_{1}

^{*}

The advent of the laser has placed stringent requirements on the fabrication, performance and quality of optical elements employed within systems for most practical applications. Their high power performance is generally governed by three distinct steps, firstly the absorption of incident optical radiation (governed primarily by various absorption mechanisms); secondly, followed by a temperature increase and response governed primarily by thermal properties and finally the elements thermo-optical and thermomechanical response, e.g., distortion, stress birefringenous fracture, etc. All of which needs to be understood in the design as efficient, compact, reliable and useful for many applications high power systems, under a variety of operating conditions, pulsed and continuous wave, rep-rated or burst mode of varying duty cycles.

Archaeologists claim that people learned to make mirrors more than five thousand years ago. At home they used polished bronze or silver plates, and only ancient Romans started to manufacture glass mirrors with a tin or lead coating. The technology of mirror production changed many times, but basically all the mirrors were either metal-coated glass or quartz. Never before, the energy of radiation fluxes was so damaging to mirrors because they reflected very little light, for example, from distant stars, from a candle flame or splinter.

Modern problems of interaction of optical mirrors with high-power radiation arose with the advent of first lasers. Virtually in all types of lasers generation occurs in a resonator, which consists of at least a pair of mirrors, and through one of them radiation is emitted. Initially, traditional quartz disks with a mirrored surface suited well the purpose, but over the years passed, the power of laser radiation fluxes has increased thousand fold. Today, the problem of making a mirror that can operate and maintain its high optical performance when exposed to intense radiation has become one of the key challenges in the improvement of high-power lasers and modern high-energy systems. Design, materials testing and technological ideas are used in power optics work and are extremely useful for specialists involved in the development of astronomical and space optics, new efficient systems based on laser diode assemblies, and in many other areas of the ever-expanding knowledge.

Back in the early 1970s, we drew attention to the phenomenon, which, of course, was to limit the further growth of the power generated by the laser system and, consequently, the intensity of the laser beam on the surface of the irradiated object [1-14]. More than twenty years of fundamental and applied research devoted to the study of this phenomenon and to the solution of problems associated with it allow a conclusion that its essence consists in the following.

The optical surface of even a very good mirror does not fully reflect radiation falling on it. A small portion of energy (fractions of a percent, depending on the wavelength) is absorbed by the mirror and turns into heat. As the laser beam power increases, even a small amount of it is sufficient to induce thermal stresses in a mirror. They distort the geometry of the reflecting surface, which influence, for example, such an important property of laser light, as the possibility of its propagation over long distances and concentration in a small volume. Now we know that thermal deformation of mirrors not only affect the main characteristics of the beam, but also lead to quenching of generation. And is it possible to manufacture a mirror, which can resist a flux of radiation power of a few megawatts? This question was probably the main one in the world laser community of the early 1970s. The General Physics Institute and enterprises of the defense complexes working under its leadership are carrying out research in the field of so-called high-intensity laser optics or power optics. The main requirement to power optics is a high threshold of optical performance. For each type of a laser system, depending on its applications the permissible value of the elastic distortion of the mirror surface is specified, which, for example, is measured in fractions of a percent of a laser wavelength of a specific laser system. For a CO_{2} laser the generated radiation lies in the infrared region of the spectrum, and the wavelength is 10.6 µm; therefore, the elastic distortion shall not exceed fractions of a micron. Even higher requirements are imposed on the optics of the short-wavelength range. The light intensity, at which the distortions of the mirror surface reach some limits for a specified laser source, is an optical performance threshold of the mirror. It means that the better the optical quality of the mirror material, the higher the threshold of optical performance at higher intensities. This threshold level cannot be violated, because elastic deformations of the mirror change the shape of the laser beam, which will ultimately affect the prospects of delivering the light energy to the interaction region. If the light load is further increased, then deformations become inelastic (plastic), which means that irreversible changes are produced on the mirror, i.e., its reflecting surface is damaged and should be replaced. Therefore, it is needed to manufacture mirrors, which could withstand high optical loads for quite a long time, i.e., up to several kilowatts per square centimeter of the mirror surface.

To understand how difficult this task is, it is enough to give two examples. The mirror, which has been in the hands of an experimenter for only a few seconds, can exhibit deformations of the optical surface that is close to the maximum permissible value, because of the uneven hand heating. However, in this case, if the mirror is left to cool, the shape will be restored. In reality mirrors of modern high-power lasers are exposed to power densities of several kilowatts per square centimeter, which is comparable to the energy flux, which is emitted into the environment directly from the unit surface of the Sun. And this means that if we somehow contrived to “put a laser mirror on the Sun”, the shape of its surface should not change significantly even in this case.

