1. Introduction

Dielectric metamaterials consisting of periodic nanoparticles or subwavelength structures on or within dielectric materials have drawn significant attention in the last decade [1–13] owing to their extraordinary features, such as negative index [1–3], directional scattering [4–6], ultra-thin flat lenses [7–10], and perfect mirrors [11,12], which cannot be realized with ordinary materials in nature. Unlike metals, dielectric materials with a high refractive index and a low extinction coefficient suppress light dissipation and exhibit strong resonances in arrays of dielectric nanoparticles, depending on the period of lattice or polarization of the incident light, leading to fluorescence enhancement [4,13], optical sensors [14–16], or color filters [17–20].

The electric and magnetic resonances in a spherical dielectric particle are calculated using the Mie theory from which electromagnetic fields are expressed as a series of electric dipole (ED), magnetic dipole (MD), and higher-order excitations [21] and can be approximated to dipole (or quadrupole) terms for nanoparticles [22–24]. The periodic arrays of spherical nanoparticles can be rationally treated as a set of dipole radiators, and the total dipole moments of the arrays are obtained by the electromagnetic fields contributed by each dipole, known as the coupled-dipole method [25]. Meanwhile, the resonance can also be interpreted as leaky Bloch modes at which the incident light is coupled with the guided mode under resonance conditions, known as guided-mode or leaky-mode resonance [26,27].

To the best of our knowledge, the first experimental demonstration was reported in [18,19], where the square lattice of Si nanostructures on a quartz substrate exhibited reflective color filters with high angular tolerance, and the resonance effects were described in terms of guided-mode waves [28]. The optical resonances in nanoparticle arrays have led to a great interest in the area of dielectric metamaterials from which the various optical properties induced by the ED and MD resonances were studied in accordance with the types of particle arrays, period of nanoparticles, and light polarization [24–27,29–32].

The resonance conditions of nanoparticle arrays with a square lattice can be tuned by varying the periodicities of nanoparticles at which significant lattice resonances occur near the wavelength of diffraction, known as a Rayleigh anomaly (RA) [24,27,32]. For normal incident light, each dipole mode (ED or MD) is separately coupled with different periodicities in the x- and y-axis directions, and the far-field resonance effects are explored with respect to the RA conditions of the grating [24,32]. The resonances can also be achieved by changing the incident angle with a constant period over the lattice, where the same ED or MD resonances are induced at certain incident angles of polarized light [27].

Herein, we have discovered the internal structures of guided-mode waves related to ED and MD resonances in a slab of dielectric nanostructures induced by polarized light at different incident angles. We also explored the conditions of the resonance appearance or disappearance in the slab through a theoretical analysis of the energy fluxes associated with the modes. The analysis provided clear interpretations of unique optical properties in a slab of dielectric nanostructures, where the symmetry of resonance modes contributes to significant effects in the stability of the resonance modes at the oblique incident of light and the stability could be analyzed by the components of Poynting vector associated with the resonance modes near the RA conditions. Specifically, we revealed that the MD resonance excited by light with TM polarization was attributed to the guided-mode wave with the energy flux in the opposite direction of the incident light, which could be effectively interpreted as a negative refractive index. Meanwhile, the MD resonance excited by light with TE polarization was attributed to the guided-mode wave with energy fluxes of two components that were almost orthogonal to the incident plane. The ED resonance was attributed to the guided-mode waves with energy fluxes in both directions––that is, in-plane and out-of-plane, for both TE and TM incident lights. The results were supported by experimental demonstrations of optical effects by RA conditions for various incident angles of TE- and TM-polarized light. These results imply the potential applications for designing of optical metasurfaces, color filters in display applications, optical switches, and high-performance sensors utilizing the polarizations.

2. Guided-mode waves in two dimensional arrays on a substrate

The arrays of nanoparticles or nanolayers with a high refractive index can be considered a wave-guiding layer with an effective refractive index n_eff. When n_eff is larger than the refractive indices of both the upper layer n_o and the bottom substrate n_s, the incident light can be trapped and guided in the layer (see Fig. 1(a)) because of the diffraction and excitation of the guided mode propagating inside the slab. The guided-mode wave with propagation vector q close to $2\pi /L$ (where L is the period of the array) can be excited by the first-order diffraction of incident light at normal incidence. The excitation of the guided modes and the trapping of light into the slab result in resonances in the reflection and transmission spectra, which can be used for various applications [14–20].

