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96 Fiber Optic Communications

Air
molecules

E 2 = mgh

h

E 1 = 0

Figure 3.5 Air molecules at ground and at height h.

where ΔE = mgh is the energy required to lift a molecule of mass m to the height h, and g is the gravitational
constant (see Fig. 3.5).
Using Eq. (3.10) in Eq. (3.8), we obtain

A ∕B 21
21
u ()= . (3.12)
s
(B ∕B ) exp (ℏ∕k T)− 1
12 21 B
According to Planck’s law, the energy spectral density per unit volume at thermal equilibrium is given by
3 3
ℏ n 0 1
u ()= , (3.13)
s 2 3
c exp (ℏ∕k T)− 1
B
where n is the refractive index and c is the velocity of light in vacuum. Consider a hollow container heated to
0
a temperature T by a furnace, as shown in Fig. 3.6. Under thermal equilibrium, if you make a very small hole
and observe the spectrum of radiation, it would look like the curves shown in Fig. 3.7. For all these curves, the
energy of low-frequency and high-frequency components of the electromagnetic waves approaches zero and
the peak of the energy spectral density increases with temperature. A similar experiment was carried out by
Rubens and Kurlbaum [6], and Planck developed a theoretical description for the enclosed radiation [7].From
his derivation it follows that the energy spectral density at thermal equilibrium is given by Eq. (3.13) and it is
shown in Fig. 3.7, which is in good agreement with the measured data of Rubens and Kurlbaum [6]. Planck
assumed that energy exchange between radiation and matter takes place as a discrete packet or quantum of

Hollow container

Spectrometer

Furnace

Figure 3.6 The radiation trapped in a hollow container under thermal equilibrium.