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Abstract
Dynamic tailoring of the propagating surface plasmon polaritons (SPPs) through incident angle modulation is proposed and numerically demonstrated. The generation and tailoring mechanism of the SPPs are discussed. The relationship formula between the incident angle and the generated SPP wave vector direction is theoretically derived. The correctness of the formula is verified with three different approaches using finite difference time domain method. Using this formula, the generated SPP wave vector direction can be precisely modulated by changing the incident angle. The precise modulation results of two dimensional Bessel-like SPP beam and SPP bottle beam array are given. The results can deepen the understanding of the generation and modulation mechanism of the SPPs.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon polaritons (SPPs) are surface electromagnetic waves that propagate tightly bound to a metal-dielectric interface, and are originated from the strong interaction between an external electromagnetic field and free electron oscillations in the metal [1,2]. Owing to its unique properties, such as the strong enhancement of local field intensity and confinement of light energy to sub-wavelength scale, SPPs are expected to be a good carrier between the photonics and electronics for applications such as on-chip photonics circuits [3], microscopy [4,5], optical tweezers [6], bio-photonics [7], lithography [8–10], etc.
However, the limited propagation distance and beam diffraction effect during SPP beam propagation are the main obstacles to the application of SPPs in photonic integrated devices. To overcome these issues, researchers have proposed a series of new types of in-plane propagating non-diffracting SPP beams (i.e., preserve their spatial shape during propagation), such as non-diffracting collimated plasmon beam [11], self-accelerating surface plasmon beams [12,13], two dimensional (2D) Bessel-like SPP beam [14,15], surface plasmonic Cosine-Gauss beam [16–18], arbitrary bending plasmonic self-accelerating beam [19,20], and 2D surface plasmonic bottle beam array (surface plasmonic void array) [21–23]. In the above mentioned literatures [14–16,22,24], a straightforward way to generate the SPP is to fabricate a plasmon launcher consisting of two metallic gratings forming an angle in or in the metal film, which under normal illumination excite in-plane SPP beams. The non-diffracting propagation characteristic of the SPP beams is controlled by the cross angle between the two gratings. However, the main focus of the above mentioned literatures are shining on the physical properties of the generated non-diffracting SPP beams in a fixed structure under normal illumination. Little literature discusses how to dynamically modulate the properties of the generated 2D SPP beams based on the proposed fixed structures (i.e., 2D intersecting metallic gratings), and what changes will be taken place in the 2D SPP field distribution under the inclined incident light?
Meanwhile, recent advances in 2D near-field SPP wave modulation, have led researchers to develop more flexible means to dynamically manipulate arbitrary forms of SPP field distribution. The SPP beams can be controlled in dynamic manners simply by adjusting the characteristics of the incident light, such as wavelength [25], polarization [26–29], phase [17,30–32], incident angle [12,21,25,33]. For example, by controlling the incident angle, the SPP beam propagation directions and intensities [12,25,31], the focus position of a silver circular slit plasmonic lens [33], the shape and area of the excited plasmonic bottle-beams together with the location of the hot spot [12,21] can be dynamically controlled. However, the quantitative relationship between the incident angle and the generated SPP wave vector direction is not given in these above literatures. In the reference [34], the full vectorial model is developed that relates any plane wave incident on any one-dimensional grating to the generated SPP, and all possible solutions are derived. However, a periodic grating is only efficiently for a specified SPP wavelength. Using a simple static one-dimensional metallic grating and a dynamic incident optical beam, the possible SPP propagation angle is shown to depend on the specific grating of choice, and only a dynamically controlled SPP hot spot result is numerically demonstrated. Compared with the grating structure, although the coupling efficiency of a single groove is lower, it can couple light of different frequencies, which means different wavelength of light can be used to excite SPP on a fixed groove coupling structure [34,35].
