Surface Plasmon Dispersion Relation Research Articles

Guiding infrared electromagnetic waves through TI nanowires with extremely large wavenumber and azimuthal index.

In this paper, the dispersion relations of the surface plasmon polaritons (SPPs) in TI nanowires have been investigated. For simplicity, TI nanowire has been modeled as a dielectric cylinder with a conductive surface, the conductivity of which is an anti-symmetric tensor. The off-diagonal terms of the conductivity tensor only slightly change the dispersion relations. Due to small conductivities, these SPPs have extremely large wavenumbers and azimuthal indices; the electric fields are tightly confined near the conductive surface. For high-order modes, cut-off phenomena have been observed. In the end, the effects of losses and much larger bulk permittivities on the dispersion relations of surface plasmons have been discussed. The simple model proposed in this paper can be directly applied to other materials with arbitrary surface conductivity. Our investigations show that TI nanostructures are promising platforms for nanophotonic applications in the future.

МОДЕЛИРОВАНИЕ ВОЗБУЖДЕНИЯ ПОВЕРХНОСТНЫХ ПЛАЗМОН-ПОЛЯРИТОНОВ В ПРЯМОУГОЛЬНОМ НАНОРЕЗОНАТОРЕ НА ОСНОВЕ ЗОЛОТА

This paper considers the modeling of surface plasmon polaritons excitation in a limited nanostructure on the basis of a two-dimensional area discrete model, the area interacts with the oscillator. The nanostructure is a rectangle defined on the metal surface at the interface of the gold–silicon oxide, the surface plasmon–polaritons are excited by an electromagnetic radiation point source located above the metal surface. The dynamics of radiation point source is described by a discrete version of the Van der Pol equation with a small nonlinearity of the source parameters. The parameters of the goldsilicon oxide structure (the wave phase speed, the radiation source frequency, the characteristic time in the system, etc.) will be received from the dispersion relation for surface plasmon–polaritons at a single metal–dielectric interface. The distributions of the wave field in the structure will be analyzed at different positions of the point oscillator and different coupling coefficients of the wave field with the oscillators. The resonant wave field mode composition at different positions of the point oscillator and different coupling coefficients will be found using the two-dimensional Fourier transform of the wave field, the time evolution of excited modes of the wave field amplitudes will be also analyzed. In conclusion, the paper gives applicability limits of the considered model for studying the surface plasmon polaritons excitation in a limited nanostructure.

Theory of strong coupling between molecules and surface plasmons on a grating.

The strong coupling of molecules with surface plasmons results in hybrid states which are part molecule, part surface-bound light. Since molecular resonances may acquire the spatial coherence of plasmons, which have mm-scale propagation lengths, strong-coupling with molecular resonances potentially enables long-range molecular energy transfer. Gratings are often used to couple incident light to surface plasmons, by scattering the otherwise non-radiative surface plasmon inside the light-line. We calculate the dispersion relation for surface plasmons strongly coupled to molecular resonances when grating scattering is involved. By treating the molecules as independent oscillators rather than the more typically considered single collective dipole, we find the full multi-band dispersion relation. This approach offers a natural way to include the dark states in the dispersion. We demonstrate that for a molecular resonance tuned near the crossing point of forward and backward grating-scattered plasmon modes, the interaction between plasmons and molecules gives a five-band dispersion relation, including a bright state not captured in calculations using a single collective dipole. We also show that the role of the grating in breaking the translational invariance of the system appears in the position-dependent coupling between the molecules and the surface plasmon. The presence of the grating is thus not only important for the experimental observation of molecule-surface-plasmon coupling, but also provides an additional design parameter that tunes the system.

Tuning surface plasmon-exciton coupling via thickness controlling of excitonic layer

We present the Rabi energy tuning effects in strong coupling effects between surface plasmon (SP) and exciton in the Kretschmann-Raether configuration by varying excitonic layer thickness. Both experimental results and theoretical calculations indicate that the anticross-like dispersion relation comes from the rapid change of permittivity near the exciton state and can be attributed to the strong coupling between SP and exciton. Our findings reveal that the excitonic layer plays a key role in the SP-exciton coupling. The increase of the excitonic layer will not only enlarge the Rabi splitting energy, but also cause the redshift of SP dispersion relation. Thus to fulfill the coupling condition, there is a critical thickness of the excitonic layer at specific incident angle . With the increase of excitonic layer thickness, the Rabi energy increases monotonously and tends to saturate. Our findings will be beneficial for the better understanding of SP-exciton strong coupling in KR configuration and can be useful in tuning the Rabi energy and resonant conditions according to practical applications.

Exact dispersion relations for the hybrid plasmon-phonon modes in graphene on dielectric substrates with polar optical phonons.

