High-Dynamic-Range Absorbance Measurement by Integrating Sphere Spectroscopy with Sample Inside Using a Brewster Cell and Multiple-Pass Model

Abstract

The integrating sphere with sample inside (ISSI) method is useful for absorption spectroscopy of scattering samples, but the measured absorbance ($A_{\mathit{meas}}$) becomes nonlinear with dye concentration (c) because the sample is placed inside the sphere. This study modeled the $A_{\mathit{meas}}-c$ relationship for ISSI using a cylindrical cell (CC) and a Brewster cell (BC) with simple analytical expressions based on the fraction of light not passing through the sample and the effective weights of light passing through it multiple times. Four aqueous dye solutions—Trypan Blue, Brilliant Blue FCF, Tartrazine, and New Coccine—were used as non-scattering samples. For CC, a single-pass model reproduced the measured relationship well for all dyes, and linearity was maintained in the low-absorbance region (up to approximately half of the saturation absorbance, $A_{\max }/2\approx 0.67$ Abs). For BC, the same low-absorbance region (up to approximately $A_{\max }/2\approx 1.21$ Abs) also exhibited practical linearity, but the full relationship including saturation required a multiple-pass model. Model selection based on adjusted RMSE and AICc identified the 3-pass model as the minimum sufficient model for BC. The saturation absorbance $A_{\mathit{max}}$ was on average 1.81 times higher for BC than for CC (corresponding to an approximately 12-fold expansion in linear intensity ratio), and the upper concentration limit of the linear approximation was on average 1.85 times higher. These results demonstrate that BC extends the measurable concentration range while preserving practical low-absorbance linearity. In addition, the wavelength dependence of $A_{\mathit{max}}$ observed at short wavelengths is attributed primarily to the reduced reflectance of the BaSO₄ integrating-sphere wall rather than to the refractive-index dispersion of the quartz cell.

1. Introduction

Ultraviolet–visible (UV-Vis) absorption spectroscopy is a fundamental analytical method for evaluating the electronic structure and composition of substances by using energy transitions associated with light absorption by atoms and molecules [1]. Because it is relatively simple, safe, and nondestructive, is widely employed in fields such as materials science [2,3], biochemistry [4,5,6,7], medicine [8,9,10], and food and agriculture [11,12,13]. However, when the sample is a scattering medium, such as a colloidal dispersion, particle suspension, or cell suspension, the attenuation of transmitted light is affected not only by absorption but also by scattering [14,15,16,17]. Therefore, in conventional transmission measurements, it is difficult to evaluate the absorption component accurately, and a measurement system capable of appropriately separating and collecting transmitted and scattered light is required.

For absorption measurements of such scattering samples, methods using an integrating sphere (IS) are widely used [18,19]. The inner wall of the IS is typically made of highly diffuse reflective materials such as barium sulfate (BaSO4), diffuse gold, and polytetrafluoroethylene (PTFE) [20,21]. In a conventional configuration (hereafter, Setup-A), the sample is placed at the entrance port of the IS, allowing transmitted and forward-scattered light to be collected [22,23]. However, this configuration cannot sufficiently capture backward-scattered light, and the measurement accuracy may decrease for samples for which the contribution of backward scattering is not negligible [24]. To address this problem, methods such as the double-integrating-sphere technique [25,26] and the integrating cavity absorption meter have also been proposed [27,28], but these generally require bulky optics and more complex instrumentation. Therefore, there remains value in developing a simple method that can collect scattered light over the full solid angle while using only a single IS.

To overcome this limitation, the IS with sample inside (ISSI) method has been proposed [6,29,30,31,32,33]. In ISSI, an optical cell containing the sample is placed inside the IS, so that scattered light from the sample can be collected by the inner wall of the sphere over the full solid angle. Figure 1b–d show schematic diagrams of the ISSI configurations treated in this study, namely, the ISSI configuration using a cylindrical cell (Setup B-Cylinder) and the ISSI configuration using a Brewster cell (Setup B-Brewster).

However, in ISSI, because light repeatedly reflects from the inner wall of the IS, the detected signal includes components that pass through the sample multiple times, as well as stray-light components that do not pass through the sample even once, such as light reflected at the cell surface. Consequently, the relationship between the measured absorbance, $A_{\mathit{meas}}$, and dye concentration, $c$, deviates from the simple linear relationship predicted by the Beer–Lambert law [31,34,35]. We previously derived an analytical expression for a configuration in which a cylindrical cell (CC) was placed inside the IS. This expression contains three parameters: the fraction of incident light transmitted through the sample, the probability that light reflected from the inner wall of the IS passes through the sample again, and the optical path length. Using this model, we reproduced the nonlinear relationship between the $A_{\mathit{meas}}$ and the absorption coefficient for both non-scattering samples (aqueous nickel nitrate and cobalt nitrate solutions) and scattering samples (polystyrene bead dispersions) [31]. We also developed a Monte Carlo ray-tracing simulation (RTS-ISSI), which successfully estimated the $A_{\mathit{meas}}$ and path-length distribution from the absorption coefficient, scattering coefficient, and anisotropy factor [34]. Furthermore, analysis using RTS-ISSI showed that the path-length distribution broadens owing to the dispersion of scatterers, and we reported its application to high-dynamic-range absorption measurements that exploit this characteristic [35].

However, ISSI has a limitation in that the saturation absorbance, $A_{\mathit{max}}$, becomes lower than the instrumental upper limit of the spectrophotometer. In the conventional configuration using a CC, light reflected at the cell surface is detected as stray light that does not carry absorption information from the sample. As a result, the $A_{\mathit{meas}}$ saturates in the high-absorbance region, and $A_{\mathit{max}}$ has been reported to be limited to approximately 1.4 [31]. To secure a sufficient measurement range for highly absorbing samples, a cell design is required that suppresses the stray-light component that does not pass through the sample and extends the upper measurement limit while retaining the advantages of ISSI.

As a solution to this problem, the present study focuses on Brewster’s law. At the Brewster angle, reflection of p-polarized light ideally becomes zero, and polarization selectivity can be achieved while suppressing reflection losses at an uncoated interface [36]. This property has been used to reduce losses in laser resonators and high-finesse optical cavities, and broadband resonator designs using Brewster-angle prism retroreflectors have been reported for spectroscopic applications [37,38,39]. More recently, resonant power enhancement and optical-feedback locking using Brewster-window cavities have also been applied to high-sensitivity mid-infrared spectroscopy, including intracavity quartz-enhanced photoacoustic spectroscopy (QEPAS) [40,41]. These studies indicate that reducing interfacial reflection remains an effective design principle.

