A. La Rosa
Lecture Notes
APPLIED OPTICS
________________________________________________________________________
POLARIZATION
Linearly-polarized light
Description of linearly polarized light (using Real variables)
Alternative description of linearly polarized light using phasors
Left circularly-polarized light
Description of left-circularly polarized light (using Real
variables)
Alternative description of left -circularly polarized light using
phases
Right circularly-polarized light
Description of left-circularly polarized light (using Real
variables)
Alternative description of left -circularly polarized light using
phasors
Pattern of electromagnetic radiation from
a dipole
Linearly polarized light
The figure shows three different cases of linearly polarized light
X
Y
Description of linearly polarized
light (using Real variables)
Z
Y
E
Ey
X
Ex
Given the following harmonic
components (of frequency ω) of the
electric field that exists at a given
position z =zo
Electric field
Ex = Em cos (ωt)
Ey = Em cos (ωt)
How does the magnitude and orientation of the electric field changes as
a function of time t?
The graph shows the
electric field E at the
time t=0
Y
Em
E
Em
X
Ex
Ey
Y
Em
E(t)
-Em
Em
X
E changes with time,
but always points
along this line
-Em
Linearly
polarized
light
The first step is to
create two phasors
Notice, the phasors are
referenced to two
different axes, X and Y
respectively
Ey
phasor
phasor
Ex
Real - space
Ex Ey
Y
E(t)
E(t
Linearly
polarized
light
X
Alternative description of linearly polarized light using
phasors
First, let's recall that given a real field
Ex = Em cos (ωt)
it can be represented as the horizontal
components of a phasor
phasor for Ex
Im
Ex
Real
E = (Ex, Ey)
Linearly
polarized
light
(in real
variables)
The first step is to
create two phasors
Notice, the phasors are
referenced to two
different axes, X and Y
respectively
Ey
phasor
phasorr
Ex
Y
Phasor
Phasor
E(t)
Linearly
Linearly
polarized
polarized
light
light
(in complex
variables)
Y
X
Circularly polarized light
Z
Y
X
Left-circularly polarized light
Task:
Given Ex = Em cos (ωt )
Ey = Em cos (ωt - π/2)
Electric field vector
E at a fixed position
z=zo
indicate how does the orientation and magnitude
of the electric
of the field vector change as a function of time
Real - space
Y
At t = 0
X
time
time
Left circularly
polarized light
Real - space
ω
Electric field vector
rotates counter-clockwise
with angular velocity ω
(when looking along the direction in which the wave
propagates and into the source).
Left circularly
polarized light
Alternative description of left-circularly polarized
Phasor of
Im
light using phasors
magnitude A
A
Ex = A cos (θ)
θ
Ex
Real
Real
Ey = B cos (γ)
Im
B
γ
Ex = Em cos (ωt)
Ey = Em cos (ωt - π/2)
Left circularly
polarized light
Ey
Ex
Right-circularly polarized light
Y
X
Task:
Given Ex = Em cos (ωt )
Electric field vector
E at a fixed position
z=zo
Ey = Em cos (ωt + π/2)
indicate how does the orientation and magnitude
indicatefield
howvector
the orientation
of the electric
change as and
a function of time
Y
At t = 0
X
Cos (ωt )
time
Cos (ωt + π/2)
time
Electric field
ω
Electric field vector
rotates clockwise
with angular velocity ω
when looking along the direction in which the
wave propagates and into the source
Right circularly
polarized light
Alternative description of right-circularly polarized
light using phasors
We create
two phasors
ω
Ex
Ey
ω
ω
Electric
field
Right circularly
polarized light
An extra simplification step is possible. Indeed, since COS is a symmetric
function, the mathematical expressions for Ex and Ey given above can be
re-written as:
Right circularly
polarized light
Right
circularly
polarized
light
Right circularly
polarized light
Both phasors and the electric field rotate together in the clockwise
direction
Summary
Left circularly polarized light
Y
Ex = Em cos (ωt )
Ey = Em cos (ωt - π/2)
X
Right circularly polarized light
Y
Ex = Em cos (ωt )
Ey = Em cos (ωt + π/2)
Ex = Em cos (- ωt )
Ey = Em cos (- ωt - π/2)
Rule of thumb for the complex variable description:
Notice in the expressions above that
The y-component phasor is
i written as lagging π/2 with
respect to the x-component phasor
;
the right or left circularly polarization character
is given
h
by the sign in front of of ω.
X