In

opticsOptics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

,

**stimulated emission** is the process by which an atomic electron (or an excited molecular state) interacting with an electromagnetic wave of a certain frequency may drop to a lower

energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

level, transferring its energy to that field. A photon created in this manner has the same

phasePhase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

,

frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

, polarization, and direction of travel as the photons of the incident wave. This is in contrast to

spontaneous emissionSpontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon...

which occurs without regard to the ambient electromagnetic field.

However the process is identical in form to atomic

absorptionIn physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...

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 inversionIn physics, specifically statistical mechanics, a population inversion occurs when a system exists in state with more members in an excited state than in lower energy states...

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

laserA laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

or

maserA maser is a device that produces coherent electromagnetic waves through amplification by stimulated emission. Historically, “maser” derives from the original, upper-case acronym MASER, which stands for "Microwave Amplification by Stimulated Emission of Radiation"...

.

Lacking a feedback mechanism, laser amplifiers and

superluminescentAmplified spontaneous emission or superluminescence is light, produced by spontaneous emission, that has been optically amplified by the process of stimulated emission in a gain medium. It is inherent in the field of random lasers....

sources also function on the basis of stimulated emission.

Stimulated emission is a

quantum mechanicalQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

phenomenon discovered (on theoretical grounds) by Einstein. Each downward atomic transition results in the creation of exactly one photon. However it can generally be understood in terms of a "classical"

fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

interacting with a quantum mechanical

atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

.

## Overview

ElectronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s and how they interact with

electromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

s are important in our understanding of

chemistryChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

and

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

.

In the

classical viewClassical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...

, the energy of an electron orbiting an atomic nucleus is larger for orbits further from the

nucleusThe nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

of an

atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

. However, quantum mechanical effects force electrons to take on discrete positions in

orbitalsAn atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

. Thus, electrons are found in specific energy levels of an atom, two of which are shown below:

When an electron absorbs energy either from

lightLight or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

(photons) or from

heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

(phonons), it will receive that incident quanta of energy. But transitions are only allowed in between discrete energy levels such as the two shown above.

This leads to emission lines and

absorption lineA spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from a deficiency or excess of photons in a narrow frequency range, compared with the nearby frequencies.- Types of line spectra :...

s.

When an electron is

excitedExcitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state....

from a lower to a higher energy level, it will not 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 emissionSpontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon...

".

The phase associated with the photon that is emitted is random.

A material with many atoms in such an excited state may thus result in

radiationIn physics, radiation is a process in which energetic particles or energetic waves travel through a medium or space. There are two distinct types of radiation; ionizing and non-ionizing...

which is very spectrally limited (centered around one

wavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

of light), but the individual photons would

have no common phase relationship and would emanate in random directions.

This is the mechanism of

fluorescenceFluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation of a different wavelength. It is a form of luminescence. In most cases, emitted light has a longer wavelength, and therefore lower energy, than the absorbed radiation...

and thermal emission.

An external electromagnetic field at a frequency associated with a transition can affect the quantum mechanical state of the atom.

The atom will act like a small electric

dipoleIn physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

oscillating in response to the external field.

A consequence of this oscillation is that

the rate of transitions between two states is enhanced beyond that due to spontaneous emission. Such a transition to the higher state is called

absorptionIn physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...

, destroying an incident photon. A transition from the higher to a lower energy state, however, produces an additional photon; this is the process of

**stimulated emission**.

## 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 emissionSpontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon...

, releasing the difference in energies between the two states as a photon. The photon will have

frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

ν and energy

*h*ν, given approximately by:

where

*h* is

Planck's constantThe Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...

.

Alternatively, if the excited-state atom is perturbed by an electric field of frequency

, 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:

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:

.

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:

.

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:

.

Thus a net power is released into the electric field equal to the photon energy

*h*ν 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:

. Otherwise there is net absorption and the power of the wave is reduced during passage through the medium. The special condition

is known as a

population inversionIn physics, specifically statistical mechanics, a population inversion occurs when a system exists in state with more members in an excited state than in lower energy states...

, 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

coherentIn physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....

