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The transmission of normally incident plane wave through an array of subwavelength metallic slits modified by perpendicular cuts has been explored. The cuts in middle of slit affect the even and odd modes of slits in different manners. To the best of our knowledge, this is the first work to investigate the influence of cuts on higher modes of slit, which is quite different from that on the fundamental mode studied in previous papers. Shifting the cuts along the vertical slits, we can get two kinds of resonances, which can be excited when the cuts locate at the center of electric or magnetic antinodes. In addition, we propose a new explanation model, in which we ascribe the resonance to three reasons: the F-P cavity theory, the surface current flow, and the surface charges. Irrespective of even and odd modes, the increase in the length of current flow or decrease in ability of accumulating charges dominates when the cut lies at different locations, which corresponds to a red or blue shift of resonant wavelengths. All calculated results are well explained by our proposed model.
©2009 Optical Society of America
1. Introduction
Enhanced transmission through subwavelength metallic openings has inspired great interest [1, 2, 3, 4] since the work of extraordinary optical transmission through (EOT) a subwavelength hole array [5]. Besides of subwavelength hole arrays [6, 7, 8, 9, 10], slit and slit array [11, 12, 13, 14, 15, 16, 17, 18, 19, 20] are topics of considerable fascination to achieve extraordinary high transmission in several applications [21, 22, 23], including superlenses and optical filters. The initial work owes the EOT to surface plasmons, which can be modulated by adjusting both surfaces of slits or apertures in optical, microwave, and terahertz regions. Slits support a propagating transverse electromagnetic wave without cutoff, and the high transmission of slits was due to the Fabry-Perot (F-P) interferences for thick plate. The F-P cavity theory emphasizes the depth of slits. Up to now, most researchers put their emphasis on the surface configuration of metal plate, periodicity or depth of slit arrays or holes. However, the dependence of transmission on conformation inside apertures is not investigated systematically.
Lockyear et al. have investigated the transmission of a normally incident plane wave through a singly stepped subwavelength slit in a metal plate [24]. The existence of the step, which locates in the middle of slits, increases the resonant wavelength substantially. The fields at resonant frequencies show all of the transmission maxima correspond to the fundamental F-P mode. In this paper, the transmission spectra of periodic slits modified by perpendicular cuts in a metal plate are studied in the near infrared region. There exist three higher order F-P modes in the considered wavelength range. The existence of cuts in the middle of slits decreases the resonant wavelength of even order F-P mode, which is quite different from the case of odd order mode (The fundamental mode with mode number 1 also belongs to odd order mode). Remarkably, the resonant wavelength does not change monotonically, which depends on the location of perpendicular cuts. Widening the cuts at the center of slits, the resonant wavelength keeps nearly fixed for even order modes, while increases for odd order modes. These findings can not be interpreted by the established theories and raise new questions about the higher order resonances.
This article is arranged as follows: Section 2 describes the the dispersion equation of the metal and the structure of periodic slits with perpendicular cuts. Section 3 presents the calculated transmission spectra by the finite-difference time-domain (FDTD) method and corresponding analysis. Finally, a conclusion is drawn in Section 4.
2. Metallic structures and numerical tools
Schematic view of a unit cell of the one-dimensional metallic grating is shown in Fig. 1. The period and thickness of the grating are p = 900nm and h 1 = 1800nm, respectively. The grating slit, with a width w 1 of 90nm, is modified by a perpendicular cut as shown in Fig. 1. The parameters of cuts in this work are chosen to disturb the grating slightly, hence the cut width w 2 and thickness h 2 are changed within a small range.
Two-dimensional electromagnetic fields inside and near the metal film are simulated by the FDTD [25, 26] in this paper. The calculated region is truncated by using perfectly matched layers on the top and bottom boundaries. The left and right boundaries are treated by periodic boundary conditions due to the periodicity of the structure. The incident light is along the y direction with TM polarization, (i.e., the incident electric field E 0 is along the x direction), as shown in Fig. 1. When a pulsed plane wave is incident upon the periodic metal structure, the time-domain field E(t) is recorded on the output plane. Then the frequency spectrum E(ω) can be easily achieved via the Fourier transform method.
The frequency-dependent permittivity of the metal is expressed by the Drude model for the dielectric response of the metal which is mainly governed by its free electron plasmon
wherewp is the plasma frequency, γ is the absorption coefficient, and ω stands for the frequency. The metal investigated is silver and the considered wavelength range is 800 to 1800nm. The corresponding parameters are ω p = 1.37×1016 rad/s and γ= 3.21×1013 rad/s [27]. When ω < ω p, the real part of the permittivity is negative, the standard time iteration scheme of the FDTD method becomes unstable. However, this problem can be solved by introducing the current density [28, 29] into Maxwell’s equations.
