## Abstract

We present an experimental and theoretical investigation of the non-linear multimode dynamics of external–cavity VCSELs emitting at 1 and 2.3 *μ*m. We account for the stable single–frequency and linearly polarized emission by these laser sources, even in the presence of quantum noise and non-linear mode interactions originating from Four–Wave–Mixing via population pulsations in the quantum-wells. This fact is a consequence of the mode antiphase dynamics. Thanks to the high-Q external cavity configuration, the laser dynamics fall into the oscillation-relaxation-free class-A regime. The characteristic time to achieve single mode emission is ~1 ms for a 15 mm long cavity with an antireflection coated structure and no spectral filter, as for an “ideal” homogeneous gain laser. The side mode suppression ratio is as high as 40 dB, close to the quantum limit. The laser linewidth is at the quantum limit, and is ~1 Hz at 1mW output. An experimental value <20 kHz has been established. Under standard conditions, without spectral filtering, the optimum cavity length for highly coherent single mode operation is expected in the range 5 to 30 mm. Finally, for cavity lengths typically shorter than 5 mm, we rather have an “ideal” homogeneous gain class-B laser, exhibiting oscillation-relaxation of the intensity in the 0.1 GHz range. These properties contrast with the intrinsic strongly non-linear dynamics of conventional semiconductor lasers.

© 2007 Optical Society of America

## 1. Introduction

Semiconductor lasers are compact systems that offer a broad homogeneous gain and capable of room-temperature (RT), continuous-wave (CW) operation anywhere from 0.4 to 3 *µ*m. Semiconductor laser technology is maturing rapidly and is finding applications in areas such as high resolution spectroscopy (trace gas analysis, in-situ H_{2}0 and CO_{2} isotope ratio measurements), medicine, optical telecommunications and radar-lidar, metrology where ultra-narrow low noise sources are required. Thus there is a strong interest in developing high quality single frequency tunable lasers operating CW at RT with 1 to 1000 mW of output power in the *λ*=0.8--2.7 *μ*m range, where III–V semiconductors (GaAs, InP, GaSb) are now available.

It has been recently shown that monolithic semiconductor devices - i.e. conventional microcavity VCSELs and DFB edge emitting lasers - need more complex and expensive technological steps to achieve the required specifications [1], compared with the “simple” Vertical- external-Cavity Surface-Emitting Laser (VeCSEL) configuration discussed here. Indeed the high-Q vertical cavity design “boosts” the coherence of the emission providing a circular TEM_{00} beam at high output power [2], with low divergence (a few degrees), ultra narrow linewidth (kHz range), high side-mode-suppression-ratio [1, 3, 4], broad continuous tunability (0.25 THz), low sensitivity to optical feedback, and good polarization stability [1, 3]. These features are intrinsic to the use of a high-Q air-gap “long” (≫1 *µ*m) stable optical cavity, where the contribution of amplified spontaneous emission is negligible. Previous publications reported on the stable single-frequency plane-concave short-cavity Quantum-Well (QW) VeCSEL s operating at RT at *λ*=1 *µ*m (GaAs-based) [5] and *λ*=2.3 *µ*m (GaSb-based) [3], with more than 20mW and 5mW output power respectively, without the need for a spectral intracavity filter.

We present here a first detailed study, both from an experimental and a theoretical point of view, of the non-linear multimode dynamics of these new source. Our specific aim is to investigate its stability and purity in single mode operation. As a starting point we will first shortly resume some of our experimental results on this system, at 1 *µ*m and 2.3 *µ*m. We will then present a modeling of the spectro-temporal dynamics of the QW VeCSEL including quantum noise. We demonstrate why theses sources operate stable single-frequency and linearly polarized, even without any intracavity selective element, in the presence of Four-Wave-Mixing (FWM) in the QWgain during the build-up transient. This is the most relevant non-linear mode interaction occurring in this laser source and is a consequence of pulsations of the carrier density at the intermode beat frequencies [6]. We show that thanks to the high-Q external cavity configuration, the oscillation-relaxation-free class-A regime is typically observed, with properties close to the quantum limit as in the case of an “ideal” homogeneously broadened laser. This contrasts with the intrinsic non-linear dynamics of edge emitting diode lasers and microcavity VCSELs. This study is valid with any VCSEL structure materials (e.g. GaAs, InP or GaSb-based semiconductors), either optically or electrically pumped, as soon as the gain medium thickness is ≪*λ*. It is valid for any linear or ring cavity of length shorter than typically 3m if the beam incidence angle on the 1/2-VCSEL is *<*λ/w_{0} (w_{0} the laser beam radius).

