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Multiple reversals of optical binding force in plasmonic disk-ring nanostructures with dipole-multipole Fano resonances Qiang Zhang and Jun Jun Xiao* College of Electronic and Information Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, Guangdong, China *Corresponding author: [email protected] Received June 17, 2013; revised August 30, 2013; accepted September 13, 2013; posted September 17, 2013 (Doc. ID 192315); published October 14, 2013 We study the optical far-field and near-field characteristics, and the optical force effects of plasmonic disk-ring nanostructures. There are multiple Fano features resulting from the scattering interferences of the hybridized modes from the disk’s dipole and the ring’s higher-order modes. In particular, it is found that the optical binding force between the disk and the ring shows multiple sign reversals spectrally, from the dipole-quadrupole regime up to the dipole-decapole regime. We show that the zero-force points can be categorized into two types: the positive-to-negative ones resulting from the Fano dip and the negative-to-positive ones associated with the transitions between dipole-multipole modes. The multiple sign reversals of the optical forces are tunable by the geometrical size and gap of the disk and ring. Such characters make it possible to organize unusual optical matters from individual plasmonic nanoparticles. © 2013 Optical Society of America OCIS codes: (250.5403) Plasmonics; (120.4880) Optomechanics; (290.0290) Scattering. http://dx.doi.org/10.1364/OL.38.004240
Fano resonance is named after Ugo Fano, who gave the first theoretical explanation for the sharp asymmetric profile of autoionization spectra [1]. The Fano line shape cannot be described by the Lorentz formula [2,3]. In plasmonic nanostructures, Fano resonance can happen due to the resonant destructive interference between a super-radiant (bright) mode and subradiant (dark) modes [4,5]. Fundamentally, Fano resonance can be understood by a classical coupled oscillator model [2]. When the frequency of the external drive crosses the Fano dip, which is near the resonant frequency of the “dark” oscillator, there is a strong phase variation between the oscillations of the two coupled oscillators [2]. Various plasmonic nanostructures featuring Fano spectra have been reported, including dolmen structures, plasmonic ring-disk cavities, nanoparticle heterodimers, plasmonic nanohole arrays, and so on [6–10]. Despite the notable asymmetric line shape on far-field spectra, it is worth mentioning that the strong Fano phase variation is also important to near-field associated properties [11]. Optical force, one of the most important optomechanical effects, is often recognized as optical radiation pressure or gradient force [12,13]. When two nanoparticles are put in an intense optical field, mutual optical force between the nanoparticles may arise due to multiple scattering. This is often referred to as the optical binding force (OBF) [14]. The OBF between plasmonic nanoparticles is substantially related to the optical near field in terms of both strength and spatial distribution, thus can be dramatically enhanced by surface plasmon resonance [15–19]. It is interesting to study the impacts on the OBF by Fano resonance in plasmonic nanostructures. Very recently, we found that in plasmonic nanorod heterodimers, the dipole-quadrupole (DQ) Fano resonance dramatically affects the OBF and can even lead to a sign reversal [20]. This would certainly make very unusual consequences in bounding stable “optical matters” from plasmonic particles. Then, questions that naturally arise 0146-9592/13/204240-04$15.00/0
are if this OBF reversal persists for higher-order Fano resonances and how it can survive under different geometrical and spectral conditions. In this Letter we deliver such a study and show that the reversal can occur several times in a system of multiple Fano resonances. We choose a silver disk-ring nanostructure (DRN) for the study, as shown in Fig. 1. The height of the DRN and the wall width of the nanoring are kept as H 60 nm and W 20 nm, correspondingly. The multiple Fano resonances of the DRN come from the interference between the dipole mode of the nanodisk and the higher-order modes of the nanoring [21,22]. We consider a normally incident plane wave with longitudinal polarization along the axis of the DRN. Using the finite integral technique to obtain the optical fields, we calculate the optical forces on the ring and the disk by a closed-surface integration of the Maxwell stress tensor [20,23]. The dielectric function of the silver is taken from Johnson and Christy [24] and the nanostructure is assumed freestanding in air. We particularly examine the OBF Fbind between the disk and the ring for different frequencies. Same as in the DQ case [20], with the defined coordinated system, positive (negative) Fbind refers to attraction (repulsion) between the disk and the ring. To show the relationship between the OBF and the multiple Fano resonances clearly, we first study the multiple Fano spectra of the DRN. As a typical example,
Fig. 1.
