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Abstract
Existing transmission type optical quarter-wave plates based on metallic sub-wavelength structures can hardly realize transmission efficiency above 60%. And their working bandwidths are still very narrow. In this paper, we demonstrate a transmission type quarter-wave plate design with efficiency above 92% over a broad wavelength range (from 1260 nm to 1560 nm). The device proposed is based on a one-dimensional metal-insulator-metal waveguide array buried in silica. Phase difference between transmitted TE and TM components can be tuned continuously. At the same time, transmission efficiency can be kept above 90% in the same spectral range for both the TE and TM incidences. The broad bandwidth and remarkable efficiency are explained with the combination of low dispersion of waveguide modes and the resonant cavity enhanced transmission effect. To give a better understanding of the structure, we also propose a modified effective medium model. The optical response of the structure can be well reproduced with the semi-analytic effective medium model.
© 2017 Optical Society of America
1. Introduction
As fundamental optical devices, wave plates are widely used in various application areas. Traditional wave plates are based on birefringence crystals in which orthogonal components of incident light experience different refractive indexes. However, the birefringence effect of optical crystals is too weak that it usually needs a relatively thick device to get enough phase difference. In addition, operation bandwidth of traditional wave-plates is limited. When facing with ever growing demands in emerging photonic integration, traditional wave plates become helpless due to their limitation in thickness, bulky configuration, and narrow bandwidth.
Recent progress in metamaterials provides unprecedented opportunities to manipulate light polarization at the nanoscale with the giant optical activity of artificial meta-atoms or meta-molecules [1–8]. In 2010, Tao Li demonstrated an optical half-wave plate with bilayer metallic nano-structures [9]. Later, Andrea Alù proposed a pair of complementary metamaterial design for optical quarter-wave plate [10]. Since then, metamaterial wave plates are realized with various nanoparticles and their Babinet inverted counterparts [11–21]. However, these early strategies suffer from low efficiency and narrow bandwidth due to the plasmon resonance involved.
To achieve high efficiency and broadband operation simultaneously, several new methods have been developed (e.g. dielectric metasurfaces, reflective metasurfaces, and waveguide transmission). Compared to plasmonic metamaterials, dielectric metamaterials can realize much higher efficiency due to the low loss Mie resonances involved. And Huygens’ surface built with dielectric materials have shown marvelous ability in phase control with near unity efficiency and ultra-broad bandwidth [22–26]. Wave plates based on dielectric metamaterials [26–29] have also been demonstrated with excellent performance. But most of these dielectric metasurfaces compose of silicon nanoparticles, which are still lossy above 300 THz. Thus, their operating wavelengths are limited. Dielectric phase control metasurfaces at visible wavelength are realized with materials with wider band gap like TiO2 [22, 30]. But the fabrication of TiO2 nanoparticles with high aspect ratio is rather challenging. In addition, the refractive index contrast between dielectric resonators and surrounding medium is crucial. Therefore, the substrate materials are limited to low refractive index materials, which hinders device integration on other substrates like Si and GaAs et al.
Reflective metasurfaces typically consist of a top antenna array layer, a dielectric spacer layer, and a metal back reflection layer [28, 31–37]. The back reflection layer is thick enough to eliminate transmission loss, and the absorption can be kept well below 20%. Therefore, the efficiency of reflective metasurface can be improved to 80% easily. In addition, the gap plasmon resonance [36–38] and the overlapping of multiple resonances [5, 39] in these metasurfaces promise much broader bandwidth. What’s more, operating wavelength range of this kind of devices can be tuned readily, and broadband high efficiency wave plates have been reported from visible spectrum to microwave range. However, optical setup involving reflective devices becomes complicated. In most cases, wave plates working in transmission mode rather than reflection mode are needed.
Recently, waveguide modes are utilized to realize the function of wave plates [40–42]. By tailoring the difference between the effective refractive indexes of TE mode and TM mode, the phase difference between these two modes can be adjusted to π/2 or π, thus, realizing the function of a quarter-wave plate or a half-wave plate. This kind of wave plates is expected to be able to achieve both high efficiency and broad bandwidth in transmission mode, considering the relatively low loss and dispersion of waveguide modes. In addition to absorption loss, reflection should also be suppressed in order to achieve high efficiency. Yet, the highest efficiency achieved so far is around 50% [40], which is still too low. With further suppression of reflection, transmission can be enhanced, then the efficiency of this kind of device can be comparable with dielectric and reflective metasurfaces.
