When light encounters a boundary between two media with different refractive indices, some of it is usually reflected as shown in the figure above. The Fresnel Equations (also known as the Fresnel coefficients) are defined as the ratio of the electric field of a reflected and transmitted wave to the electric field of the incident wave. Brewster’s angle Zero reflection for parallel r || =0! n i n t + n i t = 2n t n t + n i T = t2 cos ! Derivation of Brewster’s AngleDerivation of Brewster’s Angle () 4 222 222 222 42 2 2 22 22 2 22 2 cos sin 0 c : cos sin cos sin 0 114cos sin1141sin sin 2cos 2cos 114si os si n4si n p pp pp p p pp p p p p p p p p Brewster's angle for polarizing angle for the TM case nn nn n nn r nn θ θθ θθ θ θθ θ θθ θθ θθ θ ⇒=− −+ = ±− ±− − ⇒= = ±− + = −− == +− 4 2 1 2 2 2 2 2 n 112sin Summary 17 r = n t! Reflection and Transmission Typical reflection and transmission curves for external reflection.These curves are the graphical representation of the Fresnel equations.Note that the reflected amplitude for the light polarized parallel to the incident plane is zero for a specific angle called the Brewster angle. 2 Sir David Brewster (1781-1868) is mostly remembered for his invention of the Here we systematically derive the above relations from electromagnetic premises. polarization at: “Brewster's angle” The value of this angle depends on the value of the ratio n i /n t: Brewster = tan-1(n t/n i) Sir David Brewster 1781 - 1868 For air to glass (n glass = 1.5), this is 56.3°. Fresnel equations for transmissivity and reﬂectivity At normal incidence At Brewster’s angle the reﬂectivity of the P-polarized ﬁeld goes to zero The power reﬂectivity and transmissivity of a beam are 6. The fraction that is reflected is described by the Fresnel equations, and is dependent upon the incoming light's polarization and angle of incidence. This ratio is complex and hence, it describes relative amplitude as well as phase shifts between the waves. Writing Snell’s law at Brewster’s angle, ni B nt t nt θB nt θB π θ θ ) cos 2 sin = sin = sin( − =, or i t B n n tanθ = (17) which is Brewster’s Law2. i R = r2 value of θi at which this occurs is known as Brewster’s Angle θB. t cos ! 4 Outline • Review of Oblique Incidence • Review of Snell s Law • Fresnel Equations • Evanescence and TIR • Brewster s Angle … Fresnel Equations and EM Power Flow Reading - Shen and Kong – Ch.

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