Sketch of the disk-ring structure and the incident light.
© 2013 Optical Society of America
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS
we set the system with geometry parameters of Rdisk 70 nm, Rring 150 nm, g 20 nm, and show its optical properties in Fig. 2(a). In the visible regime, the nanodisk individually has a broadband scattering cross section (SCS, black-dashed line) that is contributed by its dipolar plasmon mode at 594 THz. For the nanoring, because of the larger size, its dipolar mode is far below this frequency range. We use a dipole source to excite its higher-order modes such as quadrupole, hexapole, octupole, and decapole [21]. These higher-order modes are regarded as “dark” since they have relatively much larger quality factors [22]. The peak positions in the local field enhancement (LFE) curve (blue dash-dotted line) are signatures of such higher-order modes and labeled with “Q,” “H,” “O,” and “D” in Fig. 2(a). The LFE is measured at a specific point in the opposite side to the dipole source, across the ring center, 10 nm away from the nanoring’s outer wall. As the two nanoparticles are put nearby, the dipolar mode of the disk and the higher-order modes of the ring couple and hybridize, leading to multiple Fano resonances that are observable as the multiple Fano dips in the SCS of the DRN (thick solid line). The numerically obtained OBF spectrum of the DRN is shown in Fig. 2(b). Interestingly, we see that inside the multiple Fano resonances band, Fbind crosses the zero point several times, which we refer to as multiple optical binding force reversals (MOBFRs). These MOBFRs indicate that, as the frequency of the incident plane wave increases from 300 to
700 THz, the nanodisk and the nanoring repeatedly experience several transitions between attraction and repulsion in their mutual interaction. It would be instructive to check when the OBF reversals would happen. There are, in fact, two classes of reversal points; one of them are the points where the OBF changes from positive to negative, as marked by the vertical dashed lines in Fig. 2(b). For short, we call them P-N points. They are at frequencies of f 392 THz, 516 THz, and 603 THz (labeled with A, B, and C), respectively. The others are the locations where the OBF changes from negative to positive (N-P points) and we have marked them with vertical dotted lines at frequencies f 456 THz, 562 THz, and 642 THz (labeled with A0 , B0 , and C0 ). It is noticed that below point A, there are two more reversal points which belongs to the dipole–dipole coupling range. Surprisingly, it is found that the reversal points A, B, and C all fall nearly at the Fano dips in the SCS of the DRN [solid line in Fig. 2(a)] and also coincide with the peaks in the LFE of the individual nanoring [dash-dotted line in Fig. 2(a)]. The LFE peaks indicate the resonant frequencies of higher-order modes of the nanoring. According to the coupled oscillator model, the Fano dip appears very close to the “dark mode” frequency [2]. These are strong evidence that the P-N reversals are closely associated with the Fano resonances. We note that the Fano dip corresponding to pinot A is hard to distinguish, since it is deeply merged in the nanoring’s dipolar resonant peak, which is at much lower frequency and is far outside the Fano resonance band [8]. The correspondence between the P-N reversal and the Fano resonance can be corroborated by the near-field distribution shown in Fig. 3. The local fields for frequencies on the opposite sides of these reversal points are plotted in Figs. 3(a)–3(c). We have deliberately chosen two frequencies near each reversal point. Here, the upper (lower) subplots are of frequencies lower (higher) than (a)
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Fig. 2. (a) SCS of the DRN (black-solid line) and the individual nanodisk (black-dashed line). The LFE of the individual nanoring (blue dash-dotted line) is also plotted. (b) OBF spectrum of the DRN. The reversal points are marked with A, B, and C with respect to the vertical dashed lines, and the other reversal points are marked with A0 , B0 , and C0 according to the vertical dotted lines.
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Fig. 3. Near-field amplitude patterns for frequencies near the six reversal points, as shown in Fig. 2, panel (a), (b), and (c) for the positive-to-negative ones (A, B, and C) while (d), (e), and (f) for the negative-to-positive points (A0 , B0 , and C0 ). The upper (lower) contour in each panel corresponds to a frequency at the lower (higher) side of the reversal points.
