Modeling Laser-Material Interactions with the Beer-Lambert Law
High-intensity lasers incident upon a material that is partially transparent will deposit power into the material itself. If the absorption of the incident light can be described by the Beer-Lambert law, it is possible to model this power deposition using the core functionality of COMSOL Multiphysics. We will demonstrate how to model the absorption of the laser light and the resultant heating for a material with temperature-dependent absorptivity.
The Beer-Lambert Law and Material Heating
When light is incident upon a semitransparent material, some of the energy will be absorbed by the material itself. If we can assume that the light is single wavelength, collimated (such as from a laser), and experiences very minimal refraction, reflection, or scattering within the material itself, then it is appropriate to model the light intensity via the Beer-Lambert law. This law can be written in differential form for the light intensity I as:
where z is the coordinate along the beam direction and \alpha(T) is the temperature-dependent absorption coefficient of the material. Because this temperature can vary in space and time, we must also solve the governing partial differential equation for temperature distribution within the material:
where the heat source term, Q, equals the absorbed light. These two equations present a bidirectionally coupled multiphysics problem that is well suited for modeling within the core architecture of COMSOL Multiphysics. Let’s find out how…
Implementation in COMSOL Multiphysics
We will consider the problem shown above, which depicts a solid cylinder of material (20 mm in diameter and 25 mm in length) with a laser incident on the top. To reduce the model size, we will exploit symmetry and consider only one quarter of the entire cylinder. We will also partition the domain up into two volumes. These volumes will represent the same material, but we will only solve the Beer-Lambert law on the inside domain — the only region that the beam is heating up.
To implement the Beer-Lambert law, we will begin by adding the General Form PDE interface with the Dependent Variables and Units settings, as shown in the figure below.
Settings for the implementing the Beer-Lambert law. Note the Units settings.
Next, the equation itself is implemented via the General Form PDE interface, as illustrated in the following screenshot. Aside from the source term, f, all terms within the equation are set to zero; thus, the equation being solved is f=0. The source term is set to Iz-(50[1/m]*(1+(T-300[K])/40[K]))*I, where the partial derivative of light intensity with respect to the z-direction is Iz, and the absorption coefficient is (50[1/m]*(1+(T-300[K])/40[K])), which introduces a temperature dependency for illustrative purposes. This one line implements the Beer-Lambert law for a material with a temperature-dependent absorption coefficient, assuming that we will also solve for the temperature field, T, in our model.
Implementation of the Beer-Lambert law with the General Form PDE interface.
Since this equation is linear and stationary, the Initial Values do not affect the solution for the intensity variable. The Zero Flux boundary condition is the natural boundary condition and does not impose a constraint or loading term. It is appropriate on most faces, with the exception of the illuminated face. We will assume that the incident laser light intensity follows a Gaussian distribution with respect to distance from the origin. At the origin, and immediately above the material, the incident intensity is 3 W/mm2. Some of the laser light will be reflected at the dielectric interface, so the intensity of light at the surface of the material is reduced to 0.95 of the incident intensity. This condition is implemented with a Dirichlet Boundary Condition. At the face opposite to the incident face, the default Zero Flux boundary condition can be physically interpreted as meaning that any light reaching that boundary will leave the domain.
The Dirichlet Boundary Condition sets the incident light intensity within the material.
With these settings described above, the problem of temperature-dependent light absorption governed by the Beer-Lambert law has been implemented. It is, of course, also necessary to solve for the temperature variation in the material over time. We will consider an arbitrary material with a thermal conductivity of 2 W/m/K, a density of 2000 kg/m3, and a specific heat of 1000 J/kg/K that is initially at 300 K with a volumetric heat source.
The heat source itself is simply the absorption coefficient times the intensity, or equivalently, the derivative of the intensity with respect to the propagation direction, which can be entered as shown below.
The heat source term is the absorbed light.
Most other boundaries can be left at the default Thermal Insulation, which will also be appropriate for implementing the symmetry of the temperature field. However, at the illuminated boundary, the temperature will rise significantly and radiative heat loss can occur. This can be modeled with the Diffuse Surface boundary condition, which takes the ambient temperature of the surroundings and the surface emissivity as inputs.
Thermal radiation from the top face to the surroundings is modeled with the Diffuse Surface boundary condition.
It is worth noting that using the Diffuse Surface boundary condition implies that the object radiates as a gray body. However, the gray body assumption would imply that this material is opaque. So how can we reconcile this with the fact that we are using the Beer-Lambert law, which is appropriate for semitransparent materials?
We can resolve this apparent discrepancy by noting that the material absorptivity is highly wavelength-dependent. At the wavelength of incident laser light that we are considering in this example, the penetration depth is large. However, when the part heats up, it will re-radiate primarily in the long-infrared regime. At long-infrared wavelengths, we can assume that the penetration depth is very small, and thus the assumption that the material bulk is opaque for emitted radiation is valid.
It is possible to solve this model either for the steady-state solution or for the transient response. The figure below shows the temperature and light intensity in the material over time, as well as the finite element mesh that is used. Although it is not necessary to use a swept mesh in the absorption direction, applying this feature provides a smooth solution for the light intensity with relatively fewer elements than a tetrahedral mesh. The plot of light intensity and temperature with respect to depth at the centerline illustrates the effect of the varying absorption coefficient due to the rise in temperature.
Plot of the mesh (on the far left) and the light intensity and temperature at different times.
Light intensity and temperature as a function of depth along the centerline over time.
Summary and Further Refinements
Here, we have highlighted how the General Form PDE interface, available in the core COMSOL Multiphysics package, can be used for implementing the Beer-Lambert law to model the heating of a semitransparent medium with temperature-dependent absorptivity. This approach is appropriate if the incident light is collimated and at a wavelength where the material is semitransparent.
Although this approach has been presented in the context of laser heating, the incident light needs only to be collimated for this approach to be valid. The light does not need to be coherent nor single wavelength. A wide spectrum source can be broken down into a sum of several wavelength bands over which the material absorption coefficient is roughly constant, with each solved using a separate General Form PDE interface.
In the approach presented here, the material itself is assumed to be completely opaque to ambient thermal radiation. It is, however, possible to model thermal re-radiation within the material using the Radiation in Participating Media physics interface available within the Heat Transfer Module.
The Beer-Lambert law does assume that the incident laser light is perfectly collimated and propagates in a single direction. If you are instead modeling a focused laser beam with gradual variations in the intensity along the optical path then the Beam Envelopes interface in the Wave Optics Module is more appropriate.
In future blog posts, we will introduce these as well as alternate approaches for modeling laser-material interactions. Stay tuned!
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