## Localized surface plasmon resonance in arrays of nano-gold cylinders: inverse problem and propagation of uncertainties |

Optics Express, Vol. 21, Issue 2, pp. 2245-2262 (2013)

http://dx.doi.org/10.1364/OE.21.002245

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### Abstract

The plasmonic nanostructures are widely used to design sensors with improved capabilities. The position of the localized surface plasmon resonance (LSPR) is part of their characteristics and deserves to be specifically studied, according to its importance in sensor tuning, especially for spectroscopic applications. In the visible and near infra-red domain, the LSPR of an array of nano-gold-cylinders is considered as a function of the diameter, height of cylinders and the thickness of chromium adhesion layer and roughness. A numerical experience plan is used to calculate heuristic laws governing the inverse problem and the propagation of uncertainties. Simple linear formulae are deduced from fitting of discrete dipole approximation (DDA) calculations of spectra and a good agreement with various experimental results is found. The size of cylinders can be deduced from a target position of the LSPR and conversely, the approximate position of the LSPR can be simply deduced from the height and diameter of cylinders. The sensitivity coefficients and the propagation of uncertainties on these parameters are evaluated from the fitting of 15500 computations of the DDA model. The case of a grating of nanodisks and of homothetic cylinders is presented and expected trends in the improvement of the fabrication process are proposed.

© 2013 OSA

## 1. Introduction

2. M. Vidotti, R. F. Carvalhal, R. K. Mendes, D. C. M. Ferreira, and L. T. Kubota, “Biosensors based on gold nanostructures,” J. Braz. Chem. Soc. **22**, 3–20 (2011). [CrossRef]

*λ*

_{0}(

*LSPR*) (Sec. 2). Then, obtained from systematic numerical experiments, the approach will be to deduce the analytical function giving the LSPR position as a function of the key parameters. This resolution can be done analytically, if the relationship between parameters and

*λ*

_{0}(

*LSPR*) is an invertible function of the key parameters (Sec. 3). The result helps to obtain an heuristic formula for

*λ*

_{0}(

*LSPR*) within the investigated domain of parameters (Sec. 4). The propagation of uncertainties and therefore the tolerance of fabrication of such nano-device are deduced. A final validation of the solution of the inverse problem with some experimental data is finally proposed (Sec. 5).

## 2. The array of nanocylinders, experimental uncertainties

17. L. Billot, M. Lamy de la Chapelle, A. S. Grimault, A. Vial, D. Barchiesi, J.-L. Bijeon, P.-M. Adam, and P. Royer, “Surface enhanced Raman scattering on gold nanowire arrays: evidence of strong multipolar surface plasmon resonance enhancement,” Chem. Phys. Lett. **422**, 303–307 (2006). [CrossRef]

4. J. Grand, M. Lamy de la Chapelle, J.-L. Bijeon, P.-M. Adam, A. Vial, and P. Royer, “Role of localized surface plasmons in surface-enhanced Raman scattering of shape-controlled metallic particles in regular arrays,” Phys. Rev. B **72**, 033407 (2005). [CrossRef]

5. N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, M. Salerno, G. Schider, B. Lamprecht, A. Leitner, and F. R. Aussenegg, “Controlling the optical response of regular arrays of gold particles for surface-enhanced Raman scattering,” Phys. Rev. B **65**, 075419–075427 (2002). [CrossRef]

8. N. Guillot, H. Shen, B. Frémaux, O. Péron, E. Rinnert, T. Toury, and M. Lamy de la Chapelle, “Surface enhanced Raman scattering optimization of gold nanocylinder arrays: influence of the localized surface plasmon resonance and excitation wavelength,” Appl. Phys. Lett. **97**, 023113–023116 (2010). [CrossRef]

14. J. Grand, Plasmons de surface de nanoparticules : spectroscopie d’extinction en champs proche et lointain, diffusion Raman exaltée, Ph.D. thesis (Université de technologie de Troyes, 2004). [PubMed]

18. M. Pelton, J. Aizpurua, and G. W. Bryant, “Metal-nanoparticles plasmonics,” Laser & Photon. Rev. **2**, 136–159 (2008). [CrossRef]

