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Categories of Optical Elements that modify states of polarization:PowerPoint Presentation

Categories of Optical Elements that modify states of polarization:

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## PowerPoint Slideshow about ' Categories of Optical Elements that modify states of polarization:' - brian-ortiz

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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)

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

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

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

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

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)

- 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:

(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

(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

(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

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:

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

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

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