Investigation of electric field enhancement between metal blocks at the focused field generated by a radially polarized beam Kyoko Kitamura,1,2,* Ting Ting Xu,2 and Susumu Noda2,3 1
The Hakubi Center for Advanced Research, Kyoto University, Kyoto 615-8510, Japan Department of Electronic Science and Engineering, Kyoto University, Kyoto 615-8510, Japan 3 Photonics and Electronics Science and Engineering Center, Kyoto University, Kyoto 615-8510, Japan * [email protected] 2
Abstract: A radially polarized beam possesses peculiar focusing properties compared with a linearly polarized beam, for example, the generation of a strong longitudinal field and zero intensity of the Poynting vector on the beam axis. In order to exploit these focusing properties, here we consider a system in which gold metal cubes are arranged along the propagation direction of the beam. An electric field enhancement of more than 20-times can be generated between two gold cubes separated by a distance λ/10 on the optical axis. This is because the energy of a radially polarized beam can propagate even if a metal cube is located on the beam axis, and a longitudinal field generated between the cubes can induce a surface plasmon mode. We show that these results are peculiar properties that cannot be produced with an incident linearly polarized beam. We also show that the beam can generate multiple regions of electrical field enhancement in the propagating direction when multiple metal cubes are arranged on the beam axis. ©2013 Optical Society of America OCIS codes: (260.1960) Diffraction theory; (260.5430) Polarization.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010). H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). D. N. Schimpf, W. P. Putnam, M. D. Grogan, S. Ramachandran, and F. X. Kartner, “Radially polarized BesselGauss beams in ABCD optical systems and fiber-based generation,” in Conference on Lasers and ElectroOptics, Technical Digest (CD) (Optical Society of America, 2013), paper JTh2A.67. K. Kitamura, M. Nishimoto, K. Sakai, and S. Noda, “Needle-like focus generation by radially polarized halo beams emitted by photonic-crystal ring-cavity laser,” Appl. Phys. Lett. 101(22), 221103 (2012). E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Photonics: lasers producing tailored beams,” Nature 441(7096), 946 (2006). D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surfaceemitting two-dimensional photonic crystal diode laser,” Opt. Express 12(8), 1562–1568 (2004). S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of twodimensional photonic crystal laser by unit cell structure design,” Science 293(5532), 1123–1125 (2001). M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316 (1999). K. Kitamura, K. Sakai, and S. Noda, “Finite-difference time-domain (FDTD) analysis on the interaction between a metal block and a radially polarized focused beam,” Opt. Express 19(15), 13750–13756 (2011). B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253(1274), 358–379 (1959).
#198087 - $15.00 USD Received 23 Sep 2013; revised 6 Dec 2013; accepted 16 Dec 2013; published 19 Dec 2013 (C) 2013 OSA 30 December 2013 | Vol. 21, No. 26 | DOI:10.1364/OE.21.032217 | OPTICS EXPRESS 32217
14. H. Landolt and R. Börnstein, Landolt-Börnstein-Tabellenwerk Zahlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik und Technik, 6th ed. (Springer, 1962), Chap. 28. 15. J. R. Wait, “Exact surface impedance for a cylindrical conductor,” Electron. Lett. 15(20), 659–660 (1979). 16. S. Kellali, B. Jecko, and A. Reineix, “Implementation of a surface impedance formalism at oblique incidence in FDTD method,” IEEE Trans. Electromagn. Compat. 35, 347–356 (1993).
