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Nanoscale high-intensity light focusing with pure dielectric nonspherical scatterer Vijay M. Sundaram and Sy-bor Wen* Department of Mechanical Engineering, Texas A&M University, College Station, Texas 77840, USA *Corresponding author: [email protected] Received October 23, 2013; revised December 16, 2013; accepted December 24, 2013; posted December 24, 2013 (Doc. ID 199186); published January 27, 2014 Light scattering from nonspherically symmetric pure dielectric structures is examined. From the finite element full-wave analysis, it is found that teardrop-shaped scatterers can focus visible light to a ∼10 nm spot with an intensity enhancement ∼105 when the incident light is radially polarized. © 2014 Optical Society of America OCIS codes: (290.0290) Scattering; (230.0230) Optical devices; (160.0160) Materials; (080.3630) Lenses. http://dx.doi.org/10.1364/OL.39.000582
Optical-based high-resolution detection/fabrication, compared with competing methods, such as electron/ ion beams and x rays, have advantages in low operating cost, no requirement of a vacuum environment, and better versatility in operating conditions. However, compared with the competing methods, spatial resolution of optical-based detection/fabrication methods is commonly limited to a few micrometers due to the characteristics of light diffraction [1,2]. To improve the optical resolution, methodologies such as (a) light leakage from a subdiffraction limit aperture [3], (b) plasmon resonance [4,5], and (c) light scattering by a metal nanostructure have been developed to achieve nanoscale light confinement [6,7]. All the existing nanoscale light confinement methodologies require utilization of metallic materials, which leads to dissipation of optical energy and the resulting strong joule heating. The joule heating in nanostructures can cause undesirable thermal expansion and thermal damage of the device and is the main obstacle in the application of existing nano-optical devices in highenergy operation [8]. In addition, the dielectric constants of common metals are strong functions of wavelength in UV to IR ranges. Therefore, it is difficult to have a broadband metallic nano-optic device, operating for a wide range of wavelengths for nanoscale detection. To prevent the above undesirable side effects, dielectric nanostructures can be better choices of material in constructing nano-optics. Compared with metal, dielectric materials having negligible imaginary part of dielectric constant cause minimum joule heating, which prevents the undesirable side effects indicated above during high-energy operation [9]. However, with the smaller dielectric constant, dielectric materials cannot achieve nanoscale light confinement through mechanisms (a) and (b) described previously for metal. On the other hand, nanoscale light confinement can be achieved with scattering by dielectric structures [10,11]. The intensity of the nanoscale forward scattering with spherical dielectric scatterers, however, is much smaller than that with spherical metal scatterers (also due to its much smaller dielectric constant). The goal of this study is to find the geometry of dielectric scatterers that can provide nanoscale forward scattering with high focusing intensity that is comparable with standard spherical metallic scatterers. The resulting dielectric scatterer, 0146-9592/14/030582-04$15.00/0
with minimum joule heating, will allow high-intensity, high-throughput nanoscale light focusing, which cannot be achieved with other nano-optical devices. Such dielectric scatterers can be applied in establishing highthroughput scanning optical probes for high-speed superresolution imaging, nanoscale direct-writing, nanoscale mask-free lithography, nanoscale laser spectroscopy, and nano-optical tweezing, which cannot be easily achieved with existing scanning optical probes [12–14]. A full-wave analysis with the finite element method (FEM) is adopted in this study to identify scattered field distribution of different structures. A radially polarized beam, which is a requirement to have forward scattering from nanostructures, is selected as the far-field illumination [15]. More specifically, radially polarized Bessel beam, which can be generated with axicons and optical phase plates is adopted [16,17]. The electric field of the radially polarized Bessel beam can be expressed as [18] E r r; z E o cosα sinαJ 1 k sinαre−jk cosα−ωt ; (1) E ϕ r; z 0; E z r; z −iE o sin2 αJ 0 k sinαre−jk cosα−ωt ;
(2)
(3)
