Optics · Upper Division

Lasers and Coherent Light

The laser — Light Amplification by Stimulated Emission of Radiation — is the quintessential device where quantum mechanics, thermodynamics, and wave optics converge. Understanding lasers requires population inversions, optical cavities, and coherence theory.

Prerequisites

Atomic structure (Ch. 21) \cdot Wave optics (Ch. W) \cdot Statistical mechanics (Ch. S) \cdot Quantum

mechanics (Ch. 20)

Learning Goals

Explain spontaneous emission, stimulated emission, and absorption using the Einstein A and B coefficients.
Describe why population inversion is required for laser gain and why it cannot occur in a two-level system.
Calculate the threshold gain coefficient from cavity mirror reflectivities.
Characterize a Gaussian beam using its waist radius, Rayleigh range, and divergence angle.
Distinguish temporal and spatial coherence and explain why laser light excels at both.

LZ.1 Einstein Coefficients and Stimulated Emission

Einstein (1917) analyzed the interaction of atoms with radiation by introducing three processes: absorption, spontaneous emission, and stimulated emission.

Definition LZ.1 — Einstein A and B Coefficients

For a two-level atom (ground |1⟩, excited |2⟩, energy gap \hbar\omega

:Absorption:

rate = B_{12} \rho(\omega) N_{1} \qquad (\rho = radiation energy density

Spontaneous emission:

rate = A_{21} N_{2}

Stimulated emission:

rate = B_{21} \rho(\omega) N_{2}

In thermal equilibrium, detailed balance + Planck distribution require: A_{21}/B_{21} = \hbar\omega^{3}/(\pi^{2}c

³)

and B_{12} = B_{21} (for non-degenerate levels). The stimulated emission rate equals the absorpt

rate when the populations are equal — this is the key to laser amplification.

Stimulated emission produces a photon identical to the stimulating one in frequency, phase, direction, and polarization. This coherence is the source of laser beam quality. Spontaneous emission is noise; stimulated emission is signal.

LZ.2 Population Inversion and Gain

In thermal equilibrium (Boltzmann): N₂/N₁ = e^(−ℏω/k_BT) < 1. More atoms are always in the ground state. For net stimulated emission (gain), we need population inversion: N₂ > N₁. This is impossible in a two-level system at equilibrium (saturation makes N₁ = N₂ at most). Real lasers use three- or four-level schemes:

Three-level laser (e.g., ruby):Pump ground → excited band → metastable level (fast decay). Inversion between metastable and ground state. Must invert >50% of atoms — threshold is high.

Four-level laser (e.g., Nd:YAG, He-Ne): Lower laser level is rapidly depopulated (fast decay to ground). Inversion maintained at any pump level — much lower threshold. Most practical lasers are four-level.

G(\nu) = \sigma(\nu)(N_{2} - N_{1}) \qquad (gain per unit length)

(LZ.1)

Here σ(ν) is the stimulated emission cross section. Laser oscillation begins when gain equals loss: G × L = 1 (round-trip condition in the cavity).

Example LZ.1 — Threshold Population Inversion

$A He-Ne laser at 632.8 nm: cavity length L = 30 cm, mirror reflectivities R_{1} = 1.0, R_{2} = 0$ .99. Gain medium fills the cavity. Find the threshold gain coefficient.

Round-trip condition:

R_{1} R_{2} e^{2gL} = 1 (gain must overcome mirror losses

Solve for g:

e^{2gL} = 1/(R_{1}R_{2}) = 1/0.99

2gL = ln(1/0.99) = 0.01005 \to g = 0.01005/(2\times0.30) = 0.0167 m^{-1} = 1.67 \times10^{-2} m^{-1}

Inversion:

\DeltaN = N_{2} - N_{1} = g/\sigma. For He-Ne at 633 nm: \sigma \approx 3\times10^{-17} m^{2}. \DeltaN = 0.0167/3\times10^{-17} \approx 5.6\times10^{14} m^{-}

³ —

extremely small compared to gas density ~10^{23} m^{-3}

LZ.3 Optical Resonators and Modes

The laser cavity (Fabry-Pérot resonator) selects discrete longitudinal modes— frequencies where the cavity forms standing waves:

\nu_{n} = nc/(2L) \qquad (mode spacing: \Delta\nu = c/2L)

(LZ.2)

For L = 30 cm: Δν = 500 MHz. The gain bandwidth of the medium (Doppler-broadened ~1.5 GHz for He-Ne) may support 3 longitudinal modes. Single-mode operation requires short cavities or intra-cavity etalons.

