The transport of intensity equation for optical path length recovery using partially coherent illumination
Spotlight summary: Propagation-based phase retrieval, namely the recovery of the phase of a forward propagating (and typically paraxial) optical field from intensity measurements over one or more planes perpendicular to the optic axis, is of importance for a variety of radiation and matter wavefields.
It may surprise you that there is some confusion as to the meaning of the term “phase”, in the context of propagation-based phase retrieval. For a mono-energetic complex scalar wave, the situation is clear: the “phase” is equal to the argument (“arg”) of the complex function describing the field. Such a field, which is perfectly spatially and temporally coherent, has wavefronts corresponding to surfaces of constant phase.
However, all realistic sources are partially coherent. In the context of second-order coherence theory for statistically stationary quasi-monochromatic complex scalar fields, the statistics of the partially coherent disturbance can be described by two-point correlation functions such as the cross-spectral density W. For the case of W, which is a complex function of two spatial variables, “phase” refers to something different from (but reducible to) the “phase” of a mono-energetic scalar wave (which is a function of one, rather than two, spatial variables).
Now, an important subset of propagation-based phase-retrieval techniques consider intensity measurements over closely spaced planes perpendicular to the optic axis, as the input data for reconstructing the phase of a paraxial optical field. When partially coherent illumination is utilized in such phase-retrieval scenarios, it is unclear what “phase” is actually measured. Many works implicitly or explicitly assume strict monochromaticity, which in light of the fact that no field is perfectly coherent leaves unclear precisely what “phase” is being measured in such techniques.
The beautiful paper by Petruccelli, Tian and Barbastathis clarifies this. Beginning with the paraxial equation for the cross-spectral density, they derive an accompanying transport equation which is then manipulated into a partially-coherent form of the so-called transport of intensity equation (TIE). This partially coherent TIE, which reduces to its coherent counterpart in the limiting case of perfect coherence, is used as the basis for phase retrieval using partially coherent optical fields. The question of what “phase” this equation measures is clarified, with the resulting phase-retrieval algorithm convincingly applied to the measurement of optical path lengths via both simulations and visible-light experiments.
Reading this paper was a revelation to me. I learned a great deal from my first reading of it, and even more from several subsequent readings. It has my strongest recommendation.
--David M. Paganin
Technical Division: Information Acquisition, Processing, and Display
ToC Category: Image Processing
|OCIS Codes:||(030.0030) Coherence and statistical optics : Coherence and statistical optics|
|(100.5070) Image processing : Phase retrieval|
|(110.4980) Imaging systems : Partial coherence in imaging|
|(350.5030) Other areas of optics : Phase|
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