Question 3 of 6: GaAs LED Emission and a GaAs Quantum-Well Fabry-Perot Laser
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. 98-Phys-B2 Electro-Optical Engineering, National
Examination May 2015 — a three-hour closed-book examination (one 8.5×11 inch
double-sided handwritten note sheet permitted). The cover page states any five of
the six questions constitute a complete paper and only the first five as they appear
in the answer book are marked; every question is nonetheless answered in full below so the
paper remains a complete study resource. Figure-based Question 4 is solved against the actual
photodiode responsivity curve printed on the exam, not an
assumed shape.
Reference texts. G. Keiser, Optical Fiber Communications, 4th ed.
(fiber modes and dispersion, link power and risetime budgets, LED/laser and photodiode
characteristics, EDFA); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics,
2nd ed. (LED spectral width, laser diode rate equations, photodiode noise); E. Hecht,
Optics, 5th ed. (waveguiding and dispersion background).
Question 3: GaAs LED Emission and a GaAs Quantum-Well Fabry-Perot Laser (equal value)
Given. Part (a): $T=27^\circ\text{C}=300.15$ K, $I=15$ mA,
$\eta_{ext}=15\%$, GaAs bandgap $E_g=1.41$ eV. Part (b): cavity length $L=200\ \mu\text{m}$,
width $w=5\ \mu\text{m}$, $\lambda=850$ nm, threshold current $I_{th}=2$ mA, material loss
$\alpha_{mat}=25\ \text{cm}^{-1}$, internal quantum efficiency $\eta_i=80\%$, operating current
$I=20$ mA, GaAs refractive index $n=3.63$.
Find. (a) LED emission wavelength, spectral (FWHM) width, and output
optical power. (b)(i) Threshold modal gain $g_{th}$ and photon lifetime $\tau_p$. (ii) The
areal photon density in the cavity. (iii) Total internally generated optical power. (iv) Power
emitted from one cleaved facet.
Approach. (a) The LED photon energy is set by the bandgap; the emission
linewidth follows from the thermal broadening of the spontaneous recombination lineshape, and
the output power from the external quantum efficiency applied to the injected current. (b) The
laser reaches threshold when modal gain equals total cavity loss (material + mirror); above
threshold, the injected carriers in excess of threshold generate photons at a rate set by
$\eta_i$ and the photon lifetime, and a fraction of the internally generated power
proportional to the mirror-loss share of the total loss escapes through the two (symmetric)
facets.
Part (a)
Emission wavelength. The dominant emission photon energy equals the
bandgap: $h\nu\approx E_g=1.41$ eV, so
$$\lambda=\frac{hc}{E_g}=\frac{1239.8\ \text{eV}\cdot\text{nm}}{1.41\ \text{eV}}=\boxed{879\ \text{nm}}.$$
Spectral width. The spontaneous-emission lineshape
$\propto(h\nu-E_g)^{1/2}\exp[-(h\nu-E_g)/kT]$ has an energy FWHM $\Delta E\approx1.8\,kT$; at
$T=300.15$ K, $kT=0.02587$ eV, so $\Delta E=0.0466$ eV. Converting to wavelength via
$\Delta\lambda=(\lambda^2/hc)\,\Delta E$,
$$\Delta\lambda=\frac{(879\ \text{nm})^2}{1239.8\ \text{eV}\cdot\text{nm}}(0.0466\ \text{eV})\approx\boxed{29\ \text{nm}}.$$
Output optical power. With external quantum efficiency $\eta_{ext}=0.15$,
a fraction 0.15 of the injected electrons yields a photon that escapes the LED, each
with energy $h\nu\approx E_g$:
$$P_{out}=\eta_{ext}\,I\,\frac{h\nu}{q}=0.15\times(15\times10^{-3})\times1.41\ \text{V}=\boxed{3.17\ \text{mW}}.$$
Part (b) — Fabry-Perot laser
GaAs quantum-well Fabry-Perot cavity: cleaved facets act as mirrors of
reflectivity $R_1=R_2$ (Fresnel reflection at the GaAs/air interface).
Facet reflectivity and mirror loss. Each cleaved GaAs/air facet reflects
$$R=\left(\frac{n-1}{n+1}\right)^2=\left(\frac{2.63}{4.63}\right)^2=0.323.$$
The mirror (end) loss for a symmetric cavity ($R_1=R_2=R$) is
$$\alpha_m=\frac{1}{2L}\ln\!\frac{1}{R_1R_2}=\frac{1}{2(200\times10^{-4}\ \text{cm})}\ln\!\frac{1}{0.323^2}=56.6\ \text{cm}^{-1}.$$
Threshold gain. Lasing threshold requires the modal gain to exactly
offset all cavity loss:
$$g_{th}=\alpha_{mat}+\alpha_m=25+56.6=\boxed{81.6\ \text{cm}^{-1}}.$$
Photon lifetime. With the group velocity approximated as
$v_g=c/n=2.998\times10^8/3.63=8.26\times10^7$ m/s,
$$\tau_p=\frac{1}{v_g\,g_{th}}=\frac{1}{(8.26\times10^7\ \text{m/s})(8160\ \text{m}^{-1})}=\boxed{1.48\ \text{ps}}.$$
Areal photon density. No active-layer thickness is given, so the photon
population is reported per unit of the cavity's top-view footprint area $A=Lw$. Above
threshold, at steady state, the rate of stimulated photon generation balances the cavity's
photon loss rate $S/\tau_p$, giving (spontaneous emission neglected, as instructed)
$$S_{areal}=\frac{\eta_i\,\tau_p\,(I-I_{th})}{qA}
=\frac{(0.80)(1.48\times10^{-12}\ \text{s})(20-2)\times10^{-3}\ \text{A}}{(1.602\times10^{-19}\ \text{C})(200\times10^{-4}\times5\times10^{-4}\ \text{cm}^2)}
=\boxed{1.33\times10^{10}\ \text{cm}^{-2}}.$$
Total internally generated power. Every carrier injected above threshold
that recombines with efficiency $\eta_i$ produces one photon of energy $h\nu=hc/\lambda$;
summing over the cavity (the $\tau_p$ and $A$ used to define $S_{areal}$ cancel out of this
energy-balance form):
$$P_{internal}=\eta_i\,(I-I_{th})\,\frac{h\nu}{q}=(0.80)(18\times10^{-3})\left(\frac{1.4593\ \text{eV}}{1}\right)=\boxed{21.0\ \text{mW}}.$$
where $h\nu=hc/\lambda=1239.8/850=1.459$ eV at 850 nm.
Power from one facet. Of the internally generated power, only the
fraction lost through the mirrors (as opposed to absorbed by $\alpha_{mat}$) escapes as usable
light, split equally between the two identical facets:
$$P_{facet}=P_{internal}\times\frac{\alpha_m}{g_{th}}\times\frac12
=21.0\ \text{mW}\times\frac{56.6}{81.6}\times\frac12=\boxed{7.28\ \text{mW}}.$$
(Equivalently, the external differential efficiency is
$\eta_d=\eta_i\,\alpha_m/g_{th}=0.555$, and $P_{facet}=\eta_d(I-I_{th})(h\nu/q)/2$.)