 

Fig. 1. Dispersion curves of ω(q) for a propagating vector q and guided modes in a planar layer (a) and in periodic arrays (b), where L, n_o, n_s, n_H, and n_L are the period and the refractive indices of the air, substrate, and high and low index media, respectively. The modes near q∼2π/L can be excited by the incident light through first-order diffraction.

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The symmetry condition for the forward and backward waves induces a bandgap, as shown in Fig. 1(b). For lower band energy, a light field is accumulated in a material with a high refractive index [33], which promotes electric or magnetic field enhancement [18]. As the period of L increases, the guided-mode waves can leak out to the substrate when $|q |\le ({2\pi /\lambda } ){n_s}$, which leads to the critical distance of the mode disappearance

The guided-mode waves in a slab of 2D arrays of nanoparticles have more complex structures than those of 1D linear arrays [18,19]. For an oblique incident angle, the guided modes may propagate not only in the forward and backward directions but also in orthogonal directions, resulting in resonances in the reflection and transmission spectra with several unexpected features. In particular, 2D arrays of dielectric particles with a high refractive index exhibit bandpass resonances in the visible wavelength with high angular tolerance [18,19] at which the guided modes associated with the ED and MD resonances can explain the reasons for the high incident angle tolerance and provide more knowledge on optical energy flows propagating inside a slab.

Let us consider 2D arrays of dielectric particles with a high refractive index on a substrate with different periods in the x- and y-directions: L_x and L_y (Fig. 2). The incident light at an angle θ generates electric and magnetic fields in the slab, which can be represented as a combination of spatial harmonics.

 

Fig. 2. (a) Schematic of 2D arrays of a nanostructure on a substrate illuminated by incident light. The wavy arrows illustrate guided-mode waves in a slab of nanostructures. L_x and L_y are the periods of the nanostructures. The TM and TE polarizations are defined as in (b) and (c).

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If the refractive index of the nanoparticles is slightly larger than that of the ambient medium, the slab of nanoparticles can be considered a medium with low perturbation of the refractive index $\varDelta n/n < < 1$. The electric and magnetic fields generated by incident light in slab ${\overrightarrow E ^{(m,m^{\prime})}}(z)$ and ${\overrightarrow H ^{(m,m^{\prime})}}(z)$ can be expressed as a combination of forward and backward waves:

The vector q = $(q_x^m,q_y^{m^{\prime}})$ is the wave vector that defines the direction of the phase propagation of each eigenmode of (m,m’) and does not need to coincide with the direction of energy propagation in the medium. Indeed, the energy flux of the guided-mode wave is a combination of fluxes of the waves propagating along the forward “+” and backward “–” waves along the z-direction and is defined as the Poynting vector:

Thus, the eigenmode ${\overrightarrow E _p}$ is contributed to by partial waves with various wave vectors ${\overrightarrow q ^{(m,m^{\prime})}}$ with $q_x^{(m)}$ and $q_y^{(m^{\prime})}$ as defined by Eq. (2): however, the energy flux of each partial component is defined as ${\overrightarrow S ^{({\pm} )(m,m^{\prime})}}$. Moreover, the eigenmode ${\overrightarrow E _p}$ in Eq. (9) becomes a guided-mode wave when the ${\overrightarrow q ^{(m,m^{\prime})}}$ in the plane components satisfies the total internal reflection condition.

 

Fig. 3. Propagation vectors q of guided-mode waves which can be excited in the slab by incident light due to first order of diffraction. The blue and red arrows correspond to the waves excited in the incident plane due to diffraction in the x-direction (i.e., m = ±1, m′ = 0) and in the y-direction (i.e., m = 0, m′ = ±1), respectively

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Thus, the guided-mode waves having partial components (m=±1, m’ = 0) (we will use further notation (±,0) for short) disappear for L_x above $RA_x^{( + )}$ (or $RA_x^{( - )}$). Moreover, if L_x exceeds RA, then

Similar RA conditions for the (0,±) partial components of the wave vectors, ${q^{(0, \pm )}} = {k_\parallel } \pm 2\pi /{L_y}$ propagating in the orthogonal y-direction are obtained from the relations of Eq. (13) as follows:

