Correction: Mori et al. Reproduction of Visible Absorbance Spectra of Highly Scattering Suspensions within an Integrating Sphere by Monte Carlo Simulation. Photonics 2023, 10, 754

Error in Figure 3

In the original publication [1], there was a mistake in Figure 3. There was an error in the y-axis values of the absorption coefficient µ_a. To correctly calculate µ_a from the absorbance per 1 cm of A₁, the formula µ_a = A₁ ln(10) should be used. However, the y-axis values in these graphs were mistakenly calculated as A₁/ln(10). This error occurred during the creation of Figure 3. While µ_a was used as an input parameter for the Monte Carlo simulations, the program internally used A₁ and calculated µ_a as µ_a = A₁ ln(10). Therefore, the incorrect µ_a was used in the simulations. The corrected Figure 3 appears below.

Error in Table 1

In the original publication, there was a mistake in Table 1 as published. In Table 1, we incorrectly used the symbols µ_s and µ_a to represent scattering efficiency and absorption efficiency, respectively. These symbols were already used in the text to denote the scattering coefficient and absorption coefficient, which could lead to confusion. We have corrected the symbol for scattering efficiency to Q_s and for absorption efficiency to Q_a. Additionally, we mistakenly labeled “Anisotropy Coefficient” as “Anisotropy Efficiency”. The corrected

Table 1. Optical properties of polystyrene microspheres (diameter: 500 nm) irradiated with 591 nm light are calculated by a numerical algorithm based on Mie scattering theory. The scattering efficiency, absorption efficiency, and anisotropy coefficient of polystyrene microspheres were calculated by a numerical calculation program based on Mie scattering theory using the wavelength of irradiated light, particle diameter and complex refractive index of polystyrene microspheres as input parameters.

Input Parameters				Output Parameters
Wavelength [nm]	Diameter [nm]	Refractive Index		Scattering Efficiency ${\mathit{Q}}_{\mathit{s}}$	Absorption Efficiency ${\mathit{Q}}_{\mathit{a}}$	Anisotropy Coefficient $\mathit{g}$
		Real	Imaginary
591	500	1.585	0.000	0.8346288	0.0000000	0.8394426

appears below. Furthermore, the following sentence regarding Q_s and Q_a was added at the end of Section 2.3:

“The scattering coefficient µs of polystyrene microspheres was calculated based on the scattering efficiency Q_s determined using Mie scattering theory. The absorption of polystyrene microspheres was neglected because the absorption efficiency Q_a = 0.0.”

Error in Equation (1)

In the original publication, there was a mistake in Equation (1). The numerator of the Henyey–Greenstein function was missing the exponent 3/2. The corrected Equation (1) appears below.

$$p_{sca}={\displaystyle \frac{1-g^2}{2{\left(1+g^2-2g\cos \theta \right)}^{\frac{3}{2}}}}$$

(1)

Appendix Section

In the original publication, an error was found in the Monte Carlo simulation code, and the code has been subsequently corrected. The results of the simulations performed under more appropriate conditions using the revised code have been added as Appendix B, as shown below:

Appendix B

The simulation presented in the published study was found to have the following issue: In this simulation, the scattering angle $\theta$ and the azimuthal angle $\varphi$ of the scattering direction were determined based on random numbers. Since $\theta$ and $\varphi$ are independent, separate random numbers should be used for their determination. However, the simulation inadvertently employed the same random number for both $\theta$ and $\varphi$. A corrected simulation, in which independent random numbers were appropriately assigned, was performed to re-estimate the absorbance of Sample M. The results revealed that the simulated absorbance values after correction were higher than those obtained from the original simulation. Nonetheless, the qualitative trend was preserved, whereby the absorbance of Sample M, which contained dispersed scatterers, was smaller than that of the non-scattering sample.

