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Photons and atoms!
2
Photons and atoms!
Energy levels in atoms, molecules and solids!
!
Occupation of energy levels!
!
Interaction of photons with atoms!
Spontaneous emission!
Absorption!
Stimulated emission!
Line broadening!
Fundamentals of Photonics, Ch. 13!
3
Light interacts with matter!
Matter contains electric charges that are affected by the time-varying
electric field of light.!
Atoms, molecules and solids have allowed energy levels. A photon
can interact with an atom of its energy matches the difference between
two levels.!
!
Atomic energy levels (Bohr)!
Boltzmann statistics (Boltzmann)!
Blackbody radiation (Planck)!
Stimulated emission (Einstein)!
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Energy levels!
The behavior of a particle of mass m subject to a potential V(r) that
does not change with time is governed by the time-independent
Schrödinger equation!
2 2
−
∇ ψ(r) +V (r)ψ(r) = Eψ(r)
2m
The solutions (eigenvalues E) provide the allowed values of the energy
of the system. These can be:!
•  discrete
(atom)!
€
•  continuous (free particle)!
•  bands (semiconductor)!
Light makes the system move from one level to another.!
Energy levels
Occupation of energy levels
Photon-atom interaction
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Energy levels in atoms!
are due to the potential energies of electrons in the presence of the
nucleus and the other electrons, orbital and spin angular momenta!
Ex.: Hydrogen-like atom in the “Bohr theory”!
!
Coulomb potential! V (r ) = −Ze 2 / r
!
Discrete energy levels!
2 4
M
Z
e
1
E n = − € r 2 2 2 , n = 1,2,3,…
(4πε 0 ) 2 n
n = principal quantum number
Energy levels
Occupation of energy levels
Photon-atom interaction
6
Multielectron atoms!
The energy levels are occupied in accordance to Pauli exclusion
principle and by filling successive shells associated to the quantum
numbers n = 1,2,3… and l = 0,1,… n-1.!
!Example: Neon (10 electrons) – 1s2 2s2 2p6!
Energy levels
Occupation of energy levels
Photon-atom interaction
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Multielectron atoms show
periodic properties!
Energy levels
Occupation of energy levels
Photon-atom interaction
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Energy levels in molecules!
•  Potential energies associated with interatomic forces!
•  Rotational (µwave, IR), vibrational (IR), electronic (V, UV)!
Energy levels
Occupation of energy levels
Photon-atom interaction
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Energy levels in dyes!
•  Dye = large, complex molecules!
•  They have a large number of energy levels and transitions!
Rhodamine 6G
Energy levels
Occupation of energy levels
Photon-atom interaction
10
Energy levels in solids!
•  Influenced by potentials of individual atoms and neighboring atoms!
•  As atoms are brought together in a solid, their individual energy levels
!broaden.!
Energy levels
Occupation of energy levels
Photon-atom interaction
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Broadening of energy levels in solids!
Energy levels
Occupation of energy levels
Photon-atom interaction
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Doped dielectric media!
Solid state lasers are typically made using doped media:!
•  a transparent dielectric media (“host”)!
•  an active laser atom or ion (“dopant”)!
Energy levels may be affected by the host media depending on the layout of the active electrons:!
•  not much in the case of lanthanide-metals (Nd:YAG, Nd:glass…)!
•  significantly in the case of transition metals (ruby = Cr:Al2O3…)!
Energy levels
Occupation of energy levels
Photon-atom interaction
13
Atomic structure of transition
and lanthanide metals!
Energy levels
Occupation of energy levels
Photon-atom interaction
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Ruby vs. alexandrite!
Energy levels
Occupation of energy levels
Photon-atom interaction
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Nd:YAG vs. Nd:glass!
Energy levels
Occupation of energy levels
Photon-atom interaction
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Occupation of energy levels!
•  Collection of a large number of atoms,
each of which is in an allowed energy
level E1, E2, … !
•  Thermal equilibrium at temperature T!
!
Boltzmann distribution!
€
Energy levels
Occupation of energy levels
P(E m ) ∝ exp(−E m / kT )
m = 1,2,3,…
probability that an
arbitrary atom is in an
energy level Em.!
Photon-atom interaction
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Occupation of energy levels!
•  In a large number of atoms Ntot, with N1
occupying energy level E1 and N2
occupying level E2 we have the
population ratio:!
N2
# E 2 − E1 &
= exp% −
(
$
N1
kT '
N1 + N 2 = Ntot
•  In the case of degenerate energy levels:!
€
N 2 g2
# E 2 − E1 &
=
exp% −
(
$
N 1 g1
kT '
Under equilibrium conditions and T > 0 we always have:
€
N2 < N1 if E2 > E1
Energy levels
Occupation of energy levels
Photon-atom interaction
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Example: energy levels in ruby!
The ruby laser has a transition ΔE
corresponding to laser emission at 0.69 µm:!
!
!
!
At a temperature of 300 K we have:!
!
€
The population ratio is!
€
c
ΔE = hν = h
λ
= 2.88 × 10 −19 J
kT = 4.14 × 10 −21J
N2
= exp(−hν / kT )
N1
= exp(−69.5)
≈ 10−30
Energy levels
Occupation of energy levels
Photon-atom interaction
€
€
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Planck’s radiation law!
spectral energy density [J/(cm3s-1)]
emitted by a blackbody:!
% 8πν2 (
hν
ρ(ν) = ' 3 *
& c ) exp(hν / kT ) − 1
radiation or energy density [J/cm3]
contained in a bandwidth Δν :!
€
ρ(ν)Δν
optical intensity [W/cm2] :!
I = cρ(ν)Δν
Energy levels
Occupation of energy levels
Photon-atom interaction
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Interaction between photons and atoms!
Three types of interaction are possible:!
Spontaneous
emission!
Energy levels
Absorption!
Occupation of energy levels
Stimulated
emission!
Photon-atom interaction
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Spontaneous emission!
The decay rate:
•  is proportional to the upper level
population N2
•  does not depend on the number of
photons of frequency ν
hν = E 2 − E 1
€
A21 spontaneous transition
probability from level 2 to 1 [s-1]
τsp spontaneous lifetime from
level 2 to 1 [s]
∂N 2 / ∂t = −A21N 2
⇒ N 2 (t ) = N 2 (0)e
τ sp = 1/ A21
In a large number of atoms emitting spontaneously, the emitted photons are incoherent.!
Energy levels
Occupation of energy levels
Photon-atom interaction
−
t
τsp
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Absorption!
The absorption rate:
•  is proportional to the lower level
population N1
•  is proportional to the radiation density
ρ(ν) at the “correct” frequency ρ(ν12)
B12
constant of proportionality
[cm3/s2J]
ρ(ν)
radiation density (energy /
volume × frequency) [J/cm3s-1]
B12ρ(ν)
probability per unit frequency
that the transition 1 → 2 is
induced by the field [s-1]
Energy levels
Occupation of energy levels
∂N1
= −B12ρ(ν12 )N1
∂t
Photon-atom interaction
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Stimulated emission!
The emission rate:
•  is proportional to the upper level
population N2
•  is proportional to the radiation density at
the “correct” frequency ρ(ν21)
B21ρ(ν) probability per unit frequency
that the transition 2 → 1 is
induced by the field [s-1]
∂N 2
= −B 21ρ(ν21)N 2
∂t
In stimulated emission, the emitted photons are coherent
€
(i.e. have the same phase).!
Energy levels
Occupation of energy levels
Photon-atom interaction
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Combining absorption and
emission in thermal equilibrium!
∂N1
∂N 2
=−
= B 21ρ(ν)N 2 − B12ρ(ν)N1 + A21N 2
∂t
∂t
stim. em.
€
absorption
spont. em.
∂N1 ∂N 2
=
= 0⇒
∂t
∂t
N 2 A21 + N 2ρ(ν)B 21 = N1ρ(ν)B12
emission (st+sp)
absorption
Now using Boltzmann’s distribution for the ratio N2 / N1:
€
A21 / B21
ρ(ν21) =
(B12 / B21) (g1 / g 2 )exp (hν21 / kT ) − 1
Energy levels
Occupation of energy levels
Photon-atom interaction
€
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Einstein’s relations and
coefficients!
Comparing the previous expression with Planck’s radiation law we
obtain the relatives magnitudes of A and B:!
A21 $ 8πν2 '
= & 3 ) hν
B 21 % c (
g1
B 21 =
B12
g2
Einstein’s relations!
Using the previous expression for A21 and the relation λν=c:!
1
λ3
A ≡ A21 =
B ≡ B21 =
τ sp
8πhτ 21
Energy levels
Occupation of energy levels
Einstein’s
coefficients!
Photon-atom interaction
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Example: rates of spontaneous
and stimulated emission!
Rate of spontaneous emission:!
!
!
Rate of stimulated emission:!
!
!
The two rates are equal when!
!
For a wavelength of λ=1 µm calculate
the threshold intensity assuming a
linewidth Δν=107 Hz!
!
Energy levels
Occupation of energy levels
1
A=
τ sp
Bρ(ν21)
ρ(ν21) = A / B = 8πh / λ 3
A / B = 1.66 × 10−14 J/m3s −1
I = cρ(ν)Δν ≈ 5 mW/cm2
Photon-atom interaction
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Atomic lineshapes and linewidths!
So far:
In reality:
!“infinitely sharp” energy gap E2–E1!
!monochromatic light of frequency ν21!
!finite transition linewidth Δν
!signal with bandwidth dν
Absorption and emission
cross-sections of
Yb:doped glasses used
in fiber lasers.
Energy levels
Occupation of energy levels
Photon-atom interaction
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The transition cross section σ(ν)
determines light-atom interaction!
It is a function that characterizes the relative magnitude of the
interaction of the atom with a photon of frequency ν.
The strength S
(cm2s-1)
of the interaction is:
S=
The lineshape function g(ν) is the normalized curve:
∞
∫ σ(ν)dν
0
g(ν) = σ(ν)/ S
Δν ∝ 1/ g(ν0 )
€
€
Energy levels
Occupation of energy levels
Photon-atom interaction
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What are the causes of line-broadening?!
Homogeneous broadening!
Inhomogeneous broadening!
Every atom has the same lineshape:
all atoms are affected equally by a
given signal.!
!
•  lifetime broadening!
•  collision broadening!
•  dipolar broadening!
•  thermal broadening!
The center frequency ν0 of each atom
is displaced individually, leading to
different lineshapes.!
!
•  Doppler broadening!
•  crystal inhomogeneities!
Lorentz distribution
Gaussian distribution
g(ν) =
Energy levels
Δν / 2π
2
(ν − ν0 ) + (Δν / 2)
2
2
,
ln 2 / π
& ν − ν0 ) /
g(ν) =
exp.− ln2(
+ 1
'
Δν / 2
Δν / 2 * 10
.-
Occupation of energy levels
Photon-atom interaction