Fig. 2. Magnitudes of electric and magnetic field distributions for three transmission peaks. The incident wavelengths are 972nm in (a) and (b), 1180nm in (c) and (d), and 1563nm in (e) and (f), respectively. The numbers of color scale bar are magnitudes of field amplitudes normalized to the illuminating electric field.
3. Analysis of transmission spectra
Three resonant wavelengths (972nm, 1180nm, 1563nm) are obtained by using the FDTD method for a bare array of slits in the considered wavelength range. As a reference, the magnitudes of static field distributions are displayed in Fig. 2. The field amplitudes are normalized to the illuminating electric field. The electric amplitudes in upper Figs. 2(a), 2(c), and 2(e) are obtained at a time T/4 (T is the time period) before the magnetic amplitudes shown in Figs. 2(b), 2(d), and 2(f). Obviously, the slits act as F-P cavities according to Ref. [30], and the mode number is N = 5 for λ= 972nm, N = 4 for λ = 1180nm, and N = 3 for λ= 1563nm.
When w 2 = 90nm and h 2 = 270nm, the dependence of transmission of the slit arrays with cuts on the vertical position h 3 of the perpendicular cuts is shown in Fig. 3. Three white dashed lines in the figure correspond to the resonant wavelengths for the bare array of vertical slits, which are 972, 1180, and 1563nm, respectively. The introduction of cuts has changed the wavelengths of transmission peaks, and formed three zigzag transmission ridges with respect to the depth h 3 and wavelength. Each turning of transmission ridges corresponds to a transmission maximum. The third transmission ridge, evolved from resonance wavelength around 1563nm, have two maxima on the left side of wavelength λ = 1563nm indicated by L1 and L2, and three on the right side (labeled by R1, R2, and R3). We can observe similar results in the first and second ridge. The transmission maxima L1 and L2 have the same resonant wavelength (1538nm) while R1, R2, and R3 have the other wavelength of 1701nm. This conclusion is also true for the first and second ridges in Fig. 3.
Fig. 3. Transmission as a function of wavelength and cuts depth h 3. The width of cuts is w 2 = 270nm, and the thickness is h 2 = 90nm. The dashed lines mark the positions of resonant wavelengths for the bare array of vertical slits. The number of the color scale bar is the transmittance through the metallic gratings.
We focus on the turnings where transmission reaches its maxima. For convenience, we just take the third ridge as an example. The turnings are marked by R1, R2, R3, L1, and L2 in Fig. 3. Figure 4 shows the electric and magnetic magnitude distributions for three turnings on the right side of the dashed line on the third zigzag transmission ridge. Figures 4(a) and 4(b) to the first transmission peak R1; 4(c) and 4(d) to R2, 4(e) and 4(f) to R3. When the cuts locate at the center of the magnetic field antinodes or the nodes of electric field, transmission reaches its maxima [24]. By contrast with Figs. 2(e) and 2(f), the existence of cuts affects electric amplitude slightly, whereas magnetic amplitude inside the cuts becomes much larger.
Magnitudes of field amplitudes for L1 and L2 are shown in Fig. 5, where the cuts locate at the nodes of magnetic field or antinodes of electric field. Because the electric antinodes are truncated by cuts, the electric field in slits becomes smaller than adjacent one. However, the magnetic field in slits has been changed slightly in comparison with Fig. 2(f).
The turnings in the other two transmission ridges have similar cut locations and field distributions. For the transmission maxima on the right side of white dashed lines, cuts lies at antinodes of magnetic field (or nodes of electric field), which influence magnetic field greatly while affect electric field trivially. For transmission maxima on the left side of the dashed lines, cuts exists at node of magnetic field (or antinodes of electric field), which influence electric field substantially while change the magnetic field mildly. Obviously, two types of turnings in the three transmission ridges correspond to two mechanisms in altering the resonant wavelength.
Fig. 4. Magnitudes of electric and magnetic field distributions for three transmission peaks labeled by R1, R2, and R3 in Fig. 3. Subgraphs (a) and (b) are field distributions for peak labeled by R1; (c) and (d) for R2; (e) and (f) for R3. The numbers of color scale bar are magnitudes of field amplitudes normalized to the illuminating electric field.
Fig. 5. Magnitudes of electric and magnetic field distributions for the two transmission peaks labeled by L1 and L2 in Fig. 3. Subgraphs (a) and (b) are field distribution for peak labeled by L1, (c) and (d) are for L2. The numbers of color scale bar are magnitudes of field amplitudes normalized to the illuminating electric field.