In response, we reviewed and revised the calculation methods for the input parameters of the simulation, namely the absorption coefficient ${\mu }_a$, the scattering coefficient ${\mu }_s$, and the anisotropy coefficient $g$, as well as the distribution of the incident light specified in the simulation. These adjustments enabled the corrected simulation (the revised Monte Carlo simulation, hereafter referred to as revised MC) to reproduce the experimental results, thereby addressing the aforementioned issue. The primary modifications are given as follows:

(1) The calculation method for the absorption coefficient ${\mathit{\mu }}_{\mathit{a}}$ derived from the absorbance measurement results: In this paper, the absorption coefficient ${\mu }_a$ of Trypan Blue was obtained from the linear relationship between the absorbance $A$ and concentration $C$ measured using the configuration shown in Figure 1a. Specifically, the relationship $A=tC+u$ (where $t$ and $u$ are positive real constants) was used to calculate ${\mu }_a=\ln \left(10\right)(tC+u)$. However, the intercept $u$ reflects losses due to the reflection during absorption measurements. Therefore, it is more appropriate to evaluate $A$ and $C$ as proportional (such that $A\to 0$ as $C\to 0$) by removing the influence of $u$. In the revised MC, the corrected relationship $A=tC$ and ${\mu }_a=\ln \left(10\right)tC$, which excludes the effect of reflection losses, was adopted as the true value.

(2) The parameter settings used to calculate the scattering coefficient ${\mathit{\mu }}_{\mathit{s}}$ and anisotropy coefficient $\mathit{g}$ of polystyrene spheres (PSs): In this paper, the scattering coefficient ${\mu }_s$ and anisotropy coefficient $g$ of the polystyrene spheres (PSs), which serve as scatterers, were determined through calculations based on Mie scattering theory using the refractive index and diameter of the PSs listed in Table 1. In contrast, for the revised MC, the real part of the refractive index of the PSs used to calculate ${\mu }_s$ and $g$ was computed using the first term of the Sellmeier equation [39,48] to account for its wavelength dependence. Furthermore, although the nominal diameter of the PSs provided by the manufacturer was 500 nm, the mean diameter of the lot used was 510 nm, which was considered more appropriate. Therefore, the diameter of the PSs used to calculate ${\mu }_s$ and $g$ in the revised MC was set to 510 nm.

(3) Incident beam distribution: In the simulation described in this paper, the incident beam distribution was assumed to follow a two-dimensional Gaussian distribution. In contrast, the revised MC adopted a flat-top distribution, which more accurately reproduced the experimental results. The flat-top distribution refers to a uniform distribution within a specified rectangular range, with the distribution dropping to zero outside this range.

Figure A2 presents a reproduction of the absorbance measurement results of the aqueous Trypan Blue solution and Sample M (Figure 5) using the revised MC.

Figure A2. The absorbance of (a) the non-scattering Trypan Blue solution and (b) scattering Sample M, measured using the setup shown in Figure 2, as reproduced by the revised MC. This figure replaces the simulation results presented in Figure 6 with those from the revised MC. As shown in Figure 6, the left panels display the absorption spectra (solid lines represent the experimental results, and cross marks represent the revised MC results), while the right panels show the relationship between the concentration and absorbance at the absorption peak wavelength of 591 nm.

Figure A2. The absorbance of (a) the non-scattering Trypan Blue solution and (b) scattering Sample M, measured using the setup shown in Figure 2, as reproduced by the revised MC. This figure replaces the simulation results presented in Figure 6 with those from the revised MC. As shown in Figure 6, the left panels display the absorption spectra (solid lines represent the experimental results, and cross marks represent the revised MC results), while the right panels show the relationship between the concentration and absorbance at the absorption peak wavelength of 591 nm.

References

With this correction, a reference has been added as follows:

48. Gienger, J.; Bär, M.; Neukammer, J. Extinction Spectra of Suspensions of Microspheres: Determination of the Spectral Refractive Index and Particle Size Distribution with Nanometer Accuracy. Appl. Opt. 2018, 57, 344–355.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.