Observation of slow light in the noise spectrum of a vertical external cavity surface emitting laser
A. El Amili, B.-X. Miranda, F. Goldfarb, G. Baili, G. Beaudoin, I. Sagnes, F. Bretenaker, M. Alouini
aa r X i v : . [ phy s i c s . op ti c s ] O c t Observation of slow light in the noise spectrum of a vertical external cavity surfaceemitting laser
A. El Amili , ∗ B.-X. Miranda , , F. Goldfarb , G. Baili , G. Beaudoin , I. Sagnes , F. Bretenaker , and M. Alouini , Laboratoire Aimé Cotton, CNRS-Université Paris Sud 11, 91405 Orsay Cedex, France Institut de Physique de Rennes, CNRS-Université de Rennes I, 35042 Rennes Cedex, France Thales Research and Technology, Campus Polytechnique, 91127 Palaiseau Cedex, France and Laboratoire de Photonique et Nanostructures, CNRS, Route de Nozay, 91460 Marcoussis, France (Dated: June 3, 2018)The role of coherent population oscillations is evidenced in the noise spectrum of an ultra-lownoise lasers. This effect is isolated in the intensity noise spectrum of an optimized single-frequencyvertical external cavity surface emitting laser. The coherent population oscillations induced by thelasing mode manifest themselves through their associated dispersion that leads to slow light effectsprobed by the spontaneous emission present in the non-lasing side modes.
PACS numbers: 42.55.Ah, 42.60.Lh, 42.50.Lc
Since the early works of Sommerfeld [1] and Brillouin[2, 3] on light propagation through resonant atomic sys-tems, slow and fast light (SFL) have been the subject ofconsiderable research efforts. To control the group ve-locity of light, various approaches have been proposedand demonstrated, such as, e. g., electromagnetically in-duced transparency [4, 5], coherent population oscilla-tions (CPO) [6, 7], and stimulated Brillouin scattering[8]. All these approaches are based on the well knownKramers-Krönig relations stating that a narrow reso-nance in a given absorption profile gives rise to verystrong index dispersion in the medium. Consequently, apulse of light can propagate through a material slower orfaster than the velocity of light in vacuum without violat-ing Einstein’s causality [9]. In this framework, the majorpart of the studies reported in the literature is devotedto single-pass propagation in the considered dispersivemedium: the pulse shape or the amplitude modulationof the light is fixed at the entrance of the SFL system.The point is then to investigate how these characteristicsevolve during propagation through the medium.Systems, such as lasers, in which the light is self orga-nized, have not attracted so much attention in this con-text. Yet, CPO, an ubiquitous mechanism inducing SFL,is present in any active medium provided that a strongoptical beam saturates this medium. Thus, CPO mustbe present in any single frequency laser since the oscil-lating beam acts as a strong pump which, by definition,saturates the active medium. This effect could be ob-served using an external probe whose angular frequencyis detuned with respect to the oscillating mode, by lessthan the inverse of the population inversion lifetime /τ c .Besides, it has been shown in semiconductor optical am-plifiers (SOAs) that CPO induced SFL leads to a signifi-cant modification of the spectral noise characteristics atthe output of the SOA [10, 11]. Consequently, this effectshould be also visible in the laser excess noise, using the ∗ Electronic address: [email protected] spontaneous emission present in the non-lasing side lon-gitudinal modes of a single-frequency laser as probe ofthe CPO effect. To reach this situation, the free spectralrange (FSR) of the laser must not be larger than /τ c .This is seldom fulfilled in most common lasers. For in-stance, in ion-doped solid-state lasers, τ c is in the rangeof µ s - 10 ms [12]. Thus, the FSR of the laser shouldbe smaller than 1 MHz, forbidding single-frequency op-eration. On the other hand, τ c in semiconductor lasersis in the ns range. Consequently, CPO effects are ef-ficient at offset frequencies below a few GHz from thelasing mode [13]. The FSR of edge emitting semiconduc-tor lasers being around 100 GHz makes them unsuitablefor this experiment. However, class-A vertical externalcavity surface emitting semiconductor lasers (VECSELs)[14] recently developed for their low noise characteris-tics exhibit i) single-frequency operation, ii) ultra-narrowlinewidth [15], iii) shot-noise limited intensity noise [16],and iv) a FSR in the GHz range. All these characteristicsmake them perfectly suited for the observation of CPOinduced SFL in their noise spectrum.The laser used in our experiment is a VECSEL whichoperates at ∼ µ m (Fig. 1). The 1/2-VCSEL gain chipis a multi-layered stack, over L m ≈ µ m length, ofsemiconductors materials. Gain is produced by six In-GaAs/GaAsP strained quantum wells grown on a high re-flectivity Bragg mirror. The Bragg mirror side is bondedonto a SiC substrate to dissipate the heat towards aPeltier cooler. The top of the gain structure is cov-ered by an anti-reflection coating. The gain is broad( ∼ L < ∼
10 cm from the gain structure. In theseconditions, / πτ c is not negligible compared with theFSR ( ∆ > ∼ . ). The laser is optically pumped at808 nm. The pump is focused to an elliptical spot on thestructure with the ellipse aligned with the [110] crystalaxis to avoid polarization flips. A − µ m thick glassétalon is inserted inside the cavity to make the laser sin-gle mode. Its spectrum is continuously analyzed with a P e lti e r E t a l on P u m p L a s e r D i od e l = n m S pec t r u m A na l yze r R F A m p li f i e r G = d B P ho t od i ode DC - GH z F ab r y - P e r o t C av it y l = µ m K n i f e e dg e R = % O p ti ca l I s o l a t o r O s c ill o s cope1 / VC S EL FIG. 1: Experimental setup. A 808 nm fibred laser is used topump the 1/2-VCSEL chip that emits around 1 µ m. The extended cavity is closed by a 99% reflection mirror. The étalonis used to obtain the single-frequency regime and a knife edgeis inserted to increase the losses in a controllable manner andthus adjust the power inside the cavity. The output light ispartly sent to a Fabry-Perot analyzer and partly to a detectorfollowed by an electrical spectrum analyzer. Fabry-Perot interferometer to ensure that the laser re-mains monomode and that there is no mode hop duringspectra acquisitions. The noise spectrum is measuredusing a setup similar to the one described by Baili et al. [14]. We use a wide bandwidth photodiode and a lownoise radio-frequency amplifier in order to reveal the ex-cess noise due to the beatnotes between the laser line andthe spontaneous emission noise at neighboring longitudi-nal mode frequencies [16]. Indeed, the laser output fieldreads E ( t ) = P p A p e − iπν p t + c . c . , where p holds for thedifferent mode orders of amplitudes A p at the cold cavityfrequencies ν p = ν + p ∆ . p = 0 corresponds to the lasingmode, and p = ± to the two closest non-lasing modes,etc... This field leads to the following photocurrent atthe output of the detector: i ph ( t ) ∝ |A | + X p =0 |A p | + X p =0 (cid:2) A A ∗ p exp ( − iπf p t ) + c . c . (cid:3) , where the side mode fields |A p | (containing only spon-taneous emission) are very small compared with the las-ing mode field |A | . Thus, the excess intensity noise,characterized by A A ∗ p , consists of peaks located at | f p | = | ν − ν p | in the Fourier space. On that account,the beat frequencies | f p | occur at harmonics of the FSRin the noise spectrum (Fig. 2). Just above threshold( η − ≪ , where η is the laser excitation ratio), theexcess noise peak exhibits a Lorentzian shape with awidth completely described by the excess of losses δγ p induced by the étalon on the p th side mode [14]. At the p th FSR frequency p ∆ , the noise spectrum is thus thesum of two Lorentzian peaks due to the beat notes of thelasing mode with the corresponding sidebands ( p th and − p th modes). By contrast, when the pumping rate is in- N o i s e po w e r ( d B m ) Noise frequency (GHz)
FIG. 2: Typical laser intensity noise spectrum. For a cavitylength L ≈
10 cm, the beat note frequency appear at the firstharmonic of the resonator FSR ∆ ≈ ∆ . The factthat this noise is composed of two Lorentzian peaks is thesignature of a CPO induced gain modulation, that leads to adispersion effect probed by the non-lasing modes located ± ∆ from the lasing frequency ν . creased, we found experimentally that the excess noiseconsists of two peaks separated by δf = f p − f − p ∼ δf ≈ ν L m L + n L m ( δn p + δn − p ) , (1)where n is the bulk refractive index of the semiconduc-tor structure. δn ± p are the modifications of the refrac-tive index of the structure experienced by the ± p sidemodes and induced by the dispersion