Abstract

Given a three-dimensional surface G, not necessarily rotationally symmetric, and away from a point source, we design a surface F such that the lens sandwiched between the two surfaces refracts radiation into a given direction or into a given point. The surface F satisfies a system of first-order partial differential equations that can be solved in terms of G and the refractive indices of the media involved.

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  1. G.-I. Kweon and C.-H. Kim, “Aspherical lens design by using a numerical analysis,” J. Korean Phys. Soc. 51, 93–103 (2007).
    [CrossRef]
  2. C. E. Gutiérrez, “Reflection, refraction and the Legendre transform,” J. Opt. Soc. Am. A 28, 284–289 (2011).
    [CrossRef]
  3. D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” Proc. SPIE 4095, 163–213 (2000).
    [CrossRef]
  4. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).
  5. C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
    [CrossRef]
  6. C. E. Gutiérrez and Q. Huang, “The near field refractor,” Geometric Methods in PDE’s, Conference for the 65th Birthday of E. Lanconelli, Vol. 7, Lecture Notes of Seminario Interdisciplinare di Matematica (Università degli Studi della Basilicata, 2008), pp. 175–188.
  7. C. E. Gutierrez and Q. Huang, “The near field refractor,” Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire (in press).

2011

2009

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[CrossRef]

2007

G.-I. Kweon and C.-H. Kim, “Aspherical lens design by using a numerical analysis,” J. Korean Phys. Soc. 51, 93–103 (2007).
[CrossRef]

2000

D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” Proc. SPIE 4095, 163–213 (2000).
[CrossRef]

Gutierrez, C. E.

C. E. Gutierrez and Q. Huang, “The near field refractor,” Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire (in press).

Gutiérrez, C. E.

C. E. Gutiérrez, “Reflection, refraction and the Legendre transform,” J. Opt. Soc. Am. A 28, 284–289 (2011).
[CrossRef]

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[CrossRef]

C. E. Gutiérrez and Q. Huang, “The near field refractor,” Geometric Methods in PDE’s, Conference for the 65th Birthday of E. Lanconelli, Vol. 7, Lecture Notes of Seminario Interdisciplinare di Matematica (Università degli Studi della Basilicata, 2008), pp. 175–188.

Huang, Q.

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[CrossRef]

C. E. Gutiérrez and Q. Huang, “The near field refractor,” Geometric Methods in PDE’s, Conference for the 65th Birthday of E. Lanconelli, Vol. 7, Lecture Notes of Seminario Interdisciplinare di Matematica (Università degli Studi della Basilicata, 2008), pp. 175–188.

C. E. Gutierrez and Q. Huang, “The near field refractor,” Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire (in press).

Kim, C.-H.

G.-I. Kweon and C.-H. Kim, “Aspherical lens design by using a numerical analysis,” J. Korean Phys. Soc. 51, 93–103 (2007).
[CrossRef]

Kweon, G.-I.

G.-I. Kweon and C.-H. Kim, “Aspherical lens design by using a numerical analysis,” J. Korean Phys. Soc. 51, 93–103 (2007).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

Shealy, D. L.

D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” Proc. SPIE 4095, 163–213 (2000).
[CrossRef]

Arch. Ration. Mech. Anal.

C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[CrossRef]

J. Korean Phys. Soc.

G.-I. Kweon and C.-H. Kim, “Aspherical lens design by using a numerical analysis,” J. Korean Phys. Soc. 51, 93–103 (2007).
[CrossRef]

J. Opt. Soc. Am. A

Proc. SPIE

D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” Proc. SPIE 4095, 163–213 (2000).
[CrossRef]

Other

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

C. E. Gutiérrez and Q. Huang, “The near field refractor,” Geometric Methods in PDE’s, Conference for the 65th Birthday of E. Lanconelli, Vol. 7, Lecture Notes of Seminario Interdisciplinare di Matematica (Università degli Studi della Basilicata, 2008), pp. 175–188.

C. E. Gutierrez and Q. Huang, “The near field refractor,” Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire (in press).

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Figures (2)

Fig. 1.
Fig. 1.

From Example 1, Section 3: Lens refracting rays from (0, 0, 0) in the direction e 1 = ( .3 , 1 ( .3 ) 2 / 2 , 1 ( .3 ) 2 / 2 ) , with a = 2 , d ( 1 , 0 , 0 ) = .1 . View 1.

