Abstract

The aberration coefficients of Buchdahl have been used almost exclusively in situations where rays are specified by their points of intersection with a pair of planes in the object space of the system. A simple method is presented in this paper whereby rays may be specified by their points of intersection with a pair of not necessarily plane surfaces situated in quite distinct parts of the system. The manner in which existing schemes for computing the aberration coefficients must be modified in order to accommodate the new coordinates is considered in detail. They stem basically from the need to reinterpret the intermediate coefficients appearing in the iteration equations and include modifications to the identities between the coefficients. Three examples of the new coordinates are discussed, namely: the case where the object surface is a surface of revolution about the optic axis, and the cases where rays are specified by their points of intersection with the (physical) aperture stop of the system (aperture coordinates) or with an ideal spherical wave surface in the image space of the system (Wo coordinates).

© 1970 Optical Society of America

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References

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  1. H. A. Buchdahl, Optical Aberration Coefficients (Dover Publications, Inc., New York, 1968).
  2. P. J. Sands, Thesis, Australian National University, 1967.
  3. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University Press, New York, 1970), Sec. 37.

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover Publications, Inc., New York, 1968).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University Press, New York, 1970), Sec. 37.

Sands, P. J.

P. J. Sands, Thesis, Australian National University, 1967.

Other (3)

H. A. Buchdahl, Optical Aberration Coefficients (Dover Publications, Inc., New York, 1968).

P. J. Sands, Thesis, Australian National University, 1967.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University Press, New York, 1970), Sec. 37.

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Equations (64)