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Fig. 4. Surface charge density for frequencies in Fig. 3. The OBF is either positive or negative, labeled by “Attraction” and “Repulsion” in each panel.
the magnitudes of the OBF. According to the classical analogue, the amplitude of the super-radiant mode of the coupled oscillators can be written as [2] c1
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that of the reversal point. It is clearly seen that the primary mode excited inside the ring is the same for each group. The most obvious change is the spatial shift of the “hot spots.” At the lower frequencies of these P-N reversal points (i.e., left side of A, B, and C in Fig. 2), the “hot spots” appear inside the gap region of the DRN. However, once the frequencies are right beyond the Fano dips (i.e., at the right side of A, B, and C in Fig. 2), the maximum local fields are pushed away from the gap region, residing more inside the ring’s inner surfaces. This redistribution of the “hot spots” leads to sharp variation of the near-field gradient across the gap. The phase difference of the coupled modes varies quickly as well near the Fano dip. Figure 4 shows the surface charge density for the cases in the corresponding panels of Fig. 3. Comparing the upper and lower panels of Figs. 4(a)–4(c), it is seen that the charge distributions on the nanoring remain almost untouched, while those on the nanodisk show visible variations, reminiscent of a phase change. In fact, the strong phase variation near the Fano dip frequencies can induce fine structures of the charge distribution [8]. Yet, it is clearly seen that, when the binding force is positive (attraction), more opposite charges accumulate over the gap side. We then focus on the N-P reversal points, e.g., points A0 , B0 , and C0 in Fig. 2. These are quite different from the P-N reversal points whose positions are close to the Fano dips of SCS in Fig. 2(a). The N-P reversal points are almost located at the dips of the LFE of the individual nanoring. Spectrally, the LFE dips must appear in the off-resonance windows between the higher-order modes of the nanoring. Concomitantly, the scattering of the DRN mainly comes from the nanodisk. For the N-P points, there is no remarkable feature in the SCS of the DRN. These indicate that this class of the reversal points must be related to the variation of the plasmon modes of the ring. Indeed, it is seen in Figs. 3(d)–3(f) that, as the frequency of the incident light sweeps through these points (A0 , B0 , and C0 ), the near-field pattern of the nanoring has apparent evolution (node number increases by one). For example, in Fig. 3(e) at f 550 THz, which is slightly less than that (562 THz) of point B0 , the nanoring shows a typical hexapolar pattern, but as the frequency increases to 570 THz, which is a little bit higher than that of the reversal point B0 , the nanoring node facing the gap splits into two parts, more like a octupolar mode. This demonstrates that B0 is a transition point between the dipole-hexapole hybridization mode and the dipole-octupole hybridization mode. Similarly, the same transitions can be seen in Figs. 3(d) and 3(f), which show that A0 and C0 are, respectively, the transition points of dipole-quadrupolar to dipole-hexapolar and dipoleoctupolar to dipole-decapolar regime. In contrast to the P-N reversal points A, B, and C, for the N-P reversal points, the “hot spots” converge much inside the gap region from the edges of the DRN. The observation that the N-P reversal is caused by mode transitions can also be seen in the charge density plots in Figs. 4(d)–4(f), where the nanodisk has no obvious change, but in the gap region of the nanoring, a new node appears. So far, it is clear that the multiple Fano resonances can lead to MOBFRs. Next, we explore how the Fano resonances affect the positions of the reversal points and
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Fig. 5. (a)–(c) OBF spectra for different g, Rdisk , and Rring . (d)–(f) Contour plots for (a)–(c), respectively.
