1. Introduction

Surface plasmon polaritons (SPPs) are evanescent electromagnetic waves coupled with collective free electron oscillations at a metal/dielectric interface and propagating along the metal/dielectric interface [1–4]. SPPs have attracted researchers’ interest because of SPPs’ special properties: a short wavelength, strong enhancement and local existence. SPPs are now widely applied in many applications: subwavelength optics [5–7], waveguide [8, 9], plasmonic splitter [10], lenses [11, 12], optical tweezers [13], microscopy [14–16], biophotonics [17], data storage [18], photonic circuitry [19,20], all-optical transistors [21, 22], etc.

Recently, research studies have been focused on achieving a unidirectional launching of SPPs for reducing the size of all-optical devices and applications to all-optical transistors [23, 24]. Many researchers have designed complex micro/nano structures for the unidirectional launching of SPPs, such as multi-materials in structures [25, 26], asymmetric structures [24, 27] and multi-groove structures [28, 29]. Later, much simpler structures were proposed, such as single slit [30], double nanogroove structures with different heights [31], and two magnetic antennas [32]. Studies have come to focus on dynamically controlling the direction of SPP waves. Hence, the concept of changing the incident light field with fixed micro/nano structures came into existence. For example, Lin et al. changed the polarization of the incident field [33], Liu et al. changed the topological charge of the incident field [34] and many others changed the incident angle [23, 35, 36] for the control of the direction of SPP waves. Dynamically controlling the direction of the SPP waves with both a simple structure and a simple light field makes plasmonic all-optical devices more accessible. In this paper, we achieve excitation of SPPs of a tunable direction and intensity ratio on a designed two thin slit structure, by simply modulating the phase difference between the incident dual fundamental Gaussian beams. That is, dynamically controlling the direction of SPPs can be achieved by dynamically modifying the phase difference of two incident fundamental Gaussian beams. The proposed method shows great potential in these applications: an SPP launcher, nanophotonic and plasmonic devices, plasmonic circuits and all-optical devices.

2. Design of the two slit structure and the formalism of dual Gaussian beams

The nano-structure this study adopts is a two thin slit structure with a designed slit separation. Figure 1 shows the simple double slit structure that this study adopts. It is composed of a silicon dioxide substrate and a thin silver layer with two slits on the substrate. As Fig. 1 shows, the incident optical field is normally incident from the substrate side, and it excites SPPs on the silver/air interface. The silver thickness should be greater than the skin depth of the incident wave to prevent strong remaining incident field energy on the silver side. Similar to Lalanne’s approaches [37], this study treats the two slits as two SPP sources emitting SPP waves on both sides of each (i.e. right side and left side). In this method, the control of the resulting SPP wave direction does not rely on the choice of slit width. However, the width of the two slits should be much smaller than the incident wavelength. When two slits have too wide a width, this will widely depart from the basic assumption of this study and cause an unwanted shift of the SPPs from the two slits. The slits’ lengths L relates to the breadth of the reactant SPP wave along the y-direction. The smaller measurement between the incident optical field’s diameter and the slits’ length L decides the breadth in which the SPP wave can be excited along the y-direction.

 

Fig. 1 Schematic diagram of a double slit structure and the definition of coordinate: (A) the top-view, and (B) the side-view. The optical field is normally incident from the substrate side and is propagating along the + z axis.

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The design of the separation between the two slits, d, is clarified in this paragraph. The mechanism of the constructive/destructive interference of SPP waves on two sides (i.e. right side and left side) achieves the resultant unidirectional propagating SPPs leaving a two- slit structure. The phase difference between the two right-propagating SPP waves and the two left-propagating SPPs waves from two SPP sources are denoted by symbols ${{\displaystyle \Delta \varphi }}_R$ and ${{\displaystyle \Delta \varphi }}_L$, respectively. They are:

In order to achieve unidirectional propagation of the resulting SPPs, it is necessary to create the situation in which SPP waves have a constructive interference in one direction and a destructive interference in another direction at the same time. When choosing the right side as the propagating direction for the resultant SPP wave, the phase difference between the two right-propagating SPP waves and the two left-propagating SPPs waves from two SPP sources should be:

When choosing left as the propagating direction of the resultant SPP wave, one can simply exchange the right-side values of Eqs. (3) and (4) and find the same d value. That is to say, no matter which direction the resultant SPP wave propagates, it is unnecessary to change the d value in Eq. (5). Referring to Eqs. (1) and (2) with fixed two-slit structures (i.e. d value), we can dynamically change the resultant SPP wave propagating direction (i.e. changing the phase difference ${{\displaystyle \Delta \varphi }}_R$ and ${{\displaystyle \Delta \varphi }}_L$) by controlling the phase difference between the two SPP sources (i.e. ${{\displaystyle \varphi }}_1-{{\displaystyle \varphi }}_2$) rather than change the slit separation d.

