## Abstract

The use of a two-dimensional (2D) high-index-contrast grating (HCG) with square periodic lattice is proposed to realize surface-emitting lasers. This is possible because the use of 2D HCG, in which multiple resonant leaky modes are excited by the 2 orthogonal directions of the grating, causes the high reflective zone to be split into two regions. Hence, a dip of the reflectivity is formed to support the excitation of a resonant cavity-mode inside the 2D HCG. With suitable design on the dimensions of the 2D HCGs, Q factor as high as 1032 can be achieved.

©2009 Optical Society of America

## 1. Introduction

Recently, extensive studies have been concentrated on the use of one-dimensional (1D) high-index-contrast sub-wavelength gratings (HCGs) as the broadband reflectors [1, 2]. This is because the HCGs can provide a high reflectivity (i.e., > 99%) over a broadband region (i.e., > 15% of the operating wavelength). Furthermore, the realization of HCGs, which consist of a single-layer 1D-grating (i.e., periodic stripes with high-index-contrast) sandwiched in-between two low-index cladding layers, can be relatively simple. In contrast to HCGs, the fabrication of highly reflective distributed Bragg reflectors (DBRs) is difficult due to the epitaxial growth constraints of semiconductor dielectric layers with low-index-contrast. Hence, the use of HCG was proposed to replace the top DBR mirror of vertical-cavity surface-emitting lasers (VCSELs) [3]. In fact, the design and fabrication of VCSELs can be further simplified if the use of DBRs can be completely avoided.

High reflectivity of 1D-HCGs can be explained by the mechanism of guided-mode resonance (GMR). The periodic modulation of refractive index generates counter-propagating leaky modes inside the grating structure. Under phase matching conditions, these counter-propagating leaky modes establish standing waves and give rise to GMR. [4–6]. As a result, the leaky modes re-radiate reflectively through the resonant interaction with the grating. It is noted that the high reflective zone of the 1D HCGs is arisen from the blend of the adjacent leaky modes [7]. Hence, if the number and spectral location of the leaky modes is controlled, it is possible to introduce a dip to the high reflective zone (i.e., a cavity mode of the HCGs). In this paper, we propose the use of a two-dimensional (2D) HCG, in which a single-layer grating is constructed by a 2D high-index-contrast of square periodic lattice, to realize a dip in the high reflective zone of the 2D HCG. Hence, it can be shown that VCSEL without using DBRs can be formed by a 2D HCG with the introduction of an active layer.

## 2. Design and analysis

Figure 1 shows the schematic of a̱ proposed 2D HCG considered in our studies. It is assumed that the square periodic lattice is constructed by a layer of high-index periodic lattice with refractive index of *n _{H}*. The square periodic lattice is supported by a layer of low-index buffer layer, which has refractive index and thickness of

*n*

_{L}and

*t*

_{buf}respectively, laid on a substrate. The top surface of the square periodic lattice is surrounded by air. In order to understand how laser cavity can be formed within the 2D HCGs, the corresponding reflection and transmission characteristics are analyzed and compared with that of the 1D HCGs. It is assumed that the structure of 1D HCGs is identical to that of the 2D HCGs except that the 2D grating (i.e., square periodic lattice) is replaced by infinite long periodic stripes with the same period.

Figure 2 shows the reflection and transmission spectra of the 1D and 2D HCGs. The period, ∧, and height, *t _{g}*, of the 1D- and 2D-gratings were set to 700 and 460 nm respectively and the corresponding thickness of buffer layer,

*t*

_{buf}, was set to infinite in the calculation. In addition, Poly-silicon (

*n*

_{H}= 3.48) and SiO

_{2}(

*n*

_{L}= 1.45) were assumed to be high-index layer and buffer layer respectively with negligible absorption loss. A 3-D finite-difference time-domain (FDTD) method was used to investigate the reflection and transmission characteristics of the HCGs [8]. In addition, periodic boundary condition was used in the FDTD calculation to approximate the periodicity of the 1D- and 2D- gratings.

