The Absorption Probability

We know from Chapter 1 that the probability P,f of indncing an optical transition from a state i to a state / is proportional to (1 //1), where in the matrix element Ip, and P f denote the eigenfnnctions of the ground and excited states, respectively, and H is the interaction Hamiltonian between the incoming light and the system (i.e., the valence electrons of the center). In general, we can assnme that // is a sinnsoidal 

The next step is to apply basic time-dependent perturbation theory to our simple two-level center that is subjected to this time-varying interaction. After solving this basic problem (Svelto, 1986), the transition probability P,/ is given by 

If the transition is of an electric dipole nature, the interaction Hamiltonian can be written as // = p E, where p is the electric dipole moment and E is the electric field of the radiation. The electric dipole moment is given by p =, where r is the 

Now we assume that the wavelength of the electromagnetic wave is much larger than the atomic dimensions. This is, of course, true for the optical range, as the shortest wavelength is around 200 nm while the atomic dimensions are of the order of 0.1 nm. In this case, the electric field does not vary within the atomic volume and so E = E (0, t) = Eo sin ojt. Therefore, we can write 

Assuming now that the incident wave interacts with centers whose vectors are randomly oriented with respect to Eo, we can average Equation (5.13) over all possible orientations. Taking into account that (cos 6 ) = 1/3 (considering that all 

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The strongest contribution to the projected mean square displacement (ku)) and therefore to the absorption probability S(E) originates from C-Fe-C and N-Fe-C bending modes (8Ai, 22E, 23E, and 24E in Table 9.2). However, the energy range of these modes (8-15 meV) strongly overlaps with that of the acoustic modes (with composition factor = 0.17, = 337, wtpe = 57) and therefore... 

Given randomly oriented dipoles, the absorption probability rate is proportional to the intensity 7, = E,tJ 2. At z = 0, the evanescent intensities are... 

By a careful inspection of Figure 4.17, we see how further transitions between bands below and above the Fermi level can also occur at energies higher than 1.5 eV. However, as these bands are not parallel, the density of states at these energies is lower than at 1.5 eV. In any case, the absorption probability is still significant, and it acconnts for the experimentally observed redaction in the reflectivity of Al in respect to the predictions from the Drude model (see Figure 4.5). 

Expression (5.14) shows that the absorption probability depends on both the incoming light intensity and the matrix element It is easy to see that /Lt,j = l/Lt = /x and so we can conclude that the absorption probability between two defined energy levels i and / is equal to the stimulated emission probability between levels / and i ... 

In the spirit of the adiabatic approximation, the transitions between two vibrational states (belonging to initial and final electronic states) must occur so rapidly that there is no change in the configurational coordinate Q. This is known as the Frank Condon principle and it implies that the transitions between i and / states can be represented by vertical arrows, as shown in Figure 5.12. Let us now assume our system to be at absolute zero temperature (0 K), so that only the phonon level = 0 is populated and all the absorption transitions depart from this phonon ground level to different phonon levels m = 0, 1, 2,... of the excited state. Taking into account Equation (5.25), the absorption probability from the = 0 state to an m state varies as follows ... 

The absorption probability is proportional to the scalar product p = IM I 1 I cos ip, where p is the transition dipole moment, C the electric vector of the exciting light wave and

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Two-photon absorption occurs when the energy of a molecular transition matches the combined energy of two photons. Quantum mechanically, the absorption probability is proportional to the two-photon transition moment from the ground state, g, to the excited state, n, via intermediate state, m, and can be expressed as follows (Boyd 1992 Abe 2001) ... 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

If, in centrosymmetric molecules, states to which a transition is forbidden in the normal absorption spectrum can make important contributions to x(3)> this suggests a strategy for enhancing the figure of merit, x(3)/a> of such a nonlinear material. Chemically introducing low lying states with gerade symmetry and thus small or zero absorption cross sections has the potential to enhance the x(3) but not increase the absorption probability, a. 

Symmetry considerations show that chemical substitution to introduce low lying, one photon forbidden states into the molecule has the potential to enhance the x(3) without increasing the absorption probability, a. 

Discussion (a) For high principal quantum numbers (n > 7) ionization is much faster than quenching (see Table I), and so the details of the possible quenching processes are unimportant. Essentially all of the atoms excited to a given nd state are ionized, and the observed signal is simply proportional to the absorption probability of the second step (12) ... 

Let us analyze how to find the excited state multipole moments bPq-As explained in the previous paragraph, at excitation by weak light the probability density pb(0, state angular momentum distribution is proportional to the absorption probability G(0,multipole moments, Pq of an excited level b can be found as... 

Using the absorption probability coefficients G(0,p), as described by Eq. (2.8), one may easily obtain the spatial distribution of the angular momenta 3a(0,p) of the ground (initial) state molecules. 

Here the proportionality factor fp ( J d J") 2/(2J +l) coincides with the dynamic part of the absorption probability introduced in Chapter 2. The sum in (5.10) presents the angular part of the absorption probability, whilst the factor (rmm + 0 %M ) 1 describes the effect upon /mm both on the part of perturbation by the external field, causing splitting of the magnetic sublevels j ujmm i and by spontaneous decay and collisions (anisotropic in the general case), together described by a set of relaxation rates Tmm Applying similar manipulations as for (5.7) to Eq. (5.6), we obtain an equation for fluorescence intensity ... 

If the absorption rate of the critical oligomers is high (spontaneous), the absorption probability for radicals stemming fh>m the initiator is... 

A light quantum of appropriate energy can be absorbed by a molecule fixed in space only if the light electric field vector has a component parallel to the molecular transition moment. If the directions of the transition moment and of the electric field vector form an angle q), the absorption probability is proportional to cos tp. (Cf. Section 1.3.5.) The light quanta of luminescence are also polarized, with the intensity again proportional to cos (p. 

Fluorescent emission will be proportional to the absorption probability and molecular parameters characterizing the fluorophore complex. Each absorbed photon can reemit energy within the emission spectrum of the dye. The fluorescent... 

The absorption probability increases if a two-photon resonance can be found. One example is illustrated in Fig. 2.42a, where the 4D level of the Na atom is excited by two-photons of a dye laser at A. = 578.7 nm and further excitation by a third photon reaching high-lying Rydberg levels nP or nF with the electronic orbital... 

Which fraction of H atoms in the i S /2 ground state that can be excited by a Doppler-free two-photon transition into the 2 5 i/2 state in a collimated H atomic beam with T = 10 m/s, when a laser with 7 = 10 W/cm and a rectangular beam cross section of 1 x 1 mm crosses the atomic beam perpendicularly and the absorption probability is P// = (ao 7) /(y hv) where ao = 10" cm and y is the linewidth. 

If the collimated molecular beam is crossed perpendicularly with a monochromatic laser beam with frequency ty propagating into the x-direction, the absorption probability for each molecule depends on its velocity component Vx. In Vol. 1, Sect. 3.2 it was shown that the center frequency of a molecular transition, which is coo in the rest frame of the moving molecule, is Doppler shifted to a frequency >q according to... 

From measurements of the absorption probability of transitions from very high levels of the ground state the upper state potential and the dependence of the transition moment on the inter-nuclear distance R can be obtained with high precision. 

V sinO < y/k, the molecule after the collision is still in resonance with the standing light wave inside the laser resonator. Such soft collisions with deflection angles 0 < therefore do not appreciably change the absorption probability of a molecule. Because of their statistical phase jumps (Vol. 1, Sect. 3.3) they do, however, contribute to the linewidth. The line profile of the Lamb dip broadened by soft collisions remains Lorentzian. 

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