# A study of enhancing the power of a CW Ti:sapphire laser by cryogenic
cooling

Simai Jia  Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1
3PU, UK

(December 2022)

###### Contents

  1. 1 Introduction
    1. 1.1 The need for high laser power
    2. 1.2 How to achieve 100 W of red laser radiation?
    3. 1.3 Cryogenic cooling of titanium-doped sapphire
  2. 2 Transverse Profile of Temperature
    1. 2.1 The problem
    2. 2.2 Circular Gaussian beam source
    3. 2.3 Beam expansion through refraction
    4. 2.4 Elliptical Gaussian beam source
  3. 3 Thermal Lens
    1. 3.1 Formula under azimuthal symmetry
    2. 3.2 Astigmatism
    3. 3.3 Enhancement of maximum power through cryogenic cooling
  4. 4 Stability Condition of the Laser Cavity
    1. 4.1 Transfer matrix and stability condition
    2. 4.2 Stability condition
    3. 4.3 Astigmatism compensation
    4. 4.4 Optimal parameters
    5. 4.5 Stability against thermal lens
  5. 5 Conclusion
  6. A Relationship of thermal conductivity of sapphire to temperature
  7. B Detailed Derivations
    1. B.1 Second order derivatives of temperature at the origin
    2. B.2 Stability condition of the laser cavity
    3. B.3 Pump power needed for 100 W output
  8. C Early work on cooled Ti:sapphire lasers in the 1990s
  9. D Related industrial companies and project
    1. D.1 Kapteyn-Murane Labs (KMLabs)
    2. D.2 Advanced Photonic Sciences (APS)
    3. D.3 FMB Oxford cryocooler
    4. D.4 Diode Pumped Optical Laser for Experiments (DiPOLE) laser project
  10. E Pump lasers for Ti:sapphire laser
    1. E.1 Existing diode-pumped solid-state (DPSS) green lasers
    2. E.2 Experimental demonstration of 130 W at 532 nm
    3. E.3 Doubled fiber lasers
  11. F ‘Cost benefit’ analysis

##  1 Introduction

### 1.1 The need for high laser power

The Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) and Atom
Interferometer Observatory and Network (AION) projects will use single-photon
transitions for atom interferometry to enable long baselines [1, 2], in a way
that is not feasible with existing two-photon techniques. The single-photon
approach requires using a clock transition from the ground state to a long-
lived excited level so that negligible spontaneous emission occurs while atoms
reside in the upper level for a short but finite time between up and down
momentum transfer pulses. This is a real population inversion whereas the
intermediate level in a Raman transition, or Bragg diffraction, is virtual.

Clock transitions are very weak, ‘forbidden’, and therefore require a strong
laser beam to give a high Rabi frequency, e.g. a few Watts with a cm-scale
laser beam waist in the first instance, and powers of 100 Watts are desirable
to realize the full advantage of this approach, as explained in this section.
This beam waist is necessary to give flat wave fronts over a long distance in
large-scale atom interferometers. Comparable power is used nowadays to obtain
good performance in the atom interferometry experiments with Rb atoms at
Stanford — a large frequency detuning is used for the Raman transitions to
reduce spontaneous emission during the process.

As well as wanting to have short pulses so that many thousands can be fitted
into the limited time of flight (e.g. $T_{\text{TOF}}=2$ s and a splitting-
deflection-recombination sequence of 4000 pulses (1:2:1) gives a maximum pulse
length of 500 $\mu$s), there is a less obvious reason for wanting a high Rabi
frequency, namely to minimize the inefficiencies in the $\pi$-pulses arising
from non-zero frequency detunings [1]. Frequency shifts of the laser radiation
relative to the transition arise from laser frequency noise or Doppler
broadening caused by a velocity spread in the atom cloud. The extreme
narrowness of the clock transition means that even small frequency detunings
affect the efficiency of $\pi$-pulses and require the laser frequency to be
stabilized to “within 10 Hz of the clock transition resonance using the
optical frequency comb” [1]. Shorter $\pi$-pulses mitigate the effect of laser
frequency shifts by increasing the Fourier-transform limited bandwidth, and it
is beneficial to have sufficient power to reach Rabi frequencies at this level
in order to minimize $\pi$-pulse inefficiencies arising from nonzero
detunings.

In the paper detailing the MAGIS proposal [1], it is assumed that large
momentum transfer (LMT) atom interferometry based on the 698 nm transition of
strontium requires 8 W of continuous wave (CW) power at this wavelength to
give a Rabi frequency of several kHz on the clock transition with a cm-scale
laser beam waist. In contrast, optical-lattice clocks use a long interrogation
time to obtain a signal that is very sensitive to the frequency difference
between the radiation and the transition. Since the Rabi frequency is
proportional to the electric field strength of the electromagnetic radiation
which scales as the square root of the intensity, in round numbers, we assume
100 W as a goal for the laser power that we want to achieve for MAGIS-100 or
AION in the future.

### 1.2 How to achieve 100 W of red laser radiation?

As outlined in the previous section, atom interferometry on the 698 nm clock
can be implemented using around 8 W of radiation with a bandwidth of 10 Hz but
increasing this to 100 W is a goal for the future.

A power of 8 W at 698 nm can be produced by a system of two titanium-doped
sapphire (Ti:sapphire or Ti:Sa) systems operated as an amplifier which is
injection-locked by a narrow-bandwidth laser (as reported by M Squared Lasers
at the first AION workshop, see Ref. [3]). In this pairing both the laser and
amplifier are pumped by separate 18 W green lasers, but a narrow bandwidth
laser can be used to injection-lock more than one amplifier, e.g. 12
amplifiers of 8 W each giving a total of 96 W could be seeded by 2 or 3
lasers, making a total of around 15 systems and pump lasers.

In principle, the output of these systems can be combined by coherent addition
on a beam splitter as illustrated in Figure 1; the coherent addition of laser
light is used to generate light for the Rb Raman experiments at Stanford (from
a few lasers) [4]. Constructive interference at the beam splitter requires
phase-locking of the two incoming beams which can be achieved by sophisticated
but standard electronics, and this ensures that the output beam maintains the
same narrow bandwidth of the incident light. Scaling to high power using 15,
or more, commercial units (Ti:Sa plus pump laser) has the advantage of being a
‘straightforward’ extension of known techniques, not requiring extensive
development, but it is worthwhile to consider alternatives (especially given
the reliability issues that have already been encountered by both MAGIS-100
and AION, with pump lasers).

Figure 1: Simplified schematic to produce high power laser by coherent
addition. PLL : phase-locked loop. The master laser and amplifiers each have a
pump laser and an optical cavity with Ti:Sa gain medium. The injection-locked
amplifiers do not need the intracavity elements (etalons etc.) necessary for a
single-frequency laser oscillator, and therefore have lower round-trip loss
and higher output power (by a factor of 2 typically). Such a system might be
expected to give a power of 48 W, extrapolating from currently achieved
results.

### 1.3 Cryogenic cooling of titanium-doped sapphire

Titanium-doped sapphire has several properties that make it a very good gain
medium for laser amplification. It provides gain over a broad spectral region,
making it very suitable for ultrafast laser systems (femtosecond pulses in the
time domain correspond to a wide range of frequencies); although we use only a
small part of that range, in the red, this means that very high-quality
crystals are available as are used in extremely high-power laser facilities
where the intensities far exceed those in CW systems, so the damage threshold
is irrelevant. Rather the limitation to the power in both pulsed and CW
applications comes from thermal effects — removing the heat dumped into the
gain medium by the pump laser beam.

Ti:Sa is a four-level laser scheme as illustrated in Figure 2 and for each
stimulated photon emitted at 698 nm the difference in energy between it and
the pump (at 532 nm) goes into vibrational energy of the crystal lattice (an
intrinsic inefficiency of 24%, or a ‘quantum efficiency’ of 76%). In practice,
the pump beam is not completely absorbed on passing through the laser crystal
and will not perfectly overlap with the laser mode but the observed efficiency
of 44% is high (8 W for 18 W input — this is the minimum specification of the
pump laser and the output may be about 20 W initially). Typically, the output
power of a laser is linearly proportional to the pump power above the lasing
threshold but this linear growth rolls off when thermal lensing starts to
affect the optical cavity.

