The resonant origins of reflectance of peak-I and peak-II were theoretically examined as a function of interparticle length between NPs. **Figure 9(a)** exhibits calculated reflectance of NP sheets with different *l* values on the basis of 20 NP layer model, revealing that reflectance monotonically enhanced with decreasing *l*. Peak-II showed a red-shift to longer wavelengths when decreasing *l* from 10 to 1 nm. In contrast, peak-I remained unchanged [**Figure 9(b)**]. These results suggest a difference in the origin of plasmon resonance between peak-I and peak-II. In general, the localized *E*-field from each metal NP usually overlaps when metal NPs are closely positioned, and plasmon coupling occurs [41]. In the plasmon hybridization process, the plasmon interaction can be categorized into bonding and anti-bonding levels. For example, the bonding level shows a red-shift of a resonant wavelength with decreasing interparticle length. On the other hand, there is a slight blue-shift a resonant peak is due to

difference of *R*/*A* ratio between experimental and calculation aspects [**Figure 8(c)**].

and absorbance, respectively (Figure 6 of Ref. [39]). Copyright 2014 by the American Chemical Society.

**5.3. Plasmon hybridization and reflectance mechanism**

184 Nanoplasmonics - Fundamentals and Applications

The character of *E*-field coupling in the NP sheets was further studied from polarized reflectance measurements. Two types of light polarized perpendicular (*s*-polarized) and parallel (*p*-polarized) to the plane of incidence were introduced the sample at an incident angle of 75°. **Figure 10** shows polarized reflectance spectra for *s*- and *p*-polarization configurations for a NP sheet. In a case of *s*-polarization, peak-I and peak-II were simultaneously obtained because the electric vector of the radiation produced electron oscillations in NPs parallel to the in-plane of the sample. On the other hand, peak-I only found for a *p*-polarization case. These optical properties concerning both polarizations are explained as follows.

The electric vector excites electron oscillations in NPs normal to the plane of the sample, and suppresses the *E*-field coupling along the in-plane direction. These results revealed that

**Figure 10.** Reflectance spectra of a 216 nm-thick NP sheet for s- and p-polarized lights. Inset indicates the direction of the electric vector of incident light in addition to an *E*-field distribution under a *p*-polarization.

peak-II was excited by the field coupling along the in-plane directions. In contrast, the reflectance at peak-I was essential for *E-*field coupling along the out-of-plane direction. That is, the near- and mid-IR reflectance of the NP sheets was attributed to the 3*D* field coupling along the out-of-plane and in-plane directions. The film thickness-dependent plasmon splitting was attributed to the formation of field coupling along the out-of-plane direction, leading to the enhanced reflectance in the near-IR range.

### **5.4. Nanoparticle gap and reflectance of ITO NP sheets**

The thermal behaviours of the NP samples were investigated by TG-DTA in an N2 atmosphere with a heating rate of 10°C/min. The weight loss up to 250°C might be related to the loss of physically or chemically absorbed water. There was an obvious weight loss in the temperature range 270–320°C because of the generation of organic species confirmed by *m*/*z* peaks at 18 (H2 O) and 44 (CO2 , C3 H8 , C2 H4 O *etc*.) [**Figure 11(a)** and **(b)**]. The decomposed species were owing to thermal removal of the surface ligands composed of fatty acids on the NP surfaces. There phenomena are identified by FT-IR measurements. The relation between surface ligand molecules and optical properties in the NP sheets was cleared by the spectral changes after annealing at different temperatures (*T*A = 150–550°C). **Figure 11(c)** shows temperature-dependent reflectance spectra taken in an inert atmosphere for a 216 nm-thick NP sheet, revealing remarkable spectral changes with increasing temperature. The two resonant peaks at 150°C were weakened gradually following the change in spectral shape with increasing temperature. Above all, the near-IR reflection at peak-I shifted to longer wavelengths at high temperatures above 300°C corresponding to the removal of the surface ligands. In addition, the annealing effects of NP sheets were checked by the electrical resistivity in the sheets. Electrical resistivity was in the order of 104 Ω.cm below annealing temperatures below 250°C because the presence of the surfactant layers on NPs seriously impedes carrier transport in the NP sheets. The surface ligands act as interparticle insulating layers in NP networks [42].

Infrared Solar Thermal-Shielding Applications Based on Oxide Semiconductor Plasmonics http://dx.doi.org/10.5772/67588 187

**Figure 11.** (a) TG-DTA curves of ITO NP samples in an N2 atmosphere. (b) TOF-Mass spectroscopy combined with TG-DTA. *M/z* signals at 18 and 44 were detected in the range 27–550°C. (c) Dependence of reflectance spectra on annealing temperature for a 216 nm-thick NP sheet.

However, this was markedly reduced at high temperatures above 350°C due to the removal of the surface ligands. Therefore, the removal of surface ligand molecules from the NPs strongly affected the whole reflective phenomena, which also clarified the importance of interparticle length in obtaining a high reflectance performance.

### **5.5. Electromagnetic responses in microwave range**

peak-II was excited by the field coupling along the in-plane directions. In contrast, the reflectance at peak-I was essential for *E-*field coupling along the out-of-plane direction. That is, the near- and mid-IR reflectance of the NP sheets was attributed to the 3*D* field coupling along the out-of-plane and in-plane directions. The film thickness-dependent plasmon splitting was attributed to the formation of field coupling along the out-of-plane direction, leading to the

**Figure 10.** Reflectance spectra of a 216 nm-thick NP sheet for s- and p-polarized lights. Inset indicates the direction of the

The thermal behaviours of the NP samples were investigated by TG-DTA in an N2

sphere with a heating rate of 10°C/min. The weight loss up to 250°C might be related to the loss of physically or chemically absorbed water. There was an obvious weight loss in the temperature range 270–320°C because of the generation of organic species confirmed by *m*/*z*

species were owing to thermal removal of the surface ligands composed of fatty acids on the NP surfaces. There phenomena are identified by FT-IR measurements. The relation between surface ligand molecules and optical properties in the NP sheets was cleared by the spectral changes after annealing at different temperatures (*T*A = 150–550°C). **Figure 11(c)** shows temperature-dependent reflectance spectra taken in an inert atmosphere for a 216 nm-thick NP sheet, revealing remarkable spectral changes with increasing temperature. The two resonant peaks at 150°C were weakened gradually following the change in spectral shape with increasing temperature. Above all, the near-IR reflection at peak-I shifted to longer wavelengths at high temperatures above 300°C corresponding to the removal of the surface ligands. In addition, the annealing effects of NP sheets were checked by the electrical resistivity in the sheets.

because the presence of the surfactant layers on NPs seriously impedes carrier transport in the NP sheets. The surface ligands act as interparticle insulating layers in NP networks [42].

atmo-

O *etc*.) [**Figure 11(a)** and **(b)**]. The decomposed

Ω.cm below annealing temperatures below 250°C

enhanced reflectance in the near-IR range.

186 Nanoplasmonics - Fundamentals and Applications

O) and 44 (CO2

Electrical resistivity was in the order of 104

peaks at 18 (H2

**5.4. Nanoparticle gap and reflectance of ITO NP sheets**

, C3 H8 , C2 H4

electric vector of incident light in addition to an *E*-field distribution under a *p*-polarization.

EM properties are shortly discussed on a NP sheet in the microwave range 0.5–40 GHz. This range is an important frequency range for telecommunications. Transparent solar-thermal shielding is effective techniques to prevent room heat in order to realize comfortable environment in vehicles. However, it is strongly required for vehicles to transmit EM waves in the microwave range through windows to carry out radio communications such as an Electronic Toll Collection System (ETC) and Information traffic system (ITS). Therefore, it is important to measure EM properties of NP sheets in addition to evaluate optical properties in the IR range.

250 nm-thick NP sheet with an A4 size was fabricated on a PET substrate (thickness: 0.2 mm) using a roll-coating method [inset of **Figure 12(a)**]. High reflectance with a close proximity of 0.6 was also obtained on a flexible PET substrate [**Figure 12(a)**]. The shielding effectiveness (SE) of the flexible NP sheet was almost zero, as different from that of a RF sputtered ITO film [**Figure 12(b)**]. The difference between the two materials related to electrical conductance (σ) in the sheets, which was in the order of 10−5 and 10−3 S/cm for the NP sheet and sputtered film, respectively. If the shielding material is thin, SE is mainly dominated by EM reflection as follows [43]:

$$\text{SE} = \text{20log}\{\sqrt{2}\,\beta\_0/2\mathbb{R}\}\tag{10}$$

where, *β*0 2 is *μ*/*ε*0 , and R is the sheet resistivity (=1/σ). The significant obstruction of electron carrier transport between NPs produced low electrical conductance because of the presence of surface ligands on the NPs, and realized the high microwave transmissions.

**Figure 12.** (a) Reflectance of a NP sheet on a PET film. Inset image represents a photograph of the fabricated sheet sample. (b) Shielding effectiveness (SB) in the microwave range 0.5–40 GHz for a NP sheet and RF-sputtered ITO film.

ITO NPs were used to obtain assembled NP sheets with small interparticle lengths by the presence of ligand molecules on the particle surfaces. This situation produced effective *E*-field coupling along the in-plane and out-of-plane directions. This caused the plasmon hybridization for the quadrupole and dipole modes, which played an important role in producing the high reflectance in the near- and mid-IR range. In addition, the *E*-field enhancements between NPs simultaneously caused a remarkable reduction of electrical contacts between the NPs, which contributed to the high microwave transmissions. The plasmonic control in assemblies of NPs represents promising potential for structural and optical designs used to fabricate a flexible thermal-shielding sheet with a reflection-type based on transparent oxide semiconductors. The knowledge gained in this study can be applied to NP sheets utilizing inexpensive ZnO and WO3 [44, 45].

### **6. Summary**

Crystallinity and local structures of oxide semiconductor NPs were conducted using ITO NPs by XRD and TEM measurements in Section 2. The plasmonic resonances of ITO NPs were clearly obtained in the near-IR range from the viewpoints of optical and EELS signals. In particular, electron-impurity scattering contributed towards plasmon damping as one of a factor that is absent in metal NPs on the basis of theoretical and experimental approaches, which was discussed in Section 3. In Sections 4 and 5, we described IR plasmonic applications in ITO NP sheets for solar-thermal shielding technology. Above all, the *E*-field coupling between NPs produced interesting plasmonic coupling because of the creation of narrow crevices in the interparticles. 3*D* field interactions along the in-plane and out-of-plane directions caused high reflectance in the near- and mid-IR regions. Finally, ITO NP sheets could be extended to obtain large-size flexible films on PET substrates, which simultaneously showed microwave transmittance essential for telecommunications.

The above results provided important insights for basic science and practical applications based on plasmonic investigations based on oxide semiconductor NPs. However, plasmonic properties and applications are stand still-points at the present time. Hereafter, it will be needed to study plasmonic phenomena on oxide semiconductor NPs towards new concepts concerning optical manipulations in the IR range.

## **Acknowledgements**

SE = 20log( √

188 Nanoplasmonics - Fundamentals and Applications

where, *β*0

2 is *μ*/*ε*0

**6. Summary**

\_\_

carrier transport between NPs produced low electrical conductance because of the presence of

ITO NPs were used to obtain assembled NP sheets with small interparticle lengths by the presence of ligand molecules on the particle surfaces. This situation produced effective *E*-field coupling along the in-plane and out-of-plane directions. This caused the plasmon hybridization for the quadrupole and dipole modes, which played an important role in producing the high reflectance in the near- and mid-IR range. In addition, the *E*-field enhancements between NPs simultaneously caused a remarkable reduction of electrical contacts between the NPs, which contributed to the high microwave transmissions. The plasmonic control in assemblies of NPs represents promising potential for structural and optical designs used to fabricate a flexible thermal-shielding sheet with a reflection-type based on transparent oxide semiconductors. The knowledge

**Figure 12.** (a) Reflectance of a NP sheet on a PET film. Inset image represents a photograph of the fabricated sheet sample. (b) Shielding effectiveness (SB) in the microwave range 0.5–40 GHz for a NP sheet and RF-sputtered ITO film.

gained in this study can be applied to NP sheets utilizing inexpensive ZnO and WO3

Crystallinity and local structures of oxide semiconductor NPs were conducted using ITO NPs by XRD and TEM measurements in Section 2. The plasmonic resonances of ITO NPs were clearly obtained in the near-IR range from the viewpoints of optical and EELS signals. In particular, electron-impurity scattering contributed towards plasmon damping as one of a factor that is absent in metal NPs on the basis of theoretical and experimental approaches, which

surface ligands on the NPs, and realized the high microwave transmissions.

, and R is the sheet resistivity (=1/σ). The significant obstruction of electron

2 *β*0 /2*R* ) (10)

[44, 45].

This research was supported in part by a grant-in-Aid from the JSPS Core-to-Core Program, A. Advanced Research Network, a grand from Toyota Physical and Chemical Research Institute, and for a grant-in-Aid for Exploratory Research and Scientific Research (B).

## **Author details**

Hiroaki Matsui1,2\* and Hitoshi Tabata1,2

## **References**

[17] A. M. Schimpf, S. D. Lounis, E. L. Runnerstrom, D. J. Milliron and D. R. Gamelin. Redox chemistries and plasmon energies of photodoped In2 O3 and Sn-doped In2 O3 (ITO) nanocrystals. Journal of American Chemistry Society. 2014;**137**:518–524. doi:10.1021/a5116953.

[4] E. Sachet, M. D. Losego, J. Guske, S. Franzen and J. -P. Maria. Mid-infrared surface plasmon resonance in zinc oxide semiconductor thin films. Applied Physics Letters.

[5] H. Matsui, A. Ikehata and H. Tabata. Surface plasmon sensors on ZnO: Ga layer surfaces: Electric-field distributions and absorption-sensitive enhancements. Applied Physics

[6] Y. Zhao, H. Pan, Y. Lou, X. Qiu, J. J. Zhu and C. Burda. Plasmonic Cu2-xS nanocrystals: Optical and structural properties of copper-deficient copper (I) sulfides. Journal of the

[7] G. Garcia, R. Buonsanti, E. L. Runnerstrom, R. J. Mendelsberg, A. Llordes, A. Anders, T. J. Richardson and D. J. Milliron. Dynamically modulating the surface plasmon resonance of doped semiconductor nanocrystals. Nano Letters. 2011;**11**(10):4415–4420. doi:10.1021/

[8] G. V. Naik, V. M. Shalaev and A. Boltasseva. Alternative plasmonic materials: Beyond gold and silver. Advanced Materials. 2013;**25**(24):3264–3294. doi:10.1002/adma201205076.

[9] T. Yamada, H. Makino, N. Yamamoto and T. Yamamoto. Ingrain and grain boundary scattering effects on electron mobility of transparent conducting polycrystalline Ga-doped ZnO films. Journal of Applied Physics. 2010;**107**:123534(1)–123534(8). doi:10.1063/1.3447981.

[10] C. Sonnichesen, T. Franzl, T. Wilk, G. von Plessen and J. Feldmann. Plasmon resonances in large noble-metal clusters. New Journal of Physics. 2002;**4**:93(1)–93(8).

