Resonance frequencies
\r\n\t• Role of technological innovation and corporate risk management
\r\n\t• Challenges for corporate governance while launching corporate environmental management among emerging economies
\r\n\t• Demonstrating the relationship between environmental risk management and sustainable management
\r\n\t• Contemplating strategic corporate environmental responsibility under the influence of cultural barriers
\r\n\t• Risk management in different countries – the international management dimension
\r\n\t• Global Standardization vs local adaptation of corporate environmental risk management in multinational corporations.
\r\n\t• Is there a transnational approach to environmental risk management?
\r\n\t• Approaches towards Risk management strategies in the short-term and long-term.
Boron-polyol interactions are of fundamental importance to human health [1], plant growth [2] and quorum sensing among certain bacteria [3]. Such diversity is perhaps not surprising when one considers boron is one of the ten most abundant elements in sea water and carbohydrates make up the planet’s most abundant class of biomass. Several boronic acids matrices are commercially available for the purification of glycoproteins by affinity chromatography [4], and boronic acids are also useful carbohydrate protecting groups.[5,6] Recently, complexes between boron and sugars have become a lynchpin for the development of synthetic carbohydrate receptors.[7] These complexes involve covalent interactions that are reversible in aqueous solution. This chapter reviews current understanding of these processes, provides a historical perspective on their discovery, identifies methods for studying these complexes and classifies these interactions by carbohydrate type. Such information is key to the design and synthesis of synthetic lectins, also termed “boronolectins” when containing boron [7].
The very nature of the reversible binding between boron acids and alcohols has been exploited in many different ways. The use of boronic acid carbohydrate recognition molecules could provide an avenue for the selective detection of specific sugars for future use in early diagnostics. By targeting cell-surface sugars, a boron-based probe could recognize particular characteristic epitopes for the identification of diseases leading to earlier treatments. In this chapter we not only review some of the fundamental aspects of boron-carbohydrate interactions but also discuss how this translates into the design of synthetic carbohydrate receptors.
The first hint of the marriage between boron and polyols was detected by Biot in his seminal studies on optical rotation. In 1832 he noted that the rotation of tartaric acid changed in the presence of boric acid.[8] It would be a century later before interaction of boron acids (boric, boronic and borinic) and monosaccharides was studied in detail. In 1913, Böeseken first noted that glucose increased the acidity of boric acid solutions.[9] It was nearly another half century before Lorand and Edwards published work quantifying the affinity of boric and phenylboronic acids for simple diols (e.g.-ethylene glycol, catechol) and common monosaccharides (i.e.-glucose, fructose, mannose, galactose).[10] The covalent product between a boronic acid and a diol is termed a boronate ester, analogous to a carboxylate ester. These interactions are favoured at basic pH ranges where the tetrahedral boronate ester is formed (Figure 1). The interchange between boron acids and divalent ligands in aqueous solution can be complex and varied depending on pH.
Boric acid interactions with vicinal diol of sugar.
There are two general organoboron families of boric acid descent that can form esters with diols through loss of water. These are boronic acids--where one hydroxy group of the parent boric acid is substituted by carbon--and borinic acids, where two hydroxy groups are substituted by carbon-based substituents.
Boron acids and possible esters with ethylene glycol.
Boric and boronic acids can form either neutral or anionic esters depending on the pH. Diol binding by boron acids is favoured at basic pH, while esterification of boron by hydroxycarboxylic acids is favoured in acidic pH ranges. Borinic acids can only form anionic borinate esters upon dehydrative condensation with a diol or divalent ligand. Boric acid can also form an anionic, tetrahedral diester with diols and related divalent ligands (Figure 2). While boronates can form neutral esters in non-polar solvents, they tend to form anionic boronate esters in water (Figure 3). Boronate ester formation is not favoured near physiologic pH and is completely cleaved under strongly acidic conditions.
Diol exchange with phenylboronic acid at varied pH.
This is because the neutral boronate ester is generally more Lewis acidic than the parent boronic acid—i.e. pKa (acid) > pKa (ester), (Scheme 1).[11] Thus, boronate ester formation is favoured at higher pH where elevated hydroxide concentrations ensure the boronate ester is “trapped” in its more stable tetrahedral form. However, depending on the specific monosaccharide, its boronate esters are not always more Lewis acidic than the free boronic acid.[12] Rate constants for esterification of simple boronates by diols fall in the range of 102-103 M-1s-1.[13] Ishihara uncovered evidence it is the trigonal boronic acid that exchanges most rapidly with diols irregardless of pH.[14] The relative affinity of boronates for diols in most carbohydrates is of the order: cis-1,2-diol > 1,3-diol >> trans-1,2-diol. Thus, certain monosaccharides have an intrinsically higher affinity for boron acids.
Multiple equilibria involved in diol exchange with phenylboronic acid.
