Four-Level Laser Systems

A four-level laser is what you get when you take the laser rate equations and the Boltzmann distribution seriously at the same time. The rate equations say that gain lives in the difference $\Delta N=N_2-(g_2/g_1)N_1$ between the upper and lower laser-level populations; Boltzmann says a level sitting an energy $E_1-E_0$ above the ground state holds only a fraction $e^{-(E_1-E_0)/k_\mathrm{B}T}$ of the atoms in equilibrium. Put those together and a design strategy falls out: build the laser transition so its lower level lies several $k_\mathrm{B}T$ above the ground state and empties faster than the upper level decays. Then $N_1$ is essentially zero at all times, the inversion collapses to $\Delta N\approx N_2$ , and every atom the pump lifts into the upper level is already an inversion. Four levels do the work. Atoms are pumped from the ground state $\lvert 0\rangle$ into a broad, short-lived band $\lvert 3\rangle$ , relax nonradiatively within nanoseconds into a long-lived metastable upper laser level $\lvert 2\rangle$ , lase on $\lvert 2\rangle\!\to\!\lvert 1\rangle$ , and drop nonradiatively from $\lvert 1\rangle$ back to $\lvert 0\rangle$ . Two fast relaxations bracket one slow one, and that asymmetry is the whole trick: the fast steps collect population into $\lvert 2\rangle$ and keep $\lvert 1\rangle$ drained. The contrast with the three-level scheme is why this topology dominates. There the lower level is the ground state, so more than half of all active atoms must be pumped up before the medium shows any gain; here the reference population is nearly zero, the ideal threshold vanishes, and a real threshold appears only through cavity loss. This article assembles the steady-state inversion $\Delta N=R\tau_2$ , the loss-limited threshold, and the quantum-defect efficiency ceiling that every diode-pumped solid-state laser lives by.

Key takeaways

A four-level laser stacks a ground state $\lvert 0\rangle$ , a lower laser level $\lvert 1\rangle$ above it, a metastable upper laser level $\lvert 2\rangle$ , and a broad pump band $\lvert 3\rangle$ ; the laser runs $\lvert 2\rangle\!\to\!\lvert 1\rangle$ .
Because $\lvert 1\rangle$ sits several $k_\mathrm{B}T$ above ground and empties fast, it is always essentially empty, so any atom pumped into $\lvert 2\rangle$ creates inversion and the ideal threshold pump rate is zero.
The defining requirement is fast nonradiative relaxation $\lvert 3\rangle\!\to\!\lvert 2\rangle$ and $\lvert 1\rangle\!\to\!\lvert 0\rangle$ , with $\tau_1\ll\tau_{21}$ ; if $\lvert 1\rangle$ empties slowly the transition self-terminates.
Steady state gives $\Delta N\approx R\,\tau_2$ , free of the $N_\mathrm{tot}/2$ penalty that burdens three-level systems; the real threshold is set by cavity loss, $R_\mathrm{th}=\gamma_\mathrm{th}/(\sigma_{21}\tau_2)$ .
Nd:YAG at $1064\,\mathrm{nm}$ ( $\tau_2\approx230\,\mu\mathrm{s}$ ) is the archetype; the quantum defect $\eta_q=\lambda_\mathrm{pump}/\lambda_\mathrm{laser}=808/1064\approx0.76$ caps efficiency and sets the heat load.
Key idea: put the lower laser level above the ground state and drain it fast, so it stays empty; then the inversion equals the pumped upper-level population $\Delta N\approx R\tau_2$ , the ideal threshold is zero, and a real threshold comes only from cavity loss.

The core insight

Put the lower laser level above the ground state and drain it fast, so it is always empty; then the inversion equals the pumped upper-level population $\Delta N\approx R\tau_2$ , the ideal threshold is zero, and a real threshold comes only from cavity loss.

