## Abstract

In this paper, systematical study on the focal shift phenomenon in planar lenses based on GaN high contrast gratings was performed using finite element method (FEM). Influence of parameters including device size, designed focal length, total phase difference on the focusing performance is presented. It shows that the focal shift is mainly determined by the Fresnel number of the lens, which is nearly equal to the total phase difference divided by π. The influence of the lens size and designed focal length can be attributed to the change of Fresnel number. Theoretical analysis based on diffraction theory is employed to predict the focal shift, which is in accordance with the numerical simulation.

© 2015 Optical Society of America

## 1. Introduction

Metasurfaces have attracted increasing attention in recent years due to their interesting physics and promising applications [1, 2
]. Comparing to three-dimensional metamaterials, metasurfaces are simpler from a fabrication point of view and can be integrated, in principle, on any planar substrate including semiconductors. Recently, a range of planar nanoscale optical components, such as antireflection coatings [3], polarization converters [4, 5
], lenses [1, 6
], and optical vortex generators [7], have been demonstrated. Particularly, planar focusing metasurface is a hot topic in the scientific community [8–17
]. Focusing element is one kind of the most ubiquitous optical components; it is widely used in applications such as communication, sensing, imaging, solar cells, light-emitting diodes (LEDs) and nanolithography, *etc*. As an important category of the focusing metasurfaces, focusing element based on semiconductor high contrast gratings (HCGs) has been widely investigated [8–14
]. The basic geometry of the HCG focusing components consists of an array of nanoscale ridges completely surrounded by low refractive index materials (e.g. air). Due to the high index contrast of the grating bars and surrounding materials, the lateral propagation length of the guided modes is short [8,11
], allowing local modulation of the transmitted or reflected phase, while maintaining a high transmittivity or reflectivity. With precise adjustment of the structural parameters of individual ridges, it is possible to create a curved phase front for the transmitted/reflected field and thus to achieve a focusing action. D. Fattal et al. [8] demonstrated flat grating reflectors with focusing abilities using non-periodic silicon gratings on glass. J. H. Lee et al. [12] proposed a semiconductor HCG metasurface that simultaneously performs focusing and polarization filtering functions. In [14], we proposed a planar lens for visible spectral region based on gallium nitride (GaN) HCGs, which is promising for the GaN-based visible LEDs for controlling the output light propagation and polarization, as well as enhancing the extraction efficiency.

Though various design principles, such as variable periods [8–10, 14
], thicknesses [12], geometries [11, 13
], have already been reported to implement HCG focusing metasurfaces, a number of basic scientific and technical problems still remain to be solved. One problem is that the actual focal lengths of the focusing HCG metasurfaces obtained from simulation and experiment usually deviate from the design (see Table 1
). For example, for the focusing polarizing beam splitter reported by J. H. Lee et al. [12], the focal lengths obtained are *f*
_{TM} = 16.76 μm for the reflected TM polarization and *f*
_{TE} = 15.81 μm for the transmitted TE polarization, while the desired focal length is 15 μm. Similar phenomenon occurs in [8, 9, 11
]. The focal shift could become a serious obstacle in the application of HCG focusing elements. Therefore, it is important to study the origin of the focal shift and the parameters that affect it.

In this paper, we present comprehensive study on the focal shift phenomenon in planar HCG lenses. The influence of lens parameters, e.g. lens size (where “size” refers to the lateral extent of the structure along the width direction of the devices), designed focal length and total phase difference, is systematically investigated using numerical simulation. Besides, theoretical analysis based on diffraction theory is employed to estimate the focal shift. By comparing simulated and analytical results, we found that the focal shift in planar HCG lens is mainly determined by the Fresnel number, which is nearly equal to the total phase difference divided by π. The influence of the lens size and designed focal length can be attributed to the change of Fresnel number. When Fresnel number is large, geometric optics is sufficient to predict the focal length desired. However, for a small Fresnel number, diffraction theory is more accurate to predict the position of the focal point.