Methods for identifying boron-polyol interactions include the following, listed in roughly chronological order of the introduction of their use in this area:
The second half of the list defines techniques that are most frequently used today in the study of boron-carbohydrate interactions. Obviously, X-ray crystallography provides the least ambiguous information about the structure of the boronate ester of interest. However, these boronate-sugar adducts are often amphiphilic in nature and do not lend themselves to the production of suitable crystals. The relatively slow exchange between boron acids and diols on the NMR time scale often makes it difficult to study by proton NMR. However, 11B-NMR can be quite useful due to the dramatic shift of the boron resonance when it is converted from its neutral, trigonal form as a boronic acid to its anionic, tetrahedral form as a boronate ester. [20-22]
Optical methods such as fluorescence and circular dichroism (CD) are powerful tools for detecting boron-carbohydrate binding interactions. Yoon and Czarnik reported the first fluorescent boronate designed to detect binding to monosaccharides.[23] James, Shinkai and co-workers reported the first fluorescent boronates to function by photoinduced electron transfer (PET) to generate an increased fluorescence output upon carbohydrate binding.[24] This type of “turn-on” system tends to be most useful in a biological setting where background fluorescence quenching can be a problem for fluorophores that respond by fluorescence quenching (“turn-off”) to ligand binding. A more complete understanding of the aminoboronate PET fluorescence mechanism has been developed by the groups of Wang [29] and Anslyn [30]. They have demonstrated that solvent insertion disrupting any dative boron-nitrogen interaction is responsible for the increased fluorescence output upon ligand binding. The Shinkai group has also designed a number of CD-active boronate receptors for oligosaccharides and have used this method to detect binding of target substrates.[25]
Advances in mass spectrometry (MS) over the past few decades, particularly electrospray ionization (ESI) and matrix-assisted laser desorption/ionization (MALDI), have revolutionized the application of this instrumental method to the study of host-guest and protein-ligand interactions. Certainly, the field of boron-based carbohydrate receptors has also benefited from the substantial improvement and refinement of these and other MS techniques. However, the tendency of boronates to dehydrate and/or oligomerize to varying degrees depending on their solvation can complicate MS analysis of boronate-carbohydrate esters. We have found that use of a glycerol matrix for fast atom bombardment (FAB) ionization is particularly useful for mass spectrometric characterization of diboronate species.[31] While other techniques are used in the study of boron-polyol complexes, those mentioned here are among the most common routinely used in the field today.
The 1992 work of Yoon and Czarnik first demonstrated the potential of boronic acids as fluorescent carbohydrate receptors for sensing applications.[23] In the past 20 years, a great deal of research has focused on the development of boron-based glucose receptors for incorporation as sensors in blood sugar monitors for diabetics.[7, 32] This has lead to the commercial development of contact lenses that can signal when circulating glucose levels drop by changing the colour of the lense to alert the wearer.[33] The affinity of mono-boronates for glucose is low at physiologic pH, but bis-boronates offer a substantial improvement in binding affinity. A landmark study from the Shinkai group involved development of a chiral glucose sensor capable of discriminating between enantiomers of glucose.[34] This utilized aminoboronates as PET sensors around a chiral binaphthol core (Figure 5). The Singaram group has developed bis-boronate bipyridinium salts (viologens) that can be tuned for selective binding of glucose (Figure 6).[35] These compounds coupled with anionic dyes are also in commercial development as blood glucose sensors.
Shinkai’s chiral binaphthol glucose sensor.
One isomer of Singaram’s family of glucose sensors.
It was initially presumed that two boronotes could bind to the C-1/C-2 diol and the C-4/C-6 diol of glucose in its hexopyranoside form.[5] However, it has been shown that boronic acids have a much higher affinity for the furanoside form of free hexoses.[36] In fact, boronates have virtually no affinity for methyl glyocsides locked in their pyranoside form at physiologic pH. This means that boronates would not be useful components in synthetic carbohydrate receptors for many cell surface carbohydrates. Mammalian cell-surface glycoconjugates, in particular, are dominated by hexopyranoside structures. The Hall group has provided an important solution to this problem when they showed that benzoboroxoles can bind methyl hexopyranosides in water at pH 7.5.[37] For glucopyranosides, the only significant binding site is the C-4/C-6 diol as all vicinal diols in this system are of a trans relationship. In galactopyranosides, there is an additional possible binding site: the C-3/C-4 cis-diol (Figure 7):
Potential binding modes between benzoboroxole (blue) and methyl-galactoside.
While a significant amount of research has been dedicated to the study of boronate-monosaccharide interactions, very little has been invested in borinate-monosaccharide exchange. Taylor has recently reported that borinic acids have substantial affinity for catechols and α-hydroxycarboxylates,[38] greater than that of 2-fluoro-5-nitrophenyboronic acid,[39] a boronate that is able to bind sugars at neutral pH. The affinity of this boronate for monosaccharides is greater than the affinity of a borinic acid for the same sugars, but this affinity in the latter case is still significant. Whether borinates can effectively bind to hexopyranosides under the same conditions still needs to be defined.
