## Abstract

A universal, geometry-independent sensitivity is derived by using a black box model of surface plasmon excitation for two-dimensional nanostructures. It is shown that the resonant wavelength of surface plasmons and dielectric property of interfacial materials dominate the sensitivity. Sensitivity data of nanostructure arrays, widely collected from independent research groups, comply well with our results. This analysis provides a conceptual and intuitive insight into the plasmonic sensing, covering various excitation arrangements under the same umbrella. The universal sensitivity offers a quantitative tool to evaluate and predict the performance of plasmonic sensors.

© 2015 Optical Society of America

## 1. Introduction

Surface plasmon resonance (SPR) is collective oscillation of electrons excited by light at the metal/dielectric interfaces [1]. Such interaction leads to significant field enhancement and SPR is extremely sensitive to dielectric properties at the interface. Moreover, the spatial and spectral properties of SPR on nanostructures can be easily tuned by controlling the geometry [2,3]. These unique aspects give rise to prosperous research and applications of plasmonic nanostructures from biosensing, molecular imaging to surface-enhanced spectroscopy [4,5]. The merit of a plasmonic sensor is determined by its sensitivity, which indicates the sensor signal variation responding to a refractive index change of the bulk environment and provides an upper bound to the biosensing. Among the most common performance indicators is the wavelength sensitivity, which has been measured by numerous experiments [6]. For example, nanohole arrays in metal films with various configurations in terms of film thickness, hole size, periodicity and pattern exhibited distinct optical response and sensitivities [7–10]. Sensitivity expressions for regular and chirped diffraction gratings were derived with wavelength shifts being a function of the local structures and diffraction orders [11,12]. However, most of current results were associated with the single arrangement of individual exciting mechanism, thereby hindering the direct comparisons across various configurations. We need a coherent framework to enable sensitivity evaluation of plasmonic nanostructures from a generality point of view.

Since plasmonic sensing is essentially the interaction between surface plasmon (SP) and matters, one question raised naturally is whether and what fundamental physical properties intrinsically and generally rule the sensitivity irrespective of individual structure geometry. Spurred by this question, in this work, a universal geometry-independent sensitivity is established for generic two-dimensional plasmonic nanostructures by using a black box model of SPR excitation. Previous theoretical efforts have been put into flat metal films [13] and nanoparticles [14,15]. Here we focus on the sensitivity analysis of plasmonic structure arrays, which denote certain nanoscale elements (i.e. holes or slits) repeated in metal films, on behalf of one main class of plasmonic objects. This universal sensitivity expression helps us clarify a series of phenomena involved in plasmonic sensing. Our expression coincides well with a considerable amount of experimental and numerical results obtained independently by other groups, confirming the validity of our analysis. This analytical outcome can be exploited for sensitivity assessment and prediction for plasmonic nanostructures with diverse geometries and arrangements.

## 2. A black box model of SP excitation

Different geometry parameters always couple together to affect the spectral features. Thus, it is difficult to establish an analytical sensitivity expression applicable to generic two-dimensional plasmonic structures. To address this dilemma, we would like to first dwell on the essential physics of SP. The dispersion curve of SPs lies on the right of light line [16], which means freely propagating light cannot directly excite the SPR due to such a momentum gap [Fig. 1(b)]. Momentum-matching techniques (e.g. grating and subwavelength holes) are required to compensate the missing momentum for the excitation of SP oscillation. Indeed, these plasmonic structures constitute very different coupling mechanisms and their geometries have substantial impacts on sensor performance [17,18]. However, in essence, plasmonic sensing is the interaction between SPs and the dielectric analyte. Thus it is rational to evaluate the sensor performance based on the property of SP itself rather than specific excitation mechanism. Therefore we propose a black box model of SP excitation [Fig. 1(a)], where the specific coupling channel is simplified into a function to provide constant momentum Δ𝑘 in the direction of SP propagation. Generally, exciting light with the frequency *ω* is input at the incident angle *θ* relative to the normal of SP plane (i.e. *k _{sp}* plane). The

*ψ*is the angle between the incident plane and SP propagating direction.

We suppose in the first approximation that appearance of plasmonic structures do not change the dispersion relation of SPs at the metal/dielectric interface. By applying the momentum-matching condition [Fig. 1(b)], we get

From this equation, we deduce the wavelength sensitivity S to refractive index,

According to this expression, an effective way to improve the sensitivity is decreasing Δ*k*. For instance, a nanohole array has been used in the configuration of attenuated total reflection [19]. In the oblique incidence, much more in-plane momentum of incident light is coupled to SPs in comparison to the case of normal incidence. As a result, its sensitivity significantly increases to the level of prism-based sensors.

