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Superfocusing of surface plasmon polaritons by metal-coated dielectric probe of tilted conical shape Ngo Thi Thu,1,* Kazuo Tanaka,1 Masahiro Tanaka,1 and Dao Ngoc Chien2 1
2
Department of Information Science, Gifu University, Yanagido 1-1, Gifu 501-1193, Japan Faculty of Electronics and Telecommunications, Hanoi University of Science and Technology, 1-DaiCoViet, HaiBaTrung, Hanoi, Vietnam *Corresponding author: [email protected] Received February 22, 2013; revised April 9, 2013; accepted April 10, 2013; posted April 10, 2013 (Doc. ID 185550); published May 14, 2013
By the volume integral equation method, the metal-coated dielectric probes of tilted conical shape were investigated for nanofocusing of surface plasmon polaritons (SPPs). We consider the cases of incident radially polarized (RP) and linearly polarized (LP) Gaussian beams and found that the tilted SPP conical probe is valid for both incident linearly and RP beams. Compared to the other asymmetric structures reported so far that are valid for incident LP waves, the structure proposed in this paper is not only simple but also straightforward to obtain the nanofocused localized and enhanced optical field on the tip of incident LP beam. © 2013 Optical Society of America OCIS codes: (230.7370) Waveguides; (240.6680) Surface plasmons; (260.2110) Electromagnetic optics. http://dx.doi.org/10.1364/JOSAA.30.001113
1. INTRODUCTION The recent explosive progress in nano-optics has been based on the nanoscale local fields that are greatly enhanced due to the resonant properties of metal nanosystems. Surface plasmon polaritons (SPPs) in nanostructured systems enable one to observe propagation, interference, and imaging on the nanoscale [1]. The strongly nanolocalized optical fields induce many enhanced nonlinear-optical phenomena and have various prospective applications. Metal-coated dielectric conical probe supporting SPPs can perform the above-mentioned function by creating enhanced local fields at the tip [2,3]. This focusing technique is often called SPP superfocusing or nanofocusing so far [1,3–9]. The metal-coated dielectric probes of conical shape will be one of the fundamental devices in nanophotonics utilizing SPPs [3–18]. There are some problems in making the nanofocused optical fields at the tip of the probe. For example, the conical probe whose shape is rotationally symmetric around a center axis is only valid for an incident radially polarized (RP) field [3,11,18]. Therefore, the structure that is suitable for the incident linearly polarized (LP) field such as HE11 mode in the optical fiber is required. In order to solve the above problem, in the previous works, a variety of asymmetric structures have been studied and investigated, including an asymmetrically corrugated fully metal-coated probe [13], rectangular slits defects in the metal coating layer of the near-field scanning optical microscopy (NSOM) probe [14], offset apertured NSOM probe [15], and uncorrugated probes with many grooves [17]. All topographies appearing in those papers are based on symmetric conical structure. The asymmetric probe is created by adding or cutting something on the original symmetric probe. Besides, the results shown in those studies have not included the concrete values of the optical intensities. In this paper, we introduce a new, simple, and straightforward shape of an 1084-7529/13/061113-06$15.00/0
asymmetric metal-coated dielectric conical probe, i.e., the tilted conical shape. Furthermore, we evaluate the basic characteristic of the probe by using concrete values of the optical intensities. The proposed tilted metal-coated dielectric conical probe supporting SPPs can make the significantly enhanced local fields at the tip for the case of an incident LP beam. The characteristics of the enhanced local fields that are nanofocused near the probe tip have been studied by many authors using numerical techniques, including the finitedifference time-domain method [11,14,15], the finite element method [12], the finite integration technique [8], and the volume integral equation (VIE) method [3–6]. In the present study, the VIE method is used to investigate nanofocusing of SPPs by metal-coated dielectric conical probe with tilted (asymmetric) shape. Based on the VIE method we show the different behavior of nanofocusing between that by symmetric and that by tilted probes. Results provide insight into the mechanism of guiding and nanofocusing in complicated SPP probes and show how such nanofocusing can be achieved using a metal-coated dielectric probe of tilted conical shape. In this paper, we analyze the basic characteristics, i.e., optical field at the tip, of the metal-coated dielectric probe of tilted conical shape. We first show the dependences of the maximum enhanced local optical intensities at the tip on the degree of asymmetry of the tilted probe for both incident RP and LP beams for various coating metals. Then we vary the polarized angle of the LP beam and show the dependence of the maximum optical intensity at the tip on the polarized angle.
