Polarization dynamics in semiconductor lasers with ...

Polarization dynamics in semiconductor lasers with incoherent optical feedback David W. Sukowa , Athanasios Gavrielidesb , Thomas Erneuxc , Michael J. Baraccoa , Zachary A. Parmentera , and Karen L. Blackburna a Department

of Physics and Engineering, Washington and Lee University, 116 N. Main St., Lexington, VA 24450 USA b Air Force Research Laboratory, Directed Energy Directorate AFRL/DELO, 3550 Aberdeen Ave. SE, Kirtland AFB, NM 87117 USA c Universit´ e Libre de Bruxelles, Optique Nonlin´eaire Th´eorique, Campus Plaine, Code Postal 231, 1050 Bruxelles, Belgium ABSTRACT The chaotic dynamics of a semiconductor laser subject to a delayed polarization-rotated optical feedback is investigated theoretically and experimentally. An extension of the usual one-polarization model is derived to account for two orthogonal polarizations of the optical field. The two-polarization model is motivated by observations of lag synchronization in our experiments using polarization-rotated optical feedback and unidirectional injection. Experimental data confirm the predictions of the two-field model. We also show that the two-polarization model can be reduced to the one-polarization model.

1. INTRODUCTION Chaos synchronization has been extensively studied for semiconductor laser (SL) systems because of their interest for optical communication systems. Chaos synchronization on unidirectionally coupled semiconductor lasers has been found in optical injection systems,1 optical feedback systems,2, 3 optoelectronic feedback systems,4, 5 and systems of two mutually coupled semiconductor lasers.6, 7 In several experiments, it was shown that chaos synchronization can exhibit perfect synchronization, as expected, but driving synchronization as well. If a master laser subject to a delayed optical feedback injects its light into a slave laser of similar wavelength, perfect chaos synchronization between master and slave lasers is possible if the injection and feedback rates are comparable. On the other hand, if the injection rate is much larger than the feedback rate, a driven response is observed where synchronization occurs with the delayed injected signal rather than with the chaotic oscillations of the master.3 An all-optical system which is dynamically equivalent to the optoelectronic feedback system is the incoherent feedback system.8 But driving synchronization has never been observed for the optoelectronic feedback system9 while it was observed for the incoherent feedback system.10 However, extensive simulations of an incoherent feedback model involving only one polarization field11 indicate that driving synchronization is not possible. This suggests that the one-polarization model might be too simple to describe the observed synchronization dynamics. In this paper, we formulate a more realistic model where the two polarizations fields are taken into account. We determine the basic properties of the polarization intensities which we then verify experimentally. Specifically, we consider a single mode diode laser subject to incoherent feedback, as described schematically in Fig. 1. It consists of a laser in an external cavity and intracavity devices that rotate the polarization state of the delayed optical feedback, adjust the feedback strength, and sample the output. We find that the steady state intensities consist of a two-polarization state as soon as the feedback is nonzero. In the experiment, the natural polarization is the horizontal polarization and the vertical polarization Please send correspondence to D.W.S. or A.G. D.W.S.: E-mail: [email protected], A.G.: E-mail: [email protected].

Laser

Rotator

Attenuator Mirror

Figure 1: Schematic diagram of a laser with delayed incoherent optical feedback.

Figure 2: Steady states of the two polarization emitted radiation from the diode laser under incoherent feedback

becomes significant only after the rotated polarization is injected into the laser. In Fig. 2 we show the steady state intensities as functions of the feedback for a laser biased at 64.0 mA and has a threshold current of 45.6 mA. These data for a single laser subject to incoherent feedback will be useful when we concentrate on the synchronization dynamics for two interacting lasers. The paper is divided into two parts. We first derive the two-polarization model and show analytically that it reduces to the usual one-polarization model plus a specific relation between the two polarization fields. This relation is then verified experimentally by studying the correlation between the two polarization intensities.

