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Stimulated emission

From Wikipedia, the free encyclopedia
Release of a photon triggered by another
Laser light is a type of stimulated emission of radiation.

Stimulated emission is the process by which an incomingphoton of a specific frequency can interact with an excited atomicelectron (or other excited molecular state), causing it to drop to a lowerenergy level. The liberated energy transfers to the electromagnetic field, creating a newphoton with afrequency,polarization, anddirection of travel that are all identical to the photons of the incident wave. This is in contrast tospontaneous 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 theAmerican Physical Society, the first person to correctly predict the phenomenon of stimulated emission wasAlbert Einstein in a series of papers starting in 1916, culminating in what is now called theEinstein B Coefficient. Einstein's work became the theoretical foundation of themaser and thelaser.[1][2][3][4] The process is identical in form to atomicabsorption 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 atthermal equilibrium, absorption exceeds stimulated emission because there are more electrons in the lower energy states than in the higher energy states. However, when apopulation inversion is present, the rate of stimulated emission exceeds that of absorption, and a netoptical amplification can be achieved. Such again medium, along with an optical resonator, is at the heart of a laser or maser.Lacking a feedback mechanism,laser amplifiers andsuperluminescent sources also function on the basis of stimulated emission.

Overview

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Electrons and their interactions withelectromagnetic fields are important in our understanding ofchemistry andphysics.In theclassical view, the energy of an electron orbiting an atomic nucleus is larger for orbits further from thenucleus of anatom. However, quantum mechanical effects force electrons to take on discrete positions inorbitals. Thus, electrons are found in specific energy levels of an atom, two of which are shown below:

When an electron absorbs energy either fromlight (photons) orheat (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 toemission lines andabsorption lines.

When an electron isexcited 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 inradiation which has a narrow spectrum (centered around onewavelength of light), but the individual photons would have no common phase relationship and would also emanate in random directions. This is the mechanism offluorescence andthermal emission.

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 electricdipole, and this dipole oscillates at a characteristic frequency. In response to the externalelectric 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

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Stimulated emission was a theoretical discovery byAlbert Einstein within the framework of theold quantum theory, wherein the emission is described in terms of photons that are the quanta of the EM field.[5][6] Stimulated emission can also occur in classical models, without reference to photons or quantum-mechanics.[7][non-primary source needed] (See alsoLaser § History.) According to physics professor and director of the MIT-Harvard Center for Ultracold AtomsDaniel Kleppner, Einstein's theory of radiation was ahead of its time and prefigures the modern theory ofquantum electrodynamics and quantum optics by several decades.[8]

Mathematical model

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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 anexcited state (2), with energiesE1 andE2 respectively.

If the atom is in the excited state, it may decay into the lower state by the process ofspontaneous emission, releasing the difference in energies between the two states as a photon. The photon will havefrequencyν0 and energyhν0, given by:E2−E1=hν0{\displaystyle E_{2}-E_{1}=h\,\nu _{0}}whereh is thePlanck constant.

Alternatively, if the excited-state atom is perturbed by an electric field of frequencyν0, 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 byN2, the rate at which stimulated emission occurs is given by∂N2∂t=−∂N1∂t=−B21ρ(ν)N2{\displaystyle {\frac {\partial N_{2}}{\partial t}}=-{\frac {\partial N_{1}}{\partial t}}=-B_{21}\,\rho (\nu )\,N_{2}}where theproportionality constantB21 is known as theEinstein B coefficient for that particular transition, andρ(ν) is the radiation density of the incident field at frequencyν. The rate of emission is thus proportional to the number of atoms in the excited stateN2, and to the density of incident photons.

At the same time, there will be a process of atomic absorption whichremoves energy from the field while raising electrons from the lower state to the upper state. Its rate is precisely the negative of the stimulated emission rate,∂N2∂t=−∂N1∂t=B12ρ(ν)N1.{\displaystyle {\frac {\partial N_{2}}{\partial t}}=-{\frac {\partial N_{1}}{\partial t}}=B_{12}\,\rho (\nu )\,N_{1}.}

The rate of absorption is thus proportional to the number of atoms in the lower state,N1. The B coefficients can be calculated using dipole approximation and time dependent perturbation theory in quantum mechanics as:[9][10]Bab=e26ϵ0ℏ2|⟨a|r→|b⟩|2{\displaystyle B_{ab}={\frac {e^{2}}{6\epsilon _{0}\hbar ^{2}}}|\langle a|{\vec {r}}|b\rangle |^{2}}whereB corresponds to energy distribution in terms of frequencyν. The B coefficient may vary based on choice of energy distribution function used, however, the product of energy distribution function and its respectiveB coefficient remains same.

