## Abstract

We use analytic expressions and simulations to examine a model laser gain element formed by integrating diamond and a solid state laser material, such as, Ti:sapphire. The gain element is designed to provide in a single composite structure the thermal management capabilities of diamond and the optical amplification of the laser material. The model results indicate low temperature and a specific radial dependence of the heat transfer coefficient at the material interfaces are needed to access the highest average powers and highest quality optical fields. We outline paths designed to increase average output power of a lowest order mode laser oscillator based on these gain elements to megawatt levels. The long term goal is economically viable solar power delivered safely from space. The short term goal is a design strategy that will facilitate “proof of principle” demonstrations using currently accessible optical pump and thermal management capabilities.

© 2003 Optical Society of America

## 1. Introduction

Transforming sunlight in space to highly coherent light and transmitting that light safely to Earth and other locations in space at megawatt power levels is of current interest [1]. Such a strategy requires lasers that can operate efficiently in the space environment. Terrestrial lasers have been constructed that operate at megawatt average power for short periods of time [2]. These existing megawatt lasers, however, do not appear to be practical for transforming sunlight in space into coherent light.

Solid state materials offer the advantages of simplicity, compact character, and durability. However, thermal shock, thermal lensing, and thermal stress induced birefringence limit the average power of high quality laser emission from solid state lasers to tens of kilowatts [3]. These thermal management problems are due in large part to a lack of laser materials having sufficiently high thermal conductivity,*κ*, low thermo-optic coefficient, dn/dT (n is refractive index and T temperature), and low thermal coefficient of expansion, *α*.

Diamond has the needed material properties, but does not now exhibit useful laser transitions and appears unlikely to do so in the future. We explore a strategy that integrates a laser gain material, e.g., Ti:sapphire, with diamond. We seek to access the thermal management capabilities of diamond and also the optical amplification capabilities of the laser material in a single composite element. Our analytical and simulation based modeling of a prototypical gain element implies that laser oscillators based on such a gain element can be tested at relatively low average power, but eventually scaled to megawatt average power. The best performance appears to call for low temperature, e.g., 100K, and novel structures.

Systems that operate at low temperature and also at megawatt average power are technologically demanding. Prototypical terrestrial systems handling this amount of power at low temperature are now, however, being constructed. A current example is a 10 MW superconducting transformer for that operates at (77K) [4].

Sunlight in space can provide the needed megawatts of optical power in the green part of the spectrum. One design of the optical pump we are considering calls for four collecting mirrors for each of 20 gain elements (see Section 7). Each mirror requires a diameter of ~12.5 m. Such an optical mirror is large, but nevertheless small compared to other large lightweight mirrors, e.g. of 30 m diameter, currently being developed for other applications in space [5].

The solution we propose calls, in effect, for more than an order of magnitude increase in *κ*, and similar decreases in *α* and dn/dT, relative to values representative of the best current laser materials. Improvements in conventional laser materials appear unlikely to offer the needed changes. We outline here a gain element design that lends itself to “proof of principle” experiments at currently accessible optical pump and thermal management capabilities.

#### 1.1 Significance and role of diamond

Diamond, and apparently only diamond at low temperatures, offers the needed multiple orders of magnitude improvement in *κ*,*α*, and dn/dT. The upper limits on average power set by thermal shock and thermal stress induced birefringence vary as *κ*/*α*, and the upper limit set by thermal lensing varies as *κ*/(*dn*/*dT*). Increases in *κ*, and reductions in * α* and dn/dT thus have a multiplicative impact on the upper limits on average power.

Not only are the upper limits on average power set by thermal shock, thermal lensing, and thermal stress induced birefringence in diamond higher than those for typical laser materials by multiple orders of magnitude at room temperature, but *α* and dn/dT also decrease for diamond by some two orders of magnitude each as the temperature is decreased from 300K to 100K. Also *κ* increases by almost an order of magntiude over this temperature range.

Diamond used at 100K consequently offers 3–4 orders of magnitude, or more, increase in the upper limits on average power set by thermal shock, thermal lensing, and thermal stress induced birefringence. Advances in growth and processing of synthetic diamond [6,7], evidence for a zero coefficient of thermal expansion in high purity single crystal diamond near 90K [8], and the advantages of isotopically enriched single crystal diamond [9] are also encouraging.

#### 1.2 Model gain element based on integrated diamond and solid state laser material

Diamond attached as a simple heat sink to a laser rod also does not appear to be a viable solution. We consequently explore and evaluate strategies where diamond is integrated with a laser gain material so that the resultant structure approximates a laser gain element composed of a *single material having the optical amplifying properties of the solid state laser material and at the same time the heat and stress management capabilities of diamond*.

