Polarization and dispersion of light
I. OBJECTIVES.
The branch of physics that studies phenomena associated with light is called optics. In
this lab we will investigate two such topics: 1) Dispersion of light . Using a diffraction
grating we will measure the wavelength  of the red line emitted by a Helium-Neon (HeNe) laser and 2) Polarization of light. Using two Polaroid sheets we will generate linealy
polarized light and explore Malus’s law.
II. EQUIPMENT.
He-Ne laser, incandescent lamp, optical bench, grating, Polaroid sheets, photometer,
component carriers, ruler.
III. INTRODUCTION.
1. Light dispersion. The term dispersion means the separation of the various
wavelengths in a light beam. This can be accomplished using one of several methods. In
this lab we will concentrate on a particular method that employs a diffraction grating. A
grating consists of a series of identical equally spaced parallel openings on an opaque
screen (see fig.1). Each opening has the form of a long slit; the separation between two
adjacent slits is equal to d. We illuminate the grating with a light beam of wavelength 
incident at right angles to the grating plane. If we assume that light is a wave it is
straightforward to show that the transmitted (also called diffracted ) light can travel only
in certain directions defined by the angle  the transmitted beam makes with the grating
normal. The latter is indicated in fig.1 by a dashed line. The angles are given by the
equation:
d sin   m
(eqs.1)
Here d is the spacing between two adjacent slits,  is the wavelength of the incident
beam, and m is an integer that can take the values: m  0 , m  1 , m  2 , …
The integer m is called the order of the diffracted beam. If the incident beam consists of
two or more wavelengths, each will be transmitted (or diffracted ) along different angles
and therefore can be easily separated. The spectrometer used in the previous experiment
used a grating to separate the various wavelengths that make up the light beam under
study.
2. Linearly polarized light. In fig.2. we show a light wave propagating along the x-axis.
In this picture the electric field vector E of the wave oscillates in the xy-plane, while the
magnetic field vector B oscillates in the xz-plane. This type of light wave is called
linearly polarized. By convention, the polarization plane of linearly polarized light is
taken to be the oscillation plane for E (xy-plane in fig.2). Light emitted by an
incandescent lamp is unpolarized. This means that the polarization plane of the beam
varies constantly while remaining perpendicular to the light beam propagation direction.
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The simplest method o generated linearly polarized light is to use a source of unpolarized
light and a Polaroid sheet as shown in fig.3. The Polaroid sheet has a particular direction
called the polarization axis indicated in fig.3 by the parallel vertical lines. The light
transmitted by the Polaroid is linearly polarized with the polarization plane parallel to the
Polaroid sheet axis. The price we pay is that we loose half of the intensity of the incident
light. In fig.4 we use one Polaroid sheet with vertical polarization axis to generate
linearly polarized light. The first Polaroid sheet is called a polarizer. We then add a
second Polaroid sheet with its polarization axis at an angle  with the axis of the first
Polaroid. The second Polaroid sheet is known as the analyzer. If we measure the
intensity I of the light beam at point P after the analyzer we find that it varies with angle
 as shown in fig.5. The intensity I is given by the equation:
I  I o cos 2 
(eqs.2)
This equation is known as Malus’s law. Here I  I o is the intensity for   0 i.e. when
the analyzer axis is parallel to the polarizer axis. We note that I  0 when   90 or
  270 i.e. when the polarizer and analyzer axes are at right angles.
IV. EXPERIMENTAL METHOD.
1. Measurement of a grating to measure the wavelength of light. The experimental setup
is shown in fig.6. The He-Ne laser beam (632.8 nm) is incident at right angles to the
grating. The spacing d  1.67 103 nm. The laser and grating are mounted on an optical
bench not shown in fig.6. The diffracted beams are observed on a ruler placed at right
angles to the incident beam at a distance L  50 cm from the grating. Five diffracted
beams result in five laser spots on the ruler as is shown in fig.7. These points are: C
( m  0 ), A ( m  1), B( m  2 ),
A ( m  1) , and B ( m  2 ). We will use spots
C, A, and B do determine  by measuring the distances x1 and x2 of points A and B from
x 
point C, respectively.
From triangle GCA we have: 1  tan 1  1  (eqs.3)
L
x 
From triangle GCB we have:  2  tan 1  2 
(eqs.4)
L
From equation 1 we have:
d sin  2
and also:  
(eqs.6)
  d sin 1 (eqs.5)
2
2. Malus’s Law is studied using the setup shown in fig.8. All optical components are
mounted on carriers that slide along the length of an optical bench. In incandescent lamp
is used as a source of unpolarized light. It is followed by the polarizer and the analyzer.
