Chapter 3 Polarization of Light Waves Lecture 1 Polarization

1. Chapter 3 Polarization of Light Waves Lecture 1 Polarization • 3.1 The concept of polarization • Introduction: • SinceF= 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

2. Ey Ex t Ey E Ex t

3. Ay E f Ax The trajectory of the end point of the electric field vector, at a fixed point as time goes, is y y' x' a b x • 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:

4. 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: Circular polarization: d = -p/2 p/2

5. Lecture 2 Jones vector 3.4 Jones vector representation A plane wave can be uniquely described by a Jones vectorin 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 normalizedJones vector, with J+J=1: • Examples: • Linearly polarized light: • Right- and left-handed circularly polarized light:

6. 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:

7. 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.

8. (*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

11. Solution 2: Let us use a little linear algebra.

12. 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

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

15. 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.