Radiometric analysis of haze in bright-annealed AISI 430 ferritic stainless steel

J. M. González-Leal; E. Gallero; A. Nuñez; J. F. Almagro

doi:10.1364/AO.451019

1. INTRODUCTION

The optical reflection of surfaces has been extensively studied in both optics and computer graphics and has led to the proposal of numerous physical approaches to model the so-called bidirectional reflectance distribution function (BRDF). Some of these approaches, such as the Rayleigh–Rice and Harvey–Shack models [1–3], are based on the Fourier analysis of the topography for very smooth surfaces, while, for rougher surfaces, models based on microfacets as Torrance–Sparrow, Cook–Torrance, He–Torrance, modified Beckmann–Kirchhoff, and Oren–Nayar [4–10] have been suggested. BRDF (in the very smooth surface approximation) is frequently used by lens and mirror manufacturers as a non-contact inspection and quality control method [11,12]. The Rayleigh–Rice model, which works for highly polished surfaces, is often used in that context. However, to characterize the BRDF of rough surfaces, this model fails, and microfacet models, typically based on the Beckmann–Kirchhoff approach [13], are more suitable. The rough surface is then modeled as a collection of specular V-cavities whose surface normals are assumed to be normally distributed. The calculation of reflected radiation is based on geometrical optics, which is plausibly valid when the surface irregularities are much larger than the wavelength of incident radiation.

The visual appearance of stainless steels demands the attention of those fields of application in which certain aesthetic criteria have to be guaranteed, ranging from household appliances and cutlery to exterior building and construction elements, coatings, and ornaments. Bright-finish AISI 430 ferritic stainless steels are ideal for these applications. These steels are manufactured in sheet form by rolling and annealing in a reducing atmosphere of ${{\rm{H}}_2}/{{\rm{N}}_2}$. To guarantee the quality of these products, reliable and robust inspection systems based on the optical reflection of their surfaces are needed, and this is the area in which this research work is carried out.

It is worth mentioning that the literature devoted to the study of the effect of roughness on the optical scattering of rough surface metallic alloys has not focused on the study of haze or other deviations in the visual appearance of industrial products. These works were mainly aimed at validating BRDF models to achieve a realistic rendering of some common metallic alloys for computer graphics [6]. Inspection and quality control of industrial products were not the aim of any of these works.

A photometric study of the visual appearance of bright-annealed AISI 430 ferritic steels has been previously reported by the authors [14]. It was concluded that there is significant variability in the photometric parameters of these products, mainly in the luminance, ${{{Y}}_{10}}$, of the CIE 1931 color space [15]. This color coordinate, also called the light reflectance value (LRV) [16], is of interest for façade builders dealing with architectural developments using stainless steel. An outstanding linear dependence was found between ${{{Y}}_{10}}$ and the ratio ${S_q}/{S_{\textit{al}}}$, with ${S_q}$ being the roughness root-mean square, and ${S_{\textit{al}}}$ the autocorrelation length [17]. The importance of considering the multiscale of the roughness [3,6] and the dependence of the roughness values with the filtering of topographic maps (typically with the cutoff wavelength of Gaussian filters [6,18]) was also shown there.

The present work is aimed at furthering the study of the optical scattering of bright-annealed AISI 430 ferritic stainless steel, through a radiometric analysis of the optical reflection to better discriminate the influence of roughness scales on the haze effect in the visual appearance of these steels. The study is based on the calculation of a functional form for total integrated scattering (TIS) [1,12,19–21] based on the integration of the He–Torrance BRDF [7]. It is worth mentioning that the He–Torrance model is a modification of the Beckmann–Kirchhoff model, which takes into account shadowing, masking, as well as all polarization and directional Fresnel effects. Other modified Beckmann–Kirchhoff models, such as the one proposed by Harvey et al. [10,22], are more physically rigorous, but more computationally demanding. This model includes a renormalization term that ensures energy conservation, but does not allow working with analytical expressions, except in the extreme cases of very smooth or very rough surfaces, which are not the case for the stainless-steel samples under study here.

In addition, the link between the haze and the optical properties of the stainless steel has been analyzed according to the total optical reflection spectra. Optical constants have been estimated based on the effective medium approximation. A mixture of the passive layer oxide and the stainless-steel base material has been considered in the calculations. The influence of both the topography and the optical constants in the optical reflection of bright-annealed AISI 430 ferritic stainless steel has been analyzed, and they have been correlated with the occurrence of the unwanted haze effect in these industrial products.

2. THEORETICAL CONSIDERATIONS

A. Total Integrated Scattering

TIS is defined as the ratio between the diffuse reflectance and total reflectance of a surface [1,12,20,21]. To obtain an analytical expression of this radiometric quantity, we use the BRDF, defined as the ratio between the radiance emitted by a surface and the irradiance on this surface:

(1)$$\begin{split}{\rm BRDF}\big({{\theta _i},{\phi _i},{\theta _r},{\phi _r};\lambda} \big) &= \frac{{d{L_r}\big({{\theta _i},{\phi _i},{\theta _r},{\phi _r};\lambda} \big)}}{{d{E_i}({\theta _i},{\phi _i};\lambda)}} \\&= \frac{{d{L_r}\big({{\theta _i},{\phi _i},{\theta _r},{\phi _r};\lambda} \big)}}{{{L_i}({{\theta _i},{\phi _i};\lambda} ){{\cos}}({{\theta _i}} )d{\omega _i}}},\end{split}$$

where ${{d}}{L_r}$ is the spectral radiance of the scattered radiation from the surface, $d{E_i}$ is the spectral irradiance of the incident radiation on the surface, ${L_i}$ is the spectral radiance of the light source, ${\theta _i}$ is the polar angle and ${\phi _i}$ the azimuthal angle of incident radiation, ${\theta _r}$ is the polar angle and ${\phi _r}$ the azimuthal angle of scattered radiation, and $d{\omega _i}$ is the solid angle of the light source.

Among the BRDF models mentioned above, the He–Torrance microfacet model stands out for surfaces with roughness values as those of the steels under study here. This model considers that the BRDF consists of three terms:

(2)$$\begin{split}{\rm BRDF}({{\theta _i},{\phi _i},{\theta _r},{\phi _r};\lambda})&= {{\rm BRDF}_{\textit{sp}}}({{\theta _i},{\phi _i},{\theta _r},{\phi _r};\lambda})\\[-3pt]&\quad + {{\rm BRDF}_{\textit{dd}}}({{\theta _i},{\phi _i},{\theta _r},{\phi _r};\lambda}) \\[-3pt]&\quad+ {{\rm BRDF}_{\textit{ud}}}(\lambda),\end{split}$$

where subscripts correspond to specular ($sp$), directional-diffuse ($dd$), and uniform-diffuse ($ud$) reflection. Expressions for these terms are in Appendix A.

