INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS
J. Opt. A: Pure Appl. Opt. 7 (2005) 286–289 doi:10.1088/1464-4258/7/7/004
Talbot effect in selfoc microlens:
application in manufacturing
M V P´ erez, C Bao, M T Flores-Arias, A Castelo and
CG´ omez-Reino
Grupo de
´
Optica GRIN, Departamento de F´ ısica Aplicada, Escola Universitaria de
´
Optica e
Optometr´ ıa and Facultade de F´ ısica, Universidade de Santiago de Compostela,
Campus Sur s/n, E15782 Santiago de Compostela (A Coru˜ na), Spain
1
E-mail: fatoni@usc.es
Received 5 November 2004, accepted for publication 24 March 2005
Published 27 May 2005
Online at stacks.iop.org/JOptA/7/286
Abstract
We present a hybrid optical device for the measurement of axial refractive
index and gradient parameter variations in a selfoc microlens by the Talbot
effect. The intrinsic nature of the device, that combines a linear grating and
an inhomogeneous medium, permits us to measure changes of the
parameters as functions of the position shift of the first self-image. The
potential accuracy of the measurement of these parameters is estimated.
Keywords: Talbot effect, selfoc microlens, GRIN optics, optical testing
1. Introduction
The Talbot effect [1–4], also called the self-imaging effect, is
a well known phenomenon in optics. This effect consists in
the reproduction of a transverse periodic field distribution at
periodical spatial intervals along the longitudinal direction as a
periodic object, in a homogeneous medium, is illuminated by
coherent light. Fundamental properties and many practical
applications have been considered [5]. The self-imaging
phenomenon has also been studied in inhomogeneous media
and recently authors have described the Talbot effect in a
tapered gradient-index (GRIN) medium [6–8]. On the other
hand, GRIN optics is an active area of research in optical
communications and optical sensing. Many devices for fibre-
optic sensors include GRIN-rod microlenses or GRIN-fibre
microlenses [8]. The purpose of this paper is to present
applications of hybrid devices formed by a periodic object
and a selfoc microlens, in order to detect small changes of
the manufacturing characteristics of selfoc microlenses.
The plan of the paper is as follows. In section 2,
we establish the statement of the problem and the results
concerning the Talbot effect in selfoc microlenses for uniform
illumination. In section 3, we present the sensitivity and
accuracy of this device in order to know the limitations
for which a position shift of the self-images is transformed
1
http://www.usc.es/grinteam
into axial refractive index and gradient parameter detectable
changes of selfoc microlenses. In section 4, the conclusions
are given.
2. Statement of the problem
Let us consider a selfoc microlens characterized by a transverse
parabolic refractive index given by
n
2
(x ) = n
2
0
(1 − g
2
0
x
2
) (1)
where n
0
is the index along the direction of propagation and
g
0
is the gradient parameter that describes the evolution of the
parabolic transverse index along the z axis. The study will
be restricted to the one-dimensional case, but extension to the
two-dimensional case is straightforward.
We assume that a one-dimensional periodic object located
at the input of the GRIN microlens is represented by
T (x
0
) =
∞
m=−∞
a
m
exp
−i
2π mx
0
p
(2)
where p is the period and a
m
is the amplitude of the mth
harmonic.
When the hybrid structure is illuminated by a uniform
plane wave of amplitude A and wavelength λ,
ψ(x
0
) = A exp(−ikz ) (3)
1464-4258/05/070286+04$30.00 © 2005 IOP Publishing Ltd Printed in the UK 286