### Time-delay signature concealment of polarization-resolved chaos outputs in vertical-cavity surface-emitting lasers with variable-polarization filtered optical feedback

Recently, abundant research has been done to the subject of semiconductor laser (SL)-based or erbium-doped fiber laser (EDFL)-based optical chaos for its various applications such as chaotic radar, secure communication, fast physical random bit generation, optical logic gates^{[1–8" target="_self" style="display: inline;">–8]}, and so on. Through introducing external perturbations such as optical injection, optoelectronic feedback, or optical feedback^{[9–12" target="_self" style="display: inline;">–12]}, SLs can be driven into a chaotic output state, where optical feedback SL has been usually regarded as a primary candidate for an optical chaos source since introducing optical feedback can relatively easily yield complex chaos. In general, an obvious time-delay signature (TDS) can be observed in a SL chaotic system with optical feedback^{[13]}. If this TDS-contained chaos signal is used as a carrier in chaos communication, the system security will be threatened since the reconstruction of the SL chaotic system can be realized via some time series analysis methods for chaotic systems^{[14]}. As a consequence, it is indispensable to search for some solutions to conceal the TDS of chaos to ensure the system security. Pre-existing research has proven that through selecting a suitable injection current and feedback strength of the SL, the TDS of chaos in an edge-emitting semiconducting laser (EESL) chaotic system with a single-mirror optical feedback can be suppressed^{[15]}. Simultaneously, through choosing appropriate feedback parameters, the TDS in a double-mirror optical feedback EESL chaotic system can be suppressed^{[16]}. Additionally, after inserting some components with chromatic dispersion such as a fiber Bragg grating (FBG) or a Fabry–Perot-type filter into the optical feedback loop, the TDS of chaos generated by EESLs can also be suppressed^{[17,18]}.

As one kind of microchip laser, vertical-cavity surface-emitting lasers (VCSELs) have some superior properties including low threshold current, low cost, single longitudinal mode operation^{[19,20]}, and so on. Distinct from EESLs, VCSELs normally emit two polarized components [namely, the x-polarized component (x-PC) and the y-polarized component (y-PC)] because of the weak anisotropies of the material and cavity, which spawns some unique feedback techniques such as polarization-preserved optical feedback (PPOF), polarization-rotated optical feedback (PROF), variable-polarization mirror optical feedback (VPMOF)^{[21–25" target="_self" style="display: inline;">–25]}, and so on. The TDS suppression of chaos generated by a VCSEL with single-VPMOF or double-VPMOF has been reported^{[2325" target="_self" style="display: inline;">–25]}. Very recently, we used a FBG as the optical feedback device to construct a variable-polarization FBG optical feedback VCSEL (VPFBGOF–VCSEL) chaos system, and then investigated the TDS concealment of chaotic outputs in this system^{[26]}. The result shows that the TDS of the chaos output from the VPFBGOF–VCSEL chaotic system is weaker than that from the VPMOF–VCSEL chaotic system with pure mirror feedback.

In this work, after inserting a filter into the mirror optical feedback loop to supply filtered optical feedback, we propose a chaotic system based on VCSELs under variable-polarization filtered optical feedback (VPFOF) and investigate numerically the TDS of chaos in this chaotic system.

Figure

Combining the spin-flip model (SFM)^{[2426" target="_self" style="display: inline;">–26]} with VPFOF^{[27]} for such a chaotic system, the rate equations of a VPFOF–VCSEL can be expressed as ^{[28]}

Several approaches can be used to quantitatively evaluate the TDS of chaotic signals, such as the mutual information (MI)^{[13]}, self-correlation function (SF)^{[15]}, and permutation entropy (PE)^{[29]}. In this Letter, we adopt SF and PE. SF could be described as

Based on the information principle, PE is proposed and owns some superior advantages such as robustness to noise, fast calculation, and simplicity. PE can be simply described as follows. The intensity time series

After considering the suggestions in Ref. [29] and the unique features in VCSELs, the

Equations (^{[30]}:

Figure

#### Fig. 2. $P\text{-}I$ curve for a solitary running VCSEL, where the dashed line represents the x-PC and the solid line represents the y-PC.

First, we concentrate on the TDS of the chaos output under fixed

#### Fig. 3. Polarization-resolved time series (first and fifth columns), power spectra (second and sixth columns), SF curves (third and seventh columns), and PE curves (fourth and eighth columns) of a VPFOF–VCSEL under $\mathrm{\Lambda}=8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GHz}$ and ${\theta}_{p}=35\xb0$ for $\eta =0\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{ns}}^{-1}$ (first row), $\eta =5\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{ns}}^{-1}$ (second row), $\eta =15\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{ns}}^{-1}$ (third row), and $\eta =28\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{ns}}^{-1}$ (fourth row).

Second, we investigate the influence of the polarizer angle

#### Fig. 4. Polarization-resolved time series (first and fifth columns), power spectra (second and sixth columns), SF curves (third and seventh columns), and PE curves (fourth and eighth columns) of a VPFOF–VCSEL under $\mathrm{\Lambda}=8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GHz}$ and $\eta =15\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{ns}}^{-1}$ for ${\theta}_{p}=0\xb0$ (first row), ${\theta}_{p}=30\xb0$ (second row), ${\theta}_{p}=60\xb0$ (third row), and ${\theta}_{p}=90\xb0$ (fourth row).

Next, we research the total evolution of the TDS of the polarization-resolved output from the VPFOF–VCSEL in the parameter space of

#### Fig. 5. Maps of $\sigma $ under different $\eta $ and $\mathrm{\Lambda}$ for a VCSEL subject to VPFOF with (a) ${\theta}_{p}=10\xb0$ , (b) ${\theta}_{p}=40\xb0$ , (c) ${\theta}_{p}=70\xb0$ , and (d) PPFOF.

Finally, we give the maps of

#### Fig. 6. Maps of $\sigma $ under different $\eta $ and ${\theta}_{p}$ for a VCSEL subject to VPFOF with (a) $\mathrm{\Lambda}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GHz}$ , (b) $\mathrm{\Lambda}=3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GHz}$ , (c) $\mathrm{\Lambda}=8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GHz}$ , and (d) VPMOF.

In conclusion, we research and numerically analyze the TDS in a chaotic system based on a VCSEL subject to VPFOF. By using the analytic techniques of SF and PE, the TDS of the chaos output can be quantitatively evaluated, and then the effects of feedback rate, polarizer angle, and filter bandwidth of the filter on the TDS of the chaotic outputs are analyzed. The results show that, through optimizing some operation parameters, the TDS of polarization-resolved outputs from the VPFOF–VCSEL can be simultaneously suppressed. In comparison with a PPFOF (or VPMOF) VCSELs chaos system, such a VPFOF–VCSEL system shows some superiority in simultaneously acquiring polarization-resolved chaotic signals with weak TDS.

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Li Zhou, Guangqiong Xia, Zhuqiang Zhong, Jiagui Wu, Shuntian Wang, Zhengmao Wu. Time-delay signature concealment of polarization-resolved chaos outputs in vertical-cavity surface-emitting lasers with variable-polarization filtered optical feedback[J]. Chinese Optics Letters, 2015, 13(9): 091401.