In this paper, the dynamic and precise tailoring of the propagating SPP beam by changing the incident angle is proposed and numerically demonstrated. A single groove is used as the element coupling structure. The relationship formula between the incident angle and the generated SPP wave vector direction is theoretically derived and numerically verified with three different approaches using finite difference time domain (FDTD) method. Using the derived relationship formula, the generated SPP wave vector direction can be precisely modulated by changing the incident angle. The dynamic precise modulation results of non-diffracting 2D Bessel-like SPP beam and 2D surface plasmonic bottle beam array are given. The generation and dynamic precise modulation mechanism of the above two plasmonic beams are mainly discussed. Both the theoretical derivation and simulation results verify the correctness and feasibility of the proposed modulating principle and can deepen the understanding of the generation and modulation mechanism of the SPPs.
2. Theoretical derivation and simulation demonstration of the modulating principle
In this section, the relationship formula between the incident angle and the generated SPP wave vector direction is theoretically derived and numerically demonstrated with three different methods.
The schematic diagram of the designed structure and the coordinate definition are shown in Fig. 1(a). Two intersecting grooves etched in a silver film are used to couple the spatial light into the SPP wave. The half cross angle of the two grooves is defined as θ (i.e. the angle between the long axis of the groove and y axis). The width and length of a rectangular groove is w = 200 nm and L = 9 um. The thickness of the silver film is 120 nm to eliminate the directly transmitted incident light [33]. The mirror symmetric two grooves are illuminated from the substrate side by a linearly polarized plane wave with an oblique angle γ (i.e., the angle between the wave vector direction of incident light ko and positive z axis). Only the two groove area (the dotted rectangle shown in Fig. 1(a)) is illuminated. The dielectric permittivity of silver is ε = −10.627 + 0.32602i at the incident wavelength λo = 514 nm, and the corresponding SPP wavelength is λSPP = 490 nm [35]. The polarization direction of incident light is shown in Fig. 1(a) with dark double arrows.
Fig. 1 (a) Schematic diagram of a double groove structure and the coordinate definition. The incident light is illuminated from the substrate side with an oblique incident angle γ. The polarization direction of the incident light is indicated with the double arrow. (b) and (c) Schematic diagrams of vector relationship between the wave vector of the incident light and the generated SPP. The SPP wave vector direction is indicated with a red arrow (OH, OF). In Fig. 1(b) ko is on the x-z plane and in Fig. 1(c) ko is on the y-z plane.
In order to theoretically derive the relationship formula between the incident angle and the generated SPP wave vector direction, the schematic diagrams of vector relationship between the in plane wave vector component of the incident light kin and the generated SPP kspp are given in Figs. 1(b) and 1(c). In Fig. 1(b), the wave vector of the incident light ko is on the x-z plane, but in Fig. 1(c), the wave vector of the incident light ko is on the y-z plane. Both in Figs. 1(b) and 1(c), the corresponding line segments denote different meanings. PO denotes a single groove. OE denotes the excited SPP wave vector direction, when the groove is illuminated at normal incidence γ = 0°.
OH denotes the excited SPP wave vector direction, when the groove is illuminated at right oblique angle γ (i.e., the ko is biased toward the positive x axis or positive y axis). β denotes the deflection angle of the wave vector direction of the SPP waves generated by the right oblique incidence relative to the normal incidence.
OF denotes the excited SPP wave vector direction, when the groove is illuminated at left oblique angle γ (i.e., the ko is biased toward the negative x axis or negative y axis). α denotes the deflection angle of the wave vector direction of the SPP waves generated by the left oblique incidence relative to the normal incidence.
PN (or PM) denotes an equivalent virtual groove. The equivalent virtual groove can be defined as: the wave vector direction of the SPP excited by PN (or PM) under normal incidence is the same as that of the SPP excited by PO under the right (or left) oblique incidence.
