Profile hyperbolic-secant

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... 

Example Parabolic profile 1-12 Example Hyperbolic secant profile 1-13 Example Clad power-law profiles... 

Graded profiles tend to equalize the transit times of different rays. There is a simple explanation for this. As a ray propagates further from the axis, (x) decreases, thus increasing the local speed of light c/n(x). This increase in speed compensates, in part, for the extra distance travelled by the ray off axis. For the hyperbolic secant profile this equalization is exact, as we show in Section 1-12. 

Fig. 1-10 Plots of the power-law profiles of Eq. (1-59) for various values of q. The clad parabolic profile corresponds lo q = 2 and the step profile to q = 00. The hyperbolic secant profile of Eq. (1-51) is shown dashed, and the dotted extension to the q = 2 core profile denotes the (infinite) parabolic profile of Eq. (1-44).

It is clear from the discussion at the beginning of Section 3-2 that there is no ray dispersion on the planar waveguide with a hyperbolic secant profile. However, on the fiber with the same profile, ray dispersion is no longer zero. There is no known profile which has zero ray dispersion for both skew and meridional rays. 

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... 

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. 

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).

Frequency-modulated (FM) pulses like the hyperbolic secant (HS) pulse are not commonly used for multislice spin-echo MRI because of a nonlinear phase of the transverse magnetization. A general theory and methods are described for conventional spin-echo imaging using a hyperbolic secant (HS) pulse for refocusing.Phase profiles produced by the HS pulse are analytically described. 

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