In this study, we applied this principle by introducing a custom-designed Brewster cell (BC), in which p-polarized light enters the cell surface at the Brewster angle. By using the BC, reflected light at the cell surface, i.e., the stray-light component that does not pass through the sample, is expected to be reduced. As a result, the $A_{\mathit{max}}$ in ISSI should increase, leading to an expanded dynamic range in the high-absorbance region. At the same time, because the BC uses an oblique-incidence optical geometry, the path-length distribution and the contribution of multiple-pass components may differ from those of the CC. Therefore, the relationship between the $A_{\mathit{meas}}$ and concentration needs to be examined within a common analytical framework.

Accordingly, the aim of this study was to describe the relationship between $A_{\mathit{meas}}$ and dye concentration c measured by the ISSI method using CC and BC with a simple analytical model. As non-scattering samples, aqueous solutions of four water-soluble dyes representing different wavelength regions in the ultraviolet–visible range were used. Because all dyes are fully dissolved molecular species in water, Rayleigh and Mie scattering are negligible at the concentrations employed. In particular, Trypan Blue has been used as a non-scattering sample in our previous ISSI study [34], confirming that dissolved dye molecules do not contribute to scattering. Trypan Blue, which is used for biological staining [42], and the food dyes Brilliant Blue FCF, Tartrazine, and New Coccine were selected [43,44], and the $A_{\mathit{meas}}-c$ relationships were obtained at multiple peak wavelengths. As analytical models, a single-pass model incorporating the fraction of light that does not pass through the sample, $P_{\mathrm{no}}$, and a multiple-pass model considering the effective weights $a_m$ of light passing through the sample up to $\mathrm{n}$ times were applied, and their suitability for CC and BC was examined. For CC, the validity of the single-pass model and the feasibility of linear approximation in the low-absorbance region were evaluated; for BC, the optimal model was selected from the 1–4-pass models based on adjusted RMSE and the corrected Akaike information criterion (AICc) [45,46,47]. Furthermore, the $A_{\mathit{max}}$ of the two cells was compared to quantitatively evaluate the improvement in dynamic range achieved by introducing the BC. Whereas the single-pass model is obtained as a simplification of the analytical expression derived in our previous work [31], the multiple-pass model introduced in the present study is a new generalization that, together with the Brewster-cell configuration, enables ISSI measurements in optical geometries where a uniform single-pass optical path can no longer be assumed. In this study, ‘high dynamic range’ refers to extending the measurable absorbance range by increasing the saturation absorbance $A_{\mathit{max}}$ through the use of a Brewster cell. Because absorbance is a logarithmic quantity, this increase from approximately 1.34 (CC) to 2.42 (BC) in $A_{\mathit{max}}$ corresponds to an approximately 12-fold expansion of the linear intensity ratio (~22 to ~263), i.e., about one order of magnitude.

2. Materials and Methods

2.1. Sample Preparation

As absorbers, four dyes were used: Trypan Blue (TB, Kanto Chemical Co., Inc., Tokyo, Japan), Brilliant Blue FCF (FCF, FUJIFILM Wako Pure Chemical Corporation, Osaka, Japan), Tartrazine (TAR, FUJIFILM Wako Pure Chemical Corporation, Osaka, Japan), and New Coccine (NCC, FUJIFILM Wako Pure Chemical Corporation, Osaka, Japan). Pure water was used as the solvent. Stock solutions were prepared by dissolving each dye in pure water at stock concentrations of 0.9 mM (=0.9 mol/m³) for TB, 1.0 mM for FCF, 2.5 mM for TAR, and 3.5 mM for NCC. The concentration series used for the measurements were prepared by volumetric dilution of these stock solutions. The dilution ratios ranged from 0.5% to 90% (v/v relative to the stock solution), and the absolute dye concentration $c$ was calculated as the product of the dilution ratio and the stock concentration.

These dyes have absorption peaks in different wavelength regions in the ultraviolet–visible range and were used as non-scattering samples for which scattering was negligible over the measured wavelength range.

2.2. Measurement Setup

2.2.1. Spectrometer and Integrating Sphere

Absorption spectroscopy measurements were performed using the integrating sphere (diameter: 60 mm; inner wall coated with BaSO₄) attached to a commercial UV–Vis–NIR spectrophotometer (SolidSpec-3700DUV, Shimadzu Corporation, Kyoto, Japan). This instrument employs a double-beam optical configuration with a deuterium lamp and a halogen lamp as light sources, and a double monochromator with diffraction gratings for wavelength selection. The spectral bandwidth was set to 8 nm. The instrument has two apertures on the side of the integrating sphere: one for the sample beam and one for the reference beam. Opposite the sample-beam entrance window, a window for diffuse-reflectance measurements is provided and closed with a white plate. The detector for the UV–Vis region is a photomultiplier tube (PMT) located at the bottom of the integrating sphere (Figure 1).

2.2.2. Setup B-Cylinder: ISSI with Cylindrical Cell (CC)

As the first configuration of the ISSI method, Setup B-Cylinder was used. In this setup, a custom cylindrical quartz-glass cell (CC) designed for placement inside the integrating sphere was used (inner diameter: 8 mm; wall thickness of the cylindrical section: 0.75 mm; sample volume: 600 µL). The CC was filled with the sample and inserted into the integrating sphere through the window for diffuse-reflectance measurements. The window was closed behind the CC with a white plate made of the same material as the integrating-sphere inner wall (Figure 1b,c). The incident beam size was 5 mm in width and 13.5 mm in height. Because the beam width was smaller than the inner diameter of the CC (8 mm), the beam cross-section was entirely contained within the cell. A schematic view of the CC is shown in Figure 2a, and its dimensions are shown in Figure 2b.

2.2.3. Setup B-Brewster: ISSI with Brewster Cell (BC)

As the second ISSI configuration, Setup B-Brewster, which uses Brewster’s law to reduce reflection at the cell surface, was employed (Figure 1d). The CC was replaced with a custom quartz-glass optical cell (Brewster cell, BC) designed so that the incident angle equals the ${\theta }_{\mathrm{B}}$ (wall thickness: 1.5 mm; sample volume: 400 µL). The refractive index of quartz glass is n = 1.46 at a wavelength of 546 nm [48], and the Brewster angle for light incident from air (refractive index: 1.0) onto quartz glass is

$${\theta }_{\mathrm{B}}=\arctan \left({\displaystyle \frac{n_2}{n_1}}\right)=\arctan \left({\displaystyle \frac{1.46}{1.0}}\right)\approx {56}^{\circ }$$

The BC was designed so that the incident light entered the cell surface at this angle. The glass wall thickness of the BC was 1.5 mm, which is twice that of the CC (0.75 mm). A schematic view of the BC is shown in Figure 2c, and its dimensions are shown in Figure 2d.

In Setup B-Brewster, the incident light was converted to p-polarized light by a polarizer. According to Brewster’s law, when p-polarized light is incident at the Brewster angle, reflection at the cell surface becomes nearly zero. This reduces the stray-light component that does not pass through the sample and is expected to increase the upper measurable absorbance limit.

