Near-infrared quarter-waveplate with near-unity polarization conversion efficiency based on silicon nanowire array Yanmeng Dai,1 Hongbing Cai,2 Huaiyi Ding,1 Zhen Ning,1 Nan Pan,2,3 Hong Zhu,1 Qinwei Shi,2 and Xiaoping Wang1,2,3,* 1 Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China 3 Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China * [email protected] 2
Abstract: Metasurfaces made of subwavelength resonators can modify the wave front of light within the thickness much less than free space wavelength, showing great promises in integrated optics. In this paper, we theoretically show that electric and magnetic resonances supported simultaneously by a subwavelength nanowire with high refractive-index can be utilized to design metasurfaces with near-unity transmittance. Taking silicon nanowire for instance, we design numerically a near-infrared quarter-waveplate with high transmittance using a subwavelength nanowire array. The operation bandwidth of the waveplate is 0.14 μm around the center wavelength of 1.71 μm. The waveplate can convert a 45° linearly polarized incident light to circularly polarized light with conversion efficiency ranging from 94% to 98% over the operation band. The performance of quarter waveplate can in principle be tuned and improved through optimizing the parameters of nanowire arrays. Its compatibility to microelectronic technologies opens up a distinct possibility to integrate nanophotonics into the current silicon-based electronic devices. ©2015 Optical Society of America OCIS codes: (160.3918) Metamaterials; (350.4238) Nanophotonics and photonic crystals; (050.5080) Phase shift; (260.5430) Polarization.
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#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8929
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1. Introduction Phase accumulation when lights propagate in a dielectric medium is frequently used to shaping the wave fronts in conventional optical components such as lenses and waveplates. For example, birefringences of crystals, which impose different propagation phases on the orthogonal components of the transmitted light, are adopted for the conventional waveplate design. However, this effect is very weak in natural crystals, resulting in waveplates with thickness of several hundred microns, impeding seriously their applications in integrated photonics. Recently, abrupt phase shifts introduced by plasmonic resonances have been proposed to design the metasurfaces [1], which can modify the wave front of light in a desired manner. A variety of optical devices, such as extraordinary refraction/reflection [1], lenses [2,3], waveplates [4,5] and holograms [6,7], have been reported with thicknesses substantially smaller than the free-space wavelengths, thus show great promises in the integrated optics and flat displayers [8]. However, there are two intrinsic limitations for the metasurfaces composed of a single plasmonic layer. The first one is the high ohmic losses at optical frequencies due to the large
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8930
imaginary part of permittivity of plasmonic materials, leading to low efficiency of metasurface devices. Accordingly, doped semiconductor plasmonic materials have been proposed to alleviate the ohmic losses [9]. Furthermore, Mie resonances associated with high refractive-index materials have also been suggested recently as an alternative choice to design metamaterial devices [10–12]. The constructed resonators have demonstrated broadband omnidirectional antireflection [13], suppressed backward scattering [14,15], zero refractive index metamterial [16], optical invisibility cloak [17], single layer perfect reflector [18,19] [20] and focusing lens [21]. The second limitation of single plasmonic layer metasurface is its symmetric forward and backward scattering, which can be solved by using multiple plasmonic layers [22]. The underlying principle is to match the impedance of vacuum by controlling the surface electric and magnetic polarizabilities of the metasurfaces [8,22]. Two layers with either electric or magnetic responses have been proposed to design such reflectionless metasurfaces [23,24]. For example, Pfeiffer and Grbic have introduced wire-capacitor circuits for electric polarized surface and split ring resonators to support the magnetic response [22]. However, the process and involved alignment for the fabrication of such multilayer structures is highly complicated and arduous [25]. Gap plasmon resonance formed by coupling a plasmonic resonator and a metal film was also reported for the high efficient metasurface [26], but can only be operated in reflection configurations. Herein, we overcome the aforementioned limitations and report a novel efficient metasurface designed with high refractive-index dielectric resonators composed of silicon nanowire arrays. We first show theoretically that a single silicon nanowire can support both electric and magnetic Mie resonances, then we demonstrate numerically that the metasurface with near-unity transmittance can be realized when the nanowires are arranged into a subwavelength array. Furthermore, a highly efficient near infrared quarter-waveplate based on silicon nanowire array is designed, which can operate over a bandwidth of 0.14 μm at the center wavelength of 1.71 μm, with polarization conversion efficiency larger than 94%. The thickness of the designed waveplate is about one third of the operating wavelengths, thus shows a great potential for integrated optics and nanophotonics. In addition, we illustrate that the center wavelength and the bandwidth of the quarter-waveplate, as well as the polarization conversion efficiency, can be further tuned and optimized through varying the parameters of the nanowire array [27]. The materials and designed structure are compatible with CMOS technology, and are thus appealing in terms of nanophotonic integration and mass-production. 