Substituting Eq. (5) into Eqs. (1) and (2), we can solve the suitable phase difference between the two SPP sources for the unidirectional launching of SPPs:

The phase of two SPP sources directly relate to the phase of the incident wave. For this method we use dual fundamental Gaussian beams as the incident waves. The transverse distribution of a dual Gaussian beam is shown in Fig. 2(A). The lateral interval between two Gaussian beams is D, and the center of the dual Gaussian beam coincides with the center of the double slit. The separation of two Gaussian beams, D, should be greater than the slit separation, d, which helps to prevent strong coupling between the left Gaussian beam and the right slit as well as the right Gaussian beam and the left slit. When describing a fundamental Gaussian beam field with $E(x,y,z;t),$ the incident field and dual Gaussian beams are expressed as:

 

Fig. 2 (A) Schematic diagram of dual Gaussian beam transverse position. (B) Simulated intensity distribution of dual Gaussian beams. (C) and (D): E-field real part and E-field imagined part of the dual Gaussian beams at the moment that the energy of one Gaussian beam is completely stored in the E-field real part, and the energy of the other one is completely stored in the E-field imagined part. The white double-headed arrows in the upper right hand corners of Figures (B), (C) and (D) indicate the beam’s polarization direction, i.e. x-polarized.

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With the help of the Mach-Zehnder interferometer [38], it is easy to construct a dual Gaussian beam with a controlled lateral interval and an adjustable phase difference between two fundamental Gaussian beams. Figure 2(B) show simulated intensity distribution of dual Gaussian beams. Figures 2(C) and 2(D) show simulated E-field real part and E-field imagined part of the dual Gaussian beams at the moment that the energy of one Gaussian beam is completely stored in the E-field real part, and the energy of the other one is completely stored in the E-field imagined part. The situation that Figs. 2(C) and 2(D) shows exists while the phase difference between the two Gaussian beams is π/2. The double-headed arrows in Figs. 2(B) to 2(D) represent the direction of the incident beam’s polarization.

3. Simulation results and discussion

This study uses the FDTD method to numerically prove the proposed method for the dynamic control of unidirectional launching of SPPs. The parameters used in the simulations are addressed as follows. The silver layer’s thickness, h, is 200 nm. The beam waist of Gaussian beams ${{\displaystyle w}}_0$ is 1.0 μm, wavelength $\lambda$ is 1.064 μm and the lateral interval between the two Gaussian beams, D, is 1.5 μm. The SPP wavelength λ_spp is obtained using the general equation for calculating SPP wavelength on the infinite planar dielectric-metal interface ${{\displaystyle \lambda }}_{spp}=\lambda \sqrt{\left({{\displaystyle \epsilon }}_d+{{\displaystyle \epsilon }}_m\right)/\left({{\displaystyle \epsilon }}_d{{\displaystyle \epsilon }}_m\right)}$ [1], where symbols ε_d and ε_m, denote the relative permittivity of dielectric and the relative permittivity of metal, respectively. In this study, the SPPs are designed in propagating across the silver/air interface. So, the ε_d value should be the relative permittivity of air, 1, and the ε_m value should be the relative permittivity of sliver. This study uses the experimental ε_m value [39], −48.809 + 3.159i. And, the calculated SPP wavelength is about 1.053 μm. The slit width b and slit length L are 80 nm and 4 μm, respectively. The separation, d, between the two slits is 750 nm (~$3{{\displaystyle \lambda }}_{spp}/4$), which is calculated using Eq. (5) with example values of M = 1 and N = 0.

