# Storage of Energy in Disk-Shaped Laser Materials

Authors: Dmitrii Kouznetsov
Journal: Research Letters in Physics
 (2008)

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## Abstract

Thin disk laser is analyzed, assuming that the heat is drained in the same direction, as the optical pulse, withdrawing the stored energy, comes.
Amplified spontaneous emission, the background loss, and overheating are taken into account with simple model. The scaling laws of the basic parameters are deduced. For the case of fixed repetition rate, the upper bound of thickness is obtained. Key parameters are suggested. The key energy parameter is promoted as criterion for evaluation of different laser materials for the high energy, high mean power disk lasers. The maximum energy per active element is estimated. For the scaling up the power and/or energy withdrawn from a single active element, the background loss should scale down inversely proportional to the cube of the background loss. This scaling law gives the criterion whether the heat should be drained in the direction orthogonal to the beam that withdraws the energy.


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## Body

## 1. Introduction

Nobody can tabulate numerically the performance of a
laser as a function of tens of variables, describing the size and shape of the
pumped region, spectra of pump, and that of signal, the spatiotemporal profile,
and so on. The qualitative analysis, that allows simple estimate of the basic
parameters in the analytic form, is an important complementary to the detailed
simulation. Such estimates cannot be substituted by the detailed simulations.
(Similarly, ultra-high-resolution objective cannot substitute a low-NA
lupa with long depth of field). The result of a qualitative
analysis can be expressed with analytic estimate, or “scaling law.”In this paper, scaling law means a mathematical
expression that shows the general trend of variation of parameters of a laser,
as it is designed for higher and higher power, or higher and higher energy.The scaling law appears, when only few physical
effects are dominant in the limit of the power or energy. For identification of
dominant mechanisms in the scaling of energy, various effects can be estimated
independently. In this paper, the limit due to the ASE, overheating, and the
background loss is considered in the simplest possible form; other effects are
neglected.Propagation of the amplified light across the thin
slab of active medium is typical for the disk lasers, whenever the active
mirror configuration [1–3] is used, or the light just crosses the set of pumped
slabs (Figure 1). In these cases, the amplification of the signal is weak
compared to that of the amplified spontaneous emission (ASE), which can have
long path in the gain medium. The special efforts are required to suppress the
ASE, and give all the advantages to the signal, propagating across the slab. In
particular, the signal, going across the medium, should have small background
loss.Set of active mirrors and set of active slabs, crossed with signal that withdraws the energy.
(a)(b)For the continuous-wave disk lasers,
roughly, at the power scaling, the loss should be reduced inversely proportional
to the cubic root of the desired power [4–6]. For power scaling at fixed value of the background
loss, the heat should be drained in the direction orthogonal to the propagation
of the laser beam (Figure 2); then other mechanisms (perhaps, deformation of
wave front due to nonlinear refraction) may become dominant.Figure 2
Robust configuration of slab laser.Figure 3
Sizes of the active slab and the longest bouncing ray.Various processes can contribute to the background
loss, including absorption and scattering at unwanted dopants inside the bulk
medium. The scattering at the surface seems to be most important among
dissipation processes that contribute to the background loss. For the estimate
of the maximal
mean power, the background loss determines minimal gain, at which
the energy still can be withdrawn from the excited medium; through ASE, this
gain limits the size of the pumped region; then the ability to drain the heat
limits the mean power.In this paper, assuming the fixed repetition rate, we
show, that similar relation keeps for pulsed operation; the background loss
should be reduced inversely proportional to the cubic root of the desired
energy. This case is important for the projects of nuclear fusion power plant
with laser driving [7–9].
This may be considered as an argument in favor of configuration shown in Figure 2, where the signal light propagates in the active medium in the direction
orthogonal to the direction of drain of heat.


## 2. Effective Path of ASE

ASE is especially important at the stage of storage of
the energy in the active medium, just before the signal pulse comes. Assume that
the ASE is efficiently absorbed (or even recycled) at the edges of the active
medium. Then, its effect on the excitation can be taken into account with scaling of
the effective lifetimeτ that becomes τexp⁡(−GS), where G is gain and S is some effective path of ASE in the gain
medium. Following [4–6], assume that the effective path S is of order of half of the maximal path of ASE
in the medium. More optimistic estimate for S (quarter of the maximal path) is suggested in
the comment [10], but
the correction coefficient is not yet justified. Due to internal
reflection (Figure 1), for the rectangular slab, the maximal path can be or
order of 2L; so, I take S=L for the estimates.The evolution of numberN of excitations in the gain medium can be
approximated with equation(1)dNdt=Pℏωp−Nexp(GL)τ,where P is power of pump, absorbed in the gain medium
and ℏωp is energy of photon of pump. Assume that pump
pulse acts during time t1, then during the short time t2−t1, most of energy stored in the gain medium is withdrawn with a by-passing pulse
at the lasing frequency, and the slab is allowed to cool during time t3 before the next pulse (Figure 4). Gain G can be estimated as follows:(2)G=NσL2h,
where σ is effective emission cross-section at the
lasing frequency. As in the case of continuous-wave operation,
it is convenient to use dimensionless variable u=GL, that has sense of the transverse-trip gain. Combining (1) and (2), I
get(3)dudt=PσAPL2−u⁡exp(u)τ,where A=L/h is an aspect ratio of the slab of the active
medium; A≫1. The steady-state solution u=u1 of (3) can be expressed as follows:(4)u1exp⁡(u1)=σAτℏωpL2P.Figure 4
Periodic evolution of numberN of excitations.For analytical estimates of orders of magnitude, I
assume that the transverse tripof
order of value u1 is achieved after time of order of(5)t1=N1ℏωpP=τexp⁡(−u1).There is no reason to keep the medium pumped much
longer than timet1, the additional increase of inversion of population is exponentially small.


