Ordinary light versus polarized light Ordinary (unpolarized) light: No fixed direction of oscillation Does not vary spatially in regular manner Produced by a number of independent atomic sources whose radiation is not synchronized Vibrations in unpolarized light viewed end-on  all planes are equally probable (rather oversimplified representation) Every vibration can be resolved into linear components Ax = A cos  and Ay = A sin which are in general unequal. As  is random, net result as if only 2 orthogonal vibrations of equal amplitudes appear but have no coherence of phase x y Ax Ay A  End-on view Side view of unpolarized beam propagating to the right

Side views of plane-polarized beam propagating to the right Ordinary light versus polarized light Polarized light: Fixed direction of oscillation Varies spatially in regular manner (e.g. elliptically polarized or circularly polarized) Only transverse wave can be polarized Side views of plane-polarized beam propagating to the right OR

Linearly polarized light (or plane polarized) It is a plane electromagnetic wave The transverse electric field wave is accompanied by an orthogonal magnetic field wave

Circularly polarized light it consists of two perpendicular electromagnetic plane waves of equal amplitude and 90° (or /2) difference in phase. source (1) (2) (3) (4) (5) (6) (7) (8) (9) Vector (1) reaches observer first, next comes (2), followed by (3), (4), … ; thus, observer sees vector tip turning clockwise

If counterclockwise, then left-circularly polarized light. If you follow the tip of the electric field vector, it would appear to be moving in a circle as it approaches you If while looking at the source, the electric vector of the light coming toward you appears to be rotating clockwise, the light is said to be right-circularly polarized. If counterclockwise, then left-circularly polarized light. The electric field vector makes one complete revolution as the light advances one wavelength toward you. Circularly polarized light may be produced by passing linearly polarized light through a quarter-wave plate at an angle of 45° to the optic axis of the plate.

Elliptically polarized light Elliptically polarized light consists of two perpendicular waves of unequal amplitude which differ in phase by 90° (or /2). right-elliptically polarized light is shown below

Vector of E-field at time t Mathematical representation of polarized light: Jones Vectors Vector E varies continuously in magnitude and changes direction every ½ - period At a particular time t, it can be written as: However, vector E should be described in space (position) and time as follows: y x Vector of E-field at time t Ey Ex Direction of propagation of light z (15-1) (15-2) which is a wave that travels in the z-direction, has amplitudes E0x & E0y and phases x & y

Rewriting equation (15-2); Mathematical representation of polarized light: Jones Vectors Rewriting equation (15-2); (15-3) Complex amplitude - the relative amplitudes and phases of the components determine the STATE of POLARIZATION; may be written as 2-element matrix (JONES VECTOR) (15-4) A x y Linearly polarized light with E-vector oriented vertically A x y Linearly polarized light with E-vector oriented horizontally

The normalized form of vector is such that Mathematical representation of polarized light: Jones Vectors General case for Linearly polarized light with E-vector inclined at angle  with respect to x-axis x y A A sin  A cos  Positive   E-vector in 1st & 3rd Quadrant; Negative   E-vector in 2nd & 4th Quadrant (15-5) Linearly polarized light requires x- and y-components to be in-phase (relative phase  = y  x = 0) The normalized form of vector is such that E.g., normalized form of Jones vector for vertically linearly polarized light is where we set A = 1

E.g., normalized form of Jones vector in (15-5) is such that Mathematical representation of polarized light: Jones Vectors E.g., normalized form of Jones vector in (15-5) is such that and we set A = 1 Alternatively, given vector where a and b are real numbers, the inclination angle is obtained from (15-6) Summary: Jones vector with both a and b real numbers, both non-zero, represents linearly polarized light at inclination angle

Mathematical representation of polarized light: Jones Vectors Similar to Lissajous figure: relative phase  = y  x for general case of E0x  E0y  = 90 E0x = E0y (circle)  =  90 E0x = E0y (circle)

Circular polarization: E0x = E0y = A and Ex lead Ey by 90 Jones Vectors for circular polarization: Circular polarization: E0x = E0y = A and Ex lead Ey by 90 As x-vibration lead y-vibration, y > x (This contradiction stems from our choice of wave representation whereby the time-dependent term in exponent is negative) For example, wave at z = 0 with x = 0 and y =  (i.e. y > x) wave function becomes: (negative sign before  indicates a lag  in v-vibration relative to the x-vibration) Taking the real parts, setting E0x = E0y = A and  = /2, we have We also have and the tip of resultant vector traces out a circle of radius A