For this discussion we regard the solid state gain material and the diamond as having the geometry of disks with the thickness of the disk small compared to the diameter. More general shapes are not ruled out. The main concern is to choose geometries and boundary conditions such that the waste heat generated in the solid state gain material flows in an axial direction from the solid state gain material into the diamond, and then radially outward in the diamond.

Good physical and thermal contact between the diamond and the solid state material are required at the planar interface. Experimental measurements of the rate of heat transfer at interfaces between solid state materials are available from prior work [10]. In particular, experiments with heat transfer between sapphire and materials other than sapphire have successfully demonstrated effective heat transfer between the sapphire and dissimilar materials [10,11]. The details of specific strategies for construction of these interfaces are, however, beyond the scope of this current paper and are not addressed here.

We assume we can specify a spatial distribution of waste heat *γ*(*r*, *z*) generated in the solid state gain material, e.g., by the choice of the optical pump distribution, where that distribution is independent of *r* and *z*. That is, *γ*(*r*,*z*)≡*γ*_{o}
where *γ*_{o}
is measured in Watts/cm^{3} and has a constant positive value for all locations within the laser gain disk. The outer rim of the laser gain disk is defined to be insulated from contact with any thermal reservoir. The outer rim of the diamond disk is defined to be in good thermal contact with a thermal reservoir at low temperature, e.g., ~100K.

We explore two cases. In the first, we choose the heat transfer coefficient at the interface, *h*(*r*), between the diamond and the gain material to yield a temperature distribution in the gain material that is invariant with radial position, *T*_{s}
(*r*,*s*)=*T*_{sz}
(*z*) (Section 2.5, Fig. 7). In the second case, explored only in the finite difference simulations, we assume *h*(*r*)=*h*_{o}
where *h*_{o}
is constant, but allow *T*_{s}
(*r*,*z*) to vary with both *r* and *z* (Section 5, Fig. 6(b))

For a gain element used in a laser oscillator it is essential to minimize the losses at the interfaces. We show in a later section (Section 6) gain element structures that orient each of the interfaces, (between a given material and air, and also between the two dissimilar materials), at Brewster angle for a linearly polarized laser field.

In our analytical and simulation based models we neglect this Brewster angle feature and regard the diamond and laser gain material as right circular cylindrical disks with planar surfaces normal to the z axis, see Fig. 1. We also develop, in Section 5, a finite difference description of gain elements composed of multiple pairs of right circular cylindrical disks. The differences between the Brewster angle structures and these right circular cylindrical disk structures, as regards thermal managment, do not appear to be of critical significance at this stage of the model development.

## 2. Analytic model

We develop in this section a relatively detailed analytic description of the temperature distribution in a pair of diamond and solid state gain material disks. We define a unit cell consisting of one gain disk and one diamond disk. For the model we consider the pair of disks to be located in the middle of a gain element composed of a sequence of a number of such pairs of disks. We consequently neglect end effects and assume a temperature distribution that is locally symmetric about the midplane of each individual disk. This midplane is defined to be oriented normal to the z axis and to bisect a given disk. Given this assumed symmetry we describe the general system by developing expressions characterizing one half of a gain disk and the adjacent half of a neighboring diamond disk.

#### 2.1 Properties of the model

As discussed above the gain element is described as composed of a sequence of pairs of right circular disks where each disk exhibits a temperature distribution as indicated schematically in Fig. 1. The point on the z axis and at the interface between the gain disk and the diamond disk defines the zero of this coordinate system, *z*=0,*r*=0. We define the temperature *on the diamond surface* at *z*=0,*r*=0as *T*_{d}
(0,0). We assume cylindrical symmetry of the temperature distribution about the z axis. While a variety of solid state gain materials could be used for the purpose of these model calculations we assume the gain material is Ti:sapphire. We anticipate using a strategy, described more fully in Section 2.5, that yields a temperature *on the sapphire surface* at z=0, *T*_{s}
(0,*r*)=Δ*T*_{ds}
+*T*_{d}
(0,0) that does not vary with radial position. For this case of radially invariant temperature in the Ti:sapphire, and given the small variation of temperature in the diamond with z, we treat the problem of describing the temperature in a given disk as two independent one-dimensional problems [12]. For this case the temperature in the sapphire disk varies primarily with *z*, while the variation with **r** is small. In the diamond, the variation of temperature with *r* is important, while the variation with z is small.