- Height (
*h*): the maximum uncertainty is*δ*= ±2 nm. This value is due to both the roughness and the process of metal deposition. The SEM images and AFM (Atomic Force Microscopy) scans reveal a RMS (root of mean square) lower than one nanometer [14_{h}].14. J. Grand, Plasmons de surface de nanoparticules : spectroscopie d’extinction en champs proche et lointain, diffusion Raman exaltée, Ph.D. thesis (Université de technologie de Troyes, 2004). [PubMed]

- Thickness (
*e*): the maximum uncertainty is also*δ*± 2 nm, but may depend on the thickness of the intermediate layer._{e} - Diameter (
*D*): the maximum uncertainty is*δ*= ±20 nm. This value is relative to both the fabrication and the resolution of the SEM [19] and to a drift of diameter and shape on the whole grating. This last source of uncertainty is evaluated through statistics on the SEM images and is compatible with that found in literature [20_{D}].20. G. Laurent, N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Evidence of multipolar excitations in surface enhanced Raman scattering,” Phys. Rev. B

**65**, 045430 (2005). [CrossRef]

*λ*

_{0}is supposed to reveal the LSPR position [21

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22. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc., New York, 1998). [CrossRef]

## 3. Model of the interaction of light with nanocylinders array

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5. N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, M. Salerno, G. Schider, B. Lamprecht, A. Leitner, and F. R. Aussenegg, “Controlling the optical response of regular arrays of gold particles for surface-enhanced Raman scattering,” Phys. Rev. B **65**, 075419–075427 (2002). [CrossRef]

6. N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticles arrays,” Appl. Phys. Lett. **82**, 3095–3097 (2003). [CrossRef]

8. N. Guillot, H. Shen, B. Frémaux, O. Péron, E. Rinnert, T. Toury, and M. Lamy de la Chapelle, “Surface enhanced Raman scattering optimization of gold nanocylinder arrays: influence of the localized surface plasmon resonance and excitation wavelength,” Appl. Phys. Lett. **97**, 023113–023116 (2010). [CrossRef]

25. Y. B. Zheng, B. K. Juluri, X. Mao, T. R. Walker, and T. J. Huang, “Systematic investigation of localized surface plasmon resonance of long-range ordered Au nanodisk arrays,” J. Appl. Phys **103**, 014308 (2008). [CrossRef]

26. H. Shen, N. Guillot, J. Rouxel, M. Lamy de la Chapelle, and T. Toury, “Optimized plasmonic nanostructures for improved sensing activities,” Opt. Express **20**, 21278–21290 (2012). [CrossRef] [PubMed]

25. Y. B. Zheng, B. K. Juluri, X. Mao, T. R. Walker, and T. J. Huang, “Systematic investigation of localized surface plasmon resonance of long-range ordered Au nanodisk arrays,” J. Appl. Phys **103**, 014308 (2008). [CrossRef]

27. A. Vial and T. Laroche, “Description of dispersion properties of metals by means of the critical points model and application to the study of resonant structures using the FDTD method,” J. Phys. D. **40**, 7152–7158 (2007). [CrossRef]

15. D. Barchiesi, D. Macías, L. Belmar-Letellier, D. Van Labeke, M. Lamy de la Chapelle, T. Toury, E. Kremer, L. Moreau, and T. Grosges, “Plasmonics: influence of the intermediate (or stick) layer on the efficiency of sensors,” Appl. Phys. B-Lasers Opt. **93**, 177–181 (2008). [CrossRef]

28. D. Barchiesi, N. Lidgi-Guigui, and M. Lamy de la Chapelle, “Functionalization layer influence on the sensitivity of surface plasmon resonance (SPR) biosensor,” Opt. Commun. **285**, 1619–1623 (2012). [CrossRef]

26. H. Shen, N. Guillot, J. Rouxel, M. Lamy de la Chapelle, and T. Toury, “Optimized plasmonic nanostructures for improved sensing activities,” Opt. Express **20**, 21278–21290 (2012). [CrossRef] [PubMed]