Transverse modes (TEM_mn): characterized by their intensity pattern in the plane perpendicular to the beam. TEM₀₀ (Gaussian beam) is the fundamental mode — smallest divergence, best focusability. Higher modes have larger diameter and spread faster.

LZ.4 Gaussian Beams

The TEM₀₀ mode is a Gaussian beam. Its intensity profile at position z:

I(r, z) = I_{0} (w_{0}/w(z))^{2} exp(-2r^{2}/w(z)^{2})

(LZ.3)

w(z) = w_{0} \sqrt(1 + (z/z_{R})^{2}) \qquad z_{R} = \piw_{0}^{2}/\lambda \qquad (Rayleigh range)

(LZ.4)

Here w₀ is the beam waist radius and z_R is the Rayleigh range — the distance over which the beam area doubles. The divergence half-angle for large z: θ ≈ λ/(πw₀) — a smaller waist means faster divergence (diffraction limit). The beam parameter product w₀ × θ = λ/π is invariant and equals ℏ/2 of the uncertainty principle (position × momentum for a photon).

Theorem LZ.1 — Coherence

Temporal coherence:

correlation between the field at one point at different times. Coherence length L_{c} = c/\Delta\nu

— the

path length difference over which interference fringes are visible. Single-mode laser: L_{c}

e kilometers.Spatial coherence:correlation between field at two points at the same time. A laser mode has high spatial coherence across the beam. Young's double slit with a laser: perfect fringes. With a thermal source: fringes only within the coherence area (related to source angular size by van Cittert-Zernike theorem).

Definition LZ.2 — Common Traps

Population inversion is required for gain: thermal equilibrium gives more atoms in the lower level.
Two-level lasers do not sustain inversion: pumping and stimulated emission compete on the same transition.
Cavity modes are frequency-selective: only resonant longitudinal modes survive repeated round trips.
Coherence is not the same as brightness: lasers are useful because phase relationships are controlled.

Exercises — LZ.1–LZ.4 Lasers and Coherent Light

A He-Ne laser has cavity length L = 0.30 m. What is the spacing between adjacent longitudi

nal modes?

MHz

Straightforward

A He-Ne laser (L = 0.30 m, R_{1} = 1.0, R_{2} = 0.99) reaches threshold. Find the minimum gain c

oefficient g (m^{-1}

m⁻¹

Straightforward

3.Derive the transmission function of a Fabry-Pérot etalon. Define the finesse F and free spectral range. What limits the spectral resolution?

Intermediate

A Gaussian beam with waist w_{0} = 5 mm is focused by a lens of focal length f = 100 mm at \lambda

= 633 nm. Find the focused spot size and depth of focus. Relate to optical storage technology.

Intermediate

5.Write rate equations for a 4-level laser. Solve for photon density above threshold. What determines the threshold and slope efficiency?

Challenging

Key Takeaways

$Einstein coefficients: B_{12}=B_{21} (absorption = stimulated emission cross-section). A_{21} = \hbar\omega^{3}$ $B/(\pi^{2}c^{3}$ .
$Population inversion N_{2} > N_{1} required for gain. Impossible in 2-level equilibrium — nee$ d 3 or 4 levels.
$Threshold: gain per round trip equals losses: R_{1}R_{2} e^{2gL} = 1$ .
$Cavity modes: \nu_n = nc/2L, spacing c/2L. TEM_{00} (Gaussian) is the fundamental transverse mo$ de.
$Gaussian beam: waist w_{0}, Rayleigh range z_{R} = \piw_{0}^{2}/\lambda, divergence \theta \approx \lambda/\piw_{0}$ .
$Coherence: temporal (L_{c} = c/\Delta\nu), spatial (Young's fringes). Laser \to very high coherence$ .