Fresnel’s formulae

We now consider a light wave with 1 = polarized in the x-direction, traveling in the waveguide between the two interfaces, waveguide-cover and waveguide-substrate. The wave will be reflected at the interfaces with angles cpc and (ps for cover and substrate, respectively. Since we assume the electric held is parallel to the interfaces, a phase shift of Tj will occur at each interface according to Fresnel s formulas, while the amplitude and polarization remain the same. Additionally, we assume for simplicity that (pc = [Pg.26]

Let us next consider a metal hoUow-optical fiber with dielectric inner coating. When the coating thickness is properly designed, the dielectric layer enhances reflection in a specific wavelength range owing to the interference effect. Let us consider a metal surface coated with a dielectric film of thickness d as shown in Fig. 2. From Fresnel s formula, the reflection coefficient of the electric field on boundary I is expressed as... 

The reflectivity / of a plane interface between two regions with complex refractive indices n n — iK and = 2 can be calculated from Fresnel s formulas [4.15]. It depends on the angle of incidence a and on the direction of polarization. Reflectivity R a) is illustrated in Fig. 4.51 for three different materials for incident light polarized parallel (/ p) and perpendicular (/ s) to the plane of incidence. 

For vertical incidence (a = 0), one obtains from Fresnel s formulas for both polarizations... 

The reflected amplitudes can be calculated from Fresnel s formulas. The total reflected intensity is obtained by summation over all reflected amplitudes taking into account the correct phase. The refractive indices are now selected such that Ai becomes a maximum. The calculation is still feasible for our example of a two-layer coating and yields for the three reflected amplitudes (double reflections are neglected)... 

Assume that a plane wave E = Aq exp[i(o — kx)] is incident at the angle a on a plane transparent plate with two parallel, partially reflecting surfaces (Fig. 4.36). At each surface the amplitude Ai is split into a reflected component Ar = A/ /R and a refracted component At = A/ / — R, neglecting absorption. The reflectivity R = /r/// depends on the angle of incidence a and on the polarization of the incident wave. Provided the refractive index n is known, R can be calculated from Fresnel s formulas [116]. From Fig. 4.36, the following relations are obtained for the amplitudes A/ of waves reflected at the upper surface, Bi of refracted waves, Q of waves reflected at the lower surface, and Dt of transmitted waves... 

Fig. 4 shows the ATR properties of the Ag sputtered films measured using He-Ne laser beam of the wavelength at 632.8 nm. The closed circles represent experimental data and the solid lines represent calculated ones. The theoretical calculations of the ATR curves were carried out from Fresnel s formula using transfer matrix method [20]. The dielectric constants and the film thicknesses obtained by curve fittings are 38.05 nm and -17.017+i 0.595 for sample A and 38.51 nm and -17.523+i 0.704 for sample B, respectively. 

Reflection of Light by a Transparent Medium in Air n (Fresnel s formulae) If i is the angle of incidence, r the angle of refraction, n the index of refraction for air (nearly equal to unity), 02 index of refraction for a medium, then the ratio of the reflected light to the incident light is. 

Next the components of Ex are examined. For a linearly-polarized incidence light beam, according to Fresnel s formula, the ratio of the amplitudes of electric field is given by... 

The transmitted field at the solid-vacuum interface is described by Fresnel s formulas and can be written as... 

Similar anisotropy should be produced in the Az surface by the slantwise eiq)osure. Using Fresnel s formulas, we can expect that polarization due to reflection at interfaces of the system is negligible. To explain, we suppose that trans-Az lying its long axis parallel to the reaction light path hardly photoisomer-ize since the incident electric vector is perpendicular to the transition vector of the Az, and would give the specific orientation axis. 

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