30. H. Aouani, J. Wenger, D. Gérard, H. Rigneault, E. Devaux, T. W. Ebbesen, F. Mahdavi, T. Xu, and S. Blair, “Crucial role of the adhesion layer on the plasmonic fluorescence enhancement,” ACS Nano **3**, 2043–2048 (2009). [CrossRef] [PubMed]

27. A. Vial and T. Laroche, “Description of dispersion properties of metals by means of the critical points model and application to the study of resonant structures using the FDTD method,” J. Phys. D. **40**, 7152–7158 (2007). [CrossRef]

30. H. Aouani, J. Wenger, D. Gérard, H. Rigneault, E. Devaux, T. W. Ebbesen, F. Mahdavi, T. Xu, and S. Blair, “Crucial role of the adhesion layer on the plasmonic fluorescence enhancement,” ACS Nano **3**, 2043–2048 (2009). [CrossRef] [PubMed]

43. H. Devoe, “Optical properties of molecular aggregates. I. Classical model of electronic absorption and refraction,” J. Chem. Phys. **41**, 393–400 (1964). [CrossRef]

44. H. Devoe, “Optical properties of molecular aggregates. II. Classical theory of the refraction, absorption, and optical activity of solutions and crystals,” J. Chem. Phys. **43**, 3199–3208 (1965). [CrossRef]

45. E. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. **186**, 705–714 (1973). [CrossRef]

*N*elements or dipoles with polarizabilities

*α*, located at

_{j}**r**

*. Each dipole has a polarization*

_{j}**P**

*=*

_{j}*α*

_{j}**E**

*, where*

_{j}**E**

*is the electric field at*

_{j}**r**

*induced by the incident wave and the sum of the dielectric fields induced by interaction with other dipoles. Consequently, a system of complex linear equations must be solved to find polarizations*

_{j}**P**

*and evaluate the extinction cross section following [36*

_{j}36. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**, 1491–1499 (1994). [CrossRef]

*Q*, the extinction efficiency [22

_{ext}22. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, Inc., New York, 1998). [CrossRef]

*k*

_{0}= 2

*π*/

*λ*

_{0}the modulus of the wave vector and

**E**the amplitude, of the illumination monochromatic plane wave. ℑ(.) is the imaginary part of a complex number.

_{0}*C*is written under the assumption of linearly polarized incident light [36

_{ext}36. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**, 1491–1499 (1994). [CrossRef]

46. V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. **39**, 853–861 (1992). [CrossRef]

41. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A **25**, 2693–2703 (2008). [CrossRef]

47. P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B **67**, 165404(1–5) (2003). [CrossRef]

48. E. Zubko, D. Petrov, Y. Grynko, Y. Shkuratov, H. Okamotot, K. Muinonen, T. Nousiainen, H. Kimura, T. Yamamoto, and G. Videen, “Validity criteria of the discrete dipole approximation,” Appl. Opt. **49**, 1267–1279 (2010). [CrossRef] [PubMed]

49. C. Ungureanu, R. G. Rayavarapu, S. Manohar, and T. G. Van Leeuwen, “Discrete dipole approximation simulations of gold nanorod optical properties: choice of input parameters and comparison with experiment,” J. Appl. Phys. **105**, 102032–102039 (2009). [CrossRef]

50. W.-H. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys. **193**, 869–875 (1995). [CrossRef]

51. H. Parviainen and K. Lumme, “Scattering from rough thin films: discrete-dipole-approximation simulations,” J. Opt. Soc. Am. A **25**, 90–97 (2008). [CrossRef]

49. C. Ungureanu, R. G. Rayavarapu, S. Manohar, and T. G. Van Leeuwen, “Discrete dipole approximation simulations of gold nanorod optical properties: choice of input parameters and comparison with experiment,” J. Appl. Phys. **105**, 102032–102039 (2009). [CrossRef]

52. B. T. Draine and P. J. Flatau, “User guide to the discrete dipole approximation code DDSCAT 7.1,” http://arXiv.org/abs/1002.1505v1 (2010).

