Abstract

A system of functional differential equations arising from Fermat’s principle is used to study qualitative questions concerning the mathematical possibility of certain rotationally symmetric piecewise homogeneous optical systems. In particular, it is shown that, given two pairs of points on the optical axis, there exist precisely two systems of single-element lenses having prescribed axial thickness and index of refraction such that these points are perfect foci. This finding sharpens an earlier result. Embedding the solution in a one-parameter family permits the construction of an asymptotic solution that requires the solution of a single nonlinear ordinary differential equation. The leading-order solution corresponds to an optical system satisfying the Herschel condition. The existence and uniqueness results are extended to optical systems having three lens boundaries such as achromatic doublets.

© 1994 Optical Society of America

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References

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  1. A. Friedman, J. McLeod, “Optimal design of an optical lens,” Arch. Rat. Mech. Anal. 99, 147–164 (1987).
    [Crossref]
  2. J. Rogers, “Existence, uniqueness, and construction of the solution of a system of ordinary functional differential equations, with application of the design of a perfectly focusing symmetric lens,” IMA J. Appl. Math. 41, 105–134 (1988).
    [Crossref]
  3. B. van-Brunt, “Functional differential equations and lens design in geometrical optics,” Ph.D. dissertation (University of Oxford, Oxford, 1989).
  4. I. Petrovski, Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1966).
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).
  6. A. Friedman, J. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rat. Mech. Anal. 101, 57–83 (1989).
    [Crossref]
  7. B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” N. Z. J. Math. 22, 101–107 (1993).
  8. B. van-Brunt, J. Ockendon, “A lens focusing light at two different wavelengths,”J. Math. Anal. Appl. 165, 156–179 (1992).
    [Crossref]
  9. E. C. Titchmarsh, The Theory of Functions, 2nd ed. (Oxford U. Press, Oxford, 1939).

1993 (1)

B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” N. Z. J. Math. 22, 101–107 (1993).

1992 (1)

B. van-Brunt, J. Ockendon, “A lens focusing light at two different wavelengths,”J. Math. Anal. Appl. 165, 156–179 (1992).
[Crossref]

1989 (1)

A. Friedman, J. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rat. Mech. Anal. 101, 57–83 (1989).
[Crossref]

1988 (1)

J. Rogers, “Existence, uniqueness, and construction of the solution of a system of ordinary functional differential equations, with application of the design of a perfectly focusing symmetric lens,” IMA J. Appl. Math. 41, 105–134 (1988).
[Crossref]

1987 (1)

A. Friedman, J. McLeod, “Optimal design of an optical lens,” Arch. Rat. Mech. Anal. 99, 147–164 (1987).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).

Friedman, A.

A. Friedman, J. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rat. Mech. Anal. 101, 57–83 (1989).
[Crossref]

A. Friedman, J. McLeod, “Optimal design of an optical lens,” Arch. Rat. Mech. Anal. 99, 147–164 (1987).
[Crossref]

McLeod, J.

A. Friedman, J. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rat. Mech. Anal. 101, 57–83 (1989).
[Crossref]

A. Friedman, J. McLeod, “Optimal design of an optical lens,” Arch. Rat. Mech. Anal. 99, 147–164 (1987).
[Crossref]

Ockendon, J.

B. van-Brunt, J. Ockendon, “A lens focusing light at two different wavelengths,”J. Math. Anal. Appl. 165, 156–179 (1992).
[Crossref]

Petrovski, I.

I. Petrovski, Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1966).

Rogers, J.

J. Rogers, “Existence, uniqueness, and construction of the solution of a system of ordinary functional differential equations, with application of the design of a perfectly focusing symmetric lens,” IMA J. Appl. Math. 41, 105–134 (1988).
[Crossref]

Titchmarsh, E. C.

E. C. Titchmarsh, The Theory of Functions, 2nd ed. (Oxford U. Press, Oxford, 1939).

van-Brunt, B.

B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” N. Z. J. Math. 22, 101–107 (1993).

