Fresnel reflection formulae

Fresnel s101 formulas [13] give the parallel ( ) and perpendicular ( L) components of the reflected and refracted light beams ... 

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... 

Similar formulae in terms of the Fresnel reflectances may be built up for the situation with a small number of discrete layers. A larger number of layers demands the use of a more general method in which the surface is considered in terms of a series of layers of slightly diflerent characteristics. The models that are used in practice consider the density variation to be a smooth function of the distance from the surface but also may include specifically a diflerent surface layer to account for surface roughness and the possibility of molecular segregation at the surface. 

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. 

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... 

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. 

Let us first consider the scheme suggested by Cook and Hill (1982). When a plane traveling light wave is totally reflected internally at the surface of a dielectric in a vacuum, a evanescent wave is generated on the surface (Fig. 7.1). By application of Fresnel s reflection formulas (Born and Wolf 1984), the intensity of evanescent wave is given by... 

The diffuse reflection coefficient of the crystal side surface was taken equal to = 0.18 for radiation coming from the gas and = 0.823 for radiation coming from the crystal. The first value was obtained by averaging of the Fresnel reflection coefficient over an incident angle, while the second follows from relation between and p, Eq. (8.8). The specular reflection coefficient was calculated by the Fresnel formulae. The reflection coefficient of a free surface of a melt is unknown at present and was taken equal to p both for diffuse and specular reflection that can be justified by a small difference between refraction index of solid and hquid phases. The latter, along with opacity of the melt, permits consideration of the... 

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. 

Since adsorbed molecules are exposed both to the incident and reflected light waves whose electric field amplitudes are interrelated by the Fresnel formulae, the vectors in s- and -polarizations appear as ... 

Physically, this formula describes the power dissipated by a harmonic oscillator (the emission dipole with moment t) as it is driven by the force felt at its own location from its own emitted and reflected electric field. PT is calculable given all the refractive indices and Fresnel coefficients of the layered model(12 33)... 

A simple and practical way to achieve the field enhancement is to use backside illumination of a dielectric plate, for instance a cover glass, in a standard DLW geometry with an oil-immersion focusing lens. According to the Fresnel formulas for the right angle incidence (0, = 0°), the coefficients of the in-plane ( ) polarized amphtudes of transmitted and reflected electric fields are, respectively ... 

Equations (2.67)-(2.70) are the Fresnel formulas for reflection and transmission of light obliquely incident on a plane boundary. 

Measurement of reflectances for incident light of various polarization states and two oblique angles of incidence the results are analyzed with the Fresnel formulas. Large angles are required for high accuracy, and this requires large sample surfaces. 

Equations (1.74) to (1.77) are the Fresnel formulae. Examination of equation (1.77) reveals that light with p polarization will not be reflected from a plane interface (/ = 0) at a specific angle of incidence, Brewster angle and, for the case... 

To get the dielectric functions e(A), since the geometrical shape of the crystal is not perfectly known, we ought to select experimental conditions so as to have the simplest possible relation between e(k) and r(A), namely, the Fresnel formula for the normal reflection amplitude of a semi-infinite dielectric ... 

Ellipsometry measures the orientation of polarized light undergoing oblique reflection from a sample surface. Linearly polarized light, when reflected from a surface, will become elliptically polarized, because of presence of the thin layer of the boundary surface between two media. Dependence between optical constants of a layer and parameters of elliptically polarized light can be found on basis of the Fresnel formulas described above. 

---