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Abstract
In this work, we propose a method of achieving quasi-continuous linear phase gradient for transmitted waves based on conformal spoof surface plasmon polariton (SSPP). To this end, a SSPP structure with high transmission is firstly designed as the unit cell of the metamaterial. To obtain the phase gradient, SSPP structures are arranged delicately in a way that they are conformal to the brachistochrone curve. In this way, quasi-continuous linear Pancharatnam-Berry (PB) phase profile can be realized strictly along one of the two transverse directions. To verify this idea, a dual-band transmissive metamaterial operating in X and Ku band was designed, fabricated and measured. Due to the phase gradient imparted by the conformal SSPP structures, high-efficiency anomalous refraction can be realized within the two bands. Different from the general PGM, the phase gradient of the conformal SSPP structure allows us to achieve the desired anomalous refraction angle without reconstructing the PB phase. Both the simulation and measurement results are well consistent with theoretical predictions. This work provides another strategy of achieving anomalous refraction and may find applications in beam steering, digital beam forming, etc.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Over the past few decades, conventional optical elements such as mirror, prisms, lens, wave plates, holograms and some diffractive elements have been designed to obtain the phase modulations of the accumulated optical path differences by designing geometric structures and refractive index profile. However, these are often hampered by heavy thickness and large loss [1–3]. The phase-gradient metamaterial (PGM) is a typical two-dimensional metamaterial composed of inhomogeneous arrays of subwavelength resonators, providing a promising candidate material [4–9]. Different from conventional optical components, PGM can provide abrupt phase changes covering 2π over the subwavelength scale [10]. These artificial sub-wavelength unit-cell-arrays can perform plenty of optical functions such as abnormal reflection/refraction, polarization conversion, surface wave excitation, beam splitting and imaging [11–13].
Surface plasmon polariton (SPP) refers to a hybrid mode of electromagnetic (EM) wave and electron density bound to a surface, having the potentials of miniaturizing photonic circuits and designing sub-wavelength devices [14–16]. Initially, SPP was primarily investigated in the field of optics, resulting in a discipline called plasmas [17]. In fact, the excellent sub-wavelength properties of SPP have attracted scientists in other fields such as microwaves, which has led to the study of SPP-like modes excited on structured metallic surfaces or metamaterials at microwave frequencies [18,19]. However, surface waves exhibit very weak confinement at terahertz and EM fields mostly reside in dielectric region because the metal can be treated as a perfect electric conductor (PEC), into which the EM fields cannot penetrate [20–22]. But it can be supported by the plasmonic metamaterials named as spoof surface plasmon polaritons (SSPP), which are usually achieved at terahertz and microwave frequencies by decorating periodic arrays of sub-wavelength slots, holes or blocks on the metal surface [23]. In addition, SSPP also possess fascinating properties, such as field enhancement and sub-wavelength transmission, and shows great application potential in miniatured microwave devices and high directional antennas [24]. Recently, digital technology has been introduced in the field of metamaterials. This concept has also been extended to the sub-wavelength range to control SSPPs, which pioneered the concept of digital SSPPs and provided the possibility of manipulation dynamic SSPPs [25,26]. It is worth mentioning that the scheme in Ref. [27] can be used to realize frequency-domain modulations, which plays a fundamental and important role in the modern information systems.
In this paper, we propose another strategy of achieving anomalous refraction by designing a quasi-continuous linear PGM based on conformal SSPP. Via aligning the SSPP structures coupling, we first designed a dual-band circular-to-circular (CTC) polarization conversion metamateiral (PCM) with high transmission as the unit cell of the metamaterial. To obtain the phase gradient, SSPP structures are arranged delicately in a way that they are conformal to the brachistochrone curve. Due to the phase gradient imparted by the conformal SSPP structures, high-efficiency anomalous refraction can be realized in X and Ku band. Unlike the general PGM, the phase gradient of the conformal SSPP structure allows us to achieve the desired anomalous refraction angle without reconstructing the PB phase. To verify this idea, a dual-band transmissive metamaterial was designed, fabricated and measured. Both the simulated and measured electric field component distribution of transmitted wave verified the high-efficiency anomalous refraction of the quasi-continuous linear PGM based on conformal SSPP.