Another possible optimization method is to modify the structures’ dispersion relation to further approach that of incident light. The guided SP modes of such structures have dispersion relations different from *k _{sp}*. However, Δ𝑘 could become quite small to still dominate the sensitivity. A plasmonic nanorod layer has been demonstrated to support a guided mode [20]. Its dispersion curve is actually designed to approach that of incident light at the resonance wavelength, so extremely small Δ𝑘 is required to excite SP and result in an extra-high sensitivity.

## 3. Analytical expression of wavelength sensitivity

To obtain an analytical sensitivity expression, Δ𝑘 and ${\epsilon}_{m}$need to be specified. Normal incidence is commonly adopted in most of experimental investigations because of its simplicity and practical consideration. In this case, the plasmonic structures provide all the necessary momentum for SP excitation, i.e.$\Delta k=\frac{\omega}{c}{\left(\frac{{\epsilon}_{m}{n}^{2}}{{\epsilon}_{m}+{n}^{2}}\right)}^{1/2}$. On the other hand, noble metals have free electron-like dielectric functions that vary quadratically with wavelength according to Drude model. At visible and near-infrared region, the real part of the dielectric function varies nearly linearly with wavelength, i.e.${\epsilon}_{m}\approx a\lambda +b$, where $a=-0.072,b=34$ [21]. Substituting for Δ𝑘 and${\epsilon}_{m}$in Eq. (2), we get an analytical sensitivity equation,

The sensitivities plotted for Au and Ag structures [Fig. 2] show a roughly linear increase as the SPR shifts to longer wavelength. Despite different dielectric properties [22], plasmonic structures with Au and Ag have almost equal sensitivities in the same dielectric (also see experimental data in Fig. 3). Given $\left|{\epsilon}_{m}\right|\gg {n}^{2}$ at visible and near-infrared region, we can safely give an approximation $S\approx {\lambda}_{m}/n$, which confirms the SPR at the same wavelength show higher sensitivity in the analyte with lower refractive index. In particular, $S\approx {\lambda}_{m}$ in air ($n=1$) and $S\approx 0.75{\lambda}_{m}$ in water ($n=1.33$), which implies that measurements in air are more sensitive compared with that in aqueous solution. In addition, this analytical format of sensitivity can give us more insight into physics behind plasmonic sensing. For example, the SP penetration modulates the interaction: the longer penetration depth at the longer wavelength [16] provides a larger sensing volume and thus a higher interaction probability.

## 4. Sensitivity quantification, comparison and prediction

The analytical sensitivity can be evaluated using those plasmonic sensors which are subject to the same principle of surface plasmon resonance. Typical embodiments of our model are plasmonic array structures including the tow-dimensional Bravais lattices of subwavelength apertures and arrays of nanoslits. They have different SPR wavelength expressions, e.g.$\frac{P}{\sqrt{{i}^{2}+{j}^{2}}}{\left(\frac{{\epsilon}_{m}{n}^{2}}{{\epsilon}_{m}+{n}^{2}}\right)}^{1/2}$for square nanohole arrays [8],$\frac{P}{\sqrt{\frac{4}{3}({i}^{2}+ij+{j}^{2})}}{\left(\frac{{\epsilon}_{m}{n}^{2}}{{\epsilon}_{m}+{n}^{2}}\right)}^{1/2}$for hexagonal nanohole arrays [8] and $\frac{P}{i}{\left(\frac{{\epsilon}_{m}{n}^{2}}{{\epsilon}_{m}+{n}^{2}}\right)}^{1/2}$ for nanoslit arrays [23], where *P* is lattice constant, *i, j* are the scattering orders in SP planes. From these wavelength expressions, we can mathematically derive their sensitivities, which are equivalent to Eq. (3).

We could validate our analytical results by using specific structures. However, this merely adds new instances of this universal model. Instead, we adopt a set of sensitivity data independently measured using metal nanostructure arrays with various geometries. A series of experimental and simulated data published by other groups (see appendix), are collected to quantify our theoretical values [Fig. 3]. It is observed that some experimental sensitivities are somewhat lower than theoretical values. This degradation can be partially attributed to coupling effects from substrates [24] and radiation damping induced by the appearance of nanostructures in real cases. Overall, our model predicts the correct range and trend of sensitivity change for plasmonic array structures.

Localized SPR (LSPR) in nanoparticles apparently has the same physical origin as those in two-dimensional plasmonic structures. The LSPR sensitivity for nanoparticles has been derived from a dipole polarizability resonance condition in the quasistatic limit [14]. This sensitivity also depends on the resonance wavelength and dielectric properties of the metal and medium. The theoretical sensitivities of both types are plotted in Fig. 4. Obviously, plasmonic array structures have much higher sensitivity in the visible range, whereas the nanoparticles’ sensitivity is approaching parallel to the former at the near-infrared regime. Their difference can be attributed to stronger confinement of SP field in nanoparticles due to its localized nature, thereby providing less sensing volume and smaller sensitivity [26].