## 3. Laser Action

Assume, that after timet1 by (5), the strong pulse comes in order to
withdraw the energy of excitation of the medium. During this pulse, the
evolution of number N of excitations in the gain medium and that of
number W of photons in the pulse can be described with
system(6)dNdt=−gF,dWdt=(g−β)F,where β is background loss, F=F(t) is flux of photons at the lasing frequency (ℏωsF is power in pulse that withdraws the energy),
and g=Gh=u/A is the single-trip gain, which is assumed to
be small, g≪1.
This assumption is justified by the condition A≫1;
at large values of g, u is huge, and the ASE becomes exponentially
strong. As the withdrawal is quick, I neglect both the ASE and the pumping
(even if it is still on until time t2)  during the laser action. Then(7)dWdN=g−β−g=−1+βL2σN.Assume, at the beginning of the
pulse (time t1), the number of excitations N=N(t1)=N1,
and at the end of the pulse, N=N(t2)=N2.
Let the number of photons in the pulse be W1 before it passes through the gain medium, and W2 after the amplification. The number of withdrawn
photons can be expressed as follows:(8)w=W2−W1=(−N+βL2σln⁡N)|N=N1N=N2.The number of excitations is related with gain;(9)N=L2σAu=L2σg.Assume the maximal excitation at
the arrival of pulse:(10)N1=L2σAu1.This is optimistic estimate,
because the maximal number of excitations is achieved only asymptotically,
while most of pump energy is dissipated with ASE.Let the pulse end, when the round-trip gaing becomes equal to loss β.
Then(11)N2=L2σβ.Substituting (10) and (11) into
(8) gives(12)w=u1L2σA(1−βAu1+βAu1ln⁡βAu1).
Expressing L2 from (4), we estimate the withdrawn energy:(13)E=ℏωsw=ωsωpPτexp⁡(−u1)η(βAu1),
where η is an elementary function,(14)η(z)=1−zln⁡ez,
shown in Figure 5. It has
sense of quantum efficiency; its argument(15)z=βAu1=βu1Lh
determines the self-similarity
of pulsed disk lasers. For the efficient operation, parameter z should be small, z≪1. For example, at z>0.01, the quantum efficiency cannot exceed 95%; and at z>0.1, the quantum efficiency cannot exceed 67%.Figure 5
Quantum efficiencyη(z) versus z by (14).At the self-similar scaling of energy, the product of
the background lossβ to the aspect ratio A=L/h should remain small constant; this should be
interpreted as law of scaling of
energy of disk lasers.


## 4. Heat Removal

The timet3 of cooling after a pulse is determined by the
repetition rate. For example, the repetition rate of several Hertz is required
for the application in the laser nuclear fusion electric plant [7–9]. There is an important
question, how much energy can be withdrawn from a single active element at given
repetition rate, without overheating. Such a maximal energy may be a limiting
factor for the application of the disk lasers, while the heat sink is realized
in the same direction, as the propagation of the signal pulse. In this section,
we estimate the maximal energy that can be withdrawn at given repetition rate 1/t3.The maximal mean power of pump, that can be delivered
to the active slab, is proportional to the area and inverse proportional to its
thickness. The coefficientR of proportionality is the thermal loading
parameter [4–6]:(16)t1t3P=RL2h.Two mechanisms may contribute to
this limitation: the thermal shock and the overheating [11, 12]. At strong pumping, the slab
may crack; and even if it does not, the laser action becomes nonefficient, if
the temperature becomes comparable with the quantum defect. There is no
established value for the coefficient R;
the estimates may vary for orders of magnitude, but this quantity is limited
for any given laser material.Using estimatet1=τexp⁡(−u1) and definition of A=L/h,
(16) can be rewritten as follows:(17)P=RAt3τLexp⁡(u1).Equations (17), (13), and (4)
determine the basic properties of the disk lasers, at the scaling up the
withdrawn power.