2. Theory and design
2.1. Theoretical analysis
The conceptual illustration of this work is shown in Fig. 1, a quasi-continuous linear PGM based on conformal SSPP, which is composed of a series of identical SSPP structures. In the case of left-handed circularly polarized (LCP) wave incidence, the transmitted waves are effectively converted into cross-polarization right-handed circularly polarized (RCP) wave with anomalous refraction, and vice versa. Moreover, in the case of linearly polarized (LP) wave incidence, the transmitted waves are divided into two beams of circularly polarized (CP) waves with anomalous refraction occurring along two opposite directions.
Fig. 1. Conceptual illustration of the designed quasi-continuous linear PGM. When LP wave is incident, the transmitted waves are divided into two beams of CP waves with anomalous refraction occurring along two opposite directions.
To verify and illustrate how our design works, we start by analyzing the characteristics of brachistochrone curve. The shape of brachistochrone curve is shown in Fig. 2(a), whose expression can be represented as
where r, t are the radius and traveled radian of the rolling circle respectively (0 ≤ t ≤ 2π). Taking the derivative of y with respect to z in this parametric equation, we can derive that the gradient of the tangent line at any point P’ on brachistochrone curve asThus, we can obtain that the corresponding angle between tangential direction and horizontal direction is
Here, the corresponding angle $\alpha$ is linear and continuous with respect to z. By the above analysis, the phase difference between z1 and z2 at any two points on brachistochrone curve can be expressed as
Therefore, the phase gradient along the z-axis is $\nabla \varphi ={\pm} 1/r$. Where “ + “ represents the incidence of LCP wave, and “ − “ represents the incidence of RCP wave. According to the general Snell’s Law, the angle of anomalous refraction can be expressed as
Fig. 2. (a) Schematic diagram of brachistochrone curve. (b, c) The structure diagram and perspective view of the proposed SSPP structures with l = 9 mm, d1 = 0.18 mm, d2 = 0.12 mm, h = 0.35 mm and T = 6 mm.
2.2 Design of the SSPP structures
The structure diagram and perspective view of the proposed SSPP structures are shown in Figs. 2(b) and 2(c). In order to achieve high transmission efficiency, the matching characteristic of the wave-vector in the air-to-dielectric and dielectric-to-air transformations need to be considered. Therefore, the height of the corrugated metallic strips needs to be modulated gradually according to the spatial distribution of SSPPs propagation constant k(z). Due to the continuous linear phase gradient of the brachistochrone curve, arranging the SSPP structure according to the brachistochrone curve can achieve an excellent matching characteristic. As a result, the corrugated metallic strips are arranged in the shape of brachistochrone curve into the planar plasmonic structure to modulate the SSPP at microwave frequencies. On account of SSPP is generated at the metal-substrate interface, the incident wave-vector along the y-polarization is greatly enhanced, resulting in the phase accumulation of the y-polarized wave along the z-axis. The y-polarized incident wave is converted into the SSPP mode with a larger k value, while the x-polarized incident wave maintains the propagation mode in the free space. Therefore, the y-polarized transmitted waves can obtain large phase accumulation. This results in a phase difference between the two orthogonal components of the transmitted wave, which is a necessary condition for polarization conversion. The phase difference between the transmitted x- and y- polarized waves can be expressed as
Therefore, the length l of the plasma structure is modulated according to the spatial distribution of SSPP propagation constant k(z). The structure is a three-layer structure with 30 metallic strips etched between two 0.5 mm-thick F4B (εr = 2.2, tanδ = 0.001) dielectric substrates. The length of the planar plasmonic structure along the z-direction is l = 2πr = 9 mm, where r = 4.5/π mm is the radius of the rolling circle, and T = 6 mm is the periodic dimension along the x- and y- directions. The geometrical parameters are designed to be: d1 = 0.18 mm, d2 = 0.12 mm, and h = 0.35 mm.
