Introduction to and

Constantin Simovski and Sergei Tretyakov

Department of Electronics and Nanoengineering School of Aalto University Finland

October 20, 2020 Contents

1 Introduction 1 1.1 Motivation ...... 1 1.2 Book content ...... 5 1.2.1 Metamaterials ...... 5 1.2.2 Nanophotonics ...... 7 Problems and control questions ...... 10 Bibliography ...... 10

2 Electromagnetic (optical) properties of materials 13 2.1 Constitutive relations and material parameters ...... 13 2.2 Frequency ...... 14 2.2.1 Linear response and causality ...... 14 2.2.2 The main dispersion mechanisms and models ...... 16 2.2.3 Kramers-Kronig relations ...... 20 2.3 Electromagnetic waves in materials ...... 24 2.3.1 Dispersion equation ...... 24 2.3.2 Dispersion diagrams and constant-frequency contours . 27 2.3.3 Phase and group velocities ...... 28 Problems and control questions ...... 29 Bibliography ...... 31

3 Metamaterials 33 3.1 Metamaterials concept ...... 33 3.2 Double-negative materials ...... 36 3.3 Perfect lens ...... 44 3.4 Engineered and extreme material parameters ...... 47 3.4.1 Artificial ...... 47 3.4.2 Wire media ...... 50

i CONTENTS ii

3.4.3 Split rings ...... 54 3.5 Engineering bianisotropy ...... 60 3.6 Modular approach in metamaterials: Materiatronics ...... 64 3.7 Tunable and programmable metamaterials ...... 69 Problems and control questions ...... 73 Bibliography ...... 74

4 Metasurfaces 80 4.1 Introduction ...... 80 4.2 Homogenization models ...... 83 4.3 Synthesis of metasurface topologies ...... 86 4.4 Examples ...... 90 4.4.1 “Extraordinary” reflection and transmission ...... 90 4.4.2 Matched transmitarrays ...... 96 4.4.3 Metamirrors ...... 99 4.5 Perfect control of transmission and reflection ...... 103 4.5.1 Perfect transmitarrays ...... 104 4.5.2 Perfect anomalous reflection ...... 108 Problems and control questions ...... 111 Bibliography ...... 113

5 Photonic crystals 118 5.1 Introduction ...... 118 5.1.1 What are photonic crystals ...... 118 5.1.2 Natural photonic crystals ...... 120 5.2 Photonic band structures ...... 122 5.2.1 Bragg phenomenon as a reason of bandgaps ...... 122 5.2.2 Bandgap structures in general ...... 127 5.2.3 Bloch’s theorem ...... 130 5.2.4 Brillouin diagrams of 2D photonic crystals ...... 135 5.2.5 Brillouin diagrams of complex three-dimensional pho- tonic crystals ...... 138 5.3 Isofrequencies of photonic crystals ...... 140 5.3.1 Isofrequencies in boundary problems ...... 140 5.3.2 Isofrequencies of photonic crystals ...... 142 5.3.3 Isofrequencies of a in a boundary prob- lem...... 145 5.4 Numerical simulations of photonic crystals and their scalability 148 CONTENTS iii

5.4.1 Numerical solution via plane-wave expansion ...... 148 5.4.2 Numerical solution of the cell problem ...... 150 5.4.3 Scalability of photonic crystals ...... 152 Problems and control questions ...... 152 Bibliography ...... 155

6 Nanophotonic applications of photonic crystals 159 6.1 Conventional applications of photonic crytals ...... 159 6.1.1 Photonic-crystal defect states ...... 159 6.1.2 Peculiarities of photonic-crystal defect . . . 161 6.1.3 Photonic-crystal fibers ...... 162 6.1.4 Photonic-crystal -emitting diodes ...... 164 6.1.5 Photonic-crystal sensors ...... 167 6.1.6 Add-drop filters ...... 168 6.2 Emerging applications of photonic crytals ...... 170 6.2.1 Dense multiplexing and demultiplexing . . 170 6.2.2 Superprism ...... 173 6.2.3 Pseudoscopic frequency selective imaging ...... 175 6.2.4 Photonic-crystal pseudolens operating at a super-quadric isofrequency ...... 180 6.2.5 Pseudolens for aberration-free imaging ...... 184 6.3 Conclusions ...... 188 Control questions ...... 188 Bibliography ...... 190

7 196 7.1 Introduction ...... 196 7.2 ...... 197 7.2.1 Dispersion equation ...... 197 7.2.2 Slab waveguides ...... 201 7.2.3 Surface waves at low frequencies ...... 205 7.2.4 Excitation of surface plasmon polaritons ...... 206 7.3 Plasmonic ...... 208 7.3.1 Polarizability ...... 208 7.3.2 Localized surface plasmons ...... 210 7.4 Chains of plasmonic nanoparticles ...... 212 7.5 Applications ...... 215 Problems and control questions ...... 215 CONTENTS iv

Bibliography ...... 217

8 Building blocks for all-optical signal processing: Metatronics220 8.1 Introduction ...... 220 8.1.1 Paradigm of all-optical signal processing ...... 220 8.1.2 Metatronics concept ...... 222 8.1.3 The problem of optical connectors in metatronics . . . 225 8.2 Epsilon-near-zero platform for metatronics . . . 227 8.2.1 Hollow with an epsilon-near-zero cladding . 228 8.3 Free space as a possible platform for metatronics ...... 232 8.3.1 Metasurface as a parallel resonant circuit ...... 234 8.3.2 Metal- metasurface as a high-order filter . . . 236 8.4 platform for metatronics ...... 238 8.5 Conclusions ...... 241 Problems and control questions ...... 242 Bibliography ...... 243

9 Selected topics of optical sensing 249 9.1 Introduction ...... 249 9.1.1 Brillouin and Raman radiations ...... 249 9.1.2 Spontaneous Raman radiation: Is it a nonlinear or a parametric effect? ...... 253 9.1.3 Surface-enhanced Raman scattering for molecular sensing255 9.2 Electromagnetic model of SERS: Local field enhancement . . . 258 9.3 ...... 264 9.3.1 Fluorescence for and imaging of small ob- jects ...... 265 9.3.2 Plasmon-enhanced fluorescence in biosensing ...... 268 9.4 Purcell effect ...... 269 9.4.1 Main physical effects in plasmon-enhanced fluorescence and their circuit model ...... 273 9.4.2 Purcell factor of a plasmonic ...... 274 9.5 Purcell effect in SERS ...... 280 9.6 Advanced SERS ...... 282 9.7 Conclusions ...... 285 Problems and control questions ...... 285 9.8 Appendix: Additional information on plasmon-enhanced flu- orescence and surface-enhanced Raman scattering ...... 288 CONTENTS v

9.8.1 On fundamental linear effects in plasmon-enhanced flu- orescence ...... 288 9.8.2 Circuit model of Rabi oscillations and transition to the ...... 291 9.8.3 SERS: physicists and chemists ...... 300 Bibliography ...... 302

