Supermode Analysis and Characterization of Triangular Vertical Cavity Surface Emitting Laser Diode Arrays

We show in simulation and experiment that the coherent coupling of electric fields between elements of a triangular photonic crystal vertical cavity surface emitting laser array can be adjusted into antiguided supermodes with approximately equal injection into each array element. The dominant supermodes are found by a complex waveguide mode simulation and can be identified by the number and arrangement of side-lobes of their Fourier-transformed far-field beam profiles. The supermode with all three elements in-phase is found to possess the minimum central lobe angular divergence. Measurements of the continuous wave coherently coupled properties of photonic crystal triangular arrays, nominally emitting at 850 nm, are compared to our simulations. We show that the array can be adjusted to lase in a desired supermode by varying the injection currents into each element of the array. The optical power confined within the central portion of the in-phase coupled beam is greater than that emitted from a single VCSEL which is advantageous for high-brightness applications.


I. INTRODUCTION
C OMBINING the beams of multiple lasers coherently has been pursued to increase the source brightness, either using multiple separate laser diodes [1] or using monolithic arrays of lasers [2]. If the individual lasers are sufficiently close together, the lasing mode is spatially present in all of the cavities, and thus a supermode analysis is warranted [3]. Coherent coupling of multiple semiconductor lasers has been pursued using both in-plane diode lasers and coherently coupled vertical cavity surface emitting lasers (VCSELs) [1], [2]. In an antiguided coupled VCSEL array [4], [5], [6], [7], [8], [9], the higher index outside of the laser cavity causes leaky mode field-propagation across the array [10]. Thus, the optical fields from the neighboring VCSELs coherently combine forming a supermode. A diode laser array can have multiple supermodes and the dominance of one particular supermode depends on the injection, array geometry or structure [6], [11]. A coupled supermode can emit an in-phase beam with an on-axis peak or an out-of-phase far-field profile with an on-axis null [6], [11], [12], [13], [14]. An in-phase supermode can confine high power within a small spot size, a desirable property for most high brightness laser applications e.g., material processing, imaging, sensing, laser pumping, optical communication and automotive applications [15], [16], [17], [18]. A photonic crystal ion implanted VCSEL array unites the stable modal discrimination from the etched photonic crystal in the top distributed Bragg reflector (DBR) mirror and the antiguiding effect from the index suppression induced by the spatially defined gain [9], [19], [20]. Previous reports of 2-dimensional (2D) coupled photonic crystal defect arrays show an in-phase supermode with a central lobe beam divergence within a range of 2.0 • to 3.5 • [9], [19], [21]. The formation of such narrow divergence on-axis beam has been attributed to the combination of separate sources of equal intensity Gaussian beams [13].
Here we analyze the on-axis beam formation as leaky-wave supermodes that overlap all the elements of a triangular coupled VCSEL array nominally emitting at 850 nm. Using independent nearly equal current injection, we show it is possible to adjust the VCSEL diode laser array into a desired supermode both in experiment and simulation. In Section II, we simulate the supermodes of this triangular array utilizing 2D complex-waveguide model and apply fast Fourier beam analysis to the far-field beam profile for supermode identification. In Section III, we characterize the observed supermodes as a function of injection current and demonstrate the coherent array can be adjusted into a desired dominant supermode identified using the Fourier transformed far-fields. We conclude in Section V that the coherently coupled in-phase supermode from triangular photonic crystal VCSEL arrays can provide narrow beam divergence for high brightness applications.  cavity. The reduced diameter holes between each cavity and the central hole promote only nearest neighbor optical coupling. The photonic crystal parameters are compliant with those found previously to achieve single mode operation from single defect photonic crystal VCSELs [22]. The detailed description of the array fabrication including the steps of anisotropic reactive-ion etching of the photonic crystal and ion-implantation are available in [23].

