1. Introduction

Surface plasmon polaritons (SPPs) are confined surface propagating electromagnetic fields at a dielectric-metal interface, which can act as energy and information carriers. Plasmonic devices and circuits provide a potential solution for miniaturized nanophotonic devices, which can, for example, help to improve the speed and bandwidth of the information processing system and the sensitivity of nano sensors. In a plasmonic circuit, the SPP launcher is crucial. Many types of SPP launchers have been proposed so far, which are designed based on different principles and to achieve specific functionalities or properties, such as the unidirectional SPP excitation [1–3], SPP beam shaping [4–8], broadband SPP excitation [9–11], and high excitation effciency [12–13].

Among these studies, special attention has been paid to the utilization of the polarization state of light to control the SPP excitation. For example, in [14], the propagation direction of SPPs excited by a nanohole array was controlled by changing the incident angle, the polarization state or the wavelength of the illumination light. In [15,16], the extinction ratio of SPPs excited by a nano slit to two opposite directions was controlled by the polarization state of an obliquely incident light or a dipole, and the mechanism was explained by superposition of the magnetic induction currents or spatial frequency spectra. In [17], a specially designed nanostructure consisting of subwavelength apertures was used as a polarization-selective SPP plane-wave source, whose principle is the interference of SPPs excited by different oriented rectangular nanoantennas. In [18,19], a geometric-phase metasurface was proposed, which can modulate the complex extinction ratio of the excited SPPs by controlling the combination of polarization helicity of the normally incident light. Despite the excellent performances of some of these SPP launchers, they utilized different specific principles and optical system configurations. Some needed oblique incident condition, some required complicated structures, and most of them did not explore the full controllability of external polarization modulation. These restrictions and limitations add difficulties to the realization of the practical optical system and restrict the applications of the proposed methods.

In this work, by thoroughly studying the polarization dependent SPP excitation from a fundamental point of view, we propose a general approach that can realize complete control of the complex extinction ratio (including amplitude and phase) of any two linearly independent SPP modes excited by any elementary SPP excitation architecture [provided that it satisfies the criterion of Eq. (2) as explained below] over a broad bandwidth. The complete control of the complex extinction ratio means that it can be tuned to any value in the extended complex plane. As an example, we construct a metaline consisting of Δ-shaped nanoantenna array, as shown in Fig. 1. Since the structure considered in this work is a linear array consisting of Δ-antenna meta units, we call it a metaline. Then the directional SPP excitation can be flexibly and dynamically tuned just by manipulating the combination of the incident eigen-polarization states, i.e., the incident polarization state. Technically, it suffices to simply tune the orientation angles of a linear polarizer (LP) and a quarter wave plate (QWP) in the optical setup, according to some trivial calculations. This may greatly simplify the design and application of SPP launchers and diversify their functionalities, such as tunable directional SPP excitation and complete unidirectional SPP excitation over a broad bandwidth. The SPP launcher with such property can be used in many situations, such as unidirectional SPP source and compact on-chip SPP interferometer, all of which can work over a broad bandwidth. Moreover, owing to the general principle of this approach, it can be easily extended to the control of the complex extinction ratio of any two linearly independent excited modes in many other linear optical system, such as two modes in a waveguide or two diffraction orders of a grating.

 

Fig. 1 Schematic illustrations of the excited SPP modes under the normal illumination of plane waves with (a) Ĵ_x polarization, (b) Ĵ_y polarization, and (c) an arbitrary combination of (a) and (b).

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2. Principle and experimental optical setup

We start with some definitions. Any polarization state of light can be decomposed into two orthogonally polarized components, whose unit vectors are denoted as Ĵ₁ and Ĵ₂. When a linear optical system is illuminated by a monochromatic plane wave with certain polarization state, there will be usually several linearly independent modes being excited. We can pick out any two of them if there are at least two modes which are presented as Ψ₁ and Ψ₂. For polarization state Ĵ₁, the modal excitation coefficients of these two modes can be expressed as α₁₁ and α₁₂. For Ĵ₂, they are α₂₁ and α₂₂. These four coefficients don’t change if the optical setup kept the same except for the polarization state of the incidence. If the incident plane wave has an arbitrary polarization state $C_1^{*}{\widehat{J}}_1+C_2^{*}{\widehat{J}}_2$, where C₁ and C₂ are both complex numbers satisfying |C₁|² + |C₂|² = 1 and the superscript * denotes the complex conjugation, then the complex extinction ratio of mode Ψ₂ over Ψ₁ can be written as

