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Thermal lensing in pulsed laser amplifiers: an analytical model

INTRODUCTION
Solid-state based laser gain media have allowed the construction
of a broad range of laser systems. For instance,
the recent development of user-friendly amplified
ultrashort-pulse laser systems has been promoted by new
and improved laser crystals, and foremost, Ti:sapphire.
However, the thermal properties of solid-state amplifiers
introduce complications: The energy deposited to excite
the gain medium is partially converted into heat, and the
subsequent cooling of the medium with the associated
thermal gradients gives rise to a spatially dependent refractive
index. For example, for a cylindrically symmetric
gain medium cooled at the outer surface, a higher temperature
in the middle of the medium results, and since
the refractive index normally increases with increasing
temperature, the gain medium exhibits an index variation
that corresponds to a focusing lens.
Ever since the first solid-state amplifiers, the issue of
thermal lensing has had to be addressed. The field was
pioneered by Koechner (see Ref. 1 and references herein),
and early papers have discussed the thermal distributions
following pumping in various geometries.2,3 In the
first Nd:glass lasers, the thermal load limited the repetition
rate of high-energy systems to the sub-hertz region.
In recent years, as the Ti:sapphire-based systems have
been driven toward higher average output power, the issue
of thermal lensing has again become of high relevance.
A variety of solutions to the problem have
emerged. First, one can try to work around the difficulties
by using compensating optics as in the thermaleigenmode
amplifier4 or by using specific cavity designs
that minimize the sensitivity to the induced thermal
lensing.5 Second, the specific thermal properties of
Ti:sapphire crystals at low temperatures have been exploited
to devise an amplifier based on a cryogenically
cooled crystal, which is, at the presently available range
of laser parameters, essentially free of thermal lensing.6–۸
In this communication, a closed analytical expression
for the thermal-lensing properties of optical crystals at
variable pump repetition rate is presented (see Sections 2
and 3). The expression is exact under the assumption of
a uniform pump-laser profile. Due to the simplicity of
the expression, it is well suited for the evaluation of the
acceptable parameter range for present and future
crystal-based laser amplifiers (see discussion in Section
4).

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