II. ARRAY DESCRIPTION AND SIMULATION
A complex dielectric waveguide model including gain and loss has been previously utilized for the modal analysis of semiconductor lasers [24], [25]. We have developed a 2-dimensional complex waveguide model to simulate the coupled supermodes of photonic crystal VCSEL arrays [21], [26]. We find more complicated 3-dimensional vectorial modeling including polarization unnecessary [27]. The real and imaginary components of the refractive index model used for triangular arrays are defined as shown in Fig. 2(a) and (b), respectively. The index and geometry parameters used in our model are included in Table I, where the rationale behind choosing the values of these parameters was previously discussed in [21]. Briefly, we include 50 cm −1 mirror and 20 cm −1 absorption loss in the imaginary field index and the hole real index accounts for the etched holes not extending into the lower mirror. At threshold, we assume a free-carrier plasma index suppression Δn ≈ 2 × 10 −3 in each element from a uniform 2 × 10 18 cm −3 carrier injection creating optical gain (negative imaginary index) corresponding to 1000 cm −1 gain [28]. With the index model shown in Fig. 2 where n(x, y) is our 2D complex index profile, n ef f is the effective modal index and U (x, y) provides the transverse mode profile. We solve for the eigen transverse modes and in Fig. 3(a) we plot the real and imaginary components of n ef f as a function of the confinement factor which is the percentage modal overlap with the spatially defined gain region [23], [25]. The three highest confined modes (labeled 1, 2, 3) have the lowest imaginary effective indices, i.e., the highest gain and nearly identical frequency (frequency difference of 710 MHz). Fig. 3 show the near field profiles of the three fundamental supermodes and their corresponding far-field intensities are shown in Fig. 3(e)-(g), respectively. Supermodes 1 and 3 with near fields shown in Fig. 3(b) and (d) have an on-axis null in the far-field beam profiles as we find in Fig. 3(e) and (g). On the other hand, supermode 2 in Fig. 3(c) with all three elements in the same phase has an on-axis peak in its far-field beam in Fig. 3(f). The structured far-fields with interference fringes of the three supermodes result in a non-zero side lobe in their fast Fourier transforms as apparent in Fig. 3(h)-(j). Supermodes 1, 2 and 3 distinctively exhibit 4, 6, and 2 dominant side lobes, respectively. A quantitative measure of the supermode profile can be defined from the fast Fourier transform [30]. The peak ratio, PR, is defined as the ratio of the magnitudes of the side peaks, I side−peak to the DC peak, I DC−peak : Supermodes 1 and 2 have a computed PR of 0.11 whereas supermode 3 has a higher PR of 0.22. With multiple supermodes overlapping in the far-field, we find lower values of PR and as the far-field beam profile tends toward an (unstructured) Gaussian profile, the PR values approach zero [30].
Increased current injection uniformly raises the refractive index throughout the array. However, within the cavities, the refractive index is suppressed due to the effect of current injection, with greater injection resulting in a higher suppression, Δn [31], [32]. In Fig. 4, we plot the variation in the imaginary effective index (imag (n ef f )) and the confinement factor for the three supermodes as a function of varying index suppression, Δn from threshold (0.002) to above threshold (0.01). Since greater negative imaginary effective index, imag (n ef f ), implies lower threshold gain [26], Fig. 4(a) shows that supermode 2 eventually becomes the supermode with the lowest threshold with increasing Δn. The dependence of the modal confinement follows a decreasing trend in Fig. 4(b). Near threshold, supermode 1 shows the lowest threshold gain with the highest confinement but with increasing Δn, the in-phase supermode 2 has the greatest confinement and lowest threshold gain. The switch from one dominant supermode to another with current injection is also observed and similarly analyzed in dual-element coherently coupled VCSEL arrays [32]. Increasing index suppression Δn  also leads to a small monotonic decrease of real(n ef f ) for all three supermodes which have nearly degenerate mode frequencies (not shown). Please note that this analysis assumes uniform index suppression in all three cavities and does not account for slight variations in injection among them.
We simulate the fundamental modes of a single-defect 7 μm diameter cavity in Fig. 5(a) and a 7-defect VCSEL with 14 μm diameter cavity [33] in Fig. 5(b) with identical pitch and hole diameter using our complex waveguide model. The beam divergence is found to decrease with the larger cavity diameter, as expected. In Fig. 5(c) we simulate a triangular array (same pitch and hole diameter) which has nearly same gain area as the 7-defect device and four added small diameter holes. The central dominant lobe in Fig. 5(c) shows a beam divergence of 3 • , narrower than both the Gaussian beam profiles of the single cavity photonic crystal VCSELs. Note that the gain regions in Fig. 5(b) and (c) are nearly equal size, yet the coherent array produces reduced beam divergence. Although the 7-defect cavity having larger near-field reduces the beam divergence, it is not practical to achieve single mode operation from a multi-defect photonic crystal cavity [33]. Therefore, the triangular array lasing in the in-phase supermode is a viable option to obtain narrow beam divergence.

