When a ray of collimated light, such as from a laser, is incident upon a semitransparent medium, it can experience both absorption and scattering. This means that the incident light is both converted to thermal energy and redirected. Under certain assumptions, these phenomena can be modeled using a diffusive approximation in the COMSOL Multiphysics^{®} software. This modeling approach has applications in laser heating of living tissue as well as materials processing. Let’s learn more!

### Defining a Semitransparent Medium

A semitransparent medium is any material through which a ray of light can travel a significant distance before being extinguished due to a combination of absorbing and scattering. Absorption is the mechanism by which the light energy is converted into thermal energy, leading to a rise in temperature. Scattering is the mechanism by which the light is redirected into other directions. The scattering of light can take many forms: At one extreme is the specular reflection and refraction that occurs at the surfaces of mirrors and dielectrics, while at the other extreme, there can be nearly isotropic scattering, as is observed within a turbid medium such as very muddy water, where the turbidity is due to small suspended particles that are randomly shaped and oriented.

*A beam of collimated light incident upon a semitransparent medium can experience isotropic scattering, meaning that the light is redirected into all directions equally. This scattering occurs everywhere along the path of the beam, and the scattered light is itself immediately re-scattered, so this image presents a simplified view of the process.*

It should be noted that essentially all real materials exhibit some degree of anisotropic scattering, meaning that light is preferentially redirected into certain directions. However, in some applications, the scattering can be approximated as isotropic, and that is the case we will address here. We will consider a ray of collimated light, a laser beam, incident upon a material, where an isotropic scattering coefficient and isotropic absorption coefficient quantify the change in the light intensity.

### Developing the Modeling Method

To understand the modeling approach, we will begin by assuming that we have a material with no scattering, only absorption. This situation is possible to model using the Heat Transfer Module’s *Radiative Beam in Absorbing Media* interface, which solves for the Beer–Lambert law within the material. When using this interface, it is assumed that the beam intensity is known at the illuminated boundary. That is, considering a beam of light of known power propagating through surrounding free space, the specified intensity is based on the fraction of the light that propagates into the material.

The equation being solved by this interface is:

where \mathbf{e}_i is the vector describing the direction of the beam and I_i is the intensity of the light as power per unit area, measured in the plane perpendicular to the beam path. There can be several different spatially overlapping incident beams, and one equation, indexed by i, is solved for each. The term \kappa is the absorption coefficient, which quantifies how these beams are absorbed. The absorbed energy is the sum from all incident beams: Q_r = \kappa \sum_i I_i. The assumption of this interface is that all absorbed light energy is converted to heat energy, but we can easily modify the interface settings to also account for scattering.

A nonzero scattering coefficient, \sigma_s, can be added to the absorption coefficient used within the *Radiative Beam in Absorbing Media* interface, so we can write \kappa_{tot} = \kappa + \sigma_s. The absorbed energy can now be decomposed in the absorbed fraction, \kappa/\kappa_{tot}, and the scattered fraction, \sigma_s/\kappa_{tot} .

We now need to compute how the scattered fraction of this light propagates through the medium, keeping in mind that it will be both absorbed and re-scattered everywhere. This is where we turn to the Heat Transfer Module’s *Radiation in Absorbing-Scattering Media* interface, which offers the P1 approximation that solves the equation:

where G is the radiant intensity of light per steradian, meaning that it accounts for light going in all directions, not just a single direction. The conversion of light to thermal energy is quantified by the term -\kappa G on the right-hand side, which leads to a decrease in radiant intensity. The source term, Q, leads to a volumetric increase in radiant intensity, and, in this situation, comes from the scattered fraction of the losses computed from the *Radiative Beam in Absorbing Media* interface; so, Q = \frac{\sigma_s}{\kappa_{tot}}Q_r.

Along with the governing equation, we also need a set of boundary conditions for the material when solving for the scattered light. Given that the incident laser light can enter the domain, it is also reasonable to assume that the scattered light can leave the modeling domain. The *Semitransparent Surface* feature is appropriate for this situation, and lets us enter an emissivity, \epsilon, and a diffuse transmissivity, \tau_d. These two quantities must be less than or equal to one and define a diffuse reflectivity \rho_d = 1-\epsilon – \tau_d. Scattered light incident upon this boundary will entirely pass through if \tau_d = 1, and, if \tau_d < 1, then the light will be partially diffusely reflected back into the domain.

### Implementation Details

To implement such a model within COMSOL Multiphysics^{®}, we can use a combination of the *Radiative Beam in Absorbing Media* interface and the *Radiation in Absorbing-Scattering Media* interface. The former interface only needs be solved for in a subdomain surrounding the path of the incident beam. Within the *Radiative Beam in Absorbing Media* interface, the absorption coefficient needs to be modified to include both the scattering and the absorption coefficients. When evaluating the results, it is thus important to reduce the absorbed heat by the absorbed fraction.

*Accounting for both absorption and scattering of collimated light via the* Absorption coefficient *of the* Radiative Beam in Absorbing Media *interface.*

The *Radiation in Absorbing-Scattering Media* interface allows us to 1) add the absorption and scattering coefficient separately and 2) add a source term using the *Radiative Source* feature, which provides the scattered fraction of the absorbed heat from the *Radiative Beam in Absorbing Media* interface.

*Coupling the scattering light from the* Radiative Beam in Absorbing Media *interface to the* Radiation in Absorbing-Scattering Media *interface.*

In terms of evaluating the results, it can be particularly insightful to evaluate the integral of the thermal losses of the incident beam, the thermal losses of the scattered light, and the fraction of the incident beam and scattered light that leaves the modeling domain. The plots and table below show the distribution of these losses as well as the integrals. The distribution of losses can subsequently be used within a heat transfer analysis to compute the variation in temperature.

*Distribution of the heat sources arising from the incident beam, left, and the scattered light, right. The sum of these sources contributes to a rise in temperature.*

Incident Beam, Absorbed Power | 0.49 W |

Scattered Light, Absorbed Power | 0.35 W |

Scattered Light, Outgoing Power | 0.14 W |

Incident Beam, Outgoing Power | 0.02 W |

Sum | 1.00 W |

*Table of integrals of heat and radiative losses. The sum of these should equal the power in the incident beam.*

### Caveats and Closing Remarks

As we have seen, it is possible to implement a model of absorption and scattering of light quite easily, but it is worth emphasizing that this method has two limitations. First, any specular reflection or refraction of light within the material, such as due to a mirror or lens, is not addressable, so only a reasonably homogeneous piece of material can be modeled. Next, the scattering within the medium is assumed to be isotropic. These limitations are counterweighted by the advantage of computational simplicity: Solving two sets of scalar equations for the collimated and scattered light intensity has very low computational cost. Furthermore, the source terms are easily combined with a thermal analysis to compute the rise in temperature. So, if you are modeling laser light interacting with a reasonably uniform sample of a semitransparent material and can assume isotropic scattering, then this approach is attractive because of its efficiency.

### Next Step

Try the model featured throughout this blog post yourself by clicking the button below, which will take you to the Application Gallery:

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