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Thermal Lensing

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Definition: a lensing effect induced by temperature gradients

German: thermische Linseneffekte

Categories: lasers, physical foundations

How to cite the article; suggest additional literature

Particularly in high-power lasers, the heating of the gain medium (e.g. a laser crystal) often causes a significant thermal lens through the following mechanisms:

Depending on the situation, these effects can have different relative strength and even sign. In many cases, the first mentioned effect (temperature dependence of the refractive index) is the most important.

temperature profile

Figure 1: Temperature profile in an end-pumped laser cylindrical crystal, seen from the pumped end. The highest temperatures occur around the beam axis. Due to cooling of the outside faces, there is a heat flow and thus inevitably a temperature gradient in the radial direction.

Challenges Resulting from Thermal Lensing

In principle, even a strong thermal lens could be compensated by adjustments of the design of the used laser resonator – for example, with a convex (defocusing) laser mirror next to the laser crystal. Nevertheless, thermal lensing can introduce various challenges to laser design:

Calculation of the Dioptric Power

In order to calculate the dioptric power of the thermal lens, one first needs to calculate the dissipated power. This is some percentage of the absorbed pump power. How large that percentage is, depends on the circumstances. In simple cases like an Yb-doped fiber laser, where the laser-active ions can hardly do anything else than absorption and emission on the pump and laser transition, and spontaneous emission normally has a minor effect on the heating, one can relatively easily calculate the heating based on the quantum defect. In other cases, where additional quenching and upconversion processes may occur, for example, the calculation is more difficult, and a substantial uncertainty may remain. Even in case of fiber amplifiers, photodarkening may substantially add to the heating, depending on the conditions.

Further, the thermal lensing effect depends substantially on the pumping and cooling geometry. As a simple example, consider an isotropic laser gain medium in which there is a uniform pump intensity and a purely radial heat flow. (This situation can occur e.g. in a side-pumped laser rod.) We then have a thermal lens with the dioptric power (inverse focal length)

dioptric power of thermal lens

where Pheat is the dissipated power, A is the pumped area, and κ is the thermal conductivity. The length of the gain medium does not matter, and the thermal lens has no aberrations in that case.

Calculator for Thermal Lensing

Thermo-optic coefficient dn/dT:
Thermal conductivity κ:
Beam radius:
Heat power:
Focal length of thermal lens: calc

Enter input values with units, where appropriate. After you have modified some inputs, click the "calc" button to recalculate the output.

The equation seems to suggest that operating with a larger pumped area and a corresponding larger resonator mode would reduce lensing effects, but a large mode is more sensitive to thermal lensing, which exactly compensates the apparent advantage. The equation also shows that a useful figure of merit for high-power gain media is the ratio of (dn / dT) / κ, which should be as small as possible. Of course, the fraction of the pump power which is converted to heat should also be as small as possible.

pump intensity and thermal profile in a laser crystal

Figure 2: Transverse pump intensity distribution (red) and thermal profile (blue), simulated for an 8 mm long end-pumped Nd:YAG rod with 14 W of dissipated power. The distribution in the beam direction is assumed to be homogeneous, which is approximately true for bidirectional pumping and not too strong pump absorption. The temperature profile is approximately parabolic only near the center of the crystal, so that a laser mode with a beam radius equal to that of the pump beam will experience some aberrations.

For a non-homogeneous distribution of dissipated power, the thermal lens will in general exhibit aberrations (see Figure 1). This is frequently a problem in end-pumped lasers. For example, when pumping with a Gaussian intensity profile, the thermal lens on the beam axis is two times stronger than calculated from the equation above based on the effective mode area, but falls off at larger radii.

Many laser crystals exhibit a positive value of dn / dT of the order of a few times 10−6 K−1, but some (e.g. Nd:YLF) have a negative temperature coefficient, leading to a defocusing lens if that effect is not overcompensated by stress effects. Also, the Raman crystal material Ba(NO3)2 (barium nitride) has a strongly negative dn / dT combined with a low thermal conductivity, leading to strong defocusing e.g. in intracavity Raman-shifted lasers.