Where this lives today

Almost every solid-state laser you meet is a four-level (or managed quasi-four-level) device, and the equations above are the ones its designers use. A green laser pointer is a diode-pumped Nd:YVO₄ crystal — a four-level cousin of Nd:YAG — lasing at $1064\,\mathrm{nm}$ and frequency-doubled to $532\,\mathrm{nm}$ ; its low threshold is why tens of milliwatts of diode pump suffice. Scale the physics up and it becomes the ytterbium-doped fibre laser of industrial cutting, where a long doped fibre gives enormous single-pass gain and the broad Yb pump band swallows the output of many diodes. At the extreme, the National Ignition Facility fires 192 Nd:glass amplifier beamlines — storing energy over the $\approx230\,\mu\mathrm{s}$ lifetime and releasing it in a nanosecond — for megajoule fusion pulses. In every case the design conversation is the same numbers: $\sigma_{21}$ and $\tau_2$ set gain and storage, cavity loss sets $R_\mathrm{th}$ , the quantum defect sets the heat load. The framework feeds forward into the threshold condition, the cavity Q factor and photon lifetime, and the resonator-mode structure.

The physical picture

Picture four buckets on a hillside connected by pipes of different widths. Pumping lifts water rapidly into the top bucket $\lvert 3\rangle$ , which has a wide hole and dumps into the metastable bucket $\lvert 2\rangle$ . Water pools in $\lvert 2\rangle$ for a long time because its outlet is a narrow trickle — the laser transition. That trickle feeds bucket $\lvert 1\rangle$ , which has an enormous hole and empties back to $\lvert 0\rangle$ almost the instant water arrives. Because $\lvert 1\rangle$ is always essentially empty, the water standing in $\lvert 2\rangle$ is the inversion — no reservoir below it to subtract. The thermodynamic advantage is that the lower laser level carries no thermal population to overcome. With $E_1-E_0\gg k_\mathrm{B}T$ the Boltzmann factor $e^{-(E_1-E_0)/k_\mathrm{B}T}$ is minuscule — for the Nd:YAG lower level at $\approx2000\,\mathrm{cm}^{-1}$ above ground, with $k_\mathrm{B}T\approx208\,\mathrm{cm}^{-1}$ at room temperature, it is about $7\times10^{-5}$ . The pump does not fight thermal equilibrium; it merely tops up a level equilibrium already leaves empty — exactly what a three-level laser cannot do, since there the lower level and ground state are one and the same.

Figure

The four-level scheme. Atoms pumped at 808 nm into the broad band ⟨3⟩ relax in nanoseconds to the metastable upper level ⟨2⟩ with τ₂ ≈ 230 μs, lase at 1064 nm to ⟨1⟩ which sits ≈2000 cm⁻¹ above ground, and ⟨1⟩ empties in nanoseconds so it stays essentially empty

Efficiency comes from that timing hierarchy. The pump band $\lvert 3\rangle$ can be broad and messy — accepting incoherent flashlamp or diode light across many nanometres — because atoms leave it so fast they forget how they arrived. The upper level $\lvert 2\rangle$ must be long-lived so population accumulates and stores energy, enabling efficient continuous-wave and Q-switched operation; the lower level $\lvert 1\rangle$ must be short-lived so it never chokes the transition. The residual cost is the quantum defect: the emitted photon $E_2-E_1$ carries less energy than the absorbed pump photon $E_3-E_0$ , and that deficit is dumped as heat — unavoidable, and it sets the slope-efficiency ceiling. The second figure shows the payoff: the four-level threshold hugs the origin while the three-level threshold sits far to the right.

Figure

Output power versus pump rate. The four-level laser reaches threshold at R_th ≈ 7×10²⁵ m⁻³ s⁻¹ set only by cavity loss, while an otherwise identical three-level medium must first pump N₂ past N_tot/2, pushing its threshold roughly 4000× higher before any light appears

The core equations

Gain requires a positive inversion — the upper-level population minus the degeneracy-weighted lower-level population. Everything specific to the four-level scheme is about making its second term negligible.

