Ellipsometry Techniques and Its Advanced Applications …
545
1.2.3 Elliptically Polarized Light
If amplitudes of electric field components along x- and y-axes are not same (i.e.
E
0
x = E
0
y ) and the phase difference is integer multiple of π/2 (i.e. θ y − θ x = ±π/2)
or if the relative phase is other than π/2 then it is called elliptically polarized light.
For instance, E
0
x = E
0
y and θ y − θ x = ±π/2 then Eq. (3) becomes:
E =
ˆ
x E
0
x ± i ˆ
y E
0
y
exp[−i(kz − ωt + θ x )],
(9)
From Eq. (9), the components of electric field along x- and y-axes can be written
as:
E x = E
0
x cos(kz − ωt + θ x ); E y = E
0
y sin(kz − ωt + θ x )
(10)
Therefore, Eq. (10) yields: E
2
x /E
02
x + E
2
y /E
02
y = 1 which is the equation of an
ellipse. If θ y − θ x = −π/2 then it is right hand (clockwise) elliptically polarized
light. Else if θ y −θ x = +π/2 then it is left hand (anticlockwise) elliptically polarized
light. Elliptically polarized light for E
0
x > E
0
y is presented in Fig. 2d.
1.2.4 Jones Vectors and Matrices
As described above, amplitudes and phase factors of the electric field components
give rise to different polarizations of light. Therefore, a mathematical representation
is required to define the polarization of light. The Jones vector [3–5] is defined by the
electric field vectors along the x- and y-axes. By expressing Eq. (3) in matrix form:
E = exp[−i(kz − ωt)]
E x
E y
,
(11)
By omitting exp[−i(kz − ωt)] term and normalizing Eq. (11), it simplifies further
to
E =
E x
E y
= E x
1
E y /E x
→
1
E 2
x + E 2
y
E x
E y
(12)
From the definition of plane polarized light, Eq. (12) can be rewritten as
E =
E x
E y
=
0
1
for horizontally polarized light,
E =
E x
E y
=
1
0
for vertically
polarized light, and
E =
E x
E y
=
1
√
2
1
1
for 45° plane polarized light. Since,