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Lecture 33 — Spontaneous and Stimulated Absorption and Emission — April 5, 1999
Physics 221 — Lecture 33 — “Atomic Absorption and Light Emission”
April 5, 1999
Reading
Problems
Reminder
Meyer-Arendt, Chapter 23; Young, Ch. 8; Hecht, Ch. 13.
Ch. 23, Problems 1, 2. Homework problem at end of notes.
Physics Colloquium today, Sergei Savikhin (Iowa State), “Studies of
Photoelectron Transfer using Ultrafast Laser Spectroscopy,” 4:00 in
4327 Stevenson
1. Basic concepts of spontaneous and stimulated emission. So now we have
finally assembled almost all the “software” we need to describe the operation of the
laser, the first important quantum optics device.
a.
The Planck distribution function for quantized atomic oscillators.
b.
The Boltzmann distribution of energy in atomic energy levels.
c.
Today we will work through the concepts of spontaneous and stimulated
emission, first propounded by Einstein in 1916-1917:
i. Spontaneous emission is just like radioactive decay, with less energetic
byproducts: an atom in an excited state has a finite probability of decay
per unit time, a decay probability characteristic of each atomic state.
ii. Stimulated absorption occurs when a photon strikes an atom with just
exactly the proper energy to induce an electronic transition between two
energy states.
iii. Einstein’s contribution was to show that there was a third process,
stimulated emission, which could cause a downward electronic transition
from an excited electronic state in response to a photon of just the same
energy needed for stimulated absorption.
d.
In this lecture, we will start out by looking at spontaneous emission, then at
stimulated absorption, and then, by combining these two, arrive at the Einstein coefficients for stimulated vs spontaneous emission probabilities. The
expression for absorption and gain of an optical medium will then be derived
in a form which lets us calculate it easily in practical case.
2. Spontaneous emission and the lineshape function. We know from the history of physics (if from nowhere else) that the classical picture of the atom proved
inadequate. However, it is possible to make some meaningful connections between
the classical oscillator and the quantum picture.
a.
In quantum physics, we know that the electrons in atoms occupy only particular states, and that only transitions between those states are allowed.
i. The energy difference between states is given by the Planck relation:
E21 = hν 21 = hω 21 = E2 − E1
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(1)
Lecture 33 — Spontaneous and Stimulated Absorption and Emission — April 5, 1999
ii. We can get exact expressions for the electronic states not from the classical equation of motion but from Schrödinger equation; the probability
of finding an electron in a particular state is its wave function ψ i( r,t ) .
iii. The probability of a transition between two states is given by an integral over all space which connects the initial and final states
∫ d r ⋅ψ 1( r,t ) (−ez)ψ 2 (r,t ) =
3
*
ψ 1(r,t ) −ezψ 2 (r,t ) = 1 −ez 2
(2)
This object is often referred to as the matrix element for an electric dipole transition between the states 2 and 1.
b.
When all of this started, however, this picture was not known, and Einstein
developed a simpler picture of spontaneous emission.
dN
N2
−t τ
− 2 = A21 N2 ≡
⇒ N2( t) = N02e spont
(3)
dt
τ spont
c.
The uncertainty principle teaches us that there is a connection between our
ability to measure the energy of a state and its lifetime:
∆E21 ⋅ ∆t = ∆( hω21 ) ⋅ ∆τ ≥ h 2
where τ is the lifetime of the state. Hence we have to think about the “fuzziness” of the transition energy and of the frequency of the radiation which accompanies it.
d.
The lineshape function describes this uncertainty in a quantitative way. It is
defined by saying that it is a function g(ω ) which has the property that
+∞
∫ g(ω )dω = 1
⇒ I(ω ) = Iog(ω )
(4)
−∞
It can be shown that for our oscillator model, the normalized lineshape function has a form which looks (not surprisingly!) a lot like the susceptibility:
g(ω ) =
∆ω
(ω − ω o )
2
+ ( ∆ω 2 )2
(5)
where ∆ω is the width of the lineshape function at half its amplitude. Such
a function is called a Lorentzian and it is also found to be the lineshape in a
full quantum-mechanical theory.
3. Transition probability for spontaneous emission. Now we can begin to make
some approximate (or semiclassical) connections between our model of the classical
electron oscillator and the quantum picture.
a.
In quantum physics, we know that the electrons in atoms occupy only particular states, and that only transitions between those states are allowed,
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Lecture 33 — Spontaneous and Stimulated Absorption and Emission — April 5, 1999
based on Planck’s conjecture that the energy difference between states is
given by:
E21 = hν 21 = hω 21 = E2 − E1
b.
(6)
For an atom which is in an excited state N2 and which then undergoes decay
or relaxation to the ground state by spontaneous emission
dN
N2
−t τ
− 2 = A21 N2 ≡
⇒ N2( t) = N02e spont
(7)
dt
τ spont
4. Induced absorption. Now consider the two-level atom in an electromagnetic
field. In the presence of an electromagnetic field, an atom which is in a state N1 can
undergo transitions to the state N2 provided that the frequency of the radiation field
satisfies Eq. (1). However, this process is not a spontaneous one, because the atom
is stimulated to absorb by the incident light field.
a.
Consider the interaction of an ensemble of identical atoms in a state E1
with a population density (per unit volume) N1.
W1→2 = B12ρ( ν)
(8)
where ρ(ν ) is the energy density of the radiation field. Thought question:
What are the units of W?
b.
If we assume that the atoms are in thermal equilibrium with the radiation
field at an absolute temperature T, then the radiation energy density is
given by the Planck distribution function:
8πn3hν 3
1
ρ(ν ) =
⋅ hν k T
3
B −1
c
e
c.
(9)
Now we borrow from Einstein’s clever treatment of the blackbody problem,
in which he noted that for the ensemble of atoms to be in equilibrium, the
average population of levels 2 and 1 must be constant. Hence:
i. The average number of atoms in state 1 is affected only by the upward
(stimulated) transitions induced by the radiation field, while the average population of atoms in state 2 is affected by both the downward
(spontaneous) emission and downward (stimulated) emission. Why?
ii. Therefore, the following equation expresses the equilibrium between the
radiation field and the two-level atoms in our hypothetical ensemble:
 8πn 3hν 3

B21
8πn3 hν3
B
N2 [ B21ρ( ν) + A21] = 
⋅ hν k BT
+ A21  = N1
⋅ hν k 12
3
3
c
e
−1
e BT − 1
 c

(10)
Note the asymmetry between the right- and left-hand-sides! What does
it mean?
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Lecture 33 — Spontaneous and Stimulated Absorption and Emission — April 5, 1999
iii. The other piece of information we have is that the atomic ensembled
must satisfy Boltzmann’s equation if it is really in thermal equilibrium:
N2 = N1e −kB T
(11)
Note again that N1 does not have to be the ground state!!
d.
Finally, we have to remember the lineshape function, which simply multiplies all of these transition probabilities. By using Eq. (11), setting B21=B12
(Why is this reasonable?), and doing some algebra, we arrive at our major
result.
c3 ρ(ν ) ⋅ g(ν )
c 3ρ(ν )
∆ν
W12 = W21 =
=
⋅
3
3
3
3
8πn hν τspont 8πn hν τ spont 2π(ν o − ν )2 + (∆ν 2)2
(12)
This completes, for all practical purposes, all we have to know about the
transition rates. Now there is only one more step: figuring out absorption
coefficients.
5. Absorption of a monochromatic wave by a two-level system. When we
deal with a light wave passing through a medium, it is most natural from the experimental point of view to ask about the total power in the beam as it enters and
exits the medium. This we can do by a qualitative argument shown below.
a. When a light wave travels through a partially absorbing medium, it deposits
a certain amount of energy per unit volume, given by
Power dI ( ν)
c 2 g( ν)
=
= ( N2 − N1 )W12 hν = ( N2 − N1 ) ⋅
I (ν )
Volume
dz
8πn2 ν2 τ spont
b.
(13)
Since this is equal to the change in intensity per unit length, we can write
down from our understanding of differential equations, what the solution is.
But see the picture below for a sketch [from Yariv, Optical Electronics] which
will help you figure it out!
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Lecture 33 — Spontaneous and Stimulated Absorption and Emission — April 5, 1999
c.
Note that the power at the exit face of the medium may be either reduced or augmented, depending on the sign of the quantity (N2-N1).
This is simply the relative population density of the two levels. So what
does it mean if this quantity is positive? Negative? Next time, we will
relate this to the susceptibilities which we had calculated earlier from
our simple electron-oscillator model.
Homework assignment. Assume that in an atom, energy levels are spaced an
equal distance apart, ∆E=1.0 eV. Calculate the relative populations of the first
four energy states.
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