Page 73 - Fiber Optic Communications Fund

54 Fiber Optic Communications

and n > n . For simplicity, let us ignore the variations with respect to the y-coordinate. The ray AB corre-
2
1
sponds to a plane wave,
E incident = Ae −it+i(x+iz) . (2.75)
There is a total internal reflection at the interface. The reflected ray BC corresponds to
E = Ae −it−ix+iz . (2.76)
ref
Note that the z-component of the wave vector does not change after the reflection, but the x-component
reverses its sign. The total field in the waveguide can be written as

E = E incident + E ref = 2A cos (x)e −i(t−z) . (2.77)

Thus, incident and reflected plane waves set up a standing wave in the x-direction. The rigorous solution to
the planar waveguide problem by solving Maxwell’s equation shows that can take discrete values and
n
2n ∕ < < 2n ∕ . In the case of an optical fiber, cos (x) is replaced by the Bessel function and the
1
0
2
0
n
rest is nearly the same. In Section 2.4 we have found that for single-mode fibers there is only one mode, with
field distribution given by
−it+i 01 z
= A R (r)e (2.78)
1 01
where is the propagation constant obtained by solving Eq. (2.57). Therefore, a guided mode of an optical
01
fiber can be imagined as a standing wave in transverse directions and a propagating wave in the z-direction
resulting from the superposition of the ray AB and the reflected ray BC. The propagation constant and
01
angle are related by
= k n sin . (2.79)
01 0 1
the discrete value of the propagation constant 01 implies that can not take arbitrary values in the interval
[ ,∕2], but only a discrete value as determined from Eqs. (2.57) and (2.79).
c
2.4.5 Radiation Modes
2 2
2
For radiation modes, < k n . In this case, the last terms on the left-hand sides of Eqs. (2.44) and (2.45) are
0 2
both positive and their solutions are given by Bessel functions,
{ ( )
C J r r ≤ a
1 m
1
R(r)= .
E J ( r)+ E Y ( r) r > a
2
1 m
2 m
2
Continuity of R(r) and dR∕dr at the core–cladding interface leads to two equations as before. But now we
have four unknowns C , E , E , and . C can be determined from the power carried by the mode and this
2
1
1
1
leaves us with three unknowns and two equations of continuity. Therefore, we can not write an eigenvalue
equation for as was done in Section 2.4.1 for guided modes. In fact, can take arbitrary values in the range
0 < < k n . A connection with ray optics can be made by defining
0 2
= k sin = k n sin , (2.80)
i
0 1
1
i
where is the angle of incidence, as shown in Fig. 2.23. The ray undergoes refraction as it goes from core
i
to cladding if the angle of incidence < . When = ,wehave
i c i c
n 2
sin = (2.81)
c
n
1