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V = β / α , W = γ / α ,
y = ( y , z ) = Y + x V .
l 0 = y 0 / υ 0 .
H = Y + l 0 V .
Λ = N υ 0 H = N ( y 0 V υ 0 Y ) ,
g j = Δ Λ j = Λ j Λ j = 0 ( 3 ) , G j = i = 1 j 1 g i = 0 ( 3 ) .
Λ j = G j + Λ 1 .
ε ˆ = G , Δ ε ˆ = g ,
Y 1 = y a 1 S + y b 1 T + Y * , V 1 = υ a 1 S + υ b 1 T + V * .
ξ = S y 2 + S z 2 , η = S y T y + S z T z , ζ = T y 2 + T z 2 .
Y = y a S + y b T + 0 ( 3 ) , V = υ a S + υ b T + 0 ( 3 ) ,
ϕ = N ( y a υ b y b υ a )
y 0 = y a s 0 + y b t 0 , υ 0 = υ a s 0 + υ b t 0 .
Λ = Λ a s 0 + Λ b t 0 ,
Λ a = N ( y a V υ a Y ) , Λ b = N ( y b V υ b Y ) .
Y = ( y b Λ a y a Λ b ) / ϕ , V = ( υ b Λ a υ a Λ b ) / ϕ .
Λ a 1 = ϕ T + N 1 ( y a 1 V * υ a 1 Y * ) ϕ T + Λ * , Λ b 1 = ϕ S + N 1 ( y b 1 V * υ b 1 Y * ) ϕ S + Λ * ,
Q = Q S + Q ¯ T ,
Y = y a ( S G b Λ * b ) + y b ( T + G a + Λ * a ) , V = υ a ( S G b Λ * b ) + υ b ( T + G a + Λ * a ) .
I 1 = 1 / l 01 , e 1 = l 01 p 1 , and e 1 = 1 / e 1 .
y 01 S = Y 1 + p 1 V 1 , ( H 0 / m ) T H 1 = Y 1 + ( l 01 + x ) V 1 .
y 01 S ( y a 1 + p 1 υ a 1 ) ( S + Λ * b ) + ( y b 1 + p 1 υ b 1 ) ( T + Λ * a ) , ( H 0 / m ) T [ y a 1 + ( l 01 + x ) υ a 1 ] ( S Λ * b ) + [ y b 1 + ( l 01 + x ) υ b 1 ] ( T + Λ * a ) .
y a 1 + p 1 υ a 1 = y 01 , y a 1 + l 01 υ a 1 = 0 , y b 1 + p 1 υ b 1 = 0 , y b 1 + l 01 υ b 1 = H 0 / m .
y a 1 = y 01 / ( 1 I 1 p 1 ) , υ a 1 = I 1 y a 1 , y b 1 = p 1 υ b 1 , υ b 1 = H 0 I 1 / m ( 1 I 1 p 1 ) .
Λ b * = 0 , Λ a * = ϕ ( υ ¯ S + T ) [ e 1 x / ( 1 + e 1 x ) ] ,
Λ a * = ϕ ( υ ¯ S + T ) [ e 1 x e 1 2 x 2 + e 1 3 x 3 + 0 ( 8 ) ] .
Λ * = n = 1 μ = 0 n ν = 0 μ [ Λ * μν ( n ) S + Λ ¯ * μν ( n ) T ] ξ n μ η μ ν ζ ν ,
Λ * μνb ( n ) = Λ ¯ * μνb ( n ) = 0 , Λ * μνa ( n ) = υ ¯ Λ ¯ * μνa ( n ) .
x = n = 1 θ n ( H y 1 2 + H z 1 2 ) n = n = 1 θ n H 01 2 n ζ n ,
Λ ¯ * 11 a ( 1 ) = ϕ H 01 2 e 1 θ 1 , Λ ¯ * 22 a ( 2 ) = ϕ H 01 4 e 1 ( θ 2 e 1 θ 1 2 ) , Λ ¯ * 33 a ( 3 ) = ϕ H 01 6 e 1 ( θ 3 2 e 1 θ 1 θ 2 + e 1 2 θ 1 3 ) , Λ ¯ * μνa ( n ) = 0 , μ υ n .
H = m H 1 ( 1 / N υ a ) ( G a + Λ a * ) ,
ε ˆ N υ a ( H m H 1 ) = G a + Λ a * .
G n n a ( n ) = Λ n n a ( n ) , G ¯ n n a ( n ) = Λ ¯ n n a * ( n ) .
y a i S y a i ( S G b i Λ b * ) + y b i ( T + G a i + Λ a * ) ,
y b i = 0 , Λ * b = G b i .
Λ * μνa ( n ) = 0 , Λ * μνb ( n ) = G μ ν b i ( n ) , Λ * μνa ( n ) = 0 , Λ ¯ * μνb ( n ) = G μ ν b i ( n ) ,
Y E = Y 1 + p 1 V 1 = ( y a 1 + p 1 υ a 1 ) ( S Λ * b ) + ( y b 1 + p 1 υ b 1 ) ( T + Λ * a ) .
Y E = y 01 S + ( 1 / N 1 υ b 1 ) G b i .
Y E = ( 1 / N 1 υ b 1 ) ( C ¯ b i ζ + S ¯ 6 b i ζ 2 + T ¯ 10 b i ζ 3 ) T + 0 ( 9 ) ,
( e x ) 2 = e 2 ( y 2 + z 2 ) + 2 ( y h y + z h z ) ,
y 0 S = y = ( y , z ) and H 0 T = h = m H 1 ,
( 1 e x ) 2 = 1 + 2 y 0 e 2 ( H 0 η 1 2 y 0 ξ ) ,
e x = e 2 y 0 ( 1 2 y 0 ξ H 0 η ) + 1 2 e 4 y 0 2 ( 1 2 y 0 ξ H 0 η ) 2 + 1 2 e 6 y 0 3 ( 1 2 y 0 ξ H 0 η ) 3 + 0 ( 8 ) .
y 0 S + ( e x ) V = ε + H 0 T ,
V = e ( ε y 0 S + H 0 T ) / ( 1 e x ) .