For the bare metallic slit array, current density standing waves are established on both metal walls of each slit by incident oscillating light [30]. The current flow on one wall of the slit must be opposite to that on the other wall. The charge density on the surfaces of slits are also formed a standing wave. Owing to the existence of current flow, the distribution of charge density has been rearranged on the both surfaces of each slit. The length of F-P cavity approximately equals to the distance of current flow on one wall, which is the extent of charge distribution in the y direction. The resonant wavelength of F-P mode in a bare slit array can be obtained by
where k = 2nπ/λ is the wave vector, n is the refractive index, L FP is the length of the cavity, θ is the total phase reflected at the ends of the slits and a N dependent value.
The existence of perpendicular cuts enlarge the length of current flow, so the effective length of F-P cavity L eff is lengthened. This increment in effective F-P cavity length can be expressed as δ J. The longer the length of current flow is, the larger δ J is.
For a bare array of vertical slits, a quasi standing wave of charge-density is established in the y direction on both metal walls of each slit by the oscillating current. The charge on one of the metal walls must be opposite to that on the other, which is like a parallel-plate capacitor. For a parallel-plate capacitor, the larger the space between two plates is, the smaller the capacitance becomes. The capacitance is the ability of accumulating charges.
Introducing cuts in slit array would diminish the ability of accumulating charges at the two metal walls, ignoring the small amount of charges on the metal surfaces of cuts in the x direction. Shortening the length of slits in the y direction also diminishes the ability of accumulating charges. From this point of view, we can regard introducing cuts to be equivalent to shortening the length of slits. The decrement of length is denoted by δ Q. The more the cuts affect the total charge quantity, the larger δ Q is.
Actually, when the cuts lengthen the effective length of current flow, they shorten the effective length of charge distribution. These two mechanisms take effect on the system simultaneously. For small cuts as perturbation, we may ascribe the effective F-P cavity length as three contributions as follows:
Substituting L eff into Eq. (2), we know that the resonant wavelength of slits with cuts becomes larger if δ J > δ Q; else, the resonant wavelength becomes smaller if δ J < δ Q.
From Maxwell’s equations, we can obtain the surface current and charge densities as:
where n is a unit normal vector directed from metal into air in slit, H and D are magnetic field and electric displacement, J and σ are surface current density and surface charge density. For a bare slit array, the J and σ are determined by Hz and Ex, respectively.
According to Eqs. (4) and (5), if the cuts locate at the center of antinode of magnetic field in slits (node of electric field), the current density affected by the cuts reaches its maximum, while the charge density influenced reaches the minimum. The resonant wavelength reaches its maximum. The transmission maxima of R1, R2, and R3 in Fig. 3 correspond to this kind of resonant peaks. On the other hand, if the cuts locate at the node of magnetic field in slits (antinode of electric field), δQ reaches its maximum and δJ has its minimum value. This kind of resonance corresponds to L1 and L2 in Fig. 3. The stronger field in the cuts location would dominate the final results. If the electric fields around the cuts are much stronger than the magnetic field, resonant wavelength will decrease, and vice versa.
Fig. 6. Dependences of transmission spectra on the thickness of cuts h 2 (a) (h 3 = 900nm, w 2 = 270nm) and on the width w 2 (b) (h 3 = 900nm, h 2 = 90nm). Each curve has been shifted vertically by +0.3 with respect to the previous one. Three dashed lines mark the resonant wavelengths of the bare slit array.
To testify the theory, we explore the transmission spectra by adjusting the thickness h 2 and width w 2 of the cuts respectively, for a fixed depth h 3 = 900nm. The results are shown in Fig. 6. The three black vertical dashed lines mark the resonant wavelengths of the bare slit array with mode number N = 5,4,3, respectively. In this case, the cuts are positioned in the middle of slits, where electric antinode for even F-P mode(here, N = 4), and node for odd mode (N = 3 and 5 in our paper). So the transmission peaks for N = 5 and N = 3 (the first and third zigzag ridges) are on the right side of the the corresponding resonant wavelengths (dashed lines in this figure) of the bare slit array; while the second peak for N = 4 is on the left side of the dashed line. The results agrees with our analysis well. Enlarging h 2 for a fixed w 2 = 270nm in Fig. 6(a), the charge density affected increases for N = 4 mode, the second transmission peak shifts to shorter wavelength. The increase of h 2 also increases the total current density, therefore the wavelengths of the first peak (N = 5) and third peak (N = 3) move to the right. Augmenting w 2 when h 2 = 90nm in Fig. 6(b), the total charge density affected does not change very much, so the wavelength of the second peak (N = 4) has slight variation; however, the first and third resonant wavelength increases greatly due to the increase in current flow length. That agrees well with our theory.
