Categories of Optical Elements that modify states of polarization:

1. Operation of a linear polarizer 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)

2. Operation of a phase retarder 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 • 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. 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. 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 bin (3): • Substitute for c in (4): • So that:

10. (phase retarder - general form) 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: x and y may be positive or negative quantities

11. (QWP, SA vertical) (QWP, SA horizontal) 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

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

13. (rotator through angle +) 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:

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