#### 2.2 Temperature distribution in the diamond disk and in the Ti:sapphire disk

The differential equation describing the z dependent component of the temperature *T*_{sz}
(*z*) in the gain material disk is [12,13]

We assume *κ*_{s}
is independent of temperature over the range of temperature variation in the specific examples examined here. We use for *κ*_{s}
a mean thermal conductivity of 4 W/(cm-K). The z dependent variation in the temperature in the sapphire is

where z varies from 0 to - *ℓ* as the observation point moves from the diamond-sapphire interface along negative z into the sapphire disk. For the case of constant heat flow across the interface the heat transfer coefficient for the interface at *z*=0,*r*=0 is *h*(0,0)=*γ*_{o}*ℓ*/Δ*T*_{ds}
_{0}, see Section 2.5 [10,12].

The temperature in the sapphire has a quadratic z dependence with a maximum at the center of the disk (*z*=-*ℓ*) of *T*_{sz}
=*T*_{d}
(0,0)+Δ*T*_{ds}
_{0}+*γ*_{o}*ℓ*
^{2}/2*κ*_{s}
, see Fig. 2. We are only concerned with the net temperature dependent differential optical phase shift experienced by the light propagating through the sapphire disk in the z direction. We use the average value Δ*T̄*_{sz}
=+*γ*_{o}*ℓ*
^{2}/3*κ*_{s}
. The net contribution to the phase shift caused by this z dependent temperature in the Ti:sapphire is Δ*$\overline{\phi}$ _{sz}*≅(2

*πℓ*/

*λ*)

*n*

_{s}(

*dn*/

*dT*)

_{s}Δ

*T̄*

_{sz}. Since this index change is independent of

*r*it will not affect the wavefront other than causing a delay uniform in

*r*. This mechanism does not influence the upper limits on average power due to thermal lensing.

For a density of waste heat *γ*_{o}
=30 kW/cm^{3} (Section 3), and *ℓ*=0.05*cm*, *κ*_{s}
=4*W*/(*cmK*), Δ*T̄*_{sz}
is small, ~6.3K. We assume that the motion of the surface of the sapphire disk in the z direction is weakly constrained and neglect stress due to this axial variation of temperature.

#### 2.3 Temperature distribution in the diamond

For the diamond disk the differential equations and solutions are

The temperature in the diamond has a quadratic dependence on *z* with a minimum value at the midplane of the diamond disk, *z*=+*ℓ*, of -*γ*_{o}*ℓ*
^{2}/2*κ*_{d}
. The integrated temperature dependent contribution to the phase shift in the diamond is proportional to Δ*T̄*_{dz}
=-*γ*_{o}*ℓ*
^{2}/3*κ*_{d}
. For *κ*_{d}
~100 W/(cm-K) this contribution is small, 0.25K, and appears to not be important in evaluating upper limits on power. The radially dependent contribution enters as a decrease in temperature with increasing radial position *r* to *T*_{d}
(*z*=0,*r*=*r*_{o}
)=-*γ*_{o}${r}_{o}^{2}$/4*κ*_{d}
at the outer edge of the diamond, Fig. 2. For *r*_{o}
=0.6 cm, *κ*_{d}
=100 W/cm-K and *γ*_{o}
=30 kW/cm^{3} (Section 3), *T*_{ds}
(*r*=*r*_{o}
)=Δ*T*_{dr}
~27 K. This approximates the finite difference model prediction, Section 5.

#### 2.4 Thermal lens compensation

An option for thermal lens compensation occurs in that one can choose a radially dependent distribution of waste heat and an interface heat transfer coefficient *h*(*r*)designed specifically to introduce a small variation in the radial temperature profile in the sapphire

This produces a long focal length negative thermal lens in the sapphire which can be adjusted to compensate the long focal length positive thermal lens in the diamond.

The thermal lens in the sapphire and the thermal lens in the diamond are both directly proportional to the waste heat density *γ*_{o}
. Thus compensation of the positive thermal lens by the negative thermal lens will scale to lowest order with varying pump power. This lens compensation is optional and may not necessarily be the best strategy since the distortion due to thermal lensing in the diamond can be unimportant for many applications.