A single groove can be regarded as a line source to excite the SPP wave propagating perpendicular to the groove, when the groove is illuminated at normal incidence γ = 0° [24]. Additional wave vector along the silver film surface can be obtained by the diffracted light from the groove, allowing a portion of the incident light to generate the SPP. The in-plane wave vector component of kin = kosin(γ) (shown with green arrow in Figs. 1(b) and 1(c)) at inclined incidence will influence the generated SPP wave vector direction. However, the magnitude of the SPP wave vector kspp is fixed, which is only determined by the dispersion curve [33]. With the fixed magnitude and direction of kin, the fixed magnitude of kspp, the relationship formula between the incident light angle and the generated SPP wave vector direction can be expressed as, in Fig. 1(b),
In Fig. 1(c),Both in Figs. 1(b) and 1(c), when the incident light is the left oblique incidence, the angle between the long axis of the equivalent virtual groove and positive y axis (i.e., the half cross angle of the two intersecting grooves) is given by,When the incident light is the right oblique incidence, the angle between the long axis of the equivalent virtual groove and positive y axis is given by,From formulas (3) and (4), one can be seen that the long axis of the equivalent virtual groove is determined not only by whether the incident light is at right oblique incidence or at left oblique incidence, but also by whether the ko is on the x-z plane or y-z plane.The FDTD software is used to analysis the modulating principle. The theoretically derived formulas are verified in three different approaches. In FDTD simulation, a perfectly matched layer absorbing boundary condition is adopted in the x-y-z directions to efficiently terminate the outer boundary of the computational region. The maximum mesh size is set to 10 nm to ensure a high simulation result. In the following discussion, in addition to the size of θ and γ, the other parameters of the grooves and incident light mentioned above remain unchanged.
Demonstration I: The simulated spatial phase distributions of the magnetic field on the x-y plane are given in Fig. 2. The deflection angle of the generated SPP wave vector direction with different incident angles can be obtained by Fig. 2. A single groove with θ = 45°, L = 10um is used as a coupling structure. The position of the groove is marked with a black line. The white solid arrow denotes the generated SPP wave vector direction at normal incidence. The white dotted arrow denotes the generated SPP wave vector direction at oblique incidence. The ko is on the x-z plane. The corresponding incident angles are: (a) normal incidence γ = 0°, (b) left oblique incidence γ = 47.9°, (c) right oblique incidence γ = 30.5°. According to formula (1), when the incident light is obliquely incident, the deflection angles of the generated SPP wave vector direction are: (b) α = 30° and (c) β = 20°, respectively. The calculation results are in good agreement with the simulation results, which proves the correctness of formulas (1).
Fig. 2 The generated SPP spatial phase distributions on x-y plane with different incident angle γ. A groove with θ = 45°, L = 10um is used as a coupling structure and is illuminated under (a) normal incidence, γ = 0°, (b) left oblique incidence, γ = 47.9° and (c) right oblique incidence, γ = 30.5°. ko is on the x-z plane.
Demonstration II: with the definition of the equivalent virtual groove mentioned previously, the simulated 2D SPP field distributions generated by the double groove structures with θ = 10° under left oblique incidence γ = 30° and by its corresponding equivalent virtual double groove structures with θ = 14.7° (calculated by formulas (1) and (3)) under normal incidence γ = 0° are given in Figs. 3(a) and 3(b), respectively. The generation mechanism of the 2D light field can be understood as the mutual interference results of two plane SPP waves. When the length of groove is larger than the wavelength of the SPP wave, the SPP wave generated by a single groove can be regarded as a plane wave [24], the two symmetrical SPP waves interfere with each other at the angle of 2θ. The 2D interference field is modulated by θ and can be indirectly modulated by the incident angle γ. The spatial period of the fringes formed by the interference of two SPP plane waves along the y axis can be expressed as:
Fig. 3 2D optical field distributions of the SPP wave generated by the double groove structure. (a) θ = 10°, under left oblique incidence, γ = 30° and (b) θ = 14.7°, γ = 0°. The normalized light intensity values versus y-axis at (c) x = 5 um, (d) x = 15 um. ko is on the x-z plane.
