**2. Analysis of laser-material interactions using Beer–Lambert law**

When a light beam incident upon a semitransparent material, part of the energy supplied by it will be absorbed by the material. This process can be described through Beer–Lambert law. If we consider monochromatic and collimated light,

*Modern Spectroscopic Techniques and Applications*

within the range of 400–650°C [7].

work for the benefit of laser-material processing.

what happens to the irradiated material [9].

log(

In 2010 Ozgowiez et al. [6] analyzed the influence of recrystallization annealing temperature on the microstructure and mechanical properties of Cu 30% Zn brass, which was subjected to cold deformation with a variable tension in the rolling process. The mechanical test showed that the properties of the brass deteriorated and the properties of the plastic increased as the recrystallization temperature increased

In 1967 Bailey studied the structure and strength of an alpha brass with 20%w Zn, 6%w Ni, and 1.5%w Al. Brass samples were subjected to solution heat treatment (SHT) at 800°C for 2 h followed by quenching with water. Subsequently, the samples were thermally treated for 2 h at 300, 400, 500, 600, and 700°C. The best

Infrared photothermal radiometry (PTR) is an optical technique used for the characterization of metallic materials such as brass, to determine the influence of the precipitation heat treatment (PHT) temperature on metallurgical microstructure, thermal properties, and microhardness. This technique is based in light absorption and consists of a laser beam incident on a sample, which will absorb the radiation and emit a thermal wave that is detected by an infrared sensor. The signals of thermal wave are translated as amplitude and phase parameters. Compared with other characterization techniques, this technique has no destructive properties and has the advantage of not having direct contact with the sample to be analyzed.

One of the most important aspects to consider in the PTR is the process of light absorption in the material to be analyzed. As trivial as this may sound, absorption very often turns out to be the most critical and cumbersome step in laser processing. An enormous amount of work has been dedicated to investigating laser absorption mechanisms under various circumstances, and a great deal can be learned from this

Absorption process can be thought of as secondary "source" of energy inside the material. While driven by the incident beam, it tends to develop its own dynamics and can behave in ways deviating from the laws or ordinary optics. It is this "secondary" source, rather than the beam emitted by the laser device, which determines

Certain wavelengths of light can be selectively absorbed by a substance or a material according to its molecular structure. The absorption of light occurs when an incident photon promotes the transition of an electron from a state of lower to higher energy. Excited electrons eventually lose this gained energy, and by a

The radiation emitted by a molecule, or an atom, after it has absorbed energy to place itself in an excited state, is defined as luminescence, depending on the nature of the excited state. All spectrophotometric methods are based on two laws that combined which are known as Beer–Lambert law. This law states that the light absorbed by a semitransparent medium is independent of the intensity of incident light, and each successive layer of the medium absorbs an equal fraction of the light passing through it. This amount of light can be calculated by Eq. (1), where *I*0 is the incident light intensity, *I* is the transmitted light intensity, l is the length through which the light passes in the spectrophotometer cell, and k is the constant of the

> \_\_ *I*0

In the absorption spectrometry, the comparison of incident light intensity before and after the interaction with a sample can be carried out nowadays, by means of different software. In the development of theoretical models, it is possible to include

*<sup>I</sup>* ) <sup>=</sup> *kl* (1)

spontaneous radiation process, they return to their initial state.

mechanical properties were obtained at 500°C PHT temperature [8].

**58**

medium:

such as laser light, and we despise losses by reflection and dispersion in the material, considering a minimum refraction and scattering, the Beer–Lambert law can be written in differential form for the light intensity **I** as

$$\frac{\partial \mathbf{I}}{\partial \mathbf{z}} = \mathbf{a}(T)\mathbf{I} \tag{2}$$

where z is the coordinate along the beam direction and α(*T*) is the temperaturedependent absorption coefficient of the material. Because the temperature can vary in space and time, we must also solve the next governing partial differential equation for temperature distribution within the material:

$$
\rho \, \mathbf{C}\_p \frac{\partial T}{\partial t} - \nabla \cdot \{\mathbf{k} \, \nabla \, \mathbf{T}\} = \mathbf{Q} = \mathbf{a} \, \mathbf{C} \mathbf{T} \mathbf{J} \tag{3}
$$

where Q is the heat source term equals the absorbed light. Eqs. (2) and (3) present a bidirectionally coupled multiphysics problem that is possible to solve within the core architecture of COMSOL Multiphysics program based in the finite element method [10].

#### **2.1 Beer–Lambert law deviations**

Beer–Lambert law states that the optical density is directly proportional to the concentration of the species they absorb. However, deviations from this law can occur due to instrumental and intrinsic causes [11–13]. Among the main deviations from Beer–Lambert law are the following:

#### *2.1.1 Light scattering*

There are two dispersion phenomena, one depends on the size of the solute particle or any suspended material. Biological samples are usually cloudy because macromolecules or other large aggregates scatter light. The optical densities resulting from the scattering of light are proportional to 1/λ<sup>4</sup> (Rayleigh scattering) and can therefore be recognized as an absorption background which increases rapidly with the decrease in wavelength [14]. The second type of dispersion is known as the Raman scattering. In this phenomenon, part of the excitation energy of light is abstracted by vibrational modes of the solvent molecules. In the case of water or hydroxyl solvents, the most dominant vibrations that absorb this energy are the OH groups, whose vibration energy is observed at a wavelength of 3300 cm<sup>−</sup><sup>1</sup> . The Raman signal of the solvents will be observed at a wavelength that is 3300 cm<sup>−</sup><sup>1</sup> less in energy than the excitation wavelength. The wavelength of the Raman scattering (λRA) can be calculated as λ RA−<sup>1</sup> = λex−<sup>1</sup> –0.00033 [15].

#### *2.1.2 Fluorescence*

If the optical density of the sample is high and if the absorbing species are fluorescent, the emitted light can reach the detector. This process will result in derivations of Beer**–**Lambert law. The effect can be minimized by maintaining the distance of the detector from the sample and decreasing the efficiency with which the emission fluorescence is collected.
