Page 284 - Fiber Optic Communications Fund

Optical Amplifiers 265

√
where |r | = R , j = 1, 2. After j round trips, the partial field at B is
j
j
√ 2j+1
j
= t t (r r ) [ G exp(i )] . (6.115)
j in 1 2 1 2 s 0
The total field output at B is the superposition of partial fields,
∞ ∞
∑ √ ∑ j
= = t t G exp(i ) h , (6.116)
out j 1 2 in s 0
j=0 j=0
where
h = r r G exp(i2 ). (6.117)
1 2
s
0
The summation in Eq. (6.116) is a geometric series and if
|h| < 1, (6.118)
we have
∞
∑ j 1
h = . (6.119)
1 − h
j=0
Therefore, Eq. (6.116) becomes
√
t t G s
in 1 2
out = exp(i ). (6.120)
0
1 − h
The overall gain G is defined as
2 2 2
1
out
2
| | |t | |t | G (f)
s
G(f)= = . (6.121)
2 ∗
| | [1 − h(f)][1 − h (f)]
in
Using Eq. (6.117 ), Eq. (6.121) can be rewritten as
(1 − R )(1 − R )G (f)
1
s
2
G(f)= √ . (6.122)
2
1 + R R G (f)− 2 R R G (f) cos (2 )
1 2 s 1 2 s 0
Using the relation
2
cos (2 )= 1 − 2sin , (6.123)
0 0
Eq. (6.122) can be put in a different form:
(1 − R )(1 − R )G (f)
1
s
2
G(f)= , (6.124)
2
2
(1 − RG ) + 4RG sin (2nfL∕c)
s s
√
where R = R R is the geometric mean of facet reflectivities. From Eq. (6.124), we see that the peak gain
1 2
occurs when
2nfL
= m, m = 0, ±1, ±2, … (6.125)
c
or
mc
f = , (6.126)
m
2nL
which is the same as the resonant frequency given by Eq. (3.44). Therefore, the cavity-type optical amplifier
amplifies any input signal whose frequency is matched to the resonant frequency f of the cavity. The peak
m