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# How to Treat Thermal Lensing in Simulations

In many laser or amplifier devices, thermal lensing plays an important rule and should therefore be taken into account in numerical simulations. In this article, I first briefly describe the origins of thermal lensing and then show you how that effect can be treated in our software.

## What is Thermal Lensing?

When a laser gain medium (e.g., a laser crystal) is pumped, we typically generate some heat, which subsequently needs to go away via thermal conduction. As a consequence of that, we inevitably obtain some temperature gradients within the gain medium. Through various physical mechanisms, those can create some lensing effect for the laser light:

• The refractive index is temperature-dependent.
• The mechanical stress inside the crystal also modifies the refractive index (photoelastic effect).
• Further, the mechanical stress can lead to bulging of the end faces, giving the laser crystal the shape of a lens.

In typical cases, the first mentioned effect is often dominant. The figure below shows the numerically calculated temperature profile in a typical case.

## Thermal Lensing in Resonator Design

Our resonator design software RP Resonator calculates modes properties of laser resonators based on the ABCD matrix algorithm. (Precisely speaking, it uses some extended matrices (ABCDEF matrices) in order to treat misalignment effects as well, but that is not relevant in our context.) Here, only lensing effects with a parabolic shape, i.e., without spherical aberrations, can be treated. The software makes it easy to introduce a distributed lensing effect. A laser crystal, for example, is defined is a “prism”, and for that one can specify a parameter <$n_2$> which is the second-order coefficient of the radial dependence of the refractive index: <$n(r) = n_0 - 0.5 n_2 r^2$>. That parameter is simply the dioptric power of the thermal lens divided by the crystal length. The dioptric power may be known from elsewhere or may at least in simple cases be calculated from the dissipated power density with a simple formula. A common case is that offer a cylindrical rod which is homogeneously pumped at least within the volume of the laser beam.

In principle, one could also insert a thin lens with a certain dioptric power to the left or right side of the laser crystal, or to the middle of the laser crystal when splitting that into two sections. In many cases, the results will be similar to those for the distributed lens. However, there are differences in some cases, e.g. when the laser crystal extends over much of the total resonator length. And it just is easy to deal with the distributed lens.

In rare cases, it may be relevant that the thermal lensing effect is not uniform along the beam direction due to the decrease of pump intensity in an end-pumped laser. You would then have to split the laser crystal into multiple sections, each one having a different strength of lensing. With a little script code, containing a loop structure, one can simply automate that, so that you won't have to manually enter multiple crystal sections.

## Thermal Lensing in Fiber and Laser Simulations

More sophisticated thermal lensing models can be used in laser simulations, as can be done with our software RP Fiber Power.

### Fiber Modes Modified by Thermal Lensing

In optical fibers, thermal lensing effects are usually negligible. That is not the case, however, for operation at very high power levels. Here, it may be appropriate to include thermal lensing in the calculation of the guided modes of the fiber. That is no problem, since you can pass an arbitrary radial refractive index profiles to the mode solver.

The radial temperature profile itself can be calculated from a simple differential equation if the radial profile of the heat generation is known. (The script language of the software offers a convenient function for solving that differential equation.) In cases with strong lensing effects, the heat generation profile may actually itself depend on the mode properties; in such a case, one iteratively approximate a self-consistent solution for the heat profile and the mode properties.

### Thermal Lensing in Beam Propagation

Our software can also be used for the numerical simulation of beam propagation in fibers – or in fact in bulk laser crystals. Here, one specify an essentially arbitrary refractive index profile, which may of course be influenced by thermal lensing. Again, the temperature distribution may be calculated from the heat profile as explained above.

For example, one may simulate the situation of a cylindrical laser rod which is end-pumped with a Gaussian or super-Gaussian beam. Here, the thermal lens will have some aberrations. It is then simple to simulate the effect of a single pass of an Gaussian input beam on the beam profile and beam quality factor. Also, one can simulate multiple resonator round trips until the beam properties have more or less reached the steady state.

## Final Remarks

Such modeling can involve a couple of non-trivial aspects. It begins with the question what level of sophistication is appropriate in a certain case – such a judgment requires some experience. Further, you may have some questions concerning some technical aspects, e.g. how exactly to calculate the temperature profile. But don't worry – when using our simulation software, you will profit from my competent and helpful technical support!

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