Abstract

We construct in dimension two a mirror that reflects collimated rays into a set of directions that amplify the image and an optical lens so that collimated rays are refracted into a set of directions with a prescribed magnifi cation factor. The profiles of these optical surfaces are given by explicit formulas involving the Legendre transformation.

© 2011 Optical Society of America

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References

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  1. R. A. Hicks and R. Bajcsy, “Reflective surfaces as computational sensors,” Image Vis. Comput. 19, 773–777 (2001).
    [CrossRef]
  2. R. A. Hicks and R. K. Perline, “Blind-spot problem for motor vehicles,” Appl. Opt. 44, 3893–3897 (2005).
    [CrossRef] [PubMed]
  3. R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, 1962).
  4. C. E. Gutiérrez and Q. Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
    [CrossRef]

2009

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

2005

2001

R. A. Hicks and R. Bajcsy, “Reflective surfaces as computational sensors,” Image Vis. Comput. 19, 773–777 (2001).
[CrossRef]

Bajcsy, R.

R. A. Hicks and R. Bajcsy, “Reflective surfaces as computational sensors,” Image Vis. Comput. 19, 773–777 (2001).
[CrossRef]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, 1962).

Gutiérrez, C. E.

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

Hicks, R. A.

R. A. Hicks and R. K. Perline, “Blind-spot problem for motor vehicles,” Appl. Opt. 44, 3893–3897 (2005).
[CrossRef] [PubMed]

R. A. Hicks and R. Bajcsy, “Reflective surfaces as computational sensors,” Image Vis. Comput. 19, 773–777 (2001).
[CrossRef]

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, 1962).

Huang, Q.

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

Perline, R. K.

Appl. Opt.

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]

Image Vis. Comput.

R. A. Hicks and R. Bajcsy, “Reflective surfaces as computational sensors,” Image Vis. Comput. 19, 773–777 (2001).
[CrossRef]

Other

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, 1962).

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Figures (6)

Fig. 1
Fig. 1

Mirror problem.

Fig. 2
Fig. 2

Graphs of f when β = 2 , 3, 4 from bottom to top.

Fig. 3
Fig. 3

Path of the reflected rays on the mirror when β = 3 .

Fig. 4
Fig. 4

Lens problem.

Fig. 5
Fig. 5

Trajectories of refracted rays when β = 2 , glass to air, κ = 2 / 3 .

Fig. 6
Fig. 6

Graphs of f when β = 2 , 3, 4 from bottom to top; glass to air, κ = 2 / 3 .

Equations (57)