## 2. Device design

The proposed lenses are designed based on GaN HCGs, which are subwavelength gratings comprised of high index GaN bars completely surrounded by low refractive index materials (air in this paper). Figure 1
shows the schematic geometry of the lens. A grating ridge array lies on the *x*-*z* plane.The grating period, thickness and fill factor (defined as the ratio of the grating bar width with respect to the grating period) are denoted by *P*, *T*, *F*, respectively, which are three structural parameters that determine the complex transmissivity of the HCG metasurface. By precisely manipulating the structural parameters, arbitrary wavefront control is possible. Considering the simplicity of final fabrication, the lenses in this paper are designed using non-periodic HCGs, which apply GaN-HCGs with different *P* and *F*, while *T* remains constant. The lenses are designed for a wavelength of λ = 460 nm and optimized for operation in transmission under normal transverse-magnetic (TM; with electric field perpendicular to the grating bars) polarized plane wave. In the analysis, the refractive indices of GaN (*n*
_{g}) and air (*n*
_{a}) are 2.47 and 1, respectively. The width of the lens, which is defined as the distance between the center of the two outermost slits, is denoted by 2*a*. The point *F* is the geometrical focus, and the designed focal length is *f*. In practice, however, because of the focal shift effect, the actual focus will be located at point *F’* with an actual focal length of *f’*. Thus we define the focal shift as the ratio of the deviation from the design to the designed focal length *f*, i.e. Δ*f/f* = (*f*-*f’)/f*.

Figures 2(a) and 2(b) show the calculated contours of the transmissivity and transmission phase as a function of grating period and fill factor for an HCG with thickness of 410 nm under normal incident TM-polarized light at a wavelength of 460 nm. The transmissivity and transmission phase contours are calculated using a rigorous coupled-wave analysis (RCWA) method [18]. The grating period varies from 260 nm to 460 nm, and the fill factor from 0.1 to 0.9. Broad high transmissivity domain is observed in Fig. 2(a). Accordingly, the transmission phase fully spans 2π phase difference, as shown in Fig. 2(b). This enables arbitrary wavefront control by precisely manipulating the spatial distribution of grating period and fill factor. While high transmissivity and 2π phase difference could also be found for thicknesses around 800 nm, gratings of this thickness are difficult to fabricate due to high aspect ratio. Therefore, the grating thickness is chosen to be 410 nm, which provides a broad area with high transmissivity and affords easy fabrication with high tolerance.

Applying grating period and fill factor variation along the subwavelength ridge array, we design our metasurface functioning as a focusing lens. According to geometric optics, in order to realize an ideally spherical wavefront for a point-like focus, the spatial phase retardation as a function of distance *x* from the center of the lens is given by

*λ*is the vacuum wavelength of the incident light,

*f*is the designed focal length,

*ϕ*(0) is the phase at the center of the lens, and the phase retardation of the plane wave propagating through the ridge located at

*x*is denoted by

*ϕ(x)*. For a focal length

*f*= 10 μm, the desired transmission phase retardation along the grating array determined by Eq. (1) is indicated by the red solid line in Fig. 3 . When the phase-difference Δϕ(

*x*) is more than 2π, it can be mapped to an equivalent value between 0 and 2π. Once the spatial phase retardation profile is determined, the geometry parameters of the grating bars at the corresponding position,

*i.e.*grating period and fill factor, can be retrieved from the transmission phase diagram shown in Fig. 2(b). The phase of each HCG bar with appropriate period and fill factor to approximate the required phase curve is indicated by the red circle in Fig. 3. Figure 3 also depicts the phase retardation of the desired lenses with focal length

*f*= 4 μm and 15 μm.