## 2. Experimental study

#### 2.1. Setup and device principles

The VeCSELs were composed of a 1/2-VCSEL structure, a 4 to 25mm air gap, and a commercial spherical concave dielectric mirror (reflectivity 99 to 99.5%–4 to 25mm radius of curvature). The laser thus operates in a standing-wave stable two-mirror plane-concave cavity (see Fig. 3). The typical 1/2-VCSEL structures were composed of an epitaxial high-reflectivity bottom Bragg mirror and an active layer on top, designed with typically 3 to 8 compressively strained type-I QWs in a 1 to 4 *λ* thick active region (depending on the pump absorption coefficient). Details of the GaAs- and GaSb-based structures, for the 1 *µm* (InGaAs/GaAs QWs) and 2.3 *µm* (InGaAsSb/AlGaAsSb QWs) spectral range can be found in [5, 7, 8]. On some samples, a Si_{3}N_{4} (*λ*/4 thick) antireflection coating was added at the air-semiconductor interface to suppress the residual microcavity effect. Even without this coating coupled-cavities effects are not relevant to the laser dynamics [9] if the reflectivity of the top surface is low compared to that of the cavity mirrors (≥99%). The only effect is to filter spectrally the modal gain bandwidth and to enhance/decrease themaximummodal gain,modifying the threshold (see section 3.1) [5, 10]. In order to increase the continuous frequency tuning range (>100GHz with a piezo), on some samples the substrate back side was polished at an angle (1–3°) and a 300 nm gold layer was evaporated. This avoids Fabry-Perot modulation of losses [5] due to the substrate (~400 *µ*m thick). Thus the typical mode number within the gain bandwidth (full-width at half-maximum, FWHM) for a 15(4)mm long cavity is 500(130) with a smooth gain profile: the laser cavity is strongly multimode. Without a wedge, single mode operation was observed to be more stable thanks to the substrate acting effectively as a weak intracavity etalon [5, 11].

The 1/2-VCSEL structures that we tested were optically-pumped in CW by a commercial low power single-transverse-mode GaAs laser diode (100mW output) emitting in the range 780–830 nm, with threshold densities below 1 kW/cm^{2} at RT. The pump beam was focused on a spot size ranging from 20 to 40 *µ*m (FWHM). In order to match this to the cavity mode waist, the cavity length must be close to the edge of stability given by the mirror radius of curvature. Typically a lasing region was obtained in the range 20 to 400 *µ*m below this length. It is interesting to remark that the use of a multimode-fibre-coupled laser diode as a pumping source is observed to hinder single mode operation of a VeCSEL in CW, unless a selective spectral element is inserted in the cavity (e.g. a 50 *µ*m thick Fabry-Perot glass etalon) [8]. This is due to the large intensity fluctuations of this source, which generate strong enough QW temperature perturbations and thus modulate the gain spectral position. Thanks to the external cavity transverse stability, the emission can be forced TEM_{00} using a tight pump beam size with respect to the laser cavity mode waist [1]. In addition, VeCSELs are usually linearly polarized even without selective intracavity elements (such as Brewster surfaces), due to circular symmetry breakdown. Indeed, about 10% gain dichroism in the QWs has been measured between the [110] and [110̄] crystal axis [1, 12], which stabilizes a linear polarization preferably along [110]. Thus, there is only one possible state with respect to transverse electric field and polarization, and only remains for the laser to select one longitudinal cavity mode thanks to the curvature of the net modal gain.

#### 2.2. Single frequency operation and transient spectral dynamics

Single frequency operation was readily observed with both samples emitting at 1 *µ*m and 2.3 *µ*m (Fig. 1) [1, 3, 5], for cavities ranging from 25 to 4mm long, with or without a wedged substrate or an antireflection coating. The side-mode-suppression-ratio was >30 dB (apparatus function limited), for a 15mm long cavity, a structure with a wedge and an antireflection coating [1] and without any spectrally selective element. The circular TEM00 beam displayed a quality factor *M ^{2}*<1.2 [1] with a divergence of a few degrees. The linear polarization extinction ratio was >45 dB. The maximum output power was 20mW at 1

*µ*m and 5mW at 2.3

*µ*m in this configuration. Using a Fagbry-Perot interferometer, we measured a laser linewidth below 20 kHz (apparatus function limited; 20

*µs*integration time) with the 2.3

*µ*m device [1, 3], without any active stabilization. The long-term stability at 1

*µ*m of one of these laser systems is shown in Fig. 2. Mode hopping arose due to temperature drift of the optical bench. This on-the-bench device showed high stability over hours, as good as commercial single-frequency laser diodes or solid-state lasers.

In order to observe the possible influence of any non-linear mode interactions, we studied the multimode laser transient after the onset of pumping, which is known to be a very sensitive tool [5, 13]. To record the spectro-temporal dynamics of a laser event before it reaches the steady state, a 15mm long VeCSEL was pumped using long constant intensity pulses, and different pump rates *η* above threshold were tested (Fig.3). The VeCSEL transient observed on the photodiode was oscillation-relaxation-free, showing that this high-Q external cavity source worked in the class-A regime. An Acousto-Optic-Modulator, triggered on the pump pulse, allowed deflecting the laser beam into a monochromator during a short time Δ*t* at a generation time *t _{g}*. Spectra were recorded with a 2.5

*µ*m extended InGaAs linear-array photodiode (Xenics, 512 pixels). Special care was taken to avoid thermal effect induced spectral shift or broadening. Absorption lines also slightly perturbed the spectral narrowing. As shown in Fig.4, the narrowing of the laser spectral width (FWHM) Δσ slowed down at short generation times for higher

*η*. But at long

*t*(≫100ms), VeCSEL emission always became single mode. In the absence of non-linear mode interactions (i.e. for an “ideal” homogeneous gain laser), Δ

_{g}*σ*

^{-2}should be proportional to

*t*[5], with no dependence on

_{g}*η*. On the other hand increasing spectral broadening with the pump rate at short generation times is a signature of non-linear mode interactions in a multimode laser [6, 13, 14]. In other words, a non-linear mode interaction may take place in a VeCSEL cavity, but it does not prevent stable single mode operation. Finally, in another experiment we built a 1m long, V-shaped plane-concave-plane VeCSEL cavity, using the same samples. This long-cavity laser did not display stable single mode operation, but the CW spectral width was still limited by the apparatus function (about four modes) when the laser setup was inserted in a closed box. This corresponds to a spectral narrowing continuing up to

*t*of 1 s [5], which shows again the high stability of this laser system, limited by technical noise.

