Performance of finite-size metal-dielectric nanoslits metasurface optical filters

Rong He; Rong He; Rong He; Cheng Chen; Xisaina Tang; Yuxiang Zheng; Yuxiang Zheng; Liangyao Chen; Junpeng Guo; Junpeng Guo

doi:10.1364/OE.498076

1. Introduction

Structural simplicity of surface subwavelength structure optical filters offers a unique advantage for device miniaturization and integrations for many integrated photonics applications such as spectral imaging, sensing, and high bandwidth optical communications [1–6]. A subwavelength period surface structure can excite guided optical resonance modes in the lateral direction with the additional photon momentum provided by the period structure. At the resonance mode frequency, the subwavelength period structure functions as an optical transmission or reflection filter. Surface plasmon resonance (SPR) filters, guide-mode resonance (GMR) filters, and hybrid GMR filters are three types of subwavelength structure optical resonance filters. SPR optical filters are made of metal periodic nanostructures, such as nanoholes or nanoslit arrays [7–15]. Because of the large energy dissipation loss in metal films, the SPR optical filters typically have broad spectral linewidth. GMR filters are traditionally made of dielectric subwavelength gratings and high-index dielectric waveguide layers [16–23]. The dielectric nanostructure gratings scatter the incident light into the high-index waveguide layer and the lateral GMR mode is excited in the structure. When the thickness of the high-index waveguide layer is large, multiple GMR modes can be excited, resulting in multiple filter wavelengths. Compared with the SPR optical filters, GMR filters have very narrow spectral linewidth and high peak transmittance. A hybrid GMR optical filter is made of deep subwavelength thin periodic metal nanograting and high-index dielectric layers [24–29]. In the hybrid metal-dielectric GMR structure, both surface plasmon mode and guided resonance mode are excited. The weak coupling between the two modes results in a high-quality factor and a narrow linewidth. A significant advantage of hybrid GMR optical filters is that the thickness of the metal nanograting is much smaller than the grating thickness of all-dielectric guided-mode resonance filters [29]. In addition, the metal structures can be used as an electrode for tunable active device applications.

In the three types of subwavelength structure resonance optical filters, periodic surface nanostructures function as a light wave coupler, coupling the free-space optical wave into the lateral resonance mode. In the past, most analyses of subwavelength nanostructure optical filters were limited to the extreme scenario where the size of the subwavelength nanostructure is infinitely large. Practically, subwavelength surface gratings have finite sizes in both lateral dimensions. The finite size affects the filter spectral linewidth and transmittance. The performance of finite-size all-dielectric GMR filters has been previously investigated [30–33]. All-dielectric GMR filters require hundreds of micrometers to achieve a narrow spectral resonance due to the high-quality factor. By using metal mirrors at the edges of the period nanostructure, a miniature GMR filter can realize a high Q-factor [34,35]. In this work, we analyze the performance of finite-size hybrid metal-dielectric nanoslits guided mode resonance metasurface optical filters by using finite-difference time-domain numerical simulations and spatial Fourier transform analysis. Our results show that in the grating period direction, the required dimension size is the product of the nanoslits period and the quality factor of the unlimited-size filter. In the nanoslits grating line direction, the required dimension size is 12 times of filter peak wavelength, which corresponds to the second zero of the Fourier transform of the aperture window function. The device size requirement in the grating line dimension is less stringent than the size requirement in the nanoslits period dimension because the filter performance is less dependent on changes in incident angles in that direction. The findings in this work are significant for integrating large numbers of metasurface filters on small surface areas.

2. FDTD simulation of finite-size nanoslits metasurface optical filters

2.1 Finite dimension size in the periodic direction of nanoslits

Figure 1(a) shows a hybrid metal-dielectric nanoslit metasurface filter structure with a limit size in the periodic direction of metal nanoslits (x-direction). The hybrid nanoslit metasurface filter consists of a thin gold nanoslit grating with a period of Λ, thickness of t₁, and slits width of w, a magnesium fluoride (MgF₂) low refractive index layer with a thickness of t₂, a silicon high refractive index layer with the thickness of t₃, and an MgF₂ substrate. A finite-difference time-domain (FDTD) software (Lumerical Solutions, Inc.) was used to calculate the transmittance spectra. In the simulations, the simulation domain is terminated with a perfect matching layer boundary in the lateral x-directions. The limited size gold nanoslits grating is terminated with perfect matching layer boundaries (PML) in the x-dimension. To maintain the simulation stability, the number of the perfect matching layers is set at 10. The simulation domain is terminated with a perfect matching layer boundary in the propagation direction and reflection direction. The number of perfect matching layers is set at 8. A plane wave of unity amplitude traveling in the negative z-direction is normally incident to the surface of the filter structure with polarization in the x-direction. Because PML absorbs energy at the boundaries, an ideal plane wave will not be produced perfectly. But far from the simulation boundary, the light field can still be considered as a plane wave. The simulation domain was a two-dimensional region with a 2 nm mesh resolution. The frequency point is 4000 and the wavelength resolution is 0.275 nm. The optical constants of gold (Au) and silicon (Si) were taken from the Ref. [36]. The refractive index of MgF₂ is fixed at 1.38. The frequency domain power transmittance is defined as the ratio of the output power and the incident power at each wavelength. The power of each wavelength is obtained from the Fourier transform of the signal in the time domain. Figures 1(b)-(e) show the calculated transmittance spectra of the hybrid metasurface optical filter with different sizes in the x-direction (L_x). The size is ∞, 480Λ, 160Λ, and 80Λ, respectively. The slit period Λ is 900 nm. The nanoslit width is 50 nm. Au grating thickness is 40 nm. The MgF₂ layer thickness is 300 nm. Si layer thickness is 215 nm. It is seen in Fig. 1(b) that the infinite-size hybrid GMR metasurface optical filter has two sharp peaks in the transmittance spectra. The peak at 1380.0 nm results from the Rayleigh anomaly [37]. The peak at 1663.8 nm originates from the weakly coupling of SPR mode and GMR mode, with a peak transmittance of 0.64 and a peak linewidth of 3.5 nm [38]. In addition, the quality factor (Q) of the infinite-size filter is 475. When the filter size decreases as seen in Figs. 1(c)-(e), the transmittance stays the same at first and then becomes smaller.

