Amplification and saturation (Ch. 14) Nov. 29, 2017

• Gain and gain coefficient: Gain `G = {I_{out}}/{I_{"in"}} = e^{gamma L}`
Variation of `I` along the lasing medium --
`{dI}/{dz} = sigma(nu) Delta N I` where `sigma = g(nu) {A_{21} lambda_o^2}/{8 pi n^2}` and `Delta N = N_2 - g_2/{g_1} N_1 `
`{d phi}/ {dz} = sigma(nu) Delta N phi` where `phi = I/{h nu}` in the textbook.
Notice that `gamma (nu) = sigma(nu) Delta N` and `sigma(nu)` expression depends on the lineshape function, i.e. `gamma(nu) = sigma(nu_o) Delta N g_1 (nu)` where `g_1 (nu)` is a lineshape function normalized to have unit at the line center --
homogeneous `sigma (nu_o) = 2 /{pi Delta nu} {A_{21} lambda_o^2}/{8 pi n^2}`and `g_1 (nu) = {({Delta nu}/2)^2)/{(nu-nu_o)^2+({Delta nu}/2)^2}`
inhomogeneous `sigma (nu_o) = ({4 ln 2)/{Delta nu_D^2 pi})^{1/2} {A_21 lambda_o^2}/{8 pi n^2}` and `g_1 (nu) = e^{-4 ln2 ({nu - nu_o}/{Delta nu_D})^2)`.

• Nonideal amplifier: Gain saturation and phase distortion (dispersion)

• Phase distortion: Recall in Chapter 5, `gamma prop chi''` and `n prop 1+ {chi'}/2 `.
Since `chi' ~~ 2 {nu-nu_o}/{Delta nu} chi''` for Lorentzian lineshape, ` n prop 1+{nu-nu_o}/{Delta nu} chi''`.
Hence, any fluctuation in gain modulates phase per unit length `phi (nu) = {nu-nu_o}/{Delta nu} gamma(nu)` and modifies resonant condition, e.g. `k_o 2d + phi (nu) 2 l_{g} = 2 pi q` for F-P cavity with gain medium length `l_g`.
Such gain dependence of distortion / dispersion lead to the shifting longitudinal modes toward the central freq., known as freq. pulling.

• Lasing starting condition and steady state: Upset thermal equilibrium with pumping to achieve population inversion. Need to set up rate equations that allow us to see startup and saturation.
Gain=losses gives the threshold gain coef. `gamma = alpha_r` which is also the steady state gain coef. and `alpha_r =alpha_s + 1/{2d} ln (1/{cc{R}_1 cc{R}_2})` for F-P cavity.
Gain saturation starts when `N_2` is depleted. The gain profile reaches steady state value when `gamma = alpha_r` at lasing frequencies.
For homogeneous case, the whole gain profile will be drawn down since gain is provided by only a single group of atoms.
Homogeneous saturation video
For Inhomogeneous case, energy is extracted from different groups of atoms at various frequencies.
Inhomogeneous saturation video,
At steady state, "hole burning" occurs in frequencies at which groups of atoms resonate and `gamma = alpha_r`. The output power is proportional to the areas burned away.

• Rate equations:
These equation give analytical steady state solution `=>` good for CW laser.
Also numerical transient solution `=>` pulse laser.
Account for --
Spon. processes (radiative and nonradiative), `1/tau = 1/t_{sp} + 1/tau_{NR}`
Stimulated processes (emission and absorption)
Atomic systems (`N_1` and `N_2`)

2 level system: Before lasing starts, consider spon processes only --
`{dN_2}/{dt} = R_2 - (1/{tau_{21}} + 1/{tau_{20}})N_2=R_2 - 1/{tau_2} N_2`
`{dN_1}/{dt} = -R_1 + 1/{tau_21} N_2 - 1/{tau_1}N_1`
At steady state, `{dN_1}/{dt} ={dN_2}/{dt}=0`, initial population difference (inversion) is `(N_2 - N_1)_0 = (Delta N)_0 = R_2 tau_2 (1 - {tau_1}/{tau_21}) + tau_1 R_1`.
Enhance `Delta N` with `R_2 uarr`, `R_1 uarr`, `tau_{21}` >> `tau_1`. Ideally, `t_{sp} ~~ tau_{21}` and `tau_{21}` << `tau_{20} => t_{sp}~~tau_2` >> `tau_1`
Consider stimulated processes when lasing starts --
`{dN_2}/{dt} = R_2 - 1/{tau_2} N_2 -W_i N_2 + W_i N_1`
`{dN_1}/{dt} = -R_1 + 1/{tau_21} N_2 - 1/{tau_1}N_1 + W_i N_2 - W_i N_1`
At steady state, `Delta N = {(Delta N)_0}/{1+ tau_s W_i}` where saturation time constant `tau_s = tau_2 + tau_1 ( 1 - {tau_2}/{tau_21})`
Population inversion is saturated at high intensity or photon flux density.

