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To create a laser, it is necessary to obtain an inversion between any pair of levels in the active medium. The mechanism by which the inversion is created is called pumping. From the conclusions obtained in the previous section, it follows that it is impossible to create an inverse population in a two-level system by exposing it to external electromagnetic radiation. Indeed, due to saturation, the inverse population will never be greater than zero. Nevertheless, the problem becomes solvable if one or two additional levels, the so-called three- and four-level pumping schemes, are introduced into consideration. In this section, we consider the mechanism for creating an inverse population for both schemes using rate equations that are derived from the balance conditions between the rates of change in the total number of particles and the total number of laser radiation photons. Using this approach gives a simple and visual description of the operation of the laser.

Three-level scheme

First, let's consider a laser operating according to a three-level scheme (fig. on the slide). Let N 1 ,N 2 ,N 3 – populations of the corresponding levels, N 0 is the total number of particles. As a characteristic of the field intensity in the resonator, we introduce the quantity q is the total number of photons in the resonator. We assume that the transitions between levels 3 and 2 are fast enough to be able to put. Let us write down the rate equations for the change in the populations and the number of photons:

In equation (4.2), the first term determines the contribution of the pump, whose rate is W n (s -1), into the change in the population of level 2. The second term reflects the change in the population of this level due to the processes of stimulated emission and absorption (for simplicity, we assumed the degrees of degeneracy of the levels under consideration to be the same).

In equation (4.3), the first term, up to sign and coefficient V coincides with the second term in the second equation. Indeed, each act of stimulated emission is accompanied by the appearance of a photon, and with stimulated absorption, the photon is absorbed. Coefficient V is called the field volume (mode volume) inside the active medium. At its core, this parameter reflects the fact that the electromagnetic field does not occupy the entire volume of the active medium in the resonator. This question will be considered in detail in the section devoted to optical cavities. Time is called the photon lifetime in the resonator and takes into account the decrease in the number of photons due to losses (for example, associated with the transmission of mirrors).

Finally, it remains to be noted that in writing (4.3) we neglected the term that takes into account spontaneous emission. Indeed, if at the zero moment of time we put q(0)=0, then we get that
, and generation cannot occur. However, at the moment we cannot correctly take into account the contribution of spontaneous emission, since for this it is necessary to have an idea of ​​the possible types of field configuration in the resonator (spatial and frequency), which can only be done with a detailed consideration of the properties of optical resonators. Nevertheless, when solving system (4.1)-(4.3), we will get the correct result if we assume that at the moment of time t=0 there is a small number of spontaneous photons in the resonator: q(0)=q 0 .

Before proceeding to further consideration of the system of equations (4.1)-(4.3), we obtain an explicit form for the coefficients B and .

Consider a resonator with length L. For simplicity, we will assume that the active medium occupies the entire space between the mirrors. Let T 1 and T 2 are the transmission coefficients of the resonator mirrors, T ext is the coefficient of internal losses per pass from one mirror to another. Then the change in intensity
for a double pass will be:

where  N=N 2 -N 1 .

For further consideration, it is convenient to introduce logarithmic losses associated with the transmission of mirrors:

Then for all types of losses we have:

(4.6a)

(4.6b)

(4.6v)

Using the expressions obtained, we determine the total losses per pass:

. (4.7)

If the levels of transmission losses and internal losses are sufficiently small (several percent), then we can assume
.

We have after substitution:

If you enter an additional condition:

<<1, (4.9)

then the exponential function can be expanded into a series and get:

. (4.10)

If we divide the resulting expression by the time interval
, corresponding to the round trip time, and use the approximation
, we get:

. (4.11)

Since the number of photons in the resonator is proportional to the intensity, the resulting expression can be compared with (4.3). In this case, the following expressions for the required quantities are obtained:

. (4.12)

If we now assume for the general case that the length of the active medium l between the mirrors is less than the resonator length L, and the refractive index of the active medium is equal to n, then, taking into account the relation obtained for the so-called optical length of the resonator L’:

, (4.13)

finally we get:

. (4.14)

If we introduce the population inversion
, then, taking into account the assumptions about the rates of transitions between levels made at the beginning of the section, it is easy to rewrite system (4.1)–(4.3) for the variables
and q:

The initial conditions for this system will be the relation we have already obtained
, as well as
.

Let us first consider the question of the magnitude of the threshold population inversion. For generation to occur, it is necessary that the quantity was positive. It can be seen from (4.16) that this condition is satisfied when
>. Hence the threshold value of the inverse population:

. (4.17)

The minimum pump power required to create a threshold population inversion is obtained from (4.15) under the conditions:
,
,q=0. This means that, on the one hand, there are still no photons in the resonator (except for a small number of spontaneous q 0), and on the other hand, the pumping rate of level 2 begins to balance the rate of spontaneous transitions from this level. Substituting (4.17) into (4.15), we obtain:

. (4.18)

If the pump power is greater than the threshold, then the number of photons will increase, and at a constant pump power it will reach some stationary value that does not change with time. The stationary values ​​of the number of photons and the population inversion are naturally obtained from the system (4.15)-(4.16), if we put in it
. Thus:

, (4.19)

. (4.20)

If you enter the coefficient
, then:

. (4.21)

Let's analyze the result. At first glance, it may seem strange that, regardless of the pump power under stationary conditions, the population inversion is always equal to the threshold value. However, it is clear that in the stationary regime the number of photons (and the field intensity) in the resonator does not change. Obviously, this condition can be satisfied only if the gain is equal to the sum of all losses. For any other ratio between gain and loss, the intensity will either increase or decrease. Since the gain is proportional to the population inversion, relation (4.19) just establishes the equality of the gain of the active medium to the total losses, which is not affected by the pump power.

At the same time, the number of photons in the resonator and, consequently, the output power of the laser radiation is directly proportional to the pump power (if, for example, mirror 2 is considered the output, then
). After substitution, we finally get:

. (4.22)

Four-level scheme

Let us now carry out a similar calculation for the case of a four-level pumping scheme (figure on the slide). Assuming that transitions between levels 3 and 2 and levels 1 and 0 are fast, i.e.
, we obtain the following system of rate equations:

After reducing this system to a system of two equations in variables
:

It can be seen that the obtained rate equation for the number of photons coincides with the analogous equation in the case of a three-level system. However, the rate equations for the inverse population differ by a factor of 2 in the second term, which is present in the case of a four-level scheme. The physical meaning of this difference is that in a three-level pumping scheme, when a photon is emitted from level 2, the population of this level decreases by one, and the population of level 1 increases by one. Therefore, the inversion decreases by 2. In the four-level scheme, the population of the 2nd level also decreases by one, but due to rapid relaxation from level 1 to level 0, the population of the 1st level does not change, that is, the inversion decreases by 1.

The values ​​of the threshold and stationary inverse populations are the same as in the case of the three-level scheme:

, (4.28)

which is a consequence of the fact that this value is determined by the level of total losses in the resonator.

For the threshold pump power, we obtain:

. (4.29)

Comparison with (4.18) shows that for the four-level scheme, the threshold pump power in
1 times less compared to the three-level scheme at the same value . This result is also explained quite clearly. In a three-level scheme, to create an inverse population, it is necessary to transfer from level 1 to level 2 at least half of the particles. In the case of a four-level scheme, the transfer of even one particle to level 2 creates an inverse population, since the population of level 1 is always practically equal to zero. This is the main advantage of the four-level scheme.

For a stationary number of photons in the resonator, the following expression is obtained:

, (4.30)

and for output power:

. (4.31)

The mechanisms we have considered for creating an inverted population are called optical pumping. In optical pumping, as a rule, high-power broadband lamps are used as a radiation source. Since the pumping efficiency is greater, the more radiation from the source is absorbed by the active medium, optical pumping is best suited for substances with strongly broadened lines, that is, for solid-state and liquid lasers.

In addition to optical pumping, there are many other ways to create a population inversion. One of the most widely used methods is electrical pumping, which is carried out by means of an electrical discharge. This mechanism is especially effective for substances with a narrow absorption line. Therefore, electrical pumping is the main method for producing inversion in gas lasers.

Among other pumping mechanisms, we note chemical pumping (which is necessary for the inversion to occur during an exothermic reaction), gas-dynamic pumping (supersonic expansion of a gas mixture), and laser pumping, when the laser beam of one laser serves to pump another.

