Section 02
The Idea, Step by Step
Start simple: light that marches in step
Shine a flashlight across a room and the bright circle spreads into a fuzzy blob. A laser pointer of the same brightness keeps a tiny, sharp dot on the far wall. The difference isn't how much light — it's that every wave in a laser beam is marching in perfect step: the same colour, the same direction, every crest lined up. An ordinary bulb throws out a jumble of colours and directions; a laser is light that has been drilled to move as one.
Build it up: one photon in, two identical photons out
Atoms hold their electrons on fixed energy "rungs." Add energy — a process called pumping — and you can lift an electron to a higher rung $E_2$. Normally it tumbles back on its own and spits out a photon in a random direction. But Einstein noticed something stranger in 1917: if a passing photon already carries exactly the right energy, $hf = E_2 - E_1$, it can tickle the excited atom into dropping early and emitting a perfect twin — same frequency, same phase, same direction. That is stimulated emission, the "se" in la‑se‑r. One photon becomes two, two become four: amplification.
The catch is that this very same photon could instead be absorbed by an atom still sitting on the lower rung. To win the race you need more atoms up top than down below — a population inversion, $N_2 > N_1$. That never happens on its own (in everyday matter $N_1 > N_2$), and creating it is exactly the pump's job.
Photon energy of red HeNe light
$$E=hf=\frac{hc}{\lambda}=\frac{1240\ \text{eV·nm}}{632.8\ \text{nm}}\approx1.96\ \text{eV}$$
That single number — the gap between two rungs — is what fixes the laser's colour.
Go deeper (AP / intro-college): gain must beat loss every round trip
Place the glowing medium between two mirrors (one of them slightly leaky) so the light bounces back and forth, picking up gain on every pass. The beam only takes off once the round-trip gain finally overtakes the round-trip losses — absorption, scattering, and the light that deliberately escapes through the output mirror:
Threshold gain condition
$$g_{th}=\alpha+\frac{1}{2L}\ln\frac{1}{R_1R_2}$$
Below threshold (pump $R<1\times R_{th}$) you just get a dim glow; above it, output power climbs almost linearly with pumping. The "purity" of the beam is set by its linewidth $\Delta\nu$ through the coherence length $l_c=c/\Delta\nu$ — how far the waves stay marching in step. In the sim, the $g$ and $\alpha$ sliders set gain and loss, $L$ sets the cavity length, the pump slider $R$ is measured in thresholds, and $\Delta\nu$ controls $l_c$.
Try this in the sim above
First, slide the pump rate $R$ below $1\times R_{th}$ and watch $N_2/N_1$ stay under 1 — no inversion, no lasing — then push it past 1 and see the readout flip to "LASING." Next, raise the loss $\alpha$ until it exceeds the gain $g$ and the output power collapses even under hard pumping: gain really must beat loss. Finally, narrow $\Delta\nu$ and watch the coherence length stretch out — the same trick that makes holograms and gravitational-wave detectors possible.
Section 03
Equations & Derivation
Stimulated Emission Rate
$$W_{21}=B_{21}\rho(\nu),\quad B_{21}=\frac{c^3}{8\pi h\nu^3}A_{21}\quad\text{(Einstein B coefficient)}$$
Population Inversion & Threshold Gain
$$\frac{dN_2}{dt}=R_p-\frac{N_2}{\tau}-W_{21}N_2,\quad N_2>N_1\text{ required},\quad g_{th}=\alpha+\frac{1}{2L}\ln\frac{1}{R_1R_2}$$
Laser Output Power
$$P_{\rm out}=\eta_{\rm sl}(R_p-R_{p,{\rm th}})\hbar\omega,\quad\eta_{\rm sl}=\frac{T}{2\alpha L+T}$$
Coherence Length
$$l_c=\frac{c}{\Delta\nu},\quad\text{(multimode HeNe: }l_c\approx20\text{–}30\;\text{cm, single-mode: }\sim\!300\;\text{m, thermal source: }\mu\text{m)}$$
Three-Level vs Four-Level Lasers
| Symbol | Quantity | Unit/Value |
|---|
| $\text{3-level (Ruby)}$ | Ground state = lower laser level; harder to achieve inversion | Requires >50% pumping |
| $\text{4-level (HeNe, Nd:YAG)}$ | Lower laser level is not ground state; quickly depopulated | Easier to achieve inversion |
1
Stimulated emission. Einstein (1917): an excited atom can be triggered to emit a photon identical (same frequency, phase, direction, polarisation) to an incident photon. This is LASER: Light Amplification by Stimulated Emission of Radiation. Key: stimulated emission probability = absorption probability, so population inversion ($N_2>N_1$) is needed for net gain.