2.2.4. Measurement Conditions

The absorbance spectrum $A_{\mathrm{meas}}\left(\mathrm{\lambda }\right)$ was measured over wavelength ranges that included the absorption band of each dye. For the baseline, a cell filled with pure water was used, and the zero-dye-concentration condition served as the reference. Since the absorption of pure water in the UV–Vis range (230–700 nm) is negligibly small, and the baseline measurement uses a water-filled cell, any contribution from the solvent is cancelled by the baseline subtraction.

2.3. Analytical Models for the A_meas–c Relationship

In the ISSI method, because the sample is placed inside the integrating sphere, optical paths different from those in conventional transmission measurements are generated. The detected light includes both multiple-pass components, in which light repeatedly reflected by the inner wall of the integrating sphere passes through the sample multiple times, and components that do not pass through the sample even once, such as light reflected at the cell surface. Consequently, the absorbance measured by ISSI, $A_{\mathit{meas}}$, shows a nonlinear relationship with the dye concentration $c$ [31]. This section describes the models used to represent the relationship between $A_{\mathit{meas}}$ and $c$.

In this study, the product $\epsilon L$ in the Beer–Lambert law ($\epsilon$: molar absorptivity; $L$: path length) was not separated; instead, it was treated as a single effective absorption parameter, $k = \epsilon L$. Accordingly, the true absorbance obeying the Beer law is expressed as $A_{\mathrm{B}}=kc$. In the following models, the fitting parameter $a_m$ is treated as the effective weight of light that passes through the sample m times.

2.3.1. Single-Pass Model and Linear Approximation

As a model for the absorbance measured by ISSI, we introduce a simple model that accounts for the effect that part of the incident light is detected without passing through the sample. The single-pass model used in the present study is derived from the analytical expression established in our previous work [31]. In Ref. [31], the detected intensity in the ISSI configuration using a cylindrical cell was modeled using two parameters: p, the probability that incident light passes through the sample cell, and x, the probability that diffusely reflected light from the integrating-sphere wall re-enters the cell (see Equation (6) of Ref. [31]). In the limit x → 0, i.e., when the contribution of re-entering light is neglected, the expression simplifies to $A_{\mathrm{meas}}=-\log \left[\left(1-p\right)+p·{10}^{-kc}\right]$. By defining $P_{\mathrm{no}}=1-p$ as the fraction of light that does not pass through the sample at all, this becomes Equation (1) below. Note that the symbol p in Ref. [31] denotes the probability that light passes through the sample, whereas ${\mathrm{P}}_{\mathrm{no}}$ in the present study denotes the complementary fraction; the two are related by $P_{\mathrm{no}}=1-p$. Let $P_{\mathrm{no}}$ denote the fraction of the incident light entering the integrating sphere that reaches the detector without passing through the sample cell. The remaining $1 - P_{\mathrm{no}}$ is assumed to pass through the sample cell at least once. For the true absorbance $A_{\mathrm{B}}=kc$, the light transmitted through the sample is attenuated by a factor of ${10}^{-A_{\mathrm{B}}}$. If scattering at the integrating-sphere inner wall is sufficiently diffusive, the difference in detection efficiency between transmitted and non-transmitted light can be neglected. Because the spectrophotometer normalizes the intensity ratio to $1$ for the blank sample ($A_{\mathrm{B}}=0$), all instrumental factors cancel. Therefore, the absorbance measured by ISSI is given by

$$A_{\mathrm{meas}}\left(A_{\mathrm{B}}\right)=-{\log }_{10}\left[P_{\mathrm{no}}+\left(1-P_{\mathrm{no}}\right){10}^{-A_{\mathrm{B}}}\right]$$

(1)

This is referred to as the single-pass model.

When the sample absorption is sufficiently large ($c \to \infty$), the second term in Equation (1) becomes negligible, yielding the saturation absorbance

$$A_{\max }=-{\log }_{10}{P}_{\mathrm{no}}$$

(2)

where $P_{\mathrm{no}}$ is the fraction of light that does not pass through the sample even once; smaller $P_{\mathrm{no}}$ gives a larger $A_{\mathit{max}}$.

Next, we consider linearity in the low-absorbance region. A Taylor expansion of Equation (1) around $A_{\mathrm{B}} = 0$ gives a first-order coefficient of 1 − $P_{\mathrm{no}}$. Therefore, in the low-absorbance region, $A_{\mathit{meas}}$ is expected to behave approximately linearly with c. Although this linearity is derived within the framework of the single-pass model, the multiple-pass model [Equation (3)] also leads to approximate linearity in the low-absorbance region ($A_B \approx 0$), because each transmitted component, ${10}^{-m{A}_{\mathrm{B}}}$ (where $m$ is the number of passes through the sample), is close to $1$. In other words, the linearity in the low-absorbance region is not contingent on the validity of the single-pass model and is an approximation that also holds in more general models.

The linearity in the low-absorbance region for each cell was evaluated by linear regression of the measured data points at or below approximately half of the saturation absorbance, $A_{\max }$. The criterion $A_{\max }/2$ does not represent a rigorous theoretical boundary; rather, it was adopted as a practical criterion for comparing the agreement between the linear approximation and the measured data. From the agreement between the fitted straight line and the measured data, the practical validity of the linear approximation in the low-absorbance region was assessed for both CC and BC.

In evaluating the linearity in the low-absorbance region, the baseline was defined using a cell filled with pure water, and therefore the zero-concentration condition was assumed to satisfy $A_{\mathrm{meas}}=0$ at $c=0$. Accordingly, the approximate straight lines shown in the insets of the concentration–absorbance plots in Section 3 and Appendix B were obtained by linear fitting using the measured data points at or below $A_{\max }/2$ together with the origin as an additional data point. In contrast, only the measured data points were used for fitting the multiple-pass model and for calculating the adjusted RMSE and AICc. For the evaluation of the practical upper concentration limit for linear approximation shown in (presented in Section 3.7), the slope of this approximate straight line was used, and the corresponding zero-intercept line, $A=kc$, was adopted to calculate the concentration at which the line reaches $A_{\max }/2$.

2.3.2. Multiple-Pass Model for the Brewster Cell

For BC, because of its cell geometry and oblique-incidence optical configuration, the path-length distribution of light that repeatedly reflects from the integrating-sphere inner wall and passes through the sample multiple times differs from that for CC. Therefore, the single-pass model [Equation (1)] may not sufficiently reproduce the measurements. We therefore introduced a multiple-pass model that considers the contribution of light passing through the sample up to $n$ times.

