Surface plasmons(SPs) are evanescent waves that propagate along the surface of dispersive media . When a light beam incidents onto the surface of metal, the electromagnetic field interacts with the free electrons inside the metal, which leads to the oscillation of free electrons to excite the SPs in the exit medium. Noble metals with nanostructure geometry have special optical properties because they can excite localized surface plasmons (LSPs) under the illumination of light field. Since the pioneer study of Ritchie , the special optical properties of SPs in nano-scale have been tightly investigated. Various metallic nano-structures are reported for nano-plasmonic devices, including thin film[3,4], nanowires[5,6], nanorods[7,8], nano-hole array[9,10], nano-slits et. al. Due to the sub-wavelength excitation range and strong field enhancement properties of SPs, they have been widely applied in super-resolution optical microscopy, nano-photonic trapping technology, biology and medical sciences and nano-photonic waveguide.
In this chapter we firstly introduce a numerical algorithm for implementing the relative permittivity model of dispersive media in finite difference time domain(FDTD)method, i.e. piecewise linear recursive revolution(PLRC) method. Next the super-resolution phenomenon derived from the surface plasmons excited at the focal region is presented and analyzed. In the third part, we explore two kinds of plasmonic nano-waveguide: parallel nanorods and metal-dielectric-metal structure. Their optical properties, such as long distance waveguiding from the focal region, turning waveguiding effect and optical switch effect, will be demonstrated in detail. Finally, we conclude our research results in recent years and look forward the applications of surface plasmons in nano-technologies.
2. FDTD method for plasmonics simulation
The FDTD method was firstly proposed by Yee in 1966. Because of its strong and precise power for simulating the propagation of the electromagnetic field, it was quickly applied in many research fields associate with the electromagnetics and optics. The detailed discrete differential equations of Maxwell equations can be found in Ref. In this section, we mainly present the FDTD algorithm for dispersive media which is referred to as piecewise linear recursive revolution(PLRC) method and the verification of the program code.
2.1. 3D FDTD algorithm for dispersive media
The recursive revolution(RC) method for implementing the relative permittivity models of dispersive media in finite difference time domain(FDTD) algorithm was first proposed by Luebbers et al.  in 1990. It had been testified to perform faster calculation speed and fewer memory space requirement than the auxiliary differential equation(ADE) method , Z-transform method  and shift operator(SO) method . In 1996, Kelley and Luebbers proposed the improved PLRC method  which remained the speed and low memory of RC method, and provided the accuracy existing in ADE method. The PLRC method for Drude model and Lorentz model has been presented respectively[22, 23]. In this section, we combine these two sets of formulations and propose the PLRC method for Drude-Lorentz model.
The curl equations of Maxwell's equations are presented as:
In dispersive medium, the displacement vector D has the linear relation with the electric field vector E in frequency domain as:
The contribution from Lorentz term for variable
where “^” means it is a complex variable.
Finally the variable
The parameters in Equ.(5) are also the sum of Drude term and the real part of Lorentz term:
From above equations we can see that the Drude and Lorentz terms are deduced respectively, so that the PLRC method can be applied for both Drude and Drude-Lorentz models. For describing the relative permittivity of silver, we adopt the values of the parameters of Drude-Lorentz model as
2.2. Verification of the program code
We adopt the classical Krestschmann-Type SPR device shown in Fig.1(a) to verify our FDTD code. The two-dimensional(2D) simulation conditions are:
3. Application of surface plasmons in super-resolution focusing
A tightly focused evanescent field can be generated by a centrally obstructed high numerical aperture objective lens , and a super-resolved evanescent focal spot of λ/3 has been obtained . The enhancement of the electromagnetic (EM) field by tight focusing enables nano-lithography using evanescent field . It has been demonstrated that a tightly focused beam can be further modulated by a negative-refraction layer together with a nonlinear layer  or a saturable absorber  to approximately λ/4~λ/5 close vicinity of the focus. Considering the difficulty in realizing a negative-refraction layer in practice, here we introduce another mechanism of light modulation in the tightly focused region. Due to the use of high numerical aperture objective lens, the focused evanescent field is highly depolarized, which offers strong transverse and longitudinal polarization components. Therefore the deployment of nano-plasmonic structure, which is polarization sensitive, offers new mechanism to modulate the focused evanescent field. In this section, we will give two research results for the application of surface plasmons in super-resolution focusing.
