space-independent model 
calculations of threshold power and slope eff.

Constants

J=1;m=1;s=1;
um=(10^(-6)) m;mm=(10^(-3))m;cm = (10^(-2)) m;nm=(10^(-9))m;
ms=(10^(-3));us=(10^(-6));ns=(10^(-9));ps=(10^(-12));
GHz=(10^9);
eV=1.6 (10^(-19)) J;
W=1;mW=0.001;
hbar=(10^(-34))J s;h=2π hbar;
c=3 (10^8) m/s;

common expressions

Logarithmic loss calculation. In Svelto’s formulation, lossFactor is the linear loss per pass, not per round trip. For example, if there is 2% loss per pass, then γLoss is defined so that Exp[-γLoss]=1-0.02

γ = overall single-pass logarithmic cavity loss, includes internal loss and OC = γInt + γOC/2. 
       Any mirror transmission other than OC is lumped into internal loss.

γLoss[lossFactor_]:=-Log[1-lossFactor];

photon energy

ePhot[λPhot_]:= hbar ((2 π c)/(λPhot));

pump absorption efficiency, longitudinal pumping, i.e. collinear with the laser mode. 
Pump transmission is Exp[-α L], so absorption is 1-Exp[-α L]

ηAbsLong[αAbs_,len_]:=1-Exp[-αAbs len];

Apump = area of pumped medium, assumed even, top hat distribution
Abeam = area of oscillator beam inside gain, assumed even, top hat distribution

space-independent model - 4 level system

threshold pump power
pTh=(1/(ηp))γ (1/σ)Aeff ((Subscript[hν, p])/τ)
another equivalent way to express this is 
pTh=(1/(ηp))γ Aeff ((Subscript[I, sat])/(Subscript[η, S])), where (Subscript[η, S])=((hbar ωLase)/(hbar ωPump))

Aeff = effective area. 
- For Gaussian pump and laser beams, this is the average of the areas of the pump and laser beams, see Svelto (6.3.20), see also (7.3.12)
- For flat-top pump, and laser mode inside pumped area, set Abeam = Apump. 

ηp = overall pumping efficiency
   must include terms that will be part of the “input power” for the measured slope efficiency, (Subscript[η, s])
   (Subscript[η, S]) is separately accounted for in the expression for the threshold power, so don’t include it as part of the pump eff. 
   ηp would include quantum efficiency in transferring pump excitation to the upper state, pump transfer efficiency from the beam to the crystal, spectral overlap efficiency for broadband lamp pumping

pTh[ηp_,γ_,Apump_,Abeam_,ePhPump_]:=(1/(ηp))γ (1/σ)((Apump+Abeam)/2) (ePhPump/τ);

slope efficiency

ηs[pThresh_,γOC_,Abeam_]:=((Abeam iSat)/pThresh) ((γOC)/2);

output power

pOut[pIn_,ηSlope_,pThresh_]:=ηSlope pThresh ((pIn/pThresh)-1);

example: you will typically be using the functions for ηSlope[ ] and pThresh[ ] defined above to calculate the output power as a function of input power for a given laser. 

x-intercept is the pump threshold
slope of line is the slope efficiency - a measure of how effective input power is converted to output power.

Plot[pOut[pIn, .2,1],{pIn,0,4}]

negative power below this is meaningless, so we could truncate:

Plot[Max[0,pOut[pIn, .2,1]],{pIn,0,4}]

Nd:YAG

define laser parameters: each laser system will be different
λLase = laser wavelength of laser being pumped
λPump = pump laser wavelength (assuming laser pumping)
σ = peak effective gain cross-section
τ = effective lifetime of upper state (not just fluorescence)
αp=absorption coefficient for pump beam
iSat = saturation intensity = ((hbar ωLase)/(σ τ))
fSat = saturation fluence ((Subscript[Γ, sat]) in class notes) = ((hbar ωLase)/σ)

Nd:YAG parameters

λLase=1064nm;
ePhotLase=ePhot[λLase];
λPump=808.nm; 
ePhotPump=ePhot[λPump];
nCry=1.76;

σ=2.8 (10^(-19)) (cm^2); (* Svelto table 2.2 *)
τ=230us;(* Svelto table 2.2 *) 
αp = 8./cm; (* absorption coefficient for pump *)
iSat=(ePhotLase/(σ τ));
fSat=(ePhotLase/σ);
Print["laser photon energy: ", ePhotLase/eV," eV"];
Print["saturation intensity: ", iSat (cm^2)," W/(cm^2)"];
Print["saturation fluence: ", fSat (cm^2)," J/(cm^2)"];

questions to answer

how thick do we need to make the crystal so that 95% of the pump beam is absorbed?

using this thickness, suppose we pump the crystal with 1W at 808nm. Approximately what pump beam diameter do we need to use to get a small-signal gain of 2? Remember to correct for absorption efficiency and for the ratio of photon energies.

representative laser system: 
match pump and laser mode sizes, with w0=140um, absorption efficiency of 95%
Assume 1.0% passive loss and 5% output coupling loss. 
Calculate the threshold pump power and the slope efficiency.

plot output power vs input power for different values of the output coupler transmission: 
0.01, 0.05 and 0.10
or set up a Manipulate[ ] plot
Also calculate the minimum power required for lasing by setting the output coupler transmission close to zero, e.g. (10^(-6))

for an input pump power of 0.5W, plot the output power as a function of the output coupling transmission to find the optimum OC