where α is a parameter determining the period of the Bessel beam. In this study, α 3.5° corresponds to an axicon angle of 173°, which is used in many Bessel beam experiments. The wavelength of the light is selected as λ 532 nm and the dielectric constant of the scatterer is ε 1.96 (close to silica) at λ 532 nm. We first examine the scattering behavior of spherical dielectric spheres to verify the accuracy of the FEM analysis. Case 1. Light scattering from dielectric nanospheres: Table 1 is the summary of the FEM results (Fig. 1) for scattering from dielectric spheres with different radii (10 nm < r < 500 nm). The spot size is defined as the FWHM at the focusing plane of maximum intensity (henceforth referred to as the focus spot). The intensity enhancement is defined as the ratio between the intensity at the focus spot and incident field. The depth of field is defined as the distance from the focus spot to © 2014 Optical Society of America
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Table 1. Scattering from Dielectric Spheres Sphere Radius (nm) 10 30 50 100 300 400 500
Intensity Enhancement
Spot Size (nm)
Depth of Field (nm)
Scattering Cross Section (μm2 )
2.18 2.28 2.45 3.26 32.5 97.5 240.0
10 30 45 80 170 180 180
10 35 50 60 160 180 220
9.5 × 10−9 6.9 × 10−6 1.47 × 10−4 7.56 × 10−3 0.872 2.03 2.96
the position along the optical axis where the spot size is doubled in scattered field. The scattering cross section is computed from the Mie solution, which governs the scattering behavior of spheres of all diameters [19,20]. The FEM results fit well with previous published data [21]. The results in Table 1, indicate that the spot size of forward scattering, depth of focus, scattering cross section, as well as the intensity enhancement, increase monotonically with respect to the radius of the sphere. Based on the results, for a dielectric sphere, a small spot size can be achieved by reducing the diameter of the spherical scatterer. On the other hand, maximum intensity enhancement at the confined spot can be achieved by choosing spherical scatterers with larger diameter. To achieve the goal of this study, which is to construct a dielectric-based scatterer that can achieve both small spot size and high focusing intensity, a nonspherically symmetric scatterer will be required. With an assumption that the spot size is mainly determined by the forward area of the scatterer and the intensity enhancement is mainly determined by the backward area when the scatterer radius is around or less than the incident wavelength, we propose that a “snowman” scatterer [Fig. 2(a)], which is a combination of a large and small spheres, can provide large backward scattering area and small forward scattering area to achieve small spot size with high focusing intensity. In the next section, we examine the snowman scatterers with different geometric configurations (i.e., different forward and backward radii) for nanoscale light focusing. Case 2: Light scattering from snowman scatterers: To verify the idea that the spot size of forward scattering with a scatterer is determined by its forward area, the snowman structure is simulated for top spheres with different radii (10 nm < r < 50 nm) when the radius of the
Fig. 2. (a) Schematic of snowman configuration of scatterer. (b) Schematic of teardrop configuration indicating the tapering angle. (c) Plot of electric field intensity of snowman scatterer with top sphere radius 10 nm and bottom 500 nm. (d) Plot of electric field intensity of teardrop scatterer with top sphere radius 10 nm and bottom 500 nm and tapering angle of 145°.
bottom sphere is fixed at 200 nm. On the other hand, to verify the idea that the intensity enhancement of forward scattering depends on its backward area, the intensity enhancement of snowman structures is simulated for different bottom sphere radius (100 nm < r < 500 nm) when the radius of the top sphere fixed at 10 nm. The results in Tables 2 and 3 are in agreement with the proposed idea: for a fixed radius of the bottom sphere (200 nm diameter), the spot size increases linearly from ∼10–50 nm when the radius of the top sphere increases Table 2. Scattering from Snowman Scatterers with Different Bottom Sphere Radii Radius of Bottom Sphere (nm) 100 200 300 400 500
Intensity Enhancement
Spot Size (nm)
Depth of Field (nm)
5.7 16.7 60 187.4 450
10 10 10 10 10
10 10 10 10 10
Table 3. Scattering from Snowman Scatterers with Different Top Sphere Radii Radius of Top Sphere (nm)
Fig. 1. Plot of electric field intensity distribution of a spherical scatterer with a radius 30 nm.