Comparing Figs. 3(a) with 3(b), the two light fields have good consistency. The normalized light intensity values versus y axis at x = 5 um and x = 15 um are given in Figs. 3(c) and 3(d), respectively. The curves quantitatively indicate that Figs. 3(a) agrees well with Fig. 3(b). The average spatial period of the interference fringes calculated through the curve of Fig. 3(d) is about 961.5 nm (corresponding to Fig. 3(a)) and 965.9 nm (corresponding to Fig. 3(b)). The corresponding relative errors are 0.4% and 0.04% to the theoretic value calculated by formula (5), which indicates that the two light fields have good consistency.
As has been experimentally demonstrated in previously reported literatures [14,15], the intensity distribution of this 2D SPP field has the characteristic of Bessel-like function (shown in Fig. 3(c)). The full width at half maximum of the central bright fringe is 638.3 nm at x = 5 um and 642.2 nm at x = 15 um, which indicates that the light field maintains a small divergence at a small propagation distance. The maximum non-diffracting distance can be estimated by using xmax = L/sin(θ), the two plane SPP waves constructively interfere within this distance [16]. Here, the xmax = 9 um/sin(10°) = 51 um. The simulated results shown in Fig. 3 demonstrate the correctness of the derived formulas (1) and (3).
Demonstration III: in this section, for the convenience of simulation, the double groove structures are set as shown in Fig. 4. The ko is still on the x-z plane, but the polarization direction is changed to y axis. This is equivalent to the incident wave vector in the y-z plane, and the polarization direction is along x axis as that shown in Figs. 1(a) and 1(c).
Fig. 4 Demonstration the correctness of formulas (2), (3) and (4). 2D optical field distributions of the SPP waves generated by double groove structures, (a) θ = 15°, under left oblique incidence γ = 5°, (b) equivalent virtual double groove structure with left groove θ = 10.4° and right groove θ = 19.6°, γ = 0°. (c) The normalized light intensity values versus x axis at y = 0 um.
The 2D SPP field distributions generated by the double groove structure with θ = 15° under left oblique incidence γ = 5° and by its corresponding equivalent virtual double groove structures with left groove θ = 10.4° and right groove θ = 19.6° (calculated by formulas (2), (4) and (3)) under normal incidence γ = 0° are shown in Figs. 4(a) and 4(b), respectively. In Fig. 4(a), although the two grooves are symmetric with respect to the y axis, due to the oblique illumination of the incident light, the wave vector directions of the two SPP waves generated by the left groove and right groove are not symmetrical about the y axis, which in turn leads to changes in the propagation direction of the interference light field. The corresponding deflection angle calculated by formula (2) with θ = 15°, γ = 5° is α = 4.603°. In Fig. 4(c), the normalized light intensity values versus x axis at y = 0 um is plotted. The curves quantitatively indicate that Fig. 4(a) and Fig. 4(b) have good consistency. Meanwhile, the position of the maximum intensity value of the central bright fringe at y = 0 um is x = −1.143 um for Fig. 4(a) and x = −1.128 um for Fig. 4(b). Using a simple geometric calculation, with respect to the y axis, the central bright fringe is deflected to the left side at an angle of αa = arctan(1.143 um/14 um) = 4.667° for Fig. 4(a) and αb = arctan(1.128 um/14 um) = 4.606° for Fig. 4(b). The corresponding relative errors are 1.4% and 0.1% to the theoretic value α = 4.603°. The simulation results are in good agreement with the theoretical ones, which proves the correctness of formulas (2), (3) and (4).
In summary, in this section, the relationship formula between the incident angle and the generated SPP wave vector direction is theoretically derived. Through the simulation results given by the above three different approaches, the correctness of the theoretically derived formulas are verified. Using this relationship formula, the feasibility of precise modulating the generated SPP wave vector direction by changing the incident angle is demonstrated.
3. Dynamic precise modulation of non-diffracting 2D SPP beams
In this section, the FDTD simulation results of dynamic precise modulation of two non-diffracting 2D SPP beams are given by changing the incident angle γ.