In the n-pass model, the detected light is decomposed into a component that does not pass through the sample and components that pass through the sample exactly $m$ times ($m = 1, 2, \dots , n$). The measured absorbance is expressed as

$$A_{\mathrm{meas}}\left(A_{\mathrm{B}}\right)=-{\log }_{10}\left[P_{\mathrm{no}}+\left(1-P_{\mathrm{no}}\right)\sum _{m=1}^n{a}_m{10}^{-m{A}_{\mathrm{B}}}\right]$$

(3)

where $a_{\mathrm{m}}$ is the effective weight of light that passes through the sample $m$ times. From energy conservation,

$$\sum _{m=1}^n{a}_m=1$$

(4)

Substituting $A_{\mathrm{B}}=kc$ gives the concentration dependence $A_{\mathrm{meas}}\left(\mathrm{c}\right)$. For $n = 1$, $a_1=1$, and the model reduces to the single-pass model of Equation (1). The explicit forms for $n =$1–4, together with the corresponding constraints and independent parameter counts, are given in Appendix A.

Here, $k = \epsilon L$ is the effective absorption parameter at the peak wavelength of each dye and represents the product of the molar absorptivity $\epsilon$ and the effective path length $L$. The $a_m$ values in each model are regarded as effective weights that empirically describe the behavior of the overall optical system without relying on a detailed geometric estimation of the path-length distribution.

2.3.3. Fitting Procedure

For the relationship between $A_{\mathit{meas}}$ and dye concentration $c$ at each absorption-peak wavelength of each dye measured with Setup B-Cylinder and Setup B-Brewster, the model parameters were determined by nonlinear least squares. The parameters were optimized using the Solver function in Microsoft Excel for Mac (Version 16.106, Microsoft Corporation, Redmond, WA, USA) so as to minimize the sum of squared errors (SSE) between the measured absorbance $A_{\exp }$ and the model absorbance $A_{\mathrm{calc}}$,

$$\mathrm{SSE}=\sum _i{\left(A_{\exp ,i}-A_{\mathrm{calc},i}\right)}^2$$

(5)

Because the model equations are nonlinear in the parameters, the optimal solution was sought by iterative calculations with Solver.

For CC, the single-pass model [Equation (1)] was fitted using $P_{\mathrm{no}}$ and $k$ as free parameters. The saturation absorbance $A_{\max }$ was then calculated from the obtained $P_{\mathrm{no}}$ using Equation (2).

For BC, the multiple-pass models from 1-pass to 4-pass [Equation (3), $n = 1, 2, 3$, and 4] were fitted separately. The free parameters in each model were $P_{\mathrm{no}}$, $a_m$ ($m = 1, \dots , n$), and $k$, under the constraint of Equation (4).

2.3.4. Evaluation of Fitting Performance

As an index of fitting accuracy, the adjusted root mean square error (adjusted RMSE) based on the SSE was used. Because the number of free parameters differs among the models, the RMSE was calculated in a degree-of-freedom-corrected form as

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{\mathrm{SSE}}{N-p}}}$$

(6)

where N is the number of data points and $p$ is the number of free parameters. Because the RMSE is expressed in absorbance units (Abs), it provides an intuitive measure of the average discrepancy between the model and the measured data.

The degree-of-freedom-corrected RMSE obtained for each sample was averaged for each cell type and number of passes and compared as the mean adjusted RMSE (±standard deviation, SD). For CC, we evaluated whether the RMSE for the 1-pass model was sufficiently small and whether the improvement obtained by introducing higher-order pass models was limited. For BC, the decrease in RMSE with increasing number of passes was quantitatively evaluated to examine whether low-order pass models were insufficient to reproduce the measured data.

2.3.5. Model Selection by AICc

If the RMSE evaluation for BC suggested that low-order pass models were insufficient to reproduce the measurements, an objective criterion was required to determine the optimal number of passes. Therefore, the corrected Akaike information criterion (AICc), which simultaneously accounts for goodness of fit and the risk of overfitting due to an increased number of parameters, was employed. The AICc is defined as

$$\mathrm{AICc}=\mathrm{AIC}+2p+{\displaystyle \frac{2p\left(p+1\right)}{N-p-1}}=N\ln \left({\displaystyle \frac{\mathrm{SSE}}{N}}\right)+2p+{\displaystyle \frac{2p\left(p+1\right)}{N-p-1}}$$

(7)

where AIC is the Akaike information criterion, $N$ is the number of data points, and $p$ is the number of independent fitting parameters after imposing the energy-conservation constraint of Equation (4). A smaller AICc indicates a better balance between goodness of fit and parsimony.

For comparison among models, the difference from the minimum AICc,

$$\mathrm{\Delta }{\mathrm{AICc}}_i={\mathrm{AICc}}_i-\min \left(\mathrm{AICc}\right)$$

(8)

was used. According to general guidelines for interpreting ΔAICc, models with $\mathrm{\Delta }\mathrm{AICc}\le 2$ have support comparable to that of the best model, models with ΔAICc of 4–7 have considerably less support, and models with $\mathrm{\Delta }\mathrm{AICc}>10$ have essentially no support [46,47].

By integrating the RMSE results for BC with model selection based on the AICc, the optimal number of passes $n$ was determined from the results over all dyes and peak wavelengths. The numbers of data points used for the AICc calculation were 21 for TB, 22 for FCF, 22 for TAR, and 23 for NCC.

3. Results

3.1. Absorbance Spectra and Concentration–Absorbance Relationships

Figure 3 and Figure 4 present, respectively, the absorbance spectra $A_{\mathrm{meas}}\left(\lambda \right)$ and the concentration dependence of $A_{\mathrm{meas}}\left(\lambda \right)$ at the absorption-peak wavelength for TB (stock concentration: 0.9 mM). The corresponding pair of plots is given in Figure 5 and Figure 6 for FCF (1.0 mM), in Figure 7 and Figure 8 for TAR (2.5 mM), and in Figure 9 and Figure 10 for NCC (3.5 mM). All solutions were prepared at dilution ratios of 0.5–90%. In the spectral figures, each curve is labeled by its dilution ratio relative to the stock solution, whereas in the concentration–absorbance plots, the horizontal axis shows the absolute dye concentration c in mM, which is the independent variable of the analytical model. The peak wavelengths used for the fitting analysis were determined from the spectra at low concentrations, where the spectral shape is not distorted by the saturation effect. At higher concentrations, the apparent peak position may shift slightly owing to the nonlinear $A_{\mathrm{meas}}-c$ relationship, but the same fixed wavelength was used across all concentrations. The fitting analysis was performed at the absorption-peak wavelength(s) specific to each dye—591 nm for TB, 408 nm and 630 nm for FCF, 258 nm and 428 nm for TAR, and 331 nm and 507 nm for NCC—so that the model could be tested across a wide range of the UV–Vis spectrum. For all four dyes, the saturation absorbance was approximately 1.4 for CC, whereas it reached approximately 2.5 for BC. For BC, increased noise was observed below 250 nm in high-concentration TAR samples; additional transmission measurements investigating this behavior are presented in Appendix C.

