Effects of laser radiation on materials. Interaction of laser radiation with matter Approximations of the thermal model of laser radiation

Laser (from the English “light amplification by stimulated emission of radiation "-" amplification of light by stimulating radiation ") or an optical quantum generator is a special type of radiation source with feedback, radiating body in which is the inversely populated environment. Laser operating principles are based on propertieslaser radiation: monochromatic and highly coherent (spatial and temporal). TAlso, among the features of radiation, a small angular divergence is often referred to (sometimes the term "high directivity of radiation" can be found), which, in turn, allows us to speak of a high intensity of laser radiation. Thus, to understand how a laser works, it is necessary to talk about the characteristic properties of laser radiation and an inversely populated medium, one of the three main components of a laser.

The spectrum of laser radiation. Monochromaticity.

One of the characteristics of the radiation of any source is its spectrum. The sun, household lighting devices have a wide spectrum of radiation, in which components with different wavelengths are present. Our eye perceives such radiation as white light, if the intensity of different components in it is approximately the same, or as light with some shade (for example, green and yellow components dominate in the light of our Sun).

In contrast, laser radiation sources have a very narrow spectrum. In some approximation, we can say that all photons of laser radiation have the same (or close) wavelengths. So, the radiation of a ruby ​​laser, for example, has a wavelength of 694.3 nm, which corresponds to the light of a red hue. The first gas laser, the helium-neon laser, also has a relatively close wavelength (632.8 nm). Argon-ion gas laser, in contrast, has a wavelength of 488.0 nm, which is perceived by our eyes as a turquoise color (intermediate between green and blue). Titanium-doped sapphire lasers have a wavelength in the infrared region (usually near 800 nm), so its radiation is invisible to humans. Some lasers (for example, semiconductor lasers with a rotating diffraction grating as an output mirror) can tune the wavelength of their radiation. What all lasers have in common, however, is that the bulk of their radiation energy is concentrated in a narrow spectral region. This property of laser radiation is called monochromaticity (from the Greek "one color"). In fig. 1 to illustrate this property, the spectra of radiation from the Sun (at the level of the outer layers of the atmosphere and at sea level) and a semiconductor laser produced by the company are shown Thorlabs.

Rice. 1. Spectra of solar radiation and semiconductor laser.

The degree of monochromaticity of laser radiation can be characterized by the spectral width of the laser line (the width can be specified as the wavelength or frequency detuning from the maximum intensity). Typically, the spectral width is set at 1/2 ( FWHM), 1 / e or 1/10 of the maximum intensity. In some modern lasers, a peak width of several kHz has been achieved, which corresponds to a laser line width of less than one billionth of a nanometer. For specialists, we note that the laser line width can be orders of magnitude narrower than the spontaneous emission line width, which is also one of the distinctive characteristics of a laser (in comparison, for example, with luminescent and superluminescent sources).

Laser coherence

Monochromaticity is an important but not the only property of laser radiation. Another defining property of laser radiation is its coherence. Usually they talk about spatial and temporal coherence.

Let us imagine that the laser beam is split in half by a semitransparent mirror: half of the beam energy passed through the mirror, the other half was reflected and went into the system of guiding mirrors (Fig. 2). After that, the second beam converges again with the first, but with a certain time delay. The maximum delay time at which the beams can interfere (i.e. interact taking into account the phase of the radiation, and not only its intensity) is called the coherence time of the laser radiation, and the length of the additional path that the second beam traveled due to its deflection is called the longitudinal coherence. The longitudinal coherence length of modern lasers can exceed a kilometer, although for most applications (e.g., industrial lasers for material processing) such a high spatial coherence of the laser beam is not required.

You can separate the laser beam in another way: instead of a semitransparent mirror, put a completely reflecting surface, but it does not cover the entire beam, but only a part of it (Fig. 2). Then the interaction of radiation will be observed, which propagated in different parts beam. The maximum distance between the points of the beam, the radiation at which will interfere, is called the length of the transverse coherence of the laser beam. Of course, for many lasers, the transverse coherence length is simply equal to the diameter of the laser beam.

Rice. 2. Towards an explanation of the concepts of temporal and spatial coherence

Angular divergence of laser radiation. Parameter M 2 .

No matter how we strive to make the laser beam parallel, it will always have a nonzero angular divergence. The smallest possible angle of divergence of laser radiationα d ("Diffraction limit") in order of magnitude is determined by the expression:

α d ~ λ / D, (1)

where λ is the wavelength of laser radiation, and D Is the width of the beam emerging from the laser. It is easy to calculate that at a wavelength of 0.5 µm (green radiation) and a laser beam width of 5 mm, the angle of divergence will be ~ 10 -4 rad, or 1/200 degree. Despite such a small value, the angular divergence may turn out to be critical for some applications (for example, for the use of lasers in combat satellite systems), since it sets the upper limit of the attainable power density of laser radiation.

In general, the quality of the laser beam can be set by the parameter M 2 ... Let the minimum achievable spot area created by an ideal lens when focusing a Gaussian beam be S ... Then, if the same lens focuses the beam from a given laser into a spot with an area S 1> S, parameter M 2 laser radiation is equal to:

M 2 = S 1 / S (2)

For the highest quality laser systems, the parameter M 2 is close to unity (in particular, lasers with the parameter M 2 equal to 1.05). However, it should be borne in mind that by far not all classes of lasers can currently achieve a low value of this parameter, which must be taken into account when choosing a laser class for a specific task.

We have briefly summarized the main properties of laser radiation. Let us now describe the main components of the laser: a medium with an inverted population, a laser cavity, a laser pump, and a laser level scheme.

Inversely populated environment. Diagram of laser levels. Quantum exit.

The main element that converts the energy of an external source (electrical, energy of non-laser radiation, energy of an additional pump laser) into light is a medium in which an inverted population of a pair of levels is created. The term "inverse population" means that a certain fraction of structural particles of the medium (molecules, atoms, or ions) is transferred to an excited state, and for a certain pair of energy levels of these particles (upper and lower laser levels), there are more particles at the upper energy level than on the bottom.

When passing through a medium with an inverted population, radiation, the quanta of which have an energy equal to the energy difference between the two laser levels, can be amplified, while removing the excitation of some of the active centers (atoms / molecules / ions). Amplification occurs due to the formation of new quanta of electromagnetic radiation, which have the same wavelength, direction of propagation, phase and state of polarization as the original quantum. Thus, the laser generates packets of identical (equal in energy, coherent and moving in the same direction) photons (Fig. 3), which determines the main properties of laser radiation.

Rice. 3. Generation of coherent photons under stimulated emission.

However, in the classical approximation, it is impossible to create an inversely populated environment in a system consisting of only two levels. Modern lasers usually have a three-level or four-level system of levels involved in lasing. In this case, excitation transfers the structural unit of the medium to the uppermost level, from which the particles relax in a short time to a lower energy value - the upper laser level. One of the lower levels is also involved in lasing — the ground state of an atom in a three-level scheme or an intermediate state in a four-level one (Fig. 4). The four-level scheme turns out to be more preferable due to the fact that the intermediate level is usually populated by a much smaller number of particles than the ground state; accordingly, it is much easier to create an inverse population (the excess of the number of excited particles over the number of atoms at the lower laser level) (to start lasing, one must inform less energy).

Rice. 4. Three-level and four-level systems of levels.

Thus, during laser generation, the minimum value of the energy imparted to the working medium is equal to the excitation energy of the uppermost level of the system, and generation occurs between the two lower levels. This explains the fact that the laser efficiency is initially limited by the ratio of the excitation energy to the laser transition energy. This ratio is called the quantum yield of the laser. It should be noted that usually the efficiency of a laser from the mains is several times (and in some cases even several tens of times) lower than its quantum efficiency.

Semiconductor lasers have a special structure of energy levels. In the process of radiation generation in semiconductor lasers, electrons of two semiconductor bands are involved, however, due to impurities that form a light-emitting p - n transition, the boundaries of these zones in different parts of the diode are shifted relative to each other. Inverse population in the area p - n transition in such lasers is created due to the flow of electrons into the transition region from the conduction band n -Part and holes from the valence band p -Plot. You can read more about semiconductor lasers in the specialized literature.

In modern lasers, various methods are used to create an inverted population, or to pump a laser.

Laser pumping. Pumping methods.

For a laser to start generating radiation, it is necessary to supply energy to its active medium in order to create an inverted population in it. This process is called laser pumping. There are several main pumping methods, the applicability of which in a particular laser depends on the type of active medium. So, for excimer and some gas lasers operating in a pulsed mode (for example, CO 2 - laser), it is possible to excite the molecules of the laser medium by an electric discharge. In cw gas lasers, a glow discharge can be used for pumping. Semiconductor lasers are pumped by applying a voltage across p - n transition of the laser. For solid-state lasers, you can use an incoherent radiation source (a flash lamp, a ruler or an array of light-emitting diodes) or another laser whose wavelength corresponds to the energy difference between the ground and excited states of an impurity atom (in solid-state lasers, as a rule, lasing occurs on atoms or ions impurities dissolved in the matrix grid - for example, for a ruby ​​laser, chromium ions are an active impurity).

Summarizing, we can say that the method of pumping a laser is determined by its type and the characteristics of the active center of the generating medium. As a rule, for each specific type of lasers there are the most effective method pumping, which determines the type and design of the system for supplying energy to the active medium.

Laser resonator. Lasing condition. Stable and unstable resonators.

The active medium and the system for delivering energy to it are still insufficient for the appearance of lasing, although it is already possible to build some devices on their basis (for example, an amplifier or a superluminescent radiation source). Laser generation, i.e. emission of monochromatic coherent light occurs only in the presence of feedback, or a laser resonator.

In the simplest case, the cavity is a pair of mirrors, one of which (the laser output mirror) is semitransparent. As another mirror, as a rule, a reflector with a reflection coefficient at the lasing wavelength close to 100% (“dull mirror”) is used in order to avoid laser generation “in two directions” and unnecessary energy loss.

The laser resonator provides a return of part of the radiation back into the active medium. This condition is important for the appearance of coherent and monochromatic radiation, since the photons returned to the medium will cause the emission of photons of the same frequency and phase. Correspondingly, the quanta of radiation that arise again in the active medium will be coherent with those that have already left the cavity. Thus, the characteristic properties of laser radiation are largely due to the design and quality of the laser resonator.

The reflectance of the output semitransparent mirror of the laser resonator is selected in such a way as to ensure the maximum output power of the laser, or based on the technological simplicity of manufacturing. For example, in some fiber lasers, an evenly cleaved fiber end face can be used as an output mirror.

An obvious condition for stable lasing is the condition of equality of the optical losses in the laser cavity (including losses due to the radiation output through the cavity mirrors) and the radiation gain in the active medium:

exp ( a× 2L) = R 1 × R 2 × exp ( g× 2L) × X, (3)

where L = active medium length,ais the gain in the active medium, R 1 and R 2 are the reflection coefficients of the resonator mirrors andg- “gray” losses in the active medium (ie, radiation losses associated with density fluctuations, defects in the laser medium, scattering of radiation and other types of optical losses that cause the attenuation of radiation when passing through the medium, except for the direct absorption of radiation quanta by the atoms of the medium). The last factor " X »Denotes all other losses present in the laser (for example, a special absorbing element can be introduced into the laser so that the laser generates short-duration pulses), in their absence it is equal to 1. To obtain the condition for the development of lasing from spontaneously emitted photons, the equality must be replaced with ">".

Equality (3) implies the following rule for choosing the output laser mirror: if the radiation gain by the active medium, taking into account gray losses (a- g) × L small, the reflectance of the output mirror R 1 should be chosen large so that the lasing does not damp due to the emission of radiation from the cavity. If the gain is large enough, it usually makes sense to choose a lower value. R 1 , since a high reflection coefficient will lead to an increase in the radiation intensity inside the cavity, which can affect the laser lifetime.

However, the laser cavity needs alignment. Let's assume that the cavity is composed of two parallel, but not aligned mirrors (for example, located at an angle to each other). In such a cavity, the radiation, having passed through the active medium several times, leaves the laser (Fig. 5). Resonators in which the radiation leaves its limits in a finite time are called unstable. Such resonators are used in some systems (for example, in high-power pulsed lasers of a special design); however, as a rule, the instability of the resonator is usually avoided in practical applications.

Rice. 5. Unstable resonator with misaligned mirrors; stable resonator and

stationary beam of radiation in it.

To increase the stability of the resonator, curved reflective surfaces are used as mirrors. At certain values ​​of the radii of the reflecting surfaces, this resonator turns out to be insensitive to small misalignments, which makes it possible to significantly simplify the work with the laser.

We have briefly described the minimum required set of elements to create a laser and the main features of laser radiation.

