Phase-conjugate Feedback Research Articles

Modulation response characteristics of a semiconductor laser with phase-conjugate feedback

Modulation response characteristics of a semiconductor laser with a phase-conjugate feedback mirror is studied. Modulation response equations about the amplitude and frequency chirping of the laser are deduced in detail. It is demonstrated that the amplitude and frequency chirping responses are the function of the modulation angular frequency, optical feedback level, and delayed time. Numerical results illustrate that the amplitude response and modulation bandwidth of the laser are larger than that of the solitary laser when the optical amplitude reflectivity is <0.0022. When the optical amplitude reflectivity is increased further, the peaks of the amplitude response of the laser show periodic variability while the modulation angular frequency at the maximal value of the amplitude response is added; however, the frequency chirping can be enlarged or reduced. The large drive current or the large detuning can enhance the modulation response and frequency chirping.

Continuous adaptive beam pointing and tracking for laser power transmission

The adaptive beam pointing concept has been revisited for the purpose of controlled transmission of laser energy from an optical transmitter to a target. After illumination, a bidirectional link is established by a retro-reflector on the target and an amplifier-phase conjugate mirror (A-PCM) on the transmitter. By setting the retro-reflector's aperture smaller than the diffraction limited spot size but big enough to provide sufficient amount of optical feedback, a stable link can be maintained and light that hits the retro-reflector's surrounded area can simultaneously be reconverted into usable electric energy. The phase conjugate feedback ensures that amplifier's distortions are compensated and the target tracked accurately.After deriving basic arithmetic expressions for the proposed system, a section is devoted for the motivation of free-space laser power transmission which is supposed to find varied applicability in space. As an example, power transmission from a satellite to the earth is described where recently proposed solar power generating structures on high-altitudes receive the power above the clouds to provide constant energy supply.In the experimental part, an A-PCM setup with reflectivity of about R(A-PCM) = 100 was realized using a semiconductor optical amplifier and a photorefractive self-pumped PCM. Simulation results show that a reflectivity of R(A-PCM)>1000 could be obtained by improving the self-pumped PCM's efficiency. That would lead to a transmission efficiency of eta>90%.

Optical phase conjugation of picosecond pulses at 106 μm in Sn_2P_2S_6:Te for wavefront correction in high-power Nd-doped amplifier systems

We report, for the first time to our knowledge, on picosecondpulse optical phase conjugation using photorefractive Sn(2)P(2)S(6) crystals. For 7.2-ps pulses at 1.06 mum, we have achieved phase-conjugate reflectivities of up to 45% with very fast build-up times, about 15 ms at an intensity of 23 W/cm(2) using Te-doped Sn(2)P(2)S(6). We furthermore demonstrate aberration-free 5 W optical output of 8-ps pulses at 1.06 mum from a side pumped Nd:YVO(4) amplifier using the Sn(2)P(2)S(6)-based phase-conjugate feedback.

Numerical Methods for Approximating Invariant Manifolds of Delayed Systems

In this paper we develop new methods for computing k-dimensional invariant manifolds of delayed systems for $k\geq2$. Our current implementation is built for $k=2$ only, but the numerical and algorithmic challenges encountered in this case will be also present for any $k>1$. For small delays, we consider methods for approximating delay differential equations (DDEs) with ordinary differential equations (ODEs). Once these approximations are made, any existing method for computing invariant manifolds of ODEs can then be used directly. We derive bounds on errors incurred by the most natural of these approximations. For large delays, we extend to DDEs the method originally introduced by Krauskopf and Osinga [Chaos, 9 (1999), pp. 768–774] for invariant manifolds of ODEs. We test the convergence of the resulting algorithms numerically and further illustrate our approach by computing two-dimensional unstable manifolds of equilibria in the context of phase-conjugate feedback lasers.

Phase-Conjugate External-Cavity Diode Laser

An external-cavity diode laser based on self-pumped phase-conjugate optical feedback is reported. An antireflection-coated diode laser is subjected to strong phase-conjugate feedback from a photorefractive Ce-doped BaTiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> crystal. This provides narrow laser linewidth and good stability. Inherent linewidth of 25 kHz and relative frequency stability of 1.5times10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-8</sup> (at 100 s) are demonstrated. Due to phase-conjugate feedback, the laser is insensitive to mechanical drifts and misalignment of the feedback element

Noise Characteristics of semiconductor Laser with Phase- Conjugate Feedback

Noise spectra of semiconductor laser subject to a phase-conjugate feedback were investigated theoretically. Time-delayed rate equations with feedback were solved and characteristic equations for relative intensity noise and frequency noise spectra were derived. Numerical results indicate that the relaxation resonance peak was moved to a new frequency region determined by the amount of feedback, This is a technological importance for the improvement of modulation bandwidth, Noise level was affected by the feedback parameters, mainly; the response time and the feedback rate.

