ISEGORIA / MATH ENCYCLOPEDIA
Laser physics: light that copies itself
A laser is an amplifier wrapped in a feedback loop. Stimulated emission copies a photon into the same mode, a pair of mirrors sends the copies back through the amplifier, and the result, above a sharp threshold, is light of extraordinary purity. The rate equations behind the threshold and the spikes on switch-on, the cavity that picks out which frequencies may lase, and mode locking, where a comb of frequencies held in phase turns into a train of femtosecond pulses.
Before you begin: Optics and quantum mechanics
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Threshold and the switch-on spikes
Two numbers describe a laser: the inversion N, how many more atoms sit in the upper level than the lower, and the number of photons φ in the cavity mode. The pump raises N, spontaneous decay lowers it with lifetime τ, and every photon in the mode stimulates more photons at a rate proportional to N. The cavity leaks photons with lifetime τ꜀. Gain balances loss at one inversion only, the threshold inversion, so above threshold the inversion is clamped there and all further pump power goes into light: the output rises in a straight line from a sharp kink. A small fraction β of spontaneous emission lands in the mode and seeds it; on a logarithmic scale the kink is a jump of about 1/β. Press Play to switch the laser on and watch it overshoot.
Worked example. A semiconductor laser has \(\tau\approx2\) ns and \(\tau_c\approx2\) ps, so \(\rho=\tau/\tau_c=1000\). Pumped at twice threshold, the inversion climbs as \(n=2(1-e^{-t/\tau})\) and crosses threshold after \(\tau\ln2=1.4\) ns; the photons then grow from the spontaneous seed and arrive as a spike several times the steady output. The ringing has \(\omega_R=\sqrt{1000}/\tau=1.6\times10^{10}\,\mathrm{s^{-1}}\), a frequency of 2.5 GHz, and dies away with time constant \(2\tau/r=2\) ns. This relaxation frequency sets how fast the laser can be modulated, which is why it matters for optical communication.
Watch out. These are the simplest four-level rate equations: one mode, a lower laser level that empties instantly, and no spatial structure. The photon number is in units of its saturation value, so \(p=1\) is the intensity at which stimulated emission depletes the inversion as fast as spontaneous decay does. Real semiconductor lasers add gain compression and carrier-dependent refractive index, which damp the oscillations faster; solid-state lasers have \(\rho\) of \(10^4\) to \(10^6\) and spike for much longer. The formula for \(\omega_R\) is a small-signal result, so the first, large spikes oscillate a little more slowly.
Why is the inversion clamped at its threshold value, however hard the laser is pumped?
In the steady state the photon number cannot change, so gain must equal loss exactly: \((n-1)p=0\) when \(\beta\) is negligible. With light in the cavity, \(p>0\), this forces \(n=1\), the inversion at which one round trip of gain exactly replaces one round trip of loss. If the inversion rose higher the photon number would grow without limit and drain it back; if it fell lower the light would die and the pump would refill it. The extra pump therefore has nowhere to go but into photons, which is why above threshold \(p=r-1\) is linear in the pump.
2. The cavity chooses the colours
Two mirrors a distance L apart support only standing waves that fit a whole number of half-wavelengths, so the cavity transmits a comb of narrow resonances spaced by the free spectral range c/2L. How narrow they are depends on the mirrors: the finesse, the ratio of spacing to width, grows like π/(1 − R) as the reflectivity R approaches 1. The gain medium supplies a much broader band; in a helium–neon tube the atoms move, and their Doppler shifts spread the gain over about 1.5 GHz. A mode lases where the comb meets gain above the loss line. Drag the mirror to change L and watch modes enter and leave, nudge it by a fraction of a wavelength to slide the comb, and lower R to see the threshold rise and the resonances broaden together.