Fig. 1. (a) Cross section of a finite-size hybrid metal-dielectric metasurface optical filter. (b)-(e) Simulated transmittance spectrum of hybrid metal-dielectric guided mode resonance metasurface optical filters with device dimension size L_x=∞, 480Λ, 160Λ, and 80Λ, respectively. The nanoslit width is 50 nm. Au grating thickness is 40 nm. The MgF₂ layer thickness is 300 nm. Si layer thickness is 215 nm. The nanoslits period Λ is 900 nm.

Download Full Size | PDF

Furthermore, the filter spectral linewidth, peak transmittance, filter wavelength, and quality factor versus the ratio of L_x to Λ are plotted in Figs. 2(a)-(b). Figure 2(a) shows the transmittance peak linewidth and transmittance versus the ratio of L_x to Λ. The black dots represent the variation trend of peak linewidth with the increase of L_x. The red diamonds represent the changes in the peak transmittance with the increase of L_x. The black dots show that, when the filter size is less than 250Λ, the peak linewidth becomes continuously narrower as the filter size increases. When L_x increases to 250Λ, the peak linewidth almost decreases to 3.5 nm and remains at 3.5 nm for larger sizes. The red diamonds show that the peak transmittance drops dramatically when L_x is less than 480Λ. When L_x is 480Λ, the peak transmittance is close to 0.58, which is 90% of the peak transmittance 0.64 of the infinite-size filter. Figure 2(b) shows the peak wavelength and quality factor versus the ratio of L_x to Λ. The blue triangles indicate the changes in the peak wavelength of the filters with different L_x. The magenta triangles represent the quality factor of the device. It is seen from the blue triangles that the peak wavelength gradually blue-shifts for the filter with L_x less than 250Λ. When L_x is larger than 250Λ, the peak wavelength is close to that of the unlimited-size filter. The magenta triangles show that the quality factor increases with the increase of L_x. When L_x reaches 480Λ, the quality factor is close to that of the infinite-size filter. In addition, the quality factor does not vary dramatically in the range of 240-480Λ. When the device size L_x exceeds 480Λ, the hybrid GMR metasurface optical filter can maintain the performance of the infinite-size hybrid GMR metasurface optical filter. Therefore, 480Λ is the critical size in the x-direction to maintain the filter performance. The result here is consistent with the result calculated from the following formula in Ref. [31],

(1)$${L_x} \simeq \Lambda \cdot ({\lambda _0}/\Delta \lambda ),$$

where Λ is the period of the nanoslits grating, λ₀ is the peak wavelength of the infinite-size hybrid optical filter, Δλ is the peak linewidth of the infinite-size hybrid optical filter, λ₀/Δλ is the quality factor (Q) of the infinite-size filter. In this work, the quality factor of the infinite-size hybrid metal-dielectric GMR filter is 475, depicted by the magenta dotted line in Fig. 2(b), which is close to the value of 480.

Fig. 2. (a) Spectral linewidth and transmittance versus the number of the grating period. (b) Peak wavelength and quality factor versus the number of the grating period. The red dotted line represents the peak transmittance of the infinite-size filter. The black dotted line indicates the linewidth of the infinite-size filter. The blue dotted line represents the peak wavelength of the infinite-size filter. The magenta dotted line represents the quality factor of the infinite-size optical filter.

Download Full Size | PDF

The electric field distribution of the infinite-size filters at resonance wavelength 1663.8 nm was calculated by FDTD simulation, as shown in Fig. 3(a). It shows the electric field is enhanced at the top and bottom surface of the Si layer. The field profiles and amplitudes of the electric field in every unit cell are the same. The electric field distribution of the filter size of 80Λ at the resonance wavelength of 1660.0 nm was calculated and plotted in Fig. 3(b). It is seen that the enhanced electric field distributes at the top and bottom surface of the Si layer when the filter size is reduced to 80Λ. However, the amplitude of the electric field decreases compared to that of the infinite filter. The field profile close to the middle of the filter is similar to that of the infinite-size filter. The amplitude of the electric field decreases from the center to the edges of the filter structure in the x-direction. It is explained that the period surface grating couples the incident light into the GMR mode in the finite-size and infinite-size filter structure. Because of the interaction between the incident light and GMR mode, a standing wave pattern resides primarily in the waveguide layer. However, when the filter size is decreased, the GMR mode fades away at the edges of the filter. Thus, the interaction length between the incident light and GMR mode decreases with the reduction of the filter size, therefore reducing the amplitude of the electric field.

Fig. 3. The electric field distribution of the hybrid metal-dielectric metasurface optical filter at resonance wavelength. (a) filter size L_x=∞, and (b) filter size L_x= 80Λ.

