## Accurate geometric characterization of gold nanorod ensemble by an inverse extinction/scattering spectroscopic method |

Optics Express, Vol. 21, Issue 18, pp. 21639-21650 (2013)

http://dx.doi.org/10.1364/OE.21.021639

Acrobat PDF (1389 KB)

### 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

## 1. Introduction

1. N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer **111**, 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. Express **21**, 2987–3000 (2013). [CrossRef] [PubMed]

*a priori*information.

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. C **115**, 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 C **112**, 12760–12768 (2008). [CrossRef]

^{3}) of sampling NRs should be characterized, which is time consuming and expensive. Therefore, in this work we aim to develop a method based on scatterometry to achieve this goal more conveniently.

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 B **109**, 20331–20338 (2005). [CrossRef]

6. S. W. Prescott and P. Mulvaney, “Gold nanorod extinction spectra,” J. Appl. Phys. **99**, 123504 (2006). [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 B **109**, 20331–20338 (2005). [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. Express **21**, 2987–3000 (2013). [CrossRef] [PubMed]

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]

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. Express **21**, 2987–3000 (2013). [CrossRef] [PubMed]

*π*solid angle [9

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 B **108**, 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 Spectroscopy **96**, 128–135 (2004). [CrossRef]

11. 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 C **114**, 2853–2860 (2010). [CrossRef]

13. M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A **8**, 871–882 (1991). [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 Spectroscopy **96**, 128–135 (2004). [CrossRef]

11. 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 C **114**, 2853–2860 (2010). [CrossRef]

14. 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]

15. B. Khlebtsov, V. Khanadeev, and B. N. Khlebtsov, “Tunable depolarized light scattering from gold and gold/silver nanorods,” Physical Chemistry Chemical Physics **12**, 3210 (2010). [CrossRef] [PubMed]

*a priori*information about the geometric values of NRs (such as the nominal width of the NRs) beforehand. When constructing the database of the scattering and extinction spectra of the NRs, the T-matrix method [12] was used to simulate the extinction and differential scattering cross sections of gold NR ensembles rigorously. By using the OESS method to characterize different NR ensemble samples experimentally, it is shown that the measurement results coincide with those of the TEM method quite well, while the OESS method is much faster and more cost effective. The OESS method is also more accurate than the OES method to determine the two critical parameters (aspect ratio and mean width) of NRs.

## 2. Methods

*p*(

*D*,

*AR*,

*e*) with respect to the three structural quantities: the width

*D*, the aspect ratio

*AR*and the end-cap factor

*e*of the NRs as well as determining the estimates of the parameters

*D*,

*AR*and

*e*. The measurement method and the numerical retrieving algorithm applied in the OESS method are presented below.

### 2.1. Measurement method and setup

**21**, 2987–3000 (2013). [CrossRef] [PubMed]

*A*(

*λ*) of the sample medium with a length

*l*, which can be expressed as where

*λ*is the wavelength of light,

*N*is the number of NRs per unit volume, 〈

_{v}*C*

_{ext}〉 is the average extinction cross section of the sample, and

*I*(

*λ*) with different subscripts represent the intensities of different light beams shown in Fig. 2(a).

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 Spectroscopy **96**, 128–135 (2004). [CrossRef]

*T*

_{0}(

*λ*) and can be expressed by where

*I*(

*λ*) with different subscripts represent the intensities of different light beams shown in Fig. 2(b).

*I*

_{r2}(

*λ*) and

*I*

_{s}(

*λ*) in Eq. (2) can be expressed as where

*T*

_{nd}(

*λ*),

*T*

_{ext}(

*λ*), and

*T*

_{lens}(

*λ*) denote the transmittances of the ND filter, the sample, and the lens, respectively,

*R*

_{1}(

*λ*) and

*R*

_{2}(

*λ*) are the reflectances of the mirrors, and

**96**, 128–135 (2004). [CrossRef]

*I*

_{r2}(

*λ*) so that the signal-to-noise ratio of the detector is poor when measuring the transmittance

*T*

_{0}(

*λ*) directly. By using the lens and the ND filter, the measurement values of

*I*

_{s}(

*λ*) and

*I*

_{r2}(

*λ*) can be adjusted to the same order of magnitude so as to get the best detection response.

