Abstract

Given a symmetrical optical system incorporating inhomogeneous media, the expressions derived in this paper may be used to compute the derivatives of its paraxial coefficients with respect to each surface curvature, refractance, the thickness of media, and parameter specifying the index distributions. Some simple examples are considered.

© 1973 Optical Society of America

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References

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  1. P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
    [Crossref]
  2. P. J. Sands, J. Opt. Soc. Am. 61, 777 (1971).
    [Crossref] [PubMed]
  3. P. J. Sands, J. Opt. Soc. Am. 61, 879 (1971).
    [Crossref]
  4. P. J. Sands, J. Opt. Soc. Am. 61, 1086 (1971).
    [Crossref]
  5. P. J. Sands, J. Opt. Soc. Am. 61, 1495 (1971).
    [Crossref]
  6. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968); Part III in particular.
  7. A. T. O’Leary, Thesis, Australian National University (1972); see Part I.
  8. D. T. Moore and P. J. Sands, J. Opt. Soc. Am. 61, 1195 (1971).
    [Crossref]
  9. D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
    [Crossref]

1971 (6)

1970 (1)

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Tables (2)

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Table I Paraxial derivatives and axial distribution.

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Table II Paraxial derivatives and cylindrical distribution.

Equations (38)

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Y j = y a j S + y b j T , V j = υ a j S + υ b j T .
Y j = N 0 i [ ( y j | υ i ) Y i ( y j | y i ) V i ] , V j = N 0 i [ ( υ j | υ i ) Y i ( υ j | y i ) V i ] .
( p j | q i ) = ( p a j q b i p b j q a i ) / g ,
( Y j / τ i ) = N 0 i [ ( y j | υ i ) ( Y i / τ i ) ( y j | y i ) ( V i / τ i ) ] , ( V j / τ i ) = N 0 i [ ( υ j | υ i ) ( Y i / τ i ) ( υ j | y i ) ( V i / τ i ) ] .
Y i = Y i , V i = ( k 1 ) I i + V i , I i = c i Y i + V i ,
( Y i / k i ) = 0 , ( V i / k i ) = I i , ( Y i / c i ) = 0 , ( V i / c i ) = ( k i 1 ) Y i .
( Y / k i ) = N 0 i ( y | y i ) I i , ( V / k i ) = N 0 i ( υ | y i ) I i ,
( Y / c i ) = ( Δ N 0 i ) ( y | y i ) Y i , ( V / c i ) = ( Δ N 0 i ) ( υ | y i ) Y i ,
N i ( x , ξ ) = N 0 i ( x ) + N 1 i ( x ) ξ ,
Y i 1 = Y i ( 0 ) V i 1 = V i ( 0 ) , N 0 i 1 = N 0 i ( 0 ) , N 1 i 1 = N 1 i ( 0 ) , Y i = Y i ( d i ) , V i = V i ( d i ) , N 0 i = N 0 i ( d i ) , N 1 i = N 1 i = N 1 i ( d i ) ,
V i ( x ) = d Y i ( x ) / d x , d ( N 0 i ( x ) V i ( x ) ) / d x = 2 N 1 i ( x ) Y i ( x )
Y i t i = V i , V i t i = N 0 i 1 ( 2 N 1 i Y i N ˙ 0 i V i ) ,
( Y / t i ) = [ N 0 i ( y | υ i ) + N ˙ 0 i ( y | y i ) ] V i 2 N 1 i ( y | y i ) Y i , ( V / t i ) = [ N 0 i ( υ | υ i ) + N ˙ 0 i ( υ | y i ) ] V i 2 N 1 i ( υ | y i ) Y i ,