2. Results and discussion Figure 1(a) shows a schematic of the cross section of the silicon nanowire, which is infinite in the direction perpendicular to the XY plane. The dimensions along the x-axis (width) and y-axis (height) are 240 nm and 340 nm, respectively. The incident light is along the negative y-axis, with the electric field is either along the x-axis (TM mode) or z-axis (TE mode). The scattering efficiency (the ratio of scattering cross section with respect to the nanowire width) of the nanowire is numerically simulated using the commercial software Lumerical FDTD Solutions and the refractive index of silicon is from [28]. Figure 1(b) shows the simulated wavelength dependence of scattering efficiency for both the TM and TE modes. As seen, the nanowire can scatter the incident light significantly over a broadband spectrum for both TM (black line) and TE (red line) modes, implying the strong Mie resonances supported by the nanowire. In particular, for the TM light, two scattering peaks emerge in the near infrared range with wavelengths of 1.06 μm and 1.4 μm, which have been suggested to be relevant to the electric dipole (ED) and magnetic dipole (MD) resonances, respectively [29]. To further identify the origination of these resonances, the distributions of magnetic field (Hz, normalized to that of the incident light in linear scale) at the wavelengths of 1.06 μm and 1.4 μm are simulated and the results are shown in Figs. 1(c) and 1(d), respectively, where the white lines indicate the boundaries of the nanowire. As seen in Fig. 1(c), two nodes located on the y-axis can be clearly
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8931
observed, which can be attributed to an ED oscillating along the x-axis. In Fig. 1(d), the magnetic field exhibits almost azimuthally symmetric distribution inside the nanowire and decays radially away from the center of the nanowire, implying a magnetic displacement current generated by a MD oscillating along the z-axis. Note that, due to coexistence of ED and MD resonances, the field distributions shown in Figs. 1(c) and 1(d) are not perfectly symmetric about the center of the nanowire.
Fig. 1. (a) Schematic of the cross section of silicon nanowire and the simulation model. Nanowire is infinite in the direction perpendicular to XY plane, and its dimensions along the x-axis (width) and y-axis (height) are 240 nm and 340 nm, respectively. TF and SF represent total field and scattering field. (b) Scattering efficiency of the nanowire for TM (black solid line) and TE (red dashed line) polarized incident light. Magnetic field distributions for TM mode at the wavelengths of (c) 1.06 μm and (d) 1.4 μm. The white lines in (c) and (d) indicate the boundaries of the nanowire.
Besides the modes and near field distribution of these resonances, the far-field scattering behavior of the nanowire can also be well interpreted by the coherent interference of fields radiated from the ED and MD. Although the analytical Mie theory is successfully applied for the calculation of ED and MD moments of the nanowire with circular cross section [29] and nanosphere [30], it is unsuitable to the nanowire with rectangle cross section. Therefore, we retrieve the ED and MD moments alternatively by fitting the numerically simulated far-field scattering patterns of the nanowire, using a correlated electric–magnetic-dipole (EMD) model. The model assumes that, for TM incident light, the scattering field of nanowire can be treated approximately as the interference pattern resulted from the z-axis orientated MD and the x-axis oscillated ED. The magnetic field H zm radiated by the MD moment m, is apparently independent of scattering angle and can be described by H zm = Am [31], in which A = iω k
8π k0 ρ ei ( k ρ −π / 4) , where ω and k0 are the frequency and wave vector of the incident
light in the vacuum, respectively, and ρ is the distance from the center of the nanowire to the #231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8932
far-field point. In the numerical simulations, we maintained ρ = 1 mm unchanged. On the other hand, the magnetic field H zp radiated by the ED moment p, is described by H zp = − Ap sin θ , demonstrating a sinusoidal dependence on the angle θ between the scattering direction and the x-axis, as shown in Fig. 1(a). The minus sign of H zp means that it is in phase (out of phase) along the negative (positive) y-axis with respect to H zm , on the condition that the ED and MD oscillate in phase with each other, resulting in the predominant forward scattering. The total magnetic field radiated by both the ED and MD, H zT , can be further obtained as: 2
H zT (θ ) = H zm + H zp
2
2 2 2 = A m 1 − 2 Re ( p m ) sin θ + p m sin 2 θ
(1)
Therefore, the ED and MD moments of nanowire can be deduced through fitting the numerically simulated far-field scattering patterns with Eq. (1). Figure 2(a) shows several representative simulation (solid lines) and fitting results (dotted line) of far field scattering patterns at 1.06 μm (black line), 1.6 μm (red line) and 1.9 μm (blue line), respectively. As seen in Fig. 2(a), the scattering field is sensitive to the scattering angle, and the fitting results are well consistent with the simulated ones, indicating the rationality of the EMDs model.
Fig. 2. Simulated (solid lines) and EMDs fitted (dashed-dotted lines) far-field scattering pattern at the wavelengths of 1.06 μm, 1.6 μm and 1.9 μm. (b) The retrieved fitting parameters of amplitude of MD moment (black line), amplitude of ED moment (red line) and the real part of ED moment (blue line). The units for ED and MD moments are C·m and A·m·s, respectively.
