Three-level Laser
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In
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, specifically
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, a population inversion occurs when a
system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its open system (systems theory), environment, is described by its boundaries, str ...
(such as a group of
atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s or
molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s) exists in a state in which more members of the system are in higher,
excited state In quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Add ...
s than in lower, unexcited
energy state A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The ...
s. It is called an "inversion" because in many familiar and commonly encountered physical systems in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in t ...
, this is not possible. This concept is of fundamental importance in
laser science Laser science or laser physics is a branch of optics that describes the theory and practice of lasers. Laser science is principally concerned with quantum electronics, laser construction, optical cavity design, the physics of producing a popula ...
because the production of a population inversion is a necessary step in the workings of a standard
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
.


Boltzmann distributions and thermal equilibrium

To understand the concept of a population inversion, it is necessary to understand some
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
and the way that
light Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
interacts with
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...
. To do so, it is useful to consider a very simple assembly of
atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s forming a laser medium. Assume there is a group of ''N'' atoms, each of which is capable of being in one of two
energy state A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The ...
s: either # The ''ground state'', with energy ''E''1; or # The ''excited state'', with energy ''E''2, with ''E''2 > ''E''1. The number of these atoms which are in the
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
is given by ''N''1, and the number in the excited state ''N''2. Since there are ''N'' atoms in total, : N_1+N_2 = N The energy difference between the two states, given by : \Delta E_ = E_2-E_1, determines the characteristic
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
\nu_ of light which will interact with the atoms; This is given by the relation : E_2-E_1 = \Delta E_ = h\nu_, ''h'' being the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
. If the group of atoms is in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in t ...
, it can be shown from
Maxwell–Boltzmann statistics In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density ...
that the ratio of the number of atoms in each state is given by the ratio of two
Boltzmann distribution In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability tha ...
s, the Boltzmann factor: : \frac = \frac\exp, where ''T'' is the
thermodynamic temperature Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expres ...
of the group of atoms, ''k'' is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
and ''g''1 and ''g''2 are the degeneracies of each state. Calculable is the ratio of the populations of the two states at
room temperature Room temperature, colloquially, denotes the range of air temperatures most people find comfortable indoors while dressed in typical clothing. Comfortable temperatures can be extended beyond this range depending on humidity, air circulation, and ...
(''T'' ≈ 300  K) for an energy difference Δ''E'' that corresponds to light of a frequency corresponding to visible light (''ν'' ≈ ). In this case Δ''E'' = ≈ 2.07 eV, and ''kT'' ≈ 0.026 eV. Since , it follows that the argument of the exponential in the equation above is a large negative number, and as such ''N''2/''N''1 is vanishingly small; i.e., there are almost no atoms in the excited state. When in thermal equilibrium, then, it is seen that the lower energy state is more populated than the higher energy state, and this is the normal state of the system. As ''T'' increases, the number of electrons in the high-energy state (''N''2) increases, but ''N''2 never exceeds ''N''1 for a system at thermal equilibrium; rather, at infinite temperature, the populations ''N''2 and ''N''1 become equal. In other words, a population inversion () can never exist for a system at thermal equilibrium. To achieve population inversion therefore requires pushing the system into a non-equilibrated state.


Interaction of light with matter

There are three types of possible interactions between a system of atoms and light that are of interest:


Absorption

If light (
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
s) of
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
ν12 passes through the group of atoms, there is a possibility of the light being absorbed by electrons which are in the ground state, which will cause them to be excited to the higher energy state. The rate of absorption is proportional to the radiation density of the light, and also to the number of atoms currently in the ground state, ''N''1.


Spontaneous emission

If atoms are in the excited state, spontaneous decay events to the ground state will occur at a rate proportional to ''N''2, the number of atoms in the excited state. The energy difference between the two states Δ''E''21 is emitted from the atom as a photon of frequency ''ν''21 as given by the frequency-energy relation above. The photons are emitted
stochastic Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; i ...
ally, and there is no fixed
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
relationship between photons emitted from a group of excited atoms; in other words, spontaneous emission is incoherent. In the absence of other processes, the number of atoms in the excited state at time ''t'', is given by : N_2(t) = N_2(0) \exp, where ''N''2(0) is the number of excited atoms at time ''t'' = 0, and ''τ''21 is the ''
mean lifetime A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and ( lambda) is a positive ra ...
'' of the transition between the two states.