\Delta N = N_2 - \frac{g_2}{g_1}\,N_1 \qquad (1)

where: $N_2,N_1$ — population densities of the upper and lower laser levels (m $^{-3}$ ); $g_2,g_1$ — their degeneracies; $\Delta N$ — inversion density, with net gain requiring $\Delta N>0$ . The lower level is a debt that must be paid before the upper level shows gain; the four-level scheme keeps that debt at zero. The defining approximation is that the lower laser level is unpopulated. Because $\lvert 1\rangle$ lies well above the ground state and decays fast, $N_1\approx0$ , and the inversion collapses to just the upper-level population.

N_1 \approx 0 \;\Rightarrow\; \Delta N \approx N_2 \qquad (2)

where: $N_1\approx0$ — the lower level is essentially empty because it lies several $k_\mathrm{B}T$ above ground and decays much faster than it fills. This single line is the entire advantage of the topology. The upper level is filled at the pump rate $R$ , drained by its finite lifetime, and depleted by stimulated emission. Below threshold the last term drops out.

\frac{\mathrm{d}N_2}{\mathrm{d}t} = R - \frac{N_2}{\tau_2} - N_2\,W_{21} \qquad (3)

where: $R$ — pump rate into $\lvert 2\rangle$ via fast decay from $\lvert 3\rangle$ (m $^{-3}\,$ s $^{-1}$ ); $\tau_2$ — total upper-level lifetime (spontaneous plus nonradiative); $W_{21}$ — stimulated-emission rate per atom, proportional to photon density. Setting $\mathrm{d}N_2/\mathrm{d}t=0$ with $W_{21}\approx0$ gives the pumped inversion. Solving that steady state is a one-line balance of pumping against decay, and it delivers the central result: the inversion is the pump rate times the upper-level lifetime, no penalty subtracted.

\Delta N_\mathrm{ss} = N_2 = R\,\tau_2 \qquad (4)

where: $\Delta N_\mathrm{ss}$ — steady-state small-signal inversion set purely by pumping. A long upper-level lifetime is an energy-storage lever: it converts a modest pump rate into a large inversion, which is why metastability is prized. The inversion times the stimulated-emission cross section gives the small-signal gain coefficient — the fractional growth of intensity per unit length the medium offers a passing wave.

\gamma(\nu) = \sigma_{21}(\nu)\,\Delta N \approx \sigma_{21}(\nu)\,R\,\tau_2 \qquad (5)

where: $\gamma$ — small-signal gain coefficient (m $^{-1}$ ); $\sigma_{21}(\nu)$ — stimulated-emission cross section on the laser transition. Gain scales linearly with pump rate — the straight output-versus-pump line above. Oscillation begins when round-trip gain balances round-trip loss. Requiring the intensity to reproduce itself after one round trip fixes the threshold gain in terms of internal loss and mirror reflectivity.

\gamma_\mathrm{th} = \alpha_\mathrm{loss} - \frac{1}{2\ell}\,\ln(R_1 R_2) \qquad (6)

where: $\alpha_\mathrm{loss}$ — distributed internal loss per unit length; $\ell$ — gain-medium length; $R_1,R_2$ — mirror power reflectivities. Since $\ln(R_1R_2)<0$ the mirror term adds loss: output coupling is a controlled leak the gain must replace. Dividing the threshold gain by $\sigma_{21}\tau_2$ gives the pump rate the source must supply — the real threshold, finite only because the cavity leaks.

R_\mathrm{th} = \frac{\gamma_\mathrm{th}}{\sigma_{21}\,\tau_2} \qquad (7)

where: $R_\mathrm{th}$ — pump rate needed to reach threshold. A better cavity or a larger $\sigma_{21}\tau_2$ product lowers it; in the lossless limit it goes to zero. Finally, the quantum defect: the ratio of laser to pump photon energy caps the fraction of pump energy that can leave as laser light.

\eta_q = \frac{E_2-E_1}{E_3-E_0} = \frac{h\nu_\mathrm{laser}}{h\nu_\mathrm{pump}} = \frac{\lambda_\mathrm{pump}}{\lambda_\mathrm{laser}} \qquad (8)

where: $\eta_q$ — quantum-defect efficiency ceiling; $\nu,\lambda$ — laser and pump frequencies and wavelengths. The complement $1-\eta_q$ is the minimum fraction of pump power that becomes waste heat.