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z = x i 1 i 2 f ( x ) .
v = i 2 ( i · N ) N .
v 1 = i 1 2 f ( x ) 1 + ( f ( x ) ) 2 ( i 1 f ( x ) i 2 ) , v 2 = i 2 + 2 ( i 1 f ( x ) i 2 ) 1 + ( f ( x ) ) 2 .
x ( k + f ( x ) ) v 1 v 2 = t ( z a ) .
v 1 v 2 = 2 i 2 f ( x ) + i 1 ( 1 ( f ( x ) ) 2 ) 2 i 1 f ( x ) i 2 ( 1 ( f ( x ) ) 2 ) ,
x ( k + f ( x ) ) ( 2 i 2 f ( x ) + i 1 ( 1 ( f ( x ) ) 2 ) 2 i 1 f ( x ) i 2 ( 1 ( f ( x ) ) 2 ) ) = t ( z a ) ,
w ( ξ ) + f ( x ) = x ξ ; ξ = f , x = w .
( k + f ( x ) ) ( 2 i 2 f ( x ) + i 1 ( 1 ( f ( x ) ) 2 ) 2 i 1 f ( x ) i 2 ( 1 ( f ( x ) ) 2 ) ) = β k ( z a ) x .
2 i 2 f ( x ) + i 1 ( 1 ( f ( x ) ) 2 ) 2 i 1 f ( x ) i 2 ( 1 ( f ( x ) ) 2 ) = β ( z a ) = β ( x i 1 i 2 f ( x ) a ) .
F ( ξ ) 2 i 2 ξ + i 1 ( 1 ξ 2 ) 2 i 1 ξ i 2 ( 1 ξ 2 ) = β ( ( 1 i 1 i 2 ξ ) w + i 1 i 2 w a ) ;
( i 2 i 1 ξ ) ( 1 i 2 i 1 ξ w ) = w + i 1 i 2 i 1 ξ w = i 2 i 2 i 1 ξ ( 1 β F ( ξ ) + a ) .
w ( ξ ) = ( i 2 i 1 ξ ) ( C + i 2 ( i 2 i 1 ξ ) 2 ( 1 β F ( ξ ) + a ) d ξ ) .
2 i 2 f ( x ) + i 1 ( 1 ( f ( x ) ) 2 ) 2 i 1 f ( x ) i 2 ( 1 ( f ( x ) ) 2 ) = β x .
F ( ξ ) = 2 i 2 ξ + i 1 ( 1 ξ 2 ) 2 i 1 ξ i 2 ( 1 ξ 2 ) = β w .
w ( 1 + i 2 i 1 ) = f ( 0 ) .
2 i 2 ξ + i 1 ( 1 ξ 2 ) 2 i 1 ξ i 2 ( 1 ξ 2 ) = i 1 i 2 + 2 ξ / i 2 i 2 ξ 2 + 2 i 1 ξ i 2 ,
x a x 2 + b x + c d x = 1 2 a ln | a x 2 + b x + c | b 2 a 1 a x 2 + b x + c d x ,
1 a x 2 + b x + c d x = 1 b 2 4 a c ln | 2 a x + b b 2 4 a c 2 a x + b + b 2 4 a c | , if     4 a c b 2 < 0.
w ( ξ ) = 1 β ( i 1 i 2 2 ξ / i 2 i 2 ξ 2 + 2 i 1 ξ i 2 ) d ξ = 1 β i 1 i 2 ξ 2 β i 2 ξ i 2 ξ 2 + 2 i 1 ξ i 2 d ξ = 1 β i 1 i 2 ξ 2 β i 2 ( 1 2 i 2 ln | i 2 ξ 2 + 2 i 1 ξ i 2 | 2 i 1 2 i 2 1 i 2 ξ 2 + 2 i 1 ξ i 2 d ξ ) + C = 1 β i 1 i 2 ξ 1 β i 2 2 ln | i 2 ξ 2 + 2 i 1 ξ i 2 | + i 1 β i 2 2 ln | 2 i 2 ξ + 2 i 1 2 2 i 2 ξ + 2 i 1 + 2 | + C = 1 β i 1 i 2 ξ 1 β i 2 2 ln | i 2 ξ 2 + 2 i 1 ξ i 2 | + i 1 β i 2 2 ln | i 2 ξ + i 1 1 i 2 ξ + i 1 + 1 | + C .
w ( ξ 0 ) = w ( 1 + i 2 i 1 ) = ( 1 + i 2 ) β i 2 1 β i 2 2 ln | 2 ( i 2 1 ) i 1 2 | + i 1 β i 2 2 ln | 1 ( i 2 + i 1 ) 1 ( i 2 i 1 ) | + C C ( i 1 , i 2 ) + C = f ( 0 ) ,
