## Abstract

We propose a planar dielectric reflector with focusing ability using concentric circular subwavelength gratings (CC-SWGs). The two-dimensional focusing ability of CC-SWGs is investigated by the rigorous coupled-wave analysis (RCWA) and finite element method (FEM). By designing the concentric circular pattern of the grating surface, a focusing reflector with high numerical aperture (NA) and high reflectivity is constructed. A CC-SWG reflector with a diameter of 32.6μm and a focal length of 6μm is investigated, which exhibits high focusing ability at normal incidence with a radially polarized plane wave. At the reflection focal plane, the full-width-half-maximum (FWHM) of the electric field intensity is 0.89μm. Numerical aperture value as high as 0.93 is achieved for the reflector with very high reflectivity of 92%.

© 2015 Optical Society of America

## 1. Introduction

Vertical cavity surface emitting lasers (VCSELs) and reflection enhanced photodetectors integrated with high index contrast subwavelength gratings (SWGs) have emerged recently [1–3]. These new types of optoelectronic devices are not only more compact and cheaper to fabricate, but also can provide better device performance. The SWG used as a reflector or a lens in these devices is a single dielectric layer grating consists of high index bars fully surrounded by a low index medium, which can provide very high reflectivity or transmissivity over a broad bandwidth. Its operation is based on a resonant phenomenon that is also known as the guided-mode resonance effect [4–8]. Owing to a simple and compact structure, SWGs are strong candidates to replace conventional distributed Bragg reflectors (DBRs) in semiconductor optoelectronic devices.

Furthermore, another kind of SWGs with non-periodic strip patterns was proposed and has been attracting people’s interest because of their salient features such as focusing ability, steering ability, high reflectivity and wafer-scale integration. This kind of grating not only maintains high reflectivity but also can achieve beam focusing or steering by phase front control, which is an important development in the application of SWGs [9–11]. It has a significant impact on low cost optoelectronic fabrication, since a more efficient integration with other optoelectronic components can be achieved by avoiding conventional curved mirrors made of glass coated with metal.

We propose a novel planar focusing reflector composed of concentric circular subwavelength gratings (CC-SWGs) on silicon-on-insulator (SOI) substrate. Different from the strip SWGs that can only achieve one-dimensional focusing, CC-SWGs can achieve two-dimensional focusing. The CC-SWGs with non-periodic ring patterns not only obtain excellent focusing ability, but also can maintain a high reflectivity. In addition, the SOI wafer not only can offer a high refractive index contrast suitable for the fabrication of SWGs, but also the fabrication process of these gratings is compatible with the complementary metal oxide semiconductor (CMOS) electronics industry.

In this Letter, we investigate the fundamental ability of CC-SWGs to manipulate the phase front of a reflected light beam and formulate useful design rules for the realization of practical structures. A CC-SWGs reflector with a diameter of 32.6μm and a focal length of 6μm is demonstrated, which exhibits high focusing ability at normal incidence with a radially polarized plane wave. At the reflection focal plane, the full-width-half-maximum (FWHM) of the electric field intensity is 0.89μm. Numerical aperture value as high as 0.93 is achieved for the reflector with very high reflectivity of 92%. The paper focus on SOI-based CC-SWGs with radially polarized incident light, but the design rules apply equally well to azimuthal polarized and unpolarized gratings.

## 2. Design and simulation

#### 2.1 Modeling of the CC-SWGs with non-periodic ring patterns

The CC-SWG structure investigated in this paper is depicted in Fig. 1. The SOI wafer consisted of a 500nm silicon layer and a 500nm buried oxide layer. The CC-SWG consists of concentric circles of Si (n≈3.48 in the near infrared) that are surrounded with air and SiO_{2} (n≈1.45) as the low-index cladding layers on the top and bottom. Our design is for radially polarization (electric field vector is perpendicular to the grating bar direction) at a wavelength of 1.55μm. Radially polarized light has its electric field vectors arranged like spokes of a wheel pointing out from the center of the beam.

