The aim of this paper is to extend the study of the diffraction patterns produced by a zone plate in a GRIN medium to off-axis points. For the sake of simplicity, we will only take into account wave fronts at a Fourier transform plane.
2. Statement of the problem
Let us begin by considering a tapered GRIN medium characterized by a transverse parabolic refractive index modulated by an axial index and whose refractive index profile is given by
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.
The disturbance produced at a point 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
) is the position of the axial ray [4
4. C. Gómez-Reino, M. V. Pérez, and C. Bao, GRIN Optics: Fundamentals and applications (Springer, Berlin, 2002), Chaps. 1 and 2.
A zone plate of infinite dimension is placed at the Fourier transform plane. This zone plate will be represented as
is the spatial period and
is the amplitude of the m
th harmonic. Period p
assumes the value given by 2
, where h
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]
Fig. 1. Geometry for the propagation of a plane wave front at a Fourier transform plane through a zone plate inside a GRIN medium.
The complex amplitude distribution at a point (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
5. J. M. Rivas-Moscoso, C. Gómez-Reino, C. Bao, and M. V. Pérez, “Tapered gradient-index media and zone plates,” J. Mod. Opt. 47, 1549–1567 (2000).
where K(r, r
0; z′) is the kernel or propagator for GRIN media in cylindrical coordinates
2(z′), Ḣ1(z′) and Ḣ2(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
2(z), Ḣ1(z) and Ḣ2(z) given in Ref. 4, which now go from z to z′.
Substitution of Eqs. (2
) and (5
) into Eq. (4
) and integration in φ provides
regardless of the value of θ, where J0 is the zero-order Bessel function of the first kind.
= 0, carrying out the integral in Eq. (6
) gives an amplitude distribution of the form
from where it follows that the irradiance distribution along the optical axis can be written as
Foci will be at positions ƒm
where the denominator in Eq. (8
) cancels, and therefore they can be obtained by solving the equation
This equation for a selfoc medium reduces to
Allowing for the oscillatory function in Eq. (10
), in a selfoc medium, foci replicate periodically at distances multiple of π/g
from their first occurrence after the zone plate.
≠ 0, integration of Eq. (6
) provides [6
6. I. S. Gradshteyn, Table of integrals, series and products (Academic, San Diego, 1980), Chap. 6.
and the irradiance can be written as
As can be observed in Fig. 2
, along the optical axis, in the portion of the medium between two Fourier transform planes, there are three peaks of irradiance for the zone plate of the amplitude, which correspond to diffractive orders 0 and ±1, and two peaks, corresponding to orders +1 and -1, for the zone plate of the phase.
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.
(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. 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.
From Eq. (7
), by making 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 ƒ0
) 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)
Now the foci have been located, we move on to analyze the diffraction patterns across the medium. Figure 3
shows the transverse irradiance evolution with the propagation length. To achieve a better knowledge of the phenomena going on, let us home in on a few representative planes. In Fig. 4(a)
we show the transverse irradiance distribution profile along the radial direction at a plane prior to ƒ-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.