2. CONFIGURATION OF THE PROBE We consider the metal-coated dielectric probe of tilted conical shape shown in Fig. 1. The dielectric conical probe has a base © 2013 Optical Society of America
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Ex r; 0 A exp−r∕w2 cos β; E y r; 0 A exp−r∕w2 sin β; Ez r; 0 jA2x∕k0 w2 exp−r∕w2 :
Fig. 1. (a) Geometry of the tilted metal-coated conical dielectric probe. The conical dielectric structure has a base radius R and a height h. The metal coating has a thickness d. The tip of the probe is located at l; 0; h in the x; y; z coordinate system. Permittivities of the surrounding free space, the coating metal, and the dielectric of the conical structure are denoted by ε0 , ε1 , and ε2 , respectively. The shape of the tilted conical structure is characterized by distance l between the z axis and the tip. Both RP and LP Gaussian beams are normally incident to the x–y plane from negative z direction. The axis of Gaussian beam coincides with the z axis. RP and LP beams propagate along the z direction and polarization of the LP beam makes an angle of β with the x axis. (b) shows the discretized structure of only 28 layers near the tip of the probe. The whole structure is composed of 328 layers. The inset shows the tiny cube with δ × δ × δ dimensions.
radius R and was made of a dielectric with a relative permittivity ε1 ∕ε0 fabricated in the x; y; z coordinate system where ε0 is the permittivity of free space. The side of conical dielectric probe is coated with the thin metal with a relative permittivity ε2 ∕ε0 and thickness d. The surrounding is the free space. The conical probe has a height h. It is apparent that the conical probe shown in Fig. 1 is not rotationally symmetric about the z axis. In order to define the degree of asymmetry of the tilted conical shape, we employ the distance l between the tip and the z axis shown in Fig. 1. The case of l 0 corresponds to the symmetric probe and the cases of nonzero values of l correspond to the tilted probes. We first divide three-dimensional models of the probe into tiny cubes having dimensions of δ × δ × δ to solve the scattering problem shown in Fig. 1 by the VIE method [3]. Then we discretize the VIE by the moment method and finally solve the resultant system of linear equations numerically by generalized minimum residual method with fast Fourier transformation. The radial and z components of electric field of the incident RP Gaussian beam at z 0 in Fig. 1(a) can be written as [3,19,20] E r r; 0 2Ar∕w exp−r∕w2 ; Ez r; 0 h i j 4A 1 − wr 2 − exp−r∕w2 : k0 w
(1)
For the incident LP Gaussian beam, the x, y, z components of electric field at z 0 can be written as [21]
(2)
In Eqs. (1) and (2), k0 is the wavenumber in free space, w is the spot size of the beam, and r x2 y2 1∕2 . We assume that the RP and LP beam propagate along the z direction and the polarization (electric field vector) of the LP beam makes an angle of β with the x axis shown in Fig. 1(a). The beam amplitude is given by A 1. In this case, the RP and LP beams given by Eqs. (1) and (2) have the same incident energies at z 0. The RP and LP optical fields of the SPPs are excited along the asymmetric surface of the probe. The optical fields of the SPPs are nanofocused by the decreasing cross section of the probe along the propagation direction. In this paper, the wavelength λ is 633 nm and δ is given by k0 δ 0.05 (δ 5 nm). The coating metals are Au, Cu, Ni, and Al whose relative permittivities are given by ε2 ∕ε0 −13.8 − j1.08, −11.6 − j1.8, −14.7 − j9.9, and −56.1 − j20.9, respectively, and the relative permittivity of the dielectric (grass) is given by ε1 ∕ε0 2.25. The beam spot size is given by w λ and the thickness of coating metal is given by k0 d ≈ 0.27 (d ≈ 27.4 nm). Base radius of the asymmetric conical structure shown in Fig. 1 is given by k0 R 7.07 (R 712 nm). The probe has a height given by k0 h 16.4 (h 1652 nm). The values of the thickness d, the base radius R, and the height h are maintained in all cases through this paper.
3. NANOFOCUSED OPTICAL INTENSITIES We discretize the whole structure using cubes arranged in 328 layers parallel to the x–y plane [28 layers near the tip are shown in Fig. 1(b)]. Here we present numerical results how a RP or LP beam is focused by the metal-coated dielectric conical probe of tilted shape and show how a highly confined electric field at the tip is generated. Figures 2(a) and 2(b) show the distributions of the optical intensities by logarithmic scale on the x–z plane [logjEx; 0; zj2 ] and y–z plane [logjE0; y; zj2 ] for the symmetric probe for incident LP and RP beams, respectively. In Fig. 2(a), we can see that the optical intensity is not focused at the tip of the symmetric probe for the case of incident LP beam in both the x–z and y–z planes. However, the optical intensity is focused at the tip of this probe for the case of the incident RP beam in Fig. 2(b). These results can be explained by Fig. 3 that shows the illustrations of the electric field vector on the x–z plane of the symmetric probe. In Fig. 3(a), we can see that, for the case of the incident LP beam, the x component of the electric vectors of the excited SPP has the same direction on the right and left surfaces of the probe. Contrarily, their z component has the opposite direction on the right and left surfaces. The main component of the nanofocused electric field on the tip has the z direction. The electric field vectors that propagate along the right and left surface have opposite direction on the tip in Fig. 3(a). Therefore, the electric field vector is canceled at the tip of the probe. For the incident RP beam, the z component of the electric vectors of the excited SPP have same direction on the right and left surfaces, as shown in Fig. 3(b). Therefore,
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Fig. 2. Optical intensity distributions on the x–z plane and y–z plane for symmetric probe (l 0 nm) and tilted probe (l 300 nm) are shown in the logarithmic scale. The optical intensities of logjEx; 0; zj2 and logjE0; y; zj2 for symmetric probe for incident LP and RP beams are shown in (a) and (b), respectively. The optical intensities of logjEx; 0; zj2 and logjEl; y; zj2 for tilted probe for incident LP and RP beams are shown in (c) and (d), respectively. The coating metal is Au (ε2 ∕ε0 −13.8 − j1.08). The dielectric is glass (ε1 ∕ε0 2.25). Other parameters keep constant (d ≈ 27.4 nm, R 712 nm, h 1652 nm).
