Download Optical fields, as complicated as it may be to grasp

Optical Fiber Communications
Assignment 1
Syed Absar Ahmed Shah
EL-543-03
A Report on
‘EM Propagation of Energy in Optical Fiber
With special reference to Maxwell’s Equations’
Optical fields, as complicated as it may be to grasp fully the
understanding of, are electromagnetic radiations of frequencies to
which the human eye is sensitive. These lie between 10 14 to 1016 Hz.
The high frequency translates into a very small wavelength. This in turn
means that optical fields should be considered as far field and weakly
diffracting limits. This implies that Maxwell’s equations can very
successfully be employed in the study of optical fields. Mostly, the
effects of diffraction can be ignored and the radiations can be
considered to consist of a single beam, thus simplifying analysis by
Maxwell’s equations. Also, optical fields are generated and detected by
atomic-scale quantum interactions rather than by antennas.
The propagation of the optical field can be described by Maxwell's
equations. Since the optical field cannot be directly detected, statistical
correlations and intensity measurements must supplement Maxwell's
equations to fully understand propagation.
The Maxwell’s equations, in differential form are given as:

B
t

H J D
t
D 
E 
B 0
Where E is the electric field, D is the electric displacement, B is the
magnetic induction,
 H is the magnetic field, J is the current density, and
 is the charge density. These are the field variables. These are
connected to the material variables via the following equations:
D  o E  P
B  o  M
Where P is the polarization of the material and M is the magnetization.
For a medium with zero conductivity, both M and P become zero.
Substituting these
 material equations into Maxwell’s equations for
isotropic media:
2
    E  o 2 E
t
2
    H  o 2 H
t
Subsequently, applying the vector identity

    A    A  2 A
Also, from the Gauss's Law we know that


  E   E      E  0
Here we have assumed for the moment that is scalar valued. Thus,
2E
 E   2
t
2
And
 2H
 H   2
t
2

These differential equations are equivalent to the wave equation:

1 2 f
 f  2
 t
2
Where v is the velocity of the wave and f is a displacement.
So notice that 
in the case of the electric and magnetic fields, the
velocity is:

1
oo
A medium in which (the permittivity is spatially constant) is
homogeneous.
Most

common
materials
are
approximately
homogeneous. This is not the case for optical fiber.
Defining light as electromagnetic radiation, we concur to the fact that it
consists of both E and H fields, as dictated by the Maxwell’s equations
and it’s derivatives shown above. These E and H fields are positioned
perpendicular to each other.
As with microwave transmission in guided media, propagation of EM
energy in optical fiber can occur in certain allowed modes. For
example, when E z = 0 , it indicates that the E component of the wave is
perpendicular to the direction of propagation. The same holds for Hz=0,
where the H component of the wave is perpendicular to the direction of
wave propagation. These two modes are known as TEm and TMm
modes respectively. However, where TEM modes are common in
microwave-guided systems, this phenomenon is seldom observed in
optical fiber media.
References
1. http://en.wikipedia.org/wiki/Electromagnetic_radiation
2. University of Waterloo, course ECE477 course presentation
http://www.ece.uwaterloo.ca/~ece477/Lectures/ECE477_2.ppt
3. SPIE page: Maxwell's Equations with Optical Fibers in Mind
4. Optical Fiber Communications – Principles and Practice (2nd
Edition) by John M. Senior. ISBN-81-203-0882-4