Abstract

It is assumed that a preliminary design of an optical system has been completed and the characteristic rays traced through it to determine the aberrations which afflict it. The present article then starts with these data and offers a set of reduction formulae. These formulae are applied, surface by surface, from the image space forward until any particular parameter is reached. This is done for every parameter in the system. These results are finally substituted in differentiated aberration formulae to give a coefficient which, when multiplied by the change in the parameter, gives the change in the aberration in the final image space. When these co-efficients are calculated for all parameters and all aberrations and recorded in a table, say, with the aberrations across the top and the parameters down the side, the designer has a complete picture of the performance of that optical system. The table then enables him to quickly adjust all parameters so as to obtain the best results possible from the set-up as given.

© 1948 Optical Society of America

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Corrections

Waldemar M. Stempel, "Erratum: A Differential Adjustment Method of Refining Optical Systems," J. Opt. Soc. Am. 39, 199-199 (1949)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-39-2-199

References

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  1. W. M. Stempel, J. Opt. Soc. of Am. 33, 278 (1943).
    [Crossref]
  2. A. L. M’Auley, Proc. Phys. Soc. LVII302, 350, 419, 435 (1945). F. D. Cruickshank, J. Opt. Soc. Am. 36, 13, 103 (1946).
    [Crossref] [PubMed]

1945 (1)

A. L. M’Auley, Proc. Phys. Soc. LVII302, 350, 419, 435 (1945). F. D. Cruickshank, J. Opt. Soc. Am. 36, 13, 103 (1946).
[Crossref] [PubMed]

1943 (1)

W. M. Stempel, J. Opt. Soc. of Am. 33, 278 (1943).
[Crossref]

M’Auley, A. L.

A. L. M’Auley, Proc. Phys. Soc. LVII302, 350, 419, 435 (1945). F. D. Cruickshank, J. Opt. Soc. Am. 36, 13, 103 (1946).
[Crossref] [PubMed]

Stempel, W. M.

W. M. Stempel, J. Opt. Soc. of Am. 33, 278 (1943).
[Crossref]

J. Opt. Soc. of Am. (1)

W. M. Stempel, J. Opt. Soc. of Am. 33, 278 (1943).
[Crossref]

Proc. Phys. Soc. (1)

A. L. M’Auley, Proc. Phys. Soc. LVII302, 350, 419, 435 (1945). F. D. Cruickshank, J. Opt. Soc. Am. 36, 13, 103 (1946).
[Crossref] [PubMed]

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Tables (2)

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Table II Summary of results.

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Form No. V Adjustment coefficients for the Cooke anastigmat series IV.

Equations (89)

