88 CHAPTER 3. OPTICAL TRANSMITTERS
Figure 3.6: Total internal reflection at the output facet of an LED. Only light emitted within a
cone of angleθcis transmitted, whereθcis the critical angle for the semiconductor–air interface.
active layer are transparent to the radiation generated. The external quantum efficiency
can then be written as
ηext=1
4 π∫θc0Tf(θ)( 2 πsinθ)dθ, (3.2.3)where we have assumed that the radiation is emitted uniformly in all directions over a
solid angle of 4π. The Fresnel transmissivityTfdepends on the incidence angleθ.In
the case of normal incidence (θ=0),Tf( 0 )= 4 n/(n+ 1 )^2. If we replace for simplicity
Tf(θ)byTf( 0 )in Eq. (3.2.3),ηextis given approximately by
ηext=n−^1 (n+ 1 )−^2. (3.2.4)By using Eq. (3.2.4) in Eq. (3.2.2) we obtain the power emitted from one facet (see
Fig. 3.6). If we usen= 3 .5 as a typical value,ηext= 1 .4%, indicating that only a small
fraction of the internal power becomes the useful output power. A further loss in useful
power occurs when the emitted light is coupled into an optical fiber. Because of the
incoherent nature of the emitted light, an LED acts as aLambertian sourcewith an
angular distributionS(θ)=S 0 cosθ, whereS 0 is the intensity in the directionθ=0.
The coupling efficiency for such a source [20] isηc=(NA)^2. Since the numerical
aperture (NA) for optical fibers is typically in the range 0.1–0.3, only a few percent of
the emitted power is coupled into the fiber. Normally, the launched power for LEDs is
100 μW or less, even though the internal power can easily exceed 10 mW.
A measure of the LED performance is the total quantum efficiencyηtot, defined as
the ratio of the emitted optical powerPeto the applied electrical power,Pelec=V 0 I,
whereV 0 is the voltage drop across the device. By using Eq. (3.2.2),ηtotis given by
ηtot=ηextηint( ̄hω/qV 0 ). (3.2.5)Typically, ̄hω≈qV 0 , andηtot≈ηextηint. The total quantum efficiencyηtot, also called
thepower-conversion efficiencyor thewall-plug efficiency, is a measure of the overall
performance of the device.