Download Module 27: Polarization-II Lecture 27: Polarization-II

Module 27: Polarization-II
Lecture 27: Polarization-II
27.1
Birefringence or double refraction
Birefringence usually occurs in anisotropic crystals. In these crystals the arrangement of atoms is different in different directions. As a consequence the
crystal properties are direction dependent. As an example consider a crystal
~ vector is along the x axis and it is
where the refractive index is ne when the E
~
no when E is perpendicular to the x axis. Here the x axis is referred to as the
optic axis. The optical properties are the same in all directions perpendicular
~ is along the optic axis. Crystals may
to the optic axis and it is different if E
have more than one optic axis. Here we only consider a situation where there
is a single optic axis. Such birefringent crystals are referred to as uniaxial
crystals. Through birefringent crystals one sees two images of the object unless the direction of viewing is particularly well chosen (in which case there is
only one image) as shown in the Figure 27.1. So there are two refracted rays
for each incident beam and hence the name. Optic axis of the crystal is in
the vertical direction in the Figure. We find that the second image is slightly
displaced from the original one. The image which is not displaced (refractive
index no ) is having a polarization perpendicular to the optic axis whereas the
displaced image (with refractive index ne ) has a polarization along the optic
axis of the crystal.
The difference ∆n = ne − no is a measure of the birefringence and is often
called the birefringence. A material is referred to as a positive material if
∆n > 0 and a negative material if ∆n < 0. We tabulate below the refractive
indices for some birefringent materials.
Crystal
no
ne
λ0 = 589.3 nm
Tourmaline
1.669 1.638
Calcite
1.6584 1.4864
Quartz
1.5443 1.5534
Sodium Nitrate 1.5854 1.3369
169
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CHAPTER 27. POLARIZATION-II
Bi refringence
ref
Double
ble re
refraction
6
Optic axis
?
Figure 27.1: Birefringence in Calcite
Nicol prism is a smart device often used in the laboratories to produce
linearly polarized beam. Here two calcite pieces cut in a special way (SQP
and PQR) are cemented in a manner shown in the left of Figure 27.2. The
two prisms are glued with a material called Canada balsam, a transparent
material having a refractive index, n = 1.55, which is between the no = 1.6584
and ne = 1.4864 of calcite. Once the unpolarized light is incident on the surface
QS of the prism as shown in the middle of the Figure 27.2, it is divided into
two rays due to birefringence. One of the rays is totally internally reflected
by the layer of Canada balsam, QP, and is absorbed by a black paint on the
wall, SP, of the crystal. The other ray transmits through the Canada balsam
to produce a plane polarized light.
The Wollaston prism is a device that uses birefringence to separate the
unpolarized incident light into two linearly polarized components. Two triangular prisms are glued together as shown in Figure 27.2. Both prisms are
made of the same birefringent material. The optic axis of the two prisms are
mutually perpendicular as shown in the Figure. We decompose the unpolar~ k and E
~ ⊥ respectively parallel and perpendicular to
ized incident light into E
~ k has refractive index ne and E
~⊥
the plane of the paper. In the first prism E
has no . The situation reverses when the light enters the second prism where
~ k has refractive index no and E
~ ⊥ has ne . Figure 27.2 shows the paths of the
E
two polarizations through the Wollaston prism. The two polarizations part
ways at the interface of the two prisms and they emerge in different directions
as shown in Figure 27.2.
