Abstract

Previous investigations involving the design of gradient lenses have been based mostly on third-order aberrations. The present paper discusses design techniques based on total aberrations. In particular, an algebraic formula is found for the total meridional curvature of radial gradient singlets. The formula obtained holds for a large class of refractive-index profiles.

© 1980 Optical Society of America

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References

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  1. D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
    [CrossRef]
  2. D. T. Moore, P. J. Sands, J. Opt. Soc. Am. 61, 1195 (1971).
    [CrossRef]
  3. D. T. Moore, P. J. Sands, U.S. Patent3,729,253 (1973).
  4. A. Gupta, K. Thyagarajan, C. Goyal, A. K. Ghatak, J. Opt. Soc. Am. 66, 1320 (1976).
    [CrossRef]
  5. S. D. Fantone, Ph.D. Thesis, University of Rochester (1979).
  6. E. Marchand, Gradient Index Optics (Academic, New York, 1978).
  7. E. Marchand, J. Opt. Soc. Am. 60, 1 (1970).
    [CrossRef]
  8. H. A. Buchdahl, J. Opt. Soc. Am. 63, 46 (1973).
    [CrossRef]
  9. D. Hamblen, U.S. Patent4,022,855 (1977).
  10. D. T. Moore, J. Opt. Soc. Am. 67, 1137 (1977).
    [CrossRef]
  11. K. Maeda, J. Hamasaki, J. Opt. Soc. Am. 67, 1672 (1977).
    [CrossRef]

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1970

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Figures (9)

Fig. 1
Fig. 1

Typical dispersion curves.

Fig. 2
Fig. 2

Notation for gradient singlet.

Fig. 3
Fig. 3

Aberrations for homogeneous spherical singlet.

Fig. 4
Fig. 4

Aberrations for homogeneous aspheric singlet.

Fig. 5
Fig. 5

Aberrations for the Wood lens.

Fig. 6
Fig. 6

Aberrations for special gradient singlet.

Fig. 7
Fig. 7

Aberrations for improved gradient singlet.

Fig. 8
Fig. 8

Aberrations for axial gradient.

Fig. 9
Fig. 9

Aberrations for image inverter.

Tables (7)

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Table I Homogeneous Spherical Lens

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Table II Homogeneous Aspheric Lens

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Table IV Special Radial Gradient

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Table V Improved Gradient Singlet

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Table VI Axial Gradient

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Table VII Image Inverter

Equations (54)

