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Abstract
We present a novel setup to study the scattering properties of mirror coatings. It is designed to measure angle-resolved scattering from coated substrates, with a special emphasis on the study of point scatterers. Near speckle-free imaging is achieved with a rotating diffuser. This setup presents advantages for the identification and analysis of individual defects, which are critical in sensitive optical applications, especially in gravitational-wave detectors.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Applications of optical cavities range from atomic clocks [1] and quantum optics [2] to gravitational-wave (GW) detectors [3,4]. The highly reflective (HR) mirrors used in such applications are usually made out of a super-polished low-absorption substrate with either an amorphous or crystalline multi-layer coating placed on top [5–8]. In many cases, the quality and properties of the mirrors are crucial to the application. Depending on the wavelength, coatings can have exceptionally low optical losses, namely absorption and scattering losses. In addition to optical losses, the mechanical losses of the mirror material have a significant impact on the performance of GW detectors and atomic clocks, and need to be minimized to reduce the Brownian thermal noise associated with them in optical cavities [5,9]. Extensive research is ongoing to find material and coating processes that achieve the lowest mechanical losses possible, with optical losses similar to the exceptionally low losses achieved by the existing state of the art [10–16].
GW detectors are especially sensitive to the optical and mechanical losses of the mirror coatings. High mechanical losses cause Brownian thermal noise which contributes significantly to the total noise in their most sensitive frequency band [17–19]. Optical absorption adds to the total cavity losses, which can deteriorate the signal amplification. More crucially, heat from absorbed light deforms the surface figure of the cavity optics which destabilizes the cavity and prevents high power operation [20]. Similar to optical absorption, scattering losses add to the total cavity losses, but they are much higher than absorption losses in present generation GW detectors [21]. In addition, scattered light from cavity mirrors can scatter off of the walls of the surrounding vacuum chamber and recombine back into the cavity causing a significant source of phase noise at some frequencies [22].
As we search for new coatings, it is important to measure and understand their scattering properties. We know scattering originates mainly from the roughness of the mirror surface or from point defects on and inside the coatings [23]. In the case of crystalline coatings, or coatings where crystallization has emerged, grain boundaries and the state of crystallization are also important [24]. The origin of surface roughness is more or less straightforward, and depends on the profile of the substrate’s surface as well as the coating deposition process [25]. It is possible to estimate the scattering intensity from a surface given the power spectral density of the surface height profile [26–31]. This calculation becomes more complex for a multi-layer coating since the height-profile correlation between layers introduces uncertainty. The situation is even more uncertain when the coating recipe requires annealing at temperatures close to the crystallization transition [32]. Annealing can induce crystalline formation, or tension between coating layers, both of which may have a significant impact on the scattering properties.
The development of defects is less understood, especially as they are likely different for different material and deposition techniques. As in the case of roughness scattering, the effects of annealing on defects is another interesting research area. Scattering defects in coatings might also be the seeds for the development of point absorbers, that can quickly degrade the performance of GW detectors [20]. Low-loss amorphous coatings currently used [21], as well as those that seem most promising [15,33], are deposited using ion-beam sputtering (IBS). IBS coatings have extremely low scattering losses, owing to their surface uniformity and low defect density. Further improvements will likely require looking into new deposition processes.
If we are to address the above, it is important to have dedicated scattering setups that study scattering and use it as a diagnostic tool. In this work, we present a novel setup to study the scattering properties of mirror coatings. Specifically, the setup is designed to measure angle-resolved scattering off of coated substrates to extract the bi-directional reflectance distribution function (BRDF), which gives information about the surface properties, such as the surface roughness and the size and location of defects. The setup uses a camera so that we not only measure the overall scattering profile as a function of angle, but also identify individual contributions to scattering from defects and surface roughness. A similar setup has been developed for the purposes of studying GW detector mirrors and is described in detail in [34] and [35]. In these two studies, images of scattered light from various coated samples were used to obtain both the scattering profile as a function of angle, and the total scattering losses. The utilization of a high-resolution CCD camera allowed for the clean identification of the region of interest on the sample, providing useful and reliable data on the optical quality of state-of-the-art low-scatter samples.
The setup presented here pushes the technique a step further, as it includes a rotating diffuser to reduce speckle from the images, which arises from the high spatial and temporal coherence of the laser used. This leads to improved imaging of the scattering area that allows for accurate BRDF measurements of individual defects, that are free of interference effects from other nearby scattering centers. Importantly, the scattering contributions of defects and surface roughness can be disentangled, when they are comparable.