1D HCG with fill factor of 75% shows broadband reflection for the illumination of TM polarized light. Two closely spaced leaky modes (at 1.49 and 1.63 μm) can be seen from the transmission spectrum. These modes interact with the waveguide grating leading to efficient reflection [9]. Each mode corresponds to a high reflective mode (i.e., transmittance approaching zero) due to GMR. For the illumination of TE polarized light, the reflection spectrum is a narrow band. This is evident from the corresponding transmission spectrum which shows wide spacing of the weak leaky modes. For 1D HCG with a fill factor of 65%, one extra isolated leaky mode appears at a shorter wavelength (1.229 μm). As a result, the wide spectral spacing of the leaky modes contributes to a shallow and wide dip in the reflection spectrum. On the other hand, for the case of 2D HCG with a fill factor of 65%, four leaky modes appeared in the transmission spectrum. It is noted that the leaky modes #1 (1.239 μm), #2 (1.306 μm) and #3 (1.48 μm), #4 (1.612 μm) contribute to the high reflective regions centered at ~1.3 and ~1.6 μm respectively. Mode #2 is an extra leaky mode appearing in the transmission spectra of the 2D HCG when compared to that of the 1D HCG with the same fill factor. Furthermore, the presence of a mode #2 leads to a significant drop of the reflectivity at ~1.343 μm (i.e., the reflection dip). In other words, the presence of multiple modes, which cause the separation of GMR location, is due to 2D nature of the grating. As a result, a dip in the reflection spectrum is obtained. This dip of the reflectivity represents the formation of a resonant cavity mode of the 2D HCG.

In order to verify that the formation of cavity mode is due to the 2D nature of the grating instead of Fabry-Perot resonance, the values of *t*
_{buf} is allowed to vary between 775 and 925 nm (see Fig. 3). I̱t is observed that when *t*
_{buf} increases from 775 to 925 nm, the reflectivity at the dip wavelength of ~1.343 μm increases from 7% to ~ 25%. However, the minimum value of the reflection dip (~3%) occurs only when *t*
_{buf} = ∞. This suggests that the dip in the reflection spectrum is due to the 2D nature of the grating and the cavity mode is not formed inside the buffer layer. However, the buffer layer has influence on the fineness of the reflection dip.

It can be shown that there is a proportional relationship between the wavelength of the reflection dip and the dimensions of the grating. For a 1D HCG, the resonant conditions of resonant leaky modes can be determined by eigen-equations [10]:

where *i* is an integral denoting the diffraction order of grating, *β _{i}* (=

*k*[

*n*

_{g}sin

*θ*-

*iλ*/∧]) is effective propagation constant of the waveguide grating,

*k*(= 2π/

*λ*) is the wavevector in free space,

*θ*is the incidence angle to the normal of the grating,

*λ*is the resonant wavelength, and

*n*

_{g}is the effective reflective index of the grating layer. The parameters ${\kappa}_{i}\left(=\sqrt{{n}_{g}^{2}{k}^{2}-{\beta}_{i}^{2}}\right),{\gamma}_{i}\left(=\sqrt{{\beta}_{i}^{2}-{k}^{2}}\right),$ and ${\delta}_{i}\left(=\sqrt{{\beta}_{i}^{2}-{n}_{L}^{2}{k}^{2}}\right)$ are defined as the wave vectors within the grating layer, air and buffer layer respectively. If the resonant wavelength

*λ*increases to

*Yλ*by a factor of

*Y*, it can also be shown that the values of

*β*

_{i},

*κ*

_{i},

*γ*

_{i}and

*δ*

_{i}can be reduced to

*β*

_{i}/

*Y*,

*κ*

_{i}/

*Y*

*γ*

_{i}/

*Y*and

*δ*

_{i}/

*Y*respectively if

*θ*= 0 and ∧ increases by a factor of

*Y*. In order to satisfy Eq. (1) for this new resonant wavelength,

*t*

_{g}has to be increased to

*Yt*

_{g}. Hence, there is a proportional relationship between resonant wavelength and the dimensions of the grating. For our 2D HCGs with the dimensions of grating identical in both

*x*and

*y*directions (see Fig. 1), the corresponding eigen-equations and phase matching conditions are similar to that of the 1D grating. Therefore, the proportional relationship should also be held for the 2D HCGs. In order to demonstrate this scalable property of the 2D HCGs numerically, a design with a reflection dip located at a wavelength of 1.343 μm was used for the calculation. The corresponding values of, ∧, fill factor,

*t*

_{g}and

*t*

_{buf}, were set to 700 nm, 65%, 460 nm and 825 nm respectively. Reflection dip at wavelength of 1.55 and 2 μm can be achieved by scaling up ∧,

*t*

_{g}and

*t*

_{buf}, altogether by a factor 1.157 and 1.489 respectively as shown in Fig. 4. It can be seen from Fig. 4 that the wavelength of reflection dips and reflection patterns are similar for all the three cases under investigation.

