Theory of optical spectroscopy by digital autocorrelation of photon-counting fluctuations
TL;DR: In this article, a detailed analysis of the relationship between the autocorrelation functions of clipped and unclipped photon counting fluctuations is given by evaluating the generating function for the joint distribution of intensity fluctuations, corrections to the earlier results arise if the time over which the signal is sampled is not negligible compared with any coherence time of the light.
Abstract: Digital autocorrelation of optical signals is discussed with particular reference to the technique of clipping, reported briefly in an earlier publication. A detailed analysis of the relationship between the autocorrelation functions of clipped and unclipped photon-counting fluctuations is given. By evaluating the generating function for the joint distribution of intensity fluctuations, corrections to the earlier results are calculated which arise if the time over which the signal is sampled is not negligible compared with any coherence time of the light. The effect of clipping in heterodyne experiments is also analysed.
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TL;DR: In this paper, the distribution function of relaxation times underlying the nonexponential relaxation function of Williams and Watts is derived and compared with the analogous Cole-Davidson distribution function, and several useful relations between relaxation and distribution functions are summarized or derived, and the limitations of deriving distribution functions from relaxation functions are discussed.
Abstract: The distribution function of relaxation times underlying the nonexponential relaxation function of Williams and Watts is derived and compared with the analogous Cole–Davidson distribution function. In order to make the comparison between the two distribution functions, a simple empirical relationship between the Cole–Davidson and Williams–Watts parameters was determined which may be used to compare data analyzed using the two fitting functions. Although the relaxation functions are similar to each other, the distribution functions are quite dissimilar. The Cole–Davidson distribution shows a sharp long time cutoff, while the Williams–Watts distribution decays approximately exponentially at long times. Finally, several useful relations between relaxation and distribution functions are summarized or derived, and the limitations of deriving distribution functions from relaxation functions are discussed.
919 citations
TL;DR: In this article, the authors use a photon detection process to measure the Doppler shifts of the scattered light in a laser beam, and then use the photon structure functions to resolve very small particle displacements.
Abstract: The motion of small scatterers in a laser beam results in Doppler shifts of the scattered light. Various homodyne or heterodyne experiments may be used to measure such shifts. The stochastic nature of spatial particle arrangements and of the photon-detection process leads to statistic data processing schemes like temporal correlation or the computation of structure functions. Photon correlation is one of these schemes and has found numerous applications in velocimetry and Brownian motion studies. Topics of current interest are dead-time corrections, the use of photon structure functions, and multiple tau measurements, which access large ranges of time constants in a single run. More recent data processing techniques are recurrence rate correlation for the immediate determination of velocity correlation functions in seeded fluid flows and the measurement of amplitude-weighted phase structure functions, which is able to resolve very small particle displacements otherwise completely obscured by random Brownian motion. Rate correlation found applications in hydrodynamic studies of the route to turbulence, while the major use of phase structure function processing is a very significant increase in the sensitivity of electrophoretic mobility measurements by light scattering.
134 citations
TL;DR: In this paper, the authors used the photon correlation technique to study the laser light scattering intensity fluctuations of supercooled liquid o−terphenyl at temperatures between 12 and −15°C.
Abstract: The laser light scattering intensity fluctuations of supercooled liquid o‐terphenyl at temperatures between 12 and −15 °C have been studied using the photon‐correlation technique. The time correlation functions obtained using the VV and VH scattering configurations at 90° scattering angle show a wide distribution of relaxation times. Angular‐dependent measurements at 45°, 90° and 135° scattering angles show that the time correlation functions for VH and for VV are independent of scattering angles. The VV and VH correlation functions cannot be fit to a bimodal distribution function; neither can they be fit to a cumulant expansion expression with less than four cumulants, but they can be satisfactorily fit to the William–Watts distribution function. The mean relaxation times derived for both isotropic and anisotropic scattering processes from the VV and VH correlation functions are nearly equal. The temperature dependence of the relaxation times cannot be described by an Arrhenius equation with a constant activation energy. The average orientational relaxation times change by more than ten orders of magnitude in the temperature range between 147 and −16 °C but the temperature dependence of the orientational relaxation times over this wide dynamic range can be excellently correlated with the modified Debye–Stokes–Einstein equation. The closeness of the mean orientational relaxation time to the mean structural relaxation time is interpreted as due to strong rotation–translation coupling in the viscoelastic state. It is found that within the dynamic range of the autocorrelator, the collective shear wave plays no role in the time dependence of the VV and VH correlation functions. Quentrec’s theory of viscoelasticity based on the coupling of the shear wave to local translational order and orientational order is found to disagree with the experimental result. Comparison with the dielectric relation time data shows that in o‐terphenyl the dielectric relaxation and the present light beating techniques probe different types of motions.
