1. Introduction

Recently metamaterials have drawn a lot of attention in the research arena [1, 2, 20]. The ability of this class of materials to respond to an electromagnetic field and bring about changes in a material’s fundamental properties has opened up doors for new applications such as flat lens, antenna miniaturization and artificial magnetic conductors to name a few [3, 16, 17]. One of the advantages of these artificial materials is that they can also be tuned over a frequency range by effectively changing the substrate properties, the geometry of the resonating metamaterial structure, or by changing the external fields with non linear dielectrics [4, 6, 7, and 8]. Another effective way of tuning resonating metamaterials such as the split ring resonators is by changing the resistance properties of the rings using capacitors that can be controlled by an applied voltage [11]. As these tunable structures are scalable with the wavelength, they can be used for modulation for any desired frequency range. In particular, a Terahertz modulator was demonstrated where the metamaterials were grown on a conductive substrate and the amplitude of the incoming wave was modulated using electrical controls to change the resistivity of the metamaterial on a GaAs substrate [9, 15]. Also recently, another research group demonstrated phase shifting with tunable negative refractive index metamaterials using variable external magnetic field [12]. Similar work was also done with negative index material by means of photoconductive coupling [10]. Complementary split ring resonators and their applicability in tuning frequency has also been studied [13]. Although these works had been successful in bringing out the idea of frequency and amplitude modulation, little investigation has been carried out in terms of phase modulation using metamaterials. Also a major challenge in working with metamaterials has been the inherent loss that arises from their resonating nature. To address such issues, in this paper we demonstrated phase only modulation using tunable metamaterials, namely the DSRRs. We mitigated the issue of material loss by employing thin resonating metamaterials operating at an off-resonant frequency to demonstrate a high degree of phase modulation with very minimal loss. Particularly, in our paper we demonstrated phase modulation using two different polarizations for the DSRRs: in the first configuration, the polarization of magnetic field, H, was perpendicular to the DSRRs as shown in Fig. 1 (a), whereas in the second configuration, shown in Fig. 1(b), the wave vector k was perpendicular to the ring. The DSRRs consisted of single split rings packed very closely where the adjacent rings’ split gaps were alternated in orientation. The key advantage of the DSRRs in the first configuration was that the rings could provide an off-resonance high permeability value when excited by the magnetic field [18]. Whereas in the second configuration the symmetric nature of the alternate rings could avoid electric coupling from the magnetic resonances that might lead to cross-polarization effects [21] when excited by the electric field .The key advantages in both of our designs were the constituent elements which could be readily manufactured using standard lithography processes and simple electrical connections could be used to regulate between the two states of the modulation using very thin layers of DSRRs in the propagation direction. Compared to conventional phase modulators, metamaterial-based phase modulators operating in transmission mode, promise to produce enhanced phase modulation in terms of exhibiting a high degree of phase change within a small volume. Moreover, being physically small it can ensure ease of structural integration and can offer scalability with wavelength that can reduce design constraints while operating in different frequencies. We believe these modulators can be the building block for spatial light modulators in the microwave frequency regime that can add significant advantage over conventional modulators by simplifying future Digital Micromirror Devices (DMD) and phased array antenna design by controlling each element of the array or pixel electronically [22, 25].

The first step in the recipe to build a tunable phase modulator was to design the DSRRs that could provide the resonant frequency around which the phase modulation would occur. Standard numerical techniques were used to retrieve the constitutive parameters and the effective refractive index information of the unit cell of the DSRRs for the two different states of modulation: electrically open and closed rings [5]. The difference in the value of the effective index for the two different states was used to calculate the phase change. We also calculated the change in phase using the volumetric field data obtained inside the computational region and explained the differences with the calculated theoretical result. In the following sections we will provide the details of the numerical setup, extraction of the relevant parameters to calculate the modulation and discuss the simulated results with a conclusion for improvement and our plan for experimental verification.

 

Fig. 1. (a). Configuration 1: H-field perpendicular to the DSRR. (b) Configuration 2: k vector perpendicular to the DSRR

Download Full Size | PDF

2. Design and modeling

The DSRRs needed to be designed in such a way so that it could operate for a particular frequency where the values of the effective permeability and permittivity of the DSRR’s would meet the following criteria: a) numerically large to create appreciable phase change through a change in the effective index, b) impedance matched interface, and c) minimize the power loss. This is formalized in terms of the Figure Of Merit (FOM) = n_real/ n_imaginary, where n_real and n_imaginary are the real and imaginary parts of the refractive index respectively. The goal will be to achieve a high FOM for each state for both of the configurations.