Fig. 2.
Fig. 2.

Example 1, Section 3: View 2.

Equations (81)

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n 1 ( x × ν ) = n 2 ( m × ν ) ,
n 1 sin θ 1 = n 2 sin θ 2 ,
x κ m = λ ν ,
λ = x · ν κ 1 κ 2 ( 1 ( x · ν ) 2 ) .
x n 2 n 1 m 1 = λ 1 ν 1 ,
λ 1 = x · ν 1 n 2 n 1 1 ( n 1 n 2 ) 2 ( 1 ( x · ν 1 ) 2 ) ,
x · ν 1 0 ,
m 1 n 3 n 2 e 1 = λ 2 ν ,
λ 1 ν 1 · e 1 x · e 1 n 3 n 1 .
d ( x ) | P ( x ) Q ( x ) | ,
f ( u , v ) G ( u , v ) + d ( x ) m 1
x λ 1 ν 1 n 3 n 1 e 1 is a multiple of the normal ν
( x λ 1 ν 1 n 3 n 1 e 1 ) · f u = 0 ,
( x λ 1 ν 1 n 3 n 1 e 1 ) · f v = 0 .
f u ( u , v ) = G u ( u , v ) + [ d ( x ) n 1 n 2 ( x λ 1 ν 1 ) ] u .
0 = [ x λ 1 ν 1 n 3 n 1 e 1 ] · [ G u ( u , v ) + [ d ( x ) n 1 n 2 ( x λ 1 ν 1 ) ] u ] = [ x λ 1 ν 1 n 3 n 1 e 1 ] · G u ( u , v ) + ( x λ 1 ν 1 ) · [ d ( x ) n 1 n 2 ( x λ 1 ν 1 ) ] u n 3 n 1 e 1 · [ d ( x ) n 1 n 2 ( x λ 1 ν 1 ) ] u = F ( u , v ) + A ( u , v ) B ( u , v ) .
A ( u , v ) = n 2 n 1 d u ( x ) .
B ( u , v ) = n 3 n 2 e 1 · [ d ( x ) ( x λ 1 ν 1 ) ] u = n 3 n 2 e 1 · [ d u ( x ) ( x λ 1 ν 1 ) + d ( x ) ( x λ 1 ν 1 ) u ] = n 3 n 2 d u ( x ) e 1 · ( x λ 1 ν 1 ) + n 3 n 2 d ( x ) e 1 · ( x λ 1 ν 1 ) u .
F ( u , v ) = ( x n 3 n 1 e 1 ) · G u ( u , v ) .
( n 2 n 1 n 3 n 2 e 1 · ( x λ 1 ν 1 ) ) d u ( x ) n 3 n 2 e 1 · ( x λ 1 ν 1 ) u d ( x ) = [ x n 3 n 1 e 1 ] · G u ( u , v ) ,
( ( n 2 n 1 n 3 n 2 e 1 · ( x λ 1 ν 1 ) ) d ( x ) ) u = [ x n 3 n 1 e 1 ] · G u ( u , v ) .
( ( n 2 n 1 n 3 n 2 e 1 · ( x λ 1 ν 1 ) ) d ( x ) ) v = [ x n 3 n 1 e 1 ] · G v ( u , v ) .
d ( x ) = 1 n 2 n 1 n 3 n 2 e 1 · ( x λ 1 ν 1 ) [ x n 3 n 1 e 1 ] · G u ( u , v ) d u = 1 n 2 n 1 n 3 n 2 e 1 · ( x λ 1 ν 1 ) ( [ x n 3 n 1 e 1 ] · G ( u , v ) + ϕ ( v ) ) , = 1 n 2 n 1 n 3 n 2 e 1 · ( x λ 1 ν 1 ) ( [ x n 3 n 1 e 1 ] · G ( u , v ) x u · G ( u , v ) d u )
( [ x n 3 n 1 e 1 ] · G ( u , v ) + ϕ ( v ) ) v = [ x n 3 n 1 e 1 ] · G v ( u , v )
d ( x ) = 1 n 2 n 1 n 3 n 2 e 1 · ( x λ 1 ν 1 ) ( [ x n 3 n 1 e 1 ] · G ( u , v ) + C ) .