[11] Z. C. Jin, I. Hamberg and C. G. Granqvist. Optical properties of sputter-deposited ZnO: Al thin films. Journal of Applied Physics. 1988;**64**:5117–5131. doi:10.1063/1.342419. [12] N. Yamamoto, H. Makino, S. Osone, A. Ujihara, T. Ito, H. Hokari, T. Maruyama and T. Yamamoto. Development of Ga-doped ZnO transparent electrodes for liquid crystal display panels. Thin Solid Films. 2012;**520**(12):4131–4138. doi:10.1016/j/tsf.2011.04.067.

[13] S. Q. Li, P. Guo, B. Buchholz, W. Zhou, Y. Hua, T. W. Odom, J. B. Ketterson, L. E. Ocola, K. Sakoda and R. P. H. Chang. Plasmon-photonic mode coupling in indium-tin-oxide

[14] D. B. Tice, S. Q. Li, M. Taglazucchi, D. B. Buchholz, E. A. Weiss and R. P. H. Chang. Ultrafast modulation of the plasma frequency of vertically aligned indium tin oxide

[15] S. Wuestner, J. M. Hamm, A. Pusch, O. Hess. Plasmonic leaky-mode lasing in active semiconductor nanowires. Laser & Photonics Reviews. 2015;**9**(2):256–262. doi:10.1002/

[16] S. D. Lounis, E. L. Runnerstrom, A. Llordés and D. J. Milliron. Defect chemistry and plasmon physics of colloidal metal oxide nanocrystals. Journal of Physical Chemistry C.

nanorod arrays. ACS Photonics. 2014;**1**(3):163–172. doi:10.1021/ph400038g.

rods. Nano Letters. 2014;**14**(3):1120–1126. doi:10.1021/nl4028044.

American Chemistry Society. 2009;**131**(12):4253–4261. doi:10.1021/ja805655b.

2013;**102**:051111(1)–051111(4). doi:10.1063/1.4791700.

nl202597n.

190 Nanoplasmonics - Fundamentals and Applications

doi:S1367-2630(02)39458-8.

por.201400231.

2014;**5**:1564–1574. doi:10.1021/jz500440e.

Letters. 2015;**106**:019905(1)–019905(5). doi:10.1063/1.4905211.

with tuned surface plasmon resonance. ACS Applied Materials & Interfaces. 2013;**5**(3):10240–10245. doi:10.1021/am403011t.

[41] P. Nordlander, C. Oubre, E. Prodan, K. Li and M. I. Stockman. Plasmon hybridization in nanoparticle dimers. Nano Letters. 2004;**4**(5):899–903. doi:10.0121/nl049681c.

with tuned surface plasmon resonance. ACS Applied Materials & Interfaces.

[29] X. Fang, C. L. Mak, J. Dai, K. Li, H. Ye and C. W. Leung. ITO/Au/ITO sandwich structure for near-infrared plasmonics. ACS Applied Materials & Interfaces. 2014;**6**(18):15743–

[30] P. Tao, A. Viswanath, L. S. Schadler, B. C. Benicewica and R. W. Siegel. Preparation and optical properties of indium tin oxide/epoxy nanocomposites with polyglycidyl methacrylate grafted nanoparticles. ACS Applied Materials & Interfaces. 2011;**3**(9):3638–3645.

[31] Z. Sun, J. He, A. Kumbhar and J. Fang. Nonaqueous synthesis and photoluminescence of

[32] H. Matsui, S. Furuta and H. Tabata. Role of electron carriers on local surface plasmon resonances in doped oxide semiconductor nanocrystals. Applied Physics Letters.

[33] J. A. School, A. L. Koh and J. A. Dionne. Quantum plasmon resonances of individual

[34] G. V. Hartland. Optical studies of dynamics in noble metal nanostructures. Chemistry

[35] D. Ross and R. Aroca. Effective medium theories in surface enhanced infrared spectroscopy: The pentacene example. Journal of Chemical Physics. 2002;**117**:8095–8103.

[36] A. Pflug, V. Sitinger, F. Ruske, B. Szyszka and G. Dittmar. Optical characterization of aluminium-doped zinc oxide films by advanced dispersion theories. Thin Solid Films.

[37] S. M. Yang, S. G. Jang, D. G. Chio, S. Kim and H. K. Yu. Nanomachining by colloidal

[38] P. Jiang and M. J. McFarland. Large-scale fabrication of wafer-size colloidal crystals, macroporous polymers and nanocomposites by spin-coating. Journal of the American

[39] H. Matsui, S. Furuta, T. Hasebe and H. Tabata. Plasmonic-field interactions at nanoparticle interfaces for infrared thermal-shielding applications based on transparent oxide semiconductors. ACS Applied Materials & Interfaces. 2016;**8**(18):11749–11759.

[40] C. Jiang, S. Markutsya and V. V. Tsukruk. Collective and individual plasmon resonances in nanoparticle films obtained by spin-assisted layer-by-layer assembly. Langmuir.

ITO nanoparticles. Langmuir. 2010;**26**(6):4246–4250. doi:10.1021/la903316b.

metallic nanoparticles. Nature. 2012;**483**:421–427. doi:10.1038/nature10904.

2014;**104**(12):211903(1)–211903(4). doi:10.1063/1.4880356.

Reviews. 2011;**111**(6):3858–3887. doi:10.1021/cr1002547.

2004;**455–456**:201–206. doi:10.1016/j.tsf.2004.01.1006.

lithography. Small. 2006;**2**(4):458–475. doi:10.1002/smll.200500390.

Chemistry of Society. 2004;**126**(42):13778–13786. doi:10.1021/ja0470923.

2013;**5**(3):10240–10245. doi:10.1021/am403011t.

15752. doi:10.1021/am5026165.

doi:10.1021/am200841n.

192 Nanoplasmonics - Fundamentals and Applications

doi:10.1063/1.1502656.

doi:10.1021/acsami6b1202.

2003;**20**(3):882–890. doi:10.1021/la0355085.

## **Surface Plasmonics and Its Applications in Infrared Sensing**

Guiru Gu, Xuejun Lu, Thitikorn Kemsri and

Yingjie Zhang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67410

### Abstract

Surface plasmonic waves have been extensively researched due to their strong surface confinement. The strong surface confinement allows high absorption in an infrared (IR) detector with a thin active absorption region. The excitation of surface plasmonic resonance (SPR) depends on the metallic structures and the interface materials. This enables engineering of plasmonic-enhanced IR detector properties (e.g. detection wavelength, polarization and angular dependence) by properly designing the plasmonic structures. This chapter first gives a brief review of the surface plasmonic waves, followed by the description of SPR excitation in a metallic two-dimensional (2D) sub-wavelength hole array (2DSHA) structure. The applications of the 2DSHA SPR in IR detector enhancement are then presented with a discussion of the polarization and angular dependence.

Keywords: surface plasmonic waves, quantum dot, infrared detection

## 1. Introduction

Surface electromagnetic waves (also called surface plasmon polaritons, SPPs), predicted by Ritchie in 1957 [1], are coherent electron oscillations at a metal surface. The coherent electron oscillations generate electromagnetic (EM) waves that travel along the dielectric/metal interface [2]. Depending on the metal structures, certain resonant modes of the surface electromagnetic waves can be strongly absorbed (or excited) [2]. This is generally referred to as excitation of surface plasmonic resonance (SPR) modes. SPRs propagate along the metal-dielectric interface. The intensity profile exponentially decreases with the distance from metal-dielectric

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

interface, that is, SPs are confined at the metal-dielectric interface. Such surface confinement effects have been used in Raman scattering, that is, surface-enhanced Raman scattering (SERS) [3–5] and spectroscopy [6, 7] and have been reported with thousands of times sensitivity enhancement [3, 4]. In addition to SERS, the SPR enhancement technology also provides a promising technique to concentrate EM energy on surface area and thus enables high quantum efficiency with a thin active absorption layer in a photodetector. Significant performance enhancements in a quantum dot infrared photodetector (QDIP) have been reported [8–15].

In addition, the SPR modes can also change EM field distribution and thus offers an effective technique for EM field engineering to achieve specific transmission and/or receiving patterns with polarization and detection spectrum engineering capability [11, 16]. In this chapter, a brief review of the fundamental concepts of SPRs will be first given and followed by the description of their applications in infrared detections. SPR-based EM field engineering will also be presented with the discussion of polarization and receiving angle control.

### 2. Review of surface plasmonic resonance

#### 2.1. Plasma frequency, complex conductivity and the Drude model

Plasmonic waves are electromagnetic waves. They follow Maxwell equations, which can be expressed as follows in the phasor domain

$$
\nabla \cdot \vec{D} = \rho,\tag{1}
$$

$$
\nabla \times \vec{E} = -\mathbf{j}\omega\mu \vec{H},
\tag{2}
$$

$$
\nabla \cdot \vec{B} = \mathbf{0},
\tag{3}
$$

$$
\nabla \times \vec{H} = \vec{J} + \mathbf{j}\omega\varepsilon \vec{E},\tag{4}
$$

where J ⇀ is the current density, D ⇀ ,E ⇀ , are the electric flux density and the electric field, respectively, and B ⇀ , H ⇀ are the magnetic flux density and the magnetic field, respectively, ω is the angular frequency, and ε and μ are the permittivity and permeability of the material, respectively. The current density J ⇀ is related to E ⇀ by:

$$
\vec{J} = \sigma \vec{E},
\tag{5}
$$

where σ is the conductivity of the material. Eq. (4) can therefore be expressed as follows:

$$\nabla \times \vec{H} = \mathbf{j}\omega\varepsilon \left( 1 + \frac{\sigma}{\mathbf{j}\omega\epsilon} \right) \overleftrightarrow{E},\tag{6}$$

where εc ¼ ε 1 þ σ jωE is the complex permittivity of the material. The Maxwell's current continuity equation is as follows:

$$
\nabla \cdot \vec{J} = -\varepsilon \frac{\partial}{\partial t} \mathfrak{n},
\tag{7}
$$

where e is the charge of an electron, e ¼ 1:6 · 10�19ðCÞ, n is the density of electrons in metal. The current density is related to the speed of the charge v by:

$$
\vec{J} = \epsilon \mathbf{n}\_0 \mathbf{v},\tag{8}
$$

where n0 is the average electron density in metal. Combining Eqs. (7) and (8), one gets:

$$
\mathfrak{n}\_{\mathsf{0}} \nabla \cdot \vec{\boldsymbol{\upsilon}} = -\frac{\partial}{\partial t} \mathfrak{n}, \tag{9}
$$

Taking another partial derivative of t, one gets:

$$
\mathfrak{n}\_{\mathsf{0}} \nabla \cdot \frac{\partial}{\partial t} \overrightarrow{\boldsymbol{\upsilon}} = -\frac{\partial^2}{\partial t^2} \boldsymbol{n}, \tag{10}
$$

Eq. (10) can be expressed as follows:

interface, that is, SPs are confined at the metal-dielectric interface. Such surface confinement effects have been used in Raman scattering, that is, surface-enhanced Raman scattering (SERS) [3–5] and spectroscopy [6, 7] and have been reported with thousands of times sensitivity enhancement [3, 4]. In addition to SERS, the SPR enhancement technology also provides a promising technique to concentrate EM energy on surface area and thus enables high quantum efficiency with a thin active absorption layer in a photodetector. Significant performance enhancements in a quantum dot infrared photodetector (QDIP) have been reported [8–15].

In addition, the SPR modes can also change EM field distribution and thus offers an effective technique for EM field engineering to achieve specific transmission and/or receiving patterns with polarization and detection spectrum engineering capability [11, 16]. In this chapter, a brief review of the fundamental concepts of SPRs will be first given and followed by the description of their applications in infrared detections. SPR-based EM field engineering will also be presented

Plasmonic waves are electromagnetic waves. They follow Maxwell equations, which can be

¼ �jωμH ⇀

> þ jωεE ⇀

are the magnetic flux density and the magnetic field, respectively, ω is the

σ jωE

E ⇀

is the complex permittivity of the material. The Maxwell's current

¼ ρ, (1)

¼ 0, (3)

, (5)

, (6)

n, (7)

, are the electric flux density and the electric field, respec-

, (2)

, (4)

∇ � D ⇀

> ∇ � B ⇀

angular frequency, and ε and μ are the permittivity and permeability of the material, respec-

∇ · E ⇀

∇ · H ⇀ ¼ J ⇀

> ⇀ by:

> > J ⇀ ¼ σE ⇀

where σ is the conductivity of the material. Eq. (4) can therefore be expressed as follows:

∇ � J ⇀

¼ jωε 1 þ

¼ �e ∂ ∂t

⇀ ,E ⇀

is related to E

∇ · H ⇀

⇀

with the discussion of polarization and receiving angle control.

2.1. Plasma frequency, complex conductivity and the Drude model

2. Review of surface plasmonic resonance

expressed as follows in the phasor domain

196 Nanoplasmonics - Fundamentals and Applications

is the current density, D

jωE

continuity equation is as follows:

where J ⇀

tively, and B

⇀ , H ⇀

tively. The current density J

where εc ¼ ε 1 þ σ

$$
\frac{\mathfrak{m}\_0}{m} \nabla \cdot \mathfrak{m} \stackrel{\star}{a} = -\frac{\partial^2}{\partial t^2} \mathfrak{n},
\tag{11}
$$

where m is the mass of an electron. The ma⇀ term is the force on the electron, which is the electric force, F ⇀ ¼ eE ⇀ . Eq. (11) can thus be changed to

$$
\frac{\epsilon\mathfrak{m}\_0}{\mathfrak{m}}\nabla \cdot \vec{E} = -\frac{\partial^2}{\partial t^2}\mathfrak{n},\tag{12}
$$

In phasor domain, Eq. (12) can be expressed as follows:

$$
\vec{\nabla}\frac{\epsilon\mathfrak{m}\_0}{m}\nabla\cdot\vec{E} = \omega\_p^2\mathfrak{n},\tag{13}
$$

where ωp is the plasma frequency. Combining Eqs. (1) and (13), one gets

$$
\omega\_{\mathfrak{p}} = \sqrt{\frac{\mathfrak{e}^2 \mathfrak{n}\_0}{m\epsilon}} \tag{14}
$$

Taking the partial derivative of t for Eq. (8), one gets

$$\frac{\partial}{\partial t}\overline{J} = \frac{e\mathfrak{m}\_0}{m}m\frac{\partial}{\partial t}\mathfrak{v},\tag{15}$$

The m ∂ ∂t v ⇀ term can be replaced by the electric force, F ⇀ ¼ eE ⇀ . Eq. (15) becomes

$$\frac{\partial}{\partial t}\vec{J} = \frac{e^2 \mathbf{n\_0}}{m}\vec{E},\tag{16}$$

In phasor domain, Eq. (16) can be rewritten as follows:

$$
\vec{j}\omega\vec{J} = \frac{e^2\mathfrak{n}\_0}{m}\vec{E},
\tag{17}
$$

Combining Eqs. (5) and (17), one gets:

$$
\sigma = \frac{e^2 n\_0}{\dot{\mu}\omega m},
\tag{18}
$$

The complex permittivity εc is thus given as follows:

$$\varepsilon\_{\varepsilon} = \varepsilon \left( 1 + \frac{\sigma}{j\omega\epsilon} \right) = \varepsilon \left( 1 - \frac{\varepsilon^2 n\_0}{\omega^2 m \epsilon} \right) = \varepsilon \left( 1 - \frac{\omega\_p^2}{\omega^2} \right) \tag{19}$$

The derivations and equations above are for the ideal lossless free-electron gas model. The loss term can be included in the model by introducing a relaxation rate γ, that is,

$$\varepsilon\_{\varepsilon} = \varepsilon \left( 1 - \frac{\omega\_p^2}{\omega^2 + j\omega\gamma} \right) \tag{20}$$

Eq. (20) is the complex permittivity given by the well-known Drude model [17].