Transcript

1 Ministry of Education and Science Russian Federation Moscow State University of Geodesy and Cartography INTERACTION OF LASER RADIATION WITH SUBSTANCE Moscow 2014

2 Ministry of Education and Science of the Russian Federation Moscow State University of Geodesy and Cartography Yu.M. Klimkov, V.S. Mayorov, M.V. Khoroshev Interaction of laser radiation with matter Moscow 2014

3 UDC Reviewers: Doctor of Phys.-Math. Sci., Professor of IPLIT RAS F.V. Lebedev; Professor of the Department of Physics, MPEI E.F. Ishchenko Compiled by: Yu.M. Klimkov, V.S. Mayorov, M.V. Horoshev Interaction of laser radiation with matter: a tutorial. M .: MIIGAiK, p. This course has been prepared in accordance with the approved by the Ministry of Education and Science of the Russian Federation a sample program disciplines for the direction "Laser technology and laser technology". Recommended by the Department of Design and Technology of Optical Instrumentation and approved for publication by the Editorial and Publishing Commission of the Faculty of Optical Information Systems and Technologies. The manual aims to help students of the 5th year of FOIST in mastering theoretical material on the course "Interaction of laser radiation with matter", as well as in the implementation of practical and laboratory work on this course. Electronic version study guide posted on the website of the MIIGAiK library

4 Introduction Interaction of laser radiation with matter is one of the most important scientific directions of modern optics and laser physics. This direction made it possible to necessary and substantially supplement the concept of fundamental photophysical processes occurring in matter (primarily in condensed media) when exposed to intense light fluxes of various durations and wavelengths. It also allowed the development of the physical foundations of numerous applications related to lasers and their applications in technology. Therefore, knowledge of the basic mechanisms and laws of laser action on matter is necessary for a full-fledged university and engineering education on laser technology and technology. The purpose of the discipline is to form students' ideas about the interaction of intense radiation with matter in the most important scientific section of optical physics. The task of the discipline is to give students modern special knowledge, taking into account the latest scientific achievements in the field of laser effects on matter and to link this knowledge with other disciplines of the specialty and general physical disciplines. The course is based on knowledge of the basic provisions and terminology of the courses "Physics", "Fundamentals of Optics", "Chemistry", "Metallurgy and technology of structural materials", "Laser technology", " Physical fundamentals quantum electronics "," Optoelectronic devices and systems "and is the base for studying the course" Laser technologies ". List of accepted designations A absorptive capacity of the medium B magnetic induction C specific heat D electric induction E electric field strength H magnetic field strength I intensity of incident radiation M mass of matter N number of photons, excited particles, population level Q heat source power R reflection coefficient 3

5 S interaction area T temperature a thermal diffusivity b lattice relief amplitude c speed of light or specific heat d lattice wave vector e emissivity h penetration depth j current density k wavenumber m imaginary part of the refractive index n refractive index q surface power density of the heat source r radius vector of spatial coordinates r, d dimensions cross sections of the laser beam t time v velocity of movement x, y, z spatial coordinates α absorption coefficient in the medium β scattering coefficient γ collision frequency of free electrons δ surface charge density ε dielectric constant ζ skin layer thickness η kinematic viscosity θ angular value λ radiation wavelength or thermal conductivity μ magnetic permeability ρ substance density σ conductivity or coefficient surface tensionτ time of exposure or duration of the process χ extinction coefficient ψ phase of electromagnetic oscillations ω frequency of electromagnetic oscillations ħ Planck's constant 4

6 1. MECHANISMS OF ENERGY ABSORPTION AND DISSIPATION IN THE ENVIRONMENT 1.1. Absorption electromagnetic waves in a medium Laser radiation is one of the types of electromagnetic energy and, therefore, the interaction of laser radiation with matter is the interaction with matter of an electromagnetic wave with its specific properties and characteristics (such as coherence, monochromaticity, etc.). Laser technological processes of material processing are primarily associated with local heating, i.e. with the transfer of energy from an electromagnetic wave to matter. All physical models of laser technology include an expression of the energy conservation law. Most often, it is the heat conduction equation in a different formulation, which for isotropic media is written in general form: T ρ c + v grad () T div λ grad () () T = qr, t, t (1.1.1) where ρ is the density ; c specific heat capacity; λ thermal conductivity; v () rt, velocity field vector; qrt (), volumetric power density of heat sources. The initial and boundary conditions for a specific heat problem are set based on the specifics of the process under consideration. A laser volumetric heat source qrt () arising in a condensed medium, in many cases, can be considered surface. Thermophysical coefficients, which are functions of temperature, are usually considered constant in engineering calculations and their averaged values ​​are used. To estimate the values ​​of the parameters of many thermal problems, the solution of the one-dimensional linear heat equation describing the heating of a semi-infinite medium (z> 0) by an unbounded homogeneous surface source is often used: AI z T () z, t = 0 4 at ierf, (1.1.2) λ 4at where A is the absorption capacity of the medium; I 0 the intensity of the incident radiation; a = λ / ρc thermal diffusivity. 5

7 The probability integral function ierf (x) erf (y) dy is tabulated y x 2 2 t. The error function erf (y) e dt π is also a table function (or is calculated numerical methods). Since ierf () 0 = 1 / π, then using formula (1.1.2), the temperature on the surface is often estimated, and in general, the value of one of the quantities T, I 0, t is found from the other two. For example, using the formula пл πλtпл I = (1.1.3) 4at, an estimate is made of the critical power density absorbed on the surface and necessary for the material to start melting in time t. Or, for example, the depth h to which heat penetrates during time τ is estimated by the expression h 2 aτ (1.1.4) The one-dimensional approximation is valid if the size of the laser beam on the surface of the material significantly exceeds the depth of penetration of heat into the material. In any case, the laser thermal effect on materials is important not just the power of the laser radiation, but the power absorbed by the material and used to obtain a useful result. The absorption capacity A, which is the coefficient at I 0 in expression (1.1.2), appears in one form or another in all laser technological processes. There are many different physical and physicochemical processes that affect the absorption capacity. The intensity of an electromagnetic wave propagating in the material being processed in the direction of the z axis changes according to the Bouguer Lambert law 6 0 () () I z = AI0 exp α z, (1.1.5) where I 0 is the intensity of the wave incident on the interface between the media; And the absorption capacity, for which A = e = 1 R (e emissivity, R reflection coefficient); α is the absorption coefficient of electromagnetic energy in the medium. The intensity of the electromagnetic wave falls 2.73 times at a distance of δ = 1 / α.

8 Absorbance A shows the fraction of the absorbed flux (energy), and the coefficient of absorption of light in the medium α, how quickly the radiation is absorbed during propagation. The heat source q, which appears in the material under the action of laser radiation, is characterized by both the total power proportional to A and localization in the volume, depending on α. Light absorption by conductors (metals) In metals (conductors), an electromagnetic wave decays exponentially in a very thin surface layer of the so-called. "Skin layer" (δ ~ cm, ie α ~ cm -1), and absorption occurs on conduction electrons. In laser processing of materials, the depth of penetration of heat into the depth of the metal, although by several orders of magnitude exceeds the thickness of the skin layer, practically adjoins the surface of the material, and therefore, in all calculations, the heat source can be considered surface. The interaction of light with metals (conductors) is determined by the presence in them of a large number of electrons, so weakly connected with the crystal lattice that these electrons can be considered practically free. The electrostatic positive charge of the ions in the metal lattice compensates for the negative charge of these electrons. Many free electrons (conduction electrons) are called electron gas. The concentration of free electrons in metals is very significant (~ cm -3). In the field of an incident electromagnetic wave, free electrons oscillate and emit secondary waves, which, when added, give a strong reflected wave. The absorption of light by conduction electrons is possible only when they interact with the metal lattice and therefore partially transforms into heat. In an ideal conductor, where there is no Joule heat loss at all, the absorption is zero, since the incident light is completely reflected. The absorption of light leads to an increase in the energy of free electrons. Since the time for establishing equilibrium in a gas of electrons is much shorter than the time for establishing equilibrium between electrons and the lattice of atoms, two thermodynamic subsystems with different temperatures, electron and phonon, appear in the metal. Part of the absorbed energy is transferred by the electrons to the lattice, but the transfer efficiency is low due to the large difference in the masses of electrons and ions. Therefore, at the first moment, the electron gas is significantly overheated in comparison with the lattice. However, an increase in the temperature of the electron gas occurs only until the amount of energy transferred to the lattice is equal to the amount of 7

9 energy received by electrons from an electromagnetic wave. In view of the fact that the characteristic time of energy exchange between these subsystems is τ ~ s, and the times of exposure of radiation to matter during laser processing of materials are almost always several orders of magnitude longer, then in what follows we will use the total temperature of the metal. For good conductors, the reflectance R is usually close to 1 and, accordingly, the absorbance A is low. Here are the values ​​of the absorption capacity of some metals (pure; at 20 ° C): Table Lasers 8 Metal Ar + λ ~ 0.488 μm Ruby λ ~ 0.69 μm Nd YAG λ ~ 1.06 μm СО 2 λ ~ 10.6 μm Al Aluminum 0 , 19 0.11 0.08 0.019 W Tungsten 0.55 0.50 0.41 0.026 Fe Iron 0.68 0.45 0.35 0.076 Cu Copper 0.56 0.17 0.10 0.015 Ni Nickel 0.40 0.32 0.26 0.03 Ag Silver 0.05 0.04 0.04 0.014 Ti Titanium 0.48 0.45 0.42 0.08 These data are valid for metals in vacuum and in most practical cases they cease to correspond to reality (for example, absorption capacity increases due to surface oxidation). Light absorption by dielectrics Light absorption by dielectrics is highly wavelength dependent. In the IR region, absorption is determined by the vibrational states of the crystal lattice, and in organic compounds, by molecular vibrations. For this region, the typical values ​​of the absorption coefficient α ~ cm -1. In the visible region, absorption can be caused by impurities (for example, transition metal ions, crystal lattice defects, etc.) or by the "tail" of strong UV absorption bands. It can also be caused by discrete electronic transitions in molecular crystals (for example, in many organic compounds). Typical coefficients in the absorption band ~ cm -1. The coefficient α can be correlated with the transmission capacity of a layer with a thickness h by the ratio

10 (I / I 0) 100 = Transmittance (%) = 100 e -αh, or (I / I 0) = e -αh, where I 0 is the intensity of the incident radiation; I is the intensity of the transmitted radiation. A useful measure of the thickness over which significant attenuation of incident radiation occurs is L = 1 / α, where L is the attenuation length. For strong absorbers α = 10 6 cm -1 and L = 10-6 cm, for relatively weak ones α = 10 cm -1 and L = 10-1 cm. The relationship between α and the refractive index is α = 4π m / λ, where m is the imaginary part of the complex refractive index ñ = n im; λ is the wavelength of the incident light. Table Material Wavelength interval at which the transmission capacity decreases to 10%, μm Al 2 O 3 0.15.6.5 As 2 S 3 0.6 13 BaF 2 0.14 15 CdSe 0.72 24 CdS 0.5 16 CdTe 0.3 30 CaF 2 0.13 12 CsBr 0.2 45 CuCl 0.4 19 Diamond (type IIa) 0.225 2.5; GaAs 1 15 Ge 1.8 23 InAs 3.8 7 PbS 3 7 MgO 0.25 8.5 Se 1 20 SiO 2, (fused) 0.2 4.5 Si 1.2 15 TiO 2 0.43 6, 2 ZnSe 0.5 22 ZnS 0,

11 In the visible region for nominally transparent materials usually k ~ 10-5 or α ~ 10 cm -1. The table shows the wavelength intervals in the IR region in which dielectrics and semiconductors are nominally transparent. In the transparency range of these materials, α can be 1 10 cm -1. Most materials with SiO bonds are relatively transparent in the visible region of the spectrum, but strongly absorb in the vicinity of λ = 10 μm. Therefore, for the processing of quartz, glass and silicate minerals, it is best to use a CO 2 laser. Organic solids absorb strongly in the infrared region, but can be transparent at shorter wavelengths (eg polyethylene). Thus, a CO 2 laser is ideal for processing these materials. Unlike metals, in which absorption of radiation occurs at the surface in the skin layer, absorption in dielectrics and most semiconductors occurs in a layer determined by the attenuation length L, which can significantly exceed the usual thickness of the skin layer. In the IR region, L> 10-4 cm, and thus, in many cases, heating can be considered volumetric. In particular, this applies to the heating of thin films, where L can exceed the film thickness. Although the reflection coefficients of polarized light depend on the angle of incidence and direction of polarization, as in metals, the dielectric constant ε is almost equal to unity during processing, and the phenomena observed when processing metals using polarized rays are not detected when processing dielectrics. Light absorption by semiconductors. The electrical and optical properties of semiconductors are related to the fact that the energy levels filled with electrons in the valence band are separated from the conduction band by a forbidden band. Accordingly, one should use a quantum approach, considering light as a flux of photons with energy ħω. Semiconductors have a low concentration of free electrons, and if the energy of the light quantum is less than the band gap (ħω

12 of the space charge region, which is due to the existence of surface local centers. Such centers can be associated both directly with the cutoff of periodicity and with adsorbed atoms and molecules. When analyzing the thermal effect of radiation on semiconductors, the following mechanisms of absorption of electromagnetic radiation are distinguished: 1. Intrinsic (interband) absorption of light. If the energy of a quantum is greater than the band gap (ħω> ezz), then due to the internal photoelectric effect, electrons from the valence band pass into the conduction band. Their lifetime until the moment of electron-hole recombination with the release of heat in the lattice is approximately s. The semiconductor begins to approach metals, and its reflectivity increases. At the same time, when radiation is absorbed by free carriers, the so-called. "Heating" (acceleration of motion) of an initially small number of electrons in the conduction band, which leads to an increase in the electron concentration as a result of thermal ionization of the valence band, i.e. a self-accelerating process of heating the substance can take place. The absorption coefficient α 1 is cm Intraband absorption (absorption by free carriers by electrons and holes). In essence, it is similar to absorption by free electrons in metals, but differs in the concentration of free carriers, which is small in the equilibrium state (cm -3). The coefficient of this absorption is α 2 ~ cm Impurity absorption. It involves carriers with energy states in the forbidden band (ħω