10 Nanostructures for enhancement of solar cells 313 10.1 Introduction ...... 313 10.1.1 A bit of history ...... 314 10.1.2 On enhancement of solar cells by passive nanostructures315 10.2 Basics of the diode physics ...... 316 10.2.1 Operation of a diode solar cell ...... 316 10.2.2 About optimization of solar cells ...... 320 10.2.3 Volt-Ampere curve of a diode-type solar cell ...... 325 10.2.4 Main parameters of diode-type solar cells ...... 326 10.2.5 On multijunction solar cells ...... 330 10.3 Optical efficiency of solar cells with an optically thick photo- voltaic layer ...... 332 10.3.1 Single-layer antireflection coating and the impact of the front glass ...... 334 10.3.2 Oblique incidence and daytime-averaged optical loss . . 338 10.3.3 Moth-eye antireflectors ...... 340 10.3.4 Black ...... 345 10.4 Antireflecting coatings formed by arrays of small spheres . . . 350 10.5 Thin-film solar cells ...... 350 10.5.1 Drawbacks, advantages and practical perspective of thin-film solar photovoltaics ...... 351 10.5.2 ARC of nanospheres for a-Si solar cells and the issue of transmission losses ...... 354 10.5.3 Light trapping in epitaxial silicon solar cells ...... 357 10.5.4 Plasmonic light trapping in thin-film solar cells . . . . 360 10.5.5 Dielectric light-trapping for amorphous-silicon solar cells365 10.5.6 Light-trapping structures integrated with the top elec- trode ...... 370 10.6 Conclusions ...... 373 Problems and control questions ...... 374 Bibliography ...... 376 CONTENTS vi

11 Nanostructures for enhancement of sys- tems 389 11.1 What are thermophotovoltaic systems ...... 389 11.2 Conventional TPV systems ...... 392 11.2.1 More about the TPV cavity and the TPV filter . . . . 392 11.2.2 Radiative heat transfer in TPV systems ...... 397 11.2.3 The optimal operation band and the Shockley-Queisser limit for TPV systems ...... 400 11.2.4 TPV generators: domestic, industrial and solar stations 402 11.3 Advanced TPV systems ...... 406 11.3.1 Solar TPV station: confined design and metasurface- based emitter ...... 406 11.3.2 Emissivity of resonant metasurfaces ...... 409 11.3.3 On near-field TPV systems ...... 412 11.3.4 Micro-TPV systems ...... 416 11.3.5 TPV generator without a TPV cavity ...... 418 11.4 Conclusions ...... 420 Problems and control questions ...... 421 Bibliography ...... 423

Index 429 Chapter 8

Building blocks for all-optical signal processing: Metatronics

8.1 Introduction

8.1.1 Paradigm of all-optical signal processing In modern telecommunication systems, transmission of signals to far dis- tances is done using optical techniques (fiber ), but processing of sig- nals is still done electronically. Thus, optoelectronic devices are needed to convert information from the optical band to the band of radio frequencies and back. An advanced optoelectronic decoder is shown in Fig. 8.1. Opto- electronic convertors, even advanced ones like that of [1], form a bottleneck in the modern telecommunication systems because [1–3]:

ˆ Best known optoelectronic convertors lose about 30% of the signal en- ergy converting electronic energy into and back.

ˆ They slow the transmission of messages. The delay restricts the fre- quency range of converted electronic signals by 60–100 GHz [2,3].

ˆ Their are comparable with electronic processors in what concerns their size, weight and power consumption [1,2].

Modern electronic processors can be mass-produced with high quality and at low cost, and their miniaturization in the past decades was very success- ful. Deeply submicron diodes, transistors, and other electronic devices make

220 CHAPTER 8. METATRONICS 221

Figure 8.1: An advanced optoelectronic convertor (decoder). Reprinted from [1] with permission of the editors. modern integrated electronic processors lightweight and compact. However, it is difficult to reduce power consumption of electronic devices. The main reason is the irreducible level of noise inherent to active devices and even to passive radio and devices at room temperatures (see Section 1.1). To ensure the sufficiently high signal to noise ratio, electronic processors as well as optoelectronic convertors have to consume significant electric power. Meanwhile, thermal optical noise at room temperatures is negligibly small in the range λ = 1260 − 1675 nm. As to external optical noises, they can be screened. As a result, electric power required to feed an operational ampli- fier based on an all- is lower by three orders of magnitude compared to that required for its electronic analogue with the same signal- to-noise ratio [4]. This estimation can be extended to an all-optical processor and even to an all-optical computer. Imagine that your computer consumes 100 mW from a small battery instead of the present 100 W taken from the electric power supply, and you will understand how important it is to develop optical analogues of electronic devices [4]. To reach this goal, one needs to create many new all-optical devices: all- CHAPTER 8. METATRONICS 222 optical switches, modulators, logic gates, memory cells, etc. This problem has been formulated in 1980s (see e.g. in [5]), and since 1980s the main progress in this field has been related to a new scientific direction called quantum com- putations [3]. Generation, transmission, reception, and storage of quantum bits of information (called qubits) have been recently demonstrated experi- mentally [6]. Quantum computers will be super-compact. Most of them will be dedicated to work with optical signals, which implies very low power con- sumption. Such devices are called linear-optics quantum computers [7,8]. In this book, we discuss only one aspect of this wide field of research and devel- opment: so-called metatronics which can potentially solve crucial problems for creation of prospective all-optical telecommunication systems. The target of metatronics is all-optical signal processing before or after computations. The idea of metatronics is that all the electronic circuitry operating with al- ternating currents and voltages at radio frequencies can be emulated by opti- cal circuitry operating with electric and magnetic fields of light. Metatronics offers an original way of tackling nano-optical data processing tasks. In prin- ciple, metatronics intersects with optical quantum computing since it may also offer possibilities for nanoscale computations and data storage [9, 10]. However, presently, work is mainly concentrated on processing of optical sig- nals. Scientifically, metatronics merges the circuit theory and design with nano-optics and uses the tools of metamaterials and plasmonics.

8.1.2 Metatronics concept The name metatronics has been recently introduced by Nader Engheta with reference to the emulation of lumped circuit elements at optical using building blocks of known metamaterials. Namely, in accordance to [10], metatronics is metamaterial-inspired optical nano-circuitry. In microelectronics, the notion of a lumped circuit is a powerful concept in which a flow (current) of charges is related to another quantity (volt- age) through the functions of lumped elements (resistor, inductor, capacitor, diode, transistor, etc.). The possibility to synthesize circuits from lumped elements is the key difference between optics/ and radio engineer- ing/electronics. Usual optical components operate with propagating waves and exploit either interference effects (lenses and phase diffraction grids) or require a long optical path for the signal (prism). One can say that the operation of optical device components is governed and mediated by wave propagation. For example, optical filters are either arrays of waveguides CHAPTER 8. METATRONICS 223 or stacks of dielectric and layers. Any layer can be consid- ered as a piece of a for an incident plane wave. Therefore, an impedance matrix of any building block of any optical filter is that of a finite-length transmission line. Recalling these formulas we may imagine how difficult it is to synthesize desired frequency dependence of the whole structure using so restrictive basic functions. Meanwhile, in radio and , lumped circuit elements allow us modularization of a system. What is happening inside the element is not relevant to the connectivity and functionality of the element to the rest of the system. The notion of lumped elements is a powerful tool in dis- coveries of new functionalities, design, and innovations. The lumped L, C, R elements have one-element impedance matrices with basic frequency depen- dencies of their impedances – direct proportionality, inverse proportionality, and uniformity. Any frequency response to input signals can be approxi- mated by polynomials engineered from these frequency dependencies. This is so because the response of lumped elements is not restricted by the phases of waves. Why it is possible in electronics? Because electronics deals with conductivity currents and voltages. Voltage is an integral of the electric field over a small path. Conductivity current is a flow of . To make the voltage relevant for the optical frequency range is possible – all we need is to make the element subwavelength. As to the conductivity currents, the in- ertia of electrons does not allow us to control them at optical frequencies in the same way as we do it at radio frequencies. However, if we recall Maxwell equations for time-harmonic fields and currents:

∇ × H(r) = J + jωD(r), ∇ × E(r) = −jωB(r), (8.1) we see (as Maxwell saw 150 years ago) that there is no principal difference be- tween the density of the conductivity current J and that of the displacement current Jd = jωD = jω0εrE, where εr is the medium relative . The flow of photons is in this meaning similar to that of electrons. Consider three types of subwavelength particles with absolute permittiv- ity ε = 0εp, shown in Fig. 8.2. The electric field inside an optically small particle determines the drop of voltage across it. The displacement current inside the particle, called polarization current, is equivalent to the conduc- tivity current. Therefore, one may introduce local complex conductivity σloc inside the nanoparticle, writing Jd = σlocE:

σloc = jω0εp. (8.2) CHAPTER 8. METATRONICS 224

Figure 8.2: Lumped optical elements emulating inductive, capacitive and resistive electronic elements. Author’s drawing after [9].

Formula (8.2) establishes the point-wise relationship between the density of the polarization current inside the nanoparticle and the electric field inside it. Integrating the displacement current density over the cross section S of our particle we may transit from Jd to the polarization current Id in it, and define the effective admittance and impedance of the particle. The current may be different in different cross sections, but we can calculate the averaged a displacement current Id integrating Id along E: Z Z a 1 Id = jω0εp E dSdz. (8.3) l z S Here l is the particle length in the z-direction and the axis z is parallel to E. Since the drop of voltage V over the nanoparticle caused by field E is equal to Z V = Ezdz, (8.4) z we obtain from (8.3) an effective complex admittance Yint which relates the a averaged polarization current Id with V . This integral admittance of the nanoparticle, evidently, depends on the shape of the particle. However, since the integral admittance is proportional to σloc (8.2), the following general observations can be done just inspecting (8.2): CHAPTER 8. METATRONICS 225

1. When εp > 0 and we can neglect the frequency dispersion of εp, the particle emulates a capacitor since Yint ∼ jω0εp. This is physically related with the electric field created inside the particle by charges accumulated in every half-period near the extremities of the nanoparticle (for simplicity, here we neglect the fringe capacitance due to the electric fields outside of the particle volume).

2. When εp is close to an imaginary value εp ≈ −j|εp| (non-plasmonic metals and some ) the particle emulates a resistor with frequency-dependent resistance Rint ∼ 1/(ω0|εp|). 3. If the particle is made of a plasmonic metal, it can be considered as an effective . Really, its permittivity (in both IR range and long-wave part of the visible range) is adequately described by the 2 2 lossless Drude formula εp ≈ 1 − ωp/ω (2.18). In the frequency range 2 2 where ω  ωp, we may approximately write εp ≈ −ωp/ω . After substitution of this εp into (8.2) we find that the integral inductance is 2 proportional to Lint = 1/(0ωp) (times a shape-dependent factor). This inductance is determined by the magnetic flux inside the nanoparticle1 and in the optics of metals it is called kinetic inductance (also here we neglect the inductance due to the magnetic flux outside of the particle volume).

The integration of the displacement current density results in additional geometrical factors for the integral values Lint,Cint and Rint, but the fre- quency dependence for small particle is still determined by the local conduc- tivity (8.2). The integral values of effective circuit parameters were derived for nanospheres, nanospheroids, nanodisks and nanorods in works [11, 12]. Thus, optical nanoparticles (as well as their dense clusters) successfully em- ulate L, C, and R electronic circuit elements.

8.1.3 The problem of optical connectors in metatronics Submicron all-optical active devices not mediated by wave interference phe- nomena have been also suggested and studied (theoretically and experimen- tally). We can mention here subwavelength all-optical switches [13, 14], log- ical gates [15, 16], and diodes [17, 18], and even so advanced nanodevices as

1See also Eq. (3.43). CHAPTER 8. METATRONICS 226 all-optical transistors [19, 20] and memory cells [21]. These lumped devices are, nowadays, under further development by several scientific groups. If we would have a set of efficient connectors for these ultimately small devices, we could realize all needed functionalities for all-optical information processing at subwavelength scale or, at least, at micron scale. The key problem of metatronics is how to obtain connecting “wires” for photons [10]. Of course, these connectors can be pieces of optical waveguides – phase delays between components will usually not influence signal process- ing. However, optical waveguides having sufficiently low dispersion and losses have substantial cross sections. Meanwhile, photonic “wires” should have subwavelength cross sections. Otherwise, the signal propagating in them will not see the subwavelength insertions in form of nanoparticles discussed above. Creation of subwavelength waveguides is a challenge for optics. Most known subwavelength optical waveguides are plasmonic ones. Plasmonic waveguides are narrowband, strongly dispersive and lossy, see Chapter 7. At room tem- peratures, optical signals in them decay over a few dozens of microns reducing below the noise level [22]. Since the key issue of metatronics is that of efficient optical connectors, and below we concentrate namely on it. Existing optical telecom operates in the band λ = 1.4 − 1.7 µm. In this band, silicon and fiber glass are practically lossless. This is the desirable frequency range for metatronics – in this case a prospective all-optical processor based on metatronics will be compatible with the existing fiber optics (i.e. with Internet). However, in principle, metatronics can work also in the visible range or in the mid-IR range where the low-loss atmospheric window enables the transfer of eye-safe signal in the form of diffraction-free beams [23]. In these casee, the frequency converters should be developed to connect processors and optical fibers. For the existing optical telecom the waveguide cross section of the order of lambda/2 and less, is submicron one. Therefore, below we call our subwavelength photonic “wires” nanoguides. The band of the efficient transmission of a nanoguide should cover the total band of optical signals under processing. Optical signals in the existing optical telecom systems require the relative bandwidth of the order of 7-10%. CHAPTER 8. METATRONICS 227 8.2 Epsilon-near-zero metamaterial platform for metatronics