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS
while the other two remain the same as in Fig. 2. These geometrical parameters can tune the Fano resonances [19] and it is shown here that the OBF spectra are also strongly affected. Nevertheless, the influences by the three geometrical parameters are quite different. The reversal points are more likely determined by Rring , analogously f 2 , which are consistent with the reversal mechanisms. The differences are more clearly demonstrated by surveying the shifts of the boundaries between the positive and the negative OBF, as shown in Figs. 5(d)–5(f). We can find that the shifts in Fig. 5(f) are much more remarkable than those in Figs. 5(d) and 5(e). In particular, in Fig. 5(e) both the P-N and N-P OBF boundaries stay almost at fixed frequencies. This is simply because the Fano spectrum is less correlated to Rdisk [e.g., f 1 in Eq. (1)] for a relatively large γ 1 . Finally, a few comments on the magnitudes of the OBF are in order. With an incident plane wave, the magnitude of the OBF is related to the spatial distance between the two coupled nanoparticles, the overlap of their optical spectra, the amplitude of each resonance, and the modes’ coupling coefficient which is related to their quality factors. The OBF at the reversal points vanishes once the amplitude of one resonant mode is strongly suppressed. Near the Fano dips (P-N points), the dipole mode of the nanodisk is frustrated by the Fano destructive interference, while, for the dips in LFE (N-P points), the frequency falls right between two high-quality resonances of the nanoring. It’s not surprising to see that, when the nanoparticles are at a larger gap, g, the magnitudes of the OBF decrease no matter if they are attractive or repulsive [see Fig. 5(a)]. The impact of particle size on the OBF magnitude is more complicated, because both spectral overlap and coupling efficiency can be altered by the particle size [see Figs. 5(b) and 5(c)]. It is noticed in Fig. 5(c) that the OBF reversal by dipole-decapole can happen for Rring 180–200 nm, which is not found in the case of Fig. 2. For example, when Rring 200 nm, the OBF becomes negative in the dipole-decapole regime, which is shifted to around 580–610 THz. In summary, we have performed systematic calculations of the OBF between two nanoparticles, with one having a dominant dipolar resonant mode and the other with higher-order modes. The plasmonic hybridization in the nanoparticle pair results in multiple Fano resonances that lead to multiple OBF reversals. The zero-force points of positive-to-negative transition are due to a maximum phase change near the Fano dip at the optical SCS, while those of negative-to-positive transition originate from the high-order mode evolution and competition as the frequency varies. These findings may be experimentally demonstrated in a plasmonically coated near-field optical tweezer and could be of interest in organizing “optical matters” of plasmonic nanoparticles.
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This work is supported by the NSFC (11004043, 11274083, and 61107036), the SMSTP (KQCX2012 0801093710373, JCYJ20120613114137248, 2011PTZZ048, and JC201105160524A), and the NSFIR (HIT.NSFIR. 2010131). References 1. U. Fano, Phys. Rev. 124, 1866 (1961). 2. Y. S. Joe, A. M. Satanin, and C. S. Kim, Phys. Scr. 74, 259 (2006). 3. B. Gallinet and O. J. F. Martin, Phys. Rev. B 83, 235427 (2011). 4. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev. Mod. Phys. 82, 2257 (2010). 5. B. Luk’yanchuk, N. I. Zheludv, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, Nat. Mater. 9, 707 (2010). 6. N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. V. Dorpe, P. Nordlander, and S. A. Maier, Nano Lett. 9, 1663 (2009). 7. F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, Nano Lett. 8, 3983 (2008). 8. L. V. Brown, H. Sobhani, J. B. Lassiter, P. Nordlander, and N. J. Halas, ACS Nano 4, 819 (2010). 9. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, Nature 391, 667 (1998). 10. A. A. Yanik, A. E. Cetin, M. Huang, A. Arter, S. H. Mousavi, A. Khanikaev, J. H. Connor, G. Shvets, and H. Altug, Proc. Natl. Acad. Sci. USA 108, 11784 (2011). 11. B. Gallinet and O. J. F. Martin, Opt. Express 19, 22167 (2011). 12. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, Opt. Lett., 11, 288 (1986). 13. K. C. Neuman and S. M. Block, Rev. Sci. Instrum. 75, 2787 (2004). 14. R. W. Bowman and M. J. Padgett, Rep. Prog. Phys. 76, 026401 (2013). 15. V. D. Miljković, T. Pakizeh, B. Sepulveda, P. Johansson, and M. Käll, J. Phys. Chem. C 114, 7472 (2010). 16. J. Ng, R. Tang, and C. T. Chan, Phys. Rev. B 77, 195407 (2008). 17. H. Liu, J. Ng, S. B. Wang, Z. H. Hang, C. T. Chan, and S. N. Zhu, New J. Phys. 13, 073040 (2011). 18. R. A. Nome, M. J. Guffey, N. F. Scherer, and S. K. Gray, J. Phys. Chem. A 113, 4408 (2009). 19. V. Demergis and E. L. Florin, Nano Lett. 12, 5756 (2012). 20. Q. Zhang, J. J. Xiao, X. M. Zhang, Y. Yao, and H. Liu, Opt. Express 21, 6601 (2013). 21. Y. Zhang, T. Q. Jia, H. M. Zhang, and Z. Z. Xu, Opt. Lett. 37, 4919 (2012). 22. J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, Phys. Rev. Lett. 90, 057401 (2003). 23. Q. Zhang, J. J. Xiao, X. M. Zhang, and Y. Yao, Opt. Commun. 301–302, 121 (2013). 24. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