_{90°}centered around the direction of scattering, the number of NRs per unit volume

*N*, and the average differential scattering cross section 〈d

_{v}*S*(

*λ*)〉, i.e., [12] Substituting Eq. (3) and Eq. (4) into Eq. (2), we can obtain where

*S*

_{90}(

*λ*) =

*N*〈d

_{v}*S*(

*λ*)〉. Eq. (5) connects the angular scattering spectra

*S*

_{90}(

*λ*) with the measurand

*T*

_{0}(

*λ*). However, it should be noted that the absolute value of

*S*

_{90}(

*λ*) cannot be measured directly but should be calibrated. A simple way is to measure a standard sample whose angular scattering cross section is known beforehand. In this work, we perform calibration by using the standard polystyrene microspheres (PS) from Thermo Fisher Scientific (China) Ltd., whose diameter is 102 nm and whose average differential scattering cross section 〈d

*S*(

*λ*)〉 can be calculated from the average scattering cross section 〈

*C*

_{sca}(

*λ*)〉 by [12, 13

13. M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A **8**, 871–882 (1991). [CrossRef]

*a*

_{1}(

*λ*, 90°) represents the element at the first row and the first column of the Mueller matrix (or phase matrix), which can be calculated conveniently by the T-matrix method [12]. The detailed calculation steps can be found in, for example, Ref. [13

13. M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A **8**, 871–882 (1991). [CrossRef]

### 2.2. Inverse-problem solution algorithm

*p*(

*D*,

*AR*,

*e*) of the gold NR ensemble can be retrieved. To solve this inverse scattering problem, we perform a similar retrieving procedure as that presented in our previous work [2

**21**, 2987–3000 (2013). [CrossRef] [PubMed]

*A*is calculated by integrating the contribution of all the composing NRs according to the PDF

*p*(

*D*,

*AR*,

*e*) [2

**21**, 2987–3000 (2013). [CrossRef] [PubMed]

*S*

_{90}(

*λ*) of a polydisperse NR ensemble can be calculated by

*AR*,

*D*, and

*e*, the condition number of the reduced linear system is usually too large to produce an accurate and stable solution. Thus in many previous works [2

**21**, 2987–3000 (2013). [CrossRef] [PubMed]

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 C **112**, 12760–12768 (2008). [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]

*D*and the end-cap factor

*e*were usually fixed as their mean values (as we have done in our previous work [2

**21**, 2987–3000 (2013). [CrossRef] [PubMed]

*AR*is the primary parameter affecting the extinction of the NR ensemble [6

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

*D*and

*e*is significant to the retrieving result of the PDF of

*AR*.

*D*is possible to be determined [5

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 B **109**, 20331–20338 (2005). [CrossRef]

*e*and discretize Eq. (9) and Eq. (10) with respect to

*AR*,

*D*, and

*λ*. Similar with the OES method [2

**21**, 2987–3000 (2013). [CrossRef] [PubMed]

*e*in the OESS method can also influence the retrieved results. Even by the OESS measurement, the mean end cap

*e*cannot be determined accurately and should be assumed or be measured by some other methods beforehand.

**A**and

**S**are

*M*× 1 vectors,

**C**and

**S**are

_{d}*M*×

*N*matrices,

**P**is a

*N*× 1 vector, and

*M*and

*N*are integer numbers. The vectors

**A**and

**S**contain the values of the measured extinction

*A*(

*λ*) and angular scattering cross section per unit volume

*S*

_{90}(

*λ*) at different

*λ*, respectively. The values of extinction cross section of gold NRs

**C**for various wavelengths

*λ*(rows), various aspect ratios

*AR*(columns) and various widths

*D*(columns) for fixed end-cap shape

*e*. By the same way, the values of differential scattering cross section d

*S*

^{g}(

*λ*,

*D*,

*AR*,

*e*) of gold NRs at the scattering angle 90° are stored in the matrix

**S**. The vector

_{d}**P**is the PDF to be solved and it has two physical constraints: the non-negativity constraint

**P**≥

**0**and the standard normalization condition

**UP**= 1 where

**P**

_{RLS}can be expressed as [17

17. J. Mroczka and D. Szczuczynski, “Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements,” Appl. Opt. **51**, 1715–1723 (2012). [CrossRef] [PubMed]

_{2}is the Euclidean norm, the superscript T means the transpose of the vectors, and the non-negative weight coefficient

*ω*

_{S}means the weight between the OES and OSS data. The value

*ω*

_{S}= max (

**A**)/max (

**S**) is used here to balance the weight of the OES data and the OSS data, or the amplitude of the OSS data would be much smaller than the amplitude of the OES data and we would not obtain a good optimization result. Note that in Eq. (12),