Synchronization performance comparison of vertical-cavity surface-emitting lasers with two different feedback chaos schemes

Most of the previous treatments of chaos synchronization use common optical feedback. In this paper, we introduce a new feedback type called phase conjugate feedback in vertical-cavity surface-emitting lasers (VCSEL) to achieve synchronization. For comparison, models of synchronization using both phase conjugated feedback (PCF) and common optical feedback (COF) are given, where the nonlinear and dispersion effects on transmitted signals in a fibre link are also considered. Numerical simulation shows two peaks when calculating the shifted correlation coefficient. In the phase conjugate feedback scheme, the larger peak always corresponds to anticipating synchronization while this does not hold true in the common optical feedback scheme. When anticipating synchronization takes place, the value of the correlation coefficient for phase conjugate feedback is larger than that for common optical feedback, and their discrimination is more than 0.1. Decoded messages in chaotic shift keying undergo less distortions when the phase conjugate feedback scheme is considered, because of the distortion correction characteristics of phase conjugate light. All this indicates that the synchronization performance of the former scheme is better than that of the latter.

One-dimensional unstable eigenfunction and manifold computations in delay differential equations

In this paper we present a new numerical technique for computing the unstable eigenfunctions of a saddle periodic orbit in a delay differential equation. This is used to obtain the necessary starting data for an established algorithm for computing one-dimensional (1D) unstable manifolds of an associated saddle fixed point of a suitable Poincaré map. To illustrate our method, we investigate an intermittent transition to chaos in a delay system describing a semiconductor laser subject to phase-conjugate feedback.

Bifurcation analysis of a semiconductor laser subject to non-instantaneous phase-conjugate feedback

We consider how the behaviour of a semiconductor laser subject to phase-conjugate feedback depends on the interaction time of the phase-conjugating mirror. We employ continuation techniques to show how the bifurcation diagram in the plane of feedback strength versus pump current changes as the interaction time of the mirror is increased. This shows that for interaction times below 0.1 ns the assumption of instantaneous feedback is justified. On the other hand, we show that increasingly larger interaction times lead to considerable changes in the size and qualitative structure of the locking region. Our study provides a more global view of the suppression of complicated and chaotic dynamics.

External cavity modes of semiconductor lasers with phase-conjugate feedback.

External cavity modes (ECMs) of a semiconductor laser with phase-conjugate feedback are defined as time-periodic pulsating intensity solutions exhibiting a frequency close to an integer multiple of the external cavity frequency. As the feedback rate progressively increases from zero, they sequentially appear as stable attractors in the bifurcation diagram. We construct a simple analytical approximation of these pulsating intensity solutions and determine their frequencies. We show that branches of ECMs are isolated. Finally, the validity of our approximation is tested by comparing numerical bifurcation diagrams obtained by simulation and continuation techniques with our analytical results.

Bifurcation Analysis of Frequency Locking in a Semiconductor Laser with Phase-Conjugate Feedback

We present a detailed study of the external-cavity modes (ECMs) of a semiconductor laser with phase-conjugate feedback. Mathematically, lasers with feedback are modeled by delay differential equations (DDEs) with an infinite-dimensional phase space. We employ new numerical bifurcation tools for DDEs to continue steady states and periodic orbits, irrespective of their stability. In this way, we show that the periodic orbits corresponding to the ECMs are connected to the steady state solution associated with the locking range of the laser. We also identify symmetric and nonsymmetric homoclinic orbits and hysteresis in the system.

Computing unstable manifolds of periodic orbits in delay differential equations

We present the first algorithm for computing unstable manifolds of saddle-type periodic orbits with one unstable Floquet multiplier in systems of autonomous delay differential equations (DDEs) with one fixed delay. Specifically, we grow the one-dimensional unstable manifold W u( q) of an associated saddle fixed point q of a Poincaré map defined by a suitable Poincaré section Σ. Starting close to q along the linear approximation to W u( q) given by the associated eigenfunction, our algorithm grows the manifold as a sequence of points, where the distance between points is governed by the curvature of the one-dimensional intersection curve W u( q)∩ Σ of W u( q) with Σ. Our algorithm makes it possible to study global bifurcations in DDEs. We illustrate this with the break-up of an invariant torus and a subsequent crisis bifurcation to chaos in a DDE model of a semiconductor laser with phase-conjugate feedback.