Equation (2) can be linked to the diffuse reflectance measured with an integrating sphere, by integrating the diffuse terms ${{\rm{BRDF}}_{\textit{dd}}}$ and ${{\rm{BRDF}}_{\textit{ud}}}$ over a hemispherical integration domain above the surface, i.e.,

(3)$$\begin{split}&{R_d}\big({{\theta _i},{\phi _i};{S_q},{S_{\textit{al}}};\lambda} \big) \\[-3pt]& = \frac{{\int_{2\pi} {L_{\textit{rd}}}\big({{\theta _i},{\phi _i},{\theta _r},{\phi _r};{S_q},{S_{\textit{al}}};\lambda} \big){\rm d}{\omega _r}}}{{d{E_i}\big({{\theta _i},{\phi _i};\lambda} \big)}}\\[-3pt] & = \frac{{\int_{2\pi} {{\rm BRDF}_d}\big({{\theta _i},{\phi _i},{\theta _r},{\phi _r};{S_q},{S_{\textit{al}}};\lambda} \big){\rm d}{E_i}({{\theta _i},{\phi _i};\lambda} ){\rm d}{\omega _r}}}{{d{E_i}\big({{\theta _i},{\phi _i};\lambda} \big)}}\\[-3pt] & = \int _0^{2\pi} \int_0^{\pi /2} {{\rm BRDF}_d}({{\theta _i},{\phi _i},{\theta _r},{\phi _r};{S_q},{S_{\textit{al}}};\lambda} )\\[-3pt]&\quad\times{\rm{\cos}}({{\theta _r}} ){{\sin}}({{\theta _r}} ){\rm d}{\theta _r}{\rm d}{\phi _r},\end{split}$$

where ${L_{\textit{rd}}}$ stands for the total diffuse radiance of the reflected light, and

\begin{split}&{{\rm BRDF}_d}\left({{\theta _i},{\phi _i},{\theta _r},{\phi _r};{S_q},{S_{\textit{al}}};\lambda} \right) \\[-3pt]&\quad= {{\rm BRDF}_{\textit{dd}}}\left({{\theta _i},{\phi _i},{\theta _r},{\phi _r};{S_q},{S_{\textit{al}}};\lambda} \right) + {{\rm BRDF}_{\textit{ud}}}(\lambda ).\end{split}

Total reflectance of the surface, ${R_t}({{\theta _i},{\phi _i};{S_q},{S_{\textit{al}}};\lambda})$, can be similarly derived by integrating all three terms of the total BRDF. It is assumed in this work that the total reflectance can be approximated by using the Fresnel reflection equation of a flat surface, i.e.,

(4)$$\begin{split}&{R_t}\big({{\theta _i},{\phi _i};{S_q},{S_{\textit{al}}};\lambda} \big) \\[-3pt]&= \int _0^{2\pi} \!\!\int _0^{\frac{\pi}{2}} {\rm BRDF}({{\theta _i},{\phi _i},{\theta _r},{\phi _r};{S_q},{S_{\textit{al}}};\lambda} )\\[-3pt]&\quad\times{{\cos}}({{\theta _r}} ){{\sin}}({{\theta _r}} ){\rm d}{\theta _r}{\rm d}{\phi _r} \\[-3pt] &= R\big({{\theta _i},n(\lambda ),k(\lambda )} \big),\end{split}$$

where $R({{\theta _i},n(\lambda),k(\lambda)})$ is the Fresnel reflection at incidence angle ${\theta _i}$, for an interface between air and a conducting medium with refractive index $n(\lambda)$, and extinction coefficient $k(\lambda)$. This assumption can be plausibly valid under the following considerations.

• Light reflected consists mainly of one-bounce reflections on the surface, and inter-reflections on the microfacets are not expected to be very significant. This assumption is plausible for the samples under study because the roughness slopes are small, as reported in Table S1 of Supplement 1.
• The roughness can be assumed to be isotropic, at least in the scale sensitive to optical scattering in the Vis-NIR. The topography of the samples can be therefore considered isotropic, and the optical scattering will not depend on the azimuthal angle ${\phi _i}$, so that ${\phi _i} = 0$ can be set to simplify Eqs. (3) and (4).
• The incidence angle in the experimental measurements of the total and diffuse reflection with that integrating sphere is near-normal, namely, 8°, and the shadowing and masking effects are not significant.
• The angular distribution of the scattered light holds near the specular direction, so the dependence of the local Fresnel reflection on ${\theta _r}$, as considered in the He–Torrance BRDF equations, is not very strong. It would be thus plausible to consider the Fresnel reflection equation for normal incidence on a flat surface corresponding to the least-square plane of the topography.

TIS is defined as the fraction of total reflected radiant power scattered out of the specularly reflected beam, or similarly, the ratio between diffuse reflectance ${R_d}$ and total reflectance ${R_t}$:

(5)$${\rm TIS}\big({{\theta _i};{S_q},{S_{\textit{al}}};\lambda} \big) = \frac{{{R_d}\big({{\theta _i},0;{S_q},{S_{\textit{al}}};\lambda} \big)}}{{{R_t}\big({{\theta _i},0;{S_q},{S_{\textit{al}}};\lambda} \big)}}.$$

The dependence of BRDF on the optical properties of the material is modeled in all three terms of the BRDF equation as a common Fresnel factor, as shown in Appendix A. Therefore, such dependence appears similar for ${R_d}$ and ${R_t}$. Hence, TIS is an indicator of the link between optical scattering and surface topography. TIS introduces optical scattering as a recognized source of metrology information, and these measurements are used as an important scattering specification. It is worth mentioning that TIS is used in the solar power industry to monitor the level of texture introduced onto the surfaces of photovoltaic collectors [1,19].

TIS can be linked to the BRDF from Eqs. (3)–(5), as follows:

(6)$$\begin{split}&{\rm TIS}\left({{\theta _i};{S_q},{S_{\textit{al}}};\lambda} \right)\\& = \frac{{\int_0^{2\pi} \int_0^{\frac{\pi}{2}} {{\rm BRDF}_d}\left({{\theta _i},0,{\theta _r},{\phi _r};{S_q},{S_{\textit{al}}};\lambda} \right)\cos ({{\theta _r}} )\sin ({{\theta _r}} ){\rm d}{\theta _r}{\rm d}{\phi _r}}}{{R({{\theta _i},n(\lambda ),k(\lambda )} )}} \\ & = \frac{1}{{R\left({{\theta _i},n(\lambda ),k(\lambda )} \right)}} \int _0^{2\pi} \int _0^{\frac{\pi}{2}} \big[{{\rm BRDF}_{\textit{dd}}}\left({{\theta _i},0,{\theta _r},{\phi _r};{S_q},{S_{\textit{al}}};\lambda} \right) \\&\quad+ {{\rm BRDF}_{\textit{ud}}}(\lambda ) \big]\cos ({{\theta _r}} )\sin ({{\theta _r}} ){\rm d}{\theta _r}{\rm d}{\phi _r} \\ & = \int _0^{2\pi} \int _0^{\frac{\pi}{2}} \frac{{G({{\theta _i};\lambda} )S\left({{\theta _i};{S_q},{S_{\textit{al}}};\lambda} \right)D\left({{\theta _i};{S_q},{S_{\textit{al}}};\lambda} \right)}}{{\pi \cos ({{\theta _i}} )}}\\&\quad\times\sin ({{\theta _r}} ){\rm d}{\theta _r}{\rm d}{\phi _r} + a(\lambda ),\end{split}$$

where factors $G,\;S,\;D$, and $a({{l}})$ are defined in Appendix A.

