February 1, 2014 / Vol. 39, No. 3 / OPTICS LETTERS
501
Interplay between absorption and radiative decay rates of surface plasmon polaritons for field enhancement in periodic arrays ZhaoLong Cao, Lei Zhang, Chung-Yu Chan, and Hock-Chun Ong* Department of Physics, The Chinese University of Hong Kong, Shatin, China *Corresponding author: [email protected] Received September 12, 2013; revised December 15, 2013; accepted December 16, 2013; posted December 16, 2013 (Doc. ID 197552); published January 23, 2014 We studied the effects of absorption and radiative decay rates of surface plasmon polaritons on the field enhancement in periodic metallic arrays by temporal coupled mode theory and finite-difference time-domain simulation. When two rates are equal, the field enhancement is the strongest and the peak height of the orthogonal reflectivity reaches 0.25. To demonstrate this fact, we fabricated two series of two-dimensional Au and Ag nanohole arrays with different geometries and measured their corresponding reflectivity and decay rates. The experimental results agree well with the analytical and numerical results. © 2014 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (260.5740) Resonance. http://dx.doi.org/10.1364/OL.39.000501
Maximizing the field enhancement in metallic structures has long been the prime question in plasmonics [1]. Applications such as fluorescence enhancement [2], energy harvesting [3], surface-enhanced Raman scattering (SERS) [4], surface plasmon resonance (SPR) sensing [4], radiation force manipulation [5], and photocatalysis [6] all depend on a strong local field. For example, a strong field increases both the generation and recombination rates of electron-hole pair in light absorbers and emitters, thus enhancing the absorption and spontaneous emission efficiencies [2,3,7,8]. Other than that, the local field senses molecules in sub-wavelength domain, enabling a route to detect molecules by SPR and SERS at extremely low concentration [9,10]. However, the general guideline to optimize the field strength is not yet available and research is still ongoing [1,11]. The main reason is that too many parameters such as geometry, material, and wavelength are needed to determine the resulting field strength and they are inextricable from each other [11]. More importantly, there seems to be a “missing link” between them. For example, it is still not quite understood what exact role geometry plays in controlling the field [12]. Periodic metallic arrays are important plasmonic systems [13]. Since their lattice and basis can be fabricated with great precision, they are expected to produce more controllable surface plasmon polaritons (SPPs), rendering more reliable performance in applications. As a result, periodic arrays have been used to make lightemitting diodes [14], solar cells [7], SERS and SPR sensors [9,15], and plasmonic tweezers [5]. It is therefore essential to develop strategy for field enhancement in periodic structures. While current approaches tend to establish a complete experimental or numerical data bank to identify the optimal condition, there is not much analytical effort [12,16]. Although analytical methods sometimes are not as rigorous as numerical simulations, they in fact provide good insight to understand the underlying physics, bridging the physical parameters and the field enhancement. 0146-9592/14/030501-04$15.00/0
In this Letter, use temporal coupled mode theory (CMT) to show that, under some circumstances, the field enhancement in periodic arrays can be optimized by matching the absorption Γabs and radiative decay Γrad rates of SPPs. In addition, when such conditions are met, the peak height of the orthogonal reflectivity reaches 0.25. To verify this concept, we performed finite-difference time-domain (FDTD) simulations on one- and two-dimensional (1D and 2D), arrays with different geometries. To demonstrate, we prepared two series of 2D Au and Ag nanohole arrays with different hole sizes by interference lithography and measured their Γabs and Γrad as well as their orthogonal reflectivity using angle-resolved reflectivity spectroscopy. Our experimental results agree well with the CMT prediction. Because it is known that decay rates are a strong function of geometry [17], the connection between field strength and geometry has been established. We began with the CMT formulation [18–20]. CMT for field enhancement has been used in nanoantennas [21] and nanoparticles [22] and is extended here for periodic arrays. For an optically thick array with m input/output ports, the time variant of a single propagating SPP mode amplitude a can be expressed as: da∕dt iωSPP − Γtot ∕2a hκj js i in Dirac’s bracket notation, where ωSPP is the resonant angular frequency (eV), Γtot is the total decay rate Γabs Γrad , and jκi κ1 κ 2 κ m T and js i s1 s2 s1m T are m × 1 column vectors signifying the complex incoupling constants and the amplitudes of the incident waves. P By applying time-reversal symmetry, we find n n hκjκi m n Γr Γrad [18,19], where Γr is the radiative decay rate of each individual port. If we consider −1 or −1; 0 SPP mode in 1D or 2D arrays [1,8,9,23], only one single kth input/output port exists and hκjκi Γkr Γrad and js i sk . Knowing the mode amplitude is harmonic with time, the above differential equation yields jaj2 Γrad jsk j2 ∕ω − ωSPP 2 Γtot ∕22 . In addition, jaj2 is related to the plasmonic field strength jE SPP j2 by jaj2 1∕2εo jE SPP j2 V eff , where εo is the permittivity © 2014 Optical Society of America
502
OPTICS LETTERS / Vol. 39, No. 3 / February 1, 2014