Derivation AWhy the four-level threshold is ideally zero

The goal is to show, side by side, why a three-level laser must pump more than half of all its atoms before showing any gain, while a four-level laser shows gain the instant the upper level holds one excess atom. We start from the same general inversion, impose the constraint that distinguishes the schemes — where the lower laser level sits — and read off each threshold, so that the disappearance of the $N_\mathrm{tot}/2$ penalty becomes a one-line consequence of moving the lower level off the ground state.

Step 1. Write the general inversion $\Delta N=N_2-N_1$ (taking $g_1=g_2$ ) and note that in a three-level laser the lower laser level is the ground state, so it and the upper level share all the atoms: $N_0+N_2=N_\mathrm{tot}$ with $N_1=N_0$ . Why: the three-level lower level starts full — it holds the entire population until the pump forcibly empties it.

Step 2. Impose $\Delta N>0$ : $N_2-N_0>0$ with $N_0=N_\mathrm{tot}-N_2$ gives $N_2>N_\mathrm{tot}/2$ . Why: more than half of every active atom must be lifted out of the ground state just to break even — a fixed cost set by the level scheme, not the cavity.

Step 3. Now the four-level constraint: $\lvert 1\rangle$ lies above the ground state and decays fast, so $N_1\approx0$ and $\Delta N\approx N_2$ . Why: the reference population is no longer $N_\mathrm{tot}/2$ but zero — an empty drain instead of a full reservoir.

Step 4. Impose $\Delta N>0$ : it requires only $N_2>0$ . Any pumped population produces gain, so the ideal threshold is zero; a real threshold enters only through the cavity, as $R_\mathrm{th}=\gamma_\mathrm{th}/(\sigma_{21}\tau_2)$ . Why: the four-level laser turns on as soon as gain overcomes loss, not when half the population is inverted — which is why low-power continuous-wave operation is routine.

Go deeper · Derivation B · 1 minSteady-state inversion with a finite lower-level lifetime

Derivation BSteady-state inversion with a finite lower-level lifetime

The idealisation $N_1\approx0$ is never perfect, so the goal here is to keep the lower-level population in the bookkeeping and find how large a lifetime $\tau_1$ the transition can tolerate before it self-terminates. We solve the coupled steady-state rate equations below threshold, express the residual $N_1$ that piles up, and read off the correction factor multiplying the ideal inversion — a factor that shows at a glance whether a material is a true four-level system or a bottlenecked one.

Step 1. Upper-level steady state below threshold ( $W_{21}\approx0$ ): $0=R-N_2/\tau_2$ , so $N_2=R\tau_2$ , where $\tau_2$ includes the decay channel $\tau_{21}$ that feeds $\lvert 1\rangle$ . Why: the upper level fills to the same $R\tau_2$ as in the ideal case; the correction lives entirely in the lower level.

Step 2. The lower level fills from above at rate $N_2/\tau_{21}$ and empties with lifetime $\tau_1$ , so $0=N_2/\tau_{21}-N_1/\tau_1$ . Why: the standing level in $\lvert 1\rangle$ is set by how fast water arrives versus how fast the wide hole lets it out.

Step 3. Solve: $N_1=N_2\,\tau_1/\tau_{21}$ . Why: the lower level holds a fraction $\tau_1/\tau_{21}$ of the upper population — small only when $\tau_1\ll\tau_{21}$ .

Step 4. Form the inversion ( $g_1=g_2$ ): $\Delta N=N_2-N_1=R\tau_2\bigl(1-\tau_1/\tau_{21}\bigr)$ . Why: the bracket measures departure from the ideal limit — essentially 1 when $\tau_1\ll\tau_{21}$ , but collapsing to zero as $\tau_1\to\tau_{21}$ , where the transition self-terminates. *The result spans both limits cleanly: as $\tau_1\to0$ it reduces to the ideal $\Delta N=R\tau_2$ , and as $\tau_1\to\tau_{21}$ the inversion collapses to zero — the self-termination bottleneck.*

Go deeper · Derivation C · 1 minThe slope-efficiency ceiling from the quantum defect

Derivation CThe slope-efficiency ceiling from the quantum defect

The four-level scheme buys its low threshold at a fixed energetic price, and the goal here is to pin that price down as a hard ceiling on optical-to-optical efficiency. We follow the energy of a single atom around the cycle — one pump photon absorbed, one laser photon emitted — form the ratio, and identify the missing energy as obligatory heat in the fast nonradiative steps, giving the maximum slope efficiency any real device can approach.