_{g}#### 3. Multimode VeCSEL non-linear dynamics theoretical study and the quantum limit

### 3.1. Laser equation formulation

As explained, there is only one possible state for the light polarization and also for the transverse electric field, preventing from transverse spatial-hole-burning. Thus only the longitudinal mode dynamics is of interest for typical VeCSEL devices. In addition, gain is provided by semiconductor QWs, which can be modeled as a homogeneous gain of bandwidth ~*k _{B}T*≃ 200cm

^{-1}at 300K [15]. Here the top of the gain profile is approximated by a parabolic function. Note that these assumptions become invalid if one considers a Quantum-Dot active region, which exhibits an inhomogeneous gain broadening due to non uniform dot size distribution. In this particular case, the laser spectrum would be multi-mode and the laser would manifest a more complex multi-mode dynamics [11, 16]. For conventional semiconductor QW lasers, the induced atomic polarization can be adiabatically eliminated from the dynamics, as the intraband relaxation time of about Γ

^{-1}

*≃10*

_{g}^{-4}ns is much shorter than the carrier lifetime

*τ*~ns and the photon lifetime. Thus the dynamics are governed only by the mode field amplitude and the carrier density. The assumptions done in this paper, as well as the parameters given values, are valid for typical (GaAs, InP, GaSb)-based VeCSELs emitting in the 0.8–3

*µ*m range, optically or electrically pumped (assuming uniform injection over the laser beam profile).

We begin with the multimode semiclassicalMaxwell-Bloch description of the laser. We make the plane-wave, slowly varying amplitude, uniformfield approximation and follow the approach of [6, 9, 14, 17]. We expand the laser field in terms of the spatial modes of the cavity,

or a standing-wave laser cavity of length *L*. Here *A _{q}* is the complex amplitude of the

*qth*cavity mode (|

*A*|

^{2}is the photon number), Δ

_{q}=

*ω*-

_{q}*ω*=

_{0}*q*Δ,

*ω*is the bare-cavity frequency of mode

_{q}*q*, and Δ=

*πc*/

*L*is the cavity mode frequency separation, assuming negligible group delay dispersion in the cavity (typical calculated value ≪1000

*fs*[10]).

^{2}*ω*is the central frequency.

_{0}In a VeCSEL, as the QWthickness is very small compared to *λ* and QWs are usually located at the antinodes of the laser electric field, all the modes are in phase on the gain medium. This prevents from longitudinal spatial-hole-burning [9, 14, 17]. In addition, carrier diffusion in the QW plane [17] can be neglected, as the transverse beam size is much larger than the typical carrier diffusion length (about $\sqrt{{D}_{h}\tau}\simeq 1.7\mu m$ for holes in GaAs, where *D _{h}* is the heavy hole diffusion coefficient and t the carrier lifetime). Note that in the case of a ring resonator or if the 1/2-VCSEL is located in the middle of the cavity, where the structure is used as a plane folding mirror with an incidence angle for the beam smaller than typically

*θ*<

_{i}*λ*/2

*w*

_{0}≃3° (for a 2

*w*

_{0}=20

*µ*m laser spot diameter at

*λ*=1

*µ*m), the model below is valid (replacing 2

*L*by

*L*′ the cavity perimeter). For larger angles, an interference pattern is created in the plane of incidence across the beam, due to standing wave distribution of each mode

*q*(with induced grating spatial frequencies sin(

*θ*)(

_{i}*w*/

_{0}*πc*+

*q*/

*L*)). It will induce spatial inhomogeneity of the carrier density in the QWplane. Thus non-linear mode interactions due to “transverse” Spatial- Hole-Burning would arise, preventing from single mode operation. Heavy hole carrier diffusion has to be taken into account as well, as it would smooth the population grating [14, 17].