To verify CTC polarization conversion transmittivities of the SSPP structures, CST Microwave Studio software was used for numerical simulation under LP wave and CP wave normal incidence from + z direction. In the simulation, the boundary conditions of x- and y-axis are set as “unit cell”, and the boundary conditions of z-axis are set as “open add space”. Figure 3(a) shows the co-polarized phase difference Δφ of the proposed SSPP structures between x- and y-polarized transmitted waves. It can be seen that the co-polarized phases in the x- and y- directions are linear and nonlinear respectively, which related to the phase in SSPP transmission and free space. When the frequencies are 11.7 GHz and 15.2 GHz corresponding to the phase differences π and 3π respectively, the transmitted waves will be cross-polarized waves. Figure 3(b) shows the simulated amplitude of the y- polarized and the x- polarized wave incidence, and CTC polarization conversion amplitude under normal incidence of LCP and RCP waves, respectively. It is observed that both the transmission amplitudes of the y- polarized and the x- polarized wave incidence are larger than −2 dB over a wide frequency band from 9.7 to 15.4 GHz. Furthermore, the CTC polarization conversion amplitude reached its peak value at 11.7 GHz and 15.2 GHz, which is due to the fact that the phase difference between x- and y-polarized directions are π and 3π. In short, the results show that the SSPP structures can realize efficient CTC polarization conversion transmission in two bands, and the cross-polarized amplitudes are larger than −1.5 dB in the frequency range: 10.5- 12.5 GHz and 14.9-15.4 GHz.
Fig. 3. (a) Co-polarized phase of the proposed SSPP structures for x- and y-polarized transmission. (b) Amplitude for x- and y-polarized co-polarized transmission, and CTC polarization conversion amplitude under left- and right-handed circularly polarized waves.
2.3 Design of the quasi-continuous linear PGM
To obtain continuous linear phase gradient, SSPP structures are bent in a way that they are conformal to the brachistochrone curve. We designed a dual-band quasi-continuous linear PGM shown in Fig. 4, which is composed of a 6×4 subunit array. Here, the radius of the rolling circle R is adjusted to 4.5mm, which is because that the arc length (S = 8R = 36 mm) of brachistochrone curve can matched exactly six conformal SSPP structures. The repetition periods of the quasi-continuous linear PGM in the x- and y- directions are L = 2πR = 9π mm and T = 6 mm, respectively.
Fig. 4. The structure diagram of the designed quasi-continuous linear PGM composed of a 6×4 subunit array.
In this way, quasi-continuous linear Pancharatnam-Berry (PB) phase profile can be realized strictly along one of the two transverse directions. Therefore, anomalous refraction can be realized at any position on the conformal SSPP structures. Unlike general PGM, the phase gradient of the conformal SSPP structures is continuous linear and defined along the y-axis strictly.
3. Simulation
In order to verify the anomalous cross-polarized transmission of the quasi-continuous linear PGM, CST microwave Studio software was used for full-wave numerical simulation under LCP wave and LP wave normal incidence from + z direction. In the simulation, LCP and LP plane waves are used as the source respectively, and the boundary conditions in x-, y- and z- directions are set as “open add space”.
In Fig. 5, the distribution of the electric field component of the incident wave from the + z direction is simulated. Figures 5(a-c) and 5(e-g) show the simulated results under LCP waves at 10.9, 11.7, 12.5, 14.9, 15.2, and 15.4 GHz, and Figs. 5(d) and 5(h) give the corresponding results under LP waves at 11.7 and 15.2 GHz. It can be seen from the figure that transmitted waves are effectively refracted and the refraction angles are $\theta $ ≈65° and 44° at the two frequencies 11.7GHz and 15.2GHz respectively, which is basically consistent with the theoretical calculation results. Moreover, the incident waves have strong refraction in the frequency bands of 10.9-12.5GHz and 14.9-15.4GHz and these bands correspond exactly to the two polarization conversion bands in Fig. 3(b).