#### 2.5 Radially varying heat transfer coefficient h(r)

As discussed above the case of heat flow across the interface between the diamond and sapphire, where the rate of heat flow is independent of radial position, is of interest. The heat flow per unit area is [12,13]

Here Δ*T*_{ds}
(*r*) is the discontinuous change in temperature at the diamond-sapphire interface, i.e. at *z*=0 and radial location *r*. We write the temperature difference between the diamond and the sapphire as Δ*T*_{ds}
(*r*)=Δ*T*_{ds}
_{0}+Δ*T*_{ds}
(*r*) where Δ*T*_{ds}
_{0} is the temperature difference between the diamond and sapphire at the location *z*=0,*r*=0. The assumption of uniform waste heat introduced at a density *γ*_{o}
, and radially independent heat flow across the interface requires that *q*_{ds}
(*r*)=*γ*_{o}*ℓ*. For constant heat flow and constant temperature in the gain disk with radial position the heat transfer coefficient then has the form

As an estimate of the maximum heat transfer coefficient at the diamond-sapphire interface we take the value *h*_{max}
~150 W/(cm^{2}-K) at reported for a high quality sapphire surface and lead as the other material at ~100K. In that experimental case good contact between the lead and the sapphire was achieved by melting the lead onto a high quality sapphire surface and then allowing the resulting structure to solidify [10]. Our finite difference simulations and plots based on the analytic results use Eq. (7) in modeling *h*(*r*).

For the model gain element explored here we postulate that, during construction of the gain element, the heat transfer coefficient at the diamond gain material interface can be given the specific radial dependence described by Eq. (7). Since the heat transfer coefficient can be decreased by decreasing the contact area at the interface [10] the radial dependence described by Eq. (7) could be achieved, e.g., by selectively maximizing the contact area at *z*=0,*r*=0and then decreasing the mean contact area at the diamond-sapphire interface as a function of radial position, while otherwise minimally altering the optical properties of the interface.

## 3. Properties of a representative gain element

For the purpose of discussion we consider a representative gain element composed of multiple pairs, e.g., ~10, of diamond and sapphire disks. The thickness of the individual disk is sufficiently small, e.g., 1 mm, that the waste heat is removed axially from the Ti:sapphire with a small variation in temperature in the z direction, as discussed above. The number of disks is made sufficiently large to access the net gain, e.g., >0.5%, needed from the gain element. Our goal is to identify a strategy based on this composite gain element that allows testing, in the short term, of the strategy by using currently available resources at moderate average power. We also seek, in the long term, to have this same strategy enable scaling to ~1 megawatt of power from a lowest order Gaussian mode laser oscillator. A number of these gain elements, e.g., 20, can be used within the laser resonator provided the total optical distortion remains within acceptable bounds, Section 7.0. We develop here values of key parameters characterizing such a set of gain elements.

#### 3.1 Optical field intensity within the laser oscillator

For efficient operation the intensity of the stimulating fields *I*_{st}
needs to be larger than *I*_{sat}
=*hν*/*στ*. Here *h* is Planck’s constant, *ν* is the optical frequency, σ is the stimulated emission cross section for the gain transition, and *τ* is the excited state lifetime for the upper laser level (for Ti:sapphire *σ*=4×10^{-19} cm^{2},*τ*=4×10^{-6} sec and *ν*~0.375×10^{14} Hz)). The saturation intensity for Ti:sapphire is consequently 2.5 MW/cm^{2} [14]. We will use *I*_{st}
=4*I*_{sat}
for this discussion, or *I*_{st}
~10 MW/cm^{2} as a guide in estimating a representative value for the density of waste heat *γ*_{o}
.

#### 3.2 Length of gain element

In general, we need an active medium having at least one dimension of the order of a reciprocal absorption length of the optical pump in the gain material. On the other hand distortion and loss typically increase with the length of the gain element. For Ti:sapphire the reciprocal absorption length for a good laser crystal is of the order of 1 cm. We will use a length L=1 cm produced by 10 disks of Ti:sapphire having a thickness of 1 mm. The disk diameter for our simulations we will use as 1.2 cm. This structure approximates the dimensions needed for either longitudinal or transverse pumping.

We will match this active material with an equal amount of diamond in the form of 11 disks of diamond of 1 cm diameter. The two end diamond disks have a thickness of 0.5 mm and the remaining 9 disks have a thickness of 1 mm. There is thus 1 cm of diamond and 1 cm of Ti:sapphire and the overall length of this nominal gain element is 2 cm. The diamond, of course, provides no gain, but serves an essential thermal management function. In general, in calculating the distortion due to a given material we will estimate the distortion caused by a particular material using 1 cm as the nominal length of that particular material.