OPTICS LETTERS / Vol. 38, No. 20 / October 15, 2013
Multiple reversals of optical binding force in plasmonic disk-ring nanostructures with dipole-multipole Fano resonances Qiang Zhang and Jun Jun Xiao* College of Electronic and Information Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, Guangdong, China *Corresponding author: [email protected] Received June 17, 2013; revised August 30, 2013; accepted September 13, 2013; posted September 17, 2013 (Doc. ID 192315); published October 14, 2013 We study the optical far-field and near-field characteristics, and the optical force effects of plasmonic disk-ring nanostructures. There are multiple Fano features resulting from the scattering interferences of the hybridized modes from the disk’s dipole and the ring’s higher-order modes. In particular, it is found that the optical binding force between the disk and the ring shows multiple sign reversals spectrally, from the dipole-quadrupole regime up to the dipole-decapole regime. We show that the zero-force points can be categorized into two types: the positive-to-negative ones resulting from the Fano dip and the negative-to-positive ones associated with the transitions between dipole-multipole modes. The multiple sign reversals of the optical forces are tunable by the geometrical size and gap of the disk and ring. Such characters make it possible to organize unusual optical matters from individual plasmonic nanoparticles. © 2013 Optical Society of America OCIS codes: (250.5403) Plasmonics; (120.4880) Optomechanics; (290.0290) Scattering. http://dx.doi.org/10.1364/OL.38.004240
Fano resonance is named after Ugo Fano, who gave the first theoretical explanation for the sharp asymmetric profile of autoionization spectra [1]. The Fano line shape cannot be described by the Lorentz formula [2,3]. In plasmonic nanostructures, Fano resonance can happen due to the resonant destructive interference between a super-radiant (bright) mode and subradiant (dark) modes [4,5]. Fundamentally, Fano resonance can be understood by a classical coupled oscillator model [2]. When the frequency of the external drive crosses the Fano dip, which is near the resonant frequency of the “dark” oscillator, there is a strong phase variation between the oscillations of the two coupled oscillators [2]. Various plasmonic nanostructures featuring Fano spectra have been reported, including dolmen structures, plasmonic ring-disk cavities, nanoparticle heterodimers, plasmonic nanohole arrays, and so on [6–10]. Despite the notable asymmetric line shape on far-field spectra, it is worth mentioning that the strong Fano phase variation is also important to near-field associated properties [11]. Optical force, one of the most important optomechanical effects, is often recognized as optical radiation pressure or gradient force [12,13]. When two nanoparticles are put in an intense optical field, mutual optical force between the nanoparticles may arise due to multiple scattering. This is often referred to as the optical binding force (OBF) [14]. The OBF between plasmonic nanoparticles is substantially related to the optical near field in terms of both strength and spatial distribution, thus can be dramatically enhanced by surface plasmon resonance [15–19]. It is interesting to study the impacts on the OBF by Fano resonance in plasmonic nanostructures. Very recently, we found that in plasmonic nanorod heterodimers, the dipole-quadrupole (DQ) Fano resonance dramatically affects the OBF and can even lead to a sign reversal [20]. This would certainly make very unusual consequences in bounding stable “optical matters” from plasmonic particles. Then, questions that naturally arise 0146-9592/13/204240-04$15.00/0
are if this OBF reversal persists for higher-order Fano resonances and how it can survive under different geometrical and spectral conditions. In this Letter we deliver such a study and show that the reversal can occur several times in a system of multiple Fano resonances. We choose a silver disk-ring nanostructure (DRN) for the study, as shown in Fig. 1. The height of the DRN and the wall width of the nanoring are kept as H 60 nm and W 20 nm, correspondingly. The multiple Fano resonances of the DRN come from the interference between the dipole mode of the nanodisk and the higher-order modes of the nanoring [21,22]. We consider a normally incident plane wave with longitudinal polarization along the axis of the DRN. Using the finite integral technique to obtain the optical fields, we calculate the optical forces on the ring and the disk by a closed-surface integration of the Maxwell stress tensor [20,23]. The dielectric function of the silver is taken from Johnson and Christy [24] and the nanostructure is assumed freestanding in air. We particularly examine the OBF Fbind between the disk and the ring for different frequencies. Same as in the DQ case [20], with the defined coordinated system, positive (negative) Fbind refers to attraction (repulsion) between the disk and the ring. To show the relationship between the OBF and the multiple Fano resonances clearly, we first study the multiple Fano spectra of the DRN. As a typical example,
Fig. 1.