With E0x = E0y = A ; x = 0 and y = /2, then Jones Vectors for circular polarization: With E0x = E0y = A ; x = 0 and y = /2, then (15-7) To get the normalized form of the vector; (each element must then be divided by 2 to get unity) And the Jones vector is; (15-8a); left-circular polarization E rotates anti-clockwise viewed head-on - i.e. wave travels toward you - left-circularly polarized light For right-circularly polarized light,Ey leads Ex by /2, resultant E rotates clockwise; we replace /2 with/2 in Eq. (15-7) and the normalized Jones vector is: (15-8b); right-circular polarization

Jones Vectors for circular polarization: However, actual character of light may not be immediately seen as in the e.g. below which is a right-circularly polarized light Prefactor of the Jones vector affects the amplitude, and thus, the irradiance of the light but NOT the polarization state (We can ignore prefactors if we want to know state of polarization and not information about energy) Jones vector For circularly polarized light, one of the elements of Jones vector is pure imaginary, and magnitudes of elements are the same.

Jones Vectors for ELLIPTICAL polarization (principal axes on xy-axes): If x- and y-vibrations are of unequal amplitudes though  = /2, then we will have (15-9a) (15-9b) For anti-clockwise rotation For clockwise rotation  = 3/2 or  /2  = /2 Normalized form of this Jones vector has prefactor of For elliptically polarized light (principal axes on xy-axes), one of the elements of Jones vector is pure imaginary, and magnitudes of elements are not equal.

Take Ex to lead Ey by angle ; its Jones vector is Jones Vectors for ELLIPTICAL polarization (but principal axes inclined to xy-axes): In this case, x- and y-vibrations are still of unequal amplitudes but phase difference at other angles [i.e.   m (linear) and   (m+ ½) (circular)] Take Ex to lead Ey by angle ; its Jones vector is (15-10a); anti-clockwise rotation whereby from Euler’s theorem we have Normalized form of this Jones vector has prefactor of For elliptically polarized light (principal axes inclined to xy-axes), one of the elements of Jones vector is a complex number (has both real and imaginary parts), and magnitudes of elements are not equal.

Elliptically polarized light oriented at angle  relative to x-axis Jones Vectors for ELLIPTICAL polarization (but principal axes inclined to xy-axes): Elliptically polarized light oriented at angle  relative to x-axis A (B2 + C2)1/2 Angle of inclination is obtained from (15-11) Ellipse occupies rectangle of sides 2E0x and 2E0y and the following relationships hold: (15-12) When Ex lags Ey, phase angle  is negative and Jones vector becomes (15-10b); clockwise rotation

Quantitative e.g.: Analyze Jones vector , what does it represent? Jones Vectors for ELLIPTICAL polarization (but principal axes inclined to xy-axes): Quantitative e.g.: Analyze Jones vector , what does it represent? Consider relative phase  = y  x =  = tan-1 (1/2) = 26.6 E0x = 3 and E0y = (22 + 12)0.5 = 5; angle of inclination is then In fact equation of the ellipse is which yields (in this case)

SUMMARY of Jones Vectors 1. Linear Polarization ( = m) 2. Circular Polarization ( = /2) Left: Right:

SUMMARY of Jones Vectors 3. Elliptical Polarization

To superpose two or more polarized modes, we add their Jones vectors, Notes on SUMMARY of Jones Vectors Any Jones vector can be multiplied by a real constant, thus, changing the amplitude but NOT the state of polarization - Vectors given have been multiplied by prefactors (when necessary) to put in normalized form - e.g. vector to represent linearly polarized light at inclined angle 45 with x-axis and has amplitude Each vector can be multiplied by factor of form , which has the effect of promoting phase difference of each element by , i.e., and (new vector represents same polarization as  is unchanged) To superpose two or more polarized modes, we add their Jones vectors, e.g. adding left- and right-circularly polarized light gives linearly polarized light of twice magnitude e.g. superpose vertically and horizontally linearly polarized light gives linearly polarized light at 45 inclination NO Jones vector to represent unpolarized or partially polarized light