# Impact of localized defects in light scattered by optical components: measurement and analysis

Adrien Bolliand, Michel Lequime, Claude Amra, Myriam Zerrad

► To cite this version:

Adrien Bolliand, Michel Lequime, Claude Amra, Myriam Zerrad. Impact of localized defects in light scattered by optical components: measurement and analysis. Optics Express, 2025, 33 (11), pp.22396. 10.1364/OE.551695 . hal-05237979

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# Impact of localized defects in light scattered by optical components: measurement and analysis

ADRIEN BOLLIAND,\* MICHEL LEQUIME, CLAUDE AMRA, AND MYRIAM ZERRAD

Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France

\*adrien.bolliand@fresnel.fr

**Abstract:** Defects and contaminants can be a critical source of performance loss. This paper presents and demonstrates the capabilities of a new SPatially and Angularly Resolved Scatterometry Equipment (SPARSE) designed to measure spatially resolved ARS of optical surfaces. Besides a calibration sample, the measurement results of three samples are detailed here. They show the amount of information accessible from a measurement, including the ability to distinguish between roughness, localized defects, and scratches. The scattered light generated by each of these three types of defects can be quantified independently.

Published by Optica Publishing Group under the terms of the [Creative Commons Attribution 4.0 License](https://creativecommons.org/licenses/by/4.0/). Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. Introduction

Scattered light induced by imperfections on optical components is one of the main causes of stray light in optical systems, which is critical to the performance of optical instruments developed for advanced metrology, such as space observation or gravitational waves detection. Scattered light from defects can be predicted using various methods [1–4]. But it is still a challenge, for a specific sample, to assess the impact of defects and contamination that are on the surface on the optical performances.

We present an original and accurate solution for quantifying the impact of surface quality of optical components on their performances with an immediate sorting of the different sources of optical scattering: intrinsic roughness, localized defects or scratches. Each of them can be routinely measured or identified by microscopy or profilometry but the analysis remains local and does not allow quantifying stray light. On the other hand, the measurement of scattered intensity can be performed with accuracy [5–12], but the discrimination between sources of scattered light is not always possible. In this context, we propose an original method to accurately measure the impact of localized defects on the scattering budgets. This will be done through the development of a dedicated instrument designed as a SPatially and Angularly Resolved Scatterometry Equipement (SPARSE). After a reminder of the required theoretical background, we present the technical challenges inherent to the design and the implementation of SPARSE. Then, we complete this work with the qualification of the performances and limitations of the resulting instrument and some examples of applications to the measurement of a calibration standard and the global analysis of two mirrors.

## 2. Light scattered by optical surfaces in presence of localized defects and particles contamination - state of the art

Light scattering [13–20] from random roughness in high precision optical components is today well understood and accurately measured. Indeed first-order electromagnetic theories [13,21,22] are perfectly suited for optical substrates and multilayer coatings, and some scatterometers
[5,23,24] record angle resolved scattering in the whole visible and near infrared spectral range with 8 decades of dynamics.

However the situation is different for defects and contaminants which cannot be predicted by first-order theories, due to their height which may approach the wavelength, or to their refractive index whose deviation from the average index can be high. Furthermore, most scatterometers do not allow to separate the contributions of local defects and random roughness in the scattering balance.

Mie theory has been used to model and anticipate issues of light scattering from contamination with spherical particles. For example, the work of Spyak and Wolf [25–28] that has been reworked by Dittman [29] and finally integrated by ESA in a software used in the early phase of Spacecraft development: Stray Lux [30]. The model developed is easily integrated into ray tracing software, allowing for the quantification of a system’s sensitivity to stray light from contamination during its early development phase. This enables designers to proactively plan the level of complexity needed to address contamination.

For defects, standard ISO 10110 [31] describe the specifications process for surface qualities in terms of scratches and digs and the standard ISO 14997 [32] describe how the before-mentioned specifications shall be verified. Form those standards, work has been done to anticipate scattering levels [33] of optical surfaces. But if these norms are well suited for cosmetic requirements, they do not enable to directly specify scattering performances, which can be the expected purpose of such requirements regarding the final use of the components [34]. Academic and industrial actors manufacturing or using high performance optical components are raising concerns that the current estimations of performances and impact from defects and contamination are inconclusive and often quite confusing.

To address this question, our group developed a CCD-ARS set-up a few years ago [35]. Based on a rotating illumination arm coupled with a fixed detection system, the bench was designed to record a spatially resolved measurement of Angular Resolved Scattering (ARS). Whereas this demonstrator validated the reciprocity between detection or illumination for the rotating axis, it didn't allow the comprehensive characterization of a component, mainly due to the presence of blind areas. Moreover, it required a rotating illumination arm, which ended up difficult to operate and align. Since then, other groups have proposed alternative solutions, such as this study performed for the Virgo Collaboration [36] by counting the local defects with dark-field microscopy added with a local ARS pattern measurement for each defect. Whereas this method has proven its effectiveness in improving the quality of the coatings, a full characterization of a component is extremely long (several days) and requires manipulation of the component due to the use of two optical benches. This makes it very difficult to implement for systematic analysis. In the same context, LIGO Collaboration developed an annealing scatterometer. The instrument has the very specific purpose of understanding defect formation during the annealing of amorphous coatings [37]. It allows for the localization of defects on the surface of the sample but the analysis is qualitative and doesn't quantify the absolute intensity of scattered light, neither its angular repartition. On the opposite, the system HOROS (Highly sensitive Optical ROughness and defect Sensor) [38,39] is a scatterometer which allows for a fast recording of the absolute ARS pattern. Coupling the detector with a robotic arm, it can perform spatial scanning and localize some defects but it seems not yet suitable for the analysis of ultra-high performance optical components. First, because the lowest level of detection is $10^{-4}$ sr$^{-1}$ [7] whereas low loss mirrors are expected to be close to $10^{-6}$ sr$^{-1}$ or less than 5 ppm [5,13,36,40]. Moreover, the diameter of the illumination spot on the sample is 2 mm [38] which does not allow isolating scattering from one defect. We can also cite [41,42], they developed a scatterometer with a CCD array as a detector for a focused illumination beam. The principle proposed in this paper is quite different from the CDD-ARS set-up previously developed in [35,43] because the imaging system used for detection is rotating around the sample which remains motionless. Due to this
relative mechanical move, there is no bijection between the CCD pixels and the sub-surfaces of the sample and the angular aperture of the detection is evolving during the angular scan. As a consequence, the recorded signal is not proportional to the ARS [44] and the movements of the camera have to be compensated by post process treatment to extract the cross-scattering section. A numerical optimization allows then to extract dimensional information on the defects.

In this context, we propose a new SPAtially and Angularly Resolved Scatterometry Equipment (SPARSE) designed to perform a fast and direct ARS measurement on a complete 1 inch diameter optical component with a 26×26 µm² spatial resolution.

