The optical transfer functions *O*_{m} and the effect of incoherence. (*a*) The location of the three illumination beams (*purple dots*) in the back focal plane of the objective lens. Each point within each beam is mutually coherent only with the corresponding point of each of the other beams (which originates from the same point of the light source); the yellow points illustrate one such coherent point triplet. (*b*) Spatial frequency components of the light amplitude. For an objective lens that satisfies the Sine Condition, the pupil plane maps directly onto the lateral spatial frequency (*dotted lines* from *a* to *b*). The difference vectors between mutually coherent amplitude components (*amber arrows* in *b*) will therefore have the same lateral extent, but will differ axially, for different source points. (*c*) Resulting spatial frequency components of the illumination intensity; the *m* = 0, ±1, and ±2 components are shown in red, green, and blue, respectively. The non-zero-*m* components, which get a contribution from each coherent point pair (*amber arrows*, corresponding to those in *b*), are axially broadened into line segments, but remain delta-function-like laterally. The raw data (*d* and *g*) is a sum of five information bands, corresponding to the five lateral illumination-intensity frequencies. Once five images are observed with different pattern phases, the bands can be separated (*e* and *h*; only the *m* = 0 (*red*), 1 (*green*), and 2 (*blue*) bands are shown), and computationally moved back to their correct lateral position (*f*). Each band has been broadened by convolution with the corresponding frequency component of the illumination intensity (corresponding colors); the order 1 component, for example, consists of two parts (*e*, *green*). The observed raw data (*g*) and separated bands (*h*) for a point-source object are in excellent agreement with the theoretical expectations.

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