Two-photon absorption tensor

In the present contribution the discussion of the NLO response is restricted to off-resonant case. The only exception is the purely resonant quantity, namely imaginary part of second-order hyperpolarizability in the resonant regime (Im-y(-tt> tu, —w, w)). This quantity describes the process of simultaneous absorption of two quanta. The two-photon absorption (TPA) process is much better understood than the three-photon absorption. The basic quantity associated with the two-photon absorption process is the two-photon absorption tensor (S ). In the most general case referring to two different photons (different polarizations and different energies is given by [75, 81] ... 

The averaging procedure of die two-photon absorption tensor over all orientations of the absorbing molecule leads to [75, 81] ... 

M is the electric dipole transition moment vector T is the two-photon absorption tensor a is the Raman scattering tensor and is a unit vector in the direction of light polarization U. 

Saupe matrix elements T = two-photon absorption tensor U V = photon electric field vector direction (polarization) x, y, = molecular orientation axes x y, z - arbitrary molecular axes X, Y, Z = laboratory axes a, P, / = Euler angles ... 

Other commonly measured x interactions are those occurring with degenerate frequencies. Such interactions include phenomena such as four-wave-mixing, self-focusing, two-photon absorption, and CARS. If a single laser beam is focused into a material, the three applied fields interacting through the nonlinear tensor are coincident, and the interpretation is very similar to that for the... 

Our present focus is on correlated electronic structure methods for describing molecular systems interacting with a structured environment where the electronic wavefunction for the molecule is given by a multiconfigurational self-consistent field wavefunction. Using the MCSCF structured environment response method it is possible to determine molecular properties such as (i) frequency-dependent polarizabilities, (ii) excitation and deexcitation energies, (iii) transition moments, (iv) two-photon matrix elements, (v) frequency-dependent first hyperpolarizability tensors, (vi) frequency-dependent polarizabilities of excited states, (vii) frequency-dependent second hyperpolarizabilities (y), (viii) three-photon absorptions, and (ix) two-photon absorption between excited states. 

Based on the MCSCF/CM quadratic response method it is possible to calculate the hyperpolarizability tensor and the two-photon absorption cross-sections. The calculated MCSCF/CM properties exhibit for all the individual tensor components substantial shifts compared with the corresponding molecular properties of the molecule in vacuum. 

This molecular response tensor is a more general form of three apparently different tensors which have featured in previous work on multiphoton processes one is the Xy2 tensor arising in the single-frequency distributive two-photon absorption (Andrews and Harlow 1984a), and another is the tensor which appears in the theory of three-photon absorption (Andrews and Wilkes 1985). It is also exactly identical to the two-frequency hyper-Raman transition tensor (Andrews 1984). 

In the two-frequency distributive case, the molecular tensor y °(a)i,o)2) has resonance conditions similar to those for )- As the single-beam case, two of the proposed resonance conditions would be likely to allow the process to be masked by single-photon absorption a third leads to the possibility of conventional two-photon absorption, and a fourth cannot be satisfied if the centers involved are initially in their ground states. The remaining condition E x E — hu>i) if a>i t02, or E x(E — ho)2) if CO2 < ft)i) remains the only truly useful resonance. Naturally, since the energetics of the excitation process are constrained only by a condition on the sum of the photon frequencies, there is a wide scope for choosing laser frequencies specifically with the aim of exploit ng this type of resonance possibility. 

Note that for H(uj + uj ) = Eb it is necessary to take the damping of the biphonon into account in the expression for A(w + u/,0). In this case, along with the real part of the tensor Xije u>,uj), an imaginary part is also present. This corresponds, as is well known, to the occurrence of two-photon absorption that is accompanied, in the given case, by the excitation of a biphonon. As applied to excitons this question has been discussed by Hanamura (51) within the framework of a somewhat different approach. We refer to it here (see also Fly-tzanis (52)) because both the generation of a second harmonic and two-photon absorption are processes that are completely described by the nonlinear polarizability of the crystal found above with bound states taken into account. Actually, these processes can be investigated by a single method. 

The two-photon tensor invariants proportional to two-photon absorptivities have been calculated for the transitions to the lowest singlet states, (BjJ and La+ (B J60 and for the prediction of the site of bromination, formylation, hydroxymethylation and nitration ofimida-zo[l,2-6][l,2,4]triazines.61... 

The CNDO/2 method has been used for the calculation of the sites of the protonation and alkylation of 1,2,4-triazines <88KGS525>. It was also applied to calculate the two-photon tensor invariants proportional to two-photon absorptivities for transitions to the lowest singlet states, Lb (B2 -) and L/ (B,/) <84CPL(107)125>. 

Figure 11.8 Variation in the absorption ratio Q and the initial anisotropy / (0) with the ratio 5 = Syf/Sxx for a diagonal planar two-photon transition tensor. The dashed line atS = 0 intersects the curves at ST = 2/3 and R p) =...

Two-Photon Processes To evaluate two-photon absorption and circular dichroism intensities, it is possible to obtain mathematical expressions similar to those of one-photon processes, assuming that the dependence of the transition tensors on the frequency is small in the energy window of interest of the spectrum. In that case, for TPA, we can fix Sejfir, ri, co) inEq. 8.23 to its value at the vertical excitation energy, a>v, and introduce FC and HT contributions through file usual linear expansion in normal coordinates [24],... 

The scattering cross section depends on the matrix element (8.9) of the polarizability tensor and contains furthermore the frequency dependence derived from the classical theory of light scattering. One obtains [8.15] analogously to the two-photon absorption cross section (Sect.7.4) ... 

Simplification occurs if the molecule has high enough symmetry to dictate the positions of the x, y, z axes. Then, only two orientation factors K and three L are independent, the tensors (i )puvuv have 21 nonzero elements of the types [((4)puvuv)] and [<(4)puvuv)] nd the quadruple sum in Equation [12] is reduced to three double sums. The two-photon absorption cross-section is proportional to... 

It is evident from the form of Equation [17] that the two-photon tensor and hence the extent of two-photon absorption, is significantly enhanced if the molecule possesses an electronic excited state of an energy close to E + fuo, for then, in the sum over r, the term that state contributes has an absolute minimum value for the denominator. This situation, typical of multiphoton processes, is known as resonance enhancement, and its physical basis can be understood in terms of the time-energy uncertainty principle ... 

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