Black body radiation and line shapes (Ch. 13) Nov. 15, 2017 |
(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}`
•
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
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)`