Laser Theory Essentials

Be able to explain every relation on this page physically, derive its leading dependence, and state where it stops being valid.

The laser in one sentence

A laser stores pump energy in a population inversion, converts it into coherent light by stimulated emission, selects self-consistent spatial and longitudinal modes with a resonator, and reaches steady state when saturated round-trip gain equals round-trip loss.

Absorption, spontaneous emission, and stimulated emission

For a transition between levels 1 and 2:

• absorption removes a resonant photon and raises an ion;

• spontaneous emission produces a photon with random emission time, direction, phase, and often polarization; and

• stimulated emission produces a photon in the stimulating optical mode, with the same frequency, phase, propagation direction, and polarization.

The small-signal gain coefficient is usefully written

(1)\[g_0(\nu)=\sigma_e(\nu)N_2-\sigma_a(\nu)N_1,\]

where \(\sigma_e\) and \(\sigma_a\) are emission and absorption cross sections. Population inversion means enough population is placed in the upper laser level that \(g_0>0\); it does not merely mean that the upper level is occupied.

Three- versus four-level media

In a three-level laser the lower laser level is the ground state. More than half of the active ions must be transferred before net gain appears, so threshold is high. In a four-level laser the lower laser level empties rapidly; even a modest upper-level population can produce inversion. Ruby is the standard three-level example, while Nd:YAG at 1064 nm is approximately four-level.

Threshold: close the round-trip balance

Let \(R_1\) and \(R_2\) be mirror power reflectivities, \(L_g\) the gain length, \(g\) the gain coefficient, and \(\mathcal L_i\) the remaining fractional round-trip power loss represented exponentially. At threshold,

(2)\[R_1R_2\exp(2g_{\rm th}L_g-\mathcal L_i)=1,\]

and therefore

\[g_{\rm th}=\frac{1}{2L_g} \left[\ln\!\left(\frac{1}{R_1R_2}\right)+\mathcal L_i\right].\]

The factor of two is the double pass through the gain medium in a linear cavity. If loss is instead defined per pass or per unit length, rewrite the balance explicitly rather than carrying this formula unchanged.

Above threshold, gain saturation clamps the round-trip gain near the loss:

(3)\[g(I)\simeq\frac{g_0}{1+I/I_{\rm sat}}, \qquad I_{\rm sat}\simeq\frac{h\nu_l}{\sigma_e\tau_f}.\]

The precise saturation factor depends on level scheme, standing versus travelling wave, and temporal regime. The design insight is robust: increasing intracavity intensity depletes inversion and reduces available gain.

Efficiency bookkeeping

Separate efficiencies so a disappointing result can be diagnosed:

(4)\[\eta_{\rm wall} =\eta_{\rm driver}\eta_{\rm coupling}\eta_{\rm absorption} \eta_{\rm quantum}\eta_{\rm overlap}\eta_{\rm extraction}.\]

The ideal Stokes limit for one pump photon producing one laser photon is

\[\eta_{\rm Stokes}=\frac{h\nu_l}{h\nu_p} =\frac{\lambda_p}{\lambda_l}.\]

The quantum-defect fraction \(1-\lambda_p/\lambda_l\) is an unavoidable heat contribution even in an otherwise ideal system. Nonradiative decay, unabsorbed pump, fluorescence, parasitic oscillation, imperfect overlap, and intracavity loss add further penalties.

Near and above threshold, many cw lasers are summarized by

(5)\[P_{\rm out}\simeq\eta_s(P_{\rm pump}-P_{\rm th}).\]

Do not report \(\eta_s\) without specifying whether pump power is electrical, incident optical, or absorbed optical power.

Longitudinal modes and linewidth

For a linear cavity with round-trip optical length \(2L_{\rm opt}\),

(6)\[\Delta\nu_{\rm FSR}=\frac{c}{2L_{\rm opt}}, \qquad L_{\rm opt}=\sum_j n_{g,j}L_j.\]

The group index belongs in a precise free-spectral-range calculation. The number of longitudinal modes is roughly gain bandwidth divided by FSR, but actual oscillation also depends on spatial and spectral hole burning, polarization, intracavity etalons, and mode competition.

A narrow gain curve does not by itself guarantee single-frequency operation. Common controls are a short cavity, unidirectional ring geometry, intracavity etalons, birefringent filters, injection seeding, and active frequency locking.

Continuous wave, Q-switching, and mode locking

Regime	Stored quantity	Release mechanism	Dominant engineering issue
Continuous wave	Nearly steady inversion and photon population	Gain balances loss continuously	Thermal lens, efficiency, noise, and mode stability
Q-switched	Inversion accumulates while cavity loss is high	Cavity Q rises rapidly	Prelasing, switching speed, damage fluence, timing jitter
Mode locked	Fixed phase relation among many longitudinal modes	Intracavity modulation or saturable absorption	Dispersion, nonlinearity, self-starting, pulse stability

Always connect pulse energy, duration, repetition rate, average power, and peak power:

(7)\[E_p=\frac{P_{\rm avg}}{f_{\rm rep}}, \qquad P_{\rm peak}\simeq\frac{E_p}{\tau_p}, \qquad F=\frac{E_p}{A_{\rm eff}}, \qquad I_{\rm peak}\simeq\frac{P_{\rm peak}}{A_{\rm eff}}.\]

Use fluence for pulse-energy-driven damage and peak irradiance for nonlinear or instantaneous effects. The temporal and spatial beam shapes change the exact peak factors.

Nonlinear conversion

For second-harmonic generation, energy conservation fixes \(\omega_{2}=2\omega_1\) while efficient build-up requires phase matching,

(8)\[\Delta k=k_{2\omega}-2k_\omega=0.\]

At low depletion, converted power grows approximately as the square of fundamental power and as \(\operatorname{sinc}^2(\Delta kL/2)\). A real design balances intensity, crystal length, walk-off, absorption, temperature or angle acceptance, focusing, coating limits, and damage. Focus as tightly as possible is therefore not a good design rule.

Fast oral checks

• More output coupling raises threshold but can improve useful extraction.

• A shorter cavity increases longitudinal-mode spacing.

• A longer upper-state lifetime helps energy storage but can constrain fast modulation.

• Pump/laser wavelength separation creates heat through quantum defect.

• Increasing intracavity power improves nonlinear conversion but tightens coating, contamination, and damage margins.