Let us further consider the guided-mode waves associated with MD and ED resonances in detail. Figure 4 shows the simulation results of reflectance intensity maps in the visible wavelength with respect to the variations of L_x and L_y, which were obtained using the rigorous coupled-wave analysis (RCWA) method [35]. TM-polarized light was set to illuminate the 2D periodic nanoparticle arrays at an incident angle of 10°. The geometries and refractive indices of the materials are described in the figure captions. In Fig. 4(a), the resonance modes of ED and MD were identified at ∼550 nm and ∼650 nm, respectively and were affected by the $RA_x^{( - )}$ and $RA_x^{( + )}$ lines of Eq. (14). Notably, the resonance intensity of the MD mode was completely suppressed above the $RA_x^{( - )}$ line, whereas the resonance intensity of the ED mode was totally suppressed immediately above the $RA_x^{( - )}$ line but existed again in the position between the $RA_x^{( - )}$ and $RA_x^{( + )}$ lines. This difference implies that the internal structures of the resonance modes of ED and MD at oblique incidences of light differed completely from those in the normal incident case. The higher excitation modes of the electric quadrupole (EQ) and magnetic quadrupole (MQ) also existed near λ = 450–500 nm: these modes were not considered in this work but rather focused on the ED and MD modes.

 

Fig. 4. Reflectance intensities in 2D arrays on a substrate with variations of period, L_x in (a) and L_y in (b), while the other period of orthogonal axis is fixed to 300 nm. The TM polarized light is illuminated with an incident angle of θ = 10°. The periodic pattern sizes are d = h = 150 nm with a refractive index of n = 4. The refractive index of the substrate is n_s = 1.45. Colored dashed lines represent RA conditions defined in Eqs. (14)–(17).

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Figure 5 represents the partial components of Poynting vectors for MD [(a) and (b)] and ED [(c) and (d)] resonances, which were estimated at the positions of white dots with λ = 658 nm, L_x = 300 nm and λ = 545 nm, L_x = 300 nm, respectively, as shown in Fig. 4(a). The magnitudes of the components of Poynting vectors, averaged to $S_x^{(m,m^{\prime})} = ({S_x^{( + )(m,m^{\prime})} + S_x^{( - )(m,m^{\prime})}} )/2$ and $S_y^{(m,m^{\prime})} = ({S_y^{( + )(m,m^{\prime})} + S_y^{( - )(m,m^{\prime})}} )/2$ are denoted as points with different sizes. The relative strengths can also be referred to with the color legend bar in each plot. In Figs. 5(a) and (b), the principal components of the Poynting vectors for the MD resonance are $S_x^{({ - ,0} )}$ = -0.62S_o and $S_x^{({ + ,0} )}$ = 0.24S_o, where S_o is the Poynting vector of the incident light. The other (m,m’) partial components are insignificant owing to the multiple scattering of the light inside the slab. This indicates that the guided-mode associated with MD resonance propagates in the x-direction and thus the resonance can be excited only at the condition of L_x below$\; RA_x^{(- )}$, but independent of the period of L_y as strongly supported by the 2D maps in Figs. 4(a) and (b). Meanwhile, Figs. 5(c) and (d) show that the ED resonance has several principal components of the Poynting vectors, ${\vec{S}_1} = S_x^{({ - ,0} )}{\vec{e}_x}$ along x direction with relative amplitude $S_x^{({ - ,0} )} ={-} 0.8{S_o}$, being the opposite in-plane direction to the incident light, and two partial components being almost orthogonal to the incident plane ${\vec{S}_2} = S_x^{({0,1} )}{\vec{e}_x} + S_y^{({0,1} )}{\vec{e}_y}$ and ${\vec{S}_3} = S_x^{({0, - 1} )}{\vec{e}_x} + S_y^{({0, - 1} )}{\vec{e}_y}$ with relative amplitudes of $S_x^{({0, \pm 1} )} = 0.04{S_o}$ and ${\; }S_y^{({0, \pm 1} )} = 0.31{S_{o,}}$ respectively. Thus, the ED mode could be excited in both x and y-directions under the conditions of L_x below$\; RA_x^{(- )}$, and L_y below RA_y, as represented in Figs. 4(a) and (b).

 

Fig. 5. Components of Poynting vectors normalized to the incident light, $S_x^{(m,m^{\prime})}$ in (a), (c) and $S_y^{(m,m^{\prime})}$ in (b), (d) of guided-mode waves excited in the slab by the incident wave of TM polarization, which are calculated at the wavelengths λ = 658 nm (MD mode) in (a) and (b), λ = 545 nm (ED mode) in (c) and (d) with L_x = L_y = 300 nm. The incident angle of light is θ = 10°. Each component is normalized to the incident wave flux. The sizes of the circles correspond to the intensity of $S_x^{(m,m^{\prime})}$ and $S_y^{(m,m^{\prime})}$. Geometrical parameters of the slab are the same as Fig. 4.

