1. Introduction

High-efficiency resonant reflection is generated by periodic particle arrays when an incident beam of light couples to a leaky Bloch mode [1–8]. These modes are excited by evanescent diffraction orders in the subwavelength regime. The applicable precise descriptor of this effect is “guided-mode resonance (GMR)” as first applied in 1990 [8]; in earlier research this phenomenon was sometimes called “anomalous reflection” [3]. The spectral properties of periodic metasurfaces, metamaterials, and photonic crystals operating under broadside incidence are governed by the fundamental physics ascribed to the guided-mode resonance effect [9]. This resonance effect is implementable in one-dimensional (1D) and 2D periodic layers across all spectral regions with pertinent low-loss materials. While the basic resonance physics is most straightforwardly understood with 1D optical lattices, it generalizes well to the 2D case by direct extension.

Particularly during the past 20 years, clusters of research groups around the world appear to have been unaware of prior work in this field. They have concocted new explanations of the operational basis while retaining all principal observations including high-efficiency, or “perfect,” reflection as well as the host of conceivable applications previously also identified. The primary misstep in these alternate explanations is attribution of the dominant effects of the assembly of particles constituting a lattice to properties and characteristics of local resonances in the isolated building-block particles. The authors of these works generally do not mention lateral modes and evanescent diffraction orders are missing as well.

Reviewing briefly, in an attempt to explain wideband reflection by high-contrast gratings, it was posited that perfect reflection was due to two ± z propagating (Fig.  1) waveguide modes residing in the grating ridges and interfering at the output high-contrast interface to cancel the transmitted wave [10–12]. Objecting to this hypothesis, it was shown that high reflection was retained on elimination of the high-contrast output interface and that the generation of a wave vector directed along –z is a diffractive effect and not related to reflection off a discrete interface [13]. Moreover, it was shown that sparse dielectric nanowire grids capable of supporting only a single ridge mode yielded wideband perfect reflection as well [14] leading to the conclusion that interference between two coexisting ridge modes is neither necessary nor causal. In addition, there is a substantial and growing body of literature citing particle-based Mie resonance as the fundamental effect enabling perfect metamaterial reflectors [15–18]. Objecting to this view, we demonstrated that Mie scattering in individual array particles is not a causal effect. By connecting the constituent lattice particles by a matched sublayer and thereby destroying the Mie cavity, we find that perfect reflection persists and the resonance bandwidth actually expands even though localized Mie resonances have been extinguished [19].

 

Fig. 1. (a) Schematic of a 2D photonic lattice composed of nanospheres where n and D label refractive index and diameter of the sphere. Each nanosphere is arranged by the lattice period (Λ = Λ_x = Λ_y). For the input beam, we use a plane wave at normal incidence with its electric-field vector along the y-axis while the magnetic-field vector is along the x-axis. (b) Total scattering efficiency (Q_scatt) spectrum of the single Si sphere (n = 3.48, D = 450 nm). At (i) λ = 1.634 µm (M₀), (ii) λ = 1.25 µm (M₁) and (iii) λ = 1.13 µm (M₂), Mie resonances appear identified as classic magnetic dipole, electric dipole, and magnetic quadrupole response, respectively. (c) Electric and magnetic field profiles of M₀, M₁ and M₂ where E and H indicate the amplitudes of electric and magnetic fields.

Download Full Size | PDF

In a few exceptional cases, authors of papers grounded in the Mie resonance viewpoint acknowledge the effects of the lattice explicitly. Babicheva and Evlyukhin study 2D periodic arrays of spherical silicon nanoparticles in the visible region [20]. It is stated that such arrays “…support lattice resonances near the Rayleigh anomaly due to the electric dipole (ED) and magnetic dipole (MD) resonant coupling between the nanoparticles.” Abujetas et al. similarly note that the “…properties of metasurfaces stem not only from the resonant properties of the meta-atoms themselves, but also from multiple scattering effects through coupling with guided-mode or lattice resonances.” [21] Additionally, it is sometimes contended that full, rigorous electromagnetic formalisms such as rigorous coupled-wave analysis (RCWA) [22] fail to bring forth the physics underlying resonant array functionality [21].