## 3. Simulation results

In order to investigate the origin of the focal shift in HCG lenses, the effect of the structural parameters including lens size, designed focal length, total phase difference is studied. The performance of all the HCG lenses is implemented using a finite element method (FEM). The simulation is two-dimensional (2D), assuming the length of the grating bars to be infinite. A plane wave with TM polarization is used as an input source. In all FEM simulations, the input source is a Gaussian beam with a waist same as 1/*e* of the lens size and is positioned at 0.8 µm below the lens. Figure 4
shows the simulation results of lenses with different sizes. In the simulations, all lenses introduce the same curvature to the incident plane wave, while the sizes of the lenses are controlled by adding/omitting grating bars with approximate periods on each side of the ridge array. Five different cases for lenses with focal length *f* = 10 μm were taken into account, described as cases *a*, *b*, *c*, *d* and *e* with lens sizes of 3.16, 4.98, 7.64, 11.69, and 16.05 µm, corresponding to total phase differences of 0.54π, 1.33π, 3.078π, 6.88π, and 12.30π, respectively. Figure 4(a) shows the E-field intensity distribution of the transmitted light for lenses with different sizes. For comparing the focal pattern, the field intensities are normalized with the maximum intensities of the foci, respectively. It can be seen that the actual focal length is to a large degree determined by the lens size. When the lens size increases, the actual focal point gradually moves away from the lens structure, approaching the designed value of 10 μm (indicated by the white line). This is in accordance with the reported results in the traditional dielectric lenses and planar plasmonic slit lenses [16, 19, 20
]. On the other hand, both the full width at half-maximum (FWHM) and depth of focus (DOF) decrease with increasing lens size, revealing that a more concentrated focal point, or a better focusing capability, can be achieved for a lens with larger size. These features can also be clearly seen in Fig. 4(b) and Fig. 4(c), which show the intensity distribution of the lenses across the actual focal point in the x direction and in the y direction, respectively.

The detailed parameters of the focusing performance of each lens, including the Fresnel number, actual focal length, focal shift, FWHM, and DOF, are presented in Table 2
. It shows that the achieved focal length increases from 4.51 μm to 9.20 µm when the lens size increases from 3.16 (case *a*) to 7.64 µm (case *c*), gradually approaching the designed focal length of *f* = 10 μm, thus the corresponding focal shift decreases from 54.9% to 8%. When the lens size increases further to 11.69 µm (case *d*), the achieved focal length will be much closer to the designed 10 µm and will not change much when the lens size continues to increase, as indicated in Table 2. Meanwhile, the FWHM (DOF) decreases from 695 nm (7.43 µm) to 297 nm (2.24 µm), which means that a focal spot of higher quality can be achieved at larger lens size.

The influence of the designed focal length on the performance of the HCG lenses are then investigated. The width of all the lenses is set as about 8 µm (for lenses with *f* = 4, 6, 10 and 15 μm, the widths are 8.05, 7.86, 7.64 and 7.84 µm, and corresponding phase differences of 7.29π, 5.10π, 3.07π and 2.19π, respectively). The simulation results are depicted in Fig. 5
. In order to compare the focus intensity, all the field intensities are normalized with the maximum intensity of the focus for lens with *f* = 4 μm. It shows that, with the increase of the designed focal length, the deviation of the actual focal length from the original design gets big, thus results in large focal shift. The maximum intensity at the focal point weakens, while both the FWHM and DOF increase, indicating larger focal spot size with lower focusing capability. Table 3
shows the detailed parameters of the focusing performance. The focal shift increases from around 1% to 15% when the designed focal length increases from *f* = 4 μm to *f* = 15 μm. The FWHM (DOF) increases from 252 nm (1.52 μm) to 630 nm (11.18 μm), while the maximum intensity at the focal point decreases from 2.84 to 1.88, which means that a focal spot of lower quality is achieved for lens with larger designed focal length.