Download Full Size | PDF

2.2 Finite dimension size in the direction parallel to nanoslits

As discussed in the previous session, the critical size in the x-direction of the hybrid metal-dielectric metasurface optical filter is related to the quality factor. Next, the transmittance spectra of hybrid nanoslits metasurface filters with finite size (D_y) in the grating line direction (y) are numerically simulated by using FDTD. The simulation domain is terminated with a perfect matching layer boundary in the lateral y-directions, and in the propagation and the reflection ± z directions. To increase the simulation stability, the number of the perfect matching layer is set at 10 in the y-direction. The number of the perfect matching layer is set at 8 in the z-direction. In the lateral x-directions, the simulation domain is set up with a periodic boundary. A plane wave of unity amplitude traveling in the negative z-direction is normally incident to the surface of the filter structure with polarization in the x-direction. The simulation domain is a three-dimensional region with a 2 nm mesh resolution. The frequency point is 4000 and the wavelength resolution is 0.275 nm. Figure 4(a) shows the 3D structure of the hybrid metal-dielectric metasurface optical filter. Figures 4(b)-(f) show the calculated transmittance curves of hybrid metal-dielectric metasurface optical filters with different sizes in the y-direction. The filter size D_y is ∞, 40Λ, 10Λ, 5Λ, and 3Λ, respectively. The nanoslit grating period Λ is 900 nm. The gold nanoslit width is 50 nm. The Au grating layer thickness is 40 nm. The MgF₂ layer thickness is 300 nm. Si layer thickness is 215 nm. It is seen that when D_y is ∞, a single and sharp peak appears at 1663.8 nm. The peak transmittance is 0.64. The spectral linewidth is 3.5 nm. When the filter size D_y decreases, as shown in Figs. 4(c)-(f), there also exists a single peak around 1663.8 nm. However, the peak transmittance is significantly reduced.

Fig. 4. (a) 3D structure of hybrid metal-dielectric guided mode resonance metasurface optical filters with finite size. (b)-(f) Simulated transmittance curves of hybrid metal-dielectric guided mode resonance metasurface optical filters with finite size D_y=∞, 40Λ, 10Λ, 5Λ, and 3Λ, respectively. The nanoslit width is 50 nm. Au grating thickness is 40 nm. The MgF₂ layer thickness is 300 nm. Si layer thickness is 215 nm. The slit period Λ is 900 nm.

Download Full Size | PDF

Next, the spectral linewidth, transmittance, filter peak wavelength, and quality factor versus the ratio of D_y to Λ are potted in Figs. 5(a)-(b). Figure 5(a) plotted the spectral linewidth and the peak transmittance versus the finite size D_y. Similarly, the black dots represent the variation of peak linewidth with the increase of D_y. The red diamond dots represent the peak transmittance versus the ratio of D_y to Λ. The black dots line in Fig. 5(a) shows that when the filter size decreases from 15Λ to 3Λ, the spectral linewidth increases slightly. The red diamond dots show that the transmittance is close to 90% of the transmittance of the infinite-size filter in the case of the D_y larger than 20Λ and decreases rapidly when the finite size D_y is below 20Λ. Figure 5(b) shows the peak wavelength and the quality factor versus finite-size D_y. The blue triangles indicate the peak wavelength versus the filter size D_y. The magenta triangles represent the quality factor versus the filter size D_y. It is seen in Fig. 5(b) that the peak wavelength is almost constant for different filter sizes. Because the optical resonance occurs in the lateral x-direction, the size in the y-direction has no effect on the peak wavelength. The quality factor increases from approximately 400 to 475, when the filter size increases from 3Λ to 20Λ. Thus, the minimal device size in the y-dimension D_y required to maintain the peak transmittance and linewidth of the infinite-size filter is 20Λ.

Fig. 5. (a) Spectral linewidth and peak transmittance as functions of the ratio of the finite size D_y to slits period. (b) Peak wavelength and quality factor as functions of the ratio of the finite size D_y to slits period. The black dotted line represents the peak linewidth of the infinite-size filter. The red dotted line represents the peak transmittance of the infinite-size filter. The blue dotted line represents the peak wavelength of the infinite-size optical filter. The magenta dotted line represents the quality factor of the infinite-size optical filter.

Download Full Size | PDF

Figure 6(a) shows the simulated electric field distribution of the filter structure with the size D_y=∞ at the resonance wavelength of 1663.8 nm. Figure 6(b) shows the simulated electric field distribution of the filter structure with the size D_y= 5Λ at the resonance wavelength of 1663.5 nm. In the simulations, an x-z plane 2D field monitor is located at y = 0. The nanoslit is located in the center of the figure. The simulation domain is terminated with perfect matching layer (PML) boundaries in the lateral y-directions. In the lateral x-directions, the simulation domain is set up with a periodic boundary. It is seen that the profile and the amplitude of the electric field of the infinite-size filter are the same as that of the filter with a size of 5Λ. The electric field enhancement is not relative to the filter size in the y-direction, because the guided wave propagates in the x-direction.

Fig. 6. The electric field distribution of the hybrid metal-dielectric metasurface optical filter at resonance wavelength. (a) filter size D_y=∞, and (b) filter size L_x= 5Λ.

Download Full Size | PDF

3. Fourier transform analysis of finite-size nanoslits optical filter

Next, an analysis based on the Fourier transform is used to explain the effects of finite size on the performance of the hybrid metasurface optical filter in the two lateral directions. Figure 7(a) shows a hybrid nanoslit grating metasurface optical filter structure with finite size. A plane wave of a unit electric field amplitude traveling in the z-direction is normally incident to the surface of the filter structure with polarization in the x-direction. When the filter sizes (L_x and D_y) are finite, the device consists equivalently of an aperture window and an infinite-size hybrid optical filter, as shown in Fig. 7(b). The aperture window and the infinite-size filter locate in the z = 0 plane. To facilitate understanding, we separate the filter away from the aperture window. The plane wave passes through the aperture window, followed by the infinite-size guide mode optical filter. At first, we analyze the propagation of a plane wave through the window aperture. According to the Fourier transform, an arbitrary geometry aperture can be written as an integral of spatial harmonic functions,

