Laser

Stimulated emission, population inversion, and operating threshold of a laser.

Conceptual context

A laser is not merely a bright lamp: it is a source that organizes light emission through an active medium, a cavity, and a threshold condition. The key process is stimulated emission, where an existing photon can induce an excited particle to emit another coherent photon. Therefore $R_est$ depends not only on stored energy but also on $\Delta_{N}$ and on enough cavity feedback. The central question is what happens to $P$ when $g$ exceeds $g_th$.

🟢 Essential level

A laser starts behaving as a laser when the active medium amplifies light more than the cavity loses it.

Below threshold, spontaneous emission and weak light can exist, but not a stable, coherent, reinforced beam. $g$ measures the ability of the medium to amplify radiation already circulating; $g_th$ marks the minimum needed to compensate $\alpha$. When $g$ is above that value, energy stored in $\Delta_{N}$ can become $P$. In the simple model, the growth is almost linear: the more net gain exceeds threshold, the more power leaves the beam. $\eta_{s}$ gives the slope of that rise and indicates how efficiently excess gain becomes optical output. Thus threshold separates luminous noise from organized oscillation. In short, the cavity is the amplifier's memory. It is like a microphone brought too close to a speaker: below a certain volume nothing special happens, but crossing the feedback threshold produces a self-sustaining squeal that organizes itself without needing any further external signal.

🔵 Formal level

Basic formulas

The model combines three layers: photon, active medium, and cavity. The individual energy of each photon is fixed by the laser-mode $f$ and by $h$.

Photon energy

The color or spectral region is connected to $\lambda$ through $c$, while $n$ matters inside the medium when the internal phase of the cavity is analyzed.

Wavelength-frequency relation

The microscopic condition for amplification is summarized by $\Delta_{N}$, the difference between $N2$ and $N1$.

Population inversion

Radiation already present, described by $\rho$, induces additional emission with a weight controlled by $B$ and by $R_est$.

Stimulated emission rate

Real student questions

Why is having many excited particles not enough? Because without useful $\Delta_{N}$, cavity feedback, and sufficient gain, emission can remain spontaneous and disordered.

Why is there a threshold? Because every cavity round trip loses energy through mirrors, absorption, and scattering; $g$ must compensate those $\alpha$.

Does $P$ grow without limit? No. The linear region is an above-threshold approximation; at high power, saturation, heating, and material limits appear.

Can changing $\lambda$ change power? It can indirectly, because it changes the active transition, cavity behavior, losses, and efficiency.

Final synthesis

A laser is a threshold machine: it turns inversion and feedback into coherent light. The central idea is to read $P$ as a response to net gain, not as mere brightness.