Abstract
We find experimentally that the relaxation oscillation peak in the relative intensity noise spectrum of a semiconductor laser has a higher damping and lower frequency when we add low frequency noise to the pump current. The broadening of the relaxation oscillation peak with increasing carrier noise level is interpreted as an increase of the nonlinear gain compression with noise strength.
| Original language | English |
|---|---|
| Article number | 071107 |
| Number of pages | 4 |
| Journal | Applied Physics Letters |
| Volume | 88 |
| Issue number | 7 |
| DOIs | |
| Publication status | Published - 24 Feb 2006 |
Funding
Van der Sande Guy a) Soriano Miguel C. Yousefi Mirvais b) Peeters Michael Danckaert Jan Verschaffelt Guy Department of Applied Physics and Photonics, Vrije Universiteit Brussel , Pleinlaan 2, B-1050 Brussels, Belgium Lenstra Daan COBRA Research Institute, Technical University-Eindhoven , The Netherlands and Laser Center, Vrije Universiteit Amsterdam , The Netherlands a) Also at: Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Campus Plaine, Code Postal 231, 1050 Bruxelles, Belgium; electronic mail: [email protected] b) Present address: COBRA Research Institute, Technical University-Eindhoven, The Netherlands. 13 02 2006 88 7 071107 27 07 2005 04 01 2006 14 02 2006 2006-02-14T09:05:31 2006 American Institute of Physics 0003-6951/2006/88(7)/071107/3/ $23.00 We find experimentally that the relaxation oscillation peak in the relative intensity noise spectrum of a semiconductor laser has a higher damping and lower frequency when we add low frequency noise to the pump current. The broadening of the relaxation oscillation peak with increasing carrier noise level is interpreted as an increase of the nonlinear gain compression with noise strength. From a fundamental point of view, two sources of noise can be distinguished in semiconductor lasers. Spontaneous recombination of electrons and holes results in spontaneous emission noise due to a fraction of spontaneously emitted photons ending up in the lasing mode (referred to as field noise ). On the other hand, the random and instantaneous character of each discrete recombination event leads to carrier inversion noise. However, in practice, external sources contribute to extra noise on the injection current of the laser, resulting in a higher carrier noise level than the fundamental shot noise level. In semiconductor lasers, only a small portion of the total spontaneously emitted photons end up in the lasing mode. These photons have random phases and henceforth lead to random fluctuations of the power and of the phase of the lasing mode. Henry showed, 1,2 as was confirmed by others later, 3,4 that the linewidth of the optical field of a cw emitting semiconductor laser is determined by these phase fluctuations and is not affected by the carrier noise. Petermann further demonstrated 3,4 that the relaxation oscillations (RO) peak of the semiconductor laser in the power spectrum is independent of the carrier noise strength. This has led to the general belief that carrier noise can be ignored in the analysis of semiconductor lasers and their dynamics. However, some of us recently found that realistic levels of carrier noise can substantially influence the dynamics of a semiconductor laser subject to optical feedback. 5 Moreover, Balle et al. reported that fluctuations in the initial conditions of the carrier inversion and in spontaneous emission have a qualitatively different importance in the switch-on statistics of class-B lasers. 6 Here, we investigate the influence of carrier noise on the RO of a vertical-cavity surface-emitting laser (VCSEL), which have the advantage of emitting in a single longitudinal mode due to their exceptionally short optical cavity. We will add low bandwidth ( 400 MHz ) noisy carriers directly to the pump current of the VCSEL, which will additively contribute to fluctuations in the inversion. As we will see, the effect of this colored noise is non trivial as it alters the RO frequency and its damping rate. We use an oxide-confined VCSEL lasing at 970 nm , which is provided by the Optoelectronics Department of the University of Ulm. 7 The threshold current of the solitary laser is I th = 0.91 mA at 25 ° C and the temperature is stabilized up to ± 0.01 K . This device emits linearly polarized light in the fundamental mode until I sw = 1.42 mA , where it switches to the other orthogonal polarization mode. The measured frequency separation between the two polarization modes is about 1.5 GHz and the switch occurs from the high-frequency mode to the low-frequency mode. Multiple transverse modes appear for currents higher than 2.4 mA . In this letter, we limit ourselves to the single transverse mode and single polarization mode regime ( I < 1.4 mA ) to minimize the effects of mode partition noise and to maximize the mode suppression ratio. The VCSEL is driven by a low-noise current source (Thorlabs