Perfect phase coherence

All these features, and particularly the extended nature of the states, their perfect phase coherence, and the sharp band edges are universal properties of the electronic structures of perfect crystals stemming from the periodicity of the lattice. 

Spin-spin relaxation is the steady decay of transverse magnetisation (phase coherence of nuclear spins) produced by the NMR excitation where there is perfect homogeneity of the magnetic field. It is evident in the shape of the FID (/fee induction decay), as the exponential decay to zero of the transverse magnetisation produced in the pulsed NMR experiment. The Fourier transformation of the FID signal (time domain) gives the FT NMR spectrum (frequency domain, Fig. 1.7). 

At low temperatures, in a sample of very small dimensions, it may happen that the phase-coherence length in Eq.(3) becomes larger than the dimensions of the sample. In a perfect crystal, the electrons will propagate ballistically from one end of the sample and we are in a ballistic regime where the laws of conductivity discussed above no more apply. The propagation of an electron is then directly related to the quantum probability of transmission across the global potential of the sample. 

In the phase-coherent, one-color pump/probe scheme (see Section 9.1.9) the wavepacket is detected when the center of the wavepacket returns to its to position, (x)to+nT — (x)to, after an integer number of vibrational periods. The pump pulse creates the wavepacket. The probe pulse creates another identical wavepacket, which may add constructively or destructively to all or part of the original pump-produced wavepacket. If the envelope delay and optical phase of the probe pulse (Albrecht, et al, 1999) are both chosen correctly, near perfect constructive or destructive interference occurs and the total spontaneous fluorescence intensity (detected after the pump and probe pulses have traversed the sample) is either quadrupled (relative to that produced by the pump pulse alone) or nulled. As discussed in Section 9.1.9, the probe pulse is delayed, relative to the pump pulse, in discrete steps of At = x/ojl- 10l is selected by the experimentalist from within the range (ljl) 1/At (At is the temporal FWHM of the pulse) to define the optical phase of the probe pulse relative to that of the pump pulse and the average excitation frequency. However, [(E) — Ev ]/K is selected by the molecule in accord with the classical Franck-Condon principle (Tellinghuisen, 1984), also within the (ojl) 1/At range. When the envelope delay is chosen so that the probe pulse arrives simultaneously with the return of the center of the vibrational wavepacket to its position at to, a relative maximum (optical phase at ojl delayed by 2mr) or minimum (optical phase at u>l delayed by (2n + l)7r) in the fluorescence intensity is observed. 

Figure 3.116 TEM images of cobalt nanoparticles, (a) Low-magnification image of e-Co. The even contrast across the individual particles implies a uniform crystalline structure for this phase (b) HR-TEM image of e-Co, showing perfect crystallographic coherence and discrete faceting (c) Low-magnification image...

FIGURE 5 Variation of the second harmonic (a) and fundamental (b) intensity with (L/L ep), where / dep is the second-harmonic depletion length, for plane waves at perfect phase matching. [Reproduced from Reintjes, J. (1985). Coherent ultraviolet and vacuum ultraviolet sources. In Laser Handbook, Vol. 5 (M. Bass and M. Stitch, eds.), North-Holland, Amsterdam.]... 

One may define the phase difference of the wave at two points on the wave front at time to as qjQ. If, then, for any time T > To the phase difference of the two points remains cpo, it is said that the wave exhibits coherence between the two points. And if this is true for any two points of the wave front, then the wave is defined as having perfect spatial coherence. In practice, spatial coherence occurs only over a limited area in an expanded laser beam. 

FIGURE 9 Second-harmonic power development during propagation for non-phase-matched, perfectly phase-matched, and quasi-phase-matched (QPM) second-harmonic generation. For QPM, the back-conversion into the fundamental is prevented by turning off the nonlinearity every other coherence length. 

Usually we are only interested in mutual intensity suitably normalised to account for the magnitude of the helds, which is called the complex degree of coherence 712 (r). This quantity is complex valued with a magnitude between 0 and 1, and describes the degree of likeness of two e. m. waves at positions ri and C2 in space separated by a time difference r. A value of 0 represents complete decorrelation ( incoherence ) and a value of 1 represents complete eorrelation ( perfect coherence ) while the complex argument represents a difference in optical phase of the helds. Special cases are the complex degree of self coherence 7n(r) where a held is compared with itself at the same position but different times, and the complex coherence factor pi2 = 712(0) which refers to the case where a held is correlated at two posihons at the same time. 

Before we state this mathematically, let us consider the implication of perfectly monochromatic illumination. We shall assume that the slit elements are illuminated by light of exactly the same wavelength and frequency. Each given pair of points in the entrance slit will then have its own fixed phase relationship. Neither point can gain or lose phase relative to the other because the fields have exactly the same frequency. The fields at these two points are perfectly coherent. Because this is true for any given pair of points in the slit, all the points have a fixed phase relationship. We call this the case of coherent irradiation. 

Now let us assume that a monochromatic source of flux is placed in the plane of the entrance slit so that there is no constant phase relationship between the fields at any two given points in the slit. This, in itself, is a contradiction, because a perfect source monochromaticity implies both spatial and temporal coherence. By definition of coherence, a constant phase relationship would result. To eliminate the possibility of such a relationship, we must require the source spectrum to have finite breadth. Let us modify the assumption accordingly but specify the source spectrum breadth narrow enough so that its spatial extent when dispersed is negligible compared with the breadth of the slits, diffraction pattern, and so on. Whenever time integrals are required to obtain observable signals from superimposed fields, we evaluate them over time periods that are long compared with the reciprocal of the frequency difference between the fields. We shall call the assumed source a quasi-monochromatic source. 

Stress builds up at a coherent interface between two phases, a and / , which have a slight lattice mismatch. For a sufficiently large misfit (or a large enough interfacial area), misfit dislocations (= localized stresses) become energetically more favorable than the coherency stress whereby a semicoherent interface will form. The lattice plane matching will be almost perfect except in the immediate neighborhood of the misfit dislocation. Usually, misfits exist in more than one dimension. Sets (/) of nonparallel misfit dislocations occur at distances... 

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