Fresnel zones and zone plates in GRIN media

The general aim of this work is the theoretical and experimental characterization of zone plates in inhomogeneous media with transversal and axial variation of the refractive index for their application to hybrid structures.

The free propagation of a wavefront in GRIN media in the parabolic approximation and the division of the wavefront into Fresnel zones are studied. We distinguish between two cases: plane wavefronts (at Fourier transform planes) and slightly curved wavefronts (off Fourier transform planes). In both cases we determine the radius and the area of each zone as well as the zone contribution to the total wave at a point inside the medium. Likewise, we find the condition that the optical path must fulfil from the upper end of each zone to the aforementioned point so that the disturbance due to successive zones will be in phase opposition. We apply the study to selfoc media and calculate an approximate expression (by way of a geometric method) for the irradiance maxima positions.

Finally, from the differences in optical path between zones we can consider the zone plate construction and establish an analogy with the conventional lens imaging formula. We apply the study to various transmission functions. In particular, we choose sinusoidal zone plates of the amplitude and Fresnel zone plates of the amplitude and phase and calculate the irradiance distribution along the optical axis after these obstacles are placed. We carry out this study for selfoc, linear divergent and parabolic divergent media.

Integer and fractional Talbot effect

The integer and fractional Talbot effect in homogeneous media (also referred to as self-imaging phenomenon) as a sequence of self-images appearing periodically along the illumination direction when periodical objects are coherently illuminated is well-known in optics and has received wide attention. Talbot, in 1836, demonstrated that a grating produces a series of self-images in planes beyond the grating when it is illuminated by a plane wave. Rayleigh explained the Talbot images in planes whose distances z from the grating are even multiples of the Talbot distances zT=p2/ l , where p is the period of the grating and l the wavelength of the light. Much later, it was observed and explained that images form at all racional multiples of zT, namely zb/a=zT, where b and a are coprime integers.

Our aim is to generalize the Talbot effect to GRIN media and the results obtained are the followings:

-A generalization of the Talbot effect to the case of a tapered gradient-index medium for nonuniform and uniform illumination has been described. Self-image positions are changed by a function depending on the taper profile, the illumination, and the periodic object. An analogy with the conventional lens-imaging formula for both types of illumination has been studied.

-The unit cell of the fractional Talbot image which contains the superposition of unit cell images of the periodic object has been described. -Irradiance at fractional and integer Talbot images in tapered gradient-index (GRIN) media for ideal periodic objects has been analyzed. Talbot effect for off-axis illumination and finite object dimension has been investigated. Results have been applied to a hybrid structure composed by a divergent linear tapered GRIN medium and a sinusoidal amplitude grating to show transverse shift and finite-aperture diffractive effect on the Talbot images as well as the walk-off effect.

-An interpretation of the Talbot effect in a tapered gradient-index medium by number theory as the output/input relationship between the integer and the noninteger difference of position and the slope of rays has been done. Unit cell and transverse magnification for Talbot images have been evaluated, and two criteria for angular magnification have been defined. The study is particularized to a finite set of diffracted rays

GRIN modelling of the human lens

Several continuous GRIN models are used to describe the human lens (cristalline). For two of them, the refractive index distribution is represented by asymetric bi-elliptical isoindicial surfaces in which the posterior curvature of any isoindicial surfaces is greather than the anterior in such a way, at the equatorial play of join. The isoindicial surface is smooth and continuous. The two model provides a closet simulation of the real situation. In the framework of GRIN modelling, the results obtained are the following:

-Gradient parameter of the human lens has been evaluated and light propagation through the lens has been described by a linear combination of two linearly independent particular solutions of the paraxial ray equation which are the axial and field rays.

-Cardinal elements of the lens have been found by the axial and field rays.

-The back refractive power of the lens, concerned with its GRIN nature, has been calculated by the slope of the field ray at the output surface. The changes in the lens thickness with age can cause the lens power to decrease or increase. "Lens paradox" is under investigation.

Planar waveguide with hyperbolic secant refractive index profile

In designing of GRIN structures it is important to select the refractive index profile that provides focusing and collimation. The simplest GRIN media with these properties are those whose refractive index varies with a parabolic function. However, GRIN media with a hyperbolic secant refractive index profile provides perfect focusing and collimation properties and present the following advantages in comparison with the parabolic ones. First, it is a more general profile, since the parabolic function can be regarded as the first order approximation of the Taylor's expansion series of the hyperbolic secant function. Second, the equations governing light propagation through these media can be analytically solved without carrying out any approximation. Finally, GRIN media with Hyperbolic Secant (HS) refractive index profile are free of aberration for meridional rays. Assuming GRIN media with this profile and a geometrical structure of planar waveguide, we have made the following work:

1. In the framework of geometrical optics.

- Ray equations have been derived and properties of focusing and collimation have been determined by the evaluation of zeros of axial and field rays

- Devices for specific purposes for integrated optics such as: focusers and collimators (for optical connections on and out of axis), deflectors and light shifters (These devices have been designed as substitutes of mirrors and conventional plane-parallel plates), and a device for controlling the beam size by longitudinal butt-joining coupling between two HS planar waveguides, have been designed

- On-axis and off-axis gaussian beam propagation, inserting a hyperbolic complex ray (analogy with the formalism of Kogelnik) have been analysed.

- Light propagation and focusing and collimation properties in planar waveguides with HS index profile axially modulated by different functions (taper waveguides, hereafter referred as HSGT) have also been analysed.

- The numerical aperture and angle and position transformations have been studied for on-axis and off-axis coupling by HSGT waveguides. A device or beam size controlling by this kind of waveguides has been designed.

2. In the framework of wave optics.

- Modal propagation from the Helmholtz's scalar equation has been studied. We obtain that guided modes have the same profile that spatial solitons, likewise, the propagation constants and the modal cut-off have been determined

- The general expressions of the optical propagator or Kernel of the integral propagation equation and its paraxial approximation have been evaluated.

- Finally, free and diffraction-limited propagation of the fundamental mode through these guides have also been analysed using the stationary phase method.