The concentration–absorbance plots show the concentration dependence of $A_{\mathrm{meas}}$ at each absorption-peak wavelength together with the fitting results of the 1- to 4-pass models. For CC, the differences among the models were small for all dyes and peak wavelengths, and the single-pass model reproduced the experimental data well. In contrast, for BC, the 1-pass and 2-pass models showed clear deviations from the experimental data, whereas the 3-pass model reproduced the data well from the low-concentration region to saturation. In all insets, the measured data in the low-absorbance region were generally consistent with the approximate straight lines obtained by linear fitting using the measured data points at or below $A_{\max }/2$ together with the origin as an additional data point (cf. Section 2.3.1).

For FCF at 630 nm [Figure 6c], only one measured point was available at or below $A_{\max }/2$. Supplementary measurements using a 0.5 mM stock solution are therefore presented in Appendix B, confirming that the low-absorbance linearity holds even with additional data points [Figure A2c].

3.2. Brilliant Blue FCF (FCF)

The results for FCF are presented below: Figure 5 shows the absorbance spectra, and Figure 6 shows the concentration dependence of $A_{\mathrm{meas}}\left(\lambda \right)$ at the absorption-peak wavelengths.

Figure 5. Absorbance spectra $A_{\mathrm{meas}}\left(\lambda \right)$ of FCF aqueous solutions. Each curve corresponds to a dilution ratio (0.5–90%, based on a stock concentration of 1.0 mM). (a) Measured with CC. (b) Measured with BC. The measured wavelength range was 380–700 nm. The legend entries are arranged in descending order of concentration (from top to bottom), corresponding to the vertical ordering of the spectra. The arrows indicate the direction of increasing dye concentration.

Figure 5. Absorbance spectra $A_{\mathrm{meas}}\left(\lambda \right)$ of FCF aqueous solutions. Each curve corresponds to a dilution ratio (0.5–90%, based on a stock concentration of 1.0 mM). (a) Measured with CC. (b) Measured with BC. The measured wavelength range was 380–700 nm. The legend entries are arranged in descending order of concentration (from top to bottom), corresponding to the vertical ordering of the spectra. The arrows indicate the direction of increasing dye concentration.

Figure 6. Concentration dependence of $A_{\mathrm{meas}}$ for FCF. (a) CC, 408 nm; (b) BC, 408 nm; (c) CC, 630 nm; (d) BC, 630 nm. “Exp.” denotes the measured values, and the colored curves represent the fitting results of the respective pass models. The insets are enlarged views of the low-concentration region. The purple dots indicate the measured data points at or below approximately $A_{\max }/2$ together with the origin, and the purple dashed lines indicate the approximate straight lines obtained by linear fitting using these points.

Figure 6. Concentration dependence of $A_{\mathrm{meas}}$ for FCF. (a) CC, 408 nm; (b) BC, 408 nm; (c) CC, 630 nm; (d) BC, 630 nm. “Exp.” denotes the measured values, and the colored curves represent the fitting results of the respective pass models. The insets are enlarged views of the low-concentration region. The purple dots indicate the measured data points at or below approximately $A_{\max }/2$ together with the origin, and the purple dashed lines indicate the approximate straight lines obtained by linear fitting using these points.

3.3. Tartrazine (TAR)

The results for TAR are presented below: Figure 7 shows the absorbance spectra, and Figure 8 shows the concentration dependence of $A_{\mathrm{meas}}\left(\lambda \right)$ at the absorption-peak wavelengths.

Figure 7. Absorbance spectra $A_{\mathrm{meas}}\left(\lambda \right)$ of TAR aqueous solutions. Each curve corresponds to a dilution ratio (0.5–90%, based on a stock concentration of 2.5 mM). (a) Measured with CC. (b) Measured with BC. The measured wavelength range was 230–550 nm. In the BC measurement, increased noise was observed below approximately 250 nm for high-concentration samples. The legend entries are arranged in descending order of concentration (from top to bottom), corresponding to the vertical ordering of the spectra. The arrows indicate the direction of increasing dye concentration.

Figure 7. Absorbance spectra $A_{\mathrm{meas}}\left(\lambda \right)$ of TAR aqueous solutions. Each curve corresponds to a dilution ratio (0.5–90%, based on a stock concentration of 2.5 mM). (a) Measured with CC. (b) Measured with BC. The measured wavelength range was 230–550 nm. In the BC measurement, increased noise was observed below approximately 250 nm for high-concentration samples. The legend entries are arranged in descending order of concentration (from top to bottom), corresponding to the vertical ordering of the spectra. The arrows indicate the direction of increasing dye concentration.

Figure 8. Concentration dependence of $A_{\mathrm{meas}}$ for TAR. (a) CC, 258 nm; (b) BC, 258 nm; (c) CC, 428 nm; (d) BC, 428 nm. “Exp.” denotes the measured values, and the colored curves represent the fitting results of the respective pass models. The insets are enlarged views of the low-concentration region. The purple dots indicate the measured data points at or below approximately $A_{\max }/2$ together with the origin, and the purple dashed lines indicate the approximate straight lines obtained by linear fitting using these points.

Figure 8. Concentration dependence of $A_{\mathrm{meas}}$ for TAR. (a) CC, 258 nm; (b) BC, 258 nm; (c) CC, 428 nm; (d) BC, 428 nm. “Exp.” denotes the measured values, and the colored curves represent the fitting results of the respective pass models. The insets are enlarged views of the low-concentration region. The purple dots indicate the measured data points at or below approximately $A_{\max }/2$ together with the origin, and the purple dashed lines indicate the approximate straight lines obtained by linear fitting using these points.

3.4. New Coccine (NCC)

The results for NCC are presented below: Figure 9 shows the absorbance spectra, and Figure 10 shows the concentration dependence of $A_{\mathrm{meas}}\left(\lambda \right)$ at the absorption-peak wavelengths.

Figure 9. Absorbance spectra $A_{\mathrm{meas}}\left(\lambda \right)$ of NCC aqueous solutions. Each curve corresponds to a dilution ratio (0.5–90%, based on a stock concentration of 3.5 mM). (a) Measured with CC. (b) Measured with BC. The measured wavelength range was 300–650 nm. The legend entries are arranged in descending order of concentration (from top to bottom), corresponding to the vertical ordering of the spectra. The arrows indicate the direction of increasing dye concentration.

Figure 9. Absorbance spectra $A_{\mathrm{meas}}\left(\lambda \right)$ of NCC aqueous solutions. Each curve corresponds to a dilution ratio (0.5–90%, based on a stock concentration of 3.5 mM). (a) Measured with CC. (b) Measured with BC. The measured wavelength range was 300–650 nm. The legend entries are arranged in descending order of concentration (from top to bottom), corresponding to the vertical ordering of the spectra. The arrows indicate the direction of increasing dye concentration.