The multimode VeCSEL semiclassical differential equations, with phase sensitive interactions, for the cavity mode amplitudes *A _{q}* and the population inversion

*N*(carrier number:

*N*=

*n*×

*V*where

_{a}*n*is the carrier density,

*V*=

_{a}*πw*

^{2}

_{0}

*L*is the active volume), valid for Δ≫

*γ*, are then given by

_{q}where *γ _{q}*=ln(1-

*T*)

_{q}*c*/2

*L*is the cavity loss rate for mode

*q*(

*T*are the total optical losses),

_{q}*N*the effective transparency carrier number, assuming a linear gain dependence with N,

_{t}*α*the linewidth enhancement factor,

the stimulated emission rate in mode *q*, where Γ_{g} is the modal gain bandwidth. Γ* _{g}* is reduced by a factor ~3 typically with a resonant λ -long micro-cavity design without coating. Δ

*=(mq) Δ. The peak stimulated emission rate is approximated as,*

_{mq}where Γ* _{c}*≃

*n*/2×|

_{o}*EQW*/

*E*|2 is the longitudinal confinement factor (

_{incident}*E*and

_{QW}*E*are the electric field on the QWs and incident on the VeCSEL structure, respectively),

_{incident}*dg/dn*the differential gain [15] taken as constant and calculated around the carrier density threshold value,

*w0*is the pump/laser radius (to simplify the pump shape is assumed to be a top hat). We calculated Γ

*=2 for an antireflection coated 1/2-VCSEL structure, and Γ*

_{c}*=7 for a resonant structure design (microcavity length set to a multiple of λ/2) without any coating [5]. Even without coating, no coupled cavities effects (feedback regime) have to be considered for the laser dynamics [9] - between the remaining microcavity and the long external cavity - as long as the reflectivity of the top surface is low compared to that of cavity mirrors. An effect of the resonant design is to induce stronger intracavity group delay dispersion (still≪5000 fs*

_{c}^{2}). Finally,

*P*is the absolute pump rate.

We will assume a constant carrier lifetime *τ *in the QW [12, 15], as the carrier density *n* is almost constant above threshold and the pump rate is not too high above threshold. *F _{q}* and

*F*are the normalized random Langevin forces for the electric field and the carrier number respectively [9, 15, 18]. These account for the quantum nature of photon emission.

_{N}Here we neglected the non-linear gain [14] (third-order susceptibility) in the QW induced by beating vibration on the spectral distributions of carriers (i.e., electrons and holes) due to lasing frequencies, which is observed as the spectral hole burning effect and whose relaxation is characterized by the intraband relaxation time, on the order of Γ^{-1}
* _{g}*≃0.1

*ps*. This effect occurs even when the number of injected carriers is constant. For a typical VeCSEL cavity Γ

*≫ Δ, the active region is very thin compared to the wavelength (in contrast to an edge emitting laser) and the electric field value is small compared to the saturation value. Thus the non-linear gain is weak and uniform over the laser bandwidth. The ratio*

_{g}*R*

_{3/1}between the third order gain and the linear gain can be evaluated [14] and is approximately given by

where *R _{cv}*≃5.3×10

^{-29}m×C (for GaAs) is the absolute value of the dipole moment of the QW,

*n*≃3.5 is the optical index,

_{o}*λ*= 1

*µ*m is the free space laser wavelength,

*η*=2 is the excess of gain over the threshold value and

*N*is the number of oscillating modes (~10).

_{mode}In the dynamical theory of lasers, models get more complicated when increasing the number of field modes and/or the number of atomic levels which are involved in the laser action. Class-B lasers will be, by definition, described by rate equations which couple the complex field amplitudes and the carrier density. The distinctive feature of semiconductor class-B lasers realized experimentally is the small value of *τ*
^{-1} with respect to *γ _{0}*, which leads to oscillatory transients in the GHz regime for usual devices. In the single-mode limit, the unique frequency of relaxation oscillations depends on the excess pump parameter η and on both relaxation constants, in the form $\sqrt{\left(\eta -1\right){\gamma}_{0}\u2044\tau}\u20442\pi $ [15, 11, 17]. This frequency manifests itself in the form of a resonant peak in the spectrum of the intensity fluctuations at laser output. This is the usual case for edge emitting diode lasers and microcavity VCSELs. The transition to the multimode regime is accompanied by the appearance of new relaxation oscillation frequencies. In multimode lasers, mode-mode coupling is an additional factor affecting the dynamics. In addition to the usual in-phase oscillation of the amplitudes of all modes, there exist other oscillation regimes such as the antiphase dynamics. In the case of high-Q VeCSELs, and for a cavity typically longer than 5mm,

*γ0*is smaller than

*τ*

^{-1}allowing for the adiabatic elimination of

*N*from the dynamics and leading to a class-A laser without relaxation oscillations of the total photon number for single mode operation. This laser system is thus of great interest for the realization of low intensity-noise sources, as is confirmed experimentally (Fig. 3, bottom) and theoretically (Fig. 6). However, FWM via population pulsation could modulate slightly the carrier density and the total intensity, as shown below, at frequencies in the sub-MHz range.

As Δ≫*τ*
^{-1}≥*γ _{0}* for a typical VeCSEL (i.e. cavity lengths 3 m≫

*L*≥ 5 mm), Eqs. (2,3) can be written in third order perturbation form by expanding the population inversion as a Fourier time series around the equilibrium value (

*N*)≃

^{st}-N_{t}*γ0/B*. We assume that the population oscillates slightly at the intermode beat frequencies (harmonics amplitude ≪

_{0}*N*), and that the population inversion as well as the fields do not change much on a time scale of Δ

^{st}-N_{t}^{-1}. We thus obtain the VeCSEL equations for the complex field amplitude,

where *M _{s}*=1/τ

*B*is the saturation photon number. In a nonselective resonator (

_{0}*γ*=

_{q}*γ*), the following relationship is valid for the saturated gain in the

_{0}*qth*mode

where *η*=(*τP-N _{t})/(τP_{th}-N_{t}*) is the excess of gain over threshold. The

