Due to selection rules, atoms of many elements have energy levels from which an electron cannot directly move to a lower level. These levels are called metastable states. An electron can move to this level by colliding with another electron or by moving from a higher level. The duration of stay of an electron in a metastable state is on the order of 10 ––3 s, while in an excited state it is 10 –8 s.
The radiation emitted during the spontaneous transition of an atom from an excited state to a ground state is called spontaneous emission. Spontaneous emission of various atoms does not occur coherently, because each atom begins and ends radiation independently of the others (Fig. 15.1a).
Emission of energy by an atom in which a transition from a metastable state to a ground state is caused electromagnetic radiation the corresponding frequency is called forced or induced, radiation (Fig. 15.1b).
The probability of stimulated emission increases sharply when the frequency coincides electromagnetic field with the natural frequency of the excited atom's radiation. Stimulated emission has the same frequency, phase, polarization and direction of propagation as the driving emission. Consequently, stimulated emission is strictly coherent with the driving emission, that is, the emitted photon is indistinguishable from the photon incident on the atom. The emitted photons, moving in one direction and encountering other excited atoms, stimulate further induced transitions, and the number of photons grows like an avalanche.
However, along with stimulated emission, a competing process, absorption, is also possible. In a system of atoms that is in thermodynamic equilibrium, absorption of incident radiation will prevail over stimulated radiation, i.e. incident radiation will be attenuated when passing through matter.
In order for a medium to enhance the radiation incident on it, it is necessary to create nonequilibrium state of the system, in which the number of atoms in excited states would be greater than their number in the ground state. Such states are called states with inverse population. The process of creating a nonequilibrium state of matter (transferring a system to a state with population inversion) is called pumped. Pumping can be done by optical, electrical and other methods. Media with inverse states are called active. They can be considered as media with a negative absorption coefficient, because the incident beam of light will be amplified when passing through these media.
For the first time, the possibility of obtaining media in which light can be amplified due to stimulated emission was pointed out in 1939 by the Russian physicist V.A. Fabrikant. He experimentally discovered stimulated emission of mercury vapor excited by an electrical discharge. The discovery of the phenomenon of amplification of electromagnetic waves and the invented method of their amplification (V.A. Fabrikant, M.M. Vudynsky, F.A. Butaeva; 1951) formed the basis of quantum electronics, the provisions of which subsequently made it possible to implement quantum amplifiers and quantum light generators.
Note: r" and k" are the vectors r and k respectively.
One of the main conclusions quantum mechanics states that each physical system (for example, an electron in an atom) can only be in one of the given energy states, the so-called eigenstates of the system. Each state (say, the electron state) can be associated with an eigenfunction
Ψ (r" , t) = U n * (r") * e -iEnt/ħ
and | Un (r") | 2 dxdydz - the probability of finding an electron in a certain state n within an elementary volume dxdydz with a center at a point determined by the radius vector r", E n - the energy of the nth state, ħ = h/2π; - Planck's constant.
Each electron in some physical system (for example, in an atom or molecule) corresponds to its own state, i.e. its own energy, and this energy has a discrete value.
In Fig. 7.1 shows a diagram of energy levels such physical system(using the example of an atom). Let us turn to two of the levels of this system - 1 and 2. Level 1 corresponds to the basic state of the physical system, where it is most likely to be found. A system (electron in an atom) can get to level 2 if some energy equal to hv = | E 2 - E 1 |.
This level 2 of the atom is an excited state. If the system (atom) is in state 2 during time t 0, then there is a finite probability that it will go into state 1, emitting a quantum of electromagnetic energy hv = E 2 - E 1. This process, which occurs randomly in time (chaotically) without the influence of an external field, is called spontaneous.
Average number of atoms undergoing a spontaneous transition from state 2 to state 1 in one second
DN 2 / dt = A 2 1 * N 2 = N 2 / (t cn) 2 1
where A 21 is the rate (probability) of a spontaneous transition, (t cn) 21 = A 21 - 1 is called the lifetime of an atom in an excited state associated with the 2→1 transition. Spontaneous transitions occur from any this state only to states that are lower in energy (for example, if an atom is in state 3, then direct transitions 3→2, 3→1 are possible, and an atom that ends up in level 2 goes spontaneously to level 1).
