Black body radiation and line shapes (Ch. 13) Nov. 15, 2017

Atomic radiation based on traditional approach from blackbody radiation (Laser Electronics 2nd Ed. Verdeyen)

• Radiation from a hot object: Black body is also a perfect radiation absorber.
Wien’s law (Classic result) gives peak emission wavelength `lambda_{peak} T = 2.8978 times 10^{-3}` (m-K)

• Relating spectral energy density `rho ( nu , T)` (J/m`{::}^-3`-Hz) and intensity spectrum `u( nu , T)` (W/m`{::}^2`-Hz): `c_o rho = u`
`u` and `rho` can be expressed in terms of `lambda`, i.e. `rho (lambda, T)`, `u(lambda, T)`.
Notice that intensity `I = int u(lambda, T) dlambda` or `int u(nu,T) dnu`
For blackbody, radiation only emit to the front making an angle with the optical axis and its intensity spectrum `u_{BB} (lambda, T) = rho (lambda, T)c_o/4`

• Rayleigh-Jeans catastrophe: Energy density `rho(nu) dnu = kT M(nu) dnu= {kT 8pi n^3 nu^2) / c_o^3 dnu` Energy density increases with freq without upper limit, not reflecting experimental results.

• Quantization of energy: Planck assigned probability density for each discrete energy following Boltzmann distribution `e^{- {h nu}/{kT}` and compute average energy level `(:E:) = (sum_{l=0}^{oo}l h nu e^{-{l hnu}/{kT}}}/{sum_{l=0}^ooe^{-{l h nu}/{kT}}} = {h nu} / (e^{{h nu}/{kT}}-1}`
Energy density `rho (nu) = {h nu} / (e^{{h nu}/{kT}}-1} ({ 8pi n^3 nu^2) / c_o^3) dnu`
Intensity `u_{BB} (lambda) d lambda = {h } / (e^{{h nu}/{kT}}-1}(2 pi n^3c_o^2)/(lambda_o^5) dlambda`

• Spontaneous emission: No input, noise-like, nondirective `{dN_1}/{dt} = - {dN_2}/{dt}=A_21 N_2`

• Stimulated emission: Input photons, direction, output following input’s polarization, freq and phase, i.e. light amplification, `{dN_1}/{dt}= - {dN_2}/{dt} = B_21 N_2 rho(nu)`

• Absorption: freq and polarization dependent, `{dN_2}/{dt}= - {dN_1}/{dt} = B_12 N_1 rho(nu)`
where `N_1` is the number density of atoms in lower state and `N_2` is the number density of atoms in upper state.

• Einstein’s approach relating radiative processes: Detailed balance -- emission rate = absorption rate under thermal equilibrium, i.e. $\frac {dN_1} {dt} |_{emission} + \frac {dN_1} {dt} |_{absorption} =0$
As `T-> oo`, `rho(nu) -> oo` and `B_21 g_2 = B_12 g_1`. where we consider `g_i` (degeneracy), the number of ways that an atom can have energy `E_i = h nu_i` and `N_i = N_0 g_i e^{- E_i / {kT}}`
`rho (nu) = {A_{21}}/{B_{21}}{1} / (e^{{h (nu_2 - nu_1 )}/{kT}}-1}`, `{A_{21}}/{B_{21}}= ({ 8pi n^3h nu^3) / c_o^3)`

Interpret `A_{21} = 1/{t_{sp}}` as spontaneous emission rate where `t_{sp}` is the spontaneous emission lifetime.
Note: `g_i = 2 J_i + 1` for `i = 1, 2` where `J_i` is the angular momentum of the energy level.

• More complete approach: The text book consider probability density or rate of spon. emission for a mode `p_{sp} = {c sigma(nu)}/V` (s`{::}^-1`) where `V` is the volume and `sigma(nu)` is the transition cross section in m`{::}^{2}`.
Probability of emission between `t` and `t+dt` is `p_{sp} Delta t`. The spatial distribution of `sigma` depending on the angle `theta` between dipole moment of the atom and the field, `sigma = sigma_{max} cos^2 theta`.
The probability density or rate of stimulated emission and absorption for a mode can be defined similarly, except that these processes have number of photons n, i.e. `P_{ab} = P_{st} ={n c sigma (nu) }/V = W_i`.

• Relationship between lines shape `g(nu)` and `sigma (nu)`: After normalizing `sigma(nu)` by oscillator strength `S= int_0^oo sigma (nu) d nu`, `g(nu) dnu = {sigma(nu)} / S dnu` will give the probability of spon. emission between `nu` and `nu + Delta nu`.
Hence, `g(nu)` is a probability density function, i.e. `int_0^{oo} g(nu) dnu = 1`. It peaks at `nu_o` (the transition freq.) and has FWHM `Delta nu`
Notice that field’s bandwidth from `rho ( nu )` is limited by that of the cavity `delta nu`.