To solve this problem and manufacture highly resistant mirrors necessary for high-power industrial lasers, it was necessary to unravel a tangle of issues related to quantum electronics, optics, thermal elasticity and heat transfer, materials science, and many other modern technologies.

It would seem that the first step on this path is obvious: to replace a semitransparent quartz disk on a metal one and couple light out by diffraction through an aperture in the mirror or over its boundary. Metals perfectly reflect laser light, have high thermal conductivity and, therefore, may well remove heat from the region of interaction of the beam with the mirror surface. But pure metals also have disadvantages: (a) high thermal expansion coefficient—they can easily change their size and shape when heated, and (b) relatively low hardness, because of which it is difficult to polish them as well as quartz.

Here it is necessary to say that the modern advances in optical polishing of metals are largely connected with the search for the solution to the problems of laser mirrors. It is funny now to recall an old episode, when physicists first came to opticians with a piece of metal. The request to polish a metal disc in the same room with quartz or calcite was simply wild because metal dust in the workshop, where ultra-precision machining of optical surfaces is performed, was inaccessible. Yet, the way out of this situation was found, i.e., a metallic mirror. And, in fact, after examining all available metals and alloys, scientists managed to increase the optical performance threshold of new mirrors over traditional quartz mirrors by an order of magnitude. But this “order of magnitude” was not enough. It became clear that the required level of light, and thus of heat loads on the mirror of a high-power laser can be achieved only by using special cooling.

When cooled with a moving liquid, the heat flow being released is directly proportional to the temperature difference between a hot body and coolant. Thermal power rates measuring kilowatts per square centimeter are not difficult to remove, if the mirror is heated to a temperature of thousands of degrees. But in this case, it is impossible to talk about the high optical quality of the mirror surface. There is a contradiction: convective heat exchange is greater at high temperatures, and for the geometric shape and other characteristics of the optical mirrors to be stable, room temperatures are needed. This contradiction could only be resolved due to more efficient heat removal; however, for room temperatures of optical surfaces of a laser mirror, neither the theory nor the practice was developed at that time.

First, scientists started experimenting with milling deep channels on the back side of a metal disk through which tap water was circulating. They were cut as close to the surface as possible, but this lead to vibration and deformation of the mirror surface due to pressure fluctuations of water. Then, there was an attempt to make the channels shallower and manufacture micro-capillary structures [15-27]. This approach led to the conclusion that it is better to use specially designed metal capillary-porous structures, similar to a sponge for washing dishes. As a result, so-called felt and powder metal-based structures have been widely adopted. Heat exchange in these structures is very intense, both due to the large surface, and due to the strong mixing of the coolant, which travels in the micro-capillaries. In addition, the matrix with a porous body skeleton functions as a supporting structure: each of its elements is linked with the mirror surface and keeps its original geometry. The coolant in such a structure behaves as a turbulent mountain stream that runs through branched and sinuous gorges. And the matrix of the porous body in this case resembles an openwork design of the bridge with a smooth road surface supported by many piers.

A thin but very hard layer, which is then polished, must be applied to a highly porous heat exchanger. Only then its surface will serve as a mirror provided that it has a highly reflective coating. The thickness of the coating separating radiation from the coolant should be small, i.e., 100 - 300 µm; otherwise, it will keep the heat absorbed by the reflecting surface of the mirror. Now there are several methods of applying thin layers on a highly porous material, but initially it was quite a challenging task. The problem was solved by using intermetallic compounds [17-19], i.e., chemical compounds of metals with each other. Intermetallic coatings are obtained, for example, by deposition from the gas phase. So it is possible not only produce a thin separating layer, but also to restore the mirror if necessary.

Intermetallic coatings have another important property: their structure allows one to manufacture mirror surfaces of high optical quality. If one looks on a polished metal through a microscope, its surface resembles an orange peel: it is covered with tiny hillocks and cavities. To reduce the micro-roughness additional measures are needed, for example, before polishing the metal is doped to make it harder and more fine-grained. But all the same the final size of micro-roughness is too large: from 0.01 to 0.1 µm. In intermetallic coatings the structure is originally very small, about 0.1 µm, and after treatment with a diamond cutter or optical polishing very good mirror surfaces with roughness less than a few thousandths of a micron can be manufactured.

Thus, the solution was found. A mirror for high-power CW and pulse-periodic lasers, where the average output power levels are high or very high, is a highly porous heat exchanger and a thin separation layer with a reflective coating. Lasers with such mirrors have been successfully used for welding, cutting and hardening of metals in industry. Besides, they are employed for addressing environmental problems and in military applications.

But it turned out that there is still room for improvement in high power optics. By changing the coolant pressure, one can make it boil at room temperature [21-35], i.e., in the same conditions under which the final optical machining of the mirror surface is carried out. When the coolant boils, a portion of the absorbed heat is spent on vapor formation; therefore, the heat exchange is tens or hundreds of times higher than in the case of the convective heat exchange. It is only need to remove bubbles from the capillary-porous structure, i.e., the size of the capillaries should be selected accordingly. And then water can be replaced with a liquid metal, for example, the eutectic alloys of sodium, potassium and cesium, which have a low melting point. The heat exchange rate becomes even higher: the heat is now not only removed by the moving fluid, but also is transferred to the coolant itself, which, like all metals, is an excellent heat conductor. Liquid metal coolants enable the removal of thermal powers from a reflecting surface, measuring ten kilowatts per square centimeter of surface.

Using the methods of metallic mirror cooling, one can solve such “non-mirror” problem as the cooling of large integrated circuits and the development of high-power anodes of X-ray lithography facilities for microelectronics. These devices must withstand the heat load measuring a few tens or hundreds of kilowatts per square centimeter. Works on creation of high-power optics for highpower lasers involve introduction of a wide range of technical, technological and design solutions from other areas of science [28,32].

Consider a strongly absorbing elastically isotropic body, which at the initial moment of time is kept at zero temperature in a relaxed state and on whose surface an axisymmetric radiative flux is incident [1-14]. The energy distribution in the laser beam cross section obeys the Gaussian where and is the radius of the irradiation region on the body surface. We assume that all the thermal and mechanical coefficients of the material do not depend on temperature, and the laser beam is absorbed directly on the irradiated surface. The nonlinear effects, as well as the energy loss by radiation and convection are neglected. Provided that the characteristic size of the beam is and the time of energy input is (L is the characteristic size of the body, and a is the thermal conductivity of the material), the problem can be solved by using the half-space model (

In this formulation, the problem of determining the temperature field is reduced to the solution of the differential equation of heat conduction for the following

initial and boundary conditions:

Here, K is the thermal conductivity and is the reflectivity of the material surface.

By applying to (1) the Hankel transform over r and the Laplace transform over t and using the initial and boundary conditions (3), we obtain:

where is the zero-order Bessel function and; is the error function.

The behavior of the temperature field on the surface of the body exposed to a laser beam is shown by the example of a quartz sample with a gold coating.

However, if the exposure time satisfies the inequality then a stationary temperature field can establish. In this case, the expression for the temperature has the form:

at the point where the maximum temperature is reached. Then, expression (5) is simplified:

For comparison, Figures 3 and 4 show the calculated distribution of the stationary temperature fields in goldcoated aluminum and quartz samples for different values of K_{0} at a given power level.

The analysis of expression (5) confirms the localization of the temperature field, the intensity of which decreases sharply with increasing distance from the center of the irradiated region on the surface and deep inside the material.

The half-width of the intensity distribution of the laser beam is several times narrower than the half-width of the corresponding temperature profile on the irradiated surface, and the broadening of the latter is determined solely by the thermal conductivity of the material:

If, during the laser exposure the melting point of the material is reached, then the optical surface is irreversibly deformed. By setting in (4) and equating the obtained expression for to the melting temperature T_{m} of the material, we can estimate the value of the maximum laser intensity, which during the exposure time t_{0} leads to the melting of the surface layer:

where is a temperature

function, which depends on the dimensionless parameter

Thus, knowing the value of the temperature function at a given F_{0} (_{пл}, K, (

From this discussion it follows that for the value of increases linearly with decreasing F_{0} and remains constant for In particular, for an aluminum sample with a reflectance, the maximum intensity, leading to the melting of the surface, is 8.6 × 10^{7} W/cm^{2} with a pulse duration t_{0} = 10^{−6} s and is virtually independent of the parameter. If the duration is t_{0} = 1 s, the values of the maximum intensity differ for different values of the parameter (W/cm^{2}, and 0.2 cm, respectively сm), which is due to the mechanism of the material conductivity in the interaction process, resulting in a significant energy exchange between neighboring areas of the irradiated body. Note that

this solution is valid for describing the processes preceding the thermal breakdown of the material. The nonstationary temperature field causes a stress in the irradiated body, which also changes with time. Let us now define thermal stresses in a solid body.

In the quasi-static formulation, the problem of determining stresses in the half-space irradiated by an axisymmetric flux leads to the equation

with the initial conditions and boundary conditions at which Here is the thermoelastic displacement potential

, is the coefficient of linear expansionand is Poisson’s ratio.