13 is carried by conduction electrons, and they make a significant contribution to the total thermal conductivity. Energy transfer in semiconductors can also be carried out using recombination radiation. Based on the foregoing, it can be concluded that by the nature of their susceptibility to laser radiation, semiconductors occupy an intermediate position between metals and transparent materials. As a result of absorption of laser radiation by semiconductors, electron-hole pairs are formed, which transfer radiation energy to the crystal lattice during recombination. Therefore, with an increase in the power of laser radiation, damage to the material will occur as a result of heating. This damage process is typical for lightly doped silicon. However, if the semiconductor is heavily doped, the damage is similar to damage in metals. The surface finish of the semiconductor also has a large influence on the damage threshold in the material. Etching increases the threshold of relatively coarsely ground crystals by more than 3 times, and for those made by chipping or chemical grinding by 10-15%. Scratches have a negligible effect, although damage in the area of ​​scratches is more noticeable Reflection and absorption of radiation by a medium with a flat surface In Section 1.1, it was said that the efficiency of using the energy of laser radiation when processing materials directly depends on their absorption capacity A. Assuming that the medium absorbs all the refracted electromagnetic wave (i.e., assuming the thickness of the medium >> 1 / α), consider the absorption capacity A (or, which is equivalent, the reflection coefficient R = 1 A) of a material with an ideal smooth flat surface. If the surface is imperfect, for example, rough, new and very nontrivial effects appear, some of which will be discussed later. Let us recall the basic concepts and properties of the electromagnetic field. The electromagnetic field is represented by two vectors: E B is the strength of the electric field; E B magnetic induction. In order to describe the effect of the field on material objects, it is necessary to introduce the second group of vectors: D Helectric induction; D H magnetic field strength. The spatial and temporal derivatives of these vectors are related by Maxwell's equations: 12

14 B rote + = 0; t D 4π roth = j t c (the first pair of Maxwell's vector equations), and divd = ρ; divb = 0 (1.2.1) (1.2.2) (1.2.3) (1.2.4) (the second pair of Maxwell's scalar equations). From equations (1.2.2) and (1.2.3), (remembering that div () 0 rot), follows the continuity equation, reflecting the law of conservation of charge: ρ + divj = 0, (1.2.5) t that is, the charges ρ and the currents j are related by this equation, and they cannot be set arbitrarily, independently of each other. In order for the Maxwell equations to have a unique solution for the field vectors for a given distribution of charges and currents, it is necessary to add relations describing the behavior of substances under the influence of the field. Such ratios are called material equations. For isotropic substances, the material equations are written in the form D = εε0 E; (1.2.6) B = µµ 0 H; (1.2.7) j = σe, (1.2.8) where ε is the dielectric constant; μ magnetic permeability; σ specific conductivity. Equation (1.2.8) is the differential form of Ohm's law. For optics, a situation is typical when there are boundaries between the media, on which the physical properties change abruptly. Consider (without derivation) the boundary conditions at the interface between two media. The normal component of the magnetic induction vector is continuous at the interface: 13

15 14 B n2 Bn 1 = 0. (1.2.9) The normal component of the electric induction vector on a surface with surface charge density ρ * experiences a jump equal to 4πρ *: Dn2 Dn 1 = 4 πρ *. (1.2.10) In the presence of a current with a surface density j *, the tangential component of the magnetic field strength experiences a jump equal to 4 π j *: c 4π Ht2 Ht1 = j *. (1.2.11) c The tangential component of the electric field strength is continuous at the interface: E E = (1.2.12) t2 t1 0. Reflection and refraction of a plane electromagnetic wave. Let a plane linearly polarized electromagnetic wave fall at an angle θ 1 onto the surface of the material (Fig). It splits into two waves: passing into the second medium and reflected. The existence of two waves follows from the solution of the problem with the given boundary conditions, since they cannot be satisfied if one does not postulate the presence of both transmitted and reflected waves. The angle of refraction is determined from the well-known expression: sin θ1 sin θ 2 = (1.2.13) n Fig Reflection and refraction of a plane electromagnetic wave from the interface between two where n = εµ is the refractive index. From Maxwell's equations and boundary conditions for the components of the electric and magnetic fields, the solution of the wave equation for the reflected and refracted waves (Fresnel's formula) is found

16 () r () (i tg θ1 θ2) // = //; (θ 1 + θ2) E E tan E () r () i sin (θ1 θ2) = E; sin (θ + θ) 1 2 (1.2.14) EE () t () i // = E // () t () i = E sin 2 sin θ cosθ 2 1 (θ + θ) (cos θ θ ) sin θ2 cosθ1. sin (θ + θ) 1 2; (1.2.15) In the general case of an absorbing medium, the refractive index is complex: () 1, n = n χ i (1.2.16) where χ is called the extinction coefficient (the attenuation of the beam during its propagation in the medium. From the Fresnel formulas, the expressions for reflection coefficients R. For transparent media (χ = 0), if the incident wave vector lies in the plane of incidence (p polarization), then (θ1 θ2), (θ + θ) 2 tan R // = (1.2.17) 2 tan 1 2 and if the vector is perpendicular to the plane of incidence (s polarization), then 2 sin R = (1.2.18) 2 sin 1 2 π At the Brewster angle θ 2 = θ 1 for p polarization of the component 2 () r E of the reflected wave becomes equal to zero (Fig, a) in ta (θ1 θ2). (θ + θ) // the case of a transparent medium and has a minimum value for an absorbing medium.For absorbing media, the angle θ 2 in expression (1.2.13) due to the complexity of the refractive index will also be complex, and this must be taken into account when substituting it into formulas (1.2.14), (1.2.15). At normal incidence (θ 1 = 0), the reflection coefficient is 15

17 a b 16 Fig Dependence of the reflection coefficient R on the angle of incidence θ for E // (p polarization, curves 1) and E (s polarization, curves 2) for the cases: a transparent medium at n = 1.5; b of the absorbing medium at n = 1.5 and χ = 1 R = () n 1 n () n 1 n χ χ. (1.2.19) If nχ >> (n + 1), then R 1; thus, at normal incidence, strong reflection is associated with large absorption of radiation in the medium. In oblique incidence, the expressions obtained are rather complex; if n 2 + n 2 χ 2 >> 1, then the following relations are valid: RR () () n 1 + χ cos θ 1 2 n cos θ + 1 1 = (p polarization), (1.2.20) n 1 + χ cos θ + 2 n cosθ + 1 // () n () n 1 1 n 1 + χ 2 cosθ + cos θ = n 1 + χ + 2 cosθ + cos θ (s polarization). (1.2.21) The component of the electric vector of the reflected wave for the polarization lying in the plane of incidence (p polarization) reaches a minimum at a certain angle of incidence (Fig. 1.2.2, b). Here are the real dependences of the reflectivity of the reflection coefficient) on the angle of incidence for iron and copper (Fig. 1.2.3, a, b). 2 4σ For metals n = ε 1 i (1.2.22) εω 0

18 and in most cases 4σ / ω >> 1 (in the optical range μ 1), therefore (1.2.19) takes the form (for normal incidence): where ω = 2πν is the circular frequency of light. 2ω R = 1 A = 1, (1.2.23) πσ a b Fig Dependence of the reflection coefficient R on the angle of incidence θ for E // (p polarization, curves 1) and E (s polarization, curves 2) for iron (a) and copper (b): solid lines at 20 C, dashed lines at 1000 C The Hagen – Rubens relationship (1.2.23) for the static conductivity is in good agreement with experimental data for IR wavelengths with λ> 5 μm. Metals are good conductors; therefore, in accordance with (1.2.23), their absorption capacity A at the emission wavelength of CO 2 lasers (λ = 10.6 μm) is small (see Table 1.1.1). It is especially low for non-ferrous metals (Al, Cu) and even more so for precious metals (Ag, Au). That is why laser processing of these materials is either difficult or practically impossible by the radiation of a CO2 gas laser. Moreover, gold coatings (less often silver due to oxidation) are often used to make mirrors for these lasers. Laser processing of non-ferrous metals is much more efficient with the radiation of solid-state YAG lasers (λ = 1.06 μm), where the absorption is much higher. The dependence of the absorption capacity A on the angle of incidence and polarization has a strong effect in laser cutting and laser welding with deep penetration, and should also be taken into account when designing various laser sensors (for example, spark gaps). 17

19 18 2. SURFACE ELECTROMAGNETIC WAVES (SEW) AND ABSORPTION OF LASER RADIATION Real surfaces of materials are never absolutely smooth, the presence of even an insignificant relief and microroughness can radically change the previously described character of interaction and absorption of laser radiation by matter. When an electromagnetic wave falls on a rough surface due to diffraction, surface electromagnetic waves of SEW (or otherwise surface polaritons) arise. SEWs propagate along the interface between two media and exist simultaneously in both of them. Interest in the study of SEWs in the optical range is due to the fact that they can be effectively excited by EM radiation on a real surface and significantly affect various processes. Among these processes: scattering of light by particles adsorbed on the surface; generation of higher harmonics when laser radiation is reflected from metals; change in absorption; photochemical reactions; the formation of surface periodic structures. SEWs are localized near the surface and decay exponentially on both sides of it (Fig. 2.1: () A = A0 exp (± ψ1,2z) exp i kx s ωt. (2.1) Fig Localization of SEW at the interface between the SEW media is not strictly transverse electromagnetic waves, but are partially longitudinal electromagnetic waves of the TM type: magnetic vector H perpendicular to the direction of SEW propagation (wave vector k s) lies in the plane of the surface. The electric vector has two components: E z perpendicular to the surface, and E x along the wave vector k s. rice Interference of SEW with the incident, reflected and

20 broken waves determines the nature of the electromagnetic field at the surface and its dissipation (absorption). As a result, any clean, unoxidized surface can have a very high absorption capacity A1, if the surface relief has a certain periodicity, modulation depth and orientation. Since an arbitrary roughness can be represented by its spatial Fourier spectrum, in principle the problem of diffraction by surface roughness is reduced to the problem of diffraction by a sinusoidal relief. The resulting electromagnetic fields are obtained by superposition of the incident and all waves diffracted by the Fourier gratings. Consider the incidence of a plane electromagnetic wave E (x, y, z, t) = E exp (ikx + ik z ω it) + kom ... iitz on the surface of a medium with a dielectric constant Fig. Components of the electric and magnetic fields of SEW (2.2) 2 ε ( ω) = ε (ω) + i ε (ω) = (n + im) (2.3) and the magnetic permeability μ = 1 filling the half-space z ξ (x, y) = 2 aq cos (q r + φ) = ξq exp (iqr) + kom flop .., (2.4) where k ω is the wave vector of the incident electromagnetic wave k 0 0 = and, accordingly, c kt = k0 sin θ x; (2.5.a) kz = k0 cos θ z, (2.5.b) where q 2π is the wave vector of the lattice q =; r = () xy, radius vector, d lying in the plane z = 0; θ is the angle of incidence of the electromagnetic wave. 19

21 As a result of the diffraction of incident radiation (2.2) at the modulated boundary (2.4), a set of diffraction fields arises both outside the medium E = E exp (ik r + γ zi ω t) + com. 20 pppp Γ = kkpp 0, and inside medium E = E exp (ikr γ zi ω t) + kom flat .. pppp γ = kk ε, pp 0 where the index p (p = 0; ± 1; ± 2;) denotes the diffraction order; k = k p q p t (2.6) (2.7) (2.8) The value p = 0 corresponds to the specularly reflected and refracted Fresnel waves. Expressions for the amplitudes of the fields outside and inside the medium are determined from the solution of Maxwell's equations and the boundary conditions for the components of the total electromagnetic field. They are described in the specialized literature and are quite complex. At the same time, some characteristics of the SEW can be obtained from general rather simple representations, for example, using vector diagrams of the law of conservation of momentum. Consider first-order diffracted waves (p = ± 1). As a result of diffraction on a sinusoidal relief, these two waves have wave vectors k1 = k t q (one hundred and fifty); (2.9) k = 1 k + t q (an stisto xso vva vow ln a), (2.10) which can be represented in the form of a vector diagram. In fig. 2.3, a circle is drawn with a radius R = k 0 equal to the value of the wave vector of the incident wave. Naturally, the incident p polarized wave will especially efficiently excite the surface wave (resonance case) when the wave vector of the Stokes and / or anti-Stokes waves is equal to the wave vector of the incident wave: kp k (2.11) 0,

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As a manuscript

Alexey Seteikin

INTERACTION OF LASER RADIATION

WITH MULTI-LAYER MATERIALS

01.04.21- laser physics

for an academic degree

Doctor of Physical and Mathematical Sciences

Saint Petersburg - 2011

The work was performed at the federal state budgetary educational institution of higher vocational education St. Petersburg State Polytechnic University

(FGBOU VPO "SPbSPU")

Scientific consultant:

Privalov Vadim Evgenievich

Official opponents: Doctor of Physical and Mathematical Sciences, Professor

Aksyonov Evgeny Timofeevich

Doctor of Physical and Mathematical Sciences, Professor

Tolmachev Yuri Alexandrovich

Doctor of Physical and Mathematical Sciences, Professor

Fedortsov Alexander Borisovich

Lead organization: Baltic State Technical University "Voenmech" named after D.F. Ustinova

The defense will take place "" 2011 in _______

at a meeting of the dissertation council D 212.229.01 at the Federal State Budgetary Educational Institution of Higher Professional Education "St. Petersburg State Polytechnic University" 195251, Russia, St. Petersburg, st. Polytechnicheskaya, d. 29, building 2, a. 470.

