Optical stabilization of an external cavity ...

Optical stabilization of an external cavity semiconductor laser by creation and destruction of external cavity modes (1)

D.W. Sukow(1), F. Rogister(2), P. M´egret(2), O. Deparis(2), and A. Gavrielides(3)

Department of Physics and Engineering, Washington and Lee University, Lexington, Virginia 24450, USA (2) Advanced Research in Optics (ARO), Service d’Electromagn´etisme et de T´el´ecommunications, Facult´e Polytechnique de Mons, 31 Boulevard Dolez, B-7000 Mons, Belgium (3) Nonlinear Optics Center, Air Force Research Laboratory AFRL/DELO 3550 Aberdeen Avenue SE, Kirtland AFB, New Mexico 87117-5776, USA

II. THEORETICAL MODEL

We present an experimental realization of an all-optical stabilization technique for a single-mode external-cavity semiconductor laser operating in the low-frequency fluctuation (LFF) regime. This technique uses a second delayed optical feedback to suppress LFF via two mechanisms: destroying the antimodes responsible for the crises that induce power dropouts, and creating new stable maximum gain modes. Both effects are observed experimentally, in agreement with numerical simulations.

A single-mode semiconductor laser subject to weak and moderate optical feedback is typically modeled with the Lang-Kobayashi equations [7]. We extend these equations to include a second delayed feedback; they are expressed as dimensionless rate equations for the complex electric field E and the excess carrier number N : (1) E˙ = (1 + iα) NE + κ1 E (t − τ1 ) exp(−iΩτ1 ) +κ2E (t − τ2 ) exp(−iΩτ2 ), (2) T N˙ = P − N − (1 + 2N ) |E |2 .

I. INTRODUCTION

The performance characteristics of semiconductor lasers often suffer when the laser is subjected to external optical feedback, which can cause a wide variety of instabilities. One such instability is LFF, characterized by an irregular sequence of brief, sudden dropouts of the laser’s average power. It is desirable to control such behavior, and various methods have been devised [1—3]. Recently, Liu and Ohtsubo [3] proposed an all-optical stabilization technique using delayed feedback for a chaotic diode laser driven well above threshold. Rogister et al. [4] have investigated this scheme at pumping levels near threshold, paying close attention to the steady-state solutions. The key concepts of this scheme are as follows. According to Sano’s [5] analysis, power dropouts occur when a trajectory collides with an unstable saddle-type antimode. However, for short cavities and weak feedback, only a few antimodes are responsible for such crises; a second delayed feedback can cause the mode-antimode pairs responsible for the crises to be shifted away from the other external cavity modes or to be destroyed, leading to LFF suppression. In addition, recent studies of short-cavity dynamics [6] have shown that the laser experiences a bifurcation cascade as mode-antimode pairs are born as the feedback strength increases, producing stable and unstable behavior in sequence. A similar cascade has been shown numerically to occur in a double-cavity system at low pump levels [4]; thus, accessing a newly created stable mode offers another route to stabilization. This scheme has practical advantages in that it does not require alteration of any parameters of the original one-cavity system (i.e. cavity length, pump current). Furthermore, it is not necessary to position the second mirror with sub-µm precision; the critical requirement is that the second cavity length be short enough to observe a bifurcation cascade. This technique is thus relatively easy to implement.

Time t is measured in units of the photon lifetime τp . The electric field amplitude E and phase φ are defined by E = E exp(iφ(t)). The external cavity round-trip times τ1 and τ2 (normalized to τp ) have associated feedback strengths κ1 and κ2 . The angular frequency of the solitary laser ω is normalized as Ω ≡ ωτ p , and P is proportional to the pumping rate above threshold. The linewidth enhancement factor α and the ratio of the carrier to photon lifetimes T ≡ τs /τp characterize the semiconductor medium. We use typical parameter values in our numerical simulations: T = 1000, α = 4, τ 1 = 1000, τ2 = 200, Ωτ1 = −1.45, and Ωτ2 = 0.8. Equations (1) and (2) have steady state solutions of the form E = Es exp [i (∆ − Ω) t] and N = Ns , where ∆ is a stationary angular frequency, and E s and Ns are

Phase Difference

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Second Feedback Strength, κ2 ( x 10 )

FIG. 1. Numerical bifurcation diagram showing cascading regions of control. The vertical axis is the phase difference function, and the bifurcation parameter is κ2 .