the electric field is not canceled and, moreover, the optical intensity is focused at the tip of probe. Figures 2(c) and 2(d) show the distributions of the optical intensities by logarithmic scale on the x–z plane [logjEx; 0; zj2 ] and y–z plane [logjEl; y; zj2 ] for the tilted probe (l 300 nm) for the incident LP and RP beams, respectively. With the tilted conical shape, we can see that the optical intensity for the incident LP beam is focused and enhanced at the tip in Fig. 2(c). Moreover, for the incident RP beam, the enhanced and localized optical intensity at the tip can be also realized for tilted probe [in Fig. 2(d)]. It means that the tilted conical shape is valid for both incident LP and RP beams.
Fig. 3. Illustration of electric field vectors on the x–z plane of symmetric probe for incident LP and RP beams are shown in (a) and (b), respectively.
4. DEPENDENCES OF MAXIMUM OPTICAL INTENSITIES ON THE DISTANCE l The dependences of the maximum enhanced optical intensities at the tip on the degree of asymmetry, i.e., distance l, are shown in Fig. 4. The nanofocused maximum intensity exists at a point just above the tip in the free space. Since the probe is composed by the cubes with size δ 5 nm in this paper, this point is located at 2.5 nm just above the surface of the tip cube. We use the value of the intensity at this point as the maximum of optical intensity in this paper. We investigated structures with four various coating metals, Au, Cu, Al, and Ni. Figure 4 reveals that the maximum optical intensity strongly depends on the distance l for all coating metals. For all cases of asymmetric structure, the distance l is varied from 0 to 600 nm under the condition that the other parameters of the shape are not changed. The results of the incident RP beam are shown in Fig. 4(a) for the cases of coating metals Au and Cu and in (b) for the cases of Al and Ni. It is found that symmetric probe (l 0 nm) is the most suitable for the incident RP beam and the maximum optical intensity decreases with the increase of distance l [shown in Figs. 4(a) and 4(b)]. For the metal Au, the maximum optical intensity is largest in all coating metals. Interestingly, Cu gives a large enhanced optical intensity of ∼2.0 × 104 , shown in Fig. 4(a) and rather large enhanced optical intensities can be also generated even by metals having large dissipations
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Maximum optical intensity (RP beam)
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Maximum optical intensity (RP beam)
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Distance l (nm)
(a)
(b)
Maximum optical intensity (LP beam)
Maximum optical intensity (LP beam)
Distance l (nm)
Distance l (nm)
Distance l (nm)
(c)
(d)
Fig. 4. Maximum of optical intensity at the tip jEj2 (a) and (b) for incident RP beam; (c) and (d) for incident LP beam. The coating metals are Au, Cu, Ni, and Al. The dielectric is glass (ε1 ∕ε0 2.25). The distance l is varied from 0 to 600 nm under the condition that the other parameters of the shape are kept constant (d ≈ 27.4 nm, R 712 nm, h 1652 nm).
that gives the largest maximum intensity increases with the increase of the dissipation of metal. It means that the dissipation of the metal is an important factor in generating large enhanced optical intensity at the tip. We show the results in Figs. 4(a) and 4(c) in the same intensity scale for Au metal coating in Fig. 5. The solid circles for RP beam
Maximum optical intensity
(∼1.0 × 103 for Ni and ∼8.2 × 102 for Al), shown in Fig. 4(b). For the incident RP beam, the maximum enhanced optical intensities in Fig. 4(a) for Au and Cu (small values of dissipation of metal) are much larger than those in Fig. 4(b) for Al and Ni (large values of dissipation of metal). For the incident LP beam, the dependences of maximum enhanced optical intensities on the distance l are shown in Figs. 4(c) and 4(d). We can see that the enhanced optical intensity at the tip cannot be realized for the symmetric probe (l 0 nm). For the case of the incident LP beam, there is an optimum distance l where the maximum optical intensity can be obtained at the tip of the tilted conical probe for all coating metals. The largest value of the maximum optical intensity jEj2max and the optimum distance l are shown in Table 1 for each coating metal. For Au, the largest maximum optical intensity can be obtained at the short distance l compared with that for Cu, Al, and Ni (see Table 1). We can see that optimum distance l
for LP beam
Table 1. Largest Value of the Maximum Optical Intensity for Incident LP Beam Coating Metal Au (−13.6 − j1.08) Cu (−11.6 − j1.8) Ni (−14.7 − j9.9) Al (−56.1 − j20.9)
jEj2max
l (nm)
5.7 × 103 2.9 × 103 1.7 × 102 1.4 × 102
290 410 430 450
Distance l (nm) Fig. 5. Maximum of optical intensity at the tip of Au-coating metal probe jEj2 for incident RP beam and incident LP beam in the same intensity scale. The distance l is varied from 0 to 600 nm under the condition that the other parameters of the shape are kept constant (d ≈ 27.4 nm, R 712 nm, h 1652 nm).