This project explores how cryogenic cooling of the laser crystal greatly
reduces thermal lensing (by over an order of magnitude) because of two
factors: (i) the increase in the thermal conductivity of the crystal (as the
phonon density decreases — the impurities from the titanium doping are not
significant), and (ii) the decrease in the temperature dependence of the
refractive index. The advantage of cooling is used in state-of-the-art pulsed
laser systems (further details in Appendix D.4). The gain curve also changes
with temperature, becoming slightly less broad, but this is not a significant
effect at 698 nm [5]. Thus, it might be possible to obtain 100 W at 698 nm
from a single amplifier with a suitable pump laser. The aim of this report is
to investigate the physics that may be a factor in choosing between having
$12\times 8$ W, $3\times 30$ W, or $1\times 100$ W etc. Other factors will be
commercial availability of components, cost and reliability (see Appendices D,
E and F for further discussion).

Figure 2: Simplified energy level structure and laser transitions of the Ti^3+
ion in Al2O3 (Ti:sapphire).

##  2 Transverse Profile of Temperature

### 2.1 The problem

Heated by the pump laser at the center and cooled at the surroundings, there
will be a temperature distribution in the Ti:sapphire crystal. At thermal
equilibrium, the heat dumped by the pump laser is dissipated by the heat flow
caused by the temperature gradient inside the crystal. The temperature field
follows the equation

$-\kappa\nabla^{2}T=q(\mathbf{r})$ (1)

where $\kappa$ is the thermal conductivity (assumed to be a constant inside
the crystal) and $q(\mathbf{r})$ is the heating power per unit volume at
position $\mathbf{r}$.

We assume that the boundary of the crystal is at the same temperature as the
environment, either at room temperature (RT) or at some cryogenic temperature.
As such, the boundary associated to Eq. (1) is that

$T=T_{0}\qquad\text{at }r=R$ (2)

where $T_{0}$ is the environment temperature and $R$ is the radius of the
laser crystal.

In the following subsections, we neglect boundary effects at the front and
rear polished surfaces, assume that the pump is uniform along the laser
crystal and solve for the temperature distribution. Under such circumstances,
the problem has translational symmetry along the central axis. We take into
account two scenarios: (i) the pump beam inside the crystal is circular; (ii)
the pump beam inside the crystal is elliptical. While analytical solution is
obtainable for the first case, the latter can only be evaluated numerically.

### 2.2 Circular Gaussian beam source

With the pump beam inside the laser crystal being a circular Gaussian beam, we
also have rotational symmetry about the central axis.

The heating power per unit volume $q(\mathbf{r})$ is a certain fraction of the
energy density of the pump beam of which the transverse profile is

$\displaystyle I(r)=I_{0}\exp\left(-\frac{2r^{2}}{w_{0}^{2}}\right)$

where $w_{0}$ is the $1/e^{2}$-beam radius. Therefore, we can write

$\displaystyle q(r)=q_{0}\exp\left(-\frac{r^{2}}{w_{1}^{2}}\right).$ (3)

where $w_{1}^{2}=w_{0}^{2}/2$.

We solve for the temperature field under cylindrical coordinates of which the
$z-$axis coincide with the central axis of the crystal. Because of the
translational symmetry along $z$ and rotational symmetry about $z$, the
temperature should be a function of the radial coordinate $r$ only. So we have
that

$-\kappa\,\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}\left(r\,\frac{\mathrm{d}T}{\mathrm{d}r}\right)=q(r)\
.$ (4)

Integrating this equation once, we get

$r\,\frac{\mathrm{d}T}{\mathrm{d}r}=-\frac{Q_{0}}{2\pi\kappa}\left[1-\exp\left(-\frac{r^{2}}{w_{1}^{2}}\right)\right]$
(5)

where $Q_{0}=\pi q_{0}w_{1}^{2}$ is the total heat dumped by the pump laser
per unit time per unit length. Solving for $T(r)$ with the boundary condition
$T(r=R)=T_{0}$, the solution can be written as

$T(r)=T_{0}+T_{p}\,g(r)$ (6)

where

$T_{p}=\frac{Q_{0}}{2\pi\kappa}=\frac{q_{0}w_{1}^{2}}{2\kappa}$ (7)

is a quantity with dimensions of temperature that characterizes heating by the
pump laser and

$g(r)=\ln\frac{R}{r}+\frac{1}{2}\left[\operatorname{Ei}\left(-\frac{r^{2}}{w_{1}^{2}}\right)-\operatorname{Ei}\left(-\frac{R^{2}}{w_{1}^{2}}\right)\right]$
(8)

is a dimensionless function that describes the radial distribution of excess
temperature, with $\operatorname{Ei}(x)$ being the exponential integral
defined as

$\operatorname{Ei}(x)=\int_{-\infty}^{x}\frac{e^{t}}{t}\,\mathrm{d}t\ .$

From this, we can calculate the temperature difference between the center and
the boundary of the crystal which is given by

$\Delta
T=T_{p}\,(g(0)-g(R))=T_{p}\left(\ln\frac{R}{w_{1}}+\frac{1}{2}\,\gamma\right)$
(9)

where $\gamma\approx 0.577$ is the Euler’s constant. For a typical laser
crystal with $R=4\,\text{mm}$, $L=20\,\text{mm}$ and
$\kappa=35\,\text{W}\text{m}^{-1}\text{K}^{-1}$ at RT, absorbing heat of
$20\,\text{W}$ and assuming $R/w_{1}=10$, we have $\Delta T=12\,\text{K}$.

To visualize the solution we get, in Figure 3, the function $g(r)$ is plotted
for different values of $w_{1}$. As $w_{1}$ increases (i.e. the pump beam
expands while remaining at the same power), the temperature near the center of
the crystal decreases but remains at the same level towards the boundary.

In Figure 4, we compare the temperature profiles under different cooling
conditions and different width of the pump beam. The crystal is supposed to be
either in room temperature environment ($T_{0}=300\,\text{K}$) or cooled
cryogenically (here we take $T_{0}=100\,\text{K}$). The thermal conductivity
vs. temperature relation of sapphire crystal in Appendix A shows that

$\kappa(T=100\,\text{K})\approx 10\,\kappa(T=300\,\text{K})\ .$

Therefore

$T_{p}(T_{0}=100\,\text{K})\approx\frac{1}{10}\,T_{p}(T_{0}=300\,\text{K})\ .$

Cooling of the crystal to cryogenic temperatures will dramatically reduce the
temperature unevenness and thus elongate the thermal lens, as can be seen by
comparing the orange (RT) and blue (cryogenic) curves. On the other hand, if
the pump beam is expanded while its total power is fixed, then the parameter
$T_{p}$ will not change but the temperature unevenness is still mitigated.
Quantitative results for the thermal lens are presented in the next section.

Figure 3: The dimensionless function $g(r)$ plotted for different $w_{1}$.

(a) $w_{1}=0.1\ R$

(b) $w_{1}=0.2\ R$

Figure 4: Radial temperature profiles under different cooling conditions and
different pump beam widths. $T_{0}=300\,\text{K}$ for normal room temperature
condition (upper) and $T_{0}=100\,\text{K}$ for cryocooled scenario (lower).

### 2.3 Beam expansion through refraction

In the previous subsection, we assumed that the pump beam is circular inside
the laser crystal. If, however, the pump beam is circular before entering the
laser crystal, then it will become elliptical because of refraction.

