1. Categories of Optical Elements that modify states of polarization:
Linear polarizers
Linear polarizer selectively removes all or most of the E-vibrations in a given direction, while allowing vibrations in the perpendicular direction to be transmitted (transmission axis)
Unpolarized light traveling in +z-direction passes through a plane polarizer, whose transmission axis (TA) is vertical
Unpolarized light represented by two perpendicular (x and y) vibrations (any direction of vibration present can then be resolved into components along these directions)
(Selectivity is usually not 100%, and partially polarized light is obtained)
Operation of a linear polarizer

2. Categories of Optical Elements:
Does not remove either of the component orthogonal E-vibrations but introduces a phase difference between them
If light corresponding to each orthogonal vibration travels with different speeds through a retardation plate, there will be a cumulative phase difference  between them as they emerge
(2) Phase retarder
Operation of a phase retarder
Vertical component travels through plate faster than horizontal component although both waves are simultaneously present at each point along the axis
Fast axis (FA) and slow axis (SA) are as indicated
Net phase difference  = 90 for quarter-wave plate;  =180 for half-wave plate

3. Categories of Optical Elements:
(3) Rotator
It has effect of rotating the direction of linearly polarized light incident on it by some particular angle
Operation of a rotator
Vertical linearly polarized light is incident on a rotator
Emerging light from rotator is a linearly polarized light whose direction of vibration has rotated anti-clockwise by an angle 

4. Jones matrix representations for
Linear polarizer :
Consider vertical linear polarizer
Let 2  2 matrix represents polarizer
Let (vertical) polarizer operate on vertically polarized light, resulting in transmitted vertically polarized light also
Writing out the equivalent algebraic equations:
we conclude that b = 0 and d = 1
Next, let (vertical) polarizer operate on horizontally polarized light, and no light is transmitted
and the corresponding algebraic equations are

5. Therefore the appropriate matrix is:
Jones matrices (for linear polarizers)
from which a = 0 and c = 0
Therefore the appropriate matrix is:
(Linear polarizer, TA vertical)
Similarly,
(Linear polarizer, TA horizontal)
For linear polarizer with TA inclined at 45 to x-axis;
allow light linearly polarized in the same direction as, and perpendicular to, the TA to pass through the polarizer one by one; we thus have
and

6. Equivalently, the algebraic equations are: From which, we obtain
Jones matrices (for linear polarizers)
Equivalently, the algebraic equations are:
From which, we obtain
Thus, the matrix is
(Linear polarizer, TA at 45)
In the same way, a general matrix representing a linear polarizer with TA at angle  can be shown to be:
(Linear polarizer, TA at )
(Proof is left as an exercise for you)

7. Transmission axis (TA) oriented at θ
If polarization is along the TA, the light is transmitted unchanged:
If the polarization is perpendicular to TA, no light is transmitted:

8. The 2 matrix eqn. can be recast as 4 algebraic eqns.:
(1)
(2)
(3)
(4)

9. Substitute for b in (3):
Substitute for c in (4):
So that:

10. Jones matrices (for phase retarders)
In order to transform the phase of the Ex-component from x to x + x and the Ey-component from y to y + y, that is,
unpolarized
we use the matrix operation as follows:
Therefore, the general form of a matrix that represents a phase retarder is:
(phase retarder - general form)
x and y may be positive or negative quantities

11. For the QWP, the phase difference = /2 distinguishing two cases:
Jones matrices (for phase retarders - QWP & HWP)
Consider Two special cases: (i) Quarter-Wave Plate (QWP) and (ii) Half-Wave Plate (HWP)
unpolarized
For the QWP, the phase difference = /2
distinguishing two cases:
(a) y  x = /2 (SA vertical)
let x = /4 and y = +/4 (other choices are also possible), we have
(b) x  y = /2 (SA horizontal), we have
(QWP, SA vertical)
(QWP, SA horizontal)

12. Correspondingly, for the QWP, the phase difference = , we have
Jones matrices (for phase retarders - QWP & HWP)
Correspondingly, for the QWP, the phase difference = , we have
(HWP, SA vertical)
(HWP, SA horizontal)
Elements of the matrices are identical because advancement of phase by  is physically equivalent to retardation by 
Difference in the prefactors

13. Jones matrices (for rotators)
For the rotator of angle , it is required that the linearly polarized light at angle  be converted to one at angle ( + )
Thus, the matrix element must satisfy: or
From trigonometric identities:
Therefore:
and the required matrix for the rotator is:
(rotator through angle +)

14. Summary of the Jones matrices

15. Production of circularly polarized light
Using Jones calculus, the QWP matrix is operated on the Jones vector for linearly polarized light:
Combination of linear polarizer (LP) inclined at angle 45 and a QWP produces circularly polarized light
unpolarized
which is a right-circularly polarized light of amplitude 1/2 times the amplitude of the original linearly polarized light
(If a QWP, SA vertical is used, left-circularly polarized light results)
Linearly polarized; inclined at angle 45; then divided equally between slow and fast axes by QWP
Emerging light has its Ex- and Ey-vectors at phase difference 90

16. Quantitative example:
What happens when we allow left-circularly polarized light to pass through an eighth-wave plate? Solution:
Let’s obtain matrix for 1/8-wave plate, i.e., a phase retarder of /4
Say we let x = 0, then
Allow it to operate on Jones vector for left-circularly polarized light:
Resultant Jones vector shows light is elliptically polarized, components are out of phase by 135
Expanding ei3/4 using Euler’s equation:
expressed in the standard notation defined earlier;
where