Section 2 describes the setup in detail. In section 3 we present the calibration procedure to convert scattering images to BRDF. The quantitative and qualitative improvement of speckle in our images is discussed in section 4. Finally, section 5 showcases how the setup performs when applied to two coating samples.
2. Experimental setup
The setup is shown in Fig. 1. Light from a CW solid state laser at $1064$ nm is transported via a single-mode fiber to a motor-operated rotating platform (gray quadrant in the figure). The wavelength was chosen to match the wavelength used in current GW detectors, as we are mainly interested in studying samples optimized for them. The light passes through a rotating diffuser (RD), and emerges diverging at an angle which depends on the diffuser design. The diffuser is used to reduce speckle in the images, as it is important for the identification of defects (see section 4). A lens close to the diffuser (L1) captures the light and in combination with a second lens (L2) focuses it on the sample. The angle of incidence onto the sample relative to the surface normal is $5^{\circ }$, but can be adjusted depending on need. Normal incidence has the complication that the reflected beam cannot be properly attenuated and leads to unwanted glares in images. In addition, due to geometric constrains, $5^{\circ }$ incidence allows for some measurements at angles closer to specular reflection. The specular reflection from the sample and the transmission through it gets attenuated using black glass (not shown in the figure). Some of the transmitted beam is collected for live monitoring of the incident power on the sample.
Fig. 1. Left: Basic layout of the setup. Light from the laser is guided through a fiber, a diffuser (RD), polarizer (P) and two lenses (L1,L2) onto the sample. Everything shown over the gray quadrant is placed on a motorized rotating mount and can rotate clockwise or counter-clockwise as viewed on the page. Scattered light from the sample surface passes through a $12$ mm iris, an objective lens (L3) and a $1064$ nm narrow-band filter, producing a focused image onto a CCD sensor. Right: Diffuser diagram depicting the relevant quantities in the speckle-reduction measurements. The red circle represents the beam position relative to the center of the diffuser. The offset is important, as for a given angular velocity $\omega$ and image exposure time, the beam sweeps a longer distance on the diffuser ($d_\textrm {opt}$).
Polarization plays an important role in scattering, and experiments can extract useful information about the surface by controlling the polarization state of the incident and detected light [23]. The rotating diffuser and the fiber significantly affect the polarization of the beam. For that reason, a polarizer (P) is placed right before the sample to fix the beam polarization vertically or horizontally, depending on the need.
The scattered light from the sample is collected at the desired angle from the camera setup. The viewing angle is chosen by rotating the platform, while the camera is fixed. The scattered light passes through a $12$ mm iris which corresponds to an angular resolution of $2^{\circ }$. A lens with $175$ mm focal length behind the iris (L3) projects an image onto the camera sensor, with a magnification of $1.15$. A bandpass filter (F) and a tube housing the imaging optics ensures that the camera only sees scattered light from the sample. The camera is a $4096\times 4096$ pixel cooled CCD sensor from Andor, with dimensions $36.8\times 36.8$ mm$^{2}$ ($9\times 9$ $\mu$m$^{2}$ each pixel). The measured imaging resolution according to the Rayleigh criterion is $\delta$x$=35\pm 1$ $\mu$m. The Rayleigh criterion defines the resolution as the distance between the maximum and the first minimum of the diffraction-induced Airy disk. The theoretical diffraction-limited resolution for our setup can be calculated using $\delta r = 1.22\lambda f/D$, where $f$ is the distance between the objective lens (L3) and the camera ($f=320$ mm), and $D$ is the diameter of the iris ($D=12$ mm). This gives $\delta r=34$ $\mu$m, in agreement with the observed resolution.
The light at the output of the fiber is linearly polarized and has power close to $80$ mW. The rotating diffuser randomly rotates the polarization, and as a result on average, half of the power is transmitted through the polarizer (P). An average of $30$ mW is delivered onto the sample. The multi-mode beam transmitted through the rotating diffuser is focused via two lenses onto the sample to a spot with a near Gaussian profile, a $4\sigma$ diameter of $1.5$ mm, and a divergence of $2^{\circ }$.
