40-7 Lasers#
Prompts
What does “laser” stand for? What is stimulated emission, and why is it the key to laser operation?
Compare absorption, spontaneous emission, and stimulated emission. In stimulated emission, how does the emitted photon relate to the stimulating photon (phase, direction, energy)?
At thermal equilibrium, what is the ratio \(N_x/N_0\) of atoms in excited vs. ground state? Why can’t thermal equilibrium produce a laser? What is population inversion?
What is a metastable state? Why are metastable states important for building a laser?
Describe how a helium–neon laser works: Which gas lases? Why is helium needed? How is population inversion achieved between the neon levels?
Lecture Notes#
Overview#
Laser = Light Amplification by Stimulated Emission of Radiation. Laser light is monochromatic, coherent, directional, and can be sharply focused — unlike the random emissions from a lightbulb.
Three processes govern light–matter interaction: absorption, spontaneous emission, and stimulated emission. The last is the basis of lasing.
Stimulated emission occurs when a photon of energy \(hf = E_{\text{high}} - E_{\text{low}}\) induces an excited atom to emit an identical photon (same phase, direction, polarization).
Population inversion — more atoms in the upper level than the lower — is required for net amplification. It cannot arise from thermal equilibrium; it must be achieved by pumping (e.g., collisions, optical pumping).
Metastable states (long-lived excited states) enable population inversion; the helium–neon laser uses them to produce laser light at 632.8 nm.
Laser Light vs. Ordinary Light#
Laser |
Lightbulb |
|
|---|---|---|
Monochromaticity |
Very sharp (e.g., 1 part in \(10^{15}\)) |
Broad, continuous spectrum |
Coherence |
Long wave trains (hundreds of km) |
Short (typically < 1 m) |
Directionality |
Very low divergence |
High divergence |
Focusability |
Can reach \(\sim 10^{17}\) W/cm² |
Much lower |
Three Ways Light Interacts with Matter#
Consider an atom with ground state \(E_0\) and excited state \(E_x\).
Absorption: Atom in \(E_0\) absorbs a photon of energy \(hf = E_x - E_0\) and jumps to \(E_x\).
Spontaneous emission: Atom in \(E_x\) decays to \(E_0\) on its own, emitting a photon. The event is random in time and direction. Ordinary light sources (e.g., lightbulb) emit this way.
Stimulated emission: Atom in \(E_x\) is struck by a photon of energy \(hf = E_x - E_0\). The atom is induced to emit a second photon identical to the first — same energy, phase, polarization, and direction. The light wave is amplified.
Why stimulated emission matters
In stimulated emission, the emitted photon is in phase with and travels in the same direction as the stimulating photon. This creates a coherent, directional beam — the essence of laser light.
Thermal Equilibrium and Population Inversion#
At thermal equilibrium at temperature \(T\), the ratio of populations in two levels is given by the Boltzmann factor:
Since \(E_x > E_0\), we always have \(N_x < N_0\) — fewer atoms in the excited state. Photons of energy \(hf\) are more likely to be absorbed than to cause stimulated emission.
For a laser, we need more stimulated emission than absorption. That requires \(N_x > N_0\) — a population inversion. This is not consistent with thermal equilibrium; we must use pumping to create and maintain it.
Why thermal equilibrium fails
For typical optical transitions (\(E_x - E_0 \sim 2\) eV) at room temperature (\(kT \approx 0.02\) eV), \(N_x/N_0 \sim e^{-100}\) is astronomically small. Achieving \(N_x/N_0 = 1/2\) would require \(T \sim 38\,000\) K — hotter than the Sun. Population inversion cannot be achieved by heating alone.
Metastable States#
Some excited states have much longer mean lifetimes (e.g., \(10^5\) times longer) than typical (\(\sim 10^{-8}\) s). These are metastable states. They are crucial for lasers because:
Atoms can accumulate in a metastable state long enough.
The lower level of the lasing transition can decay rapidly, so it stays relatively empty.
Together, these conditions allow population inversion.
The Helium–Neon Laser#
A common design uses a discharge tube filled with a 20:80 mixture of helium and neon. Neon is the lasing medium; helium is the pump.
Mechanism:
Electrons in the discharge collide with helium atoms; helium is light enough to be excited efficiently. Helium is raised to a metastable state \(E_3\) (lifetime \(\ge 1\) ms).
Helium–neon collisions: Energy of helium \(E_3\) (20.61 eV) nearly matches neon \(E_2\) (20.66 eV). Collisions transfer energy from helium to neon, populating neon \(E_2\).
Neon level E1 decays rapidly (10 ns); E2 has a longer lifetime (170 ns). So \(N_{E2} > N_{E1}\) — population inversion.
A spontaneously emitted photon triggers stimulated emission; other photons do the same. A coherent beam builds up.
Mirrors at the ends form a cavity; photons bounce back and forth, gaining more stimulated emission. One mirror is slightly leaky so part of the beam escapes as the laser output.
Output: 632.8 nm (red), continuous wave.
Population Ratio at Thermal Equilibrium#
For a laser that would emit at \(\lambda = 550\) nm (between ground and one excited state), the energy gap is \(E_x - E_0 = hc/\lambda \approx 2.26\) eV. At room temperature (\(T = 300\) K), \(kT \approx 0.026\) eV. Then
— an extremely small ratio. To get \(N_x/N_0 = 1/2\) by thermal agitation alone would require \(T \approx 38\,000\) K. Population inversion thus demands a specific pumping mechanism, not simply heating.
Summary#
Stimulated emission produces photons identical to the stimulating photon; it is the basis of laser operation.
Population inversion (\(N_{upper} > N_{lower}\)) is required for net amplification; it cannot arise from thermal equilibrium.
Metastable states and pumping (e.g., He–Ne collisions) enable population inversion.
The helium–neon laser uses helium to pump neon; neon lases between \(E_2\) and \(E_1\); mirrors form a cavity.