Figure 10. Concentration dependence of $A_{\mathrm{meas}}$ for NCC. (a) CC, 331 nm; (b) BC, 331 nm; (c) CC, 507 nm; (d) BC, 507 nm. “Exp.” denotes the measured values, and the colored curves represent the fitting results of the respective pass models. The insets are enlarged views of the low-concentration region. The purple dots indicate the measured data points at or below approximately $A_{\max }/2$ together with the origin, and the purple dashed lines indicate the approximate straight lines obtained by linear fitting using these points.

Figure 10. Concentration dependence of $A_{\mathrm{meas}}$ for NCC. (a) CC, 331 nm; (b) BC, 331 nm; (c) CC, 507 nm; (d) BC, 507 nm. “Exp.” denotes the measured values, and the colored curves represent the fitting results of the respective pass models. The insets are enlarged views of the low-concentration region. The purple dots indicate the measured data points at or below approximately $A_{\max }/2$ together with the origin, and the purple dashed lines indicate the approximate straight lines obtained by linear fitting using these points.

3.5. Fitting Performance: Adjusted RMSE

Table 1. Mean adjusted RMSE (± SD) for the pass models (1–4 passes) for CC and BC.

Cell	Model	Mean adj. RMSE	SD
CC	1	0.0203	0.0020
CC	2	0.0124	0.0043
CC	3	0.0123	0.0046
CC	4	0.0126	0.0050
BC	1	0.0929	0.0160
BC	2	0.0506	0.0117
BC	3	0.0275	0.0087
BC	4	0.0270	0.0088

Each value is the mean of the degree-of-freedom-corrected RMSE values calculated over all dyes and all peak wavelengths. CC: cylindrical cell; BC: Brewster cell; RMSE: root mean square error; SD: standard deviation.

summarizes the degree-of-freedom-corrected RMSE values [Equation (6)] obtained for each dye and each peak wavelength after averaging by cell type and number of passes.

For CC, the mean adjusted RMSE of the 1-pass model was 0.0203 ± 0.0020 Abs, which was already sufficiently small even in comparison with the models including two or more passes (0.0124–0.0126 Abs). The differences among the 2-pass, 3-pass, and 4-pass models were small, indicating that the improvement achieved by introducing higher-order pass components was limited.

In contrast, for BC, the mean adjusted RMSE of the 1-pass model was 0.0929 ± 0.0160 Abs, which was markedly larger than that for CC, indicating that the single-pass model could not reproduce the measured data sufficiently. The mean adjusted RMSE decreased substantially to 0.0506 ± 0.0117 Abs for the 2-pass model and to 0.0275 ± 0.0087 Abs for the 3-pass model. The further improvement from the 3-pass model to the 4-pass model was only marginal (0.0270 ± 0.0088 Abs), indicating that the improvement had almost saturated at the 3-pass model.

3.6. Model Selection for the Brewster Cell: ΔAICc

The results in Table 1 showed that the lower-order pass models were insufficient to reproduce the measurements for BC. Therefore, to determine the optimal number of passes objectively, the AICc values [Equation (7)] of the 1-pass to 4-pass models were calculated for each dye and each peak wavelength for BC, and the ΔAICc values [Equation (8)] were compared. Figure 11 shows the ΔAICc values for each dye and each peak wavelength in BC as a function of the number of passes.

For all samples, the 1-pass model showed ΔAICc > 10 and was statistically inappropriate.

For all samples except TAR (258 nm), the 2-pass model also gave ΔAICc > 10. Even for TAR (258 nm), the value was ΔAICc = 2.67, which was inferior to the 3-pass model. The 3-pass model was either the best model or practically equivalent to the best model (ΔAICc ≤ 2) for all samples. The 4-pass model gave the minimum AICc for some samples (TB, FCF 630 nm, and NCC 331 nm), but the differences from the 3-pass model were small (ΔAICc = 0.79–1.38), indicating essentially equivalent support. At both peak wavelengths of TAR (258 nm and 428 nm), as well as at FCF 408 nm and NCC 507 nm, the 3-pass model gave the minimum AICc, whereas the 4-pass model was slightly inferior (ΔAICc = 1.85–3.40).

Based on the above RMSE evaluation and model selection using the AICc, CC was judged to be sufficiently described by the single-pass model, whereas for BC the 3-pass model was regarded as the minimum sufficient model that balances physical plausibility and descriptive accuracy.

3.7. Saturation Absorbance A_max

Table 2. Saturation absorbance $A_{\max }$ obtained from fitting each pass model.

Cell	Pass No.	TB	FCF 408	FCF 630	TAR 258	TAR 428	NCC 331	NCC 507	Mean ${\mathit{A}}_{\mathbf{max}}$	SD
CC	1	1.41	1.33	1.41	1.15	1.37	1.31	1.41	1.34	0.09
CC	2	1.42	1.34	1.41	1.16	1.38	1.32	1.42	1.35	0.09
CC	3	1.42	1.34	1.41	1.16	1.38	1.32	1.42	1.35	0.09
CC	4	1.43	1.35	1.41	1.16	1.38	1.32	1.42	1.35	0.09
BC	1	2.45	2.32	2.50	2.11	2.42	2.28	2.46	2.36	0.14
BC	2	2.49	2.36	2.52	2.13	2.44	2.30	2.49	2.39	0.14
BC	3	2.53	2.39	2.53	2.14	2.47	2.33	2.52	2.42	0.14
BC	4	2.53	2.39	2.54	2.14	2.47	2.33	2.53	2.42	0.14

Each column corresponds to a dye and peak wavelength. Mean $A_{\max }$ and SD denote the mean value and standard deviation over all dyes and all peak wavelengths. TB: Trypan Blue; FCF: Brilliant Blue FCF; TAR: Tartrazine; NCC: New Coccine; CC: cylindrical cell; BC: Brewster cell; SD: standard deviation.

lists the saturation absorbance $A_{\max }$ obtained from fitting each pass model for each dye and each peak wavelength. $A_{\max }$ was calculated from $P_{\mathrm{no}}$ using Equation (2).

For CC, $A_{\max }$ was nearly constant regardless of the number of passes, and the mean value for the single-pass model was approximately 1.34 ± 0.09. The change in $A_{\max }$ from the single-pass model to the other models was only about 0.01, indicating that the choice of pass model had little effect on the estimated saturation absorbance.

In contrast, $A_{\max }$ for BC was markedly larger than that for CC, and the mean value for the 3-pass model was approximately 2.42 ± 0.14. Even for BC, the difference in $A_{\max }$ between the 3-pass and 4-pass models was very small. Replacing CC with BC increased $A_{\max }$ by approximately 1.08, making it about 1.81 times as high as that of CC.