Step 1. Each pumped atom absorbs one pump photon, $E_\mathrm{abs}=h\nu_\mathrm{pump}=hc/\lambda_\mathrm{pump}$ (times a pump quantum efficiency $\eta_p$ for reaching $\lvert 2\rangle$ ). Why: this is the energy input per useful excitation, the denominator of the efficiency.

Step 2. The atom emits one lower-energy laser photon, $E_\mathrm{emit}=h\nu_\mathrm{laser}=hc/\lambda_\mathrm{laser}$ . Why: the laser gap $\lvert 2\rangle\!\to\!\lvert 1\rangle$ is smaller than the pump gap, so the output photon is red-shifted.

Step 3. The ratio is $\eta_q=\lambda_\mathrm{pump}/\lambda_\mathrm{laser}$ . For Nd:YAG at $808\to1064\,\mathrm{nm}$ , $\eta_q=808/1064=0.759$ : at best $76\%$ of each pump photon leaves as light, and $24\%$ is heat. Why: the fast $\lvert 3\rangle\!\to\!\lvert 2\rangle$ and $\lvert 1\rangle\!\to\!\lvert 0\rangle$ relaxations dissipate the difference into the lattice.

Step 4. The optical-to-optical ceiling is $\eta_\mathrm{max}=\eta_p\,\eta_q$ ; multiplying by output-coupling and beam-overlap efficiencies gives the realised slope efficiency. Why: no cavity trick beats $\eta_q$ — it is fixed by the level energies alone, which is why "small quantum defect" is a headline figure of merit. *For Nd:YAG, $808/1064=0.759$ , confirming that $\approx24\%$ of each pump photon is deposited as heat.*

Limiting cases

How the idealisation holds or breaks, depending on the lower-level lifetime and its energy above ground relative to $k_\mathrm{B}T$ :

Case	Result	Meaning
Ideal four-level ( $\tau_1\to0$ )	$N_1\to0,\ \Delta N=N_2\approx R\,\tau_2$	Lower level empties instantly; inversion equals the pumped upper population; ideal threshold pump rate is zero.
Finite lower-level lifetime	$N_1=N_2\,\dfrac{\tau_1}{\tau_{21}},\ \Delta N=R\tau_2\bigl(1-\tfrac{\tau_1}{\tau_{21}}\bigr)$	A small $N_1$ accumulates; inversion is reduced by $(1-\tau_1/\tau_{21})$ . Bottleneck and self-termination as $\tau_1\to\tau_{21}$ .
Thermal repopulation (quasi-four-level)	$N_1^\mathrm{th}\propto e^{-(E_1-E_0)/k_\mathrm{B}T}$	If $E_1-E_0\sim k_\mathrm{B}T$ the lower level is thermally populated (e.g. Yb:YAG); reabsorption loss appears and threshold rises — intermediate between four- and three-level.
Degenerate limit ( $E_1\to E_0$ )	Reduces to three-level: needs $N_2>(g_2/g_1)N_1^\mathrm{ground}$	When the lower laser level merges with the ground state you recover the three-level system and its large $N_\mathrm{tot}/2$ threshold.

Common trapThe lower level empties on its own, so any four-level medium is automatically low-threshold

Not automatically — the magic requires $\tau_1\ll\tau_{21}$ , that the lower level empties much faster than the upper level feeds it. If $\tau_1$ is not short, population piles up as $N_1=N_2\,\tau_1/\tau_{21}$ , the factor $(1-\tau_1/\tau_{21})$ shrinks, and the transition self-terminates — a transient pulse followed by darkness even under strong pumping. Two related traps: do not conflate the pump band $\lvert 3\rangle$ with the upper laser level $\lvert 2\rangle$ (pump one, lase from the other); and "above the ground state" must mean by several $k_\mathrm{B}T$ , or thermal repopulation turns a four-level line into a lossy quasi-four-level one, as in Yb:YAG.

