Modeling Laser-Material Interactions with the Beer-Lambert Law

Walter Frei April 13, 2015
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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:

\frac{\partial I }{\partial z} = \alpha(T) I

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:

\rho C_p \frac{\partial T }{\partial t}-\nabla \cdot (k \nabla T)= Q = \alpha(T) I

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…

A model of laser material interactions using the Beer-Lambert law.

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.

Screenshot showing how to implement the Beer-Lambert law by setting dependent variables and units.
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.

Image showing how to set the General Form PDE interface for a simulation.
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.

Screenshot of setting the Dirichlet boundary condition for incident light intensity in a simulation.
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.

How to set the heat source term as absorbed light in COMSOL Multiphysics.
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.

Image showing how to model thermal radiation by setting a diffuse surface boundary condition.
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.

Schematic of mesh and light intensity in a Beer-Lambert simulation.
Plot of the mesh (on the far left) and the light intensity and temperature at different times.

Plotting the light intensity and temperature as a function of depth.
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!



  1. emma nkant August 17, 2015   2:50 am

    Merci beaucoup.
    Could I use this for 2 layers of material ? or I need special boundaries condition at the interface.
    Thank you for your article.

  2. Walter Frei August 17, 2015   9:02 am

    Dear Emma,
    Yes, you can use this approach for two layers of materials, simply add another “General Form PDE” interface and enter a different absorption coefficient in the different layers.
    Best Regards,

  3. Indika Perera September 11, 2015   11:26 am
  4. Natan Padoin October 5, 2015   3:12 pm

    Is there a way to extend this approach to model beer-lambert law in x, y and z directions simultaneously?

  5. Walter Frei October 5, 2015   4:09 pm

    Dear Natan,
    If you have multiple incident beams, then you can add multiple General Form interfaces for different beam directions. If you are more generally trying to model reflection and scattering, then we suggest that you contact your COMSOL Support Team for a personalized discussion about your modeling needs.

  6. Natan Padoin October 5, 2015   4:55 pm

    Thank you for your response. It does not work adding multiple General Form interfaces since a single domain is modeled. However, I will try another strategies.

  7. Natan Padoin November 11, 2015   10:16 am

    Dear Dr. Walter,
    Just a remark: We got a solution to our question. Thank you!
    Best regards,

  8. nabil belghachem January 7, 2016   4:11 am

    Dear All,

    I am a beginner on COMSOL multiphysics, and concerning this tutorial I have some question , first of all I think in the Beer law is Iz=-alpha(T)* I which will give Iz+alpha(T)*I not minus; the second question is Iz is always negative so why Q=Iz and not Q=-Iz

    finnally my last question why you used two different domains inside where the beam heats up and outside. I have tried it using only one domaine (the bigger cylinder) and the results were terrible

    thank you for your fast reply
    Best Regards,

  9. Walter Frei January 7, 2016   8:23 am

    Dear Nabil, Yes, you are correct. One term that was assumed here that is that the beam is propagating in the negative z-direction.
    Regarding your model: You would want to contact your COMSOL Support Team about this.

  10. nabil belghachem January 8, 2016   3:56 am

    Dear Dr Walter,

    thank you very much for your fast reply. Now I understand, this make sens .

    Best Regards,

  11. Caterina Gaudiuso March 4, 2016   6:05 am

    Dear all,
    I did not begin to use Comsol yet because I don’t know if this could be the right software to solve my problem. I would need to simulate the ultrashort laser pulses interaction with semicondutcor. To do it, I have two coupled equations for electron and lattice temperature evolution, a generalized Lambert-Beer law and an equation describing the evolution od electron density due to absorption processes. My problem is that the parameters in these equations depend on the electron and lattice temperature and on electron density (which are also dynamic variables). Do you suggest to use Comsol for this kind of problem and how?
    Thank you.
    Best regards.


  12. Walter Frei March 4, 2016   8:34 am

    Dear Caterina,
    What you’re describing certainly sounds like an interesting and doable problem. You may want to search on the usage of the two-temperature model in COMSOL.
    If you’re looking to engage other COMSOL users, we would encourage you to post on the discussion forum:
    And if you’d like to contact a COMSOL representative about purchasing a license for this type of work:

  13. Walter Frei March 14, 2016   8:31 am

    Dear Caterina,
    Briefly speaking, yes, what you are describing does all sound like a reasonable (albeit possibly quite complex) problem to solve. We would of course suggest that you investigate using COMSOL for this type of problem, yes.

  14. Peter Woerner May 24, 2016   3:55 pm

    Walter how did you make those plots side by side. I was able to replicate your model easily enough, but not the post processing. Can you point me in the right direction?

  15. Walter Frei May 25, 2016   9:13 am

    Hello Peter, You can add a “Deformation” subfeature to any plot and specify that the plot be moved by a fixed quantity to pattern several different cases in one image. You may also find these resources helpful:

  16. Bastian Geißler September 7, 2016   11:29 am


    thanks for the tutorial, this was very helpful. I have implemented now the Beer-Lambert-law, but now I would like to move the laser beam along a certain distance with a certain speed. How can I implement that in the existing model?

    Thanks in advance,

  17. Walter Frei September 7, 2016   11:33 am

    Hello Bastian,

    Yes, you can move the location of the laser around on the surface of the part. Within the Dirichelet boundary condition simply make the expression for incident intensity a function of position and time. For an example model that shows you how to set up such time and space-dependent expressions, please see:

    Best Regards,

  18. Bastian Geißler October 6, 2016   7:02 am

    Hello Walter,

    thanks for your help, I was able to implement the moving laser. Now I have another question:

    I have two layers of material, the upper one has a low absorption coefficient, the lower one has a high absorption coefficient. The two parts have an ideal thermal contact. So basically a small amount (about 10-20%) of the laser energy is absorbed in the upper part (which has a thickness of 2 mm), the rest reaches the lower part and is absorbed in the first 100 microns of the material. Both material interactions are with beer-lambert law.

