6. Interaction of radiation with matter

This chapter deals with the physical principles behind the operation of quantum amplifiers and generators of optical radiation. The fundamental possibility of such devices was noted in 1939 by V.A. Fabrikant, who drew attention to the ability of a medium with an inverted level population to amplify radiation passing through it. The first quantum generators, operating in the centimeter-wave range (masers), were built in 1953, and in 1960 the first similar device operating in the optical wavelength range (laser) was built. The names of these devices come from the initial letters of the English words: «maser» — Microwave Amplification by Stimulated Emission of Radiation (amplification of microwaves by stimulated emission), «laser» — Light Amplification by Stimulated Emission of Radiation (amplification of light by stimulated emission). In 1964, N. Basov, A. Prokhorov and C. Townes were awarded the Nobel Prize for fundamental work in the field of quantum electronics that led to the development of a new type of generator and amplifier — masers and lasers.

6.1. Derivation of Planck's formula following Einstein

Earlier we discussed that atoms and ensembles of atoms can exist in various states characterized by a set of quantum numbers. A transition from one energy state to another is accompanied by the absorption or emission of a quantum of electromagnetic radiation — a photon, possessing energy

where — is the frequency of the radiation. If , then we are dealing with a spontaneous (self-initiated) transition of the atom from a higher to a lower level, as a result of which a photon is emitted with energy

.

Schematically this process of spontaneous emission can be depicted as the «reaction»

where the asterisk indicates the excited state of the atom A.

If, on the other hand, , then we are dealing with a stimulated transition, accompanied by the absorption of a photon, whose energy

goes to increase the energy of the atom (transferring it to a higher, excited level):

.

In 1918, A. Einstein drew attention to the fact that there also exist «emissive» transitions of another type, which occur under the action of external electromagnetic radiation and whose probability increases with the intensity of the radiation. In such a process, a photon strikes an excited atom and causes it to transition to a lower state with the emission of another photon. In the end, the system contains two photons — the initial one and the emitted one:

.

The radiation arising as a result of such transitions is called stimulated or induced emission. Spontaneous emission is random in the directions and phases of the emitted photons, since the emitting atoms are independent. Induced emission, however, must be, in its characteristics, completely identical to the external radiation that, passing through the substance, gave rise to it. Namely: induced emission has the same frequency, direction and polarization as the inducing external radiation. The phases of the emitted photons are correlated with the inducing oscillations, that is, induced emission is coherent.

The probabilities of all three types of processes (absorption, spontaneous and stimulated emission) are calculated in quantum electrodynamics. In Einstein's time this theory had not yet been created, and he applied intuitive thermodynamic reasoning to analyze the problem. Further on, for simplicity, we will consider a set of N atoms having only two energy levels and — the so-called two-level medium. Let, at time t, some of the total number of atoms be in the higher energy state 2, and let the probability of spontaneous emission of a single atom per unit time be equal to . Then the change in the number of atoms in state 2 over a small time dt will be

The minus sign indicates a decrease in the number of atoms at level 2. The quantity is called the Einstein coefficient for spontaneous emission. Integrating now, we easily obtain

where — is the number of atoms in state 2 at the initial moment of time. By its meaning, the formula quantity

is the average lifetime of an atom in the excited state (that is, the time over which the number of excited atoms decreases by a factor of e). This parameter determines the probability of the spontaneous emission process for a given type of atom.

Let us now imagine that the atoms are in equilibrium with radiation of frequency

and spectral energy density (energy density in a unit frequency interval). The spectral energy density is proportional to the number of photons of a given frequency. The more photons there are, the more likely the absorption of one of them by an atom. Therefore, for the probability of the process of stimulated absorption of radiation by an atom per unit time we can write the expression

where the Einstein coefficient characterizes the properties of the given atom. For the number of transitions to the excited state over time dt we have