## 5. Scaling Laws

The deduction
can be drastically simplified, using the dimensionless variables. Let(18)Ed=ωsωpR2σt32ℏωp,Pd=R2σt32ℏωpτ,Ld=Rσt3ℏωp, combining (17) and (4), I get L=(RA2t3σ/ℏωpu1);P=(R2A3t32σ/ℏωpu1τ)eu1; then use of variables(19)ε=EEd,p=PPd,ℓ=LLd,allows to express all the
parameters in terms of β, z,
and u1.
Equations (17), (13), and (4) become(20)p=A3⁡exp(u1)u1ε=A3η(z)u1ℓ=A2u1. Using (15) and h=L/A gives(21)PPd=p=(zβ)3u12exp⁡(u1),(22)EEd=ε=(zβ)3u12η(z),(23)LLd=ℓ=(zβ)2u1,(24)hLd=zβ. From (24) it follows that no disk laser can
operate on thickness h≥Ld/β,
because z<1;(25)hk=Ldβ=Rσt3ℏωpβis key parameter for the
thickness of the pulsed disk laser. In order to operate with repetition rate 1/t3,
the disk laser should be thinner than hk.
Similarly,(26)Ek=Edβ3=ωsωpR2σt32ℏωpβ3,Pk=Edβ3=R2σt32ℏωpτβ3,Lk=Ldβ2=Rσt3ℏωpβ2. are key parameters for the
maximal energy, pump power, and the size. The exceed of energy above the key
parameter implies loss of efficiency and/or exponential growth of the required
pump power. For the efficient operation, parameters z and u1 should remain small. Such a scaling law is an
analogy of the scaling law by [13], that considered the optimization of maximal energy
per active element for the case when the concentration of active centers is fixed.The similar law of power scaling for CW operation is
considered in [4–6]; for that case, for the efficient operation, the
background loss should scale down inversely proportional to the cubic root of the desired output
power. We should greatly appreciate the confirmation of the scaling laws both
for pulsed and the CW disk lasers with detailed numerical simulations and/or
direct experiments. Especially important is the careful measurement of the
thermal loading parameter R and the background loss β;
the estimates of the maximal power and maximal energy are proportional to R2/β3.


## 6. Robust Configuration

The upper bound
for the energy above revers to the specific case of given repetition rate and
given background loss and given geometry (Figure1) of withdrawal of energy and
drain of heat. This upper pound is not general limit for solid-state lasers.
The energy can be withdrawn in the direction, orthogonal to the direction of
heat sink. Then, the efficiency is not so sensitive to the background loss β.Parameteru determines, how fast the energy should be
delivered to the gain medium. The pump power required grows exponentially with u1;
so, values of u1 of order of unity or even smaller look
reasonable. However, if the power available exceeds ratio E/τ for orders of magnitude, larger values of u1>1 may be used too; but in this case, it may
have sense to distribute the available power among
several lasers in such a way that each one works at small values of parameters u1 and z.Assuming fixedu1,
from (2) and (16), the required size L can be estimated as follows:(27)L=(E2σℏωp(ℏωs)2t3Ru1)1/3,at thickness of this
order(28)h=L2Rt3ℏωsEℏωp.At long interval between pulses(29)t3≥ERωsωpL,there is no need to make the
medium thin-disk shaped, it could be a cube (or cylinder) as well; the medium
has enough time to cool and the estimates above are not valid. In such a way,
the estimates (27) and (28) have sense for the configuration shown in Figure 2
at high repetition rate or high energy of pulses, that is, at high mean power.
Even in this case, the ASE, overheating, and background loss do not set general
limit of the energy that can be withdrawn from a single active element. This
may be important for application of the ceramic materials that allow to
manufacture wide pieces of active media.


## 7. Conclusions

The scaling of energy of a slab laser is considered
under assumption that the drain of heat occurs in the same direction as
propagation of the amplified light. The exponential growth of ASE is described
with effective lifetimeτexp⁡(−u), where u is transverse-trip gain. The background loss β and thermal loading parameter R are used to characterize the laser material.
Repetition rate 1/t3 is assumed to be fixed parameter, that follows
from the intended application of the pulsed disk laser. For the energy scaling
up, the background loss should be scaled down and inversely proportional to the
cubic root of the desirable energy. Over vice, the
pump power required scales up as exponential of the square root of the output
energy.The scaling laws
can be expressed in terms of the key parameters (22), (21), (23), (24).
For the efficient operation, the energy, power, size, and thickness should
remain small, compared to the key parameters.The approach suggested seems to be applicable to any
laser material, for which the thermal loadingR and cross-section σ can be estimated. No active mirror (Figure 1)
can operate at thickness larger than hk.
The key parameter Ek should be used at the evaluation of laser
materials for the energy scaling of disk lasers with fixed repetition rate.
Such evaluation is important for the design of the driver for the nuclear
fusion power plant. If the required scaling down of the background loss is not
possible, then another architecture of the laser should be considered, where
the light, withdrawing the energy, passes the slab in the direction, orthogonal
to the direction of drain of heat [14], as it is shown in Figure 2.



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*Source: 717414-2008-07-31.xml*