. When a population inversion (

) 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

corresponding to the energy of the transition. At frequencies offset from

the strength of stimulated (or spontaneous) emission will be decreased according to the so-called "line shape".

Considering only

homogeneous broadeningHomogeneous 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 Homogeneous broadening is a type of emission spectrum broadening in which all atoms radiating from a specific...

affecting an atomic or molecular resonance, the

spectral line shape functionIn physics, atomic spectral lines are of two types:* An emission line is formed when an electron makes a transition from a particular discrete energy level of an atom, to a lower energy state, emitting a photon of a particular energy and wavelength...

is described as a

Lorentzian distributionThe Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...

:

where

is the

full width at half maximumFull width at half maximum is an expression of the extent of a function, given by the difference between the two extreme values of the independent variable at which the dependent variable is equal to half of its maximum value....

or FWHM bandwidth.

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

. A line shape function can be normalized so that its value at

is unity; in the case of a Lorentzian we obtain:

.

Thus stimulated emission at frequencies away from

is reduced by this factor. In practice there may also be broadening of the line shape due to

inhomogeneous broadeningIn atomic physics, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting particles result in different shifts, the cumulative effect of which is the line broadening.The...

, most notably due to the

Doppler effectThe Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...

resulting from the distribution of velocities in a gas at a certain temperature. This has a

GaussianGaussian is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70. It has been continuously updated since then...

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

convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

of the individual line shape functions involved. Therefore optical amplification will add power to an incident optical field at frequency

at a rate given by:

.

## Stimulated emission cross section

The stimulated emission cross section (in square meters) is

where

*A*_{21} is the Einstein *A* coefficient (in radians per second),
- λ is the wavelength (in meters),
*n* is the refractive indexIn optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

of the medium (dimensionless), and
*g*(ν) is the spectral line shape function (in seconds).

## Optical amplification

Under certain conditions, stimulated emission can provide a physical mechanism for

optical amplificationAn 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 the cavity is suppressed...

. An external source of energy stimulates atoms in the ground state to transition to the excited state, creating what is called a

population inversionIn physics, specifically statistical mechanics, a population inversion occurs when a system exists in state with more members in an excited state than in lower energy states...

. 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

intensityIrradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...

.

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

where

*g*_{1} and

*g*_{2} are the

degeneraciesIn physics, two or more different quantum states are said to be degenerate if they are all at the same energy level. Statistically this means that they are all equally probable of being filled, and in Quantum Mechanics it is represented mathematically by the Hamiltonian for the system having more...

of energy levels 1 and 2, respectively.

### Small signal gain equation

The intensity (in

wattThe watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:...

s per square meter) of the stimulated emission is governed by the following differential equation:

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

where

is the

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

separation of variablesIn mathematics, separation of variables 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 on a different side of the equation....

:

Integrating, we find:

or

where

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

where

*h* is Planck's constant, and
- τ
_{S} is the saturation time constant, which depends on the spontaneous emission lifetimes of the various transitions between the energy levels related to the amplification.
- is the frequency in Hz

### 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:

where

is saturation intensity. To solve, we first rearrange the equation in order to separate the variables, intensity

*I* and position

*z*:

Integrating both sides, we obtain

or

The gain

*G* of the amplifier is defined as the optical intensity

*I* at position

*z* divided by the input intensity:

Substituting this definition into the prior equation, we find the

**general gain equation**:

### Small signal approximation

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

then the general gain equation gives the small signal gain as

or

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

### Large signal asymptotic behavior

For large input signals, where

the gain approaches unity

and the general gain equation approaches a linear

asymptoteIn analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

:

## See also

- Absorption
- Active laser medium
The active laser medium is the source of optical gain within a laser. The gain results from the stimulated emission of electronic or molecular transitions to a lower energy state from a higher energy state...

- Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

(includes a history 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 population inversion in laser media, and the temporal...

- Rabi cycle
In physics, the Rabi cycle is the cyclic behaviour of a two-state quantum system in the presence of an oscillatory driving field. A two-state system has two possible states, and if they are not degenerate energy levels the system can become "excited" when it absorbs a quantum of energy.The effect...

- Spontaneous emission
Spontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon...