## Abstract

2-dimensional simulations of high-contrast gratings (HCGs) of finite size are carried out, targeting at their applications in vertical-cavity surface-emitting lasers (VCSELs). Finite HCGs show a very different behavior from infinite grating ones. The reflectivity of a finite HCG strongly depends on the HCG size and the source size. Our simulation results predict finite reflectivity and transmission values, well consistent with reported experimental results. The band of high reflectivity (>99.5%) of finite HCGs is less broad as compared to the infinite case. Losses into a guided mode excited in the HCG plane are identified as being at the root. This guided mode is excited due to the nonzero angular components in the finite source size, and greatly enhances the transmission and the light leakage from the slab. In addition, the simulation results show that the details of the finite HCG can shape the output beam, whilst a Gaussian-like reflected wave is typically achieved. Our simulations can explain the current discrepancies between numerical predictions of reflectivities approaching 100% and working HCG-VCSELs showing finite reflectivities and nearly Gaussian-like output. Consequently, our analysis of finite HCGs is indispensable for HCG-VCSEL design.

© 2014 Optical Society of America

## 1. Introduction

Vertical-cavity surface-emitting lasers (VCSELs) are attractive low-cost light sources for optical interconnects in computer networks, because they provide additional potential for higher modulation bandwidth, lower power consumption, and symmetric beam [1–7]. High-speed, energy-efficient, and temperature-stable VCSELs for Datacom with increasing transmission distance have been progressing rapidly in the last few years [8–10]. Yet these essential parameters are still limited by the wafer design, device structure, and parasitic effects. Conventional ways of VCSEL design improvement saturate due to technological limits. Therefore, novel approaches are needed for further improvements.

Very recently a novel nanostructure, called high-contrast grating (HCG), has been attracting much attention [11–14]. The grating bars composed of the high-index material in the HCG are fully immersed in the low-index medium, resulting in a high index contrast. The grating period is in the near-wavelength regime, between the wavelength in the high-index material and in the low-index material. Theoretical analysis shows the first two waveguide array modes with real propagation constants in infinite HCGs have a π-phase difference at the output plane and cancel each other causing a nearly 100% reflection [15]. When the first two modes are very closely located in the spectrum, a high-reflectivity broad band is obtained [15]. Thus a broadband and high-reflectivity HCG can serve as a reflector and replace the top DBR to construct a HCG-VCSEL [11,14–19]. Experimentally, HCG-VCSELs show a good mode selectivity and polarization control even for large oxide apertures, and also can reach a fast tuning rate in tunable HCG-VCSELs [17–20]. In addition HCG-VCSELs reduce the mode volume because of the 100-nm scale in thickness, and realize single-mode high-power operation even at a large oxide aperture, resulting in reduced parasitic resistance [21]. Thus HCG-VCSELs are expected to achieve long-distance, high-speed modulation at high energy efficiency.

Infinite HCGs are always modelled based on rigorous coupled wave analysis (RCWA) or finite-difference time-domain (FDTD) methods with periodic boundary conditions. However, in real applications a HCG is of finite size. In this work, finite HCGs are investigated using the 2-dimensional (2D) FDTD approach. In contrast to infinite HCGs, the reflectivity and high-reflectivity band width depend on the HCG size and excited source size, and these effects are explained by the angular spectrum of the finite excited source. Here it should be noted that much work has been reported on waveguide grating structures with *low contrast index* [22–24]. These waveguide grating structures can be used as guided-mode resonance filters. The reflectivity, peak width, and incident angle dependence of these guided-mode resonance filters were studied with finite-size grating and finite-size incident wave (or oblique incidence).

## 2. Simulation model

A schematics of a HCG-VCSEL is shown in Fig.
1(a). The HCG replaces the top distributed Bragg reflector (DBR) or at
least most of it. The optical properties of HCG are determined by the refractive
index of the high-index material for grating bars (n_{H}), the refractive
index of the low-index material surrounding the grating bars (n_{L}), the
thickness of the grating (*t*), the duty cycle (*F* =
*a*/Λ), the period (Λ), as shown in Fig. 1(b). Previous calculations always treat
HCGs being periodical in the *x* direction, called *infinite
HCG* here. This means the incident wave is an infinite plane wave.
Calculations with these kinds of assumptions predict broadband and very high
reflectivity for infinite HCGs [11,14]. However, in real applications the HCGs
have a finite size in the *x* direction, called *finite
HCG*, meaning that both the HCG size and the excited source size are
finite. These assumptions are a much better match to the reality. Consequently
finite HCGs show some optical characteristics very different from infinite HCGs.

For the finite HCG, the 2D FDTD method is adopted to calculate the reflectivity spectrum with a pulse of a Gaussian source. The finite Gaussian source is placed at the center of HCG. The reflected and the transmitted waves are monitored. The reflectivity and transmittance here are defined by the ratios of the reflected power and transmitted power to the launched power, respectively. To save computational time, symmetric and perfectly matched layer (PML) boundary conditions are employed to describe the finite HCG, as shown in Fig. 1(b). The grid size here is 10 nm in both directions. The launch source is 225 nm away from the HCG. In simulations we find that the reflectivity and transmittance remain unchanged when the source is placed in the range of 2 μm far away from HCG.