Passage of radiation through matter. Inverse population of levels. Consider again a two-level medium with energy levels and . If monochromatic radiation is incident on this medium with a frequency

then when it propagates over a distance dx the change in the spectral energy density will be associated with both resonant absorption and induced (forced) emission of atoms in the system. Due to stimulated emission, the spectral energy density in the beam increases, and this increase in energy should be proportional to:

.

Here is the dimensional proportionality factor.

Similarly, due to the processes of absorption of photons, the spectral energy density in the beam decreases:

.

folding and , find the total change energy density:

Given the equality of the Einstein coefficients and introducing the absorption coefficient a, we write this equation in the form

The solution to this differential equation has the form

.

This formula gives the spectral energy density u in a beam of photons when they pass through a layer of matter with a thickness x, where corresponds to the point x = 0 .

Under conditions of thermodynamic equilibrium, in accordance with the Boltzmann distribution, , therefore, the absorption coefficient a is positive () :

Thus, the radiation energy density, as can be seen from (6.18), decreases as it passes through the substance, that is, the light is absorbed. However, if we create a system in which , then the absorption coefficient will become negative and there will be no weakening, but intensity enhancement Sveta. The state of the environment in which it is called state with inverse level population, and the environment itself is then called active medium. The inverse population of levels contradicts the equilibrium Boltzmann distribution and can be created artificially if the system is taken out of thermodynamic equilibrium.

This creates the fundamental possibility of amplifying and generating coherent optical radiation and is used in practice in the development of sources of such radiation - lasers.

The principle of operation of the laser. The creation of a laser became possible after the methods of implementing the inverse population of levels in certain substances (active media) were found. The first practical generator in the visible region of the spectrum was created in (USA by Maiman (1960)) based on ruby. Ruby is a crystal lattice containing a small ( 0,03 % – 0,05 % ) admixture of chromium ions (). On fig. 6.1 shows a diagram of the energy levels of chromium ( three-tier environment). Wide level used to excite chromium ions with the light of a powerful gas discharge lamp with a wide frequency band in the green-blue region of visible light - pump lamps. The excitation of chromium ions due to the pump energy from an external source is shown by an arrow .

Rice. 6.1. Scheme of an active three-level medium (ruby)

Electrons from the short-lived level make a fast ( c) nonradiative transition to the level (depicted by the blue arrow) . The energy released in this case is not emitted in the form of photons, but is transferred to the ruby ​​crystal. In this case, the ruby ​​is heated, so the design of the laser provides for its cooling.

Lifetime of a long-lived narrow level is c, that is, 5 orders of magnitude greater than that of the broadband level . With sufficient pump power, the number of electrons in a level (it is called metastable) becomes more than level , that is, an inverse population is created between the "working" levels and .

A photon emitted during a spontaneous transition between these levels (depicted by a dashed arrow) induces the emission of additional (forced) photons - (the transition is shown by an arrow), which in turn cause induced radiation of a whole cascade of photons with a wavelength .

Example 1 Let us determine the relative population of working levels in a ruby ​​crystal at room temperature under conditions of thermodynamic equilibrium.

Based on the wavelength emitted by the ruby ​​laser, we find the energy difference:

.

At room temperature T = 300 K we have:

From the Boltzmann distribution it now follows

.

Realization of an active medium with an inverse level population is only half the battle. For the laser to work, it is also necessary to create conditions for generating light, that is, to use positive feedback. The active medium itself can only amplify the transmitted radiation. To implement the generation regime, it is necessary to amplify the stimulated emission, which would compensate for all losses in the system. For this, the active substance is placed in optical resonator, formed, as a rule, by two parallel mirrors, one of which is semitransparent and serves to output radiation from the resonator. Structurally, the first ruby ​​lasers used cylindrical crystals with a length 40 mm and diameter 5 mm. The ends were polished parallel to each other and served as resonator mirrors. One of the ends was silver-plated so that the reflection coefficient was close to unity, and the other end was translucent, that is, it had a reflection coefficient less than unity, and was used to output radiation from the resonator. The source of excitation was a powerful pulsed xenon lamp coiled around the ruby. The ruby ​​laser device is schematically shown in fig. 6.2.

Rice. 6.2. Ruby laser device: 1- ruby rod; 2- impulse discharge lamp; 3- translucent mirror; four- mirror; 5- stimulated emission

With sufficient pump lamp power, most (about half) of the chromium ions are transferred to an excited state. After the population inversion is reached for the working levels with the energy and , the first spontaneously emitted photons corresponding to the transition between these levels do not have a preferred direction of propagation and cause stimulated emission, which also propagates in all directions in a ruby ​​crystal. Recall that the photons generated by stimulated emission fly in the same direction as the incident photons. Photons whose directions of motion form small angles with the axis of the crystalline rod experience multiple reflections from its ends. Photons propagating in other directions exit the ruby ​​crystal through its lateral surface and do not participate in the formation of the outgoing radiation. So in the resonator is generated narrow beam light, and the multiple passage of photons through the active medium induces the emission of more and more photons, increasing the intensity of the output beam.

The generation of light radiation by a ruby ​​laser is shown in fig. 6.3.

Rice. 6.3. Ruby laser generation

Thus, the optical resonator performs two functions: firstly, it creates a positive feedback and, secondly, it forms a narrow directed radiation beam with a certain spatial structure.

In the considered three-level scheme, in order to create an inverse population between the working levels, it is necessary to excite a sufficiently large fraction of atoms, which requires a significant expenditure of energy. More efficient is four-level scheme, which is applied in solid-state lasers, for example, using neodymium ions. In the most common gas laser on neutral atoms - helium- neon laser - the conditions for generation according to the four-level scheme are also satisfied. The active medium in such a laser is a mixture of inert gases - helium and neon with ground state energy (which we take as the zero level). Pumping is carried out in the process of an electric gas discharge, due to which atoms pass into an excited state with energy . Level in neon atoms (Fig. 6.4) is close to the level in helium, and in the collision of helium atoms with neon atoms, the excitation energy can be effectively transferred to the latter without radiation.

Rice. 6.4. He level scheme- Ne-laser

Thus, the level neon turns out to be more populated than the lower level . The transition between these working levels is accompanied by radiation with a wavelength 632.8 nm, which is the main in industrial Not-Ne-lasers. At the level neon atoms do not linger for a long time, quickly returning to the ground state. Note that the level in neon is extremely slightly populated, and therefore, to create an inverse population between and it is necessary to excite a small number of helium atoms. This requires much less energy for both pumping and cooling the setup, which is typical for a four-level generation scheme. Other levels of neon (not shown in Fig. 6.4) can also be used for laser generation, giving radiation both in the visible and in the IR range, and helium is used only for the pumping process.

Example 2 Let us find the relative equilibrium population of the level in neon at room temperature.

This problem differs from the previous one only in numerical values. For a change, we will carry out calculations in electron volts. Let us first express the Boltzmann constant in these units:

so at room temperature

.

Now we can easily find

From a practical point of view, such a small number does not differ from zero; therefore, even with weak pumping, an inverse population is created between the levels and .

The radiation of lasers is characterized by characteristic features:

high temporal and spatial coherence (radiation monochromaticity and low beam divergence);

high spectral intensity.

The characteristics of the radiation depend on the type of laser and the mode of operation, however, some parameters close to the limit values ​​can be noted:

Short (picosecond) laser pulses are indispensable for studying fast processes. An extremely high peak power (up to several GW) can develop in the pulse, which is equal to the power of several NPP units of a million kW each. In this case, the radiation can be concentrated in a narrow cone. Such beams allow, for example, "welding" the retina to the fundus.

Types of lasers. Within the framework of the general physics course, we cannot dwell in detail on the specific features and technical applications of various types of lasers due to their extreme diversity. We restrict ourselves to a fairly brief review of the types of lasers that differ in the characteristics of the active medium and in the methods of pumping.

solid state lasers. Usually they are pulsed, the first such laser was the ruby ​​laser described above. Popular lasers on glass with neodymium as a working substance. They generate light with a wavelength of about 1.06 µm, have large dimensions and peak power up to TW. Can be used for controlled thermonuclear fusion experiments. An example is the huge Shiva laser at the Livermore Laboratory in the USA.

