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Optical Sources, Detectors, and Systems by Robert H. Kingston

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2.2 Absorption or Amplification of Optical Waves 3 5
with B22 and
B22
the proportionality constants for the upward and
downward induced or stimulated transitions. Manipulation of the
second and fourth terms of
Eq.
(2.1) yields
(2.2)
N2
But we know from
Eq.
(1.12) that
8nhv^
"^ >(£"-/«-I)
and, therefore, to satisfy the relationship of Eq. (2.1) for all frequencies
and all temperatures,
8^2
= ^23 ~ ^/ ^^^ ^/^ -
Snhx^lc^-
Since we have
established that the upward and downward induced rates are equal, we
shall use the single constant
B,
which can be written
g^
^^ ^ ^ ^ ^ (2 3)
where
t^
is defined as the
spontaneous
emission time and is equal to J/A.
The actual values of the A and B coefficients are determined by the
specific system considered. Obviously the larger the thermal equilib-
rium absorption, as determined by the B coefficient, the smaller the
radiative or spontaneous emission time. Also, the higher the optical fre-
quency, the shorter is t^ for the same value of B or absorption
coef-
ficient.
2.2 Absorption or Amplification of Optical Waves
Since we are interested in the absorption or emission of a single-fre-
quency or
monochromatic
plane wave , we now propagate such a wave
as shown in Figure
2.2.
We assume that the intensity is small enough so
that it does not disturb the state populations and that the frequency of
the wave is v, which is at or near the "resonant" or characteristic fre-
36
Chapter 2 Interaction of Radiation with Matter
f:
'-'-..
%^:
)
/
Figure 2.2 Blackbody chamber with added plane wave of intensity /. The small numerals
represent particles in the upper, 2, or lower, 7, state.
quency of the transition, which we now call v^. Now as the wave prop-
agates it will induce upward and downward transitions and a con-
comitant decrease or increase in photons, if we imagine a thin slab of
area A and thickness dz the incident power, P = lA, will increase by
dilA) = Ad/, as shown in the figure. Then d? = Adl = dUjdi = V(du/dt) =
Adz(du/dt).
since the increase in power is equal to the added energy U
per unit time in the small volume.
dz
IF
P
+
dP
=
P
+
Adl
2.2 Absorption or Amplification of Optical Waves
37
Therefore,
f
=
f
= 7^
=
^[^^^'^^^^^^]''^^^^^^^^^"^^^"^
(2.4)
where n^ and ^2 are the number of particles per unit volume. The
bracketed quantity in the fourth term is the rate of increase in photons in
the volume. But this equation has the interesting problem that the
quantity u ^ the spectral energy density, is actually infinite since we are
talking about a monochromatic or single-frequency wave. We resolve this
dilemma by realizing that energy transitions are allowed over a small
range of frequencies about the center or resonant frequency v^. Thus
the effective spectral energy density u^ may then be written as w^ =
ugiv) = Ig(v)/c. The quantity, g(v), is known as the Uneshape function
and is sketched in Figure 2.3. It has the dimensions of inverse frequency
and the area under the curve is equal to unity, thus fg(v)dv
=
1. In our
original derivation of the A and B coefficients we used a continuum of
frequencies for the electromagnetic energy density with a single-fre-
quency particle transition, while here we have taken into account the
finite linewidth of the transition and used a single-frequency wave. The
Uneshape function describes the finite spread in the emitted spectrum
caused by the finite spontaneous emission time and also gives the pro-
portionate response of the induced process as a function of the incident
Figure 2.3 The lineshape function g(
v).

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