Figure 2(b) displays the wavelength dependences of the retrieved ED and EM moments. One can see that the absolute values of MD |m| (black line) show a Lorentz-like line shape with the resonant wavelength around 1.42 μm, thus further confirming that the scattering peak around 1.4 μm in Fig. 1(b) originates from the magnetic dipole resonance. Similarly, the absolute values of ED |p| (red line) exhibit a resonant peak around 1.08 μm, suggesting that the scattering peak around 1.06 μm in Fig. 1(b) comes from the electric dipole resonance. Another peak of |p| around 1.52 μm might be related to the Rayleigh scattering. It is worth mentioning that, for simplicity, the MD is assumed to be purely real in the above fitting process, and the real part of ED Re(p) is also plotted in Fig. 2(b) as blue line. For the wavelengths larger than 1.6 μm, it can be seen that the Re(p) is equal to |p|, implying that the ED oscillates in phase with the MD and therefore results in the predominate forward scattering. In the wavelength region of 1.1μm - 1.4 μm, Re(p) 1.6 μm.
Fig. 3. The transmittances for the Si nanowires array calculated by the coupled dipoles model (blue line), and the numerical simulation results for the arrays with periodicity of 0.4 μm (black line) and 0.6 μm (red line).
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8934
To demonstrate the applications of such high transmission array in metasurface designs, we construct a quarter-waveplate with high polarization conversion efficiency. The structure of the designed quarter waveplate is shown in the inset of Fig. 4(a), which consists of a subwavelength array of Si and SiO2 nanowires with rectangular cross sections sited on SiO2 substrate. The top SiO2 layer can also be used as an anti-reflection layer to improve the transmittance of TE light, and thus the efficiency of the waveplate. The widths of Si and SiO2 nanowires are the same of 0.2 μm, while the heights of Si and SiO2 nanowires are 0.36 μm and 0.24 μm, respectively. The periodicity of the array is 0.4 μm. The transmittances of the structure are numerically simulated and the results relevant to the TM (solid black line) and TE (dashed black line) illuminations are shown in Fig. 4(a). As seen, the transmittances are always larger than 0.65, and approaches to unity with increasing wavelength. In particular, the transmittance exceeds 90% when the wavelength λ > 1.4 μm for TM mode and λ > 1.65 μm for TE mode, indicating a broadband operation for high transmission applications. The transmission ratio between transmittances of TM illumination and TE is also shown as blue line. Figure 4(b) shows the phase difference Δφ between the transmitted lights of TM and TE modes. One can find that Δφ approaches −90° at the wavelength near 1.7 μm, indicating the character of quarter waveplate. The bandwidth criterions for the designed waveplate are normally defined as the transmission ratio ranging from 0.9 to 1.1 and Δφ satisfying −95° < Δφ < −85° [5].
Fig. 4. (a) Transmittances of the waveplate for TM mode (black solid line) and TE mode (dashed black line). Blue line indicates the transmission ratio of TM mode to TE one. (b) The phase differences Δφ between the transmitted lights for TM and TE mode.
In Fig. 4(b), a gray-shaded region illustrates that, when the wavelength ranges from 1.64 μm to 1.78 μm, Δφ satisfies the above criterion, and the corresponding transmission ratio is in the range of 1.07 to 1.01. The results indicate that the designed quarter-waveplate can well be operated in the near infrared region and with a remarkable bandwidth of 0.14 μm. Note that, because the bandwidth of quarter waveplate is dominantly limited by the criterion of phase difference, a broadband quarter waveplate can be anticipated through further optimization. Generally, a quarter-waveplate can convert a linearly polarized light into an elliptically polarized light with different ellipticity, depending on the polarization angle of the incident light. In particular, a 45° linearly polarized light can be converted into a circular polarized light with efficiency of 100% after passing through an ideal quarter-waveplate. In this context, the ellipticity of transmitted light and the polarization conversion efficiency of the quarter-waveplate depicted in Fig. 4 are characterized, in which the incident light is linearly polarized with the polarization angle of 45°, as shown in the inset of Fig. 5(a). The transmitted light is generally elliptically polarized, and its ellipticity χ can be calculated according to
χ=
1 + 1 − sin 2 ( 2 β ) sin 2 Δϕ 1 − 1 − sin 2 ( 2β ) sin 2 Δϕ
(3)
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8935
where Δφ is the phase difference of TE and TM mode shown in Fig. 4(b), and sin ( 2 β ) = 2 Ex Ez ( Ex2 + E z2 ) , here E x and Ez are the amplitudes of different components of the transmitted light. As shown in Fig. 5(a), the highest ellipticity of 0.98 can be realized at the wavelength of 1.71 μm, very close to χ = 1 for a perfect circular polarized light. Furthermore, we find that, for the wavelength ranging from 1.64 μm to 1.78 μm [gray-shaded region in Fig. 5(a)], the ellipticity of the waveplate is larger than 0.9, indicating that the designed silicon arrays can act as a quarter-waveplate with high quality.
