Lasers

A laser is a driven, dissipative oscillator that converts pump energy into a selected optical field. In the conventional picture, a pumped medium provides gain, feedback returns part of the field to that medium, and saturation establishes a steady state in which gain balances loss. The useful output is the fraction deliberately coupled out of this circulating or traveling field.

The expansion of the name, “light amplification by stimulated emission of radiation,” identifies the microscopic gain process but not the complete device. Stimulated emission can occur in a single-pass amplifier that never oscillates. A passive resonator can store a field but cannot replenish its losses. A working laser combines an energy source, an amplifying response, mode selection, feedback or an equivalent distributed return mechanism, and loss.

Enter this chapter

Helpful background. Stimulated Emission and Einstein Coefficients explain the competing emission and absorption processes. Coherent Light introduces the field-state approximation and its limits.

Question	Lessons
How does an amplifier become an oscillator?	Laser Principles, Population Inversion, Gain and Threshold
What fixes its spatial and spectral modes?	Optical Cavities, Laser Modes
How does the output evolve and fluctuate?	Rate-Equation Lasers, Linewidth and Coherence
How are pulses and frequency grids generated?	Mode Locking, Frequency Combs
How is a practical source chosen and controlled?	Semiconductor Lasers, Laser Stabilization

The Laser Operating Loop

A conventional laser closes an optical feedback loop around a pumped, saturable gain medium. Below threshold a fluctuation decays each round trip; at threshold small-signal gain balances loss. In ordinary large-occupation operation, saturation brings the gain close to the loss. The useful output is also one of the losses that the gain must replace.

The figure should be read dynamically rather than as a list of parts:

1. Pumping drives the medium away from thermal equilibrium.
2. Spontaneous emission and other fluctuations place small fields into many available modes.
3. Stimulated emission amplifies fields where the medium supplies net gain.
4. Feedback and propagation favor modes satisfying the spatial, spectral, polarization, and phase conditions of the resonator.
5. Threshold is reached when the leading mode no longer decays in the small-signal approximation.
6. Saturation and competition stop unlimited growth and redistribute the available gain.
7. Output coupling extracts a controlled traveling field.

The laser therefore selects a collective field–medium solution. Individual stimulated photons do not independently decide the beam direction, linewidth, or phase.

What Makes Laser Light Special?

No single adjective fully characterizes laser light. Choose specifications for the field that reaches the experiment:

• Spatial and modal quality. Focusing, fiber coupling, and local addressing depend on the transverse mode and beam quality. “Single mode” must distinguish longitudinal, transverse, and polarization selection; Laser Modes explains the measurements.
• Spectral quality. A linewidth needs a line shape, observation time or noise bandwidth, and drift convention. Short-term coherence does not imply long-term frequency accuracy; use Linewidth and Coherence to state a reproducible specification.
• Control. Tuning, modulation, polarization, pulse timing, and delivery stability determine how the source enters an experimental Hamiltonian. Compare Laser Stabilization with the task requirements below; quiet frequency does not guarantee quiet intensity.
• Field statistics. High occupation can justify a classical amplitude for selected observables, but does not establish classicality. Bright squeezed light is nonclassical, and a bright laser may have excess technical noise; Coherent Light sets the scope of the coherent-state approximation.

Stimulated Emission

Stimulated emission transfers energy from an excited medium into an occupied optical mode. The occupation-factor calculation separates its occupation-dependent contribution from spontaneous emission. Amplification preserves the selected mode’s spatial, spectral, and polarization character; it does not copy an identifiable photon or remove the added noise of a phase-preserving amplifier.

For laser design, the next question is whether emission exceeds absorption for the actual mode and polarization. Positive net gain does not yet imply oscillation: propagation and cavity losses remain to be overcome. Laser Principles connects transition rates and cross sections to this gain condition, including unequal absorption and emission strengths.

Gain Medium

A gain medium is matter maintained in a nonequilibrium state from which an optical field can extract energy. The pump may be optical, electrical, chemical, collisional, or supplied through another coherent field. The medium may be a dilute gas, ion-doped crystal, semiconductor junction, molecular system, fiber, or nonlinear optical device.

Three properties matter separately:

Property	Question it answers
gain magnitude	Can amplification overcome all losses?
gain spectrum	Which frequencies can grow?
gain dynamics	How quickly do populations and coherences respond?

Choose a population model from the pump and relaxation pathways, including lower-level emptying and reabsorption. Population Inversion compares two-, three-, and four-level schemes and states the limits of the inversion criterion. Specialized coherently driven gain mechanisms still require a nonequilibrium energy source.

