Lab Experiments 125
KamalJeeth Instrumentation & Service Unit
Experiment-300
A
MALUS’S LAW OF POLARISATION
Dr Jeethendra Kumar P K
KamalJeeth Instrumentation and Service Unit, Tata nagar, Bangalore-560 092. INDIA.
Email: jeeth_kjisu@rediffmail.com
Abstract
Using a polarizer-analyzer pair Malus’s law of polarization is verified. Intensity of
the light coming out from the analyzer is detected using a LDR and measured
using a current meter. The variation of the light intensity is studied with angle θ,
Cosθ and Cos2θ.
Introduction
The light coming from the Sun, candle light, and light emitted by a bulb are all known to be
un-polarized. Light consists of electromagnetic waves produced by vibrating electric charges
in all possible directions. To illustrate this, an example, is shown in Figure-1, in which two
children are holding two ends of the rope and moving their hands up and down producing
waves which traverse along the rope. Similarly, they can move their arm left and right which
will produce another pattern of waves. A different pattern of waves is produced when their
arms are moved at a different angle. Figure-2 shows the waves produced due to arm
movements in the vertical and horizontal positions. The movement of the arm in different
directions and producing waves is similar to electrons vibrations in all possible directions
producing light waves. This light is un-polarized and denoted as shown in Figure-3. The
arrow directions are three dimensional. In a two dimensional representation, the arrows
appear in the X-Y plane and the propagating light wave is called a plane polarized wave.
Figure-1: Two children moving their arms up and down to produce a wave pattern
Polarization
Certain transparent materials such as Nicol, Tourmaline are capable of filtering and allowing
light waves having vibrations in only in one plane. Such materials are called Polaroids. This
filtering is possible due the structure of the material that is having its cells arranged in a
straight line manner only in one direction as shown in Figure-4. This phenomenon of filtering
and producing light waves having vibrations confined to one particular direction is called
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polarization. Polarization is a property of a material by which light waves are filtered and
made directional.
(a)
(b)
Figure-2: (a) Up-down movement of the arm producing a wave in the vertical direction
(b) Left-right movement of arm producing a wave in horizontal position
Figure-3: Representation of a polarized and un-polarized light wave
Figure-4: Molecules arranged vertically resulting in horizontal polarization axis (b)
Molecules arranged horizontally resulting in vertical polarization direction
Polarized light is useful in many applications. Poloroids can control light intensity just like
water tap. To control light intensity one requires two Polaroids. The first Polaroid is placed in
front of the light producing polarized light. The second Polaroid, placed in front of the first,
can control the intensity of light passing through by rotating it. At a certain angle the entire
light coming out from the first Polaroid is cut off and no light passes through the second
Polaroid. The first Polaroid placed before the light source is called polarizer and second
Polaroid placed in front of the polarizer is called analyzer.
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Polarizing Materials
Naturally occurring Nicol and Tourmaline are Polaroids. They were used earlier as Polaroids.
At present Polaroid films made of plastics are available which make bulk of Polaroids for
various applications. Polyvinyl Alchohol (PVA) plastic with Iodine doping is a good
Polaroid. Stretching of the sheet during manufacturing ensures that the PVA chains are
aligned in one particular direction. Electrons from the iodine dopant are able to travel along
the chains, ensuring that light polarized parallel to the chains is absorbed by the sheet; light
polarized perpendicularly to the chains is transmitted. The durability and practicality of PVC
Polaroids make them the most common type of polarizers that are in use, for example for
sunglasses, photographic filters, and liquid crystal displays (LCDs). It is also much cheaper
than the naturally occurring Polaroids.
Figure-5: Polaroid used in this experiment (PVC is fixed in inner circle)
Figure-6: LDR sensor (fitted to the eye piece holder) and relative light intensity meter
Polarizer and Analyzer pair used in this experiment
Figure-5 shows the Polaroid used in this experiment. PVA Polaroid film is used in this. A
circular scale is attached which reads the angle in degrees. One can rotate both the analyzer
and polarizer by any angle.
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Lab Experiments 128
KamalJeeth Instrumentation & Service Unit
Malus’s Law
When light falls on a polarizer, the transmitted light gets polarized. The polarized light falling
on another Polaroid, called analyzer, transmits light depending on the orientation of its axis
with the polarizer. The intensity of light transmitted through the analyzer is given by Malus’
law. The law describes how the intensity of light transmitted by the analyzer varies with the
angle that its plane of transmission makes with that of the polarizer. The law can be stated in
words as follows:
The intensity of the transmitted light varies as the square of the cosine of the angle between
the two planes of transmission.
At= Ao Cosθ
Where
…1
At is amplitude of the light transmitted through the analyzer; and
Ao is the amplitude of the incident plane polarized light.
If At is amplitude then At2 is the intensity of the light transmitted through the analyzer.
It= At2=Ao2 Cos2θ = Io Cos2θ
Where
…2
It is the intensity of the light transmitted through the analyzer; and
Io is the intensity of the incident plane polarized light
A graph representing It/Io versus Cos2θ is a straight line with unit slope. Hence to prove
Malus’s law one has to plot a graph of It/Io versus Cos2θ.