*C*’s given by

_{m}have a direct physical meaning and, unlike the slow complex amplitudes *A _{q}*, they can be determined experimentally since they are the components of an expansion of the total field intensity as a Fourier time series, where

*C*

_{0}corresponds to the total photon number. At the onset of pumping, it represents the spontaneous radiation intensity. Then

*C*

_{0}increases exponentially before saturating and reaching the steady-state value

*C*after a build-up time

^{st}_{0}≈(η-1)M_{s}*t*≃ln[

_{b}*M*(η-1)]/(η-1)

_{s}*γ*. Neglecting the third term on the right hand side of Eq. 7, the response to a small perturbation of this class-A laser cavity is characterized by a first order low pass filter with a cut-off frequency

_{0}*f*easily calculated as

_{c}We neglected *F _{N}* since in a class-A laser the gain saturation damps instantaneously the population inversion fluctuations (adiabatic condition) [18]. We assumed negligible technical pump noise. The random Langevin force for the photons

*F*should satisfy the following conditions:

_{q}where *ζ* is the ratio of spontaneous emission to stimulated emission rate reduced to one photon [15], and is given by

where *EF _{c,v}* are the quasi-Fermi levels for electrons and heavy holes. This ratio

*ζ*is close to unity for usual VCSEL lasers, as the carrier density at threshold is well above the transparency carrier density (>1.3) if the number of QW’s is not too large (<8).

In equation 7, the third term on the right hand side characterizes the non-linear force arising because of FWM. This non-linear mode interaction occurs as a result of pulsations of the carrier density at the intermode beat frequencies [6]. As τΛ≫1, the non-linear force strength scales with $\left(\eta -1\right)\sqrt{1+{\alpha}^{2}}\u2044\tau \Delta \left(\mathrm{or}\left(\eta -1\right)L\sqrt{1+{\alpha}^{2}}\u2044\tau \right)$. Thus the longer is the cavity and/or the higher is the pump rate, the stronger are non-linear mode interactions. Without this term, equation 7 is often called the free-running approximation, and is equivalent to what one would obtain in the limit of photon rate equations without phase dependent interactions. Note that this non-linear laser system is very similar to a traveling-wave ring-cavity dye laser, in terms of t and *γ _{0}* and of the non-existence of spatial-hole-burning, except for the influence of the α parameter, which was experimentally studied in [13, 18].

### 3.2. Numerical simulations

The influence of non-linear mode interactions due to FWM on the laser dynamics and on the steady-state properties was studied quantitatively by solving numerically the system 7, 8 and 9 as a function of the generation time *t _{g}* measured from the onset of pumping. The parameters of the equations were selected to be close to those of VeCSELs used in the experiment (Tab. 1). The results of numerical calculations indicated that after a generation time of 1

*µ*s (about twice the build-up time of the total intensity), the Gaussian width of the laser spectrum is about 33 modes which we took as the initial conditions. At that generation time, even if the total intensity is almost at its steady state value

*C*, calculations show that non-linear mode interactions did not yet modified significantly the mode dynamics (see section 3.3). The equations were initially solved for 121 modes. This number was reduced at longer

^{st}0*t*to accelerate the numerical integration. At all times, the field amplitudes for the modes at the edge of the spectrum were vanishingly small, which avoided edge effects associated with the finite number of coupled equations. Since spontaneous emission constitutes the seed for lasing, the field amplitudes in the various modes are initially statistically independent:

_{g}sThese random initial conditions where employed in the solution. As in Ref. [6], the initial amplitudes *A _{q}* of the various modes had a Gaussian distribution in the complex plane with zero averages and variances 〈|

*A*|

_{q}^{2}〉, and it was assumed that ∑

*〈|*

_{q}*A*|

_{q}^{2}〉=

*C*.

^{st}_{0}Calculations were performed using a 4th-order Runge-Kutta method with finite time increments Δ*t*, whose duration satisfied the condition Δ*t*<1/2*π f _{c}* for the stability of the coupled equations. The random force

*F*simulating quantum fluctuations was added at each iteration using a two-dimensional Gaussian distribution in the complex amplitude space. The mean value of the added photon numbers is given by Eq. 12. In order to validate the third order perturbative model used, we checked that the amplitude of the carrier density oscillation harmonics were always small compared to the mean value (this ratio is ≪10

_{q}^{-2}for

*h*up to 3).

### 3.3. Influence of FWM and α on the laser dynamics and the noise properties

Figures 5 show the spectro-temporal dynamics of the multimode VeCSEL after the onset of pumping with and without (i.e. “ideal”) FWM included, taking into account *α* and quantum noise. As is experimentally observed (Fig. 4), non-linear mode interaction due to FWM slows down the spectral narrowing at short times (*t _{g}* < 100

*µ*s), when the laser is still strongly multimode. At this stage, the stronger the pump the broader the spectrum is, and the spectral width does not follow the “ideal” monotonic narrowing proportional to ${\Gamma}_{g}\u2044\sqrt{{\gamma}_{0}{t}_{g}}$ [5].