and V eff is the effective mode volume [21]. At ωSPP , we see if Γabs and V eff are slowly varying with respect to Γrad , jE SPP j2 is maximal when Γabs Γrad simply by taking ∂jE SPP j2 ∕∂Γrad 0. For periodic arrays, if the size of the basis is smaller than wavelength, Γabs can be approximated as the plain metal absorption given as Γabs ≈ ωε00m ∕ε0m 2 ε0m ∕ε0m 13∕2 , where εm0 i εm00 is the complex dielectric function of metal [17,23]. Thus, Γabs depends primarily on metal type and incident wavelength but is expected to vary slowly with geometry. In addition, assuming that V eff is proportional to the propagating length of propagating SPPs and the field decay constant in the direction normal to metal surface, we believe V eff also does not rely much on geometry when the basis is small compared to wavelength. As a result, for a given resonant wavelength, jE SPP j2 is the strongest if one varies the geometry to achieve Γabs ≈ Γrad . In addition, orthogonal, or cross-polarized, reflectivity spectroscopy can be used to probe the condition where Γabs Γrad is fulfilled. For a single port, only the specular reflection is present. If the incident polarizer is oriented at 45° with respect to the incidence plane, the Jones 1 vector of the incident field is given as E o , where 1 Eo is the field amplitude, for p- and s-polarizations. From CMT, under conservation of energy, the p-outgoing wave is given as s−k r p sk Γrad sk eiϕ ∕iω − ωSPP Γtot ∕2, where r p is the p-nonresonant reflection coefficient and ϕ is the in- and out-coupling phase-shift. One can see the wave is the interference between the reflection background and the SPP radiation damping, yielding the well-known Fano resonance [24,25]. As a result, assuming the s-incident field does not excite any resonance and no p-s polarization conversion occurs, the reflection vector after the array can be written as 2 R Eo 4 r p
3 Γrad e 5 iω − ωSPP Γtot ∕2 rs iϕ
(1)
[19,26], where r s is the s-reflection coefficient. If one places a detection analyzer orthogonally to the incident 1 −1 polarizer, the Jones operation thus is 1∕2 R −1 1 and the reflectivity Rorth measured is then given as Γ2rad ∕4ω − ωSPP 2 Γtot ∕22 , assuming r p ≈ r s over a range of wavelength. The reflectivity spectrum exhibits a Lorentzian line shape [19,26]. At resonance and Γabs Γrad , the peak height of Rorth is equal to 0.25, which thereby serves as a signature to identify the crossing of two rates. On the other hand, since jr p j is slightly less than one for most real metals, conventional p-polarized reflectivity is not necessarily equal to zero at Γabs Γrad and thus does not serve as a reliable indicator. In addition, orthogonal reflectivity enjoys higher signal-to-noise ratio than the p-polarized counterpart and is more detectable [9]. To verify the results from CMT, we have conducted FDTD. In fact, numerical simulation serves as a good tool to confirm our analytical prediction because it can construct ideal samples that are impossible or difficult
to fabricate experimentally. In particular, surface roughness and imperfect geometry are usually found in actual samples, which obscures the interpretation. Three series of 1D and 2D Ag and Au arrays with period P 670 nm, depth H 40, 60 and 150 nm, and groove width W or hole radius R varying from 30 to 230 nm at λSPP ∼ 790, 900, 900, and 910 nm, corresponding to −1 or (−1, 0) SPPs [23]. We choose shallow groove depth to avoid the emergence of waveguide mode in which its coupling with SPPs would complicate the situation [27]. We used Bloch boundary conditions at two/four sides and perfectly matched layers on the top and at the bottom of the cell. We set the wavelength of the pulsed excitation source centered either at 800 or 900 nm with bandwidth ∼100 nm to ensure that only one SPP mode is excited. For each array, we simulated the orthogonal reflectivity spectra at different incident angles for identifying the suitable angle for exciting the SPPs at λSPP [26]. Once the angle was identified, we first determined the near field pattern by placing the power monitor at 1 nm above the metal surface. We then computed Γrad and Γabs of SPPs by using the time-domain method [26]. The computation involves two steps. First, Γtot is determined by examining the transient of the field strength, which shows a single exponential decay curve typically for one resonance. We then calculated Γrad by constructing several new dielectric functions of Au and Ag by dividing their imaginary part two and five times so the absorption of metal can be proportionally reduced [26]. The decay rates simulated in these cases have identical Γrad but different Γabs . Finally, we extrapolated the y-intercept from the plot of decay rate against the reduced imaginary part since it corresponds to negligible metal absorption, thus yielding Γrad . Γabs is equal to Γtot − Γrad . The simulated Γabs and Γrad of three series are plotted in Fig. 1(a)–1(d) together with their corresponding average electric field strength and orthogonal reflectivity spectrum. The average field strength is calculated by summing up all fields from the meshes and then dividing by the mesh number. The figures show that Γabs , compared to Γrad , varies very slowly with respect to W , R, and H, except there is a slight increase at large R. The analytical Γabs for flat Ag and Au surfaces are shown as the dash lines and they compare reasonably with the simulations, justifying our assumption that Γabs behaves like a flat surface when λ > W or R. On the other hand, Γrad varies considerably with W and R, exhibiting W 1.01 and R3.8–4.8 dependences. In fact, our previous works on nanohole arrays have shown that W 1.01 depends very sensitively to hole geometry such as hole size and shape [17,28]. In particular, within the range of geometries considered here, the behavior of Γrad can be understood as the result of the scattering of SPPs by single isolated holes [17]. Our results show Γrad exhibits close to R4 dependence at a small hole size but becomes R5–8 when hole size increases [17], in agreement with the calculations here. The reflectivity peak height increases with an increasing radius due to the fact that the height is equal to Γrad ∕Γtot 2 and Γrad increases with W and R. For almost constant Γabs , the height slowly approaches one. Nevertheless, in all cases when two decay rates are matched, the
