1. Introduction

Surface plasmon polaritons (SPPs) are mixed excitations of an electromagnetic field coupled to collective electronic excitations confined to propagate at the metal-dielectric interface [1]. To allow for phase (momentum) matching, the incident polarized TM (transverse magnetic) electromagnetic wave is used to generate SPPs on a metal film incorporated with a subwavelength slit(s), hole(s) or grating(s). SPPs have been shown to play an essential role in the extraordinary transmission of light through nanoholes and apertures pre-fabricated in metal films, as was first demonstrated by Ebbesen et al. [2]. In order to develop modern plasmonic technology, capabilities to control the SPP intensity and propagation direction are needed in developing applications, such as nanoscale optical waveguides [3], nanolithography [4], and biosensing [5].

Several groups have proposed various means to allow for the unidirectional excitation to facilitate the control and splitting of the plasmonic waves. By adding different surface nano-structures on both sides of the nanoslit, Gan et al. have proposed a device capable of confining and guiding light waves of different wavelengths propagating in opposite directions [6]. Bidirectional subwavelength splitters for SPPs in THz and near infrared domain have also been studied using similar structures respectively [7,8]. Using metallic gratings with a sub-wavelength aperture, directional control of SPPs and splitting has been demonstrated [9,10]. A new class of nanoscale plasmonic sources based on subwavelength dielectric cavities embedded in a metal slab has been proposed by Lerosey et al. These authors have also demonstrated the capability of controlling the phase and amplitude of the generated plasmonic waves by utilizing strong dispersion near the Fabry-Perot resonance [11]. Using simulations, the possibility of achieving the unidirectional excitation of SPPs using two subwavelength metallic slits with different effective refractive indexes have been demonstrated by Xu et al. [12]. Opez-tejeira et al. have also demonstrated both theoretically and experimentally an unidirectional nanoslit coupler by placing nanogrooves on one side of the nanoslit to unidirectionally excite the SPPs and to focus it on a chosen location [13]. Recently, P. Lalanne et al. have discussed the interaction between nano-objects at metal surfaces, and proposed a new unidirectional SPPs coupler [14].

The structures used for controlling the SPP intensity and the excitation direction described above can be classified in two categories: one is involved with using two wavelengths in structures having asymmetrical gratings, and the other is based on SPPs interferences using a single wavelength on structures containing a nanoslit (or multiple silts). We report here, a new structure, different from the above, that can effectively control the SPP intensity and the excitation direction, and hence can be readily adopted as a tunable unidirectional SPPs source.

2. A Tunable SPPs Source

The proposed structure depicted in Fig. 1 is composed of two metallic strips (or films). A nanoslit is fabricated in the top metallic film (film A) and lying below A is another metallic film (film B). Films A and B have unequal lengths, with the length of film B much shorter than that of A. The media in the slit of film A and between the two films is air. The structure used here resembles that used by Cui and He [15]; the configuration of our structure is, however, different from that of ref. 15 in two important aspects. First, the exciting light is illuminated from the top surface of film A, whereas, in ref. 15, the exciting light is incident on the bottom surface of B. Secondly, the transmitted intensity is monitored at the exit aperture of the nanoslit placed below film B, whereas in the present configuration the transmitted intensity is monitored at the ends of film B. As shown in Fig. 1, the thickness of film A is tA ( = 390 nm) and the width of film B is Wp; other parameters used in the study are as follows: the thickness of film B is tB ( = 300 nm), the width of the nanoslit in film A is Ws ( = 25 nm), the gap between films A and B is d ( = 44 nm), and the central distance between films A and B is D, whose value depends on the location of film B, movable along the horizontal (x) axis.

 

Fig. 1 Schematic diagram of surface plasmon polaritons source. All the structure parameter symbols are shown in the figure.

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With the development of nanotechnology, tuning the position of the metal flim B is possible. For example, the flying plasmonic laser head can be used to control distance between the two metal films [16]; and the capacitance of the double-metal film capacitor is as a reference to measure the distance [17]. Here, the medium between two Ag films is air. To avoid the impractical assumption of two free-standing metal films in the proposed structure, one could insert a dielectric layer of variable refractive index between the two metal films, for example, a tunable thermo-optic layer [18].

