Question

Question # 1 Mark SLO CLO Show that for parallel polarization the reflection and transmission coefficients are m2cose n,cost, 2n,cos«, Show that for perpendicular polarization the reflection and transmission coefficients are 2T2cos η.cos0,+n,cosO 6/1/4/F158/EE/1

Answer 1

DERIVATION To derive the Fresnel equations, consider two optical media separated by an interface, as shown in Fig. 1. A plane optical wave is propagating toward the interface with wave vector k, oriented at angle θ with respect to the inter- face normal. The electric field amplitude of the wave is given by E On incidence onto the interface, this wave will be par tially transmitted and partially reflected. The transmitted

wave will propagate at angle 6 which is determined by Snell's law in n sin 0 ni where n, andn, are the refractive indices of the two media The angle θ of the reflected wave is equal to θ according to the law of reflection. We denote the amplitudes of these two waves as E, and E respectively. Our goal is to deter- mine these amplitudes To accomplish this, we apply the boundary conditions for the electric and magnetic fields at an interface between two media with different electromagnetic properties, which are known from electrostatics. Specifically, the components of the electric field E and magnetic field H, which are tan gent to the surface, must be continuous across the boundary Because the electromagnetic wave is transverse, the field incident onto the interface can be decomposed into two polarization components, one P-polarized, i.e., with the elec- tric field vector inside the plane of incidence, and the other one S-polarized, i.e, orthogonal to that plane. (Under the plane of incidence, we understand the plane that is formed by the vector k, and the normal to the interface.) We will derive the Fresnel equations for these two cases separately We begin by concentrating on the case when the inc dent wave is P-polarized (Fig. 1). Due to symmetry, the transmitted and reflected waves will have the same polari- zation. Because the E, H, and k vectors must form a right- handed triad for each of the waves, the directions of all field vectors are uniquely defined up to a sign convention which is chosen as illustrated in Fig. 1. The boundary con dition for the electric field then becomes: E, cosẻ, + E, cos®.-E, cosẻ, For the magnetic field, which is collinear in all three waves, this condition takes the form:

Incident wave Reflected wave Interface Medium1 Medium 2 NormaH wave Fig.1 Field vectors of the incident, transmitted, and reflected waves in case the electric field vectors lie within the plane of incidence (P polarization). To solve these equations, tion between the electric and magnetic field amplitudes for each wave. We know from Maxwell equations that these amplitudes in any plane electromagnetic wave must satisfy we need to incorporate the rela where ε and μ are the electric permittivity and magnetic permeability, respectively, of the material in which the wave propagates. Since the index of refraction of a material is given by n-cVeu, we have: and thus, from Eq. (3), Combining Eqs. 2 and 6, we arrive at the Fresnel equations for the P-polarized wave: where we defined the amplitude reflection and transmis- sion coefficients: E, In the case of S polarization (Fig. 2), in much the same way, we write the boundary conditions as (10) -Hi cose+ H, cose, =-H, cose, from which we derive the second pair of Fresnel equations (12)

Incident wave Reflected wave Ε, Normal Transmitted wave Fig. 2 Field vectors of the incident, transmitted, and reflected waves in case the electric field vectors are perpendicular to the plane of incidence (S polarization) 2(nilHi)cose (niju.) cos0, + (n2412) cos θ (13) Eqs. 7 and 8, as well as Eqs. 12 and 13 present Fresnel equations in their general form, which is also valid for materials with negative indices of refraction (also known as metamaterials or left-handed materials). When applying these equations to such materials, absolute values of the refractive indices must be used.I-31 Most commonly used optical materials are non magnetic, so one can approximate μί-112-μο. Under this approximation, the permeabilities in Eqs. 7, 8, 12, and 13 cancel, and the Fresnel equations can be further simplified by incorporating Snell's law tan(e- 0,) tan(e, +0,) (14) 2sin9, cos θ sine+ θ, ) cos(0,-9) (15) sin(0, -,) (16) 2sin 6, cose Ssin(,0) (17)