Wave equations in a medium The induced polarization in Maxwells - - PowerPoint PPT Presentation

Wave equations in a medium

The induced polarization in Maxwell’s Equations yields another term in the wave equation: This is the Inhomogeneous Wave Equation. The polarization is the driving term for a new solution to this equation.

2 2 2 2

E E z t µε ∂ ∂ − = ∂ ∂

2 2 2 2 2

1 E E z v t ∂ ∂ − = ∂ ∂ c n v =

2 2 2 2

E E z t µ ε ∂ ∂ − = ∂ ∂

2 2 2 2 2

1 E E z c t ∂ ∂ − = ∂ ∂

( )

( ) ( ) ( )

( )

* 1 2

, Re{ } { } | | cos

i kz t i kz t i kz t

z t e e e kz t

ω ω ω

ω

− − − −

= = + = − E E E E E

Homogeneous (Vacuum) Wave Equation

2 2 2

c n v µε µ ε = =

Wave equations

Poynting vector & Intensity of Light

( ) ( )

( )

2 2 2

| | 2 2

x y

c c I t t E E E ε ε = ≡ × = = + S E H

34

1.05457266 10

1239.85 [ ] [ ]

Js

eV nm ω λ

−

= ×

=





3

2.654 10 / c A V ε

−

≈ ×

2

1 / ? / E V m I W m = = S = E×H

• Poynting vector describes flows of E-M power
• Power flow is directed along this vector
• Usually parallel to k
• Intensity is equal to the magnitude of the time averaged Poyning vector: I=<S>

example

Reflection and Transmission @ dielectric interface

Beyond Snell’s Law: Polarization?

Reflection and Transmission (Fresnel’s equations)

Can be deduced from the application of boundary conditions of EM waves.

Reflection and Transmission of Energy @ dielectric interfaces

Energy Conservation

Normal Incidence

Reflectance and Transmittance @ dielectric interfaces