Diode lasers with optical feedback produce self-modulated square waves
David Sukow, Athanasios Gavrielides, and Thomas Erneux
Strong polarization-rotated optical feedback can activate the suppressed polarization mode in semiconductor diode lasers and lead to square-wave operation.
28 July 2006, SPIE Newsroom. DOI: 10.1117/2.1200607.0272
Semiconductor laser diodes are popular and widely used sources due to their high power, low cost, tunability, and easy integration with miniaturized electronics. They can also serve as flexible signal sources of high-frequency oscillations and pulsations. To generate waveforms while taking advantage of their high frequency capabilities, all-optical feedback methods are often simpler to use than high-speed electronics, which can be costly, inefficient, or operationally limited. In this context, a square-wave signal is a particularly useful waveform, since its rising and falling edges require high-frequency components in their creation.
One method that can generate square waves from diode lasers is polarization self-modulation, meaning that the laser output switches regularly between its two natural polarization modes rather than simply switching on and off in total intensity. This self-modulation can be achieved entirely optically by injecting light from one laser polarization mode into the orthogonal mode. This requires that the optical feedback be rotated from its original orientation by some means, typically a waveplate.1 Square-wave operation has been observed in vertical-cavity surface-emitting lasers (VCSELs), but can be difficult to achieve:2 the specific VCSEL characteristics or operating parameters that are required for such outputs are not well-understood. Another limitation is that, if the feedback path is short, the VCSEL will exhibit chaotic fluctuations rather than regular square waves.
In edge-emitting lasers however, short feedback delays still allow square-wave operation. In these devices, one polarization mode is typically suppressed, and accordingly requires relatively strong feedback to be activated.3 Our recent work has explored the dynamical mechanisms of this phenomenon,4 and helped clarify the origins of these solutions and the requirements for their observation.5
One feature of our system is unidirectional coupling from the dominant, transverse electric (TE) polarization mode to the suppressed, transverse magnetic (TM) state. We found that this type of coupling simplifies the analysis. Experimentally, it is obtained by using a Faraday rotator as the rotating element in the external cavity, rather than a waveplate.
Figure 1 shows our experimental setup. The elements of the laser feedback system are the laser diode, the rotator that changes the polarization of the beam by 45° on each pass (forward and backward), a polarizer to adjust the feedback strength, and a mirror to reflect the beam. The detection system consists of a beamsplitter to sample the light, a polarizing cube to separate the TE and TM components, and detectors to record both signals simultaneously.
The sample experimental data obtained from such a system (Figure 1, bottom) shows TE and TM waves in antiphase, exhibiting square-wave self-modulation. The waves also show rapid, damped relaxation oscillations at the start of each pulse. These are commonplace and occur whenever the laser is perturbed abruptly from its steady-state. Thus, while relaxation oscillations are expected at the onset of a square pulse, they are not a special characteristic of the square-wave state. The period of the square-wave signal is roughly twice the time of flight for photons in the external cavity, and can thus be varied simply by changing the length of the cavity.
Figure 1. Top: Schematic of our experimental apparatus showing the laser diode feedback elements (blue) and the detection system components (green). Bottom: sample square-wave data.
We explain this behavior as follows: after being fed the pump current, the laser the first emits only in the natural TE mode — since there is no feedback until the photons have had time to traverse the external cavity once. After one roundtrip of time τ in the cavity, strong optical feedback is injected into the TM mode, which overcomes its losses and drives the laser primarily in that mode. However, the resulting TM emission is extinguished by the rotator, and so, after another period of time τ, there is no longer any feedback. The laser then reverts to its normal TE emission, and the cycle repeats itself.
To understand this waveform more rigorously, we use a dimensionless mathematical model consisting of three rate equations for the complex TE field E1, the TM field E2, and the carrier density N. They are given by
where α is the linewidth enhancement factor, k the ratio of gain coefficients between TE and TM modes, β the differential losses, η the feedback strength, T the ratio of carrier to cavity lifetimes, and P the pump parameter above threshold.
The key to interpreting these equations is to note that the rotated optical feedback — delayed by one cavity roundtrip τ — appears in Equation 2 through the term ηE1(t - τ). The TM mode has greater inherent losses than the natural TE mode, as expressed by β. The TM mode does not influence the TE mode directly, but instead is mediated through the carrier equation.
Using this model, we have performed extensive numerical simulations over a wide range of parameter values. Our results show that the square-wave solution is possible over a broad range of parameters, the main requirement being that the feedback strength η must be large. Systematic studies show that, as η is increased from zero, the steady-state intensity first undergoes sustained relaxation oscillations, which are then modulated by a progressively deeper 2τ-periodic square wave. Beyond a critical value of η, the square waves become dominant and the relaxation oscillations become damped, appearing only at the rising edge of each square pulse. Numerical time series data of a fully-developed square wave, with the expected decaying relaxation oscillations, are shown in Figure 2. Note that the horizontal axis is scaled by the roundtrip time τ, illustrating the approximate 2τ periodicity.
Figure 2. Numerical square-wave data.
Our analysis also shows that the critical value of η, above which square waves occur, depends on three laser parameters: α, β, and k. Key steps in the analysis are phase and amplitude decomposition of E1 and E2, which transforms Equations (1–3) into a set of algebraic equations, provided that small terms—of the order of τ-1—multiplying time derivatives are neglected. For a solution including two successive stages, in which the two field amplitudes alternately go to zero, we obtain a relationship that is a condition for square waves to exist. The condition, illustrates quantitatively what it means for the feedback strength η to be large. Alternately, in a configuration in which the feedback strength is fixed, this relationship requires that the differential losses β must be sufficiently small.
Our work illustrates the potential of square-wave self-modulation for a variety of applications requiring high-frequency pulse rates. All-optical feedback methods are a convenient and effective way to generate such pulse trains in both VCSELs and edge-emitting semiconductor lasers. Our system, with unidirectional feedback from the TE to the TM mode, has allowed new analyses that shed light on the relationship between the operating conditions and the laser parameters for which these pulses are possible.
This research is supported by NSF CAREER grant 0239413. T.E. acknowledges the support of the Fonds National de la Recherche Scientifique and the InterUniversity Attraction Pole program of the Belgian government.
Department of Physics and Engineering, Washington and Lee University
David Sukow is an Associate Professor of Physics and Engineering at Washington and Lee University in Lexington, Virginia. He has written papers for SPIE's Physics and Simulation of Optoelectronic Devices and Semiconductor Lasers and Laser Dynamics conferences.
Directed Energy Directorate AFRL/DELO, Air Force Research Laboratory
Kirtland AFB, NM
Dr Gavrielides is the Technical Advisor of the Solid State Lasers Branch of the Air Force Research Laboratory and directs research in solid state lasers. He also performs research in the dynamics of lasers and coupled dynamical systems. He has written papers for SPIE's Physics and Simulation of Optoelectronic Devices and Semiconductor Lasers and Laser Dynamics conferences.
Optique Nonlineaire Theorique, Universite Libre de Bruxelles
Prof. Thomas Erneux is with the Theoretical Nonlinear Optics group at the Universite Libre de Bruxelles, and has previously worked at Northwestern University. He has written papers for several SPIE conferences and served on the international organizing committee of Photonics Europe.