Use small-signal gain to locate onset and saturated gain to predict operation. Saturation reduces the available amplification and couples modes competing for the same reservoir. Homogeneous steady-state saturation, spatial or spectral hole burning, and pulsed extraction need different models; see Saturation Above Threshold.

Feedback and Mode Selection

A linear cavity returns the field through two mirrors; a ring provides a traveling-wave return path. Distributed Bragg structures, whispering-gallery resonators, external cavities, and fiber loops implement feedback with other boundary conditions. Optical Cavities develops resonance, mode spacing, decay, and stability for these structures.

A candidate lasing mode must reproduce both its phase and spatial profile on a complete round trip while gaining enough power to offset loss. The gain spectrum alone cannot choose it: diffraction, pump overlap, polarization, dispersion, and competition also matter. Use the round-trip map to define the oscillator and practical mode selection to identify which controls distinguish the desired mode.

Threshold and Steady Operation

Small-signal threshold marks the point where the leading mode stops decaying. Gain and Threshold builds the loss and overlap budget. Output coupling is part of that budget: extracting a larger fraction raises the gain required to sustain the field. The worked example below shows this balance for a uniform linear cavity.

Below threshold, amplified and filtered spontaneous emission can be bright without sustaining an oscillator. Near threshold, fluctuations round the transition, so power, linewidth, photon statistics, and dynamical onset need not identify the same experimental threshold. In ordinary large-occupation operation above threshold, saturation brings stimulated gain close to loss; spontaneous emission gives a finite correction to exact equality.

For output transients or relaxation oscillations, use Rate-Equation Lasers when optical polarization relaxes fast enough to eliminate and a population reservoir plus photon number describes the system. That treatment keeps excitation counting consistent, derives the finite-spontaneous-emission crossover, and identifies when multimode dynamics, spatial structure, or coherent field–matter evolution require a richer model. Mean-population equations alone do not determine phase coherence or photon statistics.

Worked Threshold Example

Use a uniform gain region with full mode overlap, no extra lumped losses, and two passes through its one-way length $L$ per complete round trip. Here $g$ and $\alpha_{\mathrm i}$ are power gain and internal-loss coefficients per unit length, and $R_j=|r_j|^2$ are mirror power reflectivities. A field-amplitude propagation equation would use half the net power coefficient; mixing the two conventions gives an incorrect threshold.

The uniform-cavity threshold and round-trip power map give

$$\begin{aligned} g_{\mathrm{th}} &=\alpha_{\mathrm i}+\frac{1}{2L}\ln\!\left(\frac{1}{R_1R_2}\right),\\ \mathcal G_{\mathrm{rt}} &=R_1R_2\exp\!\left[2(g-\alpha_{\mathrm i})L\right]. \end{aligned}$$

Consider a uniform linear cavity with

$$\begin{gathered} L=0.30\ \mathrm m, \qquad R_1=0.999, \\ R_2=0.950, \qquad \alpha_{\mathrm i}=0.020\ \mathrm{m}^{-1}. \end{gathered}$$

The required material power-gain coefficient is

$$\begin{aligned} g_{\mathrm{th}} &= 0.020 + \frac{1}{0.60} \ln \left[ \frac{1}{ (0.999)(0.950) } \right] \\ &\simeq 0.107\ \mathrm{m}^{-1}. \end{aligned}$$

If the unsaturated gain is $g_0=0.150\ \mathrm{m}^{-1}$, then

$$\begin{aligned} \mathcal G_{\mathrm{rt}} &= (0.999)(0.950) \times \exp \left[ 2(0.150-0.020)(0.30) \right] \\ &\simeq 1.026. \end{aligned}$$

The weak field grows by about $2.6\%$ in power per round trip. That number is not the steady-state output growth. As intensity builds, gain saturation drives the net round-trip multiplier back toward one.

Coherence and Linewidth

A useful coherence specification identifies the detected mode, reference, observation time, and noise bandwidth. Frequency or phase noise, relative intensity noise, pointing, polarization, and pulse-timing fluctuations affect different AMO measurements. A single linewidth can conceal drift, modulation sidebands, or non-Lorentzian wings.

For a stationary field with a Lorentzian power-spectrum FWHM $\Delta\nu$ in hertz, the normalized first-order field correlation obeys

$$|g^{(1)}(\tau)|=e^{-\pi\Delta\nu|\tau|}, \qquad \tau_c=\frac{1}{\pi\Delta\nu}.$$

Here $\tau_c$ is the $e^{-1}$ decay time of the correlation amplitude. The Lorentzian Fourier pair derives this convention; other spectra and definitions need not give the same factor or a single coherence time. Ordinary frequency $\nu$ is related to angular frequency by $\omega=2\pi\nu$.

