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In this paper, we propose to achieve beam steering by k-dispersion engineering of spoof surface plasmon polaritons (spoof SPP) at microwave frequencies. The planar plasmonic metamaterials (PPMs) are employed to couple and guide spoof SPP. High-efficiency transmission based on spoof SPP coupling is realized via matching the wave-vectors of the spoof SPP and the space wave. The transmission phase can be modulated by k-dispersion engineering of the spoof SPP with great freedom. Due to the independent phase shift produced by the spoof SPP on the PPMs, the phase gradient achieved by using the PPMs as the sub-unit cells can be altered by changing the repetition period of the sub-unit cells. Two phase gradient materials (PGMs) are achieved by using nine different PPMs as the sub-unit cells with the repetition period q = 4mm and 4.5mm. Both the simulated and measured results demonstrated the excellent performances of the PGMs on high efficiency, wideband, tunable beam steering.
© 2016 Optical Society of America
1. Introduction
Generally, beam steering can be realized by mirror, lenses, prisms and some diffractive elements. For all of the works involving above elements, the phase modulation is accumulated by the optical path difference. Therefore, the thicknesses of the optical elements are larger than or comparable to the incident wavelength. In this regard, the concept of phase “discontinuities” is proposed quite recently. Electric/magnetic resonators and polarization converters with sub-wavelength sizes can usually provide an abrupt phase change to the co- and cross-polarization reflection/transmission, respectively. Accordingly, by an array of these sub-wavelength unit cells with spatially varying phase responses, the wave-fronts can be controlled with much more freedom. These artificial sub-wavelength unit-cell-arrays are named as phase gradient metamaterial (PGM) [1–19]. By using the PGM, a lot of optical functions can be realized including anomalous reflection/refraction [2–8], optical focusing [9–13], polarization conversion [14–16], and surface wave excitation [17–19].
Surface plasmon polaritons (SPP) are collective oscillations of free electrons trapped at metal-dielectric interfaces [20]. Natural SPP cannot exist in microwave frequency regime, but can be supported by the plasmonic metamaterials named spoof surface plasmon polaritons (spoof SPP). Plasmonic metamaterials are usually generated by decorating periodic arrays of sub-wavelength grooves, holes, or blocks on metal surface, which have been proposed to realize the spoof SPP at terahertz and microwave frequencies [21–25]. Owing the deep sub-wavelength characteristic of the spoof SPP, the continuous phase accumulation of the spoof SPP can be approximately considered as “discontinuous” compared with the space waves. Via the k-dispersion designing, the phase accumulation of the spoof SPP at a fixed distance can be freely modulated.
In this paper, beam steering based on the k-dispersion engineering of the spoof SPP mediated by planar plasmonic metamaterials (PPMs) is proposed and experimentally demonstrated. Due to the deep sub-wavelength characteristic of the spoof SPP and its highly efficient coupling on the PPMs, the PPMs with sub-wavelength length are employed to serve as the sub-unit cells of the PGM to freely modulate the transmittivity and transmission phase. The high-efficiency transmission is achieved via matching the wave-vector between the spoof SPP and the space wave. The spoof SPP mediated by the PPMs are studied by calculating the dispersions relationships and imitating the distributions of the electric fields. Using the PPMs as the sub-unit cells, PGMs with dispersive phase gradient are achieved. In addition, the achieved phase gradient can be tuned by changing the repetition period of sub-unit cells. The high-efficiency anomalous refraction of the designed PGMs is demonstrated by simulating and measuring the normal transmittivity, mirror reflectivity and the normalized transmission spectrum.
