Stimulated emissions

It should be realized that absorption and emission as discussed in Chapters 2 tuid 3 are different processes, since the former needs a radiation field and the latter not. Einstein considered the problem of transition rates in the presence of a radiation field [31]. For the transition rate from lower to upper level he wrote w = Bp, where B is the Bin.stein coefficient of (.stimulated) absorption and p is the radiation density. 

The radiation field induces also a transition from the upper to the lower state (stimulated emi.ssion). The rate is w = B p, where B is the Einstein coefficient of stimulated emission. The rate of spontaneous emission is w = A, with A the coefficient of spontaneous emission (note the absence of p in this expression). This is 

If the two levels have an energy separation of, for example, 20000 cm (visible light), the lower level will be predominantly occupied, i.e. stimulated emission can be neglected. This justifies the approach of Chapters 2 and 3. In the case of three or more levels, the situation in which one of the higher levels has a higher occupation than the ground level (population inversion) can, under special conditions, be realized. 

If one of the excited Cr ions decays spontaneously, the emitted photon will stimulate other excited ions to decay by stimulated emission,. so that there is an amplification of the original photon. This is the principle of laser action (laser stands for light amplification by stimulated emission of radiation). The amplification ends when the population inversion is over, since absorption then dominates stimulated emission. 

This book does not intend to deal with laser physics and all the possible types of lasers. TWo remarks are still worth noting  

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The resonance vector analysis has been used to explore all of the questions raised above on the fate of the polyad numbers in larger molecules, the most thoroughly investigated case so far probably being C2FI2- This molecule has been very extensively probed by absorption as well as stimulated emission pumping and dispersed fluorescence teclmiques [, 53, 70 and 71], the experimental spectra have been analysed in... 

Davis M J 1995 Trees from spectra generation, analysis, and energy transfer information Molecular Dynamics and Spectroscopy by Stimulated Emission Pumping ed H-L Dai and R W Field (Singapore World Scientific)... 

A1.6.1.3 ABSORPTION, STIMULATED EMISSION AND SPONTANEOUS EMISSION OF LIGHT... 

For themial light, the iiumber of transitions per second induced by stimulated emission integrated over solid angles, is equal to The total emission, which is the sum of the stimulated and spontaneous emission, may be obtained by letting A A + 1 in the expression for stimulated emission, giving... 

The argument is sometimes given that equation (Al.6,29) implies that the ratio of spontaneous to stimulated emission goes as the cube of the emitted photon frequency. This argument must be used with some care recall that for light at thennal equilibrium, goes as BP, and hence the rate of stimulated emission has a factor... 

The above fomuilae for the absorption spectrum can be applied, with minor modifications, to other one-photon spectroscopies, for example, emission spectroscopy, photoionization spectroscopy and photodetachment spectroscopy (photoionization of a negative ion). For stimulated emission spectroscopy, the factor of fflj is simply replaced by cOg, the stimulated light frequency however, for spontaneous emission... 

Figure Al.6.21. Bra and ket wavepacket dynamics which detennine the coherence overlap, (( ) ( ) ). Vertical arrows mark the transitions between electronic states and horizontal arrows indicate free propagation on the potential surface. Full curves are used for the ket wavepacket, while dashed curves indicate the bra wavepacket. (a) Stimulated emission, (b) Excited state (transient) absorption (from [41]).

Figure Al.6.26. Stereoscopic view of ground- and excited-state potential energy surfaces for a model collinear ABC system with the masses of HHD. The ground-state surface has a minimum, corresponding to the stable ABC molecule. This minimum is separated by saddle points from two distmct exit chaimels, one leading to AB + C the other to A + BC. The object is to use optical excitation and stimulated emission between the two surfaces to steer the wavepacket selectively out of one of the exit chaimels (reprinted from [54]).

Abramson E, Field R W, Imre D, Innes K K and Kinsey J L 1985 Fluorescence and stimulated emission... 

Stock C, Li X, Keller H-M, Schinke R and Temps F 1997 Unimolecular dissociation dynamics of highly vibrationally excited DCO x-A t- I- Investigation of dissociative resonance states by stimulated emission pumping spectroscopy J. Cham. Phys. 106 5333-58... 