In the presence of an electromagnetic field having a frequency v ~ (E 2 - E 1) / h, an atom can make a transition from state 1 to state 2, while absorbing a quantum of the electromagnetic field (photon) with energy hv. However, if the atom is already in state 2 at the moment when it is exposed to the electromagnetic field, then it can go to state 1 with the emission of a quantum with energy hv under the influence of this field. This transition corresponds induced radiation.
The process of an induced transition differs from a spontaneous one in that for an induced transition the rates of transitions 2→1 and 1→2 are equal, while for a spontaneous process the rate of transition 1→2, in which the energy of the atom increases, is zero.
In addition, induced processes have other fundamental features:
- the speed of induced processes is proportional to the intensity of the electromagnetic field, while spontaneous processes do not depend on the field;
- the wave vector k", which determines the direction of propagation of the induced radiation, coincides in direction with the corresponding vector of the driving field (spontaneous radiation has an arbitrary direction of propagation);
- the frequency, phase and polarization of the stimulated emission also coincide with the frequency, phase and polarization of the driving field, while spontaneous emission, even having the same frequency, has an arbitrary random phase and polarization.
Let us consider the case when a plane monochromatic wave with frequency v and intensity I v propagates through a medium with a volume density of atoms N 2 at level 2 and N 1 at level 1.
If we introduce the rate of transitions that are induced by a monochromatic field with frequency v, denoting it by W i (v), then we can estimate the conditions under which induced radiation will exist.
In 1 s, in a volume of 1 m 3, N 2 Wi induced transitions from level 2 to level 1 and N 1 Wi transitions from level 1 to 2 occur. Thus, the total power generated in a unit volume
The transition of an excited system (atom, molecule) from upper energy levels to lower ones can occur either spontaneously or induced.
Spontaneous is a spontaneous (independent) transition caused only by factors acting within the system and characteristic of it. These factors determine the average time the system remains in the excited state; according to the Heisenberg relation (see § 11),
Theoretically, this time can have different values within:
i.e., it depends on the properties of the system - the spread of energy values of the excited state (the average value of the time spent in excited states is usually taken as a characteristic of the system, depending on the average value. One should also take into account the effect on the system of the surrounding space (“physical vacuum”), in which even in the absence of electromagnetic waves exists, according to quantum theory, fluctuating field (“vacuum fluctuations”); this field can stimulate the transition of the awakened system to lower levels and should be included among the irremovable factors causing spontaneous transitions.
Induced is a forced (stimulated) transition in energy inferior state, caused by some external influence on the excited system: thermal collisions, interaction with neighboring particles, or an electromagnetic wave passing through the system. However, a narrower definition has been established in the literature: induced is a transition caused only by an electromagnetic wave, and of the same frequency that is emitted by the system during this transition (fields of other frequencies will not resonate with the natural oscillations of the system,
therefore, their stimulating effect will be weak). Since the “carrier” of the electromagnetic field is a photon, it follows from this definition that during induced radiation, an external photon stimulates the birth of a new photon of the same frequency (energy).
Let us consider the most important features of spontaneous and induced transitions using one simple idealized example. Let us assume that in a volume V with mirror walls there are identical systems (atoms, molecules), of which at an initial fixed moment of time some of them are transferred to an excited state with the energy the total excess energy in this volume will be equal to. For spontaneous transitions the following is characteristic:
1) the process of transition of excited systems to normal states (i.e., the radiation of excess energy is extended over time. Some systems remain in an excited state for a short time; for others, this time is longer. Therefore, the flux (power) of radiation will change over time, reaching a maximum in some moment and then will asymptotically decrease to zero. The average value of the radiation flux will be equal.
