Insights on correlation dimension from dynamics mapping of three experimental nonlinear laser systems

Abstract

BackgroundWe have analysed large data sets consisting of tens of thousands of time series from three Type B laser systems: a semiconductor laser in a photonic integrated chip, a semiconductor laser subject to optical feedback from a long free-space-external-cavity, and a solid-state laser subject to optical injection from a master laser. The lasers can deliver either constant, periodic, pulsed, or chaotic outputs when parameters such as the injection current and the level of external perturbation are varied. The systems represent examples of experimental nonlinear systems more generally and cover a broad range of complexity including systematically varying complexity in some regions.MethodsIn this work we have introduced a new procedure for semi-automatically interrogating experimental laser system output power time series to calculate the correlation dimension (CD) using the commonly adopted Grassberger-Proccacia algorithm. The new CD procedure is called the ‘minimum gradient detection algorithm’. A value of minimum gradient is returned for all time series in a data set. In some cases this can be identified as a CD, with uncertainty.FindingsApplying the new ‘minimum gradient detection algorithm’ CD procedure, we obtained robust measurements of the correlation dimension for many of the time series measured from each laser system. By mapping the results across an extended parameter space for operation of each laser system, we were able to confidently identify regions of low CD (CD < 3) and assign these robust values for the correlation dimension. However, in all three laser systems, we were not able to measure the correlation dimension at all parts of the parameter space. Nevertheless, by mapping the staged progress of the algorithm, we were able to broadly classify the dynamical output of the lasers at all parts of their respective parameter spaces. For two of the laser systems this included displaying regions of high-complexity chaos and dynamic noise. These high-complexity regions are differentiated from regions where the time series are dominated by technical noise. This is the first time such differentiation has been achieved using a CD analysis approach.ConclusionsMore can be known of the CD for a system when it is interrogated in a mapping context, than from calculations using isolated time series. This has been shown for three laser systems and the approach is expected to be useful in other areas of nonlinear science where large data sets are available and need to be semi-automatically analysed to provide real dimensional information about the complex dynamics. The CD/minimum gradient algorithm measure provides additional information that complements other measures of complexity and relative complexity, such as the permutation entropy; and conventional physical measurements.

Complexity mapping of a photonic integrated circuit laser using a correlation-dimension-based approach

“…They are almost exclusively values of 3 or less. This is consistent with prior research analyzing experimental signals to extract CD values [6,12,15,16]. We focus on two regions of mapped dynamics of the PIC device that show approximate CD value behavior ( Fig.…”

“…1 shows the average values of the minimum gradient as a function of the PICL operating conditions, for different tolerances on the spread around the average value. These values are obtained using the protocol described in [16]. The upper section of each plot is when the forward bias current to the gain/absorber (G/A) section amplifies the light fed back from the 10 mm long passive waveguide section of the device.…”

“…(Fig. 1(a) is reproduced from [16] where it is published under a Creative Commons Attribution (CC BY) license). Fig.…”

“…In this work the experimental output power time series from a PICL were considered and the complexity was quantified through the calculation of the minimum gradient values. The latter can be assigned as approximate CD values when a clear scaling region of constant value, as the embedding dimension is increased, is found [16]. The results reported here show how the minimum gradient values can be used in addition to the approximate CD values in order to evaluate the properties of time series dynamics.…”

“…We follow the same approach here and apply it for the case of the PICL system. A summarized table of the testing protocol is included in the supplementary information for reference [16].…”