• General rate equation for light amplification: `{dN_2}/{dt}= -A_{21} N_2 int_0^{oo} g(nu' ) dnu' +B_12 N_2 int_0^{oo} rho(nu' ) g(nu' ) dnu'` ` - B_21 N_2 int_0^{oo} rho(nu' ) g(nu' ) dnu' `
Einstein considered `delta nu` >> `Delta nu` and treated `g(nu)` like a delta function.
Book concentrated on the laser case `Delta nu` >> `delta nu` and `rho (nu ') = rho_{nu} delta (nu' - nu)`
`{dN_2}/{dt}= -A_{21} N_2-sigma(nu) {I_nu}/{h nu} [N_2 - {g_2}/{g_1}N_1] = -A_{21} N_2-W_i [N_2 - {g_2}/{g_1}N_1]`

where `sigma(nu)={A_{21} lambda_o^2 g(nu)}/{8 pi n^2}`,`phi = {I_{nu}}/{h nu}` and `W_i = phi sigma(nu)`.

• Total spon. emission to all modes: Consider spatial average `bar sigma (nu) = {sigma_{max} (nu) }/3`, `P_{sp}= int_0^{oo} c/V bar sigma (nu) V M(nu) dnu`. For blackbody, we have Einstein case where `g(nu)` is like a delta function.
`P_{sp} = 1 /{t_{sp}} ~~ M(nu_o) c bar S` where `bar S =int_0^{oo} bar sigma (nu)dnu = {lambda_o^2} / {8 pi t_{sp} n^2}` and `bar sigma (nu) = {lambda_o^2 g(nu)} / {8 pi t_{sp} n^2}`.

• Stimulated emission relating to blackbody radiation: `W_i=int_0^{oo} {rho(nu)V}/{h nu} {c sigma(nu)}/V dnu`
Blackbody is broadband `delta nu` >> `Delta nu -> g(nu) ~~ delta (nu - nu_o)`
`W_i = {bar n } / t_{sp}`

where number of photons per mode `bar n = {lambda_o^3 rho (nu_o)} / {8 pi n^3 h}`

Line broadening: Uncertainty of energy levels `Delta E = Delta E_1 + Delta E_2= h/{2 pi} ( 1/{tau_1} + 1/{tau_2))` where `tau_{1,2}` is transit lifetime.
In another words, `Delta nu = 1/{2 pi} ( 1/{tau_1} + 1/{tau_2))`

• Homogeneous broadening: The linewidth is widened by an effect that applies equally to all atoms, e.g. collisional (pressure) broadening in gas, lifetime broadening (radiative lifetime), and phonon broadening.
Lifetime of the excited atoms with `N_2` population is `tau_2` which follows `tau_2^{-1} = tau_{NR}^{-1} + tau_{R}^{-1}` where `tau_{NR}` is nonradiative lifetime from processes not generating light and `tau_R` is radiative lifetime from light generating processes.

• Lineshape function for homogeneous broadening: Since decay in excited atoms or power corresponds to field `e(t) = e_o e^{-t/{2 tau_2}} cos omega_o t`, spectral intensity `|E(omega)|^2` ~ `1/{(omega - omega_o)^2 + (gamma/2)^2}` from Fourier transform where `gamma = 1/tau_2`
This reminds us the Lorentzian lineshape `g(nu)={Delta nu}/{2 pi} 1/{(nu - nu_o)^2+ ({Delta nu}/2)^2}` where `Delta nu = gamma /{2 pi}` and `g(nu_o) = 2/{Delta nu pi}` Also `bar sigma_o = {lambda_o^2} / {2 pi n^2} 1 / {2 pi t_{sp} Delta nu}` If `tau_2` is entirely radiative, `bar sigma_o = {lambda_o^2} / {2 pi n^2}`

Characteristics of homogeneous broadening: Line shape can change with radiation (photons) and will scale down as a result of saturation, i.e. `g(nu_o)` decreases.

• Collisional broadening: Collision interrupts phase and decreases the coherent lifetime.
Probability of finding collision free atoms `=p(t) dt =1/{tau_c} e^{-t/{tau_c}}dt` where `tau_c` = mean time between collisions.
For a gaseous mixture of type m and n, freq of collision `f_{col} = 1/{tau_c} = N (:sigma v:)=N_m sigma [{8kT}/pi (1/{M_m}+1/{M_n})]` where v is mean velocity, `sigma` is cross section (function of v), N is the number density, `N_m` is the number density of type m, `M_m` is mass of type m atom and `M_n` is mass of type n atom.
Compute correlation of field and take Fourier transform to find intensity

`(:e(t)e(t+tau):) = e_o^2 cos omega_o tau e^{-{|tau|}/{tau_c}}`

I ~ Fourier transform of `(:e(t)e(t+tau):)=1/{(omega-omega_o)^2+(1/tau_c)^2}`

Lorentzian width = `2/ {2pi tau_c}`
Overall lineshape function (homogeneous): `g(nu) = {Delta nu_{"total"} }/{2 pi} 1/{(nu-nu_o)^2+({Delta nu_{"total"}}/2)^2}` where `Delta nu_{"total"} =1/{2pi} ( 2/tau_c+ 1/tau_2)`

Last Modified: Nov. 15, 2017
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