Resonance frequencies
\r\n\t
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Tiefenbacher",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10765.jpg",keywords:"Managing Urbanization, Managing Development, Managing Resource Use, Drought Management, Flood Management, Water Quality Monitoring, Air Quality Monitoring, Ecological Monitoring, Modeling Extreme Natural Events, Ecological Restoration, Restoring Environmental Flows, Environmental Management Perspectives",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"January 12th 2021",dateEndSecondStepPublish:"February 9th 2021",dateEndThirdStepPublish:"April 10th 2021",dateEndFourthStepPublish:"June 29th 2021",dateEndFifthStepPublish:"August 28th 2021",remainingDaysToSecondStep:"21 days",secondStepPassed:!1,currentStepOfPublishingProcess:2,editedByType:null,kuFlag:!1,biosketch:"A geospatial scholar working at the interface of natural and human systems, collaborating internationally on innovative studies about hazards and environmental challenges. Dr. Tiefenbacher has published more than 200 papers on a diverse array of topics that examine perception and behaviors with regards to the application of pesticides, releases of toxic chemicals, environments of the U.S.-Mexico borderlands, wildlife hazards, and the geography of wine.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"73876",title:"Dr.",name:"John P.",middleName:null,surname:"Tiefenbacher",slug:"john-p.-tiefenbacher",fullName:"John P. Tiefenbacher",profilePictureURL:"https://mts.intechopen.com/storage/users/73876/images/system/73876.jfif",biography:"Dr. John P. Tiefenbacher (Ph.D., Rutgers, 1992) is a professor of Geography at Texas State University. His research has focused on various aspects of hazards and environmental management. Dr. Tiefenbacher has published on a diverse array of topics that examine perception and behaviors with regards to the application of pesticides, releases of toxic chemicals, environments of the U.S.-Mexico borderlands, wildlife hazards, and the geography of wine. More recently his work pertains to spatial adaptation to climate change, spatial responses in wine growing regions to climate change, the geographies of viticulture and wine, artificial intelligence and machine learning to predict patterns of natural processes and hazards, historical ethnic enclaves in American cities and regions, and environmental adaptations of 19th century European immigrants to North America's landscapes.",institutionString:"Texas State University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"6",institution:{name:"Texas State University",institutionURL:null,country:{name:"United States of America"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"12",title:"Environmental Sciences",slug:"environmental-sciences"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"194667",firstName:"Marijana",lastName:"Francetic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/194667/images/4752_n.jpg",email:"marijana@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3621",title:"Silver Nanoparticles",subtitle:null,isOpenForSubmission:!1,hash:null,slug:"silver-nanoparticles",bookSignature:"David Pozo Perez",coverURL:"https://cdn.intechopen.com/books/images_new/3621.jpg",editedByType:"Edited by",editors:[{id:"6667",title:"Dr.",name:"David",surname:"Pozo",slug:"david-pozo",fullName:"David Pozo"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"71303",title:"Scattering from Multilayered Graphene-Based Cylindrical and Spherical Particles",doi:"10.5772/intechopen.91427",slug:"scattering-from-multilayered-graphene-based-cylindrical-and-spherical-particles",body:'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.
Extension of functional possibilities of tools is one of the main ways to increase the maintenance property of technical means. It particularly actually in modern agricultural production, based on precision agriculture technologies, where using technical means must provide: reduction of cost, conservation of ground fertility, saving energy-resources, improvement labor conditions, and increase machines capacity. One efficient way to solve these problems is using geometric modeling in designing technical means. Geometric modeling, as one of the varieties of the synthetic methods of design, is a theoretical base for different technologies of these methods, like production design and CAD technologies. In this chapter, as examples, the following case studies will be considered: development of multifunctional mold board by geometric modeling, for increasing its maintenance property; integration role of CAD technologies in PLM, including in maintenance management; and visualization of production design process of technical means according to maintenance criterions. In recent years, increased variety of applicable machines and technologies has come to exist in the world in all spheres of human activity, especially in planning their functional possibilities. So development of tools that increase their functional possibilities is one of the most important problems of modern engineering and design activity. Expanding the functional possibilities of these tools will not only increase their capacity but also reduce specific consumptions of materials. These aspects are actually in creation resource and energy saving technical facilities, that is, main engineering activity, key direction in which is considered production design. This problem is considered in the same way actual and in condition of the strategic development of the Republic of Uzbekistan [1]. The solution to these problems is directly connected with the geometric modeling, which is based on the modern problems of the production design [2, 3, 4].