2. THEORY 2.1. BASIC INCOHERENT OPTICAL FEEDBACK MODEL The description of the laser subject to incoherent optical feedback is given in terms of the natural lasing field E and the inversion of population N . The polarization of the lasing field is then rotated from the horizontal to the vertical position, delayed, and reinjected back into the laser. The main effect is an depression of the inversion of population.11 The laser rate equations for E and N are given by dE = (1 + ia)N E, dt T

dN = P − N − (1 + 2N )[I + kI(t − τ )] dt

(1) (2)

where I = |E|2 . These equations differ from the standard SL rate equations by the term multiplying I(t − τ ) in Eq. (2). The dimensionless parameters appearing in these equations are the linewidth enhancement factor a, the ratio of the carrier and photon lifetimes T , the pump parameter above threshold P , the feedback injection rate k, and the feedback delay τ . Eqs. (1) and (2) admit the steady state solution N = 0,

I=

P 1+k

(3)

while the phase φ of the laser field is arbitrary. The steady state changes stability at a Hopf bifurcation located at (1 + 2P ) k = kH ' ω (4) 2P sin(θ) p where ω = 2P/T 0, the steady state solution now is given by T

A21 A22 N

(Ω + aβ)2 + β 2 , η 2 + β 2 + (Ω + aβ)2 η2 = P 2 , η + β 2 + (Ω + aβ)2 = 0, = P

(11) (12) (13)

(14) (15) (16)

while the phase difference Φ = φ1 − φ2 is obtained from tan(Φ) =

Ω + aβ . β

(17)

The steady state intensities given by (14) and (15) are shown in Fig. (3). As the feedback strength is increased, the intensity of the horizontal polarization decreases but shares power with the vertical polarization intensity. At the critical feedback rate p ηc = β 1 + a2 , (18) the two intensities become equal. Using (18) with a = 4 and comparing with Fig. (2), we estimate that the fractional loss between the two polarizations is about 13.5%. The first Hopf bifurcation can be determined

Figure 4. Bifurcation diagram calculated numerically from the full problem for two cases. The first bifurcation is for Ω = 0. The second is for Ω = 1.4. The rest of the parameters are P = 0.5, T = 1000, τ = 1000, a = 4, and β = 0.1.

analytically by p taking advantage of the large value of T (equivalently, the small value of the relaxation oscillation frequency ω ≡ 2P/T ). Instead of (4), we now find 2 'ω ηH

(1 + 2P ) 2 [β + (Ω + aβ)2 ]. 2P sin(ωτ )

(19)

The direction of bifurcation can be either supercritical or subcritical. The two bifurcation diagrams shown in Fig. (4) for two different values of Ω exhibit a subcritical Hopf bifurcation. As the feedback rate surpasses the Hopf bifurcation point, the laser then jumps to a branch of large amplitude oscillations that overlap part of the branch of stable steady states. The numerical bifurcation diagrams have been obtained by gradually changing the feedback rate first forward and then backward to detect the possible overlap of stable branches of solutions. The bifurcation diagram with the branch of periodic solutions appearing close to η = 0.14 is for Ω = 0 while the diagram with the branch of periodic solutions appearing close to η = 0.155 is for Ω = 1.4.

2.3. MODEL REDUCTION Eqs. (5)–(7) can be reduced to a system of two equations close to Eqs. (1)–(2) by taking advantage of the relatively large value of T and τ (T ∼ 103 and τ ∼ 103 ). Introducing the new time s ≡ τ −1/2 t and inversion of population Z = τ 1/2 Z into Eqs. (5)–(7), we find dE1 = (1 + ia)ZE1 , ds τ −1/2

dE2 = −iΩE2 + (1 + ia)(τ −1/2 Z − β)E2 + ηe−i(ω1 τ +φe ) E1 (s − θ), ds T dZ 2 2 = P − τ −1/2 Z − (1 + 2τ −1/2 Z)[|E1 | + |E2 | ] τ ds

(20) (21) (22)

Figure 5. Time series of the natural horizontal polarization (a) and of the vertical (b). The parameters are P = 0.5, T = 1000, τ = 1200, α = 2, β = 0.1, Ω = 0.0, and the feedback strength is 0.132.

where θ ≡ τ 1/2 . If τ −1/2