238 CHAPTER 6. OPTICAL AMPLIFIERS
Equations (6.2.11) and (6.2.12) govern amplification of optical pulses in SOAs.
They can be solved analytically for pulses whose duration is short compared with the
carrier lifetime (τpτc). The first term on the right side of Eq. (6.2.12) can then be
neglected during pulse amplification. By introducing the reduced timeτ=t−z/vg
together withA=
√
Pexp(iφ), Eqs. (6.2.11) and (6.2.12) can be written as [18]∂P
∂z=g(z,τ)P(z,τ), (6.2.14)∂φ
∂z=−^12 βcg(z,τ), (6.2.15)∂g
∂τ=−g(z,τ)P(z,τ)/Esat. (6.2.16)Equation (6.2.14) can easily be integrated over the amplifier lengthLto yield
Pout(τ)=Pin(τ)exp[h(τ)], (6.2.17)wherePin(τ)is the input power andh(τ)is the total integrated gain defined as
h(τ)=∫L0g(z,τ)dz. (6.2.18)If Eq. (6.2.16) is integrated over the amplifier length after replacinggPby∂P/∂z,h(τ)
satisfies [18]
dh
dτ
=−
1
Esat[Pout(τ)−Pin(τ)]=−Pin(τ)
Esat(eh− 1 ). (6.2.19)Equation (6.2.19) can easily be solved to obtainh(τ). The amplification factorG(τ)is
related toh(τ)asG=exp(h)and is given by [1]
G(τ)=G 0
G 0 −(G 0 − 1 )exp[−E 0 (τ)/Esat], (6.2.20)
whereG 0 is the unsaturated amplifier gain andE 0 (τ)=
∫τ
−∞Pin(τ)dτis the partial
energy of the input pulse defined such thatE 0 (∞)equals the input pulse energyEin.
The solution (6.2.20) shows that the amplifier gain is different for different parts of
the pulse. The leading edge experiences the full gainG 0 as the amplifier is not yet sat-
urated. The trailing edge experiences the least gain since the whole pulse has saturated
the amplifier gain. The final value ofG(τ)after passage of the pulse is obtained from
Eq. (6.2.20) by replacingE 0 (τ)byEin. The intermediate values of the gain depend on
the pulse shape. Figure 6.7 shows the shape dependence ofG(τ)for super-Gaussian
input pulses by using
Pin(t)=P 0 exp[−(τ/τp)^2 m], (6.2.21)
wheremis the shape parameter. The input pulse is Gaussian form=1 but becomes
nearly rectangular asmincreases. For comparison purposes, the input energy is held
constant for different pulse shapes by choosingEin/Esat= 0 .1. The shape dependence
of the amplification factorG(τ)implies that the output pulse is distorted, and distortion
is itself shape dependent.