2
Population inversion. Under thermal equilibrium, $N_1>N_2$ (Boltzmann). To lase, $N_2>N_1$ — a non-equilibrium state created by pumping (optical, electrical, chemical). Requires a metastable upper level (long lifetime) so atoms accumulate. Three-level and four-level schemes — four-level is easier.
3
Laser cavity feedback. Two mirrors (one partially transmitting) provide feedback. Each round trip, the beam is amplified by $e^{2gL}$ and attenuated by mirror reflectivities $R_1R_2$ and internal losses $e^{-2\alpha L}$. Threshold: $R_1R_2 e^{2(g-\alpha)L}=1$. Above threshold: exponential growth limited by gain saturation.
4
Laser coherence. Temporal coherence: $l_c=c/\Delta\nu$. HeNe single-mode: $\Delta\nu\sim1$ MHz, $l_c\sim300$ m. Spatial coherence: diffraction-limited beam divergence $\theta\sim\lambda/D$. These properties enable interferometry, holography, precision cutting, and communication over optical fibres.
Ref: Yariv — Quantum Electronics (3rd Ed.), Chs. 5–6; Saleh & Teich — Fundamentals of Photonics (3rd Ed.), Ch. 15; Siegman — Lasers.
Section 05
Common Misconceptions
❌ Laser light contains only one frequency (colour).
✅ A laser has a narrow linewidth $\Delta\nu$ — much narrower than a lamp, but not zero. Even single-mode lasers have Schawlow-Townes quantum noise limit: $\Delta\nu_{\rm min}=2\pi h\nu(\Delta\nu_c)^2/P_{\rm out}$ (typically kHz–MHz). Mode-locked lasers intentionally operate with broad bandwidth ($\Delta\nu\sim100$ THz) to generate femtosecond pulses — time-bandwidth product $\Delta\nu\Delta t\geq0.44$.
📖 HRW 10th Ed., §38-4; Yariv — Quantum Electronics, Ch. 11.
❌ Lasers must be pumped with another laser.
✅ Lasers can be pumped many ways: optical (flashlamp or laser), electrical discharge (HeNe, CO₂, argon-ion), semiconductor (diode laser — current injection), chemical reaction (HF chemical laser), or even nuclear pumping. Diode-pumped solid-state (DPSS) lasers use cheaper, efficient diode lasers to pump Nd:YAG, greatly reducing size and cost.
📖 Saleh & Teich — Fundamentals of Photonics, Ch. 15.
❌ All laser beams are perfectly parallel (zero divergence).
✅ Diffraction limits beam divergence to $\theta\sim\lambda/D$ (far field, Gaussian beam: $\theta=\lambda/(\pi w_0)$). A 1 mm diameter 633 nm beam has $\theta\sim0.2$ mrad. After 1 km it's ~20 cm wide. A moonbounce laser returns ~150 m wide after 400,000 km round trip. Higher beam quality (M²≈1) gives minimum divergence.
📖 Yariv — Quantum Electronics, Ch. 6; Siegman — Lasers, Ch. 17.
Misconception research: Driver et al. — Making Sense of Secondary Science.