#### 2.2. Plasmonic waves at the dielectric and metal interface

Following Raether in Ref. [2], one can analyse plasmonic waves using a transmagnetic (TM) plane wave. Figure 1 shows the incident of a TM plane EM wave with the H-field in the y-direction. Note that the phasor expressions of the H-fields follow the convention of i k ! � R ! indicating the wave travels in the R ! direction, whereas � i k ! � R ! indicating the wave travels in the � R ! direction. A negative sign needs to be included in Eq. (6) to use this convention.

Figure 1. Scheme of TM wave incidence at the dielectric and metal interface.

Under the aforementioned convention, the E-field defined by Eq. (6) can be changed to

$$
\vec{E} = -\frac{1}{j\omega\mathfrak{x}\_c} \nabla \times \vec{H} \tag{21}
$$

From Eq. (21), the Ex can be expressed as follows:

jωJ ⇀ ¼ e2n0 m E ⇀

σ ¼ e2n0

¼ ε 1 � e2n0

The derivations and equations above are for the ideal lossless free-electron gas model. The loss

Following Raether in Ref. [2], one can analyse plasmonic waves using a transmagnetic (TM) plane wave. Figure 1 shows the incident of a TM plane EM wave with the H-field in the y-direction. Note that the phasor expressions of the H-fields follow the convention of i k

direction, whereas � i k

direction. A negative sign needs to be included in Eq. (6) to use this convention.

! � R !

εc ¼ ε 1 � ω2

ω2mE � �

> p ω2 þ jωγ

!

Combining Eqs. (5) and (17), one gets:

198 Nanoplasmonics - Fundamentals and Applications

The complex permittivity εc is thus given as follows:

εc ¼ ε 1 þ

2.2. Plasmonic waves at the dielectric and metal interface

Figure 1. Scheme of TM wave incidence at the dielectric and metal interface.

!

indicating the wave travels in the R

the � R !

σ jωE � �

term can be included in the model by introducing a relaxation rate γ, that is,

Eq. (20) is the complex permittivity given by the well-known Drude model [17].

, (17)

jωm , (18)

p ω2 !

(19)

(20)

! � R !

indicating the wave travels in

¼ ε 1 � ω2

$$\overrightarrow{E}\_{1i} = -\hat{\mathfrak{x}} \frac{\vec{j}k\_{1z}}{\dot{j}\omega\varepsilon\_{1}} H\_{1i} e^{j(k\_{1z}x + k\_{1z}z)} \tag{22}$$

$$
\overrightarrow{E}\_{1r} = \hat{\mathfrak{x}} \frac{\vec{j}k\_{1z}}{\dot{j}\omega\varepsilon\_{1}} H\_{1r} e^{j(k\_{1x}x - k\_{1z}z)} \tag{23}
$$

$$\overrightarrow{E}\_{2t} = -\hat{\mathbf{x}} \frac{\vec{j}k\_{2z}}{\dot{\mathbf{j}}u\,\varepsilon\_{2c}} H\_{2t} e^{i(k\_{2x}x + k\_{2z}z)} \tag{24}$$

where k1x, k1z are the x and z components of the propagation constant k1, respectively. Similarly, k2x, k2z are the x and z components of the propagation constant k2, respectively.

$$k\_{1x}^2 + k\_{1z}^2 = k\_1^2 = \omega^2 \epsilon\_1 \mu\_1 \tag{25}$$

$$k\_{2x}^2 + k\_{2\sharp}^2 = k\_2^2 = \omega^2 \epsilon\_{2\epsilon} \mu\_{\mathbf{2}} \tag{26}$$

where μ1 ≈ μ2 ≈ μ0 are the permeability of the material 1, 2 and vacuum, respectively. The boundary conditions are Hy and Ex continuous at the interface (i.e. z ¼ 0). Applying the boundary conditions, one gets the following:

$$H\_{1l}e^{\mathrm{jk}\_{1x}x} + H\_{1l}e^{\mathrm{jk}\_{1x}x} = H\_{2l}e^{\mathrm{jk}\_{2x}x} \tag{27}$$

$$\frac{k\_{1z}}{\varepsilon\_1} H\_{1i} e^{j k\_{1x} x} - \frac{k\_{1z}}{\varepsilon\_1} H\_{1r} e^{j k\_{1x} x} = \frac{k\_{2z}}{\varepsilon\_{2c}} H\_{2t} e^{j k\_{2x} x} \tag{28}$$

Since Eq. (27) and Eq. (28) are valid for any x, k1x ¼ k2x.

When there is no reflection, that is, H1r ¼ 0, Eqs. (27) and (28) become

$$H\_{1i} = H\_{2t} \tag{29}$$

$$\frac{k\_{1z}}{\varepsilon\_1}H\_{1i} = \frac{k\_{2z}}{\varepsilon\_{2c}}H\_{2t} \tag{30}$$

Combining Eqs. (29) and (30), one gets the following:

$$k\_{2z} = \varepsilon\_{2c} \frac{k\_{1z}}{\varepsilon\_1} \tag{31}$$

Plugging Eq. (31) into Eq. (26), one gets the following:

$$k\_{2x}^2 + k\_{2z}^2 = \frac{\varepsilon\_{2c}^2}{\varepsilon\_1^2} k\_{1z}^2 + k\_{2x}^2 = \omega^2 \epsilon\_{2c} \mu\_0 \tag{32}$$

Combing Eq. (32) with Eq. (25), one can get k1x as follows:

$$k\_{\mathbf{1}\mathbf{r}}^2 = \omega^2 \mu\_\mathbf{0} \frac{\epsilon\_{\mathbf{2}c\ell\mathbf{1}}}{(\varepsilon\_\mathbf{1} + \varepsilon\_\mathbf{2c})} \tag{33}$$

Eq. (33) can be expressed using relative permittivity ε1r, and ε2cr.

$$k\_{\mathbf{1}x}^2 = \omega^2 \mu\_0 \epsilon\_0 \frac{\epsilon\_{2cr^\ell \mathbf{1}r}}{(\varepsilon\_{\mathbf{1}r} + \varepsilon\_{\mathbf{2}cr})} \tag{34}$$

where ε1r ¼ ε1 ε0 , and ε2cr ¼ ε2c ε0 .

$$k\_x = k\_0 \text{Re}\left[\sqrt{\frac{\epsilon\_{2\alpha r \xi 1r}}{\left(\varepsilon\_{1r} + \varepsilon\_{2\alpha r}\right)}}\right] + jk\_0 \text{Im}\left[\sqrt{\frac{\epsilon\_{2cr}\epsilon\_{1r}}{\left(\varepsilon\_{1r} + \varepsilon\_{2cr}\right)}}\right] \tag{35}$$

k2z can be obtained from Eq. (34) as follows:

$$k\_{2\bar{z}} = k\_0 \text{Re} \left[ \sqrt{\frac{\epsilon\_{2cr^\xi 2cr}}{(\varepsilon\_{1r} + \varepsilon\_{2cr})}} \right] + jk\_0 \text{Im} \left[ \sqrt{\frac{\epsilon\_{2cr^\xi 2cr}}{(\varepsilon\_{1r} + \varepsilon\_{2cr})}} \right] \tag{36}$$

where Re(.) and Im(.) are taking the real and the imaginary parts. The plasmonic wave propagates in x and z directions with loss described by the imaginary parts of the corresponding directions, that is, kspp ¼ kx. Since ε2cr is generally much larger than ε1r, Eq. (36) can be simplified as follows:

$$k\_{\mathbf{x}} = k\_{\mathbf{0}} \sqrt{\epsilon\_{\mathbf{1}\mathbf{r}}} \tag{37}$$

### 2.3. Plasmonic wave excitation in a metallic two-dimensional (2D) sub-wavelength hole array (2DSHA) structure

Surface plasmonic waves can be excited by numerous structures. The metallic 2DSHA structure [18] is one of the commonly used plasmonic structures for performance enhancement in infrared photodetectors [8, 10–12]. Figure 2 shows a schematic structure with the light incident scheme of the metallic 2DSHA array structure. The 2DSHA structure is a square lattice with the period of p, and the diameter of the hole is d. The periodical structure provides a grating vector of Λ ¼ 2π p . Surface plasmonic wave can be excited when the grating vector of Λ matches the kspp, that is, kspp ¼ 2π p .

Figure 3 shows the cross section of the light incidence on the 2DSHA structure. The excited plasmonic waves and the scattered waves by the 2DSHA structure are indicated in the figure.

Plugging Eq. (31) into Eq. (26), one gets the following:

200 Nanoplasmonics - Fundamentals and Applications

k2 2x þ k2

Eq. (33) can be expressed using relative permittivity ε1r, and ε2cr.

Combing Eq. (32) with Eq. (25), one can get k1x as follows:

where ε1r ¼ ε1

as follows:

array (2DSHA) structure

vector of Λ ¼ 2π

figure.

the kspp, that is, kspp ¼ 2π

p .

ε0

, and ε2cr ¼ ε2c

k2z can be obtained from Eq. (34) as follows:

ε0 .

kx ¼ k0Re

k2z ¼ k0Re

2z ¼ ε2 2c ε2 1 k2 1z þ k2

k2 1x ¼ ω2

k2 1x ¼ ω2 μ0

μ0E0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2crE1r ðε1r þ ε2crÞ � � r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2crE2cr ðε1r þ ε2crÞ � � r 2x ¼ ω2

E2cE1

E2crE1r

þ jk0Im

þ jk0Im

ffiffiffiffiffi E1r

p . Surface plasmonic wave can be excited when the grating vector of Λ matches

where Re(.) and Im(.) are taking the real and the imaginary parts. The plasmonic wave propagates in x and z directions with loss described by the imaginary parts of the corresponding directions, that is, kspp ¼ kx. Since ε2cr is generally much larger than ε1r, Eq. (36) can be simplified

kx ¼ k0

2.3. Plasmonic wave excitation in a metallic two-dimensional (2D) sub-wavelength hole

Surface plasmonic waves can be excited by numerous structures. The metallic 2DSHA structure [18] is one of the commonly used plasmonic structures for performance enhancement in infrared photodetectors [8, 10–12]. Figure 2 shows a schematic structure with the light incident scheme of the metallic 2DSHA array structure. The 2DSHA structure is a square lattice with the period of p, and the diameter of the hole is d. The periodical structure provides a grating

Figure 3 shows the cross section of the light incidence on the 2DSHA structure. The excited plasmonic waves and the scattered waves by the 2DSHA structure are indicated in the

E2cμ0 (32)

ðε1 þ ε2cÞ (33)

ðε1r þ ε2crÞ (34)

p (37)

(35)

(36)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2crE1r ðε1r þ ε2crÞ � � r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2crE2cr ðε1r þ ε2crÞ � � r

Figure 2. Schematic structure with the light incident scheme of the metallic 2DSHA array structure. The 2DSHA structure is square lattice with a period of p, and the diameter of the hole is d.

Figure 3. Schematic of the cross-section of the light incidence on the 2DSHA structure. The 2DSHA structure generates plasmonic waves and scattering waves.

From E-M field boundary conditions, the following relations hold:

$$H\_{1s-} = H\_{2t-} \tag{38}$$

$$H\_{1s+} = H\_{2t+} \tag{39}$$

$$\frac{k\_{1z}}{\varepsilon\_1} = -\frac{k\_{2z}}{\varepsilon\_{2c}}\tag{40}$$

The excitation efficiency η can be defined by the overlap integral of the incident filed with the plasmonic waves, that is,

$$\eta = \int H(\mathbf{x}, z) e^{i(k\_{\eta\eta}\mathbf{x} + k\_{2\zeta}z)}|\_{z=\theta} d\mathbf{x} \tag{41}$$

where Hðx, zÞ is the magnetic field of the incident wave modified by the 2DSHA, that is, the periodical 2DSHA structure will introduce periodical variation in Hðx, zÞ as well.

### 2.4. Near-field E-field distribution in the 2DSHA structure

The near-field E-field component can be simulated as well. Figure 4 shows the top (x-y) and cross-section (x-z) views of the Ex component at different wavelengths. The 2DSHA structure is square lattice with a period of 2.3 µm, and the diameter of the hole is 1.15 µm. The simulation is performed using CST Microwave studio®. The incident light is linearly polarized surface normal plane wave with E-field magnitude of 1 V/m. The colour scale bar represents the magnitude of the Ex.

Figure 5 shows the top (x-y) and cross-section (x-z) views of the Ez component at different wavelengths. From Figures 4 and 5, both Ex and Ez strongly depend on the excitation wavelengths. Ez is high between 7.6 and 8.5 µm, whereas Ex is strong between 7.6 and 8.0 µm. Ez is primarily at the edge of the holes, whereas Ex is inside the holes. The period of the 2DSHA p ¼ 2:3 µm, kspp ¼ 2π p .

Figure 4. Top and cross-section views of the Ex component at different wavelengths simulated using CST Microwave studio®. The incident light is a linearly polarized surface normal plane wave with E-field magnitude of 1 V/m.

Figure 5. Top and cross-section views of the Ez component at different wavelengths simulated using CST Microwave studio®. The incident light and the simulation conditions are the same.