The solution of the problem is the sum of two fields of thermoelastic stresses: the fields whose components of the stress tensor satisfy thermoelasticity equations, but do not satisfy the boundary conditions, and the fields whose stress components characterize the temperature-free solution of the corresponding classical theory of elasticity and satisfy the boundary conditions of the thermoelastic problem. After necessary transformations, we obtain the desired stress field:

Here

(G is the displacement modulus)

The expressions obtained show the variation of the stress field of the irradiated body during exposure to laser radiation. The maximum values of the radial and circumferential stresses are at a point r = 0 on the surface, where they are equal to each other, and can be written in the form:

[is the hypergeometric function].

Expression (10) allows one to evaluate the maximum laser intensity leading during the irradiation time t_{0} to the achievement of the yield strength of the material, and to determine the maximum permissible duration of the laser pulse, corresponding to the given intensity. It is known that at intensities greater than the limit, plastic deformations arise in the surface layer, and after the termination of the laser pulse inevitable there appear residual stresses. Multiple repetition of the described process leads to a rapid destruction of the optical surface.

_{0} for a polished aluminum sample. Comparing the dependence obtained with the dependence one can see that the maximum intensities, leading to a plastic yield of the material, are by 1.5… 2 orders of magnitude lower than the values corresponding to the melting of the material. In the case of CW exposure a stationary stress field can arise, which for the halfspace has the form

Here is the degenerate hypergeometric function and E is Young’s modulus.

The components of the stress tensor and on the metal surface as a function of distance from the center of the irradiated region are shown in

Let us analyze now the other two components of the stress field in a solid body, arising during its exposure to a high-power light of finite duration, namely, and Of most interest is the consideration of the behavior of the component, because the component due to the axial symmetry of the problem everywhere on the axis is zero and has a maximum at point z_{0} > 0.

_{0}. The feature

of the behavior in the component is the fact that as the exposure time decreases the maximum stress at point z_{0} passes through a maximum, and the coordinates decreases monotonically, approaching the surface. By equating the expression for the component at point z_{0} to the ultimate tensile strength of the metal, one can estimate the intensities at which the coating starts sagging during the exposure. Thus, the destruction of the optical surface by laser radiation can occur at intensities much lower than those required for its melting.

Let us determine the value of the displacement of the solid-body surface due to the impact of laser radiation. Displacement of the body surface can be represented as the sum of the derivatives in the coordinates of the thermoelastic displacement potential and Love’ function. The expression for the normal surface displacement that matches the temperature distribution has the form

The displacement is negative, because the Z-axis is directed into the solid body. Profiles of the optical surfaces of various materials, deformed by the laser beam, are calculated at t_{0} = 1 s, K_{0} = 2 and P = 1 kW and are shown in

The graph shows that the minimum deformation

corresponds to a gold-coated quartz sample. This is explained by a slow increase in the material heating region due to the low thermal conductivity of quartz, and by the low value of its thermal expansion coefficient. However, the temperature of the surface of the quartz—gold sample is two orders of magnitude higher than the temperature of the copper sample. Equally significant is the dependence of the deformation of the irradiated surface on the energy distribution in the laser beam cross section. _{0}.

Consider the dependence of the surface deformation on the time of its irradiation. To do this, if r = 0 in (12) and the previously adopted notation is used, we obtain

where

Thus, knowing the necessary parameters one can determine the maximum deformation of the surface in the center of the irradiated region for any material and any exposure durations.

Note the following characteristics of the deformation process resulting from the consideration of the expressions obtained:

1) The process is local, and with decreasing exposure time the half-width of the deformation profile, exceeding the corresponding value of the laser beam for large t_{0}, approaches the half-width of the energy distribution in the beam cross section, which causes deformation of the irradiated surface.

2) The value of the deformation increases rapidly at

small values of the Fourier transform F_{0} and the rate of its increase is proportional to the intensity of the incident flux. When, the rate of change in the deformation decreases and the displacement of the surface continues to vary logarithmically slowly. The stage of the elastic behavior of the material is characterized by complete recovery of the initial state of the surface when terminating the laser action. However, in some cases, the deformations of the optical surface become inadmissible. Accounting for deformation is crucial for such laser systems as the quasi-optical transmission lines, telescopes, unstable resonators, and focusing systems.

The maximum of laser radiation can be estimated by equating V in (13) to the V_{ad}, corresponding to the maximum admissible deformation of the surface, at which additional losses exceed some value fixed for each system

where is the the displacement function.

Thus, knowing the values of the function,one can estimate the maximum intensity, resulting in admissible deformations of the optical surface of any material that obeys Hooke’s law. The dependence of the displacement on the dimensionless parameter F_{0} is shown in _{0}.