B. van-Brunt, J. Ockendon, “A lens focusing light at two different wavelengths,”J. Math. Anal. Appl. 165, 156–179 (1992).
[Crossref]

B. van-Brunt, “Functional differential equations and lens design in geometrical optics,” Ph.D. dissertation (University of Oxford, Oxford, 1989).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).

Arch. Rat. Mech. Anal. (2)

A. Friedman, J. McLeod, “Optimal design of an optical lens,” Arch. Rat. Mech. Anal. 99, 147–164 (1987).
[Crossref]

A. Friedman, J. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rat. Mech. Anal. 101, 57–83 (1989).
[Crossref]

IMA J. Appl. Math. (1)

J. Rogers, “Existence, uniqueness, and construction of the solution of a system of ordinary functional differential equations, with application of the design of a perfectly focusing symmetric lens,” IMA J. Appl. Math. 41, 105–134 (1988).
[Crossref]

J. Math. Anal. Appl. (1)

B. van-Brunt, J. Ockendon, “A lens focusing light at two different wavelengths,”J. Math. Anal. Appl. 165, 156–179 (1992).
[Crossref]

N. Z. J. Math. (1)

B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” N. Z. J. Math. 22, 101–107 (1993).

Other (4)

E. C. Titchmarsh, The Theory of Functions, 2nd ed. (Oxford U. Press, Oxford, 1939).

B. van-Brunt, “Functional differential equations and lens design in geometrical optics,” Ph.D. dissertation (University of Oxford, Oxford, 1989).

I. Petrovski, Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1966).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).

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Figures (6)

Fig. 1
Fig. 1

Symmetric lens.

Fig. 2
Fig. 2

General case.

Fig. 3
Fig. 3

Lens with two pairs of foci.

Fig. 4
Fig. 4

Lens with staggered foci.

Fig. 5
Fig. 5

Lens with nested foci.

Fig. 6
Fig. 6

Lens with three pairs of foci.

Equations (76)