n 3 n 1 a 2 sin u sin v a 3 cos v a 1 n 2 n 1 1 ( n 1 n 2 ) 2 ( 1 cos 2 u sin 2 v ) .
e 1 · ( x λ 1 ν 1 ) = a 1 n 2 n 1 1 ( n 1 n 2 ) 2 ( 1 cos 2 u sin 2 v ) + a 2 sin u sin v + a 3 cos v ,
[ x n 3 n 1 e 1 ] · G ( u , v ) = a { ( n 3 n 1 ) ( a 1 + a 2 tan u + a 3 cot v sec u ) + sec u csc v } .
d ( 1 , 0 , 0 ) = a ( 1 n 3 n 1 a 1 ) + C n 2 n 1 n 3 n 1 a 1 .
d ( x ) = 1 n 2 n 1 n 3 n 1 x · e 1 ( a ( 1 n 3 n 1 x · e 1 ) + C ) .
x n 2 n 1 m 1 = λ 1 ν 1 ,
λ 1 = x · ν 1 n 2 n 1 1 ( n 1 n 2 ) 2 ( 1 ( x · ν 1 ) 2 ) ,
m 1 n 3 n 2 R Q ( x ) | R Q ( x ) | = λ 2 ν ,
m 1 · ( R Q ( x ) | R Q ( x ) | ) n 3 n 2 .
d ( x ) | P ( x ) Q ( x ) | ,
f ( u , v ) G ( u , v ) + d ( x ) m 1 ,
x λ 1 ν 1 n 3 n 1 R f ( u , v ) | R f ( u , v ) | is a multiple of the normal ν
( x λ 1 ν 1 n 3 n 1 R f ( u , v ) | R f ( u , v ) | ) · f u = 0 ( x λ 1 ν 1 n 3 n 1 R f ( u , v ) | R f ( u , v ) | ) · f v = 0 .
| R f ( u , v ) | u = R f ( u , v ) | R f ( u , v ) | · f u | R f ( u , v ) | v = R f ( u , v ) | R f ( u , v ) | · f v
( x λ 1 ν 1 ) · f u = ( x λ 1 ν 1 ) · G u ( u , v ) + n 1 n 2 ( x λ 1 ν 1 ) · [ d ( x ) ( x λ 1 ν 1 ) ] u = x · G u ( u , v ) λ 1 ν 1 · G u ( u , v ) + n 1 n 2 d u ( x ) ( x λ 1 ν 1 ) · ( x λ 1 ν 1 ) + n 1 n 2 d ( x ) ( x λ 1 ν 1 ) · ( x λ 1 ν 1 ) u = g u ( u , v ) + n 2 n 1 d u ( x ) .
( x λ 1 ν 1 ) · f v = g v ( u , v ) + n 2 n 1 d v ( x ) .
g u ( u , v ) + n 2 n 1 d u ( x ) + n 3 n 1 | R f ( u , v ) | u = 0 ,
g v ( u , v ) + n 2 n 1 d v ( x ) + n 3 n 1 | R f ( u , v ) | v = 0 .
g ( u , v ) + n 2 n 1 d ( x ) + n 3 n 1 | R f ( u , v ) | = ϕ ( v ) ,
g ( u , v ) + n 2 n 1 d ( x ) + n 3 n 1 | R f ( u , v ) | = C .
n 3 n 1 | R f ( u , v ) | = C g ( u , v ) n 2 n 1 d ( x ) .
( n 3 n 1 ) 2 ( | R G | 2 2 d m 1 · ( R G ) + d 2 ) = ( C g ) 2 2 n 2 n 1 d ( C g ) + ( n 2 n 1 ) 2 d 2 .
( ( n 2 n 1 ) 2 ( n 3 n 1 ) 2 ) d 2 + 2 ( ( n 3 n 1 ) 2 m 1 · ( R G ) + n 2 n 1 ( g C ) ) d + ( C g ) 2 ( n 3 n 1 ) 2 | R G | 2 = 0 .
