Abstract

Three-dimensional focal curves generated by one-dimensional gradient-index taper have been obtained when the taper is illuminated by an off-axis plane wave. The intensity distribution along the focal curve was evaluated by taking into account the particular form of the exit pupil produced by this gradient-index medium.

© 1992 Optical Society of America

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References

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  1. J. Linares, C. Gómez-Reino, “Focal curves generated by one-dimensional GRIN tapers,” Opt. Lett. 15, 1258–1260 (1990).
    [CrossRef]
  2. C. Freré, O. Bryngdahl, “Computer-generated holograms: reconstruction of curves in 3-D,” Opt. Commun. 60, 369–372 (1980); S. Baré, C. Freré, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt. 37, 1287–1295 (1990).
    [CrossRef]
  3. M. Born, M. Wolf, Principles of Optics (Pergamon, London, 1980), Chap. 3.
  4. C. Gémez-Reino, E. Larrea, M. V. Perez, “Complex amplitude propagation through a quadratic-index medium: characteristic point function,” Appl. Opt. 22, 2927–2929 (1983).
    [CrossRef]
  5. C. Gómez-Reino, J. Linares, “Optical path integrals in gradient-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
    [CrossRef]
  6. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Chap. 4.
  7. C. Gómez-Reino, E. Larrea, “Imaging and transforming transmission through a medium with nonrotation symmetric gradient index,” Appl. Opt. 22, 387–390 (1983).
    [CrossRef] [PubMed]
  8. M. E. Harrigan, “Some first-order properties of radial gradient lenses compared to homogeneous lenses,” Appl. Opt. 23, 2702–2705 (1984).
    [CrossRef] [PubMed]
  9. E. Acosta, C. Gómez-Reino, J. Linares, “Effective radius and numerical aperture of GRIN lenses with revolution symmetry,” Appl. Opt. 26, 2952–2955 (1987).
    [CrossRef] [PubMed]
  10. C. S. Chung, H. H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485–1511 (1988).
    [CrossRef]
  11. Z. S. Hegedus, V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
    [CrossRef]

1990 (1)

1988 (1)

C. S. Chung, H. H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485–1511 (1988).
[CrossRef]

1987 (2)

1986 (1)

1984 (1)

1983 (2)

1980 (1)

C. Freré, O. Bryngdahl, “Computer-generated holograms: reconstruction of curves in 3-D,” Opt. Commun. 60, 369–372 (1980); S. Baré, C. Freré, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt. 37, 1287–1295 (1990).
[CrossRef]

Acosta, E.

Born, M.

M. Born, M. Wolf, Principles of Optics (Pergamon, London, 1980), Chap. 3.

Bryngdahl, O.

C. Freré, O. Bryngdahl, “Computer-generated holograms: reconstruction of curves in 3-D,” Opt. Commun. 60, 369–372 (1980); S. Baré, C. Freré, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt. 37, 1287–1295 (1990).
[CrossRef]

Chung, C. S.

C. S. Chung, H. H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485–1511 (1988).
[CrossRef]

Freré, C.

C. Freré, O. Bryngdahl, “Computer-generated holograms: reconstruction of curves in 3-D,” Opt. Commun. 60, 369–372 (1980); S. Baré, C. Freré, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt. 37, 1287–1295 (1990).
[CrossRef]

Gémez-Reino, C.

Gómez-Reino, C.

Harrigan, M. E.

Hegedus, Z. S.

Hopkins, H. H.

C. S. Chung, H. H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485–1511 (1988).
[CrossRef]

Larrea, E.

Linares, J.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Chap. 4.

Perez, M. V.

Sarafis, V.

Wolf, M.

M. Born, M. Wolf, Principles of Optics (Pergamon, London, 1980), Chap. 3.

Appl. Opt. (4)

J. Mod. Opt. (1)

C. S. Chung, H. H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485–1511 (1988).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

C. Freré, O. Bryngdahl, “Computer-generated holograms: reconstruction of curves in 3-D,” Opt. Commun. 60, 369–372 (1980); S. Baré, C. Freré, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt. 37, 1287–1295 (1990).
[CrossRef]

Opt. Lett. (1)

Other (2)

M. Born, M. Wolf, Principles of Optics (Pergamon, London, 1980), Chap. 3.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Chap. 4.

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Figures (6)

Fig. 1
Fig. 1

Geometrical arrangement for a 1-D GRIN taper lens illuminated by a tilted plane wave propagating along the Ý direction.

Fig. 2
Fig. 2

Focal curves generated by a linear taper: g(z) = g0/(1 − z/L)1/2, y1 = πs/g0, g0 = 0.2 mm−1, L = 60 mm. (a) focal distance; (b) the lateral displacement for a tilted plane wave.

