Modal properties of vertical cavity surface-emitting lasers (VCSELs) with holey structures are studied using a finite difference time domain (FDTD) method. We investigate loss behavior with respect to the variation of structural parameters, and explain the loss mechanism of VCSELs. We also propose an effective method to estimate the modal loss based on mode profiles obtained using FDTD simulation. Our results could provide an important guideline for optimization of the microstructures of high-power single-mode VCSELs.
©2011 Optical Society of America
Vertical cavity surface-emitting lasers (VCSELs) have unique and useful features such as low cost, low power consumption, and a small footprint. They are now key devices in local area and metropolitan networks. Consumer applications, such as laser mice and laser printers, have also grown recently.
In several applications, single transverse mode operation is highly desirable for stable output intensity and high speed modulation. For this reason, various methods have been proposed and demonstrated to obtain the high power single mode in VCSELs. As described in the literature, single-mode VCSELs have been fabricated mainly using two different techniques. One technique is to make the device small enough to support only the fundamental mode, where the best results have been achieved using oxide-confined VCSELs . However, the disadvantage of this approach is low output power due to the small active volume. The other technique is to use mode-selective loss in a larger multimode device. Metal apertures , anti-resonant reflecting elements , air hole structures , or high-contrast gratings  are introduced on top of the surface to selectively induce a large loss for the higher-order modes. Fairly good results have been reported by many groups.
The laser threshold, side-mode suppression ratio, and maximum output power are highly dependent on the configuration of the microstructures. Thus, many researchers have tried to find the optimal design to enable low threshold, high power, single-mode lasing. The typical approach includes repetition of numerical simulations of modal properties for many laser cavities with varying structural parameters [6–11] and fabrication of actual devices for several selected designs based on the simulation results, followed by experimental evaluation of the designs. Such an approach has been proven to be very successful, and great advances have been made. However, finding the best design through blind iterations require a very large number of simulations since the number of samples to be investigated increases exponentially with the number of structural parameters. The optimization process can be made more effective when an in-depth understanding of the loss and the mode selection mechanisms is provided.
Numerical analyses of modes in VCSELs have been performed using a wide variety of techniques including the effective index method [6,7], the beam propagation method , the plane wave admittance method , and the finite difference time domain (FDTD) method [8,10,11]. In this work, the full-vector three-dimensional (3-D) FDTD technique was used to analyze micro-structured VCSELs. Compared to other techniques, the FDTD method is a simple and straightforward algorithm for solving time-dependent electromagnetic problems. The algorithm requires minimal assumptions and approximations, and thus provides fairly reliable results. With this approach, it is possible to directly examine the electric field distribution at each time step, which helps to determine the physical mechanism of the device behavior.
We present an effective FDTD method to calculate the modes in a holey-structured VCSEL, and investigate the effect of structure parameters on optical properties, especially with respect to modal loss. We examined how each parameter affected modal behavior. The results of this study can provide a guideline to help optimize microstructures for a high-power single-mode VCSEL.
2. 3-D FDTD Model for Micro-Structured VCSEL
In the FDTD method, the time-dependent Maxwell's equations in partial differential form are discretized using central-difference approximations of the space and time partial derivatives . The electric field vector components in a volume of space are solved at a given instant in time, and then the magnetic field vector components in the same spatial volume are solved at the next instant in time to update the electric and magnetic fields. The process is repeated up to a designated number (typically 10,000~100,000) of iterations. Therefore, the simulation provides a view of the temporal dynamics of an optical field. The field data can be processed to obtain a field profile of a resonant mode, a resonant frequency, an optical loss, a coupling efficiency, and so on.
Figure 1 shows a schematic of the microstructured VCSEL used in this study. An oxide layer with a circular aperture was located between the active layer and the distributed Bragg reflector (DBR). Six air holes were introduced on the top DBR. The important geometrical factors that significantly affect the modal properties are the oxide aperture size, air hole size, hole depth, and hole pitch. In this study, the hole diameter was 3 μm, and the six holes were arranged at the corners of hexagon in the top DBR as shown in Fig. 2 . We found that any additional holes positioned outside the oxide aperture did not affect the modal properties since the optical field decayed very rapidly beyond the six air holes or the oxide aperture. This suggests that the air holes in the micro-structured (or photonic crystal) VCSEL worked as loss elements rather than as a photonic crystal.