(2)$$f(x,y) = \int\!\!\!\int\limits_\infty {F({k_x},{k_y})} \textrm{exp}[{ - j({k_x} \cdot x + {k_y} \cdot y)} ]\textrm{d}{k_x}\textrm{d}{k_y}, $$

with different complex amplitudes, which are the Fourier transform of the aperture function f (x, y). The incident plane wave after the aperture window is converted into multiple propagating plane waves of various k-vectors. The complex amplitude of each Fourier component is,

(3)$$F({k_x},{k_y}) = {(\frac{1}{{2\pi }})^2}\int\!\!\!\int\limits_\infty {f(x,y)} \textrm{exp}[{j({k_x} \cdot x + {k_y} \cdot y)} ]\textrm{d}x\textrm{d}y, $$

where k_x is the component of the wavevector in the x-direction, and k_y is the component of the wavevector in the y-direction. Subsequently, each Fourier component plane wave passes through an infinite GMR filter with a transmission coefficient t (k_x, k_y, λ), in which the transmission coefficient is dependent on the direction of propagation and wavelength. For each Fourier component, the transmitted plane wave after the finite size filter is

(4)$$F({k_x},{k_y}) \cdot t({k_x},{k_y},\lambda )\textrm{exp}[{ - j({k_x} \cdot x + {k_y} \cdot y + {k_z} \cdot z)} ]. $$

The total electric field after the transmission through the filter is the superposition of plane waves after the finite-size filter. At distance z, the complex optical field is

(5)$$\begin{array}{l} g(x,y,z,\lambda ) = \int\!\!\!\int\limits_\infty {F({k_x},{k_y}) \cdot t({k_x},{k_y},\lambda )\textrm{exp}[{ - j({k_x} \cdot x + {k_y} \cdot y + {k_z} \cdot z)} ]} d{k_x}d{k_y}\\ = \int\!\!\!\int\limits_\infty {F({k_x},{k_y}) \cdot t({k_x},{k_y},\lambda ) \cdot H({k_x},{k_y},z)\textrm{exp}[{ - j({k_x} \cdot x + {k_y} \cdot y)} ]} d{k_x}d{k_y} \end{array}, $$

where H (k_x, k_y, z) is the free space transfer function in the Fourier space

Fig. 7. (a) Hybrid metal-dielectric structure nanoslits grating guided-mode resonance metasurface optical filter with a finite size. (b) A schematic of finite size hybrid nanoslits grating GMR optical filter. The orange region represents the finite-size grating. The blue plane represents the infinite-size filter surface.

Download Full Size | PDF

(6)$$H({k_x},{k_y},z) = \textrm{exp}[{ - j{k_z} \cdot z} ],\;\; \textrm{where}\;\; {k_z} = \sqrt {k_0^2 - k_x^2 - k_y^2} ,\textrm{ }{k_0} = \frac{{2\pi }}{\lambda }.$$

Right after the filter at z = 0, the transmittance of the finite-size optical filter is

(7)$$T(\lambda ) = \frac{{\int\!\!\!\int {{{|{g(x,y,\lambda )} |}^2}dxdy} }}{{\int\!\!\!\int {{{|{f(x,y)} |}^2}dxdy} }}. $$

3.1 Finite dimension size in the periodic direction of the nanoslits

In section 2.1, we calculated the transmittance of the hybrid metasurface optical filters with finite size in the direction of the grating period. In this scenario, the filter has a finite size in the x-direction and an infinite size in the y-direction. When light wave incidents to this finite filter, it is equivalent to passing through a 1-dimension window aperture with size L_x, followed by an infinite-size hybrid metasurface filter. The transmission function of the 1-dimension window aperture is,

(8)$$\textrm{rect}(\frac{x}{{{L_x}}}) = \left\{ {\begin{array}{{c}} {1,}\\ {0,} \end{array}\textrm{ }} \right.\begin{array}{{c}} {\textrm{when }|x |\mathrm{\ < }{{{L_x}} / 2}}\\ {\textrm{when }|x |\ge {{{L_x}} / 2}} \end{array}. $$

The spatial Fourier transform of the 1D transmission window function of Eq. (8) is,

(9)$$F({k_x}) = A \cdot \textrm{sinc}\left( {\frac{{{L_x}{k_x}}}{{2\mathrm{\pi }}}} \right). $$

After the 1D aperture, plane waves of different propagation directions, indicated by θ, with the amplitude F (L_x, λ, θ), immediately pass through the infinite-size filter. The amplitude of the harmonic plane wave F (L_x, λ, θ) is,

(10)$$F({L_x},\lambda ,\theta ) = A \cdot \textrm{sinc}\left[ {\frac{{{L_x}}}{\lambda }\sin \theta } \right], $$

where θ is the angle between the direction of propagation and the minus z-direction in the incident plane (x-z plane) as shown in Fig. 1(a), A is a constant, and λ is the incident wavelength. Then, the multiple Fourier component plane waves are incident on the infinite-size hybrid filter at different incident angles θ. Transmittance through the infinite-size filter at different angles of incidence θ is simulated by using FDTD simulations. In the simulations, the simulation domain is a two-dimensional region with a 2 nm mesh resolution. The simulation domain is terminated with a perfect matching layer boundary in the propagation direction and reflection direction. The simulation domain is terminated with a Bloch boundary in the lateral x-directions. The incident wave using the broadband fixed angle source technique (BAFST) is obliquely incident to the surface of the filter structure with TM polarization. Figure 8(a) shows the transmittance spectra at different incident angles through the infinite-size filter in the wavelength range from 1500 nm to 1900nm. It is seen that there exist two transmittance curves in the whole transmittance spectra. Because at oblique incidence, the propagation constants of the diffraction waves along the surface in the x direction are,

