1. Introduction

Vertical External Cavity Surface Emitting Laser (VECSEL) are laser sources of growing importance because they combine the advantages of semiconductor surface emitting lasers and of high-power solid-state lasers. One end of the cavity is composed of a semiconductor gain medium grown on a high reflectivity Bragg mirror and the other end is closed by a usually spherical, high reflectivity dielectric mirror. The extent of the cavities allows for many longitudinal modes inside the gain curve. Thus, VECSELs are less sensitive to temperature effects than monolithic cavities and can be driven at higher pumping, which allows for high output powers. When combined with an intracavity saturable absorber mirror, VECSELs can be operated in the mode-locked regime to deliver ultra-fast pulses in the sub-picosecond range [1, 2] with possibly a high repetition rate. In many applications though, operation in the single fundamental transverse mode is sought together with a high output power. This is potentially a problem since higher order transverse mode can then be excited. In addition, if one wishes to control the repetition rate of pulses in the mode-locked regime, further constraints are imposed on the cavity design parameters and it may become difficult to remain in this regime. On the other hand, it is sometimes desirable to operate the laser in a highly multi-transverse mode regime. This is the case for studies on pattern formation in lasers, and more specifically for localized structures in lasers ([3, 4, 5]). Localized structures in cavities, also called cavity solitons, are independently controllable spots in the transverse plane of a system [6, 7]. Light localization in these systems can be viewed as the result of the transverse mode locking of a great number of modes. The present study was initiated in this context, but we wish to stress that transverse mode selection in VECSELs may be of great interest in other research fields too.

Pattern formation and transverse mode selection in lasers have already been the subject of many studies (see e.g. [8] for a review). Leaving aside the dynamical aspects, it has been shown that the emitted pattern is the result of a complex interplay between the cavity geometry, the boundary conditions, the material gain and its nonlinearities. In semiconductor materials, a lot of studies have taken place in Vertical Cavity Surface Emitting Lasers VCSELs ([9, 10, 11, 12, 13] to cite a few) but very few, to our knowledge, in VECSELs. VCSELs have the advantage of being single longitudinal-mode lasers with a broad range of applications. Transverse mode control can be achieved to a certain extent at the fabrication level through the design of the active zone diameter (e.g. complex transverse structures can be observed in large diameter oxide confined VCSELs or, conversely, fundamental transverse mode emission in large area photonic crystal lasers [14, 15] or by changing the operating temperature to control the detuning between the gain maximum and the cavity resonance). In other kinds of lasers, multi-transverse mode operation has been reported in solid-state micro-chip lasers [16] and in CO₂ lasers with an intracavity lens [17, 18]. In the latter study, the cavity geometry plays a great role in the laser emission profile through the Fresnel number which controls the cavity aspect ratio and hence diffraction. A very special cavity design is represented by the self-imaging cavity [19], in which the ABCD propagation matrix is unity. In this singular cavity, diffraction due to the propagation inside the cavity is almost completely canceled and is equivalent, from the transverse field point of view, to that of a zero-length planar cavity : such a cavity configuration can theoretically sustain an infinite number of degenerate transverse modes. A self-imaging cavity configuration is not stable in practice and can only be approached. In spite of this, for a cavity with curved mirrors close to a confocal or more generally self-imaging configuration, as stated in [20, 21], the Fourier mode basis is expected to be well adapted to the modes description.

In this work we study the transverse emission profile of a cavity with an intracavity lens in a configuration close to the singular self-imaging cavity. At difference with previously mentioned studies in CO₂ lasers, our gain medium is very thin and no longitudinal hole-burning effect can take place. It is then easier to decouple the propagation effects inside the cavity from the nonlinear interaction between the gain medium and the field. Moreover, the shorter wavelength emission of our system in the near-infrared allows one to study higher Fresnel number configurations. In the next sections, we first introduce theoretical considerations about our degenerate cavity. We then analyze the laser emission far-field and the RF-spectrum of the total intensity.

2. Degenerate cavities

One of the simplest degenerate optical cavity is composed of a plane mirror, a lens, and a spherical mirror as considered in [19] and sketched on Fig. 1. The self-imaging positions between these three elements are:

where d*₁ is the self-imaging distance between the plane mirror and the lens, L* is the total self-imaging cavity length, R is the end mirror radius of curvature and f the intracavity-lens focal length. We study such a cavity in a nearly degenerate configuration, i.e. where the cavity length or/and the distance between the lens and the plane mirror are close to the self-imaging distances.