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Table 2. Maximum Optical Intensities jEj2 for Incident RP and LP Beams in the Region of 350 nm < l < 410 nm Distance l
410 nm
3
3.0 × 103 4.3 × 103
5.7 × 10 5.0 × 103
Maximum optical intensity (LP beam)
RP beam LP beam
350 nm
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paper, we investigated the basic characteristics of the maximum optical intensity at the tip created by nanofocusing in the SPP metal-coated conical dielectric probe with tilted shape (asymmetric shape) by the VIE method. We consider the cases of incident RP and LP Gaussian beams and found that the tilted SPP conical probe is valid for both incident LP and RP beams. The optical intensity at the tip of the tilted metal-coated dielectric probe strongly depends on not only the degree of asymmetry but also the dissipation of coating metal for both incident RP and LP beams. Since the degree of the asymmetric shape and the coating metal of the probe affect significantly the maximum enhanced optical intensity at the tip for incident RP and LP beams, the findings of this study demonstrate that tilted conical probes must be carefully designed and fabricated.
REFERENCES Polarized angle (C degree)
Fig. 6. Maximum of optical intensity jEj2 located at 2.5 nm from the tip of the tilted probe (l 300 nm, d ≈ 27.4 nm, R 712 nm, h 1652 nm) for incident LP beam with Au metal coating depends on the polarized angle β. In this simulation, the tilted probe has been kept constant while the polarized angle is changed from 0° to 90°.
and open circles present the results of incident RP and LP beams, respectively. The value of maximum optical intensities for incident RP and LP beams are shown at l 350 and 410 nm in Table 2. From Fig. 5 and Table 2, we can see in the region of l < ∼380 nm, the maximum optical intensities for the incident RP beam are always larger than those for the incident LP beam. In the region of l > ∼380 nm the maximum optical intensities for the incident LP beam exceed those for the incident RP beam. We can see the similar behavior for other coating metals from results in Fig. 4. This means that the RP beams does not always make larger optical intensity at the tip than that made by the LP beams in the tilted probe.
5. DEPENDENCE OF MAXIMUM OPTICAL INTENSITIES ON POLARIZED ANGLE Finally, we examined the effect of changing the polarized angle β (LP beam is propagating in the z direction and its direction of polarization makes an angle of β with the x axis) on the maximum of optical intensity at the tip of probe for incident LP beam with Au metal coating in Fig. 6. For the case under study, we chose the proposed tilted conical probe (distance l 300 nm) and change the polarized angle β from 0° to 90°. As shown in Fig. 6, the maximum optical intensity strongly depends on the polarized angle β of the incident LP beam. We can see that the enhanced optical intensity at the tip can be realized for the case of β 0°. In the region of 0° < β < 45°, the maximum of optical intensity at the tip decreases with the increase of the polarized angle. When β continues to rise in excess of 45°, the maximum optical intensity changes complicatedly. However, it is found that the most suitable characteristic is obtained for the case of β 0°.
6. CONCLUSION The possibility of using a metal-coated dielectric probe of tilted conical shape utilizing SPP was demonstrated. In this
1. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 74041 (2004). 2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). 3. K. Tanaka, K. Katayama, and M. Tanaka, “Optical field characteristics of nanofocusing by conical metal-coated dielectric probe,” Opt. Express 19, 21028–21037 (2011). 4. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82, 1158–1160 (2003). 5. K. Tanaka, G. W. Burr, T. Grosjean, T. Maletzky, and U. C. Fischer, “Superfocussing in a metal-coated tetrahedral tip by dimensional reduction of surface-to edge-plasmon modes,” Appl. Phys. B 93, 257–266 (2008). 6. K. Tanaka, K. Katayama, and M. Tanaka, “Nanofocusing of surface plasmon polaritons by a pyramidal structure on an aperture,” Opt. Express 18, 787–798 (2010). 7. N. A. Janunts, K. S. Baghdasaryan, K. V. Nerkararyan, and B. Hecht, “Excitation and superfocusing of surface plasmon polaritons on a silver-coated optical fiber tip,” Opt. Commun. 253, 118–124 (2005). 8. W. Ding, S. R. Andrews, and S. A. Maier, “Internal excitation and superfocusing of surface plasmon polaritons on a silver-coated optical fiber tip,” Phys. Rev. A 75, 063822 (2007). 9. W. Liu, D. N. Neshev, A. E. Miroshnichenko, I. V. Shadrivov, and Y. S. Kivshar, “Polychromatic nanofocusing of surface plasmon polaritons,” Phys. Rev. B 83, 073404 (2011). 10. W. Chen and Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007). 11. A. V. Goncharenko, M. M. Dvoynenko, H. C. Chang, and J. K. Wang, “Electric field enhancement by a nanometer-scaled conical metal tip in the context of scattering type near field optical microscopy,” Appl. Phys. Lett. 88, 104101 (2006). 12. N. A. Issa and R. Guckenberger, “Fluorescence near metal tip: the roles of energy transfer and surface plasmon polaritons,” Opt. Express 15, 12131–12144 (2007). 13. V. Lotito, U. Sennhauser, and C. Hafner, “Effects of asymmetric surface corrugations on fully metal-coated scanning near field optical microscopy tips,” Opt. Express 18, 8722–8734 (2010). 14. W. Nakagawa, L. Vaccaro, H. P. Herzig, and C. Hafner, “Polarization mode coupling due to metal-layer modifications in apertureless near-field scanning optical microscopy probes,” J. Comput. Theor. Nanosci. 4, 692–703 (2007). 15. M. C. Quong and A. Y. Elezzabi, “Offset-apertured near-field scanning optical microscope probes,” Opt. Express 15, 10163–10174 (2007). 16. V. Lotito, U. Sennhauser, and C. Hafner, “Finite element analysis of asymmetric scanning near field optical microscopy probes,” J. Comput. Theor. Nanosci. 7, 1596–1609 (2010). 17. T. J. Antosiewicz, P. Wróbel, and T. Szoplik, “Performance of scanning near-field optical microscope probes with