1. Introduction Radially polarized beams, which possess electric field vectors arranged like the spokes of wheels pointing out from the center in the beam cross-section, have attracted much interest and are expected to be used in many applications in optics, such as improved optical data storage and optical microscopy, because of their peculiar focusing properties, for example, the generation of a strong longitudinal field on the beam axis [1], their ability to create a small focal spot size with a long depth of focus [2, 3], and zero Poynting vector on the beam axis even though the electrical field is strong [4]. Optical sources that generate radially polarized beams by using an external cavity [5], or more practical sources using optical fibers [6] and semiconductor lasers [7–11], have been developed in the last decade. Nevertheless, even though the peculiar focusing properties in free space have been shown and we can easily generate radially polarized beams due to the development of several optical sources, practical applications of radially polarized beams have not been shown yet. One of the reasons is surely that interactions between such beams and materials have not been investigated much, even though the focusing properties of these beams are very distinct from those of other beams. We have reported on novel interactions between a half-wavelength sized gold cube and a focused radially polarized beam [12]. The beam energy can propagate through the cube, even though the cube is set on the beam axis in the focal plane, and the electric field enhancement due to the cube results in interaction between the focused electric field and the cube. As mentioned before, these effects are due to the focusing property that the beam creates zero Poynting vector on the beam axis even though the electrical field is strong. In other words, all of the energy can pass through the gold cube located on the beam axis, because the cube does not disturb the Poynting vector. At the same time, the longitudinal electric field interacts with the cube and induces a surface plasmon mode. In this paper, by taking advantage of these focusing properties, we examine electric field enhancement in the tiny spaces between metal cubes placed on the beam axis along the propagating direction. In Section 2, we describe the calculation model based on the threedimensional finite-difference time-domain (3D FDTD) method, using a surface impedance method to introduce metal effects. In Section 3, we discuss the electric field enhancement corresponding to the spacing between the two gold metal cubes placed on the beam axis by comparing a focused radially polarized beam and a focused linearly polarized beam. In Section 4, we show how increasing the number of metal cubes on the beam axis affects the electric field enhancement. Finally, we give some concluding remarks in Section 5. 2. Calculation model In this study, we simulated the electromagnetic fields of focused radially polarized and linearly polarized beams in a three-dimensional finite difference time domain (FDTD) space. We excited a plane a few wavelengths away from the focal plane with initial electromagnetic fields that can be calculated by using vectorial diffraction theory [1, 13]. When both of the electric and magnetic fields were excited based on the vectorial diffraction theory over a large area with 10λ × 10λ, the electromagnetic fields would be propagated only in one direction and focused. Figure 1(a) shows the calculation model. The three-dimensional space was divided into λ/20 units, and the excitation plane was set at z/λ = −2.5 [12]. We excited the plane with focusing electromagnetic fields with NA (the numerical aperture) = 0.9, and β (the fillingfactor of the beam in the lens pupil) = 1.5. Figure 1(b) shows the time-integrated electric field intensity profile and the intensity distribution of the Poynting vector in the propagating direction (z), with exciting a focusing radially polarized beam. Note the null intensity of the Poynting vector along the beam axis, where the maximum intensity of the electric field constructed with z-polarization was generated. #198087 - $15.00 USD Received 23 Sep 2013; revised 6 Dec 2013; accepted 16 Dec 2013; published 19 Dec 2013 (C) 2013 OSA 30 December 2013 | Vol. 21, No. 26 | DOI:10.1364/OE.21.032217 | OPTICS EXPRESS 32218
We introduced two gold metal cubes with a side length of λ/2 on the optical axis, separated by distance d (the bottom of the upper cube was located at z/λ = d/2, and the top of the lower cube was located at z/λ = -d/2). We used the surface impedance method to introduce these metal cubes. The metal cubes had a complex refractive index ε = −1.88 + i3.42 at λ = 500 nm [14]. Incidentally, it is valid to use the surface impedance method because these gold cubes are much larger than the gold skin depth [15, 16]. (a)
Beam axis (x/λ=0.0, y/λ=0.0) Propagating direction
z
y
Two gold cubes (half-wavelength size) are located on the optical axis with the separation distance d (the bottom of the upper cube is located at z/λ=d/2, and the top of the lower cube is located at z/λ=-d/2) . Focal plane (z/λ=0.0)
x
3.0 2.0 1.0
|Etotal|2
3.0
|Ex|2
0.0
-1.0
-3.0 -3.0 0.0 3.0 |Ez|2
-2.0
0.0
0.0
-3.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 -3.0 -3.0 Distance from beam axis x/λ
3.0
Distance from focal plane z/λ
(b)
Distance from focal plane z/λ
Excitation plane (z/λ=-2.5) 3.0 2.0
|Sz|2
1.0 0.0
-1.0 -2.0
0.0
3.0
-3.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Distance from beam axis x/λ
Fig. 1. Calculation model for three-dimensional FDTD analysis. (a) 3D model; (b) timeaveraged distributions of the electric field intensity (Ei: i = total, x, z) and the Poynting vector (Sz) calculated by the FDTD method for a focused radially polarized beam.