February 1, 2014 / Vol. 39, No. 3 / OPTICS LETTERS
Fig. 1. One-dimensional (a) Ag array with period 670 nm, height 40 nm, and different groove widths. Two-dimensional (b) Ag and (c) Au arrays with different radii. The Ag arrays have period 670 nm and height 60 nm whereas the Au arrays have period 670 nm and height 150 nm. Each figure has three parts. The top is the absorption and radiative decay rates of −1 or −1; 0 SPPs taken at resonant wavelength ∼790, 900, and 910 nm in Γ-X direction as a function of groove width or radius. The analytical flat metal absorption rates are shown as dashed lines. The middle and the bottom are their corresponding average electric field strength and orthogonal reflectivity spectra. A dashed line at reflectivity 0.25 is shown.
average field strength is the strongest and the orthogonal reflectivity peak height reaches 0.25, unambiguously confirming the prediction from CMT. To demonstrate, we fabricated two sets of Ag and Au 2D nanohole arrays with different hole radii by interference lithography and thin film deposition [17,26,28]. The period and height of the arrays are kept at 670 and 140 nm for Ag and 650 and 270 nm for Au. By changing the exposure time, we vary R from 70 to 165 nm. Noted that we intentionally choose the hole depths in these two sets to be larger than those in simulations. In addition, the period of the Au arrays is 650 nm instead of 670 nm. Therefore, the experimental results are expected to strengthen the analytical prediction but not to verify the simulations. It also is a proof-of-concept that far-field reflectivity can identify the strongest field enhancement if near-field scanning optical microscope is not available for direct field strength measurement. As an example, a scanning electron microscopy image of one of the Ag arrays is shown in the inset of Fig. 2(a), showing the array has R 102 nm. We used orthogonal reflectivity spectroscopy and CMT to determine their absorption and radiative decay rates. For optical measurements, a collimated white light from a quartz lamp is illuminated on the sample mounted on a computer-controlled goniometer and the angle-dependent specular reflection is captured by a spectrometer–CCD detection system. A pair of incident polarizer and detection analyzer is fixed at 45° and −45° to perform orthogonal polarization [26]. Figure 2(a) shows the reflectivity mapping of the Ag sample taken in Γ-X direction, illustrating the dispersive −1; 0 SPP reflection band as confirmed by the phase-matching equation [19,26]. Instead of dips, reflection peaks are observed from the mapping [19] and several spectra
503
Fig. 2. (a) Angle- and wavelength-resolved orthogonal reflectivity mapping taken in Γ-X direction. The SEM image of the array is in the inset, showing the holes are arranged in cubic structure with period and radius 670 and 102 nm. The dashed line is deduced from phase-matching equation showing −1; 0 SPP mode is excited. (b) The corresponding reflectivity spectra (symbols) are extracted at different incident angles. The solid lines are the best fits by using CMT. (c) The deduced absorption and radiative decay rates as a function of resonant wavelength. The analytical flat Ag absorption rate is shown as the dashed line.
taken at different incident angles are extracted and plotted in Fig. 2(b), displaying Lorentzian line shapes as described by CMT [18]. We then fit the spectra by using the analytical function Rorth and the best fits are shown as the solid lines in Fig. 2(b). The deduced rates are plotted in Fig. 2(c) as a function of resonant wavelength. We see although both Γabs and Γrad decrease with increasing wavelength, Γrad declines much faster. The analytical flat surface Γabs is plotted as the dash line for reference, showing the same trend with the experimental result. The discrepancy between experiment and theory is due to surface roughness, finite hole size, and geometry imperfections. Γrad shows λ−5.6 dependence, corresponding to Mie scattering by single holes [17]. The reflectivity peak height reaches 0.25 when two rates cross at λres 940 nm. We then examined the Ag and Au samples with different hole radii so that they could be compared with theory. From the angular mappings, we extracted the −1; 0 SPP Γabs and Γrad at λres 900 nm and plotted them in Figs. 3(a) and 3(b). Their features agree well overall with the CMT and FDTD results. First, it is evident that Γabs is relatively insensitive to hole radius in both cases. Γabs of flat Ag and Au are shown as the dash lines for reference. Second, Γrad shows R2;6 and R2.7 dependences for Ag and Au, which also are consistent with our earlier and current simulation results [17]. The orthogonal reflectivity profiles, which slowly increase with increasing radius, are shown in Figs. 3(c) and 3(d). Finally, when Γrad Γabs in two cases, the peak heights of the profiles are close to 0.25, in line with the theoretical predictions. We noted the radii when Γrad Γabs are 90 and 80 nm for Ag and Au arrays and are smaller than the simulations of 187 and 145 nm. The reason could be due to the larger experimental hole depth, which shifts the radiative decay curve up with respect to
504
OPTICS LETTERS / Vol. 39, No. 3 / February 1, 2014
Fig. 3. Absorption and radiative decay rates of (a) Ag and (b) Au nanohole arrays as a function of radius for λres 900 nm. The analytical flat metal absorption rates are shown as dashed lines. The corresponding orthogonal reflectivity spectra of (c) Ag and (d) Au arrays (symbols). The peak height reaches 0.25 when Γrad Γabs . The spectra are shifted horizontally for visualization. The solid lines are the best fits by using CMT.
