112 CHAPTER 3. OPTICAL TRANSMITTERS
response of a 1.55-μm DFB laser at several bias levels [70]. The 3-dB modulation
bandwidth,f3dB, is defined as the frequency at which|H(ωm)|is reduced by 3 dB (by
a factor of 2) compared with its direct-current (dc) value. Equation (3.5.22) provides
the following analytic expression forf3dB:
f3dB=1
2 π[
Ω^2 R+Γ^2 R+ 2 (Ω^4 R+Ω^2 RΓ^2 R+Γ^4 R)^1 /^2
] 1 / 2
. (3.5.23)
For most lasers,ΓRΩR, andf3dBcan be approximated by
f3dB≈√
3 ΩR
2 π≈
(
3 GNPb
4 π^2 τp) 1 / 2
=
[
3 GN
4 π^2 q(Ib−Ith)] 1 / 2
, (3.5.24)
whereΩRwas approximated by(GGNPb)^1 /^2 in Eq. (3.5.21) andGwas replaced by
1 /τpsince gain equals loss in the above-threshold regime. The last expression was
obtained by using Eq. (3.5.7) at the bias level.
Equation (3.5.24) provides a remarkably simple expression for the modulation
bandwidth. It shows thatf3dBincreases with an increase in the bias level as
√
Pb
or as(Ib−Ith)^1 /^2. This square-root dependence has been verified for many DFB lasers
exhibiting a modulation bandwidth of up to 30 GHz [67]–[70]. Figure 3.21 shows how
f3dBcan be increased to 24 GHz for a DFB laser by biasing it at 80 mA [70]. A mod-
ulation bandwidth of 25 GHz was realized in 1994 for a packaged 1.55-μm InGaAsP
laser specifically designed for high-speed response [68].
3.5.3 Large-Signal Modulation
The small-signal analysis, although useful for a qualitative understanding of the modu-
lation response, is not generally applicable to optical communication systems where the
laser is typically biased close to threshold and modulated considerably above threshold
to obtain optical pulses representing digital bits. In this case of large-signal modulation,
the rate equations should be solved numerically. Figure 3.22 shows, as an example, the
shape of the emitted optical pulse for a laser biased atIb= 1. 1 Ithand modulated at
2 Gb/s using rectangular current pulses of duration 500 ps and amplitudeIm=Ith.
The optical pulse does not have sharp leading and trailing edges because of a limited
modulation bandwidth and exhibits a rise time∼100 ps and a fall time∼300 ps.
The initial overshoot near the leading edge is a manifestation of relaxation oscillations.
Even though the optical pulse is not an exact replica of the applied electrical pulse,
deviations are small enough that semiconductor lasers can be used in practice.
As mentioned before, amplitude modulation in semiconductor lasers is accompa-
nied by phase modulation governed by Eq. (3.5.16). A time-varying phase is equivalent
to transient changes in the mode frequency from its steady-state valueν 0. Such a pulse
is called chirped. Thefrequency chirpδν(t)is obtained by using Eq. (3.5.16) and is
given by
δν(t)=1
2 πdφ
dt=
βc
4 π[
GN(N−N 0 )−
1
τp]
. (3.5.25)
The dashed curve in Fig. 3.21 shows the frequency chirp across the optical pulse. The
mode frequency shifts toward the blue side near the leading edge and toward the red