Einstein showed from the form of Planck's law,[citation needed] that the coefficient for this transition must be identical to that for stimulated emission:B12=B21.{\displaystyle B_{12}=B_{21}.}

Thus absorption and stimulated emission are reverse processes proceeding at somewhat different rates. Another way of viewing this is to look at thenet stimulated emission or absorption viewing it as a single process. The net rate of transitions fromE2 toE1 due to this combined process can be found by adding their respective rates, given above:∂N1net∂t=−∂N2net∂t=B21ρ(ν)(N2−N1)=B21ρ(ν)ΔN.{\displaystyle {\frac {\partial N_{1}^{\text{net}}}{\partial t}}=-{\frac {\partial N_{2}^{\text{net}}}{\partial t}}=B_{21}\,\rho (\nu )\,(N_{2}-N_{1})=B_{21}\,\rho (\nu )\,\Delta N.}

Thus a net power is released into the electric field equal to the photon energyhν 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:ΔN>0{\displaystyle \Delta N>0}. Otherwise there is net absorption and the power of the wave is reduced during passage through the medium. The special conditionN2>N1{\displaystyle N_{2}>N_{1}} is known as apopulation inversion, a rather unusual condition that must be effected in thegain 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 mutuallycoherent. When a population inversion (ΔN>0{\displaystyle \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 frequencyν0{\displaystyle \nu _{0}} corresponding to the energy of the transition. At frequencies offset fromν0{\displaystyle \nu _{0}} the strength of stimulated (or spontaneous) emission will be decreased according to the so-calledline shape.Considering onlyhomogeneous broadening affecting an atomic or molecular resonance, thespectral line shape function is described as aLorentzian distributiong′(ν)=1π(Γ/2)(ν−ν0)2+(Γ/2)2{\displaystyle g'(\nu )={1 \over \pi }{(\Gamma /2) \over (\nu -\nu _{0})^{2}+(\Gamma /2)^{2}}}whereΓ{\displaystyle \Gamma } is thefull width at half maximum or FWHM bandwidth.

The peak value of the Lorentzian line shape occurs at the line center,ν=ν0{\displaystyle \nu =\nu _{0}}. A line shape function can be normalized so that its value atν0{\displaystyle \nu _{0}} is unity; in the case of a Lorentzian we obtaing(ν)=g′(ν)g′(ν0)=(Γ/2)2(ν−ν0)2+(Γ/2)2.{\displaystyle g(\nu )={g'(\nu ) \over g'(\nu _{0})}={(\Gamma /2)^{2} \over (\nu -\nu _{0})^{2}+(\Gamma /2)^{2}}.}

Thus stimulated emission at frequencies away fromν0{\displaystyle \nu _{0}} is reduced by this factor. In practice there may also be broadening of the line shape due toinhomogeneous broadening, most notably due to theDoppler effect resulting from the distribution of velocities in a gas at a certain temperature. This has aGaussian shape and reduces the peak strength of the line shape function. In a practical problem the full line shape function can be computed through aconvolution of the individual line shape functions involved. Therefore, optical amplification will add power to an incident optical field at frequencyν{\displaystyle \nu } at a rate given byP=hνg(ν)B21ρ(ν)ΔN.{\displaystyle P=h\nu \,g(\nu )\,B_{21}\,\rho (\nu )\,\Delta N.}

Stimulated emission cross section

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The stimulated emission cross section isσ21(ν)=A21λ28πn2g′(ν){\displaystyle \sigma _{21}(\nu )=A_{21}{\frac {\lambda ^{2}}{8\pi n^{2}}}g'(\nu )}where

Optical amplification

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Stimulated emission can provide a physical mechanism foroptical 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 apopulation inversion 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 inputintensity results.

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

ΔN21=N2−g2g1N1{\displaystyle \Delta N_{21}=N_{2}-{g_{2} \over g_{1}}N_{1}}

whereg1 andg2 are thedegeneracies of energy levels 1 and 2, respectively.

Small signal gain equation

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The intensity (inwatts per square metre) of the stimulated emission is governed by the following differential equation:

dIdz=σ21(ν)⋅ΔN21⋅I(z){\displaystyle {dI \over dz}=\sigma _{21}(\nu )\cdot \Delta N_{21}\cdot I(z)}

as long as the intensityI(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

dIdz=γ0(ν)⋅I(z){\displaystyle {dI \over dz}=\gamma _{0}(\nu )\cdot I(z)}

where

γ0(ν)=σ21(ν)⋅ΔN21{\displaystyle \gamma _{0}(\nu )=\sigma _{21}(\nu )\cdot \Delta N_{21}}

is thesmall-signal gain coefficient (in units of radians per metre). We can solve the differential equation usingseparation of variables:

dII(z)=γ0(ν)⋅dz{\displaystyle {dI \over I(z)}=\gamma _{0}(\nu )\cdot dz}

Integrating, we find:

ln⁡(I(z)Iin)=γ0(ν)⋅z{\displaystyle \ln \left({I(z) \over I_{in}}\right)=\gamma _{0}(\nu )\cdot z}

or

I(z)=Iineγ0(ν)z{\displaystyle I(z)=I_{in}e^{\gamma _{0}(\nu )z}}

where

Iin=I(z=0){\displaystyle I_{in}=I(z=0)\,} is the optical intensity of the input signal (in watts per square metre).