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single groove and various metal coatings,” Plasmonics 6, 11–18 (2011). 18. R. P. Zaccaria, F. De Angelis, A. Toma, L. Razzari, A. Alabastri, G. Das, C. Liberale, and E. Di Fabrizio, “Surface plasmon polariton compression through radially and linearly polarized source,” Opt. Lett. 37, 545–547 (2012).
Thu et al. 19. L. W. Davis and G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. 6, 22–23 (1981). 20. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979). 21. M. Mansuripur, Classical Optics and Its Applications (Cambridge University, 2009).
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Superfocusing of surface plasmon polaritons by metal-coated dielectric probe of tilted conical shape Ngo Thi Thu,1,* Kazuo Tanaka,1 Masahiro Tanaka,1 and Dao Ngoc Chien2 1
2
Department of Information Science, Gifu University, Yanagido 1-1, Gifu 501-1193, Japan Faculty of Electronics and Telecommunications, Hanoi University of Science and Technology, 1-DaiCoViet, HaiBaTrung, Hanoi, Vietnam *Corresponding author: [email protected] Received February 22, 2013; revised April 9, 2013; accepted April 10, 2013; posted April 10, 2013 (Doc. ID 185550); published May 14, 2013
By the volume integral equation method, the metal-coated dielectric probes of tilted conical shape were investigated for nanofocusing of surface plasmon polaritons (SPPs). We consider the cases of incident radially polarized (RP) and linearly polarized (LP) Gaussian beams and found that the tilted SPP conical probe is valid for both incident linearly and RP beams. Compared to the other asymmetric structures reported so far that are valid for incident LP waves, the structure proposed in this paper is not only simple but also straightforward to obtain the nanofocused localized and enhanced optical field on the tip of incident LP beam. © 2013 Optical Society of America OCIS codes: (230.7370) Waveguides; (240.6680) Surface plasmons; (260.2110) Electromagnetic optics. http://dx.doi.org/10.1364/JOSAA.30.001113
1. INTRODUCTION The recent explosive progress in nano-optics has been based on the nanoscale local fields that are greatly enhanced due to the resonant properties of metal nanosystems. Surface plasmon polaritons (SPPs) in nanostructured systems enable one to observe propagation, interference, and imaging on the nanoscale [1]. The strongly nanolocalized optical fields induce many enhanced nonlinear-optical phenomena and have various prospective applications. Metal-coated dielectric conical probe supporting SPPs can perform the above-mentioned function by creating enhanced local fields at the tip [2,3]. This focusing technique is often called SPP superfocusing or nanofocusing so far [1,3–9]. The metal-coated dielectric probes of conical shape will be one of the fundamental devices in nanophotonics utilizing SPPs [3–18]. There are some problems in making the nanofocused optical fields at the tip of the probe. For example, the conical probe whose shape is rotationally symmetric around a center axis is only valid for an incident radially polarized (RP) field [3,11,18]. Therefore, the structure that is suitable for the incident linearly polarized (LP) field such as HE11 mode in the optical fiber is required. In order to solve the above problem, in the previous works, a variety of asymmetric structures have been studied and investigated, including an asymmetrically corrugated fully metal-coated probe [13], rectangular slits defects in the metal coating layer of the near-field scanning optical microscopy (NSOM) probe [14], offset apertured NSOM probe [15], and uncorrugated probes with many grooves [17]. All topographies appearing in those papers are based on symmetric conical structure. The asymmetric probe is created by adding or cutting something on the original symmetric probe. Besides, the results shown in those studies have not included the concrete values of the optical intensities. In this paper, we introduce a new, simple, and straightforward shape of an 1084-7529/13/061113-06$15.00/0
asymmetric metal-coated dielectric conical probe, i.e., the tilted conical shape. Furthermore, we evaluate the basic characteristic of the probe by using concrete values of the optical intensities. The proposed tilted metal-coated dielectric conical probe supporting SPPs can make the significantly enhanced local fields at the tip for the case of an incident LP beam. The characteristics of the enhanced local fields that are nanofocused near the probe tip have been studied by many authors using numerical techniques, including the finitedifference time-domain method [11,14,15], the finite element method [12], the finite integration technique [8], and the volume integral equation (VIE) method [3–6]. In the present study, the VIE method is used to investigate nanofocusing of SPPs by metal-coated dielectric conical probe with tilted (asymmetric) shape. Based on the VIE method we show the different behavior of nanofocusing between that by symmetric and that by tilted probes. Results provide insight into the mechanism of guiding and nanofocusing in complicated SPP probes and show how such nanofocusing can be achieved using a metal-coated dielectric probe of tilted conical shape. In this paper, we analyze the basic characteristics, i.e., optical field at the tip, of the metal-coated dielectric probe of tilted conical shape. We first show the dependences of the maximum enhanced local optical intensities at the tip on the degree of asymmetry of the tilted probe for both incident RP and LP beams for various coating metals. Then we vary the polarized angle of the LP beam and show the dependence of the maximum optical intensity at the tip on the polarized angle.