In this paper, we theoretically demonstrate that transmission-type quarter-wave plate based on sub-wavelength metallic structures can also achieve high efficiency (>92%) and broad bandwidth (>300 nm) performance. The structure is based on a thin layer (600 nm) of one-dimensional metal-insulator-metal (MIM) waveguide array buried in silica. The dispersion of waveguide modes is much lower than those of the localized surface plasmon resonance and Mie resonance. We achieved 90° ± 10° phase delay between TE and TM modes from the O-band to the C-band (1260 nm to 1560 nm). At the same time, minimum transmission in the same spectral range exceeds 92% with the benefit of the resonant cavity enhanced transmission effect. To give a clear understanding of the results, we modified previous Fresnel reflection method by introducing an effective medium thickness, and proposed a semi-analytic effective medium model. Transmission properties calculated with the semi-analytic model are in excellent agreement with the FEM simulations. We also find that the resonant cavity enhanced waveguide transmission effect shows weak dependence on the metal material. Nearly the same results can be obtained with different metallic materials.
2. Structure and results
The broadband high efficiency quarter-wave plate consists of a one-dimensional array of thick silver strips buried in silica, as illustrated in Fig. 1. The periodicity of the one dimensional array is P = 800 nm. The gap between neighboring silver strips is g = 650 nm. The width and thickness of the silver strip are w = 150 nm and t = 600 nm, respectively. Wave vector of incident light is against the z-axis direction. TE (or TM) incidence represents incident light with polarization direction parallel (or perpendicular) to the strips, as shown in Fig. 1(b). The spectral response and field distribution of the structure are studied with finite element method (FEM) simulations. The region simulated is a single unit cell with periodic boundary conditions applied to x-axis direction, perfectly matched layers applied to z-axis direction. Refractive index of silica is set to be 1.46. Optical parameter of silver is taken from Rakic’s work [43].
Fig. 1 Schematic illustration of the quarter-wave plate’s structure. (a) 3D sketch of the structure. The periodicity of the structure in x-direction is P = 800 nm, the gap between neighboring silver strips is g = 650 nm, the width of the silver strip is w = 150 nm, and the height of the silver strip is t = 600 nm. (b) Profile of the structure in x-z plane.
Transmission amplitudes under TE and TM incidences and phase difference between transmitted TE and TM field are demonstrated in Fig. 2. The sharp drop around 1170 nm in Fig. 2(a) is induced by the ± 1st order Rayleigh anomalies (RA), whose spectral position can be estimated with the relation λRA = nsilica × P. For both the TE incidence and TM incidence, the minimum transmission efficiency exceeds 90% from 1200 nm to 1650 nm. And the transmittance ratio between TE and TM modes is within 1 ± 0.05 in the same wavelength range, which has already met the requirement of 1 ± 0.1 for an acceptable quarter-wave plate. To construct a quarter-wave plate, another condition should be satisfied at the same time, i.e., the phase difference between the two orthogonal field components should be within 90° ± 10° [10, 38, 44]. As can be seen from Fig. 2(b), the spectral range satisfying the condition stretches from 1260 nm to 1560 nm. Thus, the device proposed can be taken as a transmission type quarter-wave plate with a bandwidth of 300 nm and minimum efficiency above 92%. In fact, the minimum efficiency of this wave plate can be improved to an even higher value (above 95%) if the silver strip becomes narrower. But a higher aspect ratio means a more challenging fabrication process. The aspect ratio of the structure presented in this paper is t/w = 4 (600 nm/150 nm), which is still acceptable [40, 42]. And the record high performance of transmission type metallic quarter-wave plate achieved here is even better than some dielectric metasurfaces and reflective metasurfaces. The extraordinary broad bandwidth and high efficiency are the results of low dispersion of the waveguide modes and the resonant cavity enhanced transmission, as will be discussed in details in the following sections.
Fig. 2 (a) Transmittance spectra of the structure in Fig. 1 under TE incidence and TM incidence. (b) The phase difference between transmitted TE light and transmitted TM light, the shadowed region indicates the spectral range in which the phase difference is within 90° ± 10°.
3. Analyses
Different from traditional one-dimensional metallic strip grating, the silver strips here are thick enough that the dielectric spacer between neighboring strips can support waveguide modes. Therefore, the whole silver strips array can be viewed as a set of MIM waveguides in parallel connection. And the spectral response of the structure can be approximated with the collective effects of the waveguides. For TE incidence, the characteristic equation is
where βTE is the propagation constant, g is the gap between neighboring silver strips, k0 = 2π/λ, ԑd and ԑm are the dielectric constants of silica and silver, respectively. And the characteristic equation for TM mode isThe effective mode index for waveguide mode can be determined asThe mode refractive index spectra calculated with the equations above for both the TE and TM modes are displayed in Fig. 3. Real part of TM mode’s refractive index is around 1.52, the dispersion in the wavelength range interested is almost negligible. The dispersion of TE mode is higher, but still very small compared to the dispersion of resonances generated in plasmonic and dielectric nanoparticles. As a result, the relatively low dispersion of the waveguide modes promise the quarter-wave plate a much broader bandwidth than those utilizing localized surface plasmon resonances or Mie resonances. Imaginary parts of the waveguide modes’ refractive index are presented in Fig. 3(b). The very small imaginary parts together with the short propagating distance (t = 600 nm) indicate that the absorption loss in this structure can be very low. Therefore, the efficiency of this structure can be comparable with dielectric metasurfaces, though metallic components are involved.