For the probability of induced emission, Einstein proposed using an analogous formula

with some other, generally speaking, coefficient . Adding to the probability of the spontaneous transition, we obtain the total probability of a transition from state 2 to state 1 per unit time

so that the number of transitions from the excited state over time dt equals

Under thermodynamic equilibrium of matter and the electromagnetic field, a balance must be maintained between the processes of light emission and absorption, that is, an equality of the total number of acts of light emission and acts of its absorption. Such equilibrium is established in a closed cavity, the temperature of whose walls is kept constant. If, in the equilibrium state, the numbers of transitions 2–1 and 1–2 are equal

then we obtain

The distribution of atoms over energies at thermodynamic equilibrium obeys the Boltzmann law

whence

As the temperature rises, the spectral energy density must increase without limit. This will be the case only under the condition , that is, we obtain that the Einstein coefficients for stimulated absorption and induced emission of light are equal. Hence

Let us also note that the Einstein coefficients do not depend on temperature, since they relate to individual acts of absorption-emission of photons by an atom, while temperature is a characteristic of an ensemble of atoms. Then, in the limit of high temperatures, we obtain from (6.11) the expression

Comparing the previous formula with the Rayleigh–Jeans law, it follows that

Substituting (6.13) into (6.11), we obtain Planck's formula (27.26) for the spectral energy density of black-body radiation. The relations between the Einstein coefficients, which he derived from simple thermodynamic considerations, were subsequently confirmed by exact calculations.

Dividing by the number of types of oscillations per unit volume in a unit frequency interval, we obtain the average energy of one type of oscillation (photon) of frequency :

Dividing, in turn, this expression by the energy of a photon, we find the average number of photons of a given frequency at equilibrium:

We will encounter this formula and its analogues again in our course.

6.2. The concept of lasers

Passage of radiation through matter. Inverted level population. Let us again consider a two-level medium with energy levels and . If monochromatic radiation of frequency

falls on this medium, then as it propagates over a distance dx the change in the spectral energy density will be related both to resonant absorption and to induced (stimulated) emission by the atoms of the system. Due to induced emission, the spectral energy density in the beam increases, and this increase in energy must be proportional to:

• the energy density in the beam;
• the concentration of atoms in state 2;
• the distance traveled dx;
• the corresponding Einstein coefficient,

that is

.

Here — is a dimensional proportionality coefficient.

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

.

Adding and , we find the total change of the energy density:

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

The solution of this differential equation has the form

This formula gives the spectral energy density u in a beam of photons as it passes through a layer of matter of 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, as can be seen from (6.18), the radiation energy density decreases as it passes through matter, that is, the light is absorbed. However, if a system is created in which , then the absorption coefficient becomes negative, and instead of attenuation there is an amplification of the intensity of the light. The state of the medium in which is called a state with an inverted level population, and the medium itself is then called an active medium. An inverted level population contradicts the equilibrium Boltzmann distribution and can be created artificially if the system is taken out of a state of thermodynamic equilibrium.

This creates the fundamental possibility of amplification and generation of coherent optical radiation and is used in practice in the development of sources of such radiation — lasers.

Principle of laser operation. The creation of the laser became possible once ways of achieving an inverted level population in certain substances (active media) had been found. The first practical generator in the visible region of the spectrum was built in the (USA by Maiman (1960)) based on ruby. Ruby is a crystal lattice of , containing a small (0.03% – 0.05%) admixture of chromium ions (). Fig. 6.1 shows the diagram of the chromium energy levels (a three-level medium). The broad level is used to excite the chromium ions with light from a powerful gas-discharge lamp with a broad frequency band in the green-blue region of visible light — the pump lamp. The excitation of chromium ions by the pump energy from an external source is shown by the arrow .

Fig. 6.1. Diagram of the active three-level medium (ruby)

Electrons from the short-lived level undergo a fast ( s) radiationless transition to level (shown by the blue arrow). The energy released in this process is not emitted as photons, but is transferred to the ruby crystal. This causes the ruby to heat up, and so the laser design provides for its cooling.

The lifetime of the long-lived narrow level is s, that is, 5 orders of magnitude greater than that of the broadband level . At sufficient pump power, the number of electrons at level (called metastable) becomes greater than at level , that is, an inverted population is created between the «working» levels and .

A photon emitted in a spontaneous transition between these levels (shown by the dashed arrow ) induces the emission of additional (stimulated) photons — (the transition is shown by the arrow ), which in turn cause the induced emission of an entire cascade of photons with wavelength .

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

Starting from 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 that

.