TIS depends on roughness parameters ${S_q}$ and ${S_{\textit{al}}}$ via the He–Torrance microfacets BRDF Eq. (7). It is very important to note that there is not a single way of determining the values for ${S_q}$ and ${S_{\textit{al}}}$ in a way that is representative of the whole surface topography and, in fact, a number of works have been developed exploring the multiscale nature of the roughness and its effect on optical scattering. Hence, in the case of smooth surfaces fulfilling the Rayleigh criterion, i.e., ${[{4\pi \;{S_q}\cos ({{\theta _i}})/\lambda}]^2} \ll 1$, the multiscale is reflected on the power spectral density (PSD) of the surface topography [1,2], while for rough surfaces, the multiscale has been linked to the cutoff wavelength of the Gaussian filtering of topography maps [3,6,14,20,23]. The latter approach is applied in this work to analyze the TIS spectra of the bright-annealed AISI 430 ferritic stainless steel.

A least-square fitting algorithm has been defined based on the theoretical TIS spectra of Eq. (6), based on the following function to be minimized:

(7)$$\begin{split}&f\big({{c_1},{c_2}, \ldots ,{c_N},{\rm offset}} \big)\\& = \sqrt { \sum _{j = 1}^N \sum_{m = 1}^M {{\left[{{{\rm TIS}_{{\exp}}}({{\lambda _m}} ) - {c_j}{\rm TIS}\big({{\theta _i},{\phi _i};{S_{\textit{qj}}},{S_{\textit{alj}}};{\lambda _m}} \big) - {\rm offset}} \right]}^2}},\end{split}$$

where ${S_{\textit{qj}}}$ and ${S_{\textit{alj}}}$ are, respectively, the values of the roughness root-mean square and the autocorrelation length, derived after applying a Gaussian low-pass filter with cutoff wavelength ${\lambda _{\textit{cj}}}$ to the topography map, $N$ is the total number of cutoff wavelengths considered in the topographic analysis, and $M$ is the total number of points of the experimental TIS spectrum, ${{\rm{TIS}}_{{\exp}}}$, derived from the total and diffuse optical reflection spectra measured with the integrating sphere. The aim of this algorithm is to analyze the influence of the different roughness scales and assess their strength in optical scattering in the Vis-NIR and the visual appearance of these industrial products.

B. Optical Characterization

The optical characterization of the samples of bright-annealed AISI 430 ferritic stainless-steel samples is based on a best-fit algorithm approach based on the theoretical expression for the reflectance of a flat interface between air and a conductive medium with a complex refractive index, illuminated with non-polarized radiation. For this particular optical system, the reflectance, $R$, for an angle of incidence, $\theta$, coincides with the mean of Fresnel reflection factors for radiation with $s$ polarization, ${R_s}$, and for radiation with $p$ polarization, ${R_p}$, i.e.,

(8)$$R = \frac{1}{2}\left[{{R_s} + {R_p}} \right],$$

where ${R_s} = {r_s^2},{R_p} = {r_p^2}$,

(9)$${{r}_{s}}=\frac{A( {{k}^{2}}+{{n}^{2}} )-2An\cos ( \theta )+{\cos}^{2}( \theta )}{A( {{k}^{2}}+{{n}^{2}} )+2 A n\cos ( \theta )+{\cos}^{2}( \theta )},$$

(10)$${r_p} = \frac{{A - 2An\cos (\theta ) + ({{k^2} + {n^2}} ){{\cos}}^2(\theta )}}{{A + 2An\cos (\theta ) + ({{k^2} + {n^2}} ){{\cos}}^2(\theta )}},$$

(11)$$A = \sqrt {1 - \frac{{\sin^{2}(\theta )}}{n}} .$$

The assumption that the total reflection spectrum of the surface of the steels under study coincides with the Fresnel reflection of an ideally flat surface is plausible as long as the average slope of the roughness is close to zero. Therefore, the contribution of possible multiple reflections that influenced the absorption of light radiation would be very insignificant. This assumption is true for the samples under study.

The dependence of $R$ on wavelength, $\lambda$, or equivalently with photon energy, $\hbar \;\omega$, in Eq. (8), comes directly from the dispersion of the refractive index, $n(\hbar\,\omega)$, and the dispersion of the extinction coefficient, $k(\hbar\;\omega)$. Such a dependence has not been made explicit in the equations above for the sake of clarity of the expressions. Therefore, ultimately, $R$ is a function of the parameters that model the dispersion of the complex dielectric permittivity, $\tilde \varepsilon = {\varepsilon _r} - i\;{\varepsilon _i}$. Given the conductive nature of the material under study, the Drude–Lorentz classical dispersion model is considered, i.e.,

(12)$$\begin{split}&\tilde \varepsilon \left({{\varepsilon _\infty},\hbar {\omega _p},{{{\Gamma}}_d},{f_1},\hbar {\omega _{01}},{{{\gamma}}_1}, \ldots ,{f_N},\hbar {\omega _{0N}},{{{\gamma}}_N};\hbar \omega} \right)\\ &= {\varepsilon _\infty} + \frac{{\hbar {\omega _p}^2}}{{- \hbar {\omega ^2} + i{{{\Gamma}}_d}\hbar \omega }} + \mathop \sum \limits_{j = 1}^N \frac{{{f_j}\hbar {\omega _{0j}}^2}}{{\hbar {\omega _{0j}}^2 - \hbar {\omega ^2} + i{{{\gamma}}_j}\hbar \omega}}.\end{split}$$

Expressions for the refractive index and the extinction coefficient can be obtained as follows:

(13)$$\begin{split}&n\left({{\varepsilon _\infty},\hbar {\omega _p},{{{\Gamma}}_d},{f_1},\hbar {\omega _{01}},{{{\gamma}}_1}, \ldots ,{f_N},\hbar {\omega _{0N}},{{{\gamma}}_N};\hbar \omega} \right)\\& = \sqrt {\frac{1}{2}\left[{{\varepsilon _r} + \sqrt {{\varepsilon _r}^2 + {\varepsilon _i}^2}} \right]} ,\end{split}$$

(14)$$\begin{split}&k\left({{\varepsilon _\infty},\hbar {\omega _p},{{{\Gamma}}_d},{f_1},\hbar {\omega _{01}},{{{\gamma}}_1}, \cdots ,{f_N},\hbar {\omega _{0N}},{{{\gamma}}_N};\hbar \omega} \right)\\& = \sqrt {\frac{1}{2}\left[{- {\varepsilon _r} + \sqrt {{\varepsilon _r}^2 + {\varepsilon _i}^2}} \right]}.\end{split}$$

A least-square fitting algorithm has been implemented from the following function $F$ that is to be minimized:

(15)$$F\big({{\varepsilon _\infty},\hbar {\omega _p},{{{\Gamma}}_d},{f_1},\hbar {\omega _{01}},{{{\gamma}}_1}, \ldots ,{f_N},\hbar {\omega _{0N}},{{{\gamma}}_N}} \big) = \sqrt {\mathop \sum \limits_{j = 1}^M {{\left[{{R_{{\exp}}}({{\theta _i};\hbar {\omega _j}} ) - R\big({{\theta _i},{\varepsilon _s},{f_1},\hbar {\omega _{01}},{{{\gamma}}_1}, \ldots ,{f_N},\hbar {\omega _{0N}},{{{\gamma}}_N};\hbar {\omega _j}} \big)} \right]}^2}} ,$$

Where ${R_{{\exp}}}$ stands for the total optical reflection spectra, and $M$ stands for the number of points that make up the experimental total reflection spectra, measured with a spectrophotometer. It is important to note that this algorithm works with the spectra as a function of photonic energy, rather than as a function of the wavelength. Such approach allows to better highlight the characteristics (i.e., changes in slopes) of the spectra and their relationship with the oscillators of the dispersion model.

3. MATERIALS AND METHODS

A. Materials

The AISI 430 ferritic stainless-steel products studied in this work were produced in flat sheet form with a thickness range of 0.40–1.50 mm, under the manufacturing conditions reported in [24]. Bright finish was achieved by annealing in a ${{\rm{H}}_2}/{{\rm{N}}_2}$ reducing atmosphere. The gas in the bright-annealing furnace consists of ${{75}}\% \;{{\rm{H}}_2}/{{25}}\% \;{{\rm{N}}_2}$ with 6.8 ppm of ${{\rm{O}}_2}$. The control of the annealing environment is based on the dew point, which is ${-}40^\circ{\rm C}$. The optimal conditions of bright annealing were set regarding oxidation/reduction equilibrium of ${\rm{Cr}}/{\rm{C}}{{\rm{r}}_2}{{\rm{O}}_3}$ of the passive film of stainless steel. More details about the AISI 430 ferritic stainless-steel products can be found at https://www.acerinox.com.

A set of four samples, size ${{50}}\;{\rm{mm}} \times {{50}}\;{\rm{mm}}$, was taken by shear cutting from different production batches. Two of them, labeled H1 and H2, show a significant haze effect under visual inspection, while the other two, labeled M1 and M2, show a high specular mirror finish. H1 and H2 are 0.4 mm thick, M1 is 1.2 mm thick, and M2 is 0.6 mm thick.

B. Optical Reflection Spectra

The optical reflectance of the samples was measured in the wavelength range of 200–2500 nm, using a commercial double-beam dispersive UV-Vis-NIR spectrophotometer (Agilent, model Cary 5000). A 150 mm diameter integrating sphere was used to measure both the total reflection spectra according to the optical geometry 8°/di, and the diffuse reflection spectra according to the optical geometry 8°/de [15,25]. The light spot area on the surface sample was $10\;{\rm mm}_2$. Overall, the samples show relatively appreciable marks on the surface due to lamination. These marks are in the form of shallow parallel grooves with a spacing of approximately 2 µm, and they indicate the lamination direction. Their presence is not exclusive to haze-finish samples. This surface feature has been taken into account when performing the optical reflection measurements. Thus, the samples were placed in the measuring port of the integrating sphere oriented so that the direction of lamination was parallel to the plane of incidence of the light beam to reduce as much as possible its effect on the measurement of the diffuse component of the optical reflection.

C. Surface Metrology

The topography of the samples was measured using a multimode non-contact optical profilometer (Zeta Instruments, model Z300), working on confocal multi-point focus mode. A 100X lens was used to measure the height maps, covering a field of view (FOV) of ${184.2}\;\unicode{x00B5}{\rm m} \times {184.2}\;\unicode{x00B5}{\rm m}$ and an image size of ${{1439}}\;{\rm{pixels}} \times {{1439}}\;{\rm{pixels}}$. The height maps were processed and analyzed with the Mountains Map 9.0 (Digital Surf) software. The areal roughness parameters were calculated according to ISO 25178 [17]. Height profiles were additionally measured to quantify sample waviness, using a mechanical profilometer (Veeco, model Dektak 150), with a stylus with a radius of 2.5 µm, over an evaluation length of 6.5 mm. Waviness parameters were calculated according to ISO 4287 [26]. All surface topography measurements were taken at the center of the samples, in the same areas that were irradiated when recording the optical reflection spectra.

4. RESULTS AND DISCUSSION

The aim of this project is to analyze and understand the visual appearance of bright-annealed AISI 430 ferritic stainless steels. The goal is to achieve robust quality control inspection tools for these industrial products. One of the unwanted effects that reduces the quality of these steels is the presence of haze, which could be due to uncontrolled effects during processing in the manufacturing line or metallurgical effects that could make some batches more susceptible than others to the occurrence of this visual appearance effect. This work has dealt with the study of the topography and optical constants of samples of batches that present haze, to compare them with samples with high-quality bright finishes.

The total and diffuse optical reflection spectra of the bright-annealed AISI 430 ferritic stainless-steel samples under study are shown in Fig. 1. Haze-finish samples show lower total reflectance values than mirror-finish samples, over the entire spectral range. Likewise, diffuse reflection spectra also show a different behavior in samples with and without haze. In both cases, differences are linked to the optical constants and topography of the samples. Topographic maps of the samples are shown in Fig. 2. They correspond to the primary surfaces measured with the optical profilometer. The height scale highlights the coarser finish of haze-finish samples as compared to mirror-finish samples.

Fig. 1. Total reflection spectra (solid line) and diffuse reflection spectra (dashed line) of the bright-annealed AISI 430 ferritic stainless steel under study.

Download Full Size | PDF

Fig. 2. Topographic maps of the bright-annealed finish stainless-steel ferritic AISI 430 samples.

Download Full Size | PDF

It is well known that the topographic characterization of a surface is highly dependent on the spatial components or bandwidth of the features present in the sample [12,18,23,27,28]. In general, the topography of a surface includes a range of spatial wavelengths, which may fall under the categories of form, waviness, or roughness. The discrimination among them is typically performed by Gaussian filtering [29,30]. The form surface (or form profile, in the case of linear measurements), is usually called the F-surface, and it is calculated by removing the small-scale elements from the primary surface (or primary profile, in the case of linear measurements), applying a filter with a cutoff wavelength, ${\lambda _F}$ (low-pass filter in frequency). The form component is linked to the specific geometry (cylinder, sphere, etc.) or to some kind deformation of the piece. The waviness surface (or waviness profile, in the case of linear measurements) is calculated by removing the smallest-scale elements from the F-surface (or F-profile, in the case of linear measurements), applying a filter with a cutoff wavelengths, ${\lambda _c} \lt {\lambda _F}$ (low-pass filter in frequency), while the roughness surface (or roughness profile, in the case of linear measurements) is calculated by removing the largest-scale elements from the F-surface (or F-profile, in the case of linear measurements), applying a filter with the same cutoff wavelength, ${\lambda _c}$ (high-pass filter in frequency). Figure S1 of Supplement 1 illustrates the filtering process to discriminate the form, waviness, and roughness components. Roughness surfaces for the samples under study, derived from Gaussian filtering with ${\lambda _c} = {{3}}\;{\rm{mm}}$, are also shown in Fig. S2 of Supplement 1 for illustration.

It is worth mentioning that an additional filter with cutoff ${\lambda _s} \lt {\lambda _c}$ can also be applied to the surface topography to remove instrument noise and guarantee that measurements are within the resolution of the measurement instrument. This short cutoff wavelength, ${\lambda _s}$, is strongly dependent on the characteristics of the measurement instrument used. For example, for stylus profilometers, ${\lambda _s}$ is linked to the tip radius, while for optical profilometers, it is linked to the pixel resolution. The ideal Nyquist limit for the optical profilometer used in this work can be calculated to be ${\lambda _s} = {{2}} \times {{\rm{L}}_{\rm scan\:range}}\;/{{\rm{n}}_{\rm{pixels}}} = {{2}} \times {{184.2/1439}}\;{\rm{mm}} = {0.26}\;{\rm{mm}}$, and would be the theoretical minimum autocorrelation length that could be solved with this instrument. On the other hand, the long cutoff wavelength, ${\lambda _c}$, is linked to the area size of the sample, and it is reasonable to define it according to the surface performance and the phenomenon under study. It is worth noting that the distinction between surface roughness and waviness is subjective, so features that appear as roughness in one application of a surface may have the same band of frequencies as waviness in another. This is the concept of functional filtering [20,23]. Therefore, for applications linked to the visual appearance and optical scattering, it would be expected that ${\lambda _c}$ is linked to the light spectral range of interest.

In general, surfaces are the result of an infinite bandwidth process [2]. Therefore, different levels of Gaussian filtering, varying the wavelength cutoff, ${\lambda _c}$, have been considered in the determination of the values of roughness parameters ${S_q}$ and ${S_{\textit{al}}}$, as illustrated in Fig. 3. Table S1 of Supplement 1 lists numeric values, as well as the values of the ratio ${S_q} / {S_{\textit{al}}}$, which are also plotted in Fig. S3 of Supplement 1. A dependence is clearly observed in the values of ${S_q}$ and ${S_{\textit{al}}}$ on ${\lambda _c}$, and both parameters show a monotonous increase with ${\lambda _c}$, with an asymptotic trend at high ${\lambda _c}$ values. The vertical scale of the roughness is very small at low values of ${\lambda _c}$, as the distribution of the heights is mainly ascribed to the waviness and/or form component. This is clearly shown in Fig. 3(a) for ${S_q}$ and can also be observed in Fig. 3(b) for parameter ${S_{\textit{al}}}$, since the autocorrelation is calculated on a roughness with higher spectral density.

Fig. 3. Values of the root-mean square roughness, ${S_q}$ (a), and autocorrelation length, ${S_{\textit{al}}}$ (b), as a function of the Gaussian filter cutoff wavelength, ${\lambda _c}$, for the samples under study.

Download Full Size | PDF

The values of ${S_q}$ and ${S_{\textit{al}}}$ grow when the value of ${\lambda _c}$ increases, but at high values of ${\lambda _c}$, the growth rate will become lower, since the effect of filtering will become less and less significant. Eventually both parameters tend to values corresponding to the non-filtered primary area. Larger values of ${S_q}$ have been found for haze-finish samples, as compared to the values for mirror-finish samples. However, the values of ${S_{\textit{al}}}$ are very similar for all samples for ${\lambda _c}$ in the range of 2–10 mm, while out of this range, they are slightly higher for haze-finish samples.

TIS spectra were simulated based on Eq. (6), taking the values of ${S_q}$ and ${S_{\textit{al}}}$ derived from the filtered topographic maps with different cutoff wavelengths, as shown in Fig. 4. A monotonous decreasing trend is observed for each spectrum with the light wavelength. In addition, TIS values increase alongside ${\lambda _c}$. Over the whole spectrum, higher values of TIS have been systematically found for haze-finish samples as compared to mirror-finish samples. This is due to the higher values of ${S_q}$ derived from the corresponding filtered maps, which enhance the diffuse component of optical reflection, while reducing the specular component.

Fig. 4. TIS as a function of the wavelength derived from the values of ${S_q}$ and ${S_{\textit{al}}}$, for the values of ${\lambda _c}$ shown in the legend (in micrometers), for the haze-finish sample H1 (a) and for the mirror-finish sample M2 (b). The arrow shows the monotonous trend of TIS spectra when increasing ${\lambda _c}$. The TIS spectrum for the values of ${S_q}$ and ${S_{\textit{al}}}$ corresponding to the non-filtered (NF) primary height map is also included.

Download Full Size | PDF

It is important to mention that the He–Torrance equation for the ${{\rm BRDF}_{\textit{dd}}}$ cannot be written exclusively as a function of the slope ${S_q}/{S_{\textit{al}}}$. The dependency on ${S_q}$ appears exclusively in the $g$ term of the distribution function $D$, which models the angular distribution of optical scattering. Therefore, ${S_q}$ and ${S_{\textit{al}}}$ do not fulfill a reciprocity law in their effect on ${{\rm{BRDF}}_{\textit{dd}}}$, so a change of any of these parameters cannot be compensated for with a similar variation of the other parameter.

The set of TIS spectra for each sample was used in the linear combination defined in the best-fitting algorithm of Eq. (7), to analyze their contribution to the experimental TIS spectra measured for these samples. Good fits were found for all samples, as shown in Fig. 5.

Fig. 5. Fits of TIS spectra for the AISI 430 ferritic stainless steel under study from Eq. (7). Measured TIS are plotted in solid line. Fits are plotted in dotted line.

Download Full Size | PDF

Values of the fitting coefficients, ${c_j}$, of Eq. (7) are plotted in Fig. 6. The results suggest that roughness parameters calculated from a topographic map filtered with a cutoff wavelength ${\lambda _c} = {{3}}\;{\rm{mm}}$ reproduce well the experimental TIS spectra of all the bright-annealed AISI 430 ferritic stainless-steel samples under study. Said cutoff wavelength is similar to the abovementioned, with 2.5 mm and reported in [18], as the band limit for a commercial confocal optical profilometer.

Fig. 6. Values of the coefficients, ${c_j}$, for the linear combination of theorical TIS spectra simulated for the different ${\lambda _c}$, according to Eq. (7), to fit the experimental TIS spectra of the samples under study.

Download Full Size | PDF

It should be noted that the approach suggested here takes into account the possible contribution of different roughness scales to the experimental TIS spectra, as roughness parameters derived from different cutoff wavelengths have been considered in the fitting. Hence, it has been found that additional contributions to the one for ${\lambda _c} = {{3}}\;{\rm{mm}}$ are needed to build the best-fit TIS spectra of the samples. Contributions around ${\lambda _c} = {{3}}\;{\rm{mm}}$ could be due to uncertainty to establish the filter band limit. However, the contribution at ${\lambda _c} = {{100}}\;{\rm{mm}}$ found for the haze-finish samples would suggest that multiscale roughness could play some role in the experimental TIS spectra of the samples. In particular, the large values of both ${S_q}$ and ${S_{\textit{al}}}$ calculated for this ${\lambda _c}$ would point to the possible influence of surface waviness in the TIS spectra.

The waviness of the samples was evaluated from the height profiles measured with the stylus profilometer, along an evaluation length of ${L_e} = {6.5}\;{\rm{mm}}$. A Gaussian filter with 1.2 mm cutoff wavelength was used to obtain the waviness profile from the primary profile, according to the rule ${\lambda _c} = {L_e}/{{5}}$ [31]. The root-mean square height parameter, ${W_q}$, was calculated based on the filtered height profile. Three profiles were taken for each sample. Representative waviness profiles of the samples are shown in Fig. S4 of Supplement 1. Values of the offset from Eq. (7) are plotted against the ${W_q}$ in Fig. 7.