#### 3.3 Density of waste heat

A waste heat density, *γ*_{o}
≈*I*_{p}*η*_{qd}
/*L*, where *I*_{p}
is the optical pump intensity, *η*_{qd}
is the quantum defect, and *L* is the length of the active material is left in the active medium. We assume the pump intensity is related to the stimulating intensity by *I*_{p}
(1-*η*_{qd}
)=*I*_{st}
(*G*-1). That is, we assume that every pump photon, less the quantum defect, contributes to the increment in intensity experienced by the stimulating field on passing through the active medium. We neglect additional sources of waste heat such as absorption by impurities, etc. This gives *γ*_{o}
=*I*_{st}
(*G*-1))*η*_{qd}
/[*L*(1-*η*_{qd}
)] as a lower limit on the waste heat density. Using as a lower limit G=1.006 and *η*_{qd}
=1/3 we obtain an estimate of *γ*_{o}
~30 kW/cm^{3} given the values of L and *I*_{st}
calculated above.

#### 3.4 Total output power

We seek an estimate of the average total output power obtainable from a symmetric confocal resonator where the mirror spacing is twice the Rayleigh length, *z*_{o}
=${\pi w}_{\mathrm{o}}^{2}$/*λ*. Here *w*_{o}
is the beam waist and *λ* is the wavelength. The relationship *r*_{o}
=2*w*_{o}
defines the minimum value of the radius of the gain element *r*_{o}
needed to avoid diffractive distortion caused by aperturing at the edges of the lowest order Gaussian mode. We assume that all the available power in the gain region of radius *r*_{o}
, aside from the quantum defect, is extracted.

The total amount of power that can be extracted is then *P*_{tot}
=${\sum}_{\mathrm{i}=1}^{\mathrm{N}}$
*P*_{i}
where *P*_{i}
≅*γ*_{o}${\pi r}_{\mathrm{o}}^{2}$
*L*(1-*η*_{qd}
)/*η*_{qd}
and *N*≤2*z*_{o}
/*mL*. This is 60 kW for an active medium volume of 1 cm^{3} and a quantum defect of 1/3. Here N is the number of gain elements included in the resonator. The parameter m gives the number of gain element lengths that are needed in the laser oscillator to accommodate each gain element. We use a nominal value of m=100 in this discussion. This number is larger than necessary, but allows adequate space should negative dispersion produced by prisms (Section 6), be needed for dispersion compensation [15]. The total power that we can extract from this model laser oscillator is then

For a gain element radius of ~0.6 cm, m=100 and *λ*=0.8 microns Eq. (8) gives ~3.75 MW as an upper limit on average power set by this very approximate criterion.

Equation (8) illustrates the challenge of scaling solid state laser oscillators to high average power. The density of waste heat *γ*_{o}
is determined largely by the properties of the laser transition. The principal parameter available for increasing power is the radius of the gain element *r*_{o}
. Only diamond at low temperatures appears able to remove the large amount of heat over the relatively long radial distance while acceptably limiting stress and distortion.

## 4. Diamond vs. sapphire and YAG

We compare and contrast diamond with two materials commonly used for solid state lasers, sapphire and YAG. We address the degree to which the given material can remove the necessary amount of heat from the required volume while limiting the stress and distortion to the degree needed to access megawatt average power. We structure the comparisons in terms of waste heat at a density of 30 kW/cm^{3} and a gain rod of length 1 cm and radius 0.6 cm.

For diamond thermal conductivity we use *κ*_{d}
~100 W/(cm-K) throughout assuming operation near 104 K. We regard this as a conservative estimate. Values of *κ*_{d}
as large as 410 W/(cm-K) have been reported for isotopically modified single crystal diamond [9].

#### 4.1 Thermal stress fracture

The log of the thermal shock (thermal stress fracture) parameter P/L=8*πκ*(1-*ν*)*σ*
_{max}/(*αE*) is plotted in Fig. 3 for diamond, sapphire, and YAG as a function of temperature. Here P is the total power dissipated where stress fracture occurs, L is the length of the laser rod, *ν* is Poisson’s ratio, *σ*
_{max} is the tensile strength of the material, and E is Young’s modulus [3].

To avoid stress fracture the value of P/L must exceed the value indicated by the straight horizontal line (red) in Fig. 3. For this plot we assume a 1 cm length rod of radius of 0.6 cm dissipating waste heat distributed uniformly at a density of 30 kW/cm^{3}. The stress fracture parameter for diamond meets this criterion over the entire temperature range 80–300K, sapphire barely meets this criterion in the lower portion of the temperature range, and YAG does not meet this criterion at any temperature in the range 80–300K [3,9,16–18].

#### 4.2 Limits imposed by thermal lensing

Thermal lensing is caused by radial temperature gradients that combine with the thermo-optic coefficient dn/dT to cause undesirable focusing of the laser beam [3, 19–23]. An approximate expression for the thermal lens focal length is *f*_{T}
=2*κ*/[*Lγ*_{o}
(*dn*/*dT*)] where L is the length of the material [3]. The plots in Fig. 4 of log[*f*_{T}
] suggest neither sapphire nor YAG is adequate as regards the upper limit on average power set by thermal lensing. Diamond is acceptable; however, operation in the low temperature regime is required for the assumed conditions. Thermal lensing can be compensated (Section 2.4), but such strategies are preferably avoided.