Neodymium yttrium aluminum garnet (Nd:YAG) lasers are very common, emitting in the IR range at a wavelength micron. They can operate both in continuous generation mode and in pulsed, with a pulse repetition rate of up to several kHz (for comparison: a ruby ​​laser has 1 pulse every few minutes). They have a wide range of applications in electronic technology (laser technology), optical location, medicine, etc.

gas lasers. Usually these are continuous lasers. They differ in the correct spatial structure of the beam. Example: HeNe laser generating light at wavelengths 0,63 , 1,15 and 3.39 µm and having a power of the order of mW. Widely used in engineering - laser with a power of the order of kW and wavelengths 9,6 and 10.6 µm. One of the methods for pumping gas lasers is an electric discharge. A variety of lasers with an active gaseous medium are chemical and excimer lasers.

chemical lasers. Population inversion is created during a chemical reaction between two gases, such as hydrogen (deuterium) and fluorine. Based on exothermic reactions

.

molecules HF are already born with excitation of oscillations, which immediately creates an inverse population. The resulting working mixture is passed at supersonic speed through an optical resonator, in which part of the accumulated energy is released in the form of electromagnetic radiation. Using a system of resonator mirrors, this radiation is focused into a narrow beam. Such lasers emit high energy (more than 2 kJ), the pulse duration is about 30 ns, power up to Tue. Efficiency (chemical) reaches 10 % , while it is usually fractions of a percent for other types of lasers. Wavelength generated - 2.8 µm(3.8 µm for lasers on D.F.).

Of the numerous types of chemical lasers, hydrogen fluoride (deuterium) lasers have been recognized as the most promising. Problems: The radiation of hydrogen fluoride lasers with the specified wavelength is actively scattered by water molecules, which are always present in the atmosphere. This greatly reduces the brightness of the radiation. The deuterium fluoride laser operates at a wavelength for which the atmosphere is practically transparent. However, the specific energy release of such lasers is one and a half times less than that of lasers based on HF. This means that when using them in space, much more chemical fuel will have to be removed.

excimer lasers. Excimer molecules are diatomic molecules (for example,), which can only be in an excited state - their unexcited state turns out to be unstable. This is the main feature of excimer lasers: the ground state of excimer molecules is unfilled, that is, the lower working laser level is always empty. Pumping is carried out by a pulsed electron beam, which transfers a significant part of the atoms to an excited state, in which they are combined into excimer molecules.

Since the transition between operating levels is broadband, it is possible to tune the generation frequency. The laser does not produce tunable radiation in the UV region ( nm) and has high efficiency ( 20 % ) energy conversion. At present, excimer lasers with a wavelength 193 nm used in ophthalmic surgery for superficial evaporation (ablation) of the cornea.

liquid lasers. The active substance in the liquid state is uniform and can be circulated for cooling, which is an advantage over solid-state lasers. This makes it possible to obtain high energies and powers in pulsed and continuous modes. The first liquid lasers (1964–1965) used rare earth compounds. They were replaced by lasers based on organic dye solutions.

Such lasers usually use optical pumping of radiation from other lasers in the visible or UV range. An interesting property of dye lasers is the possibility of tuning the generation frequency. By selecting a dye, it is possible to obtain generation at any wavelength from the near IR to the near UV range. This is due to the wide continuous vibrational-rotational spectra of liquid molecules.

semiconductor lasers. Solid-state lasers based on semiconductor materials stand out in a separate class. Pumping is carried out by electron beam bombardment, powerful laser irradiation, but more often by electronic methods. Semiconductor lasers use transitions not between discrete energy levels of individual atoms or molecules, but between allowed energy bands, that is, sets of closely spaced levels (energy bands in crystals are discussed in more detail in subsequent sections). The use of various semiconductor materials makes it possible to obtain radiation at wavelengths from 0,7 before 1.6 µm. The dimensions of the active element are extremely small: the length of the resonator can be less than 1 mm.

Typical power on the order of several kW, pulse duration approx. 3 ns, the efficiency reaches 50 % , are widely used (fiber optics, communications). Can be used to project a TV image onto a large screen.

Free electron lasers. A beam of high-energy electrons is passed through a "magnetic comb" - a spatially periodic magnetic field that forces the electrons to oscillate at a given frequency. The corresponding device - an undulator - is a series of magnets that are located between the sections of the accelerator, so that relativistic electrons move along the axis of the undulator and oscillate transversely to it, emitting a primary ("spontaneous") electromagnetic wave. In an open resonator, where the electrons then enter, the spontaneous electromagnetic wave is amplified, creating a coherent directional laser radiation. The main feature of free electron lasers is the ability to smoothly tune the generation frequency (from the visible to the IR range) by changing the kinetic energy of the electrons. The efficiency of such lasers is 1 % at medium power up to 4 W. With the use of devices for returning electrons to the resonator, the efficiency can be increased to 20–40 % .

X-ray laser With nuclear pumping. This is the most exotic laser. Schematically, it is a nuclear warhead, on the surface of which up to 50 metal rods are fixed, oriented in different directions. The rods have two degrees of freedom and, like gun barrels, can be directed to any point in space. Along the axis of each rod is a thin wire made of a high-density material (of the order of the density of gold) - the active medium. The source of laser pumping energy is a nuclear explosion. During the explosion, the active substance passes into the plasma state. Cooling instantly, the plasma emits coherent radiation in the soft X-ray range. Due to the high concentration of energy, radiation, hitting the target, leads to explosive evaporation of the substance, the formation of a shock wave and the destruction of the target.

Thus, the principle of operation and the device of the X-ray laser make it obvious and the scope of its application. The described laser does not include resonator mirrors, which cannot be used in the X-ray range.

Some types of lasers are shown in the figure below.

Some types of lasers: 1- laboratory laser; 2- continuous laser on ;
3 - technological laser for punching holes; four- powerful technological laser

Pumping is carried out, as a rule, in one of two ways: optical or electrical. During optical pumping, the radiation of a powerful light source is absorbed by the active medium and thus transfers the atoms of the active medium to the upper level. This method is particularly well suited for solid state or liquid lasers. Line broadening mechanisms in solids and liquids lead to a very significant broadening of the spectral lines, so that one usually deals not with level pumping, but with absorption band pumping. These bands absorb an appreciable fraction of the light emitted by the pump lamp. Electric pumping is carried out by means of a sufficiently intense electric discharge, and it is especially useful for gas and semiconductor lasers. In particular, in gas lasers, due to the fact that they have a small spectral width of the absorption lines, and the pump lamps give broadband radiation, it is quite difficult to carry out optical pumping. Optical pumping could be used very effectively for semiconductor lasers. The fact is that semiconductors have a strong absorption band. However, the use of electric pumping in this case turns out to be more convenient, since an electric current passes through the semiconductor very easily.

Another pumping method is chemical. There are two noteworthy types of chemical pumping: 1) an associative reaction, leading to the formation of an AB molecule in an excited vibrational state, and 2) a dissociative reaction, leading to the formation of a B particle (atom or molecule) in an excited state.

Another way of pumping a gas molecule is the supersonic expansion of a gas mixture containing a given molecule (gadodynamic pumping). Mention should also be made of a special form of optical pumping, when a laser beam is used to pump another laser (laser pumping). The properties of a guided laser beam make it very convenient for pumping another laser, and no special brighteners are required here, as in the case of (incoherent) optical pumping. Due to the monochromaticity of the pump laser, its application is not limited to solid state and liquid lasers, but it can also be used to pump gas lasers. In this case, the line emitted by the pumping laser must coincide with the absorption line of the pumped laser. This is used, for example, to pump most far infrared lasers.

In the case of optical pumping, light from a powerful incoherent lamp is transferred to an active medium by means of an appropriate optical system. On fig. 1 shows the three most commonly used pumping schemes. In all three cases, the medium has the form of a cylindrical rod. Shown in fig. 1a the lamp has the shape of a spiral; in this case, the light enters the active medium either directly or after reflection from a mirror cylindrical surface (number 1 in Fig. 1). This configuration was used to create the first ruby ​​laser and is still sometimes used for pulsed lasers. in fig. 1b, the lamp has the shape of a cylinder (linear lamp), the radius and length of which are approximately the same as those of the active rod. The lamp is placed along one of the focal axes F1 of the mirror-reflecting elliptical cylinder (1), and the laser rod is placed along the other focal axis F2. Most of the light emitted by the lamp is reflected from the elliptical cylinder into the laser rod. On fig. 1c shows an example of a so-called close-packed configuration. The laser rod and the linear lamp are placed as close as possible to each other and are tightly surrounded by a cylindrical reflector (1). The efficiency of a close-packed configuration is usually not much lower than that of an elliptical cylinder. Often, instead of specular reflectors in the schemes in Fig. 1a and c, cylinders made of diffusely reflective materials are used. Complex types of illuminators are also used, the design of which uses more than one elliptical cylinder or several lamps in a close-packed configuration.

