Thermal lensing

When a crystal, glass or semiconductor absorbs part of the light passing through it, the heat flows out through the surfaces, so the center runs hotter than the edges. The refractive index changes with temperature, by $dn/dT$, and the resulting radial index profile acts as a lens, converging for the positive $dn/dT$ of most laser crystals. Thermal expansion adds two more contributions: the stress changes the index through the photoelastic effect, and the end faces bulge outward. The combined effect is called a thermal lens. Its strength grows with the absorbed power, and because the temperature profile is only roughly parabolic, the lens is also aberrated.

In a side- or end-pumped laser rod the heat comes from the quantum defect and other non-radiative losses of optical pumping. For a uniformly heated rod of cross-section $A$ and thermal conductivity $K$, the index term alone gives a focal length $f = 2KA/(P_h\,dn/dT)$ for a deposited heat $P_h$. For Nd:YAG, with $K$ ≈ 14 W/(m·K) and $dn/dT$ ≈ $7.3 \times 10^{-6}$ K⁻¹, a rod of 2 mm radius carrying 50 W of heat has $f$ ≈ 0.96 m from that term; the stress and end-face terms shorten it further. The lens changes the laser cavity: a resonator designed for one pump power becomes unstable, or supports a different mode size, at another, and the aberrated part of the lens degrades beam quality and couples power into higher-order modes.

The response is to design for it or avoid it. Resonators are designed to stay stable across the expected range of thermal lens power, sometimes with compensating optics. Laser geometries with large surface-to-volume ratios weaken or sidestep the transverse gradient: thin-disk lasers cool a disk a few hundred micrometers thick through its face, putting the gradient along the beam; slab lasers zigzag the beam so that it averages across the gradient; and fiber lasers spread the heat over meters of length, which is the main reason they scale to high power with near-diffraction-limited output. Crystals with lower $dn/dT$ or higher conductivity, and pumping closer to the laser wavelength to shrink the quantum defect, reduce the heat and the lens at the source.

The effect is not limited to gain media. Broad-area semiconductor lasers form a thermal waveguide along the stripe as current rises, one cause of filamentation and of kinks in their light-current curves. Windows, lenses and beam splitters in high-power beam lines absorb a small fraction of the light and shift the focus as they warm, which matters in laser machining and in the optics of gravitational-wave detectors.

References: W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006), Ch. 7; A. E. Siegman, Lasers (University Science Books, 1986).