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Abstract
Use of dipole-like localized surface plasmons (LSPs) induced on a rectangular array of silver nanodisks to excite directional surface plasmon polaritons (SPPs) on a flat silver film is demonstrated. By modifying the spectral resonance of the LSPs, an effective coupling of the incident light into the SPPs has been achieved at operational wavelength of λ0 = 628 nm. The maximum SPP intensity exceeds 25% of the incident intensity. Meanwhile, owing to the angle-dependent spatial distribution of the LSPs and the constructive interference between columns or rows of the array, the excited SPPs are supported to propagate in two orthogonal channels with a width of 1756 nm. Moreover, the propagation of the SPPs is tunable by rotating the polarization direction of the incident light between angles of 0 and π/2. This approach of launching direction-tunable SPPs shows great potential in applications of integrated plasmonic circuits.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon polaritons (SPPs) are evanescent electromagnetic waves excited at the interface between a metal and a dielectric material. They can be attributed to the resonant interaction between the free electron oscillation in the metal and the electromagnetic field of light [1]. The ability of SPPs to effectively manipulate light beyond the diffraction limit [2] inspires enormous advanced applications in micro-scale optical components such as waveguides [3–5], lenses [6,7], microscopes [8,9], spasers [10, 11] and light emitting diodes [12, 13].
One of the most fascinating prospects for SPPs is to greatly miniaturize the size of photonics circuits compared with those currently available [14–17]. Among various SPP applications in integrated photonics circuits, the fundamental and crucial technique is the excitation of SPPs and their controllable propagation. Therefore, techniques of launching SPPs have been widely studied. In general, the principle of launching SPPs by light is to compensate for the missing wave-vector between SPPs and free-space light at the same frequency, e.g., through prism coupling [14], structural scattering [18,19], the coupling of periodic corrugations on the surface of metal [20,21], and permittivity engineering of materials [22]. Furthermore, launching SPPs by an electrical source such as a dipole [23–26] or a moving electron beam [27,28] is also an effective approach. Electric dipoles offer great promise for achieving SPPs with different characteristics such as plane or spherical waves by arranging the distribution of dipoles [6, 24]. Meanwhile, the propagation of SPPs can be tunable by modulating the polarization state or phase of the incident light [26, 29]. Among these studies, nanoslits or apertures are mostly adopted to couple light to electric dipoles via plasmonic resonance to excite SPPs and control their propagation in oppositely directed channels. In addition, launching and modulating SPPs in two orthogonal channels is also an attractive approach, offering convenience and diversity in designing integrated photonics circuits. However, the structures of nanoslits or apertures proposed in the studies above are usually complex. Their use also leads to an increased proportion of the incident light being transmitted to the nonincident side, where it is treated as an undesired and non-negligible disturbance for the whole plasmonic circuit [20].
In this paper, to overcome the above shortcomings, a novel approach with a simple structure and modulation is demonstrated to launch SPPs propagated in two orthogonal channels on a flat silver film. Instead of nanoslits or apertures, a rectangular array of silver nanodisks is utilized to induce dipole-like localized surface plasmons (LSPs) as the source of the SPPs to reduce the existence of incident light on the nonincident side. This work shows that the LSPs induced by the normally incident Gaussian beam can be effectively coupled to the SPPs on the film. The field distribution of the SPPs are directional and related to the polarization direction of the LSPs. For a rectangular array with an appropriate array spacing, because of constructive interference between the SPP waves excited by columns or rows, only two orthogonal channels are allowed for the long-distance propagation of the SPPs. Moreover, an effective modulation of the propagation of the SPPs in these two channels is accessible by simply rotating the polarization direction of the incident beam between angles of 0 and π/2. Such advantages are quite suitable for integrated plasmonic circuits, especially in multichannel information transport.
2. Theoretical background
For the purpose of launching high-intensity, directional SPPs by LSPs, one must first determine the resonance wavelength of the LSPs induced on the nanodisks, the propagation constant, and the field distribution of the excited SPPs. Therefore, a single nanodisk is considered at the beginning. It is noted that the intercoupling between the LSPs will influence the spectral absorption of an array of nanodisks, leading to differences in the position of the LSP resonance peaks compared with those of a single nanodisk [30]. However, this intercoupling effect can be weak when the polarization directions of the dipole-like LSPs are in-plane and the spacing between the nanodisks is larger than the wavelength of the SPPs [12,31]. Therefore, for a nanodisk that is either single or in an array, the major resonance peaks are all around the same position, ensuring effective coupling between the light and the LSPs.
