2. Laser physics - basics Spontaneous and stimulated processes Einstein A and B coefficients Rate equation analysis Gain saturation What is a laser? LASER: Light Amplification by Stimulated Emission of Radiation "light" could mean anything from microwaves to x-rays Essential elements: 1. A laser medium - a collection of atoms, molecules, etc. 2. A pumping process - puts energy into the laser medium 3. Optical feedback - provides a mechanism for the light to interact (possibly many times) with the laser medium The two-level atom Quantum energy levels excited state ground state Absorption: promotes an electron from the ground to the excited state Emission: drops the electron back to the ground state "spontaneous emission" - the decay of an excited state to the ground state with the corresponding emission of a photon Conservation of energy: Eexcited -Eground = Ephoton Three things can occur Absorption •Promotes molecule to a higher energy state •Decreases the number of photons Spontaneous Emission •Molecule drops from a high energy state to a lower state •Increases the number of photons •This is the only one that does NOT require a photon in the initial state Stimulated Emission •Molecule drops from a high energy state to a lower state •The presence of one photon stimulates the emission of a second one Relaxation of the two-level atom An atom in the excited state can relax to the ground state by: . spontaneous emission: rate is rad . any of a variety of non-radiative pathways: rate = nr All of these processes are single-atom processes; each atom acts independently of all the others. Thus, evolution of the excited state population only depends on the number of atoms in the excited state: dN e NN N dt rad e nr e10 e 10 = total spontaneous relaxation rate from state 1 to state 0 A collection of two-level atoms "Stimulated transitions" - a collective process involving many two-level atoms stimulated absorption: light induces a transition from 0 to 1 stimulated emission: light induces a transition from 1 to 0 In the emission process, the emitted photon is identical to the photon that caused the emission! Stimulated transitions: likelihood depends on the number of photons around How did it all begin? Rayleigh-Jeans law (circa 1900): energy density of a radiation field u() = 82kT/c3 Note: the units of this expression are correct. Strictly speaking, u() is an energy density per unit bandwidth, such that the integral ud gives an answer with units of energy/volume. Total energy radiated from a black body: ud uh-oh… the "ultraviolet catastrophe" Solution: quantum mechanics Time-dependent perturbation theory As a result of a perturbation h(t), a system in quantum state 1 makes a transition to quantum state 2 with probability given by: 2 t Notation : 1 iE t ' 21 E P12 2 ehtdt'' 21 21 Harmonic perturbation Key example: suppose we subject a two-level system, initially in state 1, to a harmonic perturbation, of the form: 00t (and suppose that the frequency of ht the perturbation, , is close to 21) 2sinAtt0 0 Transition probability to state 2 is: 2 2 t A0 it21 ' it'' it Peeedt12 2 ' RWA 0 2 it/2 it /2 4A2 etet21sin / 2 21 sin / 2 0 21 21 2 21 21 2 2 sin t / 2 Note that PP12 21 4A0 21 2 2 Absorption and stimulated 21 emission are equally likely! Einstein A and B coefficients Consider a radiation field and a collection of two-level systems, in thermal equilibrium with each other. stimulated emission probability: proportional to the number of atoms in upper state N2, and also to the number of photons spontaneous emission probability: proportional to N2, but does not depend on the photon density! Note: this is the WANBuN21 2 2 same as rad stimulated absorption probability: proportional to the number of atoms in lower state N1, and also to the number of photons spontaneous absorption: there is no such thing WBuN12 1 Quantum mechanics says that these two coefficients must be equal! But: in thermal equilibrium, the upward and downward transition rates must balance: WW12 21 Einstein A and B coefficients N ABu Equate these two rates: 1 NBu2 N EEkT But Boltzmann's Law tells us that 1 e21 (in equilibrium) N2 Recognizing that E2 E1 = h, we solve for u(): A B u h e kT 1 Also has units of energy This must correspond to the Rayleigh-Jeans result in the density per unit bandwidth classical limit (h 0), which implies: A 8 h 3 B c3 c3 Since A = , we can now solve for B also: B rad rad 8 h 3 Transition rates Our expression for the downward transition rate is now: WANBuN21 2 2 Bose-Einstein distribution B AN2 1 u A A B 1 But sinceu we therefore have h WAN21 21 hkT e kT 1 e 1 In other words, W21 is proportional to: 1 + the number of photons. It is easy to see that the upward transition rate, W12, is proportional to the number of photons: 1 WBuNAN12 1 1hkT e 1 Rate equation analysis e spontaneous emission: proportional to initial state population g stimulated transitions: stimulated spontaneous proportional to initial state population proportional to photon density np the same for upwards, downwards transitions dN dN e N KnKnN N g dt eg e PPge dt dNe egNKnNN e P g e dt Note: the constant K is simply given by h·B, where B is the Einstein B coefficient Rate equation analysis, continued dNe egNKnNN e P g e dt emitted photons go emitted photons go only into the in all directions direction of the incident light So, photon number varies according to: dnP KnPg N N e Kn P N dt Number of photons grows exponentially if Ne > Ng A LASER! Rate equation analysis, part 3 In thermal equilibrium: EkT Neegg N N Population inversion is impossible in equilibrium. In a steady-state situation: dNe egNKnNN e P g e 0 dt KnP NNNegg eg Kn P Population inversion is impossible in steady-state. So how do you make a laser? Four-level system Steps 1 and 2: Combine to give an effective 3 Non-radiative pumping rate for level 2: Rp decay Step 3: 2 stimulated transitions due to np Pump Lasing transition spontaneous decay rate: 1 21 Non-radiative Step 4: decay spontaneous decay rate: 0 10 dN2 Rp 21 N2 + Kn p (N1 N2 ) dt dN 1 N Kn (N N ) N dt 21 2 p 1 2 10 1 dN 0 N R dt 10 1 p The four-level model Steady-state solution: RP 21 10 NNN12 10 21 KnP Population inversion (i.e., N < 0) is assured if 10 > 21 (even if np = 0, and even if Rp is small) A necessary condition for lasing Other necessary conditions: • a resonant cavity - provides feedback • net gain per round trip > net loss per round trip - “threshold” Saturation in the four-level atom RP 21 10 21 10 1 N RP 10 21 KnP 21 101Wsig 21 • population inversion when > 2 1 Kn 1/ • small signal inversion is proportional to N0 p 21 the pump rate • inversion level drops when Wsig > 21 • the characteristic intensity for this effect is 1 0 0.8 independent of pump rate Rp N 0.6 "gain saturation" N/ 0.4 Note: Wsig is proportional to nP and therefore to the 0.2 intensity of the light in the medium. Thus, we can 0 0 2 4 6 8 10 define a saturation intensity Isat such that: Wsig 21 WIIsig 21 sat.