Let us define the pump efficiency of a cw laser as the ratio of the minimum pump power Pm required to create a certain pump speed to the electric pump power P actually supplied to the lamp. The minimum pump power can be written as , where V is the volume of the active medium, vp is the frequency difference between the main and upper laser levels. The propagation of the pumping rate along the active rod is in many cases inhomogeneous. Therefore, it is more correct to determine the average minimum pump power , where the averaging is performed over the volume of the active medium. In this way

For a pulsed laser, by analogy, the average pump efficiency is

where the time integral is taken from the beginning to the end of the pump pulse, and E is the electrical energy supplied to the lamp.

The pumping process can be considered as consisting of 4 different stages: 1) emission of radiation from the lamp, 2) transfer of this radiation to the active rod, 3) its absorption in the rod, and 4) transfer of the absorbed energy to the upper laser level.

From expression (1) or (!а) one can find the pumping rate Wp:

Electric pumping is used in gas and p/p lasers. The electrical pumping of a gas laser is carried out by passing a direct, high-frequency (HF) or pulsed current through the gas mixture. Generally speaking, the current through the gas can flow either along the laser axis (longitudinal discharge, Fig. 2a) or across it (transverse discharge, Fig. 2b). In longitudinal discharge lasers, the electrodes are often ring-shaped, and in order to weaken the degradation of the cathode material due to collision with ions, the cathode surface area is made much larger than that of the anode. In lasers with a transverse discharge, the electrodes are extended over the entire length of the laser medium. Depending on the type of laser, a variety of electrode designs are used. Schemes with a longitudinal discharge are usually used for cw lasers, while a transverse discharge is used for pumping with constant, pulsed, and RF currents. Since the transverse dimensions of the laser are usually much smaller than the longitudinal ones, in the same gas mixture the voltage that must be applied in the case of a transverse configuration is much lower than the voltage for a longitudinal configuration. However, a longitudinal discharge, when it occurs in a dielectric (eg, glass) tube (Fig. 2a), makes it possible to obtain a more uniform and stable pump distribution.

In an electric discharge, ions and free electrons are formed, and since they acquire additional energy from an applied electric field, they can excite neutral atoms in a collision. Positive ions, due to their large mass, are accelerated much worse than electrons, and therefore do not play a significant role in the excitation process.

5.20. Optical resonators. Gaussian beams of light.

In open structures such as the Fabry-Perot interferometer, there are characteristic vibrational modes. To date, a large number of modifications of open resonators are known, which differ from each other in the configuration and mutual arrangement of the mirrors. The most simple and convenient is the resonator formed by two spherical reflectors with equal curvature, facing concave surfaces towards each other and located at a distance of a radius of curvature equal to the radius of the spheres from each other. The focal length of a spherical mirror is equal to half the radius of curvature. Therefore, the foci of the reflectors coincide, as a result of which the resonator is called confocal (Fig. 1). The interest in the confocal resonator is due to the convenience of its adjustment, which does not require parallelism of the reflectors to each other. It is only necessary that the axis of the confocal resonator intersect each reflector far enough from its edge. Otherwise, the diffraction loss may be too large.

Let us consider the confocal resonator in more detail.

Let all dimensions of the resonator be large compared to the wavelength. Then the modes of the resonator, the distribution of fields in it and the diffraction losses can be obtained on the basis of the Huygens-Fresnel principle by solving the corresponding integral equation. If the reflectors of the confocal resonator have a square section with side 2a, which is small compared to the distance between the mirrors l, equal to their radius of curvature R, and the Fresnel numbers are large, then the eigenfunctions of the Fox- and Lee-type integral equation are approximated by the products of the Hermite polynomials Hn(x) by Gaussian function.

In the Cartesian coordinate system, the origin of which is placed at the center of the resonator and the z axis coincides with the resonator axis (Fig. 1), the transverse field distribution is given by

where determines the size of the region of the cross section, at the output of which the field intensity in the resonator, proportional to S2, drops by e times. In other words, this is the width of the intensity distribution.

Hermite polynomials of several first degrees have the form:

The eigenfunctions of the equation giving the transverse distribution (1) correspond to the eigenfrequencies determined by the condition

On fig. 2 graphically presents the first three Hermite-Gauss functions for one of the transverse coordinates, constructed according to formula (1) taking into account (2). These graphs clearly show the nature of the change in the transverse distribution of the field with an increase in the transverse index n.

Resonances in a confocal resonator occur only for integer values ​​of . R.R. mode spectrum is degenerate, increasing m + n by two units and decreasing q by one gives the same frequency value. The main mode is TEM00q, the transverse distribution of the field is determined by a simple Gaussian function . The width of the intensity distribution varies along the z axis according to the law

where , and has the meaning of the beam radius in the focal plane of the resonator. The value is determined by the length of the resonator and is

On the mirror surface, the spot area of ​​the fundamental mode, as can be seen from (4) and (5), is twice as large as the cross-sectional area of ​​the caustic neck.

Solution (1) was obtained for the field inside the resonator. But when one of the mirrors is partially transparent, as is the case in the case of active laser resonators, then the outgoing wave is a traveling wave with a transverse distribution (1).

Essentially, the extraction of the fundamental mode of an active confocal resonator is a way to obtain a Gaussian beam of monochromatic light. Let's consider them in more detail.) width , which corresponds to the angular divergence

As a result, the bulk of the Gaussian trigger energy is concentrated in the solid angle

Thus, the divergence of laser radiation in the fundamental mode is determined not by the transverse, but by the longitudinal size of the laser resonator.

Essentially, formula (8) describes a diffracted wave resulting from self-diffraction of a Gaussian trigger. The diffraction pattern described by (8) is characterized by a monotonic decrease in intensity upon moving away from the axial direction, i.e. the complete absence of any oscillations in the brightness of the diffraction pattern, as well as a rapid decrease in the wave intensity on the distribution wings. This is the nature of the diffraction of a Gaussian beam at any aperture, provided that its size sufficiently exceeds the width of the beam intensity distribution.

These methods, which are widely used, include the last five groups of methods mentioned in 1. Let's consider them in order.

1. The method of external pumping or external excitation of a multilevel system. At present, this method is most widely used in quantum devices, both in masers and in solid-state and liquid lasers. It is also partially used in gas lasers. It usually uses three-level transitions or, as they say, three-level systems. The essence of the method is as follows. Let us imagine three levels (Fig. 6a), one of which (the lower one) corresponds to the normal unexcited position of the electron, and the two upper ones correspond to the excitation levels. Let us assume that it is necessary to amplify the oscillations, i.e. the working transition is the transition 3-2. To create an inverse population of levels 3,2, the medium is irradiated from the outside with energy quanta that transfer particles from level 1 to level These quanta, or, as they are called, pump quanta, create an increased population of levels 3 compared to levels 2, and therefore, when it comes signal (quanta), this signal is amplified by induced 3-2 transitions. Having passed to level 2 after the amplification event, the particle then falls back to level 1 due to a spontaneous quantum transition (the wavy arrow in Fig. 6a). In what follows, spontaneous transitions will be denoted by wavy arrows, and induced transitions by straight lines. An example of quantum devices that use this method is paramagnetic masers, which can operate only at ultralow temperatures (4.2 K) and in which energy levels 1,2,3 appear due to splitting due to the Zeeman effect of one level in external constant magnetic field, as well as a number of atomic molecular and ion gas metal vapor lasers.