*d*= 2 nm is smaller than 2.6 nm that ensured the validity of the calculations in [5

5. N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, M. Salerno, G. Schider, B. Lamprecht, A. Leitner, and F. R. Aussenegg, “Controlling the optical response of regular arrays of gold particles for surface-enhanced Raman scattering,” Phys. Rev. B **65**, 075419–075427 (2002). [CrossRef]

^{−3}. The magnitude of the electric field at distance

*r*from any dipole decreases as a polynomial function of 1/

*r*. The factor exp [−(

*γ*

*k*

_{0}

*r*)

^{4}] was introduced to vanish the coupling between remote dipoles in a periodic lattice (with

*k*

_{0}= 2

*π*/

*λ*

_{0}the modulus of the illumination wave vector) and therefore to increase the speed and accuracy of computation [41

41. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A **25**, 2693–2703 (2008). [CrossRef]

*γ*can be therefore seen as a cutoff parameter which smoothly suppresses the influence of far dipoles in periodic structures. In the present case,

*γ*= 0.1 is chosen to achieve both sufficient accuracy in the investigated size range of parameters, and a reasonable computational time (less than one hour). Indeed, the computation of spectrums in the range 550 nm to 850 nm of wavelength requires 31 evaluations of the model, if a precision of 10 nm is supposed to be sufficient for the computation of the LSPR spectral position.

*ε*[18

_{eff}18. M. Pelton, J. Aizpurua, and G. W. Bryant, “Metal-nanoparticles plasmonics,” Laser & Photon. Rev. **2**, 136–159 (2008). [CrossRef]

8. N. Guillot, H. Shen, B. Frémaux, O. Péron, E. Rinnert, T. Toury, and M. Lamy de la Chapelle, “Surface enhanced Raman scattering optimization of gold nanocylinder arrays: influence of the localized surface plasmon resonance and excitation wavelength,” Appl. Phys. Lett. **97**, 023113–023116 (2010). [CrossRef]

^{2}= 2.04) is considered as constant on the whole investigated domain of wavelengths (

*λ*

_{0}∈ [550, 850] nm).

54. A. J. Abu El-Haija, “Effective medium approximation for the effective optical constants of a bilayer and a multilayer structure based on the characteristic matrix technique,” J. Appl. Phys. **93**, 2590–2594 (2003). [CrossRef]

55. D. Barchiesi, “Numerical retrieval of thin aluminium layer properties from SPR experimental data,” Opt. Express **20**, 9064–9078 (2012). [CrossRef] [PubMed]

**65**, 075419–075427 (2002). [CrossRef]

27. A. Vial and T. Laroche, “Description of dispersion properties of metals by means of the critical points model and application to the study of resonant structures using the FDTD method,” J. Phys. D. **40**, 7152–7158 (2007). [CrossRef]

*D*= 100 nm and

*h*= 50 nm. The period of the grating was

*D*+

*P*= 300 nm with

*P*= 200 nm (Fig. 1). The relative permittivities of metals were found in the same references [56

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

*λ*

_{0}= 586 nm with FDTD and

*λ*

_{0}= 590 nm with DDA. Therefore in the investigated domain of parameters, the effective medium seems to be a usable approximation.

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

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

15. D. Barchiesi, D. Macías, L. Belmar-Letellier, D. Van Labeke, M. Lamy de la Chapelle, T. Toury, E. Kremer, L. Moreau, and T. Grosges, “Plasmonics: influence of the intermediate (or stick) layer on the efficiency of sensors,” Appl. Phys. B-Lasers Opt. **93**, 177–181 (2008). [CrossRef]

58. S. Ekgasit, C. Thammacharoen, F. Yu, and W. Knoll, “Influence of the metal film thickness on the sensitivity of surface plasmon resonance biosensors,” Appl. Spectrosc. **59**, 661–667 (2005). [CrossRef] [PubMed]

25. Y. B. Zheng, B. K. Juluri, X. Mao, T. R. Walker, and T. J. Huang, “Systematic investigation of localized surface plasmon resonance of long-range ordered Au nanodisk arrays,” J. Appl. Phys **103**, 014308 (2008). [CrossRef]