A free-running oscillator’s absolute phase diffuses. The Schawlow–Townes noise scale connects spontaneous-emission noise, resonator decay, and output power, while technical noise often dominates. A coherent-state model for a specified output packet may still reproduce the observables of interest over a declared interval. It does not guarantee a known absolute phase, independent temporal packets, or vacuum-level noise in every real source. Choose the field model by the measured correlations and the experiment’s reference and bandwidth.

Pulsed Lasers

Laser energy need not emerge as a continuous wave.

Gain switching and Q switching

Gain switching drives the gain rapidly through threshold, producing a pulse as the field builds and depletes the inversion. Q switching stores energy in the gain medium while cavity loss is high, then reduces that loss so the stored energy is released in a short, intense pulse. These mechanisms are primarily energy-storage and transient-threshold processes.

Mode locking

A laser supporting many longitudinal modes becomes mode locked only when their relative phases acquire a stable relation. Periodic constructive interference then concentrates the average power into a pulse train. The mode spacing fixes the repetition rate; coherent bandwidth and spectral phase fix the pulse shape and duration.

Mode Locking derives the finite-mode pulse train, active and passive locking mechanisms, the round-trip master-equation picture, Gaussian and $\operatorname{sech}^2$ time–bandwidth conventions, chirp, pulse characterization, and the connection to ultrafast spectroscopy.

Frequency Combs

A phase-coherent pulse train produces an optical grid controlled by two radio-frequency coordinates: tooth spacing from pulse timing and a common offset from carrier-envelope phase slip. When both are measured or stabilized, the grid links microwave references, continuous-wave optical lasers, and atomic transitions.

Frequency Combs derives the tooth equation, $f$-to-$2f$ self-referencing, tooth-index and beat-sign bookkeeping, two-degree-of-freedom stabilization, transfer-oscillator cancellation, optical frequency division, and the role of combs in optical clocks.

Role in AMO Physics

Lasers are simultaneously preparation devices, probes, force sources, local oscillators, and metrology links.

AMO task	Controlled laser quantity	Relevant failure modes
resolve a transition	carrier frequency and linewidth	drift, sidebands, power broadening
drive a quantum rotation	pulse area and optical phase	detuning, amplitude noise, spatial inhomogeneity
cool an atom	detuning, polarization, direction, intensity	frequency noise, imbalance, magnetic-field errors
trap with an AC Stark shift	intensity and spatial mode	scattering, pointing noise, differential shifts
perform Raman control	relative phase and two-photon detuning	path noise, differential Stark shifts
read out a cavity or interferometer	local-oscillator phase and mode match	shot noise, technical noise, loss
compare optical clocks	stabilized frequency and comb coherence	reference noise, cycle slips, dead time
pump a nonlinear source	pulse energy, spectrum, phase matching	multipair noise, thermal drift, modal impurity

The field model depends on the observable

The same physical laser may be represented at different levels:

1. a deterministic classical field for mean coherent dynamics;
2. a classical stochastic field when technical amplitude or phase noise matters;
3. a quantized input field when shot noise, squeezing, counting, or measurement backaction matters;
4. an explicit driven cavity–gain-medium open system when laser generation and linewidth are themselves the subject.

These descriptions are not competitors. They retain different degrees of freedom. A semiclassical drive can accurately model a Rabi experiment while vacuum modes must still be quantized to model spontaneous decay.

Exercises

1. Classify four optical systems

Classify each system as a passive resonator, optical amplifier, amplified spontaneous-emission source, or laser oscillator.

1. A lossy Fabry–Pérot cavity is illuminated by an external narrowband field.
2. A pumped rare-earth fiber amplifies an injected signal and has angled, antireflection-coated ends.
3. The same pumped fiber emits broadband light from spontaneous photons that are amplified during one pass.
4. The fiber is placed in a ring with a wavelength-selective element and an output coupler; a field persists without an injected signal.

Solution

1. The Fabry–Pérot is a passive resonator. It stores and filters the injected field but supplies no gain.
2. The seeded, reflection-suppressed fiber is an optical amplifier.
3. The unseeded one-pass output is amplified spontaneous emission.
4. The closed-loop device is a laser oscillator because a selected mode is sustained by gain and feedback without an external seed.