In our FDTD method, the VCSEL structure in the computational domain was surrounded by a perfectly matched layer (PML) with a thickness of 400 to 600 nm to absorb any outgoing waves. The spatial resolution, or the dimensions of the grids, determines the computation time and the accuracy of finite difference approximations. We determined the optimal resolutions that yielded reliable accuracy with reasonable computation time to be Δx = Δy = 0.1 μm and Δz = 0.02 μm in our VCSEL structure (Δx, Δy, and Δz are the dimensions of one grid). The small grid size along the z-axis was selected because light propagated along the z-axis, and a large variation of both the optical phase and the amplitude of the laser modes appeared.
3. Excitation of the Resonant Modes using Magnetic Field Parity
When a single dipole source is used to generate optical waves inside the computation structure, multiple resonant modes are exited at the same time, as shown by the black curves in Fig. 3 . However, to obtain mode-specific properties, the excitation of only one particular mode is required. In this case, the large number of modes densely located in a spectrum makes single mode excitation a difficult and time-consuming task. The problem can be effectively solved when using the symmetry relation of the mode to be excited. Mirror symmetry conditions or parities can be applied when the cavity structure under consideration has mirror symmetries as our VCSELs. In this study, parity was applied to the z component of the magnetic field (Hz component) with respect to the x- and y-axes. Table 1 shows the parity of three different modes in the format of (i, j), where 1/-1 denotes even/odd symmetry with respect to each axis. The z-axis parity was not used since the VCSEL structure did not have symmetry along the z axis. In each case, the dipole source was located where the mode intensity was relatively high for effective excitation of the corresponding mode. Here, the transverse modes of the VCSEL are named LPlm (linearly polarized mode) form, which is typically used for optical waveguides.
Figure 3 shows three different mode spectra obtained when the parities of Table 1 were applied. Note that only a part of the original resonant peaks (shown as black curves) was selectively excited as the parity conditions were applied. Excitation of any single mode can be performed much more easily due to the low density of the mode spectra. Successful excitation of single mode is verified by purely exponential decay of field amplitude with no fluctuation. The longest wavelength or lowest frequency peak in each result corresponds to the LP01, LP11, and LP21 modes, respectively. Each mode was excited with a dipole source with a narrow spectral band, and the resulting intensity distributions are shown in Figs. 4(a)-(c) . The mode pattern is in good agreement with those from waveguide theory. Figures 4(d)-(f) show the Hz field components of the modes, which explicitly contain the parities applied. When the parity condition was applied in our FDTD simulation, the computation was performed only for a quarter of the structure instead of the full structure. Therefore, the use of parity in mode excitation decreases the spectral density, and also reduces the memory requirement and computation time of each run.
4. Optical Loss Mechanism for Micro-Structured VCSEL
The optical loss of the excited modes in the VCSEL typically has three different origins. The first is the mirror loss due to the finite reflectance of DBR. The second is the absorption loss of the cavity material. The third is the scattering loss induced by the microstructures. In this simulation, the absorption loss was not considered since we assumed that the material was transparent for the lasing light. The mirror loss is determined basically by the wafer design, and it is not of concern because the loss was not very dependent on the modes. In this section, we consider the scattering loss induced by the air holes and explain why it shows modal dependency. Scattering loss is affected by several structural parameters including the oxide aperture size, the hole pitch (Λ), and the hole etching depth. As will be shown later, the diameter of the hole is less important than those parameters since only the inner edges of the holes contribute to the loss. The hole diameter was fixed at 3 μm throughout the study.
4.1. Variation of modal losses depending on oxide aperture size
Figure 5 shows the modal loss and resonant wavelengths for the LP01 and LP11 modes as functions of oxide aperture diameter. In this case, the hole pitch and depth were 5.0 μm and 2.2 μm, respectively. The loss is the inverse of the quality factor of each mode. The LP11 loss increased faster than the LP01 loss, and thus their difference increased as the diameter increased. This means that the larger oxide aperture is more favorable for single mode lasing. The resonant wavelengths showed negligible change beyond the oxide diameter of 8 μm.
