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-