10 20 30 40 50
Intensity Enhancement
Spot Size (nm)
Depth of Field (nm)
16.7 13.8 11.9 10.6 9.6
10 20 30 40 55
10 24 30 40 55
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from 10 to 50 nm (i.e., forward area of the scatterer increases); for a fixed radius of top sphere and the associated fixed focus spot size, the intensity enhancement increases from ∼6 times to ∼450 times when the bottom sphere radius increases from 100 to 500 nm (i.e., the backward area of the scatterer increases). The depth of field shows dependence only on the forward area but not the backward area. The depth of field increases monotonically with the diameter of the top sphere as observed for a single spherical scatterer in case 1. In conclusion, the spot size and depth of field of a snowmanshaped scatterer are mainly determined by the forward area of the scatterer when the radius of the top sphere is within the range of Rayleigh scattering (i.e., 10 nm < r < 50 nm for λ ∼ 532 nm) [22]. In this size range, the focus spot and the depth of field are both reduced by decreasing the size of the top sphere in the snowmanshaped scatterer. On the other hand, the intensity enhancement at the focused spot depends only on the backward area of the scatterer when the radius of the bottom sphere is within the range of Mie scattering (i.e., 100 nm < r < 500 nm for λ ∼ 532 nm) [22]. The intensity enhancement can be increased by increasing the diameter of the bottom sphere in this size range. Figure 2(c) shows the light intensity distribution when the radius of the top and bottom sphere of the snowman scatterer are 10 and 500 nm, respectively. Light is confined at the apex of the snowman structure to a spot size of ∼10 nm with an intensity enhancement of ∼450 times. However, the highest intensity occurs at the gap between the top and bottom sphere where the geometric angle is the smallest. This strong localized field enhancement away from the apex of the snowman structure is not desired in most applications. Therefore, a modification of the snowman configuration has to be implemented to shift this highest intensity region from the side of the scatterer to the apex of the snowman structure. A direct approach which does not affect the forward/backward radii as well as the forward/backward scattering areas of the snowman scatterer would be filling the side gap of the snowman scatterer with dielectric material. The resulting modified snowman scatterer has a teardrop shape as in Fig. 2(b), and is named as teardrop scatterer in the following discussions. In the next case, we examine the field distribution of teardrop structures under the same radially polarized illumination. Case 3: Light scattering from teardrop scatterers: To prevent the occurrence of the off-axis intensity enhancement as in the snowman-shaped scatterer, the side gaps of the snowman structure (where the highest intensity occurs) are filled with dielectric material. The filling with different tapering angles is obtained by drawing a tangent to the top sphere of the snowman scatterer from different locations on the circumference of the bottom sphere [Fig. 2(b)]. The scattered fields of the resulting teardrop structures with bottom and top sphere radii of 500 and 10 nm, respectively, and tapering angle varying from 90° to 160° are examined with FEM simulation. Figure 2(d) shows the resulting intensity distribution when the tapering angle is 145°. As observed in Fig. 2(d), the field distribution is similar to that from a snowman scatterer with the same top and bottom radii. The only difference between the two shapes of scatterers
Table 4. Scattering from Teardrop Scatterer with Different Taper Angles Tapering Angle 90° 125° 145° 155°
Intensity Enhancement
Spot Size (nm)
468 495 502 490
10 10 10 10
is the diminishing of the off-axis field enhancement in the teardrop scatterer [Fig. 2(d)] compared with a snowman scatterer [Fig. 2(c)]. Table 4 lists the spot size and intensity enhancement in the forward scattering under the different tapering angles tested in the FEM simulation. While the spot size shows no significant dependence (∼10 nm), the intensity enhancement (≳450) shows a weak dependence on the tapering angle filling the gap. To achieve high energy transport efficiency focusing with a teardrop scatterer, external illumination delivered to the teardrop scatterer should be highly focused with a high NA lens. Light focusing with the resulting two-stage optics (macro/micro