Quantifying Chaotic Unpredictability of Vertical-Cavity Surface-Emitting Lasers With Polarized Optical Feedback via Permutation Entropy

Abstract

To quantify the unpredictability of chaotic signals generated by vertical-cavity surface-emitting lasers (VCSELs) subject to polarized optical feedback, we numerically investigate the properties of recently introduced quantifier based on information theory, i.e., permutation entropy (PE) <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> . VCSEL1 subject to polarization-preserved optical feedback and VCSEL2 subject to polarization-rotated optical feedback are considered. The <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> for X-polarization (XP) mode, Y-polarization (YP) mode, and total output for both VCSELs are discussed and compared in detail. The influences of feedback strengths, feedback delays, and injection currents are focused on. For lower injection current, the 2-D maps illustrate that the trends of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> for the XP mode, YP mode, and total output are different for both VCSELs. While for higher injection current, the values of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> for both VCSELs increase with the feedback strengths, are not sensitive to feedback delays, and are higher than the counterparts for lower injection current. Moreover, the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> increases with injection current for both VCSELs with a decreasing slope. These results show that the PE is an effective tool for quantifying the degree of unpredictability of chaotic signals of VCSELs, and provides valuable information for choosing the proper chaotic carrier.