To evaluate the extension of the practically linear concentration range in BC from the viewpoint of concentration, the concentration $c\left(A_{\max }/2\right)$ at which the zero-intercept line $A=kc$, defined from the slope of the approximate straight line in the low-absorbance region, reaches $A_{\max }/2$ was calculated for each dye and each peak wavelength using the saturation absorbance $A_{\max }$. Here, $c\left(A_{\max }/2\right)$ was defined as $c\left(A_{\max }/2\right)=\left(A_{\max }/2\right)/k$.

Table 3. Ratio of the practical upper concentration limit for linear approximation in BC relative to CC.

Sample	Ratio of Upper Concentration Limit (BC/CC)
TB (591 nm)	1.81
FCF (408 nm)	1.78
FCF (630 nm)	1.81
TAR (258 nm)	1.89
TAR (428 nm)	1.81
NCC (331 nm)	1.96
NCC (507 nm)	1.90
Average	1.85

TB: Trypan Blue; FCF: Brilliant Blue FCF; TAR: Tartrazine; NCC: New Coccine; CC: cylindrical cell; BC: Brewster cell.

shows the ratio $c\left(A_{\max }/2\right)$ in BC relative to that in CC, i.e., the BC/CC ratio.

As a result, the practical upper concentration limit for linear approximation in BC was larger than that in CC for all dyes and peak wavelengths, and the arithmetic mean of the BC/CC ratio was 1.85. Therefore, although BC exhibited a more complex optical response than CC, it extended the concentration range that can still be treated as practically linear by approximately 1.85 times.

When the same evaluation was carried out using approximate straight lines with a free intercept, the BC/CC ratio changed very little. This indicates that the zero-intercept approximation adopted in this study was practically valid and that the above conclusion does not depend strongly on the treatment of the intercept.

4. Discussion

4.1. Validity of the Single-Pass Model and Linearity in the Low-Absorbance Region for CC

For CC, the single-pass model [Equation (1)] reproduced the measured results well for all dyes and all peak wavelengths, and the mean degree-of-freedom-corrected RMSE of the 1-pass model was sufficiently small at 0.0203 ± 0.0020 Abs (Table 1). Even when models with two or more passes were introduced, the improvement in RMSE was limited to about 0.008, indicating that the contribution of higher-order pass components was small. These results suggest that, for CC, the assumption of a single optical path length is generally valid, because the incident light passes through the sample with an almost constant path length (cell inner diameter ≈ 8 mm).

From the linear regression results in the insets of Figure 4a, Figure 6a,c, Figure 8a,c, and Figure 10a,c, good linearity between $A_{\mathrm{meas}}$ and the concentration c was confirmed for CC in the concentration range up to $A_{\max }/2$ (corresponding to $A_{\mathrm{meas}}\lesssim 0.67$ Abs). In the ISSI method, light diffusely reflected by the integrating-sphere wall can, strictly speaking, pass through the sample multiple times even for CC. However, as shown in Table 1, the single-pass model already provides sufficient accuracy for CC, indicating that the contribution of multiple-pass components is limited. As a result, the $A_{\mathrm{meas}}-c$ relationship for CC in the low-absorbance region agrees well with the approximate straight line obtained by linear fitting using the data points at or below $A_{\max }/2$ together with the origin as an additional data point, demonstrating that the ISSI method using CC is applicable to quantitative analysis based on a linear calibration curve derived from the conventional Beer–Lambert law.

4.2. Necessity of the Multiple-Pass Model for BC

For BC, in contrast to CC, the single-pass model could not reproduce the measured results sufficiently. The mean degree-of-freedom-corrected RMSE of the 1-pass model was 0.0929 ± 0.0160 Abs, which was approximately 4.6 times larger than that of the 1-pass model for CC (0.0203 Abs) (Table 1). This discrepancy is considered to originate from the optical configuration of BC.

CC has a cylindrical geometry, and the incident beam passes through the sample with an almost uniform path length along the diameter. In contrast, because the incident angle in BC is set to the Brewster angle (≈56°), the optical path inside the cell becomes more complex. The distance traveled by the incident light during a single pass varies with position inside the cell, and short and long path lengths coexist. Furthermore, when light diffusely reflected by the integrating-sphere wall re-enters the BC, the inclined geometry of the BC generates a wide variety of incident angles and path lengths. Therefore, in addition to light that passes through the sample only once, the contributions of light passing through the sample two or three times cannot be neglected. The multiple-pass model introduced in this study [Equation (3)] describes these multiple-pass components by the effective weight parameters $a_m$ and empirically captures the optical behavior of BC.

4.3. Optimality of the 3-Pass Model

To determine the optimal number of passes for BC, two indices were used: the degree-of-freedom-corrected RMSE [Equation (6)] and the AICc [Equation (7)]. According to the RMSE results (Table 1), the RMSE decreased markedly from 0.0929 to 0.0506 to 0.0275 Abs when the model was extended from 1 pass to 2 passes and then to 3 passes, whereas the further improvement from the 3-pass model to the 4-pass model was very small (0.0275 to 0.0270 Abs). This tendency was consistent across all dyes and all peak wavelengths, indicating that the improvement had essentially saturated at the 3-pass model.

The AICc analysis (Figure 11) showed that the 1-pass model (ΔAICc = 21–81) was statistically inappropriate for all samples (ΔAICc > 10). The 2-pass model was also inappropriate for all samples except TAR (258 nm), for which ΔAICc was 2.67. Although this value does not warrant complete rejection of the 2-pass model, it was still inferior to the 3-pass model. In contrast, the 3-pass model was either the best model or practically equivalent to the best model (ΔAICc ≤ 2) for all samples, whereas the improvement obtained by the 4-pass model was limited (ΔAICc = 0.79–3.40). According to the criteria [47], models with ΔAICc ≤ 2 receive essentially equivalent support; therefore, the 4-pass model was judged not to provide a meaningful improvement over the 3-pass model. Based on these results, the 3-pass model was concluded to be the minimally sufficient model that balances physical plausibility and descriptive accuracy for the concentration–absorbance relationship in BC. Notably, the RMSE for BC improved to 0.0275 ± 0.0087 Abs with the 3-pass model, reaching the same order of magnitude as that of the 1-pass model for CC (0.0203 ± 0.0020 Abs). This indicates that, with an appropriate pass model, the concentration–absorbance relationship in BC can also be described with accuracy comparable to that of CC.

We note that the model framework [Equation (3)] can be extended to an arbitrary number of passes n. The optimal number of passes is determined by the balance between the goodness of fit and the information-criterion penalty for additional parameters, and may change if the cell geometry or optical configuration is modified. The model structure itself is independent of the incident beam power, since it describes the spatial distribution of optical paths rather than the absolute detected intensity.

