Download FYSA220 / 2 POLARISATION OF LIGHT 1. Theory

FYSA220 / 2
POLARISATION OF LIGHT
The incoming and outgoing components of an electromagnetic radiation (e.g. visible
light) parallel and perpendicular to a reflecting surface (e.g. glass) behave differently.
Observations are used for calculating the refraction coefficient of glass based on
Brewster’s law and the accuracy of Malus’s law will be examined in the exercise.
Furthermore as an example of an optically active material a s.c. λ /2 -plate will be
studied.
Prepare for the exercise by reading:
•
•
•
Ohanian: Physics, Vol. 2, pages 806 - 810.
Alonso-Finn: Fundamental University Physics, Vol. 2, pages 778 - 791.
Young & Freedman: University Physics, 10th edition, pages 1064 – 1071, 11th
ed., pages 1262-1268.
1. Theory
Polarisation of light can happen in many ways. Here you’ll study an everyday
phenomenon, polarisation by reflection. When light hits a surface, it will reflect and
become polarised i.e. a part of its components will reflect almost perfectly and other
part will reflect only partially, or undergo a total absorption. The degree of
polarisation depends on the angle between the direction of incoming light and the
normal of the surface as well as on the material of the reflecting surface.
Light incoming from material 1 (index i, refraction index n1) into a surface of material
2 (index r, refraction index n2) partially reflects back to 1 and partially refracts into 2
(fig. 1). The angles of the incoming and refracted rays obey the Snell’s law
n1 sin θi = n2 sin θ r
(1)
If the angle between the reflected and refracted rays is right, i.e. θ ' r +θ r = π / 2 , eq. (1)
reads
sin θi
sin θi
n2 sin θi
=
=
=
'
n1 sin θ r cos θ r cos θi
(θ r' = θ i ) ,
where the incoming angle is the s.c. polarisation angle θ i = θ i . In this case the
reflected light is said to be fully planar polarised. We’ll get thus the s.c. Brewster’s
law
sin θi
= tan θi = n21 ,
cos θi
(2)
where n21 is the ratio of the refraction coefficients of the two materials n21 = n2/ n1. If
material 1 is air, n1= 1 and n21= n2.
FYS222/2 Polarisation of light
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Figure 1. Reflection and refraction of light on the surface of glass.
Plane defined by the incoming, refracted and reflected light (rays) is called in the
incoming plane. The incoming electric field (light) can be described by a vector
divided into two components Eiπ, parallel to the incoming plane, and Eiσ,
perpendicular to it. These components reflect and refract with components E'rπ and
E'rσ as well as Erπ and Erσ , correspondingly (fig. 1).
The relation between the components of the fields above can be derived from the
continuation conditions of an electric-field vector E , a displacement vector D , a
magnetic-flux density-vector B and a magnetic-field vector H at the surface of an
insulator, when the surface-charge and surface-current densities are zero (no
polarisation charges and induced currents). These conditions are as follows:
E|| (parallel to the reflecting surface) is continuous
D⊥ (normal to the reflecting surface) is continuous
B⊥ is continuous
H|| is continuous
The reflection coefficient is defined as the ratio of reflected to original components:
FYS222/2 Polarisation of light
Rπ =
Rσ =
–3–
E r' π
(3)
E r' σ
(4)
n cos θ r − n2 cos θ i
= 1
n1 cos θ r + n2 cos θ i
E iπ
n cos θ i − n2 cos θ r
= 1
n1 cos θ i + n2 cos θ r
E iσ
The transmission coefficients are defined correspondingly:
E
2n1 cos θ i
Tπ = rπ =
Eiπ n1 cos θ r + n2 cos θ i
(5)
E
2n1 cos θi
Tσ = rσ =
Eiσ n1 cos θi + n2 cos θ r
(6)
Since the intensity is proportional to the amplitude square of a field vector (in vacuum
I = cε0E²), we get the relative intensities for the both components as follows:
2
I r' π
⎛ E'
= ⎜ rπ
I iπ ⎜⎝ Eiπ
⎞
⎟ =I
π
⎟
⎠
⎛ E'
= ⎜ rσ
I iσ ⎜⎝ Eiσ
⎞
⎟ =I
σ
⎟
⎠
2
Intensity
I r' σ
(7)
Figure 2. Components of intensity as a function of the incoming angle.
(8)
FYS222/2 Polarisation of light
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Notice, that the polarisation affects the magnetic field as well because in a
electromagnetic wave the electric and magnetic field are always connected. Then
B ∝ E and E , B and the propagation direction of the wave form a right handed
system. To discuss here the electric components, only, is done for clarity. In fact, the
connection between E and B as well as the continuation conditions of the magnetic
field have been taken into account in the calculations of electric field components
above.