3. Electric field enhancement between two gold cubes Figure 2 shows the maximum intensity, which is normalized with respect to the maximum intensity in free space, as a function of the separation between the gold cubes (d). When the incident beam is linearly polarized, the maximum intensity is almost constant, irrespective of the spacing. In the case of a radially polarized beam, on the other hand, the maximum intensity is almost constant at a separation distance of more than 0.5λ, whereas, at a separation distance of less than 0.5λ, the maximum intensity increases as the distance decreases. When the separation becomes 0.1λ, the electric-field enhancement is more than 20times greater than the enhancement in free space. First, comparing a linearly polarized beam and a radially polarized beam, we consider the electric field intensity profiles of each beam when the cube separation is 0.1λ, as shown in Figs. 3(a) and 3(b). In case of the linearly polarized beam, most of the incident light is reflected by the cube placed at z/λ0, when normalized with respect to the maximum intensity of the beam in free space. This is due to the null Poynting vector in the propagating direction on the beam axis, which causes the light to propagate without being obstructed by the gold cubes placed on the beam axis. Figures 3(c) and 3(d) show magnified
#198087 - $15.00 USD Received 23 Sep 2013; revised 6 Dec 2013; accepted 16 Dec 2013; published 19 Dec 2013 (C) 2013 OSA 30 December 2013 | Vol. 21, No. 26 | DOI:10.1364/OE.21.032217 | OPTICS EXPRESS 32219
Maximum electric-field intensity （Normalized by the maximum intensity in free space)
images of the electric-field distributions with electric vectors at a certain time. The maximum electric-field intensity of the linearly polarized beam comes not from the surface plasmon mode of the gold cube but from constructive interference between the incident and reflected light, since a strong electric field is generated at a location away from the gold cube. Also, the reason why the strong intensity is not generated in the space between the cubes, in contrast to the case of the radially polarized beam, is that the electric field vectors that are diffracted at the edge of the cube become asymmetric towards x = 0 and destructively interfere, even though a surface plasmon mode is induced at the edge of the cube. On the other hand, the radially polarized beam generates the maximum intensity at precisely the center of the space between the two cubes, as shown Fig. 3(d). The electric field vectors reveal that the longitudinal polarization produced the strongest intensity.
25
Radial polarization Au Radial polarization perfect conductor x polarization Au ( z direction) x polarization Au ( x direction)
20 15 10 5 0 0
0.2
0.4
0.6
0.8
Separation distance of metal cubes d [λ]
1
Fig. 2. Maximum electric-field intensity as a function of the separation (d) between the gold cubes. The intensity is normalized with respect to the maximum intensity of the focused beam in free space.