**A**

^{T}

**CP**and

**S**

^{T}

**S**result in scalars so that their transposes are themselves. Consequently,

_{d}P**A**

^{T}

**CP**= (

**A**

^{T}

**CP**)

^{T},

**S**

^{T}

**S**= (

_{d}P**S**

^{T}

**S**)

_{d}P^{T}, and Eq. (12) can be written in another equivalent form where

**Q**= 2(

**C**

^{T}

**C**+

*ω*

_{S}

**S**

_{d}^{T}

**S**) is a symmetric matrix of size

_{d}*N*×

*N*, and

**q**= −2(

**C**

^{T}

**A**+

*ω*

_{S}

**S**

_{d}^{T}

**S**) is a

*N*-dimensional column vector. We use the active set method [16] to find the solution of Eq. (13). Then the retrieved ARD and the mean width

*D*

_{m}can be calculated easily by reshaping the vector

**P**

_{RLS}to a two dimensional matrix which consist of the PDF

*p*(

*AR*,

*D*) and averaging the row and the column of the PDF matrix, respectively.

## 3. Experimental results and discussions

### 3.1. Comparison of the OESS and TEM measurements

*p*(

*AR*) and the mean width

*D*

_{m}of gold NR ensemble samples. The results are compared with those directly obtained by the TEM method (which is considered as a benchmark). In our comparison experiment, altogether 20 NR samples were measured and analyzed, each of which contains approximately 10

^{10}NRs per millilitre. Without loss of generality, the results of four samples with different

*D*,

*AR*and

*e*are demonstrated here. The four samples, designated as NR-10, NR-20, NR-30, and NR-40, were obtained from the National Center for Nanoscience and Technology, Beijing, China, whose nominal mean width

*D*

_{m}are 10 nm, 20 nm, 30nm, and 40 nm, respectively.

*λ*was 400 nm – 850 nm, with a step of 1 nm. The extinction spectra database and the angular scattering spectra database of the gold NRs [corresponding to the matrices

**C**and

**S**in Eq. (11)] were calculated with the T-matrix method. It takes about 30 seconds to calculate a single scattering spectrum of NRs in the wavelength range of 400 nm – 850 nm, with a 1 nm spectral resolution, by using a dual-core 2.13GHz Intel Xeon CPU with 80Gb RAM. After that, the optimization procedure described above was implemented to retrieve the ARD function

_{d}*p*(

*AR*) and the mean width

*D*

_{m}of the samples, where

*AR*was discretized in the range of 1 to 5, with a step of 0.1. The inverse-problem solution algorithm was run on a 3.00GHz Intel Core2 Duo CPU with 4Gb RAM and the time consumption is about 5 seconds for each sample.

*e*

_{m}, six values with the range 0–1 and the step 0.2 were assumed beforehand. The values of

*e*

_{m}shown in the table 1 were adopted for the reason that the retrieved ARDs by the OESS method coincide with the measured ARDs by the TEM method best.

*D*

_{m}and the mean aspect ratio

*AR*

_{m}derived by the two methods coincide with each other well, with their relative difference smaller than 6% and 2%, respectively. These show that our OESS measurement results are reliable. For the standard deviation

*σ*(which represents the polydispersity of the ARD), the two sets of values also co-incide with each other relatively well, where the small difference may be owing to the deviation between the real shape of the gold NRs and the geometric model adopted in our calculation.

_{AR}*C*of the NRs can also be determined by the OESS method. It can be derived as

_{g}*C*=

_{g}*ρNv*

**V**·

**P**

_{RLS}, where

**V**is a row vector consisting of the volume of each nanorod geometry in the sample and

*ρ*is the density of bulk gold. The values of

*C*measured by the OESS method for the four samples NR-10, NR-20, NR-30, NR-40 are 14.54±0.26

_{g}*μ*g/ml, 15.78±0.17

*μ*g/ml, 18.03±0.14

*μ*g/ml, and 15.34 ± 0.24

*μ*g/ml, respectively. Since these concentration values cannot be obtained by the TEM method, we cannot make a comparison of the two methods here.

*p*(

*AR*) and the mean width

*D*

_{m}by the OESS method, as we mentioned before.

*p*(

*AR*) derived by the two methods in general coincide with each other well. With the retrieved ARD function

*p*(

*AR*) and the mean width

*D*

_{m}, the extinction spectra and 90° angular scattering spectra of the four samples were also numerically reproduced by Eq. (9) and Eq. (10), as shown in Fig. 3(a) – Fig. 6(a), which also coincide with the measured extinction spectra and the angular scattering spectra quite well. Thus Table 1 and Fig. 3 and Fig. 6 have shown that the retrieved results by the OESS method are reliable in the characterization of the gold NRs ensembles. However, there are still some small differences between the measurement results derived by the OESS method and the TEM method. In the following, we discuss in detail these differences and their possible causes.

