1. Introduction

The excitation of surface plasmon polariton on a planar metallic surface has been well studied and has many applications. Since the propagation number K_sp of surface plasmon along a air-metal interface is greater than the propagation number K ₀ in the air, using a grating coupler to couple propagating light is one of usual ways to excite surface plasmon on a metal surface. Measuring reflectance from a metal grating is also used to probe the dielectric function of the metal.[1, 2, 3] Recently, due to the discovery of extraordinary optical transmission through sub-wavelength hole arrays[4] in a metal thin film, one-dimensional metal gratings have been recognized to be a effective simplified approach to understand this phenomenon and numerous researches have been carried out to study the transmission and reflection properties of one-dimensional gratings on a metal thin film[5, 6, 7, 8, 9, 10].

Periodic or isolated structures on a metallic surface can modify the resonant condition and the surface plasmon modes. For example, to use a grating coupler to excite surface plasmon, the surface plasmon of the metal grating is not exactly the same as the surface plasmon of a planar metal surface. In the application of surface plasmon sensors in chemistry, biotechnology,and medicine, attachments of samples on the metal surface also modify the surface plasmon resonance. It is necessary to study the effects of grating structures on surface plasmon resonances. Total absorption of a shallow metal grating with subwavelength period has been described by Hutley and Maystre[11] and the reflection efficient is expressed as a simple function related with the incident angle of light and the grating depth. Surface plasmon in sinusoidal aluminum surfaces with 500 nm period at 650 nm wavelength has been studied by Popov et al.[12] and diffraction and absorption losses of surface plasmon waves propagating were investigated for various grating depths. There are also researches on the dispersion and optical properties of surface plasmon on various metallic gratings[13, 14, 15, 16, 17] and most of them concern deep-groove gratings or gratings with periods at the same order of wavelength.

With fast development of nanotechnology, periodic nanostructures with periods much less than the wavelengths of visible light become more and more common. It is essential to understand the sensitivity of the surface plasmon mode of a metal surface with nano-scale distortion. It is also interesting to have a better understanding about the localized property of surface plasmon and the effects of surface geometry on the local field enhancement. In this research, we study how nano-scale surface geometry affects the properties of the surface plasmon modes and investigated the mode properties of plasmonic structures. The coordinate-transformation differential method[18, 19, 20] is used to numerically calculate surface plasmon modes on a weakly disordered metallic surface. At such period comparable to wavelength, the complex Bloch wave number K changes little if the distortion is smaller than 20 nm. On the contrary, for a weakly distorted surface with period much smaller than wavelength, it is found that the Bloch wave numbers and field distributions of the surface plasmon modes are highly sensitive with the distortion of only several nanometers. In this situation the corresponding surface plasmon mode is highly localized compared with the surface plasmon mode of the planar silver surface. Three surface profiles, the sinusoidal profile, the groove-like profile, and the square-well-like profile, are investigated and they exhibit surface plasmon modes of wide variations. The exploration of these surface plasmon modes will result in the discovery of novel optical properties of both fundamental and practical importance and will help to manipulate the surface plasmons in metallic nanostructures.

2. Simulation method

 

Fig. 1. Three silver-air surface profiles. (a) sinusoidal profile, f(x) = h cos(2πx/d); (b)groove-like profile, f(x) = h(cos(2πx/d)+cos(3∙2πx/d)/9+cos(5∙2πx/d)/25); (c) square-well-like profile, f(x) = h (cos(2πx/d)-0.2cos(3∙2πx/d)+0.04cos(5∙2πx/d)).

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Coordinate-transformation differential method is used to simulate surface plasmon modes in periodic metallic structures. This method was originally developed by Chandezon et. al. [18, 19] to study the optical properties of the deep-groove metallic gratings and there is a detail introduction for this method in Ref. [19]. It uses a nonorthogonal curvilinear coordinate transformation to map the grating profile onto a flat plane and makes matching of the boundary conditions easier at the price of a more complex form of Maxwell equations. Fields above and below interface are represented by Fourier series of each region and are solved for their eigen-fucntions. However, in order to obtain only the bound modes, only the eigenfunctions with evanescent eigenvalues are chosen, so the matrices formed by eigenfunctions and Fourier components above and below interface are not square matrices. The fields above and below interface are represented by the sum of bound eigenfunctions and the coefficients of those eigenfunctions are obtained using singular value decomposition to match the boundary conditions. The surface plasmon mode is found by searching complex K to find the zero of the singular value. Here, we only solve for the TM mode (the magnetic-field vector is perpendicular to the plane), and the magnetic field H of a surface plasmon mode is written as H(x,y) = e^iKxH_K(x,y), where K is the Bloch wave number and H_K is the periodic magnetic field of the surface plasmon mode associated with K. The Bloch wave numbers K and field distributions H(x,y) are showed for the periodic structures of different heights, h. Since H(x,y) includes the factor of e^iKx, H(x,y) is not periodical. H(x,y) is not necessarily symmetric either, because the plasmon wave direction is predetermined in e^iKx and the coefficients of those eigenfunctions are complex numbers. Symmetric surface plasmon modes can be obtained by H_sym(x,y) = H(x,y)+H(-x,y), if they are needed. The maximum amplitudes of surface plasmon modes are normalized to one. The number of Fourier components is increased repeatedly to ensure convergence of the K (relative error < 10^-7) and surface plasmon modes.

