GRAFICA PAG 36 Optics_FresnelsEqns

# GRAFICA PAG 36 Optics_FresnelsEqns - Fresnel's Equations...

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Fresnel's Equations for Reflection and Refraction Incident, transmitted, and reflected beams at interfaces Reflection and transmission coefficients The Fresnel Equations Brewster's Angle Total internal reflection Power reflectance and transmittance Phase shifts in reflection The mysterious evanescent wave

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Reflection and transmission for an arbitrary angle of incidence at one (1) interface • Only Maxwell+Boundary conditions need. Gives Fresnels equations
Maxwell’s eqns. j dt D H dt B D B D ext r r r r r r r + = × = × = = 0 ρ

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Boundary conditions of EM wave • Tangential components of the: - E and H fields (from Gauss’ theorem) • Normal components of - D and B fields (from Stoke’s theorem) 0 ) ( 0 ) ( ) 1 ( ) 2 ( 12 ) 1 ( ) 2 ( 12 = = × B B n E E n r r r r r r 12 n r 1 2
Definitions: Planes of Incidence and the Interface and the polarizations Perpendicular (“S”) polarization sticks out of or into the plane of incidence. Plane of the interface (here the yz plane) (perpendicular to page) Plane of incidence (here the xy plane) is the plane that contains the incident and reflected k- vectors. n i n t i k r r k r t k r θ i r t E i E r E t Interface x y z Parallel (“P”) polarization lies parallel to the plane of incidence. Incident medium Transmitting medium

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Shorthand notation for the polarizations Perpendicular (S) polarization sticks up out of the plane of incidence. Parallel (P) polarization lies parallel to the plane of incidence.
Fresnel Equations We would like to compute the fraction of a light wave reflected and transmitted by a flat interface between two media with different refrac- tive indices. Fresnel was the first to do this calculation. n i n t i k r r k r t k r θ i r t E i B i E r B r E t B t Interface x y z Beam geometry for light with its electric field per- pendicular to the plane of incidence (i.e., out of the page) It proceeds by considering the boundary conditions at the interface for the electric and magnetic fields of the light waves. We’ll do the perpendicular polarization first. For example, 00 / ri rE E = / ti tE E = / rEE = ± / E = ±

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The Tangential Electric Field is Continuous In other words: The total E-field in the plane of the interface is continuous. Here, all E-fields are in the z-direction, which is in the plane of the interface (xz), so: E i ( x, y = 0, z, t ) + E r ( x, y = 0, z, t ) = E t ( x, y = 0, z, t ) n i n t i k r r k r t k r θ i r t E i B i E r B r E t B t Interface Boundary Condition for the Electric Field at an Interface x y z
The Tangential Magnetic Field* is Continuous In other words: The total B-field in the plane of the interface is continuous. Here, all B-fields are in the xy-plane, so we take the x-components: B i ( x, y=0, z, t ) cos( θ i ) + B r ( x, y=0, z, t ) cos( r ) = – B t ( x, y=0, z, t ) cos( t ) *It's really the tangential B/ μ , but we're using μ = μ 0 Boundary Condition for the Magnetic Field at an Interface n i n t i k r r k r t k r i r t E i B i E r B r E t B t Interface x y z i i

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