### 3. Metrology of scattered light - key principles

#### 3.1. Measurement of angle resolved scattering

The purpose of a scatterometer is to measure Angle Resolved Scattering patterns (ARS) [44] or Bidirectional Scattering Distribution Functions (BSDF):

$$ARS(\theta_i, \theta_s, \phi_s, \lambda) = BSDF(\theta_i, \theta_s, \phi_s, \lambda) \cos \theta_s = \frac{1}{\Phi_i} \frac{d\Phi_s}{d\Omega} \quad (1)$$

With $d\Phi_s$ the scattered power into the solid angle $d\Omega$ at direction $(\theta_s, \phi_s)$, which is normalized by the incident power $\Phi_i$ coming from direction $\theta_i$. Those elements are illustrated in Fig. 1. In other words, the ARS is a dimensionless ratio of the scattered power to the incident power, in a given direction. It is given in inverse steradians.

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Schematic representation of the geometric required configuration for the recording of ARS. $\theta_i$ is the angle of incidence, $(\theta_s, \phi_s)$ are respectively the normal and azimuthal scattering angles and $d\Omega$ is the solid angle of detection.</figcaption></figure>

To perform ARS measurements, classical scatterometers [5,6,8–12] illuminate a sample with a fixed illumination angle $\theta_i$ at one wavelength $\lambda$ and have a detector rotating around the sample to collect scattered light as a function of the scattering angles $(\theta_s, \phi_s)$ as illustrated Fig. 2(a). The angular resolution is imposed by the numerical aperture of the detection system: $NA = n_0 \sin(\Delta\alpha)$ where $\Delta\alpha$ is the half-angle of the cone of light entering the detection system. It is related to the solid angle $d\Omega$ on Fig. 1, for small values of $\Delta\alpha$, $\Delta\Omega \approx \pi(\Delta\alpha)^2$.

For optical surfaces considered in this configuration, due to first order approximation [13], the scattered intensity can be expressed as:

$$ARS(\theta_i, \theta_s, \phi_s, \lambda) = ARS(\vec{v}, \vec{v}_i) = C(\vec{v}, \vec{v}_i) \frac{1}{S} |\hat{h}(\vec{v} - \vec{v}_i)|^2 \quad (2)$$

with C an electromagnetic coefficient which does not depend on the surface topography [13], $\hat{h}(\vec{v})$ the Fourier Transform of the rough surface $h(x, y)$, and S the illuminated area. The spatial
<figure><img src="image_4.png" /><figcaption>Fig. 2. Schematic representation of the principle of (a) a classical scatterometer with a fixed illumination angle and a moving detector, (b) a symmetrical scatterometer with a moving illumination angle and a fixed detector and (c) the principle of SPARSE with a matrix detector at a fixed detection angle (0°) and a mobile illumination angle.</figcaption></figure>

frequencies $\vec{v}$ and $\vec{v}_i$ are respectively associated with the scattered and incident light in the incident medium of refractive index $n$, that is:

$$ \vec{v} = \frac{n \sin \theta_s}{\lambda} (\cos \phi, \sin \phi) \quad \text{and} \quad \vec{v}_i = \frac{n \sin \theta_i}{\lambda} (1, 0) \qquad (3) $$

This last formula is given for an incident wave vector in the incidence plane $y = 0$, with $\theta_s$ and $\phi$ the normal and polar (or azimuthal) angles. The evidence is that the argument $\vec{\beta} = \vec{v} - \vec{v}_i$ of $\hat{h}(\vec{\beta})$ in (2) can be scanned either by scanning $\vec{v}$ or $\vec{v}_i$. This reciprocity result is fundamental and very useful. Indeed it allows us to consider a symmetric configuration where the detection is fixed and the illumination is rotating around the sample [35], as illustrated Fig. 2(b). The two configurations (Fig. 2(a) and Fig. 2(b)) are then equivalent.

## 3.2. Spatially resolved measurement of angle resolved scattering

As discussed in sec. 3.1, an ARS measurement can be performed with a fixed detector and a rotating illumination angle relative to the sample. With a fixed detector, this allows using an imaging system whose optical axis is perpendicular to the surface of the component to be tested. The resulting configuration is illustrated in Fig. 2(c). Moreover, detection and illumination lines both have to be designed to meet the requirement of an angularly resolved scatterometer [44]:

* For the detection, the optical system must record the scattered light coming from each point of the sample homogeneously, under the same mean angle for each pixel, and with an angular resolution $\Delta\Omega$ (see Fig. 1) known and controlled.

* For the illumination, the angle of incidence has to be the same for each point of the sample.

The objective is the development of a spatially resolved scatterometer fulfilling all these conditions with an ability to measure ARS as close as possible to $10^{-8} \text{ sr}^{-1}$, which is the limit of measurable signals in air [45,46].

# 4. Instrument SPARSE

## 4.1. Principle

The configuration chosen for SPARSE is a double telecentric system, for the illumination and for the detection, with a Charge Coupled Device (CCD) array for the detection. A schematic representation of the set-up is given Fig. 3. It is constituted in 3 main subsystems, which are first a source block with the management of wavelength and intensity (a), and an illumination line
for the accurate control of the illumination (b). The detection system and the sample holder are
mechanically linked together on a rotating arm (c). We further detail the key principles of each
of them.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** SPAtially Resolved Scatterometry Equipment (SPARSE). The set-up is divided into 3 blocks: The source block (a), the arm for the illumination beam shaping (b) and the detection system (c).</figcaption></figure>

**4.2. Source block**

The source block is sketched Fig. 3(a). It is based on the same principle as the source block previously developed for SALSA scatterometer [6] adapted to a monochromatic-like configuration to reduce the speckle from the coherent laser, which eases the operation of the Digital Micromirrors Device (DMD). The source is a superluminescent diode (SLD635B) with a central wavelength of 635 nm and a bandwidth of 6 nm at -3 dB. The synchronization between a high-speed mechanical shutter and the detection is used as a passive lock-in detection with a minimal acquisition time of 50 ms. Throughout the measurement process, 10% of the power of the superluminescent diode is picked up and recorded with a photodiode to be used as a reference, digitized at 100 kHz over 16 bits, and account for any potential fluctuations in the source power. The use of neutral-density filters on the main beam allows for the automatic adjustment of the light beam intensity. The resulting transmission value T₀ can be selected with an attenuation from 0 to 9 Optical Densities in steps of 1. At last, light is injected in a single-mode fiber to be transported to the beam shaping arm.

