Abstract

Following the previously derived results a general condition is formulated which determines a family of the exact solutions for GRIN rod lenses with perfect (within the paraxial approximation) imaging and transforming capabilities. Using this condition a new index profile is closely evaluated, and the detailed analysis of this construction is presented.

© 1985 Optical Society of America

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References

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  1. C. Gómez-Reino, E. Larrea, “Paraxial Imaging and Transforming in a Medium with Gradient-Index: Transmittance Function,” Appl. Opt. 21, 4271 (1982).
    [CrossRef] [PubMed]
  2. C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
    [CrossRef]
  3. C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Transmittance Function and Modal Propagation in a Conical Gradient-Index Rod,” Appl. Opt. 23, 1107 (1984).
    [CrossRef] [PubMed]
  4. C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Imaging and Transforming and Modal Propagation in a Parabolic Gradient-Index Rod,” Appl. Opt. 24, 4379 (1985).
    [CrossRef] [PubMed]
  5. D. Marcuse, Light Transmission Optics (Van NostrandRein-hold, New York, 1972), Chap. 1.
  6. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chaps. 2 and 4.
  7. C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Imaging and Transforming Transmission Through Divergent Conical GRIN Rods” and “Imaging and Transforming Transmission Through Divergent Parabolical GRIN Rod,” Opt. Commun. 55, 5, 8 (1985).
    [CrossRef]
  8. C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal Propagation in Gradient-Index (GRIN) Material,” Opt. Commun. 45, 372 (1983).
    [CrossRef]

1985 (2)

C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Imaging and Transforming and Modal Propagation in a Parabolic Gradient-Index Rod,” Appl. Opt. 24, 4379 (1985).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Imaging and Transforming Transmission Through Divergent Conical GRIN Rods” and “Imaging and Transforming Transmission Through Divergent Parabolical GRIN Rod,” Opt. Commun. 55, 5, 8 (1985).
[CrossRef]

1984 (1)

1983 (1)

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal Propagation in Gradient-Index (GRIN) Material,” Opt. Commun. 45, 372 (1983).
[CrossRef]

1982 (2)

C. Gómez-Reino, E. Larrea, “Paraxial Imaging and Transforming in a Medium with Gradient-Index: Transmittance Function,” Appl. Opt. 21, 4271 (1982).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chaps. 2 and 4.

Cuadrado, J. M.

Gómez-Reino, C.

C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Imaging and Transforming and Modal Propagation in a Parabolic Gradient-Index Rod,” Appl. Opt. 24, 4379 (1985).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Imaging and Transforming Transmission Through Divergent Conical GRIN Rods” and “Imaging and Transforming Transmission Through Divergent Parabolical GRIN Rod,” Opt. Commun. 55, 5, 8 (1985).
[CrossRef]

C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Transmittance Function and Modal Propagation in a Conical Gradient-Index Rod,” Appl. Opt. 23, 1107 (1984).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal Propagation in Gradient-Index (GRIN) Material,” Opt. Commun. 45, 372 (1983).
[CrossRef]

C. Gómez-Reino, E. Larrea, “Paraxial Imaging and Transforming in a Medium with Gradient-Index: Transmittance Function,” Appl. Opt. 21, 4271 (1982).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

Larrea, E.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van NostrandRein-hold, New York, 1972), Chap. 1.

Pérez, M. V.

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Imaging and Transforming Transmission Through Divergent Conical GRIN Rods” and “Imaging and Transforming Transmission Through Divergent Parabolical GRIN Rod,” Opt. Commun. 55, 5, 8 (1985).
[CrossRef]

C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Imaging and Transforming and Modal Propagation in a Parabolic Gradient-Index Rod,” Appl. Opt. 24, 4379 (1985).
[CrossRef] [PubMed]

C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Transmittance Function and Modal Propagation in a Conical Gradient-Index Rod,” Appl. Opt. 23, 1107 (1984).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal Propagation in Gradient-Index (GRIN) Material,” Opt. Commun. 45, 372 (1983).
[CrossRef]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

Sochacka, M.

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Imaging and Transforming Transmission Through Divergent Conical GRIN Rods” and “Imaging and Transforming Transmission Through Divergent Parabolical GRIN Rod,” Opt. Commun. 55, 5, 8 (1985).
[CrossRef]

Sochacki, J.

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Imaging and Transforming Transmission Through Divergent Conical GRIN Rods” and “Imaging and Transforming Transmission Through Divergent Parabolical GRIN Rod,” Opt. Commun. 55, 5, 8 (1985).
[CrossRef]

Appl. Opt. (3)

Opt. Commun. (3)

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

C. Gómez-Reino, M. V. Pérez, J. Sochacki, M. Sochacka, “Imaging and Transforming Transmission Through Divergent Conical GRIN Rods” and “Imaging and Transforming Transmission Through Divergent Parabolical GRIN Rod,” Opt. Commun. 55, 5, 8 (1985).
[CrossRef]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal Propagation in Gradient-Index (GRIN) Material,” Opt. Commun. 45, 372 (1983).
[CrossRef]

Other (2)

D. Marcuse, Light Transmission Optics (Van NostrandRein-hold, New York, 1972), Chap. 1.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chaps. 2 and 4.