**40**, 7152–7158 (2007). [CrossRef]

59. H. Neff, W. Zong, A. Lima, M. Borre, and G. Holzhüter, “Optical properties and instrumental performance of thin gold films near the surface plasmon resonance,” Thin Solid Films **496**, 688–697 (2006). [CrossRef]

*λ*

_{0}(

*LSPR*) =

*λ*

_{0}(max(

*C*)) could depend on them [39

_{ext}39. K. S. Lee and M. A. El-Sayed, “Dependence of the enhanced optical scattering efficiency relative to that of absorption of gold metal nanorods on aspect ratio, size, end-cap shape, and medium refractive,” J. Phys. Chem. B **109**, 20331–20338 (2005). [CrossRef]

49. C. Ungureanu, R. G. Rayavarapu, S. Manohar, and T. G. Van Leeuwen, “Discrete dipole approximation simulations of gold nanorod optical properties: choice of input parameters and comparison with experiment,” J. Appl. Phys. **105**, 102032–102039 (2009). [CrossRef]

*D*,

*h*,

*e*

*λ*

_{0}) must be determined, by a first evaluation of the propagation of experimental uncertainties through the numerical model. The target is the position of the LSPR.

## 4. Discretization and key parameters

*D*,

*h*,

*e*and

*λ*

_{0}in order to maintain the computational time of the parametric study in a reasonable range and to obtain significant results. For this, we compute the propagation of experimental uncertainties through the model to check their influence on the shift of the LSPR. Two diameters are considered for this evaluation with the above described model:

*D*= 100 nm and

*D*= 200 nm, near the boundaries of the investigated domain of the parametric study. The height of cylinders is the reference in experiments

*h*= 50 nm [8

**97**, 023113–023116 (2010). [CrossRef]

*C*(

_{ext}*λ*

_{0}). The cylinders are supposed to have smooth surfaces. The corresponding uncertainty on the position of LSPR (

*λ*

_{0}(

*LSPR*)) is therefore ±5 nm in the numerical calculations. This approach corresponds to the evaluation of the partial derivative of the function

*λ*

_{0}(

*LSPR*)(

*D*,

*h*,

*e*,

*RMS*) around typical values in the hyperspace of parameters.

*h*and

*D*are key parameter for

*λ*

_{0}(

*LSPR*) as well as the roughness. The influence of the nanometric layer of chromium is less critical for the position of LSPR [27

**40**, 7152–7158 (2007). [CrossRef]

15. D. Barchiesi, D. Macías, L. Belmar-Letellier, D. Van Labeke, M. Lamy de la Chapelle, T. Toury, E. Kremer, L. Moreau, and T. Grosges, “Plasmonics: influence of the intermediate (or stick) layer on the efficiency of sensors,” Appl. Phys. B-Lasers Opt. **93**, 177–181 (2008). [CrossRef]

*u*(

_{B}*λ*

_{0}(

*LSPR*)) on the LSPR shift can therefore be evaluated from the above uncertainties of type B [61], by considering

*a priori*uniform law of probability within the above intervals and no correlation between these parameters. The following results are obtained:

*λ*

_{0}. The corresponding uncertainty on the position of LSPR (

*λ*

_{0}(

*LSPR*)) is therefore ±5 nm in the numerical calculations. No further fitting of the extinction cross section

*C*(

_{ext}*λ*

_{0}) curve is used. The evaluation of the experimental uncertainty (Eqs. (3) – (4)) is used in the following for the plot of experimental data. The computation of the position of the LSPR requires 31 computations of the extinction cross-section

*C*for

_{ext}*λ*

_{0}∈ [550;850] nm. The rough sample is considered in the following.

*D*,

*h*,

*e*,

*λ*

_{0}. Table 2 gives the intervals and the discretization step for all parameters. The total number of computations is

*N*= 15500.

