Polarization of light

1. Polarization in Waves
Dimpal Boro
Tezpur university
October 2022

2. Polarization
Polarization is a property applying to transverse waves that
specifies the geometrical orientation of the oscillations.
Figure: A vertically polarized electromagnetic wave

3. Types of Polarization
1) Linear Polarization
In Linear Polarization, the electric field vector of the
electromagnetic wave oscillates back and forth in a particular
direction.
2)Circular Polarization
In Circular Polarization, the electric field vector of the
electromagnetic wave rotates but the magnitude of the vector
remains same.
3)Elliptical Polarization
In Elliptical Polarization, the electric field vector rotates and it’s
magnitude also changes.

4. Linear Polarization
Electromagnetic wave polarized in the î direction
⃗
Ex (z, t) = îE0x cos(kz − ωt)
Electromagnetic wave polarized in the ĵ direction
⃗
Ey (z, t) = ĵE0y cos(kz − ωt)
The resultant of these two linearly polarized electromagnetic waves
is also a linearly polarized wave.
⃗
E = îE0x cos(kz − ωt) + ĵE0y cos(kz − ωt)

5. Linear Polarization
Figure: A linearly polarized electromagnetic wave

6. Complex Number Representation
Euler’s formula,
eiθ
= cos(θ) + i sin(θ)
Electromagnetic wave polarized in the ĵ direction
⃗
Ey (z, t) = ĵE0y cos(kz − ωt + ϵ) = ĵE0y Re[ei(kz−ωt+ϵ)
]
˜
⃗
Ey (z, t) = ĵE0y ei(kz−ωt+ϵ)
˜
⃗
Ey (z, t) = ĵE0y eiϵ
ei(kz−ωt)
˜
⃗
Ey (z, t) = ĵẼ0y ei(kz−ωt)
The advantage of the complex notation is that exponentials are
much easier to manipulate than sines and cosines.

7. Polarization in general
Electromagnetic wave polarized in the î direction
⃗
Ex (z, t) = îE0x cos(kz − ωt)
Electromagnetic wave polarized in the ĵ direction
⃗
Ey (z, t) = ĵE0y cos(kz − ωt + ϵ)
If ϵ is integer multiple of π, then we will get a resultant linearly
polarized light.

8. Circular Polarization
Electromagnetic wave polarized in the î direction
⃗
Ex (z, t) = îE0x cos(kz − ωt)
Electromagnetic wave polarized in the ĵ direction
⃗
Ey (z, t) = ĵE0y cos(kz − ωt + ϵ)
If
ϵ = ±
π
2
+ 2mπ
where m = 0, ±1, ±2, ±3, ... and
E0x = E0y
we will have circularly polarized light.

9. Circular Polarization
Electromagnetic wave polarized in the î direction
⃗
Ex (z, t) = îE0 cos(kz − ωt)
Electromagnetic wave polarized in the ĵ direction
⃗
Ey (z, t) = ĵE0 cos(kz − ωt −
π
2
) = ĵE0 sin(kz − ωt)
The resultant electric field is
⃗
E = îE0 cos(kz − ωt) + ĵE0 sin(kz − ωt)
This represents a right-circularly polarized wave.

10. Circular Polarization
Figure: A right-circularly polarized electromagnetic wave, Electric field
vector rotating clockwise.

11. Circular Polarization
Electromagnetic wave polarized in the î direction
⃗
Ex (z, t) = îE0 cos(kz − ωt)
Electromagnetic wave polarized in the ĵ direction
⃗
Ey (z, t) = ĵE0 cos(kz − ωt +
π
2
) = −ĵE0 sin(kz − ωt)
The resultant electric field is
⃗
E = îE0 cos(kz − ωt) − ĵE0 sin(kz − ωt)
This represents a left-circularly polarized wave.

12. Linearly polarized light as a combination of right and
left-circularly polarized light
⃗
ER = îE0 cos(kz − ωt) + ĵE0 sin(kz − ωt)
This represents a right-circularly polarized wave.
⃗
EL = îE0 cos(kz − ωt) − ĵE0 sin(kz − ωt)
This represents a left-circularly polarized wave. Adding the above
two equations
⃗
E = ⃗
ER + ⃗
EL = 2E0 cos(kz − ωt)î
Which is a linearly polarized light. Thus, a linearly polarized wave
can be synthesized from two oppositely polarized circular waves of
equal amplitude.

13. Elliptical Polarization
In elliptical polarization, the electric field vector rotates and also
it’s magnitude changes.
Ex = E0x cos(kz − ωt)
Ey = E0y cos(kz − ωt + ϵ)
Ey
E0y
= cos(kz − ωt) cos(ϵ) − sin(kz − ωt) sin(ϵ)
Ey
E0y
−
Ex
E0x
cos(ϵ) = − sin(kz − ωt) sin(ϵ)
sin(kz − wt) =
"
1 −
Ex
Eox
2
#1/2

14. Elliptical Polarization
Ey
E0y
2
+
Ex
E0x
2
− 2
Ey
E0y
Ex
E0x
cos ϵ = sin2
ϵ
This is the equation of an ellipse making an angle α with the
(Ex , Ey )-coordinate system such that
tan 2α =
2E0x E0y cos ϵ
E0x
2
− E0y
2
Figure: Elliptical light. The endpoint of the electric field vector sweeps
out an ellipse as it rotates once around.

15. Nomenclature
A linearly polarized or plane-polarized light is said to be in P-state.
Right or left-circular light is in an R and L-state, respectively.
Similarly, elliptically polarized light is said to be in E-state.
We have seen that a P-state can be written as a superposition of
R-state and L-state. An E-state can also be written as a
superposition of R and L state.
Figure: Elliptical light as the superposition of an R and L-state

16. Unpolarized light
An ordinary light source consists of a very large number of ran-
domly oriented atomic emitters. Each excited atom radiates a
polarized wavetrain for roughly 10−8s . All emissions having the
same frequency will combine to form a single resultant polarized
wave, which persists for no longer than 10−8s. New wavetrains are
constantly emitted, and the overall polarization changes in a
completely unpredictable fashion. If these changes take place at so
rapid a rate as to render any single resultant polarization state
indiscernible, the wave is referred to as natural light. It is also
known as unpolarized light.

17. Polarizers
Figure: Natural light incident on a linear polarizer tilted at an angle θ
with respect to the vertical.

18. The Wire-Grid Polarizer
Figure: A wire-grid polarizer. The grid eliminates the vertical compo-
nent (i.e., the one parallel to the wires) of the E-field and passes the hori-
zontal component. The transmission axis of the grid is perpendicular to
the wires.

19. THE END.