(11)$$\beta _{}^ \pm{=} {k_0} \cdot \sin \theta \pm \frac{{2\pi }}{\Lambda } \cdot m,\textrm{ }m = 0,\textrm{ }1,\textrm{ }2,\textrm{ }3 \ldots, $$

where k₀ is the wavevector in the free space. θ is the incident angle of incident light. Λ is the grating period. m is an integer number. The center wavelength in the left decreases with increasing incident angle and corresponds to β⁺. The center wavelength in the right increases with increasing incident angle and corresponds to β^-. In this work, we just focus on the left transmittance curve. In Fig. 8(a), the left transmittance curve has a liner blueshift at a rate of 12 nm/degree, demonstrating the dramatic sensitivity of the center wavelength to the angle of incidence. Since the amplitude of the incident plane wave is the unit in simulating the transmittance shown in Fig. 8(a), the amplitude of the wave after an aperture follows the distribution of F (L_x, λ, θ). Consequently, the transmittance of the finite-size filter is,

(12)$$T({L_x},\lambda ) = \frac{{\int\limits_{ - \mathrm{\pi }/2}^{\mathrm{\pi }/2} {{{|{F({L_x},\lambda ,\theta )} |}^2} \cdot t(\lambda ,\theta )d\theta } }}{{\int\limits_{ - \mathrm{\pi }/2}^{\mathrm{\pi }/2} {{{|{F({L_x},\lambda ,\theta )} |}^2}d\theta } }}. $$

Using the formulas (8)–(10) and (12), the size-dependent transmittance spectra of filters can be calculated theoretically. Figure 8(b) shows the transmittance T (L_x, λ) of the different size filters by using the Fourier transform analysis. It is seen that when the filter size is less than 250Λ, as the filter size decreases, the transmittance spectral linewidths increase gradually, and the center wavelength blueshifts. The results based on the Fourier transform analysis are consistent well with the results of the FDTD simulation in section 2.1.

Fig. 8. (a) The transmittance spectrum versus incident angle of the infinite-size filter in the wavelength range from 1500 nm to 1900nm. The incident plane is the x-z plane. (b) The transmittance spectrum versus filter size in the x-direction at normal incidence.

Download Full Size | PDF

To specify the transmittance at the center wavelength versus finite size L_x, the transmittance at the center wavelength 1663.8 nm versus incident angle θ is plotted in Fig. 9(a). It is seen that the small variation of the incident angle leads to a drastic drop in filter transmittance because of the wavelength variation at a high rate of 12 nm/deg. When the incident angle increases to 0.66°, the transmittance decreases to 0.01 almost close to 0. 0.66° is a critical angle of the infinite-size filter. In the case of the finite-size filter consisting of a 1D aperture and the infinite-size filter, when the plane wave passes through the 1D aperture, only the diffracted wave of the incident angle less than 0.66° can contribute to the transmittance of the filter. According to the spatial Fourier transform of the 1D aperture window function, the small size L_x of the 1D aperture broadens the angler spectra and makes the diffracted field have more high-frequency components. In our case, that means the energy proportion of the diffracted wave of the incident angle less than 0.66° decreases. There exists a critical size to make the most energy focus on the diffracted waves of the incident angle less than 0.66°. Figure 9(b) shows the light intensity versus the size of the 1D aperture at θ=0.66°. When the size of the 1D aperture increases, the Normalized light intensity decreases in fluctuations and is close to the third zero of the Fourier transform of the 1D aperture window function for the size of 480Λ. 480Λ is the critical size that makes the most energy focus on the diffracted waves of the incident angle less than 0.66°. Next, the calculated transmittance at the center wavelength 1663.8 nm versus filter size L_x by using the FDTD simulation and Fourier transform analysis was shown in Fig. 9(c). The square dots represent the FDTD simulated transmittance versus filter size L_x. The solid line represents the transmittance calculated by using Fourier transform analysis. It is seen that the transmittance obtained from FDTD simulation is consistent with that calculated using Fourier transforms analysis. The minimum size to maintain the transmittance of the infinite size filter is 480Λ. Figure 9(d) shows the transmittance curve of the filter with the size 480Λ. The dots represent the transmittance calculated using the Fourier transform analysis. The solid line represents the transmittance simulated by FDTD simulation. The two transmittance curves are almost overlapped.

Fig. 9. (a) The transmittance versus incident angle of the infinite-size filter at the center wavelength 1663.8 nm. (b) Normalized intensity versus the size of the 1D aperture at θ=0.66°. (c) The calculated transmittance versus filter size L_x at resonance wavelength 1663.8 nm. The square dots represent the FDTD simulated transmittance versus filter size. The solid line represents the transmittance calculated by using Fourier transform analysis. The incident plane is the x-z plane. (d) The simulated transmittance spectrum by using FDTD simulation (the solid line) and Fourier transform analysis (the dots).