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E = r sin I sin U .             F = N sin U N sin U .             G = sin ( I - I ) cos I cos I .
d U = U U d U + U L d L + U r d r + U N / N d N N ,
d L = L U d U + L L d L + L r d r + L N / N d N N .
d U 1 = M 1 d U 1 + O 1 d L 1 + P 1 d r 1 + Q 1 d ( N / N ) ,
d L 1 = R 1 d U 1 + T 1 d L 1 + V 1 d r 1 + W 1 d ( N / N ) .
d r 2 = ( r 2 r 1 ) 2 d r 1 .
d N 1 N 1 = - N 1 N 1 2 d N 1 , and d N 2 N 2 = d N 1 N 2 = 1 N 2 d N 1 .
d U 2 = U 2 U 2 d U 2 + U 2 L 2 d L 2 + U 2 r 2 d r 2 + U 2 - N 2 / N 2 d N 2 N 2 ,
= M 2 d U 2 + O 2 d L 2 + ( r 2 r 1 ) 2 P 2 d r 1 + Q 2 N 2 d N 1 ,
d U 2 = d U 1 = M 1 d U 1 + O 1 d L 1 + P 1 d r 1 + Q 1 d N 1 N 1 ,
d L 2 = d L 1 = R 1 d U 1 + T 1 d L 1 + V 1 d r 1 + W 1 d N 1 N 1 , d U 2 = M 2 ( M 1 d U 1 + O 1 d L 1 + P 1 d r 1 + Q 1 d N 1 N 2 ) + O 2 ( R 1 d U 1 + T 1 d L 1 + V 1 d r 1 + W 1 d N 1 N 1 ) + ( r 2 r 1 ) 2 P 2 d r 1 + Q 2 N 2 d N 1 .
d U 2 = ( M 2 M 2 + O 2 R 1 ) d U 1 + ( M 2 O 1 + O 2 T 1 ) d L 1 + ( M 2 P 1 + O 2 V 1 + ( r 2 r 1 ) 2 P 2 ) d r 1 + ( - N 1 N 1 2 M 2 Q 1 - N 1 N 1 2 O 2 W 1 + Q 2 N 2 ) d N 1 .
d L 2 = ( R 2 M 1 + T 2 R 1 ) d U 1 + ( R 2 O 1 + T 2 T 1 ) d L 1 + ( R 2 P 1 + T 2 V 1 + ( r 2 r 1 ) 2 V 2 ) d r 1 + ( - N 1 N 1 2 R 2 Q 1 - N 1 N 1 2 T 2 W 1 + W 2 N 2 ) d N 1 .
d U 2 = H 2 U d U 1 + H 2 L d L 1 + H 2 r d r 1 + H 2 N d N 1 ,
d L 2 = K 2 U d U 1 + K 2 L d L 1 + K 2 r d r 1 + K 2 N d N 1 ,
H 2 U = M 2 M 1 + O 2 R 1 , H 2 L = M 2 O 1 + O 2 T 1 , H 2 r = M 2 P 1 + O 2 V 1 + ( r 2 r 1 ) 2 P 2 , H 2 N 1 / N 1 = M 2 Q 1 + O 2 W 1 - ( N 1 N 1 ) 2 Q 2 , H 2 N 1 = - N 1 / N 1 2 ( M 2 Q 1 + O 2 W 1 ) + Q 2 N 2 .
K 2 U = R 2 M 1 + T 2 R 1 , K 2 L = R 2 O 1 + T 2 T 1 , K 2 r = R 2 P 1 + T 2 V 1 + ( r 2 r 1 ) 2 V 2 , †† K 2 N 1 / N 1 = R 2 Q 1 + T 2 W 1 - ( N 1 / N 1 ) 2 W 2 , K 2 N 1 = - N 1 / N 1 2 ( R 2 Q 1 + T 2 W 1 ) - ( N 1 / N 1 ) 2 W 2 .
d U 3 = M 3 d U 3 + O 3 d L 3 = M 3 { M 2 [ M 1 d U 1 + O 1 d L 1 + P 1 d r 1 ] + O 2 [ R 1 d U 1 + T 1 d L 1 + V 1 d r 1 ] } + O 3 { R 2 [ M 1 d U 1 + O 1 d L 1 + P 1 d r 1 ] + T 2 [ R 1 d U 1 + T 1 d L 1 + V 1 d r 1 ] } = ( M 3 M 2 M 1 + M 3 O 2 R 1 + O 3 R 2 M 1 + O 3 T 2 R 1 ) d U 1 + ( M 3 M 2 O 1 + M 3 O 2 T 1 + O 3 R 2 O 1 + O 3 T 2 T 1 ) d L 1 + ( M 3 M 2 P 1 + M 3 O 2 V 1 + O 3 R 2 P 1 + O 3 T 2 V 1 ) d r 1 .
d U a = ( M a - 2 H U a - 1 + R a - 2 H L a - 1 ) d U 1 + ( O a - 2 H U a - 1 + T a - 2 H L a - 1 ) d L 1 + ( P a - 2 H U a - 1 + V a - 2 H L a - 1 ) d r 1 .
d L a = R 3 d U 3 + T 3 d L 3 = R 3 { M 2 [ M 1 d U 1 + O 1 d L 1 + P 1 d r 1 ] + O 2 [ R 1 d U 1 + T 1 d L 1 + V 1 d r 1 ] } + T 3 { R 2 [ M 1 d U 1 + O 1 d L 1 + P 1 d r 1 ] + T 2 [ R 1 d U 1 + T 1 d L 1 + V 1 d r 1 ] } = ( R 3 M 2 M 1 + R 3 O 2 R 1 + T 3 R 2 M 1 + T 3 T 2 R 1 ) d U 1 + ( R 3 M 2 O 1 + R 3 O 2 T 1 + T 3 R 2 O 1 + T 3 T 2 T 1 ) d L 1 + ( R 3 M 2 P 1 + R 3 O 2 V 1 + T 3 R 2 P 1 + T 3 T 2 V 1 ) d r 1 = ( M a - 2 K U a - 1 + R a - 2 K L a - 1 ) d U 1 + ( O a - 2 K U a - 1 + T a - 2 K L a - 1 ) d L 1 + ( P a - 2 K U a - 1 + V a - 2 K L a - 1 ) d r 1 .
