Question 6 of 7: LED Structures, Bandwidth, GaAs Output Characteristics and Photon-Counting Statistics

Question 6: LED Structures, Bandwidth, GaAs Output Characteristics and Photon-Counting Statistics (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Find. (a) LED types, (b) bandwidth-risetime relation and numeric bandwidth, (c) $\lambda$, spectral FWHM, optical power, (d) $P(n<3\ \text{photons in 1 ns})$.

Approach. LED bandwidth is set by the spontaneous-recombination lifetime and converts to a risetime via the usual $0.35/t_r$ rule; the GaAs bandgap fixes both $\lambda$ and the thermal linewidth of spontaneous emission; the photon-counting question is a Poisson statistics problem on the attenuated mean photon rate.

1. Part (a) — LED structure types.
   Surface-emitting (Burrus-type) vs. edge-emitting LED structures.
   The two basic LED geometries are the surface-emitting (Burrus-type) LED, in which light is collected axially through a small etched well in the substrate directly above (or below) the active region, and the edge-emitting LED (ELED), in which a narrow active stripe waveguide guides light to a cleaved facet on the side of the chip, much like a laser diode without mirrors. Surface-emitting LEDs are simple to fabricate and couple reasonably well to large-core, low-NA multimode fiber, but their broad, roughly Lambertian (isotropic) emission pattern wastes most of the generated light. Edge-emitting LEDs produce a narrower, more directional beam (some waveguiding along the stripe) that couples much more power into small-core or single-mode-class fiber, at the cost of a more delicate, harder-to-align structure and lower total output power.
2. Part (b) — bandwidth and recombination; numeric bandwidth. An LED's optical output follows the excess minority-carrier population set up by spontaneous recombination; the carrier density (and hence output power) responds to a step in drive current with an exponential rise/fall governed by the spontaneous recombination lifetime $\tau$, giving a single-pole electrical response $P(f)\propto1/\sqrt{1+(2\pi f\tau)^2}$ — the faster carriers recombine, the wider the modulation bandwidth. Using the standard risetime–bandwidth relation $f_{3\text{dB}}\approx0.35/t_r$, $$f_{3\text{dB}}=\frac{0.35}{12\times10^{-9}\ \text{s}}=\boxed{29.2\ \text{MHz}}.$$
3. Part (c) — GaAs LED wavelength, spectral width and power. The peak photon energy tracks the bandgap, so $$\lambda=\frac{hc}{E_g}=\frac{1239.84\ \text{eV}\cdot\text{nm}}{1.41\ \text{eV}} =\boxed{879.3\ \text{nm}}.$$ Spontaneous emission has a thermal (Boltzmann) linewidth $$\Delta\lambda_{\text{FWHM}}\approx\frac{1.45\,kT\lambda^2}{hc} =\boxed{23.4\ \text{nm}},$$ about $1.45\,kT$ in energy (the classic $\sim1.8\,kT$ linewidth of a spontaneous-emission lineshape reduces to this coefficient once converted to a wavelength FWHM). Taking the internal quantum efficiency $\eta=18\%$ as also the (external) power efficiency and the injected carrier rate $I/q$, the emitted optical power is $$P_{\text{opt}}=\eta\,(h\nu/q)\,I=(0.18)(1.41\ \text{V})(15\ \text{mA}) =\boxed{3.81\ \text{mW}}.$$
4. Part (d) — photon-counting probability. After 40 dB of attenuation the received power is $$P_r=P\times10^{-40/10}=100\ \mu\text{W}\times10^{-4}=10.0\ \text{nW}.$$ At $0.85\ \mu\text{m}$ the photon energy is $h\nu=hc/\lambda\approx2.34\times10^{-19}\ \text{J}$, so the mean number of photons arriving in a $1\ \text{ns}$ interval is $$\bar{N}=\frac{P_r\,(1\ \text{ns})}{h\nu}=\boxed{42.79\ \text{photons}}.$$ Using the paper's own Poisson formula $P(n)=\bar{N}^n e^{-\bar N}/n!$, $$P(n<3)=P(0)+P(1)+P(2)=e^{-\bar N}\left(1+\bar N+\frac{\bar N^2}{2}\right) =\boxed{2.50e-16}.$$ With a mean of nearly 43 photons per nanosecond, the probability of receiving fewer than 3 is vanishingly small — confirmation that the link is nowhere near being photon-starved at this power/attenuation combination; a receiver failing here would be limited by electronic (thermal/shot/dark-current) noise, not by photon-arrival statistics.