## Using chaos for remote sensing of laser radiation

Optics Express, Vol. 17, Issue 9, pp. 7491-7504 (2009)

http://dx.doi.org/10.1364/OE.17.007491

Acrobat PDF (744 KB)

### Abstract

An idea is proposed for detecting a weak laser signal from a remote source in the presence of strong background noise. The scheme exploits dynamical nonlinearities arising from heterodyning signal and reference fields inside an active reference laser cavity. This paper shows that for certain reference laser configurations, the resulting bifurcations in the reference laser may be used as warning of irradiation by a laser source.

© 2009 Optical Society of America

## 1. Introduction

4. G. Vemuri and R. Roy, “Super-regenerative laser receiver: Transient dynamics of a laser with an external signal,” Phys. Rev. A **39**, 2539–2543 (1989). [CrossRef] [PubMed]

5. I. Littler, S. Balle, K. Bergmann, G. Vemuri, and R. Roy, “Detection of weak signals via the decay of an unstable state: Initiation of an injection-seeded laser,” Phys. Rev. A **41**, 4131–4134 (1990). [CrossRef] [PubMed]

6. E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A **64**, 043815–043825 (2001). [CrossRef]

7. E. Lacot, O. Hugon, and F. Stoeckel, “Hopf amplification of frequency-shifted optical feedback,” Phys. Rev. A **67**, 053806–053815 (2003). [CrossRef]

8. M. B. Spencer and W. E. Lamb Jr., “Laser with a Transmitting Mirror,” Phys. Rev. A **5**, 884–892 (1972). [CrossRef]

9. R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. **18**, 976–983 (1982). [CrossRef]

10. F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, “Deterministic chaos in lasers with injected signal,” Opt. Commun. **51**, 308–314 (1984). [CrossRef]

11. T. B. Simpson, J.M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing, “Period-doubling cascades and chaos in a semiconductor laser with optical injection,” Phys. Rev. A **51**, 4181–4185 (1995). [CrossRef] [PubMed]

12. T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, “Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection,” Phys. Rev. A **53**, 4372–4380 (1996). [CrossRef] [PubMed]

13. T. B. Simpson, “Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection,” Opt. Commun. **215**, 135–151 (2003). [CrossRef]

14. N. Shunk and K. Peterman, “Noise analysis of injection-locked semiconductor injection lasers,” IEEE J. Quantum Electron. **22**, 642–650 (1986). [CrossRef]

15. W. A. van der Graaf, A. M. Levine, and D. Lenstra, “Diode lasers locked to noisy injection,” IEEE J. Quantum Electron. **33**, 434–442 (1997). [CrossRef]

16. S. K. Hwang, J. B. Gao, and J. M. Liu, “Noise-induced chaos in an optically injected semiconductor laser model,” Phys. Rev. E **61**, 5162–5170 (2000). [CrossRef]

## 2. Theory

*E*and total carrier density,

*N*. The justification for reducing the complicated semiconductor laser device to just two system variables is based on the carrier relaxation rates being much faster than any temporal variations in laser field and total carrier population. This allows the active-medium polarization to adiabatically follow the field and population variations, with the carrier populations described by quasiequilibrium distributions. Then, semiclassical laser theory gives the following equations of motion, [9

9. R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. **18**, 976–983 (1982). [CrossRef]

*ξ*is the differential gain at threshold carrier density,

*α*is the linewidth enhancement factor, Λ is the pump rate,

*γ*and

_{E}*γ*are the photon and population decay rates, respectively,

_{N}*c*is the speed of light in vacuum and

*n*is the background refractive index. The threshold gain and carrier density in the free-running reference laser are given by

_{b}*g*=

_{th}*n*(2

_{b}γ_{E}*c*Γ)

^{-1}and

*N*=

^{th}*N*+

^{tr}*g*

_{th}ξ^{-1}, where

*N*is the transparency carrier density. We account for the effect of reference laser noise from spontaneous emission and cavity optical-path length fluctuations by the complex random number function

_{tr}*F*=

_{L}*F*+

^{′}_{L}*iF*

^{″}

_{L}with statistical properties given by

17. C. H. Henry, “Theory of the linewidth of semiconductor laser,” IEEE J. Quantum Electron. **18**, 259–264 (1982). [CrossRef]

