Mode Equations for OWLS

This assumes that the F,S interface is flat and abrupt.

Introducing the normalized propagation constant b (Ghatak and Thya-garajan 1989):

the asymmetry parameter a a = (n2 - nC)/(n| - n22), (10)

and the dimensionless waveguide parameter W:

where k = 2n/A is the wavenumber of the light in vacuo with wavelength A, the mode equations for a three layer waveguide (2,F,C) are then r , MBl /nFy (a)1/2+(ncT(a)1/2 , , tan[W(1- W I = ( nj 1-[b(b + a)]^ nc/nF)-2X1-b) (12)

with p = 0 and 1 for the TE (transverse electric, i. e. s-polarized) and TM (transverse magnetic, i. e. p-polarized) modes, respectively. The lower cutoff is given by

X 2p

and the penetration depth by

^F,a,C can be calculated if the adlayer A is assumed to be a uniform, homogeneous film (Tiefenthaler and Lukosz 1989). Measurement of two modes (typically the zeroth order TM and TE) and the simultaneous solution of the corresponding two mode equations enables the thickness and refractive index of the adlayer to be calculated. The explicit solutions for nA and dA are given by Guemouri et al. (2000). As discussed in the next subsection, these two parameters can then be combined to yield the number of particles v captured at unit area of interface according to

dn/dc where dn/dc is the refractive index increment of the particle, dependent on its polarizability and the medium in which it is dissolved or suspended (Ball and Ramsden 1998)

If the adlayer is not isotropic, then it must be characterized by two or more refractive indices. A fairly common case is for the adlayer to be uniaxial, and hence characterized by two refractive indices, no and ne (ordinary and extraordinary). They are related to the measured effective refractive indices by no = nA,TE

where nA,TE is the adlayer refractive index in the TE mode equation (Eq. 7), and ne =

Ntm,

In solving the mode equations, if there are only two of them (e.g. corresponding to the TE0 and TM0 modes), then dA must be measured independently in order to proceed. On the other hand, if thicker waveguides are used, supporting higher modes, then all the parameters maybe determined independently (albeit at lower sensitivity (Tiefenthaler and Lukosz 1989, Mann 2001).

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