Abstract

Aspect ratio, width, and end-cap factor are three critical parameters defined to characterize the geometry of metallic nanorod (NR). In our previous work [Opt. Express 21, 2987 (2013)], we reported an optical extinction spectroscopic (OES) method that can measure the aspect ratio distribution of gold NR ensembles effectively and statistically. However, the measurement accuracy was found to depend on the estimate of the width and end-cap factor of the nanorod, which unfortunately cannot be determined by the OES method itself. In this work, we propose to improve the accuracy of the OES method by applying an auxiliary scattering measurement of the NR ensemble which can help to estimate the mean width of the gold NRs effectively. This so-called optical extinction/scattering spectroscopic (OESS) method can fast characterize the aspect ratio distribution as well as the mean width of gold NR ensembles simultaneously. By comparing with the transmission electron microscopy experimentally, the OESS method shows the advantage of determining two of the three critical parameters of the NR ensembles (i.e., the aspect ratio and the mean width) more accurately and conveniently than the OES method.

© 2013 OSA

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  1. N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer111, 1–35 (2010).
    [CrossRef]
  2. N. Xu, B. Bai, Q. Tan, and G. Jin, “Fast statistical measurement of aspect ratio distribution of gold nanorod ensembles by optical extinction spectroscopy,” Opt. Express21, 2987–3000 (2013).
    [CrossRef] [PubMed]
  3. B. Khlebtsov, V. Khanadeev, T. Pylaev, and N. Khlebtsov, “A new t-matrix solvable model for nanorods: Tem-based ensemble simulations supported by experiments,” J. Phys. Chem. C115, 6317–6323 (2011).
    [CrossRef]
  4. B. N. Khlebtsov, V. A. Khanadeev, and N. G. Khlebtsov, “Observation of extra-high depolarized light scattering spectra from gold nanorods,” The Journal of Physical Chemistry C112, 12760–12768 (2008).
    [CrossRef]
  5. K.-S. Lee and M. A. El-Sayed, “Dependence of the enhanced optical scattering efficiency relative to that of absorption for gold metal nanorods on aspect ratio, size, end-cap shape, and medium refractive index,” The Journal of Physical Chemistry B109, 20331–20338 (2005).
    [CrossRef]
  6. S. W. Prescott and P. Mulvaney, “Gold nanorod extinction spectra,” J. Appl. Phys.99, 123504 (2006).
    [CrossRef]
  7. S. Eustis and M. A. El-Sayed, “Determination of the aspect ratio statistical distribution of gold nanorods in solution from a theoretical fit of the observed inhomogeneously broadened longitudinal plasmon resonance absorption spectrum,” J. Appl. Phys.100, 044324 (2006).
    [CrossRef]
  8. O. Peña, L. Rodríguez-Fernández, V. Rodríguez-Iglesias, G. Kellermann, A. Crespo-Sosa, J. C. Cheang-Wong, H. G. Silva-Pereyra, J. Arenas-Alatorre, and A. Oliver, “Determination of the size distribution of metallic nanoparticles by optical extinction spectroscopy,” Appl. Opt.48, 566–572 (2009).
    [CrossRef] [PubMed]
  9. D. D. Evanoff and G. Chumanov, “Size-controlled synthesis of nanoparticles. 2. measurement of extinction, scattering, and absorption cross sections,” The Journal of Physical Chemistry B108, 13957–13962 (2004).
    [CrossRef]
  10. V. A. Bogatyrev, L. A. Dykman, K. B. N., and N. G. Khlebtsov, “Measurement of mean size and evaluation of polydispersity of gold nanoparticles from spectra of optical absorption and scattering,” Optics and Spectroscopy96, 128–135 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  15. B. Khlebtsov, V. Khanadeev, and B. N. Khlebtsov, “Tunable depolarized light scattering from gold and gold/silver nanorods,” Physical Chemistry Chemical Physics12, 3210 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2013

2012

2011

B. Khlebtsov, V. Khanadeev, T. Pylaev, and N. Khlebtsov, “A new t-matrix solvable model for nanorods: Tem-based ensemble simulations supported by experiments,” J. Phys. Chem. C115, 6317–6323 (2011).
[CrossRef]

2010

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer111, 1–35 (2010).
[CrossRef]

B. Khlebtsov, V. Khanadeev, and B. N. Khlebtsov, “Tunable depolarized light scattering from gold and gold/silver nanorods,” Physical Chemistry Chemical Physics12, 3210 (2010).
[CrossRef] [PubMed]