In order to meet these requirements we carried out several numerical simulation using the commercial software HFSS to obtain an optimal design. Two different orientations of the DSRRs with respect to the wave vector were considered: case1) wave vector, k, parallel to the structure and case 2) wave vector, k, perpendicular to the structure. The setup in Fig. 1(a), shows the unit cell analysis for the configuration 1 which will be discussed in the following. The dimensions of the DSRR were 3 mm × 3 mm with a gap and strip width of 0.33 mm. The thickness of the metal was 0.02 mm printed on a 0.25 mm thick Duroid substrate (ε = 2.2). The unit cell of dimension 5 mm × 1 mm × 3.63 mm was placed in between the waveports with a perfect magnetic conductor (PMC) boundary condition in the y-direction and perpendicular to the ring and a perfect electric conductor (PEC) boundary condition in the z-direction and parallel to the ring. By doing so, we were able to simulate a two dimensional periodic structure along both y and z directions with minimal computation. A similar setup was used in [4]. Based on this orientation, the DSRRs were excited by the magnetic field which was perpendicular to the DSRRs. The dense stack of DSRRs acted as metalsolenoid which helped to concentrate the magnetic flux inside the structure giving rise to a strong magnetic resonance. Similar design was considered and discussed in [18] where it was highlighted that the main benefit of this design was that a high value of permeability could be obtained away from the resonance. The large off-resonance permeability value leads to a large effective index at the same frequency for the DSRRs which could be exploited for maximum phase modulation. The reflection/transmission spectra results obtained from the simulation were used to extract the effective index of the DSRR using the formulation described in [4, 5].

For a normally incident wave in free-space, at the air-sample boundary the scattering parameters, namely the S₁₁ and S₂₁ can be related to the reflection (Γ) and transmission (T) coefficients [5] as:

where,

and

Following this, the complex refractive index can be formulated as n_complex = n_real + n_imag, where $n_{\mathrm{real}}=\frac{2\mathrm{m\pi }-\arg \left(T\right)}{d{k}_0}\mathrm{and}{n}_{\mathrm{imag}}=\frac{-\log \left(\mid T\mid \right)}{d{k}_0}$. The parameter d indicates the slab thickness in the direction of propagation and $k_0=\frac{2\pi }{{\lambda }_0}$, where γ ₀ is the operational wavelength in free space. In our studies, m was set to zero as the wavelength inside the DSRRs was larger than the slab thickness. Detailed discussion of the parameter m can be found in [4]. The extracted data are shown in Fig. 2(a), where the real and imaginary parts of the effective index for the open and closed cases of the DSRRs were plotted. From the graph, at 4 GHz the real part of the effective index was 5.9 for the open state DSRRs while for the closed state the value of the real part of the effective index was 1.6. The closed state had a flat real part response and minimal loss since the DSRRs were off-resonance within the frequency range of 3.5 – 4.5 GHz. The closed state for the DSRR was achieved by effectively shorting the gaps in the DSRRs by placing a piece of metal inside the gap. Due to the large index difference between the two states, we chose a frequency band centered at 4 GHz for our phase modulation application. Although, at resonance the real index was at a maximum which would provide the maximum phase modulation, the imaginary part which indicated the loss of the structure was also very high that could potentially degrade the overall performance of the device. In particular, the transmission data as shown in Fig. 3(a) indicated that for configuration 1 the transmission was -1.6 dB and -2.8 dB for the open and short case respectively. The value of the imaginary parts for both the cases at this frequency was negligibly low and the difference in the real parts of the effective index was calculated as 5.9-1.6 = 4.3. The calculated FOM for the open state was 5.9/0.325 = 18.2 and for the closed state the FOM was 1.6/0.0003 = 5.3 × 10³. As expected the FOM was much higher for the closed state because of the absence of a resonance.

 

Fig. 2. (a) Extracted real and imaginary parts of the effective index for configuration 1. Shaded region indicates the operating region at 4 GHz. (b) Plot of the Electric field line across the computational region for configuration 1. Shaded region of 5 mm indicates the space occupied by the DSRR unit cell.

