Abstract

We give a method to construct, for a given surface and a point source, a one-parametric family of reflecting profiles, each of them with the characteristic property of producing a predetermined phase distribution of light that from the point source is incident on that surface after reflection at the profile. The profiles are constructed as the envelopes of specific families of ellipsoids of revolution. We also study the singularities of these profiles.

© 2010 Optical Society of America

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References

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  1. C. Criado and N. Alamo, “Optical properties of conics: a method for obtaining reflecting and focusing profiles,” Opt. Commun. 167, 83-88 (1999).
    [CrossRef]
  2. V. I. Oliker, “On reconstructing a reflecting surface from the scattering data in the geometric optics approximation,” Inverse Probl. 5, 51-65 (1989).
    [CrossRef]
  3. S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13, 363-373 (1997).
    [CrossRef]
  4. P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
    [CrossRef]
  5. R. Courant, Differential and Integral Calculus (Wiley-Interscience, 1968), p. 181.
  6. E. Goursat, A Course in Mathematical Analysis, Vol. I (Dover, 1959), p. 220.
  7. D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988), p. 75.
  8. V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1983), p. 33.
  9. V. I. Arnold and A. B. Givental, “Simplectic geometry,” in Dynamical Systems IV (Springer-Verlag, 1990), p. 85.

2004

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
[CrossRef]

1999

C. Criado and N. Alamo, “Optical properties of conics: a method for obtaining reflecting and focusing profiles,” Opt. Commun. 167, 83-88 (1999).
[CrossRef]

1997

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13, 363-373 (1997).
[CrossRef]

1989

V. I. Oliker, “On reconstructing a reflecting surface from the scattering data in the geometric optics approximation,” Inverse Probl. 5, 51-65 (1989).
[CrossRef]

Alamo, N.

C. Criado and N. Alamo, “Optical properties of conics: a method for obtaining reflecting and focusing profiles,” Opt. Commun. 167, 83-88 (1999).
[CrossRef]

Arnold, V. I.

V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1983), p. 33.

V. I. Arnold and A. B. Givental, “Simplectic geometry,” in Dynamical Systems IV (Springer-Verlag, 1990), p. 85.

Benítez, P.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
[CrossRef]

Blen, J.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
[CrossRef]

Chaves, J.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
[CrossRef]

Courant, R.

R. Courant, Differential and Integral Calculus (Wiley-Interscience, 1968), p. 181.

Criado, C.

C. Criado and N. Alamo, “Optical properties of conics: a method for obtaining reflecting and focusing profiles,” Opt. Commun. 167, 83-88 (1999).
[CrossRef]

Dross, O.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
[CrossRef]

Falicoff, W.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
[CrossRef]

Givental, A. B.

V. I. Arnold and A. B. Givental, “Simplectic geometry,” in Dynamical Systems IV (Springer-Verlag, 1990), p. 85.

Goursat, E.

E. Goursat, A Course in Mathematical Analysis, Vol. I (Dover, 1959), p. 220.

Hernández, M.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
[CrossRef]

Kochengin, S. A.

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13, 363-373 (1997).
[CrossRef]

Miñano, J. C.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
[CrossRef]

Mohedano, R.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
[CrossRef]

Oliker, V. I.

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13, 363-373 (1997).
[CrossRef]

V. I. Oliker, “On reconstructing a reflecting surface from the scattering data in the geometric optics approximation,” Inverse Probl. 5, 51-65 (1989).
[CrossRef]

Struik, D. J.

D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988), p. 75.

Inverse Probl.

V. I. Oliker, “On reconstructing a reflecting surface from the scattering data in the geometric optics approximation,” Inverse Probl. 5, 51-65 (1989).
[CrossRef]

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13, 363-373 (1997).
[CrossRef]

Opt. Commun.

C. Criado and N. Alamo, “Optical properties of conics: a method for obtaining reflecting and focusing profiles,” Opt. Commun. 167, 83-88 (1999).
[CrossRef]

Opt. Eng.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489-1502 (2004).
[CrossRef]

Other

R. Courant, Differential and Integral Calculus (Wiley-Interscience, 1968), p. 181.

E. Goursat, A Course in Mathematical Analysis, Vol. I (Dover, 1959), p. 220.

D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988), p. 75.

V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1983), p. 33.

V. I. Arnold and A. B. Givental, “Simplectic geometry,” in Dynamical Systems IV (Springer-Verlag, 1990), p. 85.

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Figures (3)

Fig. 1
Fig. 1

Point source F and planar section of the surface S and of the ellipsoids of the family { E a ( x ) } x whose envelope is the reflecting profile R a . Straight gray lines show the optical paths from F to S.

Fig. 2
Fig. 2

Planar section of the ellipsoid of revolution E a ( x ) with foci F and x S . The length of the optical path F y x is 2 a + f ( x ) , which coincides with the length of the major axis of the ellipse E a ( x ) . The gray curve represents the wave front W.

Fig. 3
Fig. 3

Illustration of the singularities P 1 and P 2 of the reflecting profile R a . The singularities sweep out the caustic C as the parameter a varies.

Equations (22)

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| F 1 P | + | F 2 P | = 2 a .
e a ( x , y ) = 0 ,
e a ( x ( t 1 , t 2 ) , y ) = 0 , e a t 1 ( x ( t 1 , t 2 ) , y ) = 0 , e a t 2 ( x ( t 1 , t 2 ) , y ) = 0 .
e a ( x ( t 1 , t 2 ) , y ) = | y | + | y x ( t 1 , t 2 ) | ( 2 a + f ( x ( t 1 , t 2 ) ) ) = 0
e a t i ( x ( t 1 , t 2 ) , y ) = y x ( t 1 , t 2 ) | y x ( t 1 , t 2 ) | ( x t i ) f t i = 0 , i = 1 , 2 .
{ x t 1 , x t 2 , x t 1 x t 2 }
v i = y x ( t 1 , t 2 ) | y x ( t 1 , t 2 ) | x t i = f t i , i = 1 , 2 ,
v 3 = 1 | f t 1 x t 1 + f t 2 x t 2 | 2 | x t 1 x t 2 | .
R a = x S E a ( x ) r ( x ) .
y = x + λ ( x ) v ( x ) ,
λ ( x ) = ( 2 a + f ( x ) ) 2 x 2 2 ( 2 a + f ( x ) ) + 2 x v ( x ) .
x W = x + f ( x ) v ( x ) .
x W t i v = ( x + f ( x ) v ( x ) ) t i v = x t i v + f t i | v | 2 + f v t i v = f t i + f t i = 0 ,
{ x W t 1 , x W t 2 , v } ,
( y t 1 x W t 1 y t 1 x W t 2 y t 1 v y t 2 x W t 1 y t 2 x W t 2 y t 2 v ) .
y t i x W t j = ( x W + ( λ f ) v ) t i x W t j = δ i j ( λ f ) 2 x W t i t j v ,
0 = t i ( x W t j v ) = 2 x W t i t j v + x W t j v t i .
( y t i x W t j ) = ( δ i j ( λ f ) 2 x W t i t j v )
x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) = ( 6 + 10 cos t , 7 sin t ) .
v ( t ) = ( x cos t , 10 sin t ) 49 cos 2 t + 100 sin 2 t .
ρ ( t ) = ( x 1 2 + x 2 2 ) 3 2 | x 1 x 2 x 2 x 1 | .
c ( t ) = 51 ( 1 10 cos 3 t , 1 7 sin 3 t ) ,

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