Fig. 7. Values of the offset fitting parameter from Eq. (7) versus the root-mean square height calculated from the waviness profiles of the samples. Error bars have been determined from the set of three height profiles recorded for each sample. Curve corresponds to the best fit to Eq. (16) ($\alpha = 0.1520$, $\beta = 0.6037$, $r_2 = 0.9995$).

Download Full Size | PDF

A very significant dependence between the offset values and ${W_q}$ was found, fulfilling the equation

(16)$${\rm offset} = \alpha \left[{1 - {\rm Exp}\left({- \frac{{{W_q}}}{\beta}} \right)} \right],$$

with $a$ and $b$ being the fitting parameters. This result suggests that the offset in the diffuse component of optical reflection corresponds to the specular component that has not been properly excluded by the exit port of the integrating sphere. It highlights the limitations of the integrating sphere to discriminate the specular component of optical reflection in the case of samples that are not perfectly flat.

Consequently, hypothesizing about a possible Lambertian uniform diffuse component in the optical reflection of these samples could be avoided by accepting that the sample deformation by the shear cutting of the thinner samples could explain the offset found in the TIS spectra in Fig. 5. It is worth mentioning that Lambertian reflection occurs due to multiple light reflections, typically in a layer. However, the thickness of the passive oxide layer in stainless steel is a few nanometers [32], and it could hardly be the cause of multiple reflections of radiation with a wavelength larger than 200 nm inside it.

Going one step further in the characterization of bright-annealed AISI 430 ferritic stainless steels, the total reflection spectra measured with the integrating sphere have been analyzed to obtain information on the optical constants, from the best-fit algorithm defined in Eq. (15). The number of oscillators modeling the dispersion of dielectric permittivity in Eq. (12) has been particularized to $N = {{4}}$. The results of these fits, shown in Fig. 8, are very good in all cases.

Fig. 8. Fits of the experimental total reflection spectra of the samples to the Fresnel equation for non-polarized light from Eq. (15). Fits are in solid lines. Experimental spectra are in dashed lines.

Download Full Size | PDF

The values of the fitting parameters are shown in Table S2 of Supplement 1. Based on these values, the dispersion curves for the refractive index and the extinction coefficient of the samples are plotted, as shown in Fig. 9(a). The dispersion curves of the real and imaginary parts of the complex dielectric permittivity of these materials have also been generated, as shown in Fig. 9(b). In addition, the total and diffuse reflection spectra of the samples under study have been simulated based on the fitting parameters for the TIS and for the optical constants to check the match with the corresponding experimental spectra. The results are shown in Fig. S5 of Supplement 1. Excellent agreement was found between the experimental and simulated spectra.

Fig. 9. Dispersion of the refractive index (solid lines) and extinction coefficient (dashed lines) as a function of wavelength (a) and dispersion of the real part (solid lines) and imaginary part (dashed lines) of the complex dielectric function as a function of photon energy (b), determined from the total reflection spectra of the samples, by means of the best-fitting algorithm of Eq. (15).

Download Full Size | PDF

The values of $n$ and $k$ indicate high similarities between optical constant values in the haze-finish samples. Likewise, the values of the mirror-finish samples are also very similar. This study has found that the values of $k$ are significantly lower for haze-finish samples as compared to mirror-finish samples over the whole spectrum. Similarly, the imaginary part of the dielectric function, ${\varepsilon _i} = \sigma /\omega$ ($\sigma$ being the electrical conductivity and $\omega$ the frequency of light radiation) is also lower for haze-finish samples than for mirror-finish samples, as shown in Fig. 9(b) [33].

Considering Drude and Lorentz dispersion models, the study has found that the fits of the Fresnel reflection equation to the total reflection spectra of the samples are very good in all cases, as shown in Fig. 7. This approach is plausible for the samples under study, as discussed in Section 2. The aim of this analysis is to reveal possible differences in the optical constants that could be linked to the occurrence of haze in the visual appearance of some batches of these industrial products. It is worth noting that the results for $n$ and $k$ are consistent with those reported by other authors for stainless steels of grades other than AISI 430 [34,35]. No references have been found for this particular stainless-steel grade. The fact that the passive layer is intrinsically present in stainless steels cannot be overlooked, and this means that the optical constants of the base material reported in previous literature are presumably affected by those corresponding to the phases of Fe and Cr oxides. It is worth mentioning that the optical constants of iron and chromium oxides ($n\; \approx \;{2.5},\;k \lt {{1}}$) [34,36–39] would lead to reflectance values of approximately 20%, which are significantly lower than those amounting to approximately 60% observed in the total reflectance of the samples.

Based on the abovementioned results, it could be inferred that the passive layer of haze-finish samples would have a less conductive character than that of mirror-finish samples. Our optical characterization approach does not discriminate between the passive layer and the base material, but an effective medium consisting of a mixture of Fe and Cr oxides and the base material is considered instead. Therefore, a link between the fraction of oxides in the effective medium and the thickness of the passive layer could be expected. Likewise, the smaller values of $k$ found for haze-finish samples could be linked to a passive layer thicker than in mirror-finish samples. This increase in thickness could not be considered an effect of plate rolling, as the opposite effect would then be expected as a consequence of the roll pressure on the surface. However, it could be linked to deviations in the last stage of bright annealing in the ${{\rm{H}}_2}/{{\rm{N}}_2}$ reducing atmosphere, which would enhance the growth of the oxide layer and promote the development of greater surface roughness in the sheets. Work on the production line would be necessary to verify this hypothesis, but this is difficult to manage while the factory is operative.

An air-passive layer-base metal type optical system should be considered for the rigorous characterization of these materials. The model could be further refined based on previous literature to take into consideration some gradation in the passive layer between iron and chromium oxides [32,36]. However, the rough topography of these samples makes it difficult to use high-resolution optical characterization techniques such as ellipsometry, and that is why special attention must be paid to the characteristics of the samples [40]. Although there is literature on the optical characterization of the passive layer by ellipsometry, the results reported focus mainly on the thickness and its dependence on attack by oxidizing agents [38,39]. It is worth mentioning that there are ongoing works on the characterization of the passive layer of bright-annealed AISI 430 ferritic stainless steel, concerning its composition, thickness, and optical and mechanical properties. Results will be published in due course.

5. CONCLUSION

The radiometric analysis of the optical reflection of bright-annealed AISI 430 ferritic stainless steels has been carried out based on the TIS spectra of haze-finish and mirror-finish samples from different production batches. The values of the root-mean square roughness and autocorrelation length determined from suitably filtered topographic maps have been used to simulate the theoretical spectra of TIS from the microfacets model for He–Torrance BRDF. This model has been found to be well fitting for the experimental TIS spectra. The optical constants of the samples have also been estimated based on the total reflection spectra. Lower values for both $n$ and $k$ have been found for haze-finish samples as compared to mirror-finish samples. This could be linked to a thicker dielectric passive layer, which reduces the high reflectivity expected for a bright-annealed finish. This increase observed in the oxide layer also seems to be accompanied by an increased roughness, which leads in turn to an enhanced diffusion in optical reflection. The results reported here would suggest that the occurrence of haze in bright-annealed AISI 430 ferritic steels could be linked to possible deviations in the purity of the bright-annealing ${\rm H}_2/{\rm N}_2$ reducing atmosphere that would increase the thickness of the passive layer.

