Optics and Photonics Journal, 2013, 3, 347-350
Published Online November 2013 (http://www.scirp.org/journal/opj)
http://dx.doi.org/10.4236/opj.2013.37054
Open Access OPJ
Simplified Model for Light Propagation in
Graded-Index-Medium
Rabi Ibrahim Rabady
Electrical Engineering Department, Jordan University of Science and Technology, Irbid, Jordan
Email: rrabady@just.edu.jo, rabirabady@yahoo.com
Received March 2, 2013; revised April 5, 2013; accepted May 3, 2013
Copyright © 2013 Rabi Ibrahim Rabady. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Using the ray theory of light, a simple theoretical model for the power evolution of a propagating light in graded-in-
dex-medium is presented. This work can be useful for different engineering applications that utilize graded-index-ma-
terial, and for further understanding of natural phenomena that depends on light propagation in graded-index-medium.
Keywords: Graded-Index Medium; Fresnel Reflection; Power Evolution
1. Introduction
Light propagation in graded-index-medium is found in
different natural processes and engineering applications,
such as mirage phenomenon, wavelength multiplexing
and demultiplexing, graded-index fibers and graded-in-
dex lenses [1-3].
Generally, as light propagates in graded-index-mate-
rial at large scale, it encounters two major effects; namely,
reflection and refraction. This work utilizes the simple
laws of reflection and refraction to present a useful in-
sight of the light propagation in graded-index-medium by
accounting the gradual Fresnel reflection and refraction.
Surely, using Maxwell equations leads to a more com-
prehensive and accurate solution; however, this requires
a much complicated modeling and simulation because of
the spatial dependency of the medium refractive-index.
Therefore, a simple model will be developed in order to
quantify the power evolution of propagating light in a
defined graded-index medium.
2. Theory
When light propagates in graded-index-medium it ex-
periences gradual refraction and reflection simultane-
ously. The gradual refraction behavior is modeled using
both the ray theory [4-6] and the wave theory [7,8] of
light. However, a simple modeling of the light gradual
reflection in graded-index-medium can be achieved by
considering a propagating ray of light through a graded-
index-medium that changes its refractive index in the
y-direction, as depicted in Figure 1. Therefore, as light
enters a sliced layer of the medium it travels through that
layer distance Δs, also the incident angle changes from θ1
to θ2 , whereas, the refractive index changes from n(y) =
n1 to n(y + Δy) = n2. Where, θ1 and θ2 are the incident
and transmitted ray angles with respect to norm of the
sliced layer, respectively. If light enters the graded index
medium with incident angle other than the normal inci-
dence, it would bend gradually in order to satisfy Snell’s
law. Moreover, light will experience spatial dispersion
and power reduction at each point on the ray path trajec-
tory as shown in Figure 1; where, dP is the reflected
light power because of gradual Fresnel reflection that
stems from gradual refractive index change through the
graded-index-medium. Therefore, the Fresnel reflection
coefficient of electric and magnetic polarized light can be
expressed as [7]:
12 21 2
121 22 1
sin sincoscos sin
sinsin cossincos
r
Ei
E
rE
1
(1)
12
12
tan
tan
r
Hi
E
rE
(2)
Equation (2) can be expressed in terms of angles' tan-
gents as:
12 1
1212
tantan1 tantan
1 tantantantan
H
r2
(3)
where i
E
and r
E
are the incident and reflected elec-