Creating an active medium with population inversion is only half the task. For a laser to operate, conditions for light generation must also be created — that is, positive feedback. An active medium by itself can only amplify passing radiation. To achieve the generation regime, the amplification of induced radiation must compensate for all losses in the system. For this purpose the active substance is placed in an optical resonator, usually formed by two parallel mirrors, one of which is semi-transparent and serves to output the radiation from the resonator. Structurally, the first ruby lasers used cylindrical crystals 40 mm long and 5 mm in diameter. The end faces were polished parallel to each other and served as resonator mirrors. One of the end faces was silvered so that its reflection coefficient was close to unity, while the other end face was semi-transparent, that is, had a reflection coefficient less than unity, and was used to output radiation from the resonator. The excitation source was a powerful pulsed xenon lamp wound around the ruby in a spiral. The design of the ruby laser is shown schematically in Fig. 6.2.

Fig. 6.2. Design of the ruby laser: 1 — ruby rod; 2 — pulsed gas-discharge lamp; 3 — semi-transparent mirror; 4 — mirror; 5 — induced radiation

At sufficient pump lamp power, most (about half) of the chromium ions are transferred to the excited state. Once population inversion is achieved for the working levels with energy and , the first spontaneously emitted photons corresponding to the transition between these levels have no preferred direction of propagation and induce stimulated radiation that also propagates in all directions within the ruby crystal. Recall that photons produced by stimulated emission travel in the same direction as the incident photons. Photons whose direction of travel makes small angles with the axis of the crystal rod undergo multiple reflections from its end faces. Photons propagating in other directions, however, leave the ruby crystal through its side surface and do not participate in forming the output radiation. In this way a narrow beam of light is generated in the resonator, and the repeated passage of photons through the active medium induces the emission of more and more new photons, amplifying the intensity of the output beam.

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

Fig. 6.3. Radiation generation by a ruby laser

Thus, the optical resonator performs two functions: first, it creates positive feedback, and second, it forms a narrow directed beam of radiation with a specific spatial structure.

In the three-level scheme considered above, creating population inversion between the working levels requires exciting a fairly large fraction of the atoms, which demands significant energy expenditure. A more efficient scheme is the four-level scheme, which is used in solid-state lasers, for example using neodymium ions . In the most common gas laser using neutral atoms — the 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 by an electric gas discharge, owing to which atoms are transferred to the excited state with energy . Level in neon atoms (Fig. 6.4) is close to level in helium, and upon collision of helium atoms with neon atoms the excitation energy can be efficiently transferred to the latter without radiation.

Fig. 6.4. Level diagram of the He-Ne-laser

Thus, level of neon turns out to be more populated than the lower level . The transition between these working levels is accompanied by radiation with wavelength 632.8 nm, which is the main wavelength in industrial He-Ne-lasers. At level neon atoms do not stay long, quickly returning to the ground state. Note that level in neon is populated extremely weakly, and therefore to create population inversion between and only a small number of helium atoms need to be excited. This requires far less energy expenditure both for pumping and for cooling the setup, which is characteristic of the four-level generation scheme. Other neon levels (not shown in Fig. 6.4), giving radiation in both the visible and IR ranges, can also be used for laser generation, with helium used only for the pumping process.

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

This problem differs from the previous one only in its numerical values. For variety, let us carry out the calculation in electron-volts. First let us express the Boltzmann constant in these units:

so that at room temperature

.

Now we easily find

.

Such a small number is, from a practical standpoint, indistinguishable from zero, so even with weak pumping a population inversion is created between levels and .

Laser radiation is distinguished by characteristic features:

• high temporal and spatial coherence (monochromaticity of the radiation and small beam divergence);
• high spectral intensity.

The characteristics of the radiation depend on the type of laser and its operating mode, but some near-limiting parameter values can be noted:

• generation line width — about ;
• peak intensity — about W/mm²;
• beam divergence — about 1" — angular second.

Short (picosecond) laser pulses are indispensable for studying fast processes. Extremely high peak power (up to several GW) can develop in a pulse, equal to the power of several nuclear power plant units of a million kW each. At the same time the radiation can be concentrated in a narrow cone. Such beams allow, for example, «welding» the retina to the fundus of the eye.

Types of lasers. Within the framework of a 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 will limit ourselves to a fairly brief overview of laser types, distinguished by the characteristics of the active medium and pumping methods.

