The specular component is responsible for sharp mirror-like reflection
from a surface. It can be expressed as a product:

dI_{r,sp} = |F|^{2} • e^{-g} • S • I_{i}

(5)

where |F|^{2} is the Fresnel reflectivity, which is a function of the
incidence angle
and the refractive index n of the
material, g is a function of the surface roughness, and S is a
geometrical shadowing function.

The following interactive display illustrates how the Fresnel term
|F|^{2} varies with θ_{i}
and n. The incidence angle θ_{i}
is evaluated at the center of the incident light cone on
the left. The length of the cone of reflection on the right is
proportional to the magnitude of |F|^{2.}
The solid-line semicircle
is of unit radius and is included for reference.

Figure 6: Interactive display of Fresnel term
|F|^{2.} Double click the figure to activate the
control panel; click “Play” to activate a
demonstration.

The function g, which depends on the effective surface roughness
σ, is given by:

Within the specular cone,
cosθ_{i} = cosθ_{r},
and g reduces to
g = (4πσ cosθ_{i}/λ)^{2}.
The influence of
roughness is then described by the group σ cosθ_{i}/λ,
which is the ratio of the
projected effective roughness to the wavelength, also known as the
“apparent” roughness. The roughness parameters
σ and τ
are shown schematically in Figure 7. The projection of the effective
roughness into the incident light direction is
σ cos θ_{i}.
A surface appears rough or smooth to an incident light wave depending
upon whether the apparent roughness is large or small, respectively.
Note that all surfaces appear smooth as the incident light approaches
grazing angles, that is, as
θ_{i}
approaches 90° and cos θ_{i}
approaches zero.

Figure 7: Schematic of rough surface and incident light wave

The interactive figure below illustrates how the exponential
e^{-g} varies with θ_{i}, σ, τ
and τ.
The effective roughness σ
is generally smaller than the root-mean-square vertical surface roughness
height σ_{0}
due to shadowing effects.
Although σ
is a function of
θ_{i}, θ_{r}, τ,
and σ_{0},
it is found that σ depends primarily on
σ_{0}
[HE91].
For this reason,
σ_{0}
is the parameter varied on the
“sigma” slider of the control panels. The
τ-dependence of
e^{-g}
arises from the secondary dependence of the effective
roughness σ
on τ.
In Figure 8, the solid-line semicircle
is the unit hemisphere. The length of the cone of reflection is
proportional to e^{-g}.
A statistically-generated Gaussian rough
surface topography is shown schematically, with the vertical scale
amplified with respect to the horizontal (by about a factor of ten).

Figure 8: Interactive display of
e^{-g} term. Double click the figure to activate.

The shadowing term S is illustrated below. It varies with
σ_{0} and τ.
The solid-line semicircle is the unit hemisphere,
representing the limit of no shadowing. The function S is defined in
[HE91].

Figure 9: Interactive display of shadowing term S

Finally, let's see how the entire specular component
dI_{r,sp}
varies with the parameters
θ_{i}, σ, τ, λ
material. The length of the cone of reflection on the right side in
Figure 10 is proportional to dI_{r,sp}.
The solid-line semicircle
has a radius equal to the incident radiance, I_{i}.
This limit is achieved for ideal-specular, 100%
reflection of the incident beam.

Figure 10: Interactive display of entire specular term