The dissertation can be found in the fundamental library

FSBEI HPE "St. Petersburg State Polytechnic University"

Scientific Secretary

dissertation council

Doctor of Technical Sciences, Professor A.S. Korotkov

GENERAL DESCRIPTION OF WORK

The dissertation work is devoted to the analysis of the processes of interaction of laser radiation in multilayer materials, using the methods of mathematical modeling.

Relevance of the topic. V last years, methods based on the use of laser radiation are widely used for diagnostics of the internal structure of various optically inhomogeneous objects, in particular, they find application in medicine, biology, materials science, physics of the atmosphere and ocean, and other fields of modern science.

Of particular interest are the issues of interaction of laser radiation with multilayer biological materials. Depending on the power density, three types of effects of interaction of laser radiation with biological tissue are distinguished: photochemical, at relatively low values ​​of the power density; thermal, at average values ​​of power density and photomechanical (non-linear), at very high values ​​of energy density and very short delivery time of radiation. With an increase in the energy density of the radiation delivered over a short time interval, an explosive removal of material (photoablation) occurs.

Due to the multilayer and multicomponent structure of biological tissue, the interaction of radiation with it turns out to be very complicated. For example, the stratum corneum reflects incident radiation, while the collimated light beam is converted into a diffuse one due to microscopic irregularities at the air - stratum corneum interface. Most of the light reflected by the skin is generated by backscattering from various tissue layers (stratum corneum, epidermis, dermis, microvascular system). The absorption of light by skin pigments provides quantitative information about the concentration of bilirubin, the saturation of hemoglobin with oxygen and the content of drugs in tissue and blood, which is the basis for methods for diagnosing a number of diseases.

To improve efficiency modern methods laser diagnostics, as well as for the development of new methods, it is necessary to study in detail the features of the process of light propagation in multilayer media, including biological tissues. However, at present there is no exact theory for describing the propagation of light in structurally inhomogeneous media, and experimental studies are complicated by the difficulties in maintaining the constancy of their structural and dynamic parameters. In this regard, computer modeling of the propagation of laser radiation is becoming increasingly important. It makes it possible to study more thoroughly the features of the propagation of a laser beam in model media, as well as to study the dependence of the results obtained on various parameters of the measuring system and the object under study, which is very difficult in an experiment. This allows you to develop recommendations for the most effective diagnostic measurements.

To interpret the results obtained and correctly diagnose the object under study, it is necessary to know the parameters of the propagation of light in it, which is achieved by comparing the experimental data and the results. computer simulation or theoretical calculations, if applicable. One of the main problems in calculating the propagation of radiation in biological objects is the choice of the method. In connection with rapid development computer technology often uses the Monte Carlo statistical test method. As applied to the propagation of radiation in multilayer media, this method is based on multiple repetitions of a numerical experiment to calculate a random trajectory of photons in the medium under study, followed by a generalization of the results obtained. With the accumulation of a sufficiently large amount of statistical data, the method makes it possible to make comparisons with experimental results, as well as to predict the results of experiments. The accuracy of such modeling is determined by the cost of computer time, as well as the correspondence of the model to the modeled object.

An important problem in modeling is the correct choice of the values ​​of the model parameters of the object used for the calculation, which cannot be measured explicitly. It should be noted that in a number of cases, in particular for many biological tissues, there is a significant discrepancy in the values ​​of their optical properties obtained by various authors.

All of the above confirms the relevance of the topic and allows you to formulate the purpose of this dissertation work.

The purpose of the thesis was:

Carrying out a study of the processes underlying the interaction of laser radiation of various intensities with multilayer biological media, creating models of these processes, on the one hand, important from the point of view of solving the general problem of the interaction of laser radiation with matter, and on the other hand, reflecting the specifics of multilayer biological materials.

Achieving this goal required:

1. Development of theoretical methods for studying and analyzing biological media, which involves a critical analysis of existing theories and models of light propagation in biological media and consideration of the mechanisms of interaction of laser radiation with biological tissues of complex geometry.

2. The creation of physical mathematical model propagation of laser radiation in media with arbitrary asymmetric geometry, including closed internal inhomogeneities complex shape, and methods for assessing the degree of its adequacy.

3. Analyzing the possibilities of using the developed model for solving purely practical problems and for creating new diagnostic techniques on its basis.

Scientific novelty

In the works summarized in this dissertation, the author for the first time:

1. Created scientific concept and methods of studying the interaction of laser radiation with biological tissues, arbitrary asymmetric geometry, including closed internal inhomogeneities of complex shapes.
2. A new computational domain of modeling is proposed, presented in the form of a grid with elements - tetrahedrons, which provides three-dimensional modeling of the radiation propagation process in multilayer structures, which allows working with biological media of arbitrary geometry.
3. The temperature reaction of biological tissues with the inclusion of nanoparticles to ultraviolet radiation was found. The change in the density of the absorbed light energy and temperature fields is calculated depending on the wavelength of the incident radiation, the concentration and dislocation of test nanoparticles included in the medium.

4. Developed and theoretically substantiated an original model of laser ablation of solid biological tissues, taking into account the multilayer nature of biological materials. The applicability of this model for describing the available experimental data on laser ablation of multilayer biological tissues is shown.

Reliability of results

The reliability of the results and conclusions obtained is ensured by the adequacy of the physical models used and mathematical methods, the correctness of the approximations used, the reproducibility of the calculated and experimental data, as well as their compliance with the results obtained by other authors.

Scientific and practical significance

A major scientific problem on the interaction of laser radiation with multilayer materials of any geometry has been solved. This makes it possible to generalize all the listed results and increases the scientific and practical significance not only of the results given in the dissertation, but also to make the previously obtained results more useful.

The results obtained can be used as methods for optical diagnostics of biological tissues, for example, in optical coherence tomography.

The method for calculating the temperature reaction of biological tissues using nanoparticles under irradiation with UV-A and UV-B light ranges is certified as a methodology by the State Service for Standard Reference Data (GSSSD), certificate No. 150.

Big practical use have calculations of the parameters of laser ablation of solid biological tissues. They can be used in laser surgery and dentistry.

The results obtained in the dissertation work can also be applied in the educational process - in the preparation of students, graduate students, in lecture courses in the specialty "Laser Physics".

The main provisions for the defense

1. The concept and methods of studying the interaction of laser radiation with heterogeneous multicomponent tissues and media with complex geometry, allowing to describe the processes of interaction of radiation with multilayer materials and serving as the basis for creating system software for real diagnostic techniques, devices and devices.

2. Model of the absorbed energy density distribution for different radiation ranges in multilayer media with arbitrary asymmetric geometry of the computational medium with the inclusion of closed internal inhomogeneities of complex shape, using the three-dimensional Monte Carlo method and finite element partitioning.

3. The main mechanisms of interaction of laser radiation of various intensities with multilayer tissues, which make it possible to establish the conditions for the occurrence and course of thermal processes in them and to assess the applicability of the developed model for studying the thermal loads of multilayer tissues characteristic of ablation processes in them.

4. Temperature reaction of biological tissues with the inclusion of nanoparticles to ultraviolet irradiation, which makes it possible to take into account the wavelength of the incident radiation, the concentration and dislocation of test nanoparticles included in the medium.

5. Model of laser ablation of solid biological tissues, taking into account the multilayer of biological materials.

Publication and approbation of results

The main results of the research presented in the dissertation were reported and discussed at the following scientific conferences:

I Far Eastern Conference with international and all-Russian participation. "New medical technologies in the Far East" (Khabarovsk, 1996); Regional Scientific Symposium "Ecology and Diseases of the Respiratory System, Application of New Technologies in Treatment" (Birobidzhan, 1997); II Far Eastern Scientific Conference "New Medical Technologies in the Far East" (Vladivostok, 1998); III Far East regional conference "New scientific technologies in the Far East region" (Blagoveshchensk, 1999); III International Scientific and Technical Conference “Quantum Electronics” (Minsk, 2000); III regional scientific conference “Physics: fundamental and applied research, education” (Blagoveshchensk, 2002); Regional School-Symposium "Physics and Chemistry solid"(Blagoveshchensk, 2003); International conference "Laser-optical technologies in biology and medicine" (Minsk, 2004; Fourth Asia-Pacific Conference "Fundamental Problem of Opto-and Microelectronics (APCOM 2004) (Khabarovsk, 2004); IV international conference of young scientists and specialists" Optics- 2005 "(St. Petersburg, 2005); V regional scientific conference" Physics: fundamental and applied research, education "(Khabarovsk, 2005); International symposium" Principles and processes of creating inorganic materials (Third Samson readings) "(Khabarovsk, 2006) ; VI regional scientific conference "Physics, fundamental and applied research, education" (Blagoveshchensk, AmSU, 2006); Scientific session MEPhI-2007 (Moscow, 2007); International Conference "Advanced Laser Technologies" (ALT) (Levi, Finland, 2007 ); International conference "Lasers. Measurements. Information. 2008" (St. Petersburg, 2008); XV All-Russian scientific and methodological conference "Telematics 2008" (St. Petersburg, 2008); International 1st Optical Congress "Optics - XXI century" (St. Petersburg, 2008); XVI International Scientific Conference "Laser Information Technologies in Medicine, Biology and Geoecology" (Novorossiysk, 2008); International conference “Lasers. Measurements. Information. 2009 "(St. Petersburg, 2009); VIII regional scientific conference "Physics: fundamental and applied research, education" (Blagoveshchensk, 2009); International Conference on Advanced Laser Technologies (ALT 09) (Antalya, Turkey, 2009); XX International Symposium on Bioelectrochemistry and Bioenergetics (Sibiu, Romania, 2009); International conference “Lasers. Measurements. Information "(St. Petersburg, 2010); International Conference "Laser Applications in Life Sciences" (LALS 2010) (Oulu, Finland, 2010).

All original results stated in the dissertation were obtained by the author personally or under his scientific supervision.

The structure and scope of the thesis

The dissertation consists of an introduction, five chapters and a conclusion. Contains 262 pages of typewritten text, including 105 figures, and a list of used sources, numbering 214 titles, including 35 references to the author's main publications on the topic of the dissertation.

SUMMARY OF THE DISSERTATION

In the introduction the relevance of the dissertation topic is substantiated, the goals and objectives of the work are formulated, the main provisions submitted for defense are listed, the scientific novelty of the results obtained, their practical value and reliability are noted. The main features of the interaction of laser radiation with multilayer tissues are presented.

In the first chapter a brief overview of the existing theories used to describe the propagation of laser radiation in multilayer tissues is given in order to select the most optimal mathematical approach for analyzing these processes.

The emphasis is on the analysis of two main approaches to solving problems of radiation propagation in multilayer media.

The first of them is based on the wave theory of interaction of radiation with matter, which is based on Maxwell's equations and the wave equation. The medium is characterized by the coefficients of scattering and absorption of particles, which are given in the form of random variables from spatial coordinates. As a result, corresponding integral or differential equations are obtained for statistical quantities such as variance and correlation functions. This approach is mathematically rigorous in the sense that, in principle, both the effects of multiple scattering and the effect of diffraction and interference can be taken into account. However, in this setting common decision has not yet been found, analytical solutions have been obtained only for a very narrow class of problems related mainly to highly rarefied media (biological suspensions and solutions, fog clouds in the case of line of sight of an object), which clearly complicates the possibility of analyzing processes occurring in complex multilayer wednesday

An alternative approach is based on the currently most developed analytical theory of radiation transfer (TS), which does not obviously follow from the wave equation. This theory operates directly with the transfer of energy in a medium containing particles. It assumes that each scattering particle is sufficiently distant from its neighbors, which excludes the possibility of interaction between successive scattering effects, i.e. it is assumed that there is no correlation when the fields are added, the intensities are added, but not the fields themselves. The main TP equation is the radiation transfer equation:

where is the energy brightness, is the speed of the photons, is the absorption coefficient, is the scattering coefficient, is the phase scattering function, is the function of the photon source, is the infinitesimal element of the solid angle.

It is equivalent to the Boltzmann equation used in the kinetic theory of gases and the theory of neutron transport. TP well describes many physical phenomena and is successfully applied in a wide variety of problems (atmospheric and underwater visibility, marine biology, optics of papers and photographic emulsions, in the analysis of radiation propagation in the atmospheres of planets, stars and galaxies).

It is concluded that the transfer theory is most suitable for describing the processes associated with the propagation of optical radiation in multilayer tissues of complex geometry. With its help, it is possible to solve the problems of optical diffuse tomography and spectroscopy of biological objects, to carry out reliable layer-by-layer dosimetry of laser radiation inside biological tissue. However, this requires the development and development of new methods for solving direct and inverse problems of radiation transfer for media with an arbitrary configuration and any boundary conditions. It is shown that the Monte Carlo method, which is widely used for the numerical solution of the radiation transfer equation, is promising for solving such problems.

In second the chapter, which is of an overview and analytical nature, examines the main mechanisms of interaction of laser radiation with biological tissues. Special attention is paid to the consideration of thermal effects and ablation of biological tissues.

The term "thermal interaction" describes a large group of interaction types, where the local increase in temperature is an important parameter. The thermal effect of laser radiation occurs only if the power density is> 10 W / cm2 for continuous radiation or pulsed radiation with a pulse duration of more than 1 μs. Therefore, the processes caused by the photochemical interaction of radiation with matter, occurring at very low power densities (usually 1 W / cm2) and long exposure times, are not analyzed in detail.