To get high transmission efficiency, reflection as well as absorption should be suppressed. The MIM waveguide layer is sandwiched by the upper and lower silica, forming a resonant cavity. So we can expect the characters of Fabry-Pérot (FP) resonances in the transmission spectra and standing wave patterns in the field distribution maps. As can be seen from Fig. 2(a), there is clearly a peak around 1560 nm in the TE mode’s transmission spectrum. Corresponding field distribution of this peak is shown in Fig. 4(a). Obviously, it is the field distribution of the first order FP resonance. Ordinary FP peaks are sharp. But the cavity here is too short (t = 600 nm), so its lowest order resonance has a much broader line width. Therefore, the TE incident light can realize high transmission efficiency over a broad wavelength range with the aid of the resonant cavity enhanced transmission effect. It is the same for TM incidence. But the contrast between the index of TM mode (around 1.52) and surrounding medium (nsilica = 1.46) is much smaller. As a result, the resonance peak is not so distinct, but still can be observed in Fig. 4(b) (the peak position is around 2050 nm).
Fig. 4 Field distribution at the FP resonance peak under TE (a) and TM (b) incidence. Transmittance spectra in dependence of metallic strip thickness t under TE (c) and TM (d) incidence. The transmittance spectra (e) and phase difference spectrum (f) with metallic strip thickness equal to 1500 nm.
Higher cavity modes appear for thicker waveguide layer. And the phase difference between different modes will grow larger. We calculated the spectral response for different strip thickness t. The results are shown in Figs. 4(c)-4(e). The markers (m = 1,2,3,4) on the color maps denote the order of FP cavity resonances. As expected, more higher orders appear with thicker waveguide layers. The cutoff frequency of TE mode is around 2000 nm, which is in accordance with the result calculated with Eq. (1). In our previous work, the standing-wave pattern is explained to be the field distribution of a lateral FP resonance, because the peak position is more sensitive to the gap width g rather than the strip thickness t [45]. However, with the results above, we find that the spectral response of this kind of structures can be explained better with the combination of waveguide mode and horizontal FP cavity resonance.
4. Semi-analytic effective medium model
Since the response of the structure can be well explained with the combination of the waveguide modes and resonant cavity enhanced transmission, one step further, we can take the waveguide layer as a dielectric layer with effective refractive index equal to the waveguide mode index. In this way, the whole structure can be simplified into a three-layer dielectric system as shown in Fig. 5(a). Then its spectral response can be solved analytically. The transfer matrix of this three-layer system can be written as
in which n1 and n3 represent the refractive index of the upper and lower silica, d is the thickness of the effective medium, and n2 represents the effective refractive index of the effective medium layer, i.e., the index of waveguide mode. The transmission coefficient of the three-layer system is 1/M(1,1).Fig. 5 (a) Three-layer dielectric material model for the structure in Fig. 1. Transmittance spectra (b) and phase difference spectrum (c) calculated with the transfer matrix with effective layer thickness set to be equal to the metallic strip thickness t = 600 nm.
In previous studies, the thickness of the effective medium layer is taken as the thickness of the metallic components [40, 42]. However, it’s not rigorous. We calculated the spectral response of the simplified model with the effective layer thickness taken as the strip thickness t, results of the analytic model in comparison with FEM simulations are presented in Figs. 5(b) and 5(c). As can be seen, both the transmittance spectra and phase response of the analytic model can hardly represent the FEM results. In fact we can tell from Fig. 2(a) that this model is inaccurate. The first order FP resonance under TE incidence locates at 1560 nm, corresponding refractive index of TE mode is 0.94. The dielectric layer thickness needed for a first order FP resonance should be around 830 nm rather than the silver strip thickness (600 nm). Apparently, this model needs to be modified.