Saturation intensity

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The saturation intensityIS 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

IS=hνσ(ν)⋅τS{\displaystyle I_{S}={h\nu \over \sigma (\nu )\cdot \tau _{S}}}

where

h{\displaystyle h} is thePlanck constant, and
τS{\displaystyle \tau _{\text{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.
ν{\displaystyle \nu } is the frequency in Hz

The minimum value ofIS(ν){\displaystyle I_{\text{S}}(\nu )} occurs on resonance,[11] where the cross sectionσ(ν){\displaystyle \sigma (\nu )} is the largest. This minimum value is:

Isat=π3hcλ3τS{\displaystyle I_{\text{sat}}={\frac {\pi }{3}}{hc \over \lambda ^{3}\tau _{S}}}

For a simple two-level atom with a natural linewidthΓ{\displaystyle \Gamma }, the saturation time constantτS=Γ−1{\displaystyle \tau _{\text{S}}=\Gamma ^{-1}}.

General gain equation

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The general form of the gain equation, which applies regardless of the input intensity, derives from the general differential equation for the intensityI as a function of positionz in thegain medium:

dIdz=γ0(ν)1+g¯(ν)I(z)IS⋅I(z){\displaystyle {dI \over dz}={\gamma _{0}(\nu ) \over 1+{\bar {g}}(\nu ){I(z) \over I_{S}}}\cdot I(z)}

whereIS{\displaystyle I_{S}} is saturation intensity. To solve, we first rearrange the equation in order to separate the variables, intensityI and positionz:

dII(z)[1+g¯(ν)I(z)IS]=γ0(ν)⋅dz{\displaystyle {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⁡(I(z)Iin)+g¯(ν)I(z)−IinIS=γ0(ν)⋅z{\displaystyle \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⁡(I(z)Iin)+g¯(ν)IinIS(I(z)Iin−1)=γ0(ν)⋅z{\displaystyle \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 gainG of the amplifier is defined as the optical intensityI at positionz divided by the input intensity:

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

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

ln⁡(G)+g¯(ν)IinIS(G−1)=γ0(ν)⋅z{\displaystyle \ln \left(G\right)+{\bar {g}}(\nu ){I_{in} \over I_{S}}\left(G-1\right)=\gamma _{0}(\nu )\cdot z}

Small signal approximation

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

Iin≪IS{\displaystyle I_{in}\ll I_{S}\,}

then the general gain equation gives the small signal gain as

ln⁡(G)=ln⁡(G0)=γ0(ν)⋅z{\displaystyle \ln(G)=\ln(G_{0})=\gamma _{0}(\nu )\cdot z}

or

G=G0=eγ0(ν)z{\displaystyle G=G_{0}=e^{\gamma _{0}(\nu )z}}

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

Large signal asymptotic behaviour

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For large input signals, where

Iin≫IS{\displaystyle I_{in}\gg I_{S}\,}

the gain approaches unity

G→1{\displaystyle G\rightarrow 1}

and the general gain equation approaches a linearasymptote:

I(z)=Iin+γ0(ν)⋅zg¯(ν)IS{\displaystyle I(z)=I_{in}+{\gamma _{0}(\nu )\cdot z \over {\bar {g}}(\nu )}I_{S}}

See also

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References

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  1. ^Tretkoff, Ernie (August 2005)."This Month in Physics History: Einstein Predicts Stimulated Emission".American Physical Society News.14 (8). Retrieved1 June 2022.
  2. ^Straumann, Norbert (23 Mar 2017). "Einstein in 1916: "On the Quantum Theory of Radiation"".arXiv:1703.08176 [physics.hist-ph].
  3. ^Hecht, Jeff (15 Aug 2021)."Laser".Encyclopedia Britannica. Retrieved1 June 2022.
  4. ^Stone, A. Douglas (6 October 2013).Einstein and the Quantum: The Quest of the Valiant Swabian (First ed.). Princeton University Press.ISBN 978-0691139685. Retrieved1 June 2022.
  5. ^Einstein, A (1916). "Strahlungs-emission und -absorption nach der Quantentheorie".Verhandlungen der Deutschen Physikalischen Gesellschaft.18:318–323.Bibcode:1916DPhyG..18..318E.
  6. ^Einstein, A (1917). "Zur Quantentheorie der Strahlung".Physikalische Zeitschrift.18:121–128.Bibcode:1917PhyZ...18..121E.
  7. ^Fain, B.;Milonni, P. W. (1987). "Classical stimulated emission".Journal of the Optical Society of America B.4 (1): 78.Bibcode:1987JOSAB...4...78F.doi:10.1364/JOSAB.4.000078.
  8. ^Kleppner, Daniel (1 February 2005)."Rereading Einstein on Radiation".Physics Today.58 (2):30–33.Bibcode:2005PhT....58b..30K.doi:10.1063/1.1897520. Retrieved1 June 2022.
  9. ^Hilborn, Robert (2002)."Einstein coefficients, cross sections, f values, dipole moments, and all that"(PDF).
  10. ^Segre, Carlo."The Einstein coefficients - Fundamentals of Quantum Theory II (PHYS 406)"(PDF). p. 32.
  11. ^Foot, C. J. (2005).Atomic physics. Oxford University Press. p. 142.ISBN 978-0-19-850695-9.
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