Depending on the duration of exposure and the maximum achievable tissue temperature, various thermal effects can be distinguished, such as hyperthermia, coagulation, evaporation, carbonization (charring) and melting.

Temperature is basic physical size characterizing all thermal interactions of light with tissue. To predict the thermal response, a model of the temperature distribution within the tissue must be created. Often, not one, but several thermal effects take place in biological tissues (depending on the parameters of the laser). Allocate reversible and irreversible tissue damage. Since the critical temperature of cell necrosis is determined by the time of irradiation, there is no exact temperature value at which it is possible to distinguish a reversible effect from an irreversible one. Therefore, the degree of damage to biological tissue is determined by the energy, volume and duration of irradiation. Possible thermal processes are shown in Fig. 1. Localization and spatial extent of any thermal effect depends on the biological tissue temperature during and after laser exposure.

Rice. 1. Localization of thermal effects inside biological tissue.

One of such processes is photoablation, which consists in the fact that the material decomposes under the action of high-intensity laser radiation (power density - 107-108 W / cm2, for nanosecond laser pulses). The depth of ablation, i.e. the depth of material removal in one pulse, is determined by the pulse energy up to a certain saturation limit. The geometry of the sample during ablation is determined by the spatial characteristics of the laser beam.

To create a model that describes the dependence of the ablation depth on the intensity of the incident radiation, most groups relied on the assumption that the Bouguer-Lambert law is valid for light absorption.

Photoablation will occur if:

where Iph is the threshold radiation intensity leading to photoablation. This condition indicates that in order for photoablation to occur, a certain amount of energy must be absorbed per unit volume per unit time. The threshold intensity Iph is determined minimum amount bonds that need to be broken in order to get splitting.

Ablation depth d, i.e. the depth at which I (z) = Iph:

This simple model describes the photoablation process well, except for the thresholds Iph at the onset of photoablation and Ipl at the onset of plasma formation.

The analysis of the existing mechanisms of interaction of laser radiation with multilayer biological tissues made it possible to conclude that for the study and assessment of thermal effects it is necessary to solve the nonstationary heat transfer equation taking into account the conditions of a specific problem. As such tasks in this work, the following are considered: the temperature reaction of biological tissues taking into account various inclusions and the calculation of the parameters of laser ablation of solid multilayer biological tissues, which are solved in the 4th and 5th chapters.

Chapter Three is devoted to solving the problem of constructing a mathematical model of the propagation of optical radiation in inhomogeneous biological media of complex geometry, taking into account the multilayer structure of real biological tissues, intended for calculating and analyzing the distribution of the density of absorbed light energy in its various layers.

Within the framework of this problem, special attention is paid to the development of remote optical methods for diagnostics of multilayer biological media. However, most of the known methods do not fully take into account changes in the optical and geometric parameters of the objects under study, primarily local inhomogeneities. From the point of view of modeling the visualization of such objects, the most expedient is the use of the statistical Monte Carlo method, based on the representation of radiation propagation in the form of a stream of model wave packets, each of which is formed by a set of photons of a certain "type" with a given energy and direction of propagation. This means that the model package does not exhibit such properties as phase and polarization, and is a kind of energy-carrying quasiparticle capable of forming similar less energetic particles during interaction.

The intensity distribution within biological tissue is a function of the absorption coefficient a, the scattering coefficient s, the anisotropy parameter g, and the size of the laser beam. This leads to significant difficulties in the quantitative dosimetry of radiation in laser therapy. Investigations of the distribution of light inside biological tissue with a complex multilayer structure in order to simplify the analysis can be carried out within the framework of a one-dimensional theory, which is valid when the dimensions of the laser beam are much greater than the depth of light penetration into the tissue, which is implemented for many types of phototherapy. Typical examples multilayer biological tissues are skin, walls of the bladder, uterus, blood vessels.

The application of the Monte Carlo method is based on the use of the macroscopic optical properties of the medium, which are assumed to be homogeneous within small volumes of tissue. The simulation does not take into account the details of the propagation of radiation energy within a single cell. The well-known algorithms make it possible to take into account several layers of biological tissue with different optical properties, the final size of the incident beam, and the reflection of light from the interfaces between the layers.

With its high accuracy and versatility, the main disadvantage of the Monte Carlo method is the large expenditure of computer time. Although the development of hardware and software means of computing reduces the role of the time factor, the development of new means of laser diagnostics and therapy requires the creation of effective, relatively simple and reliable algorithms for the Monte Carlo method. For example, the new condensed Monte Carlo method allows one to obtain a solution for any albedo value based on modeling for one specific albedo value, which significantly speeds up the calculations. Also developed are very economical hybrid models that combine the accuracy of the Monte Carlo method and the speed of diffusion theories or approximating analytical expressions.

Theoretical modeling allows one to investigate a set of different initial conditions and freely interpret experimental results in real time. This greatly facilitates the work and reduces the time spent in planning, preparing experiments and analyzing the results. However, the majority of modern research in this area is based on a one-dimensional or two-dimensional representation of a multiply scattering medium, which obviously imposes rather significant restrictions on the applicability of the results obtained. In this work, a mathematical model has been built that reflects the process of three-dimensional propagation of optical radiation in living tissues. It is assumed that the volume of the model environment is a set of addressable (indexed) volumetric elements of three-dimensional space. The choice of an event possible for a model package is calculated when it interacts either with an elementary volume or with its surface, if the latter is an interface between layers with different optical characteristics.

The model is based on the radiation transfer equation.

A multilayer biological medium with included inhomogeneities of arbitrary shape, to which the photon flux is directed, is considered.

The simulated medium is specified by the following parameters: thickness, scattering and absorption coefficients, average cosine of the scattering angle, relative refractive index. The medium is represented by a set of random centers scattering and absorbing photons (Fig. 2).

The incident light beam (radiation source) consists of one million packets of photons entering the medium along the z axis perpendicular to its surface (x, y) at the point with coordinates (0, 0, 0). The number of photons in the packet determines the energy of the incident beam. All calculations are performed in a 3D Cartesian coordinate system.

It is believed that the particles of the medium, on which scattering and absorption occur, are spherically symmetric. This approximation is usually used in such cases and is based on the fact that in the process of passing through a medium with strong scattering, a photon interacts with particles at different angles. Therefore, the averaged scattering indicatrix can be used. The use of this model and comparison of numerical calculations with experimental results have shown that this approximation satisfactorily describes the properties of most biological tissues.

To take into account the refraction at the interface between the two subdomains, the Fresnel law is used. In fig. 2 shows an example of the trajectory of a photon in a medium. The probability density function of the free path of a photon before the interaction - - is determined from the Bouguer-Lambert-Beer law as follows:

where a is the absorption coefficient, s is the scattering coefficient, and t is the total attenuation coefficient equal to t = a + s. When a photon is deflected by an angle, it is assumed that it is deflected symmetrically to the original direction of propagation by an azimuthal angle, the value of which lies within the interval. Asymmetric scattering is not considered.

To take absorption into account, a method called implicit photon capture is used. In modeling, the motion of not each photon separately is considered, but a packet of photons. A packet of photons (hereinafter, for simplicity, a packet) simulates the movement of many photons along a similar trajectory, as a result, when interacting with the medium, only part of the photons from the packet is absorbed, and the rest continues to move.

Rice. 2 - An example of the trajectory of a photon in a medium.

Since, when describing the propagation of laser radiation in biological tissues, it is necessary to take into account the real geometry of the medium, which can be quite complex, the multilayer of biological tissues, the size and angular distribution of the incident radiation, the Monte Carlo method was used to implement the model, which is currently the only method that makes it possible to take into account all of the above. features of the problem under consideration

The optical parameters of the biological environment are complex functions from spatial coordinates. However, this medium can be divided into sufficiently small subdomains, within which the optical properties of the medium can be set approximately, by relatively simple functions, for example, constant, linear, and quadratic functions... For Monte Carlo simulation in three-dimensional space a very important factor is how such a partition is performed.

It is shown that the finite element method seems to be the most convenient for describing complex media. The geometry of the medium is represented in the form of a grid, with the help of which the computational domain is approximated by dividing into elementary cells, the shapes of the elements of which are one of the main factors determining the accuracy and convergence rate of the numerical solution of the problem. The simpler the shape of the elements of the partition, the less computational resources are required for calculations.

It is shown that grids are considered to be of high quality, where each element is regular or close to a regular tetrahedron. The use of such an approximation of the simulated medium significantly simplifies the solution of the problem of transition between elements (going beyond the element) and finding a photon inside a grid element. A mesh is considered to be of poor quality if it contains degenerate or nearly degenerate elements.

It is concluded that with such a partition, the initial geometry of the computational domain can be arbitrary, and the simulated medium contains internal closed inhomogeneities. The model was tested on a specific medium (skin), consisting of several layers (stratum corneum, epidermis and dermis) with a closed heterogeneity in the form of a complex figure bounded by two ellipsoidal surfaces; additionally, a layer modeling air has been introduced (Fig. 3.). The center of the beam is displaced from the origin along the ox axis by 0.001 cm and is directed perpendicularly upward, its radius is 0.001 cm.

A simplified diagram of the developed simulation algorithm by the Monte Carlo method is shown in Fig. 4. The photon is initialized with a unit weight. The photon step size for the first interaction case is found and the photon is moved. If the photon has left the tissue, then the possibility of internal reflection is checked. If the photon is internally reflected, then its position is changed accordingly, and the program continues, otherwise the photon is removed and the case of reflection (or transmission) is recorded. With each step, the weight of the photon decreases. The lost weight is added to the locally bonded photon position dependent array element, which indicates the photon energy absorbed by the tissue. The remaining weight of the photon is calculated statistically, a new direction is chosen and a new step calculated.

Rice. 3. Geometry of the design environment.

Rice. 4. Algorithm for modeling by the Monte Carlo method.

The angular divergence of the beam was taken into account. The optical parameters of each layer known from the literature, in particular the absorption and scattering coefficients and the anisotropy parameter (mean cosine of the scattering angle), were used to calculate the distribution of the absorbed energy density inside the medium. In this case, the jump in the refractive index at the air - epidermis interface (n = 1.5) was taken into account. Since the refractive index of other biological tissues is 1.4, and the anisotropy parameter is greater than 0.9, i.e. At each step of the simulation, photons are scattered at small angles, then Fresnel reflections at the biological tissue - biological tissue boundaries were not taken into account.

Calculation of the distribution of the absorbed energy density makes it possible to construct a diagnostic map of the propagation of laser radiation of various spectral ranges in multilayer media with the inclusion of closed inhomogeneities in terms of known optical parameters. As an example, the wavelengths of 400 and 800 nm were chosen.

To graphically represent the propagation of radiation in the medium, the xoz section planes were chosen. In fig. 5 shows the distribution of the absorbed energy density in these planes for a wavelength of 400 nm.

Rice. 5. Distribution of the absorbed energy density in the xz section plane for a wavelength of 400 nm.

Since for infrared radiation (wavelength 800 nm) the absorption coefficient of the skin is much less than the scattering coefficient, and the medium is highly scattering, the penetration depth of the radiation should be greater in comparison with the first task. Therefore, a layer 0.5 mm thick was added to the computational domain. In fig. 6 shows the distribution of the absorbed energy density in the xz plane for a wavelength of 800 nm.

In both tasks, the laser radiation has the same power and energy. For radiation with a wavelength of 400 nm, most of the energy will be absorbed in the small volume region. Therefore, the absorbed energy density is much higher than in the case with a wavelength of 800 nm.

Fig. 6. Distribution of the absorbed energy density in the xz cross-sectional plane for a wavelength of 800 nm.

The fundamental difference between the model and the known existing models (Arridge S.R., Tuchin V.V., Prahl S.) is in the independence of the algorithm from the geometry of the medium. Using a number of tools, you can create computational domains consisting of many components of various shapes and sizes. This significantly distinguishes this model from the known ones that use plane-parallel and continuous homogeneous computational domains. Any parameters of the medium and various inclusions, for example, nanoparticles, can be used in the calculations.

Thus, the proposed model makes it possible to calculate the distribution of the density of the absorbed energy of laser radiation in multilayer materials and can be used to solve problems of analyzing the thermal fields arising from irradiation.

V fourth The chapter examines the dynamics of surface temperature fields under the action of UV radiation using the example of a multilayer medium (skin) with the inclusion of random inhomogeneities, in the form of nanoparticles. It is known that skin layers have different optical characteristics: scattering and absorption coefficients, refractive indices () and radiation scattering anisotropy factors, which was taken into account when simulating the processes of interaction of this medium with optical radiation.

Using the developed model, described in the second chapter, the densities of the absorbed light energy on the skin area containing TiO2 nanoparticles were determined. For calculations, the results of experiments on the localization of particles in the skin were used given in the literature. According to the results of these experiments, most of the spherical nanoparticles are localized at a depth of 0-3 μm from the skin surface. The selected wavelengths for consideration are 310 and 400 nm. The wavelength of 400 nm is at the border between the UV and the visible part of the spectrum, TiO2 particles are practically non-absorbing (only scattering) for such radiation. The 310 nm line is the center line in the UV-B portion of the spectrum. It is responsible for the erythemal peak of skin susceptibility, which is more or less correlated with DNA damage to cells; the dominant mechanism of interaction of radiation with TiO2 particles is absorption.

