Page 166 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Basic Description of Light
the photon anywhere is equal to one. The probability of finding a photon
⁄
⁄
in the interval k – ∆k 2 and k + ∆k 2 is then
Pk() = 2 πσ fk() 2 (4.55)
k
Macroscopic We still must create an intuitive link between electromagnetics and pho-
Electro- tons. The “trick” lies in the fact that electromagnetic radiation in vacuum
magnetics
can be brought into the exact same form as a harmonic oscillator (the har-
monic oscillator is extensively discussed in Section 3.2.5). In quantum
mechanics, the harmonic oscillator quantizes the energy as

 1 
,,,
E = n + --- —ω n, = 012 … (4.56)
 2 
The unbounded vacuum does not place any limitation on ω , which may
vary arbitrarily. We could view light in this context as an electromagnetic
wave whose energy is restricted to integer multiples of —ω starting at the
zero-point energy of —ω 2⁄ . Here a photon corresponds to an energy
step, which for visible light with a wavelength of 500 nm is the very
small quantity —ω = 1.66 × 10 – 18 J, so that for typical optical ray trans-
mission applications the fundamental step size is so insignificantly small
that the energy appears as a continuous variable.

Another possibility is to consider an electromagnetic light wave as a
superposition of photon-sized electromagnetic wave packets that are in
phase with each other. Each photon carries a fundamental quantum of
energy, —ω , and the superposition of a large quantity of such photons
appears as a macroscopic electromagnetic wave.

Cavities We now consider what happens when electromagnetic radiation is
L
trapped in a 1D cavity of length with perfectly conducting and per-
fectly reflecting walls. The electric field has a vanishing component par-
allel to a conducting surface. Clearly, the cosine functions are good
,,,
candidates for this case as long as k = n π L⁄ , where n = 123 … ,
j j
and so we use

Semiconductors for Micro and Nanosystem Technology 163