From Eq. (35), kspp is

From E-M field boundary conditions, the following relations hold:

η ¼ ð

2.4. Near-field E-field distribution in the 2DSHA structure

plasmonic waves, that is,

202 Nanoplasmonics - Fundamentals and Applications

magnitude of the Ex.

p ¼ 2:3 µm, kspp ¼ 2π

p .

k1z ε1

¼ � k2z ε2c

The excitation efficiency η can be defined by the overlap integral of the incident filed with the

Hðx, zÞeiðksppxþk2z zÞ

where Hðx, zÞ is the magnetic field of the incident wave modified by the 2DSHA, that is, the

The near-field E-field component can be simulated as well. Figure 4 shows the top (x-y) and cross-section (x-z) views of the Ex component at different wavelengths. The 2DSHA structure is square lattice with a period of 2.3 µm, and the diameter of the hole is 1.15 µm. The simulation is performed using CST Microwave studio®. The incident light is linearly polarized surface normal plane wave with E-field magnitude of 1 V/m. The colour scale bar represents the

Figure 5 shows the top (x-y) and cross-section (x-z) views of the Ez component at different wavelengths. From Figures 4 and 5, both Ex and Ez strongly depend on the excitation wavelengths. Ez is high between 7.6 and 8.5 µm, whereas Ex is strong between 7.6 and 8.0 µm. Ez is primarily at the edge of the holes, whereas Ex is inside the holes. The period of the 2DSHA

Figure 4. Top and cross-section views of the Ex component at different wavelengths simulated using CST Microwave studio®. The incident light is a linearly polarized surface normal plane wave with E-field magnitude of 1 V/m.

periodical 2DSHA structure will introduce periodical variation in Hðx, zÞ as well.

j

H1s� ¼ H2t� (38)

H1sþ ¼ H2tþ (39)

z¼0dx (41)

(40)

$$k\_{spp} = \frac{2\pi}{\lambda\_0} \text{Re}\left[\sqrt{\frac{\epsilon\_{2cr}\epsilon\_{1r}}{\left(\varepsilon\_{1r} + \varepsilon\_{2cr}\right)}}\right] + j\frac{2\pi}{\lambda\_0} \text{Im}\left[\sqrt{\frac{\epsilon\_{2cr}\epsilon\_{1r}}{\left(\varepsilon\_{1r} + \varepsilon\_{2cr}\right)}}\right] \tag{42}$$

where λ0 is the free space wavelength, εm and εd are the relative permittivity of the metal and GaAs, respectively. The dielectric constant of GaAs in the long-wave infrared region is 10.98 [19]. The dielectric constant of Au is calculated to be �1896 + i684 at 7.6 µm [1]. The calculated plasmonic resonant wavelength is λsp = 7.6 µm. At the resonant wavelength (i.e. λ ¼ 7:6 µm), the E-fields are strongly confined near the surface. Such strong surface confinement of plasmonic waves enables high absorption using a thin absorption layer and thus leads to significant enhancement in quantum dot infrared photodetectors (QDIPs) [8, 10–13, 15]. In the following sections, we will first give a brief introduction of QDIPs, followed by the description of surface plasmonic enhancement in QDIPs.

## 3. QDIPs in mid-wave infrared (MWIR) and long-wave infrared (MWIR/LWIR) detection

QDIPs are based on intersubband transitions in self-assembled InAs quantum dots (QDs). The simplified band diagram and the schematic structures of a QD infrared photodetector (QDIP) are shown in Figure 6(a) and 6(b), respectively. The s, p, d, f, represent the energy levels of a QD with the wetting layers (WL). A typical QDIP consists of vertically-stacked InAs quantum dots layers with GaAs capping layers. The electrons are excited by the normal incident light and subsequently collected through the top electrode and generate photocurrent. This is a unipolar photodetector, where only conduction band is involved in the photodetection and photocurrent generation process.

The QDIP technology offers a promising technology in MWIR and LWIR photodetection due to the advantages provided by the three-dimensional (3D) quantum confinement of

Figure 6. Schematic structure a unipolar QDIP with the intersubband transition: (a) simplified band diagram and (b) schematic of the QDIP. The s, p, d and f, represent the energy levels of a QD.

carriers—including intrinsic sensitivity to normal incident radiation [13], high photoconductive (PC) gain, high quantum efficiency [15] and photoresponsivity [16]. The normal incidence detection capability greatly simplifies the fabrication complexity for a large format (1K · 1K) FPA. The high photoconductive (PC) gain and high photoresponsivity provide a promising way for MWIR and LWIR sensing and detection.

Figure 7 shows the photodetection spectrum of a QDIP. It covers a broadband IR spectrum from 3 to 9 µm. The insert of Figure 7 shows an atomic force microscopic (AFM) picture of the QDs by self-assembled epi-growth. The growth gives high-density QDs with uniform sizes. The InAs/

Figure 7. Photodetection spectrum of a QDIP. It shows a broad IR detection spectrum from 3 to 9 µm. The insert shows an AFM image of the QDs by self-assembled epi-growth.

GaAs QD material is a mature material system with low substrate cost, high material quality and large wafer growth capability. It can offer low-cost MWIR and LWIR photodetectors and focal plane arrays (FPAs) with simplified fabrication processes and high yield. Detailed QDIP performance, such as photoconductive gains, noise, photoresponsivity and photodetectivity can be found in literature [20–27].

## 4. 2DSHA plasmonic-enhanced QDIPs

carriers—including intrinsic sensitivity to normal incident radiation [13], high photoconductive (PC) gain, high quantum efficiency [15] and photoresponsivity [16]. The normal incidence detection capability greatly simplifies the fabrication complexity for a large format (1K · 1K) FPA. The high photoconductive (PC) gain and high photoresponsivity provide a promising

Figure 6. Schematic structure a unipolar QDIP with the intersubband transition: (a) simplified band diagram and (b)

Figure 7 shows the photodetection spectrum of a QDIP. It covers a broadband IR spectrum from 3 to 9 µm. The insert of Figure 7 shows an atomic force microscopic (AFM) picture of the QDs by self-assembled epi-growth. The growth gives high-density QDs with uniform sizes. The InAs/

Figure 7. Photodetection spectrum of a QDIP. It shows a broad IR detection spectrum from 3 to 9 µm. The insert shows an

way for MWIR and LWIR sensing and detection.

204 Nanoplasmonics - Fundamentals and Applications

AFM image of the QDs by self-assembled epi-growth.

schematic of the QDIP. The s, p, d and f, represent the energy levels of a QD.

## 4.1. Backside configured 2DSHA-enhanced QDIPs

Surface confinement of plasmonic waves enables high absorption using a thin absorption layer and thus leads to significant enhancement in QDIPs [8, 10–13, 15]. We have developed a backside-configured plasmonic enhancement technology by fabricating the 2DSHA plasmonic structures on top of a QDIP and illuminating the QDIP from the opposite side of the plasmonic structure (i. e. backside illumination) [12]. Figure 8(a) and 8(b) show the schematic cross-section structures of the backside-configured plasmonic QDIP and the reference QDIP without the

Figure 8. (a) Schematic cross-section structure of the backside-configured 2DSHA plasmonic structure on the QDIP. The IR incident light illuminates from the substrate side; (b) ref-QDIP without the plasmonic structure; (c) SEM image of the 2DSHA plasmonic structure on the QDIP and (d) close-up view of the plasmonic structure (© IOP Publishing. Reproduced with permission from Ref. [12]. All rights reserved).

plasmonic structure, respectively. Figure 8(c) and 8(d) show scanning electron microscopic (SEM) images of the plasmonic structures on a QDIP and a close-up view of the plasmonic structure, respectively.

Figure 9 shows the measured photocurrent spectra of the backside-configured plasmonic QDIP compared with the top-side configured plasmonic QDIP and the reference QDIP. The backside-configured plasmonic QDIP clearly shows higher plasmonic enhancement than those of the top-side configured plasmonic QDIP and the reference QDIP.

Figure 10(a) shows the simulated E-field intensities of the top and backside-configured plasmonic structures, respectively. The backside-configured plasmonic structure can induce stronger E-field at the interface and therefore can provide larger plasmonic enhancement [12].

### 4.2. Wavelength-tuning and multispectral enhancement

From Eq. (42) and kspp ¼ 2π p , one gets:

$$
\lambda\_{sp} = \frac{p}{\sqrt{m^2 + n^2}} \text{Re} \left[ \sqrt{\frac{\epsilon\_{2cr}\epsilon\_{1r}}{(\varepsilon\_{1r} + \varepsilon\_{2cr})}} \right] \tag{43}
$$

where (m, n) are the orders of the grating vectors. Therefore, the plasmonic resonance wavelength can be tuned by changing the period of the 2DSHA plasmonic structure. The plasmon resonant

Figure 9. Measured photocurrent spectra of the backside-configured plasmonic QDIP compared with the top-side configured plasmonic QDIP and the reference QDIP. The backside-configured plasmonic QDIP clearly shows higher plasmonic enhancement.

Surface Plasmonics and Its Applications in Infrared Sensing http://dx.doi.org/10.5772/67410 207

Figure 10. Simulated E-field intensities: (a) top-configured plasmonic structure and (b) backside-configured plasmonic structure. The backside-configured plasmonic structure can induce stronger E-field at the interface and therefore can provide larger plasmonic enhancement (© IOP Publishing. Reproduced with permission from Ref. [12]. All rights reserved).

peaks λsp of different (m, n) orders for various 2DSHA periods p are shown in Figure 11 that exhibits the linear dependence of the resonant peak wavelength λsp with the period p of the 2DSHA for different (m, n) orders. As predicted by Eq. (43), good linearity is obtained for both the (1, 0)th or (0, 1)th and the (1, 1)th order plasmonic resonant peaks.

We define the photocurrent enhancement ratio R as follows:

plasmonic structure, respectively. Figure 8(c) and 8(d) show scanning electron microscopic (SEM) images of the plasmonic structures on a QDIP and a close-up view of the plasmonic structure,

Figure 9 shows the measured photocurrent spectra of the backside-configured plasmonic QDIP compared with the top-side configured plasmonic QDIP and the reference QDIP. The backside-configured plasmonic QDIP clearly shows higher plasmonic enhancement than those

Figure 10(a) shows the simulated E-field intensities of the top and backside-configured plasmonic structures, respectively. The backside-configured plasmonic structure can induce stronger E-field at the interface and therefore can provide larger plasmonic enhancement [12].

where (m, n) are the orders of the grating vectors. Therefore, the plasmonic resonance wavelength can be tuned by changing the period of the 2DSHA plasmonic structure. The plasmon resonant

Figure 9. Measured photocurrent spectra of the backside-configured plasmonic QDIP compared with the top-side configured plasmonic QDIP and the reference QDIP. The backside-configured plasmonic QDIP clearly shows higher

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2crE1r ðε1r þ ε2crÞ � � r

(43)

of the top-side configured plasmonic QDIP and the reference QDIP.

4.2. Wavelength-tuning and multispectral enhancement

p , one gets:

λsp ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ n2 p Re

respectively.

From Eq. (42) and kspp ¼ 2π

206 Nanoplasmonics - Fundamentals and Applications

plasmonic enhancement.

$$R = \frac{I\_{\text{plasmaonic}}}{I\_{\text{ref}}} \tag{44}$$

where Iplasmonic is the photocurrent of the plasmonic QDIP and Iref is the photocurrent of the reference QDIP without the 2DSHA structures.

Figure 11. (a) Plasmonic resonant peaks λsp of different (i, j) orders for various 2DSHA periods; (b) linear dependence of the resonant peak wavelength λsp with the period of different (m, n) orders.

Figure 12. Photocurrent enhancement ratio spectra for plasmonic QDIPs with different periods varying from 1.4 to 2.0 µm. The insert is a microscopic picture of a QDIP with a 2DSHA plasmonic structure (Reproduced from Ref. [28] with the permission of AIP Publishing).

Figure 12 shows the photocurrent enhancement ratio spectra for all the plasmonic QDIPs with different periods from 1.4 to 2.0 µm. The periods of the 2DSHA plasmonic structures are marked on the curves. The insert shows a microscopic picture of a QDIP with a 2DSHA plasmonic structure. The IR incidence is from the backside of the QDIPs, that is, backsideconfigured 2DSHA plasmonic structures.

As shown in Figure 12, the enhancement wavelengths can be tuned by varying the periods of the 2DSHA plasmonic structures. This allows one to achieve multispectral QDIPs without fabricating an individual filter on each photodetector [28].

### 4.3. Polarization dependence and polarimetric detection

From Eq. (43), the plasmonic resonant wavelength depends on the period p. By changing the plasmonic lattice from a square lattice to a rectangular lattice with different periods in the x and y directions, one can engineer the resonant wavelengths at different polarizations.

Figure 13(a) shows an SEM picture of a rectangular lattice 2DSHA plasmonic structure with the periods of 2.6 and 3.0 µm in the x-direction and the y-direction, respectively. Figure 13(b)–(d) shows the transmission spectra of the un-polarized, 90 polarized (i.e. polarization in y-direction), and 0 polarized (i.e. polarization in x-direction), respectively. The resonant wavelengths correspond to the orders in the x- and y- directions are marked in the figures. The unpolarized light shows transmission in both (0, 1) and (1, 0) orders. However, the 0 polarized

Surface Plasmonics and Its Applications in Infrared Sensing http://dx.doi.org/10.5772/67410 209

Figure 13. (a) SEM picture of rectangular lattice 2DSHA structure. The periods in x- and y- directions are 2.6 and 3.0 µm, respectively. (b), (c) and (d) transmission spectra at of the un-polarized, 90� polarized (i.e. polarization in y-direction), and 0� polarized (i.e. polarization in x-direction), respectively. The polarized light only shows transmission at their corresponding resonant wavelengths (© IOP Publishing. Reproduced with permission from Ref. [11]. All rights reserved).

light only shows transmission at the (1, 0) order resonant wavelength and the 90� polarized light only shows transmission at the (0, 1) order.

Figure 14 shows a microscopic picture of a QDIP with the rectangular lattice 2DSHA structure. The insert shows an SEM picture of the rectangular lattice 2DSHA structure.

Figure 15(a) and (b) shows the measured detection spectra of the plasmonic QDIP at 90� polarization and 0� polarization, respectively. The photo-response of a reference QDIP without the plasmonic structure is also shown in the figures for comparison. The plasmonic QDIP shows different detection wavelengths at different polarizations.

### 4.4. Angular dependence of the 2DSHA plasmonic enhancement

Figure 12 shows the photocurrent enhancement ratio spectra for all the plasmonic QDIPs with different periods from 1.4 to 2.0 µm. The periods of the 2DSHA plasmonic structures are marked on the curves. The insert shows a microscopic picture of a QDIP with a 2DSHA plasmonic structure. The IR incidence is from the backside of the QDIPs, that is, backside-

Figure 12. Photocurrent enhancement ratio spectra for plasmonic QDIPs with different periods varying from 1.4 to 2.0 µm. The insert is a microscopic picture of a QDIP with a 2DSHA plasmonic structure (Reproduced from Ref. [28] with the

As shown in Figure 12, the enhancement wavelengths can be tuned by varying the periods of the 2DSHA plasmonic structures. This allows one to achieve multispectral QDIPs without

From Eq. (43), the plasmonic resonant wavelength depends on the period p. By changing the plasmonic lattice from a square lattice to a rectangular lattice with different periods in the x

Figure 13(a) shows an SEM picture of a rectangular lattice 2DSHA plasmonic structure with the periods of 2.6 and 3.0 µm in the x-direction and the y-direction, respectively. Figure 13(b)–(d) shows the transmission spectra of the un-polarized, 90 polarized (i.e. polarization in y-direction), and 0 polarized (i.e. polarization in x-direction), respectively. The resonant wavelengths correspond to the orders in the x- and y- directions are marked in the figures. The unpolarized light shows transmission in both (0, 1) and (1, 0) orders. However, the 0 polarized

and y directions, one can engineer the resonant wavelengths at different polarizations.

configured 2DSHA plasmonic structures.

permission of AIP Publishing).

208 Nanoplasmonics - Fundamentals and Applications

fabricating an individual filter on each photodetector [28].

4.3. Polarization dependence and polarimetric detection

Figure 16 shows the schematic view of the top-illuminated 2DSHA plasmonic QDIP with an incident angle θ.