In this work, the sample is considered as a superposition of the stratum corneum (matrix) and TiO2 particles in it. This is possible because the cells of the layer have a thickness of about 0.5 μm and a diameter of 30 - 40 μm and, thus,

significantly exceed the size of TiO2 particles (25 - 200 nm in diameter). These particles are assumed to be nanometer-sized spheres. The scattering of radiation by such particles is described by the Mie phase function. A skin area of ​​1 cm2 was selected for modeling. The incident radiation power was 100 mW. The thickness of the simulated area of ​​the skin is about 600 microns, which sufficiently allows one to present a picture of the interaction of UV radiation with the surface layers of the skin.

The simulation uses a collimated photon beam, which corresponds to solar radiation. The sample surface is assumed to be infinite; integral (over the entire area of ​​the stratum corneum) characteristics of the recorded radiation are considered.

At the first stage, the propagation of photons in the medium, their absorption and scattering are modeled. Simulation is reduced to the launch of packets of photons, characterized by the function of heat sources (Q) and registration of events of absorption and scattering of individual photons. As a result, information is obtained on the parameters of the illumination of the medium and the absorbed power.

The desired distribution of thermal fields on the surface and along the depth of the modeled structure is determined as a solution to the differential equation of unsteady heat transfer:

where k is the coefficient of thermal conductivity, T is the temperature, Q is the function of the heat source, is the density, c is the specific heat, t is the time, r, z are the cylindrical coordinates.

It should be noted that in this problem the heat source is not localized on the surface, as is usually the case in problems of heat and mass transfer, but is volumetric and distributed over the entire volume of the medium. To solve equation (5), a finite element method was applied using triangular finite elements of the first order. A sufficiently large number of triangular finite elements of the first order, although it leads to a certain decrease in the accuracy and speed of calculations, however, has the following advantages: a large number of nodes allows one to obtain the most accurate distribution of the absorbed energy density in the medium, calculated in the previous problem; rather quickly and conveniently, you can thicken and change the given grid to meet the requirements of the task, and also, if necessary, transform these elements into elements of a higher order. To solve the problem in time, the implicit Crank-Nicholson scheme was used with the following boundary and initial conditions... On the surface where heat exchange with the environment takes place, a boundary condition of the third kind is set:

where k, A - heat transfer parameters; Text - ambient temperature. This condition takes into account the heat sink on the surface of the stratum corneum (surface heat sink).

At the lower boundary, at the depth Z1, a boundary condition of the form is set:

Studies show that for a healthy person, starting from a depth of about 450 microns, the temperature stabilizes. In addition, the simulation takes into account the heat sink caused by blood flow in small capillary vessels. Zero sinks are set on the lateral boundaries of the region:

To eliminate temperature jumps at the interlayer boundaries, the following conditions are used:

Figure 7 shows the obtained distributions of the absorbed energy density in the stratum corneum, taking into account the included inhomogeneities in the form of TiO2 nanoparticles of various concentrations. It can be seen that in the absence of particles, UV radiation at a wavelength of 310 nm is completely absorbed in the first layer (stratum corneum).

Rice. 7. Distribution of the absorbed energy density in the stratum corneum without particles and using TiO2 nanoparticles with a size of 62 nm, = 310 nm. The thickness of the surface layer containing nanoparticles is 1 μm. The thickness of the stratum corneum is 20 microns.

The introduction of titanium dioxide TiO2 nanoparticles into the stratum corneum, due to the high values ​​of the scattering coefficient of embedded particles, leads to a sharp decrease in the density of absorbed energy in the stratum corneum.

The absorption and scattering coefficients of the stratum corneum and the nanoparticle material at a wavelength of 400 nm are significantly lower than at a wavelength of 310 nm. Due to this, the density of the absorbed energy in the stratum corneum, both with and without particles, is also significantly lower (Fig. 8).

Rice. 8. Distribution of the absorbed energy density in the stratum corneum and epidermis on the skin area without particles and using TiO2 nanoparticles with a size of 122 nm, = 400 nm. The thickness of the surface layer containing nanoparticles is 1 μm. The thickness of the stratum corneum is 20 microns.

In fig. 9 shows the dynamics of temperature change on the skin surface without particles and using 1% and 5% titanium dioxide impurities in the stratum corneum. V this case the boundary condition was taken into account, which ensures the energy drain inside the tissue, due to the blood flow in the capillaries (internal heat sink) and maintains the temperature value of 37 ° C inside the skin at a depth of 500 microns.

It can be seen that already from the 10th second of exposure to radiation on the skin, the temperature stabilizes, both with the use of titanium dioxide TiO2 nanoparticles in the stratum corneum, and without them (Fig. 9).

Rice. 9. Temperature dynamics on the skin surface in the absence of particles and using TiO2 nanoparticles with a size of 62 nm in the stratum corneum, = 310 nm. The thickness of the surface layer containing nanoparticles is 1 μm. There is a drain of energy inside the skin.

The simulation results showed that high values ​​of the absorbed energy density in the upper layers of the skin lead to their significant heating. Thus, when a 5% admixture of titanium dioxide TiO2 nanoparticles is used in the stratum corneum, the value of the absorbed energy density at the skin surface reaches 1000 J / cm3 at = 310 nm. However, the thickness of this "hot" layer is only 1 µm; although this layer generates most of the heat, it is quickly transferred to other parts of the environment, and the resulting temperature decreases. The surface temperature of the skin, the stratum corneum of which does not contain nanoparticles, is formed due to the heat coming from the depth of the tissue, where most of the energy penetrates and where the amount of absorbed energy is higher. A similar effect, but to a much lesser extent, is observed at the radiation wavelength = 400 nm, which is close to the optical range (Fig. 10).

Rice. 10. Temperature dynamics on the skin surface in the absence of particles and using TiO2 nanoparticles with a size of 122 nm in the stratum corneum, = 400 nm. The thickness of the surface layer containing nanoparticles is 1 μm. There is a drain of energy inside the skin.

The developed model made it possible to analyze the effect of surface heat sink on the temperature field of the surface layer of the skin.

It is shown that without the inclusion of a drain on the skin surface, the temperature is formed mostly due to the energy absorbed in the surface layer. When a sufficiently powerful surface drain is turned on, the temperature on the surface of the fabric forms the heat coming from the underlying layers; at the same time, the maximum temperature decreases.

The error of the obtained results was calculated as a weighted difference between the maximum values ​​of the absorbed energy density and temperature in the entire region and was less than one percent.

The analysis of the obtained results of the carried out modeling of the thermal reaction of the skin to UV irradiation showed the efficiency of using nanoparticles in the development of photoprotective preparations of the skin surface.

The developed calculation model was also used to study the temperature effect of an IR laser tweezer (= 1064 nm) on a red blood cell - erythrocyte. For ease of investigation, the cell is suspended in water and is a uniform sphere with a diameter of 7 microns, completely consisting of hemoglobin. The cell membrane was not taken into account in the simulation due to its very small thickness, on the order of 10 nm. The cell is exposed to a focused laser beam with a diameter of 1 μm and a power of 100 mW. The results obtained are in good agreement with the known experimental data.

Fifth The chapter is devoted to the application of the developed model for solving a specific problem, namely, calculating thermal fields in hard tissues, in particular dentin, and determining the intensity of laser radiation to obtain the critical temperatures required for the ablation process in these media.

To implement the multidimensional mathematical model, a finite element methodology was chosen.

Dentin, the main tissue of the tooth, was chosen as the test material. Dentin is similar in composition and strength to bone tissue. Contains 72% inorganic, 28% organic matter and water.

Due to the fact that the exact physical characteristics of the presented layers have not yet been determined, then for simplicity, a two-layer model is considered. Each layer is specified by constant, independently specified optical-physical characteristics. To inflict minimal injury, use laser radiation with the smallest penetration depth. The experiment shows that this problem is solved by using lasers with infrared radiation.

We will proceed from the following assumptions:

- thermophysical characteristics for different parts of the tooth (enamel, dentin, pulp) are constant and do not depend on temperature;

- when describing the optical properties, we will assume that each part of the tooth is characterized by its own values ​​of the optical constants (absorption coefficient), which do not depend on the intensity of laser radiation.

The calculation of the light field formed during the scattering of laser radiation on the inhomogeneities of the dental tissue (microinclusions, odontoblast processes, etc.), and taking it into account when modeling the destruction process is a complex multiparametric problem. Today, such a calculation is extremely difficult due to the lack of reliable information about the optical constants of hard tissues, and it will not be taken into account when modeling the process of thermal destruction.

Therefore, it is assumed that light in biological tissue is attenuated according to Bouguer's law, while the contribution to the constant of light attenuation of the processes of scattering, absorption, waveguide effects, etc. is not detailed.

Using the algorithms described in Chapters 2 and 4, the temperature distribution was obtained. Then the amount of the removed substance was determined. According to Arrhenius's law:

where w is the frequency factor;

Ea is the activation energy;

R is the universal gas constant.

The value varies from 0 to 1. Its physical meaning- the measure of the destruction of the substance at the point (x, y, z) during the time (t-t0). The experiment shows that at, the substance can be considered remote.

In fig. 11 shows the temperature distribution on the surface of the medium, Fig. 12 - temperature distribution in the central section of the region. The intensity of laser radiation is 5 kW · cm-2.

Rice. 11. Distribution of temperature on the surface of the medium at time t = 70 ms.

The results obtained are in good agreement with the known experimental data. It is seen that an increase in temperature is not localized on the surface: a rather strong increase in temperature is observed inside the medium. Studies have shown that the laser ablation process begins at a temperature threshold of 320 ° C, and therefore a constant temperature is maintained on the surface. In fig. 13 shows the evolution of temperature at a point on the surface.

Rice. 12. Temperature distribution in the central section
area at time t = 70 ms.

Rice. 13. Time evolution of surface temperature
considered area.

The results obtained on the volume of the removed substance are shown in Fig. fourteen.

Rice. 14. Dependence of the amount of removed substance on time.

In custody the main results obtained are summarized.

The main result of the work is the creation of a new physical and mathematical model of the interaction of laser radiation with multilayer biological materials of any geometry, which allows using a number of tools to create computational domains consisting of many components of various shapes and sizes. This significantly distinguishes this model from the known ones that use plane-parallel and continuous homogeneous computational domains. Any parameters of the medium and various inclusions, for example, nanoparticles, can be used in the calculations.

A number of fundamental theoretical results have been obtained, of which the following should be noted:

A physico-mathematical model of the propagation of laser radiation in media with an arbitrary asymmetric geometry, including closed internal inhomogeneities of a complex shape, is proposed.

On the basis of this model, an algorithm was developed for calculating the distribution of the absorbed energy density for various ranges of laser radiation, when it propagates in multilayer media with an arbitrary asymmetric geometry of the computational medium with the inclusion of closed internal inhomogeneities of complex shape, using the three-dimensional Monte Carlo method and finite element partitioning.

The algorithm used in this work can be used to diagnose structural changes in biological tissue of arbitrary closed geometry, as well as to calculate the temperature fields and boundaries of the destruction area during laser therapy.

The main mechanisms of interaction of laser radiation of various intensities with multilayer biological tissues are considered and analyzed. Based on this, theoretical analysis conditions for the occurrence and course of thermal processes in them. The possibility of the applicability of the developed model for studying the thermal loads of multilayer tissues characteristic of the processes of photo- and plasma-induced ablation in them is estimated.

A model is proposed for determining the temperature response of multilayer biological tissues with the inclusion of nanoparticles to UV irradiation. The evolution of changes in the density of absorbed light energy and temperature fields is analyzed depending on the wavelength of incident radiation, concentration and dislocation of test nanoparticles incorporated into the skin.

The calculation of thermal fields in solid biological tissues arising under laser action is carried out and the intensity of laser radiation at the critical temperatures required for the ablation process in these media is determined.

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4. Tuchin V.V. Tissue Optics - Bellingham: SPIE Press, 2000.
5. Tuchin V.V. Handbook of Optical Biomedical Diagnostics — Bellingham: SPIE Press, 2002.
6. Shcherbakov Yu.N., Yakunin A.N., Yaroslavsky I.V., Tuchin V.V. Modeling of thermal processes in the interaction of non-coagulating laser radiation with multilayer biological tissue. - Optics and spectroscopy. - 1994, Vol. 76, No. 5, C. 845-850.