The optical properties of the grating are controlled by its period (Λ), duty cycle (η), and thickness (h). Since varying thickness is not feasible due to the multiple etching steps it would imply, here only variations of the period and duty cycle are considered. The reflection response of periodic CC-SWGs (the period and duty cycle is fixed), such as phase or intensity, is spatially independent. However, if the grating structure (period and duty cycle) is locally changed, these properties will gradually adapt to these variations. That is to say, the reflection properties at a given point in space depend only on the local geometry around that point. If the phase distribution along the *r* direction in CC-SWG is chosen properly, the plane wave can be focused [10–13]. The desired phase response of reflected light for focusing should be a parabolic profile:

*φ*

_{max}is the maximum phase change (the phase difference between the center and the very edge of an CC-SWG). When the phase

*φ*(

*r*) is more than 2π, it can be mapped to an equivalent value between 0 and 2π. Following this principle it is possible to contemplate a CC-SWG structure with focusing ability, which allow obtaining spatially varying phase profiles,

*φ*(

*r, θ, z*), in reflection while maintaining a high reflectivity.

#### 2.2 Radially polarized surface-normal incidence

It is difficult to directly determine the local parameters (period and duty cycle) of the CC-SWGs at normal incidence with a radially polarized plane wave, because the calculation of CC-SWGs is very complex using the analytic method. However, radial polarization for CC-SWGs, similar to TM polarization for the strip SWGs, its electric field vector is perpendicular to the grating bar direction. To simplify the calculating method, we assume that a CC-SWG illuminated by radially polarized beams can be approximated as the cylindrical analogue of a one dimensional Cartesian grating illuminated by TM polarization.

To understand or confirm this assumption, we start by calculating the reflectivity of these two structures. One is the CC-SWGs illuminated by radially polarized beams; another is the strip SWGs illuminated by TM polarized beams. For simplicity of calculation, we consider only one low-index medium surrounding the high index grating bars. Figure 2 shows the schematics of CC-SWGs and the strip SWGs, which are illuminated at normal incidence with a radially polarized beam and a TM polarized beam respectively. The colored bars represent silicon with a refractive index of 3.48, which are surrounded with air (n_{air} = 1) as the low-index cladding layers. The structural parameters for these gratings are the same, Λ = 0.77μm, η = 0.76 and h = 0.45μm. Broadband high reflection can be obtained with these parameters [6].

The reflection spectra of gratings are demonstrated in Fig. 3(a). We investigate three models. The first model is a one-dimensional periodic strip SWG, and the boundary condition is infinite. The grating is infinite in y and infinitely periodic in x direction, which is calculated by RCWA under TM polarized illumination. The second model is also a one-dimensional periodic strip SWG, but the boundary condition is semi-infinite. The grating is infinite in y direction and finitely periodic (only 60 cycles) in x direction, which is calculated by FEM under TM polarized illumination. The third model is a CC-SWG, and the boundary condition is finite. The grating is 60 cycles in r direction, which is calculated by FEM under radially polarized illumination. Calculation results of these three models are approximately the same, just there are some resonance peaks caused by the boundary conditions. It is obvious that the optical properties of CC-SWGs under radially polarized illumination are same as the strip SWGs under TM polarized illumination. Figure 3(b) shows the grating reflectivity variation with the number of cycles of the CC-SWGs. With the increase of the number of periods, reflectivity tends to saturation. When the cycle is greater than 40, the grating reflectivity reaches a steady. Though these models are different, they can be proved mathematically equivalent, and the accuracy of this assumption improves with the increase in sample size, due to the smaller curvatures. Therefore, the local parameters of the CC-SWGs can be determined by calculating one-dimensional periodic strip SWGs at normal incidence with a TM polarized plane wave.

#### 2.3 Design process

For a focusing CC-SWG design, the goal is to find the local period and duty cycle that give a particular spatial phase profile along the r direction. When the phase in this direction is given by Eq. (1), the beam can be focused on a point. As an example, we design a focusing CC-SWG here. The key parameters of the CC-SWGs were determined by calculating one-dimensional periodic strip SWGs under TM polarized illumination. The reflection properties, such as reflectivity and phase, of the periodic strip SWGs are studied using RCWA method [14,15]. In this approach the periodicity of the structure is exploited to solve Maxwell’s equations. A linear system of equations is built from the boundary conditions. The system solution yields the field distribution, as well as the reflection characteristics. Extensive simulations are performed to find sets of grating parameters that provide the targeted reflection phase and reflectivity.

