Simplified Model for Light Propagation in Graded-Index-Medium

doi:10.4236/opj.2013.37054

Paper Menu >>

Journal Menu >>

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: [email protected], [email protected]

Received March 2, 2013; revised April 5, 2013; accepted May 3, 2013

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















 





(1)



tan





 



(2)

Equation (2) can be expressed in terms of angles' tan-

gents as:

12 1

1212

tantan1 tantan

1 tantantantan





 



















(3)

where i



and r



are the incident and reflected elec-

R. I. RABADY

348

2

1

o

Δy

n(y<0)=1

)

=no

n(y)=n1

n(y+ ∆y)=n2

Δs

Δx

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.

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:

21 21

sincoscos sin

 













 

 

(4)

Dividing all terms in the numerator and the dominator

by 2

coscos 1



, Equation (4) becomes:

tan tan











 







(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:

12 12

121 2

tantan1 tantan

1 tantantantan

 

 















 

 













(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:



tan d









 

 (7)

Moreover, using the following relation:







yx x yx

yyx x









which can be rearranged into:







Therefore, the tangent of the transmittance angle 2



becomes

 

tan dyxxyx









 (8)

Substituting Equations (7) and (8) in Equations (5) and

(6) yields:

Pyx

Pyy







 







(9)

dd1

Pyxyyy

Pyyx

yyyx







 















  













 (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]:

 









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:

 

ddd







 (12)

Moreover, substituting d

dsin



 in Equation (12)

yields:

 

sin ddd









 (13)

Since the refractive index is changing only in the y-di-

rection, Equation (13) becomes:

 

sin d

ny y









 (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



xdx yxxyx



 



Open Access OPJ

R. I. RABADY 349

and the x-axis.

Substituting Equation (15) into Equation (14) yields:





 (16)

Thus,

 



  (17)

Moreover, equation 15 can be written as:

 

sin yny







Therefore,



dtan







  (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:











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:





 (20)

n(yi) is obtained by sampling n(y) every y , therefore,

the derivative



y is found numerically by:





ny ny





(21)

For a given refractive index profile n(yi), Equations

(20) and (21) are substituted in Equations (17) and (18)

to obtain i



and i



 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 mm−1. 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 mm−1).

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

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.

REFERENCES

[1] M. Heckler and A. Dreher, “Analysis of Graded-Index

Fibers Using a Full-Wave Equivalent Circuit,” Journal of

Lightwave Technology, Vol. 25, No. 1, 2007, pp. 346-349.

http://dx.doi.org/10.1109/JLT.2006.886676

[2] H. B. Salameh and M. Irshid, “Wavelength-Division

Demultiplexing Using Graded-Index Planar Structures,”

Journal of Lightwave Technology, Vol. 24, No. 6, 2006,

pp. 2401-2408.

http://dx.doi.org/10.1109/JLT.2006.874589

[3] B. Allman, A. Klein, K. Nugent and G. Opat, “Refrac-

tive-Index-Profile Determination by Using Lloyd’s Mi-

rage,” Applied Optics, Vol. 33, No. 10, 1994, pp. 1806-

1811. http://dx.doi.org/10.1364/AO.33.001806

[4] D. Bertilone and C. Pask, “Exact Ray Paths in a Graded-

Index Taper,” Applied Optics, Vol. 26, No. 7, 1987, pp.

1189-1194.

[5] E. W. Marchand, “Graded Index Optics,” Academic, New

York, 1978.

[6] A. W. Snyder and J. D. Love, “Optical Waveguide Theory,”

Chapman & Hall, New York, 1983.

[7] M. Born and E. Wolf, “Principles of Optics,” 7th Edition,

Cambridge University, Cambridge, 1999.

[8] R. Pratesi and L. Ronchi, “Wave Propagation in a Non-

parabolic Graded-Index Medium,” IEEE Transactions on

Microwave Theory and Techniques, Vol. 26, No. 11,

1978, pp. 856-858.