Three air-silver profiles are investigated in the paper to demonstrate the sensitivity of surface plasmon for different surface shapes. For simplicity, the wavelength is fixed at 650 nm and the index of refraction of silver is 0.05 + 4.4i. The simplest one is the sinusoidal profile, f(x) = h cos(2πx/d). The groove-like profile, f(x) = h(cos(2πx/d)+cos(3∙2πx/d)/9+cos(5∙2πx/d)/25), contains the first three terms of a triangular groove and it looks more realistic since it is not likely to have a very sharp tip in nano-size structure. The square-well-like profile, f(x) = h (cos(2πx/d)-0.2cos(3∙2πx/d) + 0.04cos(5∙2πx/d)), looks like a square-well with slight unevenness and is another nano-structure which is often seen in man-made metallic nanostructures. The height h is chosen to be very shallow and it is less than 20 nm. Those profiles are designed to explore the sensitivity of the surface plasmon modes to very small variations of surface profiles.

3. Results and Discussion

 

Fig. 2. Complex map of Bloch wave numbers of three air-silver surface profiles at different heights. The number near each symbol is the height of the profile in nanometer. Grating period d is 500 nm and wavelength is 650 nm. The Bloch wave number is normalized to the grating wave number K_d = 2π/d.

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For the cases in which the period d = 500 nm is comparable to wavelength (λ = 650 nm), the Bloch wave number K and the amplitudes of the plasmon modes are shown in Fig. 2 and 3. Figure 2 is the complex map of K which is normalized to the grating wave number K_d = 2π/d. For three profiles, the Bloch wave numbers have small variations, but different paths. The real parts of Bloch wave number shift 7.3%, 14.3%, and 21.8% for the sinusoidal (h = 20 nm), the groove-like (h = 19.6 nm), and the square-well-like profiles (h = 20 nm), respectively. In the groove-like case and the square-well-like case, the Bloch wave numbers curve downward and upward, and it seems to relate to the sign of the third harmonic term in the profile. The imaginary part of the Bloch wave number in the groove-like case is zero for h ≈ 19.7 nm. A real Bloch wave number indicates that, if the transverse component of the propagating vector of incident light matches the Bloch wave number, the incident energy is efficiently coupled into the surface plasmon mode and the reflection efficiency tends to be very low. It is closely related to the total absorption phenomenon described by Hutley and Maystre.[11] On the other hand, the surface plasmon modes have different shapes and behaviors for different profiles. For the sinusoidal profile, the field concentrates at center for small height, then moves over to one side with increasing height. For the groove-like profile, the field always concentrates at the tip of the profile. For the square-well-like profile, the field concentrates at the edges of both sides.

 

Fig. 3. Movies of surface plasmon modes at different heights. (a) sinusoidal profile; (b) groove-like profile;(c) square-well-like profile. Grating period is 500 nm. The length unit is nanometer. [Media 1] [Media 2] [Media 3]

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Fig. 4. Complex map of Bloch wave numbers of three silver-air surface profiles at different heights as in Fig. 2. Grating period d is 100 nm.

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For the cases with period d = 100 nm, which is smaller than one-sixth of wavelength, the Bloch wave number K and the amplitudes of the plasmon modes are shown in Fig. 4 and 5. The normalized complex maps of K has much more variations than those of long period cases. In the sinusoidal case for heights lower than 20 nm, K moves from 0.158 K_d to 0.768 K_d and there is a sharp peak at h = 11.5 nm when K ≈ K_d/2. In the groove-like case and the square-well-like case, the Bloch wave numbers move faster with increasing height. Both of them have sharp peaks when K ≈ K_d/2 (h = 6.1 nm in the groove-like case and h = 4.9 nm in the square-well-like case), but in the groove-like case K continues to increase to K ≈ K_d and in the square-well-like case, K makes a sharp turn and the real part is close to K_d/2. Their imaginary parts became large in both cases and their field distributions of surface plasmons decay fast towards +x direction. For all three profiles, small distortions less than 20 nm have significant impact on the Bloch wave numbers of surface plasmon. The field distributions of the surface plasmon modes also change very much. For the sinusoidal profile, the fields concentrate at center for small height, then move over to the right side and become a double-wing structure after K > K_d/2. Because here K are much larger than the surface plasmon wave number K_sp of a planar silver surface, the field distributions of surface plasmon are highly localized around the metal surface. For the groove-like profile, the fields concentrate at the tip of the profile for small heights, then two bright spots appear at both sides of grooves. A bright spot also appears in the bottom of the groove. The fields become to spread out for height larger than 12 nm, since K is close to the grating wave number K_d and is equivalent to K - K_d for periodicity. For the square-well-like profile, the fields concentrate at the upper edges of both sides for small height, then the second spots appear at the bottom corners of both sides for h larger than 10 nm (close to the sharp turn in the complex map of K). The surface plasmon modes remain highly localized around the surface for large real part of K and decay very fast towards +x direction for large imaginary part of K.