As shown in Figure 5, in the plane where the laser beam inside the cavity
circulates (which we call the horizontal plane), the width of the pump beam is
projected twice upon entering the laser crystal. In comparison, its width in
the direction perpendicular to this plane remains unchanged. As the beam is
incident on the crystal at Brewster’s angle $\theta_{\text{B}}=\arctan n$, if
we still record the beam width in air as $w_{1}$, then the beam widths in the
$x$ (vertical) and $y$ (horizontal) directions are given by

$\displaystyle w_{x}$ $\displaystyle=w_{1}$ (10) $\displaystyle w_{y}$
$\displaystyle=w_{1}\tan\theta_{\text{B}}=nw_{1}\ .$ (11)

Figure 5: Refraction of the pump laser beam upon entering the laser crystal.
The incident angle is Brewster’s angle $\theta_{\text{B}}=\arctan
n=60.4^{\circ}$ and the refraction angle is
$\gamma_{\text{B}}=\arctan(1/n)=29.6^{\circ}$.

The intensity profile of an elliptical Gaussian beam with widths $w_{x}$ and
$w_{y}$ is given by

$I(x,y)=I_{0}\exp\left[-\left(\frac{x^{2}}{w_{x}^{2}}+\frac{y^{2}}{w_{y}^{2}}\right)\right]$

and thus the heating power per unit volume

$q(x,y)=q_{0}\exp\left[-\left(\frac{x^{2}}{w_{x}^{2}}+\frac{y^{2}}{w_{y}^{2}}\right)\right]\
.$ (12)

This makes the laser crystal have different thermal lenses in the two
directions. Therefore, the thermal lens formed by the laser crystal is not a
circular one, but a lens with astigmatism.

### 2.4 Elliptical Gaussian beam source

With the source function given by Eq. (12), the temperature now follows

$-\kappa\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)T=q_{0}\exp\left[-\left(\frac{x^{2}}{w_{x}^{2}}+\frac{y^{2}}{w_{y}^{2}}\right)\right]$
(13)

with the boundary condition given by

$T(x,y)=T_{0}\qquad\text{at }x^{2}+y^{2}=R^{2}\ .$ (14)

Making change of variables

$u=\frac{\kappa}{q_{0}}\,(T-T_{0})\ ,$ (15)

where $u$ has the dimension of $[L]^{2}$, the problem becomes

$\displaystyle-\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)u=\exp\left[-\left(\frac{x^{2}}{w_{x}^{2}}+\frac{y^{2}}{w_{y}^{2}}\right)\right]$
(16) $\displaystyle\,u(x,y)=0\qquad\text{at }x^{2}+y^{2}=R^{2}$ (17)

which can be solved numerically.

Figure 6 shows the temperature distributions in the transverse plane with
circular and elliptical Gaussian beam sources of the same total heating power.
In the elliptical case, the ratio of the beam widths is $1:n$ and we take the
refractive index to be $n=1.76$.

(a) $w_{x}:w_{y}=1:1$

(b) $w_{x}:w_{y}=1:1.76$

Figure 6: Temperature profiles in the transverse plane of the laser crystal
heated by (a) circular Gaussian beam source, (b) elliptical Gaussian beam
source, assuming that the total heating power of the two sources are the same.

It is found in the numerical simulation that the ratio of 2${}^{\text{nd}}$
order derivatives of temperature at the center of the crystal in the $x-$ and
$y-$directions follows

$\frac{\partial_{x}^{2}\,T(0,0)}{\partial_{y}^{2}\,T(0,0)}=\frac{w_{y}}{w_{x}}$
(18)

for various ratios of $w_{x}$ to $w_{y}$. This rule can be proved for an
infinitely large crystal and will be used in the analysis of thermal lens
astigmatism in subsection 3.2.

##  3 Thermal Lens

### 3.1 Formula under azimuthal symmetry

The actual laser beam generated propagates through the crystal near its
central axis and the optical path depends on $r$. Under the paraxial
approximation, the optical path difference (OPD) is equivalent to that caused
by a thin lens of focal length $f_{T}$ (called “thermal lens” in the text to
follow)

$\text{OPD}(r)-\text{OPD}(0)=-\frac{r^{2}}{2f_{T}}\ .$ (19)

As the crystal is heated, two effects contribute to the thermal lens: (i) the
change in refractive index with temperature and (ii) the change in the length
of the crystal with temperature. Assuming that in the range of temperature
involved, the changes in refractive index and length are linear to
temperature, i.e.

$\displaystyle\frac{\mathrm{d}n}{\mathrm{d}T}$ $\displaystyle=\text{const}$
(20) $\displaystyle\frac{\mathrm{d}l}{\mathrm{d}T}$
$\displaystyle=2\alpha_{e}l_{0}=\text{const}$ (21)

where $\alpha_{e}$ is the thermal expansion coefficient and $l_{0}$ is the
length at each end section of the crystal over which expansion occurs and is
approximately equal to $R$, the radius of the crystal. Therefore, the optical
path difference can be written as

$\text{OPD}(r)-\text{OPD}(0)=\left[\frac{\mathrm{d}n}{\mathrm{d}T}\,L+(n-1)\frac{\mathrm{d}l}{\mathrm{d}T}\right](T(r)-T(0))\
.$ (22)

To evaluate the temperature difference near the center, we expand $T(r)$ in
Taylor series for which its derivatives at $r=0$ are needed. Eq. (5) gives
that

$\frac{\mathrm{d}T}{\mathrm{d}r}=-T_{p}\,\frac{1}{r}\left[1-\exp\left(-\frac{r^{2}}{w_{1}^{2}}\right)\right]$
(23)

where we have used the definition of $T_{p}$ in Eq. (7). Differentiating
again,

$\frac{\mathrm{d}^{2}T}{\mathrm{d}r^{2}}=-T_{p}\,\frac{1}{r^{2}}\left[-1+\left(1+\frac{2r^{2}}{w_{1}^{2}}\right)\exp\left(-\frac{r^{2}}{w_{1}^{2}}\right)\right]\
.$ (24)

Evaluating Eqs. (23) and (24) at $r=0$, we have

$\displaystyle\frac{\mathrm{d}T}{\mathrm{d}r}(r=0)$ $\displaystyle=0\ ,$
$\displaystyle\frac{\mathrm{d}^{2}T}{\mathrm{d}r^{2}}(r=0)$
$\displaystyle=-T_{p}\,\frac{1}{w_{1}^{2}}=-\frac{q_{0}}{2\kappa}\ .$ (25)

Therefore,

$T(r)-T(0)\simeq\frac{1}{2}\,\partial_{r}^{2}\,T(0)\,r^{2}$ (26)

and

$\text{OPD}(r)-\text{OPD}(0)\simeq\left[\frac{\mathrm{d}n}{\mathrm{d}T}\,L+2\alpha_{e}(n-1)R\right]\frac{1}{2}\,\partial_{r}^{2}\,T(0)\,r^{2}\
.$ (27)

Comparing with Eq. (19), we have that

$f_{T}^{-1}=\left[\frac{\mathrm{d}n}{\mathrm{d}T}\,L+2\alpha_{e}(n-1)R\right](-\partial_{r}^{2}\,T(0))\
.$ (28)

Thus

$f_{T}=\frac{\pi\kappa w_{0}^{2}}{\beta P_{0}}$ (29)

where

$\beta=\frac{\mathrm{d}n}{\mathrm{d}T}+\frac{2\alpha_{e}(n-1)R}{L}$ (30)

and $P_{0}=\pi w_{1}^{2}q_{0}L$ is the total heating power dissipated in the
laser crystal, with $\kappa$ being the thermal conductivity and $\alpha_{e}$
the thermal expansion coefficient.

### 3.2 Astigmatism

Though an analytical expression of the temperature distribution can no longer
be obtained when the laser crystal is heated by an elliptical Gaussian beam,
in order to calculate the thermal lens, we only need the 2${}^{\text{nd}}$
order derivatives of $T$ at the center, as shown by Eq. (28).

