Page 39 - Photonics Essentials an introduction with experiments

Electrons and Photons

Electrons and Photons 33

124
E(eV) =
(nm)
where the energy is given in electron volts and the wavelength in
nanometers.
2.5 Make a graph to scale of wavelength on the lower horizontal axis
and energy on the upper horizontal axis. The wavelength range
should vary from 200 nm to 2000 nm.
a. What is the corresponding energy range?
b. Mark the following regions:
blue light
green light
red light
1550 nm low-loss region for optical fiber telecommunications
c. Which photons have more energy, red or blue?
d. Paste a copy of this graph in your lab notebook
2.6 The energy of an electron is equal to the square of its momentum
divided by 2 times its mass. From de Broglie, we also know that
the electron behaves like a wave.
a. By taking the second derivative with respect to x of the simple
wave function (x) = A sin(kx), show that you get the follow-
ing relationship:

d 2
2
(x) = –k (x)
dx 2
b. Multiply both sides of this relationship by the appropriate
constants to derive a formula for the energy of the electron.
This formula is the basis for the Schrödinger equation, the
mathematical foundation of quantum mechanics.
2.7 Silicon has a band gap of 1.1 eV at room temperature. Using a
monochromator, you send a beam of photons with a wavelength
of 1240 nm on the surface of a silicon wafer 0.5 mm thick. Only
three things can happen: absorption, reflection, and transmis-
sion of the beam of light. Which things actually happen under
these circumstances?
2.8 When a photon passes from air into glass, its trajectory is
changed according to Snell’s law—n 1 sin( 1 ) = n 2 sin( 2 )—and the
velocity of light is reduced by the ratio of the index of refraction
of air (n 1 = 1) to that of glass (n 2 = 1.5). When the photon travels
in glass it still obeys the relationship: V = f , where V is the

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