Boron-tartaric acid interactions were studied throughout the 20th century beginning with a report in 1911 on the ability of tartrate to increase the solubility of boric acid.[40] The design of sophisticated boron based receptors for tartrate did not arise until near the end of the century when Anslyn reported the first in 1999.[41] This receptor (Figure 8) also binds citrate with what is perhaps the highest association constant reported for a small molecule with a boron-based receptor (Ka = 2x105).[42] In 2002, we showed that Shinkai’s binaphthol glucose receptor (Figure 5) has a high affinity for tartrate as well.[43] Bis-boronates such as this can bind simultaneously to both α-hydroxycarboxylates. James further showed that chiral discrimination between tartrate enantiomers can also be obtained with this receptor as was the case with monosaccharide enantiomers.[44] While the history of study surrounding boron-tartaric acid interactions is long and varied, study of the interaction of boron with sugar acid monosaccharides such as sialic acid and glucuronic acid has arisen much more recently. Fundamental to the understanding of these complexes is the fact that, unlike esterification with diols, boronate esterification by α-hydroxycarboxylic acids is favoured below pH 7.[44] We have recently provided a short review on the subject of boron:α-hydroxycarboxylate interactions used in sensing and catalysis.[45]
Shinkai first reported a boron-based sugar acid receptor containing a metal chelate that has significant affinity for glucuronic acid (log Ka = 3.4) and galacturonic acid (log Ka = 3.1) while the affinity for sialic acid was an order of magnitude lower (log Ka = 2.3).[46] Presumably the carboxylate of the sugar acid can coordinate to the chelated zinc while the boron binds to a vicinal diol on the monosaccharide. Smith and Taylor used a combination of electrostatic interaction and a boronate anchor within a polymeric system to bind to sialic acid selectively.[47] In 2004, Strongin identified a boronate that offered a colorimetric response to the presence of sialic acid.[48]
Anslyn’s guanidino-boronate receptor and high affinity ligands.
We have recently communicated the development of a bis-boronate that can bind to sialic acid at both its α-hydroxycarboxylate-type group at the anomeric centre and its glycerol tail (Figure 9).[49] Elevated levels of free sialic acid in the blood can be indicative of the presence of certain cancers. This system uses a unique combination of boronates whose esterification has an opposing affect on the overall fluorescence output of the receptor. This diminishes signals from competing ligands such as glucose that are present at much higher concentration in the blood but cannot span both binding site to strongly quench fluorescence.
As discussed in the following sections, several groups have taken advantage of the affinity of boronates for the glycerol tail of sialic acid to target glycoconjugates on cell surfaces. Several of these synthetic compounds display lectin-like biological characteristics that offer promise of the future development of bioactive boron-based molecules.
Our divergent response fluorescent receptor for sialic acid.
Creating receptors for oligosaccharides offers additional levels of complexity relative to monosaccharides. An obvious difference is the increased degrees of freedom available to oligosaccharides, particularly those that contain 1,6-linkages. In 2000, Shinkai reported development of a meso-meso-linked porphyrin scaffold where distance between two boronates was tuned to selectively bind to a tetrasaccharide of maltose (maltotetrose) over other oligomers containing from two to seven glucose units (Figure 10).[50] Binding of the two boronates must take place at both the reducing and non-reducing termini of the oligosaccharide. This is due to the fact that the C-1/C-2 diol at the reducing end and the C-4/C-6 at the non-reducing end are the only potential binding sites with an appreciable affinity for boronates. The ability to bind the tetramer selectively stems from the rigidly defined distance between the two boronates, i.e.-the tetrasaccharide offers the optimal fit to bridge these two boronates. The Shinkai group has also reported a similar strategy to bind to the important cell-surface trisaccharide, Lewis X.[51] In this case interaction is not with a reducing sugar but presumably with diols on both the galactose and fucose residues.
Heparin is a natural polysaccharide used clinically for its anti-coagulant properties. In 2002, Anslyn communicated a colorimetric sensing ensemble for detection of heparin.[52] As heparin has a high anionic charge density, the receptor was designed with a number of complementary cationic amino groups alongside boronic acids. This group has further reported success in sensing heparin within serum using a second generation receptor.[53] Schrader has recently developed a fluorescent polymeric heparin sensor that can quantify this polysaccharide with unprecedented sensitivity (30 nM).[54] Coupling of boron-carbohydrate interactions with electrostatic attraction in a multivalent manner is responsible for the high affinity of this receptor for its substrate. In spite of this avidity, the interaction can be controlled in a biologically relevant manner. Binding of the polymer to heparin can be reversed by the addition of protamine, similar to reversal of the complex between heparin and its natural target, anti-thrombin III. Other examples of biological mimicry by boron-based systems are delineated below and in the next section.
Shinkai’s oligosaccharide receptor and maltotetrose.
The demonstrated affinity of benzoboroxoles for hexopyranosides makes these boron derivatives attractive components of receptors designed to target mammalian oligosaccharides. In 2010, Hall reported development of a bis-benzoboroxole receptor for the Thomsen-Friedenreich (TF) antigen, a tumor marker composed of consecutive galactose-based residues Figure 11.[55] This receptor was optimized within a combinatorial library constructed to add additional H-bonding and hydrophobic interactions between host and its oligosaccharide guest. A natural lectin receptor for this disaccharide, peanut agglutinin lectin (PNA), binds the TF antigen quite strongly relative to other protein-carbohydrate interactions (Kd = 107). However, the synthetic bis-benzoboroxole inhibits binding of PNA to TF-antigen labelled protein at low micromolar concentrations. The bis-benzoboroxole has a higher affinity for the disaccharide than the corresponding bis-boronate although the latter still has significant affinity highlighting the importance of the additional H-bonding and hydrophobic interactions in this system. This work indicates it should be possible to target multiple hexopyranoside structures with other oligomeric benzoboroxole systems.
Hall’s bis-benzoboroxole receptor for the TF antigen.