Sketch of the disk-ring structure and the incident light.
© 2013 Optical Society of America
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS
we set the system with geometry parameters of Rdisk 70 nm, Rring 150 nm, g 20 nm, and show its optical properties in Fig. 2(a). In the visible regime, the nanodisk individually has a broadband scattering cross section (SCS, black-dashed line) that is contributed by its dipolar plasmon mode at 594 THz. For the nanoring, because of the larger size, its dipolar mode is far below this frequency range. We use a dipole source to excite its higher-order modes such as quadrupole, hexapole, octupole, and decapole [21]. These higher-order modes are regarded as “dark” since they have relatively much larger quality factors [22]. The peak positions in the local field enhancement (LFE) curve (blue dash-dotted line) are signatures of such higher-order modes and labeled with “Q,” “H,” “O,” and “D” in Fig. 2(a). The LFE is measured at a specific point in the opposite side to the dipole source, across the ring center, 10 nm away from the nanoring’s outer wall. As the two nanoparticles are put nearby, the dipolar mode of the disk and the higher-order modes of the ring couple and hybridize, leading to multiple Fano resonances that are observable as the multiple Fano dips in the SCS of the DRN (thick solid line). The numerically obtained OBF spectrum of the DRN is shown in Fig. 2(b). Interestingly, we see that inside the multiple Fano resonances band, Fbind crosses the zero point several times, which we refer to as multiple optical binding force reversals (MOBFRs). These MOBFRs indicate that, as the frequency of the incident plane wave increases from 300 to
700 THz, the nanodisk and the nanoring repeatedly experience several transitions between attraction and repulsion in their mutual interaction. It would be instructive to check when the OBF reversals would happen. There are, in fact, two classes of reversal points; one of them are the points where the OBF changes from positive to negative, as marked by the vertical dashed lines in Fig. 2(b). For short, we call them P-N points. They are at frequencies of f 392 THz, 516 THz, and 603 THz (labeled with A, B, and C), respectively. The others are the locations where the OBF changes from negative to positive (N-P points) and we have marked them with vertical dotted lines at frequencies f 456 THz, 562 THz, and 642 THz (labeled with A0 , B0 , and C0 ). It is noticed that below point A, there are two more reversal points which belongs to the dipole–dipole coupling range. Surprisingly, it is found that the reversal points A, B, and C all fall nearly at the Fano dips in the SCS of the DRN [solid line in Fig. 2(a)] and also coincide with the peaks in the LFE of the individual nanoring [dash-dotted line in Fig. 2(a)]. The LFE peaks indicate the resonant frequencies of higher-order modes of the nanoring. According to the coupled oscillator model, the Fano dip appears very close to the “dark mode” frequency [2]. These are strong evidence that the P-N reversals are closely associated with the Fano resonances. We note that the Fano dip corresponding to pinot A is hard to distinguish, since it is deeply merged in the nanoring’s dipolar resonant peak, which is at much lower frequency and is far outside the Fano resonance band [8]. The correspondence between the P-N reversal and the Fano resonance can be corroborated by the near-field distribution shown in Fig. 3. The local fields for frequencies on the opposite sides of these reversal points are plotted in Figs. 3(a)–3(c). We have deliberately chosen two frequencies near each reversal point. Here, the upper (lower) subplots are of frequencies lower (higher) than (a)
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Fig. 2. (a) SCS of the DRN (black-solid line) and the individual nanodisk (black-dashed line). The LFE of the individual nanoring (blue dash-dotted line) is also plotted. (b) OBF spectrum of the DRN. The reversal points are marked with A, B, and C with respect to the vertical dashed lines, and the other reversal points are marked with A0 , B0 , and C0 according to the vertical dotted lines.
