Optics 430530 week VII Anisotropic media Polarization This

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

Anisotropic Media II • Susceptibility tensor P. Piot, PHYS 430 -530, NIU FA 2018 2

Plane wave in crystals • Consider • Gauss’s law for D and B in charge-free medium yields so that, generally, • We conclude that the Poynting vector. is not parallel to k anymore P. Piot, PHYS 430 -530, NIU FA 2018 3

Dispersion equation in crystal (I) • Consider wave equation: • Simplify to our case • Plug complex form of E and O to yield: • Take P. Piot, PHYS 430 -530, NIU FA 2018 4

Dispersion equation in crystal (II) • To finally obtain: • Expliciting each of the components gives P. Piot, PHYS 430 -530, NIU FA 2018 5

Dispersion equation in crystal (III) • Finally the dispersion equation is • Introducing the unit vector P. Piot, PHYS 430 -530, NIU FA 2018 6

Fresnel Equation • Reckoning the index of refraction • Gives With solution of the form 7 • Combining yields the Fresnel’s equation P. Piot, PHYS 430 -530, NIU FA 2018

Biaxial & Uniaxial Crystals • There exist directions where the two possible values N (from Fresnel’s equation) are equal. • These direction are called optical axis • When a wave propagates along the optical axis all polarization components experience the same index of refraction • Biaxial crystal when nx ny and nz are unique • We write the direction as P. Piot, PHYS 430 -530, NIU FA 2018 8

Biaxial Crystals • Two solutions for n -> two direction (”optical” axis) along which all the polarization experience the same n • Only possible if nx, ny, and nz are unique and by convention nx<ny<nz. • The two axes are at directions P. Piot, PHYS 430 -530, NIU FA 2018 9

Uniaxial Crystals • Two of indexes of refraction are the same • One optical axis (taken to be z by convention) P. Piot, PHYS 430 -530, NIU FA 2018 10

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 11

Polarization: examples • Linearly-polarized waves • Elliptically-polarized waves with the special case of circularly polarized P. Piot, PHYS 430 -530, NIU FA 2018 12

Jones’ formalism (I) • Consider • Then P. Piot, PHYS 430 -530, NIU FA 2018 13

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 14

Example of special cases P. Piot, PHYS 430 -530, NIU FA 2018 15

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 16

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 17

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 18

Jones matrix of an arbitrary-direction polarizer (II) • P. Piot, PHYS 430 -530, NIU FA 2018 19

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