Hyperbolic secant

For the OS calculation, (6.58), the same optimal solution is reached by choosing a Gaussian-like hyperbolic secant function for w(Au) ... 

Hyperbolic Secant Inversion Pulses and Wideband Uniform Rate Smooth Truncation... 

Siegel et al. showed that enhancement of the CT can also be obtained using hyperbolic secant (HS) pulses to invert selectively the STs [74], Unlike the DFS waveform, whose frequency sweep is generated by a constant rf-pulse phase while modulating the amplitude, the HS pulse utilizes both amplitude and phase modulation, yielding an enhancement exceeding that obtained by DFS or RAPT [61, 74, 75]. Most recently, the pulse sequence called wideband uniform-rate smooth truncation (WURST) [76] was introduced to achieve selective adiabatic inversion using a lower power of the rf-field than that required for the HS pulses [77,78]. One of its applications involved more efficient detection of insensitive nuclei, such as 33S [79]. 

The energy of the second harmonic (SH) is found by integrating the SH intensity over space and time while assuming Gaussian spatial and hyperbolic secant temporal profiles. The normalized efficiency,, ... 

In figure 3 we compare the analytical solution of Eq. (12) and Eq. (13) with the numerical solution of Eq. (14) for the SHG conversion efficiency for three different pulse profiles (hyperbolic secant, Gaussian and rectangular). It can be seen that the simplified model perfectly describes the process of SHG when L >. Further, the analytical model of Eq. (12)... 

In modeling the SHG performanee of the bulk nonlinear optical crystals, we have assumed the incident fundamental pulses are characterized by a hyperbolic secant temporal profile, and therefore r = rg (FWHM)/l.76. For the case of KNhOs, the maximum optical-to-optical SHG efficiency achieved was 30%. The corresponding value of L/L, under these conditions, with a GVM parameter, 1.2 ps/mm, is L/L = 30. The values of beam... 

R. Siegel, T. T. Nakashima and R. E. Wasylishen, Signal enhancement of NMR spectra of half-integer quadrupolar nuclei in solids using hyperbolic secant pulses. Chem. Phys. Lett, 2004, 388,441 5. 

From equation (8) three modes of adiabatic spin inversion using RF pulses may be defined as (a) amplitude modulated pulses, e.g. I-BURP [13], G3[16], I-SNOB [17], (b) frequency modulated pulses, e.g. chirp [18,20], tangential sweep [20,21] and (c) both amplitude and frequency modulated pulses, e.g. the hyperbolic secant [22] or WURST (Wide band Uniform Rate Smooth Truncation) [23] pulse. 

The parameter gv depends on the shape of the laser pulse and the duty cycle. Pulses with a Gaussian temporal profile result in gv = 0.664 while gv = 0.588 is found for hyperbolic-secant square (sech2) pulses. Combination of Eqs. (40) and (41) results in Eq. (42), showing that pulsed lasers are more efficient excitation sources compared to lasers operating in cw mode. 

One remarkable phase- and amplitude-modulated rf pulse is the hyperbolic-secant pulse [Sil 1, Sil2]. The B field in the rotating frame field is a complex quantity modulated according to... 

Neglecting relaxation, the envelope of a strongly radiation-damped free induction decay (FID) is given by a hyperbolic secant function, ... 

Apart from the gradient itself, the degree of suppression is related to the efficiency of excitation by the selective pulse and thus the choice of selective pulse is important with, for example, hyperbolic secant pulses perform better than sine pulses." If the solvent Ti is very short and on the order of the time for the dephasing procedure, then some (unwanted) z-magnetization will be re-established prior to the excitation pulse. One method of circumventing this... 

Figure 11 Solid-state chlorine NMR spectra of i-lysine hydrochloride. Experimental spectra are shown in (A) Cl at 11.75 T and (C) Cl at 11.75 T. The Cl NMR spectrum of a stationary sample obtained at 11.75 T is shown in (E). A hyperbolic secant spin-echo sequence was used to acquire the spectra. Simulated spectra are shown above each experimental spectrum. The effects of CSA are evident in the spectrum of the stationary sample (E) shown in trace (G) is the simulated spectrum obtained with an isotropic CS tensor. The relative orientation of the EFG and CS tensor PASs are shown in the inset. From Ref. 179. Reproduced by permission of the American Chemical Society.

The other hyperbolic trigonometric functions are the hyperbolic tangent, denoted by tanh(x) the hyperbolic cotangent, denoted by coth(x) the hyperbolic secant, denoted by sech(x) and the hyperbolic cosecant, denoted by csch(x). These functions are given by the equations... 

Figure 10.8. The amplitude profiles for adiabatic pulses (a) hyperbolic secant, (b) CHIRP with smoothing of the first and last 20%, (c) WURST with n = 20 (WURST-20) and (d) WURST with n = 5 (WURST-5).

Another method of achieving a population transfer from the STs to the CT is through the use of hyperbolic secant... 

Hyperbolic secant (sech), cosecant (cosech), and cotangent (coth) are the reciprocals of cosh, sinh, and tanh, respectively. 

An incident pulse with the pulse area A = 2n propagates over considerable distances without noticeable attenuation. The time profile of a pulse with length x approaches after a pathlength Az = m/a (m = 1 2 3 ...) a hyperbolic secant profile with the electric field amplitude... 

For A = m 27T the pulse area A does not change because dA/dz = 0. It turns out, however, that for m > 1 the pulse splits into m 27T-pulses, which propagate as hyperbolic secant-pulses through the medium. The total pulse area A is still preserved. 

The full Maxwell-Bloch equations are solved by using the iterative predictor-corrector finite-diference time-domain method[76,77,78]. In what follows, we assume that the system is initially in the lowest subband. We consider a hyperbolic secant functional form for the initial... 

---