Laboratory Exercise 4.
DIFFRACTION OF LIGHT
Diffraction Gratings. Determining the Wavelength of Laser Light Using a Diffraction Grating.
Refraction. Observation of Atomic Spectra.
Theoretical background:
A) Diffraction Gratings
The principle of operation of a diffraction grating is based on the phenomena diffraction and
interference of light. Often diffraction gratings are used to split a beam of white light into its spectral
components - beams of monochromatic light , i.e., with some given wavelength λ (or color). As we
shall see, the angle at which the different monochromatic beams appear behind the grating depends
on its wavelength λ and can be used to determine the latter.
Diffraction gratings are made of transparent material with numerous parallel thin lines, etched
into their surface. To understand how a diffraction grating works, consider first a flat monochromatic
light beam falling on a screen with two slits (Fig1.) Each slit turns into a linear source of light waves.
These waves interfere with one another and the places of constructive and destructive interference
appear on a screen behind as an "interference pattern", consisting of light spots (maxima) separated
by dark regions (minima). Take a point on the screen behind. If the distance from it to the two slits
differs by a whole number of wavelengths, then whenever a crest (resp. a trough) of the wave coming
from one slit reaches that point, also a crest (resp. a trough) of the wave from the second slit will
reach it. (We say that the two waves are in phase.) The resultant wave at this point will have maximal
amplitude. The point will be illuminated. This is a point of constructive interference. For points
where the above condition does not hold waves from the two different slits arrive in different phases
and the resultant wave will have practically zero amplitude.
The diffraction grating is based on the same principle except that instead of two slits we have
a large number of parallel slits. There is one important simplification - the distance between two
neighboring slits is very small ( 0.001-0.002 mm ), while the distance to the screen is of the order of
1-2 m. Let us look at two slits and the beams coming from them and converging to a point on the very
1
distant screen (Fig. 2). The two beams are practically parallel. The difference in length of the two
beams is d. Constructive interference occurs when d = nλ, where n is an integer.
From the right triangle in Fig. 2 one sees that d = a sin ϕ , where a is the distance between two
neighboring slits. It is called the constant of the grating and is usually specified by the manufacturer.
Thus we have the following condition for the appearance of diffraction maxima:
nλ = a sin ϕ
It can be seen easily that this same condition holds for any two slits and therefore for the whole
grating. In the formula above ϕ is the angle at which diffraction maxima occur, measured from
the perpendicular to the grating and n = 0, ±1, ±2, ... Obviously for any λ there exists the central
maximum (ϕ = 0 ) corresponding to n = 0 . The angles at which the other maxima, corresponding
to n = ±1, ±2, ... appear, depend on λ . Many gratings are especially manufactured, so that only the
central maximum and the first maxima to the left and to the right appear.
B) Diffraction Through a Single Slit
Light passing through a single tiny slit produces a diffraction pattern somewhat similar to that
produced by a double slit (Fig 3). Here the light on the screen is streaked with shaded regions. A
careful mathematical analysis, using Huygen’s principle gives the following formula:
d sin θ = λ
where d is the width of the slit and θ is the angle between the perpendicular and the first dark spot
(minimum).
Since the angle θ is usually very small, we have
2
sin θ ≈ tan θ ≈ θ ≈
x
L
and so
λ=d
x
L
C) Refraction of Light in Water
Suppose we put a diffraction grating on one wall of an empty glass container and shine a laser
beam through the grating. Using a screen on the opposite wall of the container, we will observe some
distance between the central and the side maxima. If we fill the container with water and repeat the
experiment, the distances become smaller (Fig 4). One can look at this phenomenon in two seemingly
different ways. On the one hand one can say, that the laser beam was first split and then the angle
of the side beams changed on the interface between the two different media (air and water). This
bending of light is known as refraction. On the other hand, the same result would be obtained if we
had placed the grating inside the tank. We know that the distance between the maxima on the screen
is directly proportional to the wavelength λ. Thus smaller distance means that λ decreases when light
is traveling through a denser medium (water in our case).
Since λ = fv , where v is the speed of light in a certain medium and f is the frequency, we find
that v is smaller in a denser medium as well. Light travels faster in air than it does in water.
D) Atomic Spectra
When the atoms of some gas are excited in some way (say by bombarding them with electrons)
they may go into higher energy states (i.e. excited states). After some time the atoms undergo a
spontaneous transition to lower energy states emitting light photons with strictly defined energy (and
frequency). Each chemical element is characterized by its own discrete set of energy levels and
therefore the light that it emits contains specific (characteristic for this particular element) frequencies
of light. When this is passed through a diffraction grating, it is spectrally decomposed and we observe
the characteristic spectral lines for that element. We can then determine the wavelengths of the light
forming the spectral lines knowing the constant of the diffraction grating. Atomic spectra are used to
identify the presence of elements in mixtures (that we do not want to decompose chemically) or e. g.
in distant stars, intergalactic clouds, etc. On Fig. 5 a series of visible spectral lines of hydrogen is
shown, the so-called Balmer series.
3
Assignments
1. Determine λ of a laser beam using a single slit mechanism to create diffraction patterns. The
use of single slit in our lab has a practical advantage. The slit is wide enough to be measured with
a standard micrometer, while the use of a diffraction grating for the determination of λ will require
knowledge of the diffraction grating constant a.
2. Determine λ of a laser beam using a diffraction grating with a known a.
3. Determine the refraction index Nwa = vwater
.
vair
4. Observe the brightest spectral lines of H, Hg, O. Determine λ for the 3 brightest lines of
hydrogen.
4
ANSWER SHEET
Laboratory Exercise 4.
DIFFRACTION OF LIGHT
Name:
Date, hour:
Instructor’s signature:
RESULTS
Part 1:
d=
L=
x=
λ=
Part 2:
Diffr. grating constant
a=
sin ϕ =
λ=
Part 3:
sin ϕwater =
sin ϕair =
Nwa =
Part 4:
red line: λ =
blue-green line:
λ=
blue-violet line:
λ=
DISCUSSION:
Approximately how many maxima do you observe in the single slit experiment? If you make the
slit smaller, do the maxima (and minima) get closer together or further appart?
How many maxima do you observe with the diffraction grating you use? To what values of n do
they correspond?
Try to explain in your own words how we have shown that the bending of light at the interface
between air and water is due to the different velocity of light in both media.
Which of the lines from Fig. 14 have you observed in the hydrogen spectrum? Which lines
are deflected at bigger angle - the red lines or the blue lines? The wavelengths of the spectral lines
depends slightly on the temperature and pressure of the air. Why do you suppose this is so?
5