Fig. 3
Fig. 3

Focal curves generated by a parabolic taper: g(z) = g0/(l + z/L′ − z2/L2)1/2, y1 = πs/g0, g0 = 0.2 mm−1, L′ = 60 mm, L = 20 mm. (a) focal distance; (b) lateral displacement for a tilted plane wave.

Fig. 4
Fig. 4

Focal curves generated by a sinusoidal taper: g(z) = g0/[1 − sin(z)/L]1/2, y1 = πs/g0, g0 = 0.2 mm−1, L = 10 mm. (a) focal distance; (b) the lateral displacement for a tilted plane wave.

Fig. 5
Fig. 5

Ray tracing in a 1-D GRIN taper. The determination of the effective entrance pupil depends on the thickness of the medium.

Fig. 6
Fig. 6

Plots the intensity in the neighborhood of the focal curve (a) without a mask and (b) with a mask to make the width uniform along the focal curve.

Equations (63)

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n 2 = n 0 2 [ 1 + h ( z ) x - g 2 ( z ) x 2 ] ,
ψ ( x 1 , z 1 ; d ) = R 2 G ( x 1 , z 1 , x 0 , z 0 ; d ) × ψ ( x 0 , z 0 ) d x 0 d z 0 ,
G ( x 1 , z 1 , x 0 , z 0 ; d ) = F ( z 1 , d ) × exp [ i k S P ( x 1 , z 1 , x 0 , z 0 ; d ) ] ,
S P = 0 d [ n 0 2 ( x ˙ 2 + z ˙ 2 ) - n ] d y
F ( z 1 , d ) = F 0 exp ( - 1 2 n 0 0 d 2 S P d y ) ,
n 0 ( z ) = n 0 [ 1 + h 2 ( z ) 4 g 2 ( z ) ] .
{ d [ g ( z ) ] / d z g ( z ) 1 , d [ h ( z ) ] / d z h ( z ) 1 }
n 0 x ¨ + n 0 [ g 2 ( z ) x - ½ h ( z ) ] = 0 ,
z ¨ 0.
x = x - δ ( z ) ,
y = y ,
z = z ,
δ ( z ) = h ( z ) 2 g 2 ( z ) .
x ¨ + n 0 n 0 g 2 ( z ) x = 0 ,
z ¨ 0.
G ( x 1 , x 0 , z 1 , z 0 ; d ) = k n 0 / 2 π i ( H 1 d ) 1 / 2 exp ( i k n 0 ) × exp [ i k n 0 2 H 1 ( H ˙ 1 x 1 2 + H 2 x 0 2 - 2 x 1 x 0 ) ] × exp [ i k n 0 2 d ( z 1 - z 0 ) 2 ] ,
H 1 ( z 1 , d ) = sin [ g ( z 1 ) d ] / g ( z 1 ) ,
H 2 ( z 1 , d ) = cos [ g ( z 1 ) d ] ,
H 1 ( z 1 , 0 ) = H ˙ 2 ( z 1 , 0 ) = 0 , H 2 ( z 1 , 0 ) = H ˙ 1 ( z 1 , 0 ) = 1.
ψ ( x 0 ) = exp ( i k α x 0 ) ,
ψ ( x 0 ) = exp [ i k α ( x 0 + δ ) ] .
ψ ( x 1 , z 1 ; d ) = φ ( z 1 ) [ H 2 ( z 1 , d ) ] 1 / 2 exp ( i k n 0 d ) × exp { i k n 0 H ˙ 2 2 H 2 [ x 1 + ( α n 0 H ˙ 2 ) ] 2 } ,
φ ( z 1 ) = exp [ i k α 2 ( 2 δ - α n 0 H ˙ 1 H ˙ 2 ) ] .
x f ( z 1 , d ) = h ( z 1 ) 2 g 2 ( z 1 ) - α n 0 H ˙ 2 ( z 1 , d ) ,
y f ( z 1 , d ) = H 2 ( z 1 , d ) n 0 H ˙ 2 ( z 1 , d ) .
ψ ( x 1 , z 1 ; d ) = φ ( z 1 ) [ H 2 ( z 1 , d ) ] 1 / 2 exp ( i n 0 d ) × exp { i k 2 H 2 / n 0 H ˙ 2 [ x 1 + ( α n 0 H ˙ 2 ) ] 2 } ,
φ ( z 1 ) = exp ( - i k α 2 2 n 0 H ˙ 1 H ˙ 2 ) ,
x f ( z 1 , d ) = - α n 0 H ˙ 2 ( z 1 , d ) = α n 0 g ( z 1 ) sec [ g ( z 1 ) d ] ,
y f ( z 1 , d ) = - H 2 ( z 1 , d ) n 0 H ˙ 2 ( z 1 , d ) = 1 n 0 g ( z 1 ) cot [ g ( z 1 ) d ] ,
