PC2193 EXPERIMENTAL PHYSICS
BREWSTER’S ANGLE AND POLARISATION
Introduction
Light is a wave phenomenon with its electric field (or its magnetic field) vibrating in
time and in space. If the electric field of a light beam is vibrating in a fixed plane in space, such
a light beam is referred to as plane-polarised light or linearly polarised light. This fixed plane
is named as the plane of vibration. Light from most lasers is plane-polarised. On the other
hand, light form the Sun is said to be un-polarised. However, sunlight can be decomposed into
two plane-polarised light beams with their planes of vibration perpendicular to each other.
When un-polarised light is incident on the surface of a dielectric (such as a glass), it
can be decomposed into two components, denoted as “s” and “p” component. The plane of
vibration of the “s” or “p” component is perpendicular or parallel to the plane of incidence,
respectively. Their intensities depend on the angle of incidence. At a specific angle of
incidence, the intensity of “p” component of reflected light is zero; and the reflected light
becomes plane-polarised with only the “s” component. This phenomenon was discovered
by Sir David Brewster and, thus, the specific angle is called Brewster’s angle or polarisation
angle. From his experiments, Brewster confirmed that the reflected ray and the refracted ray
are 90 degrees apart when the incident angle is set at Brewster’s angle.
Figure 1. Reflection and Refraction of un-polarised Light
Snell’s law states that:
(1)
𝑛 𝑠𝑖𝑛𝜃 = 𝑛 𝑠𝑖𝑛𝜃
where 𝑛 and 𝑛 are the refractive index of the air and the dielectric, respectively; 𝜃 and
𝜃 are the angle of incidence and refraction, respectively. Because of
𝜃 + 𝜃 = 90°, 𝜃 = 90° − 𝜃 , 𝑎𝑛𝑑 𝑠𝑖𝑛𝜃 = sin 90° − 𝜃
= 𝑐𝑜𝑠𝜃 ,
(2)
we have the following equation at Brewster’s angle,
𝑛 𝑠𝑖𝑛𝜃 = 𝑛 𝑐𝑜𝑠𝜃
(3)
Finally, we arrive at
𝑡𝑎𝑛𝜃 =
𝑛
𝑛
(4)
Eqn (4) shows that Brewster’s angle depends on the refractive index of the dielectric.
Objective
1.
To determine the Brewster’s angle of acrylic material (D lens).
2.
To determine the refractive index of the acrylic.
Theory and Evaluation
The laws of reflection and refraction
The full treatment of reflection and refraction is derived from the Maxwell’s equations
of electrodynamics. It is known as Fresnel equations. First, we suppose that a plane
monochromatic wave is incident on the planar surface separating two isotropic media with
nearly equal permittivity (𝜇 = 𝜇 = 𝜇 ). Whatever the polarisation of the wave we shall
resolve its electric 𝑬 and magnetic 𝑩 fields into components parallel and perpendicular to
the plane of incidence and treat these constituents separately.
