L19 · Carriers and gain media

Semiconductor Laser Diode

Inject carriers into a quantum well, find transparency and threshold, then acquire current–power and modulation records. Follow carrier-induced chirp from the same trajectory.

Elapsed time $\text{—}$

Instantaneous current $\text{—}$

Transparency current $\text{—}$

Nominal threshold current $\text{—}$

Carrier count / transparency count $\text{—}$

Cavity photon count $\text{—}$

Photon lifetime from cavity $\text{—}$

Modal gain / total loss $\text{—}$

Front-facet output $\text{—}$

DC output reference $\text{—}$

Zero-compression slope limit $\text{—}$

DC-relative chirp $\text{—}$

Acquired output modulation depth $\text{—}$

Acquired output phase $\text{—}$

Carrier-derived chirp range $\text{—}$

Current-sweep records $\text{—}$

Unmodulated bias regime $\text{—}$

L19

When does injected current become coherent output?

Bias current

I_0

60\,\mathrm{mA}

Cavity length

L

300\,\mathrm{\mu m}

Front power reflectivity

R_2

0.32

Internal intensity loss

\alpha_i

10\,\mathrm{cm^{-1}}

Well confinement factor

\Gamma

0.3

Carrier recombination lifetime

\tau_N

2\,\mathrm{ns}

Normalized gain compression

\epsilon

2

Henry factor

\alpha_H

3

Current modulation depth

m

0

Modulation frequency

f_m

1\,\mathrm{GHz}

A pump–output knee, carrier balance and modulation record answer different questions. The nominal threshold is the zero-field gain condition.

A single-mode edge-emitting teaching diode uses total carrier pairs and cavity photons, both normalized by the transparency carrier count. That count is two hundred million at a 300-micrometre cavity and scales with length at fixed active cross-section. Time is in nanoseconds; the fixed representative wavelength is 850 nanometres. Linear carrier gain, a constant carrier lifetime and phenomenological gain compression replace microscopic band occupation and detailed recombination. Confinement changes the modal gain. Rear power reflectivity is fixed at 0.95; front reflectivity, length and internal loss set the photon lifetime. Uniform cavity escape is allocated through logarithmic facet loss, so the front power is a lumped-model readout rather than a spatial traveling-wave calculation. A deterministic spontaneous source continues below threshold and rounds the nominal threshold. Startup integrates both reservoirs with fourth-order Runge–Kutta, with time steps bounded by gain, loss and modulation. The DC line independently solves the same balances; current-sweep dots are 31 separate unmodulated startup integrations. Modulation depth and phase come from eight settled cycles sampled at 1024 uniform times. The constant Henry factor maps uncompressed carrier-gain changes to frequency shift relative to the unmodulated DC carrier state; positive carrier change means positive shift in the chosen convention. Chirp is not linewidth. Controls clear all captures. The layered chip is schematic, quantum-well thickness is enlarged, and teal represents infrared. Frequency shifts below a coherent lasing field are only formal carrier-index diagnostics. There is no microscopic band-structure solver, current transport, temperature drift, multimode competition, stochastic phase noise, real detector transfer function or commercial diode prediction.

Why the quantum well provides gain

Representative transition: $h\nu=\frac{hc}{850\,\mathrm{nm}}\simeq1.459\,\mathrm{eV}$ . Band offsets and confined-state spacing are schematic. Injecting carriers changes occupation and modal gain; this diagram is not a microscopic band solver.

Cornell: semiconductor laser physics

Start with the essentials

Focus question: When does injected current become coherent output?
One-sentence intuition: Transparency, threshold, extraction and chirp are separate consequences of one coupled trajectory.

Core mathematical model

Separate transparency from threshold

\begin{aligned}x&=N/N_{\rm tr}\\y&=S/N_{\rm tr}\\G(x,y)&=\frac{a(x-1)}{1+\epsilon y}\end{aligned}

Transparency has zero modal gain; nominal threshold requires gain to equal photon loss.

The facets set photon storage

\begin{aligned}\kappa&=v_g\left[\alpha_i+\frac{\ln(1/(R_1R_2))}{2L}\right]\\\tau_p&=1/\kappa\end{aligned}

Power reflectivities and intensity attenuation enter a round trip through both facets.

Count carriers and photons together

\begin{aligned}\dot x&=j(t)-x/\tau_N-Gy\\\dot y&=(G-\kappa)y+\beta x/\tau_N\end{aligned}

A stimulated photon removes one carrier pair; stimulated absorption reverses that transfer.

Two electrical landmarks

\begin{aligned}I_{\rm tr}&=qN_{\rm tr}/\tau_N\\I_{\rm th}&=I_{\rm tr}(1+\kappa/a)\end{aligned}

The nominal threshold is a zero-field limit. Finite spontaneous emission rounds the measured knee.

Useful output is an escape channel

\begin{aligned}P_2&=h\nu N_{\rm tr}\kappa_2y\\\kappa_2&=v_g\ln(1/R_2)/(2L)\end{aligned}

The uniform-cavity model partitions logarithmic loss. Internal absorption contributes no useful output.

Carrier change also shifts frequency

\Delta\nu=\frac{\alpha_Ha}{4\pi}(x-x_{\rm DC})

The origin is the DC carrier state. This constant-alpha model uses uncompressed carrier gain; setting alpha to zero removes chirp without changing intensity.

Common difficulties

Gain is not yet oscillation

Typical misconceptionA pumped medium must lase.

Better mental modelThe selected mode must compensate all losses and maintain its population supply.

Read the model boundary

Typical misconceptionA teaching trace predicts a commercial device.

Better mental modelThe assumptions and chosen constants define a controlled mechanism experiment, not material certification.

Run the experiment

01
Find two current landmarks
Try absorbing bias and gain without lasing, then advance each startup.
What to observe: Positive gain alone still loses to cavity escape.
02
Acquire a current sweep
Acquire the sweep, change front reflectivity, and acquire again.
What to observe: Both threshold and useful slope change.
03
Drive too fast
Acquire the slow and fast modulation presets.
What to observe: The photon response depth and phase differ although the current depth is the same.
04
Remove amplitude–phase coupling
Set the Henry factor to zero, then reacquire modulation.
What to observe: Chirp disappears while the output record stays unchanged.