Page 35 - Fiber Optic Communications Fund

16 Fiber Optic Communications

Using Eq. (1.87) in Eq. (1.91), we obtain
H
y
= E sin (t − kz). (1.92)
x0
z
Integrating Eq. (1.92) with respect to z,
E
x0
H = cos (t − kz)+ D, (1.93)
y
k
where D is a constant of integration and could depend on t. Comparing Eqs. (1.90) and (1.93), we see that D
is zero and using Eq. (1.89) we find
E x0 1
= = , (1.94)
H
y0
where is the intrinsic impedance of the dielectric medium. For free space, = 376.47 Ohms. Note that E x
and H are independent of x and y. In other words, at time t, the phase t − kz is constant in a transverse plane
y
described by z = constant and therefore, they are called plane waves.

1.6.2 Complex Notation

It is often convenient to use complex notation for electric and magnetic fields in the following forms:
̃
̃
E = E e i(t−kz) or E = E e −i(t−kz) (1.95)
x
x
x0
x0
and
̃
̃
H = H e i(t−kz) or H = H e −i(t−kz) . (1.96)
y y0 y y0
This is known as an analytic representation. The actual electric and magnetic fields can be obtained by
[ ]
̃
E = Re E = E cos (t − kz) (1.97)
x x x0
and
[ ]
̃
H = Re H y = H cos (t − kz). (1.98)
y0
y
In reality, the electric and magnetic fields are not complex, but we represent them in the complex forms
of Eqs. (1.95) and (1.96) with the understanding that the real parts of the complex fields correspond to the
actual electric and magnetic fields. This representation leads to mathematical simplifications. For example,
differentiation of a complex exponential function is the complex exponential function multiplied by some
constant. In the analytic representation, superposition of two eletromagnetic fields corresponds to addition
of two complex fields. However, care should be exercised when we take the product of two electromagnetic
fields as encountered in nonlinear optics. For example, consider the product of two electrical fields given by
E = A cos ( t − k z), n = 1, 2 (1.99)
xn n n n
A A
1 2
E E = cos [( + )t −(k + k )z]
1
1
2
x1 x2
2
2
+ cos [( − )t −(k − k )z]. (1.100)
2
1
2
1
The product of the electromagnetic fields in the complex forms is
̃
̃
E E = A A exp [i( + )t − i(k + k )z]. (1.101)
2
x1 x2
1
2
1 2
1