2009

2008

B. N. Khlebtsov, V. A. Khanadeev, and N. G. Khlebtsov, “Observation of extra-high depolarized light scattering spectra from gold nanorods,” The Journal of Physical Chemistry C112, 12760–12768 (2008).
[CrossRef]

2006

S. W. Prescott and P. Mulvaney, “Gold nanorod extinction spectra,” J. Appl. Phys.99, 123504 (2006).
[CrossRef]

S. Eustis and M. A. El-Sayed, “Determination of the aspect ratio statistical distribution of gold nanorods in solution from a theoretical fit of the observed inhomogeneously broadened longitudinal plasmon resonance absorption spectrum,” J. Appl. Phys.100, 044324 (2006).
[CrossRef]

2005

K.-S. Lee and M. A. El-Sayed, “Dependence of the enhanced optical scattering efficiency relative to that of absorption for gold metal nanorods on aspect ratio, size, end-cap shape, and medium refractive index,” The Journal of Physical Chemistry B109, 20331–20338 (2005).
[CrossRef]

A. V. Alekseeva, V. A. Bogatyrev, L. A. Dykman, B. N. Khlebtsov, L. A. Trachuk, A. G. Melnikov, and N. G. Khlebtsov, “Preparation and optical scattering characterization of gold nanorods and their application to a dot-immunogold assay,” Appl. Opt.44, 6285–6295 (2005).
[CrossRef] [PubMed]

2004

D. D. Evanoff and G. Chumanov, “Size-controlled synthesis of nanoparticles. 2. measurement of extinction, scattering, and absorption cross sections,” The Journal of Physical Chemistry B108, 13957–13962 (2004).
[CrossRef]

V. A. Bogatyrev, L. A. Dykman, K. B. N., and N. G. Khlebtsov, “Measurement of mean size and evaluation of polydispersity of gold nanoparticles from spectra of optical absorption and scattering,” Optics and Spectroscopy96, 128–135 (2004).
[CrossRef]

1991

Alekseeva, A. V.

Arenas-Alatorre, J.

Baev, A.

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

Bai, B.

Bogatyrev, V. A.

A. V. Alekseeva, V. A. Bogatyrev, L. A. Dykman, B. N. Khlebtsov, L. A. Trachuk, A. G. Melnikov, and N. G. Khlebtsov, “Preparation and optical scattering characterization of gold nanorods and their application to a dot-immunogold assay,” Appl. Opt.44, 6285–6295 (2005).
[CrossRef] [PubMed]

V. A. Bogatyrev, L. A. Dykman, K. B. N., and N. G. Khlebtsov, “Measurement of mean size and evaluation of polydispersity of gold nanoparticles from spectra of optical absorption and scattering,” Optics and Spectroscopy96, 128–135 (2004).
[CrossRef]

Cai, H.-X.

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

Cheang-Wong, J. C.

Chumanov, G.

D. D. Evanoff and G. Chumanov, “Size-controlled synthesis of nanoparticles. 2. measurement of extinction, scattering, and absorption cross sections,” The Journal of Physical Chemistry B108, 13957–13962 (2004).
[CrossRef]

Crespo-Sosa, A.

Cui, Y.

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

Dykman, L. A.

N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer111, 1–35 (2010).
[CrossRef]

A. V. Alekseeva, V. A. Bogatyrev, L. A. Dykman, B. N. Khlebtsov, L. A. Trachuk, A. G. Melnikov, and N. G. Khlebtsov, “Preparation and optical scattering characterization of gold nanorods and their application to a dot-immunogold assay,” Appl. Opt.44, 6285–6295 (2005).
[CrossRef] [PubMed]

V. A. Bogatyrev, L. A. Dykman, K. B. N., and N. G. Khlebtsov, “Measurement of mean size and evaluation of polydispersity of gold nanoparticles from spectra of optical absorption and scattering,” Optics and Spectroscopy96, 128–135 (2004).
[CrossRef]

El-Sayed, M. A.

S. Eustis and M. A. El-Sayed, “Determination of the aspect ratio statistical distribution of gold nanorods in solution from a theoretical fit of the observed inhomogeneously broadened longitudinal plasmon resonance absorption spectrum,” J. Appl. Phys.100, 044324 (2006).
[CrossRef]

K.-S. Lee and M. A. El-Sayed, “Dependence of the enhanced optical scattering efficiency relative to that of absorption for gold metal nanorods on aspect ratio, size, end-cap shape, and medium refractive index,” The Journal of Physical Chemistry B109, 20331–20338 (2005).
[CrossRef]

Eustis, S.