LDC8002) to which we externally add low-frequency noise produced by an arbitrary wave form generator of 400 MHz bandwidth (Tektronix AWG520) using a bias-T ( 100 kHz – 40 GHz ) . The low-frequency noise is passed through a high-pass filter with a 1 MHz cutoff to exclude thermal effects due to Joule heating, as they only happen in the VCSEL under test up to a few hundreds of kilohertz (kHz). Both polarization modes can be selected for detection by rotating the polarizer and are detected with a 2.4 GHz bandwidth photodiode. The converted electrical signal is analyzed using a 4 GHZ bandwidth digital scope (Tektronix CSA7404) and by a spectrum analyzer (Anritsu MS2667C) with a 9 kHz – 30 GHz bandwidth. We also pass a part of the output beam through a Fabry-Perot etalon with a free spectral range of 200 MHz and a finesse of 20. This provides us with direct access to the linewidth properties of the VCSEL. The colored noise has been characterized at two points: after the bias-T and in the relative intensity noise (RIN) spectrum of the VCSEL below 1 GHz . In these two measurements the spectrum is flat up to the 3 dB point at 400 MHz and then drops as a fifth order system. The I RMS values used in this letter can be obtained from the noise level in the RIN spectrum or using the noise level measured after the bias-T together with the VCSELs differential resistance. Both values are in accordance to within a few percent. We start by measuring the RIN spectrum of the laser as a function of the colored noise strength. These results are shown in Fig. 1 , where in Fig. 1(a) we can identify the RO peak, which changes both in position and shape as a function of the carrier noise strength [see Fig. 1(b) ]. In general, we found that the RO peak shifts to lower frequencies, while the width of the peak increases with the noise strength. Also, the intensity noise below 400 MHz is considerably enlarged due to the addition of the low frequency noise. The RIN spectra are fitted according to 3 RIN ( f ) = A f 2 + B ( Ω 2 + Γ 2 − f 2 ) 2 + 4 Γ 2 f 2 , (1) and from this we extract the RO damping rate Γ and the RO frequency Ω ( A and B are fitting parameters). The results of these fits are also shown in Fig. 1 . Figure 2(a) shows the fitted parameters as a function of the colored noise amplitude at a bias current of 1.2 mA . It is clear that Ω 2 + Γ 2 remains constant (within the error margins) with increasing noise strength, while Γ increases with the carrier noise amplitude. Thus, Ω 2 must decrease with increasing noise strength. At higher bias currents we observe similar trends, but the RO peak of these measurements is far beyond the bandwidth of our detector ( 2.4 GHz ) . These experiments show that the presence of colored carrier noise has an important influence on the dynamical behavior of a semiconductor laser, since the RO peak is a good signature of dynamical activity. Finally, we have checked for modifications to other operation characteristics of the laser due to added carrier noise (output intensity, threshold current, linewidth, etc.), but have found no changes outside the error margins of the measurements. We have also performed the same type of measurements on another type of VCSEL and also on an edge-emitting laser (EEL), and achieved qualitatively similar results. In order to understand the underlying mechanism leading to the observed increase of the RO damping rate, we use a rate equations model based on the standard Langevin equations. 6 The rate equations for the slowly varying photon number P and carrier number N read d P d t = [ g ( N ∕ V − n 0 ) 1 + ϵ s P − 1 τ p ] P + R sp + F P , (2) d N d t = I q − N τ c − g ( N ∕ V − n 0 ) 1 + ϵ s P P + F N + F C , (3) where g is the differential gain coefficient in the active region, V is the active region volume, n 0 is the transparency carrier density, and ϵ s is the gain compression factor. τ p is the photon lifetime, τ c is the electron-hole recombination lifetime (both radiative and nonradiative), I is the injected current, and q is the unit charge. The Langevin noise sources F P ( t ) and F N ( t ) account for the intrinsic noise, and satisfy ⟨ F i ( t ) F j ( t ′ ) ⟩ = 2 D i j δ ( t − t ′ ) , with D P P = R sp P and D N N = N ∕ τ c , where R sp is the rate of spontaneous emission into the lasing mode. Also, since D N P ∼ 10 − 5 , we set D N P = 0 . The low-frequency noise added experimentally to the bias current is modeled and represented by F C ( t ) . It is completely uncorrelated to F P ( t ) and F N ( t ) . This Langevin force satisfies ⟨ F C ( t ) F C ( t ′ ) ⟩ = ( D C C ∕ τ ) exp [ − ∣ t − t ′ ∣ υ ∕ τ ] . Here τ corresponds to the correlation time of the colored noise and υ determines the steepness of the noise beyond the cutoff frequency, which is obtained by numerically fitting the response of the noise generator used in the experiments. Equations (2) and (3) are to be interpreted in the Stratonovich sense. They were solved numerically