associated with theCPO effect. In a semiconductor active medium, thanksto the Bogatov effect [17], the dispersion is not an oddfunction of the frequency detuning with respect to ν .Thus, δn p = − δn − p and the two beat note frequencies f p and f − p corresponding to the p and − p modes occur atslightly different frequencies, as evidenced by the doublepeak of Fig. 2. More precisely, this CPO induced indexmodification can be derived from the gain medium rateequation. We assume that this medium can be modeledby a two-level system driven by an intracavity light field E ( t ) which is the sum of the lasing mode and the twoclosest side modes: E ( t ) = A e − iπν t + A − e − iπν − t + A e − iπν t + c . c .. As the étalon forces the laser to operate in single moderegime, one has | A | ≫ | A − | ≈ | A | . Consequently,we consider only the beat notes between the lasing andthe adjacent modes which create modulations of the pop-ulation inversion at frequencies close to ∆ . Under theseassumptions, the gain g ( ν p ) and the refractive index vari-ation δn ( ν p ) = n ( ν p ) − n seen by the side modes, thatcan be considered as weak probes, are given by [18]: g ( ν p ) = g S ( − S [(1 + S ) + α π ( ν − ν p ) τ c ](1 + S ) + [2 π ( ν − ν p ) τ c ] ) , (2) -2 -1 0 1 2012345678 (a) G R ound - t r i p g a i n ( % ) p (GHz) G ( p )G ( ) -2 -1 0 1 2-50510152025303540 (b) r t ( m r a d ) p (GHz) FIG. 3: (a) Round-trip gain versus probe frequency detuning ν − ν p . The thin line is the unsaturated gain. The dashedline is the saturated gain for the light at ν . The full anddotted-dashed lines are the gains seen by the probe for α = 0 and α = 5 , respectively. (b) Round-trip phase modificationexperienced by the side modes for α = 0 (full line) and α = 5 (dotted-dashed line). These profiles are plotted from Eqs. (2)and (3) with τ c = 2 ns , S = 0 . , and G = 2 g L m = 0 . ,which correspond to our experimental conditions. δn ( ν p ) = c πν g S S π ( ν − ν p ) τ c + α (1 + S )(1 + S ) + [2 π ( ν − ν p ) τ c ] . (3)Here g is the unsaturated gain and S the saturationparameter. α is the phase-intensity coupling coefficient(Henry’s factor) that is responsible for the Bogatov effect.Eq. (2) describes two phenomena: i) the self-saturationof the gain at ν by the field at ν [dashed line in Fig.3(a)] and ii) the modifications due to the CPO effect ofthe gains probed by the side modes at ν ± p . The evolu-tion of this gain versus probe frequency is plotted as afull line (resp. dotted-dashed line) in Fig. 3(a) for α = 0 (resp. α = 5 ). This CPO effect is also responsible forthe modification of the refractive index seen by the sidemodes which modifies the round-trip phase accumulatedby each side mode [see Fig. 3(b)]. With α = 0 , we no-tice that the phase shifts for two symmetric side modesare not opposite, restraining δf from vanishing [see eq.(1)]. Fig. 4 shows the double peak for different intracav-ity powers P circ (defined as the power of one of the twotraveling waves creating the intracavity standing wave).It should be noticed that the two excess noise peak pro-files have different widths. This is explained by the factthat at the first order, the widths depend on the lossesinduced by the intracavity étalon. These extra losses leadto the following extra loss rates for the p th side mode: δγ p = 2∆ [1 − T ( ν p )] , (4)where T ( ν p ) is the étalon intensity transmission for thatmode. When the lasing mode frequency ν coincides circ (W) 4.1 2.5 1.6 N o i s e po w e r ( d B m ) Noise Frequency (GHz)
FIG. 4: Experimental noise spectrum for different intracavitypowers. The difference between the widths of the two peaksis clearly visible. The peak widths and the spacing increasewith the intracavity power. Resolution Bandwidth=1 kHz. (cid:1) (cid:2) (cid:1) (cid:3) (cid:2)(cid:4)(cid:3) (cid:1)(cid:1) (cid:5) (cid:1) (cid:6)(cid:5) (cid:1) (cid:7) (cid:2)(cid:1) (cid:5)(cid:7)(cid:7) (cid:8)(cid:7)(cid:7) (cid:9)(cid:7)(cid:7) (cid:10)(cid:7)(cid:7) (cid:11)(cid:7)(cid:7) (cid:12)(cid:7)(cid:7) (cid:13)(cid:7)(cid:7)(cid:11)(cid:5)(cid:7)(cid:5)(cid:11) (cid:2) (cid:3) (cid:14) (cid:15) (cid:8) (cid:4) (cid:16) (cid:2) (cid:17) (cid:18) (cid:19) (cid:3) (cid:16) (cid:2)(cid:20)(cid:3) (cid:2)(cid:1) (cid:16)(cid:2)(cid:21)(cid:18)(cid:19)(cid:3)(cid:16) (cid:14)(cid:22)(cid:5)(cid:14)(cid:22)(cid:6)(cid:5)