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y ( d 1 + n d 2 ) = 0 ,
[ f ( y ) + x 0 ] f ( y ) + y d 1 - n { [ f ( ξ ) - f ( y ) ] f ( y ) - ξ - y } d 2 = 0 ,
ξ ( n d 2 + d 3 ) = 0 ;
n { [ f ( ξ ) - f ( y ) ] f ( ξ ) + ξ - y } d 2 - [ x 1 - f ( ξ ) ] f ( ξ ) + ξ d 3 = 0 ,
[ f ( y ) - x 0 ] f ( y ) + y d 1 - n [ g ( ξ ) - f ( y ) ] f ( y ) + ξ - y d 2 = 0 ,
n [ g ( ξ ) - f ( y ) ] g ( ξ ) + ξ - y d 2 + - [ x 1 - g ( ξ ) ] g ( ξ ) + ξ d 3 = 0 ,
[ f ( y ) - x 0 ] f ( y ) + y d 1 - n { [ g ( ξ ) - f ( y ) ] f ( y ) + ξ - y } d 2 = 0 ,
n { [ g ( ξ ) - f ( y ) ] g ( ξ ) + ξ - y } d 2 + - [ x 1 - g ( ξ ) ] g ( ξ ) + ξ d 3 = 0 ,
[ f ( y ) - x ^ 0 ] f ( y ) + y d ^ 1 - n { [ g ( ϕ ) - f ( y ) ] f ( y ) + ϕ - y } d ^ 2 = 0 ,
n { [ g ( ϕ ) - f ( y ) ] g ( ϕ ) + ϕ - y } d ^ 2 + - [ x ^ 1 - g ( ξ ) ] g ( ξ ) + ξ d ^ 3 = 0 ,
d 1 2 = [ f ( y ) - x 0 ] 2 + y 2 ,             d ^ 1 2 = [ f ( y ) - x ^ 0 ] 2 + y 2 , d 2 2 = [ g ( ξ ) - f ( y ) ] 2 + ( ξ - y ) 2 , d ^ 2 2 = [ g ( ϕ ) - f ( y ) ] 2 + ( ϕ - y ) 2 , d 3 2 = [ x 1 - g ( ξ ) ] 2 + ξ 2 ,             d ^ 3 2 = [ x ^ 1 - g ( ϕ ) ] 2 + ϕ 2 .
f ( 0 ) = a 0 ,             g ( 0 ) = b 0 ,
ξ ( 0 ) = 0 ,             ϕ ( 0 ) = 0
f ( y ) = a 0 + a 2 y 2 + O ( y 4 ) , g ( y ) = b 0 + b 2 y 2 + O ( y 4 ) , ξ ( y ) = c 1 y + O ( y 3 ) , ϕ ( y ) = e 1 y + O ( y 3 ) .
- 2 ( n - 1 ) a 2 l 2 l 1 + l 2 + n l 1 = n l 1 c 1 ,
2 ( n - 1 ) b 2 l 2 l 3 + l 2 + n l 3 = n l 3 c 1 ,
- 2 ( n - 1 ) a 2 l 2 l ^ 1 + l 2 + n l ^ 1 = n l ^ 1 e 1 ,
2 ( n - 1 ) b 2 l 2 l ^ 3 + l 2 + n l ^ 3 = n l ^ 3 e 1 ,
l 1 = a 0 - x 0 ,             l ^ 1 = a 0 - x ^ 0 ,             l 2 = b 0 - a 0 , l 3 = x 1 - b 0 ,             l ^ 3 = x ^ 1 - b 0 .
ξ ( y ) = F 1 [ y , ξ , f ( y ) , g ( ξ ) , f ( y ) , g ( ξ ) ] , ϕ ( y ) = F 2 [ y , ϕ , f ( y ) , g ( ϕ ) , f ( y ) , g ( ϕ ) ] ,
c 1 = e 1 - S ,             c 1 2 + S c 1 - S / T = 0 ,
S = - l 2 n ( 1 l 1 - 1 l ^ 1 ) ,             T = l 2 n ( 1 l 3 - 1 l ^ 3 ) .
S 2 + 4 S T 0.
4 n 2 l 2 2 ( 1 l 1 - 1 l ^ 1 ) ( 1 l 3 - 1 l ^ 3 ) .
Z ( y ) = H [ y , Z ( y ) , Z ( z 1 ) , Z ( z 1 ) ] ,
Z ( 0 ) = 0 ,