3.1. Simulation of radially polarized focusing through metallic thin film
As shown in Fig.2, the configuration of tightly focused evanescent field is based on the scanning total internal reflection microscopy. The refractive indices of immersion oil and the coverslip glass are 1.78. The thin silver film is coated on the surface of the coverslip glass. An annular radially polarized incident beam is focused on the upper surface of coverslip glass by an objective. We define the annular coefficient ε=d/D which produces a ring beam illumination onto the silver film. In the simulation we set ε=0.606 to make sure that the incident angle is larger than the critical angle for the total internal reflection of the ring beam.
The electromagnetic field of radially polarized focal beam is expressed as:
Under the conditions mentioned above, after 2000 time steps simulation, the time averaged intensity distributions of
It is obviously that in the total electric field distribution, the reflected field is much stronger than the transmitted field. Under the annular coefficient 0.606<ε<1.0 condition, the total ring beam includes a wide range of incident angle(34.18º<θ<67.97º). However, only the narrow angle ranges around θ=36.72° satisfy the SPs dispersion relation. We simulate the thin ring beam illumination(0.645<ε<0.65) which contains the SPR incident angle and the thin ring beam illumination(0.795<ε<0.8) which is far from the SPR incident angle in order to demonstrate the differences between the SPR focus and non-SPR focus.
The simulation results are shown in Fig.4. Comparing Fig.4(a) and (d), it can be seen that under the SPs dispersion relation, the SPR significantly enhances the focus below the silver film, which leads to a much high transmission rate of the focal field. While for the non-SPR focus, the reflection is much stronger than the transmission, so most part of the focus energy is reflected back. Comparing the
3.2. Simulation of focusing through two parallel nanorods
3.2.1. Numerical simulation model
Our simulation configuration is shown in Fig.5. Two silver nanorods are lying on the interface of two dielectric media with the separation (
If the illuminating beam polarizes along the
where i, j and k are unit vectors in the
In the simulation, the wavelength of incident focal beam is 532
3.2.2. Simulation results and discussion
The intensity distribution of a focused evanescent field under the linearly polarized illumination is shown in Fig.6 and agrees well with previous theory[26, 27]. Under the conditions described above, the intensity of
The modulation of the focused evanescent field by a pair of silver nanorods is demonstrated in Fig. 7, where the intensity distributions in planes of different distances from the interface are illustrated. In the left column, when there is no nanorod on the interface, the focal spot splits into two lobes at different distances above the interface. In the middle column, the nanorods are lying along
The detailed analysis of the LSPs effect on each polarization component at the plane 100
It is well known that the focal spot for radially polarized beam is circularly symmetrical. According to the analysis demonstrated in the previous section, it is expected that the circular symmetry would be broken due to the LSPs effect which is polarization sensitive. Fig.9 shows the intensity distributions of evanescent radially polarized focal beam at
In this section we demonstrate a new method to modulate highly focused evanescent field with a nano-plasmonic waveguide. The modulation of focus is based on the mechanism that the LSPs are polarization sensitive and the focus is strongly depolarized by a high numerical aperture objective. For a simple nano-plasmonic waveguide that consists of two silver nanorods lying on the interface between two dielectrics, LSPs effect is strongest for the polarization component perpendicular to the nano-plasmonic waveguide. A super-resolved focal spot with significantly enhanced strength can be achieved, when the nanorods are lying perpendicular to the dominant polarization component. The design of the nano-plasmonic waveguide structure gives rise to a new approach to further improve the tightly focused evanescent field to achieve the resolution beyond diffraction limit, and thus facilitates potential applications in nano-trapping and nano-lithography.
4. Plasmonic nano-waveguiding
In this section we will propose the waveguiding properties of two types of plasmonic nano-structures: parallel long nano-wires and the metal-dielectric-metal waveguide. They all present interesting optical nano-waveguiding properties and can be applied in some special research areas.