One ultimate goal of research into synthetic carbohydrate receptors is the development of compounds that can bind directly to cell surface glycocojugates. Such synthetic lectins may serve as diagnostics to monitor changes in cell surface structure associated with disease progression such as cancer. Additionally, they may be used as drug-targeting agents to deliver chemotherapeutic agents to specific cell types. An early and initially underappreciated demonstration of the targeting of cell-surface structures was the work of Hageman with fluorescent dansyl boronates shown to associate with Bacillus subtilis.[56] They were also able to show a diboronate could display other lectin like properties such as promoting the agglutination of erythrocytes. Not long after that, Gallop developed a method he defined as “boradaption” using boronates to transfer lipophilic dyes and probes into cells.[57] Although the precise mechanism was not delineated, some boronates were shown to alter the latter stages of N-linked glycoprotein processing.[58] A great deal of work has also gone into the development of lipophilic boronic acids as membrane transport agents for hydrophilic molecules such as sialic acid and its derivatives.[59] There is commercial interest in such artificial transporters for the extraction of monosaccharides, such as glucose and fructose, and disaccharides like lactose from natural sources.[28]
In 2002, Weston and Wang reported the ability to target a specific oligosaccharide epitope of a cell surface glycoconjugate.[60, 61] Use of a fluorescent bis-boronate to label the cancer-related antigen sialyl Lewis X on hepatocellular carcinoma cells was an important achievement. They used a combinatorial approach to optimize a bis-boronate in targeting the sialic acid and fucose residues on the tetrasaccharide. The receptor did not label cells that contained Lewis Y antigens lacking sialic acids, or were treated with fucosidase, to remove fucose from the cell surface. This indicates that both components are necessary for interaction with the synthetic receptor. The study marked the first time, as far as we are aware, that two different monosaccharide types had been targeted by design with a boronolectin on a cell surface (Figure 12).
Wang’s sialyl Lewis X receptor. Reprinted with permission. John Wiley & Sons, ©2010.
The Kataoka group has used boronic acids on a number of platforms to target cell surface sialic acids to engender a biological or analytical response. They have shown that polymeric boronates can cause the induction of lymphocytes in the same way as natural lectins do.[62] In addition these polyboronates can out-compete natural sialic acid-specific lectins for a cell surface. In collaboration with Miyahara, a powerful method for the direct determination of cell-surface sialic acid levels has been developed.[63, 64] Use of a self-assembled monolayer on a gold electrode allows a coating of boronates to be applied. Potentiometric measurements in the presence of cell suspensions containing either 0, 15, 30, or 100% metastatic cells are readily distinguishable (Figure 13).[64]
The study, application and manipulation of boron-carbohydrate interactions continues to expand into its third century. The properties of oligomeric and polymeric boronic acids in a cellular setting demonstrate that the terms “boronolectin” and/or synthetic lectin are appropriate. What remains for the field to advance are more examples targeting cell-surface carbohydrate structures beyond those containing sialic acid. The Hall group’s receptor for the TF antigen marks a seminal step in this direction.[54] For a more comprehensive review of boron-based carbohydrate receptors in the context of other synthetic and biologic sugar binding systems, readers are directed to the recent publication of Wang.[7]
a) Schematic representation of potentiometric SA detection with a PBA-modified gold electrode. An SEM image of a cross-section of the electrode is shown at the top next to the chemical structure of the PBA-modified self-assembled monolayer introduced onto the electrode surface. b) Change in the threshold voltage (VT) of the PBA-modified FET as a function of time upon the addition of cell suspensions (106 cells/mL) with various degrees of metastasis. [64] Reprinted with permission. John Wiley & Sons, ©2010.
In spite of its long and rich history, understanding of boron acid interactions with carbohydrates continues to increase into the 21st century. In the past 20 years, much fundamental knowledge has been gained, principally from the development of boronate-based glucose receptors for application toward blood sugar monitoring in diabetics. Currently, however, significant effort is being dedicated to the development of boron-based receptors for more complex oligosaccharides. This challenge is being undertaken by an increasing number of research groups throughout the world. These designer receptors may find application in diagnostics for cancer or infectious diseases, in drug targeting, or in providing a more fundamental understanding of the biochemical roles of cell-surface carbohydrates. The ability of boron acids to distinguish between closely related polyols either stereoselectively or chemoselectively makes them an obvious choice for anchoring synthetic carbohydrate receptors.[65] Engineering these interactions to target specific oligosaccharides is currently a difficult challenge as witnessed by the limited number of boron-based oligosaccharide receptors that have been developed at this stage. However, coupling boron-carbohydrate interactions with several additional non-covalent interactions—electrostatic, H-bonding, hydrophobic—offers the best chance of success. Future endeavours will determine the scope and limitations of boron-based carbohydrate receptors and sensors.
Cylindrically layered structures have various exotic applications. For instance, a metal-core dielectric-shell nano-wire has been proposed for the cloaking applications in the visible spectrum. The functionality of this structure is based on the induction of antiparallel currents in the core and shell regions, and the design procedure is the so-called scattering cancelation technique [1]. Experimental realization of a hybrid gold/silicon nanowire photodetector proves the practicality of these structures [2]. As an alternative approach for achieving an invisible cloak, cylindrically wrapped impedance surfaces are designed by a periodic arrangement of metallic patches, and the approach is denominated as mantle cloaking [3]. Conversely, cylindrically layered structures can be designed in a way that they exhibit a scattering cross-section far exceeding the single-channel limit. This phenomenon is known as super-scattering and has various applications in sensing, energy harvesting, bio-imaging, communication, and optical devices [4, 5]. Moreover, a cylindrical stack of alternating metals and dielectrics behaves as an anisotropic cavity and exhibits a dramatic drop of the scattering cross-section in the transition from hyperbolic to elliptic dispersion regimes [6, 7]. The Mie-Lorenz theory is a powerful, an exact, and a simple approach for designing and analyzing the aforementioned structures.