**4.3. Illumination beam shaping**

SPARSE is based on the principle of a rotating illumination. Hence by increasing the incidence
angle, the illumination pattern would grow on the sample following an extension of factor cos θᵢ
in the rotation plane. This has two main consequences: the reduction of the illumination density
and the possible illumination of the edges of the sample, which would lead to parasitic light
increasing the noise of the system. To compensate for the enlargement of the illuminated area,
the shape of the illumination is tightly controlled using a projection system composed of a Digital
Micromirror Device (DMD) and a 4F illumination set-up with a filter aperture at the Fourier
Plane, see Fig. 3(b).
The DMD is an array of 1024×768 micro-mirrors whose pixel size is 13.7 µm. A collimated beam with a diameter of 11 mm at $1/e^2$ illuminates it. Then, the 4F optical system images the pattern of the DMD onto the surface of the sample. The magnification $g = f_2/f_1 = 2.5$ is selected so that the DMD can fully illuminate the sample with a spatial resolution compatible with the resolution of the detection system. The filtering aperture is chosen to control the impulse response as defined in [47]. In this configuration, we have a bijection between 1 micro-mirror of the DMD and 1.3 pixel×1.3 pixel of the CCD array used for detection when they are in line.

The last step for the illumination is to define the command of the DMD. The DMD is used to control the illumination pattern and morphing it so that for any illumination angle, the projection pattern on the sample stays the same. Since an image of the DMD is projected on the sample and because the sample can be itself imaged by the camera, it is possible to measure the affine image transformation between the DMD and the projection on the sample. For every illumination angle, two matrices $A$ of linear transformation and $B$ of an offset can be measured to describe the affine transformation between the DMD and the sample plane. Figure 4(a) shows how matrices $A$ and $B$ are measured. Once this transformation is quantified, the inverse transformation can be defined to generate a custom illumination pattern to command the DMD see Fig. 4(b). Moreover, due to the geometrical configuration of the bench, the CCD-array avoid the specular beams reflected by the DMD and thus avoid any question regarding a possible overlapping of the detection and illumination grids. So, the sample is illuminated with the same pattern at every illumination angle. Finally, in order to have constant illumination all over the sample, the DMD is also used to temporally shape the intensity of the beam. The incoming collimated beam has a Gaussian profile and an inverted mask of this profile can be built for the DMD so that it temporally compensates for the Gaussian beam into a top-hat profile. Since the quickest exposure time is around 500 ms and the refresh rate of micro-mirrors of the DMD is around 40 µs, the constrains on the temporal average are quite weak.

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** On (a) it is shown the affine transformation morphing the DMD pattern on the sample. On (b) how this transformation can be inverted to project a custom illumination pattern on the sample. In this case, it is the "shell-like" pattern used for nominal 1 inch diameter components</figcaption></figure>

### 4.4. Detection system

The detection system combines a scientific camera with a telecentric objective, as illustrated Fig. 3(c). The camera is a PIXIS 1024B with a back-illuminated CCD array of 1024×1024 pixels. The pixels are 13 µm sided squares with 100% fill factor across the CCD and a full well capacity of $1.35 \times 10^5$ e$^-$. The temperature of the sensor is regulated to -70°C using a thermoelectric cooler. Thanks to this cooling, the dark current is limited to $2.5 \times 10^{-3}$ e$^-$/pixel$^{-1}$s$^{-1}$. The
digitization of the CCD data is over 16 bits with a low read out noise $3.9 \times 10^{-3}$ rms for a conversion gain of 2. The telecentric lens mounted on the camera is critical for this application. With it, the camera images the sample without any distortion due to a field of view. For every point on the sample, the light collected by the CCD is a cone of light scattered in the direction of all parallel chief rays, perpendicular to the CCD ($\theta_s = 0^\circ$) with a half-divergence angle $\Delta\alpha$. The half-divergence of the scattered beam collected by the system is $\Delta\alpha = \arctan(G/2N)$. The chosen objective has a magnification $G = 0.5$, the aperture number $N$ is set to 10.2 so the half-divergence $\Delta\alpha$ is around $1.4^\circ$. The working distance of the objective is 173 mm. The measurement is always done with the optical surface of the sample perpendicular to the optical axis of the telecentric lens. The detection system and sample holder are mechanically bounded. They both rotate around a vertical axis that coincides with the intersection of the optical axes of the detection and the illumination and with the front face of the sample.

# 5. Measurement method

## 5.1. Sample installation

To be characterized, a sample is positioned so that its front face is on the rotation center of the goniometric arm. The DMD is used to project a test pattern on the sample. It is considered as aligned when the center of the test pattern does not move during a rotation. SPARSE is designed to measure a complete circular 25 mm diameter sample except a 2 mm crown and 6 mm band at the bottom hidden by the holding system. In Fig. 5(a) and Fig. 5(b), we can see, respectively, a protected silver mirror and a Spectralon Diffuse Reflectance Standard mounted on SPARSE with the illuminated area highlighted red. The spatial shaping is tuned to illuminate a constant "shell-like" pattern on the sample whatever the illumination angle. Figure 5(b) is taken by the instrument's camera.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Samples mounted on SPARSE with a highlighted red pattern superimposed to visualize the illumination. (a) Picture of a protected silver mirror mounted on SPARSE taken with a cell phone. (b) Image of a Spectralon Diffuse Reflectance Standard recorded with the SPARSE camera (1024×1024 pixels) and the telecentric lens.</figcaption></figure>

## 5.2. Elementary acquisition

A classical acquisition is an angular scan of a sample for the recording of the ARS. The camera takes images of the sample while it is being illuminated with an angle of incidence $\theta_i$ in [15°, 85°] with an angular step of 1°. For every angle, a set of images is recorded, each one with the
following process for a given exposition time $\tau$ which is synthesized in the time chart given
Fig. 6:

* The shutter of the camera opens at time $t_0$ for ($\tau + 2 \times 100$) ms. Nothing is illuminated yet.

* The reference photodiode in the source block starts recording the illumination beam power with a sample rate of 100 kHz. It does not receive light yet. It will record the signal as long as the camera shutter is open for ($\tau + 2 \times 100$) ms.

* After 100 ms the shutter of the source block opens for the chosen exposition time $\tau$, illuminating the sample and the reference photodiode.

* While the photodiode records the illumination and potential laser power fluctuations, the CCD collects photons scattered from the sample.

* The shutter of the source block closes.

* After 100 ms the shutter of the camera closes.

* The photodiode stops recording.

* The image recorded by the CCD is saved.

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Chronogram of a complete acquisition for an integration time $\tau$.</figcaption></figure>

5.3. *Processing of elementary acquisition*

The data are processed with a method similar to the method developed for SALSA [6,23]. First, the dark acquisition, $m_{dark}$, which corresponds to an offset on the analog to digital converter set by the manufacturer, is subtracted to the acquisition image $m$ so that the sensors behave linearly. The signal recorded by the CCD, ($m - m_{dark}$) is proportional to the number of photons $N$ received by each pixel via the gain $G$ of the CCD and its quantum efficiency $\eta$. For the SPARSE camera, $G = 2.09 \text{ e}^{-\text{ADU}^{-1}}$ and $\eta = 0.85$ at $\lambda = 635$ nm.