In addition to the method shown, a method can also be used where the 2-1 transition serves as the working transition, when the pump quanta are still quanta, and the signal quanta are quanta (see Fig. 6b). An example of a laser operating according to the scheme of Fig. 6b can serve as a ruby ​​laser. Quantum devices often use various types of external pumping of four-level systems (Fig. 7a,b,c,d). In this case, ordinary direct or single-quantum methods can be used, illustrated by the diagrams in Fig. 7 a, b, in which the working transition is either a 4-3 transition or a 3-2 transition. (Lasers based on aluminum-yttrium garnet, on glass doped with neodymium). In addition, in four-level systems, double (or, as they are also called, sequential or two-quantum) pumping methods can be used, which can be implemented in cases where any two energy distances between the levels of the system are the same. We will consider two such methods used in paramagnetic masers:

1) Auxiliary radiation frequency doubling method.

The implementation of this method is clear from the diagram in Fig. 7,c and is possible when

where the quanta are pump quanta, and the quanta

signal quanta;

2) The method of symmetrical excitation or, as it is otherwise called, the method of push-pull pumping. Its scheme is shown in fig. 7, g. This method of double pumping is realized in ruby ​​at an angle between the symmetry axis of the crystal and the external field equal to. In this method, signal quanta are quanta, and pump quanta are quanta. The method is possible, obviously, in the case when, which takes place in a ruby ​​at a double pumping angle.

Double pumping methods usually make it possible to obtain a much larger degree of level population inversion than ordinary pumping methods. In solid-state masers, ruby, rutile, or or tungstates (salts of the where type) are most often used as paramagnetic substances, and in solid-state lasers, in addition to ruby, neodymium-activated glass and yttrium-aluminum garnet are often used.

Four-level systems have recently become widespread in liquid lasers. Liquid lasers currently have two varieties - liquid lasers based on inorganic liquid media and organic dyes. The first group is lasers that use solutions of salts of the rare earth element neodymium in inorganic liquids. They can be considered analogues of solid-state lasers using neodymium-doped glass.

The second group uses organic dye molecules. The energy structure of such a molecule contains a large number of vibrational-rotational sublevels, which are present both in the ground state of the molecule and in the excited state. Under the influence of external pumping quanta, which can be the radiation of either a flash lamp or another quantum generator, the molecules pass from level 1 of the ground state to the upper level 4 of the excited state. Then, by a nonradiative transition, the molecule enters the lower level 3 of the excited state, emits a working quantum, falling to the upper level 2 of the ground state, and then, with the help of a nonradiative transition, again finds itself at the level of the ground state. Thus, liquid lasers based on organic dye molecules operate according to a four-level system. The great advantage of such lasers is the possibility of obtaining with their help different wavelengths of generated waves from ultraviolet to near infrared. To do this, you need to use different types of dyes.

It should be noted that so far, when considering quantum transitions in multilevel systems, only useful quantum transitions have been indicated, i.e., only those transitions that directly determine the operation of quantum devices. However, in addition to them, there are a number of useless induced transitions that always accompany the mentioned useful transitions, but in most cases are inverse with respect to useful transitions and also quite significantly affect the level population and, consequently, the operation of quantum devices. The complete scheme of all transitions in a three-level system (see Fig. 7, a) has the form shown in fig. 6c, with double arrows showing useful transitions; and single ones are useless. The spontaneous transitions shown in this scheme to upper levels from lower ones are usually carried out in solids due to thermal vibrations of the lattice, considered here as a random factor, and, as a rule, have a relatively low probability.

2. Method of excitation of a multilevel system by acoustic (ultrasonic or hypersonic) oscillations. In principle, this method is no different from the previous one, only in it either one or both useful induced transitions are carried out due to the action of acoustic (usually ultrasonic or hypersonic) vibrations, and not due to electromagnetic vibrations, as in the previous case. In other words, in this method, the working quanta, or pump quanta, are not photons, but phonons.

Obviously, to implement this method, a quantum system must, firstly, transmit ultrasound or hypersound well, and secondly, it must be placed inside the corresponding ultra- or hypersonic acoustic resonator. In this case, there can be three types of quantum systems using quantum transitions due to phonons, i.e. There can be three types of systems called acoustic masers:

• 1) Systems with phonon excitation, which serve to obtain amplification of ultra- or hypersound. In these systems, the pump and signal are ultra- or hypersonic vibrations transmitted from the outside by means of appropriate piezoelectric vibrators that convert ordinary electromagnetic energy into these vibrations;
• 2) Systems with electromagnetic excitation, serving to amplify or generate ultra- or hypersonic vibrations. In these systems, pumping is carried out by photons, and the signal is a stream of phonons, and it is obvious that such a system, if it is resonant, should be placed both inside an electromagnetic resonator (by pumping) and inside an acoustic resonator (by signal).

It is these two types of systems that are often referred to as acoustic masers;

3) Systems with excitation by ultra- or hypersonic vibrations, which serve to amplify or generate electromagnetic vibrations. Such a system, which is, as it were, the inverse of the previous system, is often called a backward acoustic maser. It just represents the multilevel system of interest to us, excited by phonons.

Since phonons, like photons, are energy quanta, all those general considerations that were discussed in the past about quantum transitions associated with the action of photons also apply to the case of the action of phonons.

A method for obtaining level population inversion due to gas-discharge excitation. This method, used in lasers, despite its very wide use, is still much less studied in detail than all previous methods. Its essence lies in the fact

that atoms, ions, or molecules in a gas discharge, usually under the influence of various kinds of collisions, are obtained excited according to three-level or four-level systems. The details of the excitation circuit can be very different in different systems and for different levels, and the system can generally be multilevel. Gas-discharge lasers can be divided into atomic, ionic, and molecular lasers according to the nature of the medium used and partly according to the features of the mechanism for the formation of population inversion. Atomic lasers, with the exception of the neon-helium laser operating in the visible light range, generate generation in the infrared wavelength range. Ionic lasers, which use transitions between the energy levels of ionized gases such as argon, cadmium, selenium, mercury vapor, etc., generate generation mainly in the visible light region and are the main sources of blue and green radiation and ultraviolet lines. Molecular lasers can produce a wider spectrum of radiation, from infrared to ultraviolet lines. However, among a number of different possible types of excitation of atoms or molecules in a gas discharge, one can distinguish some basic excitation mechanisms that play the main role in various gas-discharge laser systems. We will consider three such types of excitations: 1) due to collisions; 2) due to the dissociation of the molecule; 3) electroionization and photoionization.

Collision excitations can, in turn, be divided into two groups:

a) excitation of atoms or molecules of a gas in inelastic collisions with electrons. In this case, the transition 1-3 is carried out either by a direct impact of an electron in a gas discharge, or by a series of successive excitations from one level to another, which has a large energy. Only a relatively small number of types of atoms can be excited in this way. An example is the excitation by a direct collision of one of the levels from the series in the neon atom (the level second from the top in terms of energy in the hyperfine structure, so that it can be designated.):

The working transition in this case is the transition

corresponding to the radiated length µm.

The most intense excitation of an atom by an electron impact occurs in this case, when the energy of the incident electron is slightly greater than the threshold energy of excitation of the atom. An example of excitation by a series of successive collisions with electrons is the excitation of molecules in lasers based on a mixture of u;

b) excitation by collisions in a gas discharge in the presence of impurities. The level population inversion can be obtained with a much greater intensity if a reasonably selected mixture of gases is used, such that the excitation of atoms of the main gas A occurs not only due to collisions with electrons, but also due to the resonant energy loss from impurity gas atoms excited by collisions to metastable levels B. Thus, the process of excitation of the atom proceeds to a certain extent in the following way. Atoms B due to collisions with electrons receive an excitation corresponding to the transition. It is desirable that the level be metastable and that there be no intermediate levels between and levels. This case is realized, for example, in helium atoms for parahelium-orthohelium transitions and (the latter in the presence of an intermediate level with a forbidden transition).

In addition, the energy distance should be close to. From these considerations, you need to select the gas. Due to metastability, excited atoms live for a relatively long time and, colliding with atoms, transfer their energy of their excitation to them according to the scheme

In this way, it was possible to obtain generation on a series of mixtures of atoms of inert gases and molecules, for example, on. In this case, the role of impurity atoms is played by atoms in the first two cases, and by atoms and molecules in the latter cases. This role in a number of cases turns out to be decisive in the possibility of obtaining lasing. Thus, for example, in the absence of impurities, due to purely electronic excitation by collision, it was possible to obtain generation on only three transitions, while in a mixture the number of transitions generated under various conditions reaches twenty-two. Similarly, pure generated only on two transitions, and in a mixture of seventeen transitions. And there are many such examples.

