Laser power and efficiencies (Ch. 15) Dec. 4, 2017

• Book mostly considers homogeneously broadened medium in F-P cavity. All discussions are assumed to be for this case unless further notice is given.

• Review laser requirements:
1) gain -- gamma(nu) = {gamma_o (nu)} / {1 + phi/phi_s} where gamma_o (nu)= (Delta N)_0 {lambda_o^2}/{n^2 8 pi t_{sp}} g(nu) and phi_s = 1/{tau_s sigma(nu). phi (nu) = {nu-nu_o}/{Delta nu} gamma (nu)
2) Optical feedback -- alpha_r =alpha_s + 1/{2d} ln (1/{cc{R}_1 cc{R}_2})=alpha_s + alpha_m where alpha_m is the mirror loss. tau_p = n/{alpha_r c_o}, Delta nu_{FSR} = c_o/{2nd}, cc{F} = {Delta nu_{FSR}}/{delta nu}
3) Pump -- Delta N prop R

• Lasing oscillation: Amplitude condition -- gamma_o(nu) > alpha_r => (Delta N)_0 > Delta N_t where threshold population difference Delta N_t = alpha_r/{sigma(nu)} ={8 pi n^3}/{lambda_o^2 c_o} {t_{sp}}/tau_p 1/{g(nu))
Lowest Delta N_t at nu=nu_o at which Delta N_t = {2 pi alpha_r n^2}/{lambda_o^2} if nonradiative processes are neglected Delta nu = 1/{2 pi t_{sp}}.

• Power of lasers: Initially, the laser has small signal gain coef. gamma_o (nu) > alpha_r.
When it lases, gamma (nu) drops to alpha_r at steady state with the mean flux density phi = { ( phi_s (nu) ( {gamma_o (nu)}/{alpha_r} - 1), gamma_o (nu) > alpha_r), (0, gamma_o (nu) <= alpha_r):}
OR
phi = { ( phi_s (nu) ( {(Delta N)_0 }/{Delta N_t} - 1), (Delta N)_0 > Delta N_t), (0, (Delta N)_0 <= Delta N_t):}
Notice that internal flux density inside a F-P cavity phi = phi_+ + phi_- consists of phi_+  and phi_- corresponding to flux densities from forward and backward directions. For output, we only collect phi/2 since photons escape from one of the mirrors (mirror 1 as the output coupler) with power transmittance cc{T} in one direction:
Output flux density phi_o = cc{T} phi /2
OR
Output power P_o = h nu_q cc{T} phi /2 A
A is the cross sectional area.

• Optimal output coupler: Maximize output while avoiding too large in alpha_r (limited by pumping).
phi_o = phi_s/2 cc{T} [ {gamma_o (nu)}/{alpha_s +alpha_{m2}- 1/{2d} ln(1-cc{T})} - 1] = phi_s/2 cc{T} [ {g_o }/{L- ln(1-cc{T}) } - 1] ~~phi_s/2 cc{T} [ {g_o }/{L+cc{T})] for highly reflective mirror 1.
{d phi_p}/{d cc{T}} = =>(L+cc{T})^2 = g_o L =>cc{T}_{op} = sqrt{g_o L} -L.

• Relation between output photons and pumping: Any additional pumping R beyond the threshold level R_t will appear as USEFUL loss that generates output photons measured by photon number density n_{ph} = phi/c (m{::}^-3).
Re-write equation for phi into --
n_{ph} = (n_{ph} )_s ( {(Delta N)_0 }/{Delta N_t} - 1) for  (Delta N)_0 > Delta N_t where saturation value for photon density number (n_{ph} )_s = 1/ {c tau_s sigma (nu)}
=> n_{ph} / tau_p = {(Delta N)_0 - Delta N_t}/tau_s = R - R_t for R > R_t
Notice that R t_{sp} ~~ (Delta N)_0 ~~R tau_s.