Solid-state lasers. They are usually pulsed; the first such laser was the ruby laser described above. Lasers using neodymium-doped glass as the working substance are popular. They generate light with a wavelength of about 1.06 μm, have large dimensions and peak power up to TW. They can be used for experiments on controlled thermonuclear fusion. An example is the huge «Shiva» laser at the Lawrence Livermore Laboratory in the USA.

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

Gas lasers. These are usually continuous-wave lasers. They are distinguished by a well-defined spatial beam structure. Example: the helium-neon laser, generating light at wavelengths of 0.63, 1.15 and 3.39 μm and having a power of about mW. Widely used in engineering is the — laser with power of about kW and wavelengths of 9.6 and 10.6 μm. One method of 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, for example hydrogen (deuterium) and fluorine. The basis lies in exothermic reactions

.

Molecules of HF are born already vibrationally excited, which immediately creates a population inversion. The resulting working mixture is passed at supersonic speed through the optical resonator, where part of the accumulated energy is released in the form of electromagnetic radiation. With the help of the resonator's mirror system, this radiation is focused into a narrow beam. Such lasers emit large energy (more than 2 kJ), a pulse duration of about 30 ns, and power up to W. The (chemical) efficiency reaches 10 %, whereas for other types of lasers it is usually a fraction of a percent. The generated wavelength is 2.8 μm (3.8 μm for lasers using DF).

Of the numerous types of chemical lasers, hydrogen fluoride (deuterium fluoride) lasers are recognized as the most promising. Problems: radiation from hydrogen fluoride lasers at the indicated wavelength is actively scattered by water molecules, which are always present in the atmosphere. This substantially weakens 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 lower than that of HF lasers. This means that when used in space, a much larger quantity of chemical fuel will have to be carried.

Excimer lasers. Molecules of an excimer — are diatomic molecules (for example, ) that can exist only in the excited state — the unexcited state turns out to be unstable for them. This is connected with the main feature of excimer lasers: the ground state of excimer molecules is unpopulated, that is, the lower working laser level always turns out to be empty. Pumping is carried out by a pulsed electron beam, which transfers a significant fraction of the atoms to the excited state, in which they combine into excimer molecules.

Since the transition between the working levels is broadband, tuning of the generation frequency is possible. A laser using gives tunable radiation in the UV range ( nm) and has high efficiency (20 %) of energy conversion. At present, excimer —lasers with a wavelength of 193 nm are used in ophthalmic surgery for surface evaporation (ablation) of the cornea.

Liquid lasers. The active substance in the liquid state is homogeneous and allows circulation for cooling purposes, which creates advantages over solid-state lasers. This makes it possible to obtain large energies and powers in both pulsed and continuous modes. The first liquid lasers (1964–1965) used compounds of rare-earth elements. They were superseded by lasers using solutions of organic dyes.

Such lasers usually use optical pumping by the radiation of other lasers in the visible or UV range. An interesting property of dye lasers is the ability to tune the generation frequency. By selecting the dye, generation can be obtained at any wavelength from the near IR to the near UV range. This is related to the broad continuous vibrational-rotational spectra of liquid molecules.

Semiconductor lasers. Solid-state lasers based on semiconductor materials are singled out as a separate class. Pumping is carried out by bombardment with an electron beam, 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 later sections). The use of various semiconductor materials makes it possible to obtain radiation at wavelengths from 0.7 to 1.6 μm. The dimensions of the active element are extremely small: the resonator length can be less than 1 mm.

Typical power is on the order of several kW, pulse duration is about 3 ns, efficiency reaches 50 %, and they have wide application (fiber optics, communications). They can be used for projecting a television 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 — the undulator — is a series of magnets placed between accelerator sections, so that relativistic electrons move along the undulator axis and undergo transverse oscillations, emitting a primary («spontaneous») electromagnetic wave. In the open resonator into which the electrons subsequently enter, the spontaneous electromagnetic wave is amplified, creating coherent directional laser radiation. The main feature of free-electron lasers is the possibility of smooth tuning of the generation frequency (from the visible to the IR range) by varying the kinetic energy of the electrons. The efficiency of such lasers is 1 % at an average power of up to 4 W. Using devices that return 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 oriented in different directions are mounted. The rods have two degrees of freedom and, like gun barrels, can be aimed at any point in space. Along the axis of each rod is a thin wire of a high-density material (of the order of the density of gold) — the active medium. The source of the laser's pump energy is a nuclear explosion. During the explosion, the active substance is converted into a plasma state. Cooling instantly, the plasma emits coherent radiation in the soft X-ray range. Due to the high energy concentration, the radiation, upon striking the target, leads to explosive evaporation of matter, formation of a shock wave, and destruction of the target.