H U a - 2 = M a - 2 H U a - 1 + R a - 2 H L a - 1 , H L a - 2 = O a - 2 H U a - 1 + T a - 2 H L a - 1 , H r a - 2 = P a - 2 H U a - 1 + V a - 2 H L a - 1 , K U a - 2 = M a - 2 K U a - 1 + R a - 2 K L a - 1 , K L a - 2 = O a - 2 K U a - 1 + T a - 2 K L a - 1 , K r a - 2 = P a - 2 K U a - 1 + V a - 2 K L a - 1 .
H a U = M a = U / U , K a U = R a = L / U , H a L = O a = U / L , K a L = T a = L / L , H a r = P a = U / r , K a r = V a = L / r .
H U a - 1 = M a - 1 M a + R a - 1 O a , H L a - 1 = O a - 1 M a + T a - 1 O a , H r a - 1 = P a - 1 M a + V a - 1 O a , H N / N a - 1 = Q 12 , ( See Table I ) K U a - 1 = M a - 1 R a + R a - 1 T a , K L a - 1 = O a - 1 R a + T a - 1 T a , K r a - 1 = P a - 1 R a + V a - 1 T a , K N / N a - 1 = W 12 . ( See Table I )
H U a - 2 = M a - 2 H U a - 1 + R a - 2 H L a - 1 , H L a - 2 = O a - 2 H U a - 1 + T a - 2 H L a - 1 , H r a - 2 = P a - 2 H U a - 1 + V a - 2 H L a - 1 , K U a - 2 = M a - 2 K U a - 1 + R a - 2 K L a - 1 , K L a - 2 = O a - 2 K U a - 1 + T a - 2 K L a - 1 , K r a - 2 = P a - 2 K U a - 1 + V a - 2 K L a - 1 .
H U a - 3 = h U a - 3 H U a - 1 + k U a - 3 H L a - 1 , H L a - 3 = h L a - 3 H U a - 1 + k L a - 3 H L a - 1 , H r a - 3 = h r a - 3 H U a - 1 + k r a - 3 H L a - 1 , H N / N a - 3 = h N / N a - 3 H U a - 1 + k N / N a - 3 H L a - 1 , K U a - 3 = h U a - 3 K U a - 1 + k U a - 3 K L a - 1 , K L a - 3 = h U a - 3 K U a - 1 + k L a - 3 K L a - 1 , K r a - 3 = h r a - 3 K U a - 1 + k r a - 3 K L a - 1 , K N / N a - 3 = h N / N a - 3 K U a - 1 + k N / N a - 3 K L a - 1 .
H U a - 4 = M a - 4 H U a - 3 + R a - 4 H L a - 3 , H L a - 4 = O a - 4 H U a - 3 + T a - 4 H L a - 3 , H r a - 4 = P a - 4 H U a - 3 + V a - 4 H L a - 3 , K U a - 4 = M a - 4 K U a - 3 + R a - 4 K L a - 3 , K L a - 4 = O a - 4 K U a - 3 + T a - 4 K L a - 3 , K r a - 4 = P a - 4 K U a - 3 + V a - 4 K L a - 3 .
H U a - 5 = h U a - 5 H U a - 3 + k U a - 5 H L a - 3 , H L a - 5 = h L a - 5 H U a - 3 + k L a - 5 H L a - 3 , H r a - 5 = h r a - 5 H U a - 3 + k r a - 5 H L a - 3 , H N / N a - 5 = h N / N a - 5 H U a - 3 + k N / N a - 5 H L a - 3 , K U a - 5 = h U a - 5 K U a - 3 + k U a - 5 K L a - 3 , K L a - 5 = h L a - 5 K U a - 3 + k L a - 5 K L a - 3 , K r a - 5 = h r a - 5 K U a - 3 + k L a - 5 K L a - 3 , K N / N a - 5 = h N / N a - 5 K U a - 3 + k N / N a - 5 K L a - 3 .
1 / f = ( N - 1 ) { ( 1 / r 1 ) - ( 1 / r 2 ) + [ ( N - 1 ) t / N r 1 r 2 ] } .
1 F = 1 f 1 + 1 f 2 + 1 f 3 - S 1 ( f 3 + f 2 ) + S 2 ( f 1 + f 2 ) - S 1 S 2 f 1 f 2 f 3 ,
H u l = ( L l - L u ) sin U u sin U l ( sin ( U u - U l ) ) .