( d Y / d t i ) = ( Y / t i ) + ( N ˙ 0 i / N 0 i ) ( Y / k i ) ,
( d Y / d t i ) = N 0 i ( y | υ i ) V i ( 2 N 1 i + c i N ˙ 0 i ) ( y | y i ) Y i , ( d V / d t i ) = N 0 i ( υ | υ i ) V i ( 2 N 1 i + c i N ˙ 0 i ) ( υ | y i ) Y i .
( d Y / d τ ) = ( Y / τ ) + ( k i / N 0 i ) ( N 0 i / τ ) ( Y / k i ) ( k i 1 / N 0 i 1 ) ( N 0 i 1 / τ ) ( Y / k i 1 ) = ( Y / τ ) ( y | y i ) ( N 0 i / τ ) I i + k i 1 ( y | y i 1 ) ( N 0 i 1 / τ ) I i 1 ,
( Y i / τ ) = Y i , τ ( t i ) , ( V i / τ ) = V i , τ ( t i ) ,
d d x ( Y , τ ) V , τ = 0 , d d x ( N 0 V , τ ) 2 N 1 Y , τ = 2 N 0 τ ( N 1 N 0 ) Y N 0 d d x ( N 0 , τ N 0 ) V .
N ( x , ξ ) = f ( τ ) [ g ( x ) + h ( x ) ξ ] ,
N i ( x ) V i ( x ) = N i 1 V i 1 , Y i ( x ) = Y i 1 + N i 1 V i 1 I 1 , I 1 = 0 x d s / N i ( s ) .
( V i / τ ) = Γ τ υ V i , Γ τ υ = ( N i 1 , τ / N i 1 ) ( N i , τ / N i ) , ( Y i / τ ) = Γ τ y V i , Γ τ y = N i { ( N i 1 , τ / N i 1 ) I 1 + I 1 , τ } ,
( d Y / d τ ) = N i V i [ ( y | υ i ) Γ τ y ( y | y i ) Γ τ υ ] ( y | y i ) N i , τ I i + k i 1 ( y | y i 1 ) N i I , τ I i 1 .
Y i ( x ) = C ( x ) Y i 1 + S ( x ) V i 1 , V i ( x ) = ( 2 N 0 / N 1 ) S ( x ) Y i 1 + C ( x ) V i 1 ,
C ( x ) = cosh ( α x ) , S ( x ) = ( 1 / α ) sinh ( α x ) , α 2 = 2 N 1 / N 0 ,
C ( x ) = cos ( α x ) , S ( x ) = ( 1 / α ) sin ( α x ) , α 2 = 2 N 1 / N 0 .
( Y i / N 1 ) = ( 1 / 2 N 1 ) [ t i V i S ( t i ) V i 1 ] , ( Y i / N 0 ) = ( N 1 / N 0 ) ( Y i / N 1 ) , ( V i / N 1 ) = ( 1 / N 0 ) [ t i V i + S ( t i ) Y i 1 ] , ( V i / N 0 ) = ( N 1 / N 0 ) ( V i / N 1 ) .
( d Y / d N 1 i ) = ( N 0 i / 2 N 1 i ) ( y | υ i ) [ t i V i S i V i 1 ] ( y | y i ) [ t i Y i + S i Y i 1 ] , ( d Y / d N 0 i ) = ( N 1 / N 0 ) ( d Y / d N 1 i ) ( y | y i ) I i + k i 1 ( y | y i 1 ) I i 1 ,
d d x ( Y / N 1 ) = ( V / N 1 ) , d d x [ N 0 ( V / N 1 ) ] = 2 Y ,
V ( x ) = V i 1 Y i ( x ) = Y i 1 + x V i 1 ,
( Y i / N 1 ) = ( 1 / N 0 ) ( t i 2 Y i 2 3 t i 3 V i ) , ( V i / N 1 ) = ( 1 / N 0 ) ( 2 t i Y i t i 2 V i ) ,
( d Y / d N 1 i ) = [ ( y | υ i ) t i 2 2 ( y | y i ) t i ] Y i [ 2 3 ( y | υ i ) t i 3 ( y | y i ) t i 2 ] V i ,
( N 0 j / τ ) ( g / N 0 j ) + N 0 j [ y a j ( υ b j / τ ) + ( y a j / τ ) υ b j υ a j ( y b j / τ ) ( υ a j / τ ) y b j ] = 0 .
N ( x ) = N 0 ( 1 + α x ) , I 1 = ( N 0 α ) 1 ln ( 1 + α x ) , ( I 1 / α ) = a 1 [ I 1 + x / N 0 ( x ) ] , ( I 1 / N 0 ) = I 1 / N 0 , ( N / α ) = N 0 x , ( N / N 0 ) = 1 + α x .
y a N 1 = 0.659 259 , y b N 1 = 0.148 148 , υ a N 1 = 2.000 011 , υ b N 1 = 0.674 160 ,
y a = 0.933 888 , y b = 0.659 278 , υ a = 0.1 , υ b = 1.000 197 .
y a = 0.933 70 , y b = 0.659 26 , υ a = 0.101 11 , υ b = 1.000 01 .
N fd = 20 k 2 + 48 k 36
N fd = 44 k 2 + 48 k 36 , N II = 121 k 81 .