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The internal structures of the resonance modes corresponding to the three white dots in Fig. 4 are further examined in Fig. 6, where optical energy flow at the resonance in [(a), (d), (g)], 3D cross-sectional plots in [(b), (e), (h)], and schematics of the principal component plots in [(c), (f), (i)] are compared. All the energy flows and 3D field plots were obtained with the RCWA method [35], where the electric and magnetic fields in a unit cell were calculated and converted to field vectors and magnitude plots. Figures 6(a)–(c) show the MD resonance at λ = 658 nm with L_x = L_y = 300 nm. Figures 6(d)–(f) show the ED resonance at λ = 545 nm with L_x = L_y = 300 nm. Figures 6(g)–(i) show the ED resonance at λ = 560 nm with L_x = 413 nm, L_y = 300 nm, and right below $RA_x^{( + )}$ line in Fig. 4(a). The H and E fields are localized at the center of the nanostructure in Figs. 6(a), (d), and (g), which demonstrate the resonance MD and ED modes. The optical energy flow $\vec{S}$ (averaged over y-direction) of the guided- mode wave associated with MD and ED resonances below the $RA_x^{(- )}$ line shows in the opposite direction to the parallel component of the wave vector, ${k_\parallel }$, which is consistent with the analysis of partial components in Fig. 5. Excitation of these guided-mode waves in the slab in the opposite direction of energy flow can be interpreted as negative refractions in metamaterials of nanostructures. The principal partial components of excited guided-mode waves associated with these resonances are schematically indicated by the red waved arrows in Figs. 6(c) and (f).

 

Fig. 6. Optical energy flow maps in [(a), (d), (g)], cross-sectional plots of 3D field intensities normalized to incident light in [magnetic field (b), electric field (e), (h)], and principal components of guided-mode waves in [(c), (f), (i)] of resonance modes in the slab excited by incident light with TM polarization. Parameters in (a)–(c) correspond to MD resonance at λ = 658 nm (L_x = L_y = 300 nm). Parameters in (d)–(f) correspond to ED resonance at λ = 545 nm (L_x = L_y = 300 nm). Parameters in (g)–(i) correspond to ED resonance at λ = 560 nm (L_x = 413 nm, L_y = 300 nm). The incident angle of light is θ = 10°. Geometrical parameters of the slab are the same as Fig. 4.

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Figure 6(h) represents the optical energy flow $\left\langle {\overrightarrow S } \right\rangle $ averaged over the y-direction of the guided-mode wave associated with the ED resonance at $RA_x^{( - )}$ < L_x < $RA_x^{( + )}$. Because L_x lies above $RA_x^{( - )}$, the ED mode could not hold the (-1,0) component in the structure, resulting in the disappearance of the ED resonance right above the $RA_x^{( - )}$ line. However, as L_x increases further, the ED resonance appears immediately below the $RA_x^{( + )}$ line, which corresponds to guided-mode waves with a (+,0) component in the structure. Thus, this resonance is the result of excitation of the guided-mode wave flowing in the same direction as the parallel wave vector, called the “direct mode”, as represented in Fig. 6(g). Whereas the resonance modes of ED and MD in Figs. 6(a) and (d) are excitations of the guided-mode wave flowing in the opposite direction as the parallel wave vector, called “opposite modes”.

Figure 7 shows the simulation results of the reflectance intensity maps for TE-polarized light, where the same geometries and parameters in Fig. 4 were utilized. The ED and MD resonances exist near 550 nm and 650 nm, respectively, similar to the results in Fig. 4. However, owing to the orthogonal relations between polarizations (TE and TM) and resonance modes (ED and MD), the reflectance intensities with respect to the variation of periods, L_x and L_y were opposite. The MD mode is independent of the $RA_x^{({\pm} )}$ lines and the ED mode is suppressed above $RA_x^{(+ )}$ in Fig. 7(a), whereas both the ED and MD mode are suppressed above RA_y line in Fig. 7(b).

 

Fig. 7. Reflectance intensities in the 2D arrays on a substrate with variations of period, L_x in (a) and L_y in (b), while the other period of orthogonal axis is fixed to 300 nm. The TE-polarized light is illuminated with an incident angle of θ = 10°. The periodic pattern sizes are d = h = 150 nm with a refractive index of n = 4. The refractive index of the substrate is n_s = 1.45. Colored dashed lines represent RA conditions defined in Eqs. (14)–(17).