A Two-Parameter Study of the Locking Region of a Semiconductor Laser Subject to Phase-Conjugate Feedback

We present a detailed bifurcation analysis of a single-mode semiconductor laser subject to phase-conjugate feedback, a system described by a delay differential equation. Codimension-one bifurcation curves of equilibria and periodic orbits and curves of certain connecting orbits are presented near the laser's locking region in the two-dimensional parameter plane of feedback strength and pump current. We identify several codimension-two bifurcations, including a double-Hopf point, Belyakov points, and a T-point bifurcation, and we show how they organize the dynamics. This study is the first example of a two-parameter bifurcation study, including bifurcations of periodic and connecting orbits, of a delay system. It was made possible by new numerical continuation tools, implemented in the package DDE-BIFTOOL, and showcases their usefulness for the study of delay systems arising in applications.

Bistability and torus break-up in a semiconductor laser with phase-conjugate feedback

We present a detailed study of a route to chaos via quasiperiodicity on a torus in a semiconductor laser with phase-conjugate feedback. Highlighting the use of new tools that go far beyond mere simulation, we compute bifurcation diagrams and unstable manifolds of saddle periodic orbits. In this way, we show how a torus breaks up with a final sudden onset of chaos in a crisis bifurcation. We also identify regions of bistability between periodic solutions and other attractors in the system.

Global bifurcations and bistability at the locking boundaries of a semiconductor laser with phase-conjugate feedback.

We investigate dynamics and bifurcations of a single-mode semiconductor laser subject to phase-conjugate feedback near the locking region. The system is described by rate equations which are a three-dimensional system with a delay. With tools that go much beyond mere simulation, we find and follow steady states regardless of their stability and compute unstable manifolds of saddle points. Furthermore, we identify heteroclinic bifurcations, which turn out to be responsible for bistability and excitability at the locking boundaries.

Spatio-temporal nature of the optical return field generated by four-wave mixing in photorefractive and Kerr materials

We have theoretically analyzed the return field produced by an externally pumped Kerr medium, an externally pumped photorefractive medium, and a self-pumped photorefractive medium, for an input field with arbitrary spatial and temporal characteristics. For a stationary beam with spatial structure, all three cases yield a spatial phase-conjugate return beam. For a beam with dynamic fluctuations, each case returns a field with different temporal character. For the case of a self-pumped photorefractor, it is found that the dynamics of the return beam are represented by a conventional reflection rather than a temporal phase conjugate. While this geometry may have been thought of as a simple way to provide phase-conjugate feedback for dynamic stabilization in semiconductor lasers, an in-depth analysis shows that our theoretical findings agree with previously reported experimental results.

Broad-area diode lasers with plane-mirror and phase-conjugate feedback

The output state of a broad-area diode (BAD) laser with phase conjugate feedback (PCF) from Rh:BaTiO/sub 3/ crystal is compared to the same diode laser with conventional optical feedback from a plane mirror. A number of distinct output states, including low frequency fluctuations, are observed for each feedback type. Stable single spectral mode output is achieved with phase conjugate feedback above an injection current dependent critical level (/spl sim/20%). Conventional optical feedback from a plane mirror does not induce single-mode output at any feedback level.

Low-frequency fluctuations in a semiconductor laser with phase conjugate feedback

We present numerical results obtained for a semiconductor laser with phase conjugate optical feedback. We observe both low and high frequency dynamics similar to that observed with conventional optical feedback. Here also the low frequency fluctuations correspond to power drop-outs while the high frequency fluctuations being associated with fast pulses in the power. Due to the absence of conventional external cavity modes in phase conjugate feedback, current models do not predict this behaviour.

Effects of phase-conjugate feedback on the modal content and noise characteristics of a cw argon ion laser: experimental results

Significant optical noise reduction can occur in a continuous wave argon ion laser when a very small fraction of the output beam is reinjected inside the laser cavity. Noise reduction reaching 15 dB for the low-frequency components (below 1 MHz) and reaching 58 dB at the frequencies corresponding to the beating of longitudinal modes has been obtained by optimizing the external cavity length and the feedback level. It has been possible to reduce the number of oscillating longitudinal modes from ∼40 to 2 without loss of power. We have compared the optical noise reduction for an external cavity terminated by a phase-conjugate mirror and by a conventional plane mirror.

Stability and noise properties of diode lasers with phase-conjugate feedback

For a diode laser subjected to filtered feedback from a phase-conjugating mirror, we present the first exact stability analysis and various noise spectra. The stability properties are intermediate between those of the injection laser and the laser with conventional optical feedback. The role of a finite response-time is to drastically enhance the steady-state stability For moderate feedback, the frequency noise is suppressed by several orders of magnitude, and the main relaxation frequency of the laser shows a crossover from the usual relaxation oscillation frequency to a new frequency determined by the amount of feedback. This may be of technological importance since it is expected to improve the modulation bandwidth.