with a second-order stochastic corrector-predictor integrator often called the Heun algorithm converging to the Stratonovich solution as required. The colored noise F c was accounted for as the output of a series of filtering equations with white noise as the input. These filter equations match the fifth order Butterworth of the signal generator used in the experiments. Analogous to the experiments, the colored noise F C contributes to the numerically obtained RIN at low frequencies. However, only minor changes in the RO damping rate are observed with the carrier noise strength D C C , while the RO frequency remains unaffected. This leads us to the hypothesis that a noise-induced change in one of the model parameters must be the cause of the change in the RO dynamics. In order to identify the relevant parameters, a closer look at the expressions for the RO frequency and damping rate is required. Following the procedure described in Ref. 2 , we find Γ = 1 2 [ 1 τ c + R sp P s + ( g ∕ V ) P s ( 1 + ϵ s P s ) ( 1 + ϵ s g τ p ) ] , (4) Ω 2 + Γ 2 ≈ g V τ p ( 1 + ϵ s P s ) P s , (5) with the steady state photon number P s . Even if the linear perturbation analysis used to get Eq. (4) is only valid for small values of the current fluctuations, Eqs. (4) and (5) help us to identify the parameters that can be noise dependent. By comparing the experiment in Fig. 2(a) with Eq. (4) , we can identify which parameter is most likely to be noise dependent. P s is unchanged with the noise strength. τ c cannot change with the noise strength, since no change of the threshold current was observed in the measurements. It is very unlikely that g is noise dependent, because in that case Eq. (5) indicates that Ω 2 + Γ 2 would vary stronger than Γ . Thus, the gain compression ϵ s is the best candidate for the previously mentioned noise dependency. Numerical simulations of Eqs. (2) and (3) confirm the validity of our hypothesis [see Fig. 2(b) ]. The gain compression factor ϵ s is now taken to be dependent on the strength of the colored noise in the simulations, described by the educated guess ϵ s = ϵ 0 { 1 + [ 5 I RMS ∕ ϵ 0 ( I − I th ) ] } . Note that the curve for Γ ∕ Ω 0 in Fig. 2(b) falls entirely within the error margins of the experimental curves in Fig. 2(a) . The underlying physics of the increase of the nonlinear gain compression ϵ s with the external noise amplitude must be due to the alteration of either spatial hole burning, spectral hole burning or carrier heating in the device. Generally, these three effects are identified as the cause of the nonlinear gain suppression in semiconductor lasers. From the experiments presented here, we cannot unequivocally pin down the main mechanism. However, since we observe the same effects in both, VCSELs and EELs, we believe transverse spatial hole burning can be excluded as this effect is very pronounced in VCSELs and nearly absent in EELs. Spectral hole burning and carrier heating remain as candidates. More research is necessary to conclude on this point, but this falls out of the scope of this letter. In summary, we have shown that low frequency noise on the injection current in a semiconductor laser decreases the RO frequency while it increases its damping rate. This effect has been observed in two types of VCSELs as well as in an EEL. These results clearly indicate that the presence of colored carrier noise has an important influence on the dynamical behavior of a semiconductor laser. Therefore, when describing the dynamical behavior of a semiconductor laser system on the basis of a rate equation model in the presence of noise, one should not only take into account the correct noise sources in the rate equations, but also keep in mind that the noise could alter some of the parameters in the rate equations. The authors acknowledge the Interuniversity Attraction Pole program (IAP V/18). M.Y., J.D., G.V., and G.VdS acknowledge the FWO (Fund for Scientific Research- Flanders) for their fellowships and for project support. M.C.S. acknowledges Marie Curie Fellowship HPMT-CT-2000-00063. The authors thank Professor R. Michalzik, Professor A. Barel, Professor S. Stolte, Professor F. Marin, Professor S. Balle, and Dr. I. Fischer for the interesting discussions. FIG. 1. Measured RIN spectra of the VCSEL under test at a constant current of 1.2 mA and at different carrier noise strengths together with corresponding fits. FIG. 2. The RO damping rate Γ (full, + ) and Ω 2 + Γ 2 (dashed, × ) vs the noise strength at a bias current of 1.2 mA . (a) Measurements. (b) Numerical, computed by integrating Eqs. (2) and (3) . ( g = 2.12 × 10 − 4 μ m 3 ∕ ns , V = 1.6 μ m 3 , n 0 = 4 × 10 6 μ m − 3 , τ p = 2 ps , τ c = 1.8 ns , I = 1.33 I th , I th = 0.91 mA , R sp = 1 × 10 10 s − 1 , τ − 1 = 2 π × 400 MHz , υ = 5 , ϵ s = ϵ 0 { 1 + [ a I RMS ∕ ϵ 0 ( I − I th ) ] } , ϵ 0 = 3.375 × 10 − 6 , a = 5 .) The full lines guide the eye and Ω 0 is the RO frequency in the absence of noise.