APPENDIX A

The specular term of the He–Torrance BRDF model is given by

(A1)$$\begin{split}&{{\rm BRDF}_{\textit{sp}}}\big({{\theta _i},{\phi _i},{\theta _r},{\phi _r};\lambda} \big) \\&\quad= \frac{{R\big({{\theta _i},{\theta _r},n(\lambda ),k(\lambda )} \big){e^{- g}}S}}{{{\cos}({{\theta _i}} )d{\omega _i}}}\delta \big({\theta - {\theta _i}} \big),\end{split}$$

(A2)$$R\big({{\theta _i},{\theta _r},n(\lambda ),k(\lambda )} \big) = \frac{1}{2}({r_s^2 + \;r_p^2} ),$$

(A3)$$g = {\left\{{\frac{{2\pi \!{S_q}}}{\lambda}\big[{\cos ({{\theta _i}} ) + {\cos}({{\theta _r}} )} \big]} \right\}^2},$$

(A4)$$\begin{split}&S_q^a = \frac{{{S_q}}}{{\sqrt {1 + {{\left({\frac{{{z_0}}}{{{S_q}}}} \right)}^2}}}},\\&\left\{{{z_0} \in {\mathbb R},\quad \sqrt {\frac{\pi}{2}} {z_0} = \frac{{{S_q}}}{4}({{K_i} + {K_r}} )\;{e^{- \frac{1}{2}{{\left({\frac{{{z_0}}}{{{S_q}}}} \right)}^2}}}} \right\},\end{split}$$

(A5)$$\begin{split}{K_i} &= {\tan}({{\theta _i}} )\;{\rm{erfc}}\left({\frac{{{\cot}({{\theta _i}} )}}{2}\frac{{{S_{\textit{al}}}}}{{{S_q}}}} \right),\\[-3pt]{K_r} &= {\tan}({{\theta _r}} )\;{\rm{erfc}}\left({\frac{{{\cot}({{\theta _r}} )}}{2}\frac{{{S_{\textit{al}}}}}{{{S_q}}}} \right),\end{split}$$

(A6)$$S = S({{\theta _i}} )S({{\theta _r}} ),$$

(A7)$$\begin{split}S({{\theta _i}} )& = \frac{{1 - \frac{1}{2}{\rm{erfc}}\left({\frac{{{\cot}({{\theta _i}} )}}{2}\frac{{{S_{\textit{al}}}}}{{{S_q}}}} \right)}}{{{{\Lambda}}({{\cot}({{\theta _i}} )} )}},\\S({{\theta _r}} )& = \frac{{1 - \;\frac{1}{2}{\rm{erfc}}\left({\frac{{{\cot}({{\theta _r}} )}}{2}\frac{{{S_{\textit{al}}}}}{{{S_q}}}} \right)}}{{{{\Lambda}}({{\cot}({{\theta _r}} )} )}},\end{split}$$

(A8)$$\begin{split}{{\Lambda}}({\cot (\theta )} ) &= \frac{1}{2}\left\{\frac{2}{{\sqrt \pi {\rm{cot}}(\theta )}}\frac{{{S_q}}}{{{S_{\textit{al}}}\;}}\;{e^{- {{\left({\frac{{{\cot}(\theta )}}{2}\frac{{{S_{\textit{al}}}}}{{{S_q}}}} \right)}^2}}}\right.\\& \quad-\left. {\rm{erfc}}\left({\frac{{{\cot}(\theta )}}{2}\frac{{{S_{\textit{al}}}}}{{{S_q}}}} \right) \right\},\end{split}$$

where $|{F^2}|$ is the Fresnel reflectivity for unpolarized light at the bisecting angle given by ${\rm{arccos}}({| {{{\hat k}_r} - {{\hat k}_i}} |/2})$, $\delta$ is Dirac’s delta function, $n$ the refractive index, $k$ the extinction coefficient, $S_q^a$ the apparent root-mean square roughness, ${S_q}$ the actual rms roughness, and ${S_{\textit{al}}}$ the autocorrelation length.

The directional diffuse term is given by

(A9)$$\begin{split}&{{\rm BRDF}_{\textit{dd}}}\left({{\theta _i},{\phi _i},{\theta _r},{\phi _r};\lambda} \right) \\&\quad= \frac{{R\big({{\theta _i},{\theta _r},n(\lambda ),k(\lambda )} \big)}}{\pi}\;\frac{{G\;S\;D}}{{\cos ({{\theta _i}} ){\cos}({{\theta _r}} )}},\end{split}$$

(A10)$$\begin{split}G &= {\left({\frac{\lambda}{{2\pi}}\frac{{\vec v\cdot \vec v}}{{{v_z}}}} \right)^2}\frac{1}{{{{| {{{\hat k}_r} \times {{\hat k}_i}} |}^4}}}\left[{{{\big({{{\hat s}_r}\cdot {{\hat k}_i}} \big)}^2} + {{\big({{{\hat p}_r}\cdot {{\hat k}_i}} \big)}^2}} \right]\\&\quad\times\left[{{{\big({{{\hat s}_i}\cdot {{\hat k}_r}} \big)}^2} + {{\big({{{\hat p}_i}\cdot {{\hat k}_r}} \big)}^2}} \right],\end{split}$$

(A11)$$D = \frac{{{\pi ^2}{S_{\textit{al}}}^2}}{{4\;{\lambda ^2}}}\;\mathop \sum \limits_{m = 1}^\infty \frac{{{g^m}{e^{- g}}}}{{m!m}}\;{e^{- \left({\frac{{v_{\textit{xy}}^2\;{S_{\textit{al}}}^2}}{{4\;m}}} \right)}},$$

(A12)$$\vec v = \;\frac{{2\pi}}{\lambda}\big({{{\hat k}_r} - {{\hat k}_i}} \big) = {v_x}{\hat e_x} + \;{v_y}{\hat e_y} + \;{v_z}{\hat e_z},$$

(A13)$${v_{\textit{xy}}} = \sqrt {v_x^2 + v_y^2},$$

(A14)$${\hat s_i} = \;\frac{{{{\hat k}_i} \times {{\hat e}_n}}}{{| {{{\hat k}_i} \times {{\hat e}_n}} |}},\quad {\hat p_i} = {\hat s_i} \times {\hat k_i},$$

(A15)$${\hat s_r} = \;\frac{{{{\hat k}_r} \times {{\hat e}_n}}}{{| {{{\hat k}_r} \times {{\hat e}_n}} |}},\quad {\hat p_r} = \;{\hat s_r} \times {\hat k_r},$$

where $G$ is a geometrical factor, $S$ is the shadowing/masking factor, and $D$ is a distribution function for the directional-diffuse reflection term.

Funding

Consejería de Economía, Conocimiento, Empresas y Universidad, Junta de Andalucía (FEDER-UCA18-106321).

Disclosures

The authors declare no conflict of interest.