that of the absorption as a result of larger Mie scattering, and imperfect geometries such as surface roughness, oxidation of Ag surface and irregular hole shape. In summary, we have studied the dependence of field enhancement on the absorption and radiative decay rates of SPPs in 1D and 2D periodic arrays by temporal CMT and FDTD simulation. When two rates are equal, the field enhancement is the strongest and the orthogonal reflectivity reaches 0.25. We have fabricated two series of Ag and Au with different geometries and have measured their decay rates and reflectivity. The experimental results are consistent with the theory. We believe geometry and wavelength directly control the decay rates, which subsequently affect the field strength. This research was supported by the Chinese University of Hong Kong through the RGC Competitive Earmarked Research Grants (402908, 402909, and 403310), UGC special equipment grant (SEG_CUHK07), AoE (AoE/P02/12), and CRF CUHK1/CRF/12G. References 1. S. A. Maier, Plasmonics: Fundamentals and Application, (Springer, 2007). 2. K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, Nat. Mater. 3, 601 (2004).
3. H. A. Atwater and A. Polman, Nat. Mater. 9, 205 (2010). 4. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, Nat. Mater. 7, 442 (2008). 5. M. L. Juan, M. Righini, and R. Quidant, Nat. Photonics 5, 349 (2011). 6. A. Linic, P. Christopher, and D. B. Ingram, Nat. Mater. 10, 911 (2011). 7. D. M. Schaadt, B. Feng, and E. T. Yu, Appl. Phys. Lett. 86, 063106 (2005). 8. J. Li, J. B. Xu, and H. C. Ong, Appl. Phys. Lett. 94, 241114 (2009). 9. L. Zhang, C. Y. Chan, J. Li, and H. C. Ong, Opt. Express 20, 12610 (2012). 10. S. Nie and S. R. Emory, Science 275, 1102 (1997). 11. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, Phys. Rep. 408, 131 (2005). 12. H. Paudel, K. Bayat, M. Baroughi, S. May, and D. Galipeau, Opt. Express 17, 22179 (2009). 13. T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. A. Wolff, Nature 391, 667 (1998). 14. P. A. Hobson, S. Wedge, J. A. E. Wasey, I. Sage, and W. L. Barnes, Adv. Mater. 14, 1393 (2002). 15. C. Y. Chan, J. B. Xu, M. Y. Waye, and H. C. Ong, Appl. Phys. Lett. 96, 033104 (2010). 16. H. Gao, W. Zhou, and T. W. Odom, Adv. Funct. Mater. 20, 529 (2010). 17. D. Y. Lei, J. Li, A. I. Fernandez-Dominguez, H. C. Ong, and S. A. Maier, ACS Nano 4, 432 (2010). 18. S. Wu, Z. Wang, and S. H. Fan, IEEE, J. Quantum Elect. 40, 1511 (2004). 19. H. A. Haus, Waves and Fields in Optoelectronics (PrenticeHall, 1984). 20. H. Y. Lo, C. Y. Chan, and H. C. Ong, Appl. Phys. Lett. 101, 223108 (2012). 21. T. Seok, A. Jamshidi, M. Kim, S. Dhuey, A. Lakhani, H. Choo, P. Schuck, S. Cabrini, A. Schwartzberg, J. Bokor, E. Yablonovitch, and M. Wu, Nano Lett. 11, 2606 (2011). 22. G. Sun and J. B. Khurgin, Appl. Phys. Lett. 97, 263110 (2010). 23. H. Raether, Surface Plasmons (Springer, 1988). 24. C. Genet, M. P. Exter, and J. P. Woerdman, Opt. Commun. 225, 331 (2003). 25. M. Sarrazin, J. P. Vigeron, and J. M. Vigoureux, Phys. Rev. B 67, 085415 (2003). 26. Z. L. Cao, H. Y. Lo, and H. C. Ong, Opt. Lett. 37, 5166 (2012). 27. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, Phys. Rev. Lett. 83, 2845 (1999). 28. K. C. Hui, J. T. K. Wan, J. B. Xu, and H. C. Ong, Appl. Phys. Lett. 95, 063110 (2009).