Case (1): 𝑬 perpendicular to the plane of incidence (s component)
Figure 2. An incoming wave whose 𝐸-field is normal to the plane of incidence
The most common forms of the amplitude reflection coefficient 𝑟 and the amplitude
transmission coefficient 𝑡 are simply:
𝑛 𝑐𝑜𝑠𝜃 − 𝑛 𝑐𝑜𝑠𝜃
𝑛 𝑐𝑜𝑠𝜃 + 𝑛 𝑐𝑜𝑠𝜃
2𝑛 𝑐𝑜𝑠𝜃
𝑡 =
𝑛 𝑐𝑜𝑠𝜃 + 𝑛 𝑐𝑜𝑠𝜃
𝑟 =
(5)
(6)
Case (2): 𝑬 parallel to the plane of incidence (p component)
Figure 3. An incoming wave whose 𝐸-field is in the plane of incidence
The most common forms of the amplitude reflection coefficient 𝑟 and the amplitude
transmission coefficient 𝑡 are simply:
𝑛 𝑐𝑜𝑠𝜃 − 𝑛 𝑐𝑜𝑠𝜃
𝑛 𝑐𝑜𝑠𝜃 + 𝑛 𝑐𝑜𝑠𝜃
2𝑛 𝑐𝑜𝑠𝜃
𝑡 =
𝑛 𝑐𝑜𝑠𝜃 + 𝑛 𝑐𝑜𝑠𝜃
𝑟 =
(7)
(8)
One further notational simplification can be made by using Snell’s law, where upon
the Fresnel equations for dielectric media become
𝑠𝑖𝑛(𝜃
𝑠𝑖𝑛(𝜃
𝑡𝑎𝑛(𝜃
𝑟 =−
𝑡𝑎𝑛(𝜃
𝑟 =−
2𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
𝑠𝑖𝑛(𝜃 + 𝜃 )
2𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
𝑡 =
𝑠𝑖𝑛(𝜃 + 𝜃 ) 𝑐𝑜𝑠(𝜃 − 𝜃 )
−𝜃 )
+𝜃 )
−𝜃 )
+𝜃 )
(9)
𝑡 =
(10)
Figure 4. S Polarisation (left) and P polarisation (right) of incident light
The measured reflectance 𝑅 and transmittance 𝑇 however are the ratio of flux (not
amplitude), such that
𝑅=𝑟
𝑇=
𝑛 𝑐𝑜𝑠𝜃
𝑡
𝑛 𝑐𝑜𝑠𝜃
(11)
𝑅+𝑇=1
Hence
𝑅 =𝑟
𝑅 =𝑟
(12)
𝑛 𝑐𝑜𝑠𝜃
𝑡
𝑛 𝑐𝑜𝑠𝜃
𝑛 𝑐𝑜𝑠𝜃
𝑇 =
𝑡
𝑛 𝑐𝑜𝑠𝜃
𝑇 =
In the case, when incident light has equal "p" and "s" components, the portion of "p"
component of reflected light is obtained from flux reflectances as:
𝑅
𝑐𝑜𝑠 (𝜃 + 𝜃 )
=
𝑅 +𝑅
𝑐𝑜𝑠 (𝜃 + 𝜃 ) +𝑐𝑜𝑠 (𝜃 − 𝜃 )
(13)
Equipment
Figure 5. Experiment set up
1. Laser diode: PASCO OS-8525 BASIC DIODE LASER. The power switch is on the back.
2. Polariser assembly: PASCO. There are two round polarisers. The first one (counting from
the diode) can be varied. The second one on the side with two screws should be fixed at 45°
and locked by tightening the brass screw.
3. Collimating slits: PASCO OS-8537 SPECTRO-PHOTOMETER. This slit should be
mounted on the track and adjusted by sliding so that light could pass through the #4 slits.
4. D lens: It should be mounted with its magnet side sitting on the metal lens holder. Light
will incidence and reflect on the flat surface and refract out from the semicircle surface. It
is made of materials called acrylic.
5. Square analysing polariser: PASCO. Its transmission axis is marked, placing which
vertically and horizontally can be used to exclusively transmit “s” beam and “p” beam for
light sensor to measure.
6. Light sensor: PASCO OS-8534 APERTURE BRACKET FOR LIGHT SENSOR. Its slit
should be set on #4.
7. ScienceWorkshop®750 interface: Connect the rotary motion sensor of “6” into Channel 1
and 2 of interface Connect Light Sensor into Channel A. Start the computer and open
“DataStudio” from the shortcut on the desktop. The relationship between the incident angle
𝛽 and the display value 𝑎𝑏𝑠(𝑥°) follows the equation:
𝛽=
1
1
180° − 𝑎𝑏𝑠(𝑥°) ,
2
15
1 1
𝛼=
𝑎𝑏𝑠(𝑥°)
2 15
(13)
𝑥° is from the Channel 1 and 2. The geometry demonstrates as:
Figure 6. Angle geometry illustration
Set-up and Procedure
1. Switch on the power sources (behind the table), Laser diode, and the computer.
2. With the D lens removed, calibrate the angle for the Rotary motion sensor: rotate the base
disk so the laser beam is centred on the Light Sensor slit. The angle of the base disk should
be near 180 degrees. However, it is OK if the angle is slightly off.