Consider the behavior of the maximum intensities as functions of the laser pulse duration for an aluminum sample (

kg/cm^{2} (the wavelength of the incident radiation isµm). For pulse durations and the given parameters of and, the main process, which limits the maximum value of the laser intensity, is the elastic deformation of the reflecting surface. With a pulse duration less than this, the transition of the material to the inelastic region becomes crucial, which is due the stresses (caused by irradiation) above its yield strength. It is however important to note that the pulse duration s for aluminum depends on the parameter of the incident beam and significantly decreases with its increase. Thus, for example, at cm the value of the pulse duration, for which the surface deformations should be taken into account, is s. Insignificance of the process of the reflecting surface deformation at low laser pulse durations is explained by the smallness of the effective heating of the material.

However, it should be noted that these estimates are valid for ideal surfaces in a vacuum, because the presence of air and slight contamination of the irradiated surface can lead to the breakdown of the gas near the surface and, consequently, to its destruction at laser intensities that are less than the estimated ones.

The feasibility of using structures with open porosity for cooling heat-stressed laser mirrors was justified for the first time theoretically and experimentally in [^{2} × deg. The test results of water-cooled laser mirrors that are based on the structures with open porosity first indicated the possibility of removal of high heat flows at low values of the mirror surface deformations. The maximum density of the heat flow being removed, which does not lead to destruction, is kW/cm^{2}, and at q = 2 kW/cm^{2} the µm/20, where λ value of thermal deformation was ~λ/20 µm. A further increase in the optical damage threshold of cooled mirror surfaces can be achieved by optimizing the parameters of the porous structure, the appropriate choice of the coolant and the rational design of its inlet and outlet. The development of such “optimal” laser reflectors requires a detailed study of heat and mass transfer in porous structures, which are currently insufficiently studied. Let us consider some aspects of this problem.

The temperature fields in the porous structures are calculated in the one-dimensional formulation under the following assumptions: the incident radiation is uniformly distributed over the irradiated surface; the thickness of the porous layer is much greater than the depth of heating, which makes it infinitely large and allows consideration of the half-space model; and the temperature and velocity of the flow through the thickness of the porous layer is constant.

Then the heat transfer equation, which describes the temperature distribution over the thickness of the porous layer, can be written as

where x is the coordinate; t, t_{T} is the temperature material (skeleton) of the porous structure and the coolant, respectively; h_{s} is the heat transfer coefficient between the material structure and coolant; is the skeleton thermal conductivity along the axis x; and S_{V} is heat transfer surface per unit volume. Equation (1) can be written in the dimensionless form:

where and are the dimensionless temperature, coordinates and parameters, respectively; is the average particle diameter in the structure; is the modified Nusselt number, which characterizes the ratio of convective to conductive heat transfer across the skeleton boundary; and is the dimensionless parameter characterizing the structure.

The boundary conditions of equation can be written in the form:

where is the dimensionless heat flux density; q is the heat flux density transmitted through the separating layer.

Solution to equation) with the boundary conditions (17) has the form:

It follows from (18) that the rate of temperature decrease over the thickness of the porous structure is determined by the parameter.

We write down some relations required to calculate the thermal characteristics of laser reflectors. The temperature of the separation layer at the interface with a porous substrate has the form:

The maximum heat flux density, removed from the reflector due to convective cooling, follows from the condition of equality of the coolant and the boiling point at appropriate pressure and has the form:

The degree of heat transfer in a porous structure as a result of the flow circulation and the surface development is determined by the coefficient, which characterizes the ratio of the amount of heat removed by the coolant in the structure under consideration, to the amount of heat that would be removed directly from the cooling surface of the separation layer by the coolant when it flows in a Δ channel of depth

where is the coefficient of convective heat transfer in the coolant flow in a slot gap of size Δ. For example, for the turbulent regime of the coolant flow the Nusselt number has the form

where Re, Pr are the Reynolds and Prandtl numbers.

Let us introduce the coefficient of heat transfer due to the flow turbulence where h_{s} is calculated from the condition that the flow of coolant circulating through the reflector is constant. Then, depending on the structure type and parameters and on the thermophysical properties of the coolant is calculated as follows:

where is the coefficient of ribbing of the cooled surface of the separation layer.

Equation (18) allows one to find the depth of the structure heating that defines the minimum possible thickness of the porous layer, conventionally assuming that the heating ends at the coordinates where the skeleton and the coolant temperatures differ by 1%:

In the case of removal of the maximum heat flux densities, the depth of heating is

Thus, relations (18) - (23) allow one to calculate the temperature fields and thermal characteristics of the cooled laser reflectors. In combination with the expressions describing the flow hydrodynamics, they are the basis for optimizing the parameters of porous structures, ensuring minimal thermal deformations of mirror surfaces or, if necessary, the maximum heat flux, removed in the case of convective cooling.

