Fresnel's coefficients of reflection give the ratio of reflected to incident amplitude for beams incident from the high refractive index side (labelled F) onto an interface between two transparent dielectrics F and J. Their refractive indices are nF and nj respectively, with nF > nj2,3. The coefficients are:

for the perpendicular (s, ç = 0) and parallel (p, ç = 1) polarizations, with kF = knF cos Q (2)

where k = 2n/\, the wavenumber for the light in vacuo, Q is the angle of incidence measured from the normal to the interface, and kj = ky nj - n\ sin2 9 . (3)

For a thin film A of thickness dA interposed at the interface, these expressions are modified by summing the reflections and transmissions at the two interfaces, yielding (Bousquet 1957):

where is the phase thickness of A, defined by

with kA defined by Eq. 3 with J = A. These expressions maybe straightforwardly extended to multiple thin layers in an interfacial stack (Ramsden 1993a).

Careful measurement of the amplitude of reflected light as a function of incident angle can, in principle, yield enough data to enable Eq. 1 to be fitted to the data with the unknown parameters (which are typically the opto-geometric parameters of the thin interfacial film) as fitting variables. Often the measurement is carried out with p-polarized light in the vicinity

2 In general the media may also absorb some light, i. e. nF and nj are actually complex numbers, but typically the imaginary part is very small compared to the real part, and for the convenience of writing we shall suppose it to vanish.

3 Horvath etal. (2002) have demonstrated reverse symmetry waveguide sensing with nF < nj.

of the Brewster angle for the F,S interface (at which the reflected component vanishes).

In ellipsometry, it is the change in polarization upon reflection or transmission which is measured.

The phenomenon of total internal reflection, which occurs when Q exceeds a critical angle (such that the angle of transmission would become imaginary) was known to Newton (1730), and may indeed have been discovered by him (at any rate he gives the first account published in writing). In effect, light propagating in a planar optical waveguide, i. e. a thin slab of high refractive index material sandwiched between slabs of low index material, does so via alternate total internal reflections at the two boundaries in a zig-zag path, which of course implies penetration of the light into the optically rarer medium beyond the waveguide, and is equivalent to a phase shift 0 (Eq. 8). The guided mode within the waveguide is a stationary wave, and beyond the waveguide an exponentially decaying evanescent field (Ramsden 1993a). The point at which the light turns back toward the waveguide exactly equals the characteristic decay length of the evanescent

The mode equations, which govern the relation between the effective refractive indices N and the optogeometric parameters of the waveguide, can be easily derived from Eqs. (1-3), bearing in mind that R, being in general complex, can be written as the product of its modulus and argument:

and by noting that the existence criterion for a guided mode is that the sum of the phase shifts occurring at the two reflections and upon twice traversing the width of the waveguide must sum to an integral multiple of 2n, otherwise destructive interference occurs (Tiefenthaler and Lukosz):

where m is the mode number, and the phase changes upon reflectance are derived from Eqs. 1 and 6:

field.

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