The transverse magnetic (TM, the electric component perpendicular to the HCG bars) HCG is designed to have a high reflectivity at around 850 nm. The grating bars are composed of Al_{0.6}Ga_{0.4}As (n_{H} = 3.2), and the low-index medium is air (n_{L} = 1). The other parameters are optimized using the RCWA method to obtain a broadband and high reflectivity. Here we choose Λ = 0.38 μm, *t* = 0.235 μm, *a* = 0.25 μm. The resulting reflectivity spectrum of the infinite TM-HCG is shown in Fig. 1(c).

## 3. Simulation results

In VCSELs, the current is injected laterally by a ring contact and confined by one or several aperture(s) of several micrometers in diameter. This ring contact limits the physical size of the top reflector, especially if a HCG is used. The smaller the HCG is, the lower the parasitic resistance due to current crowding is going to be. Therefore, it is crucial to know how many periods are required for a specific HCG-VCSEL. To study the effect of the HCG size, the excited source size (i.e. oxide aperture in the HCG-VCSEL) is fixed, and the HCG size is varied. The slab without grating structure and the infinite HCG are taken as references for comparison. Figure 2(a) shows the reflectivity spectrum for different HCG sizes under normal incidence (φ = 0). Compared with the slab without grating structure, the HCG greatly enhances the reflectivity around 850 nm. However, the finite HCG has a smaller band of reflectivity larger than 99.5% and the reflectivity decreases compared to the infinite HCG. The reflectivity increases with increasing HCG size. Our simulation results are consistent with the experimental results in [25], which could not be fully understood until now. Interestingly, a dip occurs at 744.7 nm in the reflectivity spectrum independent of the HCG size. This dip and its position are independent of HCG size, while it disappears in the reflectivity spectrum of the infinite HCG. The transmission as shown in Fig. 2(b) decreases with increasing HCG size. Transmission shows a maximum at 744.7 nm, same as the dip in the reflectivity spectra in Fig. 2(a). In our calculations of finite HCGs, the finite Gaussian source is used. The finite-size source can be treated as a combination of various angular components by Fourier transform [26,27]. Therefore the reflectivity depends on the sum of all the angular components of the finite source. However, just the portion of the angular components reflected by the finite HCG effectively interacting with the gain medium contributes to the confinement of the HCG-VCSELs due to the limited oxide aperture size. Thus, it is very important to know the effective reflectivity of the finite HCG in HCG-VCSELs.

Figure 2 demonstrates some of the unique properties of
finite HCGs, which do not exist for infinite HCGs. Comparing the reflectivity
spectrum of the finite HCG with the infinite HCG under normal incidence, it is
obvious that the HCG size greatly affects the reflectivity spectrum. For an infinite
HCG, the incident wave is an infinite plane wave. There exists only the zero angular
component in the angular spectrum. However, when the finite-size Gaussian source is
incident on the finite HCG, there are many nonzero angular components in the angular
spectrum of the Gaussian source. These nonzero angular components have the same
effect as the off-normal incidence in the infinite HCG. The effect of angular
contribution of finite-size sources on *infinite* HCGs is shown in
[26] using a different numerical approach
[27] and is in good agreement with our
results for large finite gratings. To investigate this effect for finite gratings we
calculate the reflectivity spectra for off-normal incidence with the RCWA method as
shown in Fig. 3.The band of high reflectivity is reduced with increasing incident angle, and a
dip appears at the wavelength of 0.744 μm, independent of the incident angle.
A similar result has been reported in [28].
The dip in the reflectivity spectrum is 744.7 nm in Fig. 2. It is obvious that the nonzero angular components in the
finite-size source cause the appearance of the dip in the reflectivity spectrum
resulting in a reduction of the high-reflectivity band width. On the other hand, the
nonzero angular components in the finite-size source are beneficial for the mode
selectivity in HCG-VCSELs, in agreement with experimental observations [25] and calculated results with another
approach [26,27].

Light leakage from the slab happens in finite HCGs, which leads to loss, as
shown in Fig. 4(a). As loss
*L* we can define *L = 1 - R - T* with
*R* and *T* as power reflectivity and
transmission, respectively. However, a monitor can be placed at the side boundary to
directly record the loss *L* in our FDTD calculations. The loss is
reduced with increasing HCG size. Interestingly, we can find a peak in the loss
spectrum also at 744.7 nm which means a considerable portion of the incident light
leaks from the edge of the finite HCG to the slab. Figure 4(b) shows the steady-state optical field at 849.54 nm,
corresponding to the reflectivity peak for finite HCG in Fig. 2(a). Because 99.7% of the incident light is reflected,
nearly no transmitted wave and leaked light are observed. However, the reflected
light becomes much weaker when the incident wave is at 744.7 nm (at the dip) as
shown in Fig. 4(c). Compared with Fig. 4(b), the transmission at this wavelength
is greatly enhanced. Interestingly, a guided mode is excited in the HCG section and
this mode can propagate transversely to the slab, enhancing the light leakage from
the slab [29,30]. This guided mode is independent of the HCG size, but disappears in
the infinite HCG, as shown in Fig. 2.

