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Stimulated emission is the process by which an incoming
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, so they alwa ...
of a specific frequency can interact with an excited atomic
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
(or other excited molecular state), causing it to drop to a lower
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
level. The liberated energy transfers to the electromagnetic field, creating a new
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, so they alwa ...
with a
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, polarization, and direction of travel that are all identical to the photons of the incident wave. This is in contrast to spontaneous emission, which occurs at a characteristic rate for each of the atoms/oscillators in the upper energy state regardless of the external electromagnetic field. According to the American Physical Society, the first person to correctly predict the phenomenon of stimulated emission was
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
in a series of papers starting in 1916, culminating in what is now called the Einstein B Coefficient. Einstein's work became the theoretical foundation of the MASER and LASER. The process is identical in form to atomic
absorption Absorption may refer to: Chemistry and biology *Absorption (biology), digestion **Absorption (small intestine) *Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials *Absorption (skin), a route by which s ...
in which the energy of an absorbed photon causes an identical but opposite atomic transition: from the lower level to a higher energy level. In normal media at thermal equilibrium, absorption exceeds stimulated emission because there are more electrons in the lower energy states than in the higher energy states. However, when a
population inversion In science, specifically statistical mechanics, a population inversion occurs while a system (such as a group of atoms or molecules) exists in a state in which more members of the system are in higher, excited states than in lower, unexcited energy ...
is present, the rate of stimulated emission exceeds that of absorption, and a net optical amplification can be achieved. Such a gain medium, along with an optical resonator, is at the heart 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" is an acronym for "light amplification by stimulated emission of radiation". The ...
or
maser A maser (, an acronym for microwave amplification by stimulated emission of radiation) is a device that produces coherent electromagnetic waves through amplification by stimulated emission. The first maser was built by Charles H. Townes, James ...
. Lacking a feedback mechanism, laser amplifiers and superluminescent sources also function on the basis of stimulated emission.


Overview

Electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
s and their interactions with
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
s are important in our understanding of
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
and
physics Physics is the natural science that studies matter, its 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 which ...
. In the classical view, the energy of an electron orbiting an atomic nucleus is larger for orbits further from the
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: * Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
of an
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
. However, quantum mechanical effects force electrons to take on discrete positions in orbitals. Thus, electrons are found in specific energy levels of an atom, two of which are shown below: When an electron absorbs energy either from
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
(photons) or
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
( phonons), it receives that incident quantum of energy. But transitions are only allowed between discrete energy levels such as the two shown above. This leads to emission lines and absorption lines. When an electron is excited from a lower to a higher energy level, it is unlikely for it to stay that way forever. An electron in an excited state may decay to a lower energy state which is not occupied, according to a particular time constant characterizing that transition. When such an electron decays without external influence, emitting a photon, that is called " spontaneous emission". The phase and direction associated with the photon that is emitted is random. A material with many atoms in such an excited state may thus result in radiation which has a narrow spectrum (centered around one
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
of light), but the individual photons would have no common phase relationship and would also emanate in random directions. This is the mechanism of
fluorescence Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation. It is a form of luminescence. In most cases, the emitted light has a longer wavelength, and therefore a lower photon energy, tha ...
and
thermal emission Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) is ...
. An external electromagnetic field at a frequency associated with a transition can affect the quantum mechanical state of the atom without being absorbed. As the electron in the atom makes a transition between two stationary states (neither of which shows a dipole field), it enters a transition state which does have a dipole field, and which acts like a small electric dipole, and this dipole oscillates at a characteristic frequency. In response to the external electric field at this frequency, the probability of the electron entering this transition state is greatly increased. Thus, the rate of transitions between two stationary states is increased beyond that of spontaneous emission. A transition from the higher to a lower energy state produces an additional photon with the same phase and direction as the incident photon; this is the process of stimulated emission.


History

Stimulated emission was a theoretical discovery by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
within the framework of the old quantum theory, wherein the emission is described in terms of photons that are the quanta of the EM field. Stimulated emission can also occur in classical models, without reference to photons or quantum-mechanics. (See also .) According to physics professor and director of the MIT-Harvard Center for Ultracold Atoms Daniel Kleppner, Einstein's theory of radiation was ahead of its time and prefigures the modern theory of quantum electrodynamics and quantum optics by several decades.