L u l = L l - ( L l - L u ) sin U u cos U l sin ( U u - U l ) ,
Long . Chro . Ab . = l y - l v ,
Trans . Chro . Ab . = ( L y p r - L ) tan U y p r - ( L v p r - L ) tan U v p r
Sph . Ab . = l - L .
Coma T = ( L p r - L u l ) tan U p r - H u l = ( L p r - L l + ( L l - L u ) sin U u cos U l sin ( U u - U l ) ) tan U p r - ( L l - L u ) sin U u sin U l sin ( U u - U l ) .
Curvature = ( L u l - L ) = L l - L - ( L l - L u ) sin U u cos U l sin ( U u - U l ) .
= H 0 N 0 u 0 N u - ( L p r - l ) tan U p r ,
= - N 0 N f tan U p r 0 - ( L p r - l ) tan U p r .
= 1 2 N k H k 2 N j - N j N j N j r surface by surface
= 1 2 H k 2 N j - 1 N j C j lens by lens .
d ( Long . Chro . Ab . ) = d l y - d l v , d ( Trans . Chro . Ab . ) = ( L y p r - L ) sec 2 U y p r d U y p r - ( L v p r - L ) sec 2 U v p r d U v p r + tan U y p r ( d L y p r - d L ) - tan U v p r ( d L v p r - d L ) ,
d ( Sph . Ab . ) = d l - d L , d ( Coma T ) = [ L p r - L l + ( L l - L u ) sin U u cos U l sin ( U u - U l ) ] sec 2 U p r d U p r + [ tan U p r ( L l - L u ) cos U u cos U l sin ( U u - U l ) - tan U p r × ( L l - L u ) sin U u cos U l cot ( U u - U l ) sin ( U u - U l ) - ( L l - L u ) cos U u sin U l sin ( U u - U l ) + ( L l - L u ) sin U u sin U l sin ( U u - U l ) cot ( U u - U l ) ] d U u + [ - tan U p r ( L l - L u ) sin U u sin U l sin ( U u - U l ) + tan U p r × ( L l - L u ) sin U u cos U l cot ( U u - U l ) sin ( U u - U l ) - ( L l - L u ) sin U u cos U l sin ( U u - U l ) - ( L l - L u ) sin U u sin U l sin ( U u - U l ) cot ( U u - U l ) ] d U l + tan U p r d L p r + [ - tan U p r sin U u cos U l sin ( U u - U l ) + sin U u sin U l sin ( U u - U l ) ] d L u + [ + tan U p r sin U u cos U l sin ( U u - U l ) - sin U u sin U l sin ( U u - U l ) - tan U p r ] d L l , d ( curvature ) = [ ( L l - L u ) sin U u cos U l cot ( U u - U l ) sin ( U u - U l ) - ( L l - L u ) cos U u cos U l sin ( U u - U l ) ] d U u + [ ( L l - L u ) sin U u sin U l sin ( U u - U l ) - ( L l - L u ) sin U u cos U l cot ( U u - U l ) sin ( U u - U l ) ] d U l - d L + [ sin U u cos U l sin ( U u - U l ) ] d L u + [ 1 - sin U u cos U l sin ( U u - U l ) ] d L l .
d ( distortion ) = - ( L p r - l p a r ) sec 2 U p r d U p r - tan U p r ( d L p r - d l p a r ) - H 0 N 0 u 0 p a r N u p a r 2 d u p a r
= - N 0 N tan U p r 0 d F - ( L p r - l p a r ) sec 2 U p r d U p r - tan U p r ( d L p r - d l p a r )
L 1 - L u tan U p r sin U u cos U u sin ( U u - U l ) sec 2 U p r sin U l cos U l cot ( U u - U l ) α = sin U u sin U l sin ( U u - U l ) , A = ( L l - L u ) sin U u sin U l sin ( U u - U l ) , β = cos U u sin U l sin ( U u - U 1 ) , B = ( L l - L u ) cos U u sin U l sin ( U u - U l ) , γ = sin U u cos U l sin ( U u - U l ) , C = ( L l - L u ) sin U u cos U l sin ( U u - U l ) , D = ( L l - L u ) cos U u cos U l sin ( U u - U l ) .