From Eq. (41), the coupling efficiency η to the plasmonic waves depends on the overlap integral of Hðx, zÞ with the plasmonic wave eiðksppxþk2zzÞ . At the incident angle θ, the Hðx, zÞ not only periodically modulated by the 2DSHA structure, but also added additional phase term eiðk0sinθxÞ , that is,

Figure 14. Microscopic picture of a QDIP with the rectangular lattice 2DSHA structure. Insert: SEM picture of the rectangular lattice 2DSHA structure (© IOP Publishing. Reproduced with permission from Ref. [11]. All rights reserved).

Figure 15. Measured detection spectra of the plasmonic QDIP at different polarizations: (a) 90 polarization and (b) 0 polarization. Different polarizations show different detection wavelengths (© IOP Publishing. Reproduced with permission from Ref. [11]. All rights reserved).

Figure 16. Schematic view of the top-illuminated 2DSHA plasmonic QDIP with an incident angle θ.

Figure 14. Microscopic picture of a QDIP with the rectangular lattice 2DSHA structure. Insert: SEM picture of the rectangular lattice 2DSHA structure (© IOP Publishing. Reproduced with permission from Ref. [11]. All rights reserved).

Figure 15. Measured detection spectra of the plasmonic QDIP at different polarizations: (a) 90 polarization and (b) 0 polarization. Different polarizations show different detection wavelengths (© IOP Publishing. Reproduced with

permission from Ref. [11]. All rights reserved).

210 Nanoplasmonics - Fundamentals and Applications

Figure 17. Measured photocurrent of the 2DSHA plasmonic QDIP at different incident angles of 0, 30, 45 and 60 compared with the reference QDIP at the same incident angles. The plasmonic resonant wavelength λ0 shift to shorter wavelengths as the incident angle increases.

$$H(\mathbf{x}, \mathbf{z}) \infty \mathbf{e}^{\pm i \Lambda \mathbf{x}} \mathbf{e}^{ik\_0 \sin \theta \mathbf{x}} \tag{45}$$

From Eq. (45), the coupling condition can be expressed as follows:

$$k\_0 \sin \theta \pm \Lambda = k\_{spp} \tag{46}$$

Taking the + sign, one can write Eq. (46) in terms of plasmonic resonant wavelength λ0 as follows:

$$\lambda\_{\mathbf{0}} = \left\{ \mathbf{Re} \left[ \sqrt{\frac{\epsilon\_{2cr}\epsilon\_{1r}}{(\varepsilon\_{1r} + \varepsilon\_{2r})}} \right] - \sin \theta \right\} \Lambda \tag{47}$$

Eq. (47) shows that the enhancement wavelength shifts to shorter wavelengths as the incident angle increases.

Figure 17 shows the measured photocurrent of the 2DSHA plasmonic QDIP at different incident angles of 0�, 30� , 45�, and 60� compared with the reference QDIP at the same incident angles. Blue shift of the plasmonic resonant wavelengths λ0 is experimentally observed as the incident angle increases.

To quantitatively analyse the relationship between the plasmonic wavelength shift and the incident angle, we plot the plasmonic resonant wavelengths λ0 versus sinθ. Figure 18 shows the λ0 versus sinθ. A linear relation is obtained, which agrees well with Eq. (47). The interception and the slope are 7.6 and �1.4 µm, respectively. The interception 7.6 µm matches the value

Figure 18. λ0 versus sinθ plot. λ0 varies linearly with sinθ. The interception and the slope are 7.6 and �1.4 µm, respectively.

Re ffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2crE1r ðε1rþε2crÞ � � q Λ in Eq. (47). However, the slope of �1.4 µm is smaller than Λ ¼ 2:3 µm as predicted in Eq. (47). This may be due to the plasmonic scattering-induced phase distribution in Hðx, zÞ in addition to the eiðk0sinθxÞ term.

## 5. Conclusion

Hðx, zÞ∝e�iΛxeik0sinθx (45)

k0sinθ � Λ ¼ kspp (46)

Λ (47)

From Eq. (45), the coupling condition can be expressed as follows:

λ0 ¼ Re

follows:

angle increases.

respectively.

incident angles of 0�, 30�

212 Nanoplasmonics - Fundamentals and Applications

incident angle increases.

Taking the + sign, one can write Eq. (46) in terms of plasmonic resonant wavelength λ0 as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2crE1r ðε1r þ ε2crÞ � � r

Eq. (47) shows that the enhancement wavelength shifts to shorter wavelengths as the incident

Figure 17 shows the measured photocurrent of the 2DSHA plasmonic QDIP at different

angles. Blue shift of the plasmonic resonant wavelengths λ0 is experimentally observed as the

To quantitatively analyse the relationship between the plasmonic wavelength shift and the incident angle, we plot the plasmonic resonant wavelengths λ0 versus sinθ. Figure 18 shows the λ0 versus sinθ. A linear relation is obtained, which agrees well with Eq. (47). The interception and the slope are 7.6 and �1.4 µm, respectively. The interception 7.6 µm matches the value

Figure 18. λ0 versus sinθ plot. λ0 varies linearly with sinθ. The interception and the slope are 7.6 and �1.4 µm,

� �

� sinθ

, 45�, and 60� compared with the reference QDIP at the same incident

In conclusion, 2DSHA plasmonic structures can effectively induce plasmonic resonant waves. The resonant wavelength depends on the period of the plasmonic structure. The surface confinement effect allows one to improve the performance of QDIPs. The plasmonic enhancement also strongly depends on the polarization and the incident angle. The polarization and incident angle dependence allow polarimetric and polarization engineering in QDIP designs.

## Author details

Guiru Gu1 \*, Xuejun Lu2 , Thitikorn Kemsri2 and Yingjie Zhang2

\*Address all correspondence to<EMAIL_ADDRESS>

## References

[20] D. Pan, E. Towe, and S. Kennerly, "Normal-incidence intersubband (In, Ga)As/GaAs quantum dot infrared photodetectors," Applied Physics Letters, vol. 73, pp. 1937–1939, 1998.

[7] H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, "Spectroscopy of single hemoglobin molecules by surface enhanced Raman scattering," Physical Review Letters, vol. 83, pp.

[8] C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, et al., "A surface plasmon enhanced infrared photodetector based on InAs quantum dots," Nano Letters,

[9] P. Senanayake, C.-H. Hung, J. Shapiro, A. Lin, B. Liang, B. S. Williams, et al., "Surface plasmon-enhanced nanopillar photodetectors," Nano Letters, vol. 11, pp. 5279–5283, 2011.

[10] P. Vasinajindakaw, J. Vaillancourt, G. Gu, R. Liu, Y. Ling, and X. Lu, "A Fano-type interference enhanced quantum dot infrared photodetector," Applied Physics Letters, vol.

[11] P. Vasinajindakaw, J. Vaillancourt, G. Gu, and X. Lu, "Surface plasmonic enhanced polarimetric longwave infrared photodetection with band pass spectral filtering," Semiconductor Science and Technology, vol. 27, pp. 65005–65009, 2012. http://dx.doi.org/10.1088/0268-

[12] G. Gu, J. Vaillancourt, P. Vasinajindakaw, and X. Lu, "Backside-configured surface plasmonic structure with over 40 times photocurrent enhancement," Semiconductor Science and Technology, vol. 28, p. 105005, 2013. http://dx.doi.org/10.1088/0268-1242/28/10/105005.

[13] G. Gu, N. Mojaverian, J. Vaillancourt, and X. Lu, "Surface plasmonic resonance induced near-field vectors and their contribution to quantum dot infrared photodetector enhance-

[14] G. Gu, J. Vaillancourt, and X. Lu, "Analysis of near-field components of a plasmonic optical antenna and their contribution to quantum dot infrared photodetector enhance-

[15] J. Vaillancourt, N. Mojaverian, and L. Xuejun, "A longwave infrared focal plane array enhanced by backside-configured plasmonic structures," IEEE Photonics Technology Let-

[16] R. Liu, P. Vasinajindakaw, G. Gu, J. Vaillancourt, and X. Lu, "Optimizing light absorption in quantum dot infrared photodetectors by tuning surface confinement of surface

[18] T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature, vol. 391, pp. 667–669, 1998. [19] Handbook of Optical Constants of Solids. Academic Press, San Diego London New York

plasmonic waves," Journal of Physics D: Applied Physics, vol. 46, p. 015102, 2013. [17] P. Drude, "Electron theory of metals," Annalen der Physik, vol. 306, pp. 566–613, 1900.

ment," Journal of Physics D: Applied Physics, vol. 47, p. 435106, 2014.

ment," Optics Express, vol. 22, pp. 24970–24976, 2014/10/20 2014.

4357–4360, 11/22/1999.

214 Nanoplasmonics - Fundamentals and Applications

1242/27/6/065005.

vol. 10, pp. 1704–1709, 2010.

98, pp. 211111–1 to 211111–3, 2011.

ters, vol. 26, pp. 745–748, 2014.

Boston Sydney Tokyo Toronto, 1998.

**Applications to Optoelectronics**

### **Equivalent Nanocircuit Theory and Its Applications** Equivalent Nanocircuit Theory and Its Applications

Qing Zhang, Jun You and Chengpu Liu Qing Zhang, Jun You and Chengpu Liu

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67681

### Abstract

A new methodology termed as equivalent nanocircuit (EN) theory is briefly introduced, and its recent important progress in designing meta-material devices with peculiar characteristics in optical or infrared frequency domain is reviewed. Three representative EN-based designs of infrared window meta-materials, such as Butterworth filter, metalinsulator-metal absorber and design-simplified TCO-based super-flat absorber, are demonstrated. All these progresses clearly indicate that the EN theory provides an inspiring advancement on the way of designing more complicated meta-devices.

Keywords: meta-material, metatronics, nanocircuit theory, surface plasmon polariton

## 1. Introduction

The functionalities of plasmonic nanodevices are dependent on their artificial structure parameters that usually are smartly engineered. To introduce convenient and feasible methods based on new paradigms to guide, the structure design is eagerly demanded. In the radio or microwave frequency domains, lumped circuit elements, e.g. resistors (R), inductors (L) and capacitors (C), can be effectively and flexibly used as blocks for designing complicated micro-electronic devices, and the conventional circuit theory is widely and successfully adopted. One question naturally arises: can such a circuit theory, along with its accompanying mathematical machinery, be extended and applied to the design of plasmonic nanostructures working at the optical domain?

Within this context, in this chapter, a new methodology, 'equivalent nanocircuit (EN) theory' is briefly introduced, and its representative applications in designing plasmonic devices with peculiar characteristics in optical or infrared frequency domains are enumerated. First, to start with the short introduction of the basis of 'metatronics', the analogy between micro-electronic lumped circuit elements (R, L, C) and optical nanocircuit elements (nanoparticles, nanoantennas, nanogratings and optical meta-surfaces) is established. Second, the method describing

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

how to use the above optical nanocircuit elements to construct a single-layer or multiple-layer complicated meta-material structure is proposed and thus the 'metatronics' concept moves forward to the multi-order circuit theory. Finally, three representative applications of multiorder EN theory to design infrared window meta-materials are demonstrated: (1) a synthesis procedure for designing a third-order Butterworth filter is proposed; (2) a metal-insulator-metal (MIM) ultra-broadband absorber is successfully designed in the infrared range; (3) with the transparent conductive oxides (TCOs) semiconductor materials as building blocks, a designsimplified broadband super-flat perfect infrared absorber is realized.

## 2. Metatronics and optical nanocircuit

### 2.1. 'Metatronics': meta-material-inspired optical circuitry

Optical meta-material bridges the gap between the conventional optics and the nanoworld, which gives rise to a diversity of surprising and profound effects fully appreciated and technologically explored in recent years [1–3]. The electromagnetic property of meta-materials is dependent on their specifically and smartly engineered artificial structures. To explore new design methodology is eagerly demanded. Historically, in electronics, basic functionalities are synthesized by 'lumped' circuit elements, such as resistors, inductors, capacitors and transistors, and more complicated operations can be realized by combining them into a complicated circuit in some specific serial or parallel ways. Now, great interests in pushing classic circuit operation to infrared or visible optical frequency range have boomed in order to achieve an optical analogy [4–7]. Generally, just simply reducing the basic unit size of circuit to the micrometre (µm) or nanometre (nm) level would be technically difficult to achieve the above goals.

The first challenge arises in the state-of-the-art nanoscale fabrication technique is its inevitable difficulty to achieve deeply sub-wavelength dimension becoming serious. The second challenge lies in material dispersion which is sometimes vital. As we know that metals such as gold, silver, aluminium and copper are highly conductive materials at RF and microwaves, commonly used in conventional circuits. However, at optical frequency, these metals would behave differently, and they exhibit plasmonic resonance instead of the usual conductivity, i.e. the coupling of optical signals with collective oscillation of conductive electrons at these metal surfaces is dominant where the real part of permittivity is negative. In other words, at optical wavelengths, the conduction current is no longer the dominant current circulating through lumped optical elements. Therefore, the traditional circuit theory and the corresponding methodology for micro-electronics lose their functionality in optical frequency domain.

In 2007, Engheta et al. have made an important breakthrough and first proposed the concept of 'metatronics' [6] which bridges the gap between low frequency circuitry design and high frequency nanodevice design. In their opinions, it is possible to realize the performance of lumped-circuit-like elements at optical frequency just by properly designing and suitably arranging plasmonic or/and non-plasmonic nanoparticles, as shown in Figure 1. The theoretical framework of 'metatronics' is very simple and is based on one Maxwell equation ∇ � H ¼ Jc þ ð�iωDÞ, and this equation has two terms on its right-side: one is the conduction

Figure 1. Analogy between sub-wavelength nanoparticles and conventional lumped nanocircuit elements at optical frequency domain. Here, one nanoparticle can take a different role when the sign of its complex permittivity ε is different. If combined in a specific way, different nanoparticles can possess some defined functionalities (adapted from Ref. [6], Figure 1).

current Jc ¼ σE and the other is the displacement current Jd ¼ �iωD. As mentioned above, the material dispersion plays a crucial role at optical wavelength, and the conduction current (originally dominant at low-frequency) is no longer dominant. Instead, the electric displacement current can be taken as the 'flowing optical current'. With the left of displacement current, the next and most important step is to re-build the formulas to calculate the 'lumped impedance' of nanoparticles as for an optical circuit.

### 2.2. Optical lumped elements

how to use the above optical nanocircuit elements to construct a single-layer or multiple-layer complicated meta-material structure is proposed and thus the 'metatronics' concept moves forward to the multi-order circuit theory. Finally, three representative applications of multiorder EN theory to design infrared window meta-materials are demonstrated: (1) a synthesis procedure for designing a third-order Butterworth filter is proposed; (2) a metal-insulator-metal (MIM) ultra-broadband absorber is successfully designed in the infrared range; (3) with the transparent conductive oxides (TCOs) semiconductor materials as building blocks, a design-

Optical meta-material bridges the gap between the conventional optics and the nanoworld, which gives rise to a diversity of surprising and profound effects fully appreciated and technologically explored in recent years [1–3]. The electromagnetic property of meta-materials is dependent on their specifically and smartly engineered artificial structures. To explore new design methodology is eagerly demanded. Historically, in electronics, basic functionalities are synthesized by 'lumped' circuit elements, such as resistors, inductors, capacitors and transistors, and more complicated operations can be realized by combining them into a complicated circuit in some specific serial or parallel ways. Now, great interests in pushing classic circuit operation to infrared or visible optical frequency range have boomed in order to achieve an optical analogy [4–7]. Generally, just simply reducing the basic unit size of circuit to the micrometre

(µm) or nanometre (nm) level would be technically difficult to achieve the above goals.

odology for micro-electronics lose their functionality in optical frequency domain.