 

Fig. 5. Movies of surface plasmon modes at different heights. (a) sinusoidal profile; (b) groove-like profile;(c) square-well-like profile. Grating period is 100 nm. The length unit is nanometer. [Media 4] [Media 5] [Media 6]

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Fig. 6. Complex map of Bloch wave numbers of three silver-air surface profiles at different heights as in Fig. 2. Grating period d is 20 nm.

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For the cases in which the period d = 20 nm is much smaller than the wavelength, the Bloch wave number K and the amplitudes of the plasmon modes are shown in Fig. 6 and 7. The general characteristics of the normalized complex map of K are similar to those of 100 nm period cases, only they are more sensitive to distortion height. Again, all of them have sharp peaks when K ≈ K_d/2, but they only takes heights h = 2.5 nm, 1.3 nm, and 1.07 nm to reach the peaks for the sinusoidal, the groove-like, and the square-well-like profiles, respectively. In the sinusoidal and the groove-like cases, Ks continue to increase near K ≈ K_d and the imaginary parts of Ks become large. Again in the square-well-like case, K makes a sharp turn and the real part is close to K_d/2 while this imaginary part becomes large. The special behavior of the square-well-like case is probably related to the geometry of the profile. There are four corners in the square-well-like profile and only two corners in the sinusoidal profile and the groove-like profile. The angular square-well-like profile is likely to result in a different trajectory. The field distributions of the surface plasmon modes also change even more dramatically and the field localization is more prominent for their Ks are larger than those in the cases of 100 nm period. It should be noted that at such small height, every kind of surface effects near metal surface plays a role in realistic situation. In this work, we only consider a clear-cut air-silver interface within the domain of classical bulk Maxwell equations. For such high-sensitivity of surface plasmon with period much less than wavelength, it may serve as an excellent tool to test the influence of surface effects of metallic nanostructures.

 

Fig. 7. Movies of surface plasmon modes at different heights. (a) sinusoidal profile; (b) groove-like profile;(c) square-well-like profile. Grating period is 20 nm. The length unit is nanometer. [Media 7] [Media 8] [Media 9]

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High sensitivity of surface plasmon for geometric structures of metallic surfaces is clearly demonstrated in the cases of the sinusoidal profile and the groove-like profile. The sinusoidal profile and the groove-like profile look similar and their difference is only on small higher harmonic terms, but the Bloch wave numbers and surface plasmon modes are quite different in all periods. The Bloch wave number K of the groove-like profile varies much faster than the K of the sinusoidal profile with increasing height. Naturally the Bloch wave number K influences the field distribution of the corresponding surface plasmon mode. The large real part of K results in high localization of the field around the metal surface and the large imaginary part results in fast decay in the +x direction. Surface plasmon modes also present rich features for nano-scale surface geometry. Conventional intuition from lightning-rod effect indicates that the enhanced field of surface plasmon tends to be on the area with sharp geometry. From our simulation results, it is clear that although surface plasmons have enhanced spots at the tip and edges, at different height enhanced spots of surface plasmons also appear at areas of smooth profile curves. Similar phenomenon is observed in the surface plasmon resonances of deep-groove metal gratings.[21, 14, 15, 16] The controllability of surface plasmon modes makes it possible to tailor the surface profiles of metallic nanostructures to specific field distributions of surface plasmon modes.

Recently, analytical equations describing the resonant transmittance and reflectance of a weakly, periodically modulated metal film have been investigated.[22, 23] The dielectric constant of the metal film is weakly modulated and the dispersion equation of surface plasmon is obtained. From equation (17) of Ref. [22] and equation (26) of Ref. [23], both works give K ≈ K_sp + c ∗ g ², where g is the amplitude of the modulation and g is very small. In order to compare with their analytical result, the grating height h in this work plays the same role as the amplitude of the modulation g. For cases in which grating period d =500 nm, the Bloch wave numbers of three profiles agree very well with the analytical result for h less than 10 nm. However, for cases with grating period d = 100 nm, the agreement becomes worse and for cases with grating period d = 20 nm, the changes of K seem to be proportional to h, not h ², for very small h. The large K_d must break the approximations leading to the analytical result.

4. Conclusion

To explore the effects of nano-scale surface geometry on the properties of surface plasmon, surface plasmon modes on a weakly disordered air-silver surface with subwavelength period are simulated by the coordinate-transformation differential method. For the long-period cases, the Bloch wave numbers are insensitive to small distortions. So for a grating coupler, which is usually with a period at the same order of incident wavelength, the Bloch wave number is little modified. On the contrary, the surface plasmon modes are highly sensitive in cases of periods much shorter than wavelength. Surface plasmon mode can be manipulated by controlling metallic surface profile to achieve highly localized field and plasmonic crystal can find numerous applications in nanophotonics. High sensitivity of the surface plasmon modes with short periods provides an approach to test the fundamental understandings, including surface effects, about metallic nanostructures.