Optics 430530 week VIII Polarization Superposition of plane

Optics 430/530, week VIII • Polarization • Superposition of plane waves This class notes freely use material from http: //optics. byu. edu/BYUOptics. Book_2015. pdf P. Piot, PHYS 430 -530, NIU FA 2018 1

Polarization: definition • Polarization refer to the direction of the E field (this is a convention). • If the direction is unpredictable the wave is said to be unpolarized • If the E-field direction is well define the wave is said to be polarized • Starting with and taking the z axis as propagation axis we can decompose E as • The relationship between the two transverse component describes the polarization P. Piot, PHYS 430 -530, NIU FA 2018 2

Polarization: examples • Linearly-polarized waves • Elliptically-polarized waves with the special case of circularly polarized P. Piot, PHYS 430 -530, NIU FA 2018 3

Jones’ formalism (I) • Consider • Then P. Piot, PHYS 430 -530, NIU FA 2018 4

Jones’ formalism (II) • The strength is unimportant for polarization considerations it only enters in the intensity as • In Jones’ formalism the polarization is represented by the vector P. Piot, PHYS 430 -530, NIU FA 2018 5

Example of special cases P. Piot, PHYS 430 -530, NIU FA 2018 6

Linear polarizers and Jones matrices • In Jones formalism the evolution of the polarization can be described by a 2 x 2 matrix (referred to as Jones’ matrix) • A simple example regards the representation of a polarizer: an optical element which only let one polarization component to pass. In such a case we have P. Piot, PHYS 430 -530, NIU FA 2018 7

Jones matrix • Generally • Note that the intensity does not remain the same as • So one always renormalized the final Jones vector as P. Piot, PHYS 430 -530, NIU FA 2018 8

Jones matrix of an arbitrary-direction polarizer (I) • Consider an incoming wave • Decompose in the. basis as • So we have where P. Piot, PHYS 430 -530, NIU FA 2018 9

Jones matrix of an arbitrary-direction polarizer (II) • P. Piot, PHYS 430 -530, NIU FA 2018 10

• P. Piot, PHYS 430 -530, NIU FA 2018 11

Waveplates • We now consider a birefringent material with its index of refraction dependent on the direction of the polarization • A waveplate is cut so that the slow and fast axis are 90 deg apart • The phase difference between the two axis is P. Piot, PHYS 430 -530, NIU FA 2018 12

Waveplates • Quarter waveplate Ca co n be n to pol ver use cir ari t lin d t o cu ze lar d w earl y ly po ave lar ize d • Half waveplate P. Piot, PHYS 430 -530, NIU FA 2018 13

Superposition of plane waves (chapt. 7) • P. Piot, PHYS 430 -530, NIU FA 2018 14

Intensity of superimposed plane waves • The Poynting vector is • So we finally get =0 is the plane waves are moving along the same direction P. Piot, PHYS 430 -530, NIU FA 2018 15

Intensity of superimposed plane waves (II) • Gathering some term we finally have • So the optical intensity is P. Piot, PHYS 430 -530, NIU FA 2018 16

Sum of two waves • P. Piot, PHYS 430 -530, NIU FA 2018 17

Group velocity • Consider the previous equation • From the argument of the cosine we can define a velocity as • this is the group velocity which describes the velocity of the wave envelope • Note that the phase velocity of the superimposed wave is P. Piot, PHYS 430 -530, NIU FA 2018 18

Frequency spectrum of light • P. Piot, PHYS 430 -530, NIU FA 2018 19

Power spectrum • P. Piot, PHYS 430 -530, NIU FA 2018 20

Fourier transforms P. Piot, PHYS 430 -530, NIU FA 2018 21

Parseval’s theorem • The Parseval theorem is a general theorem that states • Consider the example of a modulated Gaussian pulse’ • We have for the Fourier transform • So that both the time integral and frequency integral give P. Piot, PHYS 430 -530, NIU FA 2018 22