Chapter 3 Polarization of Light Waves Lecture 1 Polarization 3.1 The concept of polarization Introduction: Since F = qE, polarization determines force direction. The generation, propagation and control of lasers thus crucially depend on their state of polarization. The electric field E is used to define the polarization state of the light. In an anisotropic material the index of refraction depends on the polarization state of the light, which is used to manipulate light waves. 3.2 Polarization of monochromatic plane waves For a monochromatic plane wave propagating in the z direction, the x and y components of the electric field oscillate independently. Ex Ey

Ex Ey t E

Ellipticity (shape) a/b, and The trajectory of the end point of the electric field vector, at a fixed point as time goes, is Ax Ay E f y x x' y' a b Light is thus generally elliptically polarized. A complete description of the elliptical polarization needs Orientation angle f, Ellipticity (shape) a/b, and Handedness (sense of revolution, can be combined to show the sign of ellipticity). They can be obtained by rotating the ellipse to its normal coordinates:

Handedness (sense of revolution) of elliptical polarization: Looking at the approaching light, if the E vector revolves counterclockwise, the polarization is right-handed, with sind <0. If the E vector revolves clockwise, the polarization is left-handed, with sind >0. Many books use the opposite definition. d = -p -3p/4 -p/2 -p/4 0 p/4 p/2 3p/4 p Linear polarization: d = -p/2 p/2 Circular polarization:

Lecture 2 Jones vector 3.4 Jones vector representation A plane wave can be uniquely described by a Jones vector in terms of its complex amplitudes on the x and y axes: Normalized Jones vector: If we are only interested in the polarization state of the wave, we use the normalized Jones vector, with J+J=1: Examples: Linearly polarized light: Right- and left-handed circularly polarized light:

Light intensity: Light intensity is now defined as Orthogonal Jones vectors: Two Jones vectors J1 and J2 are orthogonal if J2+J1= 0. Obviously the orthogonal state of Theorem: An arbitrarily polarized light can be uniquely decomposed into the combination of a given pair of orthogonal polarization state. Some relations:

Examples of Jones vectors: d = -p -3p/4 -p/2 -p/4 0 p/4 p/2 3p/4 p Jones vectors are important when applied with Jones calculus, which enables us to track the polarization state and the intensity of a plane wave when traversing an arbitrary sequence of optical elements.

(. Reading) What is the orientation angle f of the ellipse (*Reading) What is the orientation angle f of the ellipse How long are the principle axes? Solution 1: Suppose the ellipse is erect after rotating the x, y axes by an angle f, that is

Solution 2: Let us use a little linear algebra.

Solution 3: The problem is: given the condition Solution 3: The problem is: given the condition for x and y, what is the maximum x2 + y2? This can be solved by the Lagrange multiplier method. Solution 4: In polar coordinates, The curve is now

Reading: Diagonalizing the tensor of a quadratic surface Question: What is the orientation angle f of the ellipse How long are the semiaxes? Solution 5: Diagonalizing the tensor of a quadratic surface f y x x' y' A-1/2 B-1/2

Therefore we conclude: If the orthogonal transformation R diagonalizes S into S', then 1) The diagonal elements of S' are the eigenvalues of S. 2) The column vectors of R-1 (or the row vectors of R) are the eigenvectors of S. 3) The new coordinate axes lie along the eigenvectors of S.