# Inversion of the Complex Refractive Index of Au-Ag Alloy Nanospheres Based on the Contour Intersection Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Contour Intersection Method

_{p}is the particle refractive index, and n

_{m}is the refractive index of the surrounding medium, as shown in Figure 2.

_{m}/λ is the size parameter, and λ is the wavelength of incident light in a vacuum. The numerical algorithm described in Bohren and Huffman’s monograph [35] was used to calculate scattering coefficients a

_{n}and b

_{n}. As seen from the Equations (1)–(4), to solve the interaction problem between light and spherical particles, Mie theory only needs to know the incident light wavelength, particle diameter, particle refractive index, and refractive index of the surrounding medium. The calculations of extinction, scattering, and absorption efficiencies of spherical particles are then straightforward.

_{r}= 1, the refractive index of the nanoparticles can be written as follows.

_{r}is the dielectric function.

_{∞}is the contribution of free electrons from high energy level transition. ω

_{p}is the plasma frequency, and Γ

_{p}is the damping coefficient of free electrons (collision frequency). ε

_{CP1}and ε

_{CP2}are the dielectric functions at two critical points. ω

_{01}and ω

_{02}are the transition thresholds at two critical points. ω

_{g1}is the transition gap at the critical point. Γ

_{1}and Γ

_{2}are the damping coefficients at two critical points. A

_{1}and A

_{2}are amplitude parameters at two critical points.

_{pAu}, ω

_{pAg}, and ω

_{pAuAg5050}represent the plasma frequencies of pure Au, pure Ag, and 50%–50% equimolar fraction Au-Ag alloys, respectively. The other parameters also have the same composition dependence. In the above dielectric function model (Equation (6)), Rioux et al. used a genetic algorithm to fit multiple sets of experimental measurement data to obtain all unknown parameters in the model. The fitting parameters are shown in Table 1.

_{p}of free electrons in metal nanoparticles is related to the average free path of free electrons. Similarly, Γ

_{p}in Au-Ag alloy nanoparticles is also affected by the mean free path of free electrons. Therefore, the dielectric function of alloy nanoparticles can be modified as follows [37].

_{pBulk}is the damping coefficient of bulk material, which can be calculated by fitting parameters. α is a dimensionless parameter close to 1. h is the Planck constant. v

_{f}is the Fermi rate of free electrons, which is 1 × 10

^{6}m/s for Au-Ag alloys [36]. L

_{eff}represents the effective mean free path of free electrons, the size of nanoparticles [38].

## 3. Results and Discussion

_{m}, the measurement error, it is assumed that the diameters of the test particles are 50 nm, the incident wavelength is 633 nm, and x is 0.62. The original value of the particle complex refractive index (n

_{o}, k

_{o}) is calculated by the modified model in the range n ∈ [n

_{o}− ∆n, n

_{o}+ ∆n]; k ∈ [k

_{o}− ∆k, k

_{o}+ ∆k]; ∆n = ∆k = 3 and with a step length of 0.001. The scattering efficiency (Q

_{sca}) and absorption efficiency (Q

_{abs}) in a given complex refractive index range are calculated by Mie theory, as shown in Figure 3a,b. Then, the contour lines are drawn using Q

_{sca}and Q

_{abs}corresponding to the original values of the complex refractive index. The two contour lines are projected onto the n-k plane to find their intersection, as shown in Figure 3c.

_{sca}and Q

_{abs}are specified in the actual operation, the contour line of Q

_{back}is added to n-k space, as shown in Figure 4a. Finally, the contours of Q

_{sca}, Q

_{abs}, and Q

_{back}are identified and projected onto the n-k space. The results show that adding Q

_{back}can acquire a unique solution, as shown in Figure 4b.

_{n}and E

_{k}are the relative errors in the real and imaginary parts of the complex refractive index. n

_{i}and n

_{0}are the inversion value and original value of the real part of the complex refractive index. k

_{i}and k

_{o}represent the inverse value and the original value of the imaginary part of the complex refractive index.

_{sca}and Q

_{abs}cannot be avoided, [−1%, 1%] random errors are added to Q

_{sca}and Q

_{abs}at different wavelengths. E

_{m}expresses random errors. The inversion results are shown in Figure 5.