Now, we discuss in detail the reason why the maximum electric field intensity is generated from the longitudinally polarized electric field. There are two ways to generate a longitudinally polarized electric field in the space between the cubes when a radially polarized beam is incident: i) One has a similar origin to the generation of longitudinal fields in free space, because the Poynting vector in the propagating direction is not obstructed by the gold cube placed at z/λ0.5. This is because no interaction occurs between the cubes, due to the sufficiently large separation between them, as shown by the intensity profile at d/λ = 0.5, as shown in Fig. 3(e). In this situation, the cubes are influenced separately by the two effects mentioned above, that is, constructive interference and surface plasmon modes. In the region d/λ
The Hakubi Center for Advanced Research, Kyoto University, Kyoto 615-8510, Japan Department of Electronic Science and Engineering, Kyoto University, Kyoto 615-8510, Japan 3 Photonics and Electronics Science and Engineering Center, Kyoto University, Kyoto 615-8510, Japan * [email protected] 2
Abstract: A radially polarized beam possesses peculiar focusing properties compared with a linearly polarized beam, for example, the generation of a strong longitudinal field and zero intensity of the Poynting vector on the beam axis. In order to exploit these focusing properties, here we consider a system in which gold metal cubes are arranged along the propagation direction of the beam. An electric field enhancement of more than 20-times can be generated between two gold cubes separated by a distance λ/10 on the optical axis. This is because the energy of a radially polarized beam can propagate even if a metal cube is located on the beam axis, and a longitudinal field generated between the cubes can induce a surface plasmon mode. We show that these results are peculiar properties that cannot be produced with an incident linearly polarized beam. We also show that the beam can generate multiple regions of electrical field enhancement in the propagating direction when multiple metal cubes are arranged on the beam axis. ©2013 Optical Society of America OCIS codes: (260.1960) Diffraction theory; (260.5430) Polarization.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010). H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). D. N. Schimpf, W. P. Putnam, M. D. Grogan, S. Ramachandran, and F. X. Kartner, “Radially polarized BesselGauss beams in ABCD optical systems and fiber-based generation,” in Conference on Lasers and ElectroOptics, Technical Digest (CD) (Optical Society of America, 2013), paper JTh2A.67. K. Kitamura, M. Nishimoto, K. Sakai, and S. Noda, “Needle-like focus generation by radially polarized halo beams emitted by photonic-crystal ring-cavity laser,” Appl. Phys. Lett. 101(22), 221103 (2012). E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Photonics: lasers producing tailored beams,” Nature 441(7096), 946 (2006). D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surfaceemitting two-dimensional photonic crystal diode laser,” Opt. Express 12(8), 1562–1568 (2004). S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of twodimensional photonic crystal laser by unit cell structure design,” Science 293(5532), 1123–1125 (2001). M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316 (1999). K. Kitamura, K. Sakai, and S. Noda, “Finite-difference time-domain (FDTD) analysis on the interaction between a metal block and a radially polarized focused beam,” Opt. Express 19(15), 13750–13756 (2011). B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253(1274), 358–379 (1959).
#198087 - $15.00 USD Received 23 Sep 2013; revised 6 Dec 2013; accepted 16 Dec 2013; published 19 Dec 2013 (C) 2013 OSA 30 December 2013 | Vol. 21, No. 26 | DOI:10.1364/OE.21.032217 | OPTICS EXPRESS 32217
14. H. Landolt and R. Börnstein, Landolt-Börnstein-Tabellenwerk Zahlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik und Technik, 6th ed. (Springer, 1962), Chap. 28. 15. J. R. Wait, “Exact surface impedance for a cylindrical conductor,” Electron. Lett. 15(20), 659–660 (1979). 16. S. Kellali, B. Jecko, and A. Reineix, “Implementation of a surface impedance formalism at oblique incidence in FDTD method,” IEEE Trans. Electromagn. Compat. 35, 347–356 (1993).