lens + teardrop scatterer) is presented next. Case 4: Light focusing with a combination of a teardrop scatterer and a far-field microlens: To provide intense illumination to induce a strong forward scattering, the teardrop scatterer is placed as the focus region of a diffraction-limited far-field ball lens with a radius 5 μm and dielectric constant ε 1.96 (i.e., dielectric microsphere). The radius of the ball lens is chosen such that the first peak of the radially polarized Bessel beam lies within the ball lens to ensure a high light intensity at the focus spot. Figure 3 shows the light intensity distribution of a teardrop scatterer with a forward radius 10 nm, backward radius 500 nm, and tapering angle 145°. The separation distance between teardrop scatterer and the ball lens (from center to center) is 7 μm in order to have the teardrop scatterer located at the focus of the ball lens. The total intensity enhancement at the apex of the teardrop scatterer is ∼150; 000 compared with the intensity of the incident light. The spot size around the apex remains ∼10 nm. In conclusion, a pure dielectric light scatterer that can achieve both high light confinement (i.e., small spot) as well as high intensity enhancement is demonstrated.
Fig. 3. Plot of electric field intensity distribution for cascade focusing configuration consisting of the teardrop scatterer placed at the focus region of 5 μm diameter microsphere.
February 1, 2014 / Vol. 39, No. 3 / OPTICS LETTERS
When the size of the dielectric scatterer is within 10 nm < r < 500 nm range, it is found that the spot size of light confinement with forward scattering depends on the forward area of the scatterer while the scattering intensity depends on the backward area of the scatterer. A corresponding teardrop scatterer composing a large bottom radius (r 200 nm) and a small top radius (r 10 nm) can serve as an efficient dielectric scatterer to achieve small spot size with high scattering intensity. The teardrop scatterer when combined with a diffractionlimited far-field lens to form a two-stage focusing device can achieve an intensity enhancement of ∼105 with a ∼10 nm spot size at the apex of the teardrop scatterer. The two-stage light focusing device constructed with pure dielectric can be valuable in high energy throughput nanoscale light detection, fabrication, and material manipulation (e.g., nano-optical tweezing). This work was supported by the National Science Foundation (CBET 0845794). The authors are pleased to acknowledge the Texas A&M Supercomputing Facility (http://sc.tamu.edu/) for providing the computing resources for this research. References 1. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002). 2. R. Heintzmann and G. Ficz, Methods Cell Biol. 81, 561 (2007). 3. E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991). 4. Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, Nano Lett. 5, 1726 (2005).
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5. P. Srisungsitthisunti, O. K. Ersoy, and X. F. Xu, Appl. Phys. Lett. 98, 223106 (2011). 6. L. Pan, Y. Park, Y. Xiong, E. Ulin-Avila, Y. Wang, L. Zeng, S. M. Xiong, J. Rho, C. Sun, D. B. Bogy, and X. Zhang, Sci. Rep. 1, 175 (2011). 7. A. Hartschuh, Angew. Chem., Int. Ed. Engl. 47, 8178 (2008). 8. A. Soni and S. B. Wen, J. Phys. D 45, 185101 (2012). 9. E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1998). 10. K. Sendur and A. Sahinoz, Opt. Express 17, 10910 (2009). 11. A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, J. Comput. Theor. Nanosci. 6, 1979 (2009). 12. Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007). 13. W. Srituravanich, L. Pan, Y. Wang, C. Sun, D. B. Bogy, and X. Zhang, Nat. Nanotechnol. 3, 733 (2008). 14. B. Bhushan, Scanning Probe Microscopy in Nanoscience and Nanotechnology (Springer, 2010). 15. B. Bennett, S. Goldberg, B. Ippolito, L. Novotny, Z. H. Liu, and M. S. Unlu, IEEE J. Sel. Top. Quantum Electron. 8, 1051 (2002). 16. M. Duocastella and C. B. Arnold, Laser Photon. Rev. 6, 607 (2012). 17. T. Watanabe, M. Fujii, Y. Watanabe, N. Toyama, and Y. Iketaki, Rev. Sci. Instrum. 75, 5131 (2004). 18. S. R. Mishra, Opt. Commun. 85, 159 (1991). 19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983). 20. G. Gouesbet, J. A. Lock, and G. Grehan, J. Quant. Spectrosc. Radiat. Transfer 112, 1 (2011). 21. S. Lecler, Y. Takakura, and P. Meyrueis, Opt. Lett. 30, 2641 (2005). 22. R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Hemisphere, 1992).