501
Interplay between absorption and radiative decay rates of surface plasmon polaritons for field enhancement in periodic arrays ZhaoLong Cao, Lei Zhang, Chung-Yu Chan, and Hock-Chun Ong* Department of Physics, The Chinese University of Hong Kong, Shatin, China *Corresponding author: [email protected] Received September 12, 2013; revised December 15, 2013; accepted December 16, 2013; posted December 16, 2013 (Doc. ID 197552); published January 23, 2014 We studied the effects of absorption and radiative decay rates of surface plasmon polaritons on the field enhancement in periodic metallic arrays by temporal coupled mode theory and finite-difference time-domain simulation. When two rates are equal, the field enhancement is the strongest and the peak height of the orthogonal reflectivity reaches 0.25. To demonstrate this fact, we fabricated two series of two-dimensional Au and Ag nanohole arrays with different geometries and measured their corresponding reflectivity and decay rates. The experimental results agree well with the analytical and numerical results. © 2014 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (260.5740) Resonance. http://dx.doi.org/10.1364/OL.39.000501
Maximizing the field enhancement in metallic structures has long been the prime question in plasmonics [1]. Applications such as fluorescence enhancement [2], energy harvesting [3], surface-enhanced Raman scattering (SERS) [4], surface plasmon resonance (SPR) sensing [4], radiation force manipulation [5], and photocatalysis [6] all depend on a strong local field. For example, a strong field increases both the generation and recombination rates of electron-hole pair in light absorbers and emitters, thus enhancing the absorption and spontaneous emission efficiencies [2,3,7,8]. Other than that, the local field senses molecules in sub-wavelength domain, enabling a route to detect molecules by SPR and SERS at extremely low concentration [9,10]. However, the general guideline to optimize the field strength is not yet available and research is still ongoing [1,11]. The main reason is that too many parameters such as geometry, material, and wavelength are needed to determine the resulting field strength and they are inextricable from each other [11]. More importantly, there seems to be a “missing link” between them. For example, it is still not quite understood what exact role geometry plays in controlling the field [12]. Periodic metallic arrays are important plasmonic systems [13]. Since their lattice and basis can be fabricated with great precision, they are expected to produce more controllable surface plasmon polaritons (SPPs), rendering more reliable performance in applications. As a result, periodic arrays have been used to make lightemitting diodes [14], solar cells [7], SERS and SPR sensors [9,15], and plasmonic tweezers [5]. It is therefore essential to develop strategy for field enhancement in periodic structures. While current approaches tend to establish a complete experimental or numerical data bank to identify the optimal condition, there is not much analytical effort [12,16]. Although analytical methods sometimes are not as rigorous as numerical simulations, they in fact provide good insight to understand the underlying physics, bridging the physical parameters and the field enhancement. 0146-9592/14/030501-04$15.00/0
In this Letter, use temporal coupled mode theory (CMT) to show that, under some circumstances, the field enhancement in periodic arrays can be optimized by matching the absorption Γabs and radiative decay Γrad rates of SPPs. In addition, when such conditions are met, the peak height of the orthogonal reflectivity reaches 0.25. To verify this concept, we performed finite-difference time-domain (FDTD) simulations on one- and two-dimensional (1D and 2D), arrays with different geometries. To demonstrate, we prepared two series of 2D Au and Ag nanohole arrays with different hole sizes by interference lithography and measured their Γabs and Γrad as well as their orthogonal reflectivity using angle-resolved reflectivity spectroscopy. Our experimental results agree well with the CMT prediction. Because it is known that decay rates are a strong function of geometry [17], the connection between field strength and geometry has been established. We began with the CMT formulation [18–20]. CMT for field enhancement has been used in nanoantennas [21] and nanoparticles [22] and is extended here for periodic arrays. For an optically thick array with m input/output ports, the time variant of a single propagating SPP mode amplitude a can be expressed as: da∕dt iωSPP − Γtot ∕2a hκj js i in Dirac’s bracket notation, where ωSPP is the resonant angular frequency (eV), Γtot is the total decay rate Γabs Γrad , and jκi κ1 κ 2 κ m T and js i s1 s2 s1m T are m × 1 column vectors signifying the complex incoupling constants and the amplitudes of the incident waves. P By applying time-reversal symmetry, we find n n hκjκi m n Γr Γrad [18,19], where Γr is the radiative decay rate of each individual port. If we consider −1 or −1; 0 SPP mode in 1D or 2D arrays [1,8,9,23], only one single kth input/output port exists and hκjκi Γkr Γrad and js i sk . Knowing the mode amplitude is harmonic with time, the above differential equation yields jaj2 Γrad jsk j2 ∕ω − ωSPP 2 Γtot ∕22 . In addition, jaj2 is related to the plasmonic field strength jE SPP j2 by jaj2 1∕2εo jE SPP j2 V eff , where εo is the permittivity © 2014 Optical Society of America
502
OPTICS LETTERS / Vol. 39, No. 3 / February 1, 2014