Therefore, if we want to control the complex extinction ratio of two excited SPP modes by incident polarization state, it is crucial to satisfy Eq. (2) in a practical optical system. There are several possible ways to achieve this goal, such as tilting the incident plane wave for highly symmetrical SPP excitation structures [15,16], or using metasurfaces under the incidence of chiral field [17–19]. Actually, the use of some complexly structured metasurfaces or the configuration of certain asymmetrical incident field are not the essential requests. Under the condition of normal incidence and line array configuration, in most cases, if the geometry of the nanoantennas do not possess C₂ symmetry, Eq. (2) can be satisfied. Such cases of broken symmetry can be found in [17–19]. As an example, here, we use Δ-shaped hole-type nanoantennas perforated in a 200 nm thick gold film, as shown in Fig. 1. The Δ-nanoantenna has a right angled isosceles triangle shape whose length of the hypotenuse is 800 nm. The spacing p between the Δ-nanoantennas is 500 nm. Under the normal illumination of a plane wave the metaline can excite two SPP modes propagating to the left and right sides, separately, which are denoted as ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$, as shown in Fig. 1. According to the symmetry in Fig. 1(a), when the incident plane wave has a polarization state of Ĵ_x (i.e., x-polarized), the modal excitation coefficients of the two excited SPP modes ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ are −α_x and α_x, respectively. In Fig. 1(b), when the incident plane wave has a polarization state of Ĵ_y (i.e., y-polarized), the coefficients of ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ are both α_y. More details about the definitions and calculations of the modal excitation coefficients of ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ are given in Section 3. In Fig. 1(c), the incident wave has an arbitrary polarization state of $C_x^{*}{\widehat{J}}_x+C_y^{*}{\widehat{J}}_y$, which is a coherent combination of Ĵ_x and Ĵ_y. C_x and C_y are both complex numbers and satisfy |C_x|² + |C_y|² = 1. So the modal excitation coefficients of ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ are −C_xα_x + C_yα_y and C_xα_x + C_yα_y, respectively. It is evident that α_x ≠ 0 and α_y ≠ 0, so that α_xα_y + α_xα_y ≠ 0, satisfying Eq. (2). Then, we can get the complex extinction ratio of the two oppositely propagating SPP waves as

Since α_x and α_y do not change for a given configuration of optical setup, we can get C_yx that contains all the information of a polarization state according to the desired Ext₊₋. The calculation procedure of the values of α_x and α_y will be explained in detail in Section 3. Then how to get the required C_yx in practice? In fact, in an optical setup it suffices to simply tune the orientation angles of a LP and a QWP to get the required C_yx. The schematic of exprimental optical setup is shown in Fig. 2(a). A laser beam first goes through a LP, and then its polarization state is modulated by a QWP to produce the required polarization state. How to calculate the required rotation angles of LP and QWP will be discussed in Section 4.

 

Fig. 2 (a) Schematic of the experimental optical setup for characterizing the SPP excitation. (b) SEM picture of a sample fabricated by focused ion beam milling.

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In order to get strong optical signal in experiment, we used a 20× objective lens [denoted as 20× Obj in Fig. 2(a)] to focus the illumination beam on the sample that was fabricated by ion beam milling. In order to get a plane wavefront and symmetrical illumination condition on the sample surface after this focusing, we placed the center of the sample as close as possible to the center of the beam waist. The scanning electron microscope (SEM) picture of a fabricated sample is shown in Fig. 2(b). To characterize the excited SPPs, two outcoupling grooves were milled symmetrically at the left and right sides of the metaline SPP launcher. Then the optical signals including the reradiated SPPs from the grooves as well as directly transmitted light through the Δ-nanoantennas were collected by another 20× Obj and recorded by a charge-coupled device (CCD). As shown in Fig. 7 and Fig. 8, the captured CCD image typically includes a central bright speckle (attributed to the directly transmitted light through the metaline) and two smaller speckles at the left and right sides (attributed to the SPPs propagating to the left and right sides of the metaline, respectively). According to the gray values of the speckles, we can quantitatively estimate the intensities and extinction ratio of the excited SPP modes.

3. Calculation of the modal excitation coefficients of ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$

SPPs can be regarded as a kind of surface guided mode on the metal-dielectric interface. So we can use the modal excitation coefficients to measure and compare the amplitude and phase of SPPs excited by a structure under certain illumination condition. In this section, we briefly introduce a method [20] used to extract the modal excitation coefficients of SPPs from the numerical result calculated by, for example, a commercial software FDTD Solutions [21]. This section is divided into three parts: the first part briefly explains the process of how to extract the modal excitation coefficients from the simulation result; the second part shows the geometric and material parameters as well as the boundary conditions set in the calculations; and the last part presents the calculated results of the modal excitation coefficients of the configurations in Figs. 1(a) and 1(b).