*v*is the lasing frequency and

*β*is the fraction of spontaneous emission energy into the lasing mode. The noise leads to a reference laser linewidth (full-width at half maximum) of Δ

*v*. The injected fields,

_{L}*E*(

_{a}*t*) = |

*E*|exp(−

_{a}*iϕ*) where

_{a}*a*=

*I*and

*B*, are from the external laser signal and background noise, respectively. Assuming that the noise in these fields arises solely from phase fluctuation, we have

*F*is a random number function with

_{a}*v*, the spectral linewidth, is an input parameter. The injected field polarizations are assumed to be same as that of the reference laser. For unpolarized injected fields,

_{a}*E*is the projection onto the reference laser polarization. There will be contributions in the orthogonal polarization, which typically sees less net gain. If this is not the case, mode competition may become important, which may result in interesting dynamics. This multimode situation is beyond the scope of the present analysis.

_{a}### 2.1. Optical injection induced instabilities in the absence of noise

13. T. B. Simpson, “Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection,” Opt. Commun. **215**, 135–151 (2003). [CrossRef]

20. S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A **72**, 033810–33818 (2005). [CrossRef]

19. E. Doedel, A. Champneys, T. Fairgrieve, Yu. Kuznetsov, B. Sandstede, and X. Wang, “AUTO 2000: Continuation and bifurcation software for ordinary differential equations,” http://sourceforge.net/projects/auto2000/.

*α*= 4, which is within the range expected for bulk or quantum-well active regions. Stable time-independent solutions correspond to phase locking of the laser field to the injected field. They occupy the shaded region bounded by saddle-node S and Hopf H bifurcation curves, that become tangent at the saddle-node-Hopf point G. Outside the shaded region are solutions corresponding to orbits by the complex electric field vector of various periods, quasiperiodic tori, and chaotic attractors, all of which describe time-dependent intensities. For non-zero detuning, the system starts out with periodic oscillation for small injection intensity and progresses through a number of instabilities and complicated nonlinear dynamics with increasing injection intensity, until locking is reached at either the Hopf bifurcation H or saddle-node bifurcation S (see, e.g., dashed line in Fig. 2). For clarity, the diagram shows only one type of bifurcation of periodic orbits namely the period-doubling (PD) bifurcation [12

12. T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, “Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection,” Phys. Rev. A **53**, 4372–4380 (1996). [CrossRef] [PubMed]

*α*= 0, a value expected for quantum dots, we found a drastic decrease in instabilities, indicating the unsuitability of a quantum-dot laser for implementing our scheme. Other factors can also influence the bifurcation curves. For example, an order of magnitude decrease in cavity decay rate increases the external-signal intensity necessary for triggering instabilities by six orders of magnitude. Additionally, the regions of complicated dynamics are appreciably smaller, and there are fewer instabilities and no chaotic dynamics.

*I*=

_{norm}*γ*

_{E}*ε*

_{0}

*ε*

_{b}

*E*

^{2}/(2

*h*̅

*v*

*Γγ*

_{N}*N*).) The figure shows that the laser undergoes several instabilities, displaying a rich range of complicated and chaotic oscillations. There is period-doubling bifurcation as indicated by the splitting of the single-maximum curve into two curves. Subsequent splittings with further increase in injection signal correspond to period-doubling cascade, eventually leading to complicated and chaotic oscillations that are separated by windows of periodic dynamics. The plot shows that only a small injection intensity (~ 300 mW/cm

_{th}^{2}) is necessary for inducing the change from stable operation to complicated dynamics involving strong intensity oscillations. Note also that further increasing the injection intensity reverses the trend, with the complicated oscillations becoming period-two oscillations (double curves), then period-one oscillation (single curve), and eventually, stationary behavior, i.e., phase locked operation.

^{2}.

## 3. Laser noise effects on system performance

*F*(

_{L}*t*) with resulting reference laser linewidths of 70MHz and 5GHz, respectively. These linewidths are achievable with edge-emitting or vertical-cavity lasers and without active stabilization. The trajectory of a noisy laser was advanced with a timestep Δ

*t*= 0.0005/

*γN*using the Runge-Kutta fourth-order routine for the deterministic calculations involving Eqs. (1) and (2) and the Euler routine for the stochastic calculations involving Eqs. (1), (2), and (6). Following such a procedure is particularly important to achieve convergence of the deterministic part for the weakly stable laser. The obtained time series of the complex laser signal were input into a Fast Fourier Transform (FFT) routine. The frequency spectra were calculated as a squared modulus of the FFT output. An individual spectrum panel in Figs. 4, 6, 8, and 9 represents the average over 500 spectra and has a resolution of ~ 4MHz.