Both films A and B are made of silver, whose permittivity is${\epsilon }_{\text{m}}=-\text{48 .8 + i 3 .16}$(λ ₀ = 1 μm), and the frequency dependent dielectric constant of Ag is given by the Drude mode [19]:

Here${\epsilon }_{\infty }=4.2$ is the relative permittivity at infinite frequency, ${\omega }_{\text{p}}=1.335\times {10}^{16}$rad/s is the bulk plasma frequency, $\gamma =1.1\times {10}^{14}$ rad/s is the collision frequency, and ω is the circular frequency of the incident light. A plane wave of TM polarization (the magnetic field is perpendicular to the x-z plane) with wavelength λ ₀ = 1 μm impinges normally on the top of film A. Then, the SPPs generated in the nanoslit in film A readily flow onto the upper surface of film B, placed below A in the direction vertical to the nanoslit. After traveling in the cavity formed by film A and B, the SPPs can be coupled out of the cavity and propagate along the interface of film A and air [20]. We consider an infinite slit along the y axis (Fig. 1), thereby reducing the configuration of the present structure to be two-dimensional. The value of the wave vector$k_{\text{sp}}$of the SPPs propagating along the interface of film A and air is given by:

Here ${\epsilon }_0$ is the vacuum permittivity; $k_0$ is the value of the wave vector of the illuminating light in vacuum. A simple calculation readily shows that the real part of $k_{\text{sp}}$ is about equal to$1.01k_0$, and the propagation distance of the SPPs is several hundreds of microns.

Numerical solutions of the Maxwell’s equations are obtained using the finite difference time-domain method (FDTD) method. Around the computational domain, an anisotropic perfectly matching layer is adopted to absorb the outgoing electromagnetic wave [21].

3. Results and Discussion

The nanoslit in film A divides film B into two individual sections: one to the left, and the other to the right of the nanoslit. As film B moves horizontally from right to left, the length of each section changes. Representative pictures depicting the time-average magnetic distribution ${\left|H_{\text{y}}\right|}^2$calculated with W _p equal to 1.75 μm for four different values of D are given in Fig. 2 . Here$H_{\text{y}}$ is the magnetic field.

 

Fig. 2 Simulated time-average intensity distributions (${\left|H_{\text{y}}\right|}^2$) for (a) D = 0 nm, (b) D = 80 nm, (c) D = 260 nm and (d) D = 340 nm.

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The frame for D = 0 nm shows the intensity distribution of ${\left|H_{\text{y}}\right|}^2$, corresponding to film B lying symmetrically below the nanoslit. The magnetic field displays five modes that distribute with a uniform density throughout the cavity. As one moves film B away from the D = 0 nm position, the mode density becomes non-uniform on both sides of the nanoslit, albeit on each side it remains uniform. Moving film B to the left of the nanoslit with D = 80 nm, three modes with strong yet uniform density are found on the left side, and two weak and uniform modes on the right side. The strong mode density observed on the left corresponds to a greater brightness in comparison with that on the right end. The opposite case is observed as film B is further moved to the location corresponding to D = 260 nm, and finally a nearly uniform distribution of the mode density is observed on each end by further moving film B to the location with D = 340 nm. The intensity distribution of ${\left|H_{\text{y}}\right|}^2$ determines the excitation directionality of the SPPs, and unidirectional SPPs source is achieved for D = 80 and 260 nm.

To calculate the SPP intensity propagating along the film A and air, we carry out Fast Fourier Transform (FFT) to extract the SPPs field on the left side and right side, respectively [12]. The intensity as a function of D for the SPPs at the left side for three representative film lengths (1.65 μm, 1.75 μm, and 1.85 μm) is shown in Fig. 3 . For each film, as D is changed, the intensity exhibits an oscillatory pattern, with a period equal to about 340 ± 10 nm. As the film length W _p increases, the intensity maximum (or the minimum) shifts to a higher D value, without changing the period.

 

Fig. 3 Variation of the intensity of the SPPs propagating to the left side of the nanoslit with D for different W _p.

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The SPPs splitting ratio is referred to as the intensity ratio of the SPPs propagating leftward to rightward. Figure 4 shows that the change of the SPPs splitting ratio with D at different W _p. One notes that, for W _p = 1.75 μm, different variations of the splitting ratio verse D are observed, and the variation range is from 1/24 to 22. When the length W _p is equal to 1.59 μm and 1.93 μm, the splitting ratio is about equal to 1 and is also not sensitive to the change of D. For all film lengths, the splitting ratio varies periodically with D with a period approximately equal to 340±10 nm, similar to that found in the SPP intensity vs. D plot (Fig. 3.). For a large intensity splitting ratio(for W _p = 1.75 μm and D = 80 nm), the structre can be used to excite the SPPs unidirectionally.

 

Fig. 4 Variation of the SPPs splitting ratio with D for different W _p.