x f ( z 1 , d ) = α n 0 g 2 ( z 1 ) d ,
y f ( z 1 , d ) = 1 n 0 g 2 ( z 1 ) d ,
y 1 ( z ) = tan - 1 { [ α / n 0 g ( z ) ] a 2 - [ α / n 0 g ( z ) ] 2 } ,
a = a - δ ( z ) ,
x 0 eff 1 ( z ) = - [ a + 2 δ ( z ) ] ,
x 0 eff u ( z ) = a cos [ g ( z ) d ] - ( α n 0 g ( z ) ) tan [ g ( z ) d ] ,
x 1 eff 1 ( z ) = ( α n 0 g ( z ) ) sin [ g ( z ) d ] - [ a + 2 δ ( z ) ] cos [ g ( z ) d ] ,
x 1 eff u ( z ) = a ,
y 2 ( z ) = tan - 1 [ a + 2 δ ( z ) α / n 0 g ( z ) ] + tan - 1 ( a { [ α / n 0 g ( z ) ] 2 + 4 δ ( z ) [ a + δ ( z ) ] } 1 / 2 ) .
x 0 eff 1 ( z ) = - [ a + 2 δ ( z ) ] ,
x 0 eff u ( z ) = { a 2 - [ α n 0 g ( z ) ] 2 } 1 / 2 ;
x 1 eff 1 ( z ) = [ α n 0 g ( z ) ] sin [ g ( z ) d ] - [ a + 2 δ ( z ) ] cos [ g ( z ) d ] ,
x 1 eff u ( z ) = [ α n 0 g ( z ) ] sin [ g ( z ) d ] - { a 2 - [ α n 0 g ( z ) ] 2 } 1 / 2 cos [ g ( z ) d ] ,
y 3 ( z ) = π - tan - 1 { [ α / n 0 g ( z ) ] a 2 - [ α / n 0 g ( z ) ] 2 } ,
x 0 eff 1 ( z ) = a cos [ g ( z ) d ] - [ α n 0 g ( z ) ] tan [ g ( z ) d ] ,
x 0 eff u ( z ) = { a 2 - [ α n 0 g ( z ) ] 2 } 1 / 2 ,
x 1 eff 1 ( z ) = [ α n 0 g ( z ) ] sin [ g ( z ) d ] - { a 2 - [ α n 0 g ( z ) ] 2 } 1 / 2 cos [ g ( z ) d ] ,
x 1 eff u ( z ) = a .
x 0 eff 1 ( z ) = - { a 2 - [ α n 0 g ( z ) ] 2 } 1 / 2 ,
x 0 eff u = { a 2 - [ α n 0 g ( z ) ] 2 } 1 / 2 ,
x 1 eff 1 ( z ) = [ α n 0 g ( z ) ] sin [ g ( z ) d ] + { a 2 - [ α n 0 g ( z ) ] 2 } 1 / 2 cos [ g ( z ) d ] ,
x 1 eff u ( z ) = [ α n 0 g ( z ) ] sin [ g ( z ) d ] + { a 2 - [ α n 0 g ( z ) ] 2 } 1 / 2 cos [ g ( z ) d ] .
x 0 eff 1 ( α < 0 ) = - x 0 eff u ( α > 0 ) ,
x 0 eff u ( α < 0 ) = - x 0 eff 1 ( α > 0 ) ,
ψ ( x , z ; y ) = i k / 2 π ( z y ) 1 / 2 exp ( i k y ) × - x 1 eff 1 x 1 eff u × exp { i k 2 y [ ( x - x 1 ) 2 + ( z - z 1 ) 2 ] } × ψ ( x 1 , z 1 ; d ) d x 1 d z 1 ,
I ( z 1 ) = ψ ( x , z ; y ) ψ * ( x , z ; y ) = ( x 1eff u - x 1eff 1 ) 2 λ y f ( z 1 ) H 2 ( z 1 , d ) sinc 2 × { k y f ( z 1 ) { x - [ α / n 0 H ˙ 2 ( z 1 , d ) ] } } × ( x 1 eff u - x 1 eff 1 ) .
I ( z 1 ) = ( x 1 eff u - x 1 eff 1 ) 2 λ y f ( z 1 ) H 2 ( z 1 , d ) .
I ( z 1 ) = n 0 g ( z 1 ) λ sin [ g ( z 1 ) d ] ( x 0 eff u - x 0 eff 1 ) 2 .
I ( z 1 ) = n 0 g 2 ( z 1 ) d λ ( 2 a - α d n 0 ) 2 .
x u ( z ) x 1 eff u ( z ) ,
x 1 ( z ) x 1 eff 1 ( z ) ,
t ( x , z ) = { 1 x 1 ( z ) x x u ( z ) , 0 otherwise ,
[ x u ( z ) - x 1 ( z ) ] - 1 n 0 g ( z ) sin [ g ( z ) d ] .
[ x u ( z ) - x 1 ( z ) ] - 2 n 0 g ( z ) sin [ g ( z ) d ] .

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