### 3.2. Discussions

*D*= 60 nm,

*AR*= 1.2, and an ensemble of gold NRs with

*D*= 30 nm,

*AR*= 1.8,

*e*= 0.8, as shown in Fig. 7(a). Indeed, we can see that the LSPR peak around 610 nm is reproduced in both spectra, where the peak of the cubic nanoparticles is even stronger.

*e*

_{m}obtained by the OESS method (

*e*

_{m}= 0.8) deviates from the value obtained by the TEM method (

*e*

_{m}= 0.4) significantly, as shown in Table 1. The possible reason is that the end cap shape of the NRs in the sample NR-30 is sharper, as shown in the inset of Fig. 5(a), which deviates significantly from our cylindrical NR model with semi-spheroidal end cap. To corroborate this, we have calculated three NR ensembles with different end-cap shapes, as shown in Fig. 7(b). It is seen evidently that for the same value of end-cap factor

*e*

_{m}= 0.4, the L-LSPR peak of gold NRs with a cone-like end cap is blue shifted compared with that of gold NRs with a semi-spheroidal end cap. As a consequence, the extinction spectrum of the NRs with cone-like end cap of factor 0.4 matches the spectrum of the NRs with semi-spheroidal end cap of factor 0.8 better. For this reason, the fitted value of the end-cap facotr that we obtained by the OESS measurement is 0.8, but not 0.4. These imply that in the practical characterization of gold NR ensembles, the database of extinction and scattering spectra should be expanded, by calculating not only NRs but also nanoparticles of other shapes (such as spheres, cubes, and NRs with special end-cap shapes). By taking these measures, the measurement accuracy of the OESS method can be further improved.

## 4. Conclusion

*AR*and

*D*, as well as the probability density functions

*p*(

*AR*) of gold NR ensembles statistically. To perform the OESS method, the angular scattering spectra at the angle of 90° are measured in addition to the extinction spectra measured by the UV-VIS spectrometer. To solve the inverse scattering problem, the extinction cross section and the differential scattering cross section of polydisperse NR ensembles are calculated rigorously by the T-matrix method. Then the critical parameters are retrieved by an optimization process with data fitting to the measured spectra.

*D*

_{m}coincide well with those obtained by the TEM method. By using the OESS method, it is also possible to determine the mass-volume concentration of NRs, which is unable to be measured by the TEM method. The comparison results indicate that the OESS method is fast, accurate, and cost effective to measure the critical geometric parameters of the gold NR ensemble. Further improvement is to enlarge the database of the extinction and scattering spectra of gold nanoparticles of different shapes (such as rectangular cuboid nanoparticles and NRs with special end-cap shapes) by considering the byproducts of practical samples, by which the retrieving accuracy can be further improved. The proposed OESS method has the potential to be developed for characterizing not only gold NR ensembles but also other non-spherical metal nanoparticles such as silver NRs.

## Acknowledgments

## References and links

1. | N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer |

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. Express |

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. C |

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 C |

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 B |

6. | S. W. Prescott and P. Mulvaney, “Gold nanorod extinction spectra,” J. Appl. Phys. |

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. |

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. |

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 B |

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 Spectroscopy |

11. | 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 C |

12. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

13. | M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A |

14. | 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. |

15. | B. Khlebtsov, V. Khanadeev, and B. N. Khlebtsov, “Tunable depolarized light scattering from gold and gold/silver nanorods,” Physical Chemistry Chemical Physics |

16. | P. Gill, W. Murray, and M. Wright, |

17. | J. Mroczka and D. Szczuczynski, “Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements,” Appl. Opt. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(290.3200) Scattering : Inverse scattering

(290.5850) Scattering : Scattering, particles

(160.4236) Materials : Nanomaterials

**ToC Category:**

Scattering

**History**

Original Manuscript: July 3, 2013

Revised Manuscript: August 19, 2013

Manuscript Accepted: August 21, 2013

Published: September 6, 2013

**Virtual Issues**

Vol. 8, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Ninghan Xu, Benfeng Bai, Qiaofeng Tan, and Guofan Jin, "Accurate geometric characterization of gold nanorod ensemble by an inverse extinction/scattering spectroscopic method," Opt. Express **21**, 21639-21650 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-18-21639

Sort: Year | Journal | Reset

### References

- N. G. Khlebtsov and L. A. Dykman, “Optical properties and biomedical applications of plasmonic nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer111, 1–35 (2010). [CrossRef]
- 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]
- 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. 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).
- M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A8, 871–882 (1991). [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]
- 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]
- P. Gill, W. Murray, and M. Wright, Numerical Linear Algebra and Optimization (Addison Wesley, 1991).
- J. Mroczka and D. Szczuczynski, “Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements,” Appl. Opt.51, 1715–1723 (2012). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.