4.4. Comparison of the Saturation Absorbance A_max and Improvement of the Dynamic Range by the Brewster Cell

As shown in Table 2, the saturation absorbance $A_{\max }$ of BC (mean 2.42 ± 0.14 for the 3-pass model) was approximately 1.08 higher than that of CC (mean 1.34 ± 0.09 for the 1-pass model). This increase in $A_{\max }$ quantitatively demonstrates that the introduction of the BC improved the dynamic range. Because absorbance is a logarithmic quantity, this increase of approximately 1.08 in $A_{\max }$ corresponds to an approximately 12-fold expansion of the linear intensity ratio (~22 to ~263), i.e., about one order of magnitude.

According to Equation (2), the saturation absorbance $A_{\max }$ is determined solely by $P_{\mathrm{n}\mathrm{o}}$, the fraction of light that does not pass through the sample. The increase in $A_{\max }$ of approximately 1.08 when CC was replaced with BC therefore indicates that $P_{\mathrm{n}\mathrm{o}}$ was substantially reduced in BC. According to Brewster’s law, when p-polarized light is incident on the cell surface at the Brewster angle, the reflectance at the surface becomes nearly zero. Because light reflected at the cell surface is detected as stray light that does not carry absorption information from the sample, a reduction in this reflected light directly leads to a decrease in $P_{\mathrm{n}\mathrm{o}}$ and thus to an increase in $A_{\max }$.

It should be noted that the difference in $A_{\max }$ between CC and BC cannot be attributed solely to the reduction in reflected light at the cell surface. In BC, the cell-wall thickness (1.5 mm) is twice that of CC (0.75 mm), and the increased thickness may cause optical losses inside the cell that differ from those in CC. However, because the internal transmittance of quartz glass in the visible region is sufficiently high (>92% for a plate thickness of 10 mm), the additional loss for a wall thickness of 1.5 mm is considered negligible. It should also be noted that $A_{\max }$ showed a slight decrease toward shorter wavelengths for both CC and BC (see Appendix C). This wavelength dependence is attributed primarily to the reduced reflectance of the BaSO₄ integrating-sphere wall in the UV region, rather than to the Brewster-angle mismatch caused by the refractive-index dispersion of the quartz cell, which has a negligible effect on p-polarization reflectance.

4.5. Comparison of the Linearity in the Low-Absorbance Region for CC and BC

For CC, good linearity between $A_{\mathrm{meas}}$ and the concentration $c$ was confirmed in the concentration range up to $A_{\max }/2$ (Section 4.1). For BC as well, the measured data points at or below $A_{\max }/2$ (corresponding to $A_{\max }/2$ ≲ 1.21 Abs) shown in the insets of Figure 4b, Figure 6b,d, Figure 8b,d and Figure 10b,d were all generally consistent with the approximate straight line obtained by linear fitting using those data points together with the origin as an additional data point. Therefore, in the low-absorbance region treated in this study, BC also exhibited practically acceptable linearity comparable to that of CC.

On the other hand, the overall $A_{\mathrm{meas}}-c$ relationship for BC could not be described sufficiently by the single-pass model, and the 3-pass model was required. This is because the cell geometry and oblique-incidence optical configuration of BC produce a broader path-length distribution and a larger contribution of multiple-pass components than in CC. In other words, BC is optically more complex than CC and, strictly speaking, exhibits somewhat greater nonlinearity. However, this difference is small in the low-absorbance region considered in the present study, and in the range shown in the insets, both CC and BC can be treated using an approximately linear relationship.

In addition, $A_{\max }$ for BC was approximately 1.08 higher than that for CC, and the mean value for the 3-pass model was approximately 2.42. Furthermore, when the concentration $c\left(A_{\max }/2\right)$ at which the zero-intercept approximate line $A=kc$ reaches $A_{\max }/2$ was calculated for each dye and each peak wavelength, the practical upper concentration limit for linear approximation in BC was, on average, 1.85 times that in CC. Thus, although BC is inferior to CC in terms of strict optical simplicity, it has the practical advantage of greatly extending the concentration range that can still be treated as nearly linear. This conclusion changed very little even when approximate straight lines with a free intercept were used, indicating that it does not strongly depend on the adoption of the zero-intercept approximation.

4.6. Physical Interpretation of the Parameters in the Multiple-Pass Model

In this study, the product $\epsilon L$ in the Beer–Lambert law was treated collectively as the effective absorption parameter $k = \epsilon L$ without separating $\epsilon$ and $L$. The multiple-pass coefficients $a_m$ were also treated empirically as effective weights of light passing through the sample $m$ times, without relying on rigorous geometric estimation of the path-length distribution. This approach is practical because it can describe the concentration–absorbance relationship of BC with sufficient accuracy by simple fitting, without requiring computationally expensive numerical analysis such as RTS-ISSI.

On the other hand, the values of $a_m$ can be discussed in relation to the geometric structure of the cell and the path-length distribution. In the present study, however, these parameters were treated empirically as effective weights of light passing through the sample $m$ times, and no rigorous geometrical interpretation was imposed. The values of $a_1$, $a_2$, and $a_3$ obtained for the 3-pass model may therefore be regarded as practical descriptors of the overall optical behavior of BC. Separating $k$ into $\epsilon$ and $L$ and quantitatively relating the fitted parameters to the geometric optical path-length distribution remain topics for future work.

The difference in the effective pass structure between CC and BC can be understood from the Fresnel reflectance at the cell surfaces. For CC at near-normal incidence, the reflectance at a single air–quartz interface is $R={\left(\left(n-1\right)/\left(n+1\right)\right)}^2\approx 0.035$ (3.5%) per surface (n = 1.46 at 546 nm [48]). Light reflected at the entrance and exit surfaces (combined ~7%) does not pass through the sample and contributes directly to $P_{\mathrm{no}}$, leading to a lower $A_{\max }$. In contrast, for BC with p-polarized light incident at the Brewster angle, the surface reflectance is ideally zero, resulting in a smaller $P_{\mathrm{no}}$ and a higher $A_{\max }$. Moreover, because the dominant optical path in CC is a single pass with nearly uniform path length along the cell diameter, the effective weights of higher-order passes are expected to be small.

In summary, CC was sufficiently described by the single-pass model, whereas BC required the 3-pass model to reproduce the full concentration–absorbance relationship. Despite its greater optical complexity, BC maintained practically acceptable linearity in the low-absorbance region while increasing $A_{\max }$ and extending the usable concentration range. Thus, the Brewster-cell design provides an effective way to expand the dynamic range of the ISSI method using a simple fitting-based approach. Future work should quantitatively relate the effective model parameters to the geometric optical path-length distribution and extend the present framework to scattering samples.