ξ = [ ξ 1 , ξ 2 , , ξ 7 ] = [ y , z 1 ( y ) , z 2 ( y ) , z 1 ( z 1 ) , z 1 ( z 1 ) , z 2 ( z 2 ) ] , ξ 0 = [ 0 , 0 , 0 , 0 , 0 , z 1 ( 0 ) , z 2 ( 0 ) ] ,             ξ = k = 1 7 ξ k , B δ = { y : y < δ } ,             Z δ = sup j = 1 , 2 sup y B δ z j ( y ) ,
Z ( 0 ) = H ( ξ 0 ) ;
N ( ξ 0 ; ) = { ξ : ξ - ξ 0 < }
h 1 ( ξ 0 ) < 1 ;
H [ y ; Z ( y ) , Z ( z 1 ) ; Z ^ ( z 1 ) ] - H [ y ; Z ( y ) , Z ( z 1 ) ; Z ( z 1 ) ] Γ Z ^ ( z 1 ) - Z ( z 1 ) δ ,
Z ( y ) = H [ y ; Z ( z 1 ) , , Z ( z m ) ; Z ( z 1 ) , Z ( z m ) ] ,
h j ( ξ 0 ) < 1 ,             j = 1 , 2 , , m .
H ( a ^ ) - H ( a ) i = 1 m L i Z ^ ( z i ) - Z ( z i ,
a = [ y ; Z ( z 1 ) , , Z ( z m ) ; Z ( z 1 ) , , Z ( z m ) ] , a ^ = [ y ; Z ( z 1 ) , , Z ( z m ) ; Z ( z 1 ) , , Z ^ ( z m ) ] ,
i = 1 m L i = θ < 1.
z 1 ( y ) = y ,             z 2 ( y ) = ξ - 1 [ ϕ ( y ) ] ,             z 3 ( y ) = ξ ( y ) , z 4 ( y ) = ϕ ( y ) ,             z 5 ( y ) = f ( y ) - a 0 ,             z 6 ( y ) = f ( y ) , z 7 ( y ) = g ( ξ ) - b 0 ,             z 8 ( y ) = g ( ξ ) .
μ [ μ 1 , μ 2 , , μ 8 ] = [ z 1 , z 2 , z 3 , z 4 , z 5 , z 5 ( z 2 ) , z 7 , z 7 ( z 2 ) ] , V i = V μ i ,             V i j = 2 V μ i μ j ,
V 1 + z 6 V 5 = 0 ,
V 3 + z 8 V 7 = 0 ,
V ^ 1 + z 6 V ^ 5 = 0 ,
V ^ 4 + z 8 ( z 2 ) V ^ 8 = 0 ,
U 2 + z 6 ( z 2 ) U 6 = 0 ,
U 4 + z 8 ( z 2 ) U 8 = 0.
h 2 = - [ U 24 + z 6 ( z 2 ) U 64 ] z 4 U 22 + 2 z 6 ( z 2 ) U 26 + z 8 ( z 2 ) U 28 + z 6 ( z 2 ) U 6 + z 6 2 ( z 2 ) U 66 + z 6 ( z 2 ) z 8 ( z 2 ) U 68 .
h 3 = - [ V 11 + 2 z 6 V 15 + z 6 V 5 + z 6 2 V 55 ] V 13 + z 6 V 53 + z 8 V 17 + z 6 z 8 V 57 ,
h 4 = - [ V ^ 41 + z 6 V ^ 45 + z 8 ( z 2 ) z 2 V ^ 48 + z 8 ( z 2 ) z 2 V ^ 8 ] V ^ 44 + z 8 ( z 2 ) V ^ 84 - z 8 ( z 2 ) [ V ^ 81 + z 6 V ^ 85 + z 6 V ^ 85 + z 8 ( z 2 ) z 2 V ^ 88 ] V ^ 44 + z 8 ( z 2 ) V ^ 84 .
h 6 = - 1 V ^ 5 { V ^ 11 + z 4 V ^ 14 + 2 z 6 V ^ 15 + z 8 ( z 2 ) z 2 V ^ 18 + z 6 [ z 4 V ^ 54 + z 6 V ^ 55 + z 8 ( z 2 ) z 2 V ^ 58 ] } ,
h 8 = - 1 V 7 { V 31 + z 3 V 33 + z 6 V 35 + 2 z 8 z 3 V 37 + z 3 ( V 17 + z 6 V 75 + z 8 z 3 V 77 ) } ,
U 22 + 2 z 6 ( z 2 ) U 26 + z 8 ( z 2 ) U 28 + z 6 ( z 2 ) U 6 + z 6 2 ( z 2 ) U 66 + z 6 ( z 2 ) z 8 ( z 2 ) U 68 = U 22 + z 6 ( 0 ) U 6 = n c 1 / l 2 > 0 , V 13 + z 6 V 53 + z 8 V 17 + z 6 z 8 V 57 = V 13 = - n / l 2 < 0 , V ^ 44 + z 8 ( z 2 ) V ^ 84 = V ^ 44 = n / l ^ 3 + 1 l ^ 4 > 0 , V ^ 5 = - V 7 = 1 - n < 0 ;