Worked examples

1 · Threshold pump rate of a real Nd:YAG rod (confidence builder)

A 10 cm Nd:YAG rod ( $\sigma_{21}=2.8\times10^{-23}\,\mathrm{m^2}$ , $\tau_2=230\,\mu\mathrm{s}$ ) sits in a cavity with $R_1=1.00$ , output coupler $R_2=0.95$ , and internal loss $\alpha_\mathrm{loss}=0.20\,\mathrm{m^{-1}}$ . Find the threshold gain, inversion, and pump rate, and compare with the three-level penalty.

Step 1. Threshold gain from equation (6). The mirror term is $-\frac{1}{2\ell}\ln(R_1R_2)=-5\times\ln(0.95)=0.2565\,\mathrm{m^{-1}}$ ; adding the internal loss, $\gamma_\mathrm{th}=0.20+0.2565=\mathbf{0.457\ m^{-1}}$ . The output coupler is the larger loss — most of the required gain just replaces the light deliberately let out.

Step 2. Threshold inversion from equation (5): $\Delta N_\mathrm{th}=\gamma_\mathrm{th}/\sigma_{21}=0.457/(2.8\times10^{-23})=\mathbf{1.6\times10^{22}\ m^{-3}}$ — a tiny fraction of the $\approx1.4\times10^{26}\,\mathrm{m^{-3}}$ Nd ions in a 1 at.% crystal.

Step 3. Threshold pump rate from equation (7): $R_\mathrm{th}=\Delta N_\mathrm{th}/\tau_2=1.6\times10^{22}/(2.3\times10^{-4})=\mathbf{7.1\times10^{25}\ m^{-3}\,s^{-1}}$ , finite only because the cavity leaks.

Step 4. The three-level contrast. A three-level laser in the same crystal would first need $N_2>N_\mathrm{tot}/2\approx6.9\times10^{25}\,\mathrm{m^{-3}}$ before any gain appears — about $4200\times$ the four-level threshold inversion. The four-level scheme lowers the required inversion by more than three orders of magnitude: $\Delta N_\mathrm{th}\approx1.6\times10^{22}\,\mathrm{m^{-3}}$ versus $\approx6.9\times10^{25}\,\mathrm{m^{-3}}$ .

2 · True four-level versus quasi-four-level: the Boltzmann test

At room temperature ( $k_\mathrm{B}T\approx208\,\mathrm{cm^{-1}}$ ), compare the thermal population of the lower laser level for Nd:YAG ( $\approx2000\,\mathrm{cm^{-1}}$ above ground) and Yb:YAG ( $\approx610\,\mathrm{cm^{-1}}$ ). Which is a genuine four-level system?

Step 1. Nd:YAG Boltzmann factor: $e^{-(E_1-E_0)/k_\mathrm{B}T}=e^{-2000/208}=e^{-9.6}=\mathbf{6.8\times10^{-5}}$ . The lower level holds essentially no thermal population — the $N_1\approx0$ approximation is excellent, and Nd:YAG is a textbook four-level laser.

Step 2. Yb:YAG Boltzmann factor: $e^{-610/208}=e^{-2.93}=\mathbf{0.054}$ , about $5\%$ — enough that the lower level reabsorbs the $1030\,\mathrm{nm}$ laser light.

Step 3. Yb:YAG is therefore a quasi-four-level system: the reabsorption is an intensity-dependent loss with a nonzero effective threshold that rises with temperature, and cooling (or hard pumping) restores near-four-level behaviour. The Boltzmann factor is the decisive test: $\sim10^{-5}$ for Nd:YAG means true four-level, $\sim5\times10^{-2}$ for Yb:YAG means quasi-four-level.

Physics of the portrait

The four-level laser was not deduced and then built — it was built, in a rush, the same year as the first laser, by physicists racing to find any medium that would oscillate. The man who found the winning topology went on to invent the tunable dye laser that four-level physics makes possible.