27.1.1
Quarter wave plate
Consider a birefringent crystal of thickness h with its optic axis along the
y direction as shown in Figure 27.3. Consider light that passes through the
crystal as shown in the figure. If we decompose the incident light into two
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27.1. BIREFRINGENCE OR DOUBLE REFRACTION
S
S
Optic
axis
Q
Q
0
1
1
0
1
0
10
0
1
10
0
1
10
0
10
1
10
0
10
1
1
0
1
0
1
0
1
0
Θ
10
0
10
10
1
10
0
10
10
10
1
Canada
balsam
O
P
O
68
P
R
R
E
δ
Figure 27.2: a) Nicol and 27.2: b) Wollaston prisms
y
z
E
x
h
Figure 27.3: A quarter wave plate
perpendicular polarizations along the x and y axes respectively, these two polarizations will traverse different optical path lengths through the crystal. The
optical path length is no h for the x component and ne h for the y component
respectively. The crystal is called a quarter wave plate if this difference in the
optical path lengths is λ/4, ie.
| n e − no | h =
λ
4
(27.1)
λ
.
2
(27.2)
and it is called a half wave plate if,
| n e − no | h =
A quarter wave plate can be used to convert linearly polarized light to
circularly polarized light as shown in the right Figure 27.3. The incident light
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CHAPTER 27. POLARIZATION-II
should be at 45◦ to the optic axis. The incident wave can be expressed as
~ t) = E0 (î + ĵ) cos(ωt − kz)
E(z,
(27.3)
The quarter wave plate introduces a phase difference between the x and y polarizations. An optical path difference of λ/4 corresponds to a phase difference
of 90◦ . The wave that comes out of the quarter wave plate can be expressed
as
~ t) = E0 [î cos(ωt − kz) + ĵ cos(ωt − kz + π/2)]
E(z,
(27.4)
which is circularly polarized light.
The output is elliptically polarized if the angle between the incident linear
polarization and the optic axis is different from 45◦ . The quarter wave plate
can also be used to convert circularly polarized or elliptically polarized light
to linearly polarized light. In contrast a half wave plate rotates the plane of
polarization as shown in the left of the Figure 27.4.
27.2
Partially polarized light
y
Θ
x
Figure 27.4: a) Half wave plate and 27.4: b) Analysis of partially polarized
light
Many of the polarizers discussed earlier produce partially polarized light.
This is a mixture of polarized and unpolarized light. Consider a mixture with
unpolarized light of intensity IU and light that is linearly polarized along the
y axis of intensity IP . A polaroid sheet, referred to as an analyzer, is used to
analyze this light (right in the Figure 27.4). The intensity of the unpolarized
light becomes IU /2 after passing through the analyzer whereas the intensity of
the polarized light is IP cos2 θ where θ is the angle between the transmission
axis of the analyzer and the y axis. The resulting intensity is
IU
I=
+ IP cos2 θ .
(27.5)
2
The intensity of the transmitted light changes as the analyzer is rotated and
the maximum and minimum intensity respectively are
IU
IU
+ IP
Imin =
.
(27.6)
Imax =
2
2
27.2. PARTIALLY POLARIZED LIGHT
173
The degree of polarization is defined as
P =
Imax − Imin
IP
=
Imax + Imin
IU + IP
(27.7)
This quantifies the fraction of the light intensity in the linearly polarized component.
Problems
~ is given to be
1. Consider polarized light whose E
~ t) = E0 [Aî cos(ωt − kz)î + B î sin(ωt − kz)ĵ] .
E(z,
Calculate the degree of polarization P for [a.] A = 1, B = 0 [b.] A =
1, B = 1 [c.] A = 1, B = 2. What are the corresponding polarization
states. What happens if these waves are sent through a quarter wave
plate that adds and extra 90◦ delay to the y component?
2. Incident light is scattered by an electron. What is the degree of polarization for scattering angles [a.] 90◦ [b.] 45◦ ?
3. A birefringent crystal of thickness d has its optic axis parallel to the
surface of the crystal. What should be the value of d (in µm ) if the
crystal is to be used as a quarter wave plate for light of wavelength
λ = 589.3 nm? (ne = 1.5334, no = 1.5443).
4. For the Wollaston prism in Figure 27.2 with θ = 30◦ , λ = 589 nm,
ne = 1.486 and no = 1.658, calculate the angle δ between the two rays
that come out.