3. Calibration
The camera records the scattering intensity in the form of counts per pixel. In order to translate these counts to BRDF, we need to take into account the solid angle covered by the camera, the transmission through the lens and filter, and the quantum efficiency of the CCD sensor. To determine all contributions experimentally, we can measure the scattering profile of a sample surface with a known BRDF. This calibration method is similar to the one described in [34]. We use a Labsphere Spectralon reflectance reference Lambertian target, which has a constant and known BRDF at $1064$ nm. Specifically, any such target with total reflectance $R$, has BRDF$=R/\pi$ sr$^{-1}$, independent of the viewing angle. The BRDF is calculated from scattering measurements via the formula:
where $P(\theta )_\textrm {scat}$ is the measured scattering power over some small solid angle at viewing angle $\theta$ relative to the surface normal, $\Omega$ is the solid angle, and $P_\textrm {i}$ is the incident power. The goal is to determine $P_\textrm {scat}$ from the CCD pixels counts, as well as the solid angle $\Omega$ of the measurement. First, we express the scattering power in terms of measured counts: where $N$ are the measured counts, $T_\textrm {exposure}$ is the image exposure time, and $r$ is the response factor of the CCD camera in units of W/(counts/s), which also accounts for the transmission of the scattering power through the imaging optics. The latter is an unknown quantity, or at least known only approximately. The BRDF formula can now be rewritten as:The second term in the parenthesis ($r/\Omega$) is an angle-independent calibration factor that we get experimentally, by measuring $N(\theta )$ for a target with known BRDF, such as the Spectralon reference target. Using this method, we determine the calibration factor to be $r/\Omega =(2.13\pm 0.10)\times 10^{-13}$ $\textrm {W}(\textrm {cts/s})^{-1}\textrm {sr}^{-1}$. Main sources of uncertainty include the determination of counts in the image, the measurement of incident power, the knowledge of reflectance of our reference target, and the standard deviation of the measurements at different angles. It is worth noting that the uncertainty in estimating the counts from scattering often dominates the total uncertainty. It arises from the difficulty in estimating the background counts on CCD, which are unrelated to the scattered light of interest. Counting statistics is usually a negligible contribution by comparison.
Figure 2 shows the BRDF measurement of the reference target, which has a calibrated value given by the manufacturer BRDF$_{\textrm {ref}}=0.185\pm 0.006$ sr$^{-1}$.
Fig. 2. Measured BRDF of the Labsphere Spectralon reflectance reference target. The absence of a trend is confirmation of the Lambertian scattering profile of the screen. The red solid line represents the mean of the data points, which is the same as the expected BRDF from the target. The dashed lines correspond to the $5\%$ uncertainty of the measurement.
4. Speckle reduction with a rotating diffuser
The samples are illuminated by light with extremely high spatial and temporal coherence. As a result, scattering images are distorted by speckle with relatively high contrast, $C$ [36]. Contrast is defined as $C=\sigma _I/<I>$, where $\sigma _I$ is the standard deviation of the intensity over some region with uniform intensity, and $<I>$ is the average intensity in the same region. $C$ can take values from $0$ to $1$, where the low end represents a speckle-free image. Speckle appears on an image as point like features on a featureless surface, with size determined by the resolution of the imaging setup. Since one of our goals is to identify point defects smaller than the resolution of our setup, it is crucial that we reduce speckle as much as possible.
There are many speckle-reduction techniques described in literature [37–40]. In most applications, it is important to obtain a low-speckle image in the shortest camera exposure time possible. In our application where exposure times are longer than $1$ second, a rotating diffuser solution is the simplest, and results in very low speckle images. The principle behind speckle reduction with a moving diffuser is that as the beam goes through the diffuser, different parts of it experience different phase shifts and arrive at the sample via different paths. The result is a beam with an irregular wavefront, that is still spatially coherent. A fixed diffuser doesn’t reduce speckle, since the phase relation between different parts of the beam is fixed and therefore interference effects are still present. A moving diffuser produces a series of images with different speckle patterns, which when averaged over the camera exposure time produce a lower speckle-contrast image.
To assess the effectiveness of our rotating diffuser setup we placed a uniformly white rough surface at the sample position. By taking images of the scattered light off the screen, we get a profile of the incident beam and the speckle contrast. Figures 3(a) and 3(b) show images of the beam spot viewed at a $20^{\circ }$ angle from the normal to the scattering surface, with a fixed diffuser and a rotating diffuser respectively. The improvement in speckle-contrast is quite significant, though the image is not fully speckle-free.
Fig. 3. Beam spot on a scattering screen positioned at the sample position. Viewing angle is $20^{\circ }$. a) The image is taken with the beam going through a fixed diffuser. b) The image is taken with the diffuser rotating. The red box in both images marks the region used for the contrast calculation, which is also the area shown in Fig. 4.