Intrinsic optical phonons and extrinsic polar optical phonons (POPs) strongly affect the graphene surface plasmons. Specifically, extraneous POPs present on the surface of an underlying substrate change the behavior of the graphene's surface plasmons sharply due to the plasmon-phonon hybridization. Here, we report modeling of exact dispersion relations for graphene's surface plasmons affected by intrinsic optical phonons and extrinsic POPs of the surface of polar dielectric substrates with one or more vibrational frequencies. In doing so, we have employed random phase approximation with modified two-dimensional polarizability (2D-Π0). The adapted Π0 addresses limitations of the previously derived plasmons dispersion, obtained using classical two-dimensional polarizability. We show the new model overcomes the unsatisfying behavior of the plasmonic dispersion relation obtained by the classical 2D-Π0 at high-wavenumbers and its inability to indicate the starting point of the mode damping. Our new simple model eliminates the complexity of the other presented models in describing the surface plasmons' behavior, specifically at high wavenumbers. Besides, we use our dispersion model to learn about the plasmon content of the hybrid modes, which is a vital value to compute output current in plasmonic graphene-based devices. The coupled-mode lifetime due to the hybrid nature depends on both plasmon and phonon lifetimes. We capture this value here. There is an excellent agreement between our theoretical results and the experimental data reported earlier. They pave the way for the exact modeling of graphene plasmons on common polar substrates and bring in the closeness of the theoretical approaches and experimental results.

Dispersion relation of graphene surface plasmon by using a quantum hydrodynamic model

Abstract A quantum hydrodynamic model is used to study the properties of surface plasmon at an interface between a monolayer graphene sheet and a Voigt substrate. The substrate is assumed as a semiconductor that applying an external magnetic field to the structure in the Voigt configuration. The dispersion relations of graphene surface plasmon are obtained analytically by solving Maxwell’s equations and the quantum hydrodynamic equations. It is found that the quantum effects significantly change the properties of graphene surface plasmon and the features of such plasmon are quite different from those in a classical hydrodynamic model. The results also show that the applied magnetic field and the graphene character greatly affect the graphene surface plasmon. Moreover, The plasmon modes exhibit distinctively different behavior for forward and backward propagating directions, which are in contrast to the cases without the quantum effects. In addition, a one-way plasmon mode is found in the lower band region. Parameter dependence of the effects is examined and discussed.

Frequency Dependent Graphene Surface Plasmon Properties on Different Dielectrics

Numerical analysis of surface plasmon behaviour is performed on graphene surface supported on different dielectric medium and with varying graphene chemical potential. The dielectric medium of graphene is varied from free standing graphene to rigid SiO2 wafer substrate to flexible PMMA polymer for diverse graphene plasmonics applications. The dispersion relation, propagation length and penetration depth of graphene surface plasmons are computed and analysed for the different dielectrics. The results show that graphene plasmonic behaviour in the various dielectrics highly depends on its chemical potential, the excitation input frequencies and produces surface plasmons with high field localization and low losses. This study of plasmonic behaviour on flexible dielectric opens up application of graphene plasmonics in the field of flexible optoelectronics.

Surface magnetoplasmons in a slit waveguide with graphene monolayers

Abstract Due to the huge potential in biosensing, imaging and spectroscopy, plasmonic components based on surface plasmons have attained stimulated intense interest, recently. This paper reports on a study regarding the properties of surface magnetoplasmons in a slit waveguide whose consists of two semi-infinite magneto-optic semiconductors separated with air layer and a graphene monolayer is placed at each interface. The dispersion relation of surface plasmons are obtained analytically and it is shown that applying an external magnetic field to the structure in the Voigt configuration greatly affects the plasmons. Results demonstrate that depends on the air thickness and the Fermi energies of the graphene monolayers, the enhancement, location and localization of the plasmons can be tuned. Magneto optical properties of such active surface magnetoplasmons could be used in the surface plasmons based devices.

Hydrodynamics of Fermi arcs: Bulk flow and surface collective modes

The hydrodynamic description of the Fermi arc surface states is proposed. In view of the strong suppression of scattering on impurities, the hydrodynamic regime for Fermi arc states should be, in principle, plausible. By using the kinetic theory, the Fermi arc hydrodynamics is derived and the corresponding effects on the bulk flow and surface collective modes are studied. For the bulk flow, the key effect of the proposed Fermi arc hydrodynamics is the modification of the corresponding boundary conditions. In a slab geometry, it is shown that, depending on the transfer rates between the surface and bulk, the hydrodynamic flow of the electron fluid inside the slab could be significantly altered and even enhanced near the surfaces. As to the spectrum of the surface collective modes, in agreement with earlier studies, it is found that the Fermi arcs allow for an additional gapless spectrum branch and a strong anisotropy of the surface plasmon dispersion relations in momentum space. The gapped modes are characterized by closed elliptic contours of constant frequency in momentum space.