Reliable data for calculating the strength of thermal lensing are often difficult to obtain, even for relatively well-known crystal materials. Mainly due to various technical difficulties related to the measurement of such properties, wrong data often occur in the literature. Further, there is a more fundamental problem related to the quantity dn / dT: in various theoretical calculations, the partial derivative of refractive index under conditions of constant strain should be used, whereas measured data usually apply to constant stress [20]. The difference between those values, which can be substantial, arises from the fact that there is also thermal expansion. It can therefore be more sensible to use an effective thermo-optic coefficient derived from measurements of thermal lensing. Even then, it is not easy to separate the effects of bulging of the end faces, the importance of which depends on the crystal geometry.

An in-depth analysis of thermal lensing often requires advanced methods such as finite element analysis (FEA) for the calculation of the temperature and stress fields in the laser crystal. This can be helpful e.g. when designing composite laser crystals (see below), where it is important to know how the effect of additional undoped parts depends on their size. In many cases, however, such calculations are plagued by a lack of data, e.g. concerning the precise pump intensity distribution, the thermal properties of media and mechanical contacts, quenching effects in the gain medium, etc., so that a high precision of the results is often difficult to achieve. Therefore, it is often advantageous to use the more pragmatic approach of working with estimates based on various simplifying assumptions, which greatly reduce the effort required and still offer useful guidance for experiments. Theoretical results may be supplemented with experimental data in order to calibrate certain parameters.

Influence of Geometry

The dependence of the dioptric power of the thermal lens on the beam area can be different for other laser geometries, such as in a thin-disk laser. An at least partially longitudinal heat flow and thus a reduction in thermal lensing may also be achieved with a transparent heat spreader, made of diamond, for example.

Such influences of geometry are very important in the context of power scaling of lasers, and their understanding can be more helpful than e.g. a sophisticated calculation of temperature profiles for any particular configuration.

Coping with Thermal Lensing in Lasers

In principle, a thermal lens can be taken into account in the design of the laser resonator. Difficulties arise from the fact that the dioptric power of the thermal lens depends on the pump power, and that the thermal lens usually has significant aberrations. This can make it difficult to achieve good beam quality. However, a good resonator design can greatly help to reduce the impact of thermal lensing. The resonator design should be made so that changes of the thermal lens do not affect too much the mode sizes in the laser resonator, and that aberrations of the thermal lens have only a small effect. It is also possible to compensate aberrations with additional optical elements in the laser resonator.

The strength of thermal lensing can be reduced in various ways:

Note that thermal lensing may occur not only in the laser gain medium, but also in other optical components, e.g. in Q switches and other optical modulators, Raman-active crystals, and even on dielectric mirrors through bulging of the substrate. If lensing effects occur and different locations in a resonator, their effects do not necessarily add up, depending on how the lenses are distributed.

A phenomenon related to thermal lensing is thermally induced birefringence, which can lead to depolarization loss and to polarization-dependent astigmatism.

Thermal Lensing Outside of a Laser

Thermal lensing can also be significant outside a laser resonator when a laser beam with high average power is sent through a transparent optical device such as a lens, a beam splitter or a protection window. As a result, the position of a focus behind such an element may depend on the optical power level. This can happen e.g. in high power laser cutting heads.

Passive Compensation of Thermal Lensing

It is possible to construct optical elements which exhibit thermal lensing with reversed sign, i.e., a defocusing action which increases with the optical power. That can be achieved e.g. with a fluid layer exhibiting a negative value of dn / dT. Such elements can be used for compensating a focusing thermal lens e.g. within a laser resonator or in some high-power focusing optics.

Thermal Lensing in Fibers

Optical fibers are largely immune to thermal lensing. This is because thermal lensing is usually too weak to substantially modify the properties of the fiber modes. In extreme situations, however, with a very high thermal load per meter of fiber and also a very large mode area, thermal lensing can significantly modify the effective mode area and even bring a fiber from the single-mode regime to a multimode regime [23].