The structure for launching SPPs by a single nanodisk is shown in Figs. 1(a) and 1(b). A nanodisk is placed a distance d above a nearby thin film. A linear-polarized Gaussian beam is normally incident to the nanodisk to induce LSPs on its surface at first. The diameter and height of the nanodisk are D and h, respectively. The thickness of the film is a. Both the nanodisk and the film are silver and embedded in a silicon dioxide medium. The dielectric constants of the local media are εj, where j = 1, m denotes the silicon dioxide medium or silver.
Fig. 1 The schematic diagrams of the designed structure for a single nanodisk in cylindrical coordinates (ρ, θ, z): (a) the view of the θ = 0 plane, and (b) the view of the z = d plane. A linear-polarized Gaussian beam represented by a yellow-color arrow, is normally incident to the nanodisk, and its polarization direction is indicated by an orange-color arrow.
If λ is the wavelength of light in the surrounding medium, then, for a small enough nanodisk (D ≪ λ), the spatial field distribution of the LSPs can be described as a dipole located at the center of the nanodisk. Therefore, actually, the nanodisk plays the role of a dipole-like LSP emitter in the process of launching SPPs on the film [32]. Generally, this approximation is adequate for the particles of dimensions below 100 nm. In this case, the phase of the harmonically oscillating electromagnetic field is practically constant over the particle volume. It ensures the validity of this approximation to describe the spatial field distribution of the particle in an electrostatic field [1]. Additionally, the resonance wavelength and intensity of the LSPs can be modified in the red-green range by altering the geometric parameters of the silver nanodisk, for example, its diameter D [33]. Moreover, the polarization direction of the LSPs is coincident with the incident beam [34].
Furthermore, the propagation constant of the excited SPPs on the film can be determined from the dispersion relation of the designed structure. Based on Maxwell’s equations and continuity boundary conditions, the dispersion relation β − ω can be described by the expression
where K0 = ω/c, and β is the SPP propagation constant.The source of the excited SPPs can be represented by a nearby in-plane dipole , located at (0, 0, d), in cylindrical coordinates (ρ, θ, z). In the quasi-static approximation, the harmonic time phase factor e−iωt can be ignored in the discussion of the spatial distribution of the SPP field. In the far-field regime where ρ is greater than a few multiples of 1/Re(β), the z component of the SPP field at the nonincident interface of the film is described by [32]
where ϕ0 is the initial phase of the dipole-like LSPs and is influenced by the volume of the nanodisk. The coefficient A(β, ω, d, a) is given by where , , the polarization angle of the incident light , and M (β, ω) is a position-independent variable.In Eq. (2), it can be seen that the z component of the SPPs is influenced by the dispersion relationship of the designed structure and the dipole strength p0. For a determined silver film, by inducing strong dipole-like LSPs at the operational wavelength, an effective coupling of the incident light to the SPPs can be achieved. It is also noted that there are other contributions to the SPP field on the incident interface between the SiO2 medium and the metal film in the case in which d is typically smaller than λ. In this situation, the spatial distribution of the SPP field can easily be influenced by the intercoupling between the LSPs and the SPPs excited at the incident interface. As d or a increases, the z component of the excited SPPs is decreased according to Eq. (3), whereas the LSP contributions is dominant [35]. Meanwhile, to reduce the influence of the incident field, the SPPs are determined at the nonincident interface of the film.
3. Results and discussion
3.1. SPPs excited by a single LSP dipole
Numerical simulations are carried out with the assistance of the commercial software FDTD Solutions. The incident Gaussian beam is x-polarized (θ0 = 0) and operated at λ0 = 628 nm, and its waist radius is 1.05 µm. In addition, the amplitude of the incident beam is set to 1 for the convenience of the simulations. The thickness of the film a is 50 nm. The diameter and height of the nanodisk D and h are 80 nm and 30 nm, respectively. The distance between the nanodisk and the film d is 50 nm. The dielectric constants of SiO2 and silver are obtained from experimental data [36].
Figure 2 shows the spectral absorption of nanodisks of three different diameters (D = 60, 70, 80 nm). The power absorbed by the nanodisk is normalized by the radiated power of the incident beam. As is shown, the resonance wavelength of the LSPs shifts in the range from 553 to 628 nm as the value of D increases. This indicates that the LSPs can be supported in the red–green range by adjusting the diameter of the silver nanodisk. For D = 80 nm, the resonance wavelength of the LSPs is located at 628 nm, which is equal to the operational wavelength λ0. This confirms that the induced LSPs have been effectively modified into the operational spectral range. Therefore, the diameter of the nanodisk is unified as 80 nm in the following studies.
