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):}`
`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`
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`
`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`.

• Radiation control:
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

Last Modified: Nov. 29, 2017
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