In Situ Characterizations of Porous Media for Applications in Biofuel Cells: Issues and Challenges - Porous Media: Applications in Biological Systems and Biotechnology - page 504

Reflectance is a ratio between the reflected and the incidence beam ampli-
tude. A ratio between the perpendicular and parallel-to-surface reflectance
components ( R p and R s ) specifies tangents of the ellipsometric angle Ψ:
tan Ψ = |

R p |

|
In complex values, the relation of ∆ and Ψ can be expressed as follows:
tan Ψ e j ∆ = R p
R s
Although both angles reflect thickness and material properties, practi-
cally ∆ reflects primarily thickness and Ψ material's properties or processes
occurred in the material.
Variations in the incident wavelength, AOI, time, and spatial references
give the basis for the spectroscopic, variable angle, dynamic, and imaging ellip-
sometry, respectively. From Ψ and ∆ and optical model analysis, the refractive
index ( N = n + ik ), thickness ( d ), and coverage (

| R s

) of the surface film can be
derived. For porous media, a parameter of interest is surface mass density
(Γ), which is reduced from thickness and refractive index into a comprehen-
sive, simple parameter.
If the film is suciently transparent, so the extinction coecient, k, is
rather constant, N = n can be assumed to make the observation of the film
thickness variation much easier directly from the change of the ellipsomet-
ric angles ∆. If the k varies with film thickness, the interpretation of the film
thickness and optical property becomes more complicated. A common practice
to deal with this situation is to use a simple volume averaging approach (Equa-
tion 13.1) or the effective-medium theory and approximation (Equation 13.2)
to assess the variations in the dielectric function. Thus,
ε = θε i +(1

θ

− θ ) ε amb (13.1)

θ ) /ε amb (13.2)
where the dielectric function ε = N 2 is for the “equivalent” interface layer,
while ε i is for the intrinsic layer composition, and ε amb is for the ambient,
respectively;

1 /ε = θ/ε i +(1 −

θ is the surface coverage for the thin film. For porous media,

θ . This simpli-

fied approach is only valid when a simple geometry is involved so the elec-
tric field can be treated fairly in a perpendicular and parallel fashion. With
oblique incidence, the use of such approximations should be cautious, espe-
cially when the pore shape is irregular or the pore size variation is significant.
The anisotropy and possible birefringence should be considered in due dili-
gence. To apply effective-medium approximation, the Bruggeman theory (von
Bruggeman 1935) is often used to give the following equation for the calcula-
tion of the surface coverage

can be replaced with a volume fraction of the solid media φ

θ or the volume fraction of the solid media φ (the

two are interchangeable in this case):
θ [( ε i − ε ) / ( ε i +2 ε )]+(1 − θ )[( ε amb − ε ) / ( ε amb +2 ε )] = 0 (13.3)