Fig. 5. (a) The ellipticity of the transmitted light. The inset shows the polarization angle of the incident light and schematic of the transmitted elliptically polarized light. (b) The polarization conversion efficiency of the waveplate.
The polarization conversion efficiency of the waveplate, defined as the ratio of the transmitted power to that of the incident light, is also calculated for the designed quarter-waveplate. As shown in Fig. 5(b), in the wavelength range of 1.64 μm to 1.78 μm, the efficiency is in the range of 94% to 98%. Again, we consider that the conversion efficiency can be further improved by optimizing the parameters of the designed waveplate. Although we have successfully demonstrated that the high efficient waveplate can be realized by silicon nanowire arrays, it is worth emphasizing that the response wavelength and the bandwidth of the waveplate can be further tuned and optimized through varying the parameters such as the thickness of top SiO2 layer, the width and height of the Si nanowire and the periodicity of the arrays. To obtain global optimized values for all these parameters, extensive optimization simulations should be done. Taking an example, we have investigated the influence of the nanowire width on the properties of the designed waveplate, while the periodicity, the heights of Si and SiO2 remain unchanged at 0.4 μm, 0.36 μm and 0.24 μm, respectively. Figures 6(a) and 6(b) show the dependence of the transmittance for TM and TE modes on the nanowire width and the wavelength of the incident light, respectively, in which the white lines correspond to the transmittances of 0.9. It can be seen from Fig. 6(a) that the transmittance of TM mode is insensitive to the variation of the nanowire width, especially in the long wavelength region. On the contrary, for TE mode in Fig. 6(b), the peak of transmittance appears and shows a parabolic dependence on the nanowire width. Moreover, the maps of phase difference Δφ versus the nanowire width is also shown in Fig. 6(c), in which the region enclosed by two white lines satisfies the criterion of −95° < Δφ < −85°. Figures 6(a)-6(c) enable us to find the optimized nanowire width for high efficient quarter-waveplate at the desired wavelength. For example, to design a quarter-waveplate working at 1.5 μm, two regions satisfied to the phase criterion [marked with black dots in Fig. 6(c)] are first chosen, and then the optimal nanowire width can be selected to maximize the transmittances of both TM and TE modes. In this case, the array with nanowire width of 0.115 μm is preferential as compared to that of 0.23 μm.
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8936
Fig. 6. The influences of the nanowire width on the transmittances of (a) TM mode, (b) TE mode, and (c) phase differences Δφ. (d) The characterization of a designed broadband waveplate. The transmittances for TM mode (solid black line), TE mode (dashed black line) and phase differences (blue line) of the transmitted light are shown.
With the aid of Fig. 6(c), we are able to design not only the quarter-waveplate with high transmittance efficiency as demonstrated above, but also the quarter-waveplate with broad phase bandwidth.To address this issue, we show such a broadband quarter-waveplate with the nanowire width of 0.23 μm in Fig. 6(d). The solid and dashed black lines indicate the transmittances for TM and TE polarized lights, respectively. Phase difference Δφ, depicted as blue line, satisfies the criterion of −95° < Δφ < −85° in the wavelength range of 1.27 μm to 1.75 μm, indicating the phase bandwidth of 0.48 μm, much larger than the result of 0.14 μm shown in Fig. 4(b). However, in this wavelength region, the transmission ratio becomes in the range of 0.93 to 1.21, implying the bandwidth of the waveplate limited by the amplitude of transmitted light. Therefore, in this case, the waveplate will convert circular polarized light into linearly polarized light with wavelength depended polarization angles, instead of 45° for an ideal quarter-waveplate. 3. Conclusion
We have shown that a single silicon nanowire can support the electric and magnetic dipole resonances simultaneously, leading to the enhanced forward scattering. The feature enables us to realize the high efficient metasurfaces with a subwavelength silicon nanowire array. We have successfully designed a quarter-waveplate working at the center wavelength of 1.71 μm with the operation band of 0.14 μm. The waveplate works in a transmission configuration with the polarization conversion efficiency exceeding 94%. We also demonstrate that the performance of waveplate can be tuned and improved through optimizing the nanowire width of the array. The waveplate thickness of 0.6 μm is only about one third of the operating wavelength, facilitating the development of integrated optical devices. Moreover, the prediction for high
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8937
efficient waveplates implemented by silicon nanowire arrays and their compatibility to the mature microelectronic technologies, offer a feasible strategy to integrate nanophotonic with silicon-based electronic devices. Acknowledgments
This work is supported by MOST of China (2011CB921403), NSFC (under Grant Nos. 11374274, 11004179 and 21421063) as well as by the Strategic Priority Research Program (B) of the CAS (XDB01020000).
