How to Treat Thermal Lensing in Simulations

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.

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 n₂ which is the second-order coefficient of the radial dependence of the refractive index: n(r) = n₀ − 0.5 n₂ r². 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.

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

Thermal Lensing in Beam Propagation

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!