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constants. It is useful to define a phase difference function φ (t) − φ (t − τ1 ) + Ωτ1 , which reduces to ∆τ1 for stationary behaviors (this function is used commonly in phase space representations of LFF). Figure 1 shows a numerically calculated bifurcation diagram for the phase difference function, with κ1 fixed at 4.6 × 10−3 and κ2 as the bifurcation parameter. When κ2 = 0, the system is in a region of LFF. When κ2 = 0.45 × 10 −3 (arrow), the mode-antimode pair responsible for the dropouts is destroyed, and LFF is suppressed. At larger values of κ 2 , regions of unstable behavior alternate with newly-born stable external cavity modes. This illustrates the two mechanisms by which the second feedback can stabilize the system.

spectrum as a function of ∆I2 , the threshold reduction due to the second cavity only. The horizontal axis has its zero at the location of the first mode that lases as κ1 increases from zero when κ 2 = 0. In the Figure, κ1 remains fixed (∆I1 = 7.1%) such that the laser displays LFF in seven external cavity modes when κ 2 = 0 [trace (a)]. Increasing κ2 only slightly [∆I2 = 0.44%, trace (b)] results in stabilization of the sixth external cavity mode of the first cavity. For ∆I2 = 1.58% [trace (c)], stability is lost, and the system once again displays LFF. Increasing κ2 further still [traces (d) and (e)], the system exhibits other complex dynamics until ∆I2 = 10.4% [trace (f)], at which point the laser again becomes stable in a newly-created external cavity mode which is wellseparated in frequency from the original seven external cavity modes of the first cavity. Further increases in κ 2 lead to a continuation of this pattern, with stabilization of increasingly distant modes interspersed with regions of complex behavior. Time series and RF spectra measured for the stabilized traces [(b) and (f)] confirm that the controlled two-cavity laser is indeed emitting a stable field of constant intensity. We observe that the control is robust, limited only by the mechanical stability of the system.

III. EXPERIMENT

In the experiment we use a laser diode (SDL-5301) with a nominal frequency of λ = 780 nm. The laser is operated at a pump current I = 25.0 mA, just below solitary threshold. The laser beam is collimated and directed to a holographic grating. The zeroth order of the grating is used to monitor the system’s behavior. The grating’s first order beam goes into the double cavity, which is formed by a beamsplitter and two 99% reflectivity mirrors placed at cavity lengths L 1 ' 23 cm and L2 ' 20 cm, respectively. The feedback strengths of each cavity are controlled independently using a pair of polarizers, and are measured as a percentage of threshold reduction ∆I = (Ith − I ) /Ith . The grating narrows the cavity bandwidth to 50 GHz, thus causing the laser to oscillate in a single solitary laser mode. The laser output is monitored using a scanning Fabry-P´erot interferometer (Newport SR-240C, free spectral range of 2000 GHz, finesse > 17000) and a fast ac-coupled photodiode (Hamamatsu C4258, 8 GHz bandwidth). The photodiode signal is amplified and connected to a 500 MHz bandwidth digitizer (Tektronix RTD 720) and an RF spectrum analyzer (HP 8596E). Figure 2 shows the experimentally observed optical

Optical power (arb. units)

(g) 11.4% (f)

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(e) 8.35% (d) 6.42%

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(b) 0.44% (a)

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We have demonstrated experimentally all-optical stabilization of LFF using a second delayed feedback. Theoretical analysis has predicted that this technique works by destroying the mode-antimode pairs responsible for the LFF-inducing crises, and by creating new maximum gain modes. This method of stabilization works without changing any parameters of the original laser system, and stabilization may be achieved regardless of the strength of the first feedback. This work is funded by the Inter-University Attraction Pole program (IAP IV/07) of the Belgian government. DWS acknowledges the support of the National Research Council and US Air Force Office of Scientific Research.

Threshold reduction for second cavity

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IV. SUMMARY

[1] J. Wieland, C.R. Mirasso, and D. Lenstra, Opt. Lett. 22, 469 (1997). [2] A. Hohl and A. Gavrielides, Opt. Lett. 23, 1606 (1998). [3] Y. Liu and J. Ohtsubo, IEEE J. Quantum. Electron. 33, 1163 (1997). [4] F. Rogister, P. M´egret, O. Deparis, and M. Blondel, submitted to Opt. Lett. [5] T. Sano, Phys. Rev. A 50, 2719 (1994). [6] A. Hohl and A. Gavrielides, Phys. Rev. Lett. 82, 1148 (1999). [7] R. Lang and K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).

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Frequency shift (GHz)

FIG. 2. Experimental optical spectra as a function of second feedback strength. Traces (b) and (f) exhibit stabilized behavior.

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