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These results are explained by the partial components of guided-mode waves associated with the MD and ED resonances in Fig. 8, which were estimated at the positions of the white dots in Fig. 7(a), at λ = 650 nm, L_x = 300 nm, and λ = 550 nm, L_x = 300 nm, respectively. The MD resonance at λ = 650 nm and L_x = L_y = 300 nm is excited by the guided-mode waves of partial components ${\vec{S}_1} = S_x^{({0,1} )}{\vec{e}_x} + S_y^{({0,1} )}{\vec{e}_y}$ and ${\vec{S}_2} = S_x^{({0, - 1} )}{\vec{e}_x} + S_y^{({0, - 1} )}{\vec{e}_y}$ with the relative amplitudes of $S_x^{({0, \pm 1} )} = 0.03{S_o}$ and ${\; }S_y^{({0, \pm 1} )} = 0.38\,{S_o}$, which represents the propagation in almost y direction and less sensitive to variation of period L_y as shown in Fig. 7(a).

 

Fig. 8. Components of Poynting vector normalized to the incident light $S_x^{(m,m^{\prime})}$ in (a), (c) and $S_y^{(m,m^{\prime})}$ in (b), (d) of guided-mode waves excited in the slab by the incident wave with TE polarization, which are calculated at the wavelengths λ = 650 nm (MD mode) in (a) and (b), λ = 550 nm (ED mode) in (c) and (d) with L_x = L_y = 300 nm. The incident angle of light is θ = 10°. Each component is normalized to the incident wave flux. The sizes of the circles correspond to the intensity of $S_x^{(m,m^{\prime})}$ and $S_y^{(m,m^{\prime})}$. Geometrical parameters of the slab are the same as Fig. 4.

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However, the ED resonance at λ = 550 nm, L_x = L_y = 300 nm is excited by the guided-mode waves of several partial components${\; }{\vec{S}_1} = S_x^{({ - 1,0} )}{\vec{e}_x}$, ${\vec{S}_2} = S_x^{({1,0} )}{\vec{e}_x}$, ${\vec{S}_3} = S_y^{({0,1} )}{\vec{e}_y}$ and ${\vec{S}_4} = S_y^{({0, - 1} )}{\vec{e}_y}$ in x and y directions, where the relative amplitudes are indicated in Fig. 8(c) and (d). Thus, the ED resonance disappears above RA_y line in Fig. 7(b) and decreases above $RA_x^{(- )}$ line in Fig. 7(a). The existence of ED resonance in the range $RA_x^{(- )} < {L_x} < \; RA_x^{(+ )}$ is ensured by the nonzero component ${\vec{S}_2} = S_x^{({1,0} )}{\vec{e}_x}$ which satisfies the condition of propagation of Eq. (14) for L_x in this range.

3. Experimental result

The above analyses were applied to 2D arrays of Si nanostructures, which were patterned with a size of d_x = d_y = 120 nm, height of h = 105 nm, and periods of L_x= L_y= 240 nm on a quartz substrate, as shown in Fig. 9.

 

Fig. 9. Scanning electron microscope image of 2D arrays of Si nanostructures, fabricated on a quartz substrate. (a) top view, and (b) side view images

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The nanostructure arrays have been verified as reflective color filters with high angular tolerances of up to 45° and analyzed by the guided-mode resonance theory in [18,19]. Herein, the changes in the resonant spectra for various incident angles θ and polarization of incident light were reinterpreted using the analytic method in Sec. 2.

Figures 10(a)–(d) show the calculation results of the reflectance intensity of the 2D Si arrays applied in Fig. 9 at an incident angle of θ = 20°. The refractive index of crystalline Si was used in [36]. The reflectance spectra appear similar to the non-absorbing 2D arrays in Fig. 4 and exhibit all features described in Section 2, except that the $RA_{ox}^{( - )}$ line is mid-positioned between the $RA_x^{( + )}$ and $RA_x^{( - )}$ lines. Owing to the smaller pattern size of 120 nm, the ED and MD modes were found near 475 nm and 525 nm, respectively, whereas no resonance modes of EQ and MQ were found [22].

 

Fig. 10. Reflectance intensities in the 2D arrays on a quartz substrate with variations of period, L_x in (a), (c) and L_y in (b), (d) while the other period of orthogonal axis is fixed to 240 nm. The polarized light is illuminated with an incident angle of θ = 20°. The periodic pattern sizes are d_x = d_y = 120 nm, and h = 105 nm. Colored dashed lines represent RA conditions as defined in Eqs. (14)–(17).