In 2007, Engheta et al. have made an important breakthrough and first proposed the concept of 'metatronics' [6] which bridges the gap between low frequency circuitry design and high frequency nanodevice design. In their opinions, it is possible to realize the performance of lumped-circuit-like elements at optical frequency just by properly designing and suitably arranging plasmonic or/and non-plasmonic nanoparticles, as shown in Figure 1. The theoretical framework of 'metatronics' is very simple and is based on one Maxwell equation ∇ � H ¼ Jc þ ð�iωDÞ, and this equation has two terms on its right-side: one is the conduction

The first challenge arises in the state-of-the-art nanoscale fabrication technique is its inevitable difficulty to achieve deeply sub-wavelength dimension becoming serious. The second challenge lies in material dispersion which is sometimes vital. As we know that metals such as gold, silver, aluminium and copper are highly conductive materials at RF and microwaves, commonly used in conventional circuits. However, at optical frequency, these metals would behave differently, and they exhibit plasmonic resonance instead of the usual conductivity, i.e. the coupling of optical signals with collective oscillation of conductive electrons at these metal surfaces is dominant where the real part of permittivity is negative. In other words, at optical wavelengths, the conduction current is no longer the dominant current circulating through lumped optical elements. Therefore, the traditional circuit theory and the corresponding meth-

simplified broadband super-flat perfect infrared absorber is realized.

2. Metatronics and optical nanocircuit

220 Nanoplasmonics - Fundamentals and Applications

2.1. 'Metatronics': meta-material-inspired optical circuitry

Based on the above 'metatronic' concept, the optical impedance (an intrinsic parameter) of nanoparticles is similar to the electrical counterpart (Z=V/I) [8], which is independent of the surrounding environments. As sketched in Figure 2, two isolated sub-wavelength nanospheres (radius R) with complex permittivity ε are immersed in homogenous space (with permittivity ε0 being real) and illuminated uniformly by a vertical electric field E0. Because of small enough size of nanoparticles (with respect to the central wavelength of illuminating field), the scattered electromagnetic (EM) fields in the vicinity of nanospheres and the total fields inside it can be obtained under suitable approximations by means of the well-known time-harmonic and quasi-static approach [9]. If integrating the flux of displacement current Jd across the sphere induced by the scattered electric field, one can get the 'total'displacement current as,

$$I\_{\rm imp} = I\_{\rm sph} + I\_{\rm fringe} = -i\omega (\varepsilon - \varepsilon\_0) \pi R^2 |E\_0|,\tag{1}$$

$$I\_{\rm sph} = -i\omega\epsilon\pi R^2(\varepsilon - \varepsilon\_0)|E\_0|/(\varepsilon - 2\varepsilon\_0),\tag{2}$$

$$I\_{\rm frings} = -i\omega \varepsilon\_0 2\pi R^2 (\varepsilon - \varepsilon\_0) |\mathbf{E}\_0|/(\varepsilon + 2\varepsilon\_0). \tag{3}$$

Figure 2. Optical nanocircuit models for (a) dielectric or (b) plasmonic nanoparticles. E0 denotes the incident electric field, and thinner field lines together with the arrows represent the electric dipolar fringe field from nanospheres. (adapted from Ref. [8], Figure 1).

Here, as shown at the bottom of Figure 2, Iimp is the 'impressed' displacement current, Isph is displacement current circulating in nanospheres and Ifringe is displacement current for the fringe (dipolar) field. Such currents are all related to the polarization charges at the surface of nanospheres induced by light source and all can be intuitively interpreted as branch currents at nodes in a parallel circuit.

The 'average' potential difference between the upper and lower hemi-spherical surfaces of the sphere is given by

$$
\langle V \rangle\_{\rm sph} = \langle V \rangle\_{\rm frings} = R(\varepsilon - \varepsilon\_0)|E\_0|/(\varepsilon - 2\varepsilon\_0) \tag{4}
$$

Thus, after having obtained the ratio between the potential difference (Eq. 4) and the effective current evaluated from Eq. (2) or (3), one can, respectively, get the equivalent impedance for the 'nanosphere' or the 'fringe' branch of the circuit as

$$Z\_{\rm sph} = \left(-i\omega\varepsilon\pi R\right)^{-1}, \quad Z\_{\rm frings} = \left(-i\omega 2\pi R\varepsilon\_0\right)^{-1}.\tag{5}$$

From Eq. (5), one can clearly see that the two parallel elements in the circuits (Figure 2) may behave differently, which is determined by the permittivity sign of the nanospheres as shown in Figure 1. For example, a non-metallic (or dielectric) sphere can be thought as a capacitor (because of (ReðεÞ > 0), parallel with a resistor (ImðεÞ 6¼ 0), indicating a loss) (left-bottom, Figure 2). In contrast, the impedance of the outside fringe is always capacitive, since its permittivity is positive. Therefore, here the equivalent circuit elements are expressed in terms of nanosphere parameters as,

$$\mathbf{C}\_{\rm sph} = \pi R \mathbf{R} \mathbf{e}[\varepsilon], \quad \mathbf{G}\_{\rm sph} = \pi \omega R \mathbf{I} \mathbf{m}[\varepsilon], \quad \mathbf{C}\_{\rm fringe} = 2\pi R \varepsilon\_0. \tag{6}$$

Instead, if such a sphere is made of plasmonic materials (e.g. Ag, Au), it behaves as an inductive element, since (ReðεÞ < 0), and similarly the equivalent inductor for such a sphere becomes,

$$L\_{\rm sph} = \left(-\omega^2 \pi R \text{Re}[\varepsilon]\right)^{-1}.\tag{7}$$

Naturally, one can implement a more complicated circuit by a series and parallel operations among different kinds of such lumped particles. If two nanoparticles share a common interface and electric field is tangential to the common interface, a parallel operation between them would be made effective (left-column, Figure 3).

In contrast, if the electric displacement vector locally is normal to that of common interface, the displacement current Jd passing across one element through the common interface must flow inside the second element under the confinement of the continuity condition, and a series cascade of these two nanoelements is realized (right-column, Figure 3). This represents an important difference with respect to those in the conventional circuit theory, and in the latter, the lumped elements are completely isolated from the external world, and their interaction with the circuit is determined by the facts that how they pass through their terminals and how they are connected with the rest of the circuit. In contrast, in the former, i.e. the configuration presented here, since the fringing and external fields (Figure 2) play dominant roles in the interaction of nanoelements with their neighbouring particles and surrounding environment,

Here, as shown at the bottom of Figure 2, Iimp is the 'impressed' displacement current, Isph is displacement current circulating in nanospheres and Ifringe is displacement current for the fringe (dipolar) field. Such currents are all related to the polarization charges at the surface of nanospheres induced by light source and all can be intuitively interpreted as branch currents

Figure 2. Optical nanocircuit models for (a) dielectric or (b) plasmonic nanoparticles. E0 denotes the incident electric field, and thinner field lines together with the arrows represent the electric dipolar fringe field from nanospheres. (adapted from

The 'average' potential difference between the upper and lower hemi-spherical surfaces of the

Thus, after having obtained the ratio between the potential difference (Eq. 4) and the effective current evaluated from Eq. (2) or (3), one can, respectively, get the equivalent impedance for

From Eq. (5), one can clearly see that the two parallel elements in the circuits (Figure 2) may behave differently, which is determined by the permittivity sign of the nanospheres as shown in Figure 1. For example, a non-metallic (or dielectric) sphere can be thought as a capacitor (because of (ReðεÞ > 0), parallel with a resistor (ImðεÞ 6¼ 0), indicating a loss) (left-bottom, Figure 2). In contrast, the impedance of the outside fringe is always capacitive, since its permittivity is positive. Therefore, here the equivalent circuit elements are expressed in terms

Instead, if such a sphere is made of plasmonic materials (e.g. Ag, Au), it behaves as an inductive element, since (ReðεÞ < 0), and similarly the equivalent inductor for such a sphere

�1

〈V〉sph ¼ 〈V〉fringe ¼ Rðε � ε0ÞjE0j=ðε � 2ε0Þ (4)

�1

: (5)

, Zfringe ¼ ð�iω2πRε0Þ

Csph ¼ πRRe½ε�, Gsph ¼ πωRIm½ε�, Cfringe ¼ 2πRε0: (6)

at nodes in a parallel circuit.

222 Nanoplasmonics - Fundamentals and Applications

of nanosphere parameters as,

becomes,

the 'nanosphere' or the 'fringe' branch of the circuit as

Zsph ¼ ð�iωεπRÞ

sphere is given by

Ref. [8], Figure 1).

Figure 3. Two conjoined two-dimensional half-spheres with different signs of permittivity, illuminated by a uniform electric field. Parallel (left) and series (right) configurations are, respectively, shown (adapted from Ref. [8], Figure 2).

the originally same connection now may consists of both series and parallel interconnections which sensitively depend on the orientation of the external applied field. This point may be taken as an important and additional degree of freedom in optical nanocircuit design.

In the above, just nanosphere is taken as an example particle. In fact, except for this, the basic 'alphabets' of metatronics can also be gratings, cylinder pillars, rectangle bricks and other antennas, and they have been widely applied to various wavelength regimes,for example in a series of theoretical analyses and numerical simulations [4, 7, 10–12].

### 2.3. Metatronics-based applications

The recent experimental progresses [7, 13] have verified the validity and potentiality of the above optical nanocircuit paradigm introduced in Section 2.2, and also demonstrated the possibility of re-configuring the circuit responses just by changing the orientation and polarization of illuminating field to induce a specific feature not available in conventional electric circuits. No doubt that this metatronics concept provides us an effective and practical tool to design optical nanodevices, such as designing and tuning of optical nanoantennas [14, 15] and meta-surfaces [16].

One representative example is shown in Figure 4 where a polarization-selective optical filter making use of a simple sub-wavelength grating is designed. Different incident illuminations and different effective optical connections between the nanoelements of this 'stereonanocircuit' are chosen to conveniently control the light transmittance, i.e. to make the circuit functions either band-stop or band-pass. This scheme may be exploited for parallel processing of multiple flows of information through a single nanostructure.

In detail, when electric field E is perpendicular to the nanorods (Figure 4a), the optical displacement current Jd 'flows' transversely across the nanorods and air gaps (in other words,

Figure 4. Sub-wavelength grating with parallel plasmonic nanorods working as a two-dimensional optical nanocircuit with a stereo-functionality. (a) The E field is perpendicular to the nanorods and nanoinductors (including nanoresistors) and nanocapacitors form a series configuration (c). In contrast, if the E field is parallel to nanorods (b), a parallel configuration (d) of lumped circuit elements is formed instead (adapted from [10], Figure 1).

the displacement current is in the plane of the arrays but perpendicular to the nanorods). Therefore, the final equivalent series impedance is written as,

the originally same connection now may consists of both series and parallel interconnections which sensitively depend on the orientation of the external applied field. This point may be

In the above, just nanosphere is taken as an example particle. In fact, except for this, the basic 'alphabets' of metatronics can also be gratings, cylinder pillars, rectangle bricks and other antennas, and they have been widely applied to various wavelength regimes,for example in a

The recent experimental progresses [7, 13] have verified the validity and potentiality of the above optical nanocircuit paradigm introduced in Section 2.2, and also demonstrated the possibility of re-configuring the circuit responses just by changing the orientation and polarization of illuminating field to induce a specific feature not available in conventional electric circuits. No doubt that this metatronics concept provides us an effective and practical tool to design optical nanodevices, such as designing and tuning of optical nanoantennas [14, 15] and

One representative example is shown in Figure 4 where a polarization-selective optical filter making use of a simple sub-wavelength grating is designed. Different incident illuminations and different effective optical connections between the nanoelements of this 'stereonanocircuit' are chosen to conveniently control the light transmittance, i.e. to make the circuit functions either band-stop or band-pass. This scheme may be exploited for parallel processing

In detail, when electric field E is perpendicular to the nanorods (Figure 4a), the optical displacement current Jd 'flows' transversely across the nanorods and air gaps (in other words,

Figure 4. Sub-wavelength grating with parallel plasmonic nanorods working as a two-dimensional optical nanocircuit with a stereo-functionality. (a) The E field is perpendicular to the nanorods and nanoinductors (including nanoresistors) and nanocapacitors form a series configuration (c). In contrast, if the E field is parallel to nanorods (b), a parallel

configuration (d) of lumped circuit elements is formed instead (adapted from [10], Figure 1).

taken as an important and additional degree of freedom in optical nanocircuit design.

series of theoretical analyses and numerical simulations [4, 7, 10–12].

of multiple flows of information through a single nanostructure.

2.3. Metatronics-based applications

224 Nanoplasmonics - Fundamentals and Applications

meta-surfaces [16].

$$\begin{array}{ll} Z\_{\perp(\text{equivalent})} = Z\_{\perp(\text{nanorod})} + Z\_{\perp(\text{airgap})}, \\ Z\_{\perp(\text{nanorod})} = \text{iw/} \omega \text{h\varepsilon}, \quad Z\_{\perp(\text{airgap})} = \text{ig/} \omega \text{h\varepsilon}\_{\text{air}}. \end{array} \tag{8}$$

In contrast, when E is parallel to the nanorods (Figure 4b), the optical displacement current Jd 'flows' along both the nanorods and air gaps. Therefore, each nanorod and its neighbouring air gap collectively have equivalent parallel impedance,

$$\begin{array}{ll} \mathbf{Z}\_{\parallel(\text{equivalent})} = \mathbf{Z}\_{\parallel(\text{nonmod})} \mathbf{Z}\_{\parallel(\text{singular})} / (\mathbf{Z}\_{\parallel(\text{nonmod})} + \mathbf{Z}\_{\parallel(\text{singular})}),\\ \mathbf{Z}\_{\parallel(\text{nonmod})} = \mathbf{i}/\omega \mathbf{u} \hbar \varepsilon, \quad \mathbf{Z}\_{\parallel(\text{singular})} = \mathbf{i}/\omega \mathbf{g} \hbar \varepsilon\_{\text{air}} \end{array} \tag{9}$$

These equivalent impedances (series or parallel,in Eqs. (8) and (9)) finally determine the transmittance of the incident optical signal, and the latter is naturally derived as,

$$\begin{aligned} T\_{\perp} &= |Z\_{\perp(\text{equivalent})} / (Z\_{\perp(\text{equivalent})} + [\eta\_0(w + \text{g})/2])|^2, \\ T\_{\parallel} &= |Z\_{\parallel(\text{equivalent})} / (Z\_{\parallel(\text{equivalent})} + [\eta\_0 / (2(w + \text{g}))])|^2. \end{aligned} \tag{10}$$

To test this optical circuit approach, one can compare the results in Figure 5 calculated from Eq. (10) with those experimentally measured (10) or exactly numerical results with the help of

Figure 5. Transmittance spectra for five different samples from A to E. Experimentally measured data for (a) perpendicular or (b) parallel polarization of the incident wave impinging on the nanorod arrays. (c and d) Full-wave simulation results and (e and f) calculation results from nanocircuit theory for polarization perpendicular (upper) or parallel (lower) to nanorods (adapted from Ref. [10], Figure 2).

a commercial software. As one expects, the grating with a 'series' configuration behaves as a band-stop filter, however the same grating with a 'parallel' configuration as a band-pass filter instead.