Then, the temporal evolution of the signal recorded by the reference photodiode is processed for every acquisition. The signal recorded while the main shutter is closed is used to evaluate the dark current, which is subtracted to the rest of the reference dataset. Then the measurement is integrated so that it becomes proportional to the illumination power used to illuminate the sample. The resulting value $S_{ref}$ is saved and used to compensate for possible fluctuations of the illumination power during measurement whatever the integration time. The signal processed for each pixel $(x_i, y_i)$ of the CCD is noted $s(x_i, y_i)$ and defined Eq. (5). Matrix $s$ is a 1024×1024 matrix
proportional to the ARS for every pixel via the calibration coefficient $\kappa$ whose determination is described in detail in Sec 5.5. In summary we have:

$$s(x_i, y_i) = \frac{1}{S_{ref}} [m(x_i, y_i) - m_{dark}(x_i, y_i)] = \frac{1}{S_{ref}} \frac{N(x_i, y_i) \times \eta}{G} \quad (4)$$

$$= \frac{1}{\kappa} \text{ARS}(x_i, y_i) \quad (5)$$

A pixel is considered valid if the signal is less than 90% of the CCD full well capacity (121 500 electrons (e⁻) per pixel). Pixels receiving more photons are considered saturated and removed from the dataset. To mitigate the consequences of the blooming effect, the eight pixels surrounding a saturated pixel are also ignored. At last, for each pixel of the CCD (xᵢ, yᵢ), the signal-to-noise ratio (SNR) is calculated as:

$$\text{SNR}(x_i, y_i) = \frac{N(x_i, y_i) \times \eta}{\sqrt{\sigma_m^2 + \sigma_{DC}^2 + \sigma_R^2}} = \frac{N(x_i, y_i) \times \eta}{\sqrt{N(x_i, y_i) + \tau i_{DC} + \sigma_R^2}} \quad (6)$$

With $\sigma_m$ the variance of the shot noise equal to the square root of the number of photons, $\sigma_{DC}$ the number of photoelectrons generated during the time $\tau$ by the dark current $i_{DC} = 2.5 \times 10^{-3} \text{ e}^{-} \text{ s}^{-1}$ and $\sigma_R = 3.89 \text{ e}^{-}$ the read-out noise.

The noise for a low photon count, near an SNR of one, is mostly due to the $\sigma_r$ then, as the photon count increases, $\sigma_m$ is predominant. For longer exposure times, which are not shown here, the dark current noise results in a less significant improvement in SNR for low photon counts, while dramatically increasing the measurement time. Hence, we have set the maximal exposure time to 500 s.

On this basis, we define, for a given ARS level, the expected signal-to-noise ratio on one pixel regarding the acquisition time. An extraction of the resulting abacus is plotted Fig. 7. First, we can see that an ARS of $10^{-7} \text{ sr}^{-1}$, corresponding to 0.1 ppm, is measurable with an SNR of 2 for an integration time of 500 s per incidence angle. Another point to notice is the accessible dynamic range. The objective is to allow the full characterization of a high quality sample, such as a superpolished mirror with a nominal ARS of $10^{-6} \text{ sr}^{-1}$ with at least one scattering defect, whose ARS is $10^{-1} \text{ sr}^{-1}$. According to Fig. 7, the measurement of the subsurfaces without and with the defects with an SNR >2 will not be possible by using the same integration time. This led us to the definition of a dedicated High Dynamic Range (HDR) procedure to allow SPARSE to cover a wider range of ARS signal values.

## 5.4. High dynamic range

As expressed in Sec. 3.2, 8 decades of dynamic can be required to measure a classical ARS on a complete sample. On the other hand, as demonstrated in Sec. 5.3, this is not possible with a single measurement, which is limited to 4 decades whatever the experimental conditions due to 16 bits digitization of the CCD signal. For this reason, we adapted the High Dynamic Range (HDR) process to SPARSE acquisitions. For each illumination angle, multiple images are recorded with different integration times to allow the collection of enough photons for every pixel with an acceptable signal-to-noise ratio.

As illustrated in Fig. 7, short exposure times allow the recording of light scattered by bright defects without saturation and the longer times allow the measurements of low scattering levels with acceptable SNR. For the acquisition of low scattering pixels, to keep a reasonable measurement duration as well as an acceptable SNR, the limit is fixed as either SNR ≥ 10 or $\tau = 500 \text{ s}$ depending on the first limit reached. A synthesis of the decision-making tree for the HDR algorithm is given Fig. 8 with three integration times: $\tau_1, \tau_2, \tau_3$. The number of shots and the corresponding acquisition times are adapted automatically by optimization software for every
<figure><img src="image_9.png" /><figcaption>Fig. 7. Signal-to-Noise Ratio (SNR) for a pixel as a function of the scattering level in terms of ARS for different integration times τ. SNR is plotted for τ =0.5 s, 5 s, 50 s and 500 s. Dashed lines are for the delimitation of the useful area upper limit (red line) is for the maximum full well capacity of a single pixel and lower limit (blue line) is for a unitary SNR. The color filled areas are for the possible configurations of measurements with acceptable SNR > 2 for the targeted ARS for a superpolished surface 10⁻⁶ sr⁻¹ and defects 10⁻¹ sr⁻¹</figcaption></figure>

acquisition. For each pixel of the three matrices, the SNR is calculated with Eq. (6). Only valid pixels are kept and they are all processed following Eq. (5). The exploitable data are averaged and the result is, for each scattering angle, a mapping $\bar{s}(x_{CCD}, y_{CCD})$ which has to be multiplied by the calibration constant of the scatterometer to be homogeneous to an ARS.

<figure><img src="image_10.png" /><figcaption>Fig. 8. Decision-making tree for the definition of HDR images</figcaption></figure>

5.5. Calibration

The calibration of SPARSE instrument is performed in a classical way for scatterometers, with the measurement of a Spectralon Diffuse Reflectance Standard whose ARS is a known Lambertian law which is defined for detection in the normal direction and a rotation of the angle of incidence as :

$$ \text{ARS}_{\text{lamb}}(\theta_i, \theta_s = 0) = \frac{\rho}{\pi} \cos \theta_i \qquad (7) $$

with $\rho = 0.98$, the albedo of our calibration sample at 635 nm.