It is shown in Appendix B.1 that, for $T(x,y)$ following Eqs. (13) and (14),
when $R\gg w_{x},w_{y}$, we have that

$\displaystyle\frac{\partial^{2}T}{\partial x^{2}}(0,0)$
$\displaystyle=-\frac{q_{0}}{\kappa}\frac{w_{y}}{w_{x}+w_{y}}$ (31)
$\displaystyle\frac{\partial^{2}T}{\partial y^{2}}(0,0)$
$\displaystyle=-\frac{q_{0}}{\kappa}\frac{w_{x}}{w_{x}+w_{y}}\ .$ (32)

With the total heating power given by $P_{0}=\pi w_{x}w_{y}q_{0}L$ in the
elliptical case, the thermal lenses in the two directions can be written as

$\displaystyle f_{T,x}$ $\displaystyle=\frac{\pi\kappa}{\beta
P_{0}}\,w_{x}(w_{x}+w_{y})=\frac{n+1}{2}\frac{\pi w_{0}^{2}\kappa}{\beta
P_{0}}$ (33) $\displaystyle f_{T,y}$ $\displaystyle=\frac{\pi\kappa}{\beta
P_{0}}\,w_{y}(w_{x}+w_{y})=\frac{n(n+1)}{2}\frac{\pi w_{0}^{2}\kappa}{\beta
P_{0}}\ ,$ (34)

from which we have that the ratio of thermal lens in the $y$ (horizontal)
direction to that in the $x$ (vertical) direction

$\frac{f_{T,H}}{f_{T,V}}=\frac{f_{T,y}}{f_{T,x}}=\frac{w_{y}}{w_{x}}=n\ .$
(35)

### 3.3 Enhancement of maximum power through cryogenic cooling

The thermal indices of sapphire at room temperature and cryogenic temperatures
are listed in Table 1, where the $\mathrm{d}n/\mathrm{d}T$ data are from Ref.
[6]. The dependence of $\alpha_{e}$ on temperature below $300\,\text{K}$ is
unclear and we assume that it decreases linearly with temperature as an upper
limit estimation.

The thermal conditions influence the optical property of the Ti:sapphire
crystal in the form of Eq. (29). It can be seen from the figures in Table 1
that if the crystal is cooled from RT to cryogenic temperatures, the thermal
conductivity $\kappa$ will increase dramatically and the factor
$\mathrm{d}n/\mathrm{d}T$ will decrease. Therefore, at the same pump power, a
cryogenically cooled crystal will have a much longer thermal lens than a
crystal at room temperature. Ideally, if $\kappa/\beta$ increases by some
factor, the pump power can be raised by the same factor while keeping the
thermal lens at the same value.

Temperature $T\ (\text{K})$ |  Thermal conductivity $\kappa\ (\text{W}\text{cm}^{-1}\text{K}^{-1})$ | $\displaystyle\frac{\mathrm{d}n}{\mathrm{d}T}\ (10^{-6}\,\text{K}^{-1})$ |  Thermal expansion coefficient $\alpha_{e}\ (10^{-6}\,\text{K}^{-1})$
---|---|---|---
77 | 10 | 2.8 |  1.4${}^{\text{a}}$
100 | 4.0 | 3.4 |  1.8${}^{\text{a}}$
300 | 0.35 | 13 | 5.5
Table 1: Thermal indices of sapphire at room temperature and cryogenic
temperatures. ${}^{\text{a}}$Upper limit estimation assuming that
$\alpha_{e}\propto T$.

By taking typical parameters of the pump laser, we can estimate the relation
of thermal lens to the pump power. Firstly, we assume that the pump has a
$1/e^{2}$ radius of $w_{0}=0.5\,\text{mm}$. Secondly, we assume that all the
pump power incident on the laser crystal is absorbed, i.e.

$P_{\text{abs}}\approx P_{\text{pump}}\ .$ (36)

This is because for a Ti^3+ doping concentration of $0.1\%$, the small-signal
absorption coefficient is $\alpha_{0}=1.5\,\text{cm}^{-1}$. For a laser
crystal with length $L\geq 2\,\text{cm}$, the transmittance
$e^{-\alpha_{0}L}\leq 5\%$. Also, the density of Ti^3+ at $0.1\%$ doping
concentration is $n_{\ch{Ti}}=4.56\times 10^{19}\,\text{cm}^{-3}$. From this,
we can deduce the absorption cross section

$\sigma^{\text{(abs)}}=\frac{\alpha_{0}}{n_{\ch{Ti}}}=3.29\times
10^{-20}\,\text{cm}^{2}\ .$ (37)

Taking the recovery lifetime for the absorption $\tau_{\text{R}}$ to be of
order $10^{2}\,\text{ps}$, then the saturation intensity of absorption

$I_{\text{sat}}^{\text{(abs)}}=\frac{hc/\lambda_{\text{pump}}}{\sigma^{\text{(abs)}}\tau_{\text{R}}}\sim
10^{11}\,\text{W}\,\text{cm}^{-2}$ (38)

which is very high ($\lambda_{\text{pump}}=532\,\text{nm}$ is the wavelength
of the pump). At a pump power of $1000\,\text{W}$, the pump intensity
$I_{\text{pump}}\sim 10^{5}\,\text{W}\,\text{cm}^{-2}$ is still negligible
compared to such a saturation intensity. So the absorption coefficient is very
close to its small-signal value even at a high pump power. Thirdly, we
estimate the fraction of absorbed pump power converted into heat by the
wavelengths of the pump and the laser.

$\eta_{\text{heat}}=\frac{P_{\text{{heat}}}}{P_{\text{{abs}}}}=1-\frac{hc/\lambda_{\text{laser}}}{hc/\lambda_{\text{pump}}}\approx
23.8\%$ (39)

where $\lambda_{\text{laser}}=698\,\text{nm}$. Additionally, in the expression
of $\beta$ (Eq. (30)), we take the refractive index $n=1.76$ and the ratio
$R/L=0.2$, consistent with a laser crystal with $R=4\,\text{mm}$ and
$L=20\,\text{mm}$. Combining Eqs. (36), (39) and (33), we have that the
thermal lens in the $x-$direction

$f_{T,x}\approx\frac{n+1}{2}\frac{\pi
w_{0}^{2}\kappa}{\beta\,\eta_{\text{heat}}\,P_{\text{pump}}}\ .$ (40)

In Figure 7, the estimated relation of $f_{T,x}$ to $P_{\text{pump}}$ is
evaluated for $T_{0}=300$, $100$ and $77\,\text{K}$. At the same pump power,
the ratio between the focal lengths of thermal lens at the three temperatures
is

$f_{T,x}(300\,\text{K}):f_{T,x}(100\,\text{K}):f_{T,x}(77\,\text{K})\approx
1:40:130\ .$ (41)

Indicated by the dashed line is $f_{T,x}=0.5\,\text{m}$, a benchmark that will
be obtained in section 4.5 below which the instability in the laser cavity
will be significant. At $300\,\text{K}$, $f_{T,x}$ reaches $0.5\,\text{m}$ at
a pump power of about $20\,$W. For an output power of $100\,$W, a pump power
of $300\,$W is needed (as given in Appendix B.3). It is clear that if the
laser crystal is cooled to $100\,\text{K}$ or $77\,\text{K}$, $f_{T,x}$ is
still above $0.5\,\text{m}$ at such a pump power.

Figure 7: Estimated relation of the focal length of thermal lens in the
$x-$direction $f_{T,x}$ to the pump power $P_{\text{pump}}$ when the laser
crystal is at $300\,\text{K}$ (red), $100\,\text{K}$ (light blue) and
$77\,\text{K}$ (dark blue). The dashed line indicates $f_{T,x}=0.5\,\text{m}$
below which the instability of the laser cavity is significant. Note that the
vertical axis is logarithmic.

##  4 Stability Condition of the Laser Cavity

### 4.1 Transfer matrix and stability condition

In this section, we calculate the stability condition of the Ti:sapphire laser
cavity, expressed in terms of the geometrical parameters. A schematic of the
bow-tie cavity is shown in Figure 8 where the components that do not affect
the stability are omitted.

As the laser beam propagates in the loop, it undergoes the following processes
(starting from the front surface of the crystal):

  1. 1.

Propagating through the crystal (experiencing refraction at two surfaces and
thermal lensing within the crystal);

  2. 2.