Our analysis reveals it is SPR that essentially determines the sensitivity of two-dimensional plasmonic sensors, whereas metal nanostructures mainly act as a coupling media to generate SPR. Beyond the Bravais lattices, quasicrystals (with long-range order but no short-range order) and aperiodic aperture arrays (without long-range or short-range order) were also demonstrated to support SPR [27–29]. Instead of reciprocal lattice vectors, quasicrystals and aperiodic structures are characterized by discrete Fourier transform vectors in their structure factors. Sharp transmission resonances appear at frequencies that closely match these discrete Fourier transform vectors [28]. These vectors in reciprocal space are in fact equal to different wave-vectors, corresponding to various SPR peak wavelengths in normal incident. In this context, our sensitivity expression is applicable to those two-dimensional aperture arrays that have discrete Fourier transform vectors in their geometrical structure factors. It is predicted that these non-periodic nanohole arrays would equally possess good optical performance thereby being used for sensing application.

## 5. Conclusions

In summary, a universal plasmonic sensitivity is established for generic two-dimensional nanostructures by using a black box model of SPR excitation. This expression defines plasmonic sensitivity based on the primary physical elements, rather than variable nanostructure geometries. The analytical model successfully explains a series of phenomena involved in plasmonic sensing. The previously published sensitivity data comply with and validate our theoretical results. This analysis provides a powerful and general tool to quantitatively evaluate and predict the performance of plasmonic nanostructure sensors.

## 6 Appendix: Sensitivity summary of two-dimensional plasmonic metal nanostructures

Publication | SPR wavelength (nm) | Metal/Dielectric | Sensitivity (nm/RIU) |
---|---|---|---|

Ref. 1 | 720 | Ag/water | 494 |

Ref. 1 | 720 | Ag/water | 524 |

Ref. 2 | 1023 | Au/water | ~700 |

Ref. 3 | 805 | Au/water | 668 |

Ref. 4 | 880 | Au/water | 615 |

Ref. 5 | 845 | Au/water | 690 |

Ref. 6 | 889 | Au/water | 630 |

Ref. 7 | 1200 | Au/alcohol | 900 |

Ref. 8 | 670 | Au/water | ~500 |

Ref. 9 | 710 | Au/water | 530 |

Ref. 10 | 650 | Ag/water | 410 |

Ref. 11 | 850 | Au/water | 600 |

Ref. 12 | 700 | Ag/water | 450 |

Ref. 13 | 710 | Au/water | 481 |

Ref. 14 | 710 | Ag/water | 470 |

Ref. 15 | 1532 | Au/ water | 1520 |

Ref. 16 | 740 | Au/ water | 495 |

Ref. 17 | 975 | Au/ water | 754 |

Ref. 18 | 1510 | Au/ water | 1022 |

Ref. 19 | 830 | Au/water | 650 |

Ref. 20 | 790 | Au/water | 575 |

Ref. 21 | 666 | Au/water | 478 |

Ref. 22 | 693 | Au/water | 451 |

Ref. 23 | 680 | Au/water | 470 |

Ref. 24 | 625 | Au/water | 409 |

Ref. 25 | 1300 | Ag/water | 1015 |

Ref. 26 | 720 | Ag/water | 513 |

Ref. 27 | 982 | Au/water | 753 |

Ref. 28 | 1035 | Au/water | 858 |

Ref. 29 | 1020 | Au/water | 788 |

Ref. 29 | 1020 | Au/water | 752 |

## Acknowledgments

This work was supported by the Natural Science and Engineering Research Council of Canada (NSERC, Grant No. 327642-2011) and Canada Foundation for Innovation (CFI, Grant No. 12928). We also thank the anonymous reviewers for their thoughtful suggestions that help us improve this work.

## References and links

**1. **S. A. Maier, *Plasmonics: Fundamentals and Applications* (Springer, 2007).

**2. **W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**(6950), 824–830 (2003). [CrossRef] [PubMed]

**3. **M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. **108**(2), 494–521 (2008). [CrossRef] [PubMed]

**4. **A. G. Brolo, “Plasmonics for future biosensors,” Nat. Photonics **6**(11), 709–713 (2012). [CrossRef]

**5. **J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. **7**(6), 442–453 (2008). [CrossRef] [PubMed]

**6. **A. Dmitriev, *Nanoplasmonic Sensors* (Springer, 2012)

**7. **T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, and T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B **16**(10), 1743–1748 (1999). [CrossRef]

**8. **S. H. Lee, T. W. Johnson, N. C. Lindquist, H. Im, D. J. Norris, and S.-H. Oh, “Linewidth-optimized extraordinary optical transmission in water with template-stripped metallic nanohole arrays,” Adv. Funct. Mater. **22**(21), 4439–4446 (2012). [CrossRef]