3.1 Dynamic modulation of non-diffracting 2D Bessel-like SPP beams
Figure 5 shows, with a fixed structure θ = 10°, the simulation results of the dynamic modulation of non-diffracting 2D Bessel-like SPP beams by varying the incident angle. From Fig. 5, one can see that as the wave vector direction of incident light ko changes from right oblique incidence (Fig. 5(a) γ = 15°) to normal incidence (Fig. 5(b) γ = 0°), and then changes to left oblique incidence (Fig. 5(c) γ = 30°), the number of interference fringes increases and spatial period of the fringes decreases. This is due to the fact that the equivalent half cross angle θ of the corresponding equivalent virtual double grooves increases from Figs. 5(a) to 5(c). According to formulas (1), (3) and (4), the equivalent half cross angle θ for Figs. 5(a) to 5(c) are 7.54°, 10°, 14.75°. According to formula (5), it is known that the fringe period decreases with the increase of θ. As previously demonstrated in Fig. 3, the average spatial period of the interference fringes can be precisely controlled by the incident light angle.
Fig. 5 Simulation results of the dynamic modulation of non-diffracting 2D Bessel-like SPP beams by varying the incident angles. The wave vector direction of incident light changes from (a) right oblique incidence γ = 15° to (b) normal incidence γ = 0°, and then changes to (c) left oblique incidence γ = 30°, respectively.
3.2 Dynamic modulation of non-diffracting 2D SPP bottle beams
Figure 6 shows the simulation results of the dynamic modulation of non-diffracting 2D SPP bottle beams by varying the incident angles. The 2D SPP bottle beams generated by using three groove structures under different incident angles: right oblique incidence γ = 30°, 20°, 10°, normal incidence γ = 0° and left oblique incidence γ = 10°, 20°, 30° are shown in Figs. 6(a) to 6(g), respectively. The three grooves are placed symmetrically about the x axis, and the half cross angle between two non-adjacent grooves is set to θ = 45°.
Fig. 6 The simulated results of the dynamic modulation of non-diffracting 2D SPP bottle beams by varying the incident angle. The half cross angle is set to θ = 45°. (a-c) right oblique incidence γ = 30°, 20°, 10°, (d) normal incidence γ = 0° and (e-g) left oblique incidence γ = 10°, 20°, 30°. (h) The curves of the normalized light intensity along x axis at y = 0 um.
The principle of generating 2D SPP bottle beams in this three groove structures can be explained by the principle given in reference [23]: the 2D bottle beam is formed by the interference of the three SPP plane waves excited by the three grooves, thus giving rise to a non-diffracting complex intensity distribution. The resulting interference pattern sets up a lattice of bottles with constant size and spacing. The size and number of the plasmonic bottles can be adjusted by the half cross angle θ between two non-adjacent grooves and the length of a groove. Thus, as indicated in reference [23], in order to obtain bottle beams with different sizes, different coupling structures with different θ must be designed and manufactured. However, in this paper, for a well-designed fixed structure, the dynamically accurate control of the bottle beam size can be achieved by simply changing the incident angle. Its basic principle can be summarized as follows: by changing the incident angle, the half cross angle θ of the two non-adjacent equivalent virtual grooves is indirectly modulated, and then the dynamic modulation of 2D SPP bottle beams is realized. The interference patterns result in a periodic modulation of the intensity along the x axis (shown in Fig. 6(h)), and the spatial period can be expressed as [23]:
For example, the half cross angles θ of the two non-adjacent equivalent virtual grooves corresponding to Figs. 6(c), 6(d) and 6(e) calculated by formulas (1), (3) and (4) are θc = 38.28°, θd = 45° and θe = 51.72°. Then the corresponding half cross angle θ is taken into the formula (6). The corresponding fringe periods are Δxc = 2.28 um, Δxd = 1.67 um and Δxe = 1.29 um. Meanwhile, the average fringe period calculated by the corresponding curves shown in Fig. 6(h) is Δxc’ = 2.33 um, Δxd’ = 1.70 um and Δxe’ = 1.30 um. The corresponding relative errors are 2.2%, 1.8% and 0.8% to the theoretic values calculated by formula (6), which indicates the simulation values agree well with the theoretical ones. Thus, with a fixed coupling structure, the SPP bottle beams with different periods can be generated by simply varying the incident angles.In summary, taking into consideration of the wave nature of the SPP, the dynamic control of 2D SPP wave can be divided into two steps in essence. The first step is to control the generated SPP wave