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Abstract
A large bandwidth and high-efficiency subwavelength quarter-wave plate (QWP) is an indispensable component of an integrated miniaturized optical system. The bandwidth of existing plasmonic quarter-wave plates with a transmission efficiency of more than 50% is less than 320 nm in the near-infrared band. In this paper, a metallic quarter-wave plate with a bandwidth of 600 nm (0.95–1.55 µm) and an average transmittance of more than 70% has been designed and shows excellent potential to be used in miniaturized optical polarization detection systems and as an optical data storage device. For TE mode incident waves, this miniaturized optical element can be equivalent to a Fabry-Pérot (FP) resonator. Meanwhile, for the TM mode incident wave, the transmission characteristics of this structure are controlled by gap surface plasmon polaritons (G-SPPs) existing in the symmetric metal/insulator/metal (MIM) configuration.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Understanding the nature and interactions of light polarization has long been a subject of discourse. Manipulating the polarization state of light is essential for many practical applications, such as photonic communications [1], satellite remote sensing [2] and bio-optical imaging and sensing [3]. One of the most fundamental methods of manipulating polarization is the quarter-wave plate, in which one component of the electric field is subjected to a phase delay ($\mathrm{\pi }/2$), relative to the orthogonal electric field component. A conventional quarter-wave plate is a birefringent crystal that allows two orthogonal components of the electric field of a ray to propagate at slightly different speeds. However, the week birefringence effect of traditional optical crystals makes it difficult to get a thin device with enough phase difference, restricting the miniaturization and integration of optical systems [4–6].
A metasurface consisting of a planar array of meta-atoms with subwavelength period can flexibly and precisely control the phase and polarization information of light [7–15], which provides a compelling idea for the design of ultra-compact optical wave plate. Reflective metasurfaces [16–20], typically consisting of a top antenna array, a dielectric spacer, and a metal back reflection layer, can function as a quarter-wave plate by optimizing the structural parameters. However, an optical setup involving reflective devices is generally complex. In many cases, the wave plate needs to work in transmission mode rather than reflection mode.
There have been many reports on the use of a metasurface to design a transmitted plasmonic waveplate. In 2012, Yu et al. experimentally demonstrated a quarter-wave plate generating light with a high degree of circular polarization ($> 0.97$) from $\mathrm{\lambda }$ = 5 to 12 $\mathrm{\mu }$m [21]. In the same year, in 2012, Roberts et al. designed a plasmonic quarter-wave plate with a bandwidth of 50 nm (710–760 nm), which was achieved by introducing an asymmetry into the lengths of the arms of the crosses [22]. Similarly, based on the anisotropic optical response induced by the arm length in the orthogonal directions, in 2013, Yang et al. propose an ultrathin quarter-wave plate with a bandwidth of 80 nm consisting of a periodic plane array of symmetrical L-shaped plasmonic antennas, whose ellipticity was 0.994 at an operating wavelength of 1550 nm [23]. In 2013, Zhao et al. used an interleaved silver nanorod array to achieve achromatic quarter-wave plate behavior for much of the visible spectrum [24]. In 2015, Li et al. presented a linear-to-circular (LTC) polarization converter composed of a single-layer gold nanorod array with a 40% transmission efficiency from $\mathrm{\lambda }$ = 1100–1700 $\textrm{n}$m [25]. In 2017, Zhu et al. used 10-nm-thick subwavelength broken rectangular annulus (BRA) arrays to increase the phase anisotropy induced by localized surface plasmons significantly. It has a 120 nm circular-to-linear (CTL) polarization bandwidth. The transmission efficiency is 0.4 at 1550 nm [26]. In addition, in 2017, Hu et al. used a thick one-dimensional metal grating forming an effective Fabry-Pérot (FP) resonator to design a quarter-wave plate with a bandwidth of 300 nm (1260–1560 nm) [27]. The transmission efficiency of the device is 0.93 at 1360 nm, which overcomes the 50% theoretical efficiency limit of ultrathin metasurfaces [28–31]. However, the fact that the entire quarter wave plate is submerged in the glass medium means that the device’s thickness can be considerable. In the same year, Li et al. proposed a quarter-wave plate consisting of a two-dimensional periodic array of cross-shaped nanobricks. The transmission efficiency and phase of the waveplate are sensitive to the interaction between double FP cavity resonances and double bulk magnetic resonances. The bandwidth with ellipticity greater than 0.9 is 140 nm, and the intensity of transmitted light is 0.85 at 1250 nm [32]. In 2019, Yu et al. designed an ultrathin quarter-wave plate with a periodic silver film $2 \times 2$ rectangular hole array with a thickness less than 30 nm. Although the transmission efficiency of the device is very low, only 40%, the effective bandwidth of the device in the near-infrared band is 525 nm [33], which is much higher than the previous work.