Figure 3(A) shows the spatial relation between the incident dual Gaussian beams and the double slits. The right beam’s intensity peak is situated on the right side of the right slit, and the left beam’s intensity peak is situated on the left side of the left slit. Figures 3(B) and 3(C) show the resultant SPP wave intensity distributions while the phase difference between the two Gaussian beams, ${{\displaystyle \varphi }}_1\text{'}-{{\displaystyle \varphi }}_2\text{'},$ are π/2 and -π/2, respectively. In simulations, this study adjusts the phase difference between dual-Gaussian beams with the same concept of the practical experiment [38]. That is, the adjustment of the phase difference between two Gaussian beams is achieved by introducing a specifying propagating distance difference between two Gaussian beams. Just as what we predicted above, when making two incident Gaussian beams possessing specific phase differences π/2 and -π/2, we can successfully achieve the unidirectional launching of SPPs. Also, the easy dynamic changing of the phase difference of two Gaussian beams makes the dynamic control of direction of the unidirectional launching of SPPs practicable. In this study, we use Zhang’s definition of the directional measurement of SPP waves [31]. The extinction ratio is defined as the intensity ratio between the SPP waves launching in two opposite directions. That is, the higher intensity divided by the lower one. Two detectors are situated next to two slits for the measuring of the SPP wave intensity. The right detector is situated on the right side of the right slit, and the left detector is situated on the left side of the left slit. i.e. positions $x=\pm 6\mu m.$ The extinction ratios in Figs. 3(B) and 3(C) reach high values around 115. The direction of the resultant SPP wave is very clear. SPP wave intensity patterns behave with no irregular divergence and no oblique propagation.

 

Fig. 3 Numerical simulation for double slits with an incident dual Gaussian beam (A) Transverse spatial relation between double slits and dual Gaussian beam, (B)(C) Simulation results of the resultant right-propagating and left-propagating SPP wave intensity distribution with a controlled phase difference between the two SPP sources. The phase difference values are labelled in the lower left hand corner of the two figures.

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There are two main reasons why the extinction ratio of this design is 115 and not the ideal value, infinity. The first reason is that the two slits are not perfect slits but are physical slits with a finite width. The second reason is that the right-propagating SPP wave from the left slit becomes weaker than the right-propagating SPP wave from the right slit. This is because the right-propagating SPP wave from the left slit propagates an additional distance for slit separation d. By the same token, so do the left-propagating SPP waves from two slits. Both reasons mean that the SPP waves cannot build totally constructive interference in one direction and totally destructive interference in another direction, so the extinction ratio can only reach 115, not the ideal value, infinity.

Adjusting the phase difference between two Gaussian beams cannot only control the resultant SPP wave direction but can also adjust the intensity ratio between two SPP waves. Figures 4(A)-4(H) show resultant SPP wave intensity distributions when ${{\displaystyle \varphi }}_1\text{'}-{{\displaystyle \varphi }}_2\text{'}$ values are 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2 and 7π/4, respectively. Adjusting the phase difference between two Gaussian beams directly influences the intensity of the two SPP waves. Figure 5 plots the intensity ratio of the right-propagating SPP wave to the left-propagating SPP wave in a decibel scale versus the phase difference ${{\displaystyle \varphi }}_1\text{'}-{{\displaystyle \varphi }}_2\text{'}.$ Fig. 5 exhibits the proposed method possessing a wide adjusting range of two resultant SPP intensity ratios: from −20 dB to 20 dB, which shows great potential in future plasmonic circuit applications.

 

Fig. 4 Resultant SPP wave intensity distributions when incident dual Gaussian beams have different phase differences: (A) 0, (B) π/4, (C)π/2, (D) 3π/4, (E) π, (F) 5π/4, (G) 3π/2 and (H) 7π/4.

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Fig. 5 The intensity ratio of the right-propagating SPP to the left-propagating SPP (in decibel units) vs. phase difference between dual Gaussian beams.

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The proposed method has high tolerance to the phase control of two Gaussian beams. Referring to Liu and Wang’s study [34], a threshold intensity ratio value for the directional SPP coupling, 2, is used. Take the threshold value as the reference. The proposed method can achieve unidirectional launching of SPPs even while it sustains a phase error greater than ± π/4.

4. Conclusion

This study provides an alternative approach to achieve tunable unidirectional launching of an SPP wave by adjusting the phase difference between the dual incident Gaussian beams. Adjusting the phase difference between two Gaussian beams cannot only control the resultant SPP wave direction but can also adjust the intensity ratio between the two SPP waves. This method exhibits a wide dynamic adjusting SPP intensity ratio range from −20 dB to 20 dB, which shows great potential in future plasmonic applications, such as SPP launchers, plasmonic logic gates, plasmonic transistors and all-optical devices.

Numerous structured beams with various intensities, phases or polarization distributions have been well developed [38, 40]. The results of this study highlights the possibility of “the dynamically control of SPP wave with adjusting incident structure beam.” With the proper design of a plasmonic coupler, according to the property of structure beams, it offers good potential to make “the dynamically control of SPP wave with adjusting incident structure beam” and this is worth further investigation.