vector direction by changing the incident angle during the process of coupling free space light into the SPP wave. The second step is to create the required light field distribution through the SPP wave interference [12,24,31]. Therefore, the precise control of the generated SPP wave vector direction is the basis for further modulation of 2D SPP field distribution. Meanwhile, by introducing a different wavelength, the interference pattern can be even more complicated. The interference pattern is not only related to the wavelength and the propagation direction of SPP waves, but also to the initial phase difference between multiple SPP waves, which can be seen by the Eq. (1) in the reference [36]. An intrinsic phase shift occurs at the single groove structure at metal-dielectric interfaces during the SPP excitation process [36]. However, the periodicity of two types of SPP wave fields given in this section is only related to the wavelength and propagation direction of SPP waves, and is independent of this phase shift. For a symmetrical coupling structure, due to the symmetry, the phase shifts generated by the two grooves in the SPP excitation process are equal, even if the incident angle changes. When two SPP plane waves interfere with each other, the two initial phase shift can cancel each other out.
4. Discussion
In this section, the relationship between the adjustment range of the generated SPP wave vector direction, the incident angle γ, the half cross angle of the two grooves θ and half cross angle of the two equivalent virtual grooves (θleft and θright) is discussed.
The SPP generation efficiency in a simple slit-groove structure at a metal and dielectric interface exhibits an intricate dependence on the groove width or slit width. The maximum generation efficiency approximately appears at w = 0.23λ [37,38]. Thus, the theoretical corresponding width of a groove for the maximum generation efficiency is w = 118nm in this paper. In our paper, the width of groove used is 200nm. Therefore, if the width of the groove remains constant, the light source with a longer wavelength can be used as the incident light to enhance the generation efficiency. Or the wavelength of the incident light remains unchanged, and the slit width can be properly reduced, which can also improve the generation efficiency. However, all the conclusions obtained in our paper, such as the relationship formula between the incident angle and the SPP propagation direction, are still correct. Reference [26] studied the polarization variant coupling efficiency for linearly polarized light under normal illumination. A narrow aperture in a metal film selectively scatters incident light that is polarized perpendicular to it, giving rise to the SPP. The coupling efficiency Ci of a narrow aperture (such as a groove) scales Ci∝cos2(θ) because the SPP emission pattern of an individual aperture is anisotropic. If the coupling efficiency is not considered, from formulas (1) and (2), when θ = 90° and γ = 90°, the maximum deflection angle of the SPP wave vector direction generated at oblique incidence with respect to normal incidence is arcsin(λspp/λo) = 72.4°, which is independent of whether the incident light is left oblique incidence or right oblique incidence. Although the coupling efficiency decreases with the increase of θ, due to the symmetry, the coupling efficiency of the two symmetrical grooves is the same under the oblique incident light. The amplitudes of the two generated SPP waves remain relatively equal, which is conducive to the formation of a stable interference field.
In order to further satisfy the concept of the equivalent virtual groove proposed in this paper, which can be used to effectively illustrate the principle of tailoring the SPP field distribution by changing the incident angle. The following additional constraints on formulas (3) and (4), θleft≤90°, θright≥0° are recommended.
Figure 7(a) shows the relation curve between the half cross angle (θleft or θright) of the equivalent virtual groove and the θ, when the incident angle γ = 90°. The ko is on the x-z plane. When the incident light is obliquely incident on the right, the condition θright≥0° is always satisfied (green curve), indicating that the incident angle can range from 0°≤γ<90°. When the incident light is obliquely incident on the left (red curve), when θ = 46.37°, θleft = 90°. When 0°≤θ<46.37°, the condition θleft≤90° is always satisfied, thus, the incident angle can range from 0°≤γ<90°. When 46.37°<θ<90°, to ensure that the condition θleft≤90° is satisfied, the incident angle γ has a maximum value and is given by γ = arcsin[λocot(θ)/λspp]. For example, when θ = 70°, the maximum incident angle γ = 22.5°.