In electronics, ideal connecting wires which carry electrons from one circuit element to another are perfectly conducting wires. Electric current in wires is confined to wires simply because wires are in a non-conducting medium (free space or a dielectric). An ideal “connecting wire” for displacement current is free space, but what can be used as an isolator for displacement current? Just looking at the basic definition of displacement current, Jd = jω0εrE, we realize that in a medium with zero permittivity Jd = 0, that is, such medium is an ideal insulator for displacement current. Medium with zero permittivity grants Jd = 0 and is an ideal insulator for the displacement current [24]. Another reason for considering the epsilon-near-zero platform is an inter- esting possibility to modify the medium’s effective permeability by immersing two-dimensional dielectric particles (infinite rods of arbitrary cross sections) a two-dimensional epsilon-near-zero medium [25]. Such “doping” does not change the effective permittivity of the host, which remains near zero. The response of a large body can be tuned with a single impurity, including cases such as engineering perfect magnetic conductor and epsilon-and-mu- near-zero media with nonmagnetic constituents. Unfortunately, this concept is valid only for 2D structures. Known microwave experiments [25, 26] use parallel-plate set-ups which emulate infinite uniform space by reflections in two parallel mirrors. Yet another interesting phenomenon associated with epsilon-near-zero media is wave tunnelling through thin channels in metal bulk (at microwave frequencies, where metals are nearly perfect conductors) if the channels are filled by ENZ media [27,28]. Media with zero permittivity do not exist in nature. However, metama- terials with a small can be, in principle, created even for the optical range. Both Lorentz and Drude types of dispersion allow the real part of the complex permittivity of a material to change the sign crossing zero at a certain frequency. Although at this frequency the imaginary part of εr is nonzero, it is not theoretically forbidden to have |Im(εr)|  1 at the frequency where Re(εr) = 0, as well as to have both |Im(εr)|  1 and |Re(εr)|  1 in a certain frequency band. Papers [29–33] present results on realizations on ENZ media for optical applications, but more progress is needed to create low-loss composites needed for effective metatronics appli- CHAPTER 8. METATRONICS 228 cations. Such prospective composite materials are called epsilon-near-zero or ENZ metamaterials.

8.2.1 Hollow waveguide with an epsilon-near-zero cladding Let us consider a free-space gap between two half spaces filled by an ENZ medium. For simplicity, let us study the ideal case when εr = 0 in the ENZ cladding. Any guided mode in this slab waveguide can be viewed as a super- position of two plane waves (Brillouin’s waves) which are totally internally reflected from the two interfaces. In usual dielectric-slab waveguides, these waves have sufficiently large incidence angles to the interface. Meanwhile, a half-space of a medium with εr = 0 completely reflects propagating plane waves for any incidence angle. This fact immediately follows from Fresnel’s reflection formulas for both TE- and TM- polarized waves if we nullify the of the reflecting medium. The only difference from a radio- frequency parallel-plate waveguide is the absence of the TEM mode in the ENZ waveguide. Working modes are either TE or TM modes. In Fig. 8.3(a) the distribution of the transverse components Ey and Hx of the lowest TM-mode is shown. The boundary conditions at the waveguide walls require Ey|y=0 = Ey|y=b = 0. The axial electric field Ez and the trans- verse magnetic field Hx penetrate into the ENZ medium but strongly decay in it being the components of the TM-polarized surface waves excited at the waveguide interfaces. Ey is also nonzero in the ENZ medium experiencing the jump at the interface. Formulas corresponding to the field distributions sketched in Fig. 8.3(a) for the domain 0 < y < b are as follows: πy ω πy E = Ae−jβ sin ,H = 0 Ae−jβ sin . (8.5) y b x β b

Here β = pk2 − (π/b)2 is the mode wave number. The cutoff wavelength λc = 2b is determined by the transverse component of the wave vector π/b. This waveguide is a single-mode one in the range b < λ < 2b, where the thickness of the waveguide is smaller than the wavelength in free space. A similar result holds for the TE-modes. If we replace the ambient with exactly zero permittivity εr = 0 by that with zero real part of εr and |Im(εr)|  1, the operation band of the waveguide does not change – in this case λc ≈ 2b. Such a waveguide can be really called a nanoguide because in the band of the optical telecom, λ = 1.4 − 1.7 µm, its thickness b is submicron. CHAPTER 8. METATRONICS 229

Figure 8.3: Two auxiliary problems: (a) where we deduce the subwavelength thickness b for a waveguide with the epsilon-zero cladding; and (b) where we prove the shunt admittance Yshunt of a particle embedded into this waveguide to be proportional jωεp.

For the main TM-mode in the epsilon-zero or ENZ waveguide the mag- netic field is symmetric with respect to the plane y = b/2, and we may iden- tify the magnetic field averaged over the waveguide cross section 0 < y < b with the effective surface current Jeff flowing on the interface in the opposite directions (because the normals to the interfaces are opposite):

b 1 Z J = H dy. (8.6) eff b x 0 The effective voltage between these two walls is, evidently, equal to

b b/2 Z Z Veff = Ey dy = 2 Ey dy. (8.7) 0 0 CHAPTER 8. METATRONICS 230

The ratio V/Jeff is a real value that can be fit for the center of the operational frequency range to the wave impedance of free space. Now let us assume that a dielectric post with the relative permittivity εp and optically small thickness ∆z  λg = 2π/β and width a  λg is introduced into such a waveguide, as it is shown in Fig. 8.3(b). Let the z- coordinates of the front face 1 of the post be z1, and for the rear face 2 of the post we have z2 = z1 + ∆z. Maxwell’s equation gives us for the fields inside the post: ∂H (y, z) x = jω ε E (y, z). (8.8) ∂z 0 p y Averaging this equation over the post thickness ∆z we obtain

H (y, z + ∆z) − H (y, z ) x 1 x 1 = jω ε Ea(y), (8.9) ∆z 0 p y

a where Ey (y) is the electric field averaged over the post thickness at the given height y. Integrating (8.9) over y from 0 to b/2 we obtain in accordance to (8.6) and (8.7):

b V [J (z + ∆z) − J (z )] = jω ε . (8.10) 2∆z eff 1 eff 1 0 p 2 Here, the voltage is averaged over the post thickness ∆z and due to its smallness it can be referred to the post center z = (z1 + z2)/2.

Figure 8.4: An ENZ nanoguide can be loaded so that it is equivalent to a two-wire transmission line shunted by a lumped parallel circuit. The cor- responding post is a dimer formed by epsilon-positive (silicon nitride) and epsilon-negative () particles. Author’s drawing after [9]. CHAPTER 8. METATRONICS 231

Now let us consider the surface current flowing along the waveguide surface within an effective strip of width a stretched along z as shown in Fig. 8.3(b). Integration of (8.10) over x within the strip (i.e. from −a/2 to a/2) gives the shunt admittance of the post:

Istrip(z1 + ∆z) − Istrip(z)(z1) = YshuntV,Yshunt = jω0εpξ, (8.11) where we have denoted