Another representative example is related to the radio-frequency (RF) antennas which have been widely investigated and applied in wireless telecommunication system in the last century. The functionality of RF antenna is information revolution. It is usually used with a radio transmitter or radio receiver. In transmission, a radio transmitter supplies an electric current oscillating at radio frequency (i.e. a high frequency alternating current (AC)) to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves (radio waves). In analogy with their RF counterparts, optical antennas made of plasmonic nanoparticles are able to efficiently coupled localized sources or guided waves at the nanoscale level to far-field radiation, and in turn, to convert the impinging radiation from the far-field into sub-wavelength localized or guided fields [17]. As the counterpart of RF antenna, an optical nanoantenna exhibits novice and interesting characteristics because of their plasmonic nature.

In detail, as depicted in Figure 6a, a conventional linear RF antenna is loaded at its feeding gap with lumped circuit elements and changing the antenna input impedance allowance to operate at a given frequency or to achieve a good match for a specific feeding network. Analogously, an optical nanocircuit opens the same possibility for an optical nanoantenna (Figure 6b), and the complex optical input impedance can be interpreted as the parallel combination of the dipole intrinsic impedance Zdip and the gap impedance Zgap [11, 15]. The former impedance is assumed to be a fixed property of the nanoantenna geometry and surrounding environment, and the latter can be engineered to a large extent by loading the gap with different materials.

Figure 6. Optical nanoantennas. RF dipole antenna (a) loaded with lumped circuit elements at its feeding gap, and analogously, a plasmonic dipole nanoantenna (b) loaded with optical nanocircuits (adapted from Ref. [15), Figure 1). The low inset shows the circuit model of antenna input impedance. (c) Tuning of nanoantenna resonance by 'gap loading' with different realistic materials (adapted from Ref. [15], Figure 3).

Such as for a cylindrical gap with height of t, radius of a and excited by an incident electric field parallel to its axis, the gap impedance is given by,

a commercial software. As one expects, the grating with a 'series' configuration behaves as a band-stop filter, however the same grating with a 'parallel' configuration as a band-pass filter

Another representative example is related to the radio-frequency (RF) antennas which have been widely investigated and applied in wireless telecommunication system in the last century. The functionality of RF antenna is information revolution. It is usually used with a radio transmitter or radio receiver. In transmission, a radio transmitter supplies an electric current oscillating at radio frequency (i.e. a high frequency alternating current (AC)) to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves (radio waves). In analogy with their RF counterparts, optical antennas made of plasmonic nanoparticles are able to efficiently coupled localized sources or guided waves at the nanoscale level to far-field radiation, and in turn, to convert the impinging radiation from the far-field into sub-wavelength localized or guided fields [17]. As the counterpart of RF antenna, an optical nanoantenna exhibits

In detail, as depicted in Figure 6a, a conventional linear RF antenna is loaded at its feeding gap with lumped circuit elements and changing the antenna input impedance allowance to operate at a given frequency or to achieve a good match for a specific feeding network. Analogously, an optical nanocircuit opens the same possibility for an optical nanoantenna (Figure 6b), and the complex optical input impedance can be interpreted as the parallel combination of the dipole intrinsic impedance Zdip and the gap impedance Zgap [11, 15]. The former impedance is assumed to be a fixed property of the nanoantenna geometry and surrounding environment, and the latter can be engineered to a large extent by loading the gap with different materials.

Figure 6. Optical nanoantennas. RF dipole antenna (a) loaded with lumped circuit elements at its feeding gap, and analogously, a plasmonic dipole nanoantenna (b) loaded with optical nanocircuits (adapted from Ref. [15), Figure 1). The low inset shows the circuit model of antenna input impedance. (c) Tuning of nanoantenna resonance by 'gap loading'

with different realistic materials (adapted from Ref. [15], Figure 3).

novice and interesting characteristics because of their plasmonic nature.

instead.

226 Nanoplasmonics - Fundamentals and Applications

$$Z\_{\text{gap}} = \frac{\text{it}}{\omega \varepsilon \varepsilon\_0 \pi a^2}. \tag{11}$$

Thus, by filling the gap with different materials (or their proper series or parallel combination), the impedance of the gap can be tailored to a large degree. As a result, one can tune the frequency response (Figure 6c) or radiation pattern easily.

Figure 7. Meta-transmit-array used for full control of nanoscale optical transmission. (a) Basic nanocircuit building block (left) and resulting meta-screen (right) with transverse inhomogeneous profiles of surface impedance. (b) Full-wave simulation of meta-transmit-array for light deflection with high efficiency and minimized reflection (adapted from Ref. [16]).

In addition, except for the above sub-wavelength grating and non-antenna, meta-surface represents another important type [18], the planar counterparts of meta-materials that provide the unprecedented control of the amplitude, phase or polarization of light waves at the subwavelength (nano) scale. For example, these two-dimensional surfaces can alter the wave-front of incident light for a widespread application in beam shaping [19–24], polarizers [25, 26] and flat lenses [27–29]. As proposed in [16], the fundamental building blocks are paired plasmonic or dielectric nanorods collectively working as an inductor–capacitor nanocircuit (Figure 7a, left), whose impedance depends directly on the filling ratio of plasmonic and dielectric materials. By suitably alternating these nanocircuit blocks on the transverse direction, one can synthesize a meta-surface with the required inhomogeneous impedance profile. The configuration composed of a stack of three meta-surfaces (Figure 7a, right), can fully control the nanoscale optical transmission, while simultaneously minimizing the reflection (impedancematching to free-space), allowing, for example light deflection with an almost ideal efficiency as shown in Figure 7b.

## 3. Equivalent nanocircuit theory for multi-layer meta-material design

In this Section, the 'design stack' is moved upward, from the 'physical layer' of optical lumped elements (pure nanoparticles, nanogratings, nanoantennas, etc.) to more complex functional devices, including: (1) infrared third-order Butterworth filters; (2) metal-insulator-metal (MIM) ultra-broadband absorbers; (3) simplified broadband super-flat perfect infrared absorbers only composed of single transparent conductive oxides (TCOs). Among these design procedures for multi-layer nanostructures in our research group, the suitability of the equivalent nanocircuit theory is confirmed once more which in turn enriches and expands the application of equivalent nanocircuit theory.

### 3.1. Infrared broadband third-order Butterworth filters

Frequency selective surfaces (FSSs) have been the subject of investigations by many researchers for decades. An FSS is a periodic structure usually composed of an assembly of identical elements arranged in one- or two-dimensional lattice. These structures are used in a variety of important applications ranging from microwave systems and antennas to radar and satellite communications. The simplest FSS device is a filter. By means of circuit elements (e.g. resistors (R), inductors (L) and capacitors (C)) FSSs can be effectively and flexibly designed to functionalize a low-pass, high-pass, band-pass or band-stop responses in the microwave or RF domain [30, 31]. The great interests in FSS's application in higher frequency range to achieve high-density and high-speed optical analogues [5, 10] have been pushed and the goals have been synthesized as realistic nanostructures at optical frequency, thanks to the optical nanocircuit theory which proves once more to be an essential design tool to construct optical FSSs or filters.

Different from those basic 'alphabets' mentioned in the previous section (nanogratings, nanoantennas, etc.), here, nanobricks are chosen as the building blocks. One reason is that they are widely used as the atoms of meta-material and plasmonic structure, and the second is that the brick's planar profile makes us easily calculate the equivalent impedance.

In addition, except for the above sub-wavelength grating and non-antenna, meta-surface represents another important type [18], the planar counterparts of meta-materials that provide the unprecedented control of the amplitude, phase or polarization of light waves at the subwavelength (nano) scale. For example, these two-dimensional surfaces can alter the wave-front of incident light for a widespread application in beam shaping [19–24], polarizers [25, 26] and flat lenses [27–29]. As proposed in [16], the fundamental building blocks are paired plasmonic or dielectric nanorods collectively working as an inductor–capacitor nanocircuit (Figure 7a, left), whose impedance depends directly on the filling ratio of plasmonic and dielectric materials. By suitably alternating these nanocircuit blocks on the transverse direction, one can synthesize a meta-surface with the required inhomogeneous impedance profile. The configuration composed of a stack of three meta-surfaces (Figure 7a, right), can fully control the nanoscale optical transmission, while simultaneously minimizing the reflection (impedancematching to free-space), allowing, for example light deflection with an almost ideal efficiency

3. Equivalent nanocircuit theory for multi-layer meta-material design

In this Section, the 'design stack' is moved upward, from the 'physical layer' of optical lumped elements (pure nanoparticles, nanogratings, nanoantennas, etc.) to more complex functional devices, including: (1) infrared third-order Butterworth filters; (2) metal-insulator-metal (MIM) ultra-broadband absorbers; (3) simplified broadband super-flat perfect infrared absorbers only composed of single transparent conductive oxides (TCOs). Among these design procedures for multi-layer nanostructures in our research group, the suitability of the equivalent nanocircuit theory is confirmed once more which in turn enriches and expands the application of equiva-

Frequency selective surfaces (FSSs) have been the subject of investigations by many researchers for decades. An FSS is a periodic structure usually composed of an assembly of identical elements arranged in one- or two-dimensional lattice. These structures are used in a variety of important applications ranging from microwave systems and antennas to radar and satellite communications. The simplest FSS device is a filter. By means of circuit elements (e.g. resistors (R), inductors (L) and capacitors (C)) FSSs can be effectively and flexibly designed to functionalize a low-pass, high-pass, band-pass or band-stop responses in the microwave or RF domain [30, 31]. The great interests in FSS's application in higher frequency range to achieve high-density and high-speed optical analogues [5, 10] have been pushed and the goals have been synthesized as realistic nanostructures at optical frequency, thanks to the optical nanocircuit theory which proves once more to be an essential design tool to construct optical

Different from those basic 'alphabets' mentioned in the previous section (nanogratings, nanoantennas, etc.), here, nanobricks are chosen as the building blocks. One reason is that they

as shown in Figure 7b.

228 Nanoplasmonics - Fundamentals and Applications

lent nanocircuit theory.

FSSs or filters.

3.1. Infrared broadband third-order Butterworth filters

As a demonstration, one layer of periodic nanosquare array based on indium-tin-oxide (ITO) used for an infrared FSS filer is presented (Figure 8). The reason why ITO material is chosen for making up of FSS filter cells is just that, it possesses low electrical resistance and high transmittance in the visible range and widely used as an electrode for displays [33]. Especially, in the infrared spectral range, ITO material can demonstrate a metallic performance and this makes it to become a counterpart of noble metal. In addition, in the practical calculations, the permittivity ITO is usually modelled by the Drude dispersion relation.

When such a nanosquare array is illuminated vertically (along z-direction) by an optical signal from the bottom side with electric field E polarized parallel to the x-direction, the optical displacement current Jd 'flows' along both the unit cells and air gaps [7, 32], the nanosquare array first acts as a 'parallel' combination of lumped elements R, L and C1, and then in series with a capacitance C2, as shown in Figure 8b. Following the general capacitor impedance formula Zc ¼ i=ðωCÞ with the capacitance C ¼ εa=b, and a, b being the element's two-dimensions, the effective lumped impedance of the unit nanosquare cells can be written as,

Figure 8. Schematic diagrams of (a) periodic nanosquare array (as for its unit cell, the width and height, as well as the gap distance between two adjacent unit cells are denoted as w, h and g), (b) equivalent lumped circuit elements and (c) whole equivalent circuit used for designing single-layer FSS filter (adapted from Ref. [32], Figure 1).

$$\begin{aligned} \mathbf{Z\_{ITO}} &= \mathbf{Z\_{R}} \| \mathbf{Z\_{L}} = \mathrm{iw/a}\hbar \mathrm{w} \varepsilon\_{\mathrm{ITO}},\\ \mathbf{Z\_{C\_{1}}} &= \mathrm{iw/a}\hbar \mathrm{g} \varepsilon\_{\mathrm{air}}, \quad \mathbf{Z\_{C\_{2}}} = \mathrm{ig/a}\hbar (\mathbf{w} + \mathbf{g}) \varepsilon\_{\mathrm{air}} \end{aligned} \tag{12}$$

Thus, the input impedance of a single layer is given by,

$$\mathbf{Z} = \mathbf{Z\_{ITO}} \| \mathbf{Z\_{C\_1}} + \mathbf{Z\_{C\_2}}.\tag{13}$$

Subsequently, the equivalent circuit model of this whole FSS system (Figure 8c), where η ¼ ffiffiffiffiffiffiffiffi μ=ε p is the intrinsic impedance of surrounding medium. Within the equivalent circuit impedance theory [34], the reflectance and transmittance of the incident optical signal are naturally obtained as,

$$\begin{array}{l} \mathbb{S}\_{11} = (\mathbb{Z} \| \boldsymbol{\eta} - \boldsymbol{\eta}) / (\mathbb{Z} \| \boldsymbol{\eta} + \boldsymbol{\eta}), \quad \mathbb{S}\_{21} = \mathbb{S}\_{11} + 1, \\\mathbb{R} = \left| \mathbb{S}\_{11} \right|^2, \quad T = \left| \mathbb{S}\_{21} \right|^2 = \left| \mathbb{Z} \mathbb{Z} / (\mathbb{Z} \mathbb{Z} + \boldsymbol{\eta}) \right|^2. \end{array} \tag{14}$$

Obviously, in order to achieve different optical responses, the impedances of nanocircuit elements can be changed, by tuning the structural size (w, h or g), the constituent material property (εITO), or the illumination light wavelength (λ ¼ 2πc=ω) and then the property of such a band-stop filter could be flexibly controlled. As a demonstration, samples from A to D with different widths of 100, 200, and 300 nm, but with fixed gap of 100 nm and height of 150 nm for simplicity are chosen.

The transmittances of the FSS filters composed of samples from A to D, shown in Figure 9a, are obtained through Eq. (14). In addition, one can employ a full-wave FDTD simulation to check the validity of such an equivalent nanocircuit theoretical model, and the corresponding transmittance spectra are shown in Figure 9b for comparison. The comparison result indicates that they are consistent well with each other.

As investigated in Figure 9c, the band-stop behaviour of the FSS filters is dependent on their geometric size. With the increment of w/g, the band-stop centre has a red-shift which can be well interpreted by the optical circuit theory [15]: the larger the width of the nanosquare unit cell, the bigger the inductor L induced between the two adjacent air gaps, hence leading to a higher resonant wavelength. In contrast, with the increment of w/g, the band-stop depth decreases and this behaviour is a result of the fact that the larger the width of nanosquare unit cell the stronger the guidance of light. From the electric field distribution for one representative sample at the resonance frequency, one can see that the unit cell structure works as an antenna and the incident light is localized at the regions between two adjacent metal unit cells, resulting in significant resonant enhancement of localized field and a guidance of most light through the air gap [35, 36].

