High-resolution Interferometric Diagnostics for Ultrashort Pulses

3. PHASE RECONSTRUCTION ALGORITHM FOR MULTIPLE SPECTRAL SHEARINGINTERFEROMETRYoping a multiple-shear algorithm, and section 3.2 presents a quantitative analysis of the precisionof SSI. Section 3.3 discusses the nature of the raw data, and then section 3.4 addresses samplingissues which immediately arise when multiple shears are being considered. Section 3.5 thenpresents the main algorithim. Section 3.6 discusses the implications for alleviating the relativephase ambiguity. Sections 3.7 and 3.8 together present a general argument that using multipleshears is always beneficial for the precision; the former considers the cost in terms of signal-tonoiseratio (SNR) of acquiring multiple shears, whilst the latter considers the benefits. Section 3.9gives some numerical examples and section 3.10 presents a summary and outlook.3.1 MotivationThis section presents several broad qualitative arguments for the use of multiple shearing spectralinterferometry.All else being equal, the noise sensitivity of SSI — that is, the precision loss due to a givendetector noise amplitude — increases with the number of sampling points across the spectrum.This introduces challenges of precision when characterising complex pulses, since in almost alldefinitions complexity is quantified by the time-bandwidth product which is proportional to therequired number of sampling points. Practically speaking, complex pulses arise frequently in ultrafastoptics, such as in coherent control [7], supercontinuum generation [268], filamentation[269], telecommunications [270] and micromachining [271]. Two factors are at play in the loss ofprecision: first, increasing the number of sampling points means increasing the effective resolutionof the spectrometer, decreasing the amount of light available and inherently reducing the precision.In practice, this is often accomplished by increasing the passband of the Fourier-domainfilter used in the processing. Second, the concatenation introduces an error at each samplingpoint; if the errors are independent (as is shown to be the usual case) then they accumulate in thefashion of a random walk. It is plausible that a scheme involving multiple shears could bypass thesecond of the aforementioned channels through which pulse complexity reduces precision; thatis, the random-walk like concatenation of errors. A small shear could resolve fine spectral details.Errors in the small shear measurement could be prevented from accumulating by the incorpo-64