Supplementary Information
Photonic simulation of topological excitations in metamaterials
Wei Tan1,2, Yong Sun1, Hong Chen1, and Shun-Qing Shen3
1Key Laboratory of Advanced Micro-structure Materials, MOE, Department of Physics, Tongji University, Shanghai 200092, China
2Beijing Computational Science Research Center, Beijing 10008, China
3Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
- Solution of the Dirac-like equation in homogeneous medium
Assume that the Dirac-like equation possesses wave-like solution, which can be written in the form of plane wave
, (S1)
where A and B are constants and k is the wave vector. Substituting Eq. (S1) into the Dirac-like equation, we get
. (S2)
This is an eigen-value problem, and the solution demonstrates the dispersion relation
. (S3)
For the conventional Dirac equation, if the mass of the system is nonzero, there would be a gap with a width of 2m. In contrast, in the Dirac-like equation derived from Maxwell’s equations, both m and V are functions of frequency, and the solution turns to
, (S4)
where and are the relative permittivity and permeability of the material, respectively, and c is the speed of light in vacuum. Usually, the EM responses of a homogeneous dielectric medium present and , which shows in the language of the Dirac-like equation. However, this homogeneous medium does not exhibit a gap with a width of 2m, but is associated with a linear gapless dispersion relation. It is because m and V in the Dirac-like equation are functions of frequency, which makes the solution differ slightly from the case with constant m and V.The absence of gap in homogeneous mediumbrings difficulties in simulating the Dirac equation in conventional dielectric materials. To overcome this problem, we propose to use metamaterials, whose permittivity and permeability can be manipulated in a controllable way in a laboratory environment.We show that the novel band structure of one-dimensional (1D) metamaterials could closely mimics the physics related to the Dirac equation.
- Dispersion relation of periodic structures
We assume that the stack layers are normal to the x axis and the wavefunction within one layer can be represented as the superposition of a left- and right-traveling wave with wave number k,
.
Here , , and C and D are constants. After propagating over a distance d into the positive x direction, the wavefunction is described by
with
.
Note that the wavefunction must be continuous across a boundary. For a system with N layers, each layer j has a transfer matrix Mj, where j increases towards higher x value. The system transfer matrix is then
.
From the system transfer matrix, the wavefunction of the system can be obtained.
For a periodic structure consisting of two components, it shows band structure. By applying Bloch-Floquet theorem and the transfer matrix method, the dispersion relation can be obtained from the equation
,
where is Bloch wave vector. Then the dispersion relation can be written as
, (S5)
where () and () are functions of , and the subscripts 1 and 2denote the two components of the structure, respectively.
- The Dirac-like gap in one-dimensional (1D) metamaterials
Metamaterials are artificially structured materials with subwavelength resonant unit cells, exhibiting complex electric and magnetic responses. Since each unit cell contains an electric and magnetic resonance, the 1D metamaterials show band structures. Take the structures in Ref. [S1] as an example, the dispersion relation can also be described by Eq. (S5), presenting a novel band gap, which is not at the Brillouin zone boundary but at the center (where ). In the subwavelength limit, i.e., , one can expand and in a Taylor series (),
, (S6a)
. (S6b)
In the region near , can also be expand in a Taylor series similar to Eq. (S6a). Substituting Eqs. (S6a) and (S6b) into Eq. (S5) and neglecting the high-order terms, we have the dispersion relationship simplified as
where ,, , , , and . Equation S7 indicates that the metamaterial-structures with subwavelength unit cell can be treated as bulk media with complex electric and magnetic responses, presented by and , respectively. One can find that has a real value when and are simultaneouslypositive or negative, whereas has a pure imaginary value when and or and , associated with the gap. Since , these two kinds of gap have distinct topological phases, and , respectively.
As in the usual case, and of metamaterials can be described by Drude models,
, (S8)
where , , , , , and are constants. Considering the lossless case, , the two band edges, and , are determined by and , respectively, as shown in Fig. S1. The frequency region between and is the gap. For the case that , the gap is characterized by , as shown in Figs. 1a(i) and S1a; for , the gap is characterized by , as shown in Fig. 1a(iii) and S1c; and if , the gap closes with , leading to the Dirac point at , as shown in Figs. 1a(ii) and S1b.
Figure S1| Effective permittivity and permeability of 1D metamaterials. a, the case for the gap corresponding to Fig. 1a(i); b, the case for the gapless structure with corresponding to Fig. 1a(ii); c, the case for the gap corresponding to Fig. 1a(iii).
