3.5. LASER CHARACTERISTICS 107
3.5.1 CW Characteristics
A rigorous derivation of the rate equations generally starts from Maxwell’s equations
together with a quantum-mechanical approach for the induced polarization (see Section
2.2). The rate equations can also be written heuristically by considering various physi-
cal phenomena through which the number of photons,P, and the number of electrons,
N, change with time inside the active region. For a single-mode laser, these equations
take the form [2]
dP
dt=GP+Rsp−P
τp, (3.5.1)
dN
dt=
I
q−
N
τc−GP, (3.5.2)
where
G=Γvggm=GN(N−N 0 ). (3.5.3)
Gis the net rate of stimulated emission andRspis the rate of spontaneous emission into
the lasing mode. Note thatRspis much smaller than the total spontaneous-emission rate
in Eq. (3.1.10). The reason is that spontaneous emission occurs in all directions over a
wide spectral range (∼30–40 nm) but only a small fraction of it, propagating along the
cavity axis and emitted at the laser frequency, actually contributes to Eq. (3.5.1). In fact,
RspandGare related byRsp=nspG, wherenspis known as thespontaneous-emission
factorand is about 2 for semiconductor lasers [2]. Although the same notation is used
for convenience, the variableNin the rate equations represents the number of electrons
rather than the carrier density; the two are related by the active volumeV. In Eq. (3.5.3),
vgis the group velocity,Γis the confinement factor, andgmis the material gain at the
mode frequency. By using Eq. (3.3.1),Gvaries linearly withNwithGN=Γvgσg/V
andN 0 =NTV.
The last term in Eq. (3.5.1) takes into account the loss of photons inside the cavity.
The parameterτpis referred to as thephoton lifetime. It is related to thecavity loss
αcavintroduced in Eq. (3.3.4) as
τ−p^1 =vgαcav=vg(αmir+αint). (3.5.4)The three terms in Eq. (3.5.2) indicate the rates at which electrons are created or de-
stroyed inside the active region. This equation is similar to Eq. (3.2.8) except for the ad-
dition of the last term, which governs the rate of electron–hole recombination through
stimulated emission. The carrier lifetimeτcincludes the loss of electrons due to both
spontaneous emission and nonradiative recombination, as indicated in Eq. (3.1.18).
TheP–Icurve characterizes the emission properties of a semiconductor laser, as
it indicates not only the threshold level but also the current that needs to be applied
to obtain a certain amount of power. Figure 3.20 shows theP–Icurves of a 1.3-μm
InGaAsP laser at temperatures in the range 10–130◦C. At room temperature, the thresh-
old is reached near 20 mA, and the laser can emit 10 mW of output power from each
facet at 100 mA of applied current. The laser performance degrades at high tempera-
tures. The threshold current is found to increase exponentially with temperature, i.e.,
Ith(T)=I 0 exp(T/T 0 ), (3.5.5)