4.1. Focusing through parallel nanorods that perpendicular to the interface
In recent years, metallic nanorods are intensely investigated and widely applied in the field of super-long waveguiding . It has been reported that the optical properties of single nanorod is sensitive to the polarization of incident wave and the rod aspect ratio . While for the two parallel nanorods and U-shaped nanorods, their optical properties are more complicated and interesting . The SPPs excited along the surface of single nanorod couple between the nanorods, which leads to the extinction spectrum whose resonance peaks are dependent on the geometry of the nanorods. The U-shaped nanorods show stronger resonance strength and more longitudinal couple modes than the two parallel nanorods.
In some cases, such as super-resolution imaging or focusing, the more complicated focusing beam with three polarization components should be considered as the incident source. However, the interaction between the focal beam and the two nanorods has not been investigated thoroughly yet. Here we fist present the SPR excitation spectrums of two parallel silver nanorods and the π-shaped silver nanorods using the FDTD method. Second we simulate the focusing process through these two structures using 3D FDTD method, respectively. The waveguide effect and electromagnetic field transfer efficiency of these two structures are compared and analyzed. Finally, the focusing process through the angular π-shaped nanorods structure is simulated which presents the ability of guiding the focal field to different directions.
4.1.1. Simulation modeling and resonance spectrums
The schemes of two silver nanorods structures are shown in Fig.10. In Fig.10(a), the length of nanorods is
As shown in Fig.11, the spectrums perform two peaks for both structures, but the positions of the peaks of the two structures under the same distance
4.1.2. Focusing through nanorods structures
Based on the results obtained above, we will investigate the focusing processes through the nanorods structures in this section. Due to the depolarization effect of high numerical aperture objective, the incident linearly polarized beam would change its polarization state after propagating through the objective, i.e. the strong longitudinal polarization component occurs. The interaction effect between the incident focal beam and the two nanorods structures are more complicated than that of the single polarized incident beam case. Here the focal beam is calculated by the vectorial Debye theory and induced into the 3D FDTD simulation region with total-field/scatter-field method. The refractive index of upper medium is 1.78 and the refractive index of lower medium is 1.0. The numerical aperture of the objective is 1.65, and the pure focused evanescent field is generated by inserting a centrally placed obstruction disk with the normalized radius
The long waveguide effect produced by the nanorods structures are clearly shown in Fig.12. Without the nano-structures, the pure evanescent focal field would decay exponentially away from the interface and only can propagate no more than 100
The two parallel nanorods structure and the π-shaped nanorods structure show almost the same SPPs resonance mode, except their excitation strength. In Fig.12(a), the electromagnetic field is strongly enhanced at the top of the two nanorods and decay exponentially very fast, which leads to four strong field points at the top of the nanorods. In Fig.12(b), the focal field is evenly enhanced by the transverse nanorod, so there is no strong field point shown at the top side. The comparisonof the electromagnetic field intensities along
Finally, we present the simulation of focusing through the angular π-shaped nanorods structure as shown in Fig.13(a). The two legs of the π-shaped nanorods are bent by the angle of 90°, so that the ends of the nanorods shift from the optical axis with the distance of
In this section, the optical properties of two silver nanorods plasmonic waveguide structures are simulated with the FDTD method. The SPR spectrums of the two parallel nanorods and the π-shaped nanorods structures are calculated, which show different SPR resonance peaks at the same distance condition for both structures. The focusing processes through these two types of structures show almost the same SPPs resonance mode with each resonance period about 150
4.2. MDM nano-waveguides
Plasmonic waveguides based on the principles of SPPs have gained great attentions in recent years due to their ability of confining and guiding optical field in sub-wavelengthscale. Among various types of plasmonic waveguides, the metal-dielectric-metal (MDM) waveguide is considered to be a key element in the fields of waveguide couplers[34, 35], sub-wavelength scale light confinement[36, 37], wavelength filters and integrated optical devices[39, 40]. The remarkable advantages of the MDM waveguide, including the strong confinement of optical field in nano-scale gaps, the high sensitivity of its transmission characteristics to the waveguide structures, and the facility of its fabrication, attracted a great deal of effort to be devoted to develop the MDM based nano-plasmonic devices. It has been reported that the transmission of MDM waveguide coupled with stub structure could be changed with the length ofthe stub. Further more, the stub filled with absorptive medium was considered to be a resonance cavity that acts as an optical switch controlled by the pumping field. An improved transmission model was also developed to describe the transmittance of multi-stubs MDM structure.