Fig. 5. Simulated Ex-component of electric field distributions at the cutting plane which parallels to yoz-plane under LCP and y-polarized waves normal incidence. (a-c, e-g) Under incidence of LCP waves at 10.9, 11.7, 12.5, 14.9, 15.2, and 15.4 GHz, respectively. (d, h) Under incidence of y-polarized wave at 11.7 and 15.2 GHz.
In addition, LCP wave and RCP wave have the same phase gradient value but opposite sign, thus the transmitted waves will be decomposed into two CP waves with anomalous refraction occurring in opposite direction under incidence of y-polarized wave. Figures 5(d) and 5(h) show the x-component distribution of the yoz-plane electric field for the y-polarized wave normal incidence with frequency at 11.7 GHz and 15.2 GHz. The distribution of the electric field shows that the transmitted waves are decomposed into two beams and refracted in opposite directions effectively.
4. Experiment
To further verify the designed dual-band quasi-continuous linear PGM, we fabricated a prototype with a size of 240mm×200mm, shown in Fig. 6(a). The plasmonic structures are fabricated using PCB photolithography. The commercial F4B dielectric substrates are used as dielectric layers, while 17-ply copper films are used as metal components. The final quasi-continuous linear PGM was obtained by bending SSPP structures. As shown in Fig. 6(b), the experimental measurements of the prototype were carried out in a microwave anechoic chamber. The near-field experimental system was used to measure the electric field distribution in the range of 100mm×200mm above the prototype, which consisted of an experimental platform, a vector network analyzer (VNA), a horn antenna, and a scanning coaxial probe. The fabricated prototype is placed between the scanning probe and a horn circularly/linearly polarized antenna. Here, the horn antenna is placed at a distance of 500 mm from the prototype to ensure quasi-plane incidence.
Fig. 6. (a) The fabricated prototype consisted of a 31×8 subunit array. (b) The near-field measurement system.
The near-field testing results are shown in Fig. 7. It is clearly indicated that the anomalous refraction angles corresponding to each frequency point are well consistent with the simulated results. In addition, in order to verify the high-efficiency transmittance of the designed PGM, we measured the transmittance under the incidence of LCP wave in a microwave anechoic chamber, and the results are shown in Fig. 8. It can be observed that the measured intensity of electric field distribution is slightly different from the simulated results. This phenomenon is mainly due to the fact that the origin of incident wave is not plane wave in the strict sense. In addition, the uncertainty of the environment may lead to the difference between the measured results and the simulated results.
Fig. 7. Measured Ex-component of electric field distributions at the cutting plane which parallels to yoz-plane under LCP and y-polarized waves normal incidence. (a-c, e-g) Under incidence of LCP waves at 10.9, 11.7, 12.5, 14.9, 15.2, and 15.4 GHz, respectively. (d, h) Under incidence of y-polarized wave at 11.7 and 15.2 GHz.
5. Conclusion
In summary, we propose a method of achieving anomalous refraction by designing a quasi-continuous linear PGM based on conformal SSPP. Bending the SSPP structures into a curved surface conformal to the brachistochrone curve, a quasi-continuous linear PGM was realized. In this way, quasi-continuous linear PB phase profile can be realized strictly. To verify this idea, a dual-band transmissive metamaterial operating in X and Ku band was designed, fabricated and measured. Significantly, the quasi-continuous linear PGM can achieve the designed angle of anomalous refraction in specific frequency band by adjusting brachistochrone curve without reconstruct the PB phase. This quasi-continuous linear PGM is demonstrated through simulation and near-field measurements. In conclusion, this work provides another strategy of achieving anomalous refraction and may find applications in beam steering, digital beam forming, etc.
Funding
Foundation for the Author of National Excellent Doctoral Dissertation of the People's Republic of China; National Natural Science Foundation of China (61601507, 61671466, 61671467, 61801509, 61901508, 61971435, 61971437);National Key Research and Development Program of China (SQ2017YFA0700201); Dr. Yuxiang Jia’s Airforce Engineering UniversityExcellent Doctoral Dissertation Support Foundation(The electromagnetic scattering modulation technology based on the spoof surface plasmon polariton).
Disclosures
The authors declare no conflicts of interest.
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