Envelope functions

Figure 3. Comparison of EXAFS envelope functions derived from the inverse transforms in Figure 2 for silica supported copper and ruthenium-copper catalysts. Reproduced with permission from Ref. 8. Copyright 1980, American Institute of Physics.

In linear imaging, these two effects can be mathematically described by damping functions, E, applied to the CTF (for details see William and Carter 1996, Spence 1988). Their combined effects are shown in Figure 2 as the envelope function to the CTF. The partial temporal coherence places a limit on the information that can be transferred in a microscope, a value called the information limit. Traditionally, the information limit is defined as the... 

Cowley 1981) ( is a convolution integral and FT is the Fourier transform). The phase-contrast imaging performance of an HRTEM is controlled by sin x, which contains the basic phase-contrast sinusoidal terms modified by an attenuating envelope function, F 9), which is essentially due to the partial coherence of the electron beam ... 

Figure 2.6. An example of a contrast transfer function (CTF). The calculated CTF of a 200CX HRTEM at Scherzer defocus and Cs = 1.2 mm. The first zero is arrowed (corresponding to 0.23 nm resolution) and the resolution in angstrdms is shown on the horizontal axis. A-D are envelope functions plotted as a function of convergence angle (0) of the beam and beam energy spread (A V). Parallel illumination is necessary for high resolution (after Boyes et al 1980).

Fig. 10.5 Experimentally measured values of bandgap of PbSe films (horizontal bars The length gives the experimental uncertainty in size, mainly due to the size distribution). The broken curve gives the theoretical relationship between bandgap and crystal size based on the hyperbohc band approximation used for PbS in Ref. 40. The room-temperature reduced effective mass (0.034) was calculated from the low-temperature value (0.022) (R. Dalven, Infrared Phys. 9 141, 1969.) according to the temperature dependence given in H. Preier, Appl. Phys. 20 189, 1979. The dotted curve is a more recent calculation based on an envelope function calculation [41].

It is interesting that envelope functions can also behave as multiperiod oscillations. This takes place if we take into account small damping. By way of an example, for the damping constant yj = y2 = 0.1, the envelope function has a feature of two period doubling oscillations. 

Consider the case when the external field is a monochromatic circularly polarized pulse, E(f) = Aexp(-iojf), where A is a slowly varying envelope function. For this pulse the phase angle of dE/df is rotated by r/2 from the direction of E. From Eqs. (5.17) and (5.18) we then find... 

Because they give access to the spectral representation of the signal, frequency-domain techniques are well suited for formant modification. The first step in frequency-domain formant modification techniques consists of obtaining a estimation of the spectral envelope. Based of the short-time representation of the signal, it. is possible to derive a spectral envelope function using a variety of different techniques. If the pitch of the signal is available, the short-time Fourier spectmm is searched for local maxima located around harmonic frequencies, then an envelope can be obtained by joining the local... 

The field is linearly polarized in the direction given by Eq. a(t) is an envelope function corresponding to a pulsed field when a(t) is constant E(t) describes a continuous wave (cw) field oscillating with the frequency w = 2nis. 

In order to apply these equations to a femtosecond pump-probe experiment, an additional assumption has to be made regarding the shape of the time resolved signal. We wish to account for the finite relaxation time of the transient polarisation and so the signal must be described by a double convolution of an exponential decay function with the pump and probe intensity envelope functions. We will assume a Gaussian peak shape so that the convolution may be calculated analytically. As we will see, the experimental results require two such contributions, and hence, the following function will be used to fit the experimental data... 

Let the laser field be expressed as s(t) — o(t) sin(coLt), where o(t) is the pulse envelope function, including polarization, and col is the central frequency. For simplicity, consider a 8 function excitation. In the rotating wave approximation, the field is expressed as e(t) = S(t) exp(—icoLt) with field strength sq. Integration over t in Eq. (25) gives... 

A strict derivation of the comb properties is not feasible as it depends on the special dispersion characteristics of the laser cavity and these data are not accessible with the desired degree of accuracy. Instead we only assume that the laser emits a stable coherent pulse train without any detailed consideration of how this is possible. Further we assume that the electric field E(t), measured for example at the output coupler, can be written as the product of a periodic envelope function A ) and a carrier wave C(t) ... 

The envelope function defines the pulse repetition time T = 27r/u>r by demanding A(t) = A(t — T). Inside the laser cavity the difference between the group velocity and the phase velocity shifts the carrier with respect to the envelope after each round trip. The electric field is therefore in general not periodic with T. To obtain the spectrum of E(t) the Fourier integral has to be calculated ... 

Fig. 1. The spectral shape of the carrier function (left) assumed to be narrower than the pulse repetition frequency Au>c

A periodic frequency chirp imposed on the pulses is accounted for by allowing a complex envelope function A(t). Thus the carrier C(t) is defined to be whatever part of the electric field that is non-periodic with T. The convolution theorem allows us to calculate the Fourier transform of E(t) from A(u>) and... 

This time dependent phase shift leads to a frequency modulation that is proportional to the time derivative of the self induced phase shift fused silica with its positive Kerr coefficient rio = 2.5 x 10 16 cm2/W [28] the leading edges of the pulses are creating extra frequencies shifted to the red ( jvi(t) < 0) while the trailing edges causes blue shifted frequencies to emerge. Self-phase modulation modifies the envelope function according to... 

The role of PolyP in the cell envelope of prokaryotes may be connected with their anionic properties, important for providing the negative charge of this compartment. In addition, PolyPs may affect the cell-envelope functions by gene activity regulation, as will be discussed below. 

Here c.c. refers to the complex conjugate and ej (t) and Uj are the temporal envelope function and unit vector of the polarization of the jth electric field. The frequencies of the five fields are assumed to be identical, i.e., ojj = co0 for j = 1, 2, 3, 4, and 5. This can be experimentally achieved by using time-delayed pulses generated from a common laser oscillator. Although the femtosecond pulses generated in the laboratory have a finite width, for the sake of simplicity the laser pulses are assumed to be impulsive in this section, i.e. ... 

Fig. 13. Plot of the amplitude (envelope) function B(Z) vs Z, for a planar interface between coexisting disordered (Z— >o) and lamellar (Z—>-< >) phases of block copolymer melts in the bulk. The midpoint of the profile (Z,B)=(0,1/2) is denoted by an open circle, while the inflection point (solid circle) is at [—(1/2) ln(3/2), 1/V3]. From Fredrickson and Binder [234]...

For D d, the envelope function is determined by the diffraction at the single slits. The width of the interferogram B is defined by the distance of the two first order minima on each side of the maximum as... 

---