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.
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).
Imaging is the estimation of the spatial distribution of a physical object by measuring the optical radiation it emits, or by making use of an optical wave that interacts with the object, via reflection or transmission, for example, before being measured by a detector. The resolution of an imaging system is limited by the inability to localize the optical field at points of the object. Under otherwise ideal conditions, resolution is limited by diffraction. The sensitivity of an imaging system is limited by the uncertainty in the measurement. Under ideal conditions, this is determined by photon noise, which depends on the statistical fluctuations of the light. In conventional imaging systems, an extended detector, such as a CCD camera or an array detector, measures the spatial distribution of the optical intensity, which is proportional to the photon flux density. In interferometric systems, the spatial distribution of the optical field is inferred from measurements of the optical intensity. With the emergence of coherence theory, imaging systems based on measurements of the second-order coherence function at pairs of points in the detection plane were developed. An example is the imaging of an incoherent object based on the van Cittert-Zernike theorem. Imaging systems based on measurement of intensity correlation, or the photon coincidence rate, at pairs of points, were developed in the 1960s. A classic example of the photon-correlation imaging of an object emitting thermal light is stellar imaging using a Hanbury-Brown-Twiss intensity-correlation interferometer. More recently, two-photon light, which may be generated via spontaneous parametric downconversion in a second-order nonlinear optical crystal, has been used for imaging. This type of two-photon (or biphoton) imaging, which has come to be called quantum imaging, is also based on the measurement of photon coincidence by the use of photon-counting array detectors or by scanning two photon-counting detectors at pairs of points. To compare the resolution and sensitivity of imaging systems based on the aforementioned types of measurements, it is necessary to derive expressions for the measured quantities in terms of the object distribution. The point-spread functions based on such expressions can be used to assess the resolution. One measure of the sensitivity of the imaging process is the signal-to-noise ratio (SNR) of the measured variables. The statistical nature of the light source must be known in order to determine the SNR. The purpose of this chapter is to compare the resolution and sensitivity of photon-correlation imaging systems that make use of thermal light and two-photon light. We will henceforth refer to these two imaging modalities as classical and quantum photon-correlation imaging, or simply classical and quantum imaging, respectively. Clearly, light in other quantum states can also be used for imaging.
The chief emphasis of the paper is given to the calculation of the information capacity of a beam of light for on-off signal modulation. A preliminary and approximate calculation in Sec. 2 is based on Gaussian modulation. On-off modulation permits an exact calculation of the information capacity; the calculation is based on the Poisson statistics of nondegenerate ensembles of Bose entities.Most of the results are given in terms of the information capacity per transmitted photon, measured in bits per photon, and called the information efficiency It. When there is no ambient light, and when the probability p that the gate is open is equal to one-half, the maximum possible information efficiency is just one bit per transmitted photon; when, however, the probability p of an open gate is less than one-half, the information efficiency It may be higher, with the upper bound log2(1/p).The calculation of the information efficiency in Sec. 3 supposes that no light reaches the receiver except that from the gated source. In Sec. 4 ambient light also is permitted to reach the receiver; extensive numerical results were obtained with a high-speed digital computer and are presented in tables and graphs.The term light is used throughout the paper instead of radiation, in order to emphasize that the results hold only for nondegenerate beams of radiation. The output of laser oscillators is therefore excluded.
The process of attenuation of electromagnetic energy when transmitted over some communication channel is stochastic in nature. Hence one can expect that, since equivocation is imposed in the transmission, in addition to reduction of average signal power, there will be a consequent degradation in channel capacity. We have attempted to calculate the effect of this phenomenon on an otherwise noiseless photon channel and when the source itself is assumed to have no undesired fluctuations. In order to do so, we have employed a straightforward variational technique and introduced the partition function method of statistical mechanics to obtain an expression for channel capacity. Using approximate methods, we find that the maximum information transfer is achieved when the probability of receiving exactlykphotons goes down, asymptotically with largek, ase^{-ck}/ \sqrt{k}, wherecis a positive constant. For very large received average energyU(in photon number), the channel capacity behaves approximately asC = \frac{1}{2} \ln U, one-half the capacity of an ideal noiseless photon channel (given by Gordon's formula for the capacity of an electromagnetic wave). For very small received average energy, we find that the (optimum) received distribution is very nearly Bose-Einstein, and the channel capacity is not degraded below its ideal value. The results are applicable to an idealized laser oscillator and also to photoelectric detection. That is, both devices by themselves may be regarded as "channels" for which the input may be modulated as desired and the output is stochastically dependent on the input.