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Journal of the Optical Society of America

Resolvable degrees of freedom in observation of a coherent object

Carl W. Helstrom

Published Jun 1, 1977• 13 citations
Publisher

Abstract

The Shannon number is shown to specify the number of degrees of freedom that can be resolved in a coherent object scrutinized by a distant optical instrument that extracts information about the object from the light passing through its aperture. The quantum-mechanical nature of the light is taken into account, and the problem of determining the object amplitude distribution is treated by means of quantum estimation theory, which defines a limitation on the accuracy of an estimate of the object distribution that cannot be surpassed by any means of processing the received light. Fundamentally, the limitation arises from the inability of more than a finite number of modes of excitation of the object to deposit a significant number of photons in the aperture.

DOI
10.1364/josa.67.000833

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The Shannon number is shown to specify the number of degrees of freedom that can be resolved in a coherent object scrutinized by a distant optical instrument that extracts information about the object from the light passing through its aperture. The quantum-mechanical nature of the light is taken into account, and the problem of determining the object amplitude distribution is treated by means of quantum estimation theory, which defines a limitation on the accuracy of an estimate of the object distribution that cannot be surpassed by any means of processing the received light. Fundamentally, the limitation arises from the inability of more than a finite number of modes of excitation of the object to deposit a significant number of photons in the aperture.
By means of the quantum-mechanical form of the Cramér-Rao inequality, a lower bound is set to the mean-square error in an unbiased estimate of a parameter of an incoherently radiating object observed in the presence of thermal background light by an optical system collecting light through a finite aperture. Estimates of absolute radiance, frequency, and position of the object are specifically analyzed. The bounds reduce in the classical limit to those previously obtained, but are valid in the quantum limit as well. When the background vanishes, the bounds depend only weakly on the effective number of independent spatial and temporal degrees of freedom of the object light at the aperture.
The detection of an incoherent object, light from which is received in the presence of thermal radiant energy, is treated by quantum detection theory under the assumption that the product of the observation time and the bandwidth of the object light is large. The threshold detector measures a quantum-mechanical operator that is a quadratic functional of the field at the aperture of the receiver. The statistics of the outcome of this measurement are approximately Poisson, and the detectability of the object depends—as under a background limitation—on the number of effectively independent spatio-temporal modes of the field. The detectabilities of objects with a rectangular and a Lorentz spectrum are compared under the assumption that the light from each possesses first-order coherence at the aperture.
We consider the physical limitations imposed on the information content of an image by the wave and quantum nature of light, when the image is obtained by illuminating a reflecting or transmitting planar object by natural---i.e., fully thermalized---light, or by observation of an object emitting incoherent (thermal) radiation. The discreteness of the degrees of freedom and the statistical properties of thermal radiation are taken into account. We derive the maximum amount of information that can be retrieved from the object. This amount is always finite and is proportional to the area of the object, the solid angle under which the entrance pupil of the receiver is seen from the object, and the time of observation. An explicit expression for the information in the case where the information recorded by the receiver obeys Planck's spectral distribution is obtained. The amount of information per photon of recorded radiation is a universal numerical constant, independent of the parameters of observation.
When any two optical fields are superposed, a fringe pattern will be formed if the observation interval is short enough (within the coherent time) and a phase value may be extracted from the fringe and assigned as the phase difference between the fields. When measured against a common reference field, phase can be defined for an optical field. However, when the intensities of the optical fields are low or the observation time is so short that only a few photons are detected, the fringe pattern is unrecognizable due to the particle nature of light. By analyzing the fringe patterns simulated by a quantum Monte Carlo method, we find that there exist uncertainties in determining the phase value. This uncertainty is universal for any optical field with a finite photon number and decreases to zero as the photon number of the field increases to infinity or the classical limit is approached. The existence of this uncertainty in determining the phase value makes it problematic to define a quantum mechanical phase. An interference experiment with coherent state is performed to confirm the theoretical predictions. The implication of the above is that phase is a classical quantity that is associated with a large number of particles and may not have a quantum mechanical analogy.