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The field distribution, as revealed by the density distribution of the modes as shown in Fig. 2, suggests that the cavity can be divided into two neighboring Fabry-Perot (F-P) cavities sharing a common face lying below the nanoslit. At resonance, each F-P cavity has a uniform field distribution of the modes as indicated in each side. For the symmetric configuration (D = 0), the two adjacent F-P cavities are equivalent, as the field distribution of the modes is uniform throughout; however, as film B is moved to D≠0 configurations, one has two nonequivalent F-P cavities. Changing D or the length of film B corresponds to tuning the F-P cavity length. The SPP intensity emerging from the left side of the cavity may be considered as the result of the interaction of the SPPs and the reflected SPPs inside the F-P cavities. For the D = 0 case, the length of each F-P cavity is W _p /2. For D≠0 configurations, the cavity lengths, W _pl and W _pr, respectively, for the F-P left and that right of the nanoslit are not equal. The sum of W _pl and W _pr is W _p, the length of film B. For a F-P interferometer with length L, the resonance condition for two waves in constructive interference is given by:

Using Eq. (3), one can calculate the effective cavity length L (or the refractive index n _eff inside the cavity). To accomplish this, we determine the W _p dependence of the modulus of the magnetic field of the $\left|H_{\text{y}}\right|$ inside the symmetrical cavity (i.e. that with D = 0). The result is shown in Fig. 5 , which exhibits two $\left|H_{\text{y}}\right|$ maxima at W _p = 0.9 and 1.59 μm, and two minima at W _p = 1.25 and 1.93 μm. The maximum $\left|H_{\text{y}}\right|$corresponds to constructive interference, and the minimum to destructive interference of the SPPs. Substituting the W _p value and other needed quantities to Eq. (3), and supposing φ unchanging with W _p, we obtain the value for n _eff to be 1.45 after a simple calculation.

 

Fig. 5 Variation of $\left|H_{\text{y}}\right|$ with the length of the film B (W _p)

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The concept of F-P interference is next invoked to calculate the period of the intensity and the splitting ratio tuning curve, as shown in Fig. 3 and Fig. 4. Assuming that the exited SPPs emitted at the bottom of the nanoslit propagate to the left, the transmission T for SPPs through the left F-P cavity can be obtained by the equation:

Here R ₁ and R ₂ are the reflectances for the two ends of the F-P cavity respectively, T ₁ and T ₂ are the transmittances, and ϕ ($=4\text{π}{n}_{\text{eff}}{W}_{\text{pl}}\text{/}{\lambda }_{\text{0}}$) is phase difference between two adjacent SPPs. It is supposed that the R ₁, R ₂, T ₁ and T ₂ are constant. Obviously, the period of T depends on that of the function ${\sin }^2(\varphi /2)$. Using the value of n _eff previously determined, we obtain the value of Λ for the period of T equal to 344.8 nm, which is in excellent agreement with the period observed in Fig. 3 and Fig. 4. The result clearly substantiates the validity of the F-P interferometer model introduced here to interpret the period of SPPs intensity data and splitting ratios.

The height of the nanoslit or the thickness of film A is expected to affect the SPP intensity. We show in Fig. 6 (a) the height (t _A) dependence of the SPP intensity (calculated at the left side, corresponding to the SPPs propagating to the left of the nanoslit for the symmetric configuration (D = 0 and W _p = 1.75 μm)). The SPP intensity reaches the maximum at t _A = 270 nm, suggesting that the t _A = 270 nm nanoslit height corresponds to a resonant cavity. At this slit height, and the SPPs is excited most efficiently. The slit height is, however, not expected to affect the splitting ratio. To verify the result we have repeated the calculation of the splitting ratio as a function of D with W_p = 1.75 μm for three different slit heights. As shown in Fig. 6 (b), the t _A dependence of the splitting ratio is negligible.

 

Fig. 6 (a) Intensity variation with t _A for the SPPs propagating to the right of the nanoslit for D = 0 and W _p = 1.75 μm (b). Variation of the SPPs splitting ratio with D for three different values of t _A with W _p = 1.75 μm

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4. Summary and Conclusion

We have introduced for the first time a new structure that can be used as a tunable unidirectional surface plasmon polaritons source. The SPP intensity and the splitting ratio are found to change periodically by tuning the position of one of the film. It is shown that using the present structure the SPPs can be unidirectionally excited with a large intensity splitting ratio. We have found that the SPP intensity depends on the slit height; however, the height dependence of the splitting ratio is negligible. We have demonstrated that the simulation results can be understood in terms of F-P interference. The new structure can provide one with an effective control over the intensity of the SPPs propagating in metal films with intended directionality.