Download Full Size | PDF

3.2 Finite size in the direction parallel to the nanoslits

In section 2.2, the transmittance of the finite-size hybrid metasurface filters is simulated. In this case, the filter size is finite in the y-direction and infinite in the x-direction. The transmission function of the 1-dimension window aperture in the y-direction is,

(13)$$\textrm{rect}(\frac{y}{{{D_y}}}) = \left\{ {\begin{array}{{c}} {1,}\\ {0,} \end{array}\textrm{ }} \right.\begin{array}{{c}} {\textrm{when }|y |\mathrm{\ < }{{{D_y}} / 2}}\\ {\textrm{when }|y |\ge {{{D_y}} / 2}} \end{array}. $$

The Fourier transform of the 1-D window aperture is,

(14)$$F({k_y}) = A \cdot \textrm{sinc}\left( {\frac{{{D_y}{k_y}}}{{2\mathrm{\pi }}}} \right). $$

After the 1D aperture, plane waves with different propagation directions, indicated by angle α, and the amplitude F (D_y, λ, α) immediately pass through the infinite-size filter. The amplitude F (D_y, λ, α) of the plane wave is,

(15)$$F({D_y},\lambda ,\alpha ) = A \cdot \textrm{sinc}\left[ {\frac{{{D_y}}}{\lambda }\sin \alpha } \right], $$

where α is the angle between the wave vector and the minus z-direction in the incident plane (y-z plane) shown in Fig. 4(a), A is a constant, and λ is the incident wavelength. The transmission efficiency after the infinite-size filter of each plane waves with different amplitudes and incident angle decides the transmittance of the finite-size filter. Next, the transmittance curves of the infinite-size filter for the different incident angles α were simulated by using FDTD simulation. In the simulations, the simulation domain is a three-dimensional region with a 2 nm mesh resolution. The simulation domain is terminated with a perfect matching layer boundary in the propagation direction and reflection direction. The simulation domain is terminated with a period boundary in the lateral x-directions. The simulation domain is terminated with a Bloch boundary in the lateral y-directions. The incident source wave using the broadband fixed angle source technique (BAFST) is obliquely incident to the surface of the filter structure with TM polarization. Figure 10(a) shows the transmittance spectra at different incident angles through the infinite-size filter in the wavelength range from 1500 nm to 1900nm. When the incident angle α increases, the center wavelength has a slight blueshift. When the incident angle varies from 0 to 10°, the center wavelength changes by 3.5 nm. The amplitude of the incident plane wave is unit in simulating the transmittance. The amplitude of the wave with different angles is different and follows F (D_y, λ, α). Thus, the transmittance of the finite-size filter is,

(16)$$T({D_y},\lambda ) = \frac{{\int\limits_{ - \mathrm{\pi }/2}^{\mathrm{\pi }/2} {{{|{F({D_y},\lambda ,\alpha )} |}^2} \cdot t(\lambda ,\alpha )d\alpha } }}{{\int\limits_{ - \mathrm{\pi }/2}^{\mathrm{\pi }/2} {{{|{F({D_y},\lambda ,\alpha )} |}^2}d\alpha } }}. $$

Fig. 10. (a) The transmittance spectrum versus incident angle α of the infinite-size filter in the wavelength range from 1500 nm to 1900nm. The incident plane is the y-z plane as shown in Fig. 3(a). (b) The transmittance spectrum versus filter size L_x in the y-direction.

Download Full Size | PDF

Using the Fourier transform analysis and the formulas (13)–(16), the transmittance spectra of different-size D_y filters were calculated. Figure 10(b) shows the transmittance curves of different-size filters by using Fourier transform analysis. It shows that when the filter size D_y surpasses 15Λ, the transmittance spectral linewidth is 3.5 nm and the center wavelength is 1663.8 nm, which are identical to the results in Fig. 5(b).

Figure 11(a) shows the transmittance versus incident angle α of the infinite-size filter at the center wavelength 1663.8 nm. It is observed that the transmittance drops slowly as increasing the incident angle α. When the transmittance decreases to 0.01, the corresponding incident angle α is 10°. 10° is the critical angle of the infinite size filter in the y-direction. In the case of the finite-size filter consisting of a 1D aperture and the infinite-size filter, when the plane wave passes through the 1D aperture, only the diffracted wave of the incident angle less than 10° can contribute to the transmittance of the infinite-size filter. If the aperture size D_y is larger, the proportion of the diffracted waves with the incident angles less than 10° is increased, which makes the larger transmittance of the finite-size filter. Hence, 10° is the critical angle of the infinite-size filter which corresponds to the critical size of the 1D aperture. To confirm the critical size, the normalized light intensity versus size D_y of the 1D aperture at α=10° is calculated in Fig. 11(b). It is seen that when the light intensity is reduced to 0, the aperture size is 12λ₀, which is the second zero of the Fourier transform of the 1D aperture function. 12λ₀ is the critical size and is close to 20Λ in section 2.2. Compared with the minimum size for maintaining the performance of the infinite filter in the x-direction, the size in the y-direction is much less. To illustrate the cause of the large difference in critical sizes between the different directions, we recall that in Fig. 8(a), the center wavelength varies at a rate of 12 nm/deg. The variation rate of the center wavelength in the x-direction in Fig. 8(a) is much larger than the one in the y-direction in Fig. 10(a). Thus, when the transmittance decreases to 0.01, the critical angle θ is 0.66° in the x-direction but 10° in the y-direction. The difference in critical angle in two lateral directions leads to the different minimum sizes. Figure 11(c) shows the calculated transmittance at the center wavelength 1663.8 nm versus filter size by using the FDTD simulation and Fourier transform analysis. The square dots represent the FDTD simulated transmittance versus the filter size D_y. The solid line represents the transmittance calculated by using Fourier transform analysis versus the filter size D_y. It is shown that the transmittance curve simulated by FDTD simulation has the same variation trend as that calculated by Fourier transforms analysis. Both results show that the critical size in the y-direction is 20Λ, which is close to 12λ₀. Only if the filter size is larger than the critical size, the transmittance is close to that of the infinite-size filter. Figure 11(d) shows the transmittance curve of the filter with the size of 12λ₀. The dots represent the transmittance calculated by the Fourier transform analysis. The solid line represents the transmittance simulated by FDTD simulation. The results show that the transmittance curve obtained by the Fourier transform analysis agrees well with the transmittance curve calculated by the FDTD numerical simulations.