4. Conclusion
Transmission of slit arrays with perpendicular cuts is investigated. Especially, the influence of the cuts on higher modes of the slit is firstly studied. When the cuts locate at magnetic field antinode, the resonant wavelength is larger than that for bare slits; while the cuts locate at electric field antinode, the resonant wavelength becomes smaller. Enlarging the width and thickening the height of cut which is positioned in the middle of slit, it is found that the cut influences the odd order modes and even order modes differently. To interpret these results, we proposed a analysis model, in which we ascribe the resonance to three contributions: the F-P cavity theory, the surface current flow, and the surface charges. The proposed model well explained all the calculated results.
Acknowledgment
This work was supported by the National Basic Research Program of China (Grant 2006CB302901) and National Natural Science Foundation of China (Grant 10604042 and 10674038).
References and links
1. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002). [PubMed]
2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003).
3. T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature (London) 446, 517–521 (2007).
4. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 445, 39–46 (2007).
5. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667–669 (1998).
6. J. Dintinger, S. Klein, F. Bustos, W. L. Barnes, and T. W. Ebbesen, “Strong coupling between surface plasmon-polaritons and organic molecules in subwavelength hole arrays,” Phys. Rev. B 71, 035424 (2005).
7. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004). [PubMed]
8. F. Miyamaru and M. Hangyo,“Anomalous terahertz transmission through double-layer metal hole arrays by coupling of surface plasmon polaritons,” Phys. Rev. B 71, 165408 (2005).
9. D. X Qu and D. Grischkowsky, “Observation of a new type of THz resonance of surface plasmons propagating on metal-film hole arrays,” Phys. Rev. Lett. 93, 196804 (2004). [PubMed]
10. J. G. Rivas, C. Schotsch, P. H. Bolivar, and H. Kurz, “Enhanced transmission of THz radiation through subwave-length holes,” Phys. Rev. B 68, 201306 (2003).
11. S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 165–173 (2000).
12. Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601–5603 (2001). [PubMed]
13. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002). [PubMed]
14. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68, 125404 (2003).
15. K. G. Lee and Q-Han Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005). [PubMed]
16. M.-W. Tsai, T.-H. Chuang, C.-Y. Meng, Y.-T. Chang, and S.-C. Lee, “High performance midinfrared narrow-band plasmonic thermal emitter,” Appl. Phys. Lett. 88, 071114 (2006).
17. J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, and J. R. Sambles, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92, 147401(2004). [PubMed]
18. Y. Xie, A. Zakharian, J. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express 13, 4485–4491 (2005). [PubMed]
19. C. Cheng, J. Chen, Q. Y. Wu, F. F. Ren, J. Xu, Y. X. Fan, and H. T. Wang, “Controllable electromagnetic transmission based on dual-metallic grating structures composed of subwavelength slits,” Appl. Phys. Lett. 91, 111111 (2007).
20. P. Ginzburg and M. Orenstein, “Plasmonic transmission lines: from micro to nano scale with l/4 impedance matching,” Opt. Express 15, 6762–6767 (2007). [PubMed]
21. H. F. Shi, C. T. Wang, C. L. Du, X. G. Luo, X. C. Dong, and H. T. Gao,“Beam manipulating by metallic nano-slits with variant widths,” Opt. Express 13, 6815–6820 (2005). [PubMed]
22. D. B. Shao and S. C. Chen,“Surface-plasmon-assisted nanoscale photolithography by polarized light,” Appl. Phys. Lett. 86, 253107 (2005).
23. Z. Sun and H. K. Kim, “Refractive transmission of light and beam shaping with metallic nano-optic lenses,” Appl. Phys. Lett. 85, 642–644 (2004).
24. M. J. Lockyear, A. P. Hibbins, and J. R. Sambles,“Transmission of microwaves through a stepped subwavelength slit,” Appl. Phys. Lett. 91, 251106 (2007).
25. D. B. Ge and Y. B. Yan, Electromagnetic Algorithm: The Finite-Difference Time-Domain method, (Electronic Science and Technology University Press, 2003)
26. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, Boston, MA, 2005, third edition).
27. F. I. Baida and D. V. Labeke, “Three-dimensional structures for enhanced transmission through a metallic film: Annular aperture arrays,” Phys. Rev. B 67, 155314 (2003).
28. S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,” Phys. Rev. B 68, 045415 (2003).
29. D. B. Shao and S. C. Chen, “Numerical simulation of surface-plasmon- assisted nanolithography,” Opt. Express 13, 6964–6973 (2005). [PubMed]
30. A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “The resonant electromagnetic fields of an array of metallic slits acting as Fabry-Perot cavities,” J. Appl. Phys. 99, 124903 (2006).