*F*(

_{L}*t*). In the case of an intensity maxima plot, the net result is a smearing of the intensity traces but with all bifurcations clearly distinguishable as in the noiseless situation. For the reference laser configuration considered, this is the case for laser linewidths up to approximately 10MHz. As the laser linewidth increases, the interplay of noise and injection-induced dynamics becomes stronger, resulting in certain bifurcations being unidentifiable in an intensity maxima plot. This is the case for the 70MHz laser, where the intensity maxima trace (Fig. 5, top plot) shows no indication of period-doubling bifurcations at low injection intensity. However, some features of noiseless injection-induced response remain, such as the large-amplitude oscillations and the inverse-period-doubling bifurcations leading to injection locking. These bifurcations also eventually disappear with further increase in reference-laser noise, as illustrated in the bottom intensity maxima trace for the 5GHz laser.

*F*(

_{L}*t*). The resonances are at ±12GHz and higher harmonics, These resonances eventually blend into the background as shown in the 5GHz spectrum.

## 4. Effects of noise presence in injection signal

^{2}and 0.1W/cm

^{2}is the injected noise level above which the low injection bifurcations become unobservable. For low injected laser intensities, both plots show a wide and uniform spread of maxima values, arising from the strong response of the reference laser to the injected noise. For the lower background level of 0.06W/cm

^{2}, the increase in intensity maxima spread around an injected laser intensity of 0.2W/cm

^{2}, suggests the presence of a stable to chaos transition. At high injection intensity, the intensity maxima spread narrows, indicating stabilization of the reference laser via injection locking. There is even a faint indication of inverse period doubling for the low injected noise case. Even so, the overall lack of abrupt changes at both background noise levels makes difficult the use of intensity maxima for estimating the locations of bifurcation boundaries.

## 5. Conclusion

## Acknowledgments

## References and links

1. | B. Krauskopf and D. Lenstra (Eds.), |

2. | D. M. Kane and K. A. Shore (Eds.), |

3. | S. Wieczorek, B. Krauskopf, T.B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. |

4. | G. Vemuri and R. Roy, “Super-regenerative laser receiver: Transient dynamics of a laser with an external signal,” Phys. Rev. A |

5. | I. Littler, S. Balle, K. Bergmann, G. Vemuri, and R. Roy, “Detection of weak signals via the decay of an unstable state: Initiation of an injection-seeded laser,” Phys. Rev. A |

6. | E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A |

7. | E. Lacot, O. Hugon, and F. Stoeckel, “Hopf amplification of frequency-shifted optical feedback,” Phys. Rev. A |

8. | M. B. Spencer and W. E. Lamb Jr., “Laser with a Transmitting Mirror,” Phys. Rev. A |

9. | R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. |

10. | F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, “Deterministic chaos in lasers with injected signal,” Opt. Commun. |

11. | T. B. Simpson, J.M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing, “Period-doubling cascades and chaos in a semiconductor laser with optical injection,” Phys. Rev. A |

12. | T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, “Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection,” Phys. Rev. A |

13. | T. B. Simpson, “Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection,” Opt. Commun. |

14. | N. Shunk and K. Peterman, “Noise analysis of injection-locked semiconductor injection lasers,” IEEE J. Quantum Electron. |

15. | W. A. van der Graaf, A. M. Levine, and D. Lenstra, “Diode lasers locked to noisy injection,” IEEE J. Quantum Electron. |

16. | S. K. Hwang, J. B. Gao, and J. M. Liu, “Noise-induced chaos in an optically injected semiconductor laser model,” Phys. Rev. E |

17. | C. H. Henry, “Theory of the linewidth of semiconductor laser,” IEEE J. Quantum Electron. |

18. | G. Vemuri and R. Roy, “Effect of injected field statistics on transient dynamics of an injection seeded laser,” Opt. Commun. |