\nWe shall consider the problem in moldboard-type tools as an example. It is well known that mold boards, as the main tool in plows, bulldozers, graders, and other specific machines, are intended for performing the preparing works in agriculture and melioration, ground works in road construction and engineering preparation of territory, as well as in municipal sphere and etc. In the classical variant bulldozer, the mold board is a frontally located cylindrical working surface, which moves earth or other mass, prism lug of the ground in the required direction and amount [5, 6]. For expansion functional possibilities of mold board, there are also development in different constructive variant, with changing location working organ and different working surface (Table 1). But these developments are basically directed at the expansion of their maintenance (functional) possibilities, for executing work of certain nature [5, 7]. The solution to these problems is directly connected to the geometric modeling, which is based on modern problems of the industrial design [2, 3, 4]. The result of the using the production design at development of mold board type tools on base of constructive geometric modeling is a “design-development” mold board, which possible produce in three types of working surface design. We shall consider the design-development working surface of mold board consists of pieces of surface. For the base of the models, we take multifunction surface consisting of linear surfaces, which are broadly used for designing mold boards (Table 2).
\nUsing linear surfaces in bulldozers’ mold board types.
No | \nGeometry of surface | \nUsing in tools | \n
---|---|---|
1. | \nFrontal planar surface | \nMoldboard of channel defogger | \n
2. | \nInclined planar surface | \nBush cutting mold board of bulldozer | \n
3. | \nFrontal cylindrical surface | \nFrontal mold board of bulldozer | \n
4. | \nInclined cylindrical surface | \nBucket mold board of scraper | \n
5. | \nFrontal conical surface | \nMoldboard of grader | \n
6. | \nInclined conical surface | \nFrontal plow’s mold board | \n
7. | \nCylindroid surface | \nUniversal plow’s mold board | \n
8. | \nConidial surface | \nHigh-speed plow’s mold board | \n
9. | \nHyperbolic-parabolic surface | \nHyperbolic body plow’s mold board | \n
10. | \nHelicoid surface | \nHelicoid body plow’s mold board | \n
11. | \nTorsos surface | \nCultural plow’s mold board | \n
12. | \nCombined surface | \nCombined body plow’s mold board | \n
Using linear surfaces in mold board-type tools.
The design-development to construct a geometric model of a mold board’s working surface applicable for work execution of the different nature raises: technical, technological and economical factors of the designed technology, allows more flexible control its functional possibility, solving constructive problems [2, 8, 9].
\nThe analysis of existing mold board designs and studies on their improvement shows that creating a new design that can increase their functional possibilities, using constructive geometric modeling method, has a broad prospect [5, 7, 8, 10, 11]. The constructive geometric design of mold board’s working surfaces can possibly be divided into three types: (1) design consisting of unbroken surface (Figure 1); (2) design consisting of surface pieces (the sections) (Figure 2); and (3) design consisting of surface elements (the plates) (Figure 3). Herewith possible creating away the prospects of the primary using these design on example: (1) unbroken design for production of polymeric mold boards; (2) sectional design for expansion of the functional possibilities and increasing the ease of manufacturing production mold boards; and (3) plate design for the best management production and functional, working, and other quality mold boards. Developmening the working surface of mold boards that can be applied in different industries needs to consider the technical, economical, and technological factors of the designed machines. So, design-development of constructive geometric model of mold board’s working surface, though exist in the designs of the considered machines, will allow more flexible control of the functional possibility of the mold board and solve the above delivered constructive problem [2, 8, 9].
\nTraditional construction of plow (a) and bulldozer (b) mold board.
Sectional construction of plow (a) and bulldozer (b) mold board: “1—wing” and “2—breast” of plow’s body; “3—frontal” and “4—side” sections of spherical mold board of bulldozer.
Plate construction of plow (a) and bulldozer* (b) mold board: 1—right, 2—middle and 3—left guiding frames, 4—right and 5—left formative plates (*construction offered by author).