Multilayered spherical structures have also attracted lots of interests in the field of optical devices. A dielectric sphere made of a high index material supports electric and magnetic dipole resonances which results in peaks in the extinction cross-section [8]. Moreover, by covering the dielectric sphere with a plasmonic metal shell, an invisible cloak is realizable, which is useful for sensors and optical memories [9]. By stacking multiple metal-dielectric shells, an anisotropic medium for scattering shaping can be achieved [10].
From the above discussions, it can be deduced that tailoring the Mie-Lorenz resonances in the curved particles results in developing novel optical devices. In this chapter, we are going to extend the realization of various optical applications based on the excitations of localized surface plasmons (LSP) in graphene-wrapped cylindrical and spherical particles. To this end, initially we introduce a brief discussion of modeling graphene material based on corresponding surface conductivity or dielectric model. Later, we extract the modified Mie-Lorenz coefficients for some curved structures with graphene interfaces. The importance of developed formulas has been proven by providing various design examples. It is worth noting that graphene-wrapped particles with a different number of layers have been proposed previously as refractive index sensors, waveguides, super-scatterers, invisible cloaks, and absorbers [11, 12, 13, 14, 15]. Our formulation provides a unified approach for the plane wave and eigenmode analysis of graphene-based optical devices.
Graphene is a 2D carbon material in a honeycomb lattice that exhibits extraordinary electrical and mechanical properties. In order to solve Maxwell’s equations in the presence of graphene, two approaches are applied by various authors, and we will review them in the following paragraphs. It should be noted that although we are discussing the graphene planar model, we will use the same formulas for the curved geometries when the number of carbon atoms exceeds 104, letting us neglect the effect of defects. Therefore, the radii of all cylinders and spheres are considered to be greater than 5 nm [16]. Moreover, bending the graphene does not have any considerable impact on the properties of its surface plasmons, except for a small downshift of the frequency. Figure 1 shows the propagation of the graphene surface plasmons on the S-shaped and G-shaped curves [17].
Propagation of graphene surface plasmons on curved structures: (a) S-shaped and (b) G-shaped [17].
Since graphene material is atomically thin, in order to consider its impact on the electromagnetic response of a given structure, boundary conditions at the interface can be simply altered. To this end, graphene surface currents that are proportional to its surface conductivity should be accounted for ensuring the discontinuity of tangential magnetic fields. In the infrared range and below, we can describe the graphene layer with a complex-valued surface conductivity
The parameters
where subscripts
Figure 2(a) and (b) shows the real and imaginary parts of graphene surface conductivity at the temperature of T = 300°K. The real part of the conductivity accounts for the losses, while the positive valued imaginary parts represent the plasmonic properties [20]. Moreover, the real and imaginary parts of the graphene equivalent bulk permittivity are shown in Figure 2(c) and (d). The negative valued real relative permittivity represents the plasmonic excitation, and the imaginary part of the permittivity represents the losses [21]. It should be noted that all of the formulas of this chapter are adapted with
(a) and (b) the real and imaginary parts of graphene surface conductivity [20] and (c) and (d) the real and imaginary parts of graphene equivalent permittivity [21].
In this section, the modified Mie-Lorenz coefficients of a single-layered graphene-coated cylindrical tube will be extracted. The formulation is expanded into the multilayered graphene-based tubes through exploiting the TMM method, and later, various applications of the analyzed structures, including emission and radiation properties, complex frequencies, super-scattering, and super-cloaking, will be explained.
Let us consider a graphene-wrapped infinitely long cylindrical tube. The structure is shown in Figure 3(a), and it is considered that a TEz-polarized plane wave illuminates the cylinder. In general, TE and TM waves are coupled in the cylindrical geometries. For the normally incident plane waves, they become decoupled, and they can be treated separately. For simplicity, we consider the normal incidence of plane waves where the wave vector
(a) A single-layered graphene-coated cylinder under TEz plane wave illumination and (b) corresponding scattering efficiency for ε1 = 3.9 and μc = 0.5 eV. The normalization factor in this figure is the diameter of the cylinder [23].
In order to obtain the modified Mie-Lorenz coefficients, the incident, scattered, and internal electromagnetic fields are expanded in terms of cylindrical coordinates special functions which are, respectively, the Bessel functions and exponentials in the radial and azimuthal directions. In order to exploit a terse mathematical notation, the vector wave functions are introduced as [22]:
The complete explanation of the above vector wave functions and their self and mutual orthogonally relations can be found in the classic electromagnetic books [22]. In the above equation,
In the graphene-based cylindrical structures, the plasmonic state is achieved via illuminating a TEz wave to the structure. Therefore, for the normal illumination, the incident, scattered, and dielectric electromagnetic fields are shown with the superscripts
where
The boundary conditions at the graphene interface at
By applying the boundary conditions in the expanded fields, the linear system of equations for extracting the unknowns can be readily obtained. The solution of the extracted equations for the scattering coefficients leads to:
The same procedure can be repeated for the TMz illumination. The normalized scattering cross-section (NSCS) reads as:
where the normalization factor is the single-channel scattering limit of the cylindrical structures. In order to have some insight into the scattering performance of graphene-wrapped wires, the scattering efficiency for ε1 = 3.9 and μc = 0.5 eV is plotted in Figure 3(b) by varying the radius of the wire. As the figure illustrates, a peak valley line shape occurs in each wavelength. They correspond to invisibility and scattering states and will be further manipulated in the next sections to develop some novel devices. The excitation frequency of the plasmons is the complex poles of the extracted coefficients [24] which will be discussed in the next subsection. Interestingly, the scattering states of graphene-coated dielectric cores are polarization-dependent. By using a left-handed metamaterial as a core, this limitation can be obviated [25].
