Question 2 of 7: GaAs Laser Diode — Longitudinal Modes, RIN, DFB Structure and Grating Period

Question 2: GaAs Laser Diode — Longitudinal Modes, RIN, DFB Structure and Grating Period (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. Gain bandwidth $\Delta\lambda_g=1.5\ \text{nm}$; cavity length $L=0.5\ \text{mm}$; GaAs bandgap $E_g=1.41\ \text{eV}$, index $n=3.63$ (part a); InGaAsP index $n=3.4$, DFB wavelength $\lambda=1550\ \text{nm}$ (part d).

Find. (a) mode spacing and number of modes under the gain curve; (b), (c) conceptual; (d) grating period $\Lambda$.

Approach. A Fabry–Perot cavity of length $L$ and index $n$ supports longitudinal modes spaced by $\Delta\lambda_m=\lambda^2/(2nL)$; the visible spectrum is that comb sampled by the gain envelope. A DFB laser instead uses a first-order Bragg grating ($\Lambda=\lambda/2n_{\text{eff}}$) to select a single mode.

1. Part (a) — output spectrum.
   Fabry-Perot output spectrum: gain envelope (dashed) modulated by the longitudinal-mode comb.
   The lasing wavelength set by the GaAs bandgap is $$\lambda=\frac{hc}{E_g}=\frac{1239.84\ \text{eV}\cdot\text{nm}}{1.41\ \text{eV}} =879.3\ \text{nm},$$ and the Fabry–Perot mode spacing for a $0.5\ \text{mm}$ cavity of index $n=3.63$ is $$\Delta\lambda_m=\frac{\lambda^2}{2nL}=\frac{(879.3\ \text{nm})^2} {2(3.63)(0.5\times10^{6}\ \text{nm})}=\boxed{0.213\ \text{nm}}.$$ With a $1.5\ \text{nm}$ gain bandwidth, the number of Fabry–Perot modes that fall under the gain envelope is $$N\approx\frac{\Delta\lambda_g}{\Delta\lambda_m}=\frac{1.5}{0.213} =\boxed{7.0\approx7\ \text{modes}}.$$ The sketch is a narrow comb of ≈7 lines spaced 0.213 nm apart, riding under a smooth Gaussian-like gain envelope centred near 879.3 nm.
2. Part (b) — relative intensity noise (RIN). RIN is the normalized power spectral density of the laser's own optical-power fluctuations, $\mathrm{RIN}=\overline{\delta P^2}(f)/\bar{P}^2$ (units $\text{Hz}^{-1}$, usually quoted in dB/Hz). It originates from spontaneous emission events coupling into the lasing mode and perturbing the photon/carrier population in the laser rate equations, producing a resonance peak near the laser's relaxation-oscillation frequency (typically a few GHz) that decays at higher and lower frequencies. In a communication link RIN adds an intensity-noise floor on top of shot and thermal receiver noise; because it scales with the square of the received signal power, it becomes the dominant noise source at high received power and sets an upper bound on the usable optical power (and hence the achievable SNR/BER) independent of how good the receiver electronics are.
3. Part (c) — DFB laser diode. A distributed-feedback (DFB) laser replaces the two cleaved Fabry–Perot mirrors with a corrugated grating etched along (or beside) the active waveguide over its whole length, so optical feedback is distributed continuously rather than concentrated at the end facets. The periodic index perturbation Bragg-reflects only a narrow band of wavelengths back into the gain medium, so a single longitudinal mode (the one nearest the Bragg wavelength) experiences far lower threshold gain than its neighbours and dominates lasing. Benefits: single-longitudinal-mode operation with a side-mode suppression ratio of 30–40 dB or more, a correspondingly narrow linewidth, greatly reduced chirp and dispersion penalty, and stable wavelength over current/temperature variations — the standard source for long-haul, high-bit-rate WDM links.
4. Part (d) — DFB grating period. A first-order Bragg grating reflects at $\lambda=2n_{\text{eff}}\Lambda$, so for an InGaAsP waveguide with $n=3.4$ at $1550\ \text{nm}$, $$\Lambda=\frac{\lambda}{2n}=\frac{1550\ \text{nm}}{2(3.4)}=\boxed{227.9\ \text{nm}}.$$ This sub-quarter-micron period is why DFB gratings are written holographically or by electron-beam lithography rather than conventional photolithography.