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Time and frequency -Domain Interpretation of Phase-lag Control01:21

Time and frequency -Domain Interpretation of Phase-lag Control

447
Phase-lag controllers are widely used in control systems to improve stability and reduce steady-state errors. A dimmer switch controlling the brightness of a light bulb serves as a practical example of phase-lag control, gradually adjusting the bulb's brightness. Mathematically, phase-lag control or low-pass filtering is represented when the factor 'a' is less than 1.
Phase-lag controllers do not place a pole at zero, but instead influence the steady-state error by amplifying any...
447

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Related Experiment Video

Updated: Apr 3, 2026

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator
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Laser diode nonlinear dynamics from a filtered phase-conjugate optical feedback.

Lionel Weicker1, Thomas Erneux2, Delphine Wolfersberger1

  • 1OPTEL Research Group, CentraleSupélec, LMOPS (EA 4423), 2 rue Édouard Belin, 57070 Metz, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 19, 2015
PubMed
Summary

Researchers studied laser diodes with filtered phase-conjugate optical feedback. Increasing filter width stabilizes laser output by merging Hopf bifurcation points, altering pulsating regimes.

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Area of Science:

  • Nonlinear dynamics
  • Laser physics
  • Optical engineering

Background:

  • Laser diodes with optical feedback exhibit complex dynamics.
  • Filtered phase-conjugate feedback introduces unique bifurcation behaviors.
  • Understanding these dynamics is crucial for laser stability and applications.

Purpose of the Study:

  • To analytically and numerically investigate laser diode rate equations under filtered phase-conjugate optical feedback.
  • To determine and analyze Hopf bifurcation conditions as a function of filter width.
  • To explain the disappearance of pulsating regimes with increasing filter width.

Main Methods:

  • Analytical methods, including asymptotic analysis, to determine Hopf bifurcation conditions.
  • Numerical simulations of laser rate equations to observe intensity dynamics.
  • Bifurcation analysis to study the evolution of steady-state and periodic solutions.

Main Results:

  • Hopf bifurcation conditions were determined and analyzed.
  • Increasing filter width leads to the disappearance of pulsating intensity regimes due to the coalescence of Hopf bifurcation points.
  • A restabilization of the steady-state solution occurs for moderate filter widths, with a bubble of periodic solutions appearing above a critical width.

Conclusions:

  • The filter width significantly influences the dynamic behavior of laser diodes with phase-conjugate feedback.
  • The observed phenomena are explained by the merging of Hopf bifurcation points.
  • In the narrow filter limit, the system exhibits bifurcations similar to those of a laser with injected signals.