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S = y / υ .
C = ( y a + y b ) / 2 y p .
D = 100 ( 1 υ 0 f / y p ) ,
Z m = y / υ ( y y p ) / ( υ y p ) ,
Z s = x / u x / u .
A = ( Z m Z s ) / 2 .
C r = υ 0 Δ f ,
= ( n n / x ) / l 2 ÿ = ( n n / y ) / l 2 ,
z = z 0 + I r 0 r d r ( m 2 / l 2 1 ) 1 / 2 θ = θ 0 + ( c / l ) z 0 z d z / r 2 m 2 = n 2 c 2 / r 2 x = r cos θ , y = r sin θ } .
x = x 0 + p 0 z 0 z d z / l , p = p 0 y = y 0 + q 0 z 0 z d z / l , q = q 0 l = ( n 2 p 0 2 q 0 2 ) 1 / 2 } ,
n = N 0 + N 1 r 2 + N 2 r 4 + . . . , r 2 = x 2 + y 2 .
n 2 = N 0 2 b 2 r 2 ,
x = c x 0 + p 0 s / b y = c y 0 + q 0 s / b p = c p 0 x 0 b s q = c q 0 y 0 b s , l = l 0 } ,
c = cost t , s = sin t , t = ( z z 0 ) b / l .
n 2 = N 0 2 [ 1 ( k r ) 2 + D ( k r ) 4 ] ,
y = Gsn ( B z + F ) ,
n = N 0 sech ( k r ) ,
n 2 = a + b z
n 2 = N 0 2 ( 1 k 2 r 2 ) ,
θ = θ 0 + I r 0 r d r r ( n 2 r 2 / e 2 1 ) 1 / 2 ,
1 / f = b s + N ¯ [ c ( c 0 c 1 ) + c 0 c 1 d 1 s 1 N ¯ ] ,
b = ( 2 N 0 N 1 ) 1 / 2 , t = b d , d 1 = d / N 0 , c = cos t s = sin t , s 1 = s / t , N ̅ = N 0 1 , N 1 < 0 .
f = 1 / 2 N 1 d .
q 0 = sin σ , l 0 = ( 1 q 0 2 ) 1 / 2 υ 0 = q 0 / l 0 , y 0 = h + υ 0 ( z 0 z e ) w = 1 c 0 z 0 = ( 1 c 0 2 y 0 2 ) 1 / 2 } .
n 0 2 = N 0 2 b 2 y 0 2 j = l 0 w c 0 y 0 q 0 g = ( n 0 2 1 + j 2 ) 1 / 2 j q = q 0 c 0 y 0 g l = l 0 + w g } .
c = cos t , s = sin t t = ( z 1 z 0 + d ) b / l . y 1 = c y 0 + q s / b q 1 = c q y 0 b s , l 1 = l w 1 = 1 c 1 z 1 = ( 1 c 1 2 y 1 2 ) 1 / 2 } .
n 1 2 = N 0 2 b 2 y 1 2 j 1 = l w 1 c 1 y 1 q 1 g 1 = ( 1 n 1 2 + j 1 2 ) 1 / 2 j 1 q 2 = q 1 c 1 y 1 g 1 l 2 = l + w 1 g 1 } .
υ = q 2 / l 2 , y = y 1 + υ ( f 1 z 1 ) .
f 1 = f ( c N ¯ c 0 s / b ) ,
z 0 = c 0 y 0 / w , w = c 0 z 0 ,
j = l 0 w c 0 q 0 g = ( g j + b 2 y 0 ) / ( g + j ) q = c 0 ( g + y 0 g ) l = g w + w g } .
c = s t , s = c t t = [ b ( z 1 z 0 ) t l ] / l w 1 = c 1 w 2 y 1 , w 2 = c 1 y 1 / w 1 } ,
y 1 = c + ( s q + q 1 t ) / b q 1 = c q b ( s + y 1 t ) l 1 = l } .
t = w 2 ( b c + s q ) b z 0 t l l w 2 q 1 ,
j 1 = l w 1 + l w 1 c 1 ( y 1 q 1 + y 1 q 1 ) g 1 = ( l 2 y 1 y 1 g 1 j 1 ) / ( g 1 + j 1 ) q 2 = q 1 c 1 ( y 1 g 1 + g 1 y 1 ) l 2 = l + w 1 g 1 + w 1 g 1 } .
υ = ( q 2 l 2 q 2 l 2 ) / l 2 2 y = y 1 + υ ( f 1 z 1 ) υ z 1 } .
Z m = z 1 f 1 z 5 z 5 = ( y 1 υ z 1 ) υ = ( 1 υ w 2 ) y 1 / υ } ,
s / b = s 1 ( z 1 z 0 + d ) / l s 1 = sin t / t = 1 t 2 / 3 ! + t 4 / 5 ! . . . } .
t / b = w 2 ( c + q s / b ) z 0 l t / b l w 2 q 1 t / b = ( z 1 z 0 + d ) / l } ,
n 0 2 = N 0 2 [ 1 ( k y 0 ) 2 + D ( k y 0 ) 4 ] n 1 2 = N 1 2 [ 1 ( k y 1 ) 2 + D ( k y 1 ) 4 ] }
z z 0 = I y 0 y d y [ a 2 ( k y ) 2 + D ( k y ) 4 ] 1 / 2 ,
a 2 = 1 l 2 / N 0 2 .
z = z 1 + d , y ̅ = k y / a ,
z 1 z 0 + d = ( I / k ) y ̅ 0 y ̅ 1 d y ( 1 y ̅ 2 + D a 2 y ̅ 4 ) 1 / 2 .
z 1 z 0 + d = I k a D 1 / 2 y ̅ 0 y ̅ 1 d y ̅ [ ( r 1 2 y ̅ 2 ) ( r 2 2 y ̅ 2 ) ] 1 / 2 .
r 1 = 1 a ( 1 + c 2 D ) 1 / 2 r 2 = 1 a ( 1 c 2 D ) 1 / 2 , c = ( 1 4 a 2 D ) 1 / 2 } .
B ( z 1 z 0 + d ) = u 0 u 1 d u [ ( 1 u 2 ) ( 1 k 1 u 2 ) ] 1 / 2 ,
B = Iak r 1 D 1 / 2 , k 1 = ( 1 c ) / ( 1 + c ) .
F ( u ) = 0 u d u [ ( 1 u 2 ) ( 1 k 1 u 2 ) ] 1 / 2 ,
B ( z 1 z 0 + d ) = F ( u 1 ) F ( u 0 ) .
u 1 = s n [ B ( z 1 z 0 + d ) + F ( u 0 ) ] ,
u 1 = y 1 / G , u 0 = y 0 / G , G = a r 2 / k ,
B ( z 1 z 0 + d ) + B ( z 1 z 0 ) = u 1 [ ( 1 u 1 2 ) ( 1 k 1 u 1 2 ) ] 1 / 2 u 0 [ ( 1 u 0 2 ) ( 1 k 1 u 0 2 ) ] 1 / 2 + k 1 k 1 [ F ( u 1 ) F ( u 0 ) ] .
q 1 = I ( n 1 2 l 2 ) 1 / 2 ,

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