2) the moment in time when the radiation of one system begins, and the location of this system is completely unrelated to the moment of radiation and the location of the other, i.e., there is no “consistency” (correlation) between the emitting systems either in space or in time. Spontaneous transitions are completely random processes, scattered in time, throughout the volume of the medium and in all possible directions; The planes of polarization and electromagnetic radiation from various systems have a probabilistic scatter, so the emitters themselves are not sources of coherent waves.
To characterize induced transitions, let us assume that one photon with an energy exactly equal to is introduced into the volume V under consideration at an instant of time. There is some probability that this photon will be absorbed by it during one of its collisions with an unexcited system; this probability will be taken into account below in a more general case (when the interaction of the systems under consideration with a photon gas occurs in volume V). We will assume that the photon is not absorbed, is repeatedly reflected from the walls of the vessel and, when colliding with excited systems, stimulates the emission of the same photons, i.e., causes induced transitions. However, each new photon that appears during these transitions will also excite induced transitions. Since the velocities of photons are high and the dimensions of the volume V are small, it will take a very short time for all excited systems present at the initial moment of time to be forced to transition to the normal state. Consequently, the following is characteristic of induced transitions:
1) the time required to emit excess energy can be adjusted and made very small, so the radiation flux can be very large;
2) in addition, the photon that caused the transition and the photon of the same energy (frequency) that appeared during this transition are in the same phase, have the same polarization and direction of movement. Therefore, the electromagnetic waves produced by stimulated emission are coherent.
However, not every collision of a photon with an excited system leads to its transition to the normal state, that is, the probability of an induced transition in each “act of interaction” of a photon with the system is not equal to one. Let us denote this probability by Let us assume that at a given moment of time there are photons in volume V and each of them, on average, can have collisions per unit time. Then the number of induced transitions per unit time, and therefore the number of photons appearing in volume V, will be equal to
Let us denote the number of excited systems in volume V by The number of collisions of photons with excited systems will be proportional to the concentration of such systems, i.e. Then it can be expressed depending on:
where Shind takes into account all other factors except the number of photons and the number of excited systems
An increase in the number of photons in volume V will also occur due to spontaneous emission. The probability of a spontaneous transition is the reciprocal of the average time spent in the excited state. Therefore, the number of photons appearing per unit time due to spontaneous transitions will be equal to
A decrease in the number of photons in volume V will occur as a result of their absorption by unexcited systems (in this case, the number of excited systems will increase). Since not every “act of interaction” of a photon with a system is accompanied by absorption, the probability of absorption should be introduced. The number of collisions per unit time of one photon with unexcited systems will be proportional to the number of such systems; therefore, by analogy with (2.83), for the loss of photons we can write:
Let us find the difference between the intensities of the processes of emission and absorption of photons, i.e., the processes of transition of systems from higher levels to lower ones and back:
Depending on the value, the following changes may occur in the volume under consideration;
1) if then in this volume there will be a gradual decrease in the density of the photon gas, i.e. absorption of radiant energy. A necessary condition for this purpose is a low concentration of excited systems: Lvozb
2) if then an equilibrium state is established in the system at a certain certain concentration of excited systems and radiant energy density;
3) if (which is possible for large values), then in the volume under consideration there will be an increase in the density of the photon gas (radiant energy).
It is obvious that a decrease or increase in radiation energy will take place not only in an isolated volume with reflective walls, but also in the case when a flow of monochromatic radiant energy (a flow of photons with a frequency propagates in a medium containing excited particles with excess energy
We'll find relative change number of photons per photon and per system; using (2.86), (2.83), (2.84) and (2.85), we obtain
Note that in the equilibrium state (which is possible only at a positive temperature according to formula (2.42) given in § 12, the ratio is equal to
The statistical sum in the denominator in this case consists of only two terms corresponding to: 1) systems in normal states with energy and 2) excited systems with energy. From this formula it follows that at an infinitely large positive temperature. This means that by increasing the temperature it is impossible to achieve state in which the number of excited systems would be more number unexcited. was greater than Mneexc, i.e. it is necessary that the number of photons appearing during transitions to lower levels should be greater than the number of photons absorbed during the same time). It was stated above that such a state cannot be achieved by increasing the temperature. Therefore, to obtain a medium capable of enhancing the radiant flux passing through it, it is necessary to use other (non-temperature) methods of excitation of atoms and molecules.