As in any resonant problem, additional information can be obtained by studying the solutions to the boundary value problem in the absence of external sources (eigenmode approach). Although, from a formal point of view, this approach has many similar aspects with those developed in previous sections, the eigenmode problem presents an additional difficulty related to the analytic continuation in the complex plane of certain physical quantities. Due to the fact that the electromagnetic energy is thus leaving the LSP (either by ohmic losses or by radiation towards environment medium), the LSP should be described by a complex frequency where the imaginary part takes into account the finite lifetime of such LSP. The eigenmode approach is not new in physics, but its appearance is associated to any resonance process (at an elementary level could be an RLC circuit), where the complex frequency is a pole of the analytical continuation to the complex plane of the response function of the system (e.g., the current on the circuit). Similarly, in the eigenmode approach presented here, the complex frequencies correspond to poles of the analytical continuation of the multipole terms (Mie-Lorenz coefficients) in the electromagnetic field expansion.
In order to derive complex frequencies of LSP modes in terms of the geometrical and constitutive parameters of the structure, we use an accurate electrodynamic formalism which closely follows the usual separation of variable approach developed in Section 2.1. We can obtain a set of two homogeneous equations for the
where the prime denotes the first derivative with respect to the argument of the function and
where
When the size of the cylinder is small compared to the eigenmode wavelength, i.e.,
Taking into account that in the non-retarded regime the propagation constant of the plasmon propagating along perfectly flat graphene sheet can be approximated by:
it follows that the dispersion relation (14) for LSPs in dielectric cylinders wrapped with a graphene sheet can be written as:
where
For large doping (
which can be analytically solved for the plasmon eigenfrequencies,
where
In the following example, we consider a graphene-coated wire with a core radius
1 | ||
2 | ||
3 | ||
4 |
Resonance frequencies
In this section, multilayered cylindrical tubes with multiple graphene interfaces are of interest. In order to ease the derivation of the unknown expansion coefficients, matrix-based TMM formulation is generalized to the tubes with several graphene interfaces. Initially, consider a layered cylinder constructed by the staked ordinary materials under TEz plane wave illumination, as shown in Figure 4. The total magnetic field at the environment can be expressed as the superposition of incident and scattered waves as in Section 2.1. The unknown expansion coefficients of the scattered wave can be determined by means of the
Multilayered cylindrical structure consisting of alternating graphene-dielectric stacks under plane wave illumination. The 2D graphene shells are represented volumetrically for the sake of illustration [31].
where C represents the core layer. In the above equation, the dynamical matrix
The argument of the above special functions is
In order to incorporate the graphene surface conductivity in the above formulas, let us consider each graphene interface as a thin dielectric with the equivalent complex permittivity defined in Eq. (3) and utilize the TMM formulation in the limiting case of a small radius at the graphene interface with the wave number of kg, i.e.,
where the free-space impedance
Widely tunable scattering cancelation is feasible by using patterned graphene-based patch meta-surface around the dielectric cylinder as shown in Figure 5. The surface impedance of the graphene patches can be simply and accurately calculated by closed-form formulas, to be inserted in the modified Mie-Lorenz theory [32].
(a) Electromagnetic cloaking of a dielectric cylinder using graphene meta-surface and (b) corresponding electric field distribution [32].
Let us consider a triple shell graphene-based nanotube under plane wave illumination, as shown in Figure 6(a). This structure is used to design a dual-band super-scatterer in the infrared frequencies. To this end, modified Mie-Lorenz coefficients of various scattering channels should have coincided with the proper choice of geometrical and optical parameters. In order to construct the Tn matrix for this geometry, one needs to multiply nine 2 × 2 dynamical matrices, which is mathematically complex for analytical scattering manipulation. Therefore, the associated planar structure, shown in Figure 6(b), is used to develop the dispersion engineering method as a quantitative design procedure of the super-scatter. The separations of the free-standing graphene layers are d1 = d2 = 45 nm in the planar structure, and the transmission line model is used to analyze it. Moreover, the chemical potential of lossless graphene material is μc = 0.2 eV in all layers. The dispersion diagram of the planar structure is illustrated in Figure 7(a), which predicts the presence of three plasmonic resonances in each scattering channel of the tube at around the frequencies that fulfill
(a) Multilayered cylindrical nanotube with three graphene shells and (b) associated planar structure [30]. R1 is denoted with Rc in the text.
(a) Dipole and quadruple Mie-Lorenz scattering coefficients for the tube of Figure 6 and (b) dispersion diagram of the associated planar structure [30]. f1p, f2p, and f3p are the plasmonic resonances of the dipole mode predicted by the planar configuration. The prime denotes the same information for the quadruple mode. f1c, f2c, and f3c are the same information calculated by the exact modified Mie-Lorenz theory of the multilayered cylindrical structure.