2. The spoof SPP mediated by PPMs
The spoof SPP mediated by the PPMs are depicted in Fig. 1. The dispersion relationships of the y-polarized wave on the PPM and the wave in free space are given in Fig. 1(a). The front view of the PPM is given in the inset in Fig. 1(a), which is consisted of periodic metallic blades. In the figure, the metallic blade structure is etched on a 0.6mm-thick F4B (εr = 2.65, tanδ = 0.001) dielectric substrate. The length and width of the metallic blade are h and w, respectively. The width of the dielectric substrate is represented by a, while p for the repetition period of the blade, and v stand for the width of the metallic connected wire in z-direction. The geometrical parameters for the PPM in the simulations are: a = 8mm, p = 0.5mm, w = 0.25mm, v = 0.3mm and h = 6.24mm. Observed from Fig. 1(a), the propagation constant k of the y-polarized wave on the PPM seriously deviates from which for the wave in free space in the whole frequency region, which is much larger than that for the wave in free space. And that the y-polarized wave is highly confined on the PPM. Therefore, the y-polarized wave on the PPM can be considered as SPP in GHz frequency region, named spoof SPP. Figure 1(b) shows the simulated distributions of the electric-field-components Ey and Ez of the spoof SPP on the PPM with the blade length h = 6.24mm at the frequency f = 11GHz. The PPM is composed of 60 blades with the total length 30mm. The y-polarized wave is feed in using a port. The boundary conditions in x-, y- and z-directions are all set to be open. It is observed from the figure that the electric field of the spoof SPP is strongly enhanced and localized on the blade structure. The practical wavelength is highly reduced compare with which in free space. Figure 1(c) gives the cutoff frequency of the spoof SPP versus the blade length h of the PPM. It is found that the cut-off frequency decreases with increasing blade length h. Figure 1(d) shows the simulated dispersion spectrum of the spoof SPP on the PPM k(f, h). In the figure, the x-axis denotes the frequency, the y-axis denotes the blade length h, and the color represents the corresponding propagation constant k. It is found that the propagation constant increases with increasing blade length h at a fixed frequency.
Fig. 1 (a) The dispersion relationships of the spoof SPP on the PPM and the wave in free space, the inset displays the front view of the PPM. (b) The distributions of the electric field components for the spoof SPP on the PPM. (c) The cut-off frequency versus the blade length h. (d) The dispersion spectrum of the spoof SPP on the PPM versus the blade length k(f, h).
3. High-efficiency transmission based on spoof SPP coupling
In order to study the transmission based on spoof SPP coupling, the PPM consisted of 60 blades in z-direction arranged periodically in xoy plane makeup the PPM array as shown in Fig. 2(a). The repetition period of the PPMs in x-direction is q. We perform full wave numerical simulations to calculate the transmission and reflection coefficients under y-polarized wave normal incidence. The boundary conditions in x- and y-directions are set to be unit cell, and open add space in z-directions. The geometrical parameters for PPM in the simulations are: a = 8mm, p = 0.5mm, h = 6.24mm, w = 0.25mm, v = 0.3mm, and q = 4mm. The simulated amplitudes of the transmission and reflection coefficients are given in Fig. 2(b), and the corresponding phases are given in Fig. 2(c). Over a wide frequency range from 4GHz to 16GHz, the incident wave transmits through the PPM array with the amplitude of the transmission above 0.7. The incident wave is partially reflected and transmitted as the frequency f<17.6GHz, and completely reflected as f>17.6GHz. This is well consistent with the dispersion relationship of the spoof SPP on the PPM with the blade length h = 6.24mm given in Fig. 1(a), in which the frequency f = 17.6GHz is just the cut-off frequency of the spoof SPP mode. This indicates that the transmission is based on the spoof SPP coupling on the PPM. The y-polarized incident wave is efficiently coupled into spoof SPP on the PPM at one side of the PPM array, and then decoupled into the y-polarized transmitted wave at the other side with high efficiency. This can also be demonstrated by the simulated nonlinear transmission phase versus frequency under y-polarized wave incidence as given in Fig. 2(c).
Fig. 2 (a) The perspective view of the PPM array with constant blade length h = 6.24mm. The (b) amplitudes and (c) phases of the transmission and reflection coefficients for y-polarized wave normal incidence. (d) The perspective view of the PPM with spatial varied blade length. The (e) amplitudes and (f) phases of the transmission and reflection coefficients for y-polarized wave normal incidence.
However, the transmission is not very close to 100%, especially for the PPM with large blade length h. In Fig. 2(b), the blade length of the PPM h = 6.24mm, the transmission is down to 0.8, and the reflection up to 0.6. This is mainly because that the propagation constant of the spoof SPP on the PPM is much larger than which for the space wave. A strong mismatching of the propagation constant at the interface results in an enhanced reflection. In order to enhance the transmission, we proposed a new type of PPM with spatial varied blade length along the propagation direction, on which the spoof SPP has a spatial varied propagation constant distribution along the propagation direction. In detail, the propagation constant has the smallest value at the two boundaries so that the incident space wave can be more efficiently coupled into the spoof SPP on the PPM. Simultaneously, the spoof SPP can also be efficiently coupled into the space wave. Therefore, we designed the PPM with spatial varied blade length along the propagation direction, on which the propagation constant of the spoof SPP firstly increases linearly from the smallest value to the largest value, then linearly falls back into the smallest value.