Stimulated emission

If an atom is already in the excited state, it may be agitated by the passage of a photon that has a
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
''ν''21 corresponding to the energy gap Δ''E'' of the excited state to ground state transition. In this case, the excited atom relaxes to the ground state, and it produces a second photon of frequency ''ν''21. The original photon is not absorbed by the atom, and so the result is two photons of the same frequency. This process is known as ''stimulated emission''. Specifically, an excited atom will act like a small electric dipole which will oscillate with the external field provided. One of the consequences of this oscillation is that it encourages electrons to decay to the lowest energy state. When this happens due to the presence of the electromagnetic field from a photon, a photon is released in the same phase and direction as the "stimulating" photon, and is called stimulated emission. The rate at which stimulated emission occurs is proportional to the number of atoms ''N''2 in the excited state, and the radiation density of the light. The base probability of a photon causing stimulated emission in a single excited atom was shown by
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
to be exactly equal to the probability of a photon being absorbed by an atom in the ground state. Therefore, when the numbers of atoms in the ground and excited states are equal, the rate of stimulated emission is equal to the rate of absorption for a given radiation density. The critical detail of stimulated emission is that the induced photon has the same
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
and
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
as the incident photon. In other words, the two photons are
coherent Coherence is, in general, a state or situation in which all the parts or ideas fit together well so that they form a united whole. More specifically, coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics ...
. It is this property that allows
optical amplification An optical amplifier is a device that amplifies an optical signal directly, without the need to first convert it to an electrical signal. An optical amplifier may be thought of as a laser without an optical cavity, or one in which feedback from ...
, and the production of a
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
system. During the operation of a laser, all three light-matter interactions described above are taking place. Initially, atoms are energized from the ground state to the excited state by a process called '' pumping'', described below. Some of these atoms decay via spontaneous emission, releasing incoherent light as photons of frequency, ''ν''. These photons are fed back into the laser medium, usually by an
optical resonator An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors or other optical elements that confines light waves similarly to how a cavity resonator confines microwaves. Optical cavities are a major component of lasers, ...
. Some of these photons are absorbed by the atoms in the ground state, and the photons are lost to the laser process. However, some photons cause stimulated emission in excited-state atoms, releasing another coherent photon. In effect, this results in ''optical amplification''. If the number of photons being amplified per unit time is greater than the number of photons being absorbed, then the net result is a continuously increasing number of photons being produced; the laser medium is said to have a gain of greater than unity. Recall from the descriptions of absorption and stimulated emission above that the rates of these two processes are proportional to the number of atoms in the ground and excited states, ''N''1 and ''N''2, respectively. If the ground state has a higher population than the excited state (''N''1 > ''N''2), then the absorption process dominates, and there is a net attenuation of photons. If the populations of the two states are the same (''N''1 = ''N''2), the rate of absorption of light exactly balances the rate of emission; the medium is then said to be ''optically transparent''. If the higher energy state has a greater population than the lower energy state (''N''1 < ''N''2), then the emission process dominates, and light in the system undergoes a net increase in intensity. It is thus clear that to produce a faster rate of stimulated emissions than absorptions, it is required that the ratio of the populations of the two states is such that ''N''2/''N''1 > 1; In other words, a population inversion is required for laser operation.


Selection rules

Many transitions involving electromagnetic radiation are strictly forbidden under quantum mechanics. The allowed transitions are described by so-called
selection rule In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in ...
s, which describe the conditions under which a radiative transition is allowed. For instance, transitions are only allowed if Δ''S'' = 0, ''S'' being the total spin angular momentum of the system. In real materials, other effects, such as interactions with the crystal lattice, intervene to circumvent the formal rules by providing alternate mechanisms. In these systems, the forbidden transitions can occur, but usually at slower rates than allowed transitions. A classic example is
phosphorescence Phosphorescence is a type of photoluminescence related to fluorescence. When exposed to light (radiation) of a shorter wavelength, a phosphorescent substance will glow, absorbing the light and reemitting it at a longer wavelength. Unlike fluor ...
where a material has a ground state with ''S'' = 0, an excited state with ''S'' = 0, and an intermediate state with ''S'' = 1. The transition from the intermediate state to the ground state by emission of light is slow because of the selection rules. Thus emission may continue after the external illumination is removed. In contrast
fluorescence Fluorescence is one of two kinds of photoluminescence, the emission of light by a substance that has absorbed light or other electromagnetic radiation. When exposed to ultraviolet radiation, many substances will glow (fluoresce) with colore ...
in materials is characterized by emission which ceases when the external illumination is removed. Transitions that do not involve the absorption or emission of radiation are not affected by selection rules. The radiationless transition between levels, such as between the excited ''S'' = 0 and ''S'' = 1 states, may proceed quickly enough to siphon off a portion of the ''S'' = 0 population before it spontaneously returns to the ground state. The existence of intermediate states in materials is essential to the technique of optical pumping of lasers (see below).