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8938
Abstract: Metasurfaces made of subwavelength resonators can modify the wave front of light within the thickness much less than free space wavelength, showing great promises in integrated optics. In this paper, we theoretically show that electric and magnetic resonances supported simultaneously by a subwavelength nanowire with high refractive-index can be utilized to design metasurfaces with near-unity transmittance. Taking silicon nanowire for instance, we design numerically a near-infrared quarter-waveplate with high transmittance using a subwavelength nanowire array. The operation bandwidth of the waveplate is 0.14 μm around the center wavelength of 1.71 μm. The waveplate can convert a 45° linearly polarized incident light to circularly polarized light with conversion efficiency ranging from 94% to 98% over the operation band. The performance of quarter waveplate can in principle be tuned and improved through optimizing the parameters of nanowire arrays. Its compatibility to microelectronic technologies opens up a distinct possibility to integrate nanophotonics into the current silicon-based electronic devices. ©2015 Optical Society of America OCIS codes: (160.3918) Metamaterials; (350.4238) Nanophotonics and photonic crystals; (050.5080) Phase shift; (260.5430) Polarization.
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#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8929
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1. Introduction Phase accumulation when lights propagate in a dielectric medium is frequently used to shaping the wave fronts in conventional optical components such as lenses and waveplates. For example, birefringences of crystals, which impose different propagation phases on the orthogonal components of the transmitted light, are adopted for the conventional waveplate design. However, this effect is very weak in natural crystals, resulting in waveplates with thickness of several hundred microns, impeding seriously their applications in integrated photonics. Recently, abrupt phase shifts introduced by plasmonic resonances have been proposed to design the metasurfaces [1], which can modify the wave front of light in a desired manner. A variety of optical devices, such as extraordinary refraction/reflection [1], lenses [2,3], waveplates [4,5] and holograms [6,7], have been reported with thicknesses substantially smaller than the free-space wavelengths, thus show great promises in the integrated optics and flat displayers [8]. However, there are two intrinsic limitations for the metasurfaces composed of a single plasmonic layer. The first one is the high ohmic losses at optical frequencies due to the large
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8930
imaginary part of permittivity of plasmonic materials, leading to low efficiency of metasurface devices. Accordingly, doped semiconductor plasmonic materials have been proposed to alleviate the ohmic losses [9]. Furthermore, Mie resonances associated with high refractive-index materials have also been suggested recently as an alternative choice to design metamaterial devices [10–12]. The constructed resonators have demonstrated broadband omnidirectional antireflection [13], suppressed backward scattering [14,15], zero refractive index metamterial [16], optical invisibility cloak [17], single layer perfect reflector [18,19] [20] and focusing lens [21]. The second limitation of single plasmonic layer metasurface is its symmetric forward and backward scattering, which can be solved by using multiple plasmonic layers [22]. The underlying principle is to match the impedance of vacuum by controlling the surface electric and magnetic polarizabilities of the metasurfaces [8,22]. Two layers with either electric or magnetic responses have been proposed to design such reflectionless metasurfaces [23,24]. For example, Pfeiffer and Grbic have introduced wire-capacitor circuits for electric polarized surface and split ring resonators to support the magnetic response [22]. However, the process and involved alignment for the fabrication of such multilayer structures is highly complicated and arduous [25]. Gap plasmon resonance formed by coupling a plasmonic resonator and a metal film was also reported for the high efficient metasurface [26], but can only be operated in reflection configurations. Herein, we overcome the aforementioned limitations and report a novel efficient metasurface designed with high refractive-index dielectric resonators composed of silicon nanowire arrays. We first show theoretically that a single silicon nanowire can support both electric and magnetic Mie resonances, then we demonstrate numerically that the metasurface with near-unity transmittance can be realized when the nanowires are arranged into a subwavelength array. Furthermore, a highly efficient near infrared quarter-waveplate based on silicon nanowire array is designed, which can operate over a bandwidth of 0.14 μm at the center wavelength of 1.71 μm, with polarization conversion efficiency larger than 94%. The thickness of the designed waveplate is about one third of the operating wavelengths, thus shows a great potential for integrated optics and nanophotonics. In addition, we illustrate that the center wavelength and the bandwidth of the quarter-waveplate, as well as the polarization conversion efficiency, can be further tuned and optimized through varying the parameters of the nanowire array [27]. The materials and designed structure are compatible with CMOS technology, and are thus appealing in terms of nanophotonic integration and mass-production. 