Varying the total phase difference in the structures, HCG lenses with designed focal length of *f* = 10 μm were simulated. As shown in Fig. 6
, the actual focal length has a large deviation from the design for a lens with a small phase difference. In the case of a lens with total phase difference of 1.03π, the focal shift is about 32%. Increasing the total phase difference, the focal length gradually approaches the design (indicated by the white line), and becomes stable and almost equals to the designed value when the phase difference is larger than 8π. In the case of phase difference of 12.30π, the focal shift is about 0.2%. The increasing total phase difference provides an accurate focal length as predicted in the design. Figures 6(b) and 6(c) show the focal shift and FWHM, DOF as a function of the total phase difference for lenses with *f* = 10 μm. With increase of the total phase difference, the focal shift rapidly decreases and is less than 0.2% when the phase difference is larger than 8π. The FWHM and DOF also reduce, indicating more confined focus both in the focal plane and the light propagating direction for lens with larger phase difference.

## 4. Discussion

Based on the simulation results described above, a straightforward understanding is that the focal performance and focal shift are highly sensitive to the size of the focusing elements. A larger size produces a better focusing capability and a smaller focal shift. This is in accordance with previous works on dielectric lenses and planar plasmonic slit lenses [16, 19, 20
]. However, device size is useful to analyse a structure with a certain designed focal length (see Table 2), but is not sufficient to compare between lenses with different designed focal lengthes (see Table 3). Like in the previous literatures [19–21
], we introduce the so-called Fresnel number *N*

*N*is the dominant factor for the focal shift effect [19–21 ].

Actually, focal shift effect, the phenomenon that the point of maximum intensity of a diffracted field locates not at the geometric focus of the focusing element, is an attractive research topic all along and has been intensively studied [19–28 ]. It was firstly reported by Y. Li et al. [22] to investigate the behavior of a monochromatic converging spherical wave diffracted by a circular aperture. Previous research relevant to plasmonic lenses also encounters this problem [16, 20, 21, 23–25 ], in which the focal shift effect is especially obvious. The focal shift phenomenon in our HCG lenses behaves exactly the same way as that in the two cases mentioned above.

Using traditional diffraction theory and taking Fresnel number *N* into account, Y. J. Li [28] investigated the focal shift phenomenon in system when a monochromatic converging spherical wave diffracted by a circular aperture, and presented a formula to predict the focal shift,

*c*is approximated to be 1.51 according to numerical experiments.

In Fig. 7
, the focal shift of HCG lenses with *f* = 10 μm are calculated using the diffraction theory described above, together with the FEM simulation results as a comparison. It shows that, for small Fresnel number, the intuitive geometrical theory does not predict the actual focal length quantitatively. Instead, the diffraction theory aforementioned has a better accuracy in estimating the focal length. This is obvious since it is the diffraction effect around the boundary of the limited lens that leads to the discrepancy between actual focal length and the geometrical prediction. Smaller Fresnel number means stronger diffraction effect. For small Fresnel numbers, Eq. (3) is more accurate to estimate the position of the focal point, while for a lens with Fresnel number larger than 8, geometric optics is sufficient to predict the focal length desired.

## 5. Conclusion

In summary, both numerical simulation and theoretical analysis about the focal shift phenomenon in planar GaN HCG lenses are presented. The focal shift depends mainly on the Fresnel number of the lens. The influence of the lens size and designed focal length can be attributed to the change of Fresnel number. Increasing the lens size or decreasing the designed focal length will increase the Fresnel number, and thus reduce the focal shift. For a lens with large Fresnel number, geometric optics is sufficient to predict the focal length desired. However, for a small Fresnel number, diffraction theory is more accurate to estimate the position of the focal point. In addition, the total phase difference of the lens is nearly equal to the Fresnel number multiplied by π. At least 8π phase difference is needed for a HCG lens to obtain a well-defined focal length.

## Acknowledgment

This work is supported by Natural Science Foundation of Jiangsu Province of China (Grant No. BK20130870, BK20130859), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 13KJB510019), and NUPTSF (Grant No. NY213008, NY213010, NY214030, NY214027).

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