Physical Model for the Object

Imagine photons to be streaming from the entrance slit of area A toward the exit slit. These of course include all wavelengths of the source. Picture next just photons of one wavelength (or wave number) xm as they flow from the entrance slit. Because these are quantum-mechanical entities, they cannot occupy continuously all positions in space during their flow. Instead, they may occupy only finite positions in space, called degrees of freedom (df) or modes. These are shown schematically as cubes in Fig. 3. Note that there are but a finite number zm of such cubes and that we must subscript z because the number of modes will vary from one wavelength to another. 

Each photon that flows from the entrance slit may exist only in such a mode. However, more than one may crowd in. This is a property of Bose-Einstein particles, to which photons belong as a class. Photons are also known to be microscopically indistinguishable, so that a given configuration of them within the modes cannot be distinguished from the same configuration where some of the photons have interchanged mode positions. As we shall see, the ML solution that we seek will depend vitally on this Bose-Einstein aspect of photon statistics. 

Consider one resolution cell xm of the spectrum. The number of df, zm, is defined as the number of independent volumes that exist over the volume swept out by photons leaving the entrance slit during one exposure time t (see Fig. 3). An independent volume is the three-dimensional space over which a photon is coherent. Hence, if it has a coherence length c Arm, with c the speed of light and Arm the coherence time, and if it has a coherence area cTm, the coherence volume is simply their product, or c Arm om. On the other hand, the total volume swept out by a wave front of photons all leaving the aperture of area A during exposure time t is ctA. Hence the number of coherence volumes is 

Of these factors, A and t are of course directly measurable and need no further consideration. 

The coherence time Axm relates to abscissa xm and resolution interval Ax as follows. By the Heisenberg uncertainty principle, Axm goes inversely with spectral purity, or 

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