Download Full Size | PDF

Thus the chosen operating frequency met the requirements outlined previously for maximum modulation with minimal loss. Intuitively the actual phase modulation that could be achieved from switching between the two states of the DSRRS is shown below:

where ∆n_eff was the effective refractive index difference, d was the lattice constant of the DSRR unit cell in terms of the wavelength, $k=\frac{2\pi }{\lambda }$, and ∆ϕ is the phase difference. For this configuration, ∆n_eff = 5.9-1.6 = 4.3, $d=\frac{5}{75}\lambda ,$.Therefore, ∆ϕ ~ 4 × 2π/γ×1/15γ ≃ π/2 As noted, a phase modulation of π/2 could be achieved. Based on the orientation, the modulation could be accounted for due to the change in the effective permeability that brought the change in the effective index from switching between the two states. In reality, the DSRRs suffered multiple reflections as a result a decrease in the phase occured as it exited the DSRRs which was not accounted for in the above formulation. This was verified by plotting the xy-plane cross-sectional volumetric electric field data at the middle of the z-axis within the computational region as shown in Fig. 2(b). The total computational region was 15 mm × 1 mm × 3.63 mm as shown in Fig. 1(a). In Fig. 2(b), the highlighted region indicated the DSRR region, where strong EM behavior could be observed for the open state. The phase was calculated at 15 mm mark in the x-direction inside the computational region, where the field was void of evanescent field for both states of the DSRRs. Note the inclined lines indicated the phase was calculated in the free space region. The difference between the two inclined lines was the actual phase change that was brought about by switching between the two states of the DSRRs which was calculated to be 173 - 103 = 70 degrees. The spatial data gave a better picture of the phase as it took into account of the phase loss due to multiple reflections within the unit cell of DSRR.

In the following we will discuss the results of phase modulation using the second configuration where the DSRR was flipped and rotated 90 . In this configuration, the k vector was perpendicular to the DSRR as shown in

 

Fig. 3. (a) Transmission data for configuration 1 at 4 GHz. (b) Transmission data for configuration 2 at 11.04 GHz.

Download Full Size | PDF

Fig. 1(b), where the electric field, aligned in the z-direction and parallel to the ring gap, excited the structure. The setup consisted of two layers of DSRRs of the same dimensions as the previous configuration. The total length of the computational region was 15 mm × 5 mm × 3.81 mm as shown in Fig. 1(b). The thickness of each DSRR slab and the substrate gap between the two DSRRs was 0.76 mm. The resonance occurred at 11.26 GHz which was contributed by the electric field parallel to the ring gaps. Such a configuration led to a loop current within a ring and coupled to the magnetic resonance. This particular orientation was studied in detail in [19, 21]. Thus in this configuration we were controlling the effective index of the DSRRs with the electric field as opposed to the previous design where the structure was excited primarily with the magnetic field. Nevertheless, a high permeability value leading to a large effective index away from the resonance of the DSRR was obtained which was our objective to demonstrate modulation for this particular orientation. The setup is shown in Fig. 1(b) where the DSRRs were excited by the waveports placed 6.36 mm away from the DSRRs and similar to the previous configuration, the effective indices for the two states were reconstructed from the scattering parameters as shown in Fig. 4(a). At 11.04 GHz, the effective index calculated for the open state DSRR was 5.32 and for the closed state was 2.47. Thus the theoretical phase modulation that could be obtained through switching could be predicted by using the Eq. (4), where $\Delta \varphi =\left(\Delta n_{\mathrm{eff}}\right)\mathrm{kd}=\left(5.32-2.47\right)\times \frac{2\pi }{\lambda }\times 2.28/27.2\lambda .$ Therefore, ∆ϕ ~ 0.47π - π/2.

 

Fig. 4. (a) Extracted real and imaginary parts of the effective index for configuration 2. The shaded region indicates the operating region of 11.04 GHz. (b) Plot of the Electric field line across the computational region for configuration 2. The shaded region indicates the space occupied by the DSRRs.