It can be shown that there can be more (i.e. N) only at a negative temperature, i.e., in a non-equilibrium state of the medium under consideration. If, in addition, this nonequilibrium state is metastable (see Part II, § 3), then with the help of a suitable external influence it is possible to cause an abrupt transition to an equilibrium state by releasing excess energy in a very short time. This idea underlies the operation of lasers.
The state of the medium in which the upper energy levels have larger filling factors compared to the lower ones is called inversion. Since in this state the medium does not weaken, as usual, but enhances the radiation passing through it, then in the formula for changing the intensity of the radiant flux in the medium
the coefficient will be a negative value (hence the exponent will be a positive value). In view of this, a medium in an inversion state is called a medium with a negative absorption index. The possibility of obtaining such media, their properties and use for amplification of optical radiation were established and developed by V. A. Fabrikant and his colleagues (1939-1951).
Rice. 1. a - spontaneous photon emission; b - stimulated emission; c - resonant absorption; E1 and E2 are the energy levels of the atom.
An atom in an excited state A, can, after a certain period of time, spontaneously, without any external influences, go into a state with lower energy (in our case, into the main one), giving off excess energy in the form of electromagnetic radiation (emitting a photon with energy h = E 2 –E 1). The process of emission of a photon by an excited atom (excited microsystem) without any external influences is called spontaneous(or spontaneous) radiation. The greater the probability of spontaneous transitions, the shorter the average lifetime of an atom in an excited state. Since spontaneous transitions are not mutually related, spontaneous emission is incoherent.
In 1916, A. Einstein, to explain the experimentally observed thermodynamic equilibrium between matter and the radiation emitted and absorbed by it, postulated that in addition to absorption and spontaneous emission, there should be a third, qualitatively different type of interaction. If on an atom in an excited state 2 , external radiation acts with a frequency that satisfies the condition hv= E 2 – E 1 , then it arises forced (induced) transition to the ground state 1 with the emission of a photon of the same energy hv= E 2 – E 1 (Fig. 309, c). During such a transition, radiation from the atom occurs photon, additionally to the photon under whose influence the transition occurred. The radiation resulting from such transitions is called forced (induced) radiation. Thus, two photons are involved in the process of stimulated emission: a primary photon that causes the excited atom to emit radiation, and a secondary photon emitted by the atom. It is important that secondary photons indistinguishable from the primary ones, being an exact copy of them.
7 Laser operating principle
Laser a device that converts pump energy (light, electrical, thermal, chemical, etc.) into the energy of a coherent, monochromatic, polarized and highly targeted radiation flux.
The physical basis of laser operation is the quantum mechanical phenomenon of forced (induced) radiation. The laser beam can be continuous, with constant amplitude, or pulsed, reaching extremely high peak powers. In some schemes, the laser working element is used as an optical amplifier for radiation from another source. There are a large number of types of lasers that use all aggregate states of matter as a working medium.
The physical basis of laser operation is the phenomenon of forced (induced) radiation. The essence of the phenomenon is that an excited atom is capable of emitting a photon under the influence of another photon without its absorption, if the energy of the latter is equal to the difference in the energies of the levels of the atom before and after the radiation. In this case, the emitted photon is coherent with the photon that caused the radiation (it is its “exact copy”). This way the light is amplified. This phenomenon differs from spontaneous emission, in which the emitted photons have a random direction of propagation, polarization and phase. The probability that a random photon will cause stimulated emission of an excited atom is exactly equal to the probability of absorption of this photon by an atom in an unexcited state. Therefore, to amplify light, it is necessary that there be more excited atoms in the medium than unexcited ones (the so-called population inversion). In a state of thermodynamic equilibrium, this condition is not satisfied, therefore various systems for pumping the laser active medium are used ( optical, electric, chemical and etc.).
