The Interaction of Light with Materials

When light strikes the boundary of two materials, a portion is reflected, transmitted, or absorbed. While reflection is the process in which electromagnetic radiation is returned at a boundary, transmittance describes the passage of electromagnetic radiation through a medium as the ratio of transmitted radiant power to incident radiant power. Absorptance is the fraction of incident light that is absorbed by a material. 

The quantities are defined as ratios and are dimensionless. The proportion of each interaction depends on the material’s physical and optical properties, surface roughness, and spectral characteristics. 

Absorptance [math]\alpha[/math] is defined as, where [math]P_i[/math] denotes the optical powers with direction as indices:  

[math]\alpha=\frac{P_{absorbed}}{P_{incident}}[/math] 

Additionally, the reflectance [math]\rho[/math]  is defined as:  

[math]\rho=\frac{P_{reflected}}{P_{incident}}[/math] 

Transmittance [math]\tau[/math]  is defined as: 

[math]\tau=\frac{P_{transmitted}}{P_{incident}}[/math] 

Due to the energy conversation of light, the sum of reflectance, transmittance, and absorptance equals one. Usually, a little of all three occurs: 

[math]\alpha +\rho \ +\ \tau \ \ =\ 1[/math] 

A physical explanation is that all photons must be accounted for when interacting with a material. Each photon, the fundamental unit of light, can follow one of three possible paths. It may be absorbed, converting its energy into heat and increasing the object’s temperature—this commonly occurs in dielectric materials and semiconductors. Alternatively, the photon may pass through the material, which happens when its energy is insufficient to excite an electron from the valence band to the conduction band, as described by the bandgap model in semiconductors. The third possibility is reflection or scattering, a behavior typical of metals. Since a light beam consists of many photons, all these interactions occur simultaneously, and the optical coefficient represents the statistical distribution of these processes. 

The refractive index n and the extinction coefficient k define a material’s optical characteristics. The refractive index defines how much light bends when entering a material by comparing its speed in a vacuum or air to its speed in the medium. In contrast, the extinction coefficient quantifies light loss due to scattering and absorption per unit volume within the material. High extinction coefficient values at shorter wavelengths indicate that these materials are opaque in that range, absorbing rather than transmitting light. If the material is absorbing [math]k\ne 0[/math], the polarization-dependent formulas must be modified to include the complex refractive index [math]\tilde{n}= n+ik[/math]. 

Some principle optical laws must be applied to calculate the optical properties. The law of reflection and Snell’s law describe the relationship between the angle of incident θi and the angle of the transmitting ray for reflection and transmission.  

[math]\phi _{in}=\phi _{out}[/math]

In the case of reflectance, the angle of incidence is equal to the angle of reflection. The variable n1 describes the refractive index of the first medium, and the refractive index n2 marks the second interacting medium. 

[math]n_{1}\sin\left(\phi_{in}\right)=n_{2}\sin\left(\phi_{out}\right)[/math]

Moreover, reflectance, transmittance, and absorptance depend on the wavelength of the affected radiation. Reflectance and transmittance can be subdivided into regular and diffuse processes. When combined, both processes equal the overall amount of reflection or transmission. 

Furthermore, every material has polarization properties when used other than at the incident angle of zero degrees. Therefore, the reflectance and transmittance coefficients also depend on the polarization state of the incoming light and the geometric distribution of the incident radiation.  

In the following, the refractive indices [math]n_1[/math] and [math]n_2[/math] are for the two media. The corresponding propagation angles [math]\phi _{in}[/math] and [math]\phi _{out}[/math]. Snell’s law must be applied to transmission as light distortion might force a deviation from a straight trajectory. Fresnel coefficients exist for the incident wave’s two different linear polarization components. Polarization directions S and P are defined by their relation to the plane of incidence. The P-polarized light is parallel, and the S-polarized light is perpendicular to the reflecting surface. Any polarization state can be resolved by combining these orthogonal linear polarizations. Figure 1 illustrates this concept schematically.