III. EXPERIMENTAL ANALYSIS
The triangular VCSEL array that we experimentally characterize has three elements with nearly equal threshold currents of approximately 2 mA [23]. We vary the supermode emission by controlling the current injection into the three elements [23], [34]. Using three model 236 Keithley precision power supplies we set approximately equal bias currents (I Right , I Center , I Left ). Above threshold, for nearly equal injection we adjust the bias currents to reduce the lasing spectrum from multiple emission lines to a single lasing peak with a minimum emission spectral linewidth on the order of 100∼110 pm (optical spectrum analyzer resolution is 20 pm or 8.3 GHz at 850 nm wavelength). Hence all three elements lasing at nearly identical frequencies produce a single narrow linewidth spectral peak, on the same order as a single VCSEL [22]. For each datapoint (I Right , I Center , I Left ) that produces a narrow single linewidth emission peak, we also measure the far-field intensity using a scientific camera. The far-field profiles and their fast Fourier transforms are compared to the simulations in Fig. 3 for supermode identification.
The coupling between array elements results in non-zero peak ratio calculated from the Fourier transform of the far-field profile. Fig. 6(a) shows the measured peak ratio values greater than 0.1 computed at points of coherent operation of the array identified by emission into a single spectral linewidth. The projection of the currents used at points of coherent operation in Fig. 6(a) approximately traces the diagonals along each current plane, where the highest peak ratio values are found for current injection roughly in the range of 3-4 mA (1.5 to 2 times threshold). To identify the possible dominant supermode at these injection points, we perform a visual inspection of their far-field intensity profiles and fast Fourier transforms. We find that counting the number of side-lobes in the Fourier transformed far fields allows identification of a dominant supermode [35]. For a subset of the coherent operation points in Fig. 6(a) we find that a dominant supermode can be identified in Fig. 6(b). The far-field beam profile and the Fourier transform of three example points of coherent operation (A, B and C in Fig. 6(b)) are shown in Fig. 6(c). The far-field profiles in Fig. 6(c) are similar to the simulated antiguided supermodes illustrated in Fig. 3(e)-(g). Fig. 6(b) also indicates that by selectively controlling the injection into the individual array elements, we can adjust the array to operate in a dominant supermode and obtain a desired beam profile. In particular, the low divergence in-phase supermode (point B) can be achieved at multiple current settings in Fig. 6(b).
The in-phase supermode divergence of the triangular VCSEL array at various approximately equal injection currents (e.g., point B in Fig. 6(b)) are compared to the divergence of a single cavity single transverse mode photonic crystal VCSEL in Fig. 7. The single cavity VCSEL has the same photonic crystal parameters as the array with a threshold of 2 mA. Both devices provide a single spectral peak shown in Fig. 7(b) with linewidth of 110 pm. Fig. 7(c) illustrates the far-field profiles of the single-element VCSEL at an injection, I = 3 mA, and Fig. 7. Comparison of (a) near-field, (b) lasing spectrum, (c) far-field profile, and (d) beam divergence as a function of nearly equal injection current for photonic crystal single cavity VCSEL and triangular array; The spectrum and far-field profiles are at I ≈ 3 mA and at point B from Fig. 6(b) respectively. the in-phase supermode at point B in Fig. 6(b). In Fig. 7(d) the single VCSEL over a range of currents exhibits a Gaussian beam divergence greater than 9°. In contrast, the coherent triangular VCSEL array provides an on-axis angular beam divergence that varies from approximately 3°to 4.5°. These findings are consistent with our simulated results shown in Fig. 5.
The power fraction of the coherent array beam delivered to a spot along the propagation axis can also be compared to a single cavity VCSEL. Adapted from the concept of power-inthe-bucket [1], we define the power fraction as the portion of the total optical power which is concentrated within an aperture of diameter D: Power fraction = D I (r, θ) rdrdθ all space I (r, θ) rdrdθ × total power The far-field intensity distributions of the in-phase supermode and the single VCSEL (see Fig. 7) are reproduced in Fig. 8(a) and (b), respectively. Fig. 8(c) compares the measured power fraction (mW) of the in-phase supermode from Fig. 8(a) with that of three independent uncoupled single-element VCSELs, each lasing at I = 3 mA. It is apparent that the supermode with on-axis beam can confine substantially more power within a small angle of divergence. In Fig. 8(d) we compare the power fraction within 10°full angle of divergence as a function of injection current. We see that at all injection currents greater power is obtained from the triangular array emitting in the in-phase supermode. This is particularly evident in Fig. 8(d) for currents greater than 3.5 mA where the largest values of peak ratio are also found. Therefore, instead of combining multiple Gaussian beams from independent lasers, it is possible to use leaky mode coherent coupling within a laser array to engineer a desired supermode with an on-axis beam having narrower angular divergence. Such narrow-divergence beams will be beneficial for achieving higher brightness.

IV. CONCLUSION
The optical fields from the three elements in an anti-guided photonic crystal VCSEL array can interfere forming three fundamental supermodes. The complex waveguide-based supermode simulation helps identify the distinctive features of these antiguided supermodes from the fast Fourier-transforms of their far-field beam profiles. Under nearly equal injection the array elements can be coherently coupled to oscillate in a quasi-single mode with narrow emission linewidth. Controlling injection to the individual array elements, it is possible to adjust the array to operate primarily in a dominant supermode. The desired supermode can be identified through Fourier analysis of the far-field beam, as shown both by simulation and experimental characterization. The in-phase supermode provides an on-axis reduced divergence beam with higher power concentrated in the central lobe, which is pertinent for high-brightness single-mode laser diode applications.