There is giving formative line—l of cylindrical surface—Φ horizontally, and P—plane of directory curve—m dispose perpendicular to these formatives on medium them. This plane crossing with working surface—Φ is divided into two equal parts, Φa and Φb, simultaneously being the symmetrical plane to these working surfaces. We shall choose line k on symmetrical plane, which will possibly conduct the bunch of the planes. These planes crossing with working surfaces Φa and Φb form curves of intersection. We shall mark these planes on both sides of the symmetrical plane P accordingly P1, P2, …, Pn and P1\', P2\', …, Pn\', as well as curves of the intersection on working surfaces Φa and Φb accordingly m1, m2, …, mn and m1\', m2\', …, mn\'. At angles between planes and symmetrical plane—P, we shall accordingly mark α1, α2, …, αn. Each pair of surface intersection curve m1, m1\'; m2, m2\'; …; mn, mn\' are formed accordingly by pair of planes P1, P1\'; P2, P2\'; …; Pn, Pn\', are symmetrical, where k is an axis of the mirror image pair of curves on working surfaces Φa and Φb (Figure 4a). So at rotation pair planes Pi and Pi\' with surfaces Φa and Φb around axis k corresponding to angle αi, planes Pi and Pi\', as well as curves mi and mi\' belonging to them, are combined. As a result of this operation, will be formed a rib on working surface, which separates the working surface into two halves. On the basis of this model, different constructive variants of the transformed mold board can possibly be developed, allowing the conversion from one working surfaces to another. It is known that when designing the complexity technical forms, considered surface mentally differs on “geometric” and “working” surfaces and from one surface possible to get different working surfaces [4, 7]. So by means of the proposed model, as a result of rotation working surfaces Φa and Φb around axis k to angle αi, a new working surface Φi is formed. Though given Φ and newly formed Φi cylindrical surfaces, they have a different working surface with different functional quality, where α enters as controlling parameter in the formation of Φi. Unlike the given surfaces Φ, a new working surface Φi promotes the improvement of directing actions of the moveable mass to the sides (Figure 4b) and from the sides (Figure 4c).
\nForming of moldboard’s working surface with bilateral action: a) transforming sheme; b) moving ground mass to the sides; c) moving ground from the sides.
The process of the formation required working surface—Φi possible to control, except parameter α, as well as position of k. In considering that the model rotation axis k is located vertically and has determined distance comparatively to Φi. However, change the position k greatly influences upon formation Φi. Here possible consider two parameters of k: change the distance—f, defined between fixed point k and m, for instance base k and sock m on horizontal plane; as well as change of the slopping angle—β to horizontal plane. Under one and same angle αi and the form of directory curve mi, change f will bring about change the mutual location pair of directory curves mi and mi\' that will bring and to change constructive parameter of mold board with working surface—Φi. From considered by author, acceptable variants (Figure 5) for given problems are chose variants (b) a chord—AB and (d) a tangent in point—С, with the result that possible neglect the parameter f that simplifies the problem. Though the other variants too have such working surface, they can bring about complication in the constructive parameter of the mold board. However, when forming the surface Φi, in variant (d) rotation is produced in inverse direction than in variant (b). With the importance of the rotation angle α, we choose within 0 < α < αmax, with the condition that planes Pi and Pi\' must cross all forming surfaces Φi, where αmax is on tgα = (l/2)/b, and overhang of curve b.
\nPosition variants of rotating axis k comparatively to m.
It is necessary to note the parameters on the form and position directory curve m of surface Φ. On condition of the problem form of directory curve—m is flat and fluent, with determined by curvature and concave side onward. Since these characteristic directory curves remain low-lying during the transformation of the surfaces, they shall select as topological parameters of curve, defining its form. Consequently, such parameters of surfaces, as their type and curvature also remain low-lying and when forming the new surface Φi. The position of curve is assigned two parameters: overhang b and height h of curve. They shall be marked as constructive parameters, since they define the design of the mold board. The following variants possibly select the relative position constructive parameters of m, defined by typical point positions (Figure 6): lower (A) and upper (B) points define h, and extreme left and right (the pair from points А, В, C) points define b. These variants directory curves are possible to choose when designing the mold board depending on execution of its work. When changing f, in the vertical position of k, the dimension height of mold board h\' in the same way remains low-lying. The parameter δbmax = bi-b derived after forming rib of surfaces Φi is situated opposite, for points, on which pass the rotation axis k (right/left—on bosom or upper/lower—on carrying).
\nRelative position variants of directory curve’s constructive parameters.