*N*computations being realized on a computer with 30 processors, a database of results can be exploited. This approach is far from the optimization of the nanostructures which requires a specific optimization method for computational time sparing, to get the best parameters set [42

42. S. Kessentini and D. Barchiesi, “Quantitative comparison of optimized nanorods, nanoshells and hollow nanospheres for photothermal therapy,” Biomed. Opt. Express **3**, 590–604 (2012). [CrossRef] [PubMed]

62. D. Macías, A. Vial, and D. Barchiesi, “Application of evolution strategies for the solution of an inverse problem in Near-Field Optics,” J. Opt. Soc. Am. A **21**, 1465–1471 (2004). [CrossRef]

63. T. Grosges, D. Barchiesi, T. Toury, and G. Gréhan, “Design of nanostructures for imaging and biomedical applications by plasmonic optimization,” Opt. Lett. **33**, 2812–2814 (2008). [CrossRef] [PubMed]

## 5. Heuristic law for *λ*_{0}(*LSPR*)

**103**, 014308 (2008). [CrossRef]

38. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B **107**, 668–677 (2003). [CrossRef]

40. P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. **110**, 7238–7248 (2006). [CrossRef]

**105**, 102032–102039 (2009). [CrossRef]

64. D. Macías, A. Vial, and D. Barchiesi, “Evolution strategies approach for the solution of an inverse problem in near-field optics,” in *Lecture notes in computer science (6e European Workshop on Evolutionary Computation in Image Analysis and Signal Processing)*, vol. 3005 / 2004, G. Raidl, S. Cagnoni, J. Branke, R. Corne, D. W. Drechsler, Y. Jin, C. Johnson, P. Machado, E. Marchiori, F. Rothlauf, G. Smith, and G. Squillero, eds. (Springer-VerlagHeidelberg, Germany, 2004), 329 –338.

*e*lower than 6 nm produces only a shift of the LSPR lower than 20 nm. A theoretical study [27

**40**, 7152–7158 (2007). [CrossRef]

*D*= 100 nm). Increasing the diameter

*D*or decreasing the height

*h*of cylinders redshift the LSPR [18

18. M. Pelton, J. Aizpurua, and G. W. Bryant, “Metal-nanoparticles plasmonics,” Laser & Photon. Rev. **2**, 136–159 (2008). [CrossRef]

**103**, 014308 (2008). [CrossRef]

*D*= 175 nm,

*h*ranged from 10 nm to 150 nm [25

**103**, 014308 (2008). [CrossRef]

*D*and

*h*for a given LSPR wavelength. The selection of the pairs of (

*D,h*) corresponding to a given

*λ*

_{0}(

*LSPR*) =

*λ*

_{0}(max(

*C*)) are fitted with an affine function by minimizing the least square residue [65

_{ext}65. D. Barchiesi and T. Grosges, “Measurement of the decay lengths of the near field signal in tapping mode,” Curr. Appl. Phys. **9**, 1227–1231 (2009). [CrossRef]

*λ*

_{0}(

*LSPR*):

*D*(nm) and

*h*(nm) is satisfied. This law is deduced from the least square fitting of the results of the parametric study, simply by computing the slope and the intercept of

*D*as a function of

*h*for each LSPR position, and then by finding the linear dependence of

*a*and

*b*on the LSPR position

*λ*

_{0}(

*LSPR*). A similar approach was used to characterize the near-field optical microscopes and the evanescent near-field around nanostructures [13

13. D. Barchiesi, “Pseudo modulation transfer function in reflection scanning near-field optical microscopy,” Opt. Commun. **154**, 167–172 (1998). [CrossRef]

66. D. Barchiesi, O. Bergossi, M. Spajer, and C. Pieralli, “Image resolution in reflection scanning near-field optical microscopy (R-SNOM) using shear-force (ShF) feedback: characterization using spline and Fourier spectrum,” Appl. Opt. **36**, 2171–2177 (1997). [CrossRef] [PubMed]

*D*for a target wavelength of LSPR. If

*h*should be calculated as a function of

*λ*

_{0}(

*LSPR*), Eq. (6):

*D*as a function of

*h*is dependent on the LSPR position. Equation (5) helps to determine the geometrical parameters (

*D*and

*h*) of the biosensor to adjust the LSPR to a given wavelength. The domain of validity of this law is the hypercube defined in Table 2 considering rough structures. Equation (5) gives a function which reciprocal can be calculated for deducing the position of the LSPR as a function of

*D*and

*h*.