Two parameters play an important role for transverse-mode selection. The first one is the Fresnel number N which controls the cavity aspect ratio and hence the number of transverse modes allowed in the cavity, and the second one is the transverse mode spacing Δν_T which governs the coupling strength between the modes of the active cavity and thus the influence of the nonlinearities on the dynamical behavior of the system [18]. N is evaluated by computing the effective Fresnel number N_e as the ratio of the area of the diffracting aperture that limits the system’s transverse dimension to the area of the fundamental Gaussian-mode along the cavity such that:

 

Fig. 1. Sketch of the near self-imaging cavity composed of a half-VCSEL mirror, a spherical output mirror (radius of curvature R) and an intracavity lens of focal f.

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where w(z) is the beam-waist of the fundamental mode along the propagation axis z and d(z) the size of the corresponding diffracting aperture. In our case, the aperture that limits the system is generally the size of the optical pump spot. Hence the Fresnel number is easily evaluated as the ratio of the square of the pump radius to the waist of the fundamental stable mode. In the degenerate configuration, this number can become very large but is then limited by the other optical elements. In our experiment, the Fresnel number can be changed either by moving the optical components of the cavity, by means of an iris inserted inside the cavity or by changing the pump beam diameter. It can be shown [8] that the number of allowed transverse modes is proportional to N ² _e, whereas the maximum order of an allowed transverse mode in the cavity is proportional to N_e.

If we consider the round-trip ABCD propagation matrix inside the cavity, the maxima of the Fresnel number follows a curve defined by B=0. This corresponds to an effective diffraction coefficient equal to zero. As noted in [22], after one round trip the intensity image is recovered but not the phase of the initial plane, since C, the cavity total curvature, is generally not zero. Conversely, when the Fresnel number tends towards zero, w ₀ on the plane mirror tends towards infinity, C=0 but since B is generally not zero, the cavity is equivalent to a planar cavity with diffraction. Fig. 2 shows the beam-waist at the half-VCSEL mirror end for the cavity shown on Fig. 1. The self-imaging position is marked (S point), and the white zones correspond to an unstable cavity. The self-imaging point S is at the intersection of two lines. One line corresponds to an infinite waist w ₀ of the fundamental stable mode inside the cavity at the plane mirror (C=0) whereas the other one corresponds to a zero fundamental mode size (B=0). Given these considerations for an empty cavity, we shall next describe our experimental setup and discuss what kind of transverse structure is selected by the cavity when we are close to the self-imaging configuration.

3. Experimental set-up and VECSEL structure

The experimental set-up is sketched on Fig. 3. The VECSEL structure is composed of a half-VCSEL closed by a spherical mirror with an intracavity lens L₁. The half-VCSEL is grown by Metal Organic Chemical Vapor Deposition (MOCVD) and designed for lasing around 1010nm with an optical pump. It is composed of a high-reflectivity (27.5 pairs) AlAs/GaAs Bragg mirror. On top of the Bragg mirror is a 3.75 λ-thick cavity with five, 7nm-thick compressively strained InGaAs/GaAs quantum wells (QWs). The QWs are distributed such that the carrier densities remain almost equal in all QWs to counteract pump depletion. The number of QWs was optimized to obtain a low laser threshold and a large differential gain at 300 °K, with total cavity losses of about 1 to 1.5%. The half-VCSEL mirror is bonded onto a SiC substrate for better heat removal and lower thermal resistance of the structure. It is maintained at a constant temperature thanks to a Peltier cooler monitored by a control-loop. The intra-cavity lens L₁ is anti-reflection coated and has a focal length of 38.1 mm. The spherical mirror has a 75mm radius of curvature and a reflectivity higher than 99% at 1020 nm. Optical pumping is achieved thanks to a high-power, fiber-coupled laser diode delivering up to 35W of optical power around 808nm. The output of the multimode fiber is focused onto the half-VCSEL with a telescope composed of two microscope objectives so as to form a 200µm diameter pump spot. The VECSEL output field and the image of the active zone are captured by two charge-coupled device cameras CCD1 and CCD2. These images will be referred to in the following as far-field and near-field of the laser, although these two planes are not strictly speaking Fourier transforms of one another. However they constitute obvious physical planes of interest since the near-field plane contains the gain medium and acts as the transverse-field limiting element and in the far field, light emitted at the laser output propagates without noticeable change of structure in its transverse profile. The total intensity of the laser is detected thanks to a 90ps rise-time avalanche photodiode (APD) and analyzed on a RF-spectrum analyzer (SA) or a 6GHz bandwidth sampling oscilloscope.

