In optics, stimulated emission is the process by which, when perturbed by a photon, matter may lose energy resulting in the creation of another photon. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena Matter is commonly defined as being anything that has mass and that takes up space. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός The perturbing photon is not destroyed in the process (cf. absorption), and the second photon is created with the same phase, frequency, polarization, and direction of travel as the original. In Physics, absorption of electromagnetic radiation is the process by which the Energy of a Photon is taken up by matter typically the electrons of an The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 Frequency is a measure of the number of occurrences of a repeating event per unit Time. Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations Stimulated emission is really a quantum mechanical phenomenon but it can be understood in terms of a "classical" field and a quantum mechanical atom. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The electromagnetic field is a physical field produced by electrically charged objects. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The process can be thought of as "optical amplification" and it forms the basis of both the laser and maser. A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission. A maser is a device that produces coherent Electromagnetic waves through amplification due to Stimulated emission.

## Overview

Electrons and how they interact with each other and electromagnetic fields form the basis for most of our understanding of chemistry and physics. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The electromagnetic field is a physical field produced by electrically charged objects. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Electrons have energy in proportion to how far they are on average from the nucleus of an atom; however quantum mechanical effects force electrons to take on quantized positions in orbitals. The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny Thus, electrons are found in specific energy levels of an atom, as shown below:

The Pauli exclusion principle forces some electrons to be farther from the nucleus than others, which is why all the electrons in an atom do not simply occupy the 1s orbital. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 In Atomic physics and Quantum chemistry, electron configuration is the arrangement of Electrons in an Atom, Molecule, or other When electrons absorb energy either from light (photons) or from heat (phonons), they move farther away from the atomic nuclei but they are only allowed to absorb energy that will land them into specific energy levels. Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 In Physics, heat, symbolized by Q, is Energy transferred from one body or system to another due to a difference in Temperature In Physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the Atomic lattice of a Solid The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom A quantum mechanical system or particle that is bound, confined spacially can only take on certain discrete values of energy as opposed to classical particles which This leads to emission lines and absorption lines. A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range compared A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range compared

When an electron is excited, it will not stay that way forever. Excitation is an elevation in energy level above an arbitrary baseline energy state On average there is a lifetime for any particular energy level after which half of the electrons initially in that state will have decayed into a lower state. Given an assembly of elements the number of which decreases ultimately to zero the lifetime (also called the mean lifetime) is a certain number that characterizes the rate A quantum mechanical system or particle that is bound, confined spacially can only take on certain discrete values of energy as opposed to classical particles which Radioactive decay is the process in which an unstable Atomic nucleus loses energy by emitting ionizing particles and Radiation. When such a decay occurs, the energy difference between the level the electron was at and the new level must be released either as a photon or a phonon. When an electron decays due to "timeout" it is said to be due to "spontaneous emission. Spontaneous emission is the process by which a light source such as an Atom, Molecule, Nanocrystal or nucleus in an Excited state " The phase associated with the photon that is emitted is random and has to do with some quantum mechanical ideas concerning the atom's internal state. If a bunch of electrons were put into an excited state somehow and then left to relax, the resulting radiation would be very spectrally limited (only one wavelength of light would be present) but the individual photons would not be in phase with one another. Radiation, as in Physics, is Energy in the form of waves or moving Subatomic particles emitted by an atom or other body as it changes from a higher energy In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. This is also called fluorescence. Fluorescence is a Luminescence that is mostly found as an

Other photons (i. e. an external electromagnetic field) will affect an atom's state. The quantum mechanical variables mentioned above are changed. Specifically the atom will act like a small electric dipole which will oscillate with the external field. In physics there are two kinds of dipoles ( Hellènic: di(s- = two- and pòla = pivot hinge An electric dipole is a Oscillation is the repetitive variation typically in Time, of some measure about a central value (often a point of Equilibrium) or between two or more different states One of the consequences of this oscillation is it encourages electrons to decay to the lower energy state. When it does this due to the presence of other photons, the released photon is in phase with the other photons and in the same direction as the other photons. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 This is known as stimulated emission.

Stimulated emission can be modelled mathematically by considering an atom which may be in one of two electronic energy states, the ground state (1) and the excited state (2), with energies E1 and E2 respectively.