H ( a ^ ) - H ( a ) L 2 Z ^ 2 - Z 2 .
h 2 z 2 ( z 2 ) = ( n - 1 ) e 1 l 2 n c 1 2 ,
0 < λ 2 < n c 1 2 ( n - 1 ) e 1 l 2 ,
Z ( 0 ) ( y ) = y H ( ξ 0 ) , Z ( k + 1 ) ( y ) = 0 y H [ ξ ; Z ( k ) ( z 1 ) , Z ( k ) ( z 2 ) ; Z ( k ) ( z 1 ( k ) ) , Z ( k ) ( z 2 ( k ) ) ] d ξ .
4 n 2 - 2 l 2 2 α 0 α 1 l 3 l 1 ( l 1 - α 0 ) ( l 3 + α 1 ) .
V ξ = 0 ,
V = l 1 + n l 2 + l 3 ,
V ^ y = 0 ,
V ^ = l 1 - α 0 + n l 2 + l 3 + α 1 ,
d j = d j 0 + d j 1 + 2 d j 2 + ,             d ^ j = d ^ j 0 + d ^ j 1 + 2 d ^ j 2 + ,
Δ 0 = 0 ,
Δ 1 = - [ f 0 ( y ) - x 0 ] α 0 d 10 + [ x 1 - g 0 ( ξ 0 ) ] α 1 d 30 + ( ϕ 1 - ξ 1 ) V 0 ξ | ξ 0 ,
Δ 2 = Γ 2 [ y , ξ 0 , ξ 1 , ϕ 1 , f 0 ( y ) , f 1 ( y ) , g 0 ( ξ 0 ) , g 0 ( ξ 0 ) , g 0 ( ξ 0 ) , g 1 ( ξ 0 ) ] + ( ϕ 2 - ξ 2 ) V c ξ | ξ 0 ,
1 - [ f 0 ( y ) - x 0 ] d 10 = M H [ 1 - x 1 - g 0 ( ξ 0 ) d 30 ] ,
Γ 2 = 0 ,
[ f 0 ( y ) - x 0 ] f 0 ( y ) + y d 10 - n { [ g 0 ( ξ 0 ) - f 0 ( y ) ] f 0 ( y ) + ξ 0 - y } d 20 = 0 ,
d 10 + n d 20 + d 30 = l 1 + n l 2 + l 3 .
ξ 0 ( y ) = ± [ x 1 - g 0 ( ξ 0 ) ] × [ ( 1 - 1 M H { 1 - [ f 0 ( y ) - x 0 ] d 10 } ) - 1 - 1 ] 1 / 2 ,
V j y = 0 ,             V j ξ j = 0 ,             V j ϕ j = 0 ,
f j ( y ) = a j 0 + a j 2 y 2 + ,             ξ j ( y ) = b j 1 y + b j ξ y 3 + , ϕ j ( y ) = c j 1 y + c j 3 y 3 + ,             l j 0 = a j 0 - x j 0 , l j 1 = x j 1 - a 30 ,             τ 1 = a 20 - a 10 ,             τ 2 = a 30 - a 20 ,
2 a 12 ( 1 - n 1 ) l j 0 τ 1 + τ 1 - n 1 ( b j 1 - 1 ) l j 0 = 0 ,
2 a 22 b j 1 τ 1 τ 2 ( n 1 - n 2 ) + n 1 τ 2 ( b j 1 - 1 ) - n 2 τ 1 ( c j 1 - b j 1 ) = 0 ,
2 a 32 c j 1 τ 2 l j 1 ( n 2 - 1 ) + n 2 l j 1 ( c j 1 - b j 1 ) + c j 1 τ 2 = 0.
n 1 b 11 - τ 1 l 10 = n 1 b 21 - τ 1 l 20 = n 1 b 31 - τ 1 l 30 , n 1 τ 2 b 11 + n 2 τ 1 c 11 b 11 = n 1 τ 2 b 21 + n 2 τ 1 c 21 b 21 = n 1 τ 2 b 31 + n 2 τ 1 c 31 b 31 , n 2 b 11 c 11 - τ 2 l 11 = n 2 b 21 c 21 - τ 2 l 21 = n 2 b 31 c 31 - τ 2 l 31 .
[ n 2 b 11 + τ 2 c 11 ( 1 l k 1 - 1 l 11 ) ] × [ ( n 1 τ 2 n 2 τ 1 + c 11 ) ( b 11 + β k ) - n 1 τ 2 b 11 ] - n 2 c 11 b 11 ( b 11 + β k ) = 0 ,             k = 2 , 3 ,
β k = τ 1 n 1 ( 1 l k 0 - 1 l 10 ) .

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