Creating a population inversion

A population inversion is required for
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
operation, but cannot be achieved in the above theoretical group of atoms with two energy-levels when they are in thermal equilibrium. In fact, any method by which the atoms are directly and continuously excited from the ground state to the excited state (such as optical absorption) will eventually reach equilibrium with the de-exciting processes of spontaneous and stimulated emission. At best, an equal population of the two states, ''N''1 = ''N''2 = ''N''/2, can be achieved, resulting in optical transparency but no net optical gain.


Three-level lasers

To achieve lasting non-equilibrium conditions, an indirect method of populating the excited state must be used. To understand how this is done, consider a slightly more realistic model, that of a ''three-level laser''. Again consider a group of ''N'' atoms, this time with each atom able to exist in any of three energy states, levels 1, 2 and 3, with energies ''E''1, ''E''2, and ''E''3, and populations ''N''1, ''N''2, and ''N''3, respectively. Assume ''E''1 < ''E''2 < ''E''3; that is, the energy of level 2 lies between that of the ground state and level 3. Initially, the system of atoms is at thermal equilibrium, and the majority of the atoms will be in the ground state, i.e., ''N''1 ≈ ''N'', . If the atoms are subjected to light of a frequency \scriptstyle\nu_ \,=\, \frac\left(E_3 - E_1\right), the process of optical absorption will excite electrons from the ground state to level 3. This process is called '' pumping'', and does not necessarily always directly involve light absorption; other methods of exciting the laser medium, such as electrical discharge or chemical reactions, may be used. The level 3 is sometimes referred to as the ''pump level'' or ''pump band'', and the energy transition as the ''pump transition'', which is shown as the arrow marked P in the diagram on the right. Upon pumping the medium, an appreciable number of atoms will transition to level 3, such that . To have a medium suitable for laser operation, it is necessary that these excited atoms quickly decay to level 2. The energy released in this transition may be emitted as a photon (spontaneous emission), however in practice the transition called the
Auger effect The Auger effect (; ) or Meitner-Auger effect is a physical phenomenon in which atoms eject electrons. It occurs when an inner-shell vacancy in an atom is filled by an electron, releasing energy that causes the emission of another electron from a ...
(labeled R in the diagram) is usually ''radiationless'', with the energy being transferred to vibrational motion (
heat In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
) of the host material surrounding the atoms, without the generation of a photon. An electron in level 2 may decay by spontaneous emission to the ground state, releasing a photon of frequency ''ν''12 (given by ), which is shown as the transition L, called the ''laser transition'' in the diagram. If the lifetime of this transition, ''τ''21 is much longer than the lifetime of the radiationless transition ''τ''32 (if , known as a ''favourable lifetime ratio''), the population of the ''E''3 will be essentially zero () and a population of excited state atoms will accumulate in level 2 (). If over half the ''N'' atoms can be accumulated in this state, this will exceed the population of the ground state ''N''1. A population inversion (''N''2 > ''N''1 ) has thus been achieved between level 1 and 2, and optical amplification at the frequency ''ν''21 can be obtained. Because at least half the population of atoms must be excited from the ground state to obtain a population inversion, the laser medium must be very strongly pumped. This makes three-level lasers rather inefficient, despite being the first type of laser to be discovered (based on a
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laser medium, by
Theodore Maiman Theodore Harold Maiman (July 11, 1927 – May 5, 2007) was an American engineer and physicist who is widely credited with the invention of the laser.Johnson, John Jr. (May 11, 2008). "Theodore H. Maiman, at age 32; scientist created the first L ...
in 1960). A three-level system could also have a radiative transition between level 3 and 2, and a non-radiative transition between 2 and 1. In this case, the pumping requirements are weaker. In practice, most lasers are ''four-level lasers'', described below.