Origins · 1 minPeter Sorokin and how this was discovered

Peter Sorokin

1931–2015 · solid-state and dye lasers, IBM Research

Peter Sorokin belongs to the small group who turned the laser from a single demonstration into a family of devices. In November 1960, months after Theodore Maiman fired the first (three-level) ruby laser, Sorokin and Mirek Stevenson at IBM operated the second laser ever built — trivalent uranium in calcium fluoride — and almost at once a samarium-doped crystal. These were the first four-level lasers, and they mattered out of proportion to their modest output: by placing the lower laser level above the ground state they showed the punishing three-level threshold was not a law of nature but a choice of level scheme, and that a laser could turn on with vanishingly little pumping. The four-level idea became the template for the low-threshold, continuous-wave solid-state lasers that followed — Nd-doped crystals above all. Sorokin's second great contribution came in 1966, when he and John Lankard demonstrated the first organic-dye laser: a broad vibronic manifold behaving as an effectively four-level system, whose empty lower sublevel gives the tunable, low-threshold gain that opened up tunable and, later, ultrafast lasers. Drain the lower level, and the physics becomes forgiving — the through-line of his work.

History

1917

— Einstein derives the A and B coefficients, predicting stimulated emission

1958

— Schawlow and Townes set out the principle of the optical maser

1960

— Sorokin and Stevenson operate the first four-level laser, trivalent uranium in CaF₂ at IBM

1964

— Geusic, Marcos and Van Uitert demonstrate the Nd:YAG laser at 1064 nm

1966

— Sorokin and Lankard build the first organic-dye laser, a tunable effectively four-level system

Going further

Texts. Svelto, Principles of Lasers, 5th ed., ch. 1 and 7 (three- versus four-level, rate equations) · Siegman, Lasers (1986), ch. 6 (pumping and population inversion) · Saleh and Teich, Fundamentals of Photonics, 3rd ed., ch. 15 (laser amplifiers and four-level pumping) · Koechner, Solid-State Laser Engineering, ch. 1 and 2 (Nd:YAG parameters) · Foot, Atomic Physics (2005), ch. 7 (laser action, rate-equation picture) · Hecht, Optics, 5th ed., ch. 13 (laser principles). The key results in this article are: $N_2=R\tau_2$ and $N_1=N_2\,\tau_1/\tau_{21}$ ; $\gamma_\mathrm{th}=\alpha_\mathrm{loss}-\tfrac{1}{2\ell}\ln(R_1R_2)=0.457\,\mathrm{m^{-1}}$ and $R_\mathrm{th}\approx7.1\times10^{25}\,\mathrm{m^{-3}\,s^{-1}}$ for the quoted rod; quantum defect $808/1064=0.759$ ; Boltzmann factors $e^{-2000/208}\approx6.8\times10^{-5}$ (Nd:YAG) and $e^{-610/208}\approx0.054$ (Yb:YAG). Nd:YAG parameters used throughout: $\tau_2\approx230\,\mu\mathrm{s}$ , $\sigma_{21}\approx2.8\times10^{-19}\,\mathrm{cm^2}$ , pump 808 nm, laser 1064 nm, lower level $\approx2000\,\mathrm{cm^{-1}}$ above ground.

Prerequisites. Population Inversion and Optical Gain · Stimulated Emission and the Gain Coefficient · Laser Rate Equations · Three-Level Laser Systems. Leads to. Laser Threshold Condition · Cavity Q Factor and Photon Lifetime · Optical Resonators and Cavity Modes.

Ask the tutor: Ask me to run the numbers for any gain medium you like — give me its pump and laser wavelengths, upper-level lifetime, and lower-level energy, and we will work out its threshold, quantum defect, and whether it is truly four-level.

Why is the four-level threshold called "zero" when real lasers clearly have a threshold?

