Stimulated emission 28469 213973252 2008-05-21T16:16:10Z JAnDbot 1725149 robot Modifying: [[de:Stimulierte Emission]] 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. The perturbing photon is ''not'' destroyed in the process (cf. [[absorption (optics)|absorption]]), and the second photon is created with the same [[phase (waves)|phase]], [[frequency]], [[polarization]], and [[direction]] of travel as the original. '''Stimulated emission''' is really a [[quantum mechanics|quantum mechanical]] phenomenon but it can be understood in terms of a "classical" [[electromagnetic field|field]] and a quantum mechanical [[atom]]. The process can be thought of as "optical [[amplification]]" and it forms the basis of both the [[laser]] and [[maser]]. ==Overview== [[Electron]]s and how they interact with each other and [[electromagnetic field]]s form the basis for most of our understanding of [[chemistry]] and [[physics]]. Electrons have energy in proportion to how far they are on average from the [[atomic nucleus|nucleus]] of an [[atom]]; however quantum mechanical effects force electrons to take on quantized positions in orbitals. Thus, electrons are found in specific energy levels of an atom, as shown below: <center>[[image:Stimulated_Emission.svg|550px]]</center><br> <!--<center>[[image:stimulatedemission.png]]</center><br>--> <center><!-- Unsourced image removed: [[Image:Laser4.gif]] --></center> 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 1[[electron configuration|s orbital]]. When electrons absorb energy either from [[light]] (photons) or from [[heat]] ([[phonon]]s), they move farther away from the [[Atomic nucleus|atomic nuclei]] but they are only allowed to absorb energy that will land them into specific [[energy level]]s. This leads to [[emission line]]s and [[Spectral line|absorption line]]s. When an electron is [[Excited state|excited]], it will not stay that way forever. On average there is a [[mean lifetime|lifetime]] for any particular [[energy level]] after which half of the electrons initially in that state will have [[Radioactive decay|decay]]ed into a lower state. 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]]." 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. This is also called [[fluorescence]]. 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. 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. 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 ''E''<sub>1</sub> and ''E''<sub>2</sub> 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. The photon will have [[frequency]] ν and energy ''h''ν, given by: :<math>E_2 - E_1 = h \nu</math>, where ''h'' is [[Physical constant|Planck's constant]]. 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: :<math>\frac{\partial N}{\partial t} = - B_{21} \rho (\nu) N </math>, where ''B''<sub>21</sub> is a [[proportionality constant]] for this particular transition in this particular atom (referred to as an ''[[Atomic spectral line#The Einstein coefficients|Einstein B coefficient]]''), and ρ(ν) is the radiation density of photons of frequency ν. 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 [[coherence (physics)|coherent]]. 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 [[spectrum]] of the photons, then, will not be an infinitesimally thin line, but will be a distribution. 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 [[Atomic spectral line|spectral line shape function]] as a [[Cauchy distribution|Lorentzian distribution]]: :<math> g(\nu) = {1 \over \pi } { (\Gamma / 2) \over (\nu - \nu_0)^2 + (\Gamma /2 )^2 }</math> where :<math> \Gamma \, </math> is the [[full width at half maximum]], or FWHM, in [[hertz]]. This model is generally valid as long as :<math> |\nu - \nu_0| << \nu_0 \, </math> and :<math>\Gamma << \nu_0 \, </math> The line shape function, regardless of the form that it takes, must satisfy the normalization condition of any probability distribution: :<math>\int_{-\infty}^{\infty} g(\nu) \cdot d \nu = 1</math> which the Lorentzian satisfies. The peak value of the Lorentzian line shape occurs at the line center: :<math> g(\nu = \nu_0) = {2 \over \pi \Gamma}</math> It is also convenient to define the '''normalized line shape function''': :<math>\bar{g}(\nu) = { g(\nu) \over g(\nu_0) } = { (\Gamma / 2)^2 \over (\nu - \nu_0)^2 + (\Gamma /2 )^2 } </math> which is dimensionless, and which has a peak value, also at the line center, of :<math>\bar{g}(\nu = \nu_0) = 1</math> ==Stimulated emission cross section== The stimulated emission cross section (in [[square meter]]s) is :<math>\sigma_{21}(\nu) = A_{21} { \lambda^2 \over 8 \pi n^2} g(\nu)</math> where :''A''<sub>21</sub> 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). ==Optical amplification== Under certain conditions, stimulated emission can provide a physical mechanism for [[optical amplifier|optical amplification]]. An external source of energy stimulates atoms in the ground state to transition to the excited state, creating what is called a [[population inversion]]. 