Average Lifetime
Atoms stay in an excited level only for a short
time (about 10-8 [sec]), and then they return
to a lower energy level by spontaneous
emission.
Every energy level has a characteristic average
lifetime, which is the average time the
electron exists in the excited state before
making a spontaneous transition. Thus, this is
the time in which the excited atoms returned
to a lower energy level.
• According to the quantum theory, the transition
from one energy level to another is described by
statistical probability.
• The probability of transition from higher energy
level to a lower one is inversely proportional to
the lifetime of the higher energy level.
• In reality, the probability for different transitions is
a characteristic of each transition, according to
selection rules.
• When the transition probability is low for a specific
transition, the lifetime of this energy level is longer
(about 10-3 [sec]), and this level becomes a "metastable" level. In this meta-stable level a large
population of atoms can assembled. As we shall
see, this level can be a candidate for lasing process.
• When the population number of a higher
energy level is bigger than the population
number of a lower energy level, a condition of
"population inversion" is established.
• If a population inversion exists between two
energy levels, the probability is high that an
incoming photon will stimulate an excited
atom to return to a lower state, while emitting
another photon of light.
• The probability for this process depend on
the match between the energy of the
incoming photon and the energy difference
between these two levels
The Einstein Relation
• To introduce probabilities for these emission and
absorption phenomena, let Ni be the number of
atoms (or molecules) per unit volume that at time t
occupy a given energy level, Ni
• For the case of spontaneous emission, the
probability that the process occurs is defined by
stating that the rate of decay of the upper state
population (dN2/dt)sp must be proportional to the
population N2. We can therefore write
1.2
the minus sign accounts for the fact that the time
derivative is negative
• The coefficient A, introduced in this way, is a
positive constant called the rate of spontaneous
emission or the Einstein A coefficient.
• The quantity τsp = 1/A is the spontaneous
emission (or radiative) lifetime. Similarly, for
nonradiative decay, we can generally write
1.3
τnr is the nonradiative decay lifetime, depends only
on the particular transition considered.
For nonradiative decay, on the other hand, τnr
depends not only on the transition but also on
characteristics of the surrounding medium.
For stimulated emission we can write
1.4
(dN2/dt)st is the rate at which transitions 2→ 1 occur as a
result of stimulated emission and W21 is the rate of
stimulated emission.
the coefficient W21 also has the dimension of (time)-1
Unlike A, however, W21 depends not only on the
particular transition but also on the intensity of the
incident em wave. More precisely, for a plane wave,
we can write
1.5
F is the photon flux of the wave and σ21 is a quantity
having the dimension of an area (the stimulated
emission cross section) and depending on
characteristics of the given transition.
we can define an absorption rate W21 using the
equation:
1.6
where (dN1/dt)a is the rate of transitions l→2 due to
absorption and N1 is the population of level 1.
1.7
where σ12 is some characteristic area (the absorption
cross section), which depends only on the particular
transition. It can be given in general farm:
1.7a
• Function g(ν — νo), symmetric about ν = ν 0, again of
unit area, i.e., such that ʃg(ν — νo)dν = 1, and
generally given by:
Einstein showed that for nondegenerate energy levels,
has W21 = W12 and thus σ21 = σ12
If levels 1 and 2 are g1-fold and g2-fold degenerate
(quantum physics describes a system in which
different quantum states have equal energy),
respectively, one then has
,
The corresponding expression for the absorption or
simulated rate W= σF can then be written as:
ρ =(nFhv/c) is the energy density of the em wave and μ
is the electric dipole moment. also we can write
• stimulated emission, and absorption can be
described in terms of absorbed or emitted photons
as follows
a) In the spontaneous emission process, the atom
decays from level 2 to level 1 through the emission
of a photon,
b) In the stimulated emission process, the incident
photon stimulates the transition 2→ 1, so that there
are two photons (the stimulating one and the
stimulated one),
c) In the absorption process, the incident photon is
simply absorbed to produce transition 1→2. Thus
each stimulated emission process creates a photon,
whereas each absorption process annihilates a
photon.
Einstein Thermodynamic Treatment
In Einstein treatment the concept of stimulated emission
was first clearly established, and the correct
relationship between spontaneous and stimulated
transition rates was derived well before the formulation
of quantum mechanics and quantum electrodynamics.
Let us assume that the material is placed in a blackbody
cavity whose walls are kept at a constant temperature T.
Once thermodynamic equilibrium is reached, an em
energy density with a spectral distribution ρv given by
Plank Eq.
1.10
is established and the material is immersed in this
radiation.
• In this material, both stimulated emission and
absorption processes occur in addition to the
spontaneous emission process.
• Since the system is in thermodynamic equilibrium, the
number of transitions per second from level 1 to level 2
must be equal to the number of transitions from level 2
to level 1. We now set
,
• Where W21 stimulation rate of transition from level 2 to
1,W12 absorption rate of transition from level1to2, B21
the Einstein stimulation coefficients, B12 the Einstein
absorption coefficients. Let Ne1 and Ne2 be the
equilibrium populations of levels 1 and 2, respectively.
We can then write
1.12
• From Boltzmann statistics we also know that, for
nondegenerate levels:
1.13
Using eq. 1.1.12 and 1.1.13.in eq. 1.1.10 we get
1.14
When ν=νo we get the following relations:
1.15 ,
1.16
This equations show the relation between Einstein
coefficients it can be measured laboratory for any
element. Integration of this equation with the
assumption that gt(v — v0) can be approximated by a
Dirac δ function in comparison with ρv (see Fig. 2.3),
we obtain
1.17
• A comparison between Eqs. (1.1.11a) and
(1.1.11b) or (1.1.17) then gives
1.18,
1.19
Plot of the function ρv(v, T) as a function of
frequency v at two values of temperature T.
• We can therefore write for the expression:
1.20
The normalized function [g(ν — νo)∆νo] is plotted in Fig. 2.6 versus
the normalized frequency difference (ν — νo)/(∆vo/2). The full
width of the curve between the two points having half the
maximum value [full width at half-maximum (FWHM)] is simply
∆v0. The symbol gt(v - v0) for the total lineshape function, which
can be expressed as:
Using Fourier analysis to obtain line profile, where gt is a function
give the probability of this excitation transition.
Emission line is described by plotting
spontaneous emission radiation
intensity as a function of frequency
(or wavelength), for the specific
lasing transition.
• A curve of the general form described by Eq.
is referred to as Lorentzian after H. E. Lorentz fig
2.6, who first derived it in his theory of the
electron oscillator.
A curve of the general form described by Eq.
is referred to as Gaussian fig 2.8
• The ratio for the spontaneous emission to the
stimulated emission can be written as
• Example:
• Calculate the ratio of spontaneous emission to
stimulated emission for a tungsten filament operating
at a temperature of 2000K taking the average
frequency to be 5x1014Hz.
• Solution
• The ratio R = exp[(6.6x10-34*4x1014)/(1.38x10-23*2000)]
• By neglecting the flux density,
R = 1.5x105
• This confirms that under normal condition of thermal
equilibrium stimulated emission is not an important
process