Assuming that the designed PPM consisted of 2m blades with spatial varied blade length h(z), the mediated spoof SPP has spatially linearly varied propagation constant k as shown in Fig. 3. The propagation constant for the first blade is fixed to be β0, which is designed to be close to the propagation constant of the space wave to the greatest extent. The change of the propagation constant between adjacent blades is dk. Accordingly, the total phase accumulation of the spoof SPP on the PPM can be modulated by altering the propagation constant difference between adjacent blades dk. The total phase accumulation of the spoof SPP on the PPM can be calculated by Ф = mp(2β0 + (m-1)dk). To achieve the phase shift △Ф, the desired propagation constant difference between adjacent blades dk(i) for the PPMs can be derived by
where i denotes the serial number of the PPMs. The corresponding spatial distributions of the blade length h(i) of the PPMs can be derived from the dispersion spectrum (given in Fig. 1(d)). Accordingly, we designed a PPM with spatial varied blade length as shown in Fig. 2(d), on which the total phase accumulation of the spoof SPP has the same value with that on the PPM with constant blade length h = 6.24mm given in Fig. 2(a) at the frequency f = 11GHz. The reflection and transmission coefficients under y-polarized wave normal incidence are simulated and the results are given in Figs. 2(e) and 2(f), where Figs. 2(e) and 2(f) give the amplitudes and phases of the reflection and transmission coefficients, respectively. Observed from the figures, we can find that the transmission of the PPM with spatial varied blade length is highly improved, which is greater than 0.95 over a wide frequency range of 4-12GHz.Fig. 3 The spatial distribution of the propagation constant k for the spoof SPP on the PPM with spatial varied blade length.
4. Achievement of the phase gradient metamaterial (PGM)
We propose to achieve the phase gradient metamaterials by using the PPMs with spatial varied blade length distributions as the sub-unit cells. The PPMs are composed of 60 blades with the total length l = 30mm. The phase shift between adjacent sub-unit cells is designed to be ΔФ = 40° at the central frequency f = 11GHz. Nine sub-unit cells are need to exactly span a total 2π phase shift. The propagation constant for the first blade of the sub-unit cells is fixed to be β0 = 230rad/m, which is approximately equal to the propagation constant of the space wave at the frequency f = 11GHz. The propagation constant difference between adjacent blades for the first sub-unit cell dk(1) is fixed to be 0.5. According to expression (1), the desired propagation constant difference for the other eight sub-unit cells can be derived. They are dk(2) = 2.1049, dk(3) = 3.7098, dk(4) = 5.3147, dk(5) = 6.9196, dk(6) = 8.5245, dk(7) = 10.1294, dk(8) = 11.7343, and dk(9) = 13.3392. From the dispersion spectrum of spoof SPP k(h, f), we can obtain the corresponding spatial distributions of the blade length h(z) for the nine different sub-unit cells. As shown in Fig. 4, (a) gives the desired spatial distributions of the propagation constant k(z) for spoof SPP on the nine sub-unit cells at the frequency f = 11GHz, (b) gives the corresponding spatial distributions of the blade length h(z) for the nine sub-unit cells. Figure 5 gives the perspective view of the “super unit” for the phase gradient metamaterial (PGM) consisted of the designed nine PPMs. The repetition period of the sub-unit cells in the phase gradient direction (x-direction) is q. The achieved phase gradient will have different values while the repetition period q is changed.
Fig. 4 (a) The spatial distribution of the propagation constant k(z). (b) The spatial distribution of the blade length h(z) for the nine sub-unit cells at the frequency f = 11GHz.
To verify the phase gradient achieved by the nine sub-unit cells, we perform numerical simulations to calculate the transmission coefficients of the nine sub-unit cells under y-polarized wave normal incidence. The simulated results are given in Fig. 6, where (a) gives the amplitudes of the transmission coefficients, and (b) gives the phases. It is found that the transmissions of the nine sub-unit cells are all greater than 0.95 in the frequency ranging from 8GHz to 13GHz. In Fig. 6(b), the phase difference between adjacent sub-unit cells are all about 40° at the frequency f = 11GHz. Most importantly, the phase difference between adjacent sub-unit cells is frequency dependent, and thus the achieved phase gradient is frequency dispersive. Moreover, phase gradient metamaterial with no chromatic aberration at several frequencies is expected to be achieved. Figure 7 gives the simulated distributions of the electric field y-components of the spoof SPP on the nine sub-unit cells. The boundary conditions in x-, y- and z-directions are all set to be open. The y-polarized wave is feed in from + z direction using a port. Observed from the simulated electric field distributions, the phase differences between adjacent sub-unit cells are 40° indeed.
