Coherence temporal

If the phase differences A0 = (pn P, i) -0 (P, 2) at a given point P between two different times t, t2 are nearly the same for all partial waves, the radiation field at P is temporally coherent. The maximum time interval At = t2 —1 for which A0 for all partial waves differ by less than tc is termed the coherence time of the radiation source. The path length Asc = cAt traveled by the wave during the coherence time At is the coherence length. 

If a constant time-independent phase difference A0 = 0(Pi) — 0( 2) exists for the total amplitudes A = Aqc at two different points Pi, P2 the radiation field is spatially coherent. All points P, P that fulfill the condition that for all times r, 0(P, t) — (j) Pn, 01 tt have nearly the same optical path difference from the source. They form the coherence volume. 

The superposition of coherent waves results in interference phenomena that, however, can be observed directly only within the coherence volume. The dimensions of this coherence volume depend on the size of the radiation source, on the spectral width of the radiation, and on the distance between the source and observation point P. 

The following examples illustrate these different expressions for the coherence properties of radiation fields. 

Consider a point source PS in the focal plane of a lens forming a parallel light beam that is divided by a beam splitter S into two partial beams (Fig. 2.23). They are superimposed in the plane of observation B after reflection from 

They are superimposed in the plane of observation B after reflection from the mirrors Mi, M2. This arrangement is called a Michelson interferometer (Sect. 4.2). The two beams with wavelength A travel different optical path lengths SMiSB and SM2SB, and their path difference in the plane B is 

The mirror M2 is mounted on a carriage and can be moved, resulting in a continuous change of AIn the plane B, one obtains maximum intensity when both amplitudes have the same phase, which means A.y = mA, and minimum intensity if = (2m + l)A/2. With increasing A.y, the contrast V = (I ax /min)/(/max + /min) decreases (Fig. 2.31) and vanishes if A.y becomes larger than the coherence length Asc (Sect. 2.9.4). Experiments show that A. c is related to the spectral width Aco of the incident wave by 

This observation may be explained as follows. A wave emitted from a point source with the spectral width Aco can be regarded as a superposition of many quasi-monochromatic components with frequencies co within the interval Aco. The superposition results in wave trains of finite length A c = = cj Aco because 

A low-pressure mercury spectral lamp with a spectral filter that only transmits the green line A = 546 nm has, because of the Doppler width Atwo = 4 X 10 Hz, a coherence length of A c — 8 cm. 

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Whereas temporal coherence is important for spectroscopy, spatial coherence is important for imaging. Consider the disturbance at two points Pi and P2 due to a finite sized source S (Fig. 2). If the source is small and distant... 

What Is Interferometry (1.3) Interferometry deals with the physical phenomena which result from the superposition of electromagnetic (e.m.) waves. Practically, interferometry is used throughout the electromagnetic spectrum astronomers use predominantly the spectral regime from radio to the near UV. Essential to interferometry is that the radiation emerges from a single source and travels along different paths to the point where it is detected. The spatio-temporal coherence characteristics of the radiation is studied with the interferometer to obtain information about the physical nature of the source. 

In the hrst case, the degree of self coherence depends on the spectral characteristics of the source. The coherence time Tc represents the time scale over which a held remains correlated this hme is inversely proportional to the spectral bandwidth Au) of the detected light. A more quantitative dehnition of quasi-monochromatic conditions is based on the coherence time all relevant delays within the interferometer should be much shorter than the coherence length CTc. A practical way to measure temporal coherence is to use a Michel-son interferometer. As we shall see, in the second case the spatial coherence depends on the apparent extent of a source. 

The fundamental quantity for interferometry is the source s visibility function. The spatial coherence properties of the source is connected with the two-dimensional Fourier transform of the spatial intensity distribution on the ce-setial sphere by virtue of the van Cittert - Zemike theorem. The measured fringe contrast is given by the source s visibility at a spatial frequency B/X, measured in units line pairs per radian. The temporal coherence properties is determined by the spectral distribution of the detected radiation. The measured fringe contrast therefore also depends on the spectral properties of the source and the instrument. 

Of course, the role of the artificially introduced stochastics for mimicking the effect of all eddies in a RANS-based particle tracking is much more pronounced than that for mimicking the effect of just the SGS eddies in a LES-based tracking procedure. In addition, the random variations may suffer from lacking the spatial or temporal correlations the turbulent fluctuations exhibit in real life. In RANS-based simulations, these correlations are not contained in the steady spatial distributions of k and e and (if applicable) the Reynolds stresses from which a typical turbulent time scale such as k/s may be derived. One may try and cure the problem of missing the temporal coherence in the velocity fluctuations by picking a new random value for the fluid s velocity only after a certain period of time has lapsed. 

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... 

Its temporal coherence, causing spectral linewidths of the induced emission to be smaller by several orders of magnitude than those of fluorescence lines emitted by spectral lamps. 

In order to realize the extremely small linewidths, attainable because of the temporal coherence of induced emission, care must be taken to ensure that the threshold condition for induced oscillation is fulfilled for only one mode. The transverse modes can be eliminated by an appropriate choice of the cavity dimensions, introducing... 

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. 

Special Issue, Spatio-temporal Coherence and Chaos in Physical Sytems, Pity sic a D 23 (1986). 

Conventional sources of electromagnetic radiation are incoherent, which means that the waves associated with any two photons of the same wavelength are, in general, out-of-phase and have a random phase relation with each other. Laser radiation, however, has both spatial and temporal coherence, which gives it special importance for many applications. 

