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-
R. I. RABADY
348
x
2
1
o
Δy
n(y<0)=1
n
(y
=0
)
=no
n(y)=n1
n(y+ y)=n2
Δs
Δx
dP
P
Figure 1. Ray optics of light propagation in graded-refrac-
tive-index-medium.
tric fields for the electric polarized light; whereas,
and are the incident and reflected electric fields for
the magnetic polarized light, respectively.
i
E
r
E
The reflected power dP, which is cut from the incident
power P, to the incident power ratio can be found for the
electric polarized light by:
22
21 21
21 21
sincoscos sin
d
sincoscos sin
r
i
E
E
P
PE
 
 





 
(4)
Dividing all terms in the numerator and the dominator
by 2
coscos 1
, Equation (4) becomes:
2
21
21
tan tan
d
tan tan
E
P
P




(5)
Moreover, using Equation (3) a similar equation for
the reflected power “dP” to the incident power “P” ratio
can be obtained for the magnetic polarized light:
22
12 12
121 2
tantan1 tantan
d
1 tantantantan
r
i
H
E
P
PE
 
 







 
 


(6)
Given y(x) is the path trajectory of the propagating
light in the graded-index-medium, as shown in Figure 1,
the tangent of the incident ray angle could be expressed
by:

1
1
d
tan d
y1
x
yx



 (7)
Moreover, using the following relation:


2
2
d
d
d
d
yx x yx
yyx x
x




which can be rearranged into:


d
Therefore, the tangent of the transmittance angle 2
becomes
 
2
1
tan dyxxyx

(8)
Substituting Equations (7) and (8) in Equations (5) and
(6) yields:
2
dd
2d
E
Pyx
Pyy



 

x
(9)
2
2
2
dd1
2d
1d
H
Pyxyyy
Pyyx
yyyx
dx

 





  




(10)
Equations (9) and (10) represent nonlinear differential
equations that permits predicting the power evolution as
light propagate in graded-index-medium for both polari-
zations. In order to solve Equations (9) and (10), the path
trajectory y(x) of light inside the graded-index-medium
need to be found since it decides the right hand side of
Equations (9) and (10).
Obviously, y' and y'' need to be found before deter-
mining the right hand side of Equations (9) and (10). In
order to achieve this goal, we start with the eikonal equa-
tion [7]:
 
dd
dd
r
n
ss




rnr
(11)
where, s is the distance along the ray path, and r is the
position vector of any point on the ray path.
Since the refractive index changes with respect to the
y-axis only, Equation (11) reduces to:
 
d
dd
ddd
ny
y
ny
s
sy


 (12)
Moreover, substituting d
dsin
x
s
in Equation (12)
yields:
 
2d
dd
sin ddd
ny
y
ny
x
xy


 (13)
Since the refractive index is changing only in the y-di-
rection, Equation (13) becomes:
 
2
2
2
d
d
sin d
d
ny
y
ny y
x


 (14)
Introducing the ray invariant
, which is defined as:
 
sin0 sin0nyyn


(15)
where n(0) and
(0) are the refractive index and the angle
of the ray with respect to x-axis at the entrance of the
graded-index-medium, respectively. Whereas, n(y) is the
refractive index) and
(y) is the angle between the ray
y
xdx yxxyx

 
Open Access OPJ
R. I. RABADY 349
and the x-axis.
Substituting Equation (15) into Equation (14) yields:

2
2
2
2
d
d
2d
d
ny
y
y
x
(16)
Thus,
 
2
22
d
dd
d
ny
ny
yy
yx
  (17)
Moreover, equation 15 can be written as:
 
sin yny
Therefore,

22
d1
dtan
ny
y
yx

  (18)
Given that n(0),
(0), P(0), and n(y) are all defined, it
would be straight forward to solve Equations (9) and (10)
numerically after substituting Equations (17) and (18);
hence, it would be possible to predict the power evolu0
tion of light as it propagate inside a graded-index-me0
dium for electric and magnetic polarized light.
It would be useful to mention here that similarly the
same results can be used for the slanted incident rays; the
only difference is to consider a new x-axis that comes
from the projection of the slanted ray on interface surface
between the homogeneous and graded-index-medium.
3. Simulation and Results
Equations (9) and (10) can be solved numerically by
writing them as difference equations using the following
iterative form:


11
ii
i
P
yPyRHy
 (19)
where RH(yi) is the right hand side of Equations (9) and
(10), and “i" is the iterative running index; whereas, yi,
and xi are obtained by:
1
d
di
ii
i
y
y
yy
y
xy
x

 (20)
n(yi) is obtained by sampling n(y) every y , therefore,
the derivative

d
d
ny
y is found numerically by:


1
d
d
i
i
y
ny ny
ny
yy
i
(21)
For a given refractive index profile n(yi), Equations
(20) and (21) are substituted in Equations (17) and (18)
to obtain i
y
and i
y
numerically, respectively. Con-
sequently, RH(yi) in Equation (19) can also be deter-
mined numerically from Equations (9) and (10).
In the following we consider a numerical example in
order to quantify the power evolution as light propagates
in a graded-index-medium. For the sake of simplicity we
consider a linearly decreasing refractive index profile for
the graded-index-medium such that n(y) = no
sy, where,
no is the refractive index at the entrance of the graded-
index-medium (y = x= 0), and s represents the gradient
of the refractive index in the y-direction as assumed in
the theory section.
Figure 2 shows the evolution of the normalized power
with respect to the power of the entering light power P(0)
in dB versus the horizontal distance x with the following
parameters:
o = /4, no = 2, and s = 0.01 mm1. Obvi-
ously, the magnetic polarization curve comes above the
electric polarization curve; this is attributed to the fact
that the Fresnel reflection of the electric polarization is
greater than the Fresnel reflection of the magnetic po-
larization at all incident angles. The knee shape in the
magnetic polarization curve is due to the light passing
through theBrewster angle as it bends inward in the de-
creasing-refractive-index-medium. The turning point,
which is the point where light experience total internal
reflection, was found to be at (x = 117.05 mm, y =
129.29 mm). It is important to emphasize here that the
gradual reflection of light while propagating through a
decreasing-refractive-index-medium not only reduces the
propagating power, but also leads to a spatial dispersion
of light because the reflected portions of light follow
different paths from the path of the main ray. Figure 3 is
a zoom on of Figure 2 for the last two millimeters along
Figure 2. Evolution of the normalized power, in dB, as light
propagates in a linearly decreasing graded-index medium in
the y-direction versus the horizontal distance x (
o = /4, no
= 2, and s = 0.01 mm1).
Open Access OPJ
R. I. RABADY
Open Access OPJ
350
115115.2 115.4 115.6 115.8116116.2 116.4 116.6116.8117
-12
-10
-8
-6
-4
-2
0
x (mm)
P (x)/ P (0 ) (dB )
Figure 3. Zoom on the last two millimeters of Figure 2 be-
fore the light total internal reflection.
the x-direction before light total internal reflection, which
also shows that only half of the power is contained
within the last 0.8 mm. This effect is critical since it de-
cides the devise spectral/spatial resolution when graded-
index material is used for wavelength multiplexing and
demultiplexing.
4. Conclusion
In this paper, the power evolution of a propagating light
inside a graded-refractive-index-medium was modeled
and numerically simulated for both the electric and the
magnetic polarizations using the simple ray theory of
light. Therefore, it would be possible to predict the light
intensity in the graded-index-medium. Moreover, it was
shown that the spatial dispersion and the associated
power reduction of the light main ray are more pro-
nounced for the electric polarized light than that of the
magnetic polarized light; this can be attributed to the fact
that the angular Fresnel reflection for the electric polar-
ized light is greater at any incident angles. This work can
be useful for the different engineering applications that
utilize graded-index-medium; further more, it could pro-
vide more insight of natural phenomena that depend on
the light propagation in graded-index-medium.
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