3. To align laser beam with light sensor, remove Collimating slits, D lens and Square
analysing polariser. Set the arm at 180°. Turn on the Laser diode and use the vertical adjust
on the laser to get the laser beam at the centre of the light sensor slit. If the laser spot is not
still centred on the slit of the Light sensor, slide the D lens to centre it. Make sure the lens
is mounted firmly against the step of the metal lens holder.
4. In DataStudio, click “Create Experiment” in the welcome window. Choose both “Light
Sensor” and “Rotary Motion Sensor” by double click. From the “Display” panel, double
click “Digits” or “Graph” to monitor the values. Click the Start button. Rotate the disk
while watching the reading on the digits display for the light intensity. Stop at the position
that gives the maximum intensity by clicking on the Stop button.
5. Mount the D lens back on the metal lens holder. Set the angle of metal lens holder to 85°.
Turn off the room lights. Click on the Start button. Rotate the base disk by monitoring the
digits display for the light intensity (the display will be about 150° so that 𝛽 is 85°). Stop
at the position that gives the maximum intensity. The Square analysing polariser should not
be in place. Write down the measured angle (𝛽) and the light intensity 𝐼 in %.
6. Insert the Square analysing polariser between the D lens and the Light sensor silt with its
transmission axis horizontal, and then write down the reading for the intensity of “p” beam
of reflected light. Then change the Square analysing polariser with its transmission axis
vertical and then write down the reading for the intensity of “s” beam of reflected light.
Stop the measurement.
7. Remove the Square analysing polariser and turn on the room light, rotate the metal lens
holder to smaller angles with an interval of 5 degrees, and repeat Steps 5 and 6 above. When
approaching Brewster’s angle, the interval should be set at one degree (and switch the gain
of the Light sensor to 10 or 100 if the reading for the light intensity is too low).
Points to note:
1. Avoid direct eye exposure with laser diode.
2. Do not rotate the second polariser fixed by screws at 45. Keep it fixed.
3. Do not touch the D lens but rotate it through the metal lens holder below.
4. Do not move the Arm until the program is started to take the actual data run. This ensures
the angle you read from the digits display is the actual angle.
5. Do not click on the Stop button until all the procedure steps are completed.
6. If the display intensity reaches 100%, rotate the first polariser to moderate to a lower
value and redo the step. Otherwise it does not give a correct ratio of intensity of light
measured with and without the square analysing polariser.
Data Analysis
Fraction of "p" polariasation
0.35
Experiment
0.3
Model
0.25
0.2
0.15
0.1
0.05
0
20
30
40
50
60
Incidence angle (deg)
70
80
Figure 7. Example data of p-polarised light versus incident angle (as example of format)
For each pair of the intensity data, calculate the ratio of the light intensity measured with
the polariser to the light intensity measured without the polariser. Use Microsoft Excel to plot
this ratio as function of the incident angle from 30 to 80 for "p" and "s" polarisation and fit
using equation (13). Examine, whether same value of refractive index fit angular dependences
of both "p" and "s" polarisations.
Then, determine Brewster’s angle at which the ratio is a minimum. Considering refractive
index of air at room temperature n1 = 1, substitute it into equation (4) for determination of the
refractive index of the D lens.
*Using the light sensor to detect the refracted light by following the way above, find out
the relationship between incident and refracted angles. Substitute them into Snell’s law,
equation (1), to determine the refractive index of the D lens. Compare with the result above.
*In Step 6 of procedure, think about the possible reasons why the sum of "s" and "p"
beam intensity cannot reach the total reflected light.
Summary
The experiment helps students understanding the law of light propagation and the concept of
Brewster angle. The ability of setup and measurement of the optical devices could be enhanced.