67. T. Grosges, D. Barchiesi, S. Kessentini, G. Gréhan, and M. Lamy de la Chapelle, “Nanoshells for photothermal therapy: a Monte-Carlo based numerical study of their design tolerance,” Biomed. Opt. Express **2**, 1584–1596 (2011). [CrossRef] [PubMed]

### 5.1. Validation by comparison with experimental data

**97**, 023113–023116 (2010). [CrossRef]

*h*for the red curve are slightly lower than the experimental expected value by considering free height

*h*. This behavior is coherent with the fact that the shape of nanoparticles is not exactly a cylinder but rather a smooth island of gold, especially for small diameters and heights.

4. J. Grand, M. Lamy de la Chapelle, J.-L. Bijeon, P.-M. Adam, A. Vial, and P. Royer, “Role of localized surface plasmons in surface-enhanced Raman scattering of shape-controlled metallic particles in regular arrays,” Phys. Rev. B **72**, 033407 (2005). [CrossRef]

*D*= 100 nm (Fig. 7). The red curve is obtained for the best fitting of the experimental data by considering free diameter

*D*. The corresponding values of

*D*are close to the experimental value of

*D*according to its uncertainty.

*h*and

*D*can be considered as hard tests of validity. The agreement of numerical and experimental results is satisfying and is coherent with the properties of the process of deposition of gold on substrate. Moreover the results show that the effective permittivity of the surrounding medium and the gold and chromium permittivities can be used to describe accurately the biosensor within the range of parameters.

### 5.2. Case of homothetic cylinders

*D*and

*h*, for a grating of rough gold cylinders with adhesion layer (

*e*= 2 nm) and

*P*= 200 nm (Fig. 1). The specific case of homothetic cylinders with

*h*=

*D*, removes a degree of freedom and leads to a simpler formula, with

*a*

_{1},

*a*

_{2},

*b*

_{1},

*b*

_{2}given in Table 3:

*h*=

*D*ranging from 50 to 70 nm, by cons it should still be checked for larger heights.

### 5.3. Case of nanodisks

*λ*

_{0}(

*LSPR*) could be of interest especially for specific aspect ratio of the cylinder. For example, for nanodiscs (

*h*<<

*D*), the position of the LSPR can be deduced from the series of Eq. (7). where

*o*[(

*h/D*)

^{2}] represents omitted terms of order higher than 2 in the series and

*a*

_{1},

*a*

_{2},

*b*

_{1},

*b*

_{2}are given in Table 3.

*h/D*is a blueshift of the LSPR, whatever are the diameters

*D*. These formula can be used to select the best parameters for a given LSPR position. Figure 8 shows the plots of the position of the LSPR as a function of the aspect ratio

*D/h*of the cylinder. The linear behavior can be compared to that observed for nanorods [68] for a wider range of aspect-ratio. This confirms the good behavior of the model in the range of parameters.

*h*and

*D*can be deduced, at least within the investigated domain of geometrical parameters and wavelengths. This is investigated in what follows.

### 5.4. Sensitivity of LSPR to uncertainties on size parameters

*λ*

_{0}(

*LSPR*)(

*D*,

*h*) (Eq. (7)). The uncertainty on

*λ*

_{0}(

*LSPR*) is deduced form the experimental uncertainties on

*D*(

*u*(

*D*)) and

*h*(

*u*(

*h*)) [61, 5.1.2]: where the partial derivatives are called sensitivity coefficients and

*a*

_{1},

*a*

_{2},

*b*

_{1},

*b*

_{2}are given in Table 3 Knowing the experimental uncertainties on

*h*and

*D*, the uncertainty on the LSPR can be deduced. The sensitivity coefficients

*S*and

_{h}*S*can be used to evaluate the effect of the experimental dispersion of values of

_{D}*D*and

*h*around a mean value. They contribute to the improvement of the uncertainty evaluation of the LSPR position given in Eqs. (3) – (4). For

*D*= 100 nm and

*D*= 200 nm,

*u*(

*λ*

_{0}(

*LSPR*)) ≈ 27 nm and 28 nm respectively. The study of the properties of

*S*and

_{D}*S*gives insight on the sensitivity of the uncertainty

_{h}*u*(

*λ*

_{0}(

*LSPR*)) to uncertainties on

*D*and

*h*. The balance between the two terms of the sum in Eq. (11) could give information on the critical process that needs to be improved in priority.