Figure 9(a) shows the measurement performed on the central pixel ARS(510, 510) = $\kappa(510, 510) \times s(510, 510)$ fitted with its theoretical response. The calibration constant $\kappa$ is
tuned on each pixel to optimize the fit on the basis of least square criteria. The indicator for the adjusted regression $R_{adj}^2$ defined by Eq. (8) quantifies the quality of the fit over the illuminated surface. If the fit is perfect, $R_{adj}^2 = 1$. Its spatial evolution is plotted Fig. 9(b). We can see that the quality of the calibration is very good: $R_{adj}^2 > 98\%$ on more than 95% of surface the calibration standard.

$$R_{adj}^2 = 1 \left( \frac{N-1}{N-P} \right) \frac{SS_{res}}{SS_{tot}} \qquad (8)$$

With:

* $SS_{res}$ the sum of squared regression $SS_{res} = \sum_{n=0}^{N}(y_n - f_n)^2$

* $SS_{tot}$ the sum of squared total $SS_{tot} = \sum_{n=0}^{N}(y_n - \bar{y})^2$

* N the number of observations. In our case $n = 37$

* P the number of regression coefficients. In our case $P = 2$

<figure><img src="image_11.png" /><figcaption>Fig. 9. (a) Comparison of the calibrated ARS measured on a Spectralon standard (blue) and its theoretical response (red). The data are for a central pixel: (510,510). (b) Mappings of the adjusted r-square indicator $R_{adj}^2$, see Eq. (8), representative of the quality of the calibration. The 400 000 pixels are calibrated independently.</figcaption></figure>

The resulting calibration constant $\kappa$ is homogeneous with a variability of less than 10% for 96% of the surface. This variability is the major source of possible errors on the calibrated ARS. As it is directly related to the inhomogeneity of the calibration sample, it can be compensated by multiple calibration measurements with different orientations of the Spectralon when it is required by the application.

To give an overview of the operations required for a SPARSE measurement presented in section 5, a simplified block diagram of data acquisition and treatment is given in Fig. 10.

To summarize, SPARSE is a spatially resolved scatterometer designed to perform $ARS(\theta_i)$ measurements with high sensitivity down to $10^{-7} \text{ sr}^{-1}$. It is optimized to measure the ARS of 25 mm diameter area whatever the shape and thickness of a plane sample with HDR. The scatterometer records up to 400 000 ARS curves corresponding to the same number of $26 \times 26 \text{ µm}^2$ subsurfaces on the component. The illumination wavelength is 635 nm using a superluminescent diode. The measurement sequence as well as the data processing are fully automated. We can notice, at this step, that the gap is huge in terms of performances between SPARSE and its ancestor, the CCD-ARS set-up [35]. Thanks to the DMD, there are no blind areas, neither on the
<figure><img src="image_12.png" /><figcaption>Fig. 10. Simplified block diagram of a data acquisition and treatment of SPARSE for a single illumination angle.</figcaption></figure>

edges, nor close to the defects. Then, thanks to the upgraded source block and the HDR procedure, we can retrieve ARS signals from every illuminated element, including highly scattering defects and low scattering roughness.

## 6. Experimental results

In this section, we illustrate the capabilities of SPARSE instrument on three representative samples: a plate designed to calibrate defects on optical surfaces following the norm ISO 10110, then two mirrors with different categories of defects.

### 6.1. Calibrated dig on ISO plate for the analysis of optical surfaces quality

The norm ISO 10110 [31] normalizes the specification of surface quality with a focus on the description of acceptable digs and scratches on optical surfaces. The norm ISO 14997 [32] normalizes the verification process of ISO 10110 specifications. This verification can be performed by visual inspection comparing the identified defects to evaluation targets with known feature size. The considered sample is one of these NIST traceable scratches and digs standards, as described in [48] and shown in Fig. 11(a). The standard is divided into six regions, referenced by a grade, which is the square root of the total defect area for a region. For each of them, the three circles on the left are three chrome plots with a calibrated scratch, dig and grid, respectively. The ARS of grade #0.006 region is measured with SPARSE between 15° and 85° with an angular step of 1°. To locate a defect, we can compute the Integrated Scattering (IS) of a microsurface.

$$IS(x_i, y_i) = \int_{15^\circ}^{85^\circ} \mathrm{ARS}(x_i, y_i) \sin(\theta_s) d\theta_s \quad (9)$$

It is a dimensionless value that indicates the microsurface's overall scattering intensity. The mapping of the resulting IS is plotted Fig. 11(b) and (c). Thanks to the DMD, the illumination shape is chosen to not illuminate the grating in order to avoid the high intensity signal generated by the diffraction orders that would induce parasitic light. Although the studied defect in the center of Fig. 11(c) is smaller than a pixel, the IS increase is visible across multiple pixels. This is due to the instrument's Point Spread Function (PSF). Since the image is displayed on a log scale, the impact of the PSF is prominently displayed.
<figure><img src="image_13.png" /><figcaption>**Fig. 11.** (a) Photography of the first three grades of the ISO 10110-7 & ISO 14997 plate 1 (NIST traceable scratch & dig Standards). The grade #0.006 region is highlighted in red. (b) Mapping of the Integrated Scattering (IS) measured for the grade #0.006 regions with SPARSE excluding the grating. (c) zoom centered on the dig feature marked with the black cross. The ARS measurement of the pixel behind the black cross is plotted Fig. 12.</figcaption></figure>

The ARS recorded for the subsurface which includes the calibrated dig is plotted Fig. 12 in blue. The ripples in the angular response are representative of the presence of the diffractive object. The expected ARS for a singular feature is proportional to the diffraction pattern of the defect in the incidence plane with slight perturbation due to additional roughness (see Fig. 6 in [35] for the numerical proof).

<figure><img src="image_14.png" /><figcaption>**Fig. 12.** The ARS measured from the dig feature on the ISO plate (blue), see Fig. 11, compared to another region free of features (black), and the theoretical Airy pattern from a cylinder with an arbitrary height (red).</figcaption></figure>

For comparison, the ARS measured on a nearby elementary surface is plotted in black. This second pattern is characteristic of an optical surface with random nanometric roughness. In this case, the ARS is similar to the expected ARS for a larger surface with the same roughness spectrum (see Fig. 4(a) in [35]).

The dig considered is a chrome-deposited cylinder with a height H around 35 nm (measured with optical profilometry) and a diameter 2R = 7 µm. As described in [4,35], for a circular localized defect of radius R on an optical surface, the ARS is proportional to the Airy pattern of
<figure><img src="image_15.png" /><figcaption>Fig. 13. (a) Integrated Scattering (IS) mapping measured on mirror #A. (b) Three detailed view of mapping (a) corresponding to the squared areas. (c) ARS measured on the three pixels from the middle of the snapshots (b) marked by a black cross.</figcaption></figure>

the corresponding circular aperture. So, the scattered intensity can be written as:

$$ARS = \beta \left| \frac{J_1(R 2\pi\nu)}{R 2\pi\nu} \right|^2 \qquad (10)$$

with $\beta$ a scattering coefficient, which is a function of the geometrical parameters of the rod, height and diameter, and its optical index [4]. As the standard is calibrated for the diameter of the rod but not its height, we have chosen to consider only the angular ripples for the curve fitting and not the absolute scattering level. The pattern calculated for a 7 µm diameter rod and an arbitrary coefficient $\beta$ is plotted Fig. 12 in red. The agreement between measurement and simulation is high and confirms that, thanks to the telecentric system, SPARSE is able to measure light scattered by a micrometric dig on a rough surface. Note that the differences at large angles result from the fact that the Airy pattern is not an exact prediction of the rod ARS. To conclude this section, we stress on the fact that the level of an ARS curve measured on a single defect with SPARSE does not depend on the pixel size. In other words, if we had a unique single isolated defect over the whole sample area, this ARS curve would be identical to that measured with a classical scatterometer. Hence the strength of SPARSE lies in its ability to observe defects as if they were alone on the sample, which is not possible with a classical scatterometer.