Figure 6 shows the time dependence of the Fourier coefficients of the total field intensity *C _{m}*. With or without FWM, the total photon number

*C*is almost constant and does not show any oscillation-relaxation, as expected for a class-A laser. In contrast to an “ideal” homogeneous laser showing a monotonic decay/increase of the mode amplitudes, and thus a monotonic decay of the

_{0}*C*, one can observe that the lasing intensities in the various modes are subject to random fluctuations when FWM is switched on. However, the variables

_{m≠0}*C*fluctuate almost harmonically in the frequency range 10–100 kHz (several order of magnitude lower than

_{m≠0}(t)*γ0*<τ

^{-1}<Δ), but with decaying amplitudes as for an “ideal” laser (Fig. 6). The fact that the variables

*C*decay almost to zero, indicates that phase relationships are established between the modes, in other words, the random phase distribution of the modes is lost, this decay also slows down the mode amplitude fluctuations. This effect is the opposite of passive mode-locking achieved by means of a saturable absorber [10]. The decay of

_{m≠0}*C*is due to the saturable amplifier; it can be called mode anti-locking and does not prevent spectral narrowing to occur. This can be considered as a typical signature of FWM in a saturable amplifier. Following the same procedure as in [6], assuming a strongly multimode laser, the oscillation frequencies

_{m≠0}*f*and the decay time

^{NL}_{m}*τ*of the Fourier coefficient

^{NL}_{m}*C*are given by

_{m≠0}and are independent of a. For generation times shorter than the fastest period 1/*f ^{NL}_{1}*≈21

*µ*s (with

*τ*≈430

^{NL}_{m}*µ*s for

*η*=1.7), the free running approximation (“ideal” laser, no FWM term in Eq. 7) is valid for the rate equations: FWM does not yet modify the mode dynamics (Fig. 5- bottom and Fig. 6). The transition from pure quantum-noise driven dynamics to non-linear deterministic dynamics can be found by setting the time

*t*(Eq. 18) equal to the fastest period 1/

_{c}*f*

^{N}L_{1}, which leads to a non-linear threshold pump rate

*ηNL*

too close to 1 to be observable. This non-linear threshold scales with 1/Γ^{2}
* _{g}L^{3}*. If an intracavity filter is used to make Γ

*narrower (e.g.~1/3 for a resonant design) and the cavity is twice shorter, this threshold would increase up to*

_{g}*ηNL*≃2, and becomes thus observable.

At generation time longer than 100 *µ*s (few times 1/*f ^{NL}*

_{1}) when the laser spectrum width is only a few modes, the mode intensity fluctuations show decaying oscillations and typically only

*C*

_{1}and

*C*

_{2}still endure. Note also the spectrum red shift - or pulling - of the maximum amplitude, induced by the coupling between the parameter a and FWM. At longer times, one also sees clearly that the strong modes show an antiphase dynamics, due to FWMand the mode anti-locking process as explained above. Indeed, the strong adjacent mode oscillations have a relative phase shift of p. Thus, the fluctuations observed on the total photon number

*C*are due to white noise arising from quantum noise as in an “ideal” laser (Fig. 6, and 7-b and Eq. 19).

_{0}Although at short times several tens of longitudinal modes are amplified by the gain medium, in agreement with the experiment the laser collapses to a stable single mode operation at long time scales for any *η*. The characteristic time for this is about 1–2ms for *η*=1.7. At these long times, FWM does not prevent from stable single mode operation, even taking into account quantum noise. FWM even accelerates slightly the dynamics to reach the steady state due to spontaneous emission, compared to an “ideal” homogeneous laser (Fig. 7). A large side-mode suppression ratio of 34 dB is achieved without any intracavity filter, similar to the “ideal” laser (Eq. 20) and in relatively good agreement with the experiment. Note however that, even if the *C _{m≠0}* coefficients almost vanish at the steady state, both phase sensitive interactions via FWM and spontaneous emission drive the mode dynamics in continuous wave. The side mode suppression ratio still increases at larger pump rates (40 dB for

*η*=3) like for an “ideal” laser, but the time to reach the steady state only increases slightly (2–3ms for η=3). Thus any technical noise (acoustic, thermal) in the apparatus should not prevent from single mode operation for cavities shorter than typically L=30mm and an antireflection coated structure, as the acoustic noise frequency span stops below the kHz level for a standard optical bench.

Thus as FWM does not modify significantly the laser dynamics with respect to the single mode stability, collapse time and total photon noise, a VeCSEL approaches an “ideal” homogeneous gain laser for which the FWM term in Eq. 7 vanishes. For this “ideal” laser [5], the characteristic time *t _{c}* needed for the laser spectrum to collapse to a width of one mode separation D, after a strong perturbation (e.g. 100% pump modulation) is

for a 15mm long cavity, an antireflection coating and a wedged substrate, as shown in Figs. 5- top and 7-a. This time is pump independent and scales with Γ^{2}
_{g}×*L*^{3}, assuming *γ _{q}=γ_{0}*. Note that reducing the gain or losses bandwidth has theoretically the same effect. Using an intracavity filter to narrow G

*(e.g. ~1/3 for a resonant design) and a cavity twice shorter,*

_{g}*t*would increase up to 70 kHz, much faster than standard acoustic or thermal fluctuation frequencies. Note that the frequency response 1/

^{-1}_{c}*t*of this single mode system is several order of magnitude slower than the cutoff frequency

_{c}*f*(≃6.2MHz here, see Fig. 7-b and Eq. 10) of this class-A laser cavity.