 

Fig. 2. Plot of the fundamental stable beam-waist size versus d ₁ and L=d ₁+d ₂ for a cavity with f=3.81cm and R=7.5cm. The S point corresponds to a self-imaging cavity.

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4. Experimental results

Images recorded at different positions of the intracavity lens and for a fixed total cavity length are displayed on figs.4,5. The pump power is slightly above laser threshold, but we do not note a substantial qualitative change in the emitted spatial structure when the pump is varied within the (limited) available range of optical powers. Let us first focus on the far-field images. As d ₁ decreases, the output field evolves from low-order, on-axis patterns to an off-axis, annular structure with azimuthal modulations. This corresponds to a decrease in the diffraction length and hence to passing from a low-Fresnel number to a high-Fresnel number configuration. At low Fresnel numbers, a small order transverse mode appears (typically a Gaussian mode or a doughnut mode as in the two images in the first two columns of Fig. 4). The near-field in this case is very similar in shape to the far-field. When the Fresnel number increases, the near and far-fields display different structures. Whereas the far-field displays an annular structure whose diameter grows with the Fresnel-number, the near field shows a complex structure composed of bright spots. In the far field (output of the VECSEL), off-axis emission is favored when the Fresnel number is high. On-axis emission is then totally suppressed. This behavior is similar to what has been predicted from numerical modeling of the mode distribution in a gain guided VCSEL by [23], and experimentally confirmed in broad area VCSELs and in oxide confined VCSELs by [24, 13]. However we are dealing here with an extended cavity with many longitudinal modes. On-axis emission from the cavity modes whose frequency is higher than the one of the gain maximum always compete with off-axis emission of the modes on the other part of the gain. However, since these modes have higher gain, they seem to dominate and suppress on-axis modes. In this context, the annular structure may be understood in terms of a superposition of plane, tilted waves that are equally possible on a whole range of orientations, giving rise to the annular structure. This interpretation is strengthened when looking at the same kind of situation but for a misaligned system (Fig. 5). The rotational symmetry of the system no-longer holds and we recover field structures observed in systems with square boundary conditions (see e.g. [11]). We are thus able to completely monitor the passage of a system best described in terms of Gaussian transverse modes to a system best described in terms of plane-wave (Fourier) modes. As for the near-field images, as the lens is moved closer to the gain section, we notice that the field structure becomes increasingly complex and contains finer and finer structures.

 

Fig. 3. Experimental set-up. SM: spherical mirror; CS: Cube splitter; M: Mirror; L1,L2,L3: Lenses; OP: Optical pump; SA: Spectrum analyzer; APD: avalanche photo diode. CCD1,2: CCD cameras.

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The same scenario occurs when moving the mirror alone and letting the lens position fixed. Suppose by example that the lens is at the self-imaging position. As the self-imaging cavity is approached (e.g. by decreasing the total cavity length), a similar annular structure whose diameter increases is formed. Emission stops close to the self-imaging point for several reasons. First of all, the self-imaging configuration is only marginally stable and any small deviation from it make the cavity unstable. Hence it is a point that in practice one cannot reach with a laser. Second, since off-axis emission occurs at an increasing angle, the intracavity lens diameter will ultimately limit the possible propagation angle of light in the cavity. When this critical angle is reached, emission stops as we could verify.

 

Fig. 4. Near-field (upper row) and far-field (lower row) images of the laser for different positions of the intracavity lens : 5.323, 5.306, 5.297 and for a fixed cavity length (⋍18.6 cm). The field of view of the bottom row images is 3.2°.

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Fig. 5. Same as Fig. 4 for an intracavity-lens position: 5.246, 5.132cm, 5.087cm. Last image on the right is for a slightly misaligned cavity.