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Fig. 3. Near-field amplitude patterns for frequencies near the six reversal points, as shown in Fig. 2, panel (a), (b), and (c) for the positive-to-negative ones (A, B, and C) while (d), (e), and (f) for the negative-to-positive points (A0 , B0 , and C0 ). The upper (lower) contour in each panel corresponds to a frequency at the lower (higher) side of the reversal points.
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650 THz
465 THz
550 THz
445 THz
0
Fig. 4. Surface charge density for frequencies in Fig. 3. The OBF is either positive or negative, labeled by “Attraction” and “Repulsion” in each panel.
the magnitudes of the OBF. According to the classical analogue, the amplitude of the super-radiant mode of the coupled oscillators can be written as [2] c1
f 21
f 22 − f 2 iγ 2 f A; − f iγ 1 f f 22 − f 2 iγ 2 f − μ2 2
(1)
where f 1 and f 2 are the intrinsic resonant frequencies of the individual oscillators, γ 1 and γ 2 are correspondingly their dissipations, μ is the coupling strength, and A is a constant for the driving amplitude. Here, we are only concerned with three of them, e.g., μ, f 1 , and f 2 since they are relatively easy to control in the DRN. In the DRN the coupling strength, μ, is mainly determined by the gap, g, while f 1 and f 2 are mainly determined by the radii of the nanodisk Rdisk and the nanoring Rring , respectively. Figures 5(a)–5(c) show the OBF spectra in the parameter space (g, Rdisk , Rring ), each panel has one of them varying
F
bind
0.8
g=20 nm
(a)
(d)
g=40 nm
0.6 0.4 0.2
35 0
25
0.0 15
1.0
(b)
R
=70 nm
− +
(e)120
disk
Rdisk=90 nm =110 nm
110 100
disk
disk
(nm)
R
R
1.0 0.8 0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6
0
90 80
Rring=180 nm
(c)
0.8
R
0.6
Rring=200 nm
0.4 0.2
180 170
0
160
0.0 −0.2 350
190
=190 nm
ring
− +
(f)200
Rring (nm)
Fbind (pN/mW/µm2) 2
+
45
−0.2
(pN/mW/µm )
55
g=60 nm
g (nm)
(pN/mW/µm2)
1.0
bind
that of the reversal point. It is clearly seen that the primary mode excited inside the ring is the same for each group. The most obvious change is the spatial shift of the “hot spots.” At the lower frequencies of these P-N reversal points (i.e., left side of A, B, and C in Fig. 2), the “hot spots” appear inside the gap region of the DRN. However, once the frequencies are right beyond the Fano dips (i.e., at the right side of A, B, and C in Fig. 2), the maximum local fields are pushed away from the gap region, residing more inside the ring’s inner surfaces. This redistribution of the “hot spots” leads to sharp variation of the near-field gradient across the gap. The phase difference of the coupled modes varies quickly as well near the Fano dip. Figure 4 shows the surface charge density for the cases in the corresponding panels of Fig. 3. Comparing the upper and lower panels of Figs. 4(a)–4(c), it is seen that the charge distributions on the nanoring remain almost untouched, while those on the nanodisk show visible variations, reminiscent of a phase change. In fact, the strong phase variation near the Fano dip frequencies can induce fine structures of the charge distribution [8]. Yet, it is clearly seen that, when the binding force is positive (attraction), more opposite charges accumulate over the gap side. We then focus on the N-P reversal points, e.g., points A0 , B0 , and C0 in Fig. 2. These are quite different from the P-N reversal points whose positions are close to the Fano dips of SCS in Fig. 2(a). The N-P reversal points are almost located at the dips of the LFE of the individual nanoring. Spectrally, the LFE dips must appear in the off-resonance windows between the higher-order modes of the nanoring. Concomitantly, the scattering of the DRN mainly comes from the nanodisk. For the N-P points, there is no remarkable feature in the SCS of the DRN. These indicate that this class of the reversal points must be related to the variation of the plasmon modes of the ring. Indeed, it is seen in Figs. 3(d)–3(f) that, as the frequency of the incident light sweeps through these points (A0 , B0 , and C0 ), the near-field pattern of the nanoring has apparent evolution (node number increases by one). For example, in Fig. 3(e) at f 550 THz, which is slightly less than that (562 THz) of point B0 , the nanoring shows a typical hexapolar pattern, but as the frequency increases