Fig. 11. (a) The transmittance versus incident angle of the infinite-size filter at center wavelength 1663.8 nm. (b) The normalized light intensity versus size D_y of the 1D aperture at α=10°. (c) The calculated transmittance at resonance wavelength 1663.8 nm versus filter size D_y. The square dots represent the FDTD simulated transmittance versus filter size. The solid line represents the transmittance calculated by using Fourier transform analysis. The incident plane is the y-z plane. The critical size is 20Λ which is close to 12λ₀. (d) The transmittance spectrum calculated by using FDTD simulation (the solid line) and Fourier transform analysis (the dots).

Download Full Size | PDF

4. Summary

In this work, we investigated the performance of finite-size metal-dielectric nanoslits metasurface optical filters and present an analysis on the performance of finite-size filters by using spatial Fourier transform and FDTD numerical simulations. Both FDTD simulations and Fourier transform analysis have shown that in the nanoslits periodic direction, the required size for maintaining the performance is the product of the period and the quality factor of the corresponding infinite size filter. In the orthogonal direction, the minimum size is 12 times the filter peak wavelength, which corresponds to the second zero of the Fourier transform of the aperture window function. The size requirement for maintaining infinite-size filter performance in the grating line direction is less stringent than that in the direction of the grating period. It is shown and explained that the slow change and less dependence of transmittance of the infinite-size device on the angle of incidence in the orthogonal direction to the structure period enables the less stringent requirement for the device size in that direction. The findings in this work are significant for integrating large numbers of metasurface optical filters on small surface areas. The conclusions in this work can also apply to other types of metasurface optical filters.

Funding

Yiwu Research Institute of Fudan University Research Fund;Fudan University-Changguang Research Fund; Yanchang Petroleum-Fudan University Research Fund

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. Z. Y. Yang, T. A. Owen, H. X. Cui, J. A. Webber, F. X. Gu, X. M. Wang, T. C. Wu, M. H. Zhuge, C. Williams, P. Wang, A. V. Zayats, W. W. Cai, L. Dai, S. Hofmann, M. Overend, L. M. Tong, Q. Yang, Z. P. Sun, and T. Hasan, “Single-nanowire spectrometers,” Science 365(6457), 1017–1020 (2019). [CrossRef]

2. Z. Wang, S. Yi, A. Chen, M. Zhou, T. S. Luk, A. James, J. Nogan, W. Ross, G. Joe, A. Shahsafi, K. X. Z. Wang, M. A. Kats, and Z. F. Yu, “Single-shot on-chip spectral sensors based on photonic crystal slabs,” Nat. Commun. 10(1), 1020 (2019). [CrossRef]

3. B. Lu, P. D. Dao, J. Liu, Y. He, and J. Shang, “Recent advances of hyperspectral imaging technology and applications in agriculture,” Remote Sens. 12(16), 2659 (2020). [CrossRef]

4. S. P. Burgos, S. Yokogawa, and H. A. Atwater, “Color imaging via nearest neighbor hole coupling in plasmonic color filters integrated onto a complementary metal-oxide semiconductor image sensor,” ACS Nano 7(11), 10038–10047 (2013). [CrossRef]

5. P. Ge, X. Ling, J. Wang, X. Liang, S. Li, and C. Zhao, “Optical filter bank modeling and design for multi-color visible light communications,” IEEE Photonics J. 13(1), 1–19 (2021). [CrossRef]

6. C. T. Hsiung and C. S. Huang, “Refractive index sensor based on gradient waveguide thickness guided-mode resonance filter,” IEEE Sens. Lett. 2(4), 1–4 (2018). [CrossRef]

7. T. Ellenbogen, K. Seo, and K. B. Crozier, “Chromatic plasmonic polarizers for active visible color filtering and polarimetry,” Nano Lett. 12(2), 1026–1031 (2012). [CrossRef]

8. Y. D. Shah, J. Grant, D. Hao, M. Kenney, V. Pusino, and D. R. S. Cumming, “Ultra-narrow line width polarization-insensitive filter using a symmetry-breaking selective plasmonic metasurface,” ACS Photonics 5(2), 663–669 (2018). [CrossRef]

9. B. Zeng, Y. Gao, and F. J. Bartoli, “Ultrathin nanostructured metals for highly transmissive plasmonic subtractive color filters,” Sci. Rep. 3(1), 2840 (2013). [CrossRef]

10. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]

11. J. Guo and H. Leong, “Investigation of surface plasmon resonance in super period gold nanoslit arrays,” J. Opt. Soc. Am. B 29(7), 1712–1716 (2012). [CrossRef]

12. X. L. Tian, H. Guo, K. H. Bhatt, S. Q. Zhao, Y. Wang, and J. Guo, “Super-period gold nanodisc grating-enabled surface plasmon resonance spectrometer sensor,” Appl. Spectrosc. 69(10), 1182–1189 (2015). [CrossRef]

13. N. Nader, S. Vangala, J. R. Hendrickson, K. D. Leedy, D. C. Look, J. Guo, and J. W. Cleary, “Investigation of plasmon resonance tunneling through subwavelength hole arrays in highly doped conductive ZnO Films,” J. Appl. Phys. 118(17), 173106 (2015). [CrossRef]

14. H. Leong and J. Guo, “A surface plasmon resonance spectrometer using a super-period metal nanohole array,” Opt. Express 20(19), 21318–21323 (2012). [CrossRef]

15. S. Yokogawa, S. P. Burgos, and H. A. Atwater, “Plasmonic color filters for CMOS image sensor applications,” Nano Lett. 12(8), 4349–4354 (2012). [CrossRef]

16. L. Macé, O. Gauthier-Lafaye, A. Monmayrant, S. Calvez, H. Camon, and H. Leplan, “Highly-resonant two-polarization transmission guided-mode resonance filter,” AIP Adv. 8(11), 115228 (2018). [CrossRef]