_{c}At the steady state, the Relative-Intensity-Noise spectrumof the total photon number (Fig. 7-b) reaches the quantum limit [15] even with FWM. It displays a second order low-pass filter behavior as for a single mode class-A laser, amplifying any noise source only below 10MHz, thus of great interest for application to low intensity noise sources. If any process disturbs the laser dynamics (e.g. gain or mirror position jitter) at frequencies in between 1/*t _{c}* to

*f*, the emission will remain multimode in CW operation. In this case if any spectral filter is inserted in the cavity, reducing the modal gain bandwidth Γ

_{c}*, a single mode solution will arise more quickly and will be more stable, with a larger side-mode suppression-ratio. For a cavity longer than typically L=30mm, technical noise falling in the kHz range will prevent from single mode operation if no intracavity filter is inserted and/or large pump noise is present (e.g. when using a high power fiber-coupled pump laser diode) [8].*

_{g}At the steady state, if technical noise is negligible, in this “ideal” homogeneous class-A laser the *rms* relative photon noise for the total intensity is set by quantum noise as [11, 18]

In this case, the side-mode suppression-ratio limited by spontaneous emission is expressed for the *qth* side mode as

*SMSR _{q}* scales with

*P*×(Γ

_{e}^{2}

_{g}*L*)

^{-1}. These values are in agreement with the simulations. From the simulations of the mode phase dynamics at long

*t*with FWM (in the frequency range 100Hz to 40MHz), the power spectrum of the strong laser mode frequency (

_{g}*q*=-2) fluctuations displayed a white frequency noise at the quantum limit [15]. Thus the laser mode linewidth

*δνL*reach the quantum limit of about 1Hz for

*P*=1mW of total emitted power at 2.3 mm, as given by the modified Schawlow-Townes fundamental limit [15], which scales with

_{e}*P*

^{-1}

*×*

_{e}*L*

^{-2},

We also carried out a simulation for a cavity of length L=1m, showing the same behavior, in agreement with the experiment, but exhibiting a longer non-linear multimode transient.

As a last remark, we want to point out that if the cavity length is shorter than typically L=5mm (Δ≫γ0≫τ^{-1}), non-linear mode interactions are vanishing since the phase sensitive interaction terms in Eqs. (2,3), at frequencies *D _{mq}*≠0, are too fast for the gain medium. Thus the carrier equation can not be adiabatically eliminated from the dynamics. This situation leads to an “ideal” homogeneous multimode laser dynamics of class-B [11, 15], exhibiting relaxation oscillations of the intensity (in the 0.1GHz frequency range for L=1mm) without phase sensitive interactions. A class-B VeCSEL is thus theoretically single mode in CW. However, this class-B laser exhibits a resonant peak in the spectrum of intensity fluctuations, amplifying any noise source or fluctuations (in

*N*or

*A*), and is much more sensitive to optical feedback.

_{q}## 4. Conclusion

We presented an experimental and theoretical investigation of the non-linear multimode dynamics of QW external-cavity VCSELs, including quantum noise. We showed why these sources operate single-frequency and linearly polarized with a TEM_{00} beam, even without any intracavity selective spectral element, and in spite of the presence of FWMvia population pulsations in the QWs. FWMis slowing down spectral narrowing during the multimode transient, but does not prevent from single mode operation with high side mode suppression ratio and a linewidth <20 kHz. The physical explanation lies in the mode antiphase dynamics (called mode anti-locking). These highly coherent sources reach the quantum limit and the dynamics fall into the oscillation-relaxation-free class-A regime, similarly to an “ideal” homogeneous laser. The laser is single mode after typically *t _{g}*=1ms for L=15mm without intracavity spectral filter.

The optimum external cavity length, for stable single mode operation and high coherence, falls in the range 5mm≤*L*<30mm for typical parameters (*T _{q}*~1%, τ~3ns). A simple intracavity spectral filter (a thin glass etalon) will allow using longer cavities for stable single mode operation [8], in the limit of

*L*≪3m, as it shortens the characteristic time

*t*(∝ Γ

_{c}^{2}

*×*

_{g}*L*

^{3}) needed to stabilize the single mode solution and plays against technical instabilities (acoustic, thermal, pump injection), having frequencies falling in/below the kHz range [4]. For longer cavities, strong QW population pulsations will drive the mode build-up, leading to a strongly non-linear dynamics and preventing from stable single mode operation. Finally, for cavities typically shorter than 5mm we have an “ideal” homogeneous gain class-B laser. These properties contrast with the intrinsic strongly non-linear dynamics of conventional semiconductor lasers.

Work is in progress to investigate the bimode stability and noise properties of these sources, as well as the sensitivity limit for application to intracavity laser absorption spectroscopy [5].

The authors would like to thank L. Cerutti at IES, I. Sagnes and V. Thierry-Mieg at LPN-CNRS-Marcoussis (France) for growth of the VCSEL structures, E. Lacot and A. Kachanov at LSP for helpful discussions. This work was supported by the French ANR MIREV program.