OPTICS LETTERS / Vol. 39, No. 3 / February 1, 2014
Nanoscale high-intensity light focusing with pure dielectric nonspherical scatterer Vijay M. Sundaram and Sy-bor Wen* Department of Mechanical Engineering, Texas A&M University, College Station, Texas 77840, USA *Corresponding author: [email protected] Received October 23, 2013; revised December 16, 2013; accepted December 24, 2013; posted December 24, 2013 (Doc. ID 199186); published January 27, 2014 Light scattering from nonspherically symmetric pure dielectric structures is examined. From the finite element full-wave analysis, it is found that teardrop-shaped scatterers can focus visible light to a ∼10 nm spot with an intensity enhancement ∼105 when the incident light is radially polarized. © 2014 Optical Society of America OCIS codes: (290.0290) Scattering; (230.0230) Optical devices; (160.0160) Materials; (080.3630) Lenses. http://dx.doi.org/10.1364/OL.39.000582
Optical-based high-resolution detection/fabrication, compared with competing methods, such as electron/ ion beams and x rays, have advantages in low operating cost, no requirement of a vacuum environment, and better versatility in operating conditions. However, compared with the competing methods, spatial resolution of optical-based detection/fabrication methods is commonly limited to a few micrometers due to the characteristics of light diffraction [1,2]. To improve the optical resolution, methodologies such as (a) light leakage from a subdiffraction limit aperture [3], (b) plasmon resonance [4,5], and (c) light scattering by a metal nanostructure have been developed to achieve nanoscale light confinement [6,7]. All the existing nanoscale light confinement methodologies require utilization of metallic materials, which leads to dissipation of optical energy and the resulting strong joule heating. The joule heating in nanostructures can cause undesirable thermal expansion and thermal damage of the device and is the main obstacle in the application of existing nano-optical devices in highenergy operation [8]. In addition, the dielectric constants of common metals are strong functions of wavelength in UV to IR ranges. Therefore, it is difficult to have a broadband metallic nano-optic device, operating for a wide range of wavelengths for nanoscale detection. To prevent the above undesirable side effects, dielectric nanostructures can be better choices of material in constructing nano-optics. Compared with metal, dielectric materials having negligible imaginary part of dielectric constant cause minimum joule heating, which prevents the undesirable side effects indicated above during high-energy operation [9]. However, with the smaller dielectric constant, dielectric materials cannot achieve nanoscale light confinement through mechanisms (a) and (b) described previously for metal. On the other hand, nanoscale light confinement can be achieved with scattering by dielectric structures [10,11]. The intensity of the nanoscale forward scattering with spherical dielectric scatterers, however, is much smaller than that with spherical metal scatterers (also due to its much smaller dielectric constant). The goal of this study is to find the geometry of dielectric scatterers that can provide nanoscale forward scattering with high focusing intensity that is comparable with standard spherical metallic scatterers. The resulting dielectric scatterer, 0146-9592/14/030582-04$15.00/0
with minimum joule heating, will allow high-intensity, high-throughput nanoscale light focusing, which cannot be achieved with other nano-optical devices. Such dielectric scatterers can be applied in establishing highthroughput scanning optical probes for high-speed superresolution imaging, nanoscale direct-writing, nanoscale mask-free lithography, nanoscale laser spectroscopy, and nano-optical tweezing, which cannot be easily achieved with existing scanning optical probes [12–14]. A full-wave analysis with the finite element method (FEM) is adopted in this study to identify scattered field distribution of different structures. A radially polarized beam, which is a requirement to have forward scattering from nanostructures, is selected as the far-field illumination [15]. More specifically, radially polarized Bessel beam, which can be generated with axicons and optical phase plates is adopted [16,17]. The electric field of the radially polarized Bessel beam can be expressed as [18] E r r; z E o cosα sinαJ 1 k sinαre−jk cosα−ωt ; (1) E ϕ r; z 0; E z r; z −iE o sin2 αJ 0 k sinαre−jk cosα−ωt ;