Mathematical model

Stimulated emission can be modelled mathematically by considering an atom that may be in one of two electronic energy states, a lower level state (possibly the ground state) (1) and an ''excited state'' (2), with energies ''E''1 and ''E''2 respectively. If the atom is in the excited state, it may decay into the lower state by the process of spontaneous emission, releasing the difference in energies between the two states as a photon. The photon will have
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
''ν''0 and energy ''hν''0, given by: E_2 - E_1 = h \, \nu_0 where ''h'' is Planck's constant. Alternatively, if the excited-state atom is perturbed by an electric field of frequency ''ν''0, it may emit an additional photon of the same frequency and in phase, thus augmenting the external field, leaving the atom in the lower energy state. This process is known as stimulated emission. In a group of such atoms, if the number of atoms in the excited state is given by ''N''2, the rate at which stimulated emission occurs is given by \frac = -\frac = - B_ \, \rho (\nu) \, N_2 where the proportionality constant ''B''21 is known as the '' Einstein B coefficient'' for that particular transition, and ''ρ''(''ν'') is the radiation density of the incident field at frequency ''ν''. The rate of emission is thus proportional to the number of atoms in the excited state ''N''2, and to the density of incident photons. At the same time, there will be a process of atomic absorption which ''removes'' energy from the field while raising electrons from the lower state to the upper state. Its rate is given by an essentially identical equation, \frac = -\frac = B_ \, \rho (\nu) \, N_1 . The rate of absorption is thus proportional to the number of atoms in the lower state, ''N''1. Einstein showed that the coefficient for this transition must be identical to that for stimulated emission: B_ =B_ . Thus absorption and stimulated emission are reverse processes proceeding at somewhat different rates. Another way of viewing this is to look at the ''net'' stimulated emission or absorption viewing it as a single process. The net rate of transitions from ''E''2 to ''E''1 due to this combined process can be found by adding their respective rates, given above: \frac = - \frac = B_ \, \rho(\nu) \, (N_2-N_1) = B_ \, \rho(\nu) \, \Delta N . Thus a net power is released into the electric field equal to the photon energy ''hν'' times this net transition rate. In order for this to be a positive number, indicating net stimulated emission, there must be more atoms in the excited state than in the lower level: \Delta N > 0. Otherwise there is net absorption and the power of the wave is reduced during passage through the medium. The special condition N_2 > N_1 is known as a
population inversion In science, specifically statistical mechanics, a population inversion occurs while a system (such as a group of atoms or molecules) exists in a state in which more members of the system are in higher, excited states than in lower, unexcited energy ...
, a rather unusual condition that must be effected in the gain medium of a laser. The notable characteristic of stimulated emission compared to everyday light sources (which depend on spontaneous emission) is that the emitted photons have the same frequency, phase, polarization, and direction of propagation as the incident photons. The photons involved are thus mutually
coherent Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deriv ...
. When a population inversion (\Delta N > 0) is present, therefore, optical amplification of incident radiation will take place. Although energy generated by stimulated emission is always at the exact frequency of the field which has stimulated it, the above rate equation refers only to excitation at the particular optical frequency \nu_0 corresponding to the energy of the transition. At frequencies offset from \nu_0 the strength of stimulated (or spontaneous) emission will be decreased according to the so-called line shape. Considering only homogeneous broadening affecting an atomic or molecular resonance, the spectral line shape function is described as a Lorentzian distribution g'(\nu) = where \Gamma is the full width at half maximum or FWHM bandwidth. The peak value of the Lorentzian line shape occurs at the line center, \nu = \nu_0. A line shape function can be normalized so that its value at \nu_0 is unity; in the case of a Lorentzian we obtain g(\nu) = = . Thus stimulated emission at frequencies away from \nu_0 is reduced by this factor. In practice there may also be broadening of the line shape due to
inhomogeneous broadening Homogeneous broadening is a type of emission spectrum broadening in which all atoms radiating from a specific level under consideration radiate with equal opportunity. If an optical emitter (e.g. an atom) shows homogeneous broadening, its spectra ...
, most notably due to the Doppler effect resulting from the distribution of velocities in a gas at a certain temperature. This has a
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...
shape and reduces the peak strength of the line shape function. In a practical problem the full line shape function can be computed through a
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of the individual line shape functions involved. Therefore, optical amplification will add power to an incident optical field at frequency \nu at a rate given by P =h\nu \, g(\nu) \, B_ \, \rho(\nu) \, \Delta N .


Stimulated emission cross section

The stimulated emission cross section is \sigma_(\nu) = A_ \frac g'(\nu) where *''A''21 is the Einstein ''A'' coefficient, *''λ'' is the wavelength in vacuum, *''n'' is the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, ...
of the medium (dimensionless), and *''g(''ν'') is the spectral line shape function.


