Page 290 - Fiber Optic Communications Fund

Optical Amplifiers 271

The reflectivity of the AR coating is quite sensitive to the width and refractive index n of the AR coating.
2
To increase the tolerance, a multilayer AR coating can be used. The experimental results of Ref. [8] show that
the tolerances in and n of a single-layer AR coating for realizing the power reflectivity R ≤ 1 × 10 −3 are
2
∘ ∘
about ±60 A and ±0.05, respectively. The double-layer AR coating has larger tolerances, ±90 A in widths
and ±0.3 in refractive index, for the same power reflectivity as in the single-layer AR coating. In this section,
we have assumed that the optical field is a plane wave and as a result, we obtained a simple expression for
the width of the AR coating. However, in a waveguide, plane waves should be replaced by the modes of the
waveguide and the reflectivity should be calculated [8].

6.6.4 Gain Saturation
As the input signal power increases beyond a certain threshold, the gain G decreases for both cavity-type
FPA and TWA. This is known as gain saturation. This phenomenon can be explained as follows. When the
population inversion is achieved, the stimulated emission dominates the absorption. Since the stimulated
emission rate is proportional to photon density, a larger input signal power enhances the stimulated emission
and, therefore, the excited carriers are depleted and the gain decreases. Under steady-state conditions, we can
set dN ∕dt to zero in Eq. (3.123) to obtain
e
N e I
G (N − N )N + = , (6.139)
0 e e,0 ph
qV
e
where
G =Γ . (6.140)
0 g
Simplifying Eq. (6.139), we find
I∕qV + G N N
0 e,0 ph
N = , (6.141)
e
G N + 1∕ e
0 ph
(I∕qV − N ∕ )
e,0
e
e g
g = (N − N )= . (6.142)
g e e,0
G N + 1
0 ph e
Eq. (6.142) can be rewritten as
Γg 0
Γg = , (6.143)
1 + N ∕N ph,sat
ph
where
g =(I∕qV − N ∕ ) , (6.144)
0
e,0
e
g e
1
N = . (6.145)
ph,sat
Γ
g e
The optical power P and photon density N ph are related by (Eq. (3.136))
P = N ℏ A. (6.146)
ph
0
So, Eq. (6.143) may be rewritten in terms of P as
Γg 0
Γg = , (6.147)
1 + P∕P
sat
ℏ A
0
P sat = . (6.148)
Γ
g e