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Proceedings of the Physical Society

The Measures of Bandwidth and Coherence Time in Optics

L. Mandeļ, E. Wolf

Published Oct 1, 1962• 65 citations
Publisher

Abstract

Several different definitions of the coherence time Δτ can be found in the literature and it is generally assumed that, in cases of practical interest, the different definitions give values of the same order of magnitude. The same remark applies also to definitions of the effective bandwidth Δν. Moreover it is generally supposed that these two quantities are related by the order of magnitude formula Δτ similar 1/Δν. It is shown from a detailed examination of one particular case of practical interest that the bandwidths as defined by two of the previously proposed formulae are of quite different orders of magnitude, although the correspondingly defined coherence times differ only by a factor of order unity. Similarly it may happen that the two values of Δ r differ greatly, whereas the corresponding values of Δν are nearly equal. This discussion shows that the reciprocity relation Δτ similar 1/Δν must be treated with caution, especially in the case of multiple-peaked distributions, such as arise in optical masers.

DOI
10.1088/0370-1328/80/4/311

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•Jan 1, 1986•0 citations
Several different definitions of the coherence time Δτ can be found in the literature and it is generally assumed that, in cases of practical interest, the different definitions give values of the same order of magnitude. The same remark applies also to definitions of the effective bandwidth Δν. Moreover it is generally supposed that these two quantities are related by the order of magnitude formula Δτ similar 1/Δν. It is shown from a detailed examination of one particular case of practical interest that the bandwidths as defined by two of the previously proposed formulae are of quite different orders of magnitude, although the correspondingly defined coherence times differ only by a factor of order unity. Similarly it may happen that the two values of Δ r differ greatly, whereas the corresponding values of Δν are nearly equal. This discussion shows that the reciprocity relation Δτ similar 1/Δν must be treated with caution, especially in the case of multiple-peaked distributions, such as arise in optical masers.
This paper is concerned with establishing a rigorous reciprocity relation of the Heisenberg type between the coherence time and the bandwidth of polychromatic radiation. Following upon an observation of Gabor (1946) that the usual formulation of the reciprocity inequality which relates the effective duration of a signal and its effective bandwidth is unsatisfactory for signals that are intrinsically real (e.g., an electric signal) a modified version is here presented. Our analysis is closely related to that of Gabor and supplements it in several respects. In particular we show that his formulation, like our own, is restricted to signals with zero mean value. Contrary to general belief, there is no evidence that the inequality holds when this restriction is removed. A definition of the coherence time of light is then proposed in terms of the (measurable) correlation function that plays a central part in the theory of partial coherence. The coherence time obeys rigorously a reciprocity inequality of the required type and is shown to have a simple interpretation in terms of averages involving the correlations in fluctuations of the instantaneous intensity. It is also shown that the coherence time may be determined experimentally from Michelson's visibility curve of interference fringes.
First, the effect of detuning on squeezing of coherent light is studied by Jaynes-Cummings model in a large N (initial mean photon number) limit. Analytical formulae for quadrature variances are obtained both in one-photon and two-photon transition cases. These formulae show that the effect is quite different in these two cases. In addition, explicit expressions for the k-th minimum of the suppressed variance are given when k is fairly large, which obviously shows it tends to zero according to 1/k2. Second, squeezing of coherent light field in a cavity by repeated interaction with atoms is studied. Numerical calculation shows the variance (delta) can be reduced by this process to a certain extent for suitably chosen interaction time t. But the squeezing becomes worse if the number of times K is too large. In the two-photon transition case, with a favorably chosen value of t, analytical formulae for (delta) (N, K) are obtained in large N approximation. These formulae show definitely that the least variance (delta) goes down when K increases up to quite a large value, and the achievable minimum value of (delta) 2 is in the order of 1/N1/2. Besides, the light field so obtained is nearly in a pure state with a minimum uncertainty product (delta) 1(delta) 2 equals 1/4.
Abstract The modern theory of coherence is based on correlation functions. A generic example could be written 〈 V ∗ ( t 1 ) V ( t 2 ) 〉 , denoting an average of products of the values of a signal V ( t ) at two specified times. Here we infer that t is a degree of freedom that the signal depends on. Typically, physical variables depend on more than one degree of freedom, and recognition of this has prompted attention to some interesting questions for the correlation functions and the several coherences that can be attributed to the same optical field. We examine some of the questions arising from the standpoint of experimental contexts. Degree of polarizability and degree of entanglement (classical non-separability) can serve as starting points for quantitative assignments.
Squeezing of a coherent light field in a cavity by repeated interaction with atoms is studied in the resonance Jaynes-Cummings model. The least quadrature variance \ensuremath{\delta} is calculated for different values of N (mean photon number), t (interaction interval for each time), and K (number of times of interaction). In the one-photon-transition case, \ensuremath{\delta} could be reduced by this process to a certain extent for suitably chosen t. The squeezing becomes worse if K is too large. \ensuremath{\delta} as a function of t and K (for fixed N) has a minimum, which becomes smaller when N increases. The situation of the two-photon-transition case is similar, the only new feature is that \ensuremath{\delta}(K,t) varies with t periodically when N is large. With a favorably chosen value of t (half of the period), analytical formulas for \ensuremath{\delta}(N,K) are obtained in the large-N approximation. These formulas show definitely, both for atoms initially in the upper or lower level, the least quadrature variance \ensuremath{\delta} goes down when K increases up to a quite large value (for atoms in the upper level it is of order N3/2{\mathit{N}}^{3/2}N3/2, while for atoms in the lower level it is of order N) and the achievable minimum value of ${\mathrm{\ensuremath{\delta}}}^{2}$ is of order 1/ \ensuremath{\surd}N . Moreover, the light fields so obtained are nearly in pure states with minimum uncertainty product ${\mathrm{\ensuremath{\delta}}}_{1}$${\mathrm{\ensuremath{\delta}}}_{2}$=1/4.
The complex degree of coherence γ(r1, r2, t1 − t2) of a stationary optical field is defined as the normalized cross-correlation function (|γ| ≤ 1) of the light disturbances at two space-time points (r1, t1), (r2, t2), the disturbances being represented by means of Gabor's analytic signals. In the present paper the general form of γ is examined under the condition that |γ| takes on the extreme value unity for all possible time differences τ = t1 − t2 (− ∞ < τ < ∞). Several cases are distinguished, depending on whether this condition is satisfied for some points or for all points in some fixed domain of space. It is pointed out, that the previously published derivations of the relevant theorems contain serious errors. The methods employed in the present paper make use of the property of nonnegative definiteness which the complex degree of coherence is shown to obey. The results have a bearing on the important but as yet unsolved ``phase problem'' of optical coherence theory.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.