Optical Invariants

A weakness of the method outlined in Sect. 2.2 for evaluating OWLS data is that the results depend in an ill-defined fashion on the model chosen to describe the adlayer A. For many adlayers encountered in practice, especially those formed from macromolecules captured at the solid-liquid interface, the assumption of a uniform, homogeneous film is unrealistic.

The use of optical invariants enables key parameters characterizing the adlayer to be calculated without the need for possibly unwarranted assumptions. The difference between the optical responses of the idealized Fresnel interface and the real interface is given in terms of surface excess polarization densities (Mann 2001) (cf. Gibbs' surface excess quantities in his treatment of the thermodynamics of thin films). Any measurable quantities should be independent of the position of the (fictitious) Fresnel interface: such optical invariants are obtained by combining the polarizabilities obtained by a multipole expansion of the surface excess polarization densities

5 More rigorously, this implies that nA and dA are not optical invariants of the system. The optical invariants arise analogously to the Gibbs formulation of interfacial thermodynamic properties: they are such quantities that are invariant under small movements of the interfacial boundary (see Sect. 2.2.4).

such that the combinations are invariant with respect to displacement of the idealized Fresnel interface. These optical invariants allow determination of the maximum information available from the data with the minimum of ambiguity.

Mann (2001) has derived the optical invariants for OWLS and given analytic equations relating the parameters obtained from OWLS measurements to the optogeometric parameters of uniaxially anisotropic adlayers. The first order non-invariant excess polarization density parallel to the interface is

where n is the excess parallel refractive index, i. e.

where ny is the refractive index parallel to the interface, i. e. the ordinary refractive index, and we can call n 2/n2C the increment of the relative dielectric constant of the adlayer.

The first and second order (in dA/X) invariants are J1, J21, J22 and J23:

nC/n2 + 1 + a where the optical anisotropy of the adlayer is

where n± is the refractive index perpendicular to the interface, i. e. the extraordinary refractive index;

where

i. e. depending only on polarization density parallel to the interface;

a and

22 nFnC

Table 3. Comparison of different techniques in terms of obtainable quantities

Technique

Non-

Optical invariants

invariant

1st ordera 2nd ordera

OWLS

Ye

J1 J22> J23

Ellipsometry

/1 /23

SAR

The power of the optical waveguide technique is that four opto-geometric parameters characterizing an (anisotropic) adlayer can be extracted. Table 3 compares the available parameters with those from other optical techniques.

If dA/X is small (< 0.01), then fe and J1 dominate the waveguide optical response, and (Mann 2001)

Nfe2

where

N2 NTM

NtmNcf/nf where

F"C

N1/2 NFM

nm kdFNF1/2

"FM

The adlayer mass M per unit area, for thin films, i. e. dA/X < 0.01, is given by

M « -fe(dc/dn2)/k, i. e. depending only on the TE modes.

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