to 570 THz, which is a little bit higher than that of the reversal point B0 , the nanoring node facing the gap splits into two parts, more like a octupolar mode. This demonstrates that B0 is a transition point between the dipole-hexapole hybridization mode and the dipole-octupole hybridization mode. Similarly, the same transitions can be seen in Figs. 3(d) and 3(f), which show that A0 and C0 are, respectively, the transition points of dipole-quadrupolar to dipole-hexapolar and dipoleoctupolar to dipole-decapolar regime. In contrast to the P-N reversal points A, B, and C, for the N-P reversal points, the “hot spots” converge much inside the gap region from the edges of the DRN. The observation that the N-P reversal is caused by mode transitions can also be seen in the charge density plots in Figs. 4(d)–4(f), where the nanodisk has no obvious change, but in the gap region of the nanoring, a new node appears. So far, it is clear that the multiple Fano resonances can lead to MOBFRs. Next, we explore how the Fano resonances affect the positions of the reversal points and
F
4242
150 400
450 500 550 Frequency (THz)
600
390
440 490 540 590 Frequency (THz)
640
−
Fig. 5. (a)–(c) OBF spectra for different g, Rdisk , and Rring . (d)–(f) Contour plots for (a)–(c), respectively.
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS
while the other two remain the same as in Fig. 2. These geometrical parameters can tune the Fano resonances [19] and it is shown here that the OBF spectra are also strongly affected. Nevertheless, the influences by the three geometrical parameters are quite different. The reversal points are more likely determined by Rring , analogously f 2 , which are consistent with the reversal mechanisms. The differences are more clearly demonstrated by surveying the shifts of the boundaries between the positive and the negative OBF, as shown in Figs. 5(d)–5(f). We can find that the shifts in Fig. 5(f) are much more remarkable than those in Figs. 5(d) and 5(e). In particular, in Fig. 5(e) both the P-N and N-P OBF boundaries stay almost at fixed frequencies. This is simply because the Fano spectrum is less correlated to Rdisk [e.g., f 1 in Eq. (1)] for a relatively large γ 1 . Finally, a few comments on the magnitudes of the OBF are in order. With an incident plane wave, the magnitude of the OBF is related to the spatial distance between the two coupled nanoparticles, the overlap of their optical spectra, the amplitude of each resonance, and the modes’ coupling coefficient which is related to their quality factors. The OBF at the reversal points vanishes once the amplitude of one resonant mode is strongly suppressed. Near the Fano dips (P-N points), the dipole mode of the nanodisk is frustrated by the Fano destructive interference, while, for the dips in LFE (N-P points), the frequency falls right between two high-quality resonances of the nanoring. It’s not surprising to see that, when the nanoparticles are at a larger gap, g, the magnitudes of the OBF decrease no matter if they are attractive or repulsive [see Fig. 5(a)]. The impact of particle size on the OBF magnitude is more complicated, because both spectral overlap and coupling efficiency can be altered by the particle size [see Figs. 5(b) and 5(c)]. It is noticed in Fig. 5(c) that the OBF reversal by dipole-decapole can happen for Rring 180–200 nm, which is not found in the case of Fig. 2. For example, when Rring 200 nm, the OBF becomes negative in the dipole-decapole regime, which is shifted to around 580–610 THz. In summary, we have performed systematic calculations of the OBF between two nanoparticles, with one having a dominant dipolar resonant mode and the other with higher-order modes. The plasmonic hybridization in the nanoparticle pair results in multiple Fano resonances that lead to multiple OBF reversals. The zero-force points of positive-to-negative transition are due to a maximum phase change near the Fano dip at the optical SCS, while those of negative-to-positive transition originate from the high-order mode evolution and competition as the frequency varies. These findings may be experimentally demonstrated in a plasmonically coated near-field optical tweezer and could be of interest in organizing “optical matters” of plasmonic nanoparticles.
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