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

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

2 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

3 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

4 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!

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

6 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.

7 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.

8 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).

9 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.

10 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

11 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!x k ( x x 0 ) 2 which have quantized energy levels: E n =!ω n + 1 2, n = 0,1,2,3... ω k / m

12 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

13 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.

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