w ( ξ ) = f ( 0 ) C ( i 1 , i 2 ) + 1 β i 1 i 2 ξ 1 β i 2 2 ln | i 2 ξ 2 + 2 i 1 ξ i 2 | + i 1 β i 2 2 ln | i 2 ξ + i 1 1 i 2 ξ + i 1 + 1 | .
F ( ξ ) = 2 i 2 ξ + i 1 ( 1 ξ 2 ) 2 i 1 ξ i 2 ( 1 ξ 2 ) = β x .
( i 2 + i 1 β x ) ± 1 + ( β x ) 2 i 2 β x i 1 .
ξ = g ( x ) ( i 2 + i 1 β x ) + 1 + ( β x ) 2 i 2 β x i 1 ,
f ( x ) = x g ( x ) w ( g ( x ) ) ,
i κ v = λ N .
λ = i · N κ v · N .
v · N = 1 κ 2 ( 1 ( i · N ) 2 ) ,
λ = i · N κ 1 κ 2 ( 1 ( i · N ) 2 ) .
i · N = 1 D ,
| f ( x ) | κ 1 κ 2 .
λ = 1 D κ 1 κ 2 ( 1 ( 1 D ) 2 ) = 1 D ( 1 κ D 2 κ 2 ( D 2 1 ) ) = 1 1 + ( f ( x ) ) 2 ( 1 κ 1 + ( 1 κ 2 ) ( f ( x ) ) 2 ) .
Φ ( ξ ) = 1 1 + ξ 2 ( 1 κ 1 + ( 1 κ 2 ) ξ 2 ) ,
λ = Φ ( f ( x ) ) .
v = 1 κ ( i λ N ) ,
v 1 = 1 κ λ f ( x ) D = 1 κ Φ ( f ( x ) ) f ( x ) 1 + ( f ( x ) ) 2 v 2 = 1 κ ( 1 λ D ) = 1 κ ( 1 Φ ( f ( x ) ) 1 + ( f ( x ) ) 2 ) .
( m f ( x ) ) v 1 v 2 = t ( x ) x ;
( m f ( x ) ) f ( x ) Φ ( f ( x ) ) 1 + ( f ( x ) ) 2 Φ ( f ( x ) ) = t ( x ) x .
w ( ξ ) + f ( x ) = x ξ ; ξ = f , x = w .
( m ( ξ w w ) ) ξ Φ ( ξ ) 1 + ξ 2 Φ ( ξ ) = α w .
w h ( ξ ) ξ h ( ξ ) + α w = m h ( ξ ) ξ h ( ξ ) + α .
h ( ξ ) = ξ ( 1 κ 1 + ( 1 κ 2 ) ξ 2 ) ξ 2 + κ 1 + ( 1 κ 2 ) ξ 2 = ξ ( 1 κ 2 ) κ ( κ + 1 + ( 1 κ 2 ) ξ 2 ) .
ξ h ( ξ ) + α = ξ 2 Δ ξ 2 Δ + 1 + α = ξ 2 ( α + Δ ) + α ( 1 Δ ) ξ 2 Δ + 1 ,
h ( ξ ) ξ h ( ξ ) + α = ξ Δ ξ 2 ( α + Δ ) + α ( 1 Δ ) .
w ( ξ ) = m + K exp ( h ( ξ ) ξ h ( ξ ) + α d ξ ) .
h ( ξ ) ξ h ( ξ ) + α = ξ ( 1 κ 1 d ξ 2 ) ξ 2 ( α + 1 κ 1 d ξ 2 ) + α ( κ 1 d ξ 2 ) .
( m f ( x ) ) f ( x ) Φ ( f ( x ) ) 1 + ( f ( x ) ) 2 Φ ( f ( x ) ) = ( β m 1 ) x ,
f ( x ) Φ ( f ( x ) ) 1 + ( f ( x ) ) 2 Φ ( f ( x ) ) = β x .
ξ Φ ( ξ ) 1 + ξ 2 Φ ( ξ ) = β w ,
w ( ξ ) = 1 β h ( ξ ) .
w ( ξ ) = 1 β 0 ξ h ( s ) d s .
κ 1 + ( 1 κ 2 ) ξ 2 + κ 2 ln [ κ + 1 + ( 1 κ 2 ) ξ 2 ] + κ κ 2 ln ( 1 + κ ) .
h ( ξ ) = κ 2 ( 1 + ξ 2 ) Δ ( 1 Δ ) ( ξ 2 + 1 Δ ) 2 ,
h ( ± κ 1 κ 2 ) = ± 1 κ 2 κ .
g :     [ 1 κ 2 κ , 1 κ 2 κ ] [ κ 1 κ 2 , κ 1 κ 2 ] ,
g ( x ) = { κ 2 x + κ 2 ( x 2 + x 4 ) 1 κ 2 + x 2 , for     0 x 1 κ 2 κ κ 2 x κ 2 ( x 2 + x 4 ) 1 κ 2 + x 2 , for     1 κ 2 κ x 0 ,
f ( x ) = x g ( β x ) w ( g ( β x ) ) .

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