Considering polarization effects, Fresnel’s law of reflection expands to the terms with subindex that are the respective polarization directions. The respective power coefficient for the reflectance must be obtained from the reflection coefficient, which calculates their square magnitude. In the following, the complex refractive index of the second material must be considered, which is [math]\tilde{n_{2}}= n_{2}+ik[/math]. 

[math]\rho_{s}=\mid r_{s}\mid^{2}=\mid\frac{n_{1}\cos\left(\phi_{in}\right)-\tilde{n_{2}}\cos\left(\phi_{out}\right)}{n_{1}\cos\left(\phi_{in}\right)+\tilde{n_{2}}\cos\left(\phi_{out}\right)}\mid^{2}[/math]

[math]\rho_{p}=\mid r_{p}\mid^{2}=\mid\frac{\tilde{n_{2}}\cos\left(\phi_{in}\right)-{n_{1}}\cos\left(\phi_{out}\right)}{\tilde{n_{2}}\cos\left(\phi_{in}\right)+{n_{1}}\cos\left(\phi_{out}\right)}\mid^{2} [/math]

Although these relationships provide fundamental physics insights, most practical applications emphasize “natural light,” which is defined as unpolarized. This signifies an equal distribution of power in the s and p polarizations, meaning that the effective reflectivity of the material is essentially the average of the two reflectivities.
[math]\rho =\frac{1}{2}\left(\rho _S\ +\ \rho _P\right)[/math] 

In the case of normal incidence, there is no distinction between s and p polarization. Thus, the reflectance simplifies to the following equation. 

[math]\rho\left(\phi_{in}=0°\right)=\frac{\left(n_{1}-n_{2}\right)^{2}+k^{2}}{\left(n_{1}+n_{2}\right)^{2}+k^{2}}[/math] 

The next set of equations describes Fresnel’s law of transmission coefficients, including polarization effects. The calculation of the power transmission coefficient is less straightforward since the light travels in different directions in the two media.  

[math]\tau_{s}=\frac{n_{2}\cos\left(\phi_{in}\right)}{n_{1}\cos\left(\phi_{out}\right)}\mid t_{s}\mid^{2}=\frac{n_{2}\cos\left(\phi_{in}\right)}{n_{1}\cos\left(\phi_{out}\right)}\mid\frac{2n_{1}\cos\left(\phi_{in}\right)}{{{n_{1}}\cos\left(\phi_{in}\right)+\tilde{n_{2}}\cos\left(\phi_{out}\right)}}\mid^{2}[/math] 

[math]\tau_{p}=\frac{n_{2}\cos\left(\phi_{in}\right)}{n_{1}\cos\left(\phi_{out}\right)}\mid t_{p}\mid^{2}=\frac{n_{2}\cos\left(\phi_{in}\right)}{n_{1}\cos\left(\phi_{out}\right)}\mid\frac{2n_{1}\cos\left(\phi_{in}\right)}{{{\tilde{n_{2}}}\cos\left(\phi_{in}\right)+{n_{1}}\cos\left(\phi_{out}\right)}}\mid^{2}[/math] 

Like effective reflectance, the effective transmittance can be assumed by taking the mean of both polarization directions. 

Under normal incident, there is no deflection of the beam, and the equation is simplified to:   

[math]\tau\left(\phi_{in}=0°\right)=\frac{4n_{1}n_{2}}{\left(n_{1}+n_{2}\right)^{2}+k^{2}}[/math]

Increasing the incident angle modifies the polarization states of both refracted and reflected light. At Brewster’s angle, unpolarized incident light becomes linearly S-polarized upon reflection. A negative reflection coefficient indicates light reflection from an optically thinner medium to a denser one. 

Fresnel equations can also be applied to absorbing media, such as metals, by incorporating a complex refractive index. In this case, the imaginary component represents absorbance. Consequently, the reflection and transmission coefficients become more complex than real values. Calculating power transmission for absorbing media at non-normal incidence is particularly challenging, as the required correction factor for varying propagation angles is inherently complex. In contrast, the power transmission factor itself must remain an actual quantity. 