_{m}is 1%, the incident light wavelength is 633 nm, x is 0.62, and particle sizes are 20 nm, 40 nm, 60 nm, 80 nm, and 100 nm. The selection of the Au molar fraction is based on the analysis of the variation of the scattering and absorption efficiencies of Au-Ag alloy nanoparticles, considering the Au component ratio and the experimental results of Besner, Meunier et al. [21], as shown in Figure 7.

_{sca}of alloy nanoparticles increases with particle size and is larger than that of pure gold and pure silver. The influence of particle size on the Q

_{sca}of alloy nanoparticles is more significant, resulting in a large relative error. Figure 8c shows that the imaginary part of the complex refractive index changes minutely with increasing particle size.

_{m}of 0%, 1%, 5%, and 7% were inverted by the iterative and contour intersection methods, and the relative errors were analyzed.

_{m}= 5%, the relative error in the real part of the complex refractive index is about 10%. The relative error fluctuates when E

_{m}> 5%, and the inversion results become unstable. The possible reason is the smaller wavelength, which dramatically influences the particles’ scattering and absorption characteristics. Figure 11d manifests the relative error in the imaginary part of the complex refractive index, which has a trend similar to that of Figure 11b. The relative error in the imaginary part of the complex refractive index is below 3% when E

_{m}< 5%.

_{m}(measurement errors). It can be seen from Figure 12a that for the real part of the complex refractive index, the mean relative error of the inversion results from the iterative method increases with the increase in the measurement error. The mean relative error of the inversion results from the contour intersection method also has the same trend when E

_{m}< 5%. However, when E

_{m}> 5%, the mean relative error of the real part of the complex refractive index decreases. It can be seen from Figure 12b that for the imaginary part of the complex refractive index, the trend is similar to the real part of the refractive index. The overall analysis shows that the average relative error of the inversion results from the contour intersection method is less than that of the inversion results from the iterative method. When E

_{m}> 5%, with the sharp increase in the number of intersection points, the inversion result values show unpredictable volatility. An iterative method is a method of checking for the solutions against output conditions in a specific range. The increase in the measurement error largely depends on the selection of the starting parameters. Therefore, the influence of measurement error in the iterative method is stronger than that of the contour intersection method.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The complex refractive index of Au nanoparticles with diameters of 50 nm at a wavelength of 633 nm. (

**a**) Scattering efficiency (red). (

**b**) Absorption efficiency (blue). (

**c**) Scattering and absorption efficiencies after isometric projection on the n-k plane are inverted by the contour intersection method.

**Figure 4.**The complex refractive index of Au nanoparticles with diameters of 50 nm at 633 nm. (

**a**) Backscattering efficiency (green). (

**b**) Projection of contour lines in the n-k plane inverted by the contour intersection method.

**Figure 5.**Effect of Au component proportion on the real part (n) and imaginary part (k) of the inverted refractive index (

**a,c**) with the corresponding relative errors (

**b,d**).

**Figure 6.**The relative errors in the real part (

**a**) and imaginary part (

**b**) of the complex refractive index inverted by the contour intersection method and the iterative method with respect to the change in the Au molar fraction.

**Figure 7.**Scattering and absorption efficiencies of Au-Ag alloy nanosphere with respect to the Au molar fraction.

**Figure 8.**The effect of particle size on the real part (n) and imaginary part (k) of the refractive index inverted by the contour intersection method. (

**a**) The inversion results of the real part of the refractive index where molar fraction of Au is 0, 0.62, and 1, respectively. (

**b**) The relative errors in the inversion of the real part of the refractive index. (

**c**) The inversion results for the imaginary part of the refractive index where molar fraction of Au is 0, 0.62, and 1, respectively. (

**d**) The relative errors in the inversion of the imaginary part of the refractive index.

**Figure 9.**The relative error in the inversion results of the contour intersection and iterative methods with respect to particle sizes. The relative errors of the real part (

**a**) and imaginary part (

**b**) of the complex refractive index. The molar fraction of Au is 0.62.

**Figure 10.**Relative errors in the refractive indices of Ag nanoparticles, Au-Ag alloy nanoparticles, and Au nanoparticles with respect to incident wavelengths after inversion by the contour intersection method, where x is 0.62 for Au-Ag alloy. (