The primary source of generation is the process of spontaneous emission, therefore, to ensure the continuity of generations of photons, the existence of a positive feedback is necessary, due to which the emitted photons cause subsequent acts of induced emission. To do this, the laser active medium is placed in an optical cavity. In the simplest case, it consists of two mirrors, one of which is translucent - through it the laser beam partially exits the resonator. Reflecting from the mirrors, the radiation beam passes repeatedly through the resonator, causing induced transitions in it. The radiation can be either continuous or pulsed. At the same time, using various devices (rotating prisms, Kerr cells etc.) to quickly turn the feedback off and on and thereby reduce the period of the pulses, it is possible to create conditions for generating radiation of very high power (the so-called giant pulses). This mode of laser operation is called the modulated mode. quality factor.
The radiation generated by the laser is monochromatic (one or a discrete set wavelengths), since the probability of emission of a photon of a certain wavelength is greater than that of a closely located one, associated with the broadening of the spectral line, and, accordingly, the probability of induced transitions at this frequency also has a maximum. Therefore, gradually during the generation process, photons of a given wavelength will dominate over all other photons. In addition, due to the special arrangement of the mirrors, only those photons that propagate in a direction parallel to the optical axis of the resonator at a short distance from it are retained in the laser beam; the remaining photons quickly leave the resonator volume. Thus, the laser beam has a very small divergence angle ] . Finally, the laser beam has a strictly defined polarization. To do this, various polaroids are introduced into the resonator, for example, they can serve as flat glass plates installed at a Brewster angle to the direction of propagation of the laser beam
The internal energy of atoms, molecules, ions, various compounds and media formed by these particles is quantized. Each molecule (atom, ion) can interact with electromagnetic radiation, making a transition from one energy level to another. In this case, the internal energy changes from one value corresponding to a certain movement and orientation of electrons and nuclei to another value corresponding to other movements and orientations.
The energy of the radiation field is also quantized, so that the exchange of energy between the field and the particles interacting with it can occur only in discrete portions.
The frequency of radiation associated with the transition of an atom (molecule, ion) between energy states is determined by Bohr's frequency postulate
Where E 1U E 2- respectively, the energy of a particle (atom, molecule, ion) in the upper and lower energy states, N- Planck's constant, V - frequency.
Not all transitions between energy states are possible. If the particle is in the upper state, then there is a certain probability that after a certain period of time it will go into the lower state and a change in energy will occur. This transition can be either radiative or non-radiative, both under the influence of external influences and without it. In a medium with discrete energy levels, there are three types of transitions: induced spontaneous And relaxation.
During induced transitions, a quantum system can be transferred from one energy state to another both with the absorption of external field energy quanta and with the emission of an electromagnetic energy quantum. Induced, or stimulated, radiation is stimulated by an external electromagnetic field. The probability of induced transitions (both radiative and non-radiative) is non-zero only for an external field of a resonant frequency, the quantum energy of which coincides with the difference in the energies of the two states under consideration. Induced radiation is completely identical to the radiation that causes it. This means that the electromagnetic wave created by induced transitions has the same frequency, phase, polarization and direction of propagation as the external radiation that caused the induced transition.
If the quantum system under consideration has two energy levels E 2 > E x(Fig. 17.1), during transitions between which a quantum of energy Lu is emitted or absorbed, then the particles of the system under consideration are in the field of their own radiation, the spectral volumetric energy density of which at the transition frequency is equal to p h>. This field causes transitions both from the lower state to the upper and from the upper to the lower (Fig. 17.1, a). The probabilities of these induced
Rice. 17.1
transitions FOR absorption AND radiation 1^,2 and IV 21 per unit time are respectively proportional to p y:
Where B 12, B 21 - Einstein coefficients respectively for induced absorption and emission.
Spontaneous transitions (Fig. 17.1, b) originate from a higher energy state E 2 to the bottom E x spontaneously - without external influence - with the radiation of the Lu quantum, i.e. they are radiative. The probability of such transitions does not depend on the external electromagnetic field and is proportional to time. During the time
where L 21 is the Einstein coefficient for spontaneous emission.