d ( Long . Chro . Ab . ) = d l y - d l v .
d ( Trans . Chro . Ab . ) = ( L y p r - L ) sec 2 U y p r d U y p r - ( L v p r - L ) sec 2 U v p r d U v p r + tan U y r ( d L y p r - d L ) - tan U v p r ( d L v p r - d L ) .
d ( Sph . Ab . ) = d l - d L .
d ( Coma T ) = [ L p r - L l + C ] sec 2 U p r d U p r + [ D tan U p r - C tan U p r cot ( U u - U t ) - B + A cot ( U u - U l ) ] d U u + [ - A tan U p r + C tan U p r cot ( U u - U t ) - C - A cot ( U u - U l ) ] d U l + tan U p r d L p r + [ - γ tan U p r + α ] d L u + [ γ tan U p r - α - tan U p r ] d L i ,             when             U b > U a ,
= [ L p r - L u + B ] sec 2 U p r d U p r + [ - A tan U p r - B tan U p r cot ( U u - U l ) - B + A cot ( U u - U l ) ] d U u + [ D tan U p r + B tan U p r cot ( U u - U l ) - C - A cot ( U u - U l ) ] d U l + tan U p r d L p r + [ - tan U p r - β tan U p r + α ] d L u + [ β tan U - α ] d L i ,             when             U a > U b .
d ( curvature ) = [ C cot ( U u - U l ) - D ] d U u + [ A - C cot ( U u - U l ) ] d U l - d L + γ d L u + [ 1 - γ ] d L l .
d ( distortion ) = - ( L - l ) sec 2 U p r d U p r - tan U p r ( d L p r - d l ) - H 0 N u p a r N u p a r d u p a r , for near objects ,
= - ( N 0 / N ) tan U 0 p r d f - ( L p r - l ) sec 2 U p r d U p r - tan U p r ( d L - d l ) , for distant objects.
d ( Long. Chro. Ab. ) = ( K l y x - K l v x ) d x ,
d ( Trans. Chro. Ab. ) = { ( L y p r - L ) sec 2 U y p r H x y p r - ( L v p r - L ) sec 2 U p r H x v p r + tan U y p r ( K x y p r - K x L ) - tan U v p r ( K x v p r - K x L ) } d x .
d ( Sph. Ab. ) = ( K x l - K x L ) d x .
d ( Coma T ) = { [ L p r - L l + C ] sec 2 U p r H x p r + [ D tan U p r - C tan U p r cot ( U u - U l ) - B + A cot ( U u - U l ) ] H x u + [ - A tan U p r + C tan U p r cot ( U u - U l ) - C - A cot ( U u - U l ) ] H x l + tan U p r K x p r + [ - γ tan U p r + α ] K x u + [ γ tan U p r - α - tan U p r ] K x l } d x .
d ( curvature T ) = { [ C cot ( U u - U l ) - D ] H x u + [ A - C cot ( U u - U l ) ] H x l - K x L + γ K x u + [ 1 - γ ] K x l } d x .
d ( distortion ) = { - ( L p r - l ) sec 2 U p r H x u - tan U p r ( K x p r - K x l ) - H N u p a r N u 2 p a r H x u } d x , for near objects ,
= - N 0 N tan U 0 p r d F - { ( L p r - l ) sec 2 U p r H x p r + tan U p r ( K x p r - K x l ) } d x , for distant objects.
U = U + I - I ,
L - r = ( N / N ) ( sin U / sin U ) ( L - r ) .
d ( L - r ) = ( L - r ) N / N d N N + ( L - r ) sin U d sin U + ( L - r ) ( L - r ) d ( L - r ) - ( L - r ) sin U d sin U = ( L - r ) sin U sin U d N N + N N ( L - r ) sin U d sin U + N N sin U sin U d ( L - r ) - N N ( L - r ) sin 2 U d sin U .
d ( L - r ) / L - r = d N N / N N + d sin U / sin U + d ( L - r ) / L - r - d sin U / sin U ,
1 1 E L - r = 1 1 E N / N + 1 1 E sin U + 1 1 E L - r - 1 1 E sin U .
