Second-order response

The second-order nonlinear optical processes of SHG and SFG are described correspondingly by second-order perturbation theory. In this case, two photons at the drivmg frequency or frequencies are destroyed and a photon at the SH or SF is created. This is accomplished tlnough a succession of tlnee real or virtual transitions, as shown in figure Bl.5.4. These transitions start from an occupied initial energy eigenstate g), pass tlnough intennediate states n ) and n) and return to the initial state g). A fiill calculation of the second-order response for the case of SFG yields [37]... 

The second-order response equations for the cluster amplitudes and the Lagrangian multipliers are ... 

To derive working expressions for the dispersion coefficients Dabcd we need the power series expansion of the first-order and second-order responses of the cluster amplitudes and the Lagrangian multipliers in their frequency arguments. In Refs. [22,29] we have introduced the coupled cluster Cauchy vectors ... 

To find the power series expansion of Eq. (30) in ub, ojc, u>d we can thus replace the first-order responses of the cluster amplitudes and Lagrangian multipliers and the second-order responses of the cluster amplitudes by the expansions in Eqs. (37), (39) and (44) and express OJA as —ojb ojc — ojd- However, doing so starting from Eq. (30) leads to expressions which involve an unneccessary large number of second-order Cauchy vectors C m,n). To keep the number of second-order... 

Hirose et al. [26] proposed a homodyne scheme to achieve the background-free detection of the fourth-order field. With pump irradiation in a transient grating configuration, the fourth-order field propagates in a direction different from that of the second-order field because of different phase match conditions. The fourth-order field is homodyned to make ffourth(td. 2 D) and spatially filtered from the second-order response hecond td, 2 D). 

A semi-empirical, second-order response lag is used. This consists of a first-order lag equation representing the diffusion of oxygen through the liquid film on the surface of the electrode membrane... 

Second order response Third order response Fourth order response Fifth order response... 

Figure 4.12 A solid-state electrode showing a second-order response. The electrode shown in Figure 4.11 can be modified by the incorporation of silver chloride into the membrane to enable the activity of chloride ions in a sample to be measured. A surface reaction between the test chloride ions and the membrane silver ions alters the activity of the latter, resulting in a change in the potential difference across the membrane.

Myers, R.H., Vining, G.G., Giovannitti-Jensen, A., and Myers, S.L. (1992), Variance Dispersion Properties of Second-Order Response Surface Designs . J. Qual. Technol., 24, pp. 1-11. 

ALTERNATING CURRENT PERTURBATION. SECOND-ORDER RESPONSES 2.4.1 The second-order effects... 

Fig. 21. Input signals and corresponding second-order responses for (a) rectification, (b) demodulation, DSB mode, and (c) demodulation, AM mode.

Fig. 22. Schematic representation of the electrochemical cell connected to a perturbation circuit and a detection circuit for second-order responses. In this case, it is supposed that the perturbation signal has a high frequency and the response signal a low frequency.

Assuming that the detection device in Fig. 22 selectively measures the desired second-order response as a voltage A EM, this quantity is related to the second-order current Aj2 that flows externally by... 

In general, x<3> is several orders of magnitude smaller than x(2) [78]. However, this contribution can be comparable to the second-order response when the dc field is large (i.e. 104 — 105 V/cm) as is the case for a metal electrode in solution held at a bias on either side of the PZC. The SH fields can be viewed as arising from a sum frequency mixing process where two incident ac fields at frequency co are mixed with a dc field at frequency co = 0 to produce a reflected and transmitted wave at frequency 2co. 

We used the model of the fast dynamics of the system in Equation (4.36) to design a nonlinear input-output linearizing output feedback controller with integral action (Daoutidis and Kravaris 1992) for x. The controller was designed to produce the critically damped second-order response... 

Table 3 Spin-orbit coupling matrix elements between singlet and triplet states form second order response theory calculations on H20++ (10-6 au). A 6ai, 3b2, 3bj, la2 active space B 4ai, 2b2, 2bi active space. From Ref. [55],...

We conclude that, for a fully variational wave function, only the first-order response of the wave function dX/dx is required to calculate the energy to second order. In particular, for the evaluation of polarizabilities and magnetizabilities, the second-order response of the wave function d2X/dx2 is not needed. 

Table 1. Calculated second-order responses (P) at 0.5 eV for selected NLO clusters. 1 ...

Combining Eqs. (177) and (179) and evaluating all functionals at the ground-state density o. we obtain a Dyson-type relation for the second-order response... 

The main remark is that the ratio step responses are very different from the simple flow rate step responses of Fig. 2. Second order responses are present y lu2,y luf), some step responses show low magnitudes (yfu/, yfuy, yfuf). Also, some time constants are relatively different. When ratios are manipulated, several flow rates are simultaneously manipulated which makes the total character complex and unpredictable. 

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