2. CONFIGURATION OF THE PROBE We consider the metal-coated dielectric probe of tilted conical shape shown in Fig. 1. The dielectric conical probe has a base © 2013 Optical Society of America
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Ex r; 0 A exp−r∕w2 cos β; E y r; 0 A exp−r∕w2 sin β; Ez r; 0 jA2x∕k0 w2 exp−r∕w2 :
Fig. 1. (a) Geometry of the tilted metal-coated conical dielectric probe. The conical dielectric structure has a base radius R and a height h. The metal coating has a thickness d. The tip of the probe is located at l; 0; h in the x; y; z coordinate system. Permittivities of the surrounding free space, the coating metal, and the dielectric of the conical structure are denoted by ε0 , ε1 , and ε2 , respectively. The shape of the tilted conical structure is characterized by distance l between the z axis and the tip. Both RP and LP Gaussian beams are normally incident to the x–y plane from negative z direction. The axis of Gaussian beam coincides with the z axis. RP and LP beams propagate along the z direction and polarization of the LP beam makes an angle of β with the x axis. (b) shows the discretized structure of only 28 layers near the tip of the probe. The whole structure is composed of 328 layers. The inset shows the tiny cube with δ × δ × δ dimensions.
radius R and was made of a dielectric with a relative permittivity ε1 ∕ε0 fabricated in the x; y; z coordinate system where ε0 is the permittivity of free space. The side of conical dielectric probe is coated with the thin metal with a relative permittivity ε2 ∕ε0 and thickness d. The surrounding is the free space. The conical probe has a height h. It is apparent that the conical probe shown in Fig. 1 is not rotationally symmetric about the z axis. In order to define the degree of asymmetry of the tilted conical shape, we employ the distance l between the tip and the z axis shown in Fig. 1. The case of l 0 corresponds to the symmetric probe and the cases of nonzero values of l correspond to the tilted probes. We first divide three-dimensional models of the probe into tiny cubes having dimensions of δ × δ × δ to solve the scattering problem shown in Fig. 1 by the VIE method [3]. Then we discretize the VIE by the moment method and finally solve the resultant system of linear equations numerically by generalized minimum residual method with fast Fourier transformation. The radial and z components of electric field of the incident RP Gaussian beam at z 0 in Fig. 1(a) can be written as [3,19,20] E r r; 0 2Ar∕w exp−r∕w2 ; Ez r; 0 h i j 4A 1 − wr 2 − exp−r∕w2 : k0 w
(1)
For the incident LP Gaussian beam, the x, y, z components of electric field at z 0 can be written as [21]
(2)
In Eqs. (1) and (2), k0 is the wavenumber in free space, w is the spot size of the beam, and r x2 y2 1∕2 . We assume that the RP and LP beam propagate along the z direction and the polarization (electric field vector) of the LP beam makes an angle of β with the x axis shown in Fig. 1(a). The beam amplitude is given by A 1. In this case, the RP and LP beams given by Eqs. (1) and (2) have the same incident energies at z 0. The RP and LP optical fields of the SPPs are excited along the asymmetric surface of the probe. The optical fields of the SPPs are nanofocused by the decreasing cross section of the probe along the propagation direction. In this paper, the wavelength λ is 633 nm and δ is given by k0 δ 0.05 (δ 5 nm). The coating metals are Au, Cu, Ni, and Al whose relative permittivities are given by ε2 ∕ε0 −13.8 − j1.08, −11.6 − j1.8, −14.7 − j9.9, and −56.1 − j20.9, respectively, and the relative permittivity of the dielectric (grass) is given by ε1 ∕ε0 2.25. The beam spot size is given by w λ and the thickness of coating metal is given by k0 d ≈ 0.27 (d ≈ 27.4 nm). Base radius of the asymmetric conical structure shown in Fig. 1 is given by k0 R 7.07 (R 712 nm). The probe has a height given by k0 h 16.4 (h 1652 nm). The values of the thickness d, the base radius R, and the height h are maintained in all cases through this paper.