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Figure 11 shows the reflectance spectra of the Si nanostructures at incident angles of up to 60°, where the measured (solid lines) and simulated (dashed lines) reflectance spectra are in good agreement for both (a) TE and (b) TM polarization. The reflectance profile in the range of 450–550 nm is superposed with ED mode (∼475 nm) and MD mode (∼525 nm), and the modes exhibit different angle dependences. In TE polarization, the peak intensities of the ED mode are approximately 60%, which is 10–15% lower compared to the MD mode at incident angles of θ = 0–20°. However, at an angle above θ = 50°, the peak intensities of the two modes were similar (∼80%). Meanwhile, in TM polarization, both the ED and MD modes decrease with an increase in the incident angle from θ = 0 to 20°: thus, the two peak intensities decrease to ∼20% at incident angles above 50°. Considering that the intensity of the resonance mode is substantially changed near the RA lines, and the RA lines are functions of the incident angle, as shown in Eq. (14)–(17), the different angle dependences can be explained.

 

Fig. 11. Reflectivity spectral data for different incident angles of TE light (a) and TM incident light (b). The solid (dashed) lines represent the measured (calculation) results.

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Figure 12 shows the calculation results of the reflectance intensities of 2D Si arrays with respect to the variations in L_x at the incident angle of θ = 50°. The orthogonal period L_y was fixed at 240 nm. The period of L_x, corresponding to the RA conditions at the ED and MD resonances, is ∼200–300 nm in Fig. 12, which is comparable to the sample period of L_x = 240 nm. According to the results in Figs. 11 and 12, the changes in the reflectance spectra are consistent with the behaviors of the guided-mode waves of the ED and MD resonances described in Sec. 2. Here, the guided-mode wave associated with MD resonance for TE polarization contains (0,+) and (0,-) components (see Fig. 8(b)), which is not affected by $RA_x^{( - )}$ condition as represented in Fig. 12(a), while the guided-mode wave associated with MD resonance for TM polarization contains only the (-,0) component (see Fig. 5(а)), which disappears at the L_x above $RA_x^{( - )}$ condition. The guided-mode wave associated with ED resonance for TM polarization is a “direct mode” in the range of $RA_x^{( - )}$ < L_x < $RA_{ox}^{( - )}$ and is represented as the marginal intensities of the ED mode (∼450 nm) in the range of 200 nm < L_x < 250 nm in Fig. 12(b). The results are also verified by the experimental results in Fig. 11(b), where the reflectance spectra for incident angles θ > 50° show non-zero intensities at ∼475 nm.

 

Fig. 12. Reflectance intensities in the 2D Si arrays on a quartz substrate with variations of period L_x. The other period of orthogonal axis is fixed to 240 nm. The light is illuminated with an incident angle of θ = 50° for TE (a) and TM (b). The periodic pattern sizes are d = 120 nm, h = 105 nm with the refractive index of crystalline Si. The refractive index of quartz substrate is n_s = 1.5. Colored dashed lines represent the RA lines of Eq. (14) and (15).

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4. Conclusions

We analyzed the internal structures of guided-mode waves associated with ED and MD resonances excited in a layer of dielectric nanoparticles with a high refractive index under the incident light. The guided-mode wave associated with MD optical resonances are induced by the guided-mode wave with energy flux in the opposite direction, parallel to the in-plane direction of incident light for TM polarization, and by two guided-mode waves with energy fluxes orthogonal to the in-plane direction for TE polarization. Owing to the relevant components of Poynting vectors propagating in a close orthogonal direction for TE polarization, the resonance modes are significantly stable up to a large incident angle. Meanwhile, the guided-mode waves associated with ED resonance consist of partial components propagating in both in-plane and orthogonal directions of the incident light and are thus affected by RA conditions due to the diffraction in both directions. This results in the suppression of resonances at specific periods of L_x and L_y or certain incident angles with constant periods. The analyses of guided-mode waves associated with ED and MD resonances successfully explain the stability or vanishing of resonance modes with variations in the inter-distance between nanostructures near the RA condition. The aforementioned theoretical analyses were experimentally verified with changes in the reflectance spectra of Si arrays on a quartz substrate at an oblique incident angle. These works could be used to the potential applications for designing of optical devices, structural color filters, optical switches, and high-performance sensing devices.