Fig. 6 The simulated transmission coefficients of the nine sub-unit cells for y-polarized wave normal incidence, (a) amplitudes, and (b) phases.
Fig. 7 The simulated distributions of the electric field y-components for the spoof SPP on the nine sub-unit cells of the PGM at the frequency f = 11GHz.
In additions, as for the designed PGMs, the modulation of the transmission phase is achieved by the phase accumulation of the spoof SPP on the sub-unit cells and has nothing to do with the repetition of the sub-unit cells q. Therefore, the phase differences between adjacent sub-unit cells will remain unchanged if the repetition period q is changed. Accordingly, the phase gradient of the PGM will be altered by varying the repetition period of the sub-unit cells q. The refraction angle will be tuned via changing the repetition period q. Therefore, this PGM can be considered to be tunable.
To verify the anomalous refraction of the designed PGMs, we performed full wave numerical simulations to calculate the distributions of the electromagnetic fields for y-polarized wave illuminating onto the designed PGMs with the repetition period q = 4mm and q = 4.5mm, respectively. Figure 8 shows the distributions of the electromagnetic field components (Ey, Hx, Hz) at the central frequency f = 11GHz, where (a) is for the PGM with q = 4mm, and (b) is for the PGM with q = 4.5mm. It can be found that the incident waves are efficiently abnormally transmitted. The observed refraction angles are about 49.45° and 42.33°, which are well consistent with the theoretically designed anomalous refraction angles. The electric field for the incident wave is in y-direction and the magnetic field in x-direction. As for the anomalously refracted wave, the electric field is still in y-direction, but the x- and z-components of the magnetic field are all existed. The anomalous refraction angle can be easily derived from the distributions of the z-components of the magnetic fields.
Fig. 8 The distributions of the electric and magnetic field components (Ey, Hx, and Hy) at the frequency f = 11GHz for y-polarized wave illuminating onto the two PGMs with the sub-unit cell repetition period (a) q = 4mm, and (b) q = 4.5mm.
Figure 9 gives the simulated normalized transmittivities versus the refraction angles θt at the frequency f = 11GHz. It is found that the incident wave is anomalously transmitted with the refraction angles θt = 49° and 42°, respectively, for the PGMs with the sub-unit cell repetition period q = 4 and 4.5mm. Figure 10 gives the normalized transmission spectrums t(f, θt) for y-polarized wave normal incidence, where (a) is for the PGM with q = 4mm, and (b) is for the PGM with q = 4.5mm. In the figures, the x-axis denotes the frequency, the y-axis denotes the refraction angle θt, and the color represents the normalized transmittivity. It can be found that the transmittances in the normal direction are almost zero. The incident wave is efficiently anomalously refracted over a wide frequency range from 9GHz to 12.5GHz. The anomalous refraction angles are approximately linearly changed with the frequency, which is attributed to the achieved nonlinearly dispersive phase gradient due to the nonlinearly dispersive characteristic of the spoof SPP. In addition, because of the different sub-unit cell repetition period q, the achieved phase gradients for the two PGMs are different. And thus different anomalous refraction angles will be resulted. By comparison, we can find that the anomalous refraction angles for the PGM with q = 4.5mm are all less than that for the PGM with q = 4mm.
Fig. 9 The simulated normalized transmittivities versus the refraction angles for y-polarized wave normal incidence onto the designed PGMs with the repetition period q = 4 and 4.5mm.
Fig. 10 The simulated normalized transmission spectrums t(f, θt) for the y-polarized wave normal incidence onto the PGMs with different repetition period (a) q = 4mm and (b) q = 4.5mm.
In order to further verify the tunability of the PGM, we simulated the normalized transmission spectrum of the designed PGM with different sub-unit cell repetition period t(q, θt) at the central frequency f = 11GHz. The simulated result is shown in Fig. 11. The x-axis represents the repetition period q, the y-axis denotes the refraction angle θt, and the color represents the normalized transmittivity. The theoretically calculated anomalous refraction angle versus the sub-unit cell repetition period q is also given in Fig. 11 using the white circle. Observed from the figure, it is found that the simulated normalized transmission spectrum agrees well with the theoretical anomalous refraction angles. At the frequency f = 11GHz, the anomalous refraction angle is altered from 25° to 75° while the sub-unit cell repetition period of the PGM q is changed from the 3mm to 6.6mm.