4 level system: `tau_{32}` << `tau_{21} => N_3 ~~0`, `tau_1` << `tau_{21} => N_1~~0` and `R_1 =0`
4 level dynamics
`=> (Delta N)_0 = R tau_2 ( 1- {tau_1}/{tau_{21}}) ~~ R t_{sp}` and `tau_s ~~ tau_2`
`=> Delta N ~~ {R t_{sp}}/{1+ t_{sp} W_i}`
Consider `R` depends on `N_2` and `N_1` -- `N_{"total"} = N_{g} +N_1 + N_2+N_3 ~~ N_{g} + N_2 =>Delta N = N_2 -N_1 ~~ N_2`
`W` = Transition probability between levels 0 and 3 and ` R=(N_{g} - N_3)W ~~ (N_{"total"} - Delta N)W`
`=> Delta N ~~ {t_{sp} N_{"total"} W}/{1+t_{sp}W+t_{sp} W_i}`
Identify `(Delta N)_0 = {t_{sp} N_{"total"}W}/{1+ t_{sp} W}`and `tau_s ={t_{sp}} / (1+t_{sp}W}`

3 level system: `tau_{32}` << `tau_{21} => N_3 =0` and `N_{"total"} ~~ N_1 + N_2` Note: `Delta N=0` when `N_1=N_2={N_{"total"}}/2`, i.e. overhead with min. pump of `E_3 N_{"total"}/2 1/{tau_{sp}}`.
3 level dynamics
Here ` R_1 = R_2 =R`, `tau_1 -> oo` and `tau_2 = tau_{21}`
At steady sate, `0 = R_2 - 1/{tau_2} N_2 -W_i N_2 + W_i N_1` `=> N_2 = {tau_{21} (R+W_i N_{"total"})}/(1+2 W_i tau_{21}}`
`Delta N = N_2 - N_1 = 2N_2 -N_{"total"} = {2 tau_{21}R - N_{"total"}}/{1+2 tau_{21} W_i}`
Consider `R=(N_1 - N_3)W ~~ {N_{"total"} - Delta N}/2 W`
Identify `(Delta N)_0 = {N_{"total"} (t_{sp} W -1)}/{1+t_{sp}W}` and `tau_s = {2 tau_{21}}/{1+t_{sp}W}` `=> Delta N = {(Delta N)_0}/(1+tau_s W_i)`
Note: Saturation in population inversion and `tau_s` (relating to `Delta nu`).

• Gain saturation: Express `Delta N = (Delta N)_0/{1+ phi/{phi_s (nu)}}` where `tau_s W_i = tau_s phi sigma(nu) =>` saturation photon flux density `phi_s =1/{tau_s sigma(nu)}`.
Since `gamma = Delta N sigma(nu)`, `gamma(nu) = {gamma_o (nu)} /{1+phi/{phi_s(nu)}` where small signal gain coef. `gamma_o(nu) =(Delta N)_0 sigma(nu)`.
In terms of intensity `I=h nu phi`, `gamma(nu) = {gamma_o (nu)} /{1+(I/I_s) g_1 (nu)}` where `I_s = {h nu}/{tau_s sigma(nu_o))` is defined as the line center.
Owing to freq. dependent ` g_1 (nu)`, the homogeneous lineshape has photon flux (intensity) dependent width of `Delta nu_s = Delta nu sqrt {1+phi/{phi_s(nu_o)}}` or `Delta nu_s = Delta nu sqrt {1+I/I_s}`.
`{dphi}/{dz} = {gamma_o phi} /{1+phi/{phi_s(nu)}` `=> ln({phi(z)}/{phi(0)}) + {phi(z)-phi(0)}/{phi_s}=gamma_o z`
Let `z=d`, `X={phi(0)}/{phi_s}`, `Y={phi(d)}/{phi_s}`
`ln(Y) + Y = ln (X) + X + gamma_o d`
For `X ~~ 0`, ` Y~~0=> Y=X e^{gamma_o d}` (small signal `Y=XG`)
For `X` >> 1, `Y ~~ X+ gamma_o d` (saturation `G ~~1`)

• Gain saturation in inhomogeneously broadened media: `bar gamma_o (nu) = (Delta N)_o {lambda^2}/{8 pi t_{sp}} bar{g}(nu)`.
`bar gamma(nu)` is difficult to compute since `phi_s prop 1/{g(nu)` --
`bar gamma(nu) = (:gamma_{beta} (nu):)` where `gamma_{beta} (nu) = {(Delta N)_0 sigma(nu_o) g_{beta} (nu)}/ (1+ phi sigma(nu_o) tau_s g_{beta} (nu)}` and `g_{beta} (nu)` is the lineshape function for group `beta` of atoms.
For Doppler broadened medium, `gamma_{beta} (nu) = {(Delta N)_0 sigma(nu_o){Delta nu}/{2 pi}}/{(nu - nu_{beta}-nu_o)^2+ ({Delta nu_s}/2)^2}`
`bar gamma(nu) = int_{-oo}^{oo} gamma_{beta} (nu) p( nu_{beta}) d nu_{beta}` where `p(nu_{beta}) = (2 pi sigma_D^2)^{-1/2} e^{-{nu_{beta}^2}/{2 sigma_D^2}` and `sigma_D = {Delta nu_D}/(8ln2)^{1/2}`.
For `Delta nu_D` >> `Delta nu_s` and `nu=nu_o`, `bar gamma(nu_o) = {(Delta N)_0 sigma(nu_o)} / sqrt{1 + phi/{phi_s (nu_o)}}={(Delta N)_0 sigma(nu_o)} / sqrt{1 + I/I_s}`
Inhomogeneous broadening case, the effect of saturation is less severe owing to increases in both width and depth of the hole burning.

• Noise: Amplified spon. emission (ASE)
Number of photons per unit area over a solid angle `dOmega` and for propagation of one polarization over distance `dz` = `N_2 1/{t_{sp}} g(nu) B 1/2 {d Omega}/{4 pi}`
Since optical system obeys Poisson statistics, it has shot noise with variance of `sigma_s^2` that is proportional to average signal `bar n_s`.
Optical amplifier obeys Bose-Einstein statistics with ASE variance `sigma_{ASE}^2 = bar n_{ASE} + bar n_{ASE}^2`.
The convolution of both systems has a variance of `sigma_n^2 = bar n_s + bar n_{ASE} +bar n_{ASE}^2 + 2bar n_s bar n_{ASE}`.

Last Modified: Nov. 29, 2017
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