For a typical FP resonance, the standing wave pattern should be within the cavity. However, the pattern in Figs. 4(a) and 4(b) extends to the upper and lower silica region. That’s to say the effective cavity length should be longer than the strip thickness t. In fact, the mode expansion effect and the scattering of the metallic components should be taken into account when determining the thickness of the effective medium layer. For the first order cavity resonance at 1560 nm, the effect of mode expansion and scattering can be represented by the extra cavity length (830nm-600nm). The mode expansion and scattering effects are different at different spectral positions. Therefore, the extra cavity length should be different, too. For a different metallic strip thickness t, the FP resonance appears at a different wavelength. The spectral responses for different strip thicknesses have already been shown in Figs. 4(c) and 4(d). The extra cavity length Δd(λ) for the first order FP resonance (m = 1) can be determined by the relation
where n2(λ) is the effective refractive index at wavelength λ, and tFP(λ) is the actual metallic strip thickness to form the first order FP resonance at wavelength λ, which can be estimated from Figs. 4(c) and 4(d). The extra cavity length spectra calculated for TE and TM incidences are displayed in Figs. 6(a) and 6(b), respectively.Fig. 6 Extra cavity length Δd estimated from Eq. (5) for TE incidence (a) and TM incidence (d), respectively. Transmittance (c) and phase difference (d) calculated with the semi-analytic model in comparison with the results from FEM simulations.
For the structure presented in Fig. 1(a) with t = 600 nm, the effective medium layer thickness of the simplified three-layer model should be d(λ) = Δd(λ) + t. We calculated the spectral response with the modified model. The semi-analytic results in comparison with FEM simulation results are presented in Figs. 6(c) and 6(d). For TE incidence the analytic results are in excellent agreement with the FEM results. Their maximum difference is below 1%. As for TM incidence, the maximum error is about 6%. The relatively larger discrepancy might be induced by the localized surface plasmon resonance (LSPR) excited on the metallic strips, which has not been taken into account in the simplified effective medium model. For the strip width of w = 150 nm here, the LSPR is stronger at shorter wavelength side, and its coupling with the scattering of the periodic grating will also be stronger at shorter wavelength (the ± 1st order RA locate at 1168nm). As a result, the discrepancy grows larger as the wavelength gets shorter. It is the same for the phase response shown in Fig. 6(d). The maximum phase discrepancy is about 16° at the short wavelength side, and the minimum discrepancy is only about 3° at long wavelength side. Despite the LSPR effect, the semi-analytic effective medium model still can represent the structure very well.
5. Discussion
Plasmonic metasurfaces have strong dependence on metallic materials. The spectral responses are different for devices built with different metals, though their structures are identical. Usually, a certain kind of metallic material shows much better performance. Therefore, the structures are often optimized with respect to a certain kind of material to achieve desired spectral response. As a result, the applications of some existing designs are limited to specified conditions. However, the waveguide mode and cavity resonance discussed in this paper is not so sensitive to the dielectric constants of metallic materials. The waveguide layer is finally simplified into an effective medium layer. The key parameters of the effective medium layer is its effective refractive index and thickness. In fact, the dielectric constants of copper, silver, gold, and aluminum are similar to each other within the spectral range under consideration. And the effective refractive index is dominated by the gap width and the dielectric spacer. Thus, the effective refractive index of the waveguide layer built with these kind of metals are almost the same. So we can expect same performance for devices built with these different metallic materials. We replaced silver with different metallic materials (gold, copper, and aluminum), the transmission and phase response spectra calculated with FEM simulations are presented in Fig. 7. As expected, the spectra lines are almost the same for different metals. The relatively free choice of material promises a much wider application area. In addition, the continuous metallic strip array can even act as functionalized transparent electrodes for opto-electronic devices, like polarization converting transparent electrodes for VCSEL, or polarization independent antireflection contact for photodiode, which is advantageous for opto-electronic integration.
Fig. 7 Spectral response for devices built with different metallic materials. (a) Transmittance under TE and TM incidence. (b) Phase difference between transmitted TE and TM field.
6. Conclusion
In conclusion, we have demonstrated a high efficiency (above 92%) and broadband (>300 nm) transmission-type quarter-wave plate based on one-dimensional thick metallic strip array. The broad bandwidth is the result of the low dispersion of waveguide modes. And the high efficiency arises from the resonant cavity enhanced transmission. We also designed a high efficiency and broad band half-wave plate design with the same strategy. To give a better understanding of the mechanics involved, we modified the effective medium model by introducing an effective cavity thickness. Transmittance spectra and phase response calculated with the new semi-analytic model are in excellent agreement with FEM simulations. The performance of the proposed device is comparable with previous high efficiency and broad bandwidth wave plates based on dielectric and reflective metasurfaces. In addition, the requirements on background material and metallic material are not so strict compared to dielectric and plasmonic metasurfaces, the transmission type optical setup is also simpler than reflection arrangements. And the metallic strips can even serve as ohmic contacts for opto-electronic devices, which is beneficial for opto-electronic integration.
Funding
National Basic Research Program of China (973 Program) (2015CB932402); National Key Research and Development Program of China (2016YFB0402401); Open Fund of IPOC (BUPT) (IPOC2016B006).
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