First-Order Optical Polarization

In the last chapter, we used a steady-state treatment to relate the shape of an absorption band to the dynamics of relaxations in the excited state. Because a period on the order of the electronic dephasing time will be required to establish a steady state, Eqs. (10.43) and (10.44) apply only on time scales longer than this. We need to escape this limitation if we hope to explore the relaxation dynamics themselves. Our first goal in this chapter is to develop a more general approach for analyzing spectroscopic experiments on femtosecond and picosecond time scales. This provides a platform for discussing how pump-probe and photon-echo experiments can be used to probe the dynamics of structural flucmations and the transfer of energy or electrons on these short time scales. 

To start, consider an ensemble of systems, each of which has two states (m and n) with energies and E , respectively. In the presence of a weak radiation field E(t) = Eo[exp(icot) + exp(—ia)t)], the Hamiltonian matrix element H ,) can be written// ,( ) = // -f where is the matrix element in the absence 

We can simplify matters if we adjust aU the elements of p by subtracting the Boltzmann-equilibrium values in the absence of the radiation field (0 for off-diagonal elements, andp as prescribed by Eq. (10.26) for diagonal elements). The rate constant for stochastic relaxations of the adjusted density matrix element then can be written as where Yrm. = i iotn = m and 1/T2 for n w. To simplify the notation further, let E /ti = 

11 Pump-Probe Spectroscopy, Photon Echoes and Vibrational Wavepackets 

With these definitions and adjustments, the stochastic Liouville equation (Eq. 10.30) for the ensemble of two-state systems in the presence of the radiation field takes the form 

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Fig. 11.1 Pathways in Liouville space. The circles labeled a,a and b,b represent the diagonal elements of the density matrix (populations) for a two-state system those labeled a,b and b,a represent off-diagonal elements (coherences). Lines represent individual interactions with a radiation field, with vertical lines denoting interactions that change the left-hand (bra) index of the density matrix and horizontal lines those that change the right-hand (ket) index. In the convention used here, the zero-order density matrix (p ° ) is at the lower left, and time increases upwards and to the right downward or leftward steps are not allowed. The coherences in the shaded circles are endpoints of the two one-step pathways [pa,a — pb (B) and pa — Paj, (C)] that contribute to the first-order density matrix (p< )) and the first-order optical polarization (P ). A second interaction with the radiation field (dotted line) can convert a coherence to the excited state (Pbb) Of the ground (paa) state. The pathways in (B) and (C) are described as complex conjugates because one can be generated fi om the other by interchanging the two indices at each step...

Now let s see if we get the same result by evaluating the first-order polarization. Because p aa and are zero, the first-order optical polarization at time t has only... 

This is followed by two field actions which again create a vibrational coherence but, now, with opposite phase to the first coherence. Hence one obtains a partial rephasing, or echo, of the macroscopic polarization. The final field action creates the seventh order optical polarization which launches the signal field (the eighth field). Just as for the spin echo in NMR or the electronic echo in 4WM, the degree of rephasing (tlie... 

For high-pressure Raman scattering study, the Ge NCs and the Si substrate have different first-order optical phonon modes (Ge mode at 300 cm and Si mode at 521 cm ), making it much easier to study the strain effects on the NCs and the substrate than in the Si NCs/Si02/Si nanosystem, where the Raman modes of Si NCs and substrate Si overlap. We have shown in Section 12.3 and Figure 12.4 that the Si-2TA at 300 cm can be eliminated by specific polarization configuration. Moreover, our experiments [1-5] suggested that the Raman... 

The macroscopic optical analysis of these effects requires the introduction of two complex indexes of refraction for the ferromagnetic material, one for lefr-circu-larly polarized light and another for right-circularly polarized light, which to first order, are given by... 

Nonlinear second order optical properties such as second harmonic generation and the linear electrooptic effect arise from the first non-linear term in the constitutive relation for the polarization P(t) of a medium in an applied electric field E(t) = E cos ot. 

The first and third order terms in odd powers of the applied electric field are present for all materials. In the second order term, a polarization is induced proportional to the square of the applied electric field, and the. nonlinear second order optical susceptibility must, therefore, vanish in crystals that possess a center of symmetry. In addition to the noncentrosymmetric structure, efficient second harmonic generation requires crystals to possess propagation directions where the crystal birefringence cancels the natural dispersion leading to phase matching. 

Experimental verification of the ISRS generation can be primarily given by the pump polarization dependence. The coherent phonons driven by ISRS (second order process) should follow the symmetry of the Raman tensor, while those mediated by photoexcited carriers should obey the polarization dependence of the optical absorption (first order process). It is possible, however, that both ISRS and carrier-mediated generations contribute to the generation of a single phonon mode. The polarization dependence is then described by the sum of the first- and second-order processes [20-22], as shown in Fig. 2.3. 

The first issue of the construction of noncentrosymmetric LB films with highly efficient optical nonlinearity is how one overcome the difficulty in realizing a high-degree orientational order of polar molecules, which possess high... 

In fact, because the integrated first-order rate equation (Equation (8.24)) is written in terms of a ratio of concentrations, we do not need actual concentrations in moles per litre, but can employ any physicochemical parameter that is proportional to concentration. Obvious parameters include conductance, optical absorbance, the angle through which a beam of plane-polarized light is rotated (polarimetry), titre from a titration and even mass, e.g. if a gas is evolved. 

Second-order optical nonlinearities result from introduction of a cubic term in the potential function for the electron, and third-order optical nonlinearities result from introduction of a quartic term (Figure 18). Two important points relate to the symmetry of this perturbation. First, while negative and positive p both give rise to the same potential and therefore the same physical effects (the only difference being the orientation of the coordinate system), a negative y will lead to a different electron potential than will a positive y. Second, the quartic perturbation has mirror symmetry with respect to a distortion coordinate as a result, both centrosymmetric and noncentrosymmetric materials will exhibit third-order optical nonlinearities. If we reconsider equation 23 for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric), we see that polarization at a given point in space is ... 

Many of the different susceptibilities in Equations (2.165)-(2.167) correspond to important experiments in linear and nonlinear optics. x<(>> describes a possible zero-order (permanent) polarization of the medium j(1)(0 0) is the first-order static susceptibility which is related to the permittivity at zero frequency, e(0), while ft> o>) is the linear optical susceptibility related to the refractive index n" at frequency to. Turning to nonlinear effects, the Pockels susceptibility j(2)(- to, 0) and the Kerr susceptibility X(3 —to to, 0,0) describe the change of the refractive index induced by an externally applied static field. The susceptibility j(2)(—2to to, to) describes frequency doubling usually called second harmonic generation (SHG) and j(3)(-2 to, to, 0) describes the influence of an external field on the SHG process which is of great importance for the characterization of second-order NLO properties in solution in electric field second harmonic generation (EFISHG). 

The optical behavior of asbestos fibers viewed with crossed polars has been described. Crystalline fibers have positions of extinction 90° apart. The fact that crystalline fibers have retardation has also been mentioned. With crossed polars and a first order red plate in place, asbestos fibers will go from yellow to extinction to blue, back to yellow, etc., upon rotation of the stage. If the fiber bends, this is equivalent to a rotation of the stage and the color will change. If the fiber... 

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