By sweeping the values of the period and duty cycle, we obtain the whole maps of reflectivity and phase change of the periodic strip SWGs with TM-polarized light at normal incidence, as shown in Fig. 4. It should be noted that although the reflectance and phase maps are calculated for one-dimensional periodic strip gratings, their use in arriving at a design for a CC-SWGs is excellent, which we have previously introduced briefly. The thickness of the gratings is fixed at 500nm, while the period varies from 0.3μm to 1.2μm and the duty cycle varies from 0.2 to 0.8. Figure 4(a) shows the reflectivity of the reflected light. Figure 4(b) exhibits that the phase of the reflected light cover a full 2π range of variation within the high-reflectivity region, realizing a full 2π is important, because it enables arbitrary phase front control. In general, this result serves as a look-up table to select a set of discrete data (Λ_{n}, η_{n}) that correspond to the maximum reflectivity *R*_{max}(Λ_{n}, η_{n}) for any phase *φ*_{n}. The subscript n is an integer which corresponds to one point in the map. Then the local period and duty cycle are found, in the one by one optimization process, according to the target phase distribution *φ*(*r*) from Eq. (1).

Figure 5 displays a set of useful structural parameters, the local period and duty cycle, to design CC-SWGs reflector with focusing abilities. As illustrated in Fig. 5(a), the green line is the ideal phase distribution along *r* direction for a CC-SWG focusing reflector calculated by Eq. (1) with a focal length of 6μm, and the blue circles are the each designed grating bars chosen from Fig. 4. The actual dimensions of these grating bars, as well as their corresponding periods, chosen for a CC-SWG reflector design are shown in Fig. 5(b). The blue triangles show the widths of the grating bars, and the red squares show the periods of the grating bars.

## 3. Results and discussion

The structural parameters of a CC-SWG are shown in Fig. 5(b), obtained by using RCWA method. The parameters are locally varied in order to master the phase response. This process creates a final structure that is no longer periodic. Therefore, to investigate this model RCWA is no longer useful because it requires a periodic grating structure. The method used, instead, is finite element method (FEM). The commercial software COMSOL implementing the FEM is used to simulate the non-periodic structure. Considering the spatial rotation symmetry of structure and incident light field, a two-dimensional axis symmetry model is used to simplify such three-dimensional numerical calculation.

Here we have used a FEM to simulate a CC-SWGs reflector of large NA (0.93) with a diameter of 32.6μm and a focal length of 6μm. The design of the device requires a total phase variation of 14π from center to edge, which is obtained by modulation of the local period and duty cycle using the reflection map of Fig. 4. In Fig. 6(a), a FEM simulation of the CC-SWG designed to have focusing ability is shown. The light source is radially polarized plane wave and it has a wavelength of 1.55μm. It is obvious to see the focal spot appears at z = 6μm. As can be seen in Fig. 6(b), the transverse shape of the focal spot is symmetrically uniform and the FWHM is about 0.89μm at the focal plane. Numerical aperture of 0.93 is achieved for the reflector with very high reflectivity of 92%.

To further reveal the focusing ability, the changes of the focusing power with the focal length are investigated. With the increase of the focal length, the peak electric field intensity gradually declines at the focal plane, while the FWHM generally increases, as shown in Fig. 7(a). This is because of NA decreases with the increase of the focal length. On the other hand, the reflected light field consists of two parts: radial and longitudinal electric fields, as shown in Fig. 7(b). If the radial component turns to be the dominant, the FWHM will also increase.

The characterizations of non-periodic CC-SWGs with wave-front manipulation ability are performed on a grating implemented on a SOI wafer. The design parameters, the local period and duty cycle, of the non-periodic gratings calculated by RCWA are useful to design CC-SWGs that have to fulfill a certain set of requirements. The focal length depends on the curvature of the parabolic phase profile given by the non-periodic CC-SWGs. Varying this parameter the focus can be moved closer or further from the grating. The planar non-periodic CC-SWGs will have a significant impact on low cost integrated optoelectronic devices since a more efficient integration of optoelectronic modules can be achieved by avoiding expensive external lens systems.