Whereas these works [20, 21] are valuable contributions to the field, we do not agree with some of their conclusions. First, as shown here and in many of our prior works [6, 9, 19], lattice resonances do not necessarily reside near the Rayleigh anomaly but appear generally at wavelengths away from it. The resonance wavelength is not specifically connected to the Rayleigh wavelength. Second, we feel that lattice resonance described as arising via “resonant coupling” between particles does not reveal the true resonance physics. The full description is in terms of evanescent-wave-excited lateral Bloch modes as provided in the present contribution. Third, as we show here and many times in the past, full numerical RCWA treatment is capable of bringing out all key features of the various forms of resonant lattices including the one treated presently.

Accordingly, in this article, with focus on the spectral properties of resonance reflection, we treat a classic 2D periodic array consisting of spherical particles or nanospheres. To disable Mie resonance, we apply an optimal antireflection (AR) coating to the spheres. We compute reflectance maps for coated and uncoated spheres and show that perfect reflection persists across sizeable bandwidths in both cases. We compute Mie-resonance spectra of both particle types and quantify the reduction in the Mie scattering efficiency for the AR-coated spheres. The reflectance properties of AR-coated spherical arrays have not appeared in the literature previously. From this novel viewpoint, these results illustrate high-efficiency resonance reflection in Mie-resonance-quenched particle arrays. In the explosive deployment and analysis of strictly periodic optical lattices, it is of fundamental importance that the attendant published works contain the proper, correct physical explanations of the root causes of the observed spectra and local electromagnetic field distributions. This work is a contribution to substantiate the proper physical description of these key effects.

2. Arrays of uncoated silicon spheres

We model a periodic array composed of nanospheres where n and D denote refractive index and diameter as illustrated in Fig.  1(a). The particles are arrayed in a 2D lattice with period Λ embedded in air with refractive index n_air = 1. The input beam is modeled as a plane wave at normal incidence with fixed polarization (electric-field vector along the y-axis while the magnetic field points along the x-axis). Figure  1(b) shows the total scattering efficiency (Q_scatt) of an isolated silicon sphere with a fixed, nondispersed refractive index n = 3.48 and diameter D = 450 nm. It is calculated using analytical formulas from Mie scattering theory as [23]

We calculate reflectance spectra of this photonic lattice versus Λ by performing rigorous coupled-wave analysis [22]. When this lattice is in the subwavelength regime, only zero-order reflectance (R₀) remains and perfect reflection is possible. Figure  2(a) shows the R₀ color map as a function of Λ from 0.5 to 2 µm. For comparison, the wavelengths corresponding to M₀, M₁ and M₂ are also indicated by vertical lines. The perfect reflection loci (displayed in dark red color) are controlled by the period of the lattice. This is because the period strongly affects the homogenized effective-medium refractive index of the lattice which, in turn, defines the character and properties of the lateral leaky Bloch modes [19, 24]. As Λ increases, the reflection band changes in resonance wavelength position up to the Rayleigh line (Λ=λ, displayed by a white dashed line). Beyond the Rayleigh line, R₀ < 1 because higher propagating diffraction orders draw power. For points (λ, Λ) under the Rayleigh line, subwavelength conditions prevail and no higher-order diffracted waves propagate external to the lattice. We note that the Mie lines are not correlated directly with the reflection band.

 

Fig. 2. Perfect reflection bands generated by a resonant photonic lattice. (a) Calculated R₀ spectral map as a function of Λ. For comparison, Mie resonance locations (M₀-M₂) are displayed by vertical lines. The white dashed line indicates the Rayleigh line (Λ=λ). (b)-(d) Representative E and H profiles associated with the photonic lattice at (i) Λ = 1.25 μm, λ = 1.522 μm, (ii) Λ = 1.25 μm, λ = 1.372 μm and (iii) Λ = 1.58 μm, λ = 1.62 μm. Shown are localized fields at the nanospheres on a linear scale (upper plots) as well as standing-wave interference patterns of lateral counter-propagating Bloch modes on a log scale (lower plots) at these points as marked in (a). Point (i) and fields in (b) pertain to TM₀ modes propagating along ± y. Point (ii) and fields in (c) correspond to TE₀ modes propagating along ± x. Point (iii) locates in the reflection null on the merged mode line where these TM₀ and TE₀ modes overlap. The fields in (d) correspond to this point. The lateral Bloch modes exhibit standing waves along the x direction (TE modes) and y direction (TM modes). These modes travelling along orthogonal directions are copolarized for the excitation shown in Fig.  1 and thus the reflected waves due to each mode can interfere effectively to cancel the reflection and to yield perfect transmission at that point as shown in the spectrum. We note that the local fields in (d) are those of the overlapping TE₀ and TM₀ modes and cannot represent these modes individually. Additionally, we note that the log-scale amplification of the standing waves makes them look artificially sharp.