Optical amplification

Stimulated emission can provide a physical mechanism for optical amplification. If an external source of energy stimulates more than 50% of the atoms in the ground state to transition into the excited state, then what is called a
population inversion In science, specifically statistical mechanics, a population inversion occurs while a system (such as a group of atoms or molecules) exists in a state in which more members of the system are in higher, excited states than in lower, unexcited energy ...
is created. When light of the appropriate frequency passes through the inverted medium, the photons are either absorbed by the atoms that remain in the ground state or the photons stimulate the excited atoms to emit additional photons of the same frequency, phase, and direction. Since more atoms are in the excited state than in the ground state then an amplification of the input intensity results. The population inversion, in units of atoms per cubic meter, is :\Delta N_ = N_2 - N_1 where ''g''1 and ''g''2 are the degeneracies of energy levels 1 and 2, respectively.


Small signal gain equation

The intensity (in
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James ...
s per square meter) of the stimulated emission is governed by the following differential equation: : = \sigma_(\nu) \cdot \Delta N_ \cdot I(z) as long as the intensity ''I''(''z'') is small enough so that it does not have a significant effect on the magnitude of the population inversion. Grouping the first two factors together, this equation simplifies as : = \gamma_0(\nu) \cdot I(z) where : \gamma_0(\nu) = \sigma_(\nu) \cdot \Delta N_ is the ''small-signal gain coefficient'' (in units of radians per meter). We can solve the differential equation using
separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
: : = \gamma_0(\nu) \cdot dz Integrating, we find: :\ln \left( \right) = \gamma_0(\nu) \cdot z or : I(z) = I_e^ where : I_ = I(z=0) \, is the optical intensity of the input signal (in watts per square meter).


Saturation intensity

The saturation intensity ''I''S is defined as the input intensity at which the gain of the optical amplifier drops to exactly half of the small-signal gain. We can compute the saturation intensity as :I_S = where :''h'' is Planck's constant, and :\tau_\text is the saturation time constant, which depends on the spontaneous emission lifetimes of the various transitions between the energy levels related to the amplification. :\nu is the frequency in Hz The minimum value of I_\text(\nu) occurs on resonance, where the cross section \sigma(\nu) is the largest. This minimum value is: :I_\text = \frac For a simple two-level atom with a natural linewidth \Gamma, the saturation time constant \tau_\text=\Gamma^.


General gain equation

The general form of the gain equation, which applies regardless of the input intensity, derives from the general differential equation for the intensity ''I'' as a function of position ''z'' in the gain medium: : = \cdot I(z) where I_S is saturation intensity. To solve, we first rearrange the equation in order to separate the variables, intensity ''I'' and position ''z'': : \left 1 + \bar(\nu) \right = \gamma_0(\nu)\cdot dz Integrating both sides, we obtain :\ln \left( \right) + \bar(\nu) = \gamma_0(\nu) \cdot z or :\ln \left( \right) + \bar(\nu) \left( - 1 \right) = \gamma_0(\nu) \cdot z The gain ''G'' of the amplifier is defined as the optical intensity ''I'' at position ''z'' divided by the input intensity: :G = G(z) = Substituting this definition into the prior equation, we find the general gain equation: :\ln \left( G \right) + \bar(\nu) \left( G - 1 \right) = \gamma_0(\nu) \cdot z


Small signal approximation

In the special case where the input signal is small compared to the saturation intensity, in other words, :I_ \ll I_S \, then the general gain equation gives the small signal gain as : \ln(G) = \ln(G_0) = \gamma_0(\nu) \cdot z or : G = G_0 = e^ which is identical to the small signal gain equation (see above).


Large signal asymptotic behavior

For large input signals, where :I_ \gg I_S \, the gain approaches unity :G \rightarrow 1 and the general gain equation approaches a linear asymptote: :I(z) = I_ + I_S


See also

*
Absorption Absorption may refer to: Chemistry and biology *Absorption (biology), digestion **Absorption (small intestine) *Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials *Absorption (skin), a route by which s ...
* Active laser medium *
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" is an acronym for "light amplification by stimulated emission of radiation". The ...
(includes a
history History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the invention of writing systems is considered prehistory. "History" is an umbrella term comprising past events as well ...
section) *
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 popu ...
* Rabi cycle * Spontaneous emission * STED microscopy


References

* * . .3 Laser Fundamentals, William T. Silfvast {{DEFAULTSORT:Stimulated Emission Electromagnetic radiation Physical phenomena Laser science