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  • Open Access

Pulsed optical injection steering in multistable semiconductor laser arrays under correlated noise

Max M. Chumley* and Herbert G. Winful†

  • *Contact author: mchumley@umich.edu
  • †Contact author: arrays@umich.edu
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Phys. Rev. Research 8, 033112 – Published 27 July, 2026

DOI: https://doi.org/10.1103/pjln-px5d

Abstract

We demonstrate robust programmable state preparation in small Vertical Cavity Surface Emitting Laser (VCSEL) arrays with optical feedback using transient optical injection in the form of Gaussian pulses. In Lang-Kobayashi-type models of delay-coupled two- and three-laser arrays, multistability gives rise to coexisting synchronized and symmetry-broken equilibrium branches. We show that short injection pulses applied to one or more lasers can steer the system from free-running operation to any stable equilibrium branch in the absence of noise by appropriate choice of pulse detuning and amplitude, after which the selected state persists without continued forcing. With correlated noise, injection steering remains effective, but branches with small basins of attraction are effectively destabilized by noise. These results validate pulsed injection as a practical mechanism for attractor selection in multistable VCSEL arrays and point to a feasible route toward experimental realization of programmable collective-state control.

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References (41)

  1. D. K. Serkland, K. D. Choquette, G. R. Hadley, K. M. Geib, and A. A. Allerman, Two-element phased array of antiguided vertical-cavity lasers, Appl. Phys. Lett. 75, 3754 (1999).
  2. A. C. Lehman, D. Siriani, and K. Choquette, Two-dimensional electronic beam-steering with implant-defined coherent VCSEL arrays, Electron. Lett. 43 1202 (2007).
  3. M. Xun, Y. Sun, J. Zhou, C. Xu, Y. Xie, H. Wang, Q. Kan, X. Liu, Z. Jin, and D. Wu, Nineteen-element in-phase coherent vertical-cavity surface-emitting laser array with low side lobe intensity, Opt. Express 27, 774 (2019).
  4. H. Dave, Z. Gao, S. T. M. Fryslie, B. J. Thompson, and K. D. Choquette, Static and dynamic properties of coherently-coupled photonic-crystal vertical-cavity surface-emitting laser arrays, IEEE J. Sel. Top. Quantum Electron. 25, 1 (2019).
  5. J. Mulet, C. Masoller, and C. Mirasso, Modeling bidirectionally coupled single-mode semiconductor lasers, Phys. Rev. A 65, 063815 (2002).
  6. H. Erzgräber, B. Krauskopf, and D. Lenstra, Compound laser modes of mutually delay-coupled lasers, SIAM J. Appl. Dyn. Syst. 5, 30 (2006).
  7. S. Yanchuk, A. Stefanski, T. Kapitaniak, and J. Wojewoda, Dynamics of an array of mutually coupled semiconductor lasers, Phys. Rev. E 73, 016209 (2006).
  8. E. Clerkin, S. O’Brien, and A. Amann, Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers, Phys. Rev. E 89, 032919 (2014).
  9. A. Prasad, Y.-C. Lai, A. Gavrielides, and V. Kovanis, Complicated basins in external-cavity semiconductor lasers, Phys. Lett. A 314, 44 (2003).
  10. J. Mercier and M. McCall, Injection-locking of one and two dimensional vertical cavity semiconductor laser arrays, in The European Conference on Lasers and Electro-Optics (Optica Publishing Group, Washington, DC, 1996).
  11. T. Fishman and A. Hardy, Injection-locking analysis of vertical-cavity laser arrays, J. Opt. Soc. Am. B 16, 38 (1999).
  12. B. Lücke, G. Hergenhan, U. Brauch, M. Scholl, A. Giesen, H. Opower, and H. Hügel, Autostable injection locking of a 4×4 VCSEL array with on-chip master laser, in Vertical-Cavity Surface-Emitting Lasers IV, Proceedings of SPIE, edited by K. D. Choquette and C. Lei (SPIE, Bellingham, WA, 2000), Vol. 3946, pp. 240–245.
  13. G. Hergenhan, B. Lücke, M. Scholl, U. Brauch, A. Giesen, H. Hügel, and H. Opower, Fast coherence switching of an injection-locked vertical-cavity laser array, in Conference on Lasers and Electro-optics, OSA Technical Digest (Optica Publishing Group, Washington, DC, 2000), p. CMS5.
  14. B. J. Thompson, Z. Gao, S. T. M. Fryslie, and K. D. Choquette, Mode engineering in linear coherently coupled vertical-cavity surface-emitting laser arrays, IEEE J. Sel. Top. Quantum Electron. 25, 1 (2019).
  15. N. Jahan, W. North, P. Strzebonski, and K. D. Choquette, Supermode switching in coherently-coupled vertical cavity surface emitting laser diode arrays, IEEE J. Sel. Top. Quantum Electron. 28, 1 (2022).