- Experimental realization of band inversion in the microwaveregime
We employed the scheme of composite right/left-handed (CRLH) transmission line (TL) to realize 1D metamaterials in the microwave regime. The segments are all fabricated on copper-clad 1.57-mm thick Rogers RT5880 substrates, whose relative permittivity and tangent loss are , , respectively. Two 50- Subminiature version A (SMA) connectors are used as the input and output ports. The configuration of the unit cell of this structure is shown in Fig. S2, consisting of a shunt inductance Lin parallel with a capacitance C0 brought by the TL segment and a capacitance C in series with an inductance L0 attributed to the TL segment.The unit cell has a length of d=8 mm (which is less than 1/10 wavelength in the microstrip), and an electromagnetic wave does not “see” discontinuities of the structure. Thus the CRLH TL can be considered effectively homogenous. The effective permittivity and permeability of the CRLH TL is given as [S2,S3]
(S9)
where () is the permittivity (permeability) of environment media, and denote losses, is the length of a unit cell, and is geometric factor. For simplicity here, we assume , and the band edges is obtained as
. (S10)
It is clearly that the frequencies can be readily tuned by tailoring , , , and . If we can adjust the band edges from to or vice versa, the band inversion would be achieved.
Figure S2| Configuration of the unit cell of CRLH TL. The unit cell is much smaller than the operating wavelength, consisting of a shunt inductance Lin parallel and a capacitance C in series.
Interestingly, we find that and vary with the width of the TL, . With an increase of , increases and decreases. As a result, one band edge moves to high frequency, whereas the other moves to low frequency. This property provides us great convenience and flexibility in inverting the bands. As shown in Fig. S1, the structure with w=10 mm has presenting a gap, whereas that with w=2 mm has presenting a gap. Five samples were fabricated with decreased width to demonstrate this phenomenon, as shown in Fig. S3. The measured DOS spectra of these samples are shown in Fig. 1c.
Figure S3| Samples with decreased width for realizing band inversion.Each sample contains 24 units, and each unit has a length of d=8 mmwith a series capacitorC=3.3 pF and a shunt inductor L=10 nH. The measured DOS spectra of these samples are shown in Fig. 1c.
- Domain wall and bound state
These two kinds of gaps have distinct topological phases, which can be distinguished from the mass. Interestingly, the combination of these materials forms a domain wall at the interface and would generate a bound state [S4,S5]. In the following, we derive this bound state from the Dirac-like equation. Consider an example
(S11)
( and , and in the gap region and ). The wavefunction of the left and the right part can be respectively written as
, (S12a)
. (S12b)
The wavefunction is required to be continuous at the interface, and thus we get
. (S13)
By solving Eq. (S13), one can obtain the eigen-frequency of the bound state. In solids, for the Dirac system with , the solution is E=0, which is called the Jackiw-Rebbi solution [S4,S5]. However, in photonic systems, since m and Vare functions of frequency, the solution of eigen-frequency is none-zero.
We fabricated a sample consisting of two TL components, wA=2.5 mm and wB=8.5 mm, to demonstrate the domain wall and bound state, as shown in Fig. S4. From the results of Figs. S1 and S3, one can find that component A has a gap and component B has a gap. The combination of these two components shows well-defined domain wall and bound state, as shown in Fig. 2. The parameters are given as , , , , and , which approximately satisfy Eq. (S13).
Figure S4| Fabricated samples demonstrating the domain wall and bound state. The widths of two components with distinct topological phases are w=2.5 mm and w=8.5 mm, respectively. Each component contains 12 units and structural parameters are of the same with those in Fig. S3.
- Periodic lattice structure
We design a periodic stack of two TL blocks with () and (), which also show interesting topological properties. The unit of the samples has a length of d=7 mm with a series capacitorC=1.0 pF and a shunt inductor L=3.3 nH, leading to . Although the two TLs have band gaps, the interface of the two TLs forms a domain wall and consequently generates a bound state nearby. The overlapping of the periodic bound states would lead to a band structure. In the periodic boundary condition, the numerical simulation demonstrates that the dispersion relations have similar band structures of a diatomic chain, presenting a gap between the “acoustic” and “optical” branches as in Fig. 3b. Note that the gap closeswhen . In our experiment, we fabricated two structures with broken periodic boundary by removing one block of or , as shown in Fig. S5.
The measured DOS for microwave show that the first case in Fig. 3c exhibits a clear gap between band edges at and, which value is close to the calculated value of the loop. The non-zero DOS is attributed to the loss of the metamaterials, which is characterized by the parameters and in Eq. (4). They are fitted to be 0.24 from the measured data. The second case in Fig. 3d exhibits a similar band structure as in Fig. 3c, but presents an additional peak at between the two peaks at and . A more detailed analysis indicates that the two peaks at and corresponds to theband edges as shown in Fig. 3b, in which slight shifts of the position are caused by the finite size effect. The resonant peak at corresponds to two bound states at the ends, as shown in Figs. 3e and 3f. This indicates that the topological properties of the two designed chains in Fig. 3c and 3d are topologically distinguished, although they are constructed by the same blocks of and . Thus our measurement demonstrates explicitly the existence of the end states in 1D topological phase.
