Light Propagation in Media Boundary problems Absorption coefficient, a, and refractive index, n. Reflected and refracted beams from surfaces – the Snell’s law Incident, transmitted, and reflected beams determined by boundary conditions The "Fresnel Equations" 3/30/2016 Light is an Electromagnetic Wave ~ i ( kx t ) E y (r , t ) E y e ~ i ( kx t ) Bz ( r , t ) Bz e 2 E 2 E 2 0 t 2 B 2 B 2 0 dt 1. The electric field, the magnetic field, and the k-vector are all perpendicular: EB k 2. The electric and magnetic fields are in phase. 3/30/2016 c 1 0 0 3 *108 m / s Wave Fronts At a given time, a wave's "wave-fronts" are the planes where the wave has its maxima. A plane wave's wave-fronts are equally spaced, a wavelength apart. They're perpendicular to the propagation direction, and they propagate with time. Wave-fronts are helpful for drawing pictures of interfering waves. 3/30/2016 Wave Propagation in Media Typically, the speed of light, the wavelength, and the amplitude decrease. Vacuum (or air) Medium n=1 n=2 Absorption depth = 1/ Absorptive nk k n E(x,t) = E0 exp[i(kx – t)] Wavelength decreases Dispersive E(x,t) = E0exp[(–/2)x]exp[i(nkx– t)] where is the "absorption coefficient" and n is the "refractive index." 3/30/2016 I(z) = I(0) exp(- x) Polarization Notation Parallel ("p") polarization Perpendicular("s") polarization Note the little lines and circles. “P” polarization is the parallel polarization, and it lies parallel to the 3/30/2016 plane of incidence. y z x “S” polarization is the perpendicular polarization, and it sticks up out of the plane of incidence Light Encounters A Surface Constructive interference occurs for a reflected beam if the angle of incidence = the angle of reflection. Constructive interference occurs for a transmitted beam if the sine of the angle of incidence = sine of the angle of "refraction." (Snell's Law) 3/30/2016 Refraction and Snell's Law AD = BD/sin(qi) AD = AE/sin(qt) qi So: BD/sin(qi) = AE/sin(qt) But: BD = vi Dt = (c0/ni) Dt & AE = vt Dt = (c0/nt ) Dt So: (c0/ni) Dt / sin(qi) = (c0/nt) Dt / sin(qt) Or: ni sin(qi) = nt sin(qt) qt 3/30/2016 Snell's Law Explains Many Effects 3/30/2016 Reflection and Transmission of Waves y A yB y A y B x x y1 t x c A y3 t x cB apply continuity conditions for separate components y2 t x c A y x x y A y1 t y2 t cA cA x y B y3 t cB 3/30/2016 A Reflection Coefficient Transmission Coefficient combine forward and reflected waves to give total fields for each region B x hence derive fractional transmission and reflection y2 t cB c A r12 y1 t c A cB y3 t 2c B t12 y1 t cB c A 9 Fresnel's Equations for Reflection and Refraction Wave equations Boundary conditions ki Er Ei Bi kr ni Br qi qr Interface qt Et Bt 3/30/2016 nt kt Practical Applications of Fresnel’s Equations Lasers use Brewster’s angle components to avoid reflective losses: R = 100% 0% reflection! Laser medium R = 90% 0% reflection! Optical fibers use total internal reflection. Hollow fibers use highincidence-angle near-unity reflections. 