Effect of left-handed materials in surface plasmon excitation and propagationlength

We investigated the dispersion relation of surface plasmons (SPs) excited in a medium comprised of a left-handed material (LHM) layer. We investigated different light polarizations in SP excitation along an LHM-metal interface. We studied SP excitation through a classical Kretschmann geometry, employing an LHM interlayer between a dielectric and a metal layer. Using this three-layer configuration, we investigated the effect of LHM medium on characterizing SP propagation length, which is directly related to the energy of the surface waves.

Channel surface plasmons in a continuous and flat graphene sheet

We derive an integral equation describing surface-plasmon polaritons in graphene deposited on a substrate with a planar surface and a dielectric protrusion in the opposite surface of the dielectric slab. We show that the problem is mathematically equivalent to the solution of a Fredholm equation, which we solve exactly. In addition, we show that the dispersion relation of the localized surface plasmons is determined by the geometric parameters of the protrusion alone. We also show that such system supports both even and odd modes. We give the electrostatic potential and the stream plot of the electrostatic field, which clearly show the localized nature of the surface plasmons in a continuous and flat graphene sheet.

Surface waves guided by metamaterials with rotational disorder

The analytical analysis of the metamaterial boundary with the rotational disorder reveals both bound and leaky surface plasmon (SP) modes. The dispersion relations of SPs propagating on a surface of these metamaterials are presented along with the propagation lengths. The rigorous modeling and analysis of surface waves at the boundary of two metamaterials possessing rotational disorder are presented. Dispersion properties of two different metamaterial boundaries have been investigated. The results show that the boundary of the metamaterials having different dielectrics employed allows for the presence of the particular modes crossing the light line with the significant portion at lower frequencies lying above the free space light line.

Dual tunable plasmon-induced transparency based on silicon-air grating coupled graphene structure in terahertz metamaterial.

A graphene plasmonic structure consists of three graphene layers mingled with a silicon-air grating is proposed. We theoretically predict and numerically simulate the plasmon-induced transparency effect in this system at terahertz wavelengths, and a dual plasmon-induced transparency peaks can be successfully tuned by virtually shifting the desired Fermi energy on graphene layers. We investigate the surface plasmon dispersion relation by means of analytic calculations, and we can achieve the numerical solution of propagation constant got by the dispersion relation. A suitable theoretical model is established to study spectral features in the plasmonic graphene system, and the theoretical results agree well with the simulations. The proposed model and findings may provide guidance for fundamental research of highly tunable optoelectronic devices.

Surface plasmon dispersion relation at an interface between thin metal film and dielectric using a quantum hydrodynamic model

The plasmon excited at the surface of a thin metal film covering on a semi-infinite dielectric substrate is investigated in detailed. The surface plasmon dispersion relation is derived and presented by using the quantum hydrodynamic theory with taking into the quantum statistical and quantum diffraction effects. The effects of the thin metal film character rs and the dielectric constant ε1 on the dispersion relations are shown and discussed, i.e., without and with quantum effects. The results show that increasing rs weakens the quantum effects while increasing ε1 can enhance the quantum effects. In addition, the plasmon dispersion relation with quantum effects is also compared to a classical model. This simple structure proposed in the paper with quantum hydrodynamic theory can be used for yielding meaningful results for studying more complex systems related to surface plasmons.

Research progress of plasmonic structure illumination microscopy

Structure illumination microscopy (SIM) is a novel imaging technique with advantages of high spatial resolution, wide imaging field and fast imaging speed. By illuminating the sample with patterned light and analyzing the information about Moir fringes outside the normal range of observation, SIM can achieve about 2-fold higher in resolution than the diffraction limit, thus it has played an important role in the field of biomedical imaging. In recent years, to further improve the resolution of SIM, people have proposed a new technique called plasmonic SIM (PSIM), in which the dynamically tunable sub-wavelength surface plasmon fringes are used as the structured illuminating light and thus the resolution reaches to 3-4 times higher than the diffraction limit. The PSIM technique can also suppress the background noise and improve the signal-to-noise ratio, showing great potential applications in near-surface biomedical imaging. In this review paper, we introduce the principle and research progress of PSIM. In Section 1, we first review the development of optical microscope, including several important near-field and far-field microscopy techniques, and then introduce the history and recent development of SIM and PSIM techniques. In Section 2, we present the basic theory of PSIM, including the dispersion relation and excitation methods of surface plasmon, the principle and imaging process of SIM, and the principle of increasing resolution by PSIM. In Section 3, we review the recent research progress of two types of PSIMs in detail. The first type is the nanostructure-assisted PSIM, in which the periodic metallic nanostructures such as grating or antenna array are used to excite the surface plasmon fringes, and then the shift of fringes is modulated by changing the angle of incident light. The resolution of such a type of PSIM is mainly dependent on the period of nanostructure, thus can be improved to a few tens of nanometers with deep-subwavelength structure period. The other type is the all-optically controlled PSIM, in which the structured light with designed distribution of phase or polarization (e.g. optical vortex) is used as the incident light to excite the surface plasmon fringes on a flat metal film, and then the fringes are dynamically controlled by modulating the phase or polarization of incident light. Without the help of nanostructure, such a type of PSIM usually has a resolution of about 100 nm, but benefits from the structureless excitation of plasmonic fringes in an all-optical configuration, thereby showing more dynamic regulation and reducing the need to fabricate nanometer-sized complex structures. In the final Section, we summarize the features of PSIM and discuss the outlook for this technique. Further studies are needed to improve the performance of PSIM and to expand the scope of practical applications in biomedical imaging.

