spontaneous stimulated emission and lasers

An atom has discrete energy levels; a transition between levels $E_2$ and $E_1$ involves absorption or emission of a photon with energy

$$h\nu=E_2-E_1.$$

In spontaneous emission, an excited atom decays randomly without an external photon; the emitted photon has arbitrary direction, phase, and polarization, so an ordinary light source is incoherent. In stimulated emission, a photon with energy $h\nu=E_2-E_1$ induces an excited atom to decay and emit a second photon with the same frequency, direction, phase, and polarization; these photons are coherent and reinforce each other. In thermal equilibrium, level populations obey the Boltzmann distribution

$$\frac{N_2}{N_1}=e^{-(E_2-E_1)/(k_B T)}<1,$$

so absorption dominates over stimulated emission. To amplify light, a population inversion $N_2>N_1$ is needed; it is created by pumping through a metastable level, and an optical resonator provides multiple passes and mode selection. This is how a laser works: stimulated emission in an inverted medium produces a coherent, monochromatic, and directional beam.

An atom transitions from level $E_2$ to level $E_1$; the energy difference is $E_2-E_1=1.96$ eV. Find the wavelength of the emitted photon. Use $hc=1240$ eV·nm.

At room temperature $T=300$ K ($k_B T\approx 0.0259$ eV), estimate the equilibrium population ratio $N_2/N_1$ for a transition with $E_2-E_1=1.96$ eV. Can such a transition provide laser gain without pumping? Justify your answer.

A three-level laser is pumped from ground level 0 to level 2 by photons of energy 3.0 eV. The atom quickly decays nonradiatively to metastable level 1, which lies 1.0 eV above level 0; stimulated emission occurs on transition $1\to 0$. Calculate the laser photon energy and the fraction of pump energy converted into laser light; explain where the remaining energy goes and why a four-level laser can have a lower threshold.

Use a safe interactive laser simulation (for example, PhET "Lasers" or an equivalent open model). Do not use real lasers and do not point a beam at people. Turn on the pump and record: (1) whether population inversion $N_2>N_1$ is achieved; (2) whether the photons become coherent and directional. Draw an energy-level diagram and label spontaneous emission, stimulated emission, absorption, pumping, and the resonator. Conclude: without population inversion there is no gain.

Show the difference between spontaneous and stimulated emission and the role of population inversion in a laser.

Diagram: three atomic energy levels; an upward arrow — absorption, a wavy arrow in an arbitrary direction — spontaneous emission, two parallel arrows in the same direction — stimulated emission. Next to it, a graph of $N_2/N_1=e^{-(E_2-E_1)/(k_B T)}$ versus $T$: in thermal equilibrium the curve lies below unity, and after pumping a point $N_2>N_1$ (inversion) is marked.

A cheap red laser pointer emits at wavelength $\lambda = 650\ \text{nm}$ with power $P = 1{,}0\ \text{mW}$. The energy of one photon of that wavelength is $E = \frac{hc}{\lambda} = \frac{6{,}63\cdot10^{-34}\ \text{J}\cdot\text{s}\times 3{,}00\cdot10^{8}\ \text{m/s}}{650\cdot10^{-9}\ \text{m}} = 3{,}06\cdot10^{-19}\ \text{J} = 1{,}91\ \text{eV}.$ In one second the pointer emits $N = P/E = 1{,}0\cdot10^{-3}/3{,}06\cdot10^{-19} \approx 3{,}3\cdot10^{15}$ photons, and nearly all of them carry the same energy, which is why the beam looks purely red with no admixture of other colours.

Stimulated emission is at work here: a photon passing an excited atom of the semiconductor triggers the release of a second photon with the same energy, phase and direction, and the number of such copies grows like an avalanche. Spontaneous emission supplies the very first photons that start this avalanche, while the pump current maintains the population inversion — more atoms excited than not. The fact that every photon carries the energy of the level gap, $h\nu = E_2 - E_1$, is what makes the beam monochromatic, and the shared phase is what makes it coherent.

A coach measures the distance from the goal to the corner spot with a laser rangefinder and reads $L = 100{,}0\ \text{m}$. Light covers the round trip $2L = 200{,}0\ \text{m}$ in $t = \frac{2L}{c} = \frac{200{,}0}{3{,}00\cdot10^{8}} = 6{,}67\cdot10^{-7}\ \text{s} = 667\ \text{ns},$ so the device's electronics must count intervals of a few hundred nanoseconds. The beam divergence of such a device is about $1\ \text{mrad}$, which means that at $100\ \text{m}$ the light spot is only $0{,}10\ \text{m}$ across — smaller than a football.

Such a short, well-shaped pulse is possible only because every act of stimulated emission creates a photon copy: the same energy, phase and direction as the photon that travelled along the resonator. That is why the pulse does not spread out over 200 m and stays inside a narrow beam. The rangefinder creates the population inversion for a brief moment with a strong pump flash, and the monochromaticity lets the receiver cut off sunlight with a narrow-band filter.

An airport ceilometer lidar sends a pulse into the sky at wavelength $\lambda = 1064\ \text{nm}$ with energy $10\ \mu\text{J}$. The energy of a single photon is $E = \frac{hc}{\lambda} = \frac{1240\ \text{eV}\cdot\text{nm}}{1064\ \text{nm}} = 1{,}17\ \text{eV} = 1{,}87\cdot10^{-19}\ \text{J},$ so one pulse carries $10\cdot10^{-6}/1{,}87\cdot10^{-19} \approx 5{,}4\cdot10^{13}$ photons. If the base of the cloud lies at height $H = 2000\ \text{m}$, the reflected signal comes back after $t = \frac{2H}{c} = \frac{4000}{3{,}00\cdot10^{8}} = 1{,}33\cdot10^{-5}\ \text{s} = 13{,}3\ \mu\text{s},$ and from that time the instrument computes the height of the cloud layer.

The energy of each photon is fixed by the level gap of the working transition, $h\nu = E_2 - E_1$, so a lidar chooses its wavelength by choosing the active medium rather than by using filters. Stimulated emission copies the direction, so the whole pulse flies upwards within a solid angle of about a milliradian and does not light up the entire sky; the monochromaticity lets the receiver recognise 'its own' photons by wavelength among scattered sunlight. Spontaneous emission is the noise source here: some photons escape sideways and do not amplify the pulse.