The dynamics of an extended, spatiotemporally chaotic system might appear extremely complex. Nevertheless, the local dynamics, observed through a finite spatiotemporal window, can often be thought of as a visitation sequence of a finite repertoire of finite patterns. To make statistical predictions about the system, one needs to know how often a given pattern occurs. Here we address this fundamental question within a spatiotemporal cat, a one-dimensional spatial lattice of coupled cat maps evolving in time. In spatiotemporal cat, any spatiotemporal state is labeled by a unique two-dimensional lattice of symbols from a finite alphabet, with the lattice states and their symbolic representation related linearly (hence ‘linear encoding’). We show that the state of the system over a finite spatiotemporal domain can be described with exponentially increasing precision by a finite pattern of symbols, and we provide a systematic, lattice Green’s function methodology to calculate the frequency (i.e., the measure) of such states.

We introduce a simple delay-scanned complete spatiotemporal intensity-and-phase measurement technique based on wavelength-multiplexed holography to characterize long, complex pulses in space and time. We demonstrate it using pulses emerging from multi-mode fiber. This technique extends the temporal range and spectral resolution of the single-frame STRIPED FISH technique without using an otherwise-required expensive ultranarrow-bandpass filter. With this technique, we measured the complete intensity and phase of up to ten fiber modes from a multi-mode fiber (normalized frequency V ≈10) over a ~3ps time range. Spatiotemporal complexities such as intermodal delay, modal dispersion, and material dispersion were also intuitively displayed by the retrieved results. Agreement between the reconstructed color movies and the monitored time-averaged spatial profiles confirms the validity to this delay-scanned STRIPED FISH method.

The past 30 years have seen spectacular progress in the development of techniques for measuring the complete temporal field, and even the complete spatiotemporal field, of ultrashort laser pulses. The challenge has been to measure a pulse without the use of a shorter event or an independent known reference pulse, neither of which is typically available. We begin with autocorrelation, the first such “self-referenced” pulse-measurement method ever proposed, which measures only a rough pulse length, and we describe its limitations. One such limitation is the presence of a somewhat unintuitive “coherent artifact,” which occurs for complicated pulses and also when averaging over a train of pulses whose shapes vary from pulse to pulse. We then describe the most important modern techniques capable of measuring the complete temporal intensity and phase of even complicated ultrashort pulses, as well as their ability (or inability) to measure such unstable pulse trains. A pulse reliably measured with such a device can then be used as a reference pulse in conjunction with another technique, such as spectral interferometry or holography, to measure pulses otherwise unmeasurable by a self-referenced technique. Examples include techniques for measuring low-intensity pulse(s) and for measuring the complete spatiotemporal intensity and phase of arbitrary pulse(s). This Tutorial is limited to well-established, proven methods, but other methods whose description proves instructive will be discussed.

No abstract