Contour maps of the Re2(Ez) field on the z = −a surface of the silver film are shown in Figs. 3(a) and 3(b). The field sources are the LSPs induced on the nanodisk and an equivalent electric dipole located at the center of the nanodisk, respectively. Meanwhile, to enhance the visibility, the Re2(Ez) field shown in Figs. 3(a) and 3(b) are normalized by the maximum values of each, respectively. It can be seen that these two fields are highly similar. Both of them are mainly distributed along the x-direction and are gradually suppressed as the value of θ increases from 0 to π/2, and they even vanished along y-direction.
Fig. 3 (a) The contour map of the Re2(Ez) field on the nonincident surface (z = −a) of the film for a single nanodisk, the operational wavelength λ0 = 628 nm and the polarization angle of the incident beam θ0 = 0. (b) Similar result to that in (a) except that the nanodisk is replaced by an x-polarized dipole located at the central position of the nanodisk. (c) and (d) The corresponding Re2(Ez) fields for the nanodisk and the dipole at ρ = 2058.5 nm and ρ = 2119.7 nm, respectively. (e) and (f) The corresponding θ = 0 profiles of the Re(Ez) fields for the nanodisk and the dipole, respectively.
According to Eq. (2), in this simulated case, the spatial distribution of the Ez field in the far-field regime can be simplified as
where A0 is a constant in this situation. The corresponding Re2(Ez) fields for the nanodisk and the dipole at ρ = 2058.5 nm and ρ = 2119.7 nm are shown in Figs. 3(c) and 3(d), respectively. For the positions with a constant distance away from the nanodisk, the variation of the corresponding Re2(Ez) with different values of θ can be well described with the factor cos2θ according to Eq. (4). Moreover, the corresponding wavelength of the SPPs on the film is 397 nm, as obtained from the propagation constant β given in Eq. (1). As Figs. 3(e) and 3(f) show, in the far-field region (ρ > 700 nm), the wavelengths of the Re(Ez) fields for the nanodisk and the electric dipole are 393.6 nm and can be well described by a damped cylindrical wave . The SPP field excited by the LSPs is obviously different with the electric dipole where ρ is smaller than 204.4 nm for the nanodisk and the dipole, respectively. This disagreement is caused by the difference of the initial phase ϕ0 between the nanodisk and the dipole. It can be attributed to the difference of the volume between the nanodisk and the equivalent dipole.Except for the spatial distribution of the SPP field, the local Abs2(Ez) field enhancement is also investigated to confirm the coupling effect between the incident light and the SPPs. Figure 4(a) shows the non-normalized field distribution of the Abs2(Ez) corresponding to the case in Fig. 3(a). The maximum Abs2(Ez) is < 0.02, which indicates that the coupling effect between the incident light and the SPPs is weak in the situation of a single nanodisk. This weak coupling effect can be attributed to the low-level absorption of the incident light (~ 0.03 at 628 nm) by the nanodisk. Additionally, by calculating the Abs2(Ez) field on the same surface without the nanodisk, the local Abs2(Ez) enhancement is presented in Fig. 4(b). The value of the local Abs2(Ez) enhancement is displayed on a log10-scale to enhance its visibility. It shows that Abs2(Ez) has been significantly enhanced by a couple of orders of magnitude, especially in the area far away from the nanodisk. Meanwhile, owing to the LSPs induced on the nanodisk, an obvious enhancement can be also found in the central area.
Fig. 4 (a) The contour map of the non-normalized Abs2(Ez) field corresponding to the case in Fig. 3(a). (b) The enhancement of the corresponding local Abs2(Ez) field (on a log10-scale) compared with the absence of the nanodisk.
3.2. SPPs excited by a rectangular array of LSP dipoles
In Section 3.1, it was shown that the spatial distribution of the SPP field excited by a single nanodisk is angle-dependent. According to Eq. (2), the direction of the SPPs can be also modulated by altering the polarization angle of the incident light θ0. As the value of θ0 increases from 0 to π/2, the z component of the SPP intensity Abs2(Ez) is decreased in the x-direction (θ = 0), whereas it is increased in the y-direction (θ = π/2). Such a characteristic makes this approach suitable for designing a direction-tunable SPP launcher. Especially, it is accessible and simple to achieve continuous modulation by changing the polarization direction of linear-polarized light between the angles of 0 and π/2 by using electrooptic crystals, for example, ferroelectric crystals [37]. Generally, a directional SPP wave can be achieved by arranging many nanodisks into one column. Meanwhile, an appropriate spacing between the nanodisks ensures a weak intercoupling effect between them to influence the LSP resonance wavelengths of the whole array. Furthermore, long-distance propagation of SPPs on the film is available by placing multiple columns based on the constructive interference between SPP waves excited by each column.