The degree of polarisation can be measured using another polarising plane called
analyser, here (fig. 3). When the analyser is rotated around its axis with an angle θ, it
transmits only the component EA = Ecosθ of the plane-polarised light. Since the
intensity of light is proportional to the amplitude square, we get the Malus’s law for
the intensity of the light passing the analyser:
I = I 0 cos 2 θ ,
(9)
where I0 is the intensity of the plane-polarised light before the analyser.
The degree of polarisation of the reflected light is defined as follows:
I
I −I
−I
P = max min = σ π
I max + I min Iσ + Iπ
(10)
The degree of polarisation of linearly-polarised light is 1 and of non-polarised 0. Eq.
(10) shows that with a complete polarisation a component vanishes.
Figure 3. Analysing linearly-polarised light.
λ /2 -plate
To affect the polarisation of a laser light we use a s.c. λ/2 -plate (or a λ/4 -plate). This
active element can be built e.g. of a quartz-crystal fabricated so that the optical axis is
parallel to its surface. When a linearly-polarised light passes the sheet, it divides into
FYS222/2 Polarisation of light
–5–
two components called by the ordinary and extra-ordinary ones. These components
are polarised perpendicular to each other and have different velocities (nord ≠ next i.e.
the refraction coefficients of the crystal are different for the two components).
Figure 4. Working principle of a λ/2 -crystal. a) The direction of the electromagnetic
–field vector changes an angel 2θ when passing the crystal. b) A close-up from the
phenomenon showing the phase difference created.
The
phase shift for a distance d travelled in the crystal is equal
2π
d (nord − next ) . If the phase shift is π i.e. 180°, one can show that the out
to δ =
λ
going light is linearly polarised as well (fig. 4). The direction of the electric-field
vector rotates by 2θ. If θ = 45°, the plane of polarisation is rotated by 90°. The
thickness of the sheet must then be d =
λ
2
nord − next
.
2. Equipment
Equipment used is shown in figure 5. Non-polarised light from the light source is
directed via a collimator slit and a light guide to a dark polarising plate. The plate is
inserted to a support fixed to a turntable with a scale (not precise for absolute values)
to register the entrance angle desired. The analyser can be moved around the turntable
at a spot where it collects the reflected light. The analyser has a build-in polariser the
transmission direction of which can be continuously changed turning the analyser.
with a scale for determining the angle between incoming and reflected light. The
analyser has an angle scale as well.
FYS222/2 Polarisation of light
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Figure 5. The equipment used in the work. V = light source (led), S = light guide, P =
polarizer, A = analyser with a photo detector, V = voltmeter and T = power source.
The photo detector is a led (light emitting diode, IPL530) equipped with an amplifier.
The output voltage of the detector is directly proportional to the intensity of incoming
light. The operating power is supplied to the detector by two 9 V batteries placed in a
box with a switch.
3. Measurements
Since the straight determination of the entrance angle θi is inaccurate, set first the
detector angle (that is double the entrance angle). This can be set quite precisely with
help of the angle scale of the turntable. Then turn the polarising plate so that the
reading of the voltmeter is at maximum. Then the adjusting inaccuracies of the
entrance angle remain as small as possible. Now, the entrance angle can be taken as
one half of the total angle.
Start the measurements by removing the polariser (reflector) and measure the
intensity of a light entering straight to the detector (or more precisely the output
voltage). The brightness of the led is adjusted so that the output voltage of the detector
is about 1 V for the straight incoming light. Check that the intensity maximum is at
the reading 180 on the angle scale. Then the polariser is put back on its place and one
makes sure that the rotational angle of the analyser in respect of the polariser is
correct, see the instruction in the lab.
Then the Iσ – and Iπ –components of the reflected light are measured as a function of
the entrance angle. The measurement should be done for suitably many entrance
angles in the range 85°–15°. The angle of the polariser in the analyser is 0° when
measuring Iσ and 90° when measuring Iπ. Close to the polarisation angle measure Iπ
with finer steps.
In the second part, set the incoming angle equal to the polarisation angle. Measure the
intensity of polarised light passing through the analyser as a function of the angle of
the polarizer in the analyser in steps of 10°.
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In the end, set the λ/2 -plate into the holder in the analyser tube and repeat the
measurements of the second part with three angles differing with 45º.
4. Results
Display drawings of Iπ and Iσ as a function of the entrance angle and interpret the
results. For Iπ present also a close-up near to the polarisation angle (between 50°-60°)
and define the polarisation angle. In addition to these curves, make a drawing of the
degree of polarisation and report the value for the polarisation angle. Calculate the
refraction index from the polarisation angle using the Brewster’s law.
Compare experimental and theoretical results for the Malus’s law drawing them in the
same figure. Use the experimental maximum intensity for I0.
In the report, the functioning of the λ/2 –plate and its role in the measurements should
be explained. There is plenty of information of λ/2- and λ/4 -plates in the Internet, try
out search words “wave plate” or “polarization rotator”, for example.