Einstein derived the relationship between spontaneous emission rate and the absorption intensity or stimulated emission rate in 1917 using a thennodynamic argument [13]. Both absorption intensity and emission rate depend on the transition moment integral of equation (B 1.1.1). so that gives us a way to relate them. The symbol A is often used for the rate constant for emission it is sometimes called the Einstein A coefficient. For emission in the gas phase from a state to a lower state j we can write... 

The fluorescence signal is linearly proportional to the fraction/of molecules excited. The absorption rate and the stimulated emission rate 1 2 are proportional to the laser power. In the limit of low laser power,/is proportional to the laser power, while this is no longer true at high powers 1 2 <42 j). Care must thus be taken in a laser fluorescence experiment to be sure that one is operating in the linear regime, or that proper account of saturation effects is taken, since transitions with different strengdis reach saturation at different laser powers. 

Berry M T, Loomis R A, Gianoarlo L C and Lester M I 1991 Stimulated emission pumping of intermoleoular vibrations... 

Figure C 1.5.10. Nonnalized fluorescence intensity correlation function for a single terrylene molecule in p-terjDhenyl at 2 K. The solid line is tire tlieoretical curve. Regions of deviation from tire long-time value of unity due to photon antibunching (the finite lifetime of tire excited singlet state), Rabi oscillations (absorjDtion-stimulated emission cycles driven by tire laser field) and photon bunching (dark periods caused by intersystem crossing to tire triplet state) are indicated. Reproduced witli pennission from Plakhotnik et al [66], adapted from [118].

The acronym LASER (Light Amplification via tire Stimulated Emission of Radiation) defines the process of amplification. For all intents and purjDoses tliis metliod was elegantly outlined by Einstein in 1917 [H] wherein he derived a treatment of the dynamic equilibrium of a material in a electromagnetic field absorbing and emitting photons. Key here is tire insight tliat, in addition to absorjDtion and spontaneous emission processes, in an excited system one can stimulate tire emission of a photon by interaction witli tire electromagnetic field. It is tliis stimulated emission process which lays tire conceptual foundation of tire laser. 

The light emitted in the spontaneous recombination process can leave tire semiconductor, be absorbed or cause additional transitions by stimulating electrons in tire CB to make a transition to tire VB. In tliis stimulated recombination process anotlier photon is emitted. The rate of stimulated emission is governed by a detailed balance between absorjDtion, and spontaneous and stimulated emission rates. Stimulated emission occurs when tire probability of a photon causing a transition of an electron from tire CB to VB witli tire emission of anotlier photon is greater tlian that for tire upward transition of an electron from tire VB to tire CB upon absorjDtion of tire photon. These rates are commonly described in tenns of Einstein s H and 5 coefficients [8, 43]. For semiconductors, tliere is a simple condition describing tire carrier density necessary for stimulated emission, or lasing. This carrier density is known as... 

Here, Ri f and Rf i are the rates (per moleeule) of transitions for the i ==> f and f ==> i transitions respeetively. As noted above, these rates are proportional to the intensity of the light souree (i.e., the photon intensity) at the resonant frequeney and to the square of a matrix element eonneeting the respeetive states. This matrix element square is oti fp in the former ease and otf ip in the latter. Beeause the perturbation operator whose matrix elements are ai f and af i is Hermitian (this is true through all orders of perturbation theory and for all terms in the long-wavelength expansion), these two quantities are eomplex eonjugates of one another, and, henee ai fp = af ip, from whieh it follows that Ri f = Rf i. This means that the state-to-state absorption and stimulated emission rate eoeffieients (i.e., the rate per moleeule undergoing the transition) are identieal. This result is referred to as the prineiple of microscopic reversibility. 

If the temperature were raised, more molecules would attain the excited state, but even at 50,000°C there would be only one excited-state atom for every two ground-state atoms, and stimulated emission would not produce a large cascade effect. To reach the excess of stimulated emissions needed to build a large cascade (lasing), the population of excited-state molecules must exceed that of the ground state, preferably at normal ambient temperatures. This situation of an excess of excited-state over ground-state molecules is called a population inversion in order to contrast it with normal ground-state conditions. 

The timing of the emission is clearly dependent on the system in use. For example, if pumping is relatively slow and stimulated emission is fast, then the emergent beam of laser light will appear as a short pulse (subsequent lasing must await sufficient population inversion). This behavior is... 

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