In order to design a dual-band super-scatterer, the plasmonic resonances of two scattering channels have coincided by fine-tuning the results of the Bohr’s model. The optimized geometrical and constitutive parameters are Rc = 45.45 nm, d1 = 45.05 nm, d2 = 43.23 nm, ε1 = 3.2, ε2 = 2.1, ε3 = 2.2, and ε4 = 1. Figure 8 shows the NSCS and magnetic field distribution for the dual operating bands of the structure. It is clear that NSCS exceeds the single-channel limit by the factor of 4, and in the corresponding magnetic field, there is a large shadow around the nanometer-sized cylinder at each operating frequency. Other designs are also feasible by altering optical and geometrical parameters. Furthermore, the far-field radiation pattern is a hybrid dipole-quadrupole due to simultaneous excitation of the first two channels. It should be noted that an inherent characteristic of the super-scatterer design using plasmonic graphene material is extreme sensitivity to the parameters. Moreover, in the presence of losses, the scattering amplitudes do not reach the single-channel limit anymore, and this restricts the practical applicability of the concepts to low-frequency windows.
(a) and (b) The NSCS of dual-band super-scatterer respectively, in the first and second operating frequencies and (c) and (d) corresponding magnetic field distributions [30].
As another example, the dispersion diagram of Figure 7(a) along with Foster’s theorem has been used to conclude that each scattering channel of the triple shell tube contains two zeros which are lying between the plasmonic resonances, predicted by the Bohr’s model. Later, we have coincided the zeros and poles of the first two scattering channels in order to observe super-scattering and super-cloaking simultaneously [33]. The optimized material and geometrical parameters are εc = 3.2, ε1 = ε2 = 2.1, Rc = 45.45 nm, d1 = 46.25 nm, and d2 = 46.049 nm. The NSCS curves corresponding to the super-cloaking and super-scattering regimes are illustrated in Figure 9(a) and (b), as well as the expected phenomenon, is clearly observed. The corresponding magnetic field distributions, shown in Figure 9(c) and (d), also manifest the reduced and enhanced scatterings in the corresponding operating bands, respectively. Similar to the dual-band super-scatterer of the previous section, the performance of this structure is very sensitive to the optical, material, and geometrical parameters. By further increasing the number of graphene shells, other plasmonic resonances and zeros can be achieved for the manipulation of the optical response.
Simultaneous super-scattering and super-cloaking using the structure of Figure 6. NSCS for (a) super-cloaking and (b) super-scattering regimes and corresponding magnetic field distributions, respectively, in (c) and (d) [33].
In this section, multilayered graphene-coated particles with spherical morphology are investigated, and corresponding modified Mie-Lorenz coefficients are extracted by expanding the incident, scattered, and transmitted electromagnetic fields in terms of spherical harmonics. It is clear that by increasing the number of graphene layers, further degrees of freedom for manipulating the optical response can be achieved. For the simplicity of the performance optimization, an equivalent RLC circuit is proposed in the quasistatic regime for the sub-wavelength plasmons, and various practical examples are presented.
In this section, the most general graphene-based structure with N dielectric layers, as shown in Figure 10, is considered, and plane wave scattering is analyzed through extracting recurrence relations for modified Mie-Lorenz coefficients. It should be noted that since, in the TMM method, multiple matrix inversions are necessary, unlike the cylindrically layered structures of the previous section, the spherical geometries are analyzed through recurrence relations. Also, scattering from a single graphene-coated sphere has been formulated elsewhere [16], and it can be simply attained as the special case of our formulation.
Spherical graphene-dielectric stack (a) 2D and (b) 3D views [34]. Please note that the numbering of the layers is started from the outermost layer in order to preserve the consistency with the reference paper [35].
The scattering analysis is very similar to that of the single-shell sphere [16], unless the Kronecker delta function is used in the expansions in order to find the electromagnetic fields of any desired layer with terse expansions. Therefore [34]:
By considering
where super-indices (1) in the vector wave functions show that the Hankel functions are used in the field expansions. The boundary conditions at the interface of adjacent layers read as:
Therefore, the linear system of equations resulting from the above conditions is:
where
where the sub/superscripts H and V represent the TE and TM waves, respectively. The directions of propagation of these waves are realized thought the subscripts F (outgoing waves) and P (incoming waves). The effective reflection coefficients are extracted as:
Moreover, it can be readily shown that the transmission coefficients read as:
where
where symbol
The extinction efficiencies of graphene-based particles with different number of layers: (a) two, (b) three, and (c) four [34].
In order to realize the priority of the closed-form analytical formulation with respect to the numerical analysis, the simulation times of both methods are included in Table 2. Considerable time reduction using the exact solution is evident. Moreover, since 3D meshing and perfectly matched layers are not required in this method, it is efficient in terms of memory as well.
Structure | Simulation time | |
---|---|---|
Analytical | CST | |
Figure 11(a) | 0.053214 s | 32 h, 50 m, 18 s |
Figure 11(b) | 0.045831 s | 33 h, 45 m, 25 s |
Figure 11(c) | 0.151555 s | 33 h, 34 m, 55 s |
Comparing the simulation time of CST and our codes [34].