FIG. 5: (a) Etalon transmission versus frequency. When thelasing mode frequency is shifted by δν from the maximum ofétalon transmission, the transmissions for the side modes at ν ± are no longer equal. (b) Extra loss rates δγ ± versus δν . with a maximum of the transmission spectrum, both sidemodes transmissions are equal: T ( ν ) = T ( ν − ) < and the peak widths are also equal: δγ = δγ − . Butif ν is shifted by δν > from the étalon resonance fre-quency [see Fig. 5(a)], the étalon transmission for mode p = +1 (resp. p = − ) decreases (resp. increases) with δν . Figure 5(b) shows the effect of such a detuning onthe extra loss rates δγ ± . We check from the experi-ment whether this evolution of the extra losses experi-enced by the side modes correctly explains the widthsof the two peaks, like in the simple model of Ref. [16].The two peaks of Fig. 4 are fitted by two Lorentzians inwhich some asymmetry is included to take into accountthe Bogatov effect induced by the Henry factor [17]. Fig.6(a) reproduces the evolution of the peak widths versusintracavity power. This intracavity power is varied by (a) P circ (W) p / ( k H z ) (b) P circ (W) f ( k H z ) FIG. 6: (a) Peak widths δγ ± versus intracavity power. (b)Peak spacing δf versus intracavity power. Squares: measure-ments. Full line: prediction obtained from eqs. (1) and (3)with the same parameters as in Fig. 3 introducing controlled diffraction losses inside the cavityusing a knife edge for a constant pump power, in orderto keep g constant. The variation of the intracavitypower modifies the laser frequency shift δν , leading todifferent evolutions of δγ and δγ − , as expected fromFig. 5(b). However, the magnitudes of the experimen-tally observed variations of these widths are significantlylarger than those calculated in the simple linear modelof Eq. (4), suggesting the enhancement of this effect bynonlinear contributions. Moreover, it is expected thatincreasing the intracavity power, and thus the gain satu-ration, leads to an increase of δf . Fig. 6(b) clearly showsthat the frequency shift δf between the two peaks in-creases with the intracavity power, evidencing the non-linear origin of the double peak noise spectrum expectedfrom Eq. (3). The full line in Fig. 6(b) is obtained fromeqs. (1) and (3) with our experimental parameters. Itshows that our simple model based on a two-level systemincluding Henry’s factor gives the good order of magni-tude for δf and the correct sign for its evolution versus intracavity power. One should not be surprised by thefact that the agreement with the measurements is notperfect: the model of eqs. (2) and (3) is too crude tofully describe the gain and index saturation in strainedquantum wells. Moreover, we overlooked many effectsthat may lead to a discrepancy with respect to our sim-ple approach such as i) the variation of α with the car-rier density, ii) the thermally induced variations of theindex and of the laser mode diameter, iii) the variationsof τ with the carrier density, iv) the possible existenceof an offset in δf due to the linear dispersion of the gainmedium and the étalon. Notice also that since the cavityFSR ∆ is larger than the width of the CPO dip of Fig.3(a), we are probe the wings of the dispersion profile ofFig. 3(b), i. e., in the slow light regime. Moreover, wehave checked that this phenomenon is not related to acoupled cavity effect since we observed exactly the samebehavior of the noise spectrum with another 1/2-VCSELwithout any anti-reflection coating. If the splitting be-tween the two peaks were due to a coupled cavity effect,it should be completely different in the absence of theanti-reflection coating, contrary to our observations.In conclusion, we experimentally evidenced the exis-tence of intracavity slow light effects in a laser inducedby the CPO mechanism. These effects are probed bythe laser spontaneous emission noise present in the nonlasing modes. We have shown that this noise is a veryefficient probe to explore the intracavity CPO effects andtheir evolution with the laser parameters such as the in-tracavity power. Moreover, we have predicted that thisfirst observation of slow light inside a laser cavity shouldbe able to lead to intracavity fast light if the side modefrequencies are closer to the lasing mode frequencies, i. e.,for a longer cavity. 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