To understand these phenomena, we plotted the optical intensity distributions of the LP01 and LP11 modes in a cross-section of the yz-plane, as shown in Fig. 6 . The color bar denotes the optical intensity on a log scale. The figure shows two main loss channels: the optical transmission through the top DBR (the top DBR has fewer layers than the bottom DBR), and optical leakage (or scattering) through air holes. An animation built with multiple shots of field profiles taken at successive times clearly shows that the optical wave inside the air holes was a propagating wave, whereas the optical wave inside the cavity was a standing wave . When the oxide aperture was small enough and no leakage loss occurred, both the LP01 and LP11 modes experienced the same loss of 2.9x10−4; this corresponds to the top DBR loss.
The LP11 loss was larger than the LP01 loss since its field spread wider into the air holes. The oxide layer with a low refractive index (shown in a short white line in the middle level) provided transverse confinement for the lasing mode to reduce the mode size. The large difference in the transverse mode size of the LP11 mode shown in Figs. 6(a) and (b) proves this concept, and it is responsible for the large variation of LP11 loss shown in Fig. 5. In contrast, the variation of both the modal loss and the mode size was relatively small for the LP01 mode because its mode profile was already well defined by the air holes rather than the oxide aperture.
For high quality single mode lasing, a large difference between the LP01 and LP11 loss is preferred. Low loss for the LP01 mode is also important for a low threshold. To achieve high output power of the VCSEL, the ratio of DBR loss to total loss should be considered since the leakage loss does not contribute to a laser emission with a Gaussian profile. Based on this reasoning, a desirable cavity structure is, roughly speaking, one with a large value of (LP11 loss – LP01 loss)/(LP01 loss). In this sense, the large oxide aperture is preferred, as shown in Fig. 5. However, since the oxide aperture diameter also plays a role as a current channel and significantly affects the threshold current, we selected a relatively small diameter of 8 μm for this study.
It is not straightforward to find the best sets of structural parameters producing low-threshold, high-power, single-mode lasers with only the simulation data. Instead, it will help analyze the experimental data, and provide an intelligent feedback to the next design for better performances.
4.2. Variation of modal losses depending on air hole pitch
We fixed the oxide aperture and hole etching depth to 8.0 μm and 2.2 μm, respectively, and changed the air hole pitch from 3.5 to 5.5 μm. Figure 7 shows the simulation results. Both LP01 and LP11 losses increased as the pitch decreased. When comparing the profiles in Figs. 8(a) and (b) , the overall mode patterns did not change very much according to the positions of air holes. Note that this is true only when the perturbation induced by the air holes is not strong enough to shape the mode profile. The situation changes when the hole etching is deep, as will be shown in next section. The inward shift of air holes results in a large leakage of light through the air holes. The leakage effect is more dominant for the LP11 mode since it has stronger field intensity near the air hole. The lasing wavelength of each mode decreased as the pitch shrank. This is reasonable since the presence of air holes resulted in a decrease in the effective refractive index of the cavity.
When the pitch was large than 5 μm and thus the air holes were located outside of the oxide aperture, both of the modal losses converged to the mirror loss resulting in no mode selectivity. It corresponds to the case of the oxide-VCSEL with no air holes. Therefore the graph of Fig. 7 demonstrates the role of air holes as mode-selective loss elements.
4.3. Variation of modal losses depending on air hole depth
In this study, the air hole depth was varied while keeping the oxide aperture and hole pitch at 8.0 μm and 5.0 μm, respectively. Figure 9 shows the simulation results for the modal loss and resonant wavelengths. The monotonic decrease in the resonant wavelength is well understood, since a shift of the air hole to the cavity center with the highest optical intensity in either the transverse or vertical direction will reduce the effective refractive index. However, the modal loss showed peaks for both modes at a hole depth of 1.8 μm. When the depth went beyond 2.0 μm, the loss decreased quickly and reached a minimum of 2.9x10−4. This behavior was contrary to what we expected.
The underlying mechanism was found when observing the field profiles shown in Fig. 10 . When the depth increased from 1 to 1.8 μm, the hole induced a larger loss while retaining the overall distributions of the modes, which is similar to the change from (b) to (a) in Fig. 8. However, when the depth increased to 3 and 3.4 μm, a dramatic change occurred in the mode profiles. The lateral mode size shrank to greatly reduce the leakage loss. This means that the air holes started to act both as an optical enclosure and as a loss element when the hole was deep enough.