The regime of the coolant flow in porous materials, which are of interest for high-power optics, is a transition between laminar and turbulent regimes; in this case, the calculation of the heat transfer coefficient between the skeleton and the coolant is very difficult despite numerous experimental data. The lack of a unified approach to processing of the experimental results and the inability to accurately determine such important characteristics as the S_{V}, d_{s} etc. make the calculation of heat transfer of liquid drops in various types of porous structures with geometrical parameters that vary widely. Therefore, these equations can only be recommended as a first approximation to calculate the volumetric heat transfer coefficient in structures that are identical to the tested ones, and to gas coolants. It is generally accepted that the criterion equation of the interporous convective heat transfer for gases and droplets has the form:

where c and n are the constants depending only on the structural characteristics of the porous material.

Based on the experimental data of papers we analyzed the dependences of c and n on the structural characteristics of porous materials for which the latter are quite authentically known; as a result, we found that c and n depend mainly on the bulk porosity. Thus, relation for the dimensionless Nusselt number with account for correlation expressions с(Π_{V}) and n(Π_{V}) allows one to calculate the coefficient of convective heat transfer in a porous material.

The temperature field and thermal deformation of the reflector are largely determined by the discharge of the coolant pumped through a porous layer, which depends on the hydrodynamic characteristics and conditions of the coolant inlet and outlet. Hydrodynamic characteristics of a single-phase fluid flow in porous materials, mainly in the region were studied in many experimental works. The expressions proposed by different authors can be used to calculate the hydrodynamics of the flow in materials that has an identical structure to the samples studied.

In general, the hydrodynamics of the flow in porous materials is described by the modified Darcy’s equation (Dupuit – Reynolds – Forchheimer):

where p is the flow pressure; u is the filtration rate, equal to the ratio of the specific coolant discharge G to the density ρ; and β are the viscous and inertial is coefficient of dynamic μ resistance coefficients, respectively; and viscosity of the coolant.

From Equation (25) we obtain the equation for the coefficient of friction, C_{f}, in the form

where (the characteristic size β/α).

Practical use of relation (26) for the calculation of the hydrodynamics of the flow in different types of porous materials, whose structural characteristics vary widely, is hampered by lack of information about the coefficients α and β, which are usually determined experimentally.

Well known a slightly different approach to the calculation of C_{f}: as a characteristic size they used, where K is the permeability coefficient, characterizing the hydrodynamics of the flow according to Darcy’s law, then

The relationship parameters α and β between the coefficients c and K can be represented as and.

Parameter c is a universal constant for identical porous structures. For example, for all the materials made of metal powders with spherical or close-to-spherical particles we have, and for materials made of powders of arbitrary shape we have. Thus, in the future when calculating the hydraulic characteristics of the structures we assume, although in our case this provides a somewhat higher value of the coefficient of friction.

Research conducted with fibrous materials, foams and structures of the layers of particles of arbitrary shape has shown that the parameter c for these materials is relatively small. This is explained by the fact that c, characterizing the fraction of inertial losses, depends largely on the nature of the wake formed behind solid particles due to the flow, and the wake depends on the type of porous structure.

The presence of the free ends (further turbulizing the flow) in metal-fibrous structures increases c; therefore, for a relatively short fibers, l/d = 50 - 100, с ≈ 0,132.

The permeability coefficient K, which is a structural characteristic of a porous material, does not depend on the flow regime and is determined experimentally from Darcy’s law. In connection with the development of works in the field of heat pipes, many experimental data are currently available to determine K for powder and metal fibrous structures. According to our experimental data, the relation of the permeability coefficient for metal fibrous structures with bulk porosity has the form:

where A and m are the coefficients depending on the relative length of the fibers l/d. Similar expressions can be obtained for powder materials.

In addition, the permeability coefficient is calculated from the known Carman-Kozeny relation:

where is a constant depending on the structure.

Expressions (27) - (29) were used to determine the hydraulic characteristics of the power optics elements, made of porous materials based on metal powders and metal fibrous structures.

Usually, in cooled laser reflectors the coolant is supplied to and removed from the porous structure through evenly distributed alternating channels, which are made in the form of slots or holes on the cooling surface. With a slotted inlet, a more uniform velocity field of the coolant and a minimum differential pressure required for its pumping are provided. In the case of inlets and outlets in the form of alternating holes we may deal with a significant non-uniformity of the velocity field in calculating the flow in radial directions. This leads to additional pressure drops in the circulation of the coolant, which are considered by the coefficient K_{г}; in this case, the total pressure drop in a porous layer has the form

where Δр is the pressure in the case of a uniform velocity field.