LIST OF MAIN PUBLICATIONS

1. Seteikin A.Yu., Gavrilenko V.N. ., Krokhina N.A. Dosimetry of low-intensity laser beams in biomaterials. // Bulletin of the AmSU. - Blagoveshchensk. - 1999. - Issue 6. - S. 24-27.
2. Seteikin A.Yu., Gavrilenko V.N. . Features of the distribution of laser radiation in multilayer biomaterials. Preprint. - Blagoveshchensk: AmSU. - 2000 .-- 60 p.
3. Seteikin A.Yu., Gavrilenko V.N. . Automated system of dosimetry of laser beams when interacting with multilayer materials // Proceedings of the III International Scientific and Technical Conference “Quantum Electronics” - Minsk, 2000. - pp. 193-194.
4. Seteikin A.Yu., Gershevich M.M. Modeling of propagation processes of laser radiation in multilayer materials // Blagoveshchensk: Bulletin of the AmSU. - 2001. - No. 11. - S. 26-28.
5. Seteikin A.Yu., Ershov I.A. Model of the structure of the transparent lens of the eye for the problem of laser radiation scattering in transparent biological tissues // Materials of the first Amur interregional scientific and practical conference"Chemistry and chemical education at the turn of the century." - Blagoveshchensk: BSPU. - 2001 .-- S. 110-111.
6. Seteikin A.Yu., Ershov I.A. Modeling of light scattering in transparent biological tissues with spherical inhomogeneities // Bulletin of AmSU. - 2001. - No. 13. - S. 18-20.
7. Seteikin A.Yu., Ershov I.A. Effects of multiple scattering in the transparent lens of the eye during laser diagnostics // Bulletin of the AmSU. - 2001. - No. 15. - S. 29-30.
8. Seteikin A.Yu., Ershov I.A., Gershevich M.M. Modeling the processes of interaction of low-intensity laser beams with multilayer biomaterials // Technical Physics Journal. - 2002. - T. 72. - Issue 1. - S. 110-114.
9. Seteikin A.Yu. Application of the Monte Carlo method for modeling the reflection spectra of optical radiation from randomly inhomogeneous multilayer strongly scattering and absorbing light media // Bulletin of AmSU. - 2002. - No. 19. - S. 24-27.
10. Seteikin A.Yu. Simulation of the processes of scattering of laser radiation in multilayer biomaterials by the Monte Carlo method // Informatics and Control Systems. - 2003. - No. 2. - S. 31-37.
11. Seteikin A.Yu. Analysis of the propagation of laser radiation in multilayer biomaterials by the Monte Carlo method // Bulletin scientific communications... - Khabarovsk: Publishing house of the Far Eastern State University of Economics. - 2004. - No. 8. - S. 22-28.
12. Seteikin A.Yu. The use of the Monte Carlo method for modeling the reflection spectra of optical radiation from randomly inhomogeneous multilayer strongly scattering and absorbing light media // Sbornik scientific papers"Optics of Crystals". - Khabarovsk: Publishing house of the Far Eastern State University of Economics. - 2004 .-- S. 34-43
13. Gavrilenko V.N., Seteikin A.Yu. Accounting for scattering processes in the interaction of laser radiation with transparent biological materials // Proceedings of the International Conference "Laser-optical technologies in biology and medicine" - Minsk: Institute of Physics of the National Academy of Sciences of Belarus, 2004. –S.245-249.
14. Seteikin A.Yu. Calculation of the temperature fields with it the laser irradiation on biomaterials // PROCEEDINGS of the Fourth Asia-Pacific Conference "Fundamental Problem of Opto-and Microelectronics" (APCOM 2004). - DVGUPS: Khabarovsk. - 2004. - P. 459-464.
15. Seteikin A.Yu. Analysis by the Monte Carlo method of the propagation of laser radiation in multilayer biomaterials // Izvestiya vuzov. Physics.- 2005. - No. 3.- P.53-57.
16. Seteikin A.Yu. A model for calculating the temperature fields arising from the action of laser radiation on a multilayer biological tissue // Optical magazine.- 2005. - T.72. - No. 7. - S.42-47.
17. Seteikin A.Yu. Optical-thermophysical model of interaction of laser radiation with multilayer materials // Proceedings of universities. Physics.- 2005. - No. 6. Application. - S. 99-101.
18. Seteikin A.Yu., Krasnikov I.V. Calculation of temperature fields arising from the interaction of laser radiation with a multilayer biomaterial // Proceedings of the fifth regional scientific conference "Physics: fundamental and applied research, education". - Khabarovsk: Publishing house of the Pacific State. University, 2005. - S.32-33.
19. Seteikin A.Yu., Krasnikov I.V. Thermophysical model of interaction of laser radiation with multilayer biological tissue // Bulletin of AmSU. Series "Natural and Economic Sciences". - 2005. - Issue 31. - P.13-15.
20. Seteikin A.Yu. Analysis by the Monte Carlo method of the propagation of laser radiation in multilayer biomaterials // Optics and spectroscopy.– 2005. - T.99. - Issue 4. - S.685-689.
21. Seteikin A.Yu., Krasnikov I.V. Calculation of the temperature effect of low-intensity laser radiation on multilayer biological tissue // Materials of the International Symposium "Principles and Processes of Creation of Inorganic Materials (Third Samson Readings)". - Khabarovsk: Pacific State Publishing House. un-that. - 2006.- S.304-306.
22. Seteikin A.Yu., Krasnikov I.V. Calculation of temperature fields arising from the interaction of laser radiation with a multilayer biomaterial // Optical magazine.- 2006. - T.73. - No. 3. - S.31-34.
23. Seteikin A.Yu., Krasnikov I.V. Analysis of thermal effects arising from the interaction of laser radiation with a multilayer biomaterial // Proceedings of universities. Physics.– 2006. - No. 10. - S. 90-94.
24. Seteikin A.Yu., Krasnikov I.V. On thermal effects under the action of laser radiation on biological tissue // Materials of the sixth regional scientific conference “Physics, fundamental and applied research, education”. - Blagoveshchensk: AmSU. - 2006 .-- S. 104-106.
25. Seteikin A.Yu., Krasnikov I.V., Fogel N.I. Description of the effect of laser radiation on the skin using the Monte Carlo method // Proceedings of the scientific session MEPhI-2007. - M .: MEPhI. - 2007 .-- S. 117-118.
26. Seteikin A.Yu., Krasnikov I.V. Research an thermal influence of laser radiation an skin with non-trivial geometry // Proceedings of SPIE. – 2007. - Vol. 6826. - P. 127-131.
27. Seteikin A.Yu., Krasnikov I.V., Fogel N.I. Modeling of temperature fields taking into account the propagation of light in biological tissue // Proceedings of universities. Instrumentation... –2007. –Т.50. - No. 9. - P.24-28.
28. Seteikin A. Yu., Krivtsun AM Modeling the propagation of optical radiation in media with spatially variable parameters // Bulletin of the Amur State University. - 2008. - Issue. 41 .-- S. 12-13.
29. Minailov A.V., Seteikin A. Yu. On the study of liquid multicomponent biological media by optical-acoustic methods // Bulletin of the AmSU. - 2008. - Issue. 41 .-- S. 14-15.
30. Averyanov Yu. G., Seteikin A. Yu. Laser ablation of biological tissues // Bulletin of the AmSU. - 2008. - Issue. 41 .-- S. 31-32.
31. Seteikin A. Yu., Krasnikov I. V., Foth H.-J. Analysis of the arising thermal loads in biological tissue irradiated by laser radiation in the infrared range // Proceedings of the International Optical Congress "Optics - XXI century". - Vol. 1. "Fundamental Problems of Optics - 2008". - SPb., 2008 .-- S.119-120.
32. Pavlov M.S., Seteikin A. Yu. Application of a three-dimensional modification of the Monte Carlo method for modeling the propagation of light in biological tissues. // Proceedings of the International Optical Congress "Optics - XXI century". - Vol. 1. "Fundamental Problems of Optics - 2008". - SPb., 2008. - S. 120-121.
33. Khramtsov I.I., Seteikin A. Yu. Modeling of the process of laser ablation of a tooth based on a thermal model. // Proceedings of the International Optical Congress "Optics - XXI century". - Vol. 1. "Fundamental Problems of Optics - 2008". - SPb., 2008. - P.248.
34. Seteikin A. Yu., Krasnikov I. V., Foth H.-J. Experimental research thermal effect of laser radiation on biological tissues. // Bulletin of SPBO AIN. - SPb .: Publishing house of the Polytechnic University. - 2008. - Issue. 4. - S.273-277.
35. Seteikin A. Yu., Krasnikov I. V., Pavlov M. S. Three-dimensional model of the propagation of light in biological tissues. // Scientific and technical statements of SPbSPU. Series of Physics and Mathematics, 2008. - Issue 6. - S. 120-123.
36. Seteikin A.Yu., Krivtsun A.M. Investigation of the process of interaction of radiation with biological tissues containing optical inhomogeneities // Collection of reports of the 19th International conference “Lasers. Measurements. Information. 2009 ", SPb .: Publishing house of polytechnic. un-ta, 2009. –T 1. - С.245-254.
37. Seteikin A.Yu., Krasnikov I.V., Popov A.P. Investigation of the thermal effects of UV radiation on human skin with the inclusion of titanium oxide nanoparticles // Proceedings of the 19th International Conference “Lasers. Measurements. Information. 2009 ", SPb .: Publishing house of polytechnic. University, 2009. –T 1. - S.254-268.
38. Seteikin A.Yu., Khramtsov I.I. Investigation of the process of laser ablation of biological tissue under the influence of ultrashort laser pulses // Materials of the VIII regional scientific conference "Physics: fundamental and applied research, education". - Blagoveshchensk: Amur state. un-t., 2009 - p. 250-253.
39. Seteikin A.Yu., Pavlov M.S. Modeling of the propagation of laser radiation in biological multicomponent tissues // Materials of the VIII regional scientific conference "Physics: fundamental and applied research, education". - Blagoveshchensk: Amur state. un-t., 2009 - S. 307-310.
40. Seteikin A.Yu., Krasnikov I.V., Popov A.P. Investigation of the thermal protective properties of TiO2 nanoparticles introduced into the skin upon irradiation with UV-A and UV-B light ranges // Proceedings of the VIII Regional Scientific Conference "Physics: Basic and Applied Research, Education". - Blagoveshchensk: Amur state. un-t., 2009 - pp. 322-326.
41. Seteikin A.Yu., Krasnikov I.V., Popov A.P. Method for calculating the temperature reaction of biological tissues using nanoparticles under irradiation with UV-A and UV-B light ranges // Methodology GSSSD MR 150-2009. Ross. scientific and technical information center for standardization, metrology and conformity assessment. - M., 2009 .-- 40 p .: ill. 18.bibliogr. name 24 - Rus. name Dep. at FSUE "Standartinform".
42. Seteikin A. Yu., Krasnikov I. V., Popov A. P., Fotiadi A. E. Temperature response of biotissues containing nanoparticles to irradiation with UV-A and UV-B light. // SPbSPU Scientific and Technical Bulletin, Physics and Mathematics Series.– 2009. - Issue 1. - S. 113-118.
43. Krasnikov I.V., Seteikin A.Yu., Popov A.P. Changing the sun and heat-shielding properties of human skin by introducing nanoparticles of titanium dioxide // Optics and spectroscopy.– 2010. - T. 109, No. 2. - S. 332-337.
44. Seteikin A.Yu., Privalov V.E. Photoablation of biological tissues // Bulletin of St. Petersburg University.– 2010. - Ser. 11. Issue 2. - S. 225-237.
45. Fadeev D.A., Seteikin A.Yu. Analysis of multiple scattering of laser radiation in biological media with spatial fluctuations of optical parameters // SPbSPU Scientific and Technical Bulletin, Ser. "Physical and mathematical sciences".– 2010. - Issue 2. - S. 102-106.
46. Krasnikov I., Seteikin A., Bernhardt I. Thermal processes in red blood cells exposed to infrared laser tweezers (= 1064 nm) // Journal of Biophotonics. - 2011. - Vol. 4., No. 3. - P. 206-212.
47. Seteikin A. Yu., Krasnikov I. V., Pavlov M. S. Simulation of the propagation of optical radiation by the Monte Carlo method in biological media with closed internal inhomogeneities // Optical magazine - 2010. - Issue 77., No. 10. - S. 15-19.

49. Krasnikov I., Seteikin A., Bernhardt I. Simulation of laser light proropagation and thermal processes in red blood cells exposed to infrared laser tweezers (= 1064 nm) // Optical Memory and Neural Networks (Information Optics) - 2010. - Vol. 19., No. 4. - P. 330-337.

50. Krivtsun A.M., Seteikin A.Yu. Analysis of the propagation of optical radiation in biological media using calculations on graphic processors // Scientific and technical statements of SPbSPU, Series of physical and mathematical sciences, 2011, Issue 1, pp. 55-61.

51. Seteikin A.Yu., Popov A.P. Interaction of light with biological tissues and nanoparticles // LAP Lambert Academic Publishing - 2011-212 pp.

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The purpose of optical systems in laser systems is as follows:

- manufacturing of optical resonators and obtaining laser radiation,

- transmission of laser radiation energy to the treatment site,

- regulation of radiation parameters,

- formation of a light beam with a high power density (focusing),

- aiming radiation at the processed point,

- control over the processing process and evaluation of its results.

Optical systems contain the following main elements:

- focusing - lenses, objectives,

- reflective elements - mirrors, scanners,

- refractive - total reflection prisms, optical deflectors (devices allowing to split one beam into several beams),

- regulating radiation - optical shutters, etc.,

- transmitting light guides.

Focusing elements serve to change the diameter of the laser emitter beam in order to change the radiation power density. In technological installations, as a rule, it is required to reduce the beam diameter and increase the radiation power density, i.e. focus the radiation.

The simplest and most widely used method of focusing radiation is the use of a single lens (Fig.), Where f is the focal length, F is the focal plane of the optical system.

Due to the fact that laser radiation has a certain divergence (albeit very small), it can be focused (reduced) to a certain size. The diameter of the light spot of radiation has the smallest value in the focal plane F and is determined by the formula:

Substituting the expression for θ, we obtain

(2.38)

In practice, focusing distortion (aberration) is observed

Taking into account spherical aberrations

, (2.39)

where P * is a calculated parameter (determined by the size and shape of the lens).

Knowing the energy or power of the laser radiation W and, P and, it is possible to calculate the energy density or power in the focused spot:

; . (2.40)

Earlier (see the properties of laser radiation), these values ​​were estimated based on the diameter of the laser radiation. When focusing, these parameters increase by several orders of magnitude. In practice, one usually strives to reduce the diameter of the radiation spot.