3. NANOFOCUSED OPTICAL INTENSITIES We discretize the whole structure using cubes arranged in 328 layers parallel to the x–y plane [28 layers near the tip are shown in Fig. 1(b)]. Here we present numerical results how a RP or LP beam is focused by the metal-coated dielectric conical probe of tilted shape and show how a highly confined electric field at the tip is generated. Figures 2(a) and 2(b) show the distributions of the optical intensities by logarithmic scale on the x–z plane [logjEx; 0; zj2 ] and y–z plane [logjE0; y; zj2 ] for the symmetric probe for incident LP and RP beams, respectively. In Fig. 2(a), we can see that the optical intensity is not focused at the tip of the symmetric probe for the case of incident LP beam in both the x–z and y–z planes. However, the optical intensity is focused at the tip of this probe for the case of the incident RP beam in Fig. 2(b). These results can be explained by Fig. 3 that shows the illustrations of the electric field vector on the x–z plane of the symmetric probe. In Fig. 3(a), we can see that, for the case of the incident LP beam, the x component of the electric vectors of the excited SPP has the same direction on the right and left surfaces of the probe. Contrarily, their z component has the opposite direction on the right and left surfaces. The main component of the nanofocused electric field on the tip has the z direction. The electric field vectors that propagate along the right and left surface have opposite direction on the tip in Fig. 3(a). Therefore, the electric field vector is canceled at the tip of the probe. For the incident RP beam, the z component of the electric vectors of the excited SPP have same direction on the right and left surfaces, as shown in Fig. 3(b). Therefore,
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Fig. 2. Optical intensity distributions on the x–z plane and y–z plane for symmetric probe (l 0 nm) and tilted probe (l 300 nm) are shown in the logarithmic scale. The optical intensities of logjEx; 0; zj2 and logjE0; y; zj2 for symmetric probe for incident LP and RP beams are shown in (a) and (b), respectively. The optical intensities of logjEx; 0; zj2 and logjEl; y; zj2 for tilted probe for incident LP and RP beams are shown in (c) and (d), respectively. The coating metal is Au (ε2 ∕ε0 −13.8 − j1.08). The dielectric is glass (ε1 ∕ε0 2.25). Other parameters keep constant (d ≈ 27.4 nm, R 712 nm, h 1652 nm).
the electric field is not canceled and, moreover, the optical intensity is focused at the tip of probe. Figures 2(c) and 2(d) show the distributions of the optical intensities by logarithmic scale on the x–z plane [logjEx; 0; zj2 ] and y–z plane [logjEl; y; zj2 ] for the tilted probe (l 300 nm) for the incident LP and RP beams, respectively. With the tilted conical shape, we can see that the optical intensity for the incident LP beam is focused and enhanced at the tip in Fig. 2(c). Moreover, for the incident RP beam, the enhanced and localized optical intensity at the tip can be also realized for tilted probe [in Fig. 2(d)]. It means that the tilted conical shape is valid for both incident LP and RP beams.
Fig. 3. Illustration of electric field vectors on the x–z plane of symmetric probe for incident LP and RP beams are shown in (a) and (b), respectively.
4. DEPENDENCES OF MAXIMUM OPTICAL INTENSITIES ON THE DISTANCE l The dependences of the maximum enhanced optical intensities at the tip on the degree of asymmetry, i.e., distance l, are shown in Fig. 4. The nanofocused maximum intensity exists at a point just above the tip in the free space. Since the probe is composed by the cubes with size δ 5 nm in this paper, this point is located at 2.5 nm just above the surface of the tip cube. We use the value of the intensity at this point as the maximum of optical intensity in this paper. We investigated structures with four various coating metals, Au, Cu, Al, and Ni. Figure 4 reveals that the maximum optical intensity strongly depends on the distance l for all coating metals. For all cases of asymmetric structure, the distance l is varied from 0 to 600 nm under the condition that the other parameters of the shape are not changed. The results of the incident RP beam are shown in Fig. 4(a) for the cases of coating metals Au and Cu and in (b) for the cases of Al and Ni. It is found that symmetric probe (l 0 nm) is the most suitable for the incident RP beam and the maximum optical intensity decreases with the increase of distance l [shown in Figs. 4(a) and 4(b)]. For the metal Au, the maximum optical intensity is largest in all coating metals. Interestingly, Cu gives a large enhanced optical intensity of ∼2.0 × 104 , shown in Fig. 4(a) and rather large enhanced optical intensities can be also generated even by metals having large dissipations
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Maximum optical intensity (RP beam)
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Maximum optical intensity (RP beam)
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Distance l (nm)
(a)
(b)
Maximum optical intensity (LP beam)
Maximum optical intensity (LP beam)
Distance l (nm)
Distance l (nm)
Distance l (nm)
(c)
(d)
Fig. 4. Maximum of optical intensity at the tip jEj2 (a) and (b) for incident RP beam; (c) and (d) for incident LP beam. The coating metals are Au, Cu, Ni, and Al. The dielectric is glass (ε1 ∕ε0 2.25). The distance l is varied from 0 to 600 nm under the condition that the other parameters of the shape are kept constant (d ≈ 27.4 nm, R 712 nm, h 1652 nm).