## References and links

**1. ** A. Garnache, A. Ouvrard, L. Cerutti, D. Barat, A. Vicet, F. Genty, Y. Rouillard, D. Romanini, and E. Cerda- Méndez, “2–2.7mm single frequency tunable Sb-based lasers operating in CW at RT: Microcavity and External- cavity VCSELs, DFB,” Proc. SPIE Photonics Europe, Semiconductor lasers and laser dynamics pp. 6184–23 (2006).

**2. **S. Lutgen, T. Albrecht, P. Brick, W. Reill, J. Luft, and W. Spath, “8-W High-Efficiency Continuous-Wave Semiconductor Disk Laser at 1000 nm,” Appl. Phys. Lett. **82**, 3620–3622 (2003). [CrossRef]

**3. **A. Ouvrard, A. Garnache, L. Cerutti, F. Genty, and D. Romanini, “Single Frequency Tunable Sb-based VCSELs emitting at 2.3mm,” IEEE Photon. Technol. Lett. **17**, 128–134 (2005). [CrossRef]

**4. **M. Holm, D. Burns, and A. Ferguson, “Actively Stabilized Single-Frequency Vertical-External-Cavity AlGaAs Laser,” IEEE Photon. Technol. Lett. **11**, 1551–1553 (1999). [CrossRef]

**5. ** A. Garnache, A. Kachanov, F. Stoeckel, and R. Houdré, “Diode-pumped broadband Vertical-External-Cavity Surface-Emitting semiconductor Laser. Application to high sensitivity intracavity laser absorption spectroscopy,” J. Opt. Soc. Am. B **17**, 1589 (2000). [CrossRef]

**6. **S. Kovalenko, S. Semin, and D. Toptygin, “Influence of the Raman mode interaction on the lasing kinetics of a wide-band ring laser,” Sov. J. Quantum Electron. **21(4)**, 407–411 (1991). [CrossRef]

**7. **L. Cerutti, A. Garnache, A. Ouvrard, and F. Genty, “High Temperature CW Operation of Sb-Based Vertical External Cavity Surface Emitting Laser near 2.3mm,” J. Cryst. Growth **268**, 128 (2004). [CrossRef]

**8. **M. Jacquemet, M. Domenech, G. Lucas-Leclin, P. Georges, J. Dion, M. Strassner, I. Sagnes, and A. Garnache, “Single-Frequency High-Power CW Vertical External Cavity Surface Emitting Semiconductor Laser at 1003 nm and 501nm by Intracavity Frequency Doubling,” Appl. Phys. B In press (2006).

**9. **S. Hodges, M. Munroe, J. Cooper, and M. Raymer, “Multimode laser model with coupled cavities and quantum noise,” J. Opt. Soc. Am. B **14**, 191–199 (1997). [CrossRef]

**10. **A. Garnache, S. Hoogland, A. Tropper, I. Sagnes, G. Saint-Girons, and J. Roberts, “Sub-500-fs soliton-like pulse in a passively mode-locked broadband surface-emitting laser with 100-mW average power,” Appl. Phys. Lett. **80**, 3892–3894 (2002). [CrossRef]

**11. **A. Garnache, “Study and realization of new types of near-IR lasers for high sensitivity intra-cavity-laser- absorption-spectroscopy application. Strongly multi-mode laser dynamics.” Ph.D. thesis, Joseph Fourier University, Grenoble (1999).

**12. **A. Garnache, A. Bouchier, E. K. Attarbaoui, A. Ouvrard, L. Cerutti, and E. Cerda-Méndez, “Sb-based type-I Quantum-Well Gain and Quantum Efficiency study. Application to 2.3mm VCSELs,” Proc. EOS annual meeting, Paris, Photonic Devices in Space (TOM 5) (2006).

**13. **S. E. Vinogradov, A. A. Kachanov, S. A. Kovalenko, and E. A. Sviridenkov, “Nonlinear dynamics of a multimode dye laser with an adjustable resonator dispersion and implications for the sensitivity of in-resonator laser spectroscopy,” JETP Lett. **55**, 581–585 (1992).

**14. **M. Yamada, “Theoretical analysis of nonlinear optical phenomena taking into account the beating vibration of the electron density in semiconductor lasers,” J. Appl. Phys. **66**, 81–89 (1989). [CrossRef]

**15. **L. A. Coldren and S. W. Corzine, *Diode lasers and Photonic Integrated Circuits* (Wiley, New York, 1995).

**16. **M. Grundmann, “How a quantum-dot laser turns on,” Appl. Phys. Lett. **77**, 1428–1430 (2000). [CrossRef]

**17. **P. A. Khandokhin, I. V. Koryukin, Y. I. Khanin, and P. Mandel, “Influence of carrier diffusion on the dynamics of a two-mode laser,” IEEE J. Quantum Electron. **31**, 647–652 (1995). [CrossRef]

**18. **S. A. Kovalenko, “Quantum intensity fluctuations in multimode cw lasers and maximum sensitivity of intracavity laser spectroscopy,” Sov. J. Quantum Electron. **11**, 759–762 (1981). [CrossRef]