# Principles of optics for paper surfaces

Principles of light, its nature and interaction with surfaces and material is explained briefly in this article.

## Light, refractive index and Fresnel’s equations

Light can be described either as a small energy particle (photon) or as an electromagnetic wave. The photon concept explains how light interacts with electron structures so that colours or fluorescence arise. The concept of magnetic waves helps us understand the most basic mechanism in paper optics. Electromagnetic waves can be of different kinds, from short wave, high-frequency X-ray radiation to long wave, slowly-oscillating radio waves. The visible colour spectrum stretches from the blue region of 380 nm to red light of 730 nm.

The velocity of light is slightly lower inside a material than in a vacuum. The ratio of velocity in material and a vacuum defines the refractive index. According to Huygen’s principle, the velocity decrease leads to a change in direction of the propagation of light, so the light is refracted. Figure 1 illustrates the refracted and reflected light in the interface between a material and a vacuum. The refraction of light is described by Snell’s law according to equation (1):

(1)

where θ_{i} is the angle of incident,

θ_{r} the angle of refraction,

*n _{0}* the refractive index of vacuum, and

*n*the refractive index of the material.

*Figure 1. Light refracted and reflected in the interface between a material of refractive index n and vacuum (refractive index n _{0}) according to Snell’s law.*

The refractive index also determines the reflection of light from surfaces. ^{1} According to Fresnel’s equations, the electromagnetic waves are divided into two plane polarised components *R _{s}* and

*R*, of which

_{p}*R*oscillates perpendicular to the surface and

_{s}*Rp*parallel to the surface.

(2)

(3)

where θ is the angle of incidence,

*n _{0}* the refractive index in vacuum, and

*n*the refractive index of the material.

For non-polarised light, the reflectance R is given by the mean of the two plane polarised components.

(4)

Perpendicular to the surface (zero degrees incident angle), about 4% of the light is reflected. When the angle of incidence increases, the reflectance rises for the s-polarised part. For the second part (*R _{p}*), the reflectance decreases in the beginning, goes to zero at the so-called Brewster angle, and then increases rapidly at high angles. For cellulose and glass, both of which have a refractive index of 1.5, the Brewster angle is about 57 degrees. This behaviour is shown in Fig 2.

Natural light which has been specularly reflected in e.g. water will thus be partly s-polarised. By filtering away this component with a polarisation filter, disturbing gloss reflections can be strongly reduced (polarised sunglasses). The polarised light depolarises when it penetrates a material like paper where it is exposed to multiple refractions and reflections.

*Figure 2. Specular reflection vs. the angle of incidence of the plane polarised components R _{s} and R_{p} and depolarised light R according to Equations 2, 3 and 4.*

The different properties of the two polarised components can be utilised to separate surface reflection from bulk reflection ^{2,3}. In the graphical industry, polarisation filters are used to reduce the influence of specular reflections when measuring the print density. Figure 3 illustrates how the two plane polarised components can be extracted with a polarisation filter. The theory of polarised light (Equations 1–3) is also applied to determine the refractive index of the surface of coatings ^{4}, using Equations 5 and 6.

(5)

where

(6)

Fresnel’s equations are only valid for dielectric or non-conductive media. Metallic materials have high light absorption and for these the theories of complex refractive indexes must be applied. The high light absorption of inks means that the theories of complex refractive indexes need to be considered. One example of this is determining the refractive index of ink films based on ellipsometry ^{5}.

*Figure 3. With a polarisation filter the two plane polarised components R _{s} and R_{p} can be extracted.*