• Efficiency: Output photon flux Phi_o = eta_e (R - R_t ) V where V is the volume and extraction (coupling) efficiency eta_e = alpha_{m1} / {alpha_r} = c tau_p alpha_{m1} = {c tau_p} / {2d} ln (1/cc{R}_1) ~~ {c tau_p} / {2d} cc{T}for cc{T} << 1.
In terms of power, output power P_o = h nu Phi_o.
A few figures of merit for efficiency --
$-$ Best scenario in terms of pump energy and output energy, quantum efficiency eta_q = nu / nu_{pump}.
$-$ Worst scenario in terms of optical power and electrical power power P_p, power conversion also known as overall or wall-plug) efficiency eta_c = P_o/P_p.
Notice that pumping rate R = {eta P_p} / {h nu V} where eta is the conversion efficiency of electrical power to pump photons.
$-$ In between measurement in terms of the slope of the optical output characteristic, differential power conversion (or slope) efficiency eta_s = {dP_o} /{d R}.
Notice that eta_s prop eta_q eta_e and may not be unitless.

• More accurate power description for F-P cavity: Recall internal intensity of a F-P cavity consists of forward wave intensity I_+ and backward wave intensity I_- $-$ {dI_+}/{dz} = gamma I_+ and {dI_-}/{dz} = - gamma I_- where gamma (z) = gamma_o / {1+ {I_+ + I_-}/I_s}.
Notice that {d(I_+ I_-)}/{dz} = I_{-} {dI_+}/{dz} + I_+ {dI_-}/{dz}=0 => I_+ I_{-} = constant =c. Hence, gamma (z) = gamma_o / {1+ {I_+ + c/{I_+}}/I_s} or gamma (z) = gamma_o / {1+ {c/{I_-} + I_-}/I_s}
Solution for the ODE $-$
ln (I_2 / I_1) + {I_2 - I_1} / I_s - c /I_s (1/I_2 - 1/I_1) = gamma_o d
and
ln (I_4 / I_3) + {I_4 - I_3} / I_s - c /I_s (1/I_4 - 1/I_3) = gamma_o d
where forward wave intensities I_1 = I_+ (0) and I_2 = I_+ (d) relate to backward wave intensities I_3 = I_+ (d) and I_4 = I_+ (0) by the boundary conditions I_3 = cc{R}_2 I_2 and I_1 = cc{R}_1 I_4 that replace the constant with c ={I_1^2}/{cc{R}_1} = cc{R}_1 I_4^2 and c = {I_3^2}/{cc{R}_2} = cc{R}_2 I_2^2
Finally, I_2 = I_s / {(1 - sqrt{cc{R}_1 cc{R}_2})(1+ sqrt{cc{R}_2 / cc{R}_1})} (gamma_o d - ln (1 / sqrt{cc{R}_1 cc{R}_2})) and P_o = I_2 A cc{T}.

• Spectral distribution: For homogeneous broadened medium, initial number of mode M = B/ {Delta nu_{FSR}} where B is the gain bandwidth at which gamma > alpha_r.
At steady state, the whole gain curve drops so that only a few modes or only one mode can be sustained.
However, multimode lasers prevail since high order modes have different spatial extents from TEM{::}_00 mode in the gain medium that can support other modes in the low intensity regions of TEM{::}_00. This can be explained in terms of spatial hole bunking and consider gamma (nu) = {gamma_o (nu)} / {1+ sum_{j=1}^M phi_j/{phi_s (nu_j)}}
For inhomogeneously broadened medium, each group of atoms interact with certain cavity mode. Particularly for Doppler-broadened medium, each mode interact with 2 groups of atoms when velocity v ne 0 in a F-P cavity nu_q = nu_o +- v/c nu_p.
For v=0, there is only one group providing energy. As a result, Lamb dip occurs at nu_o.

Spatial filter for transverse modes $-$ pin holes
Polarization filter $-$ Brewster window
Spectral filter for longitudinal modes $-$ intracavity etalon, externally coupled cavity
Tuning these filter allows limited freq. tuning.

• Pulse lasers: Store energy and release it in a burst, i.e. high peak power but usually low average power.
Gain switching -- modulator controls pump power.
Q-switching -- an absorber decreases the Q of a cavity by introducing high loss and building up Delt N.
Cavity dumping -- the output coupler becomes completely reflective to build up photons by stopping their escape and completely remove a mirror for a short period to output.
Mode locking -- a saturable absorber is used to gate all modes and lock on their phases so that they can be added coherently to form a short pulse, similar to the Fourier series.

• Transient analysis