a/2 Z a∆z I = J dx, ξ = . (8.12) strip eff b −a/2

From (8.11) we can easily find the effective capacitance of the particle, if it is performed of an epsilon-positive material: Ceff = 0εrξ. Epsilon-negative materials in the optical range are metals for which the Drude frequency dis- persion holds. Neglecting the lossy factor and the relatively small term (−1) 2 2 in the Drude permittivity formula (2.18), we may write εr ≈ −ωp/ω , that 2 gives for the inductance length Leff = 1/ξ0ωp. It is also easy to show that taking into account dielectric or metal losses makes the embedded particle equivalent to a parallel connection of either an inductor or a capacitor and a resistor. Because the nanoparticle is effectively excited by fields created by currents flowing right at the position of the particle, it is not essential that the ENZ cladding plate has infinite extent. If the ENZ slab has a finite width w (which may be smaller than the wavelength and even close to a), the result (8.11) remains valid with a rather small correction to the value of ξ. Thus, we can state that an ENZ waveguide loaded by a small post in the optical range is equivalent to a two-wire line (as it is thought for low frequencies, where metal wires are close to perfect conductors) loaded by a shunt admittance: Yeff = jωεpξ. Here the coefficient ξ is a positive value depending on the post geometry and the waveguide width. If we want to load an ENZ nanoguide by a resonant circuit we can stack an epsilon-positive post with an epsilon-negative (plasmonic) one. The ca- pacitance of the epsilon-positive part (dielectric) can resonate with the in- ductance of the plasmonic part of the dimer in the near- and even in the visible range if the permittivity of the dielectric is high enough. In Fig. 8.4 there is an example of such load experiencing resonance in the blue range, reproduces that from paper [9], where the epsilon-positive material CHAPTER 8. METATRONICS 232 is silicon nitride (20 nm thick post) and epsilon-negative material is silver (also 20 nm). In the resonance band the impact of the optical loss in sil- ver is significant, therefore, the corresponding equivalent scheme comprises a resistor. The series loading of an ENZ waveguide is also possible. For example, a small part of the epsilon-zero waveguide of length ∆z can be made thinner than λ/2. Then in this part the waveguide is below the cutoff and the main mode cannot propagate in it. However, if the length ∆z is small enough (∆z  λg) the eigenmode can tunnel through this squeezed part, and it is possible to show that such waveguide patterning is equivalent to a capacitive load connected in series to an effective transmission line. Moreover, one can transform this series load into a resonant circuit filling the corresponding part of the waveguide with a plasmonic metal (negative-epsilon medium). Although this approach is promising and opens up many unique possibil- ities for light processing at subwavelength scale, there are significant techno- logical problems on the route towards practical implementations. Probably the most challenging issue is creation of low-loss epsilon-near-zero claddings which would allow propagation of waves along such ENZ waveguides to long distances. For example, ENZ photonic wires for the use in mid infrared range described in paper [34] allow propagation only at distances of the order of the wavelength.

8.3 Free space as a possible platform for meta- tronics

The most straightforward solution for an optical connector is a wave beam propagating in free space. Gaussian beams in the visible range and near- IR ranges can be obtained with the effective width about 2 − 3 µm using conventional optics. If we drop the requirement of submicron sizes for all the components of metatronics, we can keep the main idea: Optical signal processing at subwavelength scale without using optical interference phe- nomena. A few-micron collimation still grants to an all-optical processor a miniaturization compared to possible analogues utilizing conventional wave- interference approaches and compared to existing optoelectronic convertors. Though this miniaturization will be not so drastic as that potentially granted by hypothetical lossless broadband ENZ media, it is still offering compatibil- CHAPTER 8. METATRONICS 233 ity with prospective all-optical devices having the area of few square microns and subwavelength thickness. Let us explain why and how it is possible. First, consider the example where such the device is a filter. A sub- stantial width of the wave beam impinging this filter means that we cannot use nanoparticles for filtering. We need to find objects with corresponding transversal dimensions which are optically thin along the beam propagation direction (otherwise the operation will be mediated by the wave interference) and can be described as lumped impedances: , , and resistances. These objects are known – they are metasurfaces, see Chapter 4. Filtering can be performed by periodic metasurfaces such as grids of parallel metal or semiconductor nanobars/.

Figure 8.5: A metasurface of parallel nanowires may filter the signal with both polarizations of light whereas the size of the illuminated spot restricts the minimal area of the metasurface. Left panel: sheet impedance of the metasurface is inductive. Right panel: sheet impedance of the same meta- surface is capacitive.

In Fig. 8.5 we reproduce a typical atomic-force microscope picture of such a metasurface illustrating two cases of the wave beam transmission. In the left panel the light of the normally incident wave beam is polarized in parallel to the nanowires and in the right panel the light is polarized transversely. The interaction of the illuminated part of the metasurface with a typical Gaussian wave beam is similar to that of the whole metasurface with a normally incident plane wave. The sheet impedance (also called grid impedance) of an optically dense grid of parallel metal wires is inductive when the electric field is polarized along the wires. When it is polarized CHAPTER 8. METATRONICS 234 across them, the sheet impedance is capacitive. Moreover, if the nanowires are performed of a material experiencing resonant dispersion, such grid will be equivalent to a resonant circuit. If the electric field is polarized along the wires, this is a parallel circuit. Next, we analyse this case as an example.

8.3.1 Metasurface as a parallel resonant circuit Figure 8.6(a) shows a cross-section view of a metasurface operating as a stopband filter at wavelengths λ = 11 − 13 µm [39]. In this example the metasurface is a grid of parallel silicon nitride nanowires of thickness 250 nm and width w = 225 nm. The gap between adjacent nanowires is g = 75 nm. Let us show that its equivalent scheme is that presented in Fig. 8.6(b) for the case when the electric field is polarized along nanowires. Here the equivalent transmission line is free space that can be treated as a a TEM waveguide with the wave impedance Z0 = η and the wave number k0 = ω/c.

Figure 8.6: (a) Sectional view of the metasurface with resonant transmittance and reflectance. (b) Equivalent circuit of the metasurface.

Consider the normal incidence of a plane wave on the grid depicted in Fig. 8.6(a). Let the wave be polarized so that H = Hy0 is directed across the wires and E = Ex0 is along them. The wires are optically thin and have a rectangular cross section. The grid period a = w +g is also optically small. Therefore, we may homogenize this grid replacing it with a layer of the same thickness h and the averaged relative permittivity ε · w + 1 · g ε = r , (8.13) ef a CHAPTER 8. METATRONICS 235

where εr is the relative permittivity of silicon nitride. Then the jump of the magnetic fields across the grid can be found from Maxwell’s equation for the effective dielectric layer with permittivity εef : ∂H H| − H| ≈ z=+h/2 z=−h/2 = −jω ε E. (8.14) ∂z h 0 ef Since h is optically small, the variation of E across the grid is negligible, and E in the right-hand side of (8.14) can be identified as the mean field at the central plane z = 0 of the effective layer. Meanwhile, the jump of the magnetic field across the grid in the left-hand side of (8.14) is substantial (due to large |εr| in the right-hand side). In accordance to the Maxwell boundary conditions, the jump of the tangential magnetic field across a thin sheet is equal (with sign minus) to the surface current in this sheet. Here we attribute this current to the central plane (z = 0) of the metasurface and substituting (8.13) into (8.14) obtain ε w + g J = Y E,Y = jω hE r . (8.15) g g g 0 a

The effective current Jg flows along the nanowires and the scalar sheet ad- 2 mittance Yg relates it to the electric field E having the same direction. It is clear that the sheet admittance Yg determined by (8.15) can be interpreted as Yg = Ynw + Yag, i.e., a parallel connection of two admittances – that of a and that of the air gap between two adjacent nanowires: jω ε wh jω gh Y = 0 r ,Y = 0 . (8.16) nw a ag a

Here, Yag is capacitive and Ynw is also capacitive if εr is non-resonant and positive. However, silicon nitride is a semiconductor whose permittivity εr in the band λ = 10 − 13 µm is essentially complex and experiences a resonance, approximated as follows [39]: 0.17 ω ε = 1 + , 0 = 25 THz, γ = 0.14. (8.17) r  2 2π 1 − ω + jγ ω ω0 ω0

2The same result was obtained in [38] in a more strict and difficult way. CHAPTER 8. METATRONICS 236