In this article, we numerically demonstrate a plasmonic quarter-wave plate composed of a silicon substrate and silver metal grating whose lateral gap is filled with silicon. The operation bandwidth is 600 nm (0.95–1.55 $\mathrm{\mu }\textrm{m}$). Moreover, the average transmission efficiency and circular polarizability in the bandwidth range are 0.7 and 0.97, respectively. The excellent broadband performance is illustrated by the interaction of two resonance modes, which consist of an effective FP resonance mode [34–38] for TE incident wave and G-SPPs existing in the symmetric metal/insulator/metal (MIM) configuration for TM incident wave. In addition, the substrate of the silver grating is silicon, which means that the manufacturing process of the metasurface is compatible with the Complementary Metal Oxide Semiconductor (CMOS) process.
2. Structure and analysis
As shown in Fig. 1(a), the silver grating is grown on the silicon substrate, and the gap between adjacent metal strips is filled with silicon. In addition, the optical medium above the silver grating is air. The incident light is directed from the silicon medium to the air. The optical constants of silver come from Ref [39]., and the refractive index of silicon is from the literature [40]. Meanwhile, the polarization directions of TE mode and TM mode are illustrated in Fig. 1(b). The full-wave numerical simulation model, called optical model A, is based on the commercial software COMSOL Multiphysics, which is used to calculate the intensity and phase information of the transmission mode of the miniaturized quarter-wave plate. In addition, periodic boundary conditions are applied on both sides of the X direction to truncate the electromagnetic field. A perfect matching layer (PML) is added on both sides of the Z direction to absorb electromagnetic reflection wave.
Fig. 1. (a). 3D view of the quarter-wave plate composed of a silicon substrate and silver metal grating whose lateral gap is filled with silicon. The lights are normally incident from the bottom. Figure 1(b). A numerical simulation model calculating the intensity and phase information of the transmission mode of the device.
The design tunability of the device by scaling up or down the dimensions is particularly important to note. The effects of different structure parameters on the performance of different linear polarizations in the wavelength range of 0.6–2.6 $\mathrm{\mu }$m is investigated. Figure 2(a) shows the effect of the width of the grating on the transmission of the TE incidences. It is seen that the transmission peaks corresponding to the MIM waveguide fundamental mode in the range of 1.3–1.9 $\mathrm{\mu }$m intensely blueshifts. In contrast, the transmission peaks in the range of 0.85–1.1$\mathrm{\mu }$m , caused by the higher-order waveguide mode, shifts slightly to a shorter wavelength as the metal width increases. The blue shift phenomenon shown in Fig. 2(a) may be due to the change of coupling between adjacent lateral waveguide modes with the increase of metal width a. On the contrary, the resonant peak in Fig. 2(b) red shifts as the grating height h increases. As shown in Fig. 2(c), when the period (P) is less than 300 nanometers, the resonance peak red shifts as P increases. However, when P is greater than 300 nanometers, the number of resonance peaks has increased. These phenomena may be due to the excitation of high-order FP cavity resonant modes. Figure 2(d,e,f) shows the effect of some geometric parameters on the transmission of TM incidence. As shown in Fig. 2(d), an increase in the grating width makes the transmission peak red shift. The transmission peak in Fig. 4(e) is also red shifted with increased grating height. Figure 2(f) shows that the corresponding transmission spectra remain almost unchanged except for the period of 360 nm, which may be explained by the excitation of high order mode in the case of TM incidence. Figure 2(g,i) shows that the phase difference increases sharply with the increase of metal width a, but decreases significantly with the increase of period P. This means that the device performance will deteriorate significantly when the metal width and period slightly deviate from the theoretical value. However, Fig. 2(h) shows that the phase difference increases very slowly with increased grating height h. Finally, an optimal quarter-wave plate whose duty cycle, thickness, and height are 0.87, 225 nm and 240 nm, respectively, is achieved.
