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Physical Review Letters

Measuring the Phase of Spatially Coherent Polychromatic Fields

Aristide C. Dogariu, Gabriel Popescu

Published Nov 25, 2002• 15 citations
Publisher

Abstract

Realistic optical fields are never completely monochromatic and the interpretation of the concept of phase is not straightforward. It has been pointed out by Wolf that, if a strict condition of spatial coherence is imposed, the statistically averaged behavior of a polychromatic field can be described by an associated monochromatic wave. We show that, for spatially coherent polychromatic optical fields, the measurement of the phase of the second-order correlations determines the phase of this associated, spatially coherent field. We verify this prediction using a novel interferometric technique for measuring the cross-spectral density of a steady-state optical field.

DOI
10.1103/physrevlett.89.243902

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Realistic optical fields are never completely monochromatic and the interpretation of the concept of phase is not straightforward. It has been pointed out by Wolf that, if a strict condition of spatial coherence is imposed, the statistically averaged behavior of a polychromatic field can be described by an associated monochromatic wave. We show that, for spatially coherent polychromatic optical fields, the measurement of the phase of the second-order correlations determines the phase of this associated, spatially coherent field. We verify this prediction using a novel interferometric technique for measuring the cross-spectral density of a steady-state optical field.
Light is known to exhibit quantum uncertainty in terms of its amplitude, phase, and polarization. However, quantum uncertainty related to coherence, which is also a fundamental physical property of light, remains to be considered. Here, we formulate and explore the concept of quantum optical coherence uncertainty. We focus on the first-order coherence of the simplest possible light field, a purely monochromatic plane wave, which is classically completely stable. Starting from a scalar treatment, we show that the field displays zero coherence uncertainty only for a number state. We then proceed to the vectorial regime and establish that any state leads to coherence fluctuations, governed by a set of uncertainty relations depending on the polarization state and space-time points. Our work thus provides fundamental insights into the quantum character of optical coherence, with potential benefits in applications using highly sensitive interferometric and polarimetric techniques.
A new representation is obtained for the cross‐spectral density function of stationary optical wavefields. We show that under very general conditions it is possible to construct an ensemble of monochromatic wave functions, all of the same frequency ω, with the property that the cross‐spectral density function W(r−1,r−2,ω) of the field is expressible as a correlation between the values of the wave functions at the points r−1 and r−2. The representation is intimately related to a new mode expansion of the field and has a number of potential applications to various problems that involve wavefields of any state of coherence.
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.
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.
Interferometric photon-correlation measurements, which correspond to the second-order intensity cross correlations between the two output ports of an unbalanced Michelson (or Mach-Zehnder) interferometer, are sensitive to both amplitude and phase fluctuations of an incoming beam of light. Here, we present the theoretical framework behind these measurements and show that they can be used to unambiguously differentiate a coherent wave undergoing dynamical amplitude and phase fluctuations from a chaotic state of light. This technique may thus be used to characterize the output of nanolasers and monitor the onset of coherent emission.