Download Full Size | PDF

Figures  2(b)–2(d) provides representative electric (E) and magnetic (H) field profiles associated with the photonic lattice at (i) Λ = 1.25 μm, λ = 1.522 μm, (ii) Λ = 1.25 μm, λ = 1.372 μm and (iii) Λ = 1.58 μm, λ = 1.62 μm. These panels illustrate the localized fields at the nanospheres on a linear scale (upper plots) as well as the standing-wave interference patterns of the lateral counter-propagating Bloch modes on a log scale (lower plots) at these points as marked in Fig.  2(a). Point (i) and fields in Fig.  2(b) pertain to TM₀ modes propagating along ± y. Point (ii) and the fields in Fig.  2(c) correspond to TE₀ modes propagating along ± x. Point (iii) locates in the reflection null on the merged mode line where the TM₀ and TE₀ modes overlap. The fields in Fig.  2(d) correspond to this point. The coexisting Bloch modes exhibit standing waves along the x direction (TE mode) and y direction (TM mode). These modes travelling along orthogonal directions are copolarized for the excitation field shown in Fig.  1 that has an electric-field vector along the y direction. Therefore, the reflected waves due to each mode can interfere effectively to cancel the reflected wave and to yield perfect transmission at that point as shown in the spectrum. In [20], a reflection minimum, analogous to that at point (iii) here, is attributed to a Kerker effect.

3. Arrays of AR-coated silicon spheres

Now we apply a single-layer antireflection (AR) coating to the silicon particles to diminish the local cavity by suppressing surface reflections at the high-index air-Si interface. Perfect quarter-wavelength single-layer AR pertains to a flat surface at a single wavelength. Here, we approximate the AR-coated particle first by applying films with thickness d=λ/4n₂ and index n₂. The layer model and reflectance map are shown in Fig.  3(a). There appears a good AR region at the quarter-wave thickness as measured by the scale bar in Fig.  3(a). The model for the AR-coated sphere and a map of the backscattering efficiency response Q_back are shown in Fig.  3(b) where the d is the thickness of the AR shell. The backscattering efficiency is given by [23]

 

Fig. 3. Quenching of Mie resonance with AR coatings. (a) Classic thin-film AR effects. Thickness and refractive index of the inner layer are the same as for the Si sphere (D = 450 nm and n₁ = 3.48) with ${n_2} = \sqrt {{n_1}} \approx 1.865$. The R₀ map is calculated by varying the thickness of the AR layers denoted by d. The white dashed line indicates the quarter-wavelength locus ($d = \lambda /4{n_2}$). (b) Structure of the AR-coated sphere where the D and d are defined by r₁ (inner radius) and r₂ (outer radius). Refractive indices (n₁ and n₂) of the core and shell are the same as in (a). The color map represents Q_back spectra as a function of d.

Download Full Size | PDF

Figure  4 confirms that Mie scattering resonance is subdued by the AR effect where the AR film thickness (d = 220 nm) is chosen for the wavelength of M₀ (λ = 1.634 µm) of the original Si sphere. Viewing the computed fields in Fig.  4(a), the localized signature is greatly diminished relative to the strongly-confined fields in the uncoated sphere as in Fig.  1(c). Owing to the AR effect and attendant low backward scattering, the light passes through the particle with dominant forward scattering. In addition, as seen in the Q_scatt spectrum of Fig.  4(b), the two resonance peaks from Fig.  1(b) are significantly broadened and the total scattering efficiency is reduced. As expected, relative to the M₀ position of the original uncoated Si sphere, the Mie scattering resonance is red-shifted and the quality (Q) factor of the resonance decreases.

 

Fig. 4. Effects of AR coating on the scattering properties of silicon spheres. (a) Field profile at λ = 1.634 µm corresponding to the M₀ position of the bare nanosphere in Fig.  1(b) and the AR-coating design wavelength. (b) Q_scatt spectra of the AR-coated Si sphere.

