Operation of a linear polarizer
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Operation of a linear polarizer. Categories of Optical Elements that modify states of polarization:. Linear polarizers.

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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)

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

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


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)

  • If the polarization is perpendicular to TA, no light is transmitted:

  • Substitute for c in (4):

  • So that:

(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,


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 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)


  • 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 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

(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:


From trigonometric identities:


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


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

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;