S. Eustis and M. A. El-Sayed, “Determination of the aspect ratio statistical distribution of gold nanorods in solution from a theoretical fit of the observed inhomogeneously broadened longitudinal plasmon resonance absorption spectrum,” J. Appl. Phys.100, 044324 (2006).
[CrossRef]

Evanoff, D. D.

D. D. Evanoff and G. Chumanov, “Size-controlled synthesis of nanoparticles. 2. measurement of extinction, scattering, and absorption cross sections,” The Journal of Physical Chemistry B108, 13957–13962 (2004).
[CrossRef]

Gill, P.

P. Gill, W. Murray, and M. Wright, Numerical Linear Algebra and Optimization (Addison Wesley, 1991).

He, G. S.

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

Hu, R.

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

Jin, G.

Kellermann, G.

Khanadeev, V.

B. Khlebtsov, V. Khanadeev, T. Pylaev, and N. Khlebtsov, “A new t-matrix solvable model for nanorods: Tem-based ensemble simulations supported by experiments,” J. Phys. Chem. C115, 6317–6323 (2011).
[CrossRef]

B. Khlebtsov, V. Khanadeev, and B. N. Khlebtsov, “Tunable depolarized light scattering from gold and gold/silver nanorods,” Physical Chemistry Chemical Physics12, 3210 (2010).
[CrossRef] [PubMed]

Khanadeev, V. A.

B. N. Khlebtsov, V. A. Khanadeev, and N. G. Khlebtsov, “Observation of extra-high depolarized light scattering spectra from gold nanorods,” The Journal of Physical Chemistry C112, 12760–12768 (2008).
[CrossRef]

Khlebtsov, B.

B. Khlebtsov, V. Khanadeev, T. Pylaev, and N. Khlebtsov, “A new t-matrix solvable model for nanorods: Tem-based ensemble simulations supported by experiments,” J. Phys. Chem. C115, 6317–6323 (2011).
[CrossRef]

B. Khlebtsov, V. Khanadeev, and B. N. Khlebtsov, “Tunable depolarized light scattering from gold and gold/silver nanorods,” Physical Chemistry Chemical Physics12, 3210 (2010).
[CrossRef] [PubMed]

Khlebtsov, B. N.

B. Khlebtsov, V. Khanadeev, and B. N. Khlebtsov, “Tunable depolarized light scattering from gold and gold/silver nanorods,” Physical Chemistry Chemical Physics12, 3210 (2010).
[CrossRef] [PubMed]

B. N. Khlebtsov, V. A. Khanadeev, and N. G. Khlebtsov, “Observation of extra-high depolarized light scattering spectra from gold nanorods,” The Journal of Physical Chemistry C112, 12760–12768 (2008).
[CrossRef]

A. V. Alekseeva, V. A. Bogatyrev, L. A. Dykman, B. N. Khlebtsov, L. A. Trachuk, A. G. Melnikov, and N. G. Khlebtsov, “Preparation and optical scattering characterization of gold nanorods and their application to a dot-immunogold assay,” Appl. Opt.44, 6285–6295 (2005).
[CrossRef] [PubMed]

Khlebtsov, N.

B. Khlebtsov, V. Khanadeev, T. Pylaev, and N. Khlebtsov, “A new t-matrix solvable model for nanorods: Tem-based ensemble simulations supported by experiments,” J. Phys. Chem. C115, 6317–6323 (2011).
[CrossRef]

Khlebtsov, N. G.

N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer111, 1–35 (2010).
[CrossRef]

B. N. Khlebtsov, V. A. Khanadeev, and N. G. Khlebtsov, “Observation of extra-high depolarized light scattering spectra from gold nanorods,” The Journal of Physical Chemistry C112, 12760–12768 (2008).
[CrossRef]

A. V. Alekseeva, V. A. Bogatyrev, L. A. Dykman, B. N. Khlebtsov, L. A. Trachuk, A. G. Melnikov, and N. G. Khlebtsov, “Preparation and optical scattering characterization of gold nanorods and their application to a dot-immunogold assay,” Appl. Opt.44, 6285–6295 (2005).
[CrossRef] [PubMed]

V. A. Bogatyrev, L. A. Dykman, K. B. N., and N. G. Khlebtsov, “Measurement of mean size and evaluation of polydispersity of gold nanoparticles from spectra of optical absorption and scattering,” Optics and Spectroscopy96, 128–135 (2004).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Lee, K.-S.

