Physics · Optics · Grade 11

Laser physics and holography

When does a laser “switch on”, and why does it first emit strong spikes?

The situation

In a university optics laboratory a student is assembling a small Nd:YAG laser: a crystal rod, two mirrors and a pump lamp. When the lamp is switched on, the output power does not settle at once — first a series of short, very powerful “spikes” appears, and then they die away. At the next table a hologram is being recorded with a He–Ne laser — for this the beam must be highly coherent. When does a laser “switch on”, where do the spikes come from, and how many frequencies live inside the cavity?

Topic

Laser physics and holography. Laser cavity and the hologram

Key ideas

Inversion and threshold
Relaxation oscillations
Longitudinal modes

Formulas

\frac{d N}{d t} = R_{p} - \frac{N}{τ} - G N q, \frac{d q}{d t} = G N q - \frac{q}{τ_{c}} + β \frac{N}{τ}

Rate equations: $N$ is the inversion (atoms in the upper level), $q$ is the number of photons in the cavity, and $β N / τ$ is the spontaneous-emission “seed”.

N_{th} = \frac{1}{G τ_{c}} = \frac{γ A}{σ}, τ_{c} = \frac{L_{e}}{c γ}, γ = γ_{i} + \frac{1}{2} \ln \frac{1}{R_{1} R_{2}}

Lasing threshold: the gain equals the losses. $τ_{c}$ is the photon lifetime in the cavity, $σ$ is the stimulated-emission cross-section.

ω = \sqrt{ω_{R}^{2} - Γ^{2}}, ω_{R}^{2} = \frac{x - 1}{τ τ_{c}}, Γ = \frac{x}{2 τ}

Frequency and damping of the relaxation oscillations; $x = R_{p} / R_{p, th}$ shows how many times the pumping exceeds the threshold.

Δ ν = \frac{c}{2 L_{e}}, N_{modes} \approx \frac{Δ ν_{g}}{Δ ν}

The frequency difference between longitudinal modes, and the number of modes that fit within the gain width $Δ ν_{g}$ .

Λ = \frac{λ}{2 \sin (θ / 2)}

The spacing of the interference fringes on the hologram; $θ$ is the angle between the reference and object beams.

Theory

Three- and four-level lasers

Pumping raises atoms to the upper level; they quickly drop to the upper laser level and stay there a long time (in Nd:YAG $τ = 230$ µs). In a four-level laser (Nd:YAG, He–Ne) the lower laser level empties quickly, so inversion is easy to achieve. When a photon passes close to an excited atom, it “knocks out” a second photon exactly like itself — stimulated emission; the first photons come from spontaneous emission (on the stage, flashes in random directions). The ruby laser (T. Maiman, 1960) is a three-level laser: Cr³⁺ ions in a synthetic ruby crystal (Al₂O₃) absorb the green-blue light of a flash lamp, quickly drop to a metastable level ( $τ \approx 3$ ms) and emit red light at 694.3 nm. Here the lower laser level is the ground state, so until more than half of the ions are excited the crystal absorbs its own light: the threshold is much higher, and the ruby laser usually works in pulses. The ends of the rod are coated with mirrors, or the rod is placed between external mirrors — choose the “Ruby” medium and look at the 0.5 limit on the N₂/N graph.

Threshold and output power

A laser “switches on” only when the gain in one round trip makes up for the losses at the mirrors and the internal losses. Below the threshold pumping there is only weak spontaneous emission. Above the threshold the inversion is clamped at $N_{th}$ , and the excess pumping turns entirely into photons: $P \propto R_{p} - R_{p, th}$ . If you reduce the $R$ of the output mirror, more light comes out, but the threshold rises.

Why do spikes appear?

In Nd:YAG the lifetime of the upper level $τ$ is thousands of times longer than the photon lifetime $τ_{c}$ (tens of ns). When the pump is switched on, the inversion rises above the threshold, the photons multiply like an avalanche, give a short powerful spike and “eat up” the inversion. Then the process repeats and gradually dies away — these are relaxation oscillations. The first big spikes are non-linear and have a longer period; for small oscillations the period tends to $2 π / ω$ . In He–Ne $τ \approx τ_{c}$ , so the oscillations die away almost at once.