In the resonance band unity can be neglected compared to the resonant term in the expression for εr. After the substitution of (8.17) into (8.16) we obtain the effective impedance of the nanowire Znw = 1/Ynw in the form jωa γa Znw = 2 + = jωL + R, (8.18) 0whω0 0whω0 that is a series connection of the effective inductance L and effective resistance R of the grid. It explains the equivalent circuit in Fig. 8.6(b). The amplitude transmission coefficient T = 1 + R of the metasurface can be easily expressed through Yg:

Y0 − Yinput T = 1 + R = 1 + ,Yinput ≡ Y0 + Yg, (8.19) Y0 + Yinput p where Y0 = 1/η0 = 0/µ0 is the admittance of free space and Yinput is the input admittance of the metasurface, which evidently results from a parallel connection of the sheet admittance and the admittance of free space. From (8.19), we obtain for the power transmittance of the metasurface:

2 r 2 1 µ0 τ = |T | = , η0 ≡ . (8.20) 1 + Ygη0/2 0

The minimum of τmin = 0.16 is achieved when the value |εr| in Yg attains its maximum – at the resonance of the circuit. Beyond the resonance band τ ≈ 0.9. Of course, the performance of this stopband filter is not stellar, but it is a good conceptual example explaining the equivalence of a resonant metasurface in free space to a resonant lumped load.

8.3.2 Metal-dielectric metasurface as a high-order fil- ter

From Eq. (8.16) we see that the sheet impedance (Zg ≡ 1/Yg) of a solid (g = 0) dielectric (εr > 1) sheet is capacitive. If we substitute for εr the permittivity of a plasmonic metal adopting the lossless approximation 2 2 2 2 (1 − ωp/ω ) and neglecting unity compared to ωp/ω , we obtain an inductive grid impedance. Stacking a dielectric sheet and a plasmonic sheet results in a parallel connection of effective inductive and capacitive loads. Alternat- ing dielectric and plasmonic strips in a single composite sheet we obtain a CHAPTER 8. METATRONICS 237 series connection of L and C [40]. All these equivalent schemes are shown in Fig. 8.7. Using this approach one can engineer passband or stopband filters of the second or third orders, i.e., with practically acceptable slope of the amplitude-frequency characteristics at the band edges. An in-plane

Figure 8.7: (a) The equivalent scheme for a dielectric sheet in free space. (b) The same for a plasmonic sheet. (c) The same for a stack of the dielectric and plasmonic strips. (d) The same for a metasurface of alternating plasmonic and dielectric strips. Author’s drawing after [40]. isotropic variant of a similar filter – with square plasmonic patches inserted into a high-permittivity sheet – was studied in [41]. Thus, the filtering func- tionality is really achievable with filtering metasurfaces using free space as metatronics platform instead of nanoguides. Finally, let us note that tightly focused wave beams (for optics, with the effective cross section about 2 − 3 µm) needed for metatronics devices based on the free-space platform, can be formed also by metasurfaces (see review [42] and Chapter 4). Using this approach, the need of conventional bulky lenses for beam focusing can be eliminated. CHAPTER 8. METATRONICS 238 8.4 Graphene platform for metatronics

A prospective all-optical processor utilizing wave beams in free space should comprise a lot of empty gaps inside it and the level of miniaturization granted by such platform is not comparable with that promised by ENZ waveguides used as optical connectors. Since the progress in this field has been mod- est and the desire to engineer small all-optical chips is strong, the group of Engheta has theoretically (as of the time of writing) developed another plat- form for metatronics – that based on graphene – a stable 2D crystal lattice (monolayer of carbon atoms) [43]. Due to its remarkable electronic, optical and mechanical properties, many new applications have been suggested [44]. The use of graphene for metatronics was suggested in paper [45]. This appli- cation is based on the tunability of the surface conductivity of graphene in the terahertz and far-IR frequency regions. Thus this type of metatronics (if whenever implemented) will obviously demand frequency conversion of the near infrared signals carried by optical fibers. Surface conductivity of graphene σg is a complex number defined by re- lation J = σgE, where E is the complex amplitude of the tangential elec- tric field in the graphene sheet plane and J is the surface current density. The term surface conductivity may be misleading, because the surface ad- mittance/impedance usually means the input admittance/impedance seen at the surface of a bulk body. In this case, however, surface conductivity mod- els the response of a negligible-thickness sheet, relating the electric field at the sheet with the jump of the tangential magnetic field across it. It other words, this parameter is what we called above the grid admittance or sheet admittance of a metasurface. However, here we will adopt the terminology commonly used in the graphene research and call this parameter the surface conductivity. For metatronics it is important that σg is complex-valued and frequency dependent, and this dependence is controllable via the so-called chemical po- tential µc of graphene. Here we will not deviate into into solid-state physics and discuss the nature of the relationship between the chemical potential and the Fermi energy level EF . For our purposes it is enough to know that the chemical potential depends on the external static electric field (both normal and tangential components) which can be applied to graphene sheets using a DC voltage source. In Fig. 8.8 the surface conductivity of graphene is de- picted for several values of the Fermi level at zero absolute temperature [46]. These three values lie in the range of Fermi levels achievable due to prac- CHAPTER 8. METATRONICS 239

Figure 8.8: Imaginary and real parts of σg versus frequency for different values of the Fermi energy (i.e. different chemical potentials). Reprinted from [46] with permission of the authors. tically applicable static electric fields. From Fig. 8.8 we can expect that σg can be tuned to respond as an inductive or capacitive sheet by properly changing the bias voltage. For example, at 25 THz the sheet is inductive and moderately lossy if EF = 0.06 eV and capacitive and moderately lossy for EF = 0.08 eV. Note, however, that the plot is Fig. 8.8 is an idealization assuming zero temperature and approximate models of the graphene prop- erties. In practice, it is usually easier to realize low-loss inductive response, and for realization of capacitive properties metal-graphene metasurfaces may be preferable. Potential use of non-homogeneously biased graphene sheets for meta- tronics applications was discussed in paper [45]. The main idea is to bias different areas of a continuous graphene sheet with different bias voltages, creating areas with desired sheet impedances, either inductive or capacitive. Figure 8.9 shows a conceptual illustration of a graphene-sheet transmission line (a “wire”). Although the concept is rather attractive and promising, CHAPTER 8. METATRONICS 240

Figure 8.9: (a) An effective wire of a graphene printed circuit board is an SPP waveguide. (b) Simulated map of the electric field. Both figures are reprinted from [45] with permission of AAAS. practical realizations, unfortunately, are very challenging. To the best of our knowledge, at this time there are no experimental demonstration of such devices. The main problems are in difficulties to realize local biasing (the approach of varying substrate thickness, illustrated in Fig. 8.9, requires ex- cessively high bias voltages) and high level of resistive loss in graphene sheets. Note also that the initial simulations made in [45] assume cryogenic temper- atures. Later studies have shown that is possible to fabricate on a dielectric substrate so-called graphene nanoribbons operating at room temperatures, whose inductive, capacitive and resistive properties can be controlled via bias voltage [47–49]. In fact, these nanoribons are essentially non-linear, however, their non-linearity seems to be controllable [50,51]. Experiments reported in works [50–52] promise that graphene-based metatronics devices may possi- bly work at room temperatures. In the near future we will see if it indeed realistic. CHAPTER 8. METATRONICS 241 8.5 Conclusions

