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
OR
{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.

For Inhomogeneous case, energy is extracted from different groups of atoms at various frequencies.
,
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

=> (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}}.

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}.