Brightness or directionality alone did not decide the classification.

2. Change the output coupler

In the uniform linear-cavity model, the output-coupler reflectivity is reduced while all other quantities are held fixed. Explain why the threshold rises, why the extractable fraction per encounter rises, and why choosing the smallest possible reflectivity does not generally maximize useful output.

Solution

Reducing $R_2$ increases the logarithmic mirror-loss term,

$$\frac{1}{2L} \ln \left( \frac{1}{R_1R_2} \right),$$

so more material gain is required at threshold. If absorption is negligible, the transmitted fraction $T_2\approx1-R_2$ increases, so a larger fraction of the intracavity power is extracted on each encounter.

These trends compete. Excessive output coupling can raise the required gain beyond what the pump supplies and can greatly reduce circulating power. Excessively weak coupling traps power while internal loss consumes it. The optimum depends on available saturated gain, internal loss, mode overlap, and the desired output rather than on transmissivity alone.

3. Infer a coherence time

A stationary laser has a Lorentzian power spectrum with FWHM $\Delta\nu=100\ \mathrm{kHz}$. Under the convention $|g^{(1)}(\tau)|=\exp(-\pi\Delta\nu|\tau|)$, find the $e^{-1}$ coherence time and the corresponding vacuum propagation length $c\tau_c$.

Solution

The coherence time is

$$\tau_c = \frac{1}{ \pi(100\times10^3\ \mathrm{s}^{-1}) } \simeq 3.18\ \mu\mathrm s.$$

Using $c\simeq2.998\times10^8\ \mathrm{m\,s}^{-1}$,

$$c\tau_c \simeq 9.54\times10^2\ \mathrm m.$$

This is about $954\ \mathrm m$. It is a correlation length for the declared stationary Lorentzian model, not necessarily the distance over which an unstabilized experiment retains a known absolute phase.

4. Distinguish pulsed-laser mechanisms

For gain switching, Q switching, and mode locking, identify the principal quantity being controlled and one diagnostic that distinguishes the regime.

Solution

Gain switching drives gain through threshold and observes the transient field buildup and inversion depletion. The pulse timing follows the pump drive; a time-resolved output versus pump waveform tests the mechanism.

Q switching stores inversion while cavity loss is high and releases it after the loss is reduced. Its pulses are typically separated by an energy-storage timescale rather than one cavity round trip; simultaneous measurement of intracavity loss, inversion proxy, and output pulse distinguishes it.

Mode locking establishes a stable relative phase among many longitudinal modes. A radio-frequency line at the pulse repetition rate together with a broad coherent optical spectrum and phase-sensitive pulse characterization tests this regime. A pulse train alone is insufficient because Q-switched mode locking can place short pulses inside a slower envelope.

References

• S. T. Cundiff and J. Ye, “Colloquium: Femtosecond Optical Frequency Combs,” Reviews of Modern Physics 75, 325–342 (2003).
• S. T. Diddams, “The Evolving Optical Frequency Comb,” Journal of the Optical Society of America B 27, B51–B62 (2010).
• A. Einstein, “Zur Quantentheorie der Strahlung,” Physikalische Zeitschrift 18, 121–128 (1917). The source of the absorption, spontaneous-emission, and stimulated-emission coefficient framework.
• R. J. Glauber, “The Quantum Theory of Optical Coherence,” Physical Review 130, 2529–2539 (1963).
• H. Haken, Laser Theory, Springer (1984), doi:10.1007/978-3-642-45551-7.
• H. A. Haus, “Mode-Locking of Lasers,” IEEE Journal of Selected Topics in Quantum Electronics 6, 1173–1185 (2000).
• T. H. Maiman, “Stimulated Optical Radiation in Ruby,” Nature 187, 493–494 (1960).
• L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press (1995), doi:10.1017/CBO9781139644105.
• P. W. Milonni and J. H. Eberly, Laser Physics, Wiley (2010).
• K. Mølmer, “Optical Coherence: A Convenient Fiction,” Physical Review A 55, 3195–3203 (1997).
• M. Sargent III, M. O. Scully, and W. E. Lamb Jr., Laser Physics, Addison–Wesley (1974).
• A. L. Schawlow and C. H. Townes, “Infrared and Optical Masers,” Physical Review 112, 1940–1949 (1958).
• A. E. Siegman, Lasers, University Science Books (1986).
• Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical Frequency Metrology,” Nature 416, 233–237 (2002).
• H. M. Wiseman, “Defining the (Atom) Laser,” Physical Review A 56, 2068–2084 (1997).