(2)
(3)
where α is a parameter determining the period of the Bessel beam. In this study, α 3.5° corresponds to an axicon angle of 173°, which is used in many Bessel beam experiments. The wavelength of the light is selected as λ 532 nm and the dielectric constant of the scatterer is ε 1.96 (close to silica) at λ 532 nm. We first examine the scattering behavior of spherical dielectric spheres to verify the accuracy of the FEM analysis. Case 1. Light scattering from dielectric nanospheres: Table 1 is the summary of the FEM results (Fig. 1) for scattering from dielectric spheres with different radii (10 nm < r < 500 nm). The spot size is defined as the FWHM at the focusing plane of maximum intensity (henceforth referred to as the focus spot). The intensity enhancement is defined as the ratio between the intensity at the focus spot and incident field. The depth of field is defined as the distance from the focus spot to © 2014 Optical Society of America
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Table 1. Scattering from Dielectric Spheres Sphere Radius (nm) 10 30 50 100 300 400 500
Intensity Enhancement
Spot Size (nm)
Depth of Field (nm)
Scattering Cross Section (μm2 )
2.18 2.28 2.45 3.26 32.5 97.5 240.0
10 30 45 80 170 180 180
10 35 50 60 160 180 220
9.5 × 10−9 6.9 × 10−6 1.47 × 10−4 7.56 × 10−3 0.872 2.03 2.96
the position along the optical axis where the spot size is doubled in scattered field. The scattering cross section is computed from the Mie solution, which governs the scattering behavior of spheres of all diameters [19,20]. The FEM results fit well with previous published data [21]. The results in Table 1, indicate that the spot size of forward scattering, depth of focus, scattering cross section, as well as the intensity enhancement, increase monotonically with respect to the radius of the sphere. Based on the results, for a dielectric sphere, a small spot size can be achieved by reducing the diameter of the spherical scatterer. On the other hand, maximum intensity enhancement at the confined spot can be achieved by choosing spherical scatterers with larger diameter. To achieve the goal of this study, which is to construct a dielectric-based scatterer that can achieve both small spot size and high focusing intensity, a nonspherically symmetric scatterer will be required. With an assumption that the spot size is mainly determined by the forward area of the scatterer and the intensity enhancement is mainly determined by the backward area when the scatterer radius is around or less than the incident wavelength, we propose that a “snowman” scatterer [Fig. 2(a)], which is a combination of a large and small spheres, can provide large backward scattering area and small forward scattering area to achieve small spot size with high focusing intensity. In the next section, we examine the snowman scatterers with different geometric configurations (i.e., different forward and backward radii) for nanoscale light focusing. Case 2: Light scattering from snowman scatterers: To verify the idea that the spot size of forward scattering with a scatterer is determined by its forward area, the snowman structure is simulated for top spheres with different radii (10 nm < r < 50 nm) when the radius of the
Fig. 2. (a) Schematic of snowman configuration of scatterer. (b) Schematic of teardrop configuration indicating the tapering angle. (c) Plot of electric field intensity of snowman scatterer with top sphere radius 10 nm and bottom 500 nm. (d) Plot of electric field intensity of teardrop scatterer with top sphere radius 10 nm and bottom 500 nm and tapering angle of 145°.