Dispersion relation and self-collimation frequency of spoof surface plasmon using tight binding model

The analytical dispersion relation of spoof surface plasmon (SSP) is known only in the low-frequency limit and thus cannot be used to describe various practically important characteristics of SSP in the high-frequency limit (such as multimodal nature, anisotropic propagation, self-collimation). In this article, we consider a square lattice of holes made on a perfect electric conductor and derive a closed form expression of the SSP dispersion relation in the high-frequency limit using a tight binding model. Instead of using prior knowledge of the band diagram along the entire first Brillouin zone (BZ) edge, we analytically determine the hopping parameters by using the eigenfrequencies only at the three high-symmetry points of the square lattice. Using this dispersion relation, we derive an expression for the self-collimation frequency of SSP. We show that this analytical formulation is also applicable to dielectric photonic crystals and can be used to predict the frequencies corresponding to centimetre-scale supercollimation and second band self-collimation in these structures. Finally, we show that our analytical results are in agreement with the simulation results for both SSP and photonic crystals.

Plasmon Hybridization in a Symmetry-Broken Metallic Nanotube Above a Substrate

An explicit form of the surface plasmon dispersion relation of a metallic nanotube with a nonconcentric core above a metallic substrate is presented, based on the separability of Laplace equation in bipolar coordinates. Three distinct plasmon mode frequencies (ω 0, ω +, and ω −) are found. Numerical results show ω + and ω − are variable with the geometrical characteristics of the system, while the value of ω 0 is always stable, when the dielectric permittivity of nanotube core, and dielectric between the nanotube and substrate are same.

Analytical Approach of the Nonlinear Surface Plasmon at a Left-Handed Material

The surface wave dispersion relations of surface Plasmon at the interface of a left-handed material and a non-linear Kerr medium of arbitrary nonlinearity are derived based on a generalized first integral approach. The normalized power flow is also investigated for various values of frequency. The above study is conducted for both cases: self-focusing (αp0) and de-focusing (αs0) nonlinear Kerr coefficient.

Extraordinary transmission through gain-assisted silicon-based nanohole arrays in telecommunication regimes.

Extraordinary gain-assisted transmission in telecommunication regimes through circular nanohole arrays drilled on a metallic film is investigated theoretically. Silicon-compatible Er-Yb silicate, which has a photoluminescence peak in the telecommunication regime, was selected for optical amplification purposes. Geometrical parameters were optimized analytically in order to present transmission resonances at telecommunication regions. The required gain value for lossless propagation was determined by considering the surface-plasmon dispersion relation. Simulation results show that the predicted gain for lossless propagation cannot completely compensate the loss. By increasing gain value, absorption becomes zero and transmission approaches unit through a laser with a pumping power of 372mW at 1480nm.

Surface plasmons of a graphene parallel plate waveguide bounded by Kerr-type nonlinear media

The exact dispersion relations of the transverse magnetic surface plasmons (SPs) supported by a graphene parallel plate waveguide (PPWG), surrounded on one or both sides by Kerr-type nonlinear media, are obtained analytically. It is shown that if self-focusing nonlinear materials are chosen as the surrounding media, the SPs localization length (LL) is decreased, while their propagation length (PL) remains unchanged, as compared to those of a typical graphene PPWG. Moreover, PL and LL of the SPs are considerably affected by adjusting nonlinear parts of the dielectric permittivities of the nonlinear media. It is found that using an appropriate defocusing nonlinear material as a substrate of the graphene PPWG, which is surrounded on one side by the nonlinear medium, leads to noticeable enhancement of the propagation and localization characteristics of the surface plasmons. The results presented here can be useful for enhancing capabilities of plasmonic devices based on the graphene PPWG for sensing and waveguide applications.