Worked example. A 30 cm He–Ne cavity has \(\nu_F=c/2L=499.7\) MHz. With mirrors of \(R=0.98\) the finesse is \(\pi\sqrt{0.98}/0.02=156\), so each resonance is \(3.2\) MHz wide and a photon survives \(\tau_c=1/(2\pi\,\delta\nu)=50\) ns, about 25 round trips; the quality factor is \(\nu/\delta\nu=1.5\times10^8\). Threshold needs a single-pass gain of \(-\ln0.98=2.0\%\). With 4% at line centre the gain stays above threshold over \(\pm0.75\) GHz, a window 1.5 GHz wide that holds two or three modes, depending on where the comb sits (three in the starting figure). Shorten the cavity below about 10 cm and \(\nu_F\) exceeds that window: the laser runs on a single frequency.
Watch out. The lasing modes are marked wherever the unsaturated gain exceeds the loss, which is right for a Doppler-broadened gas, where each mode draws on its own velocity class of atoms. In a homogeneously broadened medium the modes share the same atoms, and the strongest one saturates the gain for all the others, so such lasers tend to run on fewer modes than this picture suggests. Mirror transmission is taken as the only loss and both mirrors are equal; the standing wave is drawn with 12 half-wavelengths, not the million or so of a real cavity, and the mode heights are not powers.
Why does moving a mirror by only 316 nm shift every resonance by a whole free spectral range?
A resonance needs \(2L=q\lambda\) for an integer \(q\). Near 632.8 nm a 30 cm cavity has \(q\approx948{,}000\). Lengthening the cavity by \(\lambda/2=316.4\) nm makes room for exactly one more half-wavelength, so the mode that sat at a given frequency now has index \(q+1\), and each resonance has slid down by one spacing \(c/2L\) to take the place of its neighbour. That is why single-frequency lasers need their cavity length held to a few nanometres, and why a laser left to drift hops from mode to mode as it warms up.
3. Mode locking: a comb becomes a pulse
A laser running on many modes emits a sum of sinusoids at frequencies spaced by exactly 1/T, where T is the round-trip time. If their phases are random the sum is a noisy pattern that repeats every round trip. If the phases are locked together, the fields add constructively at one instant per round trip and cancel almost everywhere else: N modes produce a pulse N times the mean intensity and about T/N long, circulating in the cavity and leaking out once per trip. The mean power is the same in both cases; locking only rearranges it in time. Slide the locking, change the number of modes, and watch the speckle condense into a pulse.
Worked example. With 15 equal modes locked, the peak is \(15\) times the mean and the pulse is \(0.886/15=0.059\) of a round trip long. A titanium-sapphire oscillator with an 80 MHz repetition rate has \(T=12.5\) ns; a 40 nm bandwidth at 800 nm is \(\Delta\nu=c\,\Delta\lambda/\lambda^2=19\) THz, about 230,000 modes. For a Gaussian spectrum the shortest pulse has \(\Delta t\,\Delta\nu=2\ln2/\pi=0.441\), here 24 fs: the same average power as the free-running laser, squeezed into bursts half a million times brighter.
Watch out. The figure imposes the phases by hand. In a real laser locking comes from a mechanism that favours pulses: a modulator driven at the round-trip frequency (active locking), or a saturable absorber or Kerr lens whose loss is lower at high intensity (passive locking), and the phases settle into step because a pulse then loses less than a continuous wave. Dispersion in the cavity makes the mode spacing slightly uneven, which chirps and lengthens the pulse unless it is compensated. The Gaussian comb here is cut off at about 2.4 standard deviations, so its pulse is a few per cent longer than the ideal 0.441 product.
Why does locking the phases raise the peak intensity without changing the average power?
Averaged over a round trip, the cross terms \(a_qa_{q^{\prime}}e^{i(\cdots)}\) between different modes oscillate at multiples of \(1/T\) and average to zero, so \(\langle I\rangle=\sum a_q^2\) whatever the phases. The phases decide only where in the round trip the energy sits. Locked, the \(N\) amplitudes add to \(N\) times one at \(t=0\), giving \(N^2\) in intensity, and energy conservation then demands that the pulse occupy only about \(1/N\) of the period. Random phases add like a random walk, with typical intensity \(N\) spread over the whole period.