## 4. Conclusion

A planar focusing reflector using SOI-based CC-SWGs is presented. We investigated and explained how to obtain focusing performance by phase front control, and calculated the structure parameters of CC-SWGs by using the RCWA and finite element method as well. Under the incident radially polarized (1.55μm) illumination, the CC-SWGs reflector with a diameter of 32.6μm can generate a focal spot with 6μm focal length and 0.89μm FWHM by rationally selecting parameters. Numerical aperture of 0.93 is achieved for the reflector with very high reflectivity of 92%. Such planar focusing reflectors with CC-SWGs structure can be fabricated with standard photolithography, and can be integrated in photonic devices easily. In addition, this approach can be generalized to obtain different phase profiles from the gratings, broadening a series of possible applications in integrated optoelectronic devices.

## Acknowledgments

This work is supported by the Natural Science Foundation of Beijing, China (4132069), the National Natural Science Foundation of China (61274044), the National Basic Research Program of China (2010CB327601), Program of Key International Science and Technology Cooperation Projects (2011RR000100), 111 Project of China (B07005).

## References and links

**1. **M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics **1**(2), 119–122 (2007). [CrossRef]

**2. **T. Gründl, K. Zogal, P. Debernardi, M. Müller, C. Grasse, K. Geiger, R. Meyer, G. Böhm, M. Amann, F. Küppers, and P. Meissner, “Continuously tunable, polarization stable SWG MEMS VCSELs at 1.55μm,” IEEE Photon. Technol. Lett. **25**(9), 841–843 (2013). [CrossRef]

**3. **X. Duan, Y. Huang, X. Ren, Y. Shang, X. Fan, and F. Hu, “High-efficiency InGaAs/InP photodetector incorporating SOI-based concentric circular subwavelength gratings,” IEEE Photon. Technol. Lett. **24**(10), 863–865 (2012). [CrossRef]

**4. **S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. **32**(14), 2606–2613 (1993). [CrossRef] [PubMed]

**5. **C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62μm) using a subwavelength grating,” IEEE Photon. Technol. Lett. **16**(7), 1676–1678 (2004). [CrossRef]

**6. **C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photon. **4**(3), 379–440 (2012). [CrossRef]

**7. **A. Ricciardi, S. Campopiano, A. Cusano, T. F. Krauss, and L. O’Faolain, “Broadband mirrors in the near-infrared based on subwavelength gratings in SOI,” IEEE Photon. J. **2**(5), 696–702 (2010). [CrossRef]

**8. **D. Zhao, H. Yang, Z. Ma, and W. Zhou, “Polarization independent broadband reflectors based on cross-stacked gratings,” Opt. Express **19**(10), 9050–9055 (2011). [CrossRef] [PubMed]

**9. **D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics **4**(7), 466–470 (2010). [CrossRef]

**10. **F. Lu, F. G. Sedgwick, V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Planar high-numerical-aperture low-loss focusing reflectors and lenses using subwavelength high contrast gratings,” Opt. Express **18**(12), 12606–12614 (2010). [CrossRef] [PubMed]

**11. **L. Carletti, R. Malureanu, J. Mørk, and I. S. Chung, “High-index-contrast grating reflector with beam steering ability for the transmitted beam,” Opt. Express **19**(23), 23567–23572 (2011). [CrossRef] [PubMed]

**12. **Y. Yu and H. Zappe, “Effect of lens size on the focusing performance of plasmonic lenses and suggestions for the design,” Opt. Express **19**(10), 9434–9444 (2011). [CrossRef] [PubMed]

**13. **G. M. Lerman, M. Grajower, A. Yanai, and U. Levy, “Light transmission through a circular metallic grating under broadband radial and azimuthal polarization illumination,” Opt. Lett. **36**(20), 3972–3974 (2011). [CrossRef] [PubMed]

**14. **M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A **71**(7), 811–818 (1981). [CrossRef]

**15. **M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**(5), 1068–1076 (1995). [CrossRef]