78 citations
TL;DR: The role of the phase autocorrelation function in the generation of speckle, focusing and other scintillation effects produced when radiation is scattered by a deep random phase screen is investigated in this article.
Abstract: The role of the phase autocorrelation function in the generation of speckle, focusing and other scintillation effects produced when radiation is scattered by a deep random phase screen is investigated. New analytical formulae are obtained which reveal the mathematical origin of the observed physical phenomena, and these are supported by a range of numerically computed results.
65 citations
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TL;DR: In this article, the photon statistics of arbitrary fields in fully quantum-mechanical terms are discussed, and a general method of representing the density operator for the field is discussed as well as a simple formulation of a superposition law for photon fields.
Abstract: Methods are developed for discussing the photon statistics of arbitrary fields in fully quantum-mechanical terms. In order to keep the classical limit of quantum electrodynamics plainly in view, extensive use is made of the coherent states of the field. These states, which reduce the field correlation functions to factorized forms, are shown to offer a convenient basis for the description of fields of all types. Although they are not orthogonal to one another, the coherent states form a complete set. It is shown that any quantum state of the field may be expanded in terms of them in a unique way. Expansions are also developed for arbitrary operators in terms of products of the coherent state vectors. These expansions are discussed as a general method of representing the density operator for the field. A particular form is exhibited for the density operator which makes it possible to carry out many quantum-mechanical calculations by methods resembling those of classical theory. This representation permits clear insights into the essential distinction between the quantum and classical descriptions of the field. It leads, in addition, to a simple formulation of a superposition law for photon fields. Detailed discussions are given of the incoherent fields which are generated by superposing the outputs of many stationary sources. These fields are all shown to have intimately related properties, some of which have been known for the particular case of blackbody radiation.
5,372 citations
01 Jan 1966
TL;DR: In this paper, the authors calculate the intensity spectrum to be expected for clipped (also called "limited") noise, where the limiting amplitude is very small compared to the rms amplitude before clipping.
Abstract: The present report calculates in some detail the (intensity) spectrum to be expected for clipped (also called "limited") noise Two cases are considered: (A), the clipping of an unmodulated noise band (DINA) [1] and (B) a carrier modulated by clipped noise The computations are made for various shapes of noise bands before clipping, viz, 1) a uniform or rectangular structure, 2) a Gaussian distribution, 3) an "optical" (Lorentzian) shape factor of the type 1/[(v-v 0 )2+ δ2] The simplest type of calculation to make is that for what we term "extreme clipping," wherein the limiting amplitude is very small compared to the rms amplitude before clipping The mathematical theory for this is given in Section III, while Section IV develops the theory for clipping at an arbitrary level The basic mathematical method, which is rather general and is useful, we believe, for a variety of noise problems, is presented in Section II and consists in utilizing a relation between the correlation function and the normal surface, along lines suggested by Rice [1] The results of the calculation are discussed in Section I and are displayed in Figures 4-10 and Tables I and II If the clipping is not down to more than the rms level before limiting (equivalent to clipping at about 14 times the rms level after clipping), there is practically no distortion of the spectrum Even in the case of extreme clipping the wastage of power due to spoiling of the spectrum's uniformity is small, amounting to only 31 percent in (A) and 24 percent in (B) Of the 31 percent loss in (A), 19 percent is due to production of harmonics of the central frequency Corresponding harmonics are absent in (B) Clipping is beneficial for jamming purposes in either (A) or (B) since it reduces the peak power requirements In addition, in (B) it materially diminishes the wastage of power in the carrier frequency These facts are demonstrated