Figure S5| Fabricated samples for the structures with with broken periodic boundary. The two TL blocks have the width of and , respectively, and have the same length of 14 mm. Each TL block contains two units, and the lumped elements include a series capacitorC=1.0 pF and a shunt inductor L=3.3 nH.
- Measurements
A network analyzer (Agilent PNA N5222A) was used to characterize our samples in frequency domain. Transmission and reflection properties were obtained directly, and the density of states could be derived from the measured group delay. In theory, the DOS of a lossless optical system is proportional to the group delay , (whereD is the total length of the sample). The group delay is a measure of the time delay of the amplitude envelopes of the various sinusoidal components of a wave package through a device under test, and is a function of frequency for each component, and can be measured directly. In practice, dissipation is inevitable in the present experiments. In this case, the DOS is not very well-defined. However, the group delay is still measurable. When the energy loss is not very large, it is a good approximation to use the group delay to define the DOS. In the present work, the imaginary part of the effective mass in Eq. (3) is comparably small when the gap is large (with large absolute value of effective mass), which is confirmed by the extracted parameters from Fig. 2. Although it lowers the transmission and broadens the resonant states, the main feature of band inversion and domain wall remains. Since the study of band inversion and end states requires a large gap, we believe that the concept of DOS can be used in our measurements to a certain extent.
Then we carried out some simple microwave experiments in time domain to investigate the field distribution. At the particular frequency, a monochromatic wave generated from Agilent E8267D is input to the sample. After that, voltage signals at the different positions along the sample are picked and recorded, using the high-impedance active probe (Tektronix P7260) and the oscilloscope (Tektronix TDS7704B). In each unit cell, only one position which is near the shunt inductor is probed, and thus the LC resonances within a unit are not detected.Numerical simulations were obtained using a commercial software package (CST Microwave Studio).
- Validity of 1D CRLH TL model
The validity of the model has been studied extensively, and many novel phenomena based on CRLH TL metamaterials, such as negative refraction and super-resolving lens, are successfully described by the model as summarized in Refs. [S2]and[S3]. The parameters are determined by the circuit theory. It gives the effective permittivity and permeability of the CRLH TL in the long-wavelength limit as shown in Equation (5). We can also fit the model parameters from the experimental data, such as transmission or the density of states. For example from the measured density of states we can deduce the losses of a sample. We adopt the value of the loss, 0.24, which was fitted from the experimental data.
In our work, besides the analytical method where the TL formula of permittivity and permeability are used, we have also performed full-wave simulations considering detailed structures of each unit cell by using CST Microwave Studio (see for example Figs. 2b&c). All the results show well-defined features supporting theoretical predictions. To show how well the analytical method works, we plot the transmission spectra of the five samples in Fig. 1, as shown in Fig. S6. The red line is analytical calculations based on equation (5) with experimental parameters such as L=10 nH, C=3.3 pF, and d=8 mm, and the black line is experimental measurements. For the two samples of w=7 mm and w=4.5 mm, the theoretical predictions are in very good agreement with the experimental observations. For larger or smaller w, some deviations are shown in the band-gap region, which are coming from mismatches of impedance between the TL and the ports. In these cases, some adjustments of the impedance may be needed, because in the formula of permittivity and permeability, and vary with the impedance[S2][S3]. To fit the experimental results, we do here small adjustments as , , and , while the form of the formula remains. The results are presented in Fig. S6 (blue line). Take for example the case of w=2 mm, the parameters are
before adjustment, and
after adjustment. The comparison between the theoretical calculations and the experimental results implies that equation (5) can provide a good description for effective permittivity and permeability of the TL structures in our work.
Figure S6|Transmission spectra of five fabricated samples. Black line: experimental measurements; red line: analytical calculations; blue line: calculations with adjusted impedance.
References
[S1]Jiang, H., Chen, H., Li, H., Zhang, Y., Zi, J. Zhu, S. Properties of one-dimensional photonic crystals containing single-negative materials. Phys. Rev. E.69, 066607(2004).
[S2]Negative-Refraction Metamaterials: Fundamental Principles and Applications, edited by Eleftheriades, G. V. Balmain K. G. (John Wiley & Sons, Inc., New Jersey, 2005).
[S3]Caloz, CItoh, T. Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, New York, 2006).
[S4]Jackiw, R. Rebbi, C. Solitons with fermion number ½.Phys. Rev. D13, 3398–3409 (1976).
[S5]Shen, S. Q., Shan, W. Y. & Lu, H. Z. Topological insulator and the Dirac equation. Spin1, 33–44 (2011).