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.
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.
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.
A most common way to store information is to encode it in the optical properties of an object and to retrieve it by viewing the object by reflected and transmitted natural (thermalized) light - or even by light emitted by the object itself - for a specified time interval. The discreteness of the radiation degrees of freedom and the statistical properties of thermal (incoherent) radiation impose limitations on the amount of the retrieved information. We derive the maximum 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.
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.
In the classical theory of light, an observation can be made as small as we please. However, for some very small objects this assumption does not hold true. For example, let us take a Mach-Zehnder interferometer [1], as shown in Fig. 6.1. It can be seen that only a very small fraction of light is required to illuminate the object transparency s(x, y). This small fraction of light, however, carries all the information that we intended. However, the other path of light carries almost all the energy of the light source. The corresponding irradiance distributed on the photographic emulsion is [2] /(/?, q) = R2 + \S(p, q)\2 + 2RS(p, q)cos[oc0p + >(/?, q)] (6.1) where R is the background (reference) beam, S(p, q) is the corresponding Fourier spectrum of the object transparency, (/?, q) are the corresponding spatial frequency coordinates, ao is an arbitrary constant, and S(p, q) = \S(p, q)\ exp [/>(/>, q)] (6.2) In our example, it is assumed that R » \S(p, q)\. Thus the interference term (the last term) is considered very small in comparison with the background irradiance. In principle, it is possible to enhance the weak interference term by means of a coherent optical processor [2], so it is simply a matter of making s(x, y) observable. In the classical theory, it is possible to observe with as weak an object beam as we wish. However, we know instinctively that this is not practical, since an increase in background light would at the same time increase background fluctuation. Thus the informational term (the interference term), could be completely buried in the background fluctuation. So in practice there exists a practical lower limit. Beyond this limit it is not possible to retrieve the information (the object).
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.