that gives the largest maximum intensity increases with the increase of the dissipation of metal. It means that the dissipation of the metal is an important factor in generating large enhanced optical intensity at the tip. We show the results in Figs. 4(a) and 4(c) in the same intensity scale for Au metal coating in Fig. 5. The solid circles for RP beam
Maximum optical intensity
(∼1.0 × 103 for Ni and ∼8.2 × 102 for Al), shown in Fig. 4(b). For the incident RP beam, the maximum enhanced optical intensities in Fig. 4(a) for Au and Cu (small values of dissipation of metal) are much larger than those in Fig. 4(b) for Al and Ni (large values of dissipation of metal). For the incident LP beam, the dependences of maximum enhanced optical intensities on the distance l are shown in Figs. 4(c) and 4(d). We can see that the enhanced optical intensity at the tip cannot be realized for the symmetric probe (l 0 nm). For the case of the incident LP beam, there is an optimum distance l where the maximum optical intensity can be obtained at the tip of the tilted conical probe for all coating metals. The largest value of the maximum optical intensity jEj2max and the optimum distance l are shown in Table 1 for each coating metal. For Au, the largest maximum optical intensity can be obtained at the short distance l compared with that for Cu, Al, and Ni (see Table 1). We can see that optimum distance l
for LP beam
Table 1. Largest Value of the Maximum Optical Intensity for Incident LP Beam Coating Metal Au (−13.6 − j1.08) Cu (−11.6 − j1.8) Ni (−14.7 − j9.9) Al (−56.1 − j20.9)
jEj2max
l (nm)
5.7 × 103 2.9 × 103 1.7 × 102 1.4 × 102
290 410 430 450
Distance l (nm) Fig. 5. Maximum of optical intensity at the tip of Au-coating metal probe jEj2 for incident RP beam and incident LP beam in the same intensity scale. The distance l is varied from 0 to 600 nm under the condition that the other parameters of the shape are kept constant (d ≈ 27.4 nm, R 712 nm, h 1652 nm).
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Table 2. Maximum Optical Intensities jEj2 for Incident RP and LP Beams in the Region of 350 nm < l < 410 nm Distance l
410 nm
3
3.0 × 103 4.3 × 103
5.7 × 10 5.0 × 103
Maximum optical intensity (LP beam)
RP beam LP beam
350 nm
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paper, we investigated the basic characteristics of the maximum optical intensity at the tip created by nanofocusing in the SPP metal-coated conical dielectric probe with tilted shape (asymmetric shape) by the VIE method. We consider the cases of incident RP and LP Gaussian beams and found that the tilted SPP conical probe is valid for both incident LP and RP beams. The optical intensity at the tip of the tilted metal-coated dielectric probe strongly depends on not only the degree of asymmetry but also the dissipation of coating metal for both incident RP and LP beams. Since the degree of the asymmetric shape and the coating metal of the probe affect significantly the maximum enhanced optical intensity at the tip for incident RP and LP beams, the findings of this study demonstrate that tilted conical probes must be carefully designed and fabricated.
REFERENCES Polarized angle (C degree)
Fig. 6. Maximum of optical intensity jEj2 located at 2.5 nm from the tip of the tilted probe (l 300 nm, d ≈ 27.4 nm, R 712 nm, h 1652 nm) for incident LP beam with Au metal coating depends on the polarized angle β. In this simulation, the tilted probe has been kept constant while the polarized angle is changed from 0° to 90°.
and open circles present the results of incident RP and LP beams, respectively. The value of maximum optical intensities for incident RP and LP beams are shown at l 350 and 410 nm in Table 2. From Fig. 5 and Table 2, we can see in the region of l < ∼380 nm, the maximum optical intensities for the incident RP beam are always larger than those for the incident LP beam. In the region of l > ∼380 nm the maximum optical intensities for the incident LP beam exceed those for the incident RP beam. We can see the similar behavior for other coating metals from results in Fig. 4. This means that the RP beams does not always make larger optical intensity at the tip than that made by the LP beams in the tilted probe.
5. DEPENDENCE OF MAXIMUM OPTICAL INTENSITIES ON POLARIZED ANGLE Finally, we examined the effect of changing the polarized angle β (LP beam is propagating in the z direction and its direction of polarization makes an angle of β with the x axis) on the maximum of optical intensity at the tip of probe for incident LP beam with Au metal coating in Fig. 6. For the case under study, we chose the proposed tilted conical probe (distance l 300 nm) and change the polarized angle β from 0° to 90°. As shown in Fig. 6, the maximum optical intensity strongly depends on the polarized angle β of the incident LP beam. We can see that the enhanced optical intensity at the tip can be realized for the case of β 0°. In the region of 0° < β < 45°, the maximum of optical intensity at the tip decreases with the increase of the polarized angle. When β continues to rise in excess of 45°, the maximum optical intensity changes complicatedly. However, it is found that the most suitable characteristic is obtained for the case of β 0°.
6. CONCLUSION The possibility of using a metal-coated dielectric probe of tilted conical shape utilizing SPP was demonstrated. In this
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