bottom sphere is fixed at 200 nm. On the other hand, to verify the idea that the intensity enhancement of forward scattering depends on its backward area, the intensity enhancement of snowman structures is simulated for different bottom sphere radius (100 nm < r < 500 nm) when the radius of the top sphere fixed at 10 nm. The results in Tables 2 and 3 are in agreement with the proposed idea: for a fixed radius of the bottom sphere (200 nm diameter), the spot size increases linearly from ∼10–50 nm when the radius of the top sphere increases Table 2. Scattering from Snowman Scatterers with Different Bottom Sphere Radii Radius of Bottom Sphere (nm) 100 200 300 400 500
Intensity Enhancement
Spot Size (nm)
Depth of Field (nm)
5.7 16.7 60 187.4 450
10 10 10 10 10
10 10 10 10 10
Table 3. Scattering from Snowman Scatterers with Different Top Sphere Radii Radius of Top Sphere (nm)
Fig. 1. Plot of electric field intensity distribution of a spherical scatterer with a radius 30 nm.
10 20 30 40 50
Intensity Enhancement
Spot Size (nm)
Depth of Field (nm)
16.7 13.8 11.9 10.6 9.6
10 20 30 40 55
10 24 30 40 55
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from 10 to 50 nm (i.e., forward area of the scatterer increases); for a fixed radius of top sphere and the associated fixed focus spot size, the intensity enhancement increases from ∼6 times to ∼450 times when the bottom sphere radius increases from 100 to 500 nm (i.e., the backward area of the scatterer increases). The depth of field shows dependence only on the forward area but not the backward area. The depth of field increases monotonically with the diameter of the top sphere as observed for a single spherical scatterer in case 1. In conclusion, the spot size and depth of field of a snowmanshaped scatterer are mainly determined by the forward area of the scatterer when the radius of the top sphere is within the range of Rayleigh scattering (i.e., 10 nm < r < 50 nm for λ ∼ 532 nm) [22]. In this size range, the focus spot and the depth of field are both reduced by decreasing the size of the top sphere in the snowmanshaped scatterer. On the other hand, the intensity enhancement at the focused spot depends only on the backward area of the scatterer when the radius of the bottom sphere is within the range of Mie scattering (i.e., 100 nm < r < 500 nm for λ ∼ 532 nm) [22]. The intensity enhancement can be increased by increasing the diameter of the bottom sphere in this size range. Figure 2(c) shows the light intensity distribution when the radius of the top and bottom sphere of the snowman scatterer are 10 and 500 nm, respectively. Light is confined at the apex of the snowman structure to a spot size of ∼10 nm with an intensity enhancement of ∼450 times. However, the highest intensity occurs at the gap between the top and bottom sphere where the geometric angle is the smallest. This strong localized field enhancement away from the apex of the snowman structure is not desired in most applications. Therefore, a modification of the snowman configuration has to be implemented to shift this highest intensity region from the side of the scatterer to the apex of the snowman structure. A direct approach which does not affect the forward/backward radii as well as the forward/backward scattering areas of the snowman scatterer would be filling the side gap of the snowman scatterer with dielectric material. The resulting modified snowman scatterer has a teardrop shape as in Fig. 2(b), and is named as teardrop scatterer in the following discussions. In the next case, we examine the field distribution of teardrop structures under the same radially polarized illumination. Case 3: Light scattering from teardrop scatterers: To prevent the occurrence of the off-axis intensity enhancement as in the snowman-shaped scatterer, the side gaps of the snowman structure (where the highest intensity occurs) are filled with dielectric material. The filling with different tapering angles is obtained by drawing a tangent to the top sphere of the snowman scatterer from different locations on the circumference of the bottom sphere [Fig. 2(b)]. The scattered fields of the resulting teardrop structures with bottom and top sphere radii of 500 and 10 nm, respectively, and tapering angle varying from 90° to 160° are examined with FEM simulation. Figure 2(d) shows the resulting intensity distribution when the tapering angle is 145°. As observed in Fig. 2(d), the field distribution is similar to that from a snowman scatterer with the same top and bottom radii. The only difference between the two shapes of scatterers