Four-level laser

Here, there are four energy levels, energies ''E''1, ''E''2, ''E''3, ''E''4, and populations ''N''1, ''N''2, ''N''3, ''N''4, respectively. The energies of each level are such that ''E''1 < ''E''2 < ''E''3 < ''E''4. In this system, the pumping transition P excites the atoms in the ground state (level 1) into the pump band (level 4). From level 4, the atoms again decay by a fast, non-radiative transition Ra into the level 3. Since the lifetime of the laser transition L is long compared to that of Ra (), a population accumulates in level 3 (the ''upper laser level''), which may relax by spontaneous or stimulated emission into level 2 (the ''lower laser level''). This level likewise has a fast, non-radiative decay Rb into the ground state. As before, the presence of a fast, radiationless decay transition results in the population of the pump band being quickly depleted (''N''4 ≈ 0). In a four-level system, any atom in the lower laser level ''E''2 is also quickly de-excited, leading to a negligible population in that state (''N''2 ≈ 0). This is important, since any appreciable population accumulating in level 3, the upper laser level, will form a population inversion with respect to level 2. That is, as long as ''N''3 > 0, then ''N''3 > ''N''2, and a population inversion is achieved. Thus optical amplification, and laser operation, can take place at a frequency of ''ν''32 (''E''3 − ''E''2 = ''hν''32). Since only a few atoms must be excited into the upper laser level to form a population inversion, a four-level laser is much more efficient than a three-level one, and most practical lasers are of this type. In reality, many more than four energy levels may be involved in the laser process, with complex excitation and relaxation processes involved between these levels. In particular, the pump band may consist of several distinct energy levels, or a continuum of levels, which allow optical pumping of the medium over a wide range of wavelengths. Note that in both three- and four-level lasers, the energy of the pumping transition is greater than that of the laser transition. This means that, if the laser is optically pumped, the frequency of the pumping light must be greater than that of the resulting laser light. In other words, the pump wavelength is shorter than the laser wavelength. It is possible in some media to use multiple photon absorptions between multiple lower-energy transitions to reach the pump level; such lasers are called ''up-conversion'' lasers. While in many lasers the laser process involves the transition of atoms between different
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
ic energy states, as described in the model above, this is not the only mechanism that can result in laser action. For example, there are many common lasers (e.g.,
dye lasers A dye laser is a laser that uses an organic dye as the lasing medium, usually as a liquid solution. Compared to gases and most solid state lasing media, a dye can usually be used for a much wider range of wavelengths, often spanning 50 to 100 n ...
,
carbon dioxide laser The carbon-dioxide laser (CO2 laser) was one of the earliest gas lasers to be developed. It was invented by C. Kumar N. Patel, Kumar Patel of Bell Labs in 1964 and is still one of the most useful types of laser. Carbon dioxide, Carbon-dioxide lase ...
s) where the laser medium consists of complete molecules, and energy states correspond to vibrational and rotational modes of oscillation of the molecules. This is the case with water masers, that occur in nature. In some media it is possible, by imposing an additional optical or microwave field, to use quantum coherence effects to reduce the likelihood of a ground-state to excited-state transition. This technique, known as
lasing without inversion Lasing without inversion (LWI), or lasing without population inversion, is a technique used for light amplification by stimulated emission without the requirement of population inversion. A laser working under this scheme exploits the quantum int ...
, allows optical amplification to take place without producing a population inversion between the two states.


Other methods of creating a population inversion

Stimulated emission was first observed in the microwave region of the electromagnetic spectrum, giving rise to the acronym
MASER A maser is a device that produces coherent electromagnetic waves ( microwaves), through amplification by stimulated emission. The term is an acronym for microwave amplification by stimulated emission of radiation. Nikolay Basov, Alexander Pr ...
for Microwave Amplification by Stimulated Emission of Radiation. In the microwave region, the Boltzmann distribution of molecules among energy states is such that, at room temperature, all states are populated almost equally. To create a population inversion under these conditions, it is necessary to selectively remove some atoms or molecules from the system based on differences in properties. For instance, in a
hydrogen maser A hydrogen maser, also known as hydrogen frequency standard, is a specific type of maser that uses the intrinsic properties of the hydrogen atom to serve as a precision frequency reference. Overview Both the proton and electron of a hydrogen ato ...
, the well-known 21cm wave transition in atomic hydrogen, where the lone electron flips its spin state from parallel to the nuclear spin to antiparallel, can be used to create a population inversion because the parallel state has a magnetic moment and the antiparallel state does not. A strong inhomogeneous magnetic field will separate atoms in the higher energy state from a beam of mixed-state atoms. The separated population represents a population inversion that can exhibit stimulated emissions.


See also

*
Laser construction A laser is constructed from three principal parts: *An energy source (usually referred to as the ''Laser pumping, pump'' or ''pump source''), *A ''gain medium'' or ''Active laser medium, laser medium'', and *Two or more mirrors that form an ''o ...
*
Negative temperature Certain system (thermodynamics), systems can achieve negative thermodynamic temperature; that is, their Thermodynamic temperature, temperature can be expressed as a negative number, negative quantity on the Kelvin or Rankine scale, Rankine scale ...
*
Quantum electronics Quantum optics is a branch of atomic, molecular, and optical physics and quantum chemistry that studies the behavior of photons (individual quanta of light). It includes the study of the particle-like properties of photons and their interaction w ...


References

* Svelto, Orazio (1998). ''Principles of Lasers'', 4th ed. (trans. David Hanna), Springer. {{Lasers Laser science Statistical mechanics