Let us consider the method of excitation due to the dissociation of molecules. This method is based on the following process. A molecule consisting of two atoms and, under the influence of a collision with an electron or with another molecule, or with an atom, or with a photon, turns out to be in an excited state, from which it emerges by dissociation into atoms, and one of them turns out to be excited. The process is described by the equation

However, as a rule, a quantum of light, a photon, acts as a particle hitting a molecule, and the process is called photodissociation and has a high efficiency. Since the dissociation method can be implemented in the absence of a gas discharge, this method is often referred to as a chemical method for obtaining population inversion. In one of the first lasers using this method, the gas was irradiated with light from a high-power flash lamp, causing photodissociation in a pattern, and then excited iodine atoms lased at a wavelength of µm. Since large volumes of gas can be subjected to photodissociation, iodine lasers can produce high pulsed and continuous output power. Assuming that the dissociation process is described by a system of transformations of the molecule and writing down two equations of the kinetics of this process for the corresponding concentrations of the considered particles

where is the probability per unit time of photoexcitation of the molecule; - the corresponding probability of its formation in the collision of one atom and an atom;

and are the probabilities of spontaneous and induced transitions per unit time, it is possible, taking into account (4), from the stationary version (24) to obtain an analog of formula (9):

where is the intensity (power flux) of the radiation, and the approximate value for was obtained under the assumption of a sufficiently fast process of molecule recovery, when their total concentration is so high that and.

Let us consider the method of electroionization and photoionization excitation of gas-discharge lasers, the first of which was already mentioned in Sec. 2. when describing the method for obtaining excimer molecules.

One of the main tasks of laser technology is the problem of increasing the radiation energy taken from a unit volume of the excited gas. To solve this problem, it is necessary to increase the gas pressure. In this case, the energy of the electrons in the discharge is spent, firstly, on the creation of plasma conductivity (ionization) and, secondly, on the excitation of active gas particles. However, the optimal values ​​of the electron energy required to perform each of these functions are different, which significantly reduces the efficiency of the system. To separately perform these functions (ionization and excitation), in order to increase the efficiency of the system, an electroionization method is used, which consists in the fact that an additional flow of electrons is injected into the discharge region, which serve to ionize gas atoms, i.e. to create plasma conductivity. In this case, the voltage on the electrodes can be reduced so that it becomes optimal for excitation of gas atoms.

In a device using the electroionization method, through a hole in the cathode of the discharge gap, electrons enter the region between the discharge electrodes, coming from a vacuum volume separated from the discharge region, in which the pressure is close to atmospheric, by thin aluminum foil. The electrons created by the electron gun or gun system bombard this foil with high energy (of the order of 100 keV) and penetrate through it into the discharge region at speeds that are optimal for ionization. Since the system operates in a pulsed mode, the foil does not have time to burn out. Special mirrors form a Fabry-Perot resonator in the discharge gap, and one of the mirrors releases generation quanta.

The photoionization method differs from the electroionization method in that ionization in the discharge gap is carried out by external irradiation with light, and not by fast electrons.

Gas-dynamic method for obtaining inverse population. This method was proposed by Soviet physicists V. K. Konyukhov and A. M. Prokhorov in 1966. Its idea is as follows. If we heat a gas consisting of atoms or molecules having a three-level system (Fig. 8), in which the probability of spontaneous transition is much greater than the probability of spontaneous transition and greater than the transition probability, then upon heating the number of excited molecules located at levels 2 will be greater than the number of molecules that are at levels 3, because .

However, if this gas is then rapidly cooled, then more molecules will be retained at levels 3 than at levels 2 due to the fact that an inverse population at the transition will also be created in this way for some time. On fig. Figure 8 shows the change in time t, which has elapsed since the moment of gas cooling, in the number of excited molecules that are at levels u. It can be seen that at. The setup that implements this method based on the use of molecules is shown in Fig.

Liquid fuel enters the combustion chamber 1 through tube 2, and through tubes 3 and 4 oxygen and molecules and serving as impurities. With the help of the ignition device 5, the fuel is ignited, and a hot mixture of gases is formed, having a relative composition

enters at a temperature under high pressure into the nozzle 6, from where this mixture enters a large volume 7 at supersonic speed, where there is a rapid expansion and, consequently, a rapid cooling of the gas. In this case, the cooled gas finds itself in the region of the Fabry-Perot resonator formed by mirrors 8 and 9, where the induced deexcitation of molecules and laser generation take place.

Such gas-dynamic lasers currently make it possible to obtain a continuous power of the order of 500 kW.

5. Plasma methods for obtaining population inversion are based on the fact that in a cold plasma (in contrast to a hot gas-discharge plasma) electrons have low velocities and therefore recombine intensively with ions in the bulk. At the same time, they occupy the upper unfilled energy levels of the atom and thus form atoms excited at the upper level, creating an inverse population with respect to the lower levels of excitation of atoms. If and is the concentration of ions and atoms excited to the upper and lower levels, then the equations of the kinetics of the processes will be:

where the probability of an ion per unit time to recombine with an electron by landing it on the upper level, is the probability of spontaneous purification of the lower level per unit time; and are the corresponding probabilities of spontaneous and induced transitions. From the stationary versions of equations (26.), taking into account (4.), we have an expression like (9.):

From (27) it follows that in order to increase it is necessary to increase, i.e. clear the lower level as quickly as possible. The problem of cleaning the lower working level is one of the main problems in plasma and gas-discharge methods for obtaining population inversion. There are four main mechanisms for this cleansing:

• 1. due to spontaneous transition to a lower (or main) energy level (radiation purification);
• 2. by transferring the excitation energy of the lower level to cooled free plasma electrons by colliding with them;
• 3. due to inelastic collisions with specially added impurity gas atoms, and the lower level excitation energy can go either to resonant excitation transfer to a neighboring impurity atom, or to its ionization, or to increase the kinetic energy of its motion (impact of the second kind). By adding the required number of successfully found impurity atoms, it is possible to significantly increase u;
• 4. chemical, when specially added impurity atoms actively enter into a chemical reaction with atoms located precisely at the lower levels of excitation, forming new molecules and thus reducing the plasma volume.

According to the implementation methods, plasma (recombination) lasers are divided into pulsed, electron-beam, nuclear-pumped, plasmodynamic and plasmochemical. In pulsed lasers, generation is carried out after the passage of a powerful pulsed discharge in a gas consisting of a mixture of working and buffer gases, the latter also serving to rapidly cool electrons during the afterglow of the discharge, when laser generation is taking place. (An example is lasers on ionized vapors of alkaline earth metals :). In electron-beam lasers and nuclear-pumped lasers, either a beam of fast gas-ionizing electrons or gas-ionizing fragments of nuclear reactions obtained from stationary nuclear reactors or during specially created nuclear explosions are introduced into the cold working gas from the outside (it is in this latter way that they try to implement a laser, generating X-rays).

In plasmodynamic lasers, generation is carried out in the cooling sections of a freely moving plasma jet, previously formed with the help of a gas discharge, in a gas jet passing through the discharge section, or formed in some other way. In this case, the jet can be rapidly cooled due to expansion, its density can be increased by compression in a longitudinal magnetic field, either external or implemented due to the pinch effect, etc.

Plasma-chemical lasers are characterized by various chemical methods of cleaning the lower working level.

4. Equations for the kinetics of changes in the population of levels in multilevel quantum systems and conditions for inverse population

An analysis of the conditions for obtaining an inverse population in multilevel systems and the kinetics of the processes of this obtaining can be carried out with varying degrees of approximation. Three different approaches to this analysis will be discussed below.

