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Chapter 23: Fresnel equations Recall basic laws of optics normal i Law of reflection: i r Law of refraction “Snell’s Law”: sin i n2 sin t n1 r n1 n2 t Incident, reflected, refracted, and normal in same plane Easy to derive on the basis of: Huygens’ principle: every point on a wavefront may be regarded as a secondary source of wavelets Fermat’s principle: the path a beam of light takes between two points is the one which is traversed in the least time Today, we’ll show how they can be derived when we consider light to be an electromagnetic wave E and B are harmonic E E0 sin( k r t ) B B0 sin( k r t ) Also, at any specified point in time and space, E cB where c is the velocity of the propagating wave, c 1 0 0 2.998 108 m/s We’ll also determine the fraction of the light reflected vs. transmitted external reflection, i R 1.0 T .5 T R 0 0° 30° 60° 90° Incidence angle, i and the change in the phase upon reflection Let’s start with polarization… y light is a 3-D vector field z x linear polarization circular polarization …and consider it relative to a plane interface Plane of incidence: formed by and k and the normal of the interface plane normal E B k Polarization modes (= confusing nomenclature!) always relative to plane of incidence TE: transverse electric s: senkrecht polarized (E-field sticks in and out of the plane) perpendicular ( ), horizontal TM: transverse magnetic p: plane polarized (E-field in the plane) parallel ( || ), vertical Plane waves with k along z direction oscillating electric field y y x x Any polarization state can be described as linear combination of these two: i kz t y i kz t x ˆ E E0 x e x E0 y e yˆ i E E0 x eix xˆ E0 y e y yˆ ei kz t “complex amplitude” contains all polarization info Derivation of laws of reflection and refraction using diagram from Pedrotti3 boundary point At the boundary point: phases of the three waves must be equal: (k i r it ) (k r r r t ) (k t r t t ) true for any boundary point and time, so let’s take r 0 i t r t t t or i r t hence, the frequencies are equal t 0 ki r k r r kt r and if we now consider which means all three propagation vectors lie in the same plane Reflection ki r k r r kt r focus on first two terms: ki r k r r ki r sin i kr r sin r incident and reflected beams travel in same medium; same l ki k r hence we arrive at the law of reflection: i r Refraction ki r k r r kt r now the last two terms: k r r kt r kr r sin r kt r sin t reflected and transmitted beams travel in different media (same frequencies; different wavelengths!): kr / vr n1 / c kt / vt n2 / c which leads to the law of refraction: n1 sin r n2 sin t Boundary conditions from Maxwell’s eqns for both electric and magnetic fields, components parallel to boundary plane must be continuous as boundary is passed TE waves electric fields: parallel to boundary plane E 0i Ei yˆ E 0 r Er yˆ E 0t Et yˆ complex field amplitudes continuity requires: Ei Er Et Boundary conditions from Maxwell’s eqns for both electric and magnetic fields, components parallel to boundary plane must be continuous as boundary is passed TE waves magnetic fields: i ( k i r t ) B i ( B cos i xˆ B sin i zˆ )e i ( k r r t ) B r ( B cos r xˆ B sin r zˆ )e i ( k t r t ) B t ( B cos t xˆ B sin t zˆ )e continuity requires: Bi cos Br cos Bt cos same analysis can be performed for TM waves Summary of boundary conditions Ei n1 n2 TE waves Er Bt Bi Ei TM waves Er Bt Bi Br Et Br Et Bi Br Bt Ei Er Et Bi cosi Br cos r Bt cost Ei cos i Er cos r Et cos t amplitudes are related: E vB nc B Fresnel equations TE waves TM waves Get all in terms of E and apply law of reflection (i = r): Ei Er Et ni Ei ni Er nt Et ni Ei cosi ni Er cosi nt Et cost Ei cosi Er cosi Et cost For reflection: eliminate Et , separate Ei and Er , and take ratio: cos i nnti cos t E rTE r Ei cos i nnti cos t rTM E r Ei nt ni nt