Based on the results of Section 3.1, the modified Mie-Lorenz coefficients of the graphene-based spherical particles form infinite summations in terms of spherical Bessel and Hankel functions. In general, graphene plasmons are excited in the sub-wavelength regime, and only the leading order term of the summation is sufficient for achieving the results with acceptable precision. In this regime, the polynomial expansion of the special functions can also be truncated in the first few terms [22]. Later, the extracted modified Mie-Lorenz coefficients can be rewritten in the form of the polynomials. To further simplify the real-time monitoring and performance optimization of the graphene-coated nanoparticles, an equivalent RLC circuit can be proposed by representing the rational functions in the continued fraction form as [36]:
The equivalent circuit corresponding to the above representation is shown in Figure 12.
The proposed equivalent circuit for the scattering analysis of electrically small graphene-coated spheres [36].
The continued fraction representation for the TM coefficients is:
where
The elements of the equivalent circuit for the TM coefficients read as:
In order to illustrate the application of Mie analysis for the graphene-wrapped structures, let us consider vertical and horizontal dipoles in the proximity of a graphene-coated sphere, as shown in Figure 13. Although in the Mie analysis, the excitation is considered to be a plane wave, by using the scattering coefficients, the total decay rates can be calculated for the dipole emitters, and it can be proven that the localized surface plasmons of the graphene-wrapped spheres can enhance the total decay rate, which is connected to the Purcell factor [16, 37]. The amount of electric field enhancement for the radial-oriented and tangential oscillating dipoles with the distance of xd, respectively, read as:
(a) Vertical and horizontal dipole emitters in the proximity of the graphene-coated sphere and (b) the local field enhancement for various dipole distances with averaged orientation [37].
Figure 13(b) shows the local field enhancement for the average orientation of the dipole emitter in the vicinity of the sphere with R1 = 20 nm, coated by a graphene material with the chemical potential of μc = 0.1 eV. As the figure shows, an enhanced electric field in the order of ∼104 is obtained for the dipole distance of 1 nm with averaged orientation, and it decreases as the dipole moves away from the sphere.
The possibility of a super-scatterer design using graphene-coated spherical particles is illustrated in Figure 14. The design parameters are ε1 = 1.44, R1 = 0.24 μm, and μc = 0.3 eV. The structure can be simply analyzed by the modified Mie-Lorenz coefficients. The general design concepts are similar to their cylindrical counterparts, namely, dispersion engineering using the associated planar structure, as shown in the inset of the figure. Due to the excitation of TM surface plasmons, the normalized extinction cross-section is five times greater than the bare dielectric sphere. Moreover, similar to the cylindrical super-scatterers, by considering a small amount of loss for the graphene coating by assigning
(a) Atomically thin super-scatterer and associated planar structure shown in the inset and (b) corresponding normalized scattering cross-sections by considering lossless and lossy graphene shells [38].
By pattering graphene-based disks with various radii around a dielectric sphere, it is feasible to design a wide-band electromagnetic cloak at infrared frequencies. The geometry of this structure is illustrated in Figure 15. In order to analyze the proposed cloak by the modified Mie-Lorenz theory, the polarizability of the disks can be inserted in the equivalent conductivity method. The extracted equivalent surface conductivity can be used to tune the surface reactance of the sphere for the purpose of cloaking [39].
Wide-band cloaking using graphene disks with varying radii [39].
The other application that can be adapted to our proposed formulation of multilayered spherical structures is multi-frequency cloaking. As Figure 16 shows, by proper design, a single graphene coating can eliminate the dipole resonace in a single reconfigurable frequency. The radius of the sphere is R1 = 100 nm and its core permittivity is ε1 = 3. It can be concluded that double graphene shells can suppress the scattering in the dual frequencies since each graphene shell with different geometrical and optical properties can support localized surface plasmon resonances in a specific frequency. By further increase of the graphene shells, other frequency bands can be generated. Figure 16(b) shows the cloaking performance of a spherical particle with multiple graphene shells. The radii of the spheres are 107.5, 131.5, and 140 nm, and the corresponding chemical potentials are 900, 500, and 700 meV, respectively. The permittivity of the dielectric filler is 2.1 [21].
(a) Single and (b) multi-frequency cloaking using single/multiple graphene shells around a spherical particle [21].
As another example, a dielectric-metal core-shell spherical resonator (DMCSR) with the resonance frequency lying in the near-infrared spectrum is considered. In order to increase the optical absorption, the outer layer of the structure is covered with graphene. The localized surface plasmons of graphene are mainly excited in the far-infrared frequencies and in the near-infrared and visible range; it behaves like a dielectric. By hybridizing the graphene with a resonator, its optical absorption can be greatly enhanced. Figure 17 shows the performance of the structure for various core radii [15].
Strong tunable absorption using a graphene-coated spherical resonator with fixed dielectric core refractive index of n and silver shell thickness of t [15].
The provided examples are just a few instances for scattering analysis of graphene-based structures. Based on the derived formulas, other novel optoelectronic devices based on graphene plasmons can be proposed. Moreover, since assemblies of polarizable particles fabricated by graphene exhibit interesting properties such as enhanced absorption, negative permittivity, giant near-field enhancement, and large enhancements in the emission and the radiation of the dipole emitters [40, 41, 42, 43], the research can be extended to the multiple scattering theory.
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