( H 0 / m ) T ( y a 1 + l 0 υ a 1 ) ( S Λ * b ) + ( y b 1 + l 01 υ b 1 ) ( T + Λ a * ) ,
y a 1 + l 01 υ a 1 = 0 , y b 1 + l 01 υ b 1 = H 0 / m , Λ * a = 0 .
V = [ υ a + ( G | υ ) Λ * b υ a ] S + [ υ b + ( G ¯ | υ ) Λ ¯ * a υ a ] T ,
υ a = e y 0 , υ b = e H 0 , Λ * b = [ e x / ( 1 e x ) ] ( H 0 G a / y 0 ϕ ) G b , Λ ¯ * b = e x / ( 1 e x ) ( H 0 / y 0 ) ( G ¯ a + ϕ ) G ¯ b .
e x / ( 1 e x ) = e 2 y 0 ( 1 2 y 0 ξ H 0 η ) + 3 2 e 4 y 0 2 ( 1 2 y 0 ξ H 0 η ) 2 + 5 2 e 6 y 0 3 ( 1 2 y 0 ξ H 0 η ) 3 + 0 ( 8 ) .
Λ * b ( 1 ) = G b ( 1 ) ϕ e 2 y 0 ( 1 2 y 0 ξ H 0 η ) , Λ * b ( 2 ) = G b ( 2 ) 3 2 ϕ e 4 y 0 2 ( 1 2 y 0 ξ H 0 η ) 2 + e 2 H 0 ( 1 2 y 0 ξ H 0 η ) G a ( 1 ) , Λ * b ( 3 ) = G b ( 3 ) 5 2 ϕ e 6 y 0 3 ( 1 2 y 0 ξ H 0 η ) 3 + 3 2 e 4 y 0 H 0 × ( 1 2 y 0 ξ H 0 η ) 2 G a ( 1 ) + e 2 H 0 ( 1 2 y 0 ξ H 0 η ) G a ( 2 ) .
Λ * 00 b ( 1 ) = A b 1 2 ϕ e 2 y 0 2 , Λ * 10 b ( 1 ) = B b + ϕ e 2 H 0 y 0 , Λ * 11 b ( 1 ) = C b ,
Λ * 00 b ( 2 ) = S 1 b 3 8 ϕ e 4 y 0 4 + 1 2 e 2 H 0 y 0 A a , Λ * 10 b ( 2 ) = S 2 b + 3 2 ϕ e 4 y 0 3 H 0 + e 2 H 0 ( 1 2 y 0 B a H 0 A a ) , Λ * 11 b ( 2 ) = S 3 b + 1 2 e 2 H 0 y 0 C a , Λ * 20 b ( 2 ) = S 4 b 3 2 ϕ e 4 y 0 2 H 0 2 e 2 H 0 2 B a , Λ * 21 b ( 2 ) = S 5 b e 2 H 0 2 C a , Λ * 22 b ( 2 ) = S 6 b .
E x * = N α ( Z V Y W )
Δ E * x E * xj E * x 1 = 0.
Y = ( Λ | y ) S + ( Λ ¯ | y ) T , V = ( Λ | υ ) S + ( Λ ¯ | υ ) T ,
E * x = ( 1 / ϕ ) ( S z T y S y T z ) α ( Λ | Λ ¯ ) .
Δ α ( Λ | Λ ¯ ) 0.
Λ a = G a , Λ ¯ a = ϕ + G ¯ a , Λ b = ϕ + G b , Λ ¯ b = G ¯ b ,
Δ α [ ( G | G ¯ ) + ϕ ( G ¯ a G b ) + ϕ 2 ] 0.
Δ ( G ¯ a G b ) Δ ( G ¯ | G ) + ( α 1 α 1 ) ϕ + ( α 1 α 1 ) [ G ¯ a 1 G b 1 ( G ¯ 1 | G 1 ) ] .
Δ ( A ¯ a A b ) = 1 2 ϕ Δ υ a 2 , Δ ( B ¯ a B b ) = ϕ Δ υ a υ b , Δ ( C ¯ a C b ) = 1 2 ϕ Δ υ b 2 .
Δ ( S ¯ 1 a S 1 b ) = Δ ( A ¯ | A ) + ϕ Δ υ a A υ 1 8 ϕ ( υ a 2 υ a 1 2 ) ( υ a 2 + 3 υ a 1 2 ) + 1 2 ( A ¯ a 1 A b 1 ) Δ υ a 2 , Δ ( S ¯ 2 a S 2 b ) = Δ [ ( A ¯ | B ) + ( B ¯ | A ) ] + ϕ Δ ( υ a B υ + υ a A ¯ υ + υ b A υ ) 1 2 ϕ ( υ a 3 υ b + υ a 2 υ a 1 υ b 1 + υ a υ b υ a 1 2 3 υ a 1 3 υ b 1 ) + 1 2 ( B ¯ a 1 B b ) Δ υ a 2 + ( A ¯ a 1 A b 1 ) Δ υ a υ b , Δ ( S ¯ 3 a S 3 b ) = Δ [ ( A ¯ | C ) + ( C ¯ | A ) ] + ϕ Δ ( υ a C υ + υ b A ¯ υ ) 1 4 ϕ ( υ a 2 υ b 2 + υ b 2 υ a 1 2 + υ a 2 υ b 1 2 3 υ a 1 2 υ b 1 2 ) + 1 2 ( C ¯ a 1 C b 1 ) Δ υ a 2 + 1 2 ( A ¯ a 1 A b 1 ) Δ υ b 2 , Δ ( S ¯ 4 a S 4 b ) = Δ ( B ¯ | B ) + ϕ Δ ( υ a B ¯ υ + υ b B υ ) 1 2 ϕ ( υ a 2 υ b 2 + 2 υ a υ b υ a 1 υ b 1 3 υ a 1 2 υ b 1 2 ) + ( B ¯ a 1 B b 1 ) Δ υ a υ b , Δ ( S ¯ 5 a S 5 b ) = Δ [ ( B ¯ | C ) + ( C ¯ | B ) ] + ϕ Δ ( υ a C ¯ υ + υ b C υ + υ b B ¯ υ ) 1 2 ϕ ( υ a υ b 3 + υ b 2 υ a 1 υ b 1 + υ a υ b υ b 1 2 3 υ b 1 3 υ a 1 ) + 1 2 ( B ¯ a 1 B b 1 ) Δ υ b 2 + ( C ¯ a 1 C b 1 ) Δ υ a υ b , Δ ( S ¯ 6 a S 6 b ) = Δ ( C ¯ | C ) + ϕ Δ υ b C ¯ υ 1 8 ϕ ( υ b 2 υ b 1 2 ) ( υ b 2 + 3 υ b 1 2 ) + 1 2 ( C ¯ a 1 C b 1 ) Δ υ b 2 .
i = 1 j 1 g μ ν i ( n ) + Λ * μν ( n ) ,

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