The criterions of the choice variant relative position of typical point of directory curve m on h and b, when designing mold board possible to explain, linking these points with typical positions of formatives l. For example, we shall select the following position formatives l, getting through typical points m on width b in respect to h, upper, lower, front, back, as well as average (on h or on b), and define their influence upon nature of the moving the moveable mass on working surface of the mold board (Table 3). From given table, it can be understood that the nature of the moveable mass on working surface is possible to control, having changed relations h and b, by changing the slopping angle β to axis k. Unlike vertical position, the slopping k on angle β onward or will back add the working surface except improvements of the shift of the moveable mass aside under its horizontal trimming (Figure 7a), as well as perfects the functional quality on shaping tilted lowering (Figure 7b) and ascent (Figure 7c) from moveable mass. This is the positions reached by change forming l, which present as well as plowshare, for horizontal plane on angle—φ, after forming Φi. The angle φ possible define by projection model, on base of descriptive geometry rules [12], using joining method (Figure 7d). Turning the horizontal plane on 90°, to joint it with frontal projection combine the projections k and l. Rotating l on angle αi, marked its l’, easy find the frontal projection lv\'. Since l revolves on frontal projection plane, perpendicular to k, circle of the rotation l projects on the horizontal plane as an ellipse. By means of projection beams, find lh\' and define φ angle of the slopping l on horizontal plane using the square-wave triangle, also considered as the corner of the slopping of the plowshare. After transformation working surface Φ on Φi under inclined k, will increase the height dimension h\' of mold board though h decreasing on δh. At, the higher part rib bends over onward or for lower part back, daring on distance δb\'. As a result of transformation, working surface changes the lengths corresponding to forming li within 0 < δb < δbmax, offset end forming belonging to rib to surfaces. In point, on which pass the rotation axis k, length li is equal δb = 0, but in nose (upper or lower) of a part it is equal δb = δbmax.
\nNo | \nOn width b, in respect to h and through points | \nNature of moving the moveable mass on worker of the surfaces | \n||
---|---|---|---|---|
Anterior | \nAverage | \nPosterior | \n||
1. | \nSuperior—В | \nNot available | \nInterior—А | \nPowerfully postponed in before. | \n
2. | \nSuperior—В | \nInterior—А | \nAverage—С | \nPartly is taken on breast and powerfully postponed in before. | \n
3. | \nSuperior/interior—В/А | \nNot available | \nAverage—С | \nCompletely taken on breast and powerfully postponed in before. | \n
4. | \nInterior—А | \nSuperior—В | \nAverage—С | \nCompletely taken on breast and weakly postponed in before. | \n
5. | \nInterior—А | \nNot available | \nSuperior—В | \nCompletely taken on bosom. | \n
Positions of formatives and their influence to working surface nature.
Determination of inclined working surface geometric parameters.
The definable parameters got Φi on two variants, and on positions of the descriptive geometry, make sure that αi parameters and Φi are alike, but are mutually negative (Figure 8a and b) [12]. This allows to combine two variants in one design and as a result enlarges the functional possibilities of the designed mold board (Figure 8c). We can select five compartments of working surfaces on intersection lines. Alternate switching-on or switching-off of corresponding compartments will enable the mold board to work in three modes: moving the mass frontal, moving ground mass to the sides, or from the sides. The proposed device of geometric modeling-transformed working surface allows to develop a constructive geometric model of a multifunctional mold board. This development is intended for organizations to produce specific machines. Parameterization of mold board’s working surface relieves designers’ work, increases the choice a variant under development mold board’s working surface, and allows effectively to solve the constructive problems.
\nForming of working surface with bilateral action.
The modern production is founded on using information science and communication technologies as CALS-technology (continuous acquisition and lifecycle support) or PLM-technology (information support of the product lifecycle management processes). PLM is an approach to designing and producing high-tech and scientifically based product, using information science and computer technology at all stages of the product life cycle [13].
\nThis aspect actual in condition of developing countries, like Uzbekistan, where using these technologies is innovative process in production. One of the problems in this process is adapting them in production, that is, translating the engineering data to PLM system, by way of integrating PLM and CAD/CAE/CAM systems, using the product’s engineering database at the base of PDM-technology (product data management).
\nThe product’s engineering data are possibly divided into three groups: structural (constructional), functional, and technological. Let me present to you the structural data, which we can call the geometric data, that are necessary for integrating CAD and PDM systems. The product’s geometric data are used not only in enterprise where they are produced but at all stages of the product life cycle from designing to maintenance. So, creating the geometric database, using different forms of the geometric data (Figure 9a), is very important in the product life cycle.
\nTypes and levels of geometric data (a). Integration role of geometrical modeling system in PLM (b).
As is well known, the product lifecycle includes the period from origin necessity for creating the product up to its liquidations in consequent exhaustion of consumer characteristic. Primary stages of product life cycle are selecting four main stages: designing, producing, maintaining, and utilizing.