**a**) Relative errors in the inversion of the real part of the refractive index. (

**b**) Relative errors in the inversion of the imaginary part of the refractive index.

**Figure 11.**The effect of measurement error on the inversion of refractive index by contour intersection method. (

**a**) Inversion of the real part of the refractive index with a molar fraction of Au 0.62. (

**b**) The relative error in the inversion of the real part of the refractive index. (

**c**) The inversion of the imaginary part of the refractive index. (

**d**) The relative error in the inversion of the imaginary part of the refractive index.

**Figure 12.**The average relative errors in the inversion results of the contour intersection method and the iterative method with respect to E

_{m}. The average relative errors in the real (

**a**) and imaginary (

**b**) parts of the refractive index at an Au molar fraction of 0.62.

Metal | Au | AuAg_{5050} | Ag |
---|---|---|---|

ω_{p}/eV | 8.9234 | 9.0218 | 8.5546 |

Γ_{p}/eV | 0.042389 | 0.16713 | 0.022427 |

ε_{∞} | 2.2715 | 2.2838 | 1.7381 |

ω_{g1}/eV | 2.6652 | 3.0209 | 4.0575 |

ω_{01}/eV | 2.3957 | 2.7976 | 3.9260 |

Γ_{1}/eV | 0.1788 | 0.18833 | 0.017723 |

A_{1} | 73.251 | 22.996 | 51.217 |

ω_{02}/eV | 3.5362 | 3.3400 | 4.1655 |

Γ_{2}/eV | 0.35467 | 0.68309 | 0.18819 |

A_{2} | 40.007 | 57.540 | 30.770 |

**Table 2.**The inversion results from the contour intersection and the iterative methods with a wavelength of 633 nm and nanoparticle diameters of 50 nm. The value of x is 0.62.

Method | n_{o} | k_{o} | n_{i} | k_{i} | E_{n} (%) | E_{k} (%) |
---|---|---|---|---|---|---|

Contour intersection method | 0.34677 | 3.96138 | 0.34639 | 3.96082 | 0.11 | 0.1 |

Iterative method | 0.34677 | 3.96138 | 0.33956 | 3.94333 | 2.08 | 0.46 |

**Table 3.**The inversion results and times of the contour intersection method with different complex refractive index steps for an incident wavelength of 633 nm, diameters of 50 nm, and x equal to 0.62.

Step Size | n_{o} | k_{o} | n_{i} | k_{i} | E_{n} (%) | E_{k} (%) | Time (s) |
---|---|---|---|---|---|---|---|

0.0005 | 0.34677 | 3.96138 | 0.34684 | 3.96167 | 0.021 | 0.007 | 632.37 |

0.001 | 0.34677 | 3.96138 | 0.34831 | 3.96601 | 0.444 | 0.117 | 50.81 |

0.005 | 0.34677 | 3.96138 | 0.34927 | 3.96617 | 0.720 | 0.121 | 1.40 |

0.01 | 0.34677 | 3.96138 | 0.35052 | 3.96992 | 1.079 | 0.216 | 0.81 |

0.05 | 0.34677 | 3.96138 | 0.33770 | 3.93061 | 2.616 | 0.777 | 0.73 |

0.1 | 0.34677 | 3.96138 | 0.35882 | 3.99743 | 3.473 | 0.910 | 0.66 |

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**MDPI and ACS Style**

Cheng, L.; Tuersun, P.; Ma, D.; Wumaier, D.; Li, Y.
Inversion of the Complex Refractive Index of Au-Ag Alloy Nanospheres Based on the Contour Intersection Method. *Materials* **2023**, *16*, 3291.
https://doi.org/10.3390/ma16093291

**AMA Style**

Cheng L, Tuersun P, Ma D, Wumaier D, Li Y.
Inversion of the Complex Refractive Index of Au-Ag Alloy Nanospheres Based on the Contour Intersection Method. *Materials*. 2023; 16(9):3291.
https://doi.org/10.3390/ma16093291

**Chicago/Turabian Style**

Cheng, Long, Paerhatijiang Tuersun, Dengpan Ma, Dilishati Wumaier, and Yixuan Li.
2023. "Inversion of the Complex Refractive Index of Au-Ag Alloy Nanospheres Based on the Contour Intersection Method" *Materials* 16, no. 9: 3291.
https://doi.org/10.3390/ma16093291