Both Polaroid sheets are mounted on a 360 goniometric circle with the zero
corresponding to the polarization axis (see fig.9). A lens collects the light transmitted
through the analyzer and focuses it on the tip of an optical fiber bundle. The optical
fibers guide the light to the photometer which measures the light beam intensity I. Thus I
2
can be measured as function of the angle  . Here is the angle between the polarization
axes of the two Polaroid sheets.
V. PROCEDURE.
V-1: Using the setup of fig.6 observe the first (m = 1) and second ( m = 2) order laser
diffraction spots on the ruler (see fig.7). The distance L is setup at 50 cm from the
grating.
Measure the distance x1 of point A from point C and record the value of x1 in the data
sheet.
Measure the distance x2 of point B from point C and record the value of x2 in the data
sheet.
V-2: Use the setup shown in fig.8. Here  P and  A are the angles of the polarizer and
analyzer axes with the vertical indicated by a white line on the component carrier).
Set P  0 and  A  0 . Measure the light intensity I o using the photometer. Record I o
in the data sheet.
Set  A at 10 leaving P  0 . Thus   A  P  A . Measure the light intensity I and
record the value in the data sheet.
Repeat process (b) for  A  20, 30, 40, ... ,350 , and 360 .
VI. FOR THE REPORT.
VI-1. Diffraction grating
a. Calculate the angle 1 using eqs. 3.
b. Calculate the angle 2 using eqs. 4.
c. Calculate He Ne using eqs.5.
d. Calculate He Ne using eqs.6.
VI2. Malus’s law.
a. Tabulate the values of I ( ) and  collected in section V-2. Enter a third column in
I ( )
the table in which you enter the ratio
.
Io
I ( )
b. Plot
versus  . The plot should look like that shown in fig.5.
Io
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VII. QUESTIONS.
VII-1. In section V-1 you measured the wavelength He Ne of the red line of the He-Ne
laser using the first (m = 1) and second (m =2 ) order laser diffraction spots. The
accepted value for He Ne  632.8 nm. Calculate the differences 1 and 2 between
your experimental values and the accepted value. Which difference is larger?
VII-2. Calculate the maximum value mmax the integer m can have in the experiment of
section V-1. Hint: sin   1 .
VII-3. Calculate the angles 1 and 2 for the two wavelengths 1  588.99 nm and
2  589.59 nm of the bright yellow doublet emitted by sodium.
VII-4. In section VI-2 you measured I versus the angle  . Suggest a method to
convince yourselves that Malus’s law  I  I o cos 2   is obeyed.
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Incident
beam
Transmitted
beam
Grating
Fig.1: Schematic diagram of a grating.
Light is incident at right angles to the
grating. Light is diffracted at an angle
 with the grating normal. After
Halliday, Resnick and Walker
Fundamentals of Physics.
Fig.2: Diagram of an electromagnetic
wave traveling along the x-axis. This
wave is linearly polarized with the
electric field vector E oscillating in the
xy-plane, and the magnetic field vector
in the xz-plane. After Halliday,
Resnick and Walker Fundamentals of
Physics.
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Fig.3: Unpolarized light is incident on a
Polaroid sheet. The transmitted beam
emerges linearly polarized with the electric
field E parallel to the Polaroid axis. After
Halliday, Resnick and Walker Fundamentals
of Physics
Analyzer axis
φ
Polarizer
Analyzer
P
Fig.4: Schematic diagram of the setup used to
study Malus’s law. An unpolarized light beam
passes through a polariser and then through an
analyzer. The axis of the two Polaroid sheets are at
an angle . After Halliday, Resnick and Walker
Fundamentals of Physics.
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I/Io
1
φ
O
90°
180°
270°
360°
Fig.5: Plot of the light intensity I at point P
in fig.4 as function of the angle .
x2
x1
m = -2 m = -1
B'
m=1
m=0
A'
A
C
θ1
L
G
m=2
ruler
B
θ2
Grating
He-Ne
laser
Fig.6: Diffraction of a laser beam by a grating. The diffracted beams
are observed on a ruler placed at right angles to the laser beam
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m = -2
m = -1
B'
A'
m=0
m=1
A
C
m=2
ruler
B
x1
x2
Fig.7: Diffraction pattern of the laser beam in fig.6
The diffracted laser spots are observed on a ruler at right
angles to the laser beam
20 cm
Lens
f =127mm
25 cm
Incandescent
light source
bulb
Optical bench
Fiber optic cable
Analyzer Polarizer
Photometer
Fig.8: Schematic diagram of the experimental setup for the study of Malus’s
law
Fig.9: Polaroid sheet used as a
polariser and as an analyser
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