Figure 5(a) shows nanodisks (D = 80 nm and h = 30 nm) arranged into a 5 × 5 rectangular array to launch directional SPPs. The spacing of the array is set to be the wavelength of the SPPs, thereby ensuring that the constructive interference condition is fulfilled in both directions along the x- and y- axes. Therefore, two orthogonal channels along these directions are established for supporting the propagation of the SPPs. For the convenience of evaluating the directionality of the SPPs, the orthogonal channels are represented by the white dashed line. The width of these channels is set to 1756 nm, which is close to the size of the array. In Fig. 5(b), one can see that the absorption peak position of the whole array has slightly shifted from 628 to 635 nm because of the weak LSP intercoupling effect between the nanodisks. However, the peak level of the array still exceeds 0.37 at 628 nm. Compared with the single nanodisk, more nanodisks have engaged in the absorption of the incident light in the case of the array, increasing the efficiency of power absorption more than 12-fold. Therefore, an effective coupling of the incident light to the LSPs can be realized at the operational wavelength.
Fig. 5 (a) The distribution of the designed rectangular array of the nanodisks for launching directional SPPs in two orthogonal channels. λspp is the wavelength of the SPPs. (b) The spectral absorptions of a single nanodisk and a rectangular array of nanodisks, respectively.
The process of continuously modulating the propagation of the SPPs on the film is shown in Figs. 6(a)–6(e). In Fig. 6(a), it can be seen that the Abs2(Ez) field is propagating along the x-direction in channel 1 when θ0 = 0. Meanwhile, the maximum Abs2(Ez) has exceeded 0.25 in this case. This indicates that the maximum SPP intensity is above the level of 25% of the incident intensity. Except for the constructive interference in the x-direction, this effective coupling of the incident light to the SPPs is expected because the absorption of the array has greatly increased compared with that of the single nanodisk, significantly strengthening the strength of the LSPs and resulting in the enhancement of the SPP intensity. As Figs. 6(a)–6(e) show, by increasing the value of θ0 from 0 to π/2, the component of the Abs2(Ez) is gradually decreasing in channel 1, whereas it is increasing in channel 2. Finally, the propagation direction of the SPPs has a π/2 rotation when θ0 = π/2. The presented modulation process indicates that the propagation direction of the SPPs can be controlled by rotating the polarization direction of the normally incident light. Because the constructive interference condition is only fulfilled in the two orthogonal channels (represented by the white dashed lines) in Figs. 6(a)–6(e), long-distance propagation of the SPPs is only supported in these channels. Moreover, the SPP wave has good directionality owing to the width of these narrow channels being < 2µm. Such a characteristic makes this approach suitable for plasmonic-based information transportation because of the significant reduction of the potential signal noise caused by the un-intentional propagation of the SPPs.
Fig. 6 (a)–(e) The contour maps of the Abs2(Ez) fields on the z = −a surface of the silver film for a rectangular array of the nanodisks. The incident wavelength λ0 = 628 nm. The polarization angle of the incident beam θ0 = 0, π/6, π/4, π/3 and π/2, respectively.
4. Conclusion
In summary, a novel approach is demonstrated to effectively launch direction-tunable SPPs on a flat silver film by using a rectangular array of silver na nodisks. A linear-polarized Gaussian beam is normally incident to the array to induce the dipole-like LSPs that play the role of the source of the excited SPPs. The strong resonance of the LSPs at the operational wavelength is accessible by adjusting the geometric parameters of the nanodisks. By arranging the nanodisks into a rectangular array, the launched SPPs are supported to propagate in two orthogonal channels based on the constructive interference between the SPP waves excited by the columns or rows of the array. The maximum SPP intensity can exceed 25% of the incident intensity, confirming the effective coupling between the incident light and the SPPs. Furthermore, owing to the angle-dependent field distribution of the LSPs, the propagation of the SPPs can be modulated by simply rotating the polarization direction of the incident light between the angles of 0 and π/2. Such advantages are considerable for integrated plasmonic circuits, especially in multichannel information transport, for example, for a direction-tunable or dual-channel SPP launcher.
Funding
Science and Technology Program Project for the Innovation of Forefront and Key Technology of Guangdong Province, China (2014B010119004, 2014B010121001); Institute of Science and Technology Collaborative Innovation Major Project of Guangzhou, China (201604010047); Innovation Project of Graduate School of South China Normal University, China (2017LKXM002); Special Fund for Scientific and Technological Innovation and Development of Guangzhou-Foreign Science and Technology Cooperation Project, China (201807010083).
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