## Zone-plate diffraction patterns in gradient-index media

Optics Express, Vol. 11, Issue 2, pp. 81-86 (2003)

http://dx.doi.org/10.1364/OE.11.000081

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### 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

## 1. Introduction

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

*et al.*studied the free propagation of a wave front in tapered gradient-index media with parabolic refractive index and the division of the wave front into Fresnel zones, thereby obtaining the zone contribution to the disturbance at observation points at the optical axis and the total disturbance produced by the summation of a number of such zones [3

3. J. M. Rivas-Moscoso, C. Gómez-Reino, and M. V. Pérez, “Fresnel zones in tapered gradient-index media,” J. Opt. Soc. Am. A **19**, 2253–2264 (2002). [CrossRef]

## 2. Statement of the problem

*n*

_{0}is the index at the

*z*optical axis and

*g*(

*z*) is the taper function that describes the evolution of the transverse parabolic index distribution along the

*z*axis.

*r*

_{0}on a Fourier transform plane at

*z*′ =

*z*by an on-axis point source at

*z*′ = 0 can be expressed, in parabolic approximation, by the following plane wave front

*p*is the spatial period and

*a*

_{m}is the amplitude of the

*m*th harmonic. Period

*p*assumes the value given by 2

*h*

_{1}is the radius of the first Fresnel zone for a wave front at the Fourier transform plane [3

3. J. M. Rivas-Moscoso, C. Gómez-Reino, and M. V. Pérez, “Fresnel zones in tapered gradient-index media,” J. Opt. Soc. Am. A **19**, 2253–2264 (2002). [CrossRef]

*r, z*’) inside the medium after the zone plate (see Fig. 1) may be calculated from the wave front in Eq. (2) and the zone-plate transmission function in Eq. (3) by solving the diffraction integral [5]

*K*(

*r*,

*r*

_{0};

*z*′) is the kernel or propagator for GRIN media in cylindrical coordinates

*H*

_{1}(

*z*′),

*H*

_{2}(

*z*′), Ḣ

_{1}(

*z*′) and Ḣ

_{2}(

*z*′) are, respectively, the position and the slope of the axial and field rays at

*z*′ after propagating from

*z*. This involves changing the integration limits in the argument of the sine and cosine functions in the expressions of

*H*

_{1}(

*z*),

*H*

_{2}(

*z*), Ḣ

_{1}(

*z*) and Ḣ

_{2}(

*z*) given in Ref. 4, which now go from

*z*to

*z*′.

_{0}is the zero-order Bessel function of the first kind.

*r*= 0, carrying out the integral in Eq. (6) gives an amplitude distribution of the form

_{m}where the denominator in Eq. (8) cancels, and therefore they can be obtained by solving the equation

*g*

_{0}from their first occurrence after the zone plate.

## 3. Discussion

*m*in the summation going from -1 to +1, which is equivalent to having a sinusoidal zone plate) and selfoc media. These simplifications do not restrict the study; rather the conclusions drawn may be straightforwardly extended to other types of media and zone-plate transmission functions. Calculations are made for refractive index at the optical axis

*n*

_{0}= 1.5, gradient parameter

*g*(0) =

*g*

_{0}= 0.1 mm

^{-1}, wave-front position

*z*= π/(2

*g*

_{0}) and zone-plate period

*p*= 0.1216 mm

^{2}.

*m*= 0 in the summation, which is tantamount to removing the zone plate, we can see that the free propagation of a plane wave front at the Fourier transform plane would produce a singularity in the irradiance distribution at the image plane (position ƒ

_{0}) and therefore there would be a focus even without the zone plate. This singularity disappears when placing a zone plate of the phase, as can be stated from the inspection of Fig. 2(b).

_{-1}for a zone plate of the amplitude (green-line graph) and of the phase (blue-line graph). The irradiance distribution consists of a series of maxima and minima of irradiance, which gives place to bright and dark rings, whose brightness varies according to a pattern (observe the graph for the zone plate of the amplitude in Fig. 3). Figure 4(b) shows the transverse irradiance distribution at a plane halfway between foci -1 and +1. By comparison with Fig. 4(a), we observe that there is a central minimum of irradiance before focus -1 for the zone plate of the phase and this minimum switches to a maximum at this plane. Likewise, the reverse transition, i.e. from maximum to minimum, occurs at ƒ

_{+1}. For the zone plate of the amplitude, there is no such change, the irradiance presenting a central minimum all along. On the other hand, in Figure 4(b) the irradiance pattern for the zone plate of the amplitude looks as though alternate maxima (and minima) were modulated by different enveloping waves. This happens at planes different from the ones that correspond to diffractive orders 0 and ±1. At these planes maxima of irradiance smash together and cannot be told apart. Of the three of them, the order zero is especially interesting. To show what happens at this plane, Fig. 4(c) is provided. For a zone plate of the amplitude the undistinguishable maxima form a figure as if “modulated” by the irradiance pattern for a zone plate of the phase. Similarly, at foci ±1, apart from the change for the zone plate of the phase mentioned above, the irradiance for the zone plate of the amplitude is “modulated” by the square modulus of the contributions of orders 0 and ∓1 to the complex amplitude distribution.

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. Ojeda-Castañeda and C. Gómez-Reino, eds., |

2. | Yu. A. Kravtsov and Yu. I. Orlov, |

3. | J. M. Rivas-Moscoso, C. Gómez-Reino, and M. V. Pérez, “Fresnel zones in tapered gradient-index media,” J. Opt. Soc. Am. A |

4. | C. Gómez-Reino, M. V. Pérez, and C. Bao, GRIN Optics: |

5. | J. M. Rivas-Moscoso, C. Gómez-Reino, C. Bao, and M. V. Pérez, “Tapered gradient-index media and zone plates,” J. Mod. Opt. |

6. | I. S. Gradshteyn, |

**OCIS Codes**

(110.2760) Imaging systems : Gradient-index lenses

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 20, 2002

Revised Manuscript: January 9, 2003

Published: January 27, 2003

**Citation**

Jose Rivas-Moscoso, Carlos Gomez-Reino, and Maria Perez, "Zone-plate diffraction patterns in gradient-index media," Opt. Express **11**, 81-86 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-2-81

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### References

- 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]
- 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]
- C. Gomez-Reino, M. V. Perez and C. Bao, GRIN Optics: Fundamentals and applications (Springer, Berlin, 2002), Chaps. 1 and 2.
- 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).
- I. S. Gradshteyn, Table of integrals, series and products (Academic, San Diego, 1980), Chap. 6.

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