1. Introduction Radially polarized beams, which possess electric field vectors arranged like the spokes of wheels pointing out from the center in the beam cross-section, have attracted much interest and are expected to be used in many applications in optics, such as improved optical data storage and optical microscopy, because of their peculiar focusing properties, for example, the generation of a strong longitudinal field on the beam axis [1], their ability to create a small focal spot size with a long depth of focus [2, 3], and zero Poynting vector on the beam axis even though the electrical field is strong [4]. Optical sources that generate radially polarized beams by using an external cavity [5], or more practical sources using optical fibers [6] and semiconductor lasers [7–11], have been developed in the last decade. Nevertheless, even though the peculiar focusing properties in free space have been shown and we can easily generate radially polarized beams due to the development of several optical sources, practical applications of radially polarized beams have not been shown yet. One of the reasons is surely that interactions between such beams and materials have not been investigated much, even though the focusing properties of these beams are very distinct from those of other beams. We have reported on novel interactions between a half-wavelength sized gold cube and a focused radially polarized beam [12]. The beam energy can propagate through the cube, even though the cube is set on the beam axis in the focal plane, and the electric field enhancement due to the cube results in interaction between the focused electric field and the cube. As mentioned before, these effects are due to the focusing property that the beam creates zero Poynting vector on the beam axis even though the electrical field is strong. In other words, all of the energy can pass through the gold cube located on the beam axis, because the cube does not disturb the Poynting vector. At the same time, the longitudinal electric field interacts with the cube and induces a surface plasmon mode. In this paper, by taking advantage of these focusing properties, we examine electric field enhancement in the tiny spaces between metal cubes placed on the beam axis along the propagating direction. In Section 2, we describe the calculation model based on the threedimensional finite-difference time-domain (3D FDTD) method, using a surface impedance method to introduce metal effects. In Section 3, we discuss the electric field enhancement corresponding to the spacing between the two gold metal cubes placed on the beam axis by comparing a focused radially polarized beam and a focused linearly polarized beam. In Section 4, we show how increasing the number of metal cubes on the beam axis affects the electric field enhancement. Finally, we give some concluding remarks in Section 5. 2. Calculation model In this study, we simulated the electromagnetic fields of focused radially polarized and linearly polarized beams in a three-dimensional finite difference time domain (FDTD) space. We excited a plane a few wavelengths away from the focal plane with initial electromagnetic fields that can be calculated by using vectorial diffraction theory [1, 13]. When both of the electric and magnetic fields were excited based on the vectorial diffraction theory over a large area with 10λ × 10λ, the electromagnetic fields would be propagated only in one direction and focused. Figure 1(a) shows the calculation model. The three-dimensional space was divided into λ/20 units, and the excitation plane was set at z/λ = −2.5 [12]. We excited the plane with focusing electromagnetic fields with NA (the numerical aperture) = 0.9, and β (the fillingfactor of the beam in the lens pupil) = 1.5. Figure 1(b) shows the time-integrated electric field intensity profile and the intensity distribution of the Poynting vector in the propagating direction (z), with exciting a focusing radially polarized beam. Note the null intensity of the Poynting vector along the beam axis, where the maximum intensity of the electric field constructed with z-polarization was generated. #198087 - $15.00 USD Received 23 Sep 2013; revised 6 Dec 2013; accepted 16 Dec 2013; published 19 Dec 2013 (C) 2013 OSA 30 December 2013 | Vol. 21, No. 26 | DOI:10.1364/OE.21.032217 | OPTICS EXPRESS 32218
We introduced two gold metal cubes with a side length of λ/2 on the optical axis, separated by distance d (the bottom of the upper cube was located at z/λ = d/2, and the top of the lower cube was located at z/λ = -d/2). We used the surface impedance method to introduce these metal cubes. The metal cubes had a complex refractive index ε = −1.88 + i3.42 at λ = 500 nm [14]. Incidentally, it is valid to use the surface impedance method because these gold cubes are much larger than the gold skin depth [15, 16]. (a)
Beam axis (x/λ=0.0, y/λ=0.0) Propagating direction
z
y
Two gold cubes (half-wavelength size) are located on the optical axis with the separation distance d (the bottom of the upper cube is located at z/λ=d/2, and the top of the lower cube is located at z/λ=-d/2) . Focal plane (z/λ=0.0)
x
3.0 2.0 1.0
|Etotal|2
3.0
|Ex|2
0.0
-1.0
-3.0 -3.0 0.0 3.0 |Ez|2
-2.0
0.0
0.0
-3.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 -3.0 -3.0 Distance from beam axis x/λ
3.0
Distance from focal plane z/λ
(b)
Distance from focal plane z/λ
Excitation plane (z/λ=-2.5) 3.0 2.0
|Sz|2
1.0 0.0
-1.0 -2.0
0.0
3.0
-3.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Distance from beam axis x/λ
Fig. 1. Calculation model for three-dimensional FDTD analysis. (a) 3D model; (b) timeaveraged distributions of the electric field intensity (Ei: i = total, x, z) and the Poynting vector (Sz) calculated by the FDTD method for a focused radially polarized beam.