If the atom is in the excited state, it may decay into the ground state by the process of spontaneous emission, releasing the difference in energies between the two states as a photon. Spontaneous emission is the process by which a light source such as an Atom, Molecule, Nanocrystal or nucleus in an Excited state The photon will have frequency ν and energy hν, given by:

E2E1 = hν,

where h is Planck's constant. Frequency is a measure of the number of occurrences of a repeating event per unit Time. A physical Constant is a Physical quantity that is generally believed to be both universal in nature and constant in time

Alternatively, if the excited-state atom is perturbed by the electric field of a photon with frequency ν, it may release a second photon of the same frequency, in phase with the first photon. The atom will again decay into the ground 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, the rate at which stimulated emission occurs is given by:

$\frac{\partial N}{\partial t} = - B_{21} \rho (\nu) N$,

where B21 is a proportionality constant for this particular transition in this particular atom (referred to as an Einstein B coefficient), and ρ(ν) is the radiation density of photons of frequency ν. This article is about proportionality the mathematical relation In Physics, atomic Spectral lines are of two types An emission line is formed when an electron makes a transition from a particular discrete The rate of emission is thus proportional to the number of atoms in the excited state, N, and the density of the perturbing photons.

The critical detail of stimulated emission is that the emitted photon is identical to the stimulating photon in that it has the same frequency, phase, polarization, and direction of propagation. The two photons, as a result, are totally coherent. In Physics, coherence is a property of waves that enables stationary (i It is this property that allows optical amplification to take place.

Although most directly related to the discussion of how lasers work, stimulated emission touches on some of the most basic concepts in physics and the interaction of light and matter. It is a very important topic, and key to the understanding of optics specifically and physics in general.

For various reasons, the frequencies of the various photons emitted will not be exactly the same. For example, since the individual atoms in a laser medium are typically at some finite temperature, the Doppler effect will cause the photon wavelengths to vary from atom to atom (although the actual mechanism involved is more complex because of the more complex relationship between relative wavelength of stimulating photon and emitted photon). The Doppler effect (or Doppler shift) named after Christian Doppler, is the change in Frequency and Wavelength of a Wave for The spectrum of the photons, then, will not be an infinitesimally thin line, but will be a distribution. A spectrum (plural spectra or spectrums) is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. This distribution in the spectrum of emitted photons is called "line shape".

Although there are many possible line shapes, it is common to model the spectral line shape function as a Lorentzian distribution:

$g(\nu) = {1 \over \pi } { (\Gamma / 2) \over (\nu - \nu_0)^2 + (\Gamma /2 )^2 }$

where

$\Gamma \,$ is the full width at half maximum, or FWHM, in hertz. In Physics, atomic Spectral lines are of two types An emission line is formed when an electron makes a transition from a particular discrete The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous Probability distribution. A full width at half maximum ( FWHM) is an expression of the extent of a function given by the difference between the two extreme values of the Independent variable The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second.

This model is generally valid as long as

$|\nu - \nu_0| << \nu_0 \,$

and

$\Gamma << \nu_0 \,$

The line shape function, regardless of the form that it takes, must satisfy the normalization condition of any probability distribution:

$\int_{-\infty}^{\infty} g(\nu) \cdot d \nu = 1$

which the Lorentzian satisfies.

The peak value of the Lorentzian line shape occurs at the line center:

$g(\nu = \nu_0) = {2 \over \pi \Gamma}$

It is also convenient to define the normalized line shape function:

$\bar{g}(\nu) = { g(\nu) \over g(\nu_0) } = { (\Gamma / 2)^2 \over (\nu - \nu_0)^2 + (\Gamma /2 )^2 }$

which is dimensionless, and which has a peak value, also at the line center, of

$\bar{g}(\nu = \nu_0) = 1$

## Stimulated emission cross section

The stimulated emission cross section (in square meters) is

$\sigma_{21}(\nu) = A_{21} { \lambda^2 \over 8 \pi n^2} g(\nu)$

where

A21 is the Einstein A coefficient (in radians per second),
λ is the wavelength (in meters),
n is the refractive index of the medium (dimensionless), and
g(ν) is the spectral line shape function (in seconds). M^2 redirects here For other uses see M². CM2 redirects here The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium

## Optical amplification

Under certain conditions, stimulated emission can provide a physical mechanism for 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 external source of energy stimulates atoms in the ground state to transition to the excited state, creating what is called a population inversion. In Physics, specifically Statistical mechanics, a population inversion occurs when a system (such as a group of Atoms or Molecules exists in state When light of the appropriate frequency passes through the inverted medium, the photons stimulate the excited atoms to emit additional photons of the same frequency, phase, and direction, resulting in an amplification of the input intensity. Irradiance, radiant emittance, and radiant exitance are Radiometry terms for the power of Electromagnetic radiation at a surface per unit

The population inversion, in units of atoms per cubic meter, is

$\Delta N_{21} = \left( N_2 - {g_2 \over g_1} N_1 \right)$

where g1 and g2 are the degeneracies of energy levels 1 and 2, respectively. CM3 redirects here If you were looking for the 3rd game in the Cooking Mama series abbreviated as CM3 see here. This article refers to physical states having the same energy

### Small signal gain equation

The intensity (in watts per square meter) of the stimulated emission is governed by the following differential equation:

${ dI \over dz} = \sigma_{21}(\nu) \cdot \Delta N_{21} \cdot I(z)$

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. The watt (symbol W) is the SI derived unit of power, equal to one Joule of energy per Second. M^2 redirects here For other uses see M². CM2 redirects here Grouping the first two factors together, this equation simplifies as

${ dI \over dz} = \gamma_0(\nu) \cdot I(z)$

where

$\gamma_0(\nu) = \sigma_{21}(\nu) \cdot \Delta N_{21}$

is the small-signal gain coefficient (in units of radians per meter). We can solve the differential equation using separation of variables:

${ dI \over I(z)} = \gamma_0(\nu) \cdot dz$

Integrating, we find:

$\ln \left( {I(z) \over I_{in}} \right) = \gamma_0(\nu) \cdot z$

or

$I(z) = I_{in}e^{\gamma_0(\nu) z}$

where

$I_{in} = I(z=0) \,$ is the optical intensity of the input signal (in watts per square meter). In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an

### Saturation intensity

The saturation intensity IS 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

$I_S = {h \nu \over \sigma(\nu) \cdot \tau_S }$

where

h is Planck's constant, and
τS is the saturation time constant, which depends on the spontaneous emission lifetimes of the various transitions between the energy levels related to the amplification. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta.

### 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:

${ dI \over dz} = { \gamma_0(\nu) \over 1 + \bar{g}(\nu) { I(z) \over I_S } } \cdot I(z)$

where IS is intensity. The active laser medium or gain medium is the source of optical Gain within a Laser. To solve, we first rearrange the equation in order to separate the variables, intensity I and position z:

${ dI \over I(z)} \left[ 1 + \bar{g}(\nu) { I(z) \over I_S } \right] = \gamma_0(\nu)\cdot dz$

Integrating both sides, we obtain

$\ln \left( { I(z) \over I_{in} } \right) + \bar{g}(\nu) { I(z) - I_{in} \over I_S} = \gamma_0(\nu) \cdot z$

or

$\ln \left( { I(z) \over I_{in} } \right) + \bar{g}(\nu) { I_{in} \over I_S } \left( { I(z) \over I_{in} } - 1 \right) = \gamma_0(\nu) \cdot z$

The gain G of the amplifier is defined as the optical intensity I at position z divided by the input intensity:

$G = G(z) = { I(z) \over I_{in} }$

Substituting this definition into the prior equation, we find the general gain equation:

$\ln \left( G \right) + \bar{g}(\nu) { I_{in} \over I_S } \left( G - 1 \right) = \gamma_0(\nu) \cdot z$

### Small signal approximation

In the special case where the input signal is small compared to the saturation intensity, in other words,

$I_{in} << I_S \,$

then the general gain equation gives the small signal gain as

$\ln(G) = \ln(G_0) = \gamma_0(\nu) \cdot z$

or

$G = G_0 = e^{\gamma_0(\nu) z}$

which is identical to the small signal gain equation (see above).

### Large signal asymptotic behavior

For large input signals, where

$I_{in} >> I_S \,$

the gain approaches unity

$G \rightarrow 1$

and the general gain equation approaches a linear asymptote:

$I(z) = I_{in} + { \gamma_0(\nu) \cdot z \over \bar{g}(\nu) } I_S$

## References

• Saleh, Bahaa E. An asymptote of a real-valued function y=f(x is a curve which describes the behavior of f as either x or y goes to infinity A. and Teich, Malvin Carl (1991). Fundamentals of Photonics. New York: John Wiley & Sons. ISBN 0-471-83965-5.