Thus, the operating principle and design of the X-ray laser also make its area of application obvious. Resonator mirrors are not provided for in the laser described, since their use in the X-ray range is not feasible.

Some types of lasers are shown in the figure below.

Some types of lasers: 1 — laboratory laser; 2 — continuous-wave laser using ;
3 — technological laser for drilling holes; 4 — high-power technological laser

6.3. Nonlinear Effects in Optics

The propagation of light waves emitted by ordinary light sources in a medium is described by linear differential equations (linear optics), which means that the optical characteristics of the medium are independent of light intensity. The creation of lasers made it possible to obtain light waves with an electric field strength comparable in magnitude to the strength of the microscopic intra-atomic field (of order V/m). In such fields the refractive index and other optical characteristics of the medium exhibit a dependence on the field strength of the light wave. In this case the principle of superposition of fields is violated, that is, the principle of independent propagation of electromagnetic waves in matter, and the corresponding differential equations become nonlinear. This leads to a substantial change in the character of known optical phenomena occurring in the medium, as well as to the emergence of entirely new phenomena known as nonlinear effects in optics. These include:

• nonlinear absorption of light (self-induced transparency and self-darkening of the medium);
• frequency conversion (harmonic generation, parametric generation, and stimulated light scattering);
• self-action effects of light (self-focusing and self-defocusing of beams);
• multiphoton processes.

The nonlinear effect of saturation of light absorption was observed long before the advent of lasers. In 1923, S.I. Vavilov and V.L. Levshin discovered a decrease in the light absorption coefficient of uranium glass with increasing irradiation intensity. This first nonlinear-optical effect is explained by the fact that, owing to strong irradiation, a significant fraction of the absorbing particles of the medium pass into the excited state and can no longer absorb light.

The nonlinear effects of frequency conversion and self-action of light arise because a nonlinear component appears in the polarization of the medium, growing with increasing intensity of the electromagnetic wave. The presence of such a nonlinear component is explained by the anharmonicity of the oscillations of the medium's particles in the field of a powerful light wave. Because of this anharmonicity, the medium's response to a harmonic external electric field becomes nonlinear, that is, it ceases to reproduce the shape of the external influence. Thus, new harmonic components appear in the electric field re-radiated by the atoms and molecules of the substance — generation of higher optical harmonics of the optical radiation incident on the medium occurs. The process of second-harmonic generation in «nonlinear crystals» with specially selected properties is the most widespread in practical applications. Thus, for example, the IR radiation of an Nd:YAG laser, invisible to the eye, with wavelength μm is converted in a KDP crystal (potassium dihydrogen phosphate) into green light with wavelength μm.

The nonlinear polarization of the medium is also the cause of the dependence of the refractive index on the intensity of the electromagnetic wave. If the refractive index increases with increasing intensity of the light wave, the rays bend toward the beam axis, and when a certain critical power is exceeded, the phenomenon of self-focusing of light is observed — a diverging light beam becomes converging. The opposite self-action process of light is also possible — self-defocusing of the beam, if the refractive index decreases with increasing wave intensity.

In multiphoton processes of nonlinear interaction between optical radiation and matter, the simultaneous absorption (or emission) of two or more photons occurs in a single elementary act. The probability of such processes increases with increasing wave intensity. The multiphoton photoeffect or photoionization is accompanied by the absorption of several photons, the energy of each of which is less than the work function or the ionization energy, respectively. The phenomenon of Raman scattering of light also belongs to the nonlinear processes of interaction with a medium.

The list of processes could be continued with such things as nonlinear scattering of light and so on, occurring during the interaction of powerful laser radiation with a medium. Thus, with the advent of lasers a new field of physics arose — nonlinear optics.