#### 4.3 Limits imposed by thermal stress induced birefringence

A third mechanism that hinders scaling of solid state lasers to high average power with good beam quality is thermal stress induced birefringence [24–28]. Differential thermal expansion due to the radial temperature gradients causes a birefringence of the laser crystal. This induced birefringence depolarizes the laser field and results in loss. The fraction of the power in the beam lost due to such depolarization for diamond, given a lowest order Gaussian mode, is

where *p*
_{11} and *p*
_{12} are the photoelastic coefficients for diamond, *c*
_{11} and *c*
_{12} are the elastic stiffness coefficients for diamond, *ν* is Poisson’s ratio, *E* is Young’s modulus, and *P*_{h}
is the total heat dissipated in the rod [3]. Similar expressions can be derived or found in the literature for sapphire [26] and YAG [3]. Results are shown in Fig. 5.

The curves given in Figs. 3,4, and 5 imply that diamond, and only diamond at lower temperatures, is adequate to access the 3–4 orders of magnitude, or more, improvement needed in each of the upper limits set by thermal shock, thermal lensing, and thermal birefringence. The reason diamond does so well is that: (1) diamond typically offers better performance at room temperature than any other material as regards *α,ν* and dn/dT, (2) the upper limits imposed by thermal shock, lensing, and birefringence scale as the ratios, *κ*/*α* and *κ*/(*dn*/*dT*), and hence an order of magnitude advantage in each parameter leads to multiple orders of magnitude improvement in the upper limits on accessible power and (3) each of the parameters *α*,*κ* and dn/dT for diamond improve by an order of magnitude or more with decreasing temperature.

## 5.0 Finite difference model of multilayer gain element

We use finite difference modeling to examine the properties of a structure that approximates an actual gain element. We compare and contrast several gain element designs each performing the same heat dissipation task. One design is that of a conventional cylindrical rod composed solely of Ti:sapphire having a radius of 0.6 cm, a length of 1 cm, and dissipating a uniform density of waste heat 30 kW/cm^{3} solely through outward flow to the cylinder surface, Fig. 6(a).

A second design is a gain element composed of alternating Ti:sapphire and diamond disks having, at the diamond-sapphire interfaces, a heat transfer coefficient *h*(*r*) that is constant with radial position *r*, Fig. 6(b). The third design is the preferred gain element where *h*(*r*) has been given the specific radial dependence derived in Section 2, Eq. (7), that yields a temperature distribution in the Ti:sapphire constant with radius, Fig. 7.

#### 5.1 Gain elements having radially constant thermal conductivity

The case of Ti:sapphire alone is illustrated in Fig. 6 (a). The thermal conductivity of the sapphire near 100K is large, 4 W/cm-K, but not as large as the thermal conductivity of diamond, 100 W/cm-K. Here we neglect the variation of the thermal conductivity of sapphire with temperature. The large radial temperature gradient implies an unacceptably short focal length thermal lens.

The case of a gain element that integrates diamond with the solid state gain material and has a heat transfer coefficient at the interfaces that is constant with radial position is illustrated in Fig. 6 (b). Here we used *h*_{o}
=150 Watts/(cm^{2}-K) [10]. The diamond reduces the temperature excursion as compared to sapphire alone. The temperature excursion in the sapphire in Fig.6 (b) is~27 K as opposed to 660K as shown in Fig. 6(a).

This reduced radial temperature variation of 27K in the sapphire of the composite diamond-sapphire gain element is a drastic improvement as compared with sapphire alone. However, given the relatively large dn/dT for sapphire 4.4×10^{-6}K^{-1}, vs. 7×10^{-8} K^{-1} for diamond, this structure fails by a factor ~50 as regards adequately reducing thermal lensing.

#### 5.2 Gain element including diamond and having a radially varying heat transfer coefficient

Further reductions of the variation of temperature with radial position in the solid state gain material, in this example, Ti:sapphire, will assist in transferring more of the thermal management task to diamond and will further reduce loss and distortion of the optical fields. We show in Fig. 7 a finite difference simulation for a gain element where the radial variation in the heat transfer coefficient, see Eq. (7), is designed to yield a radially independent temperature variation in the gain material.

