L19 · Carriers and gain media

Semiconductor Laser Diode

When does injected current become coherent output? When injected carriers supply enough optical gain to balance all cavity losses. Weak spontaneous light already appears below threshold; stimulated emission becomes strong above it. The useful output depends on how much light escapes through the front facet instead of being absorbed inside or leaving through the rear.

Drive current into a quantum-well diode and find where it becomes transparent and where it starts to lase. Record a power–current curve and the response to current modulation, and follow the frequency chirp that the changing carriers cause in the same run.

Record end time $\text{—}$

Instantaneous current $\text{—}$

Transparency current $\text{—}$

Nominal threshold current $\text{—}$

Final carriers / transparency $\text{—}$

Cavity photon count $\text{—}$

Photon lifetime from cavity $\text{—}$

Modal gain / total loss $\text{—}$

Front output: final / cycle mean $\text{—}$

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}

The knee in output versus pump current, the carrier balance and the modulation record each answer a different question. The nominal threshold is where gain equals loss with no light in the cavity.

The model is a single-mode, edge-emitting teaching diode. It tracks the total number of carrier pairs and of cavity photons, both divided by the carrier count at transparency. That count is two hundred million for a 300-micrometre cavity and grows in proportion to length when the active cross-section is fixed. Time is in nanoseconds, and the wavelength is fixed at a representative 850 nanometres. Gain rises linearly with carriers, the carrier lifetime is constant, and gain compression is an empirical term; these stand in for microscopic band filling and detailed recombination. The confinement factor changes the modal gain. The rear facet reflects 0.95 of the power; front reflectivity, cavity length and internal loss set the photon lifetime. Escape is treated as uniform along the cavity and split between the facets by their logarithmic losses, so the front power comes from this lumped model, not from a traveling-wave calculation along the chip. A steady, noise-free spontaneous source keeps running below threshold and rounds off the nominal threshold. Startup integrates carriers and photons together with fourth-order Runge–Kutta, using time steps limited by the gain, loss and modulation. The DC line solves the same balance equations separately; the 31 dots of a current sweep are separate startup runs without modulation. Modulation depth and phase come from eight cycles after settling, sampled at 1024 evenly spaced times. A constant Henry factor turns changes in the uncompressed carrier gain into a frequency shift, measured from the unmodulated DC carrier state; in the convention used here, more carriers give a positive shift. Chirp is not the same thing as linewidth. Changing a control replaces the startup or modulation record automatically and clears the optional current sweep. The output card is the final startup power for DC, or the mean of 1024 uniform samples over eight modulation cycles. The layered chip is schematic. Length changes linearly with the cavity-length control, but layer thicknesses and contacts are enlarged. Teal stands for infrared light. The separate leads end on opposite contacts; electrical transport through the layers is not solved. The facet coatings are partly transmitting. The two receivers are ideal whole-port collectors with no aperture loss, not images of a computed far field. Carrier markers show a capped fraction of the nominal threshold population, not individual carriers or a spatial density solution. Light guides use a fixed logarithmic mapping of front output; rear leakage has its own logarithmic-loss power. Their positions and speed are illustrative, with no optical phase or travel-time meaning. Below threshold there is no coherent laser field, so the frequency shift there is only a formal measure of the carrier-induced index change. The model has no microscopic band-structure calculation, current transport, temperature drift, multimode competition, random phase noise or real detector response, and it does not predict any commercial diode.

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

Physics tutorial

From current injection to light and chirp

BackgroundCurrent pushes electrons and holes into a quantum well. At transparency, stimulated absorption and emission balance. At the nominal threshold, small-signal modal gain balances the cavity loss. The same simulation gives the power–current curve, the response to modulation and the frequency chirp.

Why it mattersSpontaneous emission already produces weak light below threshold. Feedback lets stimulated emission dominate once the supplied carriers can compensate the cavity loss.

Start with the essentials

Focus question: When does injected current become coherent output?
One-sentence intuition: Carriers accumulate before the light turns on; once stimulated emission grows, it consumes the stored carriers and the two populations can ring toward a steady state.

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}

Both populations are divided by the transparency carrier count. The gain slope is a rate per normalized carrier population, including optical confinement; compression reduces the gain as photon population grows. Transparency means zero net stimulated gain, not zero spontaneous light.

The facets set the photon lifetime

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

One round trip includes the internal intensity loss and one reflection from each facet, counted with power reflectivities.

Rate equations for carriers and photons

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

The injection rate is current divided by charge and the transparency carrier count. Each stimulated photon consumes one electron–hole pair; absorption reverses that transfer. Adding the two equations cancels the stimulated terms. Only recombination outside the mode and photon escape or absorption remove the combined population.

Transparency and threshold currents

\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 the zero-light, small-signal limit. Spontaneous emission rounds the measured knee. Holding the active cross-section fixed, a longer chip needs more carriers: its lower mirror loss does not guarantee a lower total threshold current.

Useful output leaves through the front facet

\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 splits the escape between the facets according to their logarithmic losses. Light lost to internal absorption never reaches the output.

Carrier change also shifts frequency

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

The shift is measured from the DC carrier state. The Henry factor is constant and acts on the uncompressed carrier gain. Setting it to zero removes the chirp and leaves the intensity unchanged.

Common difficulties

Gain alone does not make a laser

Typical misconceptionOnce current flows through the diode, it has to lase.

Better mental modelThe mode lases only if its gain makes up for all the losses and the carriers feeding it keep being resupplied.

Know what the model leaves out

Typical misconceptionThis teaching curve predicts how a commercial diode behaves.

Better mental modelThe assumptions and chosen constants make this a controlled test of mechanisms. It does not certify how any real material or device performs.

Run the experiment

01
Find transparency and threshold
Choose Absorbing bias, then open Advanced controls and choose Gain without lasing. Each selection loads a completed startup automatically.
What to observe: Positive gain is not enough: photons still escape the cavity faster than the gain replaces them.
02
Record a current sweep
Open Advanced controls and press Acquire current sweep. Change Front power reflectivity and acquire again; compare the dots with the separately solved gray DC line.
What to observe: Both the threshold current and the slope of the useful output change.
03
Modulate too fast
Choose Slow direct modulation and Fast direct modulation. The records update automatically; open All numerical results to compare depth and phase.
What to observe: The current is modulated by the same depth, yet the light responds with a different depth and phase.
04
Turn off the amplitude–phase coupling
Choose Slow direct modulation, open Advanced controls and set Henry factor to zero. Compare the frozen power trace and chirp plot.
What to observe: The chirp vanishes, but the output power trace stays the same.