D = 4 ( ( ( n 3 n 1 ) 2 m 1 · ( R G ) + n 2 n 1 ( g C ) ) 2 ( ( n 2 n 1 ) 2 ( n 3 n 1 ) 2 ) ( ( C g ) 2 ( n 3 n 1 ) 2 | R G | 2 ) ) = 4 ( ( n 3 n 1 ) 4 ( m 1 · ( R G ) ) 2 + 2 ( n 3 n 1 ) 2 m 1 · ( R G ) n 2 n 1 ( g C ) + ( n 3 n 1 ) 2 ( g C ) 2 + ( ( n 2 n 1 ) 2 ( n 3 n 1 ) 2 ) ( n 3 n 1 ) 2 | R G | 2 ) = 4 ( n 3 n 1 ) 2 ( ( n 3 n 1 ) 2 ( m 1 · ( R G ) ) 2 + 2 m 1 · ( R G ) n 2 n 1 ( g C ) + ( g C ) 2 + ( ( n 2 n 1 ) 2 ( n 3 n 1 ) 2 ) | R G | 2 ) = 4 ( n 3 n 1 ) 2 ( ( n 2 n 1 ) 2 ( m 1 · ( R G ) ) 2 + 2 m 1 · ( R G ) n 2 n 1 ( g C ) + ( g C ) 2 + ( ( n 2 n 1 ) 2 ( n 3 n 1 ) 2 ) ( | R G | 2 ( m 1 · ( R G ) ) 2 ) ) = 4 ( n 3 n 1 ) 2 ( ( n 2 n 1 m 1 · ( R G ) + g C ) 2 + ( ( n 2 n 1 ) 2 ( n 3 n 1 ) 2 ) ( | R G | 2 ( m 1 · ( R G ) ) 2 ) ) 0
m 1 · ( R G ( u , v ) ) n 1 n 2 ( C g ( u , v ) ) .
Δ 1 ( x ) = 2 ( ( n 3 n 1 ) 2 m 1 · ( R G ) + n 2 n 1 ( g C ) ) , Δ 2 ( x ) = ( C g ) 2 ( n 3 n 1 ) 2 | R G | 2 .
m 1 = n 1 n 2 ( x λ 1 ν 1 ) = n 1 n 2 ( n 2 n 1 1 ( n 1 n 2 ) 2 ( 1 cos 2 u sin 2 v ) , sin u sin v , cos v ) ,
Δ 1 ( x ) = 2 n 2 n 1 ( a cos u sin v C ) + 2 n 3 2 n 1 n 2 ( ( b a ) n 2 n 1 1 ( n 1 n 2 ) 2 ( 1 cos 2 u sin 2 v ) a ( cos u sin v 1 cos u sin v ) ) ,
Δ 2 ( x ) = ( C a cos u sin v ) 2 ( n 3 n 1 ) 2 | a cos u sin v x R | 2 .
Δ 1 Δ 1 ( 0 , π / 2 ) = 2 n 2 n 1 ( a C ) + 2 ( n 3 n 1 ) 2 ( b a )
Δ 2 Δ 2 ( 0 ) = ( C a ) 2 ( n 3 n 1 ) 2 ( b a ) 2 .
Δ Δ 1 2 4 ( ( n 2 n 1 ) 2 ( n 3 n 1 ) 2 ) Δ 2 > 0 .
Δ = 4 ( n 3 n 1 ) 2 ( a C + n 2 n 1 ( b a ) ) 2 > 0 ,
d ( 1 , 0 , 0 ) = Δ 1 ± Δ 2 ( ( n 2 n 1 ) 2 ( n 3 n 1 ) 2 ) ,
Δ 1 ± Δ = 2 n 2 n 1 ( a C ) 2 ( n 3 n 1 ) 2 ( b a ) ± 4 ( n 3 n 1 ) 2 ( a C + n 2 n 1 ( b a ) ) 2 = 2 n 2 n 1 ( a C ) 2 ( n 3 n 1 ) 2 ( b a ) ± 2 n 3 n 1 | a C + n 2 n 1 ( b a ) | .
a + n 2 n 1 d ( 1 , 0 , 0 ) + n 3 n 1 ( b f ( 0 , π / 2 ) ) = C ,
a + ( n 2 n 1 n 3 n 1 ) d ( 1 , 0 , 0 ) + n 3 n 1 ( b a ) = C .
a + n 3 n 1 ( b a ) C a + n 2 n 1 ( b a ) ,
a C + n 3 n 1 ( b a ) 0 a C + n 2 n 1 ( b a ) .