For this integrated diamond and Ti:sapphire structure designed so that the temperature in the sapphire is constant with radial position, the diamond alone exhibits a radial variation in temperature and consequently the thermally induced shock, lensing, and stress birefringence are those characteristic of diamond. The upper limits on the accessible average power are consequently increased by 3–4 orders of magnitude, or more, as compared to the case of a typical solid state gain material used alone and at room temperature.

Thermal shock in the diamond is not a concern given the extraordinary robustness of diamond against thermal shock. The magnitude of the residual thermal lensing and birefringence effects in diamond at low temperature appears to be sufficiently small to allow total average power of a megawatt as high quality optical fields. Further reductions in the undesirable consequences of the remaining small radial variation of temperature in the diamond may be accessible through compensation strategies or the use of high purity single crystal diamond in the temperature region where the coefficient of thermal expansion passes through zero, ~90K [8] or through use of isotopically modified single crystal diamond [9].

## 6. Brewster surface structures

The right circular disks oriented normal to the optical axis used for these finite difference simulations would produce unacceptable reflection losses for a laser oscillator. This problem can be avoided by constructing the gain element so that a linearly polarized laser field encounters each interface between different materials and each interface between a material and the surrounding space at Brewster’s angle. We illustrate such gain elements in Fig. 8.

#### 6.1 Integrated diamond and solid state material laser element

The goal of having all interfaces at Brewster’s angle and using two different materials calls for structures of the type shown in Fig. 8. The views shown here represent a planar cross section taken through the gain element where the intersecting plane includes the axis of the gain element and also the linear polarization vector of the laser field. The various interfaces are oriented so that the linearly polarized laser field always intersects each interface (between different materials, or between material and air) at Brewster’s angle. The wedged sections at the ends of each element are needed to satisfy the Brewster angle criterion at both the interfaces between different materials and the interfaces between material and free space.

The complete gain element is regarded here as cylindrical, with the cylinder axis approximately collinear with the direction of propagation of the laser field within the gain element. The angular orientations of the interfaces in Fig. 8 (a) for the “Brewster” configuration were calculated for diamond and sapphire as the two materials. The laser wavelength was assumed to be 0.8 microns. The difference in index between doped and undoped sapphire is regarded as unimportant.

This design strategy can be applied to other gain materials and other wavelengths with appropriate changes for the different wavelengths and different refractive indices. The version of the gain element shown in Fig. 8 (b) illustrates one strategy that, in effect, combines a prism shape with a parallelepiped shaped gain element so as to provide means for dispersion compensation [15]. A similar consequence, with an additional set of interfaces between the solid state material and air, could be achieved by using a stand alone prism in combination with the gain element shown in Fig. 8(a).

## 7. Modular approach to megawatt total power

For the model explored here multiple gain elements are required in the laser oscillator to access a total average power of a megawatt. We illustrate a possible strategy in Figs. 9 and 10. We refer to this structure as the “backbone laser” configuration. The term is derived from the multiple similar gain elements as illustrated in Fig. 9. This set of regularly spaced elements supports, e.g., the lowest order Gaussian mode of an approximately confocal resonator.

In this model system the individual gain element is regarded as: (1) held at a temperature of ~100K, (2) having the optimal radial dependence of *h*(*r*), (3) containing approximately ten pairs of diamond and Ti:sapphire disks of radius *r*_{o}
=0.6 cm, and (4) dissipating a density of waste heat, γ=30 kW/cm^{3}. The waste heat is generated initially in the Ti:sapphire, flows into the diamond in an axial direction, and is then removed via outward radial flow in the diamond. This strategy keeps thermal shock, lensing, and birefringence loss within acceptable ranges as discussed in Section 4 for each of the multiple, e.g., 20, gain elements. In principle, for this model system, this allows a total average power out of the resonator of ~1 MW. Experiments on the use of cooled mirrors to handle the intraresonator intensities, larger than 1 MW/cm^{2}, are encouraging [29].

The prism shaped element in Fig. 8 and Fig. 10 suggests a possible strategy for adding dispersion compensation to the laser oscillator. In general, net negative dispersion in the resonator using prisms requires a spacing between elements of some two orders of magnitude times the length of the material in the element [15]. In this case for our nominal element length of 2 cm the element spacing required would be ~2 m. For 20 elements this calls for a resonator length of 40 m which is satisfied by our 70 m long resonator.

The presence of the nonlinear gain medium internal to the prism shaped element affects the analysis of the system in ways we have not attempted to address in this present work. The consequence may be favorable or not depending on the particulars of a given application. The prism element can be separated from the gain element and used in a more traditional manner [15] as a strategy for avoiding this situation.