Fig. 2. (a, b, c): The influence of the metal width a, grating height h and grating period P on the transmission spectrum for the TE mode incidence. Figure 2(d, e, f): The influence of the geometric structure parameter on the transmission spectrum for the TM mode incidence. Figure 2(g, h, i): The influence of the geometric parameter on the phase difference spectrum. The yellow area covers a phase difference space of 80 to 100 degrees. In the simulations, P=240 nm, a=32 nm, h=225 nm, unless specific indication
There are two transmission peaks at 1 $\mathrm{\mu }$m and 1.6 $\mathrm{\mu }$m in the case of the TE mode incidence, as shown in Fig. 3(a). Figure 3(b) shows a color map of the transmission intensity depending on wavelength $\mathrm{\lambda }$ and thickness h, in which there are three bending transmission peaks. Meanwhile, it is speculated that the lateral FP resonance modes (with m = 0, 1, and 2) can explain these three significant transmission peaks. The resonance modes in the optical MIM configuration composed of a metal strip, silicon medium and metal strip satisfy $2{k_z}{h_1} + 2{\mathrm{\Phi }_R} = 2m\pi $, where m is an integer and ${\mathrm{\Phi }_R}$ is the phase picked up by the waveguide mode by reflection at each of the openings. The fact that the mode order of the FP resonator is sensitive to the thickness h in the MIM waveguide structure has been reported in the literature [27,34]. It is necessary to introduce the electric field distribution of the device to understand the mechanism of optical resonance more intuitively. Figure 3(c) plots the distribution of the absolute value of the electric field intensity of the quarter-wave plate in the x-z section in the case of TE mode incidence at 1.6 $\mathrm{\mu }$m resonant wavelength, where there is a half period in each silicon medium in the z-axis direction (The absolute operator can reduce the period size by half). Furthermore, ${\mathrm{\Phi }_R}$ is supposed to be $- \mathrm{\pi }/2$, so we can naturally conclude that ${k_z}{h_1}$ is equal to 1/2*1/2 * 2$\mathrm{\pi }$ and m is 0. Figure 3(d) plots the distribution of the absolute value of the electric field intensity at 1 $\mathrm{\mu }$m resonant wavelength. There are 1.5 periods, which means ${k_z}{h_1}$ = 1/2*3/2 * 2$\mathrm{\pi }$ and m = 1. It is concluded that the transmittance of the quarter-wave plate is manipulated by a lateral FP resonance mode in case of TE mode incidence.
Fig. 3. Transmission spectrum and electric field intensity distribution of the metasurface. (a): Transmission spectrum in the case of the TE mode incidence. (b): Transmittance spectra depending on metallic strip thickness h. (c, d): electric field intensity distribution of the metasurface in x-z cross section for 1.6 $\mathrm{\mu }$m and 1 $\mathrm{\mu }$m, respectively. The white dotted line is the boundary of the metal.
Figure 4(a) shows the transmission spectrum of the device in the case of the TM mode incidence, and there are transmission peaks at 0.88$\mathrm{\mu }$m, 1 $\mathrm{\mu }$m, and 1.8 $\mathrm{\mu }$m. Considering the metasurface as a one-dimensional array composed of many isolated MIM waveguides is helpful to analyze its transmission spectrum. A two-dimensional numerical simulation model, called simulation model B, is used to analyze the propagation modes supported by an isolated MIM waveguide. In this model, the numerical port boundary is used on both sides of the Z direction. A perfect electric conductor (PEC) or perfect matching layer (PML) condition is used inside the metal, which is five wavelengths away from the interface between the metal and the silicon filling layer, as shown in Fig. 4(b). Moreover, surface plasmon polaritons (SPPs) excited at the interface between the metal and the silicon satisfy $\mathrm{\beta } = {k_0}\sqrt {\frac{{{\varepsilon _1}{\varepsilon _2}}}{{{\varepsilon _1} + {\varepsilon _2}}}} $, where ${\varepsilon _1}$ and ${\varepsilon _2}$ are the electric permittivity of dielectric and metal, respectively, and ${k_0}$ is the vacuum wave vector. Significantly, the SPPs cannot be directly excited by an incident plane wave because of the wavevector mismatch. However, it can be overcome by using a prism above the surface [41,42] and by periodically corrugating the interface [43] or by waveguide [44]. Figure 4(c) illustrates the dispersion curve in the near-infrared band. The dotted line represents SPPs existing on the surface between silver and lateral silicon. The asterisk (circle) dot line indicates the fundamental mode supported in the MIM waveguide, which is obtained using model B for the PEC (PML) boundary condition. The solid line corresponds to G-SPP existing in the symmetric MIM configuration [45,46]. The G-SPP dispersion relation are:
Fig. 4. (a): Transmission spectrum for the TM incidence. Figure 4(b): A numerical simulation model calculating waveguide mode. The thickness of silver and silicon is $5{\lambda _0}$ and 208 nm, respectively. Figure 4(c): Dispersion relation. The dotted line represents SPPs. The boundary condition of the model B corresponding to the asterisk is Perfect electric conductor (PEC), and circle corresponds to Perfect matching layer (PML). The solid line corresponds to the G-SPP, and the thickness of the silicon dielectric layer is 208 nm, 50 nm and 20 nm, respectively. Figure 4(d): the magnetic field distribution of the fundamental mode obtained using the model B, the thickness of whose dielectric layer is equal to the thickness of the silicon gap in the metasurface of Fig. 1(a). The positive direction of z-axis is the propagation direction of waveguide. Figure 4(e, f): magnetic field intensity distribution of the metasurface in x-z cross section for 1000 nm and 1800nm, respectively. The white dotted line is the boundary of the metal.
