Fresnel reflection and transmission amplitude coefficients and Brewster's angle. (a) Case II: Consider light polarized parallel...

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Fresnel reflection and transmission amplitude coefficients and Brewster's angle. (a) Case II: Consider light polarized parallel to the plane of incidence (p-polarized). Starting from the appropriate boundary conditions given by Maxwell's equations, derive the expressions for the Fresnel reflection and transmission amplitude coefficients in p-polarized light, r, and to Ts = 1₂ = + = sin(0₁ - 0,) sin(0; + 0,) 2 sin(8,) cos(8₂) sin(0; + 0,) Eip = Include a sketch showing the relationship between k, E, and B for the incident, reflected, and transmitted light. (b) The Fresnel amplitude expression may be rewritten in terms of the incident and transmitted angles 0; and 0, only as n, cos(0) - n, cos(8₁) n; cos(8₁) + n, cos(0) 2n; cos(0;) Ei.pn; cos(0,) + n, cos(0;) rp = + = + tan(0₁ - 0₁) tan(0; + 0,) 2 sin(8,) cos(0;) sin(0; + 0,) cos(0,0₁) Derive the equations for the for the perpendicular (s-polarized) Fresnel coefficients only. Bonus: Derive for the parallel (p-polarized) Fresnel coefficients, which are trigonometrically messier. You might first try working backwards from the answer. (c) Describe physically what happens to light incident on a boundary at Brewster's angle B. Using the expressions for the Fresnel coefficients in part (b), show that for light incident at Brewster's angle, 0, +0,= π/2. Also, show that tan 0g = n₁/n₁. (d) Show that the polarization angles for internal and external reflection at a given interface are complementary, that is, 0 + 0 = π/2. Fresnel reflection and transmission amplitude coefficients and Brewster's angle. (a) Case II: Consider light polarized parallel to the plane of incidence (p-polarized). Starting from the appropriate boundary conditions given by Maxwell's equations, derive the expressions for the Fresnel reflection and transmission amplitude coefficients in p-polarized light, r, and to Ts = 1₂ = + = sin(0₁ - 0,) sin(0; + 0,) 2 sin(8,) cos(8₂) sin(0; + 0,) Eip = Include a sketch showing the relationship between k, E, and B for the incident, reflected, and transmitted light. (b) The Fresnel amplitude expression may be rewritten in terms of the incident and transmitted angles 0; and 0, only as n, cos(0) - n, cos(8₁) n; cos(8₁) + n, cos(0) 2n; cos(0;) Ei.pn; cos(0,) + n, cos(0;) rp = + = + tan(0₁ - 0₁) tan(0; + 0,) 2 sin(8,) cos(0;) sin(0; + 0,) cos(0,0₁) Derive the equations for the for the perpendicular (s-polarized) Fresnel coefficients only. Bonus: Derive for the parallel (p-polarized) Fresnel coefficients, which are trigonometrically messier. You might first try working backwards from the answer. (c) Describe physically what happens to light incident on a boundary at Brewster's angle B. Using the expressions for the Fresnel coefficients in part (b), show that for light incident at Brewster's angle, 0, +0,= π/2. Also, show that tan 0g = n₁/n₁. (d) Show that the polarization angles for internal and external reflection at a given interface are complementary, that is, 0 + 0 = π/2.