Generalized Projections
Although it is theoretically somewhat complex, the method of Generalized Projections has proven to be an extremely reliable method for retrieving pulses from FROG traces. Unfortunately, its sophistication is the source of some misunderstanding and mistrust from scientists in the optics community. Hence, this section will attempt to give some insight into the basic philosophy and implementation of the method, if not its detailed workings.
First, imagine a space that contains all possible signal electric fields. For a given measurement, there is a set of these fields that will satisfy the measured FROG trace. We refer to these fields as satisfying the data constraint. There is another set that consists of the signal fields that can be expressed using the form for the nonlinear interaction used in the measurement. For SHG, this is the set of fields that can be expressed in the form . This is referred to as satisfying the mathematical form constraint.
These two sets intersect at exactly one point. There is only one possible signal field that both has the correct intensity to match the data trace and fits the mathematical form dictated by the nonlinear interaction. To find that point, which will give the pulse we’re trying to measure, Generalized Projections is used. The Generalized Projections Algorithm operates in this electric field space. At each step, we find the closest point to the current guess point that will satisfy the constraint for the other set. That is, the current guess is “projected” onto the other set. This closest point becomes the new current guess, and the closest point on the first set is found. By alternating between projecting onto the mathematical constraint set and projecting onto the data constraint set, we eventually end up at the solution.
Projecting onto the data constraint set is simple. To be in that set, the magnitude squared of the signal field has to match the intensity measured by the trace. The signal field is Fourier transformed to . The closest point in the data constraint set is found by replacing the magnitude of by the magnitude of the data, leaving the phase of intact.
Projecting onto the mathematical constraint set is not simple. Unlike the data constraint, there is not an easy way to tell which point in the mathematical constraint set is closest. A general expression for the distance between the current point and any point in the mathematical constraint set is created, and then that expression is minimized by taking the gradient of that distance with respect the current field guess. This process is discussed in more detail in this paper.
This cycle is repeated until the error between the signal guess and the data constraint (after applying the mathematical constraint) reaches some target minimum value. can be found by simply integrating with respect to delay . A second FROG trace is usually constructed mathematically from the solution and compared with the original measurement.
Read more about this topic: Frequency-resolved Optical Gating, Retrieval Algorithm
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