A planar metal-dielectric interface supporting SPPs is shown in Fig. 3. The SPP modes propagating along the positive direction and negative direction of x-axis can be denoted as ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$, respectively. They have only three electromagnetic components, which can be written as [20]

 

Fig. 3 Schematic of a dielectric-metal interface. The dielectric and metal are both infinitely thick. ɛ_d and ɛ_m represent the permittivities of dielectric and metal, respectively. The dielectric-metal interface is located at z = z′.

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If we compare Eqs. (6) and (7), we may find that the E fields of ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ are symmetric about yz plane. Since the metaline considered in our work is symmetric, the excited E field is anti-symmetric about yz plane under x-polarized illumination; and the excited E field is symmetric about yz plane under y-polarized illumination. That is why the modal excitation coefficients of ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ are opposite under x-polarized illumination and are the same under y-polarized illumination as shown in Fig. 1. Here, we denote ${\alpha }_x^{\pm }$ and ${\alpha }_y^{\pm }$ as the modal excitation coefficients of under the x- and y-polarized illuminations, respectively.

We should get the excited fields first if we want to calculate α_x and α_y. The electromagnetic field including SPP modes excited by the metaline was numerically calculated by a commercial software FDTD Solutions [21].

The simulation area is a cuboid. The x minimum, x maximum, y minimum, y maximum, z minimum and z maximum of the cuboid are −10 μm, 10 μm, −0.25 μm, 0.25 μm, −3 μm and 5 μm, respectively. The boundary conditions of x minimum, x maximum, z minimum and z maximum are set to PML (Perfect Matching Layer) that can absorb outgoing waves, so that the finite simulation area along the x and z directions can be regarded infinitely large in practice. The boundary conditions along the y axis are set to periodic since the structure is periodic along the y direction. The mesh order defined in the software of the model is set to 5, which means that the size of each grid is about 6 nm. Such mesh accuracy is enough to get accurate simulation results. As shown in Fig. 1, the gold-silica interface is located in the xy plane, and the gold-air interface is located in the z=200 nm plane. The center of the height of the Δ-nanoantenna’s hypotenuse is located on the z-axis, and the hypotenuse is located in the y=200 nm plane. The thickness of the gold film is 200 nm, which is much larger than the skin depth of SPPs whose vacuum excitation wavelength is larger than 700 nm. So the SPP modes supported by the gold-air interface in the structure shown in Fig. 1 can be regarded approximately the same as in the structure with infinitely thick gold shown in Fig. 3. The dispersive refractive index data of gold used in the simulations were chosen from the build-in experimental data called ’Au (Gold) - Johnson and Christy’ in the software. The data are also shown in Figs. 5(e) and 5(f).

We use a total-field scattered-field source (TFSFSource) in the model as the incident plane wave. The amplitude of the electric field of the source in silica is 1V/m. The polarization state of the TFSFSource is set to have x-polarization or y-polarization state to simulate the corresponding excited electromagnetic fields. The intensity distributions |E|² of the fields under these two illuminations are shown in Figs. 4(a) and 4(b). The plane of the field distributions is located at the xz plane, where the x minimum, x maximum, z minimum and z maximum are −10 μm, 10 μm, 0 μm and 5 μm, respectively. In Eq. (8), the bounds of z are infinite. However, the field of SPPs is very weak at z far away from the gold-air interface. For the excitation wavelengths ranging from 710 nm to 1000 nm, the z values ranging from 0 μm to 5 μm is enough for the integrals in Eq. (8). The calculated amplitudes and phases of modal excitation coefficients of ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ are shown in Figs. 4(c)–4(f).

 

Fig. 4 (a) |E|² distribution of the electromagnetic field in the xz plane excited by an x-polarized illumination. (b) |E|² distribution of the electromagnetic field in the xz plane excited by a y-polarized illumination. (c) The amplitudes of the modal excitation coefficients ${\alpha }_x^{+}$ and ${\alpha }_x^{-}$ of the excited ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ modes under x-polarized illumination. (d) The amplitudes of the modal excitation coefficients ${\alpha }_y^{+}$ and ${\alpha }_y^{-}$ of the excited ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ modes under y-polarized illumination (b). (e) The same as (c), but for the phases of the modal excitation coefficients. (f) The same as (d), but for the phases of the modal excitation coefficients. The red lines stand for ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and the blue lines stand for ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$.