Under the assumption of uniform injection (outflow) of the coolant in the radial directions in the regions of supply (outlet) restricted by the region (s is a step between alternating holes, r_{0} is the radius of the hole), the coefficient K_{г}, characterizing the influence of collector effects on hydraulic resistance during the motion of the fluid in a porous layer, can be written as:

Here G is the mass flow rate of the coolant, F is the area of the exposed surface, n is the number of channels for the supply (outlet) of the coolant, and is a relative spacing between the holes. One can see from Equation (31) that K_{г} depends both on the geometric characteristics of the supply and removal of the coolant (on a) and on G; with increasing a and G, the coefficient K_{г} increases. Thus, the coefficient K_{г} is characterized by design excellence of the inlet and outlet system of the mirror coolant.

With the known K_{г} the total pressure drop in the porous layer is calculated by the formula (30), taking into account the expression for calculating р:

where v = Gs/(ρFΔ) is the coolant filtration rate.

As for the problems of cooling of power optics elements, of interest is to study the thermal conductivity of a porous material skeleton without account for the thermal conductivity of the coolant. Taking into account the summarized the results of numerous experimental studies of thermal conductivity of materials of various structures and showed that the effective thermal conductivity depends not only on the volume porosity, but also on such factors as the sintering behavior of the material, the size and shape of the primary particles, pressing and sintering technologies, etc., which leads to a large scatter in the data and makes it difficult to generalize the results. Practical data are summarized in the form of the dependence (П_{V}) for the samples, manufactured using the single technology and the same type of material. Analysis of experimental results shows that for the calculation of a wide range of porous powder materials can be based on the Odolevsky equation:

where λ_{к} is the thermal conductivity of a compact material.

Effective thermal conductivity of metal fibrous felt structures may have a considerable anisotropy depending on the direction of the fibers in the felt. Thus, on the basis of is generalized by the relations:

where is the effective thermal conductivity in the direction parallel to (34a) and perpendicular (34b) to the felt-making plane. In the latter case one can also use the expression

Expressions (34b) and (35) satisfactorily approximate the experimental data and can be used in the determination of the thermal characteristics of cooled laser mirrors, made of metal fibrous structures. In this case, expression (34b) describes the upper limit of the experimental data (an optimistic estimate), and (35)—the lower (a pessimistic estimate).

In general, thermal deformations of a mirror are calculated by using the temperature field of the separating layer and the porous base and the known conditions of their attachment. The exact solution of this problem of thermal strength under real conditions of the reflector exposure is very difficult.

To estimate the low distortions of the optical surface, which are characteristic of deformation of power-optics elements, one can make the assumption of free expansion of the porous base and the separating layer according to the temperature fields. Then, the thermal deformation of the mirror surface W is the sum of extensions of the separating and porous layers:

where α_{p} and α_{n} are the temperature coefficients of linear expansion of the separating and porous layers, respectively; is the dimensionless temperature; t_{0} is the coolant temperature at the inlet of the reflector; and t_{1} and t_{2} are the temperatures of the outer and inner surfaces of the separating layer, respectively.

It follows from equation (36) that the higher the coolant flow rate, the lower the W. The maximum deformations of the optical surface are realized in the region of the outflow of the coolant from a porous layer, the heating value of the coolant through the slot channels having the form

where G_{T} is the heat capacity of the coolant.

The above-derived expressions describing the processes of heat and mass transfer in porous structures, were used to calculate the characteristics of cooled laser mirrors with the coolant inlet and outlet system in the form of uniformly (with a step s) alternating holes; the distribution of the heat load on the surface of the reflector was assumed uniform.

_{V} at a constant d_{s}) for two selected regions of the reflector, corresponding to the regions of injection W_{1}(q_{max}) and outflow W_{2}(q_{max}) of the coolant. Varying the average grain size d_{s} (or the diameter of the fiber), we can construct a family of curves, characterized by a constant value of d_{s} and variable porosity П_{V}, for reflectors with the same type of capillary structure. The curves are plotted at a constant pressure drop and by taking into account the coolant heating in the porous layer; the temperature of the coolant at the inlet temperature was assumed equal to the temperature of the final finishing of the mirror. The deformation of the optical surface in the coolant outflow region obeys W_{2} > W_{1}. Crucial to the selection of the structural characteristics of the porous material is curve 2, and the difference between curves 1 and 2 characterizes the degree of the cooling system perfection. The curves are the envelopes of the working thermal deformation characteristics of a family of reflectors with this type of structure. Performance characteristic of the reflector with a given porosity structure П_{V} is obtained by connecting the

straight line from point C with the origin of the coordinates. Point C corresponds to the maximum heat flow, removed due to convective cooling, and the line segment OC is a dependence of thermal deformations of the mirror surface on the heat load. In general, curve 2 has two main points: point A corresponding to optimal porosity which facilitates removal of the maximum heat flux density for the selected grain size, coolant pressure drop, and coolant supply and removal conditions; and point B (the point of tangency of curve 2 with the straight line from the origin of the coordinates) corresponding to the porosity, for which in the porous structure the optimal thermal distortions of the mirror surface are realized.