2. Results and discussion Figure 1(a) shows a schematic of the cross section of the silicon nanowire, which is infinite in the direction perpendicular to the XY plane. The dimensions along the x-axis (width) and y-axis (height) are 240 nm and 340 nm, respectively. The incident light is along the negative y-axis, with the electric field is either along the x-axis (TM mode) or z-axis (TE mode). The scattering efficiency (the ratio of scattering cross section with respect to the nanowire width) of the nanowire is numerically simulated using the commercial software Lumerical FDTD Solutions and the refractive index of silicon is from [28]. Figure 1(b) shows the simulated wavelength dependence of scattering efficiency for both the TM and TE modes. As seen, the nanowire can scatter the incident light significantly over a broadband spectrum for both TM (black line) and TE (red line) modes, implying the strong Mie resonances supported by the nanowire. In particular, for the TM light, two scattering peaks emerge in the near infrared range with wavelengths of 1.06 μm and 1.4 μm, which have been suggested to be relevant to the electric dipole (ED) and magnetic dipole (MD) resonances, respectively [29]. To further identify the origination of these resonances, the distributions of magnetic field (Hz, normalized to that of the incident light in linear scale) at the wavelengths of 1.06 μm and 1.4 μm are simulated and the results are shown in Figs. 1(c) and 1(d), respectively, where the white lines indicate the boundaries of the nanowire. As seen in Fig. 1(c), two nodes located on the y-axis can be clearly
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8931
observed, which can be attributed to an ED oscillating along the x-axis. In Fig. 1(d), the magnetic field exhibits almost azimuthally symmetric distribution inside the nanowire and decays radially away from the center of the nanowire, implying a magnetic displacement current generated by a MD oscillating along the z-axis. Note that, due to coexistence of ED and MD resonances, the field distributions shown in Figs. 1(c) and 1(d) are not perfectly symmetric about the center of the nanowire.
Fig. 1. (a) Schematic of the cross section of silicon nanowire and the simulation model. Nanowire is infinite in the direction perpendicular to XY plane, and its dimensions along the x-axis (width) and y-axis (height) are 240 nm and 340 nm, respectively. TF and SF represent total field and scattering field. (b) Scattering efficiency of the nanowire for TM (black solid line) and TE (red dashed line) polarized incident light. Magnetic field distributions for TM mode at the wavelengths of (c) 1.06 μm and (d) 1.4 μm. The white lines in (c) and (d) indicate the boundaries of the nanowire.
Besides the modes and near field distribution of these resonances, the far-field scattering behavior of the nanowire can also be well interpreted by the coherent interference of fields radiated from the ED and MD. Although the analytical Mie theory is successfully applied for the calculation of ED and MD moments of the nanowire with circular cross section [29] and nanosphere [30], it is unsuitable to the nanowire with rectangle cross section. Therefore, we retrieve the ED and MD moments alternatively by fitting the numerically simulated far-field scattering patterns of the nanowire, using a correlated electric–magnetic-dipole (EMD) model. The model assumes that, for TM incident light, the scattering field of nanowire can be treated approximately as the interference pattern resulted from the z-axis orientated MD and the x-axis oscillated ED. The magnetic field H zm radiated by the MD moment m, is apparently independent of scattering angle and can be described by H zm = Am [31], in which A = iω k
8π k0 ρ ei ( k ρ −π / 4) , where ω and k0 are the frequency and wave vector of the incident
light in the vacuum, respectively, and ρ is the distance from the center of the nanowire to the #231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8932
far-field point. In the numerical simulations, we maintained ρ = 1 mm unchanged. On the other hand, the magnetic field H zp radiated by the ED moment p, is described by H zp = − Ap sin θ , demonstrating a sinusoidal dependence on the angle θ between the scattering direction and the x-axis, as shown in Fig. 1(a). The minus sign of H zp means that it is in phase (out of phase) along the negative (positive) y-axis with respect to H zm , on the condition that the ED and MD oscillate in phase with each other, resulting in the predominant forward scattering. The total magnetic field radiated by both the ED and MD, H zT , can be further obtained as: 2
H zT (θ ) = H zm + H zp
2
2 2 2 = A m 1 − 2 Re ( p m ) sin θ + p m sin 2 θ
(1)
Therefore, the ED and MD moments of nanowire can be deduced through fitting the numerically simulated far-field scattering patterns with Eq. (1). Figure 2(a) shows several representative simulation (solid lines) and fitting results (dotted line) of far field scattering patterns at 1.06 μm (black line), 1.6 μm (red line) and 1.9 μm (blue line), respectively. As seen in Fig. 2(a), the scattering field is sensitive to the scattering angle, and the fitting results are well consistent with the simulated ones, indicating the rationality of the EMDs model.
Fig. 2. Simulated (solid lines) and EMDs fitted (dashed-dotted lines) far-field scattering pattern at the wavelengths of 1.06 μm, 1.6 μm and 1.9 μm. (b) The retrieved fitting parameters of amplitude of MD moment (black line), amplitude of ED moment (red line) and the real part of ED moment (blue line). The units for ED and MD moments are C·m and A·m·s, respectively.