17. M. Niraula, J. W. Yoon, and R. Magnusson, “Single-layer optical bandpass filter technology,” Opt. Lett. 40(21), 5062–5065 (2015). [CrossRef]

18. H. Hemmati, Y. H. Ko, and R. Magnusson, “Fiber-facet-integrated guided-mode resonance filters and sensors: experimental realization,” Opt. Lett. 43(3), 358–361 (2018). [CrossRef]

19. H. J. Shin and G. Ok, “Terahertz guided mode resonance sensing platform based on freestanding dielectric materials: high Q-factor and tunable spectrum,” Appl. Sci. 10(3), 1013 (2020). [CrossRef]

20. A. Chu, C. Gréboval, N. Goubet, B. Martinez, C. Livache, J. L. Qu, P. Rastogi, F. A. Bresciani, Y. Prado, S. Suffit, S. Ithurria, G. Vincent, and E. Lhuillier, “Near unity absorption in nanocrystal based short wave infrared photodetectors using guided mode resonators,” ACS Photonics 6(10), 2553–2561 (2019). [CrossRef]

21. Y. Khorrami, D. Fathi, and R. Rumpf, “Guided-mode resonance filter optimal inverse design using one- and two- dimensional grating,” J. Opt. Soc. Am. B 37(2), 425–432 (2020). [CrossRef]

22. C. Maës, G. Vincent, F. G. Flores, L. Cerutti, R. Haïdar, and T. Taliercio, “Infrared spectral filter based on all-semiconductor guided-mode resonance,” Opt. Lett. 44(12), 3090–3093 (2019). [CrossRef]

23. R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992). [CrossRef]

24. A. F. Kaplan, T. Xu, and L. J. Guo, “High efficiency resonance-based spectrum filters with tunable transmission bandwidth fabricated using nanoimprint lithography,” Appl. Phys. Lett. 99(14), 143111 (2011). [CrossRef]

25. X. Chong, E. Li, K. Squire, and A. X. Wang, “On-chip near-infrared spectroscopy of CO₂ using high resolution plasmonic filter array,” Appl. Phys. Lett. 108(22), 221106 (2016). [CrossRef]

26. X. Chong, E. Li, F. Ren, and A. X. Wang, “Broadband on-chip near-infrared spectroscopy based on a plasmonic grating filter array,” Opt. Lett. 41(9), 1913–1916 (2016). [CrossRef]

27. D. B. Mazulquim, K. J. Lee, J. W. Yoon, L. V. Muniz, B. H. V. Borges, L. G. Neto, and R. Magnusson, “Efficient band-pass color filters enabled by resonant modes and plasmons near the Rayleigh anomaly,” Opt. Express 22(25), 30843–30851 (2014). [CrossRef]

28. Y. Yang, L. Guo, M. Y. Lyu, and Z. Sun, “Resonances of hybridized bound plasmon modes in optically-thin metallic nanoslit arrays for narrow-band transmissive filtering,” Optik 127(5), 2784–2788 (2016). [CrossRef]

29. R. He, C. Chen, R. Zhang, L. Chen, and J. Guo, “Dual dielectric cap gold nanoslits array optical resonance filter with large figure-of-merit,” Opt. Express 28(22), 32456–32467 (2020). [CrossRef]

30. R. R. Boye and R. K. Kostuk, “Investigation of the effect of finite grating size on the performance of guided-mode resonance filters,” Appl. Opt. 39(21), 3649–3653 (2000). [CrossRef]

31. S. T. Thurman and G. M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt. 42(16), 3225 (2003). [CrossRef]

32. D. Fleischman, K. T. Fountaine, C. R. Bukowsky, G. L. Tagliabue, L. A. Sweatlock, and A. A. Harry, “High spectral resolution plasmonic color filters with subwavelength dimensions,” ACS Photonics 6(2), 332–338 (2019). [CrossRef]

33. W. X. Liu, Z. Q. Lai, H. Guo, and Y. Liu, “Guided-mode resonance filters with shallow grating,” Opt. Lett. 35(6), 865–867 (2010). [CrossRef]

34. R. Sayeed, M. Mamun, V. Avrutin, and U. Ozgur, “Pixel-scale miniaturization of guided mode resonance transmission filters in short wave infrared,” Opt. Express 30(8), 12204–12214 (2022). [CrossRef]

35. R. C. Ng, J. C. Garcia, J. R. Greer, and K. T. Fountaine, “Miniaturization of a-Si guided mode resonance filter arrays for near-IR multi-spectral filtering,” Appl. Phys. Lett. 117(11), 111106 (2020). [CrossRef]

36. E. D. Palik, Handbook of Optical Constants of Solids, 1st ed. (Academic, 1998).

37. L. Rayleigh, “III. Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14(79), 60–65 (1907). [CrossRef]

38. R. He, C. Chen, R. Shen, E. T. Hu, R. Zhang, L. Chen, and J. Guo, “Weakly coupled hybrid guided mode resonance optical transmission filter,” J. Opt. Soc. Am. B 39(4), 925–933 (2022). [CrossRef]

Performance of finite-size metal-dielectric nanoslits metasurface optical filters

Abstract

1. Introduction

2. FDTD simulation of finite-size nanoslits metasurface optical filters

2.1 Finite dimension size in the periodic direction of nanoslits

2.2 Finite dimension size in the direction parallel to nanoslits

3. Fourier transform analysis of finite-size nanoslits optical filter

3.1 Finite dimension size in the periodic direction of the nanoslits

3.2 Finite size in the direction parallel to the nanoslits

4. Summary

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Equations (16)

Optics Express