3/30/2016 Fresnel's Equations for Reflection and Refraction Wave equations Boundary conditions: tangential fields are continuous ki Er Ei Bi kr ni Br qi qr Interface qt Et Bt 3/30/2016 nt kt 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 refractive indices. Fresnel was the first to do this calculation. ki Er Ei Bi kr Br qi qr Interface Beam geometry for light with its electric field perpendicular to the plane of incidence (i.e., out of the page) 3/30/2016 y ni z qt Et Bt nt kt x Boundary Condition for the Electric Field at an Interface The Tangential Electric Field is Continuous ki The total E-field in the plane of the interface is continuous. Er Ei Bi qi qr y x z kr ni Br Interface qt Et Here, all E-fields are in the z-direction, Bt kt which is in the plane of the interface (xz), so: Ei(x, y = 0, z, t) + Er(x, y = 0, z, t) = Et(x, y = 0, z, t) 3/30/2016 nt Boundary Condition for the Electric Field at an Interface The Tangential Magnetic Field* is Continuous ki 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: Er Ei Bi Interface x z kr ni Br qi qr qt Et Bt nt kt –Bi(x, y = 0, z, t) cos(qi) + Br(x, y = 0, z, t) cos(qr) = –Bt(x, y = 0, z, t) cos(qt) *It's really the tangential B/, but we're using 0 3/30/2016 y Reflection and Transmission for Perpendicularly Polarized Light Ignoring the rapidly varying parts of the light wave and keeping only the complex amplitudes: E0i E0 r E0t B0i cos(qi ) B0 r cos(q r ) B0t cos(qt ) But B E /(c0 / n) nE / c0 and qi q r : ni ( E0 r E0i ) cos(qi ) nt E0t cos(qt ) Substituting for E0t using E0i E0 r E0t : ni ( E0 r E0i ) cos(qi ) nt ( E0 r E0i ) cos(qt ) 3/30/2016 Reflection & Transmission Coefficients for Perpendicularly Polarized Light Rearranging ni ( E0 r E0i ) cos(qi ) nt ( E0 r E0i ) cos(qt ) yields: E0 r ni cos(qi ) nt cos(qt ) E0i ni cos(qi ) nt cos(qt ) Solving for E0 r / E0i yields the reflection coefficient : r E0 r / E0i ni cos(q i ) nt cos(q t ) / ni cos(q i ) nt cos(q t ) Analogously, the transmission coefficient, E0t / E0i , is t E0t / E0i 2ni cos(q i ) / ni cos(q i ) nt cos(q t ) These equations are called the Fresnel Equations for perpendicularly polarized light. 3/30/2016 Fresnel Equations—Parallel E Field y ki kr Ei Bi Br qi qr ni × z Er Interface Beam geometry for light with its electric field parallel to the plane of incidence (i.e., in the page) qt Et Bt nt kt Note that the reflected magnetic field must point into the screen to achieve E B k . The x means “into the screen.” 3/30/2016 x Reflection & Transmission Coefficients for Parallel Polarized Light For parallel polarized light, and B0i - B0r = B0t E0icos(qi) + E0rcos(qr) = E0tcos(qt) Solving for E0r / E0i yields the reflection coefficient, r||: r|| E0 r / E0i ni cos(qt ) nt cos(qi ) / ni cos(qt ) nt cos(qi ) Analogously, the transmission coefficient, t|| = E0t / E0i, is t|| E0t / E0i 2ni cos(qi ) / ni cos(qt ) nt cos(qi ) These equations are called the Fresnel Equations for parallel polarized light. 3/30/2016 Reflection & Transmission Coefficients for an Air-to-Glass Interface nair 1 < nglass 1.5 Total reflection at q = 90° for both polarizations Zero reflection for parallel polarization at 56.3° “Brewster's angle” (For different refractive indices, Brewster’s angle will be different.) Reflection coefficient, r Note: 1.0 Brewster’s angle .5 r||=0! 0 r ┴ -.5 -1.0 0° 3/30/2016 r|| 30° 60° 90° Incidence angle, qi Reflection Coefficients for a Glassto-Air Interface 1.0 Note: Total internal reflection above the "critical angle" qcrit arcsin(nt /ni) (The sine in Snell's Law can't be > 1!) 3/30/2016 Reflection coefficient, r nglass 1.5 > nair 1 Critical angle r ┴ .5 Total internal reflection 0 Brewster’s angle -.5 Critical angle r|| -1.0 0° 30° 60° 90° Incidence angle, qi

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