Figure 2(b) displays the wavelength dependences of the retrieved ED and EM moments. One can see that the absolute values of MD |m| (black line) show a Lorentz-like line shape with the resonant wavelength around 1.42 μm, thus further confirming that the scattering peak around 1.4 μm in Fig. 1(b) originates from the magnetic dipole resonance. Similarly, the absolute values of ED |p| (red line) exhibit a resonant peak around 1.08 μm, suggesting that the scattering peak around 1.06 μm in Fig. 1(b) comes from the electric dipole resonance. Another peak of |p| around 1.52 μm might be related to the Rayleigh scattering. It is worth mentioning that, for simplicity, the MD is assumed to be purely real in the above fitting process, and the real part of ED Re(p) is also plotted in Fig. 2(b) as blue line. For the wavelengths larger than 1.6 μm, it can be seen that the Re(p) is equal to |p|, implying that the ED oscillates in phase with the MD and therefore results in the predominate forward scattering. In the wavelength region of 1.1μm - 1.4 μm, Re(p) 1.6 μm.
Fig. 3. The transmittances for the Si nanowires array calculated by the coupled dipoles model (blue line), and the numerical simulation results for the arrays with periodicity of 0.4 μm (black line) and 0.6 μm (red line).
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8934
To demonstrate the applications of such high transmission array in metasurface designs, we construct a quarter-waveplate with high polarization conversion efficiency. The structure of the designed quarter waveplate is shown in the inset of Fig. 4(a), which consists of a subwavelength array of Si and SiO2 nanowires with rectangular cross sections sited on SiO2 substrate. The top SiO2 layer can also be used as an anti-reflection layer to improve the transmittance of TE light, and thus the efficiency of the waveplate. The widths of Si and SiO2 nanowires are the same of 0.2 μm, while the heights of Si and SiO2 nanowires are 0.36 μm and 0.24 μm, respectively. The periodicity of the array is 0.4 μm. The transmittances of the structure are numerically simulated and the results relevant to the TM (solid black line) and TE (dashed black line) illuminations are shown in Fig. 4(a). As seen, the transmittances are always larger than 0.65, and approaches to unity with increasing wavelength. In particular, the transmittance exceeds 90% when the wavelength λ > 1.4 μm for TM mode and λ > 1.65 μm for TE mode, indicating a broadband operation for high transmission applications. The transmission ratio between transmittances of TM illumination and TE is also shown as blue line. Figure 4(b) shows the phase difference Δφ between the transmitted lights of TM and TE modes. One can find that Δφ approaches −90° at the wavelength near 1.7 μm, indicating the character of quarter waveplate. The bandwidth criterions for the designed waveplate are normally defined as the transmission ratio ranging from 0.9 to 1.1 and Δφ satisfying −95° < Δφ < −85° [5].
Fig. 4. (a) Transmittances of the waveplate for TM mode (black solid line) and TE mode (dashed black line). Blue line indicates the transmission ratio of TM mode to TE one. (b) The phase differences Δφ between the transmitted lights for TM and TE mode.
In Fig. 4(b), a gray-shaded region illustrates that, when the wavelength ranges from 1.64 μm to 1.78 μm, Δφ satisfies the above criterion, and the corresponding transmission ratio is in the range of 1.07 to 1.01. The results indicate that the designed quarter-waveplate can well be operated in the near infrared region and with a remarkable bandwidth of 0.14 μm. Note that, because the bandwidth of quarter waveplate is dominantly limited by the criterion of phase difference, a broadband quarter waveplate can be anticipated through further optimization. Generally, a quarter-waveplate can convert a linearly polarized light into an elliptically polarized light with different ellipticity, depending on the polarization angle of the incident light. In particular, a 45° linearly polarized light can be converted into a circular polarized light with efficiency of 100% after passing through an ideal quarter-waveplate. In this context, the ellipticity of transmitted light and the polarization conversion efficiency of the quarter-waveplate depicted in Fig. 4 are characterized, in which the incident light is linearly polarized with the polarization angle of 45°, as shown in the inset of Fig. 5(a). The transmitted light is generally elliptically polarized, and its ellipticity χ can be calculated according to
χ=
1 + 1 − sin 2 ( 2 β ) sin 2 Δϕ 1 − 1 − sin 2 ( 2β ) sin 2 Δϕ
(3)
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8935
where Δφ is the phase difference of TE and TM mode shown in Fig. 4(b), and sin ( 2 β ) = 2 Ex Ez ( Ex2 + E z2 ) , here E x and Ez are the amplitudes of different components of the transmitted light. As shown in Fig. 5(a), the highest ellipticity of 0.98 can be realized at the wavelength of 1.71 μm, very close to χ = 1 for a perfect circular polarized light. Furthermore, we find that, for the wavelength ranging from 1.64 μm to 1.78 μm [gray-shaded region in Fig. 5(a)], the ellipticity of the waveplate is larger than 0.9, indicating that the designed silicon arrays can act as a quarter-waveplate with high quality.