Metatronics, which can be briefly defined as all-optical nano-circuitry, is a very exciting scientific direction. It promises us a novel class of ultra-fast, low-power, broadband and extremely small chips and processors. Its target is all-optical telecom without optoelectronic conversion and all-optical com- puters including those based on quantum effects. This target it is still far from practical implementation, although theoretically it is achievable and significant progress has been achieved on this way since the initial paper [9] was published. In this book, we have not reviewed non-linear optical de- vices for metatronics – subwavelength optical diodes, transistors, modulators, etc. The essential point of metatronics is the possibility to properly connect these devices. These connectors must also be compatible with all-optical filters. Thus, the all-optical chip must be compact – grant miniaturization as compared to conventional optical devices employing wave interference in dielectrics or in free space. We have seen that currently one develops three main platforms for connecting submicron all-optical devices:

ˆ Nanoguides formed as free-space channels in ENZ background media. Filtering functionality is offered by nanoparticles inserted into these nanoguides.

ˆ Free space where strongly collimated light beams propagate. Filter- ing functionality is offered by metasurfaces, and better connection to submicron devices is granted by focusing metasurfaces.

ˆ Graphene sheets or graphene nanoribbons.

Among them, the most realistic platform is evidently the second one. How- ever, it offers only limited miniaturization and the use of conventional optics cannot be completely avoided. It also requires solving a lot of practical prob- lems, especially on the compatibility of different devices in the presence of diffraction and other parasitic effects. Probably, this is the reason why this platform develops slowly. The third platform in what concerns the graphene nanoribbons develops fast. If this development is not blocked by any un- expected physical effect, one may possibly expect practical all-optical chips created in the next decade. As to the first platform, it seems for the instance more challenging. It is more difficult to believe into commercialization of metatronics based on this platform in the next future. CHAPTER 8. METATRONICS 242

In any case, at the present stage of these studies, we cannot reject any of the three possible technological platforms of metatronics. Perhaps, alterna- tive – more practical or more advantageous – platforms for metatronics will appear in the future. In our opinion, the target of metatronics – a practical, compact and energy-efficient all-optical signal processor – definitely deserves more efforts and resources than those invested into this science nowadays.

Problem and control questions

Problem Using the formula (8.15) for the grid admittance of an optically dense grid (metasurface) of parallel infinitely long dielectric nanobars of a material whose complex permittivity εr is described by formulas (8.17), design a stop- band filter performed as a bilayer grid – two parallel identical metasurfaces separated by a certain gap d. The stopband is defined as the range of wave- lengths in which the power transmittance does not exceed one half of the maximal one. The explicit task is as follows:

1. Neglecting near-field interactions, i.e., assuming that in the space be- tween the metasurfaces there are only two plane waves (the fundamen- tal Floquet harmonic), which is an adequate approximation if d ≥ h, derive the transmittance τ of a bilayer filter expressing it through Yg and d.

2. Find analytically the optimal value for d which grants the minimal transmittance at λ0, the wavelength at the center of the stopband.

3. Show numerically (e.g. in Matlab) that the resonant (λ0 = 12 µm) transmittance for silicon nitride bilayer metasurface is lower than that for a single grid. Let the geometric parameters be equal g = 75 nm, w = 225 nm, h = 250 nm (see Fig. 8.6).

4. Can the stopband of this bilayer filter be equal to that of the monolayer one? If yes, for which value of d?

5. Show numerically that the frequency dependence of the transmittance of the bilayer filter is closer to the ideal one (zero in the stopband and unity outside it) than that of the monolayer grid. What is the reason CHAPTER 8. METATRONICS 243

of this improvement? Hint – recall the resonance curve for two weakly coupled identical RLC-circuits.

Questions 1. Metatronics is an optical circuitry analogous to electronic circuits. What is an analogue of an in metatronics: a light wave, a phonon, or a ?

2. What is the main advantage of metatronics compared to the older idea of all-optical signal processing governed by wave processes?

3. Why metatronics needs ENZ nanoguides instead of already known plas- monic nanoguides? Why a parallel-plate waveguides of silver plates cannot operate in the optical range similarly to its larger radio-frequency analogue?

4. What are the advantages and disadvantages of graphene-based meta- tronics compared to metatronics based on wave beams in free space?

Bibliography

[1] V. J. Sorger, R. F. Oulton, R.-M. Maa, and X. Zhang, “Toward inte- grated plasmonic circuits,” MRS Bulletin 37, 728-738 (2012).

[2] D. D. Nolte, Mind at light speed: A new kind of intelligence, The Free Press (Simon and Schuster, Inc.), NY (2001).

[3] R. S. Tucker, “The role of optics in computing,” Nature Phot. 4, 405-410 (2010).

[4] E. Yablonovitch, “Why we need to replace the transistor, and what would be the newly required material properties?” International Congress and Photonics’2016, St. Petersburg, June 27-29, Uni- versity ITMO, Book of Abstracts, p. 46.

[5] D. G. Feitelson, : A survey for computer scientists, The MIT Press, Cambridge, MA (1988). CHAPTER 8. METATRONICS 244

[6] J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, D. N. Jamieson, A. S. Dzurak, and A. Morello, “A single-atom electron spin qubit in silicon,” Nature 489, 541-545 (2012).

[7] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135-174 (2007).

[8] D. Gevaux, “Optical quantum circuits: To the quantum level,” Nature Phot. 2, 337-337 (2008).

[9] N. Engheta, “Circuits with light at nanoscales: optical nanocircuits in- spired by metamaterials,” Science 317, 1698-1702 (2007).

[10] N. Engheta, “Taming light at the nanoscale,” Physics World 23, 31-34 (2010).

[11] N. Engheta, A. Salandrino, and A. Al´u,“Circuit elements at optical frequencies: Nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).

[12] A. Polemi, A. Al´u,and N. Engheta, “Nanocircuit loading of plasmonic waveguides,” IEEE Trans. Antennas Propag. 60, 4381-4390 (2012).

[13] I. Glesk, P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, and S. Janz, “All-optical switching using nonlinear subwavelength Mach-Zehnder on silicon,” Optics Express 19, 14031-14039 (2011).

[14] B. Piccione, C. Cho, L. K. Van Vugt, and R. Agarwal, “All-optical active switching in individual semiconductor nanowires,” Nature Nanotechn. 7, 640-645 (2012).

[15] Z. Yang, Y. Fu, J. Yang, C. Hua and J. Zhang, “Spin-encoded sub- wavelength all-optical logic gates based on single-element optical slot nanoantennas,” Nanoscale 10, 4523-4527 (2018).

[16] P. Singh, D. K. Tripathi, S. Jaiswal, and H. K. Dixit, “All-optical logic gates: designs, classification, and comparison,” Adv. Opt. Technol. 14, 275083 (2014). CHAPTER 8. METATRONICS 245

[17] W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14, 955-958 (2014).

[18] Q. Yun-Ping, N. Xiang-Hong, B. Yu-Long, W. Xiang-Xian, “All-optical diode of subwavelength single slit with multi-pair groove structure based on SPPs-CDEW hybrid model,” Acta Physica Sinica 66, 117102 (2017).