As for a more practical and wide application, a flatter and broader band-stop filtering response curve with a fast roll-off would be much advantageous [37]. To gain this aim, as the general FSS-based filter design scheme does [38, 39], a third-order Butterworth band-stop filter is realized by cascading triple-layer of nanosquare unit cells with a specific separation distance D between the consecutive layers (Figure 10a). The corresponding equivalent circuit is modelled in Figure 10b. Then, the whole structure is separated into four regions along the

Figure 9. (a) Theoretical and (b) numerical transmittance spectra based on equivalent circuit theory and FDTD simulation for samples from A to D, respectively. (c) The resonance wavelength and transmittance dip as a function of w/g. (d) The representative electric field distribution for sample B at resonance wavelength (adapted from Ref. [32], Figure 2).

propagation direction (Figure 10c). Finally, along the similar procedure, the transmittance T after the third layer can be obtained step by step.

(1) As for the first layer, the theoretical S-parameters are

ZITO ¼ ZRkZL ¼ iw=ωhwεITO,

Thus, the input impedance of a single layer is given by,

230 Nanoplasmonics - Fundamentals and Applications

R ¼ jS11j

2

, T ¼ jS21j

η ¼ ffiffiffiffiffiffiffiffi

naturally obtained as,

nm for simplicity are chosen.

through the air gap [35, 36].

they are consistent well with each other.

ZC1 ¼ iw=ωhgεair, ZC2 ¼ ig=ωhðw þ gÞεair

Subsequently, the equivalent circuit model of this whole FSS system (Figure 8c), where

S11 ¼ ðZkη � ηÞ=ðZkη þ ηÞ, S21 ¼ S11 þ 1,

Obviously, in order to achieve different optical responses, the impedances of nanocircuit elements can be changed, by tuning the structural size (w, h or g), the constituent material property (εITO), or the illumination light wavelength (λ ¼ 2πc=ω) and then the property of such a band-stop filter could be flexibly controlled. As a demonstration, samples from A to D with different widths of 100, 200, and 300 nm, but with fixed gap of 100 nm and height of 150

The transmittances of the FSS filters composed of samples from A to D, shown in Figure 9a, are obtained through Eq. (14). In addition, one can employ a full-wave FDTD simulation to check the validity of such an equivalent nanocircuit theoretical model, and the corresponding transmittance spectra are shown in Figure 9b for comparison. The comparison result indicates that

As investigated in Figure 9c, the band-stop behaviour of the FSS filters is dependent on their geometric size. With the increment of w/g, the band-stop centre has a red-shift which can be well interpreted by the optical circuit theory [15]: the larger the width of the nanosquare unit cell, the bigger the inductor L induced between the two adjacent air gaps, hence leading to a higher resonant wavelength. In contrast, with the increment of w/g, the band-stop depth decreases and this behaviour is a result of the fact that the larger the width of nanosquare unit cell the stronger the guidance of light. From the electric field distribution for one representative sample at the resonance frequency, one can see that the unit cell structure works as an antenna and the incident light is localized at the regions between two adjacent metal unit cells, resulting in significant resonant enhancement of localized field and a guidance of most light

As for a more practical and wide application, a flatter and broader band-stop filtering response curve with a fast roll-off would be much advantageous [37]. To gain this aim, as the general FSS-based filter design scheme does [38, 39], a third-order Butterworth band-stop filter is realized by cascading triple-layer of nanosquare unit cells with a specific separation distance D between the consecutive layers (Figure 10a). The corresponding equivalent circuit is modelled in Figure 10b. Then, the whole structure is separated into four regions along the

2 ¼ j2Z=ð2Z þ ηÞj2

μ=ε p is the intrinsic impedance of surrounding medium. Within the equivalent circuit impedance theory [34], the reflectance and transmittance of the incident optical signal are

Z ¼ ZITOkZC1 þ ZC2 : (13)

: (14)

(12)

$$\mathbf{S}\_{11} = \frac{\mathbf{Z}\_1 \| \boldsymbol{\eta} - \boldsymbol{\eta}}{\mathbf{Z}\_1 \| \boldsymbol{\eta} + \boldsymbol{\eta}} \quad , \quad \mathbf{S}\_{21} = \mathbf{S}\_{11} + 1, \; \mathbf{S}\_{22} = \mathbf{S}\_{11}, \; \mathbf{S}\_{12} = \mathbf{S}\_{21}.\tag{15}$$

Here, the impedance of the surrounding medium above or between the neighbouring layers is η and that of each layer is Zi (i = 1, 2, 3). As for the second or third layer, the S-parameters (S33, S43, …) can be get by just replacing Z1 in Eq. (15) with Z2 or Z3.

(2) The reflected power PiR and transmitted power PiF from each layer are evaluated as follows,

$$\begin{aligned} P\_{1R} &= \left| \mathbf{S}\_{11} \right|^2 P\_{1F} + \left| \mathbf{S}\_{12} \right|^2 P\_{2R}, & P\_{2F} &= \left| \mathbf{S}\_{21} \right|^2 P\_{1F} + \left| \mathbf{S}\_{22} \right|^2 P\_{2R}, \\ P\_{2R} &= \left| \mathbf{S}\_{33} \right|^2 P\_{2F} + \left| \mathbf{S}\_{34} \right|^2 P\_{3R}, & P\_{3F} &= \left| \mathbf{S}\_{43} \right|^2 P\_{2F} + \left| \mathbf{S}\_{44} \right|^2 P\_{3R}, \\ P\_{3R} &= \left| \mathbf{S}\_{55} \right|^2 P\_{3F}, & P\_{4F} &= \left| \mathbf{S}\_{65} \right|^2 P\_{3F}. \end{aligned} \tag{16}$$

For simplicity, the absorption loss caused by the surrounding medium is assumed negligible.

Figure 10. Schematic diagrams of (a) triple-layer FSS filter, (b) equivalent circuit and (c) S-parameters (adapted from Ref. [32], Figure 3).

(3) The total transmittance after the third layer is then written as

$$T = P\_{45}/P\_{1F} \ = \frac{|\mathcal{S}\_{21}|^2 |\mathcal{S}\_{43}|^2 |\mathcal{S}\_{65}|^2}{1 - |\mathcal{S}\_{11}|^2 |\mathcal{S}\_{33}|^2 - |\mathcal{S}\_{33}|^2 |\mathcal{S}\_{55}|^2 + |\mathcal{S}\_{11}|^2 |\mathcal{S}\_{33}|^4 |\mathcal{S}\_{55}|^2 - |\mathcal{S}\_{11}|^2 |\mathcal{S}\_{43}|^2 |\mathcal{S}\_{55}|^2} . \tag{17}$$

As for any specific third-order Butterworth filter with desired central band-stop frequency and band-stop width, one can easily get the suitable choice of geometrical parameters for design. For example, if the band-stop edge frequencies are ω1 and ω2, the central frequency can be defined as ω0 ¼ ffiffiffiffiffiffiffiffiffiffiffi ω1ω2 p and band-stop width as ωb ¼ ðω2 � ω1Þ=ω0. The reactance slope parameters of this circuit model in terms of low-pass prototype parameters G0, G1, …, Gnþ1 and cut-off frequency ωc are given as [34]

$$\frac{Z\_i}{Z\_0} = \frac{G\_0}{\omega\_c G\_l \omega\_b} \text{ if } n = \text{even}; \quad \frac{Z\_i}{Z\_0} = \frac{1}{\omega\_c G\_0 G\_l \omega\_b}, \text{ if } n = \text{odd}. \tag{18}$$

Here, Gi is the normalized prototype element values and n is the order of the Butterworth filter, that is the number of resonators. As for a third-order (n = odd) band-stop filter (G1 = 1, G2 = 2, G3 = 1) with the height of nanosquare cells at each layer h1 ¼ 1=2h2 ¼ h3 fixed, the impedance of each layer (Figure 10b) can be obtained as Z1 ¼ 2Z2 ¼ Z3 ¼ η=ðωcωbÞ.

As for this triple-layer third-order filter, if adopting parameters listed in Table 1, the theoretical and numerical transmittance spectra can be directly obtained (Figure 11a) based on Eq. (17) from an equivalent circuit theory and FDTD simulation, respectively. The comparison between

Table 1. Element values for low-pass prototype circuit and geometrical parameters of third-order band-stop FSS filter.

Figure 11. (a) Theoretical and numerical transmittance spectra for single-layer (first-order) or triple-layer (third-order) filter. (b) One representative electrical field distribution for a third-order filter at a wavelength of 2000 nm (adapted from Ref. [32], Figure 4).

single-layer and triple layer filters indicates that, for a triple-layer filter, its band-stop width and depth both become larger, and moreover, its band-stop bottom is much flatter. In addition, the transmittance is nearly zero, indicated by the electric field distribution in Figure 11b.

It is necessary to point out that the corresponding FDTD calculation results show a good agreement with for single-layer case, but more obvious deviation from the equivalent circuit theoretical calculation. The reason for deviation arises from the simple assumption that the triple-layer of nanosquare array is independent of each other. If the layer-layer coupling is considered the deviation maybe reduced. However, anyway, the proposed synthesis procedure is confirmed helpful to design a Butterworth band-stop filter.

#### 3.2. Infrared broadband multi-layer MIM absorbers

(3) The total transmittance after the third layer is then written as

2 � jS33j 2 jS55j

if n ¼ even;

of each layer (Figure 10b) can be obtained as Z1 ¼ 2Z2 ¼ Z3 ¼ η=ðωcωbÞ.

2 jS43j 2 jS65j 2

As for any specific third-order Butterworth filter with desired central band-stop frequency and band-stop width, one can easily get the suitable choice of geometrical parameters for design. For example, if the band-stop edge frequencies are ω1 and ω2, the central frequency can be

Figure 10. Schematic diagrams of (a) triple-layer FSS filter, (b) equivalent circuit and (c) S-parameters (adapted from Ref.

parameters of this circuit model in terms of low-pass prototype parameters G0, G1, …, Gnþ1

Here, Gi is the normalized prototype element values and n is the order of the Butterworth filter, that is the number of resonators. As for a third-order (n = odd) band-stop filter (G1 = 1, G2 = 2, G3 = 1) with the height of nanosquare cells at each layer h1 ¼ 1=2h2 ¼ h3 fixed, the impedance

As for this triple-layer third-order filter, if adopting parameters listed in Table 1, the theoretical and numerical transmittance spectra can be directly obtained (Figure 11a) based on Eq. (17) from an equivalent circuit theory and FDTD simulation, respectively. The comparison between

Zi Z0

2 þ jS11j 2 jS33j 4 jS55j

p and band-stop width as ωb ¼ ðω2 � ω1Þ=ω0. The reactance slope

¼ 1 ωcG0Giωb 2 � jS11j 2 jS43j 2 jS55j

, if n ¼ odd: (18)

2 : (17)

T ¼ P4F=P1F ¼ jS21j

1 � jS11j 2 jS33j

232 Nanoplasmonics - Fundamentals and Applications

ω1ω2

¼ G0 ωcGiωb

and cut-off frequency ωc are given as [34]

Zi Z0

defined as ω0 ¼ ffiffiffiffiffiffiffiffiffiffiffi

[32], Figure 3).

Meta-material absorbers are used broadly in thermal detectors [40], imaging [41], security detection [42] and stealth devices [43]. In 2008, Landy et al. first proposed a thin perfect metamaterial absorber simultaneously exciting electric and magnetic resonances (MRs) to realize the impedance match with the surrounding medium and thus eliminating any reflection and perfectly absorbing the incident waves at microwave bands [44]. Since then, applications to various wavelength regimes have been demonstrated widely by numerical simulations and experiments [45, 46]. However, the application of these perfect absorbers is limited for their narrowband and simple resonant behaviours.

The most widely used scheme [47–49] instead is to slow down the incident wave or totally absorb them by a gradually changed pyramid-shaped metal-insulator-metal (MIM) topology (Figure 12a). In fact, an absorber can also be thought as a filter, by operating the frequency response of the absorptance with A = 1 – T – R. Furthermore, each layer of the multi-layer MIM absorber projecting to the bottom plane is a nanosquare structure (Figure 12b), same as basic building blocks for third-order Butterworth filter (Section 3.1). Thus, naturally, the equivalent nanocircuit procedure can be directly transplanted to the design of MIM multi-layer absorber.

The first step is to evaluate the impedance Z (Figure 12c) of each layer, following the same procedure in Section 3.1 which can be obtained as,

$$\begin{array}{ll} \mathbf{Z\_{ITO}} = \mathbf{Z\_{R}} \| \mathbf{Z\_{L}} = \text{iw/a\text{\textdegreeline\varepsilon\_{ITO}}}, & \mathbf{Z\_{C\_{1}}} = \text{iw/a\text{\textdegreeline\varepsilon\_{air}}},\\ \mathbf{Z\_{C\_{2}}} = \text{ig/}\omega\text{h}(\mathbf{w} + \mathbf{g})\varepsilon\_{\text{air}}, & \mathbf{Z} = \text{Z\_{ITO}} \| \mathbf{Z\_{C\_{1}}} + \mathbf{Z\_{C\_{2}}}. \end{array} \tag{19}$$

The second step is to cascade the impedance of each layer into a whole circuit, for example that of a triple-layer structure. Each layer is separated by one quarter of central wavelength of the incident light (Figure 12a). The corresponding equivalent circuit can be modelled as in Figure 12b, where η ¼ ffiffiffiffiffiffiffiffi μ=ε p is the intrinsic impedance of SiO2 dielectric layers, and Zi (i = 1, 2, 3) is the impedance of each NS layer. Within transmission line theory [34], Z(3) is the effective impedance

Figure 12. Schematic diagrams of (a) MIM absorber, (b) equivalent lumped circuit elements and (c) whole equivalent circuit. Here, only single nanosquare patch layer is considered (adapted from Ref. [50], Figure 1).

at the top interface of the third NS layer, given by Zð3Þ ¼ Z3kη: Similarly, for the interface at the second or first layer, the effective impedances can be written as,

$$Z(2) = Z\_2 \| \left( \eta^2 / Z(3) \right), \quad Z(1) = Z\_1 \| \left( \eta^2 / Z(2) \right) \tag{20}$$

Then, the reflectance of this pyramid triple layer structure is given by,

The most widely used scheme [47–49] instead is to slow down the incident wave or totally absorb them by a gradually changed pyramid-shaped metal-insulator-metal (MIM) topology (Figure 12a). In fact, an absorber can also be thought as a filter, by operating the frequency response of the absorptance with A = 1 – T – R. Furthermore, each layer of the multi-layer MIM absorber projecting to the bottom plane is a nanosquare structure (Figure 12b), same as basic building blocks for third-order Butterworth filter (Section 3.1). Thus, naturally, the equivalent nanocircuit procedure can be directly transplanted to the design of MIM multi-layer absorber. The first step is to evaluate the impedance Z (Figure 12c) of each layer, following the same

ZITO ¼ ZRkZL ¼ iw=ωhwεITO , ZC1 ¼ iw=ωhgεair,