In practical applications, the Fresnel equations may not fully capture the optical behavior of an interface if the surface quality is inadequate. Even minor surface irregularities, particularly those on the scale of the incident wavelength, can introduce significant wavefront distortions. Therefore, in summary, the optical properties of materials are not constant since they depend on many parameters such as surface conditions, angle of incidence, polarization effects, and temperature. 

When infrared radiation strikes a material surface, it penetrates the material and is either transmitted or absorbed within its bulk. Inside the medium, the infrared signal progressively weakens with depth due to absorption by the medium.  

According to Kirchhoff’s law, the depth from which infrared emission originates is governed by the material’s optical attenuation characteristics. This relationship is crucial for engineers and researchers to consider when analysing and interpreting infrared measurements. 

An attenuation coefficient [math]\mu[/math] quantifies how quickly the intensity decays per unit length. The assumption is that the attenuation coefficient corresponds to the material emission. The attenuation coefficient is directly related to the material’s optical properties, linking the material’s extinction coefficient [math]k[/math], which is the imaginary part of its refractive index, to the wavelength λ of the infrared light. For short propagation distances z, where absorption remains minimal, the absorbed power decreases approximately linearly with length. Over longer propagation distances, however, transmittance follows an exponential decay. 

[math]\mu =4\pi \frac{k}{\lambda }[/math] 

The absorption coefficient in the following equation defines the absorbance for the length [math]z[/math]. 

[math]\alpha =1-\ e^{-\mu z}[/math] 

The intensity of the infrared light describes the exponential decay of intensity with depth. 

[math]I=I_0\cdot \ e^{-\mu z}[/math] 

According to the Beer–Lambert law, the intensity decays exponentially such that the penetration depth (on the order of 1/μ) is the characteristic length scale over which the intensity drops by a factor of e. For instance, one absorption length (1/μ) reduces the intensity I to ~37% of the initial intensity.  

A useful rule-of-thumb for penetration depth. In practice, for sufficiently short path lengths μz ≪ 1 (such as very thin films), absorption remains small and the absorbed power increases approximately linearly with depth. However, beyond this initial regime, transmittance follows an exponential decay – meaning each additional equal thickness absorbs an equal fraction of the remaining light, not an equal amount. This transition from near linear to exponential behavior with increasing thickness is well-documented in the optics literature.  

Figure 2 illustrates the modelled absorptance vs. depth behavior in a semitransparent material such as plastic across multiple infrared wavelength bands. Each curve shows the fraction of incident infrared power absorbed between the surface and a given depth z in the material. At the surface (z = 0), absorptance is 0% (no depth, no absorption). As z increases, the absorbed fraction approaches 100%, meaning that deeper layers cumulatively absorb essentially all the incoming infrared radiation. 

The Beer–Lambert exponential law governs the shape of each curve. For shorter wavelengths, μ is larger, so the curve rises sharply – nearly all the infrared is absorbed within only a few micrometers of depth. In contrast, for longer,  μ is smaller and the absorptance increases more gradually with depth, reaching close to 100% only after tens of micrometers. In other words, short-wavelength infrared light is absorbed in a much shallower region of materials, whereas long-wavelength IR penetrates further before being fully absorbed.  

Figure 3 presents the complementary perspective to Figure 1 by showing how the local infrared intensity (or equivalently, the contribution to emission) decays with depth for the same material and wavelengths. Here, the intensity [math]I(z)[\math] is normalized to 100% at the surface [math](z = 0)[\math] and plotted as a function of depth. As expected, all curves start at 100% and decrease in a monotonic manner. The short-wavelength IR intensity drops off extremely rapidly. In contrast, the long-wave IR band retains a significant fraction of intensity even tens of micrometers deep.  

According to Kirchhoff’s law of thermal radiation, a material’s emissivity equals its absorptance (for a given wavelength and temperature) in thermal equilibrium. Thus, the high emissivity here is consistent with the plastic’s strong absorption of infrared light; conversely, its reflectance is low. This point highlights that materials that absorb IR well are also good IR emitters.