The choice of material and the basic parameters of the structure (d_{s}, П_{V}) must be based on a comparison of a family of curves 2 with possibilities of obtaining the desired porous structures and separating layers.

Figures 14 and 15 present the results of numerical cal-

culations of thermal deformation characteristics of a family of water-cooled reflectors with porous structures made of the widely used copper powders (Figures 14(a) and (b)) and molybdenum powders (Figures 14(c) and (d)), illustrating the capabilities of the method. The average grain size and bulk porosity varied between 20 ≤ d_{s} ≤ 200 µm and 0, 1 ≤ П_{V} ≤ 0,9.

One can see from Fig. 14 that the maximum heat flux densities are as follows: in the region of the coolant inlet q_{пр1} = 12.8 kW/cm^{2}, in the region of the coolant outlet q_{пр2} = 3.75 kW/cm^{2}, with W_{1} = 0.1 - 0.4 µm, and W_{2} = 0.4 - 3.5 µm. In the region of the maximum heat flux density W_{2} = 1.5 - 2 µm, which is in most cases much higher than the maximum allowable deformation of the reflectors for CO_{2} lasers.

With the increase in coolant flow (_{2} can be dramatically reduced by using the materials with high bulk porosity (0.7 - 0.8) and the transition to a larger grain size. In this case, q_{mах} = 2 kW/cm^{2} at W = 0.5 µm. Powder-material structures with П_{V} > 0,75 for the power optics are very difficult to produce. With a decrease in porosity to the real values (~0.6) the heat flux densities being removed will be ~ 1.4 kW/cm^{2} at a 0.5 µm deformation. Note that increasing the coolant flow rate is also achieved through rational design of the coolant inlet and outlet system, providing a more uniform distribution of the coolant along the cooling surface.

One can see from Figures 14(c) and (d) that the use of porous molybdenum allows an approximately four-fold decrease in the level of thermal deformations of mirror surfaces in both the first and second regions; in this case q_{пр} somewhat decreases: q_{пр1} = 7,8 kW/cm^{2}, (q_{пр2} = 2,8 kW/cm^{2}). For this structure (_{2} lasers. For example, a structure with Π_{V} = 83% и d_{s} = 20 µm facilitates the removal of q_{max} = 2.8 kW/cm^{2} at W = 0.45 µm. With increasing the coolant flow (_{max} decreases. In this region the level of heat flows to be removed can be increased by the use of materials with the larger size of the particles; for example, the molybdenum powder structure with Π_{V} = 83% и d_{s} = 20 µm facilitates the removal of q_{max} = 2.1 kW/cm^{2} at W = 0.23 µm.

_{пр}, which is associated with a decrease in the effective thermal conductivity of these structures compared to powder structures.

In regions of the coolant outflow (Figures 15(b) and (d)) the maximum heat flux densities are on the same level as for the mirrors with powder structures; however, an increase in the coolant flow leads to a strong dependence of W on П_{V} and to less sensitivity to the diameter of the initial fibers. Deformations corresponding q# are approximately two times lower compared to the powder structures: for copper and molybdenum reflectors W = 1.2 and 0.3 µm at q_{пр} = 3,9 kW/cm^{2} and q_{пр} = 2.6 kW/cm^{2}, respectively. For mirrors with a copper fiber structure the heat flux densities, which can be realized with W = λ/20 (where λ = 10.6 µm), of 2.4 kW/cm^{2}, are 2.4 kW/cm^{2}, which exceeds the corresponding level of q_{max} for powder materials. Thus, the use of metal fibrous structures for producing power optics elements is very promising; for example, the structure of molybdenum fibers with a diameter of 100 - 200 µm and a porosity of 50% - 60% q_{max} = 2.2 kW/cm^{2} at W ≈ 0.15 µm. In addition, the possibility of obtaining materials with high П_{V} (70% - 80%) makes it possible to manufacture optimal reflectors.

We have compared the calculation results of thermal mirror-surface deformations of the laser reflector based on a porous copper-powder material with the experimental data.