## 6.2. Application to the analysis of mirrors

### 6.2.1. Localized defects

To illustrate the capabilities of SPARSE instrument, we now consider two silver protected mirrors #A and #B with a 25 mm diameter. For each of them, the angular acquisition range is from 15° to 85° with an angular step of 1°. For every illumination angle, an HDR acquisition has been made.

Figure 13(a) shows a 2D rendering of the IS map for the mirror #A. The color map is in log scale, so a value at -2 means a IS of $10^{-2}$. Three representative areas of 1.1×1.1 mm² have been chosen and plotted Fig. 13(b). Each of them counts 1 764 subsurfaces (42×42) and the ARS measured for the central subsurface marked with a black cross is plotted Fig. 13(b) with the same color as the limitation of the corresponding square. The green ARS is representative of the presence of a localized defect with a clear diffraction pattern. The red plot indicates the presence
<figure><img src="image_16.png" /><figcaption>**Fig. 14.** (a) IS mapping measured with SPARSE on mirror #A. (b) Topography measured on mirror #A by optical profilometry on a 4×4 mm² area with a spatial resolution of 1.08 µm, the measured surface is the same as (a). (c) Histogram of the local IS measured by SPARSE and plotted on (a) with visualization of the threshold. (d) Histogram of the local heights measured by optical profilometry and plotted on (b) with visualization of the threshold. (e) Localization of the defects identified by the profilometer (red) and the scattering areas identified by SPARSE (dark contours)</figcaption></figure>

of a defect, which generates a lot of scattered light. It could be a cluster, a big dig or a particle with irregular shape. The blue curve is a classical ARS representative of an optical surface without any defect. As expected, the IS value allows identifying the presence of a defect on the measured subsurface, and the angular signature of the ARS is characteristic of the nature of the defect. This allows extracting the localization of the scattering points from a single measurement. Such extraction requires defining a threshold above which light is considered to originate from random roughness or localized defect. For stray light analysis, this gives results more suited to the application than the processing of microscope or profilometer images.

To illustrate this point, we measured the surface profile of mirror #A, with an optical profilometer (White Light Interferometer Zygo New View 7300), on a stitched area of 4 mm×4 mm with a resolution of 1.08 µm. Each pixel measured by SPARSE is in bijection with 24×24 pixels of the topography measured by profilometer and the two resulting mappings are proposed Fig. 14(a) and Fig. 14(b).

The histograms of IS values measured with SPARSE, and local heights measured by profilometry, are plotted respectively Fig. 14(c) and Fig. 14(d). The mean value measured for IS is 2.65 × 10⁻⁵ with a standard deviation of 3.22 × 10⁻⁵ on the 22 801 subsurfaces measured. We define a threshold above which scattering is considered to originate from local defects that may be problematic. For this illustration, this value is arbitrary chosen as 8 × 10⁻⁴ though it can be modified regarding the application. The areas where scattering exceeds the threshold are delimited with black lines Fig. 14(e). Regarding the profilometric image, the points are considered as defects if their height is more than 5 times the root mean square roughness which is 0.6 nm. Hence, every point with a local height higher than 3 nm (|h| ≥ 3 nm) is marked with a red cross Fig. 14(e). We note again that this threshold is arbitrary chosen to allow the comparison with the defects detected by SPARSE, though it is very low regarding classical microscopic analysis [36,49,50].
<figure><img src="image_17.png" /><figcaption>**Fig. 15.** IS in log scale extracted from the ARS measurements of mirror #B with SPARSE for two perpendicular orientations of the component (a) and (b). Measurement (b) is numerically rotated *a posteriori* with an angle of 95°, defined to superimpose the three defects marked with triangles. (c) is the half sum of the common areas of (a) and (b). The color scale is weighted by the origin of scattered light between (a) and (b). A pixel with the same IS on (a) and (b) is plotted with a gray scale, while a pixel visible with a scan in the horizontal plane and not visible in the vertical plane is plotted in magenta.</figcaption></figure>

Looking accurately at the comparative map Fig. 14(c), we observe that the majority of defects detected by the profilometer are also detected with SPARSE, though not all of them. On the other hand, a large number of scattering microsurfaces are detected by SPARSE but not by profilometry. In a first step this statement could be quantified as follows: a qualification of the component based on the profilometer may highlight the fact that 0.046% of the surface is covered with defects, whereas SPARSE indicates that more than 2.52% (that is, an increase by a factor 60) of the surface generates additional scattered light. However this statement must be corrected by the fact that each SPARSE pixel includes 24×24=576 Zygo pixels. Taking account of this last remark, one would expect the profilometer to detect more defects than SPARSE, which is not the case. We do not go any further in this analysis, as it depends on the threshold; future work will follow on this key topic.

To be complete, the mean IS of the scattering points is $1.06 \times 10^{-3}$ and the total surface area stripped of defects has a mean IS of $1.44 \times 10^{-4}$. This value has to be compared with the initial mean IS of $1.67 \times 10^{-4}$ which takes into account the whole surface, including the scattering points. In this case, 2.52% of the surface is responsible of an increase 16.2% of the mean IS value.

At this stage, we can deal with the specific case of defects larger than the pixel size. By definition, these are no longer diffracting objects at the wavelength at which the instrument is used. They are therefore measured as local sub-surfaces with an ARS different from the background of the global surface and the ARS of the whole defect can be extracted as the mean value of the ARS of all the considered sub-surfaces. In addition, in the specific case where there is an edge on the imaged subsurface, the ARS measured is proportional to the diffraction pattern of the edge. This is the case, for example, with scratches, whose analysis also requires specific processing because of their asymmetry.

