Page 158 - Principles and Applications of NanoMEMS Physics

146 Chapter 3

)
V ≅ ( 2 =λ ) 3 ( 2c ν , then the density of photon states per unit frequency,
3
per unit volume, 1( mode ( / ) ∆ ν V ), may be expressed in terms of the cavity
Q as follows,

1 mod e = 1 mod e = 8ν 2 Q = ρ . (204)
( ν∆ )( ) § ν § · c · 3 c 3 c
V
¨ ¨ ¸ ¨ ¸ ¸
2 ¹
¹
© Q © ν
Comparing (203) and (204) it is seen that they are related by,

⋅
ρ ≅ § 2 · Q ρ . (205)
¸
¨
c s
© π ¹
Thus, a cavity enclosure of quality factor Q increases the effective density of
photon states in free space by the factor of ( Q2 ) π . In turn, since the
spontaneous emission rate is proportional to this density of photon states,
this rate is increased, in particular, to [172],
A ≅ QA . (206)
c
The larger issue elicited by Purcell’s observation was that the spontaneous
emission rate of an atom may be modified according to the properties of the
surroundings. In particular, as Kleppner [172] pointed out, the spontaneous
emission of an atom in a cavity may be inhibited if the cavity has dimensions
smaller than the radiaton wavelength, but it may be enhanced (increased), as
6
in (20 ), if the cavity resonates at this wavelength.
This realization that the spontaneous emission rate of an atom may be
suppressed or enhanced by modifying the properties of the radiation field in
the surroundings, has many practical applications. For instance, in solid-state
electronics it is well known that spontaneous emission is fundamentally
responsible for non-radiative recombination processes, which limit the
performance of semiconductor lasers, heterojunction bipolar transistors, and
solar cells [51]. How would one apply the cavity QED concept to inhibit the
spontaneous emission in these situations, where one is dealing not with
single atoms, but with entire devices, is not at all obvious. The answer to this
question was advanced by Yablonovitch in 1987 [51] with his photonic
band-gap crystal (PBC) idea. Indeed, by surrounding the devices in question
with a PBC exhibiting a band gap which overlaps the electronic band edge
(across which the non-radiative transitions would occur) the spontaneous