The phase difference of quarter-wave plate $({\Delta \phi } )$ depends on the difference of refractive index of the material in two orthogonal directions, and it satisfies:
where ${\beta _1}\;\textrm{and}\;{\beta _2}$ are the propagating wave vectors for two orthogonal modes, and d is the propagation length. Furthermore, in the isolated MIM waveguide composed of infinite thickness silver layer and a 208 nm thick silicon, ${\beta _1}$ and ${\beta _2}$ represent the fundamental modes corresponding to the cutoff frequency of the TE and TM mode, respectively, and d is the thickness of the metal grating. Figure 5(a) shows the phase difference spectrum for different materials covering the infinite space above the metasurface. The solid line represents the phase difference information extracted by the port boundary conditions in model A, verifying that phase difference is independent of optical materials. The green triangle dotted line corresponds to the phase difference calculated by Eq. (3), where the fundamental mode wave vector $\mathrm{\beta }$ of the isolated MIM waveguide comes from optical model B. An almost accurate curve shape matching is shown in Fig. 5(a) in the wavelength range of 0.95–1.7 $\mathrm{\mu }$m, which means that the quarter-wave plate has such a large bandwidth explained by the collective action of an infinite number of laterally isolated waveguides. Although there is an ∼18 degrees error between the two curves, this can be explained by the weak coupling between adjacent MIM waveguides. In sacrificing the device’s compactness and replacing the upper air medium with silica or silicon medium, the transmittance of the device increases with the increase of the refractive index of the upper medium, as witnessed in Fig. 5(b). It is worth noting that the transmittance of a quarter -wave plate is close to 100% in the wavelength range of 0.95–1.38 $\mathrm{\mu }$m when the upper medium is silicon.Fig. 5. (a): The phase difference spectrum obtained by FEM and Eq. (3). Figure 5(b): transmission spectrum for different upper media
3. Result and discussions
In the preceding sections, we have thoroughly studied the optical properties of the nano-strip configuration in the TE and TM incidence, emphasizing parameter regions for which an efficient quarter-wave plate in transmission can be constructed. In this section, we describe a transmissive metasurface using the nano-strip dimensions of Sec. 2 with an average efficiency of 70%, which exhibits the characteristics of an optical quarter-wave plate in the near-infrared (0.95–1.55 $\mathrm{\mu }$m) band. The transmission spectrum of the plasmonic quarter-wave plate in the near-infrared band is shown in Fig. 6(a). It is seen that a sharp drop induced by the ± 1st order Rayleigh anomalies (RA) occurs at about 0.93 $\mathrm{\mu }$m, which can be predicted by the relation ${\lambda _{RA}} = {n_{Silicon}} \times P$. The transmittance $|{{T_{xx}}} |$ of the metasurface is greater than 0.6 for $\mathrm{\lambda } > 0.95\; \mathrm{\mu }\textrm{m}$ in the case of TM incidence, and average transmittance $|{{T_{yy}}} |$ is greater than 0.8 in the case of 0.95–1.7 $\mathrm{\mu }$m TE incident light. The efficiency of the metallic plasmonic quarter-wave plate breaks through the limit of 50% transmission of the thin metal metasurface. The phase difference between the projection components of the transmitted beam in two orthogonal polarization directions must satisfy the requirement of ${90^{\circ}} \pm {10^{\circ}}$ for an acceptable quarter-wave plate. Figure 6(a) also shows that the spectral range meeting the condition stretches from 0.95 to 1.55 $\mathrm{\mu}$m. Thus, the device we designed with a bandwidth of 600 nm and an average transmittance above 70% embodies the function of a quarter-wave plate.
Fig. 6. (a): Transmission coefficients for normal incident TE and TM mode and phase difference between x- and y-polarized light. Figure 6(b): Degree and angle of linear polarization (DoLP and AoLP, respectively) for normal incident left and right circularly polarization polarized light (LCP and RCP, respectively).