The investigation approaches for the transmission characteristics of MDM waveguide include theoretical transmission line theory(TLT) and FDTD method . In this section, we first provide the formulas of the optical transmission characteristics of the MDM waveguide with a stub structure deduced by TLT. And then the FDTD method will be employed to numerically study the optical switch effect of the stub structure in terms of changing the length and the refractive index of the stub, respectively. The simulation results coincide with the calculation results of TLT and the physical mechanisms of the optical switch effect are analyzed and discussed.
4.2.1. Scheme modeling and transmission line theory
The scheme of the MDM waveguide coupled with a single stub structure is shown in Fig.14(a). Firstly we assume that the medium in the stub is air, i.e.
According to the transmission line theory, the stub can be considered as admittance. Assume that the phase shift only occurs when the SPPs are reflected by the end of the stub. Z0 and Zs are the characteristic impedances of the loss-free transmission lines corresponding to the MDM waveguide and the stub, respectively. Their relation is expressed as:
where L is the length of the stub and λSP is the propagating wavelength of the SPPs. From Fig.14(b), the amplitude transmission of the electric field can be expressed as:
where Y0=1/Z0 and YS=1/ZS. Therefore, the energy transmission of the MDM waveguide coupled with a single stub is finally expressed as:
From Equ.(29) we can see that the energy transmission is the function of L and λSP. In the stub, the SPPs wavelength λSPS can be changed by the refractive index of the medium in the stub. As a result, the transmission would be modulated periodically by changing the length L and the refractive index n2 of the stub linearly.
4.2.2. Numerical simulation modeling and results
We simulated the transmission of the MDM waveguide with L ranging from 0 to 1
From Fig.15(a) we can see that the simulation results agree wellwith the theoretical data calculated by Equ.(29). The SPPs excited along the surfaces of the metal layers propagate in the waveguide with a resonance mode. When passing through the stub, a part of the SPPspropagate into the side-coupled stub.The SPPs reflected from the end of the stub interfere with the passing SPPs, which leads to a modulation of the superposition wave.If we change the length of the stub, the phase change of the interference would cause the transmission vary from 0 to 1, so that the MDM waveguide coupled with a single stub structure presents the optical switch effect on the incident wave. As the incident wavelength
Another approach for modulating the phase of the reflected wave from the end of the stub is changing the refractive index of the stub. We carried out a series of simulations with the refractive index
We have numerically studied the transmission characteristics of the MDM waveguide coupled with stub structure as functions of the length and the refractive index of the stub, respectively. The 2D FDTD simulation results show that the transmission rates obtained by both approaches change as periodical distributions, which implies that the MDM waveguide can be treated as an optical switch device controlled by the length and the refractive index of the stub. The physical mechanism of this phenomenon is the phase modulation of the interference of the reflected SPPs wave from the end of the stub and the passing SPPs wave in the waveguide. The results help us to further apply the MDM waveguide as an optical switch element in nano-scale optical chips and optical integrated devices.
The importance of surface plasmons in the applications of nano-photonics has been proved in many examples. In this chapter we focused on two fields: super-resolution focusing and nano-waveguiding. Super-resolution focusing is the key element for nano-lithography, high density optical data storage, and super-resolution imaging. We have presented that the high density focal field can be re-distributed by some specified metallic nano-structures such as nano-film and parallel nanorods, so that a super-resolution focusing can be generated in some particular space areas. This method can be further applied in optical nano-trapping due to the small and enhanced plasmonic focus. While for nano-waveguiding there have been many nano-structures and methods reported previously. Here we have presented the parallel nanorods and MDM structures, respectively. When the parallel nanorods are perpendicularly put on the interface of two materials, the incident focal field generates surface plasmons along their surfaces which form a nano-waveguide. The fields of focus can propagate along the waveguide for a long distance. The MDM with a stub structure showed a optical switch effect with the altering of the length and the refractive index of the stub, which has a potential application in optical communications and optical sensing.
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