Page 547 - Fiber Optic Communications Fund

528 Fiber Optic Communications

When L is finite, and if we assume that the walls of the cube are perfectly conducting, the field should vanish
at the walls. In this case k , k , and k take discrete values given by
z
y
x
2n x 2n y 2n z
k = , k = , and k = , (A.7)
z
y
x
L L L
where n , n , and n are integers. In other words, they are the standing waves formed by the superposition
y
z
x
of plane waves propagating in opposite directions (cos(t − k x − k y − k z) and cos(t + k x + k y + k z)).
y
z
y
z
x
x
In this case, spontaneous emission occurs at discrete angles in the direction of k = k ̂x + k ̂y + k ,̂z with k ,
x
y
z
x
k , and k given by Eq. (A.7). We wish to find the number of modes per unit volume, with angular frequencies
z
y
ranging from to + d. This corresponds to wave numbers ranging from k (= |k|)to k + dk. For the given
value of k, there can be a number of modes with different values of k , k , and k satisfying Eq. (A.6). For
z
y
x
√ √
example, k = k, k = k = 0 is a mode propagating in the x-direction and k = k∕ 2, k = k∕ 2, k = 0is
x y z x y z
∘ ∘
another mode propagating at angle 45 to the x-axis and 45 to the y-axis, and so on. The wave numbers
ranging from k to k + dk correspond to modes in the intervals [k , k + dk ],[k , k + dk ], and [k , k + dk ]
x x x y y y z z z
with
2
2
2
k = k + k + k 2 z (A.8)
y
x
and
2 2 2 2
(k + dk) =(k + dk ) +(k + dk ) +(k + dk ) . (A.9)
x x y y z z
From Eq. (A.7), we have
2
dk = dn , (A.10)
x
x
L
where dn is the number of modes in the interval [k , k + dk ]. The total number of modes with the
x
x
x
x
x-component of the wave vector ranging from k to k + dk ,the y-component ranging from k to k + dk ,
x
y
y
x
y
x
and the z-component ranging from k to k + dk is
z
z
z
L 3
dn dn dn = dk dk dk , (A.11)
x y z 3 x y z
(2)
where dk dk dk represents the volume of the spherical shell enclosed between two spheres with radii k and
z
y
x
k + dk, as shown in Fig. A.1. Therefore,
dk dk dk =(area of the sphere with radius k)× dk
z
y
x
2
= 4k dk. (A.12)
Substituting Eq. (A.12) into Eq. (A.11), we find that the total number of modes per unit volume with angular
frequency ranging from to + d is
2 3
dn dn dn 2 n d
x y z 4k dk 0
= = , (A.13)
2 3
L 3 (2) 3 2 c
where
n ∕c = k. (A.14)
0