19. | E. Doedel, A. Champneys, T. Fairgrieve, Yu. Kuznetsov, B. Sandstede, and X. Wang, “AUTO 2000: Continuation and bifurcation software for ordinary differential equations,” http://sourceforge.net/projects/auto2000/. |

20. | S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A |

**OCIS Codes**

(140.1540) Lasers and laser optics : Chaos

(140.5960) Lasers and laser optics : Semiconductor lasers

(280.3420) Remote sensing and sensors : Laser sensors

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: March 3, 2009

Revised Manuscript: March 27, 2009

Manuscript Accepted: April 6, 2009

Published: April 22, 2009

**Citation**

Weng W. Chow and Sebastian Wieczorek, "Using chaos for remote sensing of laser radiation," Opt. Express **17**, 7491-7504 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7491

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### References

- B. Krauskopf and D. Lenstra (Eds.), Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, vol. 548, 2000.
- D. M. Kane and K. A. Shore (Eds.), Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, (Wiley, 2005, pp. 147-183).
- S. Wieczorek, B. Krauskopf, T.B. Simpson, and D. Lenstra, "The dynamical complexity of optically injected semiconductor lasers," Phys. Rep. 416, 1-128 (20050.
- G. Vemuri and R. Roy, "Super-regenerative laser receiver: Transient dynamics of a laser with an external signal," Phys. Rev. A 39, 2539-2543 (1989). [CrossRef] [PubMed]
- I. Littler, S. Balle, K. Bergmann, G. Vemuri, and R. Roy, "Detection of weak signals via the decay of an unstable state: Initiation of an injection-seeded laser," Phys. Rev. A 41, 4131-4134 (1990). [CrossRef] [PubMed]
- E. Lacot, R. Day, and F. Stoeckel, "Coherent laser detection by frequency-shifted optical feedback," Phys. Rev. A 64, 043815-043825 (2001). [CrossRef]
- E. Lacot, O. Hugon, and F. Stoeckel, "Hopf amplification of frequency-shifted optical feedback," Phys. Rev. A 67, 053806-053815 (2003). [CrossRef]
- M. B. Spencer and W. E. Lamb, Jr., "Laser with a Transmitting Mirror," Phys. Rev. A 5, 884-892 (1972). [CrossRef]
- R. Lang, "Injection locking properties of a semiconductor laser," IEEE J. Quantum Electron. 18, 976-983 (1982). [CrossRef]
- F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, "Deterministic chaos in lasers with injected signal," Opt. Commun. 51, 308-314 (1984). [CrossRef]
- T. B. Simpson, J.M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing, "Period-doubling cascades and chaos in a semiconductor laser with optical injection," Phys. Rev. A 51, 4181-4185 (1995). [CrossRef] [PubMed]
- T. Erneux, V. Kovanis, A. Gavrielides, and P. M. Alsing, "Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection," Phys. Rev. A 53, 4372-4380 (1996). [CrossRef] [PubMed]
- T. B. Simpson, "Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection," Opt. Commun. 215, 135-151 (2003). [CrossRef]
- N. Shunk and K. Peterman, "Noise analysis of injection-locked semiconductor injection lasers," IEEE J. Quantum Electron. 22, 642-650 (1986). [CrossRef]
- W. A. van der Graaf, A. M. Levine, and D. Lenstra, "Diode lasers locked to noisy injection," IEEE J. Quantum Electron. 33, 434-442 (1997). [CrossRef]
- S. K. Hwang, J. B. Gao, and J. M. Liu, "Noise-induced chaos in an optically injected semiconductor laser model," Phys. Rev. E 61, 5162-5170 (2000). [CrossRef]
- C. H. Henry, "Theory of the linewidth of semiconductor laser," IEEE J. Quantum Electron. 18, 259-264 (1982). [CrossRef]
- G. Vemuri and R. Roy, "Effect of injected field statistics on transient dynamics of an injection seeded laser," Opt. Commun. 77, 471-493 (1990). [CrossRef]
- E. Doedel, A. Champneys, T. Fairgrieve, Yu. Kuznetsov, B. Sandstede, and X. Wang, "AUTO 2000: Continuation and bifurcation software for ordinary differential equations," http://sourceforge.net/projects/auto2000/.
- S. Valling, T. Fordell, and A. M. Lindberg, "Maps of the dynamics of an optically injected solid-state laser," Phys. Rev. A 72, 033810-33818 (2005). [CrossRef]

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