### 6.2.2. Scratches

The second mirror (#B) presents some scratches so that a specific processing is required to take account of the anisotropic nature of the sample. Scratches are expected to scatter more in the direction perpendicular to their long dimension. In order to study the two largest scratches visible to the naked eye, two scans were performed with the instrument, where the sample in the sample
<figure><img src="image_18.png" /><figcaption>**Fig. 16.** Left: Zoom of the superposition of the IS data plotted Fig. 15(c). Middle: zoomed out snapshot from the left. Right: ARS of the center pixels of the snapshot in green and magenta for the two acquisitions in different orientations. In the referential of the sample, vertical and horizontal angular scans are respectively plotted in green and magenta.</figcaption></figure>

holder was rotated by around 95° between the two scans. The corresponding IS maps are plotted respectively Fig. 15(a) and Fig. 15(b). For mapping (b), the plot is reoriented to facilitate visual comparison, but the scan was performed along the indicated arrow. Three markers on each map were chosen as references for the *a posteriori* numerical rotation of the sample (b); they mark pixels with higher IS values that are easily identifiable on both measurements.

On the Fig. 15(c) we see the IS values of the two dataset recombined by software with the addition of the two colormaps (green and magenta). The aim here is to provide a qualitative map that shows whether or not there is a preferred direction of scattering. The quantitative analysis will then be carried out by comparing the ARS. On Fig. 15(c), if the IS values of both measurements are similar, the pixels appear gray. Indeed, magenta and green of equal intensity produce a shade of gray. If a IS value is higher in one measurement than in another, then the corresponding pixel appears green or magenta, depending on which measurement it comes from. For example, if the ARS measured with the horizontal scan (Fig. 15(a)) is two orders of magnitude higher than the ARS measured on through the vertical axis by the same pixel (Fig. 15(b)), the corresponding subsurface is plotted in magenta Fig. 15(c).

We can notice that the background IS, which is representative of the random roughness, presents comparable values for the two orientations. It can also be noted that various defects have similar IS values in both measurements. They appear as shades of black on the combined image. Scratches, however, scatter more in one direction than the other and therefore appear either green or magenta.

Complementary information can be provided by the analysis of the angular patterns of each subsurface. A zoom of Fig. 15(c) is plotted in Fig. 16 with additional zoomed-in views of three areas representative of three configurations: a large and punctual defect (1), a scratch scattering with the green scan (2) and the background roughness (3). The ARS measured, in both orientation, of the central pixel of the zoomed areas are plotted on Fig. 16-right. As expected, the ARS measured for the large defect (1) and for the roughness (3) are similar in the vertical and horizontal planes; this is representative of isotropic scattering. On the other hand, a scratch (2) generates the largest amount of scattered light when its longest axis is perpendicular to the plane of the angular scan. At the opposite, it cannot be discriminated from the background roughness
if the long axis is parallel to the angular scan. This is due to the anisotropic nature of the scratch, which results in a non-symmetrical diffraction pattern in space. To be quantitative, for this scratch, the ARS for $\theta_i = 20^\circ$ is $1.5 \times 10^{-2} \text{ sr}^{-1}$ in the horizontal plane and $3.0 \times 10^{-4} \text{ sr}^{-1}$ in the vertical plane. We can see here that, in the presence of scratches, the distribution of stray light depends on their orientation. Consequently, to be relevant, the characterisation of a sample must be carried out for the position in which it will be used for the application.

## 7. Conclusion

We proposed a new and original instrumentation dedicated to the spatially resolved metrology of light scattered by optical components. SPARSE, the SPatially and Angularly Resolved Scatterometry Equipment developed by our group, can now record more than 400 000 ARS patterns corresponding to the same number of subsurfaces with area $26 \times 26 \ \mu\text{m}^2$. After a presentation of the principle of the set-up and its description, we used SPARSE for the optical analysis of representative optical components. The results highlight the instrument's ability to measure and discriminate between scattering losses due to roughness and those due to localized defects. If a defect is detected, a complementary measurement in a second orientation then allows to classify the defect as a dig or a scratch. In addition to the bench's capabilities, this study highlights the need to adapt the specifications of the optics in terms of scattering and/or contamination to the future use of the components. SPARSE appears to be a powerful tool for establishing these criteria. The upcoming evolution of the bench will allow scanning samples at different wavelengths in order to quantify the chromaticity of defects. Another improvement will be the increase of the illumination power to gain one order of magnitude on the level of the floor measurement. Finally, we presented a comparison with results issued from optical profilometry. Some questions remain open and will be analyzed with respect to the scattering and height thresholds that were used for both techniques.

**Funding.** Agence Nationale de la Recherche (IDEC ANR-21-ESRE-0002, X-LOSM ANR-23-CE08-0027); Thales Group; Commissariat à l'Énergie Atomique et aux Énergies Alternatives.

**Acknowledgments.** The authors gratefully acknowledge the support of the European Gravitational Observatory (EGO) within the context of the Virgo Collaboration. SPARSE is part of the DIFFUSIF platform (https://fresnel.fr/diffusif) as scattered light metrology instrument. The authors thank Matthis Bourgeois for his contributions in the processing of the profilometer images.

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** 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.

## References

1. C. F. Bohren and D. R. Huffman, *Absorption and scattering of light by small particles* (Wiley, New York, 1983).

2. A. Egel, K. M. Czajkowski, D. Theobald, et al., “SMUTHI: A python package for the simulation of light scattering by multiple particles near or between planar interfaces,” *J. Quant. Spectrosc. Radiat. Transf.* **273**, 107846 (2021).

3. E. C. Fest, *Stray light analysis and control* (SPIE Press, Bellingham, Washington, 2013).

4. S. Maure, G. Albrand, and C. Amra, “Low-level scattering and localized defects,” *Appl. Opt.* **35**(28), 5573 (1996).

5. M. Fouchier, M. Zerrad, M. Lequime, et al., “Wide-range wavelength and angle resolved light scattering measurement setup,” *Opt. Lett.* **45**(9), 2506 (2020).

6. M. Lequime, M. Zerrad, and C. Amra, “Breakthrough spectrophotometric instrument for the ultra-fine characterization of the spectral transmittance of thin-film optical filters,” *Opt. Express* **26**(26), 34236 (2018).

7. M. Trost and S. Schröder, “Roughness and Scatter in Optical Coatings,” in *Optical Characterization of Thin Solid Films*, (2018).

8. D. R. Cheever, F. M. Cad, K. A. Klicker, et al., “Design Review Of A Unique Complete Angle Scatter Instrument (CASI),” (San Diego, CA, United States, 1987), p. 13.

9. C. Amra, D. Torricini, and P. Roche, “Multiwavelength (045–106 µm) angle-resolved scatterometer or how to extend the optical window,” *Appl. Opt.* **32**(28), 5462 (1993).

10. T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” *Rev. Sci. Instrum.* **70**(9), 3688–3695 (1999).
11. A. Von Finck, T. Herffurth, S. Schröder, et al., "Characterization of optical coatings using a multisource table-top scatterometer," Appl. Opt. 53(4), A259 (2014).

12. A. Von Finck, T. Herffurth, A. Duparré, et al., "International round-robin experiment for angle-resolved light scattering measurement," Appl. Opt. 58(24), 6638 (2019).