Small oxide apertures are preferred to achieve a small footprint and low threshold current for VCSELs and are beneficial to realize high-speed and energy-efficient VCSELs for data transmission. Hence compact HCG-VCSELs are very attractive. Figure 5 shows the reflectivity spectra for different source sizes with a HCG size of 8.36 μm (22*Λ). The reflectivity at around 850 nm increases and saturates at 99.7% as the beam size increases. Starting from the source size of about 3 μm, the reflectivity at around 850 nm can reach 99.5% which is good for HCG-VCSELs to easily achieve the threshold. However, for a beam size of 1 μm, the reflectivity is less than 90%, because there are more nonzero angular components as compared to a beam size of 5 μm, thus causing lower reflectivity. With reducing the source size, more nonzero angular components occur. A smaller source leads to a lower reflectivity and narrower high-reflectivity band. Thus, for 850-nm HCG-VCSELs with a small oxide aperture (e.g. 1 μm) threshold is difficult to reach because of the too low reflectivity. This is again in excellent agreement with the experimental results in [25], and 3D calculations with infinite gratings reported in [26]. The reflectivity peak shifts to shorter wavelengths as the beam size increases. The reflectivity dip also occurs at 744.7 nm, corresponding to the enhanced transmission peak and the guided mode in the finite HCG.

Field profiles of the reflected waves for different HCG sizes are shown in Fig. 6(a). The reflected waves are Gaussian-like, because of the zero-order diffraction in the finite HCG. Thus, the reflected waves interact strongly with the gain medium in HCG-VCSELs thanks to the Gaussian-like reflected wave. A threshold current of below mA can be realized [25]. This means that finite HCGs can well confine the optical wave in a HCG-VCSEL cavity. As the finite HCG has losses into the slab and scattering at the edges occurs, the reflected and the transmitted field are no longer tightly connected. The output beam profiles affect coupling to the fiber and consequently high-speed link performance. Such far fields are shown for different configurations in Figs. 6(b) and 6(c). Figure 6(b) shows the profiles for different HCG sizes with a fixed-size source. There is a main lobe located in the center of the far field, and depending on the dimensions, several side lobes surrounding the main lobe can occur. The intensity of the side lobes decreases with increasing the HCG size. Figure 6(c) shows the far field profiles for a quite large HCG for different source sizes. Again, the far-field can vary from a well-focused Gaussian-like beam to a quite distorted far field. The intensity of the side lobes are best suppressed for a source size of 7 μm. Here the far field profile is very close to a Gaussian pattern, well consistent with the experimental results in [31]. From Fig. 6(c), we can find that the optimum source size for a fixed HCG size. This result shows that a comprehensive modelling as presented here is definitely needed to understand HCG-based devices. With its nontrivial behavior, HCG technology opens up now many new degrees of freedom in device-design.

## 4. Conclusions

In this work HCGs for VCSELs are studied using RCWA and FDTD methods. In real world HCG-VCSELs, both the HCG size and the oxide aperture are finite. Finite HCGs shows a clearly different behavior compared to infinite HCGs. The HCG size and the source size strongly affect the reflectivity and the high-reflectivity band, consistent with previously reported experimental results. With increasing HCG size, the reflectivity and high-reflectivity band of the finite HCG increase and saturate. Therefore, a practical size that is large enough can be clearly identified. Our calculations explain the experimental results that HCG reflectivities larger than 99.99% are not observed in practical VCSEL devices. The reflectivity and high-reflectivity band width of the finite HCG increases with the source size, but also reaches saturation. The high-reflectivity band is finite because of a guided mode in the HCG plane, which is nonexistent for infinite boundary conditions. This is a very crucial point for practical HCG design, as finite HCGs demonstrate some unique optical properties. The guided mode is caused by the nonzero angular components in the finite source, which can be confirmed by the reflectivity spectra of the off-normal incidence in the infinite HCG. The guided mode also contributes to the enhanced transmission and enhanced leakage from the slab. In addition, the finite HCG can shape the incident beam. When the Gaussian source is incident on the finite HCG, the reflected wave is Gaussian-distributed. However, in the field profile of the transmitted wave, there are side lobes surrounding the main lobe in the center, with their intensities depending on the HCG size and source size. With optimization, a Gaussian-like profile of the transmitted wave can be achieved. The simulations of the finite HCGs presented here are showing qualitative differences compared to infinite HCG models published previously. As HCG-VCSELs are very sensitive to HCG parameters, this more comprehensive model is indispensable in HCG-VCSEL design, and gives valid predictions for HCG-devices in good agreement with experimental findings.

## Acknowledgment

We gratefully acknowledge the German Research Foundation (DFG) for funding via the collaborative research center 787 and the Alexander von Humboldt Foundation for supporting Anjin Liu by a Postdoc fellowship.

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