ni cos i cos t cos i cos t Apply law of refraction ni sin i nt sin t and let n rTE cos i n 2 sin 2 i cos i n sin i 2 2 rTM nt : ni n 2 cos i n 2 sin 2 i n 2 cos i n 2 sin 2 i Fresnel equations TE waves TM waves For transmission: eliminate Er , separate Ei and Et , take ratio… tTE Et 2 cos i Ei cos i n 2 sin 2 i tTM Et 2n cos i Ei n 2 cos i n 2 sin 2 i And together: tTE 1 rTE ntTM 1 rTE Fresnel equations, graphically External and internal reflections External and internal reflections occur when external reflection: n1 n2 internal reflection: n1 n2 n2 n 1 n1 n2 n 1 n1 External reflections (i.e. air-glass) n = n2/n1 = 1.5 RTM = 0 (here, reflected light TE polarized; RTE = 15%) at normal : 4% 1 n Rr 1 n 2 normal grazing - at normal and grazing incidence, coefficients have same magnitude 2 - negative values of r indicate phase change Pr Er 2 R r - fraction of power in reflected wave = reflectance = Pi Ei - fraction of power transmitted wave = transmittance = T cos t 2 Pt t n Pi cos i Note: R+T = 1 2 Glare http://www.ray-ban.com/clarity/index.html?lang=uk Internal reflections (i.e. glass-air) n = n2/n1 = 1.5 total internal reflection - incident angle where RTM = 0 is: p 2 - both RTE rTE2 and RTM rTM reach values of unity before =90° total internal reflection c sin 1 (n) sin 1 n2 n1 Internal reflections (i.e. glass-air) Conservation of energy it’s always true that and Pi Pr Pt R T 1 in terms of irradiance (I, W/m2) I i Ai I r Ar It At using laws of reflection and refraction, you can deduce 2 Pr I r E0 r r 2 R Pi I i E0i and cos t 2 t T n cos i Summary: Reflectance and Transmittance for an Air-to-Glass Interface Perpendicular polarization 1.0 Parallel polarization 1.0 T T .5 .5 R 0 R 0 0° 30° 60° Incidence angle, i 90° 0° 30° 60° Incidence angle, i 90° Summary: Reflectance and Transmittance for a Glass-to-Air Interface Perpendicular polarization 1.0 Parallel polarization 1.0 R R .5 .5 T T 0 0 0° 30° 60° Incidence angle, i 90° 0° 30° 60° Incidence angle, i 90° Back to reflections Brewster’s angle or the polarizing angle is the angle p, at which RTM = 0: p tan 1 n tan 1 n2 n1 Brewster’s angle for internal and external reflections at p, TM is perfectly transmitted with no reflection at Brewster’s angle, “s skips and p plunges” s-polarized light (TE) skips off the surface; p-polarized light (TM) plunges in Brewster’s angle Punky Brewster (1984-1986) Sir David Brewster (1781-1868) Brewster’s other angles: the kaleidoscope http://www.brewstersociety.com/cbs_sundaymorning_09.html Phase changes upon reflection -recall the negative reflection coefficients -indicates that sometimes electric field vector reverses direction upon reflection: Er r E ip r E e r E0e i ( k r t ) r E0e i ( k r t p ) -p phase shift external reflection: all angles for TE and at p for TM internal reflection: more complex… Phase changes upon reflection: internal in the region c , r is complex rTE rTM cos i sin 2 n 2 cos i sin 2 n 2 n 2 cos i sin 2 n 2 n 2 cos i sin 2 n 2 reflection coefficients in polar form: r r e phase shift on reflection i Phase changes upon reflection: internal depending on angle of incidence, p p sin 2 n 2 TE tan cos 2 sin 2 n 2 TM p tan n 2 cos 2 Exploiting the phase difference circular polarization -consists of equal amplitude components of TE and TM linear polarized light, with phases that differ by ±p/2 -can be created by internal reflections in a Fresnel rhomb each reflection produces a π/4 phase delay http://www.halbo.com/fr_rhmb.htm Summary of phase shifts on reflection external reflection TE mode TM mode TE mode TM mode air glass internal reflection air glass A lovely example How do we quantify beauty? Case study for reflection and refraction Exercises You are encouraged to solve all problems in the textbook (Pedrotti3). The following may be covered in the werkcollege on 21 September 2011: Chapter 23: 1, 2, 3, 5, 12, 16, 20