Δ 1 ± Δ = 2 n 2 n 1 ( a C ) 2 ( n 3 n 1 ) 2 ( b a ) ± 2 n 3 n 1 ( a C + n 2 n 1 ( b a ) ) = 2 ( n 2 n 1 ± n 3 n 1 ) ( a C ) + 2 ( ( n 3 n 1 ) 2 ± n 3 n 2 n 1 2 ) ( b a ) .
Δ 1 Δ = 2 ( n 2 n 1 n 3 n 1 ) ( a C ) + 2 ( ( n 3 n 1 ) 2 n 3 n 2 n 1 2 ) ( b a ) = 2 ( n 2 n 1 + n 3 n 1 ) ( C a ) + 2 ( ( n 3 n 1 ) 2 n 3 n 2 n 1 2 ) ( b a ) 2 ( n 2 n 1 + n 3 n 1 ) n 3 n 1 ( b a ) + 2 ( ( n 3 n 1 ) 2 n 3 n 2 n 1 2 ) ( b a ) = 2 ( ( n 2 n 1 + n 3 n 1 ) n 3 n 1 + ( ( n 3 n 1 ) 2 n 3 n 2 n 1 2 ) ) ( b a ) = 0 .
d ( 1 , 0 , 0 ) = C a n 3 n 1 ( b a ) ( n 2 n 1 n 3 n 1 ) = 2 ( n 2 n 1 ± n 3 n 1 ) ( a C ) + 2 ( ( n 3 n 1 ) 2 ± n 3 n 2 n 1 2 ) ( b a ) 2 ( n 2 n 1 n 3 n 1 ) ( n 2 n 1 + n 3 n 1 ) ,
C a n 3 n 1 ( b a ) = ( n 2 n 1 ± n 3 n 1 ) ( a C ) + ( ( n 3 n 1 ) 2 ± n 3 n 2 n 1 2 ) ( b a ) ( n 2 n 1 + n 3 n 1 ) ,
d ( x ) = Δ 1 ( x ) Δ 1 ( x ) 2 4 ( ( n 2 n 1 ) 2 ( n 3 n 1 ) 2 ) Δ 2 ( x ) 2 ( ( n 2 n 1 ) 2 ( n 3 n 1 ) 2 ) ,
x n 2 n 1 m 1 = λ 1 ν 1 ,
m 1 n 3 n 2 e 1 = λ 2 ν 2 ,
( m 1 n 3 n 2 e 1 ) ν 2 ,
( m 1 n 3 n 2 e 1 ) · [ ρ ( x ) x + d ( x ) m 1 ] u = 0 ( m 1 n 3 n 2 e 1 ) · [ ρ ( x ) x + d ( x ) m 1 ] v = 0 .
0 = m 1 · ( ρ ( x ) x ) u n 3 n 2 e 1 · ( ρ ( x ) x ) u + m 1 · ( d ( x ) m 1 ) u n 3 n 2 e 1 · ( d ( x ) m 1 ) u ,
m 1 · ( ρ ( x ) x ) u = n 1 n 2 ( x λ 1 ν 1 ) · ( ρ ( x ) x ) u = n 1 n 2 x · ( ρ ( x ) x ) u n 1 n 2 λ 1 ν 1 · ( ρ ( x ) x ) u = n 1 n 2 x · ( ρ ( x ) x ) u , since ( ρ ( x ) x ) u ν 1 = n 1 n 2 x · ( ρ u ( x ) x + ρ ( x ) x u ) = n 1 n 2 ρ u ( x ) .
m 1 · ( d ( x ) m 1 ) u = m 1 · d u ( x ) m 1 + m 1 · d ( x ) ( m 1 ) u = m 1 · m 1 d u ( x ) + m 1 · ( m 1 ) u d ( x ) = d u ( x ) , since | m 1 | = 1 .
m 1 = f ( x ) ρ ( x ) x | f ( x ) ρ ( x ) x |
e 1 · ( d ( x ) m 1 ) u = e 1 · ( f ( x ) ρ ( x ) x ) u .
n 1 n 2 ρ u ( x ) n 3 n 2 e 1 · ( ρ ( x ) x ) u + d u ( x ) n 3 n 2 e 1 · ( f ( x ) ρ ( x ) x ) u = 0 ,
n 1 n 2 ρ u ( x ) + d u ( x ) n 3 n 2 e 1 · ( f ( x ) ) u = 0 .
n 1 n 2 ρ v ( x ) + d v ( x ) n 3 n 2 e 1 · ( f ( x ) ) v = 0 .

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