## 3. Surface emitting laser with 2D HCG

Conventional 1D 2^{nd} order grating structure using low-index contrast has been used to fabricate surface emitting lasers [11]. The proposed 2D grating can also be applied to realize surface-emitting lasers by inserting a thin active layer between the grating and buffer layers. However, the 2D grating structure, which need not to be a 2^{nd} order grating, is necessary to have high-index-contrast. This is because high-index-contrast between the grating and buffer layers is essential to achieve high reflective zones and a sharp reflection dip. Reduced index contrast below 2 (i.e., *n*
_{H}/*n*
_{L} < 2) may lead to loss in reflectivity and poor Q factor of the device. Hence, refractive index and peak gain wavelength of the active layer should be close to the buffer layer and wavelength of the reflection dip respectively. If the buffer layer was assumed to be made by SiO_{2}, which has a refractive index of 1.45 at a wavelength of 1.343 μm, Nd:YAG can be used as an active layer. Hence, the refractive index, *n*
_{act}, and the peak gain wavelength of the active layer can be set to be 1.82 and 1.343 μm respectively [12] in the calculation. In addition, the thickness of the active layer, *t*
_{act}, was assumed to be 100 nm. In our analysis, a Lorentz model [8] was used to implement the gain material into FDTD algorithm. In this case, the corresponding permittivity, *ε*, can be expressed as

where ε_{o} = 3.3124, ω_{o} = 1.4×10^{3} THz (~1.343 μm), and δ = 31.4 THz (~ 50 nm). The emission spectra observed from the surface of the 2D HCGs can be calculated by a Fourier transform of the electric fields.

Two configurations of the 2D-HCG laser, i.e., with 1) *t*
_{buf} = 725 nm and 2) *t*
_{buf} = ∞, are considered in the studies. Fig. 5 plots the emission spectra of the 2D-HCG laser at threshold. Sharp resonance is observed at ~1.34 μm for the case *t*
_{buf} = ∞. However, the resonance wavelength shifted to ~1.342 μm for the case *t*
_{buf} = 725 nm. This is because the effective refractive index of the buffer layer is slightly increased due to the influence of substrate (which is assumed to have refractive index larger than the buffer layer) so that the resonant frequency of the standing wave inside the 2D grating is slightly shifted. For the case of *t*
_{buf} = ∞, Q factor is found to be ~925. This large value of Q is expected from the reflection spectrum of the 2D HCG with *t*
_{buf} = ∞ as the corresponding reflectivity is the lowest at the dip wavelength (see Fig. 3). On the other hand, although the presence of buffer layer will increase the reflectivity of the reflection dip, the reduction of 10% linewidth of the reflection dip (see Fig. 3) can compensate for the increase of reflectivity so that the value of Q factor can be improved. The inset of Fig. 5 shows the Q factor of the 2D HCGs with different values of *t*
_{buf}. The corresponding reflectivity and 10% linewidth of the reflection dip are also plotted in the figure. It is observed that the magnitude of reflection increases with *t*
_{buf} but the linewidth of the reflection dip reaches a minimum for *t*
_{buf} between 725 and 775 nm. As a result, a maximum value of Q factor equal to 1032 can be obtained at *t*
_{buf} = 725 nm. However, the Q factor does not improve for the value of *t*
_{buf} other than 725 nm.

## 4. Conclusions

In conclusion, the reflection characteristics of 2D HCGs were investigated. With suitable selection of ∧, fill factor and *t*
_{buf} of the 2D HCGs, cavity resonant (i.e., a dip of the reflectivity) can be observed from the corresponding reflection spectrum. The excitation of cavity mode (i.e., the dip) in 2D HCGs can be explained by the excitation of extra resonant leaky mode (#2 mode in Fig. 2(b)) causing the formation of two high reflective zones in the reflection spectrum. It is noted that there is a very narrow range of ∧, fill factor and *t*
_{g} to generate the cavity mode. By inserting a thin active layer between the grating and buffer layers, a 2D HCG surface-emitting laser with Q factor as high as 1032 can be realized and the value of Q factor is dependent on thickness of the buffer layer. In addition, in contrast to 1D HCGs, symmetric nature of 2D HCG makes it insensitive to the incident polarization. Such property may also be well suited for applications in amplifiers and optical filters.

## Acknowledgments

This work was supported by Singapore Ministry of Education Academic Research Fund Tier 2, grant no. ARC 2/06 and A*Star SERC grant 072 101 0023.

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