and V eff is the effective mode volume [21]. At ωSPP , we see if Γabs and V eff are slowly varying with respect to Γrad , jE SPP j2 is maximal when Γabs Γrad simply by taking ∂jE SPP j2 ∕∂Γrad 0. For periodic arrays, if the size of the basis is smaller than wavelength, Γabs can be approximated as the plain metal absorption given as Γabs ≈ ωε00m ∕ε0m 2 ε0m ∕ε0m 13∕2 , where εm0 i εm00 is the complex dielectric function of metal [17,23]. Thus, Γabs depends primarily on metal type and incident wavelength but is expected to vary slowly with geometry. In addition, assuming that V eff is proportional to the propagating length of propagating SPPs and the field decay constant in the direction normal to metal surface, we believe V eff also does not rely much on geometry when the basis is small compared to wavelength. As a result, for a given resonant wavelength, jE SPP j2 is the strongest if one varies the geometry to achieve Γabs ≈ Γrad . In addition, orthogonal, or cross-polarized, reflectivity spectroscopy can be used to probe the condition where Γabs Γrad is fulfilled. For a single port, only the specular reflection is present. If the incident polarizer is oriented at 45° with respect to the incidence plane, the Jones 1 vector of the incident field is given as E o , where 1 Eo is the field amplitude, for p- and s-polarizations. From CMT, under conservation of energy, the p-outgoing wave is given as s−k r p sk Γrad sk eiϕ ∕iω − ωSPP Γtot ∕2, where r p is the p-nonresonant reflection coefficient and ϕ is the in- and out-coupling phase-shift. One can see the wave is the interference between the reflection background and the SPP radiation damping, yielding the well-known Fano resonance [24,25]. As a result, assuming the s-incident field does not excite any resonance and no p-s polarization conversion occurs, the reflection vector after the array can be written as 2 R Eo 4 r p
3 Γrad e 5 iω − ωSPP Γtot ∕2 rs iϕ
(1)
[19,26], where r s is the s-reflection coefficient. If one places a detection analyzer orthogonally to the incident 1 −1 polarizer, the Jones operation thus is 1∕2 R −1 1 and the reflectivity Rorth measured is then given as Γ2rad ∕4ω − ωSPP 2 Γtot ∕22 , assuming r p ≈ r s over a range of wavelength. The reflectivity spectrum exhibits a Lorentzian line shape [19,26]. At resonance and Γabs Γrad , the peak height of Rorth is equal to 0.25, which thereby serves as a signature to identify the crossing of two rates. On the other hand, since jr p j is slightly less than one for most real metals, conventional p-polarized reflectivity is not necessarily equal to zero at Γabs Γrad and thus does not serve as a reliable indicator. In addition, orthogonal reflectivity enjoys higher signal-to-noise ratio than the p-polarized counterpart and is more detectable [9]. To verify the results from CMT, we have conducted FDTD. In fact, numerical simulation serves as a good tool to confirm our analytical prediction because it can construct ideal samples that are impossible or difficult
to fabricate experimentally. In particular, surface roughness and imperfect geometry are usually found in actual samples, which obscures the interpretation. Three series of 1D and 2D Ag and Au arrays with period P 670 nm, depth H 40, 60 and 150 nm, and groove width W or hole radius R varying from 30 to 230 nm at λSPP ∼ 790, 900, 900, and 910 nm, corresponding to −1 or (−1, 0) SPPs [23]. We choose shallow groove depth to avoid the emergence of waveguide mode in which its coupling with SPPs would complicate the situation [27]. We used Bloch boundary conditions at two/four sides and perfectly matched layers on the top and at the bottom of the cell. We set the wavelength of the pulsed excitation source centered either at 800 or 900 nm with bandwidth ∼100 nm to ensure that only one SPP mode is excited. For each array, we simulated the orthogonal reflectivity spectra at different incident angles for identifying the suitable angle for exciting the SPPs at λSPP [26]. Once the angle was identified, we first determined the near field pattern by placing the power monitor at 1 nm above the metal surface. We then computed Γrad and Γabs of SPPs by using the time-domain method [26]. The computation involves two steps. First, Γtot is determined by examining the transient of the field strength, which shows a single exponential decay curve typically for one resonance. We then calculated Γrad by constructing several new dielectric functions of Au and Ag by dividing their imaginary part two and five times so the absorption of metal can be proportionally reduced [26]. The decay rates simulated in these cases have identical Γrad but different Γabs . Finally, we extrapolated the y-intercept from the plot of decay rate against the reduced imaginary part since it corresponds to negligible metal absorption, thus yielding Γrad . Γabs is equal to Γtot − Γrad . The simulated Γabs and Γrad of three series are plotted in Fig. 1(a)–1(d) together with their corresponding average electric field strength and orthogonal reflectivity spectrum. The average field strength is calculated by summing up all fields from the meshes and then dividing by the mesh number. The figures show that Γabs , compared to Γrad , varies very slowly with respect to W , R, and H, except there is a slight increase at large R. The analytical Γabs for flat Ag and Au surfaces are shown as the dash lines and they compare reasonably with the simulations, justifying our assumption that Γabs behaves like a flat surface when λ > W or R. On the other hand, Γrad varies considerably with W and R, exhibiting W 1.01 and R3.8–4.8 dependences. In fact, our previous works on nanohole arrays have shown that W 1.01 depends very sensitively to hole geometry such as hole size and shape [17,28]. In particular, within the range of geometries considered here, the behavior of Γrad can be understood as the result of the scattering of SPPs by single isolated holes [17]. Our results show Γrad exhibits close to R4 dependence at a small hole size but becomes R5–8 when hole size increases [17], in agreement with the calculations here. The reflectivity peak height increases with an increasing radius due to the fact that the height is equal to Γrad ∕Γtot 2 and Γrad increases with W and R. For almost constant Γabs , the height slowly approaches one. Nevertheless, in all cases when two decay rates are matched, the