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5. Spectral analysis

In order to understand how complexity increases in the patterns, it is interesting to analyze the RF power spectrum of the total intensity. This measure gives information about the beating of the phase locked lasing modes. At each longitudinal mode of the cavity, one can associate a family of transverse modes whose separation is given by the Gouy phase accumulated in the propagation through the cavity. Depending on the separation between the longitudinal and transverse modes, several situations may appear [22]. In the general case, far from the cavity stability limits, the longitudinal mode spacing is of the same order of magnitude as the transverse mode spacing. However, when diffraction vanishes (B→0), the transverse mode separation goes to zero and the cavity is infinitely degenerate. It is important again here to stress the exact correspondence between the cavity at study in a diffractionless configuration and the zero-thickness cavity. Similarly, when the system departs slightly from this position with a distance d, from a transverse point of view it is equivalent to a plane cavity with thickness d. The RF-spectra are obtained by acquiring a sequence on a 6GHz bandwidth digital oscilloscope and by subsequently calculating the power spectrum. They are made close to the laser threshold with a pulsed pump (200µs duration, 4.2ms repetition rate) in order to prevent thermal lensing as much as possible. By combining several spectra obtained at different cavity configurations, we obtained a map as shown on Fig. 6. It has been obtained at a fixed cavity length, varying the lens position d ₁. The central peak around 793MHz arises from the longitudinal mode beating and allows us to extract the cavity length rather precisely. The d ₁ interval spans the entire set of configurations from a fundamental Gaussian mode (d1=54mm) to a wide-angle flower-like pattern. Side-modes are present almost right from the fundamental Gaussian mode configuration with a beating frequency of roughly 40MHz (Fig. 6(a)). The other peaks separated by 120MHz can be readily attributed to the back-reflection of the second surface of the output coupler. A closer look at the central peak reveals a complex structure (Fig. 6(b)). In the low-Fresnel number configuration, only a single peak is present. When the lens is moved towards higher Fresnel number configurations, the peak broadens and closely spaced side-modes appear. The mode spacing is decreasing as expected when going close to a degenerate cavity. This feature can be attributed to the transverse structure of the laser emission. A qualitative change in the spectrum accompanies the change of the transverse structure from a low-order Gaussian structure to the annular structure, namely the appearance of the several side-modes. However, contrarily to what one may expect in a highly multimode system, a structure appears in the spectrum and the on-axis and off-axis emissions display clearly qualitatively different spectra. When the laser cavity is close to its unstable configuration, and hence has reached a high Fresnel number, we see only a few peaks. However in this configuration the laser becomes very sensitive to mechanical noise that tends to increase the laser threshold. The complete understanding of the spectral feature certainly calls for a deeper analysis and would benefit from a theoretical description not available at the moment and that will be undertaken in a future work. However a simple analysis can already give good insights into the mode beating spectra. Indeed, considering that the laser modes frequencies are only slightly different from those of the cold-cavity modes, we can plot the mode beating spectra of the transverse modes corresponding to the linear case when varying the lens position (see Fig. 7). The frequencies of the various transverse modes can be extracted from the Gouy phase, easily calculated knowing the ABCD matrix [25]. The beat spectra is simulated taking into account the beating of several transverse modes arising from possibly several families of transverse modes. In principle the number of modes to be taken into account is itself a function of the lens position d ₁ through the Fresnel number dependence but we took it constant here for simplicity. We see a good agreement between the experimental and the simulated spectra. In particular, the beating frequencies collapse to a single frequency when approaching the high Fresnel number situation, a feature that can be seen in the experimental result in Fig. 6(b) and is related in the transverse plane to the appearance of the structures in Fig. 5.

 

Fig. 6. Left figure (a) : map of RF-spectra of the total laser intensity for L=18.9cm versus the lens position d ₁. The right figure (b) is a zoom around the central peak.

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Fig. 7. Beat spectra of the cold cavity modes when varying the lens position. Ten transverse modes are taken into account in this simulation. The beat frequencies are plotted rescaled to the free spectral range spacing of the longitudinal modes ν ₀.

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6. Conclusion

In conclusion we have shown the transformation and the selection of the transverse structure emitted by a VECSEL with an intracavity lens : from a Gaussian transverse mode structure the field evolves toward a configuration best described in terms of a combination of Fourier modes. This feature is reminiscent of the high-Fresnel number configuration of the system with the novelty here that the system is also highly multimode. Despite this, clear structures are formed with an ordered RF spectrum. A theoretical description of the system is complex and will be undertaken. We note that in a three-mode quasi-self-imaging system, some recent numerical modeling [20] taking into account the full diffraction integral in the cavity exhibited patterns similar to those we obtained. This is encouraging since it underlines the generality of the results presented here and also shows that it may be feasible to model our system with a limited number of longitudinal modes at least as far as transverse patterns are concerned.

This work has been carried out in the framework of the FunFACS European project (www.funfacs.org) and benefited also of the support from the Région Île-de-France.