    Is it possible to do this with one heat source?
    At the moment I implemented simply two heat sources at the top of each layer and two dirichlet boundary conditions with the different absorptions coefficients. It would be nice to implement it in a way that if the heat reaches the lower material the beer-lambert law is calculated with the new absorption coefficient.

    Thanks in advance!
    Best regards,

  19. Walter Frei October 10, 2016   10:31 am

    Hello Bastian,
    What you could do, in the situation that you’re describing, is assume that the incoming light in incoherent (probably fair to do if your first layer is 2mm, and if we assume an effective wavelength of around 1um) Under those assumptions, you could implement a bi-directional Beer-Lambert law approach, solving for both forward- and backward-propagating light.

  20. Bastian Geißler November 3, 2016   8:05 am

    Dear all,
    by the way is the equation for the gaussian intensity distribution correct? Because in literature I found the following equation: I_Gauss = I_max*exp((-2*(x^2+y^2))/r_g^2), where I_max is the maximum intensity at r = 0, and r_g is the radius of the gaussian beam. I_max is then defined as I_max = (2*P)/(pi*r_g^2), where P is the power of the laser.
    Thansk in advance

  21. SEN WANG November 17, 2016   11:48 pm

    Hi Walter,

    I am a beginner with comsol multiphonics. I am trying to model sunlight absorption in side a 50 micro layer of material using beer Lambert’s law. Is it appropriate to use beer Lambert’s law in my case?

    My second question is about the heat source. Is it always Iz, the differential form? regardless of having the general form PDE or not? I have created a 2 D model , I don’t have the general form PDE, can the software still solve the differential form equation?


  22. Ciprian Plostinar February 6, 2017   5:19 pm

    Thank you for the blog! The model appears listed in the Application Gallery but cannot be downloaded due to an html error which redirects to the current web page. Please, could you make it availbale there? Cheers!

  23. Walter Frei February 7, 2017   9:07 am

    Hello Ciprian,
    This should now be addressed.
    Best Regards,

  24. Ciprian Plostinar February 8, 2017   6:05 pm

    Hello again Walter, and thank you for making the model available. It is not clear for me yet where does the 40[K] come from in the expression of the absroption coefficient. Would you clarify it for us, please? Many thanks! Ciprian

  25. Walter Frei March 1, 2017   12:03 pm

    Hello Ciprian,
    The 40[K] term is merely there to demonstrate a temperature dependence of the absorption coefficient. It is arbitrary, and meant just for demonstration purposes.

  26. George Aloyan March 20, 2017   5:58 am

    I have a question. Downloaded the model from a given link, but when I look at heat source equation, I only can choose Q=Q0, but not the one that in on the screenshot. What don’t I understand?
    Thanks in advance,
    George Aloyan

  27. Caty Fairclough March 20, 2017   8:45 am

    Hi George,

    Thanks for your comment! For questions related to your modeling, please contact our support team.

    Online support center:


  28. George Aloyan March 20, 2017   4:57 pm

    Thank you for response.
    And question about this model – why Iz is used, but not I?
    Thanks in advance,
    George Aloyan

  29. Caty Fairclough March 22, 2017   10:44 am

    Hi George,

    Thank you for your additional comment. When you contact our support team, they can help address this question as well.


  30. Saurabh Singh May 3, 2017   11:18 am

    This model is provided in the application gallery but the version is 5.2a. Can you please provide it in older version?

  31. Caty Fairclough May 15, 2017   4:46 pm

    Hi Saurabh,

    Thanks for your comment.

    For your question, please contact our Support team.

    Online Support Center:

  32. Changmu Han August 1, 2018   9:43 pm

    Hello Walter,

    Thank you for the good example of laser-matter interaction with Beer-Lambert law.
    I have two questions. Can a pulsed laser be applied in this example? If yes, could you please elaborate it? Also, I have the same question as Bastian has but I don’t understand what you mean by bi-directional Beer-Lambert law. Could you please also illustrate it? Thanks.

  33. Walter Frei August 2, 2018   8:56 am

    Hello Changmu,
    Yes, of course, you can pulse any source. See, for example:
    Also, please note that the Heat Transfer Module ( now includes a “Radiative Beam in Absorbing Media” interface that will simplify this type of modeling.
    Best Regards,

  34. Sushil Kumar September 15, 2018   2:58 pm

    Hi Walter,

    Thank you for the post ! I’ve some doubts please try to help me.
    1. How one can used this model for multi-layer semiconductor (say 3 layer) without
    considering symmetry in structure ?
    2. In “General Form PDE 1” and “Dirichlet Boundary Condition” node,
    “Source Term: Iz-(50[1/m ]* (1+(T-300[K])/40[K]))*I (W/m^2)” and
    “Intensity Term: 0.95*3[W/mm^2]*exp(-(x^2+y^2)/(1[mm])^2)”, how and at what basis
    they define like this ?

  35. benchikh hanane May 19, 2019   7:49 am

    I am a biginner on comsol, i want to help me for calculate of transmitted , reflected and absorbed fraction of incident solar radiation on

  36. benchikh hanane May 19, 2019   7:51 am

    i am a biginner on comsol , i want to help me to caculate of transmitted , reflected and bsorbed fraction of incident solar radiation on a flat glass,
    Best Regards,

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