L = L sin U cos 1 / 2 ( I - U ) sin U cos 1 / 2 ( I - U ) ,
1 1 E L - r = 1 1 E sin U + 1 1 E L - r - 1 1 E sin U
U = U + I - I
d U = d U + d I - d I ,
d U U = U U d U U + I U d I I - I U d I I ,
1 1 E U = U U 1 1 E U + 1 U 1 1 E I - I U 1 1 E I .
d sin x = cos x d x , 1 1 E sin x = x cot x 1 1 E x .
d log sin x = ( log a e / sin x ) d sin x , 1 1 E log sin x = log a e log sin x 1 1 E sin x ( a is usually 10 ) .
1 1 E sin U = U tan U 1 1 E U + I tan U 1 1 E I - I tan U 1 1 E I = tan U tan U 1 1 E sin U + tan I tan U 1 1 E sin I - tan I tan U 1 1 E sin I = log sin U tan U log 10 e tan U 1 1 E log sin U + log sin I log 10 e tan I tan U 1 1 E log sin I - log sin I log 10 e tan I tan U 1 1 E log sin I .
1 1 E log sin U = log sin U log sin U 1 1 E log sin U + log sin I log sin U tan I tan U 1 1 E log sin I - log sin I log sin U tan I tan U 1 1 E log sin I = E log sin U log sin U + E log sin I log sin U tan I tan U - E log sin I log sin U tan I tan U .
d log x = ( d x / x ) log a e .
d ( dist. ) = - N 0 N tan U 0 p r d F - { ( L p r - l ) sec 2 U p r + tan U p r ( K p r r 3 - K p r r 3 ) } d r 3 .
tan U 0 p r = + 0.4610063 , tan U p r = + 0.4800550 , sec 2 U p r = + 1.230453.
H p r r 3 = - 0.00047664 , K p r r 3 = + 0.013898 , K l r 3 = + 1.30567.
1 f 2 = ( N - 1 ) { 1 r 3 - 1 r 4 + ( N - 1 ) t N r 3 r 4 } , f 2 r 3 = f 2 2 r 3 2 { ( N - 1 ) [ 1 + ( N - 1 ) t N r 4 ] } .
1 F = 1 f 1 + 1 f 2 + 1 f 3 - { S 1 ( f 3 + f 2 ) + S 2 ( f 1 + f 2 ) - S 1 S 2 f 1 f 2 f 3 }
F f 2 = F f 2 2 { 1 + S 1 S 2 - S 1 f 3 - S 2 f 1 f 1 f 3 } .
F r 3 = F r 3 2 { 1 + S 1 S 2 - S 1 f 3 - S 2 f 1 f 1 f 3 } × { ( N - 1 ) [ 1 + ( N - 1 ) t N r 4 ] } .
F = chosen height at first surface sin U for the image space = 9 5 ° 4 36.7 6 = 101.3048 , F r 3 = ( when indicated substitutions are made ) = 0.01251840.
d ( distortion ) = { - 0.4610063 × 0.01251840 - 0.05703884 + 0.6201221 } d r 3 = + 0.5573122 d r 3 .
% change in aberration = 0.65 × number of surfaces × percent change in the parameter.
Q 12 = U 2 / N 1 / N 1 = Q 1 + [ ( N 1 N 1 cos I 2 - cos I 2 ) / N 1 N 1 cos I 2 cos I 2 ] [ Q 1 sin I 2 cot U 2 + W 1 sin U 2 r 2 ] + [ N 1 N 1 ] 2 sin I 2 cos I 2 . ( entire element ) W 12 = L 2 / N 1 / N 1 = ( L 2 - r 2 ) ( W 1 sin U 2 r 2 sin I 2 + Q 1 cot U 2 - Q 12 cot U 2 - N 1 N 1 ) ( entire element )
Q 12 par = 0 , W 12 par = + W 1 ( l 2 - r 2 ) l 2 - r 2 - N 1 N 1 l 1 - r 1 r 1 ( l 2 - r 2 ) + l 1 - r 1 r 1 ( l 2 - r 2 ) 2 l 2 - r 2 + N 1 - N 1 N 1 l 1 - r 1 r 1 ( l 2 - r 2 ) 2 r 2 - N 1 - N 1 N 1 r 2 ( l 2 - r 2 ) 2 l 2 - r 2 W 1 - N 1 N 1 ( l 2 - r 2 ) 2 r 2 - N 1 N 1 ( l 2 - r 2 ) .
U / U = + ( N / N ) ( cos I / cos I ) for the marginal ray , N / N for the paraxial ray , L / U = - ( N / N ) ( cos I / cos I ) [ r cos I + ( l - r ) cos U ] / sin U for the marginal ray , = - ( N / N ) ( L / u ) for the nominal paraxial ray.