Propagating for a distance of $L_{1}/2$;

  3. 3.

Reflection at mirror of radius of curvature $R$ with incident angle $\phi$;

  4. 4.

Propagating for $L_{2}$;

  5. 5.

Reflection at mirror of radius of curvature $R$ with incident angle $\phi$;

  6. 6.

Propagating for $L_{1}/2$.

The overall transfer matrix can be found by multiplying the following transfer
matrices for each step.

  1. 1.

Propagation for a distance of $L$

$M_{\text{trans}}(L)=\begin{pmatrix}1&L\\\ 0&1\end{pmatrix}\ .$ (42)

  2. 2.

Reflection from a curved mirror

$M_{\text{mirror}}(R_{\text{eff}})=\begin{pmatrix}1&0\\\
-2/R_{\text{eff}}&1\end{pmatrix}$ (43)

where $R_{\text{eff}}$ is the effective radius of curvature.

  3. 3.

Propagation in the crystal which has a length $l$, refractive index $n$ and a
thermal lens $f_{T}$

$M_{\text{crystal}}(l_{\text{eff}},f_{T})=\begin{pmatrix}1&l_{\text{eff}}\\\
-1/f_{T}&1\end{pmatrix}$ (44)

where $l_{\text{eff}}$ is the effective length of the crystal.

In these expressions, there are a few parameters that differ for light in the
horizontal and vertical planes which are summarized in Table 2. Because the
ratio between the thermal lens in the $y$ and $x$ directions is equal to the
refractive index, we refer to $f_{T,x}$ as $f_{T}$ and $f_{T,y}$ as $nf_{T}$.

|  Effective radius of curvature $R_{\text{eff}}$ |  Effective crystal length $l_{\text{eff}}$ |  Thermal lens $f_{T}$
---|---|---|---
$x-$direction vertical | $R/\cos\phi$ | $l_{x}=l/n$ | $f_{T,x}=f_{T}$
$y-$direction horizontal | $R\cos\phi$ | $l_{y}=l/n^{3}$ | $f_{T,y}=nf_{T}$
Table 2: Parameters that have different values in the vertical ($x$) and
horizontal ($y$) directions.

Based on the matrices above, we can define the overall transfer matrix as

$\begin{split}M(L_{1},L_{2},R_{\text{eff}},l_{\text{eff}},f_{T})&=M_{\text{trans}}(L_{1}/2)M_{\text{mirror}}(R_{\text{eff}})M_{\text{trans}}(L_{2})\\\
&\quad\
M_{\text{mirror}}(R_{\text{eff}})M_{\text{trans}}(L_{1}/2)M_{\text{crystal}}(l_{\text{eff}},f_{T}).\end{split}$
(45)

Therefore, the transfer matrices in the $x-$ and $y-$directions can be written
as

$\displaystyle M_{x}$ $\displaystyle=M(L_{1},\ L_{2},\ R/\cos\phi,\ l/n,\
f_{T})\ ,$ (46) $\displaystyle M_{y}$ $\displaystyle=M(L_{1},\ L_{2},\
R\cos\phi,\ l/n^{3},\ nf_{T})\ .$ (47)

Figure 8: Simplified schematic of the Ti:sapphire laser cavity.

### 4.2 Stability condition

If we write the overall transfer matrix as

$M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix},$ (48)

then the complex radius of the Gaussian beam mode is given by

$q=\frac{1}{2C}\left[(A-D)\pm\sqrt{(A-D)^{2}+4BC}\right]$ (49)

and the condition for the laser cavity to be stable is

$\Delta=(A-D)^{2}+4BC<0$ (50)

(specific derivation is given in Appendix B.2).

Since

$q=z-z_{0}+\mathrm{i}z_{\text{R}}\ ,$ (51)

we can define the squared Rayleigh range as

$z_{\text{R}}^{2}=-\frac{1}{4C^{2}}\left[(A-D)^{2}+4BC\right]\ .$ (52)

In this way, the condition for the laser cavity being stable is equivalent to

$z_{\text{R}}^{2}>0\ .$ (53)

### 4.3 Astigmatism compensation

With the transfer matrices in the $x-$ and $y-$directions given by Eqs. (46)
and (47), the squared Rayleigh ranges in the two directions can be written as
functions of 6 parameters

$\displaystyle z^{2}_{\text{R},x}$
$\displaystyle=\zeta_{x}(L_{1},L_{2},R,\phi,l,f_{T})$ (54) $\displaystyle
z^{2}_{\text{R},y}$ $\displaystyle=\zeta_{y}(L_{1},L_{2},R,\phi,l,f_{T})\ .$
(55)

In general, $z^{2}_{\text{R},x}$ and $z^{2}_{\text{R},y}$ are not the same and
the range of parameters for $z_{\text{R}}^{2}>0$ are different.

For example, for a particular bow-tie cavity with parameters given in Ref.
[7],

$\displaystyle L_{2}$ $\displaystyle=170\,\text{cm}$ $\displaystyle R$
$\displaystyle=10\,\text{cm}$ $\displaystyle l$ $\displaystyle=2\,\text{cm}\
,$

if we do not consider the thermal lens (i.e. taking $f_{T}\rightarrow\infty$),
then at $\phi=0^{\circ}$, the ranges of $L_{1}$ for the cavity to be stable in
the $x-$ and $y-$directions do not overlap and the cavity can never function
properly, as shown in Figure 9(a). Instead, $\phi$ needs to be chosen
carefully so that the centers of the two ranges coincide and this is found at
$\phi=15.31^{\circ}$, as shown in Figure 9(b).

We refer to the range of $L_{1}$ where $z_{\text{R}}^{2}>0$ as the working
region. In general, the $z_{\text{R}}^{2}$ vs. $L_{1}$ curve within the
working region is approximately parabolic with a single maximum. We refer to
the value of $L_{1}$ where $z_{\text{R}}^{2}$ takes this maximum as the
optimal working region. This is mathematically defined by

$\displaystyle L_{1,x}\quad\text{is such
that}\quad\frac{\partial\zeta_{x}}{\partial
L_{1}}(L_{1}=L_{1,x},L_{2},R,\phi,l,f_{T})=0$ (56) $\displaystyle
L_{1,y}\quad\text{is such that}\quad\frac{\partial\zeta_{y}}{\partial
L_{1}}(L_{1}=L_{1,y},L_{2},R,\phi,l,f_{T})=0$ (57)

for the $x-$ and $y-$directions. We want the laser cavity to operate near the
optimal working region so that the influence of tiny changes in $L_{1}$ upon
the stability of the laser is minimized.

The optimal working regions of the $x-$ and $y-$directions are also different
in general. To compensate the astigmatism is to choose a proper set of
parameters such that $L_{1,x}$ and $L_{1,y}$ coincide, and we choose to do so
at zero power (i.e. when $f_{T}\rightarrow\infty$). This is given by

$\displaystyle L_{1,x}=L_{1,y}=L_{1}$
$\displaystyle\Leftrightarrow\quad\frac{\partial\zeta_{x}}{\partial
L_{1}}(L_{1},L_{2},R,\phi,l,\infty)=\frac{\partial\zeta_{y}}{\partial
L_{1}}(L_{1},L_{2},R,\phi,l,\infty)=0\ .$ (58)

Eq. (58) imposes two restrictions. If we take the crystal length $l$ and the
second path length $L_{2}$ as given parameters, then we can determine the
three remaining parameters $L_{1}$, $R$ and $\phi$ based on one of them. In
practice, $L_{1}$ and $\phi$ can be adjusted precisely but only mirrors with
particular $R$ are available. Therefore, we calculate $L_{1}$ and $\phi$ from
$R$. Though the expression of $z_{\text{R}}$ is complicated, Eq. (58) can be
simplified into 2 polynomials in terms of $L_{1}$ and $\cos\phi$ which can be
solved numerically.

To sum up, at no thermal lens, $L_{1}$ and $\phi$ can be determined from given
$L_{2}$, $R$ and $l$ according to Eq. (58) such that the optimal working
regions of the $x-$ and $y-$directions are coincident.

