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Fresnel Equations Electromagnetism II (spring term 2020) Lecture 15 Fresnel equations E Goudzovski [email protected] http://epweb2.ph.bham.ac.uk/user/goudzovski/Y2-EM2 Previous lecture For monochromatic waves, there are two independent boundary conditions. We will use these: Geometric laws of reflection and refraction laws follow from the existence of linear boundary conditions. In particular, Snell’s law: Total internal reflection for n1>n2 for angles larger than the critical angle: Total internal reflection gives rise to an evanescent field. 1 This lecture Lecture 15: Fresnel equations S and P polarizations of plane waves Derivation of the Fresnel equations Main properties of the Fresnel coefficients Brewster’s angle Energy transport at boundary 2 S and P polarizations The plane of incidence contains the incident and reflected k-vectors. A plane EM wave can be presented as superposition of two waves: 1) E is perpendicular to the plane of incidence (perpendicular, ⊥, s polarization); 2) E lies in the plane of incidence (parallel, ∥, p polarization); The amount of reflected (and transmitted) light is different for the two incident polarizations, and is determined by the boundary conditions. reflected light Recall that the B field is perpendicular to the E field 3 The S polarisation (1) Electric field pointing up out of the plane of incidence Plane of (i.e. out of the page) interface: y=0 Magnetic field directions are determined using Assuming 1=2, boundary conditions lead to the following relations for the field amplitudes: 4 The S polarisation (2) Using the relation (lecture 11), The reflection coefficient for the s-polarized (⊥) wave: The transmission coefficient for the s-polarized (⊥) wave: 5 The P polarisation (1) Electric field parallel to the plane of incidence Plane of (i.e. in the page) interface: y=0 Note: the B’ vector points into the page due to The boundary conditions lead to 6 The P polarisation (2) Using the relation (lecture 11), (multiply by ) (multiply by n2) The reflection coefficient for the p-polarized (∥) wave: The transmission coefficient for the p-polarized wave: 7 Fresnel equations for the amplitude reflection and transmission factors For s-polarized wave, For p-polarized wave, For both polarizations, If the incident wave is polarized, then the reflected and refracted waves are also polarized 8 Air to glass interface n1 = 1; n2=1.5 The two polarizations are indistinguishable at 1=0. At grazing incidence (1=/2), total reflection for both polarizations: For a wave incident at the Brewster’s angle, no reflection of p-polarized light: The signs of r and t are determined by the choice of the axis directions, and have no physical meaning. 9 Glass to air interface n1 = 1.5; n2=1 Main features: 1) Brewster’s angle: 2) Total internal reflection above the critical angle of incidence: (lecture 14) 10 Brewster’s angle: (from Fresnel’s equations) (Snell’s refraction law) Multiply the two equations: (trivial solution, n =n ), or 1 2 1 1 n1 i.e. n2 2 Brewster’s angle: 11 Energy transport at boundary Reflectance (proportion of energy reflected off surface): Transmittance (proportion of energy transmitted): Perpendicular (s) polarization Parallel (p) polarization Natural light reflected off an air/dielectric boundary is largely s-polarized, Air to glass Air except for very large and very small angles of incidence. Glass to air Glass 12 Energy conservation (not discussed in the lecture) Reflection and transmission coefficients r, t are ratios of wave amplitudes, not energies. Reflectance R and the reflection coefficient r: Transmittance T and the transmission coefficient t: accounting for the expansion (contraction) the beam on refraction, Fresnel coefficients r, t satisfy energy conservation, separately for the s- and p-polarizations 13 Reflection at normal incidence From Fresnel’s equations, for 1=2=0, and For air-glass interface (n1=1, n2=1.5), R=4% and T=96%. For air-water interface, R=2% for visible light (n=1.33); R=64% for radiowaves (n=9). Reflectance at normal incidence does not depend on the direction of wave propagation. Implications for photography: lens flare. 14 Examples Reflected natural light is largely s-polarized, except for very large and very small angles of incidence. Polarizer filters are used in photography to remove reflected sunlight. Polaroid sunglasses principle: s-polarized light reflected off a horizontal surface is polarized mainly horizontally. 15 Summary Dynamic properties of reflection at the boundary of dielectrics: The reflection and transmission (Fresnel) coefficients are different for p (∥) and s (⊥) polarized waves. Reflectance and transmittance at normal incidence: For grazing incidence, . For incidence at the Brewster’s angle , the p-polarized wave is fully transmitted: . 16 .
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