"Zero" refers to the ideal threshold set by the level scheme alone. In a four-level system the lower laser level is empty, so you need only $N_2>0$ to have gain — there is no population penalty to pay before the medium becomes an amplifier. Every real laser still has a finite threshold, but it comes entirely from the cavity: the round-trip gain must overcome mirror transmission, scattering, and absorption. So the real threshold pump rate is $R_\mathrm{th}=\gamma_\mathrm{th}/(\sigma_{21}\tau_2)$ , which shrinks as you improve the mirrors and lower the internal loss, and would vanish in a perfect lossless cavity.

Why does the upper level need to be long-lived but the lower level short-lived?

They play opposite roles. The upper level $\lvert 2\rangle$ is where population accumulates and energy is stored, so it must be metastable — a long $\tau_2$ means a modest pump rate builds a large inversion ( $\Delta N=R\tau_2$ ) and lets the medium store energy for Q-switching. The lower level $\lvert 1\rangle$ is a drain that must never fill, so it must be short-lived: it has to dump its atoms back to the ground state faster than the laser transition feeds it, keeping $N_1\approx0$ . If the lower level lingers, population piles up, the factor $(1-\tau_1/\tau_{21})$ shrinks toward zero, and the transition self-terminates. Storage above, fast drainage below.

What is the difference between the pump band and the upper laser level?

They are distinct levels with distinct jobs, and confusing them is a common slip. The pump band $\lvert 3\rangle$ is what the pump light excites; it is deliberately broad so it can absorb messy, incoherent flashlamp or diode light across many nanometres, and short-lived so atoms leave it fast. The upper laser level $\lvert 2\rangle$ is where those atoms land after a fast nonradiative drop from $\lvert 3\rangle$ ; it is narrow and metastable, and the laser transition starts from it. So pumping targets $\lvert 3\rangle$ but lasing originates from $\lvert 2\rangle$ . The fast $\lvert 3\rangle\!\to\!\lvert 2\rangle$ relaxation is what lets a broad, cheap pump feed a clean, narrow laser line.

Where does the quantum-defect energy actually go?

Into heat in the crystal lattice. A pumped atom absorbs a high-energy pump photon ( $E_3-E_0$ ) but emits a lower-energy laser photon ( $E_2-E_1$ ); the difference is released as phonons in the two fast nonradiative steps, $\lvert 3\rangle\!\to\!\lvert 2\rangle$ and $\lvert 1\rangle\!\to\!\lvert 0\rangle$ . For Nd:YAG pumped at 808 nm and lasing at 1064 nm this is about $24\%$ of every absorbed pump photon — roughly 0.37 eV per cycle. In a high-power head that adds up to many watts deposited in the rod, driving thermal lensing, stress birefringence, and ultimately fracture. Minimising the quantum defect by pumping closer to the laser wavelength is a major theme in high-power laser design.

What exactly is a "quasi-four-level" laser like Yb:YAG?

It is a system that would be four-level except that its lower laser level sits too close to the ground state — a few $k_\mathrm{B}T$ up rather than many. Boltzmann statistics then give it a nonzero thermal population: for Yb:YAG at room temperature about 5% of the atoms occupy the lower level, and those atoms reabsorb the laser wavelength. This reabsorption is a loss that must be bleached before net gain appears, so the effective threshold is nonzero and rises with temperature — intermediate between the four- and three-level extremes. Cooling the crystal depopulates the lower level and restores near-four-level performance, which is why cryogenic Yb lasers are so efficient. The Boltzmann factor at the operating temperature is the clean diagnostic.

Why can four-level lasers run continuous-wave so much more easily than three-level ones?

Because continuous-wave operation means continuously maintaining an inversion, and the cost of that inversion is what differs. A three-level laser must keep more than half of all its active atoms pumped out of the ground state at all times — a relentless pumping demand that heats the medium and usually forces pulsed operation. A four-level laser only has to maintain the small inversion needed to overcome cavity loss, $\Delta N_\mathrm{th}=\gamma_\mathrm{th}/\sigma_{21}$ , which for a real Nd:YAG rod is thousands of times smaller. That modest, sustainable pumping load is what makes low-power, efficient, continuous-wave solid-state lasers — from laser pointers to fibre lasers — practical.