The light intensity cannot be measured directly. The light energy is converted into electrical
energy using photo detectors such as a photo cell or light dependent resistor (LDR). In such
photo detectors the current produced is directly proportional to the light intensity.
It α current
It α Icurrent
It = K Icurrent
The constant K appearing here is nothing but the conversion efficiency of photo detector. If
we take the ratio of two transmitted light intensities at θ=0° and θ=50°then we have
intensities
I10° = K Icurrent at 10° and I50° = K Icurrent at 50°
Taking the ratio of the two intensities
‫ܫ‬ଵ଴ ‫ܫܭ‬௖௨௥௥௘௡௧ ଵ଴ ‫ܫ‬௖௨௥௥௘௡௧ଵ଴
=
=
‫ܫ‬ହ଴ ‫ܫܭ‬௖௨௥௥௘௡௧ ହ଴ ‫ܫ‬௖௨௥௥௘௡௧ ହ଴
This indicates that by taking the ratio of two currents transmitted through the analyzer at two
different angles is the same as the light intensity ratio. Hence ratios of the two currents
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KamalJeeth Instrumentation & Service Unit
represent comparative light intensity. In this experiment we have taken the ratio with respect
to Io, the current obtained when axis of the analyzer is parallel to the plane polarized axis of
the light.
Using this concept Malus’s law (equations 1 and 2) is verified in this experiment. The angles
are noted experimentally from the dial fitted to the Polaroids. Figure-5 shows the photo
detector (LDR) used in this experiment. A multi-range current meter and 5V power supply
are used in the measurement. Figure-7 shows the complete experimental setup mounted on a
spectrometer.
Figure-7: Experimental Set-up for verifying Malus’s law
Apparatus Used
The apparatus consists of a 6˝ spectrometer, a polarizer-analyzer pair, halogen light source,
relative light intensity meter (multi-range current meter), and power supply (5V). The
complete experimental set-up is shown in Figure-7. The same experiment mounted on an
optical bench is shown at the cover page of this issue of LE.
Experimental Procedure
The experiment consists of two parts
Part-A: Orienting the polarizer toward polarization axis
Part-B: Determination of transmitted light intensity
Part-A: Orienting the polarizer toward polarization axis
1. The polarizer is fitted onto the collimator of the spectrometer and light is allowed to
fall on the collimator lens directly, removing the adjustable slit of the collimator.
2. The analyzer is fitted to the telescope of the spectrometer and its angle is set to 0°.
The eye piece is replaced by LDR, as shown Figures 6 and 7.
3. The LDR sensor is connected to the relative light intensity meter and the current is
noted. Current in the meter shows that light transmitted through the analyzer is falling
on the sensor.
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KamalJeeth Instrumentation & Service Unit
4. The two arms of the spectrometer (collimator and telescope) are aligned in a straight
line and collimator and telescope are focused so that current in the meter is maximum.
θ°
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
330
340
350
360
Table-1
I(mA) IoCos2θ(mA)
I/Io
Cosθ
Cos2θ
1
1
0.175
0.175
1
0.984
0.97
0.159
0.169
0.965
0.939
0.883
0.152
0.154
0.88
0.866
0.75
0.140
0.131
0.75
0.766
0.586
0.125
0.102
0.58
0.642
0.413
0.106
0.072
0.41
0.50
0.25
0.085
0.043
0.25
0.342
0.167
0.068
0.029
0.16
0.173
0.030
0.050
0.005
0.03
0
0
0.042
0
0
-0.173
0.030
0.046
0.005
0.262
-0.342
0.167
0.059
0.029
0.337
-0.5
0.25
0.072
0.043
0.411
-0.642
0.413
0.091
0.072
0.52
-0.766
0.586
0.111
0.102
0.63
-0.866
0.75
0.130
0.131
0.75
-0.939
0.883
0.145
0.154
0.82
-0.984
0.97
0.155
0.169
0.88
-1
1
0.160
0.175
0.91
-0.984
0.97
0.159
0.169
0.90
-0.939
0.883
0.153
0.154
0.88
-0.866
0.75
0.140
0.131
0.75
-0.766
0.586
0.125
0.102
0.58
-0.642
0.413
0.106
0.072
0.41
-0.5
0.25
0.085
0.043
0.24
-0.342
0.167
0.066
0.029
0.16
-0.173
0.030
0.051
0.005
0.02
0
0
0.042
0
0
0.173
0.030
0.043
0.005
0.02
0.342
0.167
0.055
0.029
0.16
0.5
0.25
0.070
0.043
0.24
0.642
0.413
0.092
0.072
0.41
0.766
0.586
0.112
0.102
0.58
0.866
0.75
0.128
0.131
0.75
0.939
0.883
0.144
0.154
0.88
0.984
0.97
0.154
0.169
o.96
1.000
1
0.160
0.175
1
LDR current variation with polarization angle
5. Now the analyzer is rotated slowly until the current in the meter attains maximum
value. The angle of rotation of the analyzer, Φ , is noted.