February 1, 2014 / Vol. 39, No. 3 / OPTICS LETTERS
Fig. 1. One-dimensional (a) Ag array with period 670 nm, height 40 nm, and different groove widths. Two-dimensional (b) Ag and (c) Au arrays with different radii. The Ag arrays have period 670 nm and height 60 nm whereas the Au arrays have period 670 nm and height 150 nm. Each figure has three parts. The top is the absorption and radiative decay rates of −1 or −1; 0 SPPs taken at resonant wavelength ∼790, 900, and 910 nm in Γ-X direction as a function of groove width or radius. The analytical flat metal absorption rates are shown as dashed lines. The middle and the bottom are their corresponding average electric field strength and orthogonal reflectivity spectra. A dashed line at reflectivity 0.25 is shown.
average field strength is the strongest and the orthogonal reflectivity peak height reaches 0.25, unambiguously confirming the prediction from CMT. To demonstrate, we fabricated two sets of Ag and Au 2D nanohole arrays with different hole radii by interference lithography and thin film deposition [17,26,28]. The period and height of the arrays are kept at 670 and 140 nm for Ag and 650 and 270 nm for Au. By changing the exposure time, we vary R from 70 to 165 nm. Noted that we intentionally choose the hole depths in these two sets to be larger than those in simulations. In addition, the period of the Au arrays is 650 nm instead of 670 nm. Therefore, the experimental results are expected to strengthen the analytical prediction but not to verify the simulations. It also is a proof-of-concept that far-field reflectivity can identify the strongest field enhancement if near-field scanning optical microscope is not available for direct field strength measurement. As an example, a scanning electron microscopy image of one of the Ag arrays is shown in the inset of Fig. 2(a), showing the array has R 102 nm. We used orthogonal reflectivity spectroscopy and CMT to determine their absorption and radiative decay rates. For optical measurements, a collimated white light from a quartz lamp is illuminated on the sample mounted on a computer-controlled goniometer and the angle-dependent specular reflection is captured by a spectrometer–CCD detection system. A pair of incident polarizer and detection analyzer is fixed at 45° and −45° to perform orthogonal polarization [26]. Figure 2(a) shows the reflectivity mapping of the Ag sample taken in Γ-X direction, illustrating the dispersive −1; 0 SPP reflection band as confirmed by the phase-matching equation [19,26]. Instead of dips, reflection peaks are observed from the mapping [19] and several spectra
503
Fig. 2. (a) Angle- and wavelength-resolved orthogonal reflectivity mapping taken in Γ-X direction. The SEM image of the array is in the inset, showing the holes are arranged in cubic structure with period and radius 670 and 102 nm. The dashed line is deduced from phase-matching equation showing −1; 0 SPP mode is excited. (b) The corresponding reflectivity spectra (symbols) are extracted at different incident angles. The solid lines are the best fits by using CMT. (c) The deduced absorption and radiative decay rates as a function of resonant wavelength. The analytical flat Ag absorption rate is shown as the dashed line.
taken at different incident angles are extracted and plotted in Fig. 2(b), displaying Lorentzian line shapes as described by CMT [18]. We then fit the spectra by using the analytical function Rorth and the best fits are shown as the solid lines in Fig. 2(b). The deduced rates are plotted in Fig. 2(c) as a function of resonant wavelength. We see although both Γabs and Γrad decrease with increasing wavelength, Γrad declines much faster. The analytical flat surface Γabs is plotted as the dash line for reference, showing the same trend with the experimental result. The discrepancy between experiment and theory is due to surface roughness, finite hole size, and geometry imperfections. Γrad shows λ−5.6 dependence, corresponding to Mie scattering by single holes [17]. The reflectivity peak height reaches 0.25 when two rates cross at λres 940 nm. We then examined the Ag and Au samples with different hole radii so that they could be compared with theory. From the angular mappings, we extracted the −1; 0 SPP Γabs and Γrad at λres 900 nm and plotted them in Figs. 3(a) and 3(b). Their features agree well overall with the CMT and FDTD results. First, it is evident that Γabs is relatively insensitive to hole radius in both cases. Γabs of flat Ag and Au are shown as the dash lines for reference. Second, Γrad shows R2;6 and R2.7 dependences for Ag and Au, which also are consistent with our earlier and current simulation results [17]. The orthogonal reflectivity profiles, which slowly increase with increasing radius, are shown in Figs. 3(c) and 3(d). Finally, when Γrad Γabs in two cases, the peak heights of the profiles are close to 0.25, in line with the theoretical predictions. We noted the radii when Γrad Γabs are 90 and 80 nm for Ag and Au arrays and are smaller than the simulations of 187 and 145 nm. The reason could be due to the larger experimental hole depth, which shifts the radiative decay curve up with respect to
504
OPTICS LETTERS / Vol. 39, No. 3 / February 1, 2014
Fig. 3. Absorption and radiative decay rates of (a) Ag and (b) Au nanohole arrays as a function of radius for λres 900 nm. The analytical flat metal absorption rates are shown as dashed lines. The corresponding orthogonal reflectivity spectra of (c) Ag and (d) Au arrays (symbols). The peak height reaches 0.25 when Γrad Γabs . The spectra are shifted horizontally for visualization. The solid lines are the best fits by using CMT.