1. Analysis based on taking into account only two working levels of a multilevel system. Such a scheme, shown in Fig. 10 has already been used in the analysis of plasma methods for obtaining population inversion, and if in equations (26.) we replace (external pumping rate of level 2), then these equations will describe the kinetics of processes in the considered approximation, and the stationary version of the solution of these equations gives the expression (27 .), which is an analogue of the general relation (9.), and has the form

from which it follows that the stationary inverse population of the working levels cannot be obtained at. Such a working transition, in which, is called self-locking. An example of such a transition is a copper vapor laser. It is possible to obtain an inverted population in such a laser only at the initial stage of the transient process corresponding to the leading edge of the discharge current pulse. Let us analyze this transient process on the basis of equations (26.), in which we set (there is no external signal). In this case, from the first equation under the initial conditions

; it turns out

that after substitution into the second equation (26.) and integration under the initial conditions

Gives an expression

defining the process of change. It follows from (29.) that, therefore, the course of the function for various ratios between and will be as shown in Fig. 11, and from equations (26) at it can be obtained that this course is described by the relation and has a maximum at

It follows from Fig. 11 that there is indeed an inequality in the self-closing transition during the initial period. Since it follows from the stationary version of equations (36)

then, subtracting the second equation (36) from the first one and substituting from this approximate (for non-stationary mode) equalities, we can obtain the equation

approximately describing the kinetics of the process in the case and at. This equation is often used for approximate analysis of transients in laser systems.

• 2. Analysis of a three-level system with an upper working transition, taking into account the spontaneous filling of the upper levels. Such filling must be taken into account in the case of paramagnetic masers, when thermal spontaneous transitions significantly affect the behavior of the system, especially at temperatures other than cryogenic ones. The scheme under consideration corresponds to Fig. 6, a, c and in the case of pumping by light quanta, the equations of the kinetics of changes in populations (concentrations of the corresponding atoms) , and levels 1,2 and 3 have the form

moreover, as the resulting concentration of active atoms

• (In (31) and in (32) the quantities are the probabilities of spontaneous transitions per unit time from the i-th level to the j-th, a are the corresponding probabilities of induced transitions).

If we find from (34), (31), and (32), then subtracting all terms (32) from (34), we can obtain an equation for the difference that determines dy/dt . If all terms of this equation are differentiated with respect to time, substituting

it is possible, after determining from (34), (3l) and (32) and substituting instead of its value from the equation for dy/dt , to obtain the final equation that determines in the general case the dependence y= f(t)

From relations (З5) - (41) one can obtain a stationary value, and the parameters included in these relations have a clear physical meaning. So, in the absence of pumping, when, the expression is obtained

from which it follows that

there is a value in the absence of signal and pumping. Comparison of (42) with (3)-(5) shows that - is the spontaneous relaxation time (excitation lifetime) of the signal transition 32 in the absence of pumping. It can be shown that there is a similar relaxation time of the pump transition 31 in the absence of a signal, when. From (33) and (39) one can obtain the relation

which determines the population of level 1 at.

The stationary value can be represented in a form similar to expression (9):

from which it follows that, in the general case, the population inversion (i.e.) can be obtained only when (), and in the presence of a sufficiently large pump, such that

Comparing Expressions

(42) and (44), (45), one can verify that the effective relaxation time of excitation of signal transition levels

decreases with increasing pumping at, . It follows from (44) that the population inversion of the levels of the signal transition () is proportional to the value

which can be estimated assuming that in the absence of external influence of the population, obey the Boltzmann law:

Whence it follows that for masers with few

compared with kT at room temperature (at = 10 GHz and at T = 300 K), to increase () it is necessary to decrease T. Therefore, masers can operate normally only at cryogenic temperatures. Physically, this is explained by the fact that thermal motion throws particles to higher levels, equalizing particle concentrations at different levels and thereby reducing. In lasers, where the energy interval is sufficiently large, there is usually no need to lower the temperature.

Analysis of three-level and four-level systems without taking into account the spontaneous filling of the upper levels. For masers at cryogenic temperatures and for lasers at room temperatures, spontaneous transitions to upper levels can be neglected with a good degree of approximation; assume at, so that, as follows from (37)-(41), (43), (46), the considered parameters have the values

so that the stationary population difference is also obtained in the form of expression (9)

It can be seen from expression (52) that at when the working transition 32 becomes self-locking. Simple parameter calculations

basis (50), (5l) show that

when changing over a wide range.

In amplifying systems (especially in masers) the signal is usually small and it can be assumed that from (52) follows the expression

which shows , that at, when, the inversion of the population levels of the signal transition 32 occurs at an arbitrarily small pumping. We will see that this is not the case for the case of working transition 12. At very strong pumping (), the populations of levels 1 and 3 are equalized (which will be shown below) and from (55) it follows that the parameter is a two-level multilevel kinetics equation

determines the largest relative population inversion that occurs at, . In addition, because in this case

then the relations

determine the populations of levels in a three-level system in the absence of a signal and at very strong pumping.

Let us consider a three-level system with a working transition 2I , a typical application of which is a ruby ​​laser. In this case, for for , when the scheme in Fig. 6b is valid, the kinetic equations similar to (31) and (32) will have the form

and their stationary version after the replacement gives a solution in the form of relation (9):

is still determined from (50) and (51). It follows from (60) and (61) that the population inversion in this case can take place only for, when, and for such a large pumping that

(in contrast to the case of using transition 32 as a working one). For the case of no signal () it can be obtained from (61) and (55) that

so that at what was mentioned above.

Thus, when using transition 32 for , the population inversion usually occurs at a lower pumping than when using transition 12 for .

Let us consider a four-level quantum system with a working transition 32 as a signal one (see Fig. 7b). Such a system is implemented in a neodymium-activated glass laser, in liquid dye lasers, etc. The equations for the kinetics of changes in the populations of quantum levels have in this case the form

From the stationary version () of these equations, it follows that the inverse population difference of the working transition, written in the form (9), has the value:

From (66) it follows that in this system, as well as in a three-level system with a working transition 32, the population inversion sets in at an arbitrarily small pumping (), but only if the inequality

If this inequality is violated, the transition 32 in the four-level system will be self-closing and the system is capable of operating only in the initial periods of pulsed excitation.

Consideration in sections 2-4 of stationary modes of various types of quantum systems shows that they all have the same type of nonlinearity, which determines the dependence of the gain on the intensity I of the light wave field in accordance with the general and identical expressions (8), (9), (11), (14), (20), (22), (27), (28), (44), (60), (65).

This makes it possible to build a theory of various types of quantum self-oscillators according to a single plan, analyze their behavior, and optimize their parameters according to a scheme common to all these devices.

Lecture 1 2 .

The nature of the world. Spontaneous and forced emission. Energy level population inversion. The principle of operation of the laser.

1. Atoms can be in stationary states with discrete energy values ​​for an arbitrarily long time without radiating energy.

1.1. The transition from one stationary state to another stationary state is accompanied by the absorption or emission of a quantum of electromagnetic radiation.

1.2. When absorbing a quantum of electromagnetic radiation, the electron passes to a level with a higher energy value, and the atom itself passes into a higher-energy excited state, in which it can only stay for 10-8 s.

1.2.1. Since a strictly defined energy value is necessary for the transition to a higher energy level, when atoms are excited by electromagnetic radiation quanta, only those quanta are absorbed whose energy is equal to the difference between the energies of the initial and final states.

1.2.2. If a substance is excited by radiation with a continuous spectrum, then only those quanta will be absorbed, the energies of which correspond to the energies of the electron's transition to higher energy levels. As a result of the passage of such radiation through a substance, dark lines appear in the spectrum of this radiation, which are called absorption spectrum .

1.3. The transition of an atom to the ground state can occur either directly or by sequentially moving an electron to lower energy levels.

1.4. The transition of an electron to a level with a lower energy is accompanied by the emission of a quantum of electromagnetic radiation, the energy of which is equal to the difference between the energies of the levels of the initial and final states.

1.5. Since there can be quite a lot of excited states, the emitted quanta have different energies, and, consequently, different wavelengths.

1.6. Since the excited states have discrete energy values, the set of emitted quanta forms a line spectrum.

1.6.1. Transitions of electrons from high energy levels to one level form series of lines in the spectrum, the parameters of which are characteristic of a given element and differ from the parameters of a similar series of another element.

1.6.2. The set of series forms a spectrum characteristic radiation substance, which is an unambiguous characteristic of this substance.

1.6.3. On the basis of measurements of the parameters of the characteristic spectrum, methods of spectral analysis have been developed.

2. The emission of quanta by an excited atom in the absence of external action usually occurs spontaneously, and the resulting radiation is called spontaneous emission .

2.1. In spontaneous emission, each quantum occurs randomly and has its own phase of oscillations, and therefore spontaneous emission does not have temporal coherence .