13. C. Amra, M. Lequime, and M. Zerrad, *Electromagnetic optics of thin-film coating* (Cambridge university press, 2021), 1st ed.

14. C. Amra, "Light scattering from multilayer optics I Tools of investigation," J. Opt. Soc. Am. A 11(1), 197 (1994).

15. S. Schröder, A. Duparré, and A. Tünnermann, "Roughness evolution and scatter losses of multilayers for 193 nm optics," Appl. Opt. 47(13), C88 (2008).

16. J. E. Harvey, "Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles," Opt. Eng. 51(1), 013402 (2012).

17. C. Amra, "From light scattering to the microstructure of thin-film multilayers," Appl. Opt. 32(28), 5481 (1993).

18. C. Amra, "First-order vector theory of bulk scattering in optical multilayers," J. Opt. Soc. Am. A 10(2), 365 (1993).

19. M. Zerrad, C. Deumié, M. Lequime, et al., "An alternative scattering method to characterize surface roughness from transparent substrates," Opt. Express 15(15), 9222 (2007).

20. S. Schröder, T. Herffurth, H. Blaschke, et al., "Angle-resolved scattering: an effective method for characterizing thin-film coatings," Appl. Opt. 50(9), C164 (2011).

21. J. M. Elson, J. P. Rahn, and J. M. Bennett, "Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties," Appl. Opt. 22(20), 3207 (1983).

22. C. Amra, C. Grèzes-Besset, and L. Bruel, "Comparison of surface and bulk scattering in optical multilayers," Appl. Opt. 32(28), 5492 (1993).

23. S. Liukaityte, M. Lequime, M. Zerrad, et al., "Broadband spectral transmittance measurements of complex thin-film filters with optical densities of up to 12," Opt. Lett. 40(14), 3225 (2015).

24. S. Schröder, T. Herffurth, M. Trost, et al., "Angle-resolved scattering and reflectance of extreme-ultraviolet multilayer coatings: measurement and analysis," Appl. Opt. 49(9), 1503 (2010).

25. P. R. Spyak and W. L. Wolfe, "Scatter from particulate-contaminated mirrors. part 1: theory and experiment for polystyrene spheres and lambda=0.6328 um," Opt. Eng. 31(8), 1746 (1992).

26. P. R. Spyak and W. L. Wolfe, "Scatter from particulate-contaminated mirrors. part 2: theory and experiment for dust and lambda=0.6328 um," Opt. Eng. 31(8), 1757 (1992).

27. P. R. Spyak and W. L. Wolfe, "Scatter from particulate-contaminated mirrors. part 3: theory and experiment for dust and lambda=10.6 um," Opt. Eng. 31(8), 1764 (1992).

28. P. R. Spyak and W. L. Wolfe, "Scatter from particulate-contaminated mirrors. part 4: properties of scatter from dust for visible to far-infrared wavelengths," Opt. Eng. 31(8), 1775 (1992).

29. M. G. Dittman, "Contamination scatter functions for stray-light analysis," (Seattle, WA, 2002), p. 99.

30. F. Mariani, J. Caron, M. Taccola, et al., "StrayLux: an efficient tool for stray-light modelling in optical instruments," in *International Conference on Space Optics — ICSO 2018*, N. Karafolas, Z. Sodnik, and B. Cugny, eds. (SPIE, Chania, Greece, 2019), p. 296.

31. "ISO-10110-7:2017 Optics and photonics — Preparation of drawings for optical elements and systems Part 7: Surface imperfections," (2017).

32. "ISO-14997:2017 Optics and photonics — Test methods for surface imperfections of optical elements," (2017).

33. G. L. Peterson, "A BRDF model for scratches and digs," (San Diego, California, USA, 2012), p. 84950G.

34. E. Herman, D. M. Aikens, and R. N. Youngworth, *Modern Optics Drawings The ISO 10110 Companion* (SPIE Press, 2021).

35. M. Lequime, M. Zerrad, C. Deumié, et al., "A goniometric light scattering instrument with high-resolution imaging," Opt. Commun. 282(7), 1265-1273 (2009).

36. S. Sayah, B. Sassolas, J. Degallaix, et al., "Point defects in IBS coating for very low loss mirrors," Appl. Opt. 60(14), 4068 (2011).

37. M. Rezac, D. Martinez, A. Gleckl, et al., "Imaging scatterometer for observing in situ changes to optical coatings during air annealing," Appl. Opt. 62(7), B97 (2023).

38. T. Herffurth, S. Schröder, M. Trost, et al., "Comprehensive nanostructure and defect analysis using a simple 3D light-scatter sensor," Appl. Opt. 52(14), 3279 (2013).

39. T. Herffurth, M. Trost, M. Beier, et al., "Assessing surface imperfections of freeforms using a robotic light scattering sensor," Opt. Eng. 58(09), 1 (2019).

40. G. D. Cole, W. Zhang, M. J. Martin, et al., "Tenfold reduction of Brownian noise in high-reflectivity optical coatings," Nat. Photonics 7(8), 644-650 (2013).

41. B. Loglia, E. Derrick, L. Ingraham, et al., "Defect characterization in optical coatings using scattered light," Appl. Opt. 62(22), 6046 (2023).

42. A. Kontos, B. Loglia, B. King, et al., "Near speckle-free imaging setup for scattering studies of mirror coatings," Opt. Express 29(26), 44012 (2021).

43. M. Zerrad, M. Lequime, and C. Amra, "Far-field spatially angle-resolved scattering measurements: practical way to recover surface topography," Opt. Eng. 53(9), 092012 (2014).

44. "ISO-19986:2020 Lasers and laser-related equipment — Test method for angle resolved scattering," (2020).
45. C. Asmail, J. Hsia, A. Parr, et al., "Rayleigh scattering limits for low-level bidirectional reflectance distribution function measurements," *Appl. Opt.* **33**(25), 6084 (1994).
46. C. Asmail, A. Parr, and J. Hsia, "Rayleigh scattering limits for low-level bidirectional reflectance distribution function measurements: corrigendum," *Appl. Opt.* **38**(28), 6027 (1999).
47. J. Liang, "High-Precision Laser Beam Shaping and Image Projection," Ph.D. thesis, University of Texas at Austin (2012).
48. M. Levy, "ISO 10110-7 / ISO 14997 Plus Surface Scratch & Dig Evaluation Targets," Tech. rep. (2007).
49. M. W. Davis, G. R. Gladstone, T. K. Greathouse, et al., "Contamination control processes and results from the JUICE ultraviolet spectrograph," (Antibes, France, 2024).
50. A.-S. Munser, M. Trost, S. Schröder, et al., "Analysis of very low bacterial counts in small sample volumes using angle-resolved light scattering," *Appl. Opt.* **62**(2), 411 (2023).