[19] X. Fang, K.F. MacDonald, and N. I. Zheludev, “Controlling light with light using coherent metadevices: all-optical transistor, summator and invertor,” Light: Sci. Applic. 4, 292 (2015).

[20] W. Chen, K.M. Beck, R. Bucker, M. Gullans, M.D. Lukin, H. Tanji- Suzuki, and V. Vuletic, “All-optical switch and transistor gated by one stored photon,” Science 341, 768-770 (2013).

[21] E. Kuramochi and M. Notomi, “Optical memory: Phase-change mem- ory,” Nature Phot. 9, 712-714 (2015).

[22] Plasmonic nanoguides and circuits, S. Bozhevolnyi, Ed., Pan Stanford Publishing, Singapore (2009).

[23] N.S. Prasad, “Optical communications in the mid-wave IR spectral band,” J. Opt. Fiber Commun. Rep. 2, 556-602 (2005)

[24] B. Edwards and N. Engheta, “Experimental verification of displacement- current conduits in metamaterials-inspired optical circuitry,” Phys. Rev. Lett. 108, 193902 (2012).

[25] I. Liberal, A. M. Mahmoud, Y. Li, B. Edwards, N. Engheta, “Photonic doping of epsilon-near-zero media,” Science 355, 1058-1062 (2017).

[26] Z. Zhou, Y. Li, H. Li, W. Sun, I. Liberal, and N. Engheta, “Substrate- integrated photonic doping for near-zero-index devices,” Nature Comm. 10, 4132 (2019).

[27] M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using -near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006). CHAPTER 8. METATRONICS 246

[28] R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).

[29] A. V. Goncharenko and K.-R. Chen, “Strategy for designing epsilon- near-zero nanostructured metamaterials over a frequency range,” J. Nanophoton. 4, 041530 (2010).

[30] A. Monti, F. Bilotti, A. Toscano, and L. Vegni, “Possible implementa- tion of epsilon-near-zero metamaterials working at optical frequencies,” Optics Communications 285, 3412-3418 (2012).

[31] P. Pinchuk and K. Jiang, “Broadband epsilon-near-zero metamaterials based on metal-polymer composite thin films,” Proc. of SPIE 9545, 95450W (2015).

[32] R. Maas, J. Parsons, N. Engheta, and A. Polman, “Experimental re- alization of an epsilon-near-zero metamaterial at visible wavelengths,” Nat. Photonics 7, 907-912 (2013).

[33] E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n = 0 structures for visible light,” Nat. Photonics 7, 907-912 (2013).

[34] R. Liu, C. M. Roberts, Y. Zhong, V. A. Podolskiy, and D. Wasser- man, “Epsilon-near-zero photonics wires,” ACS Photonics 3, 1045-1052 (2016).

[35] M. H. Javani and M. I. Stockman, “Real and imaginary properties of epsilon-near-zero materials,” Phys. Rev. Lett. 117, 107404 (2016).

[36] A. J. Labelle, M. Bonifazi, Y. Tian, C. Wong, S. Hoogland, G. Favraud, G. Walters, B. Sutherland, M. Liu, J. Li, X. Zhang, S. O. Kelley, E. H. Sargent, and A. Fratalocchi, “Broadband epsilon-near-zero reflectors enhance the quantum efficiency of thin solar cells at visible and infrared wavelengths,” ACS Appl. Mater. Interfaces 9, 5556-5565 (2017).

[37] W. Lewandowski, M. Fruhnert, J. Mieczkowski, C. Rockstuhl, and E. Gorecka, “Dynamically self-assembled silver nanoparticles as a thermally ,” Nature Comm. 6, 6590 (2015). CHAPTER 8. METATRONICS 247

[38] Y. Sun, B. Edwards, A. Al`u,and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nature Mat.11, 208-2012 (2012).

[39] H. Caglayan, S.-H. Hong, B. Edwards, C. R. Kagan, and N. Engheta, “Near-infrared metatronic nanocircuits by design,” Phys. Rev. Lett. 111, 073904 (2013).

[40] Y. Li, I. Liberal, and N. Engheta, “Engineering spectral dispersion with multi-ordered optical metasurfaces using insertion-loss method,” Pro- ceedings of the CLEO International Conference on Quantum Electron- ics and Laser Science QELS’2016, San Jose, California, US, 5–10 June, 2016, paper FM2D.5(1-8).

[41] J. Cheng and H. Mossalaei, “Truly achromatic optical metasurfaces: a filter circuit theory-based design,” J. Opt. Soc. America B 32, 2115-2120 (2015).

[42] S. B. Glybovski, S. A. Tretyakov, P. A. Belov, Y. S. Kivshar, and C. R. Simovski, “Metasurfaces: From to visible,” Phys. Rep. 634, 1-72 (2016).

[43] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666-669 (2004).

[44] M. I. Katsnelson, Graphene: carbon in two dimensions, Cambridge Uni- versity Press, Cambridge, UK (2012).

[45] A. Vakil and N. Engheta, “ using graphene,” Sci- ence 332, 1291-1294 (2011).

[46] Y. Fan, F. Zhang, Q. Fu and H. Li, “Harvesting plasmonic excitations in graphene for tunable terahertz/infrared metamaterials,” in: Recent Advances in Graphene Research, P.K. Nayak, Ed., InTech Open (2016), doi:10.5772/61909.

[47] Q. Bao and K.P. Loh, “Graphene photonic, plasmonic, and broadband optoelectronic devices,” ACS Nano 6, 3677-3694 (2012). CHAPTER 8. METATRONICS 248

[48] K. J. A. Ooi, H. S. Chu, L. K. Ang, and P. Bai, “Mid-infrared active graphene nanoribbon plasmonic waveguide devices,” JOSA B 30, 3111- 3116 (2013).

[49] A. Tavousi, M. A. Mansouri-Birjandi, and M. Janfaza, “Optoelectronic application of graphene nanoribbon for mid-infrared bandpass filtering,” Applied Optics 57, 5800-5804 (2018).

[50] E. Carrasco, M. Tamagnone, J.R Mosig, T. Low and J. Perruisseau- Carrier, “Gate-controlled mid-infrared light bending with aperiodic graphene nanoribbons array,” 26, 134002 (2015).

[51] S. Zamani and R. Farghadan, “Graphene nanoribbon spin- ,” Phys. Rev. Applied 10, 034059 (2018).

[52] Q. Guo, R. Yu, C. Li, S. Yuan, B. Deng, F. J. Garcia de Abajo, and F. Xia, “Efficient electrical detection of mid-infrared graphene plasmons at room temperature,” Nature Mat. 17, 986–992 (2018). Index

all-optical chip, 241 computer, 221 devices, 221 processor, 221, 226, 232, 238 telecommunication systems, 222 transistor, 221 chemical potential, 238 ENZ metamaterial, 228 epsilon-and-mu-near-zero, 227 Fermi level, 238 fiber optics, 220 grid admittance, 238 linear-optics quantum computers, 222 nanoguides, 226 nanoribbons, 240 optical nano-circuitry, 222 optoelectronic devices, 220 perfect magnetic conductor, 227 quantum computations, 222 sheet admittance, 235 sheet impedance, 233 surface conductivity, 238 thermal optical noise, 221

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