The question of whether pulse stretching and compression are needed depends on the properties of the resonator and the peak intraresonator power. The diamond and sapphire both have a damage threshold that is high, ~10 terawatts/cm^{2}, and the beam area of the megawatt class system is relatively large ~1cm^{2}. For our model system, if no other factors must be considered, intra-resonator peak powers up to 10 terawatts are, in principle, allowed. This suggests the possibility of high peak output powers as well as high average output powers, possibly without a need for stretching and compression of pulses within the resonator.

## 8. Conclusions

We conclude that, within the framework of this model system, diamond and a solid state laser gain material, such as Ti:sapphire, can be integrated so as to obtain the gain characteristics of the laser material and *also* the heat and stress management capabilities of the diamond in a *single* composite gain element. A laser oscillator operating in the lowest order Gaussian mode of a confocal resonator using a number of these gain elements, e.g., 20, appears to offer high quality optical power at average powers of the order of a megawatt.

Modelocking of this laser oscillator to achieve megawatt average power and also terawatt peak power from the same laser oscillator appears to be allowed by fundamental laws, and by the properties of this integrated diamond and Ti:sapphire laser element.

#### 8.1 Low temperature operation

This approach to megawatt average power appears to require operation of the diamond-sapphire gain element at low temperature, e.g., ~100K. Room temperature operation is allowed and is predicted to produce high quality optical fields at higher output power than conventional laser materials, but is not expected to produce the 3–4 orders of magnitude, or more, improvement expected near 100K.

#### 8.2 Long resonator

This strategy for producing high average power and high quality optical fields calls for the use of a relatively long laser resonator, e.g., 70 m. This long resonator is important both in producing a large beam radius needed for the relatively large area gain element and also for providing a long intra-resonator path that allows adequate room for multiple, e.g., 20 gain elements and the associated optical pump and thermal management components.

Maintaining the alignment of a 70 m long laser resonator is challenging. However, more stringent alignment conditions are being met in existing systems, such as the 4 km long LIGO interferometer used for gravitational wave observation. This latter system, of course, has the advantage of being located in a specifically stabilized environment[30].

Mechanical stability of relatively large optical structures in space, such as the Hubble space telescope, has been demonstrated. The design and realization of the large optical structures in space with the mechanical stability needed for the proposed laser oscillator will be challenging. We see, however, no fundamental barriers to constructing such structures and maintaining the needed degree of mechanical stability.

#### 8.3 Structured interface between the diamond and the solid state gain material

In this model system the optimum performance, e.g., the highest average power, with the lowest optical distortion and loss call for a particular radial dependence of the heat transfer coefficient. Constructing such an interface with low optical loss and the particular radial dependence appears to be a demanding practical task.

The physical differences in the transmission of heat across a solid state interface and the transmission of optical fields across such an interface suggests achievement of a uniform low loss optical transmission, and also a specific, radially dependent, heat transfer is possible. Constructing such an interface with sufficiently high damage threshold and sufficiently low optical loss, however, appears challenging. We cannot predict the time required for, or even the feasibility of, mastering these practical tasks.

#### 8.4 High peak power

The same design features that facilitate obtaining high average power from a solid state laser oscillator also facilitate producing high peak powers. The increase in mode area by some 3–4 orders of magnitude as compared with current solid state lasers allows, in principle, a corresponding increase in peak power. Also the higher damage thresholds of sapphire and diamond, 10 terawatts/cm^{2}, for pulses of 100 fsec duration, as compared with conventional laser material further facilitates high peak power. This may also reduce the need for pulse stretching and chirping.

#### 8.5. High peak power, high average power and propulsion

The possibility of high peak power and high average power in a single system is interesting as regards optically based propulsion. There is reason to anticipate single cycle optical pulses [31]. Such single cycle pulses offer a potential to produce unipolar pulses at a fixed location [32]. Such unipolar pulses are fundamentally different from typical optical fields in that they do not change polarity during an optical period.

Such unipolar pulses, approaching terawatt peak powers and produced at megawatt average power could provide a valuable means of accelerating ions to velocities, e.g., 10^{4} m/s, needed for high specific impulse propulsion. Such a technology could be useful for lift-to-orbit applications and for other space based applications [33].

## Acknowledgments

We thank Willis Lamb, Jr., in particular, for his lifelong guidance and inspiration as a leader, researcher, and educator, especially in the field of laser physics. We also thank the Joint Technical Office (JTO) for support, NASA for support under grant NCC8-200, ARO for support under grant DAAD19-02-1-007, and AMCOM for support under contract DAAH01-01-C-R160. We also thank Jim Butler, Donna Fork, Ken Howell, Gerald Karr, and Dane Phillips for helpful discussions.

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