Download Full Size | PDF

The transmission loss is shown in Fig. 3(b), where the S₂₁ loss was -1.45 dB for the open case and -2.3 dB for the closed case. Furthermore, the FOM calculated for the open state was 5.32/ 0.26 = 20.4 and for the closed state it was 2.47 /0.01 = 247. This configuration showed that a phase modulation of π/2 could also be achieved with the structures perpendicular to the excitation. Following this, we used the xy-plane cross-sectional electric field spatial data in the middle of the z-axis of the computational domain as shown in Fig. 1(b), to extract the phase difference between the two states of the DSRRs. At the 15 mm mark in the x direction, the phase calculated for the open DSRR was 40 degrees, while at the same mark the phase for the closed DSRR was calculated to be 120 degrees as shown in Fig. 4(b). Thus the phase modulation that could be produced from switching between the states was 120-40 = 80 degrees, which was very close to the theoretical prediction. We also performed numerical analysis for a single pair of DSRRs where the thickness of the metmaterial was 1.14mm. We observed that the resonance frequency shifted to 12.4 GHz from our previously operating frequency at 11.04 GHz for the double layers of DSRRs. In particular, for the single layer of DSRRs at 11.04 GHz, the real index was 3.67 and 2.58 as compared to 5.32 and 2.47 for the open case and closed case respectively for the double layers. Thus the ∆n_eff for the single layer at 11.04 GHz was considerably low producing a phase variation of only 18°. This could be possibly accounted for the frequency shift of the resonant open rings for which the real index was noticeably different for the single and double layers although the real index for the non-resonant closed rings was quite similar.

3. Discussion

The primary difference between the two configurations is the nature of exciting the structures, where magnetic field was used to excite configuration 1 and electric field was used to excite configuration 2. The proposed DSRRs had relatively large index values away from the resonance for both of the configurations. This catered to our requirements as we could obtain appreciably large real part of indices with low transmission loss as large losses are noticed at the resonance in a typical resonating structure. The phase change was calculated using both the spectral and spatial data and it was noted that the difference between the two sets of data was higher for configuration 1. This might be explained by the index contrast which was higher (4.3) in configuration 1 compared to a smaller index contrast of 2.85 in configuration 2. The FOM was similar for both of the configurations; however the loss was lower for both the states for configuration 2. Furthermore, in configuration 1, we noted that for the open rings, even though the FOM was lower indicating a higher material loss compared to the closed rings, the transmission was higher for the open rings in contrast to the closed rings. We believe this was because of the reflection loss that was smaller for the open rings than it was for the closed rings. This was verified by looking at the impedance values of the rings at open and closed state. We observed that the open rings were better impedance matched than the closed rings. In particular, the impedance of the open rings was 1.9Z_o, where Z_o is the free-space impedance and impedance for the closed rings was 0.31Z_o. Similar observation was made for configuration 2 where impedance for open rings was 1.96Z_o and for closed rings it was 0.5Z_o. A higher reflection loss in closed rings in contrast to open rings before the onset of the resonance was also noted in [6]. Of the two designs, configuration 2 demonstrated a higher phase modulation than configuration 1. One of the key features of our modulators was that we could maintain minimal variation in amplitude while producing a large change in phase. In particular, the variation in amplitude for configuration 1 was calculated as -2.8-(-1.6) = -1.2 dB and for the configuration 2 it was -2.3-(-1.45) =-0.85 dB. Moreover the thicknesses of the DSRR layers were only 5 mm and 2.28 mm for configuration 1 and 2 respectively. Thus in terms of modulation, amplitude variation and ease of excitation, configuration 2 would be the preferred design. In our study we demonstrated the highest modulation that could be obtained with the least amount of layers in the propagation direction. Although not presented in this paper, the modulation can be brought about using electrical connections such as varactors to switch between the two states of the DSRRs [23]. In our future work we plan to implement a common electrical connection to a set of DSRRs to simplify the switching operation. Along that line, in a recent study, tuning of a set of SRRs loaded with varactors was demonstrated [26]. To date, one limitation of this design is the narrow bandwidth of operation. We plan to address this issue through the investigation of large bandwidth metamaterials [24].

4. Conclusion

In this paper, we demonstrated phase modulation by using resonating metamaterials and operating at an off-resonance frequency. Phase modulation using DSRRs with two different configurations was presented. Configuration 1 was excited by the H-field perpendicular to the ring and configuration 2 was excited by the E-field which was parallel to the splits. It was shown that a high degree of phase modulation can be achieved by switching between the short and open case of the DSRRs with very thin layers of metamaterials while maintaining a very high transmission. The results were verified with numerical simulations using spectral and spatial data. These types of phase modulators can have variety of applications in RF frequency such as in RADAR and LIDAR in optical frequency. Metamaterial phase modulators can also play a role in millimeter wave holography applications. We are currently studying how to dynamically modulate the phase with our proposed structures. Modulators have already been fabricated and measured data will be presented soon.