  16. T. Mori, Y. Yamayoshi, and H. Kawaguchi, Low-switching-energy and high-repetition-frequency all-optical flip-flop operations of a polarization bistable vertical-cavity surface-emitting laser, Appl. Phys. Lett. 88, 101102 (2006).
  17. N. Brandonisio, P. Heinricht, S. Osborne, A. Amann, and S. O’Brien, Bistability and all-optical memory in dual-mode diode lasers with time-delayed optical feedback, IEEE Photon. J. 4, 95 (2012).
  18. H. Erzgräber, S. Wieczorek, and B. Krauskopf, Locking behavior of three coupled laser oscillators, Phys. Rev. E 80, 026212 (2009).
  19. R. Lang and K. Kobayashi, External optical feedback effects on semiconductor injection laser properties, IEEE J. Quantum Electron. 16, 347 (1980).
  20. I. Fatadin, D. Ives, and M. Wicks, Numerical simulation of intensity and phase noise from extracted parameters for CW DFB lasers, IEEE J. Quantum Electron. 42, 934 (2006).
  21. W. Ma, B. Xiong, C. Sun, X. Ke, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, J. Yu, and Y. Luo, Linewidth narrowing of mutually injection locked semiconductor lasers with short and long delay, Appl. Sci. 9, 1436 (2019).
  22. N. Nair, K. Hu, M. Berrill, K. Wiesenfeld, and Y. Braiman, Using disorder to overcome disorder: A mechanism for frequency and phase synchronization of diode laser arrays, Phys. Rev. Lett. 127, 173901 (2021).
  23. L.-L. Ye, N. Vigne, F.-Y. Lin, H. Cao, and Y.-C. Lai, Optimal sparse networks for synchronization of semiconductor lasers, arXiv:2511.03205.
  24. T. Yang, Y.-H. Chen, Y.-C. Wang, W. Ou, L.-Y. Ying, Y. Mei, A.-Q. Tian, J.-P. Liu, H.-C. Guo, and B.-P. Zhang, Green vertical-cavity surface-emitting lasers based on InGaN quantum dots and short cavity, Nano Micro Lett. 15, 223 (2023).
  25. W. Liu, X. Hu, L. Ying, S. Chen, J. Zhang, H. Akiyama, Z. Cai, and B. Zhang, On the importance of cavity-length and heat dissipation in GaN-based vertical-cavity surface-emitting lasers, Sci. Rep. 5, 9600 (2015).
  26. H. Erzgräber, D. Lenstra, B. Krauskopf, E. Wille, M. Peil, I. Fischer, and W. Elsäßer, Mutually delay-coupled semiconductor lasers: Mode bifurcation scenarios, Opt. Commun. 255, 286 (2005).
  27. M. M. Chumley, vcsel_lib: Simulation code for pulsed optical injection steering in semiconductor laser arrays, GitHub repository (2026), https://github.com/maxchumley/vcsel_lib.
  28. V. Flunkert, Delay-Coupled Complex Systems: And Applications to Lasers (Springer, Berlin, 2011).
  29. dde23, Solve delay differential equations (DDEs) with constant delays, MATLAB, https://www.mathworks.com/help/matlab/ref/dde23.html.
  30. G. Ansmann, Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE, Chaos 28, 043116 (2018).
  31. J. C. Butcher, General linear methods, Acta Numer. 15, 157 (2006).
  32. E. Hairer and C. Lubich, Numerical solution of ordinary differential equations, The Princeton companion to applied mathematics (2012), pp. 293–305.
  33. A. E. D. Barioni, A. N. Montanari, and A. E. Motter, Interpretable disorder-promoted synchronization and coherence in coupled laser networks, Phys. Rev. Lett. 135, 197401 (2025).
  34. root(method=‘hybr’), SciPy v1.17.0 Manual, https://docs.scipy.org/doc/scipy/reference/optimize.root-hybr.html#optimize-root-hybr.
  35. H. Smith, Linear systems and linearization, in An Introduction to Delay Differential Equations with Applications to the Life Sciences, edited by H. Smith (Springer, New York, NY, 2011), pp. 41–59.
  36. G. G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Pitman Research Notes in Mathematics Series (Longman Scientific & Technical and Wiley, 1989).
  37. Z. Wu and W. Michiels, Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method, J. Comput. Appl. Math. 236, 2499 (2012).
  38. W. Michiels and S.-I. Niculescu, Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach, 2nd ed. (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2014).
  39. E. Jarlebring, K. Meerbergen, and W. Michiels, A Krylov method for the delay eigenvalue problem, SIAM J. Sci. Comput. 32, 3278 (2010).
  40. D. Breda, S. Maset, and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput. 27, 482 (2005).
  41. DDE-BIFTOOL: Bifurcation analysis of DDEs, https://twr.cs.kuleuven.be/research/software/delay/ddebiftool. shtml.

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Phys. Rev. Research 8, 033112– Published 27 July, 2026
  • Received 1 May 2026
  • Accepted 2 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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