2.2. According to quantum theory, the probability pν finding an atom in a state with energy εν obeys the Boltzmann distribution

which allows, for a given value of the energy supplied to the atom, to determine the ability of an electron to occupy one or another energy level.

2.3. The number of electrons that are simultaneously in the energy level is called level population .

2.4. In the absence of external influences, the equilibrium population of levels at a given temperature is maintained by spontaneous emission of quanta.

3. The form of the spontaneous emission spectrum depends on the state of the atom emitting this spectrum.

3.1. Isolated atoms emit radiation with atomic spectrum .

3.1.1. The composition of the atomic spectrum for the hydrogen atom and hydrogen-like ions can be easily calculated using the Balmer-Rydberg formula.

3.1.2. For other atoms and ions, the calculation of atomic spectra is a more difficult problem.

3.2. If atoms form a molecule, then there is molecular spectrum (striped spectrum ). Each band in this spectrum is a collection of closely spaced spectral lines.

3.2.1. As in atomic spectra, each line in the molecular spectrum results from a change in the energy of the molecule.

3.2.2. The energy of a molecule can be represented as

where is the energy of the translational motion of the molecule; is the energy of the rotational motion of the molecule; is the energy of vibrational motion of the atoms of the molecule relative to each other; is the energy of the electron shell of the molecule; is the intranuclear energy of the molecule.

3.2.3. The energy of the translational motion of a molecule is not quantized and its changes cannot lead to the appearance of a molecular spectrum, and the effect on the molecular spectrum can be ignored in the first approximation.

3.2.4. According to Bohr's frequency rule

where , , are the changes in the corresponding parts of the energy of the molecule.

3.2.5. The formation of bands is due to the fact that

3.2.6. Molecular spectra have a rather complex form.

3.2.6.1. The spectrum due only to the transition from one rotational level to another rotational level ( rotational spectrum ), located in the far infrared region (wavelength 0.1 ¸ 1 mm).

3.2.6.2. The spectrum due only to the transition from one vibrational level to another vibrational level ( vibrational spectrum ), located in the infrared region (wavelength 1 ¸ 10 microns).

3.2.6.3. The spectrum due only to the transition from one electronic level to another electronic level ( atomic spectrum ), located in the visible, ultraviolet and X-ray regions of the spectrum (wavelength 0.8 microns ¸ 10-10 m).

3.2.6.4. When the energy of vibrational motion of a molecule changes, the energy of rotational motion can also change. This gives rise to vibrational-rotational spectrum , which is a vibrational spectrum, each line of which is accompanied by closely spaced lines of rotational transitions.

3.2.6.5. Transitions between the electronic levels of a molecule are often accompanied by transitions between vibrational levels. The result is a spectrum called electronic-oscillatory , and, since vibrational transitions are accompanied by rotational transitions, the vibrational levels in the electronic-vibrational spectrum appear as smeared bands.

3.3. Raman scattering ( independent study).

4. The transition of atoms from a more excited state to a less excited state under the influence of an external quantum of electromagnetic radiation is called stimulated emission .

4.1. The probability of stimulated emission depends on the energy of the quantum acting on the excited atoms. The maximum probability of the occurrence of stimulated emission will be when the energy of the exciting quantum of the transition energy is equal.

4.2. When a quantum passes through a system of excited atoms, a flow of quanta arises, the energy of which is equal to the energy of the exciting quantum ( optical amplification effect ).

4.3. The absorption of light in matter occurs in accordance with the Bouguer-Lambert law

where is the natural absorption rate, and X is the thickness of the absorbing layer.

The amplification of the flux of quanta when passing through matter is similar to negative absorption coefficient (negative adsorption of light ).

4.4. For a medium with a negative absorption coefficient, the Bouguer-Lambert-Fabrikant law is valid

The light intensity sharply increases with increasing layer thickness.

4.5. A medium with a negative absorption coefficient is called active medium .

5. Three types of transitions are possible between two energy levels

the transition of an electron to a higher energy state upon absorption of a quantum (1); spontaneous transition of an electron to a less high-energy state (2); forced transition of an electron to a less high-energy state (3).

5.1. The number of electrons at excited levels obeys the Boltzmann distribution and is called level population .

5.2. In the usual scheme of radiation, the population N the higher energy level is less than the population of the lower energy level.

5.3. The number of quantum absorption events is proportional to the population N 1 less high-energy level, and the number of emission events is proportional to the population N 2 higher energy levels.

5.4. The natural absorption index in the Bouguer-Lambert law is proportional to the difference between the number of absorption and emission events

where k- coefficient of proportionality.

5.5. With the usual radiation scheme, the Boltzmann distribution of electrons due to spontaneous transitions ().

5.6. Due to the intense excitation of the system of atoms ( pumping ) it is possible to achieve such a violation of the Boltzmann distribution that N 2 will get bigger N 1 (population inversion ). Then the natural absorption index becomes less than zero and we get the Bouguer-Lambert-Fabrikant law.

6. The appearance of stimulated emission is implemented in lasers .

6.1. Initially, to obtain stimulated emission, a three-level scheme was used in ruby, the crystal lattice of which contains an impurity of Cr, which creates a narrow double additional level AT in the zone of excited states.

6.1.1. When an atomic system is excited by the light of a xenon lamp ( optical pumping ) a large number of electrons during the absorption of quanta (1) is transferred from the ground level BUT to excited levels C and D .

6.1.2. Electrons from these levels, through spontaneous transitions (2) without radiation, populate a lower energy level AT , creating an inverse population on it. In this case, the transition energy is transferred to the crystal lattice and increases the temperature of the substance.

6.1.3. Transitions from the inverse level B to the main level A are carried out under the action of quanta with an energy corresponding to the energy difference between the inverse level and the main level.

6.2. The hardware circuit of the laser is a rod BUT from the active substance, limited at the ends by two mirrors - opaque AT and translucent FROM.

6.2.1. After pumping the active substance, the very first transition from the inverse level to the ground level leads to the formation of a quantum that triggers the process of laser radiation.

6.2.2. The propagation of a quantum in an active medium leads to the initiation of forced transitions. According to the Bouguer-Lambert-Fabrikant law, quanta propagating along the rod have the highest efficiency.

6.2.3. When reflected from a semi-transparent mirror, a part of the quantum flux goes out of the active medium, which is laser radiation. The rest of the quantum flux returns to the active medium to initiate forced transitions.

6.2.4. A slight deviation of the direction of propagation of quanta from the axis of the crystal is eliminated using the curved surface of the reflecting mirrors AT and FROM.

6.2.5. The effect of quantum amplification increases significantly with multiple passage of initiating quanta through the active medium.

6.2.6. The inverted chromium level consists of two sublevels, and therefore the radiation of a ruby ​​laser consists of quanta with two wavelengths (0.6927 nm and 0.6943 nm).

7. Currently, the following are used as an active medium in lasers:

solids (ruby; neodymium-activated yttrium-aluminum garnet; neodymium-activated glass); gases and gas mixtures (N2; CO; CO2; metal vapours); liquids (solutions of organic dyes); semiconductors.

7.1. Laser radiation in solids arises during transitions between the energy levels of impurity atoms. Wavelength within 0.35¸1.06 µm at power up to 1 kW.

7.2. Laser radiation in gases most often arises during electronic-vibrational transitions between different electronic states (N2-laser, excimer lasers) or vibration-rotational transitions within the same electronic state (CO2-, CO-lasers). Wavelength within 5¸11 microns at power up to 15 kW.

7.3. Laser radiation in liquids during electronic transitions between the energy levels of dyes. Wavelength within 0.2¸5 µm at power up to 1.5 W. A smooth tuning of the wavelength is possible.

7.4. The population inversion in semiconductor lasers is created on transitions between states in the valence bands of a semiconductor crystal, and not between discrete levels. Wavelength within 0.75¸30 µm at power up to 0.5 W.

8. The main characteristics of laser radiation are:

Spatial and temporal coherence of radiation . The coherence time reaches 10-3 s. This corresponds to a coherence length of approximately 105 m. Good monochromaticity of radiation . The impurity levels are much narrower than the levels of the main substance, and therefore the spectral width of the radiation may not exceed 10-11 ± 10-10 m. Small beam divergence :

0.5¸10 mrad for gas lasers;

0.2¸5 mrad for solid state lasers.

High power density in a focused beam (up to 1010 W/m2).

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