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Figures (1)

Fig. 1
Fig. 1

Geometry of the GRIN rod with paraboloidal surfaces of constant index: a family of (shifted) paraboloids has a common apex located on the z axis at a distance L from the system origin; d determines the location of the output plane.

Equations (45)

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n 2 = n 0 2 [ 1 g 2 ( z ) ( x 2 + y 2 ) ] ,
2 E + k 2 n 2 E = ( 1 n 2 E n 2 ) ,
2 = 2 x 2 + 2 y 2 + 2 z 2 .
2 E + k 2 n 2 E = 0 .
E ( r ) = ψ ( r ) exp ( i k n 0 z ) .
2 ψ + 2 i k n 0 ψ z + k 2 ( n 2 n 0 2 ) ψ = 0 ,
2 = 2 x 2 + 2 y 2 .
H ¨ ( z ) + g 2 ( z ) H ( z ) = 0 ,
| g ˙ ( z ) | g 2 ( z ) 1 .
H 1 ( z ) = [ g 0 g ( z ) ] 1 / 2 sin [ 0 z g ( z ) d z ] ,
H 2 ( z ) = [ g 0 / g ( z ) ] 1 / 2 cos [ 0 z g ( z ) d z ] + 1 2 g ˙ ( 0 ) g 0 [ g 0 g ( z ) ] 1 / 2 sin [ 0 z g ( z ) d z ] ,
H 1 ( 0 ) = H ˙ 2 ( 0 ) = 0 ; H ˙ 1 ( 0 ) = H 2 ( 0 ) = 1 ,
H 1 ( z ) H 2 ( z ) H 1 ( z ) H 2 ( z ) = 1 ,
g ¨ g 3 2 g ˙ 2 = 0 .
g 3 d ( g ˙ / g 2 ) d z + g 2 ( g ˙ / g 2 ) 2 .
H 1 ( z ) = L b [ g 0 / g ( z ) ] 1 / 2 sin [ b g 0 L 0 z g ( z ) d z ] ,
H 2 ( z ) = [ g 0 / g ( z ) ] 1 / 2 cos [ b g 0 L 0 z g ( z ) d z ] + L 2 b g ˙ ( 0 ) [ g 0 g ( z ) ] 1 / 2 sin [ b g 0 L 0 z g ( z ) d z ] ,
g ¨ g 3 2 g ˙ 2 + 2 ( b 2 g 0 2 L 2 1 ) g 4 = 0 .
g 1 ( z ) = g 0 1 z / L , b 2 = g 0 2 L 2 1 4 > 0 conical rod ,
g 2 ( z ) = g 0 1 z 2 / L 2 , b 2 = g 0 2 L 2 1 > 0 parabolidal rod ,
g 1 + ( z ) = g 0 1 + z / L reversal conical rod ,
g 2 + ( z ) = g 0 1 + z 2 / L 2 reversal paraboloidal rod ,
b 2 = g 0 2 L 2 1 4 > 0 ,
b 2 = g 0 2 L 2 + 1 ,
g ¨ g 3 2 g ˙ 2 = 0 .
d F d z = 1 2 F 2 ,
F g ˙ / g .
g 3 ± ( z ) = g 0 ( 1 ± z / L ) 2 .
g ( z ) = g 0 ( 1 z / L ) 2 , z < L .
H 1 ( z ) = 1 g 0 ( 1 z / L ) sin ( g 0 L z L z ) ,
H 2 ( z ) = ( 1 z / L ) cos ( g 0 L z L z ) + 1 g 0 L ( 1 z / L ) × sin ( g 0 L z L z ) .
z l = L 1 + g 0 L l π , M = ( 1 ) l ( 1 z i L ) .
z ˜ m = L 1 + g 0 L m π + arccot ( 1 / g 0 L ) .
z ˜ m = L 1 + g 0 L ( m + 1 2 ) π
M ˜ = H 1 ( z ˜ m ) = ( 1 ) m g 0 ( 1 z ˜ m L ) .
t ( x , y , d ) = exp ( i k n 0 d ) H 2 ( d ) exp [ i k n 0 H ˙ 2 ( d ) 2 H 2 ( d ) ( x 2 + y 2 ) ] ,
f = H 2 ( d ) n 0 H ˙ 2 ( d ) .
f ( d ) = ( L d ) 2 n 0 L [ 1 + 1 g 0 L tan ( g 0 L d L d ) d L 1 d / L + g 0 2 L 2 g 0 L tan ( g 0 L d L d ) ] .
f ( d ) = L n 0 ( 1 + g 0 2 L 2 ) [ 1 + g 0 L tan ( g 0 d ) ] .
β p q = k n 0 ( p + q + 1 ) z 0 z g ( z ) d z = k n 0 ( p + q + 1 ) g 0 L L z .
g 0 L z r L z r = r π
β p q = k n 0 [ 1 + 2 ( p + q + 1 ) g 0 L k n 0 ( L z ) ] 1 / 2 ,
β p q = β p q ( p + q + 1 ) 2 g 0 2 L 2 2 k n 0 ( L z ) 2 + .
| β p q β p q | z π ,
z max r max 2 λ ( p max + q max + 1 ) 2 4 n 0 ,

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