K.-S. Lee and M. A. El-Sayed, “Dependence of the enhanced optical scattering efficiency relative to that of absorption for gold metal nanorods on aspect ratio, size, end-cap shape, and medium refractive index,” The Journal of Physical Chemistry B109, 20331–20338 (2005).
[CrossRef]

Melnikov, A. G.

Mishchenko, M. I.

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A8, 871–882 (1991).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Mroczka, J.

Mulvaney, P.

S. W. Prescott and P. Mulvaney, “Gold nanorod extinction spectra,” J. Appl. Phys.99, 123504 (2006).
[CrossRef]

Murray, W.

P. Gill, W. Murray, and M. Wright, Numerical Linear Algebra and Optimization (Addison Wesley, 1991).

N., K. B.

V. A. Bogatyrev, L. A. Dykman, K. B. N., and N. G. Khlebtsov, “Measurement of mean size and evaluation of polydispersity of gold nanoparticles from spectra of optical absorption and scattering,” Optics and Spectroscopy96, 128–135 (2004).
[CrossRef]

Oliver, A.

Peña, O.

Prasad, P. N.

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

Prescott, S. W.

S. W. Prescott and P. Mulvaney, “Gold nanorod extinction spectra,” J. Appl. Phys.99, 123504 (2006).
[CrossRef]

Pylaev, T.

B. Khlebtsov, V. Khanadeev, T. Pylaev, and N. Khlebtsov, “A new t-matrix solvable model for nanorods: Tem-based ensemble simulations supported by experiments,” J. Phys. Chem. C115, 6317–6323 (2011).
[CrossRef]

Rodríguez-Fernández, L.

Rodríguez-Iglesias, V.

Silva-Pereyra, H. G.

Szczuczynski, D.

Tan, Q.

Trachuk, L. A.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Wright, M.

P. Gill, W. Murray, and M. Wright, Numerical Linear Algebra and Optimization (Addison Wesley, 1991).

Xu, N.

Yong, K.-T.

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

Zhang, X.-H.

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

Zhu, J.

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

S. W. Prescott and P. Mulvaney, “Gold nanorod extinction spectra,” J. Appl. Phys.99, 123504 (2006).
[CrossRef]

S. Eustis and M. A. El-Sayed, “Determination of the aspect ratio statistical distribution of gold nanorods in solution from a theoretical fit of the observed inhomogeneously broadened longitudinal plasmon resonance absorption spectrum,” J. Appl. Phys.100, 044324 (2006).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. Chem. C

B. Khlebtsov, V. Khanadeev, T. Pylaev, and N. Khlebtsov, “A new t-matrix solvable model for nanorods: Tem-based ensemble simulations supported by experiments,” J. Phys. Chem. C115, 6317–6323 (2011).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer111, 1–35 (2010).
[CrossRef]

Opt. Express

Optics and Spectroscopy

V. A. Bogatyrev, L. A. Dykman, K. B. N., and N. G. Khlebtsov, “Measurement of mean size and evaluation of polydispersity of gold nanoparticles from spectra of optical absorption and scattering,” Optics and Spectroscopy96, 128–135 (2004).
[CrossRef]

Physical Chemistry Chemical Physics

B. Khlebtsov, V. Khanadeev, and B. N. Khlebtsov, “Tunable depolarized light scattering from gold and gold/silver nanorods,” Physical Chemistry Chemical Physics12, 3210 (2010).
[CrossRef] [PubMed]

The Journal of Physical Chemistry B

D. D. Evanoff and G. Chumanov, “Size-controlled synthesis of nanoparticles. 2. measurement of extinction, scattering, and absorption cross sections,” The Journal of Physical Chemistry B108, 13957–13962 (2004).
[CrossRef]

K.-S. Lee and M. A. El-Sayed, “Dependence of the enhanced optical scattering efficiency relative to that of absorption for gold metal nanorods on aspect ratio, size, end-cap shape, and medium refractive index,” The Journal of Physical Chemistry B109, 20331–20338 (2005).
[CrossRef]

The Journal of Physical Chemistry C

B. N. Khlebtsov, V. A. Khanadeev, and N. G. Khlebtsov, “Observation of extra-high depolarized light scattering spectra from gold nanorods,” The Journal of Physical Chemistry C112, 12760–12768 (2008).
[CrossRef]

