Page 50 - Fiber Optic Communications Fund

Electromagnetics and Optics 31

y
a
a y
θ
x
a x
√
2
2
Figure 1.28 The x-and y-polarization components of a plane wave. The magnitude is |a| = a x + a y and the angle is
−1
= tan (a ∕a ).
y x
1.11 Polarization of Light

So far we have assumed that the electric and magnetic fields of a plane wave are along the x- and y-directions,
respectively. In general, an electric field can be in any direction in the x–y plane. This plane wave propagates
in the z-direction. The electric field intensity can be written as
E = A x + A y, (1.188)
y
x
A = a exp [i(t − kz)+ i ], (1.189)
x
x
x
A = a exp [i(t − kz)+ i ], (1.190)
y
y
y
where a and a are amplitudes of the x- and y-polarization components, respectively, and and are the
y
x
x
y
corresponding phases. Using Eqs. (1.189) and (1.190), Eq. (1.188) can be written as
E = a exp [i(t − kz)+ i ], (1.191)
x
a = a x + a exp (iΔ)y, (1.192)
y
x
where Δ = − . Here, a is the complex field envelope vector. If one of the polarization components
y
x
vanishes (a = 0, for example), the light is said to be linearly polarized in the direction of the other polarization
y
component (the x-direction). If Δ = 0or , the light wave is also linearly polarized. This is because the
−1
2
2
magnitude of a in this case is a + a and the direction of a is determined by =±tan (a ∕a ) with respect
x y y x
to the x-axis, as shown in Fig. 1.28. The light wave is linearly polarized at an angle with respect to the x-axis.
A plane wave of angular frequency is characterized completely by the complex field envelope vector a.It
can also be written in the form of a column matrix, known as the Jones vector:
[ ]
a x
a = . (1.193)
a exp (iΔ)
y
The above form will be used for the description of polarization mode dispersion in optical fibers.
Exercises
1.1 Two identical charges are separated by 1 mm in vacuum. Each of them experience a repulsive force
of 0.225 N. Calculate (a) the amount of charge and (b) the magnitude of electric field intensity at the
location of a charge due to the other charge.
7
(Ans: (a) 5 nC; (b) 4.49 ×10 N/C.)