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It is obvious that the modal excitation coefficients of ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ are zero in the half plane where y<0 and y>0, respectively. So we only plot the modal excitation coefficients of ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ for y>0 and y<0, respectively. For x-polarized illumination, we can see that the amplitudes of the modal excitation coefficients ${\alpha }_x^{+}$ and ${\alpha }_x^{-}$ of ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ are the same and their phases differ by π. It means that the modal excitation coefficients are opposite, and we can use ${\alpha }_x={\alpha }_x^{+}=-{\alpha }_x^{-}$ to simplify the notation. For y-polarized illumination, we can see that both the amplitudes and the phases of the modal excitation coefficients of ${\mathrm{\Psi }}_{\mathit{SPP}}^{+}$ and ${\mathrm{\Psi }}_{\mathit{SPP}}^{-}$ are the same. It means that the modal excitation coefficients are the same, and we can use ${\alpha }_y={\alpha }_y^{+}={\alpha }_y^{-}$ to simplify the notation. These two comparisons verify our statement above.

Using the method explained above, we can calculate α_x and α_y for all the wavelengths. The amplitudes and phases of the modal excitation coefficients for wavelengths ranging from 710 nm to 1000 nm with a step of 10 nm are shown in Fig. 5.

 

Fig. 5 (a) Amplitudes of α_x, (b) amplitudes of α_y, (c) phases of α_x, and (d) phases of α_y at different wavelengths. Real parts (e) and imaginary parts (f) of gold index at different wavelengths.

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4. Calculation of the rotation angles of the LP and QWP

The incident plane wave, which is the beam waist of a focused laser beam in our experiment, is propagating along positive z direction. So its Jones vector can be written as

 

Fig. 6 (a) Schematic illustration of ψ, where x and y axes represent the original coordinate system, and a and b represent the rotated coordinate system. The long axis of the vibration ellipse of the electric field is along a, and the short axis is along b. (b) Schematic illustration of the angle φ formed by the polarizing direction of the LP and the fast axis of the QWP, and the angle ψ_t formed by the fast axis of the QWP and the long axis of the vibration ellipse

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The real part of the electric field containing time harmonic part can be expressed in the rotated coordinate system as follows

The working wavelength of the QWP is denoted as λ_s. If the wavelength of the incident light is λ, then the polarization state of the light after going through the QWP can be expressed as

According to Eq. (12), we can get

At the working wavelength λ_s of a QWP, the combined polarization state could be any point on the surface of the Poincare sphere; however, at other illumination wavelengths, the farther it is away from λ_s, the narrower the area its polarization state could be on the surface of the Poincare sphere. However, there are other optical elements such as Pockels cell and liquid crystal variable retarder that could replace the QWP to break this limitation. Hence, with this method, it is obvious that one can completely control and dynamically tune the complex extinction ratio Ext₊₋ just by tuning the LP and QWP in an optical system. This may greatly simplify the design of SPP launchers and diversify their functionalities and tunability.

5. Results and discussion

In this section, we demonstrate that we can realize dual functionalites by applying this method to the sample shown in Fig. 2(b). The first function is to achieve tunable directional SPP excitation at a given wavelength, i.e., to realize arbitrary real extinction ratio of the two SPP modes. Here, we choose Arg(Ext₊₋)=0 because we cannot characterize the phase of Ext₊₋ in experiment. However, achieving arbitrary real extinction ratio is enough to validate the effectiveness of our approach. The second one is to realize complete unidirectional excitation of SPPs over a broad bandwidth.