Table 4. Scattering from Teardrop Scatterer with Different Taper Angles Tapering Angle 90° 125° 145° 155°
Intensity Enhancement
Spot Size (nm)
468 495 502 490
10 10 10 10
is the diminishing of the off-axis field enhancement in the teardrop scatterer [Fig. 2(d)] compared with a snowman scatterer [Fig. 2(c)]. Table 4 lists the spot size and intensity enhancement in the forward scattering under the different tapering angles tested in the FEM simulation. While the spot size shows no significant dependence (∼10 nm), the intensity enhancement (≳450) shows a weak dependence on the tapering angle filling the gap. To achieve high energy transport efficiency focusing with a teardrop scatterer, external illumination delivered to the teardrop scatterer should be highly focused with a high NA lens. Light focusing with the resulting two-stage optics (macro/micro lens + teardrop scatterer) is presented next. Case 4: Light focusing with a combination of a teardrop scatterer and a far-field microlens: To provide intense illumination to induce a strong forward scattering, the teardrop scatterer is placed as the focus region of a diffraction-limited far-field ball lens with a radius 5 μm and dielectric constant ε 1.96 (i.e., dielectric microsphere). The radius of the ball lens is chosen such that the first peak of the radially polarized Bessel beam lies within the ball lens to ensure a high light intensity at the focus spot. Figure 3 shows the light intensity distribution of a teardrop scatterer with a forward radius 10 nm, backward radius 500 nm, and tapering angle 145°. The separation distance between teardrop scatterer and the ball lens (from center to center) is 7 μm in order to have the teardrop scatterer located at the focus of the ball lens. The total intensity enhancement at the apex of the teardrop scatterer is ∼150; 000 compared with the intensity of the incident light. The spot size around the apex remains ∼10 nm. In conclusion, a pure dielectric light scatterer that can achieve both high light confinement (i.e., small spot) as well as high intensity enhancement is demonstrated.
Fig. 3. Plot of electric field intensity distribution for cascade focusing configuration consisting of the teardrop scatterer placed at the focus region of 5 μm diameter microsphere.
February 1, 2014 / Vol. 39, No. 3 / OPTICS LETTERS
When the size of the dielectric scatterer is within 10 nm < r < 500 nm range, it is found that the spot size of light confinement with forward scattering depends on the forward area of the scatterer while the scattering intensity depends on the backward area of the scatterer. A corresponding teardrop scatterer composing a large bottom radius (r 200 nm) and a small top radius (r 10 nm) can serve as an efficient dielectric scatterer to achieve small spot size with high scattering intensity. The teardrop scatterer when combined with a diffractionlimited far-field lens to form a two-stage focusing device can achieve an intensity enhancement of ∼105 with a ∼10 nm spot size at the apex of the teardrop scatterer. The two-stage light focusing device constructed with pure dielectric can be valuable in high energy throughput nanoscale light detection, fabrication, and material manipulation (e.g., nano-optical tweezing). This work was supported by the National Science Foundation (CBET 0845794). The authors are pleased to acknowledge the Texas A&M Supercomputing Facility (http://sc.tamu.edu/) for providing the computing resources for this research. References 1. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002). 2. R. Heintzmann and G. Ficz, Methods Cell Biol. 81, 561 (2007). 3. E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991). 4. Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, Nano Lett. 5, 1726 (2005).
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