The cumulative envelope function, E s), can be complex and is attributable to a number of instrumental and experimental effects, such as spatial and temporal coherence and specimen motion. It has been shown that in practice a simple Gaussian function with width B adequately describes the cumulative envelope function (Saad et al, 2001) ... 

The new theory places the pioneer organism in a locally and temporally coherent volcanic flow setting. Therefore, all its assumptions must be compatible with each other. Such a comprehensive theory cannot be conceived in one stroke. It has to evolve. Indeed, the theory presented here has been evolving over the past 20 years, its progress guided by chemical experiments and by the increase of its relative explanatory power (Popper). The partial aspects of the theory as it now stands shall be addressed one at a time in a manner that makes their coherence within the theory apparent. 

The above-reported chemical reactions proceed under conditions that are compatible with an origin of life under the locally and temporally coherent conditions of a volcanic flow system. Therefore, the discovered reactions may well be components of the metabolic system of the pioneer organism. As additional components come into experimental view, the theory is expected to evolve. So far we have addressed the notions of growth and reproduction as aspects of one unitary chemical system. We now show that this unitary system is also the physical basis for the earliest mechanism of evolution and that it constitutes in fact the evolutionary Aiflage for the emergence of the cellular and genetic features of extant forms of life. 

As mentioned in Section 3.1.2, attractive UV sources for lithography are those that produce high power and poor spatial and temporal coherence. Jain and co-workers (13-15) demonstrated that excimer lasers provide excellent quality, speckle-free images with resolution to 0.5 xm in a contact mode. The images were obtained in l- xm-thick diazoquinone photoresists such as AZ2400 with a XeCl laser at 308 nm and a KrF laser at 248 nm... 

F(t)) corresponds to the time-averaged photon flux. The latter is proportional to (/o(f)2) because most detectors provide a response proportional to (Io(t)2). fluorescence quantum yield and t]col is the collection efficiency of the optical setup used. The parameter gc corresponds to the second-order temporal coherence that is, gc = I0(t)2)/ I0(t))2. Therefore, Eq. (37) represents all experimental quantities needed for quantitative evaluation of TPA. These are the spatial distribution of the incident light (Jv, S1 (r) dV), the degree of the second-order temporal coherence (gc), the fluorescence collection efficiency (jjco1), and the fluorescence quantum yield (r). Details are compiled in the literature [85, 86, 366, 368, 373]. 

Quantitative measurements of S require additional screening of temporal fluctuations of the photon flux—the so-called temporal coherence gc. Because the chromophore is excited in the first cycle of the pulse train, evaluation of gc is needed for only one excitation cycle. Defining th as the excitation pulse width and/itH as the duty cycle, gc is expressed by Eq. (41). 

Lasers are unique energy sources characterized by their spectral purity, spatial and temporal coherence, and high average peak intensity. Each of these characteristics has led to applications that take advantage of these qualities ... 

If a structural reaction is rapidly and uniformly initiated in a crystal, what will the likely effects be on the Laue x-ray intensities The very weak intermolecular interactions characteristic of protein crystals make it implausible that existence of a certain tertiary structure in one molecule would favor a particular tertiary structure in its neighbors. That is, all molecules in the crystal are likely to behave independently of each other, as they do in solution. Their populations will evolve smoothly in time in a manner governed by the reactant concentrations and the rate constants. To preserve the x-ray diffraction pattern, spatial coherence must be maintained between molecules, but temporal coherence need not be. Thus, at any instant the crystal will contain many different conformations, each representing a different structural intermediate. The total structure factor for a particular reflection will be a weighted vector sum of the time-independent structure factors of each conformation, where the weights are the time-dependent fractional occupancies. In favorable cases [26), it may be possible to extract the individual time-independent structure factors from this sum, and hence to obtain directly the structure of each intermediate. 

In order to quantify the temporal coherence of patterns arising at different correlation lengths, we introduce the quantity... 

On the other hand, the correlation time versus noise intensity in Fig. 5.22(c) shows that the temporal coherence of the system in contrast to the spatial ordering decreases rapidly with increasing noise. 

We have seen that delayed feedback can be an efficient method for manipulation of essential characteristics of chaotic or noise-induced spatiotem-poral dynamics in a spatially discrete front system and in a continuous reaction-diffusion system. By variation of the time delay one can stabilize particular unstable periodic orbits associated with space-time patterns, or deliberately change the timescale of oscillatory patterns, and thus adjust and stabilize the frequency of the electronic device. Moreover, with a proper choice of feedback parameters one can also effectively control the coherence of spatio-temporal dynamics, e. g. enhance or destroy it. Increase of coherence is possible up to a reasonably large intensity of noise. However, as the level of noise grows, the efficiency of the control upon the temporal coherence decreases. 

Study of the Spatial and Temporal Coherence of High-Order Harmonics, Pascal Salieres,... 

The varieties of exposure sources that have found applications in UV and visible light optical lithography can be broadly divided into two groups (i) high-pressure arc lamp or incoherent sources and (ii) laser sources or temporally coherent sources. In the laser-type sources, we include all techniques and devices for radiation generation that have their basis in stimulated emission of radiation. 

The key operational parameters of exciplex and excimer lasers used in optical lithographic applications include exposure-dose-related parameters comprising average power, pulse energy, repetition rate, and pulse width temporal coherence spatial coherence including beam dimensions, beam divergence, and beam uniformity and maintenance and reliability. Table 13.2 lists some of the key operational parameters of KrF, ArF, and F2 laser systems used in optical lithography. 

The spectral bandwidth of the radiation emitted by a laser determines its temporal coherence, expressed as a coherence length 1, and given by... 

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