Fig. 5. (a) The ellipticity of the transmitted light. The inset shows the polarization angle of the incident light and schematic of the transmitted elliptically polarized light. (b) The polarization conversion efficiency of the waveplate.
The polarization conversion efficiency of the waveplate, defined as the ratio of the transmitted power to that of the incident light, is also calculated for the designed quarter-waveplate. As shown in Fig. 5(b), in the wavelength range of 1.64 μm to 1.78 μm, the efficiency is in the range of 94% to 98%. Again, we consider that the conversion efficiency can be further improved by optimizing the parameters of the designed waveplate. Although we have successfully demonstrated that the high efficient waveplate can be realized by silicon nanowire arrays, it is worth emphasizing that the response wavelength and the bandwidth of the waveplate can be further tuned and optimized through varying the parameters such as the thickness of top SiO2 layer, the width and height of the Si nanowire and the periodicity of the arrays. To obtain global optimized values for all these parameters, extensive optimization simulations should be done. Taking an example, we have investigated the influence of the nanowire width on the properties of the designed waveplate, while the periodicity, the heights of Si and SiO2 remain unchanged at 0.4 μm, 0.36 μm and 0.24 μm, respectively. Figures 6(a) and 6(b) show the dependence of the transmittance for TM and TE modes on the nanowire width and the wavelength of the incident light, respectively, in which the white lines correspond to the transmittances of 0.9. It can be seen from Fig. 6(a) that the transmittance of TM mode is insensitive to the variation of the nanowire width, especially in the long wavelength region. On the contrary, for TE mode in Fig. 6(b), the peak of transmittance appears and shows a parabolic dependence on the nanowire width. Moreover, the maps of phase difference Δφ versus the nanowire width is also shown in Fig. 6(c), in which the region enclosed by two white lines satisfies the criterion of −95° < Δφ < −85°. Figures 6(a)-6(c) enable us to find the optimized nanowire width for high efficient quarter-waveplate at the desired wavelength. For example, to design a quarter-waveplate working at 1.5 μm, two regions satisfied to the phase criterion [marked with black dots in Fig. 6(c)] are first chosen, and then the optimal nanowire width can be selected to maximize the transmittances of both TM and TE modes. In this case, the array with nanowire width of 0.115 μm is preferential as compared to that of 0.23 μm.
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8936
Fig. 6. The influences of the nanowire width on the transmittances of (a) TM mode, (b) TE mode, and (c) phase differences Δφ. (d) The characterization of a designed broadband waveplate. The transmittances for TM mode (solid black line), TE mode (dashed black line) and phase differences (blue line) of the transmitted light are shown.
With the aid of Fig. 6(c), we are able to design not only the quarter-waveplate with high transmittance efficiency as demonstrated above, but also the quarter-waveplate with broad phase bandwidth.To address this issue, we show such a broadband quarter-waveplate with the nanowire width of 0.23 μm in Fig. 6(d). The solid and dashed black lines indicate the transmittances for TM and TE polarized lights, respectively. Phase difference Δφ, depicted as blue line, satisfies the criterion of −95° < Δφ < −85° in the wavelength range of 1.27 μm to 1.75 μm, indicating the phase bandwidth of 0.48 μm, much larger than the result of 0.14 μm shown in Fig. 4(b). However, in this wavelength region, the transmission ratio becomes in the range of 0.93 to 1.21, implying the bandwidth of the waveplate limited by the amplitude of transmitted light. Therefore, in this case, the waveplate will convert circular polarized light into linearly polarized light with wavelength depended polarization angles, instead of 45° for an ideal quarter-waveplate. 3. Conclusion
We have shown that a single silicon nanowire can support the electric and magnetic dipole resonances simultaneously, leading to the enhanced forward scattering. The feature enables us to realize the high efficient metasurfaces with a subwavelength silicon nanowire array. We have successfully designed a quarter-waveplate working at the center wavelength of 1.71 μm with the operation band of 0.14 μm. The waveplate works in a transmission configuration with the polarization conversion efficiency exceeding 94%. We also demonstrate that the performance of waveplate can be tuned and improved through optimizing the nanowire width of the array. The waveplate thickness of 0.6 μm is only about one third of the operating wavelength, facilitating the development of integrated optical devices. Moreover, the prediction for high
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8937
efficient waveplates implemented by silicon nanowire arrays and their compatibility to the mature microelectronic technologies, offer a feasible strategy to integrate nanophotonic with silicon-based electronic devices. Acknowledgments
This work is supported by MOST of China (2011CB921403), NSFC (under Grant Nos. 11374274, 11004179 and 21421063) as well as by the Strategic Priority Research Program (B) of the CAS (XDB01020000).
#231209 - $15.00 USD Received 22 Dec 2014; revised 18 Mar 2015; accepted 21 Mar 2015; published 31 Mar 2015 (C) 2015 OSA 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008929 | OPTICS EXPRESS 8938
