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() = 82kT/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 it21 ' it'' it Peeedt12'  2   RWA  0 2 4A2  etetit 21/2sin  / 2 it 21 /2 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  e21 (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  21 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  1hkT 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 dN e  NKnNN   dt eg e P g e Note: the constant K is simply given by h·B, where B is the Einstein B coefficient Rate equation analysis, continued

dN e  NKnNN   dt eg e P g e

emitted photons go emitted photons go only into the in all directions direction of the incident light

So, photon number varies according to: dn P  Kn N N Kn N dt Pg e P

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: dN e   NKnNN 0 dt eg e P g e

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 101Wsig 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