\nThough life cycles of old and new products always form the unceasing cycle, because of brightly not images, traditionally life cycles of each product were considered separately, which during the initial stage was designing but finally cutting. However, author, founding on his conducting researches, offers to consider that that beginning of PLM is from creation an instrument from stone, bones and wood by primitive man. Since no one can reject that the base of modern industrial robot is an instrument of the stone age, the end of “old” product is a beginning of “new” product. It is possible to say that the beginning and the end of PLM connect with beginning and end of mankind on land. The present production conditions, in which production design steel play one of solving roles, relationship between “old” and “new” product lifecycle become reveals itself all more brighter. Coming from author’s offers separate stage of the designing on two: conceptual and engineering design. Conceptual design stage is founded on the basis of geometric modeling and it is the closing stage of the product life cycle, having a causal relationship between “maintenance-utilizing” and “designing” stages. Nowadays, producing geometric modeling has become a primary method, facilitating designing. In this stage, the product will be designed on the basis of the relationships between exhaustion of consumer characteristic of old product and necessity for creating the new (innovative) product.
\nThe need for geometric data arises at all stages of the product life cycle, particularly in the initial stage—“conceptual design” stage, when it is very high. Created at this stage, geometric database is directly or indirectly used also in the other stages of the product life cycle, by integrating CAD and PDM systems. It is necessary to note that the need to create a “new product” is basically formed in the maintenance of “old” product. Since it at this stage is not only used Geometric data of “old” product in maintenance but arises Geometric data of “new” product in designing.
\nAs example, we shall consider creating the agricultural machinery tools’ geometric database, which is necessary to enterprises that participate in these products life cycle [14]. Creating this database needs the review, classification, and analysis of appropriate information about agricultural machinery tools from a geometric standpoint. This will allow us to reveal general and individual geometric features of these tools that assist in an efficient management of product data for all participating enterprises in this process (Figure 9b). At present, the author is the leading researcher on development of theoretical bases and applying aspects of the geometric modeling of agricultural machinery tools. With the results from research, models, algorithms, and methods of designing these tools with mold board surface by geometric modeling were worked out.
\nAs a result, information sufficient for creating the geometric database in different forms was accumulated. We shall select the following forms of geometric data: verbal, graphical, parametric, algorithmic, 2D and 3D model, and methodical (that may include all geometric data forms). All elaborations are executed in AutoCAD 2013 system.
\nThe present pace of industry development requires the development and introduction in production innovation designing technologies. Using the methods of the production design gives the essential result in the process of developing technical object on different criterions engineers play one of key role in this, since saving to energy and resource, improvement to ease of manufacturing and functionality, as well as capacity, mainly depend on under development them technology and technical facilities [15]. Using of these methods require from constructors revealing the problems, which decisions are connected with geometrical modeling, on which is founded production design, as well as way of their decision. We shall consider using one of the methods of the production design—“Concept selection,” choosing the models to design. The moldboard have a complex technical form, long period of change on improvement their design, and have a universal geometric model. These factors allow using the production design in development mold board on geometric features, influencing on technician-technological features. As is well known, application in agricultural production plows has a different mold board constructions in accordance with their destination. The combination, on advantage of different criterions, considered constrictions in one new constriction, with necessary changes, by screening-method of “Concept selection” will allow to choose the models for development. The development conducting on the main types design moldboard plows, in which is taken into account row of the main criterion of the choice to models on geometric features (based on geometric data and parameters), presented by requirements of the producers and consumers (Table 4). As base, at estimations of the criterion, is chose design A. Geometric features are valued upon their advantage (+) and defect (−). The features design, obviously not by specialists as advantage or defect, are conditionally evaluated neutral (0), coming from that considerations that they specifically do not influence upon these features. The visualization of the qualitative estimation and analysis of the features in such a way allow to choose the directions of modeling of designs on advantage, and then to combination of design. They are hereinafter offered development of the models, on basic model, occupied 1–3 places with provision for their advantage.
\nScreening of model selection process for designing mold board.
The proposed constructive geometric model of mold board’s working surface allows to develop the multifunctional tools applicable in agricultural, engineering, road building, mining, and municipal service industries and in other branches of machinery use. Parameterization of mold board’s working surface relieves designers’ work, increases the variants choice under developing mold board’s working surface, and allows effectively to solve the constructive problems. The integration of geometric modeling systems (CAD) and methods allows the efficient use not only in designing process and but also in production and maintenance processes of the technical facilities. Creating the product’s geometric database by CAD technologies became one of the necessary tasks of production, particularly engineering products. In contemporary conditions of using CALS technologies, “conceptual design” stage of innovative product by methods and facilities of geometric modeling is the defining stage of the product life cycle. The visualization of the design-development process allows to develop a new product according to maintenance criterions. Efficient use of this method reduces design process time, labor, and material costs during the development of a new product.
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