To realize the first function of tunable directional SPP excitation at a given wavelength, it is aimed to achieve arbitrary real extinction ratio Ext₊₋ at this wavelength. The wavelength of incident light is chosen as 810 nm. To better demonstrate the tunability of the extinction ratio Ext₊₋, we plot arctan(|Ext₊₋|²) in Fig. 7. Then, when arctan(|Ext₊₋|²) is 0 or π/2, it refers to the situations of unidirectional SPP excitation to the left or right sides, respectively; when arctan(|Ext₊₋|²) is π/4, it means symmetrical SPP excitation to both sides; and any other value of arctan(|Ext₊₋|²) means asymmetrical excitation of SPPs. We increase linearly arctan(|Ext₊₋|²) from 0 to π/2 with an increment of π/60, i.e., with 31 steps. The required polarization states of the incident light and the corresponding rotation angles of the LP and QWP can be calculated trivially in the way explained in detail in Section 4. Fig. 7 show the comparison of the desired and experimental results. The blue line with triangular marks represents the desired values of arctan(|Ext₊₋|²), each of which corresponds to certain asymmetrical or symmetrical SPP excitation state. The red ellipses with arrows are visual illustrations of the required polarization states of incident light to achieve the desired Ext₊₋ of seven typical experimental points: 1, 6, 11, 16, 21, 26 and 31. The corresponding rotation angles of the LP and QWP to achieve these desired polarization states are calculated and given below the red ellipses. The green line with circular marks represents the experimentally measured arctan(|Ext₊₋|²). The insets in Fig. 7 show the CCD images of three special cases when the unidirectional or symmetrical SPP excitations happen. In our experiment, the intensities of the excited SPPs are estimated by summing up the gray values of the pixels inside each light spot at the location of the outcoupling groove. Then the intensities of the SPPs are used to calculate |Ext₊₋|². If we take a close look at Fig. 7, we can see that the polarization states of the points whose point number are N₁ and N₂ satisfying N₁ + N₂ = 32 are symmetrical and the rotation angles of LP and QWP of them are also symmetrycal about the middle point, 16, which represents y-polarization. This is because the extinction ratio of each point-pair is inverse, which also means that they satisfy arctan(|Ext_+−N1|²)+arctan(|Ext_+−N2|²) = π/2 or |Ext_+−N1||Ext_+−N2| = 1. These facts and the good agreement between the theoretical and experimental results well validate the effectiveness of our approach.

With the same metaline, we can also realize a broadband complete unidirectional SPP launcher. Theoretically, without the loss of generality, we demonstrate the realization of Ext₊₋(λ) = 0 in the wavelength range from 710 nm to 1000 nm with a step of 10 nm. The calculation of the rotation angles of LP and QWP for each wavelength can be carried out similarly as mentioned above. Fig. 8 shows the comparison of the desired and experimental results. The red ellipses, blue lines with triangular marks and green lines with circular marks have the same meaning as those in Fig. 7 except that the blue and green lines represent |Ext₊₋|² instead of arctan|Ext₊₋|². The two CCD images in the bottom of Fig. 8 show the unidirectional SPP excitation effect at wavelengths 710 nm and 1000 nm, respectively. This expriment also validates the effectiveness of our approach quite well.

Although the above two experiments have realized the desired functionalities relatively well, we may still find some small deviations between the experimental and theoretical results. This may be caused by several practical reasons, such as the deviation of the practical permittivity of gold film from the adopted values in calculation, the fabrication errors of the Δ-nanoantennas, and the alignment error of the sample center with respect to the center of the incident beam waist. But all of these problems can be overcome by improving the experimental techniques.

 

Fig. 7 Comparison of the desired (blue triangles) and the experimentally measured (green circles) arctan|Ext₊₋|² of the tunable directional SPP excitation. The required polarization states of incident light and the corresponding rotation angles of the LP (θ_LP) and QWP (θ_QWP) for seven typical experimental points are also given below the curves. The inset CCD images show the unidirectional and symmetrical SPP excitations at three special points.

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Fig. 8 Comparison of the desired (blue triangles) and experimentally measured (green circles) |Ext₊₋|² in a broadband range for wavelength from 710 nm to 1000 nm with a step of 10 nm for complete unidirectional SPP excitation. The required incident polarization states at different wavelengths are illustrated by red ellipses. The two CCD images in the bottom show the unidirectional SPP excitation effect at wavelengths 710 nm and 1000 nm. The y-axis stands for |Ext₊₋|².

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6. Conclusion

In summary, we have proposed a general polarization modulation method that can achieve complete control of the complex extinction ratio of two linearly independent SPP modes excited by an elementary SPP excitation architecture over a broad bandwidth. Just by adjusting the LP and QWP in an optical system to control the polarization state of incident light, the directional SPP excitation can be flexibly and dynamically tuned. By applying this method, we have constructed a metaline consisting of Δ-nanoantennas as a dual-function SPP launcher. Just by controlling the incident polarization state, the same device can work either as a tunable directional SPP launcher at a given wavelength or a broadband complete unidirectional SPP launcher. The experimental results agree well with the desired ones, validating the effectiveness and simplicity of our approach. More importantly, the basic model of the method reveals that the polarization state of the incident light can be used to control the complex extinction ratio of any two linearly independent excited modes in a linear optical system, such as two modes in a waveguide or two diffraction orders in a grating, under certain condition described by Eq. (2). Therefore, this method may have wide applications not only in plasmonics but also in many other applicational fields of optics.