To investigate this effect in detail, we plotted the mode size as a function of hole depth as shown in Fig. 11 . The mode size was defined by the positions where the optical intensity dropped by 1/e at the level of the oxide layer. The mode size of 4 μm, which was determined by the oxide aperture, was retained until the depth increased to 1.8 μm. It started to shrink after 1.8 um, and reached the bottom line at a hole depth of 3.0 μm. In the final stage, the mode was well confined in the GaAs-air waveguide, as shown in Fig. 10 (d). We do not yet understand why the mode size changed significantly at a depth of 1.0 to 1.4 μm.
As the hole deepened, the air layer encountered light with higher intensities; thus, the leakage loss increased. This effect can be represented by the reflection loss (or transmittance) of the DBR assuming that a plane wave is incident on the DBR, as shown by the blue dots in Fig. 11. The gradual increase in transmittance was responsible for the initial increase of modal loss shown in Fig. 9, up to a hole depth of 1.8 μm before the mode size began to shrink. The reflection loss continuously increased as the hole deepened, but now the effect of mode size reduction was more dominant and the modal loss decreased. The two factors shown in Fig. 11 successfully explain the loss behavior shown in Fig. 9. This result supports the experimental results and the discussions described in , regarding the mode size variation depending on the hole depth.
5. Estimation of Modal Loss Based on Mode Intensity Profile
As previously discussed above, the scattering (or leakage) loss of each mode depends on the optical intensity distribution on the surface of the holes, especially on their bottom surfaces. This suggests that the modal loss can be estimated from a mode intensity profile instead of a direct measurement of the time-dependent attenuation of optical intensity. We numerically investigated the relation between the modal loss and the mode intensity profile, and propose an effective tool to optimize the cavity structure.
First, we assumed that the leakage loss was proportional to the optical energy density integrated over the bottom surface of the holes as described by Eq. (1). Integration over the oxide layer was used to represent the total optical energy of the mode since the maximum energy density is found there.
Figures 12 and 13 show the ratio calculated using the mode profiles for each hole pitch and depth. The right axes denote total loss from the previous simulation results excluding the DBR loss of 2.9x10−4. The good agreement between the two data sets verifies our proposal. In Figs. 12 and 13, note that the two data sets keep the same relative scale; thus, the relative scale can be roughly determined using any single data point.
Our proposed loss estimation technique has important advantages over the conventional loss simulation approach. First, it requires less computation time for a FDTD run compared to the conventional approach. Note that the mode profile can be taken at an instant during FDTD simulation to be used for loss estimation, whereas a conventional loss calculation requires a certain amount of time to measure the temporal decay of optical energy. Secondly, the proposed method is able to provide not only the loss for the current cavity structure, but also for other structures slightly perturbed from the original. The first-order change in loss due to the perturbation of the hole structure can be calculated using the original mode profile overlapped on the perturbed hole structure. Such a calculation can be performed very quickly without the need for a new FDTD computation. Loss comparisons for several variations in cavity structures will yield the desirable direction of the hole shift either in transverse or vertical plane, to produce a modal loss favorable for our purpose. This technique should be very useful in the optimization of hole structures for a high-power single-mode VCSEL.
FDTD techniques were applied to the analyses of modal properties of micro-structured VCSELs. We showed that an appropriate selection of spatial resolutions and magnetic field parity enabled highly accurate simulation with a reasonable computation time and memory size. Based on the simulation of several structural variations, we examined the mechanism of loss induced by the air holes and how the loss became mode-dependent. The introduction of air holes caused the creation of a loss channel through the holes and, at the same time, reshaped the mode profile. We found that leakage loss could be successfully estimated using the mode intensity distribution at the bottom surfaces of the hole. The proposed loss estimation technique could be used for the fast evaluation of a given cavity structure, and to provide a guideline when searching for optimal cavity structures.
This work was supported by Regional Innovation Center for Photonic Materials and Devices at Chonnam National University under grant R12-2002-054, and by a Korea Research Foundation grant funded by the Korean government (MOEHRD, Basic Research Promotion Fund) (KRF-2008-331-C00115).
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