that of the absorption as a result of larger Mie scattering, and imperfect geometries such as surface roughness, oxidation of Ag surface and irregular hole shape. In summary, we have studied the dependence of field enhancement on the absorption and radiative decay rates of SPPs in 1D and 2D periodic arrays by temporal CMT and FDTD simulation. When two rates are equal, the field enhancement is the strongest and the orthogonal reflectivity reaches 0.25. We have fabricated two series of Ag and Au with different geometries and have measured their decay rates and reflectivity. The experimental results are consistent with the theory. We believe geometry and wavelength directly control the decay rates, which subsequently affect the field strength. This research was supported by the Chinese University of Hong Kong through the RGC Competitive Earmarked Research Grants (402908, 402909, and 403310), UGC special equipment grant (SEG_CUHK07), AoE (AoE/P02/12), and CRF CUHK1/CRF/12G. References 1. S. A. Maier, Plasmonics: Fundamentals and Application, (Springer, 2007). 2. K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, Nat. Mater. 3, 601 (2004).
3. H. A. Atwater and A. Polman, Nat. Mater. 9, 205 (2010). 4. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, Nat. Mater. 7, 442 (2008). 5. M. L. Juan, M. Righini, and R. Quidant, Nat. Photonics 5, 349 (2011). 6. A. Linic, P. Christopher, and D. B. Ingram, Nat. Mater. 10, 911 (2011). 7. D. M. Schaadt, B. Feng, and E. T. Yu, Appl. Phys. Lett. 86, 063106 (2005). 8. J. Li, J. B. Xu, and H. C. Ong, Appl. Phys. Lett. 94, 241114 (2009). 9. L. Zhang, C. Y. Chan, J. Li, and H. C. Ong, Opt. Express 20, 12610 (2012). 10. S. Nie and S. R. Emory, Science 275, 1102 (1997). 11. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, Phys. Rep. 408, 131 (2005). 12. H. Paudel, K. Bayat, M. Baroughi, S. May, and D. Galipeau, Opt. Express 17, 22179 (2009). 13. T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. A. Wolff, Nature 391, 667 (1998). 14. P. A. Hobson, S. Wedge, J. A. E. Wasey, I. Sage, and W. L. Barnes, Adv. Mater. 14, 1393 (2002). 15. C. Y. Chan, J. B. Xu, M. Y. Waye, and H. C. Ong, Appl. Phys. Lett. 96, 033104 (2010). 16. H. Gao, W. Zhou, and T. W. Odom, Adv. Funct. Mater. 20, 529 (2010). 17. D. Y. Lei, J. Li, A. I. Fernandez-Dominguez, H. C. Ong, and S. A. Maier, ACS Nano 4, 432 (2010). 18. S. Wu, Z. Wang, and S. H. Fan, IEEE, J. Quantum Elect. 40, 1511 (2004). 19. H. A. Haus, Waves and Fields in Optoelectronics (PrenticeHall, 1984). 20. H. Y. Lo, C. Y. Chan, and H. C. Ong, Appl. Phys. Lett. 101, 223108 (2012). 21. T. Seok, A. Jamshidi, M. Kim, S. Dhuey, A. Lakhani, H. Choo, P. Schuck, S. Cabrini, A. Schwartzberg, J. Bokor, E. Yablonovitch, and M. Wu, Nano Lett. 11, 2606 (2011). 22. G. Sun and J. B. Khurgin, Appl. Phys. Lett. 97, 263110 (2010). 23. H. Raether, Surface Plasmons (Springer, 1988). 24. C. Genet, M. P. Exter, and J. P. Woerdman, Opt. Commun. 225, 331 (2003). 25. M. Sarrazin, J. P. Vigeron, and J. M. Vigoureux, Phys. Rev. B 67, 085415 (2003). 26. Z. L. Cao, H. Y. Lo, and H. C. Ong, Opt. Lett. 37, 5166 (2012). 27. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, Phys. Rev. Lett. 83, 2845 (1999). 28. K. C. Hui, J. T. K. Wan, J. B. Xu, and H. C. Ong, Appl. Phys. Lett. 95, 063110 (2009).