(a)

(b)

Figure 9: The relation of $z_{\text{R}}^{2}$ to $L_{1}$ in the $x$ (vertical)
and $y$ (horizontal) directions, for the incident angle on the spherical
mirror $\phi$ being (a) $0^{\circ}$ where the working regions do not overlap
and (b) $15.31^{\circ}$ which gives astigmatism compensation. Figure 10: The
relations of $z_{\text{R}}^{2}$ to $L_{1}$ under the same conditions as in
Figure 9(b) except that $f_{T}$ is now $5\,\text{cm}$. Astigmatism
compensation is broken by the thermal lens.

### 4.4 Optimal parameters

Using the protocol introduced in the previous subsection, we can obtain
several sets of cavity parameters which give astigmatism compensation. These
are chosen for $R=10$, $20$ and $30\,\text{cm}$ respectively, listed in Table
3.

$L_{1}$ (cm) |  $L_{2}$ (cm) |  $R$ (cm) |  $\phi$ (∘) |  $l$ (cm)
---|---|---|---|---
9.568 | 170.0 | 10.00 | 15.31 | 2.00
19.975 | 240.0 | 20.00 | 11.96 | 2.50
31.207 | 240.0 | 30.00 | 9.53 | 2.50
Table 3: Three sets of cavity parameters with astigmatism compensation where
the radii of curvature of the spherical mirror $R$ are $10$, $20$ and
$30\,\text{cm}$ respectively.

### 4.5 Stability against thermal lens

With increasing effect of thermal lens ($f_{T}$ decreasing from $\infty$), the
optimal working regions are shifted and the $z_{\text{R}}^{2}$ vs. $L_{1}$
curves are distorted. Figure 10 shows the $z_{\text{R}}^{2}$ vs. $L_{1}$
curves for the astigmatism compensated design at a strong thermal lens of
$f_{T}=5\,\text{cm}$.

The response to $f_{T}$ for optimized designs with different $R$ are similar
but differ in magnitude. To characterize the influence of thermal lens
accurately, we plot the optimal working regions of the $x-$ and
$y-$directions, i.e. $L_{1,x}$ and $L_{1,y}$ as defined in Eqs. (56) and (57),
and their relative difference $(L_{1,x}-L_{1,y})/L_{1,y}$ against $f_{T}$ for
the first two sets of parameters in Table 3, shown in Figure 11. In these
graphs, optimal working regions are evaluated for $f_{T}$ below
$2.5\,\text{m}$. Comparing the behavior of the two designs, we can find that
the one with larger $R$ (as shown in Figs. 11(b) and 11(d)) has both a smaller
percentage increase in optimal working regions and a smaller relative
difference between optimal working regions in the two directions.

Besides, since $f_{T}$ enters the expressions of transfer matrix and Rayleigh
range in the form of $1/f_{T}$, all the curves in Figure 11 have the shape of
the inverse scale function. It can be estimated that in order for the relative
difference between the optical working regions in the $x-$ and $y-$directions
to be below $0.1\%$, that is $(L_{1,x}-L_{1,y})/L_{1,y}\leq 0.1\%$ in Figs.
11(c) and 11(d), the focal length of thermal lens needs to be

$f_{T}\gtrsim 0.5\,\text{m}\ .$ (59)

This is how the benchmark we used in section 3.3 is derived. As shown in
Figure 7, Eq. (59) is satisfied for pump power up to $500\,$W, given that the
laser crystal is cooled to temperatures of $100\,\text{K}$ or lower.
Therefore, we can expect the instability of the laser cavity to be
insignificant if the laser crystal is cryogenically cooled.

(a) $R=10\,\text{cm}$, $l=2.0\,\text{cm}$

(b) $R=20\,\text{cm}$, $l=2.5\,\text{cm}$

(c) $R=10\,\text{cm}$, $l=2.0\,\text{cm}$

(d) $R=20\,\text{cm}$, $l=2.5\,\text{cm}$

Figure 11: (a), (b) Relation of optimal working regions in the $x$ (vertical)
direction $L_{1,x}$ and in the $y$ (horizontal) direction $L_{1,y}$, as
defined in Eqs. (56) and (57), to the focal length of thermal lens $f_{T}$ for
two astigmatism compensated designs with $R=10$ and $20\,\text{cm}$. (c), (d)
Relative difference between $L_{1,x}$ and $L_{1,y}$ in (a), (b) as functions
of $f_{T}$. The laser cavity parameters are listed at the right of each figure
where $L_{1}\,:$ first path length, $L_{2}\,:$ second path length, $R\,:$
spherical mirror radius of curvature, $\phi\,:$ incident angle on the
spherical mirror and $l\,:$ laser crystal length. $L_{1}$ and $\phi$ are
chosen such that $L_{1,x}=L_{1,y}=L_{1}$ when $f_{T}\rightarrow\infty$. Note
that (a), (c) characterize the same design and (b), (d) characterize the same
design. For both designs, as $f_{T}$ decreases, $L_{1,x}$ and $L_{1,y}$
increase and their relative difference also increases.

##  5 Conclusion

In this study, we analyzed the practicality of enhancing the power of
Ti:sapphire laser by cryogenically cooling the laser crystal. We obtained
analytical expressions of the focal length of thermal lens for the pump inside
the laser crystal being a circular Gaussian beam or an elliptical Gaussian
beam. In order for the relative difference between the optimal laser cavity
path lengths in the horizontal and vertical directions to be less than
$0.1\%$, the focal length of thermal lens should be longer than
$0.5\,\text{m}$. Such a requirement can be met if the laser crystal is cooled
to temperatures below $100\,\text{K}$ while being pumped at a power that is
enough to produce $100\,$W output. In conclusion, it is very likely that an
output power of $100\,$W can be achieved on a single Ti:sapphire laser by
cryogenic cooling of the laser crystal.

## References

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##  Appendix A Relationship of thermal conductivity of sapphire to temperature

Figure A.1: Thermal conductivity of sapphire (Al2O3) vs. temperature. Note
that the vertical axis is logarithmic. After [8].

##  Appendix B Detailed Derivations

### B.1 Second order derivatives of temperature at the origin

In Eqs. (16) and (17), when the radius of the laser crystal is much larger
than the widths of the beam, i.e. $R\gg w_{x},w_{y}$, the boundary condition
can be approximated by

$u\rightarrow 0\qquad\text{at }x^{2}+y^{2}\rightarrow\infty\ .$

Under such circumstance, $u(x,y)$ can in principle be calculated through
Fourier transform. While the analytical expression of $u(x,y)$ is hard to get,
its second order derivatives at the origin can be obtained. The field $u(x,y)$
and its Fourier transform $\hat{u}(k_{x},k_{y})$ are linked by

$u(x,y)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\hat{u}(k_{x},k_{y})\exp[\mathrm{i}(k_{x}x+k_{y}y)]\,\mathrm{d}k_{x}\,\mathrm{d}k_{y}\
.$

Therefore,

$\displaystyle\partial_{x}^{2}\,u$
$\displaystyle=\frac{1}{2\pi}\iint(-k_{x}^{2})\,\hat{u}\exp[\mathrm{i}(k_{x}x+k_{y}y)]\,\mathrm{d}k_{x}\,\mathrm{d}k_{y}$
(B.1) $\displaystyle\partial_{y}^{2}\,u$
$\displaystyle=\frac{1}{2\pi}\iint(-k_{y}^{2})\,\hat{u}\exp[\mathrm{i}(k_{x}x+k_{y}y)]\,\mathrm{d}k_{x}\,\mathrm{d}k_{y}$
(B.2)

where the integration limits on $(-\infty,\infty)$ are dropped. We also have
that the Fourier transform of the source function

$\hat{q}(k_{x},k_{y})=\frac{1}{2}\,w_{x}w_{y}\exp\left[-\frac{1}{4}\left(w_{x}^{2}k_{x}^{2}+w_{y}^{2}k_{y}^{2}\right)\right]\
.$

