Optics 430530 week II Plane wave solution of

Optics 430/530, week II • Plane wave solution of Maxwell’s equations • Plane waves • Refractive index & dispersion This class notes freely use material from http: //optics. byu. edu/BYUOptics. Book_2015. pdf P. Piot, PHYS 430 -530, NIU FA 2018 1

Wave equation in vacuum • Consider the case when the LHS=0 then the wave equation reduces to • The solution is of the form E(r, t) it can describe an optical “pulse” of light. • A subclass of solution consists of “traveling” wave where the field dependence is of the form E(. ) • • specifies the direction of motion is the velocity of the wave P. Piot, PHYS 430 -530, NIU FA 2018 2

Plane solution of the Wave equation • A class of solution has the functional form Wave vector: Constant “phase” term k and w are not independent they are related via the dispersion relation P. Piot, PHYS 430 -530, NIU FA 2018 3

What about the magnetic field? • A similar wave equation than the one for E can be written for B with solution Same parameters as for E • The field amplitude is related to the E-field amplitude via • B and E are perpendicular to each other • Using Gauss law one finds that k and E are also perpendicular • The field amplitudes are related via P. Piot, PHYS 430 -530, NIU FA 2018 4

Complex notation • It is more convenient to write the e. m. field as the real part of the complex number where (in general • In Physics, the written as ) is often omitted (for simpler notation) and the field (Complex notation of the plane-wave solution) • Note that the above notation implies this is a complex notation and that the real part would have to be consider at the end. P. Piot, PHYS 430 -530, NIU FA 2018 5

Index of refraction I • Consider a isotropic, homogeneous and non-conducting medium (e. g. a dielectric). Then the wave equation simplifies to • Take (“weak-field” regime take P proportional to E) • Substituting in the wave eqn: P. Piot, PHYS 430 -530, NIU FA 2018 6

Index of refraction II • Introduce -- remember relation between D, P, E: – this is the constitutive equation. • explicit in previous eqn to yield • Using we finally get Susceptibility Usually a Complex number… • Introduce the permittivity of the material as P. Piot, PHYS 430 -530, NIU FA 2018 7

Index of refraction III • The complex index of refraction is defined by Real part Imaginary part (losses) If loss/absorption negligible: • Accordingly the wavevector is P. Piot, PHYS 430 -530, NIU FA 2018 8

Plane wave in a medium • Expliciting k in the complex form of the plane-wave solution gives: • Considering the real part lead to P. Piot, PHYS 430 -530, NIU FA 2018 9

Lorentz model of dielectrics I • We need to understand the connection between polarization and applied (external) electric field • Classical model of an atoms (Lorentz): atoms are surrounded by a cloud of electron at rest • Lorentz model uses non-relativistic Newton’s Not equations: e: the her e B (we -field we ig ak f n ield contrib ore Hookean’s assu utio Pulling damping n mp tion “restoring force” force ) P. Piot, PHYS 430 -530, NIU FA 2018 10

Lorentz model of dielectrics II • The latter equation can be rewritten as • Explicit the E field in its complex form. Consider k. r<<1 and that re assumes the same temporal dependence as E Lore dist n ribu tzian tion in w P. Piot, PHYS 430 -530, NIU FA 2018 11

Polarization Plasma frequency • Recalling that • gives: P. Piot, PHYS 430 -530, NIU FA 2018 12

Real & Imaginary part of N • Recall • Hence • Generalization accounting for multiple atoms species this is a variant of Sellmeier’s equation. P. Piot, PHYS 430 -530, NIU FA 2018 13

Energy considerations • Electromagnetic wave do carry energy • Measuring, e. g. , the properties of a laser often rely on detecting energy deposited by the pulse into various detectors: • CCD camera • Photo-diode, etc… • Description of energy flow is described by the Poynting vector and the Poynting’s theorem… P. Piot, PHYS 430 -530, NIU FA 2018 14

Poynting’s theorem • From Maxwell equations: . (1. 3) and. . (1. 4) and substrating these two equations • Gives • Which simplified to Poynting theorem P. Piot, PHYS 430 -530, NIU FA 2018 15

Poynting’s Vector • The Poynting theorem can be rewritten • Note that a volume integral yields P. Piot, PHYS 430 -530, NIU FA 2018 16

Computing the irradiance of a plane wave • Irradiance is the power received per unit surface (W/m 2) • It is an important parameter to assess, e. g. whether the optical pulse is above the damage threshold of an optical element (e. g. a crystal or mirror). • So the irradiance is related to the Poynting vector • Note of caution about complex form of the field: when we take product of field we have to make sure we have a prescription to ensure the field is real valued P. Piot, PHYS 430 -530, NIU FA 2018 17

Complex forms • Consider • When multiplying complex representation of the field it is more convenient (and still “economical”) to consider P. Piot, PHYS 430 -530, NIU FA 2018 18

Complex forms • Consider • When multiplying complex representation of the field it is more convenient (and still “economical”) to consider P. Piot, PHYS 430 -530, NIU FA 2018 19

Poynting vector for a plane wave • The vector is • Time averaging • And expliciting k Modulus is irradiance (also intensity) Dissipated power due to losses in the mdeium P. Piot, PHYS 430 -530, NIU FA 2018 20