Fig. 7 The relation curves between the maximum incident angle γ and the half cross angle of the symmetrical grooves. (a) and (c) The relation curve between the half angle θleft (or θright) of the equivalent virtual groove with the θ. (b) The relation curve between the maximum incident angle γ with the θ. (a) and (b) ko is on the x-z plane. (c) and (d) ko is on the y-z plane.
Figure 7(b) shows the relation curve between the maximum incident angle γ and the half cross angle θ. The ko is on the x-z plane. From the curve one can see that the maximum incident angle can be taken to 90°, when 0°≤θ<46.37°. However, when 46.37°<θ<90°, the maximum incident angle γ decreases with the increase of the θ.
Figure 7(c) shows the relation curve between the half cross angle θleft (or θright) of the equivalent virtual groove and the θ, when γ = 90°. The ko is on the y-z plane. When the incident light is obliquely incident on the left, the condition θleft≤90° is always satisfied (red curve), indicating that the incident angle can range from 0°≤γ<90°. When the incident light is obliquely incident on the right (green curve), when θ = 43.63°, θright = 0°. When 43.63°≤θ<90°, the condition θright≥0° is always satisfied, indicating that the incident angle can range from 0°≤γ<90°. When 0°≤θ<43.63°, to ensure that the condition θright≥0° is satisfied, the incident angle γ has a maximum value and is given by γ = arcsin[λotan(θ)/λspp.]. For example, when θ = 30°, the maximum incident angle γ = 37.24°.
Figure 7(d) shows the relation curve between the maximum incident angle γ and the half cross angle θ. The ko is on the x-z plane. From the curve one can see that the maximum incident angle can be taken to 90°, when 43.63°≤θ<90°. However, when 0°≤θ<43.63°, the maximum incident angle increases with the increase of the θ.
5. Conclusion
In conclusion, the relationship formula between the incident angle and the generated SPP wave vector direction is theoretically derived. The correctness of the derived formula is verified with three different approaches based on FDTD methods. By using this relationship formula, the generated SPP wave vector direction can be precisely modulated by changing the incident angle with a fixed structure. The relationship between the adjustment range of the generated SPP wave vector direction, the incident angle γ, the half cross angle of the two grooves θ is given and discussed.
Meanwhile, using the derived relationship formulas, the dynamic and precise tailoring of the field distributions of two 2D non-diffracting SPP beams (i.e., the Bessel-like SPP beam and plasmonic bottle beam array) by changing the incident light angle are realized and simulated demonstrated. The generation and the dynamic modulation mechanism of the above two 2D SPP beams without necessitating any structural changes are detailed discussed. The basic principle is as follows: the change of incident angle will cause the change of SPP wave vector direction generated by a single groove, and then affect the 2D light filed distribution generated by the interference of the two or three (or even more) SPP waves, and finally precise and dynamic control of 2D SPP light field distributions are achieved. Both the theoretical derivation and simulation results verify the correctness and feasibility of the proposed modulation principle and can deepen the understanding of the generation and modulation mechanism of the SPPs.
For a groove structure, SPP can be coupled towards two directions, approximately fifty percent incident light cannot be used to generate SPP wave. Unidirectional coupling is important to enhance efficiency [39–41]. Thus, in the following research work, a special micro/nano structure device can be designed, which cannot only realize the unidirectional excitation of SPP, but also can flexibly control the SPP light field.
Finally, this dynamic modulation approach can be used in planar plasmonic circuit or in near filed optical trapping, such as, routing optical signals to multichannel sub-wavelength wave guides [17], sorting and manipulating 2D near filed particle with specific size [22,23].
Funding
Natural National Science Foundation of China (NSFC) (61775140).
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