00403129 = - 2.4872912 , h = h 0 = y ¯ - ū d = 0.00050135 - ( 0.00403129 ) ( - 2.4872912 ) = 0.0105284 , u = - 0.24788263 , S I = 0.02878781 , S II = - 0.00022930.
D = y ¯ u - y x u x = 0.00050135 0.00403129 - 0 - 0.005 = 0.1243650 , S II * = [ ( h + θ D + θ d ) / h 0 ] S II + [ θ D / u d ] S I , S II 3 * = [ ( 0.0105284 ) + ( 0.05 ) ( 0.1243650 ) + ( 0.05 ) ( - 2.4872912 ) ( 0.0105284 ) ] ( - 0.00022930 ) + [ ( 0.05 ) ( 0.1243650 ) ( - 0.24788263 ) ( - 2.4872912 ) ] ( 0.02878781 ) , S II 3 * = 0.0026341.
d = y u - y ¯ ū = 0.56230854 - 0.05854937 - 0.00151065 0.00252324 = - 10.2026945 , h = h 0 = y ¯ - ū d = 0.00151065 - ( 0.00252324 ) ( - 10.2026945 ) = 0.0272545 , u = - 0.05854937 , S I = 0.00351993 , S II = 0.00006653 , θ = - u x = 0.03033536 , D = y ¯ ū - y x u x = 0.00151065 0.00252324 - - 0.01213503 - 0.03033536 = 0.1986643 , S II 4 * = [ ( 0.0272545 ) + ( 0.03033536 ) ( 0.1986643 ) + ( 0.03033536 ) ( - 10.2026945 ) 0.0272545 ] ( 0.00006653 ) + [ ( 0.03033536 ) ( 0.1986643 ) ( - 0.05854937 ) ( - 10.2026945 ) ] ( 0.00351993 ) , S II 4 * = - 0.0006388.
d = y u - y ¯ ū = 0.58911324 0.04873583 - 0.00401377 0.00455113 = 11.2059663 , h = h 0 = y ¯ - ū d = 0.00401377 - ( 0.00455113 ) ( 11.2059663 ) = - 0.0469860 , u = 0.04873583 , S I = - 0.00045658 , S II = - 0.00001835 , θ = - u x = 0.00314272 , D = y ¯ ū - y x u x = 0.00401377 0.00455113 - - 0.03384513 - 0.00314272 = - 9.8874469 , S II 5 * = [ ( - 0.0469860 ) + ( 0.00314272 ) ( - 9.8874469 ) + ( 0.00314272 ) ( 11.2059663 ) ( - 0.0469860 ) ] ( - 0.00001835 ) + [ ( 0.00314272 ) ( - 9.8874469 ) ( 0.04873583 ) ( 11.2059663 ) ] ( - 0.00045658 ) , S II 5 * = 0.0000093.
d = y u - y ¯ ū = 0.58990988 0.00036211 - 0.00974881 0.00260684 = 1625.3506237 , h = h 0 = y ¯ - ū d = 0.00974881 - ( 0.00260684 ) ( 1625.3506237 ) = - 4.2272779 , u = 0.00036211 , S I = - 0.00793893 , S II = - 0.00002914 , θ = - u x = 0.00023237 , D = y ¯ ū - y x u x = 0.00974881 0.00260684 - - 0.03435628 - 0.00023237 = - 144.1119086 , S II 6 * = [ ( - 4.2272779 ) + ( 0.00023237 ) ( - 144.1119086 ) + ( 0.00023237 ) ( 1625.3506237 ) ( - 4.2272779 ) ] ( - 0.00002914 ) + [ ( 0.00023237 ) ( - 144.1119086 ) ( 0.00036211 ) ( 1625.3506237 ) ] ( - 0.00793893 ) , S II 6 * = 0.0004249.