particularly clearly by Tables I and II For instance, Table II shows us that in (B) the ratio of the energy in the noise sideband to that in the carrier is only 023 when the clipping level is twice the original rms noise level, but increases to 052 when these two levels are the same, and to 10 for extreme clipping It is to be cautioned that the present report calculates only the (intensity) spectrum of the clipped noise, and does not deal with its effectiveness on a receiver, which we hope to discuss later from a quantitative standpoint [2] We can, however, say qualitatively that if the receiver breadth is very small compared to the noise band, the received disturbance will have the same type of Gaussian fluctuation, and hence the same effectiveness as unclipped noise with the same spectral distribution On the other hand, if the receiver is comparable with the noise band in width, there will be, due to the clipping, a tendency for a "ceiling" in the resultant deflection of the recording device, and under these conditions the utility of clipped noise for jamming is materially diminished
493 citations
01 Sep 1959
TL;DR: In this article, the probability distribution p(n, T) of the number of counts n from a photoelectric detector illuminated by coherent light for a time T is studied, by associating photons stochastically with Gaussian random waves.
Abstract: The probability distribution p(n, T) of the number of counts n from a photoelectric detector illuminated by coherent light for a time T is studied, by associating photons stochastically with Gaussian random waves. The cumulants of the distribution are derived and it is shown to be of the expected form for a boson assembly in a limited volume of phase space. The distribution depends strongly on the degeneracy of the light beam. It approaches the Poisson form for classical particles at low degeneracies and the distribution characteristic of classical waves at high degeneracies. The analysis leads, incidentally, to an expression for the extent of the unit cell of phase space in the direction of the beam. It is argued that this should be adopted as the measure of coherence length.
386 citations
TL;DR: In this paper, the authors summarize significant developments in radar meteorology since 1958 and integrate them properly within the framework of knowledge existing prior to that time, and the emphasis in this chapter is on the basic principles and concepts of radar meteorologists.
Abstract: Publisher Summary In this chapter there were two main objectives: To summarize significant developments in radar meteorology since 1958 and to integrate them properly within the framework of knowledge existing prior to that time. The emphasis in this chapter is on the basic principles and concepts of radar meteorology. In order to relate the measured reflectivity to the size and concentration of the particles in the scattering volume, and thus to the conventional meteorological parameters, such as liquid water content and precipitation rate, this chapter is first to determine the radar cross section of an individual scatterer as a function of its refractive index, size, shape, and orientation. Because of the importance of precipitation rate in a variety of meteorological problems, ranging from flood warning to research on the water budget of a storm, extensive consideration is given to its measurement by radar. Radar measurements of storm structure and dimensions suffer from beam width and side lobe effects. In particular, the problem of estimating true storm height is discussed in some detail in the next section, and it is shown how the heights of intense storms may be greatly overestimated by the effects of side lobes, while the visible tops of some weaker storms may be underestimated. The use of Doppler or coherent radar techniques permits the measurement of the velocities of the scatterers, and so, provides a vital new dimension in radar probing of the atmosphere. The basic Doppler theory and the relation of the Doppler spectrum of the scatterers to the fluctuation spectrum of echo intensity on incoherent (conventional) pulse radar are reviewed in some detail. The final section is devoted to a comprehensive review of the long-elusive phenomenon of “angel” echoes.
246 citations
TL;DR: In this paper, the analogue of Van Vleck's theorem on the spectrum of clipped noise was formulated for use in optical spectroscopy, and the analogue was used for optical spectrograms.
Abstract: The analogue of a theorem of Van Vleck on the spectrum of clipped noise is formulated for use in optical spectroscopy.
97 citations