Specular reflection from a rough surface

Anyone who has made perception studies of gloss on a coated or printed surface realises that this is a complex task. When one looks at samples at different angles to the light, a range of different gloss phenomena will be visible: from very fine-scale gloss variations that may create a pleasant appearance to very inhomogeneous, disturbing gloss spots or speckles of different lateral sizes. Sometimes one realises that topography lies behind the disturbance. Orange peel and fibre rising are such examples. For an overview of paper gloss, see e.g. Oittinen ^1-3.

The specular reflection described by Fresnel’s equations is thus idealised, since it assumes a totally smooth and flat surface. This is not the case with paper. The impact of surface roughness on the reflection of light is the subject of extensive research. There is a lack of theories on how roughness influences the reflection. The equation of Beckmann and Spizzichino ⁴, see Equation 1, is however often used to describe how very fine-scale roughness reduces the specular reflectance. This equation is valid for micro roughness where the variation of height is less than the wavelength of light. If the height variation is smaller than 1/16 of the wavelength, the surface is said to be optically flat. The calculation using Equation 1 is shown in Figure 1. It shows that surface micro roughness has a larger damping effect on gloss at lower angles (closer to normal) than at higher angles (more flat viewing) ⁵.

(1)

where I is the intensity of reflected light,
I₀ the intensity of incident light,
θ_i the angle of incidence,
σ the standard deviation of the surface micro roughness, and
λ the wavelength of incident light.

Besides micro roughness, a paper also displays waviness or macro roughness, with variations that exceed the wavelength of light. Macro roughness makes specular reflection deviate in different direction whereas micro roughness has the damping effect (Equation 1) already discussed. Due to the heterogeneity of the surface, there is of course a vague limit between macro and micro roughness.

Figure 1. Damping of specular reflection caused by different surface roughness in μm according to Equation 1. Four different standard deviations (σ) in μm and wavelength of light 0.56 μm.

The topography is often approximated with facet maps (see Figure 2). The size of the facets is from about 7 μm to about 50 μm, depending on the resolution of the measuring method. The facets are approximated as a mirror ⁶ or assigned reflectance values ^7,8 or micro roughness values ⁹. Extensive and very fine-scaled measurements of topography of a printed surface have been performed. Confocal Laser Scanning Microscope (CLSM) was used to calculate the facets’ slope distribution and Atomic-Force Microscope (AFM) to measure the micro roughness of each facet. Both parameters helped to explain the observed gloss ⁹. When the slopes of a facet are steep and high, multiple surface scattering can be an essential part of the reflection from the surface ¹⁰. Extremely fine-scale roughness, much less than the wavelength of light, can give rise to Raleigh light scattering phenomena. Gloss and gloss variation can also be related to the refractive index and the void fraction of the surface of coated or printed papers ¹¹. Many of these phenomena are probably involved in the gloss of coated papers illustrated in Figure 3, which shows gloss measured for different incident angles of a range of coated papers.

Figure 2. In the facet model of paper, surface light is reflected in different directions depending on the tilt angle of the facets ¹².

Figure 3. Mean gloss level vs. angle of incidence for different papers. The gloss level 100% is defined as the gloss value measured for black glass at 70° angle of incidence ⁵.

Diffuse reflectance from a fibre structure

Paper is a complex structure consisting mainly of fibres, filler pigment particles and air. Light is reflected at the fibres and particle surfaces in the surface layer but also inside the paper structure. The light also penetrates the fibres and filler particles and changes direction. Some light is absorbed but the remainder goes into the air and is again reflected and refracted by other fibres and pigments. After a number of interactions, a certain proportion of the light reaches the upper surface, leaving the surface at different angles. We perceive this as a matt white surface and define it as diffuse surface reflection. Some of the light will find its way through the paper, coming out at the back-side as transmitted light ¹³. Thus, light scattering from a pigment-filled fibre structure comprises multiple refractions, multiple reflections and diffraction that occur in the fibre filler matrix. Figure 4 attempts to illustrate this complex set of interactions.

Figure 4. Schematic illustration of the different optical phenomena that occur when light strikes paper.

Optical homogeneity is also important in the case of diffuse reflections from the paper surface. Very limited research has been reported in this field, possibly because properties like formation are closely related to optical homogeneity and because we are less sensitive to variations of high reflectance. For a printed surface, the evenness is critical (e.g. mottling).