3. Electric field enhancement between two gold cubes Figure 2 shows the maximum intensity, which is normalized with respect to the maximum intensity in free space, as a function of the separation between the gold cubes (d). When the incident beam is linearly polarized, the maximum intensity is almost constant, irrespective of the spacing. In the case of a radially polarized beam, on the other hand, the maximum intensity is almost constant at a separation distance of more than 0.5λ, whereas, at a separation distance of less than 0.5λ, the maximum intensity increases as the distance decreases. When the separation becomes 0.1λ, the electric-field enhancement is more than 20times greater than the enhancement in free space. First, comparing a linearly polarized beam and a radially polarized beam, we consider the electric field intensity profiles of each beam when the cube separation is 0.1λ, as shown in Figs. 3(a) and 3(b). In case of the linearly polarized beam, most of the incident light is reflected by the cube placed at z/λ0, when normalized with respect to the maximum intensity of the beam in free space. This is due to the null Poynting vector in the propagating direction on the beam axis, which causes the light to propagate without being obstructed by the gold cubes placed on the beam axis. Figures 3(c) and 3(d) show magnified
#198087 - $15.00 USD Received 23 Sep 2013; revised 6 Dec 2013; accepted 16 Dec 2013; published 19 Dec 2013 (C) 2013 OSA 30 December 2013 | Vol. 21, No. 26 | DOI:10.1364/OE.21.032217 | OPTICS EXPRESS 32219
Maximum electric-field intensity （Normalized by the maximum intensity in free space)
images of the electric-field distributions with electric vectors at a certain time. The maximum electric-field intensity of the linearly polarized beam comes not from the surface plasmon mode of the gold cube but from constructive interference between the incident and reflected light, since a strong electric field is generated at a location away from the gold cube. Also, the reason why the strong intensity is not generated in the space between the cubes, in contrast to the case of the radially polarized beam, is that the electric field vectors that are diffracted at the edge of the cube become asymmetric towards x = 0 and destructively interfere, even though a surface plasmon mode is induced at the edge of the cube. On the other hand, the radially polarized beam generates the maximum intensity at precisely the center of the space between the two cubes, as shown Fig. 3(d). The electric field vectors reveal that the longitudinal polarization produced the strongest intensity.
25
Radial polarization Au Radial polarization perfect conductor x polarization Au ( z direction) x polarization Au ( x direction)
20 15 10 5 0 0
0.2
0.4
0.6
0.8
Separation distance of metal cubes d [λ]
1
Fig. 2. Maximum electric-field intensity as a function of the separation (d) between the gold cubes. The intensity is normalized with respect to the maximum intensity of the focused beam in free space.
Now, we discuss in detail the reason why the maximum electric field intensity is generated from the longitudinally polarized electric field. There are two ways to generate a longitudinally polarized electric field in the space between the cubes when a radially polarized beam is incident: i) One has a similar origin to the generation of longitudinal fields in free space, because the Poynting vector in the propagating direction is not obstructed by the gold cube placed at z/λ0.5. This is because no interaction occurs between the cubes, due to the sufficiently large separation between them, as shown by the intensity profile at d/λ = 0.5, as shown in Fig. 3(e). In this situation, the cubes are influenced separately by the two effects mentioned above, that is, constructive interference and surface plasmon modes. In the region d/λ