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 [[irradiance|intensity]]. The population inversion, in units of atoms per [[cubic meter]], is :<math>\Delta N_{21} = \left( N_2 - {g_2 \over g_1} N_1 \right)</math> where ''g''<sub>1</sub> and ''g''<sub>2</sub> are the [[degenerate energy level|degeneracies]] of energy levels 1 and 2, respectively. ===Small signal gain equation=== The intensity (in [[watt]]s per [[square meter]]) of the stimulated emission is governed by the following differential equation: :<math>{ dI \over dz} = \sigma_{21}(\nu) \cdot \Delta N_{21} \cdot I(z) </math> 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 :<math>{ dI \over dz} = \gamma_0(\nu) \cdot I(z) </math> where :<math> \gamma_0(\nu) = \sigma_{21}(\nu) \cdot \Delta N_{21} </math> is the ''small-signal gain coefficient'' (in units of radians per meter). We can solve the differential equation using [[separation of variables]]: :<math>{ dI \over I(z)} = \gamma_0(\nu) \cdot dz </math> Integrating, we find: :<math>\ln \left( {I(z) \over I_{in}} \right) = \gamma_0(\nu) \cdot z </math> or :<math> I(z) = I_{in}e^{\gamma_0(\nu) z} </math> where :<math> I_{in} = I(z=0) \, </math> is the optical intensity of the input signal (in watts per square meter). ===Saturation intensity=== The saturation intensity ''I''<sub>S</sub> 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 :<math>I_S = {h \nu \over \sigma(\nu) \cdot \tau_S }</math> where :''h'' is [[Planck's constant]], and :τ<sub>S</sub> is the saturation time constant, which depends on the spontaneous emission lifetimes of the various transitions between the energy levels related to the amplification. ===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]]: :<math>{ dI \over dz} = { \gamma_0(\nu) \over 1 + \bar{g}(\nu) { I(z) \over I_S } } \cdot I(z) </math> where <math>I_S</math> is intensity. To solve, we first rearrange the equation in order to separate the variables, intensity ''I'' and position ''z'': :<math>{ dI \over I(z)} \left[ 1 + \bar{g}(\nu) { I(z) \over I_S } \right] = \gamma_0(\nu)\cdot dz </math> Integrating both sides, we obtain :<math>\ln \left( { I(z) \over I_{in} } \right) + \bar{g}(\nu) { I(z) - I_{in} \over I_S} = \gamma_0(\nu) \cdot z</math> or :<math>\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</math> The gain ''G'' of the amplifier is defined as the optical intensity ''I'' at position ''z'' divided by the input intensity: :<math>G = G(z) = { I(z) \over I_{in} } </math> Substituting this definition into the prior equation, we find the '''general gain equation''': :<math>\ln \left( G \right) + \bar{g}(\nu) { I_{in} \over I_S } \left( G - 1 \right) = \gamma_0(\nu) \cdot z</math> ===Small signal approximation=== In the special case where the input signal is small compared to the saturation intensity, in other words, :<math>I_{in} << I_S \, </math> then the general gain equation gives the small signal gain as :<math> \ln(G) = \ln(G_0) = \gamma_0(\nu) \cdot z</math> or :<math> G = G_0 = e^{\gamma_0(\nu) z}</math> which is identical to the small signal gain equation (see above). ===Large signal asymptotic behavior=== For large input signals, where :<math>I_{in} >> I_S \, </math> the gain approaches unity :<math>G \rightarrow 1 </math> and the general gain equation approaches a linear [[asymptote]]: :<math>I(z) = I_{in} + { \gamma_0(\nu) \cdot z \over \bar{g}(\nu) } I_S</math> ==References== *{{cite book | title = Fundamentals of Photonics | author = Saleh, Bahaa E. A. and Teich, Malvin Carl | publisher = John Wiley & Sons | location = New York | year = 1991 | id= ISBN 0-471-83965-5 }} ==See also== *[[Absorption (optics)|Absorption]] *[[Spontaneous emission]] *[[Active laser medium]] *[[Laser science]] *[[Rabi cycle]] [[Category: Electromagnetic radiation]] [[Category:Fundamental physics concepts]] [[Category:Laser science]] [[ca:Emissió estimulada]] [[cs:Stimulovaná emise]] [[de:Stimulierte Emission]] [[fr:Émission stimulée]] [[ko:자극 방출]] [[it:Emissione stimolata]] [[he:פליטה מאולצת]] [[nl:Gestimuleerde emissie]] [[ja:誘導放出]] [[pl:Emisja wymuszona]] [[ru:Вынужденное излучение]] [[uk:Вимушене випромінювання]]