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In mathematics and physics, a phase space of a dynamical system is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. In partially coherent optical systems, the phase space is known as the Wigner distribution, representing simultaneously the values of the optical field in spatial and spatial frequency domains. Related to the notion of "etendue", the value of the optical field in phase-space is conserved and represents optical information.
The world studied by mathematical projections is a process of linearization. At some level of approximation, the structure of details collapses into an average behavior, that is the linear relationship between two variables in a system. Hence cause-effect laws that are so common in the physical sciences are also seen in optics as theories of diffraction. A common approximation in denoting phase objects, is replacing them with purely imaginary value, according to the first term of the Taylor series approximation therein. This is of relevance in theories of phase contrast (eg. Zernike phase contrast), where the object is assumed to be purely imaginary, in scattering theories such as for photomasks in lithography, where scattering from thick mask edges is assumed to be imaginary valued, or in theories of weak speckle, where the speckle is used to image the contrast transfer function of the system. This approximation is akin to the weak object approximation, of Max Born and Rytov. A phase space description of linear assumptions in scattering is demonstrated, which is sensitive to both the coherence of light and the structure of the scatterer. The method can be extended to volumetric scattering, where light propagation is simply a shear in phase space, hence all optical processes are reduced to topological transforms of the underlying Wigner distribution. Using a weak phase object, interactions of symmetries of the imaging system corresponding to the symmetries of the scattered speckle shown by theory are also seen in experiments at optical and EUV wavelengths.
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