Fig. 11 The normalized transmission spectrum of the designed PGM with different sub-unit cell repetition period q t(q, θt) at the frequency f = 11GHz.
5. Experiment verification
The PGM samples were fabricated using the print circuit board (PCB) technique. The photograph of the PGM samples with the sub-unit cell repetition period q = 4 and 4.5mm together with the nine sub-unit cells (PPMs) are given in Fig. 12(a). The experimental measurement setup is given in Fig. 12(b), in which a revolving stage is employed. Two horn antennas with standard gain are fixed at the two spiral arms of the revolving stage. One is used as a transmitter, and the other is used as a receiver. The x- and y-polarized waves can be transmitted or received by placing the short side of the horn antenna in its direction. The PGM sample is located at the center of the revolving stage. The two horn antennas are directly opposite on the two sides of the PGM samples to measure the normal transmittivity. The mirror reflectivity is measured by placing the two horn antennas forming a 4° intersection angle at one side of the samples. The measured mirror reflectivities and normal transmittivities for y-polarized wave normal incidence are given in Figs. 12(c) and 12(d), where (c) is for the PGM sample with q = 4mm, (d) for the PGM sample with q = 4.5mm. In the figures, the normal transmittivity and mirror reflectivity are all less than −10dB in the frequency ranging 11-13GHz for the PGM with q = 4mm and 11.6-13GHz for the PGM with q = 4.5mm. As for the measurement of the transmission spectrums, the transmitter is directly head upon the PGM sample. The receiver can receive the transmitted wave at different directions by rotating the spiral arm. Figures 12(e) and 12(f) give the measured normalized transmission spectrums of the PGM samples with q = 4mm and 4.5mm, respectively. It is observed from the figures that the measured anomalous refraction angles of the two PGM samples reveal good accordance with the simulated results given in Fig. 10 except a wider beam width. This is mainly because that the limited distance between the sample and the receiver and large aperture of the receiver make the receiver have a low angular resolution. In any refractive direction, the receiver can simultaneously receive the transmitted waves in a wide angle domain about 15°. Besides, compared with the simulated transmission spectrums given in Fig. 10, the measured anomalous refraction angles are slightly less than the simulated refraction angles in the whole frequency range. Possible reasons mainly include: (a) the mismachining tolerance of the fabricated PGM samples influences the experimental results; (b) the normal direction of the PGM sample is not directly facing the transmitter in the measurement, therefore, the readings of the refraction angle in the experiment is not equal to the actual refraction angle.
Fig. 12 (a) The photograph of the fabricated PGM samples. (b) The experimental measurement setup. The experimental measured mirror reflectivity and normal transmittivity for y-polarized wave normal incidence onto the PGMs with the repetition period (c) q = 4mm and (d) q = 4.5mm. The measured normalized transmission spectrums for y-polarized wave normal incidence onto the PGMs with the repetition period (e) q = 4mm and (f) q = 4.5mm.
6. Conclusion
In conclusion, the spoof SPP mediated by the PPM is studied by calculating the dispersion relationships and simulating the distributions of the electric fields. The transmission of the PPM based on spoof SPP coupling is proposed and demonstrated. The transmission is improved via matching the propagation constant of the spoof SPP and the space wave at the interface by using the blade length spatial varied PPM. Tunable PGMs achieved by using the PPMs as the sub-unit cells was proposed and realized. The transmission phase is modulated by the phase accumulation of the deep sub-wavelength spoof SPP. A dispersive phase gradient can be achieved owing to the nonlinearly dispersive characteristic of the spoof SPP. The phase gradient can be tuned by changing the repetition period of the PPMs. Two phase gradient metamaterials with different repetition period q = 4mm and 4.5mm are designed and fabricated. The transmission spectrums and the distributions of the electric fields were simulated. The mirror reflectivity, normal transmittivity, and the transmission spectrums were experimentally measured. Both the simulations and the experiments demonstrated the wideband, high efficiency, tunable PGM based on spoof SPP.
Acknowledgments
The authors are grateful to the supports from the National Natural Science Foundation of China (NSFC) under Grant Nos. 61331005, 11204378, 61302023, the National Science Foundation for Post-doctoral Scientists of China under Grant Nos. 2013M532131, 2013M532221, and the Special Funds for Authors of Annual Excellent Doctoral Degree Dissertations of China under Grant No. 201242.
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