Abstract

Diffraction patterns inside gradient-index media in which a complex zone plate is placed at a Fourier transform plane are studied by making use of the optical propagator. The results are illustrated with an example corresponding to a selfoc medium and a Fresnel zone plate of the amplitude and of the phase.

© 2002 Optical Society of America

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References

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  1. J. Ojeda-Castaneda and C. Gomez-Reino, eds., Selected Papers on Zone Plates, Vol. MS 128 of SPIE Mileston Series (SPIE Press, Bellingham, Wash., 1996), and references therein.
  2. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), Sec. 2.10.2.
    [CrossRef]
  3. J. M. Rivas-Moscoso, C. Gomez-Reino and M. V. Perez, �??Fresnel zones in tapered gradient-index media,�?? J. Opt. Soc. Am. A 19, 2253-2264 (2002).
    [CrossRef]
  4. C. Gomez-Reino, M. V. Perez and C. Bao, GRIN Optics: Fundamentals and applications (Springer, Berlin, 2002), Chaps. 1 and 2.
  5. J. M. Rivas-Moscoso, C. Gomez-Reino, C. Bao and M. V. Perez, �??Tapered gradient-index media and zone plates,�?? J. Mod. Opt. 47, 1549-1567 (2000).
  6. I. S. Gradshteyn, Table of integrals, series and products (Academic, San Diego, 1980), Chap. 6.

J. Mod. Opt.

J. M. Rivas-Moscoso, C. Gomez-Reino, C. Bao and M. V. Perez, �??Tapered gradient-index media and zone plates,�?? J. Mod. Opt. 47, 1549-1567 (2000).

J. Opt. Soc. Am. A

Other

I. S. Gradshteyn, Table of integrals, series and products (Academic, San Diego, 1980), Chap. 6.

J. Ojeda-Castaneda and C. Gomez-Reino, eds., Selected Papers on Zone Plates, Vol. MS 128 of SPIE Mileston Series (SPIE Press, Bellingham, Wash., 1996), and references therein.

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), Sec. 2.10.2.
[CrossRef]

C. Gomez-Reino, M. V. Perez and C. Bao, GRIN Optics: Fundamentals and applications (Springer, Berlin, 2002), Chaps. 1 and 2.

Supplementary Material (2)

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Figures (4)

Fig. 1.
Fig. 1.

Geometry for the propagation of a plane wave front at a Fourier transform plane through a zone plate inside a GRIN medium.

Fig. 2.
Fig. 2.

Irradiance along the optical axis for (a) a Fresnel zone plate of the amplitude and (b) of the phase. Shown are enlargements of the foci regions in both cases.

Fig. 3.
Fig. 3.

(both 1.48 MB) Evolution of the zone-plate diffraction patterns in a selfoc medium with (a) a Fresnel zone plate of the amplitude and (b) of the phase (pace: every 0.1 mm). [Media 2]

Fig. 4.
Fig. 4.

Irradiance at transverse planes (a) z′ = 29.4409 mm, (b) z′ = 30.9840 mm and (c) z′ = 31.4159 mm for Fresnel zone plates of the amplitude (green line) and of the phase (blue line) with period p = 0.1216 mm2 in a selfoc medium with parameters n 0 = 1.5 and g 0 = 0.1 mm-1. For the sake of reproducibility, in graphs (b) and (c) the irradiance for the zone plate of the phase was shifted upwards and/or enlarged by the amounts shown in the respective graphs.

Equations (16)

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n 2 ( r ; z ) = n 0 2 [ 1 - g 2 ( z ) r 2 ] ; r 2 = x 2 + y 2 ,
ψ ( r 0 ; z ) = kn 0 2 πi H 1 ( z ) exp ( i k n 0 z ) ,
T ( r 0 ) = m = −∞ a m exp { i 2 π m p r 0 2 } ,
ψ ( r ; z ) = K ( r , r 0 ; z ) T ( r 0 ) ψ ( r 0 ) r 0 d r 0 d φ ,
K ( r , r 0 ; z ) = i kn 0 2 π H 1 ( z ) exp [ i kn 0 ( z - z ) ]
× exp { i kn 0 2 H 1 ( z ) [ H ˙ 1 ( z ) r 2 + H 2 ( z ) r 0 2 2 rr 0 cos ( φ - θ ) ] }
ψ ( r ; z ) = k 2 n 0 2 2 π H 1 ( z ) H 1 ( z ) exp ( i kn 0 z ) exp [ i kn 0 2 H ˙ 1 ( z ) H 1 ( z ) r 2 ]
× m = 0 a m exp { i [ kn 0 H 2 ( z ) 2 H 1 ( z ) π m h 1 2 ] r 0 2 } J 0 [ kn 0 rr 0 H 1 ( z ) ] r 0 d r 0 ,
ψ ( z ) = i k 2 n 0 2 exp ( i kn 0 z ) 2 π H 1 ( z ) H 1 ( z ) m = a m [ kn 0 H 2 ( z ) H 1 ( z ) 2 π m h 1 2 ] 1 ,
I ( z ) = k 4 n 0 4 4 π 2 H 1 2 ( z ) H 1 2 ( z ) m , l = a m a 1 * { [ kn 0 H 2 ( z ) H 1 ( z ) 2 π m h 1 2 ] [ kn 0 H 2 ( z ) H 1 ( z ) 2 π l h 1 2 ] } 1 .
tan [ z f m g ( z ¯ ) d z ¯ ] = k n 0 g 0 h 1 2 2 π m .
f m = ( 1 g 0 ) tan 1 [ k n 0 g 0 h 1 2 ( 2 π m ) ] + z .
ψ ( r ; z ) = i k 2 n 0 2 exp ( i kn 0 z ) 2 π H 1 ( z ) H 1 ( z ) exp ( i kn 0 2 H 1 ( z ) H ˙ 1 ( z ) r 2 ) m = a m 2
× [ kn 0 2 H 1 ( z ) H 2 ( z ) 2 π m p ] 1 exp { i k 2 n 0 2 r 2 4 H 1 2 ( z ) [ kn 0 2 H 1 ( z ) H 2 ( z ) 2 π m p ] 1 } ,
I ( z ) = k 4 n 0 4 4 π 2 H 1 2 ( z ) H 1 2 ( z ) m , l = a m a l * { [ kn 0 H 2 ( z ) H 1 ( z ) 4 π m p ] [ kn 0 H 2 ( z ) H 1 ( z ) 4 π l p ] } 1
× exp { i k 2 n 0 2 r 2 4 H 1 2 ( z ) [ ( kn 0 2 H 1 ( z ) H 2 ( z ) 2 π m p ) 1 ( kn 0 2 H 1 ( z ) H 2 ( z ) 2 π l p ) 1 ] } .

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