II.C-1
II. Light is a Ray (Geometrical Optics)
II.C. Prisms and Dispersion 1. Refraction Through a Prism
A prism is just a block of glass or some other transparent optical material with flat sides. The most common prism is a cylinder with a triangular cross section. If we shine light upon one of the flat sides of the prism and vary the angle of incidence, θ, we find that the angle of deviation δ (the total angular deviation of the incident light ray caused by the prism) has a minimum value δmin
at a certain angle θ that depends only on the prism index n and the
prism angle A.
Since we can measure δmin and A, we can write the index n as a function of these parameters, or n(δmin,A), to determine the index of refraction of a piece of glass or other unknown optical material shaped like a prism. Let’s see what this functional dependence is explicitly.
a A
n air
δ θ1
B
δ1 θ2
δ2 θ1′ θ2′
Now the total angular deviation of the incident ray is the sum of the angular deviation of the ray caused by the 1st surface and the angular deviation caused by the 2nd surface, or
δ δ= 1+δ2.
Furthermore, recognizing that opposite angles at intersecting lines are equal, we find θ1= ′ +θ δ1 1 and θ2−δ2 = ′θ2,
or, δ1= − ′θ θ1 1 and δ2 =θ2− ′θ2.
Therefore we see δ =
(
θ θ1+ 2)
− ′ + ′(
θ θ1 2)
.Although we don’t go through the details of the proof, it turns out that δ = δmin when θ1 = θ2
(which also means θ1′ = θ2′ according to Snell’s Law). This result can be shown by writing δ as a function of θ1, taking the derivative, and then setting it equal to zero to find the minimum value.
Thus δmin =2
(
θ θ1− ′1)
. Now if we note thatA= ′ + ′θ θ1 2
a A
n air θ δ
II.C-2
(a result which can be shown by recognizing that the sum of the internal angles of a 4-sided
polygon is 360˚, i.e., A + B + 90˚ + 90˚ = 360˚, and that the sum of the internal angles of a triangle is 180˚, i.e., B + θ1′ + θ2′ = 180˚),
then we see that A=2θ1′ when δ = δmin.
Finally, using Snell’s Law, sinθ1=nsinθ1′, we can write the index in terms of δmin and A:
Index of a Prism : n A A
min A
δ δmin
, sin
( )
=[ (sin + ) ]
[ ]
2
2 .
Notice that for very small prism angles A, the deviation δ is also small, and we can thus apply the small-angle approximation for the sine function,
sin
( )
φ ≅φ whenφis tiny.With this approximation we can estimate the minimum deviation angle δmin knowing only A and n:
Approximate Minimum Deviation Angle of a Prism : δmin ≅ A n
(
−1 .)
The minimum deviation angle can be measured directly by carefully rotating the prism with respect to a unidirectional beam of incident light (as from a laser) and then measuring the direction of the output beam.
2. Dispersion as Observed in a Prism
You have probably all seen a prism separate, or disperse , white light into its many colors. The same phenomenon is responsible for the separation of colors you see in a rainbow, where tiny nearly spherical water droplets act like prisms. Why do different colors of light experience different deviation angles?
“white light”
red orange
yellow green
blue indigo
violet
As we will see in more detail later in the course, the different colors are associated with different
wavelengths of light, or, equivalently, with photons that carry different energies . The diagram below provides a little more quantitative information:
II.C-3
400 nm* 500 nm 600 nm 700 nm
5.0×10–19 J** 4.0×10–19 J 3.3×10–19 J 2.8×10–19 J violet blue green yellow orange red
WAVELENGTH
ENERGY OF A SINGLE PHOTON
* 1 nanometer (nm) = 1 billionth of a meter (m)
** 1 Joule (J) = energy used in 1 second (s) by 1 Watt (W) of power
Furthermore, it turns out that the refractive index of a material is not really a constant. This variation is called:
Material Dispersion : The refractive index of a material varies with the wavelength of light (and therefore with photon energy).
a n(λ)
λ
Angular dispersion in a prism (or water droplet) thus occurs because, as we just showed in the previous section, the deviation angle δ is a function of the refractive index n.
