90 CHAPTER 3. OPTICAL TRANSMITTERS
injection [5]. The result is
Rspon(ω)=A 0 ( ̄hω−Eg)^1 /^2 exp[−(h ̄ω−Eg)/kBT], (3.2.7)whereA 0 is a constant andEgis the bandgap. It is easy to deduce thatRspon(ω)
peaks when ̄hω=Eg+kBT/2 and has a full-width at half-maximum (FWHM)∆ν≈
- 8 kBT/h. At room temperature (T=300 K) the FWHM is about 11 THz. In practice,
the spectral width is expressed in nanometers by using∆ν=(c/λ^2 )∆λand increases
asλ^2 with an increase in the emission wavelengthλ. As a result,∆λis larger for In-
GaAsP LEDs emitting at 1.3μm by about a factor of 1.7 compared with GaAs LEDs.
Figure 3.7(b) shows the output spectrum of a typical 1.3-μm LED and compares it
with the theoretical curve obtained by using Eq. (3.2.7). Because of a large spectral
width (∆λ=50–60 nm), the bit rate–distance product is limited considerably by fiber
dispersion when LEDs are used in optical communication systems. LEDs are suit-
able primarily for local-area-network applications with bit rates of 10–100 Mb/s and
transmission distances of a few kilometers.
3.2.3 Modulation Response
The modulation response of LEDs depends on carrier dynamics and is limited by the
carrier lifetimeτcdefined by Eq. (3.1.18). It can be determined by using arate equation
for the carrier densityN. Since electrons and holes are injected in pairs and recombine
in pairs, it is enough to consider the rate equation for only one type of charge carrier.
The rate equation should include all mechanisms through which electrons appear and
disappear inside the active region. For LEDs it takes the simple form (since stimulated
emission is negligible)
dN
dt
=
I
qV−
N
τc, (3.2.8)
where the last term includes both radiative and nonradiative recombination processes
through the carrier lifetimeτc. Consider sinusoidal modulation of the injected current
in the form (the use of complex notation simplifies the math)
I(t)=Ib+Imexp(iωmt), (3.2.9)whereIbis the bias current,Imis the modulation current, andωmis the modulation
frequency. Since Eq. (3.2.8) is linear, its general solution can be written as
N(t)=Nb+Nmexp(iωmt), (3.2.10)whereNb=τcIb/qV,Vis the volume of active region andNmis given by
Nm(ωm)=τcIm/qV
1 +iωmτc. (3.2.11)
The modulated powerPmis related to|Nm|linearly. One can define the LED transfer
functionH(ωm)as
H(ωm)=Nm(ωm)
Nm( 0 )=
1
1 +iωmτc