So $\hat{u}(k_{x},k_{y})$ follows the equation

$(k_{x}^{2}+k_{y}^{2})\,\hat{u}=\frac{1}{2}\,w_{x}w_{y}\exp\left[-\frac{1}{4}\left(w_{x}^{2}k_{x}^{2}+w_{y}^{2}k_{y}^{2}\right)\right]$

of which the solution is

$\hat{u}=\frac{1}{2}\,w_{x}w_{y}\,\frac{1}{k_{x}^{2}+k_{y}^{2}}\exp\left[-\frac{1}{4}\left(w_{x}^{2}k_{x}^{2}+w_{y}^{2}k_{y}^{2}\right)\right]\
.$

Therefore, $u(x,y)$ can formally be written as

$u(x,y)=\frac{1}{4\pi}\,w_{x}w_{y}\iint\frac{1}{k_{x}^{2}+k_{y}^{2}}\exp\left[-\frac{1}{4}\left(w_{x}^{2}k_{x}^{2}+w_{y}^{2}k_{y}^{2}\right)\right]\exp[\mathrm{i}(k_{x}x+k_{y}y)]\,\mathrm{d}k_{x}\,\mathrm{d}k_{y}$

but the integral can hardly be evaluated.

Substituting into Eq. (B.1) and evaluating at $(x,y)=(0,0)$,

$\begin{split}\frac{\partial^{2}u}{\partial
x^{2}}(0,0)&=-\frac{1}{4\pi}\,w_{x}w_{y}\iint\frac{k_{x}^{2}}{k_{x}^{2}+k_{y}^{2}}\exp\left[-\frac{1}{4}\left(w_{x}^{2}k_{x}^{2}+w_{y}^{2}k_{y}^{2}\right)\right]\,\mathrm{d}k_{x}\,\mathrm{d}k_{y}\\\
&=-\frac{w_{y}}{w_{x}+w_{y}}\end{split}$ (B.3)

where the second equality is given by MATHEMATICA as shown in Figure B.1.
Swapping $w_{x}\leftrightarrow w_{y}$, we can get

$\frac{\partial^{2}u}{\partial y^{2}}(0,0)=-\frac{w_{x}}{w_{x}+w_{y}}$

which can also be obtained through the relation that

$\frac{\partial^{2}u}{\partial x^{2}}(0,0)+\frac{\partial^{2}u}{\partial
y^{2}}(0,0)=-1$

which is Eq. (16) evaluated at $(x,y)=(0,0)$.

According to the manner of variable changing in Eq. (15), we thus have

$\displaystyle\frac{\partial^{2}T}{\partial x^{2}}(0,0)$
$\displaystyle=-\frac{q_{0}}{\kappa}\frac{w_{y}}{w_{x}+w_{y}}$ (B.4)
$\displaystyle\frac{\partial^{2}T}{\partial y^{2}}(0,0)$
$\displaystyle=-\frac{q_{0}}{\kappa}\frac{w_{x}}{w_{x}+w_{y}}\ .$ (B.5)

Figure B.1: MATHEMATICA’s evaluation of Eq. (B.3).

### B.2 Stability condition of the laser cavity

As for a Gaussian beam, the transfer matrix acts on the vector

$\begin{pmatrix}q\\\ 1\end{pmatrix}$

where $q$ is the complex radius. After one round-trip, the complex radius
should be unchanged. Therefore

$\displaystyle\begin{pmatrix}q\\\
1\end{pmatrix}=\mathcal{UF}\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\begin{pmatrix}q\\\ 1\end{pmatrix}\ .$ (B.6) where
$\mathcal{UF}$ is an uninteresting factor, $\displaystyle\Rightarrow\quad$
$\displaystyle q=\frac{Aq+B}{Cq+D}$ $\displaystyle\Rightarrow\quad$
$\displaystyle Cq^{2}+(D-A)q-B=0$ $\displaystyle\Rightarrow\quad$
$\displaystyle q=\frac{1}{2C}\left[(A-D)\pm\sqrt{(A-D)^{2}+4BC}\right]\ .$
(B.7)

In order for the mode to be a Gaussian beam, $q$ must be complex, therefore
the discriminant must be negative, i.e.

$\Delta=(A-D)^{2}+4BC<0\ .$ (B.8)

### B.3 Pump power needed for 100 W output

Figure B.2 shows the measured relation of output power to pump power for a M
Squared SolsTiS laser operating at $698\,\text{nm}$ and room temperature. This
relation can be fitted with a piecewise linear function

$\displaystyle P_{\text{out}}=$ $\displaystyle
k(P_{\text{pump}}-P_{\text{thresh}})\quad$ $P_{\text{pump}}\geq
P_{\text{thresh}}$ (B.9) $\displaystyle P_{\text{out}}=$ $\displaystyle
P_{\text{out}}=$ $\displaystyle 0\quad$ $P_{\text{pump}}<P_{\text{thresh}}$
(B.10)

where $P_{\text{thresh}}$ is the threshold pump power. The fitting parameters
are $k=0.377$ and $P_{\text{thresh}}=6.32\,$W. The good linear relation
between output power and pump power above lasing threshold means that the
laser cavity is stable and that the effect of thermal lens is insignificant.

As we showed in Figure 7, if the laser crystal is cooled to cryogenic
temperatures, the thermal lens will not break the stability of the laser
cavity up to $500\,$W pump power. Under such circumstance, we can expect the
output power to continue growing with the pump power linearly according to Eq.
(B.9) and $P_{\text{out}}$ will reach $100\,$W when $P_{\text{pump}}=249\,$W.
So, in round numbers, $300\,$W pump power is enough to produce $100\,$W
output.

Figure B.2: Measured relation of output power to pump power for a M Squared
SolsTiS laser operating at $698\,\text{nm}$ and room temperature. Red line
shows piecewise linear fit as defined by Eqs. (B.9) and (B.10).

##  Appendix C Early work on cooled Ti:sapphire lasers in the 1990s

The advantages of cryocooling have been recognized since the early days of
Ti:Sa lasers. A paper from 1991 [9] states that “At 93K, sapphire (Al2O3) has
much better thermal properties than at room temperature. By taking advantage
of the improved thermal properties, a Ti:Al2O3 laser is operated with 350 W of
output power under thermally steady-state conditions. Model calculations of
thermal lensing and higher order thermo-optic aberrations indicate that the
output power capability is 200 times larger at 77 K than at 300 K”, where
‘thermally steady-state conditions’ means experimental measurements using so-
called macro pulses of 170 microseconds (longer than the thermal response
time). This quasi-CW operation demonstrated that high-power operation would be
sustainable continuously (actual ‘steady-state’) if a high-power pump laser
was available. Good thermal contact between the sapphire and the surrounding
copper block, attached to a dewar, was achieved by a thin sheet of indium
squeezed between them (a standard cryogenic technique, and other experimental
aspects are also still relevant for the more powerful pump laser systems now
available). A $43\,\mathrm{W}$, CW Ti:sapphire laser was reported at the CLEO
conference in 1991 [10].

##  Appendix D Related industrial companies and project

### D.1 Kapteyn-Murane Labs (KMLabs)

This spin-out from University of Colorado, Boulder111www.kmlabs.com
manufactures short pulse systems with cryocooled Ti:sapphire laser oscillators
(femtosecond pulses and products to generate vacuum ultraviolet (VUV) and
extreme ultraviolet (EUV or XUV) radiation from them). This company produces
cryocooled modules but the Ti:Sa crystals are generally shorter (to reduce
dispersion) than optimal for CW operation, however they “see no fundamental
issue in making a $>30$ W at 698 nm Ti:Sa CW laser” but it would require R&D
to implement this. KMLabs has delivered custom CW systems but the development
of a new product is a costly project [email communication]. The company’s
standard cryocooler can handle 90 W of excess thermal energy. Interestingly, a
Pulse-pumped CW tunable Ti:sapphire laser was reported at the CLEO conference
(2009) by Hsiao-hua Liu from KMLabs [11].