G. S. He, J. Zhu, K.-T. Yong, A. Baev, H.-X. Cai, R. Hu, Y. Cui, X.-H. Zhang, and P. N. Prasad, “Scattering and absorption cross-section spectral measurements of gold nanorods in water,” The Journal of Physical Chemistry C114, 2853–2860 (2010).
[CrossRef]

Other

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

P. Gill, W. Murray, and M. Wright, Numerical Linear Algebra and Optimization (Addison Wesley, 1991).

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Figures (7)

Fig. 1
Fig. 1

Geometric model of the NR. Several NRs with the same width D and aspect ratio AR but different end-cap factor e are demonstrated.

Fig. 2
Fig. 2

Optical setups used in the OESS method for measuring (a) the optical extinction spectra and (b) the angular light scattering spectra at an angle of 90° of the gold NRs.

Fig. 3
Fig. 3

Measurement results of the sample NR-10: (a) the absorbance A measured by the OES method and the 90° scattering intensity S90 (cm−1) measured by the OSS method; (b) the ARD function retrieved by the OESS method; (c) the ARD function measured by the TEM method. In (a), the original measurement data of the extinction spectra A-Exp. (circle dots) and the scattering spectra S90-Exp. (square dots, multiplied by 60), as well as the corresponding numerically reproduced extinction spectra A-Fit (solid line) and scattering spectra S90-Fit (dashed line, multiplied by 60) according to the retrieved NR parameters are given. The inset in (a) shows the TEM image of the sample. In (b) and (c), both the discrete ARD and a Gaussian fit of it are given; the values in parentheses give the mean AR and the standard deviation of the ARD.

Fig. 4
Fig. 4

The same as Fig. 3, but for sample NR-20.

Fig. 5
Fig. 5

The same as Fig. 3, but for sample NR-30.

Fig. 6
Fig. 6

The same as Fig. 3, but for sample NR-40.

Fig. 7
Fig. 7

Comparison of the extinction cross sections of different gold nanoparticles: (a) rectangular cuboid gold nanoparticles and gold NRs, (b) gold NRs with different end caps.

Tables (1)

Tables Icon

Table 1 Measurement results of the four gold NR ensemble samples

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

A ( λ ) = log [ I e ( λ ) I r 0 ( λ ) I r 1 ( λ ) I m 0 ( λ ) ] = l N v ln 10 C ext ( λ ) ,
T 0 ( λ ) = I s ( λ ) I r 2 ( λ ) I r 0 ( λ ) I m 0 ( λ ) ,
I r 2 ( λ ) = I r 0 ( λ ) T nd ( λ ) T ext ( λ ) , I s ( λ ) = I m 0 ( λ ) R 1 ( λ ) R 2 ( λ ) T lens ( λ ) T ext ( λ ) α sca 90 ° ( λ ) ,
α sca 90 ° ( λ ) N v Ω 90 ° d S ( λ ) .
T 0 ( λ ) S 90 ( λ ) ,
d S ps ( λ ) = a 1 ps ( λ , 90 ° ) C sca ps ( λ ) 4 π ,
C sca ps ( λ ) = C ext ps ( λ ) C abs ps ( λ ) = C ext ps ( λ ) = ln 10 l N v ps A ps ( λ ) .
S 90 g ( λ ) = N v g d S g ( λ ) = ln 10 l T 0 g ( λ ) T 0 ps ( λ ) A ps ( λ ) a 1 ps ( λ ) 4 π ,
A g ( λ ) = l N v g ln 10 C ext g ( λ ) = l N v g ln 10 D min D max AR min AR max e min e max p ( D , A R , e ) C ext g ( λ , D , A R , e ) d D d A R d e .
S 90 g ( λ ) = N v g D min D max AR min AR max e min e max p ( D , A R , e ) d S g ( λ , D , A R , e ) d D d A R d e .
A = C P , S = S d P ,
P RLS = min P { A C P 2 2 + ω S S S d P 2 2 } = min P { ( A C P ) T ( A C P ) + ω S ( S S d P ) T ( S S d P ) } = min P { P T ( C T C + ω S S d T S d ) P A T C P ( A T C P ) T ω S S T S d P ω S ( S T S d P ) T } ,
P RLS = min P { P T ( C T C + ω S S d T S d ) P 2 ( A T C + ω S S T S d ) P } = min P { 1 2 P T Q P + q T P } ,

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