Total number of transitions per unit time from the energy state E 2("upper") to "lower" state E x(transition 2 - - 1) is equal to the product of the number of particles n 2 in state 2 on the probability of transition 2 -* 1 per unit time for one particle.
In thermodynamic equilibrium, the ensemble of particles does not lose or gain energy, i.e., the number of emitted quanta (the number of transitions from the upper energy state E 2 to the bottom E x state) must be equal to the number of absorbed quanta (the number of transitions from the state E x V E 2).
At thermal equilibrium, the distribution of particle populations across energy levels obeys Boltzmann's law
Where p 19 p 2 - respectively, the number of particles in states E x And E 2 е 1У § 2- statistical weights (multiplicities of degeneracy) of levels 2 and 1. The proportionality of the populations of levels to their statistical weights is due to the fact that the probability of a particle being in a certain quantum state is determined only by the energy of this state, and different quantum states, entirely determined by the full set of quantum numbers, can have the same energy.
At thermodynamic equilibrium, the number of radiative transitions FROM the upper STATE to the lower (N2) equal to the number of transitions from the lower state to the upper state (A^,) occurring with the absorption of radiation. The number of LG 2 transitions is determined by the probability of one transition multiplied by the population of the level C energy Eow i.e.
Similarly, the number of induced transitions from the lower state to the upper state, which determine energy absorption, is equal to
The relationship between the coefficients A 21, -B 21, AT 12 is found from the condition of thermodynamic equilibrium, at which LG 1 = A^. Equating expressions (17.4) and (17.5), we can determine the spectral field density of the intrinsic (equilibrium) radiation of the equilibrium system under consideration
(which is true for an equilibrium system) and use the Bora Lu frequency condition = E 2 - E x, then, making the assumption that the probabilities of induced absorption and emission are equal, i.e. 8V U2 =£2^21" we obtain the relation for the Einstein coefficients for spontaneous and stimulated emission:
The probability of radiative transitions per unit time (with the emission of quanta of spontaneous and stimulated emission) is equal to
Estimates show that for microwave and optical ranges L 21 <£ В 21 , т. е. вероятность спонтанного излучения много меньше, чем индуцированного, а поскольку спонтанное излучение определяет шумы, то в квантовых приборах роль шумов незначительна.
It should be noted that the equilibrium radiation of the entire system of particles in relation to each of the particles is an external electromagnetic field that stimulates the absorption or emission of energy by the particle, depending on its state. The quantity 8tsu 2 /c 3 included in expressions (17.7) and (17.8) determines the number of types of waves or oscillations in a unit volume and in a unit frequency interval for a region whose dimensions are large compared to the wavelength X = c/.
In addition to induced and spontaneous transitions in quantum systems, non-radiative relaxation transitions are of significant importance. Non-radiative relaxation transitions play a dual role: they lead to additional broadening of spectral lines (see Section 17.3) and establish thermodynamic equilibrium of the quantum system with its environment.
Relaxation transitions occur, as a rule, due to the thermal motion of particles. Heat absorption is accompanied by transitions of particles to a higher level and, conversely, the conversion of particle energy into heat occurs when it transitions to a lower energy level. Thus, relaxation transitions lead to the establishment of an equilibrium energy distribution of particles that is quite specific for a given temperature.
In real systems, the influence of spontaneous emission on the natural width of spectral lines can be neglected in comparison with relaxation processes, which more effectively reduce the lifetimes of excited states, which leads to broadening of spectral lines (as follows from the uncertainty relation for energy-time). The mechanism of these relaxation processes is highly dependent on the specific system. For example, for paramagnetic crystals, in particular in the case of electron paramagnetic resonance, a significant contribution to the broadening of emission lines is made by spin-spin And spin-lattice interactions and related relaxation processes with characteristic times of the order of 10_1 ..A0_3 s and 10~ 7 ...10~ k s, respectively.
Thus, relaxation processes that contribute to the establishment of thermal equilibrium in the environment ensure the continuity of the process of absorption of the energy of external electromagnetic radiation.
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