An alternative way to depict the figure of merit of the quarter-wave plate is through the degree of linear polarization (DoLP) and the angle of linear polarization (AoLP) defined by the Stokes parameters:
where ${s_0} = {|{E_x^T} |^2} + {|{E_y^T} |^2}$, ${s_1} = {|{E_x^T} |^2} - {|{E_y^T} |^2}$, and ${s_2} = E_x^T{({E_y^T} )^\ast } + E_y^T{({E_x^T} )^\ast }$. . Here, “*” represents the complex conjugate operator, the superscript “T” means the transmission electric field, and subscript “x(y)” refers to the component of the electric field in the x (y) direction. The DoLP of the transmitted light in the entire operation bandwidth (0.95–1.55 $\mathrm{\mu }$m) for circularly polarized incident light is larger than 0.97, as shown in Fig. 6(b), and whose polarization characteristic is equivalent to linear polarization. Figure 6(b) also illustrates the angle of linear polarization (AoLP), describing the polarization angle of the linearly transmitted light with respect to the x-axis for circularly polarized incident light [16]. Figure 6(b) shows that AoLP almost does not change in the operating bandwidth. AoLP is almost constant at ${\pm} {45^0}$ (the variation is less than ${3^0}$), where the sign of AoLP depends on the handedness of the incident wave. Table 1 shows a comparison of the characteristics of transmissive quarter-wave plates appeared in several recent articles. The quarter-wave plate in this article has the largest bandwidth and average circularity for the requirement that the device’s efficiency should exceed 0.5.
Table 1. Comparison of the characteristics of quarter-wave plates in the visible and near-infrared band
For a quarter-wave plate, a linearly polarized wave can also be converted into a circularly polarized wave. Figure 7(a,b) showshe Stokes parameters of the output light for normal incident 45 degree and −45 degree linearly polarized light, respectively. In addition, ${S_1}$, ${S_2}$, and ${S_3}$ are normalized to ${S_0}$ such that their values change between −1 and +1. A quantitative comparison is necessary by extracting the average errors for ${S_1}$, ${S_2}$, and ${S_3}$. Errors for the degree of linear and circular polarizations are defined as ${\left|{\left|{\sqrt {S_1^2 + S_2^2} /{S_0} - \sqrt {D_1^2 + D_2^2} /{D_0}} \right|} \right|^2}$ and ${|{|{{S_3}/{S_0} - {D_3}/{D_0}} |} |^2}$, respectively. Here, $|{|\textrm{x} |} |$ indicates that the absolute value operator acts on the element x, where (${S_1},{S_2},{S_3}$) are the Stokes parameters of transmitted light obtained using full -wave numerical simulations, and (${D_1},{D_2},{D_3}$) representing the theoretical Stokes parameter are (0,0,1) and (0,0,-1) for 45 degree and −45 degree linearly incident polarized light, respectively.
Fig. 7. (a,b): The Stokes parameters of transmitted light, Error of the degree of linear polarization (DOLP), and Error of the degree of circular polarization (DOCP) for normal incident 45 and −45 degree linearly polarized light. The yellow shadow spans the operating bandwidth. Figure 7(c): Transmission intensity for different incident polarizations.
Moreover, the average errors of the degree of linear and circular polarization are less than 0.03 and 0.0005, respectively, at operation bandwidth for ${\pm} {45^{\circ}}$ polarized incident light. Figure 7(c) shows that the output intensity of the device is weakly dependent on the orientation of the incident linear polarization, which is advantageous for practical application.
4. Conclusion
In conclusion. Numerically realizable metal grating configurations composed of a silicon substrate and silver metal strip whose lateral gap is filled with silicon are considered. For the TE mode incidence, the metasurface can be equivalent to a FP cavity. Meanwhile, for the TM incidence, its optical properties are determined by G-SPP existing in the symmetric metal/insulator/metal (MIM) configuration. The device performs a quarter-wave plate function over a bandwidth of 950–1550 nm, and the average transmission efficiency and circular polarizability in the bandwidth range are 0.7 and 0.97, respectively. This ultracompact quarter -wave plate has excellent potential to become an optical component of the miniaturized optical path.
Funding
Strategic Priority Research Program of Chinese Academy of Sciences (XDB43010000); National Natural Science Foundation of China (61835011, 12075244); Key Research Program of Frontier Science, Chinese Academy of sciences (QYZDY-SSW-JSC004); National Science and Technology Major Project ((2018ZX01005101-010);National Key Research and Development Program of China (2020YFB2206103).
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.
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