### D.2 Advanced Photonic Sciences (APS)

This US company222www.apslasers.com produces cryogenic lasers but mainly
Yb:YAG (not Ti:Sa) but their closed-cycle cooling system (not requiring liquid
nitrogen refrigerant) may be of interest for the long term since it gives 1 kW
of cooling power at around 100 K, which would otherwise imply a high usage of
liquid nitrogen (LN) refrigerant. The company claims “world-record
performance” for a cryogenic laser with a CW sustained average power of 963 W.
A system producing 300 W is almost a standard product (Thor-300). The company
personnel, with others, have written a 74-page open-access review in 2016
[12]. This gives an historical perspective, from their point of view, and
details of the physics underlying cryogenic lasers in general (but sapphire
and Ti:Sa are incidental to the main theme and described as a ‘legacy laser
material’!).

### D.3 FMB Oxford cryocooler

The cooler from this company333fmb-oxford.com provides 3 kW of cooling power
at 100 K. The companies KMLabs and APS favor cryocoolers, presumably because
they do not want to restrict the use to customers that have suitable
refrigeration supplies. (The vibrations associated with mechanical cooling
engines can be circumvented by switching them off temporarily to interleave
periods of cooling and operation.) For MAGIS-100 and AION, however, the supply
of refrigerant is probably an incidental expense and the infrastructure for LN
storage and transport would not be a major issue at a large facility such as
Fermilab. Thus, it may be worthwhile to investigate this FMB Oxford cryocooler
— this company originated as part of Oxford Instruments (Synchrotron Group,
sold in 2001) and is located near Oxford. The system is designed to cool
crystals in monochromators for X-rays from synchrotrons with special attention
to low-vibration operation (suppressing mechanical resonances and low
turbulence flow). Thus, it is potentially suitable for laser crystals,
although the tolerable level of vibration when amplifying narrow-bandwidth
radiation would need to be quantified. Mechanical vibrations of relatively
large objects (a few cm in size) are typically at frequencies that fall within
the range that can be suppressed using fast negative feedback (bandwidths of
MHz with suitable electronics). However, this requires either optical
components that can handle hundreds of Watts (large diameter beams and
suitable crystals) or negative feedback on the light before amplification
(accounting for time delays).

### D.4 Diode Pumped Optical Laser for Experiments (DiPOLE) laser project

A kilowatt-average-power pulsed system Yb:YAG laser cooled by recirculating
helium gas was designed and constructed at the Central Laser Facility of the
Science and Technology Facilities Council (STFC) at the Rutherford Appleton
Laboratory444https://www.clf.stfc.ac.uk/Pages/D100.aspx. It produces pulse
energy of 150 J at 1 Hz, or 75 J at 10 Hz which corresponds to 750 W of
average optical power output. The system uses a slab geometry and has a large
beam size ($75\times 75$ mm), as is common for such pulsed laser facilities,
and it is noteworthy for being cooled by high-pressure helium (10 bar at 150
K) from a specially designed cryocooler.

The recirculating helium gas is cooled by LN, via a heat exchanger, and is
commercially available from GRE Ltd, Devon, UK555gre-ltd.co.uk. This design
with the crystal(s) in flowing helium gas is unlikely to be directly
applicable to a narrow bandwidth CW case, but is included here as an example
of an alternative approach to extracting the heat from the crystal surface.
People from the Physics Department in Oxford were involved in the early
stages.

##  Appendix E Pump lasers for Ti:sapphire laser

### E.1 Existing diode-pumped solid-state (DPSS) green lasers

Currently, the laser systems in MAGIS-100 and AION are pumped by a newly
developed Equinox system from M Squared Lasers (which has reliability issues).
The Verdi laser from Coherent Inc. has been used successfully for many years
(in Oxford and elsewhere). Whatever configuration is chosen, either coherent
combination of many sources or a few much higher power amplifiers, it seems
likely that a narrow bandwidth master laser will be required with the power
that goes beyond what is available from diode lasers. The operation of a diode
laser with extremely high-powered amplifiers would be a fragile setup.
Therefore, at least one Verdi laser, or otherwise, is required to provide 18 W
at 532 nm. (A 25 W DPSS laser is available from Spectra-Physics: Millennia
laser666https://www.spectra-physics.com/en/f/millennia-ev-cw-dpss-laser.)

### E.2 Experimental demonstration of 130 W at 532 nm

Extremely efficient generation of CW radiation at a wavelength of 532 nm has
been demonstrated that produced 134 W from a fundamental power of 149 W by
second-harmonic generation in an external optical resonator around a lithium
triborate crystal [13]. This was limited by the available fundamental power
and could be pushed to even higher powers.

Nowadays, the company Azurlight Systems produces $>100$ W fiber lasers and
lower power green lasers, but has a strong interest in developing higher power
systems in the green [private communication]. This company has a good
reputation for stable laser systems for use in cold-atom experiments, whereas
other fiber laser companies are more focused on industrial applications.

### E.3 Doubled fiber lasers

Fiber lasers commercially available from the company
IPG777https://www.ipgphotonics.com/en can reach 100 W of CW output and are
marketed mainly for industrial applications such as cutting and welding; they
are not commonly used for pumping Ti:Sa so their suitability would have to be
tested. Interestingly, IPG produces a quasi-CW laser that outputs 1 kW of
green light. At first sight, this pulsed laser might seem unsuitable, but the
frequency of the pulses can be up to 250 MHz; a repetition time of 4 ns is
much less than the response time of a laser cavity which therefore smooths the
output intensity. A demonstration of this Pulse-pumped CW tunable Ti:sapphire
laser was reported at the CLEO conference (2009) by Hsiao-hua Liu from KMLabs
[11], as noted in Appendix D.1. In Ti:Sa the upper laser level has a lifetime
of around $3.2\,\mu\text{s}$ which is relatively short for solid-state laser,
making relaxation oscillations less likely. Broadly speaking, although the
lifetime for spontaneous emission is critical for determining whether CW
population inversion can be obtained, it is not so important under operating
conditions where the simulated emission rate greatly exceeds that of the
spontaneous process. An important parameter is the lifetime of the photon in
the optical cavity (cavity lifetime $\tau_{\text{cav}}$). The round-trip time
is given by the speed of light divided by the length of a round trip, e.g.
$L/c=2$ ns for $L=0.6$ m. The lifetime for photon in the cavity equals this
multiplied by the number of round trips which can be approximated as the
finesse, e.g. for an amplitude reflectivity of $R=0.85$, the finesse

$\mathcal{F}=\pi\sqrt{R}/(1-R)=20\ ,$

hence $\tau_{\text{cav}}=40$ ns.

This system with a storage time long compared to the period between pump laser
pulses could be analyzed using the usual set of coupled rate equations for the
photon number (or intensity) in the cavity and population inversion density
used to describe Q-switching in laser physics texts [14, pp. 188–193], but
that is beyond the scope of this report. The likely conclusion of such an
analysis is that there is a benefit in making the cavity longer to increase
the photon lifetime. A (folded) cavity length of tens of meters could be
achieved, increasing the cavity lifetime to $2\,\mu\text{s}$ which smooths out
fluctuations on shorter timescales. Even with such a long response time
compared to the smoothing, however, this seems a risky approach for a system
that must also has a narrow bandwidth, although the bandwidth requirement
becomes less stringent in inverse proportion to the pulse length.

##  Appendix F ‘Cost benefit’ analysis

This preliminary survey of various possibilities indicates possible choices
such as the coherent addition of many tens of standard room temperature laser
systems ($<10$ W each) or higher power ($>100$ W) systems (e.g. cryocooled
lasers). Currently it appears that the former approach is the default in
MAGIS-100 and AION and it remains to be seen how far it can be pushed in terms
of reliability and cost (any of these approaches is likely to exceed 1 million
currency units: $, £ or €). Cryocooling is a ‘standard’ technique in kilowatt
laser systems but requires a bespoke design and the associated research and
development effort.