## 1 Introduction  
Metamaterials have stimulated significant research activity in the field of metallic micro- and nanostructures, from the viewpoints of their exotic electromagnetic responses, including magnetic resonance [1] and negative refraction effect [2, 3]. Generally, three-dimensional photonic metamaterials operating at optical wavelengths are in high demand in nanofabrication, except for multilayer structures [4, 5, 6, 7, 8, 9]. Therefore, photonic metamaterials in experiment have been mostly limited to single-layer or a-few-layer stacking structures [10, 11]. Photonic metamaterials working at optical wavelengths have been extensively investigated to realize unconventional or extreme optical constants [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], such as a negative refractive index. They were mostly designed and fabricated using noble metals, such as gold [12, 13, 14] and silver [15, 16, 17, 18, 19, 20, 21, 22]. Recently, metasurfaces which were originally conceived as single-layered metallic structures [23, 24, 25] have been attracting great interest. To date, the diversity includes a-few-layer surface structures on or connected to bulk substrates [26, 27, 28, 29, 30, 31, 32, 33]. Many of the numerous papers on metasurfaces have been complied in review papers [34, 35]. The single-layer metasurfaces tend to have more elaborated structures than most photonic metamaterials reported so far. Currently, all-dielectric metasurfaces are a significant subgroup of metasurfaces [36]. One of their features is a series of Mie resonances induced in high refractive index nanostructures [37, 38]. Thus, these metasurfaces were found rich in their physical properties. Further diverse optical functions, such as phase-and-polarization control [39], flat lenses [40, 41, 42, 43], and holograms [44, 45], have been realized based on the dielectric metasurfaces. In addition to the search for novel optical devices and light-wave manipulation, light-matter interaction in artificial configurations has been attracting growing interest, ranging from fundamental investigations to practical applications. One of the practical issues is fluorescence (FL) enhancement because FL-intensity enhancement is directly linked to diverse FL-sensing applications in biotechnology. To date, several papers [46; 47; 48; 49] have reported on highly successful FL-intensity enhancement exceeding two-thousand-fold in sparsely FL-molecule dispersed configurations, among the numerous papers on FL enhancement. These very large FL enhancements were achieved by exploiting metallic metasurfaces. To the best of our knowledge, there are no published reports on highly FL-enhancing dielectric metasurfaces in FL-molecule sparsely dispersed configurations. However, it was reported that photonic crystal slabs enabled hundred-fold FL enhancement [50], and that wavelength-selective photoluminescence (PL) extraction from semiconductor quantum dots (QDs) can be very effective in a dielectric metasurface [51]; thus, an opportunity exists to explore highly FL-enhancing dielectric metasurfaces. Here, we aim at examining Si-rod-array metasurfaces in terms of FL enhancement. We note that, at present, there is no theory that is able to predict highly FL-enhancing artificial nanostructures; therefore, experimentis the most direct and reliable way to find them. Before delving into the details of this investigation, we begin by describing the typical properties of a Si-rod-array metasurface, and briefly survey the main characteristics. Figure 1 illustrates a typical metasurface of cylindrical Si rods with Mie resonances [52; 53]. Figure 1a illustrates a metasurface on a SiO\({}_{2}\) substrate that is transparent in the wavelength range of interest. The unit structure is a cylindrical Si rod of diameter \(D\) and height \(H\), and arrayed with a periodic length \(P\) in a square lattice. The periodic length \(P\) is defined to be the center-to-center distance between the nearest-neighbor Si rods.