Data availability

The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

REFERENCES

1. J. C. Stover, Optical scattering. Measurement and Analysis, 3rd ed. (SPIE, 2012).

2. A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28, 1121–1138 (2011). [CrossRef]

3. T. V. Vorburger, E. Marx, and T. R. Lettieri, “Regimes of surface roughness measurable with light scattering,” Appl. Opt. 32, 3401–3408 (1993). [CrossRef]

4. R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” ACM Trans. Graph. 1, 7–24 (1982). [CrossRef]

5. K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1114 (1967). [CrossRef]

6. H. Li and K. E. Torrance, “An experimental study of the correlation between surface roughness and light scattering for rough metallic surfaces,” Proc. SPIE 5878, 58780V (2005). [CrossRef]

7. X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection, in SIGGRAPH ’91,” in Proceedings of the 18th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1991), pp. 175–186.

8. B. Walter, S. Marschner, H. Li, and K. Torrance, “Microfacet models for refraction through rough surfaces,” in Eurographics Symposium on Rendering (The Eurographics Association, 2007), pp. 195–205.

9. M. Oren and S. K. Nayar, “Generalization of Lambert’s reflectance model,” in Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques–SIGGRAPH (ACM, 1994), pp. 239–246.

10. J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46, 078002 (2007). [CrossRef]

11. J. C. Stover, S. Schroeder, C. Staats, V. Lopushenko, and E. Church, “Linking Rayleigh-Rice theory with near linear shift invariance in light scattering phenomena,” Proc. SPIE 9961, 996102 (2016). [CrossRef]

12. J. E. Harvey, S. Schröder, N. Choi, and A. Duparré, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012). [CrossRef]

13. P. Beckmann, “Scattering of light by rough surfaces,” Prog. Opt. 6, 53–69 (1967). [CrossRef]

14. J. M. González-Leal, E. Gallero, E. Blanco, M. R. del Solar, A. Nuñez, and J. F. Almagro, “Analysis of the visual appearance of AISI 430 ferritic stainless steel flat sheets manufactured by cool rolling and bright annealing,” Metals 11, 1058 (2021). [CrossRef]

15. Technical report: colorimetry CIE 15:2004 (International Commission on Illumination, 2004), 3rd ed.

16. “Light reflectance value (LRV) of a surface – method of test,” BS 8493:2008+A1:2010 (British Standard Institution, 2010).

17. “Geometrical product specifications (GPS) — surface texture: areal — part 2: terms, definitions and surface texture parameters,” ISO 25178-2:2012 (International Organization for Standardization, 2012).

18. N. A. Feidenhans’L, P. E. Hansen, L. Pilný, M. H. Madsen, G. Bissacco, J. C. Petersen, and R. Taboryski, “Comparison of optical methods for surface roughness characterization,” Meas. Sci. Technol. 26, 085208 (2015). [CrossRef]

19. J. C. Stover and E. L. Hegstrom, “Scatter metrology of photovoltaic textured surfaces,” Proc. SPIE 7771, 777109 (2010). [CrossRef]

20. T. R. Thomas, Rough Surfaces, 2nd ed. (Imperial College, 1999).

21. H. Davies, “Reflection of electromagnetic waves from a rough surface,” in Proceeding of the Institution of Electrical Engineers (Institution of Electrical Engineers, 1954).

22. S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Comparison of microfacet BRDF model to modified Beckmann-Kirchhoff BRDF model for rough and smooth surfaces,” Opt. Express 23, 29100–29112 (2015). [CrossRef]

23. T. Y. Lin, L. Blunt, and K. J. Stout, “Determination of proper frequency bandwidth for 3D topography measurement using spectral analysis. Part i: isotropic surfaces,” Wear 166, 221 (1993). [CrossRef]

24. I. C. García, A. N. Galindo, J. F. Almagro Bello, J. M. González Leal, and J. F. Botana Pedemonte, “Characterisation of high temperature oxidation phenomena during AISI 430 stainless steel manufacturing under a controlled H₂ atmosphere for bright annealing,” Metals 11, 191 (2021). [CrossRef]

25. “Practical methods for the measurement of reflectance and transmittance,” CIE 130:1998 (International Commission on Illumination, 1998).

26. “Geometrical product specifications (GPS)–surface texture: profile method - terms, definitions and surface texture parameters,” ISO 4287:1997/Amd 1:2009 (International Organization for Standardization, 1997).

27. P. Pawlus, “Digitisation of surface topography measurement results,” Measurement 40, 672 (2007). [CrossRef]

28. C. Y. Poon and B. Bhushan, “Comparison of surface roughness measurements by stylus profiler, AFM and non-contact optical profiler,” Wear 190, 76–88 (1995). [CrossRef]

29. “Geometrical product specifications (GPS) - surface texture: profile method - metrological characteristics of phase correct filters,” ISO 11562:1996/Cor1:1998 (International Organization for Standardization, 1998).

30. “Geometrical product specifications (GPS) — filtration — part 21: linear profile filters: Gaussian filters,” ISO 16610-21:2011 (International Organization for Standardization, 2011).

31. “Geometrical product specifications (GPS) - surface texture: profile method - rules and procedures for the assessment of surface texture, ISO 4288:1996/Cor 1:1998 (International Organization for Standardization, 1998).

32. C. O. A. Olsson and D. Landolt, “Passive films on stainless steels - chemistry, structure and growth,” Electrochim. Acta 48, 1093–1104 (2003). [CrossRef]

33. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

34. B. Karlsson and C. G. Ribbing, “Optical constants and spectral selectivity of stainless steel and its oxides,” J. Appl. Phys. 53, 6340 (1982). [CrossRef]

35. M. Seo, J. Lee, and M. Lee, “Grating-coupled surface plasmon resonance on bulk stainless steel,” Opt. Express 25, 26939–26949 (2017). [CrossRef]

36. R. A. Castelli, P. D. Persans, W. Strohmayer, and V. Parkinson, “Optical reflection spectroscopy of thick corrosion layers on 304 stainless steel,” Corros. Sci. 49, 4396–4414 (2007). [CrossRef]

37. M. D. Scafetta, T. C. Kaspar, M. E. Bowden, S. R. Spurgeon, B. Matthews, and S. A. Chambers, “Reversible oxidation quantified by optical properties in epitaxial Fe₂CrO₄+Î films on (001) MgAl₂O₄,” ACS Omega 5, 3240–3249 (2020). [CrossRef]

38. H. Xu, L. Wang, D. Sun, and H. Yu, “The passive oxide films growth on 316L stainless steel in borate buffer solution measured by real-time spectroscopic ellipsometry,” Appl. Surf. Sci. 351, 367–373 (2015). [CrossRef]

39. S. V. Phadnis, M. K. Totlani, and D. Bhattacharya, “Ellipsometric study of passive films on the stainless steel SS304,” Trans. Inst. Met. Finish. 76, 235–237 (1998). [CrossRef]

40. M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986). [CrossRef]

Radiometric analysis of haze in bright-annealed AISI 430 ferritic stainless steel

Abstract

1. INTRODUCTION

2. THEORETICAL CONSIDERATIONS

A. Total Integrated Scattering

B. Optical Characterization

3. MATERIALS AND METHODS

A. Materials

B. Optical Reflection Spectra

C. Surface Metrology

4. RESULTS AND DISCUSSION

5. CONCLUSION

APPENDIX A

Funding

Disclosures

Data availability

Supplemental document

REFERENCES

Supplementary Material (1)

Data availability

Cited By

Figures (9)

Equations (32)

Applied Optics