**9. **L. Zhang, C. Y. Chan, J. Li, and H. C. Ong, “Rational design of high performance surface plasmon resonance sensors based on two-dimensional metallic hole arrays,” Opt. Express **20**(11), 12610–12621 (2012). [CrossRef] [PubMed]

**10. **K. Nakamoto, R. Kurita, O. Niwa, T. Fujii, and M. Nishida, “Development of a mass-producible on-chip plasmonic nanohole array biosensor,” Nanoscale **3**(12), 5067–5075 (2011). [CrossRef] [PubMed]

**11. **J. Homola, I. Koudela, and S. S. Yee, “Surface plasmon resonance sensors based on diffraction gratings and prism couplers: sensitivity comparison,” Sensor. Actuat. Biol. Chem. **54**(1–2), 16–24 (1999).

**12. **W. H. Yeh, J. Kleingartner, and A. C. Hillier, “Wavelength tunable surface plasmon resonance-enhanced optical transmission through a chirped diffraction grating,” Anal. Chem. **82**(12), 4988–4993 (2010). [CrossRef] [PubMed]

**13. **J. Homola, *Surface Plasmon Resonance Based Sensors* (Springer, 2006).

**14. **M. M. Miller and A. A. Lazarides, “Sensitivity of metal nanoparticle surface plasmon resonance to the dielectric environment,” J. Phys. Chem. B **109**(46), 21556–21565 (2005). [CrossRef] [PubMed]

**15. **J. Yang, H. Giessen, and P. Lalanne, “Simple analytical expression for the peak-frequency shifts of plasmonic resonances for sensing,” Nano Lett. **15**(5), 3439–3444 (2015). [CrossRef] [PubMed]

**16. **H. Raether, *Surface Plasmons* (Springer, 1988).

**17. **K.-L. Lee, W.-S. Wang, and P.-K. Wei, “Comparisons of surface plasmon sensitivities in periodic gold nanostructures,” Plasmonics **3**(4), 119–125 (2008). [CrossRef]

**18. **T. Sannomiya, O. Scholder, K. Jefimovs, C. Hafner, and A. B. Dahlin, “Investigation of plasmon resonances in metal films with nanohole arrays for biosensing applications,” Small **7**(12), 1653–1663 (2011). [CrossRef] [PubMed]

**19. **M. Couture, L. S. Live, A. Dhawan, and J.-F. Masson, “EOT or Kretschmann configuration? Comparative study of the plasmonic modes in gold nanohole arrays,” Analyst (Lond.) **137**(18), 4162–4170 (2012). [CrossRef] [PubMed]

**20. **A. V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G. A. Wurtz, R. Atkinson, R. Pollard, V. A. Podolskiy, and A. V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nat. Mater. **8**(11), 867–871 (2009). [CrossRef] [PubMed]

**21. **M. M. Miller and A. A. Lazarides, “Sensitivity of metal nanoparticle plasmon resonance band position to the dielectric environment as observed in scattering,” J. Opt. A, Pure Appl. Opt. **8**(4), S239–S249 (2006). [CrossRef]

**22. **P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

**23. **K.-L. Lee and P.-K. Wei, “Optimization of periodic gold nanostructures for intensity-sensitive detection,” Appl. Phys. Lett. **99**(8), 083108 (2011). [CrossRef]

**24. **B. Brian, B. Sepúlveda, Y. Alaverdyan, L. M. Lechuga, and M. Käll, “Sensitivity enhancement of nanoplasmonic sensors in low refractive index substrates,” Opt. Express **17**(3), 2015–2023 (2009). [CrossRef] [PubMed]

**25. **J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**(5685), 847–848 (2004). [CrossRef] [PubMed]

**26. **M. A. Otte, B. Sepúlveda, W. Ni, J. P. Juste, L. M. Liz-Marzán, and L. M. Lechuga, “Identification of the optimal spectral region for plasmonic and nanoplasmonic sensing,” ACS Nano **4**(1), 349–357 (2010). [CrossRef] [PubMed]

**27. **F. Przybilla, C. Genet, and T. W. Ebbesen, “Enhanced transmission through Penrose subwavelength hole arrays,” Appl. Phys. Lett. **89**(12), 121115 (2006). [CrossRef]

**28. **T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature **446**(7135), 517–521 (2007). [CrossRef] [PubMed]

**29. **J. Bravo-Abad, A. I. Fernández-Domínguez, F. J. García-Vidal, and L. Martín-Moreno, “Theory of extraordinary transmission of light through quasiperiodic arrays of subwavelength holes,” Phys. Rev. Lett. **99**(20), 203905 (2007). [CrossRef] [PubMed]