Today: general condition for threshold operation physics of atomic, vibrational, rotational gain media intro to the Lorentz model

Today: general condition for threshold operation physics of atomic, vibrational, rotational gain media intro to the Lorentz model

Laser operation Simplified energy conversion processes in a laser medium: a) pump excitation (net: may involve other levels) b) spontaneous emission c) stimulated emission d) absorption (also a stimulated process) e) non-radiative deexcitation (may involve other levels) These all operate dynamically in a laser

Operation at threshold: onset The number of photons in a laser cavity changes for two main reasons: 1. photons are added to the laser mode by stimulated emission 2. photons leave the laser mode by losses : emission through a partial mirror, absorption, scattering For photon number q(t), we can represent these as: dq dt = anq bq with pumping p, we can write the number n of excited atoms as: dn dt = anq fn + p

Operation at threshold: onset These are distinct, but coupled, equations: dq dt dn dt = anq bq At steady state, there s no time change, so n t n = b a = anq fn + p Threshold number of excited atoms using this in the number equation, q = p b f a Threshold for pumping rate: q cannot be less than zero at onset, with q =0 we find the pumping threshold p t = fb a = fn t Must pump equal to losses, at threshold excitation!

The vast majority of light in the universe comes from molecular vibrations emitting light. Electrons vibrate in their motion around nuclei High frequency: ~10 14-10 17 cycles per second. Nuclei in molecules vibrate with respect to each other Intermediate frequency: ~10 11-10 13 cycles per second. Nuclei in molecules rotate Low frequency: ~10 9-10 10 cycles per second.

Atomic and molecular vibrations correspond to excited energy levels in quantum mechanics. Energy levels are everything in quantum mechanics. Excited level Energy ΔE = hν Ground level The atom is vibrating at frequency, ν. The atom is at least partially in an excited state.

Excited atoms emit photons spontaneously. When an atom in an excited state falls to a lower energy level, it emits a photon of light. Excited level Energy Ground level Molecules typically remain excited for no longer than a few nanoseconds. This is often also called fluorescence or, when it takes longer, phosphorescence.

Different atoms emit light at different widely separated frequencies. Each colored emission line corresponds to a difference between two energy levels. These are emission spectra from gases of hot atoms. Frequency (energy) Atoms have relatively simple energy level systems (and hence simple spectra).

Atoms and molecules can also absorb photons, making a transition from a lower level to a more excited one. Energy Excited level Ground level This is, of course, absorption. Absorption lines in an otherwise continuous light spectrum due to a cold atomic gas in front of a hot source.

Atomic energy levels: H Hydrogen & hydrogenic ions, have Bohr levels: 1 E n = 4πε 0 me 4 1 2! 2 n 2 so transition energies for radial states are: E n E n = 1 4πε 0 more generally there are angular momentum states s, p, d, f and multi-electron atoms are even more complex, so real atomic states have more complex distributions me 4 1 2! 2 n 1 2 n 2

Molecular vibrational energy levels molecules have stable binding distances, thus a potential minimum where net force is zero Taylor series expansion around that separation has a leading quadratic term. thus for small oscillations, molecules are simple harmonic oscillators: E = 1 2 m!x2 + 1 2 k ( x x 0 ) 2 which have quantized energy levels: E n =!ω n + 1 2, n = 0,1,2,3... ω k / m

Molecule rotational energy levels the classical energy of a dumbbell in rotation is: rotational energy states are likewise quantized, so that state levels make a manifold: E J =!2 2I E = 1 2 Iω 2 R = L2 2I J(J +1), J = 0,1,2,... Together with vibrational states, the rovibrational states are a complex manifold, leading to elaborate spectra https://en.wikipedia.org/wiki/rotational_spectroscopy

Molecules have many energy levels. A typical molecule s energy levels: 2 nd excited electronic state 1 st excited electronic state Energy E = E electonic + E vibrational + E rotational Lowest vibrational and rotational level of this electronic manifold Excited vibrational and rotational level Ground electronic state Transition There are many other complications, such as spin-orbit coupling, nuclear spin, etc., which split levels. As a result, molecules generally have very complex spectra.