Φ = 24°
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6. Now the polarizer is adjusted to 24° and the analyzer is set to 0°. This ensures that the
plane of the polarized light coming out of the polarizer is parallel to the polarization
axis. After this the polarizer is not disturbed further throughout the experiment
Note: If the polarizer is not adjusted parallel to the polarization axis, there will be no
minimum current at θ=0°.
7. With polarizer angle set to 24° and analyzer angle set to 0°, the maximum current in
the relative light intensity meter is noted. This is Io
Io=0.175mA =175µA.
8. The analyzer is now rotated by 10° and the current is noted. This is ‘I’, for θ=10°
It = 0.154, for θ=10°
9. Theoretical value of I is calculated using the equation
It = Io Cos2θ = 0.175 Cos2 10 = 0.175 (0.984)2= 0.175x 0.9698= 0.169mA.
10. The value of the current 0.169mA is nearly equal to 0.154mA; for θ=10°
11. The trial is repeated by rotating the analyzer in steps to 200, 300 up to 3600. In each
case the current is noted and tabulated in Table-1.
A graph is drawn taking the current ‘I’ along Y-axis and angle of rotation of analyzer
on the X-axis, as shown in Figure-8. From the graph the cosine nature of the curve
(A= Ao Cosθ) is clearly evident, validating the Malus’s law.
Current (mA)
Measured
Calculated Equation-2
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
50
100
150
200
250
300
350
400
Polarization Angle (Degree)
Figure-8: Cosine nature of current variation with analyzer rotation angle
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13. Cosθ, Cos2θ, I/Io and IoCosθ are calculated and presented in Table-1. Two more
graphs showing the variation of I/Io versus Cosθ and I/Io versus Cos2θ are shown in
Figures 9 and 10 respectively.
14. The slope of straight line in graph shown in Figure-10 is calculated.
Slope=0.992≈1
Normalized Intensity (I/Io)
1.2
1
0.8
0.6
0.4
0.2
0
-1
-0.5
0
0.5
1
Cosθ
Figure-9: Amplitude variation of the light transmitted through the analyzer
NNormalized intensity (I/Io)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Cos2θ
Figure-10: Normalized current variation with analyzer angular position
Results
Slope of the straight line = 0.992 ≈ 1
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Discussion
1. The current (proportional to light intensity), noted for different angles of rotation of
the analyzer, follows a cosine curve for 3600 of rotation, indicating the validity of
equation-1. The experimentally measured current and that calculated using equation-2
agree within the limits of the experimental error.
2. The relative or normalized light intensity (It/Io) versus Cos2θ curve is a straight line,
as expected, with unit slope indicating the correctness of the Malus’s law.
3. The relative intensity of the light emerging from analyzer is maximum at 0° and 180°
and it attains minimum value at 90° and 270°. In between it varies as a Cosine
function as indicated by the graph in Figure-8 and Figure-9.
Etienne Louis Malus (1775–1812)
Etienne Louis Malus (1775–1812)
Picture courtesy: chemistry.about.com/.../Etienne-Louis-Malus.htm
Etienne Louis Malus was a French military engineer and physicist. Malus, who was born in
Paris, attended the military school in Mezières (1793) and the newly established Ecole
Polytechnique (1794–96) where he received his basic science education. He was
commissioned in 1796 and served as a military engineer in Napoleon's expedition to Egypt
and Syria. After his return to France he held various military engineering appointments. He
became an examiner in geometry and analysis in 1805 and an examiner in physics in 1806 at
the Ecole Polytechnique; these posts brought him in contact with other physicists.
Malus carried out many researches in optics, which was his main scientific interest. He is
remembered for his discovery made in 1808 that light rays may be polarized by reflection. He
found this while looking through a crystal of Iceland spar at the windows of a house
reflecting the rays of the Sun. ( Iceland spar is a doubly refracting transparent calcite used in
optical instruments) He noticed that on rotating the crystal the light got extinguished in
certain positions. In explaining his observations Malus, a Newtonian and believer in
corpuscular theory of light, argued that light particles have sides or poles and used for the
first time the word ‘polarization’ to describe the phenomenon. Malus's work in optics gave
considerable impetus to investigations on polarization and the optical properties of crystals.
References
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[1]
Malus law, PHYWE series of publication, Laboratory Experiment Publisher.
[2]
D Amrani and P Paradis, Malus’s law of light polarization using a computer based
laboratory; http://www.journal.lapen.org.mx
[3]
A Jenkins and H E White; Fundamentals of Optics, 4th Edition Page-503. McGrawHill , International Edition, 1981.
[4]
Polarization, Light waves and color-lesson-1, http://physicsclassroom.com/class/light
[5]
Polarization: Polaroid and Malus’ law; Malus law:
http://scholar.hw.ac.uk/site/physics/topic
[6]
Biography http://www.answers.com/topic/tienne-louis-malus
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