Download Full Size | PDF

Finally, we demonstrate perfect reflection with the AR-coated Si spheres in the photonic lattice. Thus, Fig.  5(a) shows a calculated R₀ (λ, Λ) map for a 2D array of spheres configured as in Fig.  4(a). Two perfect reflection bands appear in the spectral region displayed contributed by lattice-generated guided-mode resonances. The appearance of a reflection null near λ ∼2 µm and Λ ∼1.8 µm in Fig.  5(a) can be explained similarly to the analogous null at point (iii) in Fig.  2(a). As can be seen in the field profiles of Fig.  5(b), standing wave patterns form due to interference between counterpropagating lateral Bloch modes. Lattice resonance induces strong optical confinement even if the local cavity of each individual particle is eliminated. This is because the lattice, in spite of the AR coat on the particles, still exhibits finite values of effective refractive index on which to support the modes. We remark that the AR-coat design wavelength is 1.634 μm and efficient reflectance occurs near this value of wavelength.

 

Fig. 5. Perfect reflection bands generated by a 2D resonant photonic lattice of AR-coated Si spheres. (a) R₀ (λ, Λ) map. (b) Field profiles at point (i) Λ = 1.05 µm, λ = 1.634 µm and (ii) Λ = 1.48 µm, λ = 1.484 µm. For point (ii) close to the Rayleigh line, the field extends significantly out of the guiding lattice as seen in the figure.

Download Full Size | PDF

As seen in Fig.  3(b), the backscattering efficiency of an isolated, coated nanosphere is reduced significantly near the AR design wavelength even though the quarter-wave AR concept belongs to the domain of flat thin-film optics. Concomitantly, we see in Fig.  4(b) that the total scattering efficiency Q_scatt is significantly reduced relative to that of the uncoated sphere. Even in spectral bands with zero reflectance in the case of AR-coated substrates, there are still electromagnetic processes occurring providing interference effects that work to sustain zero and low reflectance in a neighborhood around the design wavelength. Similarly, the localized Mie resonances in the AR-coated sphere still exist to support the extensive, low backscattering efficiency Q_back that is shown in Fig.  3(b). To quantify and compare the contributions of the individual Mie dipoles and quadrupoles, we compute the effect of these distinct Mie resonances on the total efficiency. Figure  6(a) provides the results for the uncoated model sphere displaying explicitly the contributions of the MD, ED, and MQ to Q_scatt. For the AR-coated sphere in Fig.  6(b), Q_scatt is relatively lower and now built with contributions from four Mie resonance. We note that the amplitudes of these resonances are all lower than that of the lowest resonance (ED) in Fig.  6(a). The more complex appearance of the Mie resonances for the AR-coated sphere, in particular the near overlap of ED and MD as well as of EQ and MQ, may arise from the fact that the coated sphere contains effectively four cavities, or potential resonance paths, versus the single cavity of the uncoated sphere.

 

Fig. 6. Multipole decomposition and corresponding total scattering efficiency of uncoated and AR-coated silicon spheres. (a) Isolated bare sphere with n=3.48 and D=450 nm. (b) The same sphere with a quarter-wave AR coat centered at λ = 1.634 µm which is the peak wavelength of M₀ in (a). Shown are spectra for the magnetic dipole (MD), electric dipole (ED), magnetic quadrupole (MQ) and electric quadrupole (EQ).

Download Full Size | PDF

4. Conclusions

In summary, we treat fundamental aspects of resonant reflection induced by periodic photonic lattices. Of chief interest is the behavior of the optical lattice relative to the Mie resonance response of the building-block particles composing the lattice. We select a classic 2D lattice of Si spheres for the study as the Mie resonance response of spherical particles has been known for more than 100 years [25]. Coating the Si sphere with an AR film optimized at the bare-sphere magnetic-dipole wavelength, we quantify greatly diminished Mie-scattering efficiency and Q factor. On forming a lattice with the AR-coated spheres, we find extensive spectral regions exhibiting perfect reflection. Illustrative comparisons of the near-field response of coated and uncoated single particles with the lattice-induced near fields are provided. Thus, a clear differentiation of Mie resonance and guided-mode resonance in mediating perfect reflection in periodic particle assemblies is provided. The clarity and simplicity of the present exposition may contribute to dispelling misconceptions on fundamental operational principles of resonant optical lattices.