It can be seen from formula (2.39) that in order to decrease the diameter of the focused radiation spot, it is necessary to decrease the focal length. However, this can only be done up to certain limits, since if the distance between the lens and the focusing surface is too small, there is a risk of damage to the lens (for example, vapors and liquid particles of the processed material).

Therefore, to obtain a spot with a diameter of several microns, another method is used - increase beam diameter using a telescopic system - see (2.39).

The diameter of the light spot in this case is determined taking into account (2.39) by the following formula:

,

where Г> 1 is the magnification of the telescopic system.

The optimal focal length of the lens (at which the smallest diameter of the focused spot is achieved) can be determined by the formula:

(2.41)

When laser radiation passes through, the lenses of the optical system are heated due to partial absorption of radiation. This can lead to thermal deformation and damage to the optical system. Therefore, the radiation power density should not exceed certain values ​​that allow long-term normal operation of parts of the optical system.

The allowable power density depends on the material from which the spokes are made and the wavelength of the radiation.

- to focus radiation with a wavelength of 0.4 - 2 microns (visible and near infrared spectra), lenses made of various types of optical glass are used. The permissible power density is ~ 10 3 W / cm 2.

- for radiation with a wavelength of 10.6 μm

(CO 2 lasers) Conventional optical materials are opaque. Materials for the manufacture of lenses are:

- single crystals of salts of hydrohalic acids - NaCl, KBr, KCl, etc.

Allowable power density ~ 10 3 W / cm 2. They are highly hygroscopic and have a short service life.

- semiconductor crystals - germanium, gallium arsenide, etc. The permissible power density is 100 W / cm 2.

At a radiation power exceeding the permissible one, either forced air or liquid cooling of the lenses, or focusing systems made of mirrors with metal coatings on a metal base (for the purpose of better cooling) are used. The basis is glass, copper, silicon. Plating - gold, silver, copper, nickel, molybdenum, aluminum, etc.

Reflective and refractive elements optical systems serve to change the direction of laser radiation. They are used in optical resonators and systems for transporting laser radiation.

At a laser radiation wavelength of 0.4 - 2 μm, total internal reflection prisms and mirrors with a multilayer dielectric coating (to increase the reflection coefficient and decrease the distance) are used for this purpose.

At a radiation wavelength of 10.6 μm. they use flat, convex, concave mirrors with a metal coating (made of gold and aluminum), which have a high reflectance (~ 1). By changing the density of the coatings, you can change the reflectance, i.e. make translucent mirrors.

In practice, the problem often arises of moving the laser beam along an arbitrary contour. For this, a system of movable flat mirrors is used (see Fig.).

1 - laser emitter

2,3 - movable mirrors

4 - lens

5 - material

Mirrors 2 and 3 and lens 4 move along the X-axis together, and only mirror 3 and lens 4 can move along the Y-axis.

Simultaneous movement along the X and Y axes allows you to get any beam path.

With the use of mirrors, laser beam scanning systems are manufactured, i.e. periodically moving it along the same trajectory.

Regulating elements optical systems are designed to change the energy, power of laser radiation, its spatial and temporal characteristics. These include

- optical quantum amplifiers - devices that increase the energy of laser pulsed radiation. In fact, these are lasers, in which they are not generated spontaneously, but under the influence of radiation from another laser. As a result, the radiation energy of the optical amplifier is added to the energy of the initiating radiation pulse.

- devices for adjusting the radiation power from zero to the nominal value - diagrams with a variable aperture diameter, replaceable light filters with different absorption coefficients, optical shutters, modulators, shutters.

The following types of gates are used as gates for modulators.

- electro-optical (Ponkels effect), based on the phenomenon of the plane of polarization by some substances under the influence of high constant voltage up to 5kV.

- mechanical shutters - rotating mirrors up to 30,000 rpm.

- Saturable shutter gates are based on the phenomenon: at a certain value of the radiation intensity, some organic dyes become transparent.

- acousto-optic shutters, quartz glass and germanium (for the IR range) when exposed to ultrasonic waves are accompanied by large losses (scattering) for laser radiation and its generation stops.

The shutters are installed in the resonator. In addition, mechanical shutters are used at the exit of laser radiation from the resonator.

Transmitting elements optical systems are designed to transmit laser radiation over distances up to several tens of kilometers. - for this use fiber light guides.

A large number of light guides are currently known. The most widely used optical fibers are of the following design

The optical fiber consists of a core 1 with a refractive index n 1, a cladding 2 with a refractive index n 2> n 1 and a protective cladding 3. Materials used for manufacturing: a core, for example, quartz with an addition of titanium to increase the refractive index, a cladding made of pure quartz ... In general, for the manufacture of these optical fiber elements, a large number of different types of glasses and polymers are currently used; Various varnishes, polymers, metals are used for the protective shell, it protects the fiber from the external environment (moisture), increases mechanical strength, and improves optical characteristics. The fiber diameter ranges from several tens to several hundred microns. The core has a diameter ranging from a few microns. up to 1000 microns. (1mm.).

In optical fibers, the phenomenon of internal total reflection is used (Fig.). At the interface between the two media, the phenomenon of refraction and reflection of light occurs. When the luminous flux passes from a medium with a high refractive index n 1 to a medium with n 2

. (2.42)

Thus, if, when the light flux enters the fiber core, it falls on the interface with the cladding at an angle ≥ θ cr, then this flux propagates only within the core.

An important characteristic of the fiber is the attenuation of the efficiency of the light flux when propagating along the fiber. At present, optical fibers with an attenuation of ~ 1 dB / km have been created.

The question we get asked all the time is: Is it possible to simulate the heating of substances due to their interaction with laser radiation in COMSOL Multiphysics? The answer, of course, depends on what kind of problem you are going to solve, since different modeling methods are suitable for different problems. Today, we will discuss various approaches to simulate the heating of materials illuminated by laser radiation.

Introduction to Modeling the Interaction of Laser Radiation with Matter

While there are many different types of laser light sources, they are all similar when viewed in terms of what they output. Laser radiation is concentrated near one wavelength and is coherent. As a rule, the output radiation is also focused into a narrow collimated beam. This collimated, coherent and monochromatic light source can be used as an extremely accurate heat source in a wide range of applications including, and.

When laser radiation hits a solid, some of its energy is absorbed, leading to local heating. Liquids and gases (and plasma), of course, can also be heated by lasers, but heating liquids is almost always accompanied by strong convection effects. In this article, we ignore convection and focus on heating solids.

Solids can be partially or completely opaque to radiation at the laser wavelength. Depending on the degree of transparency, different approaches will be applicable to simulate a laser heat source. In addition, it must be remembered that all scales must be compared with the radiation wavelength. Different approaches are required to describe focused radiation and for a relatively wide beam. If the material interacting with the incident beam has geometric features comparable to the wavelength, it is necessary to additionally consider how the beam will interact with these fine structures.

Before you start modeling any interactions of laser radiation with matter, you must first determine the optical properties of the material, both at the laser wavelength and in the infrared range. You also need to know both the relative sizes of the objects that are being heated and the laser wavelength and beam parameters. This information will be useful to you when choosing the appropriate approach for modeling your problem.

Surface Heat Sources

In the case of materials opaque at the laser wavelength, or close to this, laser radiation can be considered as a surface heat source. The easiest way to do this is by using the function Deposited Beam Power(shown below), which is available in the Heat Transfer Module version 5.1 of COMSOL Multiphysics. In addition, it is also easy to manually define the surface heat source using only the COMSOL Multiphysics core like.

A surface heat source assumes that the beam energy is absorbed in a layer of negligible thickness compared to the dimensions of the object being heated. The step of dividing the finite element mesh should be sufficient only to take into account changes in the temperature field and the size of the laser spot. The laser radiation itself is not modeled explicitly, and it is assumed that the part of the laser radiation reflected from the material does not return back. When using a surface heat source, you need to manually set the absorption coefficient of the material at the laser wavelength and scale the released beam power accordingly.

Deposited Beam Power function in the Heat Transfer Module, used to simulate two crossed laser beams. The resulting surface heat source is shown.

Bulk Heat Sources

In the case of partially transparent materials, the bulk of the laser energy will be released inside the region, not on the surface, and any approach must be appropriately tied to the relative geometric dimensions of the objects and the wavelength.

Geometric Optics

If the size of the heated objects is much larger than the wavelength, but the laser radiation converges and diverges as it propagates through a series of optical elements and, possibly, is reflected by mirrors, then functionality will be the best choice. In this approach, light is viewed as a beam propagating through an absorbing, homogeneous and inhomogeneous medium.
As radiation propagates through absorbing materials (i.e. optical glasses) and intersects interfaces, some of the energy will be spent on heating the material. The absorption in the volume of the region is modeled using a complex refractive index. At the interface, you can use the reflectance or absorption coefficient. All of these properties can be temperature dependent. For those interested in this approach, from our Application Gallery, will provide a good starting point.

Laser beam focused by a system of two lenses. Heating of the lenses due to the propagation of high-intensity laser radiation shifts the focus point.

Bouguer-Lambert-Beer law

If the size of the heated objects and the laser spot is much larger than the wavelength, then the Bouguer - Lambert - Beer law is suitable for modeling the absorption of radiation in the material. This approach assumes that the laser beam is completely parallel and unidirectional.

When using the Bouguer-Lambert-Beer law, the absorption coefficient of the material and the coefficient of reflection from the surface must be known. Both of these coefficients can be functions of temperature. The corresponding setting of the parameters of such a model was described earlier in our blog article “Modeling the Interaction of Laser Radiation with Matter on the Basis of the Bouguer-Lambert-Beer Law”.

You can use the Bouguer-Lambert-Beer approach if you know the intensity of the incident laser radiation and there are no reflections of light inside the material and / or off the object's boundaries.

Laser heating of semitransparent solids simulated using the Bouguer-Lambert-Beer law.

Beam envelope method

If the heated region is large, but the laser beam is sharply focused inside it, neither geometric optics nor the approach based on the Bouguer – Lambert – Beer law can accurately calculate the fields and energy losses near the focus. These methods do not directly solve Maxwell's equations, but interpret light as a set of rays. available in is the most suitable choice in this case.

The Beam Envelope Method solves the Maxwell system of equations for the case when the amplitude of the wave packet is a slowly varying function of coordinates. The approach works if the value of the wave vector in the simulated medium and the approximate direction of radiation propagation are approximately known. This case corresponds to modeling, as well as waveguide structures such as or a ring resonator. Since the direction of the beam is known, the finite element mesh can be rather coarse in the direction of propagation, thereby reducing computational costs.

Focused laser beam propagating in the region of a substance with cylindrical symmetry. The intensity at the input surface and along the optical axis within the region is graphically displayed according to the mesh grid.

Beam envelope method can be combined with the interface via multiphysics connection Electromagnetic Heat Source... This connection is established automatically when you add an interface. on the menu Add Physics.

Interface Laser Heating adds interfaces Beam Envelopes and Heat Transfer in Solids and establishes a multiphysics connection between them.

Full Wavelength Approach

Finally, if the structure to be heated has dimensions comparable to the wavelength, it is necessary to solve the Maxwell system of equations without any assumptions regarding the direction of propagation of laser radiation in the simulated space. In this case, we need an interface Electromagnetic Waves, Frequency Domain which is available in both the Wave Optics Module and. In addition, the RF module contains an interface Microwave Heating(similar to interface Laser Heating described above) and binds the interface with interface Heat Transfer in Solids... Despite the name, the RF module and interface Microwave Heating suitable for modeling.

The full-wave approach requires the partitioning of the finite element mesh necessary for the resolution of the laser wavelength. Since the beam can be scattered in any direction, the mesh must be sufficiently uniform with respect to the mesh size. A good example of using the interface Electromagnetic Waves, Frequency Domain is: as demonstrated below.

Heating a gold nanosphere by laser radiation. The radiation loss in the sphere and the magnitude of the surrounding electric field are displayed according to the grid of the partition.

Modeling Heat Transfer, Convection, and Reradiation In and Around Material

You can use any of the five previous approaches to simulate the energy release from a laser source in a solid material. Simulation of temperature rise and heat flux in and around the material additionally requires an interface Heat Transfer in Solids... Available in the core of the software package, this interface is designed to simulate heat transfer in solids and set the appropriate boundary conditions: a fixed temperature, a thermally insulated boundary, or the presence of heat flow through it. The interface also includes various boundary conditions for simulating convective heat transfer to the surrounding atmosphere or liquid, as well as radiative cooling (due to radiation) into an environment with a known temperature.

If the material in question is transparent to laser radiation, then most likely, it is also partially transparent to thermal radiation (infrared). This infrared radiation will neither be coherent nor collimated, so we cannot use any of the above approaches to describe re-radiation in semitransparent media. Instead, we can use the distributed media radiation approach. This method is intended to simulate heat transfer in materials in which there is a significant heat flux inside the material due to the radiation process. An example of such an approach from our Application Gallery might be.

Conclusion

In this article, we looked at the various methods available in COMSOL Multiphysics for simulating laser heating of solid materials. Surface and volumetric heating approaches were presented, along with a brief overview of heat transfer modeling capabilities. Until now, we have considered only the heating of a solid material, which does not undergo a change in its phase state. Heating liquids and gases - and phase transition modeling - will be covered in subsequent articles on this blog. Stay tuned!