Modes and the hologram

Only standing waves that satisfy the condition $L = q λ / 2$ live in the cavity — they are spaced in frequency by $c / 2 L$ . The Doppler width of He–Ne is ≈ 1.5 GHz; for $L = 30$ cm the mode spacing is ≈ 500 MHz, so 3 modes fit inside. In a hologram the reference and object beams record interference fringes on the plate: at $θ = 30 °$ , $Λ \approx 1.2$ µm, i.e. more than 800 lines/mm — ordinary photographic film cannot resolve this. When illuminated, the hologram diffracts the reference beam and restores the wave that came from the object — its amplitude and phase — so the image looks three-dimensional. Model: a single generalised mode, uniform gain; the gain width affects only the spectrum; the pumping is switched on suddenly at $t = 0$ and stays constant (the ruby's flash-lamp pulse is treated the same way).

Quantum transitions: the Einstein coefficients

An atom makes three kinds of transitions between levels $E_{1}$ and $E_{2}$ : absorption (a photon is absorbed, at a rate $B_{12} ρ (ν) N_{1}$ ), spontaneous emission (on its own, in a random direction and with a random phase, at a rate $A_{21} N_{2} = N_{2} / τ$ ) and stimulated (induced) emission (under the action of an incoming photon, at a rate $B_{21} ρ (ν) N_{2}$ ). In 1916 Einstein showed from thermal equilibrium that $g_{1} B_{12} = g_{2} B_{21}$ and $A_{21} / B_{21} = 8 π h ν^{3} / c^{3}$ . A stimulated photon has the same frequency, phase, direction and polarisation as the incoming one — hence the coherence of laser light. At equilibrium $N_{2} < N_{1}$ , so matter absorbs light; for amplification a population inversion $N_{2} > N_{1}$ is needed. In quantum mechanics these rates follow from time-dependent perturbation theory — the Fermi golden rule $W = \frac{2 π}{ℏ} | ⟨ 2 | \hat{V} | 1 ⟩ |^{2} ρ (E)$ . An electric dipole transition occurs only if the matrix element is non-zero — the selection rules ( $Δ l = \pm 1$ , $Δ m = 0, \pm 1$ ). “Forbidden” transitions are very slow: this is why the laser level of Cr³⁺ and Nd³⁺ ions lives for hundreds of microseconds to milliseconds (a metastable level).

Types of laser, properties of the light and applications

By active medium, lasers are solid-state (ruby, Nd:YAG, fibre lasers), gas (He–Ne, CO₂ — 10.6 µm, argon-ion), semiconductor (laser diodes: pointer, fibre-optic communication, CD/DVD reader) and dye (the wavelength is tunable); by mode of operation, continuous and pulsed. Properties of laser light: monochromaticity — the line width is set by the cavity modes; coherence — the coherence length is $l_{c} \approx c / Δ ν$ : ≈ 20 cm for a multimode He–Ne, millimetres for Nd:YAG (see “Coherence length” in the measurements); directionality — the beam spreads only because of diffraction, $θ \approx λ / (π w_{0})$ , where $w_{0}$ is the radius of the beam waist; polarisation — Brewster windows pass one polarisation without loss, so such a He–Ne beam is linearly polarised. The spectral content is the set of lasing longitudinal modes (the spectrum graph). Applications: in medicine — “welding” the retina, correcting the cornea, the CO₂ laser scalpel; in industry — cutting, welding and marking metal; in technology — fibre-optic communication, barcode readers, laser rangefinders, holography.

Did you know?

The first laser was started by the American physicist Theodore Maiman on 16 May 1960: it used a synthetic ruby crystal and a spiral flash lamp, and the wavelength of its light was 694.3 nm.
Holography was invented by the Hungarian-British physicist Dennis Gabor in 1947 in order to improve the image of the electron microscope; he received the Nobel Prize in 1971. Bright three-dimensional holograms began to be made only after the laser appeared.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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Interference and diffraction — Grade 11
Polarisation of light — Grade 11
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Infrared, UV and X-rays — Grade 11
Scattering of light — Year 1
Crystal optics — Year 1
Interferometers and LIGO — Year 1