Bibliography

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 [2]T. Kimura and K. Otsuka, “Thermal effects of a continuously pumped Nd3+:YAG laser”, IEEE J. Quantum Electron. 7 (8), 403 (1971)
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[4]B. Bendow and P. D. Gianino, “Optics of thermal lensing in solids”, Appl. Opt. 12 (4), 710 (1973)
[5]V. Magni, “Multielement stable resonators containing a variable lens”, J. Opt. Soc. Am. A 4 (10), 1962 (1987)
[6]D. Metcalf et al., “Laser resonators containing self-focusing elements”, Appl. Opt. 26 (21), 4508 (1987)
[7]S. De Silvestri et al., “Rod thermal lensing effects in solid-state laser ring resonators”, Opt. Commun. 65 (5), 373 (1988)
[8]M. E. Innocenzi et al., “Thermal modeling of continuous-wave end-pumped solid-state lasers”, Appl. Phys. Lett. 56, 1831 (1990)
[9]W. Winkler et al., “Heating by optical absorption and the performance of interferometric gravitational-wave detectors”, Phys. Rev. A 44 (11), 7022 (1991)
[10]H. J. Eichler et al., “Thermal lensing and depolarization in a highly pumped Nd:YAG laser amplifier”, J. Phys. D: Appl. Phys. 26, 1884 (1993)
[11]L. Yan and C. H. Lee, “Thermal effects in end-pumped Nd:phosphate glasses”, J. Appl. Phys. 75 (3), 1286 (1994)
[12]J. L. Blows et al., “Heat generation in Nd:YVO4 with and without laser action”, IEEE Photon. Technol. Lett. 10 (12), 1727 (1998)
[13]N. Hodgson et al., “High power TEM00 mode operation of diode-pumped solid-state lasers”, Proc. SPIE 3611, 119 (1999)
[14]P. J. Hardman et al., “Energy-transfer upconversion and thermal lensing in high-power end-pumped Nd:YLF laser crystals”, IEEE J. Quantum Electron. 35 (4), 647 (1999)
[15]R. Paschotta et al., “Thermal effects in high-power end-pumped lasers with elliptical-mode geometry”, IEEE J. Sel. Top. Quantum Electron. 6 (4), 636 (2000)
[16]E. Wyss et al., “Thermooptical compensation methods for high-power lasers”, IEEE J. Quantum Electron. 38 (12), 1620 (2002)
[17]S. Chénais et al., “Thermal lensing in diode-pumped ytterbium lasers – Part I: theoretical analysis and wavefront measurements”, IEEE J. Quantum Electron. 40 (9), 1217 (2004)
[18]R. L. Aggarwal et al., “Measurement of thermo-optic properties of Y3Al5O12, Lu3Al5O12, YAlO3, LiYF4, LiLuF4, BaY2F8, KGd(WO4)2, and KY(WO4)2 laser crystals in the 80–300 K temperature range”, J. Appl. Phys. 98 (10), 103514 (2005)
[19]G. Wagner et al., “Simulations of thermal lensing of a Ti:sapphire crystal end-pumped with high average power”, Opt. Express 13 (20), 8045 (2005)
[20]S. Chénais et al., “On thermal effects in solid-state lasers: the case of ytterbium-doped materials”, Prog. Quantum Electron. 30 (12), 89 (2006)
[21]A. Soloviev et al., “Experimental study of thermal lens features in laser ceramics”, Opt. Express 16 (25), 21012 (2008)
[22]P. A. Loiko et al., “Thermo-optic coefficients and thermal lensing in Nd-doped KGd(WO4)2 laser crystals”, Appl. Opt. 49 (34), 6651 (2010)
[23]K. R. Hansen et al., “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers”, Opt. Express 19 (24), 23965 (2011)
[24]I. I. Kuznetsov et al., “Thermal effects in end-pumped Yb:YAG thin-disk and Yb:YAG/YAG composite active element”, IEEE J. Quantum Electron. 50 (3), 133 (2014)
[25]W. Koechner, Solid-State Laser Engineering, 6th edn., Springer, Berlin (2006)

(Suggest additional literature!)

See also: lasers, gain media, lenses, beam quality, depolarization loss, composite laser crystals, thin-disk lasers, cryogenic lasers, power scaling of lasers, Spotlight article 2006-07-10, Spotlight article 2007-01-15
and other articles in the categories lasers, physical foundations

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