*S*(Fig. 9) show a slow decreasing of the sensitivity with

_{D}*h*. The mean sensitivity of

*u*(

*λ*

_{0}(

*LSPR*)) on

*D*is therefore about 1.45 in the considered range of parameters. The same behavior is exhibited in Fig. 10, |

*S*| decreases with

_{h}*h*. The mean value of the sensitivity is around −4 and decreases with

*D*. In terms of uncertainties, the weight of each term in the sum in Eq. (11) is of the same order of magnitude. Therefore, the uncertainty on the shape of cylinders (and on the diameter) should be decreased in priority in order to get a better control of the biosensor efficiency. This improvement is critical for cylinders with smaller height.

*h*∈ [10; 70] nm (Fig. 11). The relative uncertainty is about two times greater for small diameters and the absolute uncertainty decreases slightly as

*h*increases.

*D,h*) does hardly differ from that on bigger ones. Therefore, the whole process of fabrication, whatever is the LSPR position, will take advantage of the improvement of the lateral control of metal deposition.

*D*and

*h*of the cylinders. This law seems to be valid even if an adhesion layer of chromium of thickness

*e*= 2 − 6 nm is used. Moreover, the simplicity of this law helps to determine a first approximation of the size of the cylinder for tuning the LSPR position, as well as the effect of the propagation of experimental uncertainties on the position of the LSPR.

## 6. Conclusion

*h*, the diameter

*D*, the thickness

*e*of the chromium adhesion layer, and the roughness of the nanocylinders. Thin adhesion layers slightly modify the position of the LSPR. Roughness induces a redshift which enables to fit experimental results despite the use of bulk permittivities for chromium and gold and an effective index model for the system substrate-air. The relative influence of

*h*and

*D*is more complex but can be deduced from an heuristic law for the LSPR position. This law is obtained through an affine fitting of the proposed numerical experience plan. The agreement with experimental results [4

4. J. Grand, M. Lamy de la Chapelle, J.-L. Bijeon, P.-M. Adam, A. Vial, and P. Royer, “Role of localized surface plasmons in surface-enhanced Raman scattering of shape-controlled metallic particles in regular arrays,” Phys. Rev. B **72**, 033407 (2005). [CrossRef]

**97**, 023113–023116 (2010). [CrossRef]

## Acknowledgments

*Nanoantenna*European Project (FP7 Health-F5-2009-241818).

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61. | Working Group 1, |

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66. | D. Barchiesi, O. Bergossi, M. Spajer, and C. Pieralli, “Image resolution in reflection scanning near-field optical microscopy (R-SNOM) using shear-force (ShF) feedback: characterization using spline and Fourier spectrum,” Appl. Opt. |

67. | T. Grosges, D. Barchiesi, S. Kessentini, G. Gréhan, and M. Lamy de la Chapelle, “Nanoshells for photothermal therapy: a Monte-Carlo based numerical study of their design tolerance,” Biomed. Opt. Express |

68. | K. J. Prashant, X. Huang, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape and composition: Application in biological imaging and biomedicine,” Accounts Chem. Res. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(100.3190) Image processing : Inverse problems

(130.6010) Integrated optics : Sensors

(350.4600) Other areas of optics : Optical engineering

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: November 7, 2012

Revised Manuscript: December 21, 2012

Manuscript Accepted: January 9, 2013

Published: January 23, 2013

**Virtual Issues**

Vol. 8, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Dominique Barchiesi, Sameh Kessentini, Nicolas Guillot, Marc Lamy de la Chapelle, and Thomas Grosges, "Localized surface plasmon resonance in arrays of nano-gold cylinders: inverse problem and propagation of uncertainties," Opt. Express **21**, 2245-2262 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-2-2245

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