To calculate the speckle-contrast, $C$, we can’t apply the contrast formula over our beam spot area, since the intensity is not uniform. We first fit a 2D Gaussian profile and normalize the central part of the intensity so that we get a uniform profile. Then we calculate $C$ over a central region that is large enough for an accurate estimation, but small enough so the normalized profile is indeed uniform. The area is marked by the red rectangle in Fig. 3. Fig. 4 is the normalized intensity profile of the data in Fig. 3, zoomed around the central region.
Fig. 4. Same data as in Fig. 3, only the intensity profile is normalized to a fitted Gaussian profile and zoomed in the central area marked by the red rectangle. The uniformity of the resulting image allows us to calculate the speckle-contrast.
We follow this procedure for the evaluation and comparison of different diffuser configurations, as described in the following section.
4.1 Diffuser configuration tests
The goal of the diffuser configuration tests is to find the setup that results in the lowest speckle-contrast, while at the same time keeping the beam spot $4\sigma$ diameter at the sample position smaller than $2$ mm and the beam divergence lower than $2^{\circ }$.
We tested 2 diffuser types. A $20^{\circ }$ flat-top engineered diffuser from Thorlabs, and a $5^{\circ }$ holographic diffuser from Edmund Optics. For each diffuser type, we adjusted the lenses to get the best beam spot at the sample position. For the $20^{\circ }$ diffuser, this corresponded to a $3.4$ mm $4\sigma$ diameter and a $2^{\circ }$ beam divergence. For the $5^{\circ }$ diffuser, we obtained a $1.5$ mm diameter with the same beam divergence. This is expected, since a more aggressive diffuser ($20^{\circ }$ versus $5^{\circ }$) mixes more spatial modes and increases the beam diameter for a given divergence.
In the case of the $20^{\circ }$ diffuser, we varied the distance between the rotation axis of the diffuser and the spot where the incoming beam passed through it (henceforth referred to as offset). In addition, we varied the angular velocity of the diffuser, while keeping the camera exposure time constant at $10$ seconds. The speckle-contrast is expected to depend on the distance, $d_{\textrm {opt}}$ the beam spot sweeps on the diffuser during the exposure time (see Fig. 1). The sweep distance depends on the offset, angular velocity and exposure time simply by $d_\textrm {opt}=\omega R T_\textrm {exp}$. That expectation is confirmed in the data shown in Fig. 5(a), where the contrast is plotted as a function of the sweep distance. The results of different offsets (denoted as “R” in the legend) and angular velocities are plotted together and all follow the same trend. The solid lines serve to guide the eye. The best contrast achieved for this diffuser was $0.05$, with the plateau starts around $3$ mm sweep distance. For comparison, images with a non-rotating diffuser have contrast close to $0.6$. Generally, contrast lower than $0.03$ is considered near speckle-free [41].
Fig. 5. Contrast results as a function of the sweep distance $d_{\textrm {opt}}$ (see Fig. 1) for two different diffusers. a) For the $20^{\circ }$ diffuser, we tested various rotation speeds and beam offsets ($R$). The figure shows that the only relevant parameter is the sweep distance, as expected. b) For the $5^{\circ }$ diffuser we only varied the rotation speed, keeping the beam offset constant. The contrast measurements showed similar trends with the $20^{\circ }$ diffuser, but produced the lowest-speckle images. The solid lines serve to guide the eye.
For the $5^{\circ }$ diffuser we did not do measurements for different offsets, and only varied the angular velocity of the diffuser, keeping the exposure time at $10$ seconds. The results are shown Fig. 5(b). From the figure we can see that a plateau is reached after $10$ mm sweep distance, which is longer than in the case of the $20^{\circ }$ diffuser, but with a lower contrast, reaching $C=0.038$.
The images shown in Figs. 3 and 4 correspond to our optimal and final configuration, which corresponds to the $5^{\circ }$ holographic diffuser, with a $3$ mm beam offset and $100^{\circ }/s$ diffuser angular velocity. As already mentioned, the resulting beam spot at the sample position in a 2-lens configuration has a Gaussian profile, with a $4\sigma$ diameter of $1.5$ mm, and a divergence of $2^{\circ }$.
Here, it is interesting to consider what is the shortest exposure time ($T_{min}$) we can use and still obtain an image with $C=0.05$. As already mentioned, this application requires long exposure times in order to measure extremely low scattering levels from super-polished mirrors. We only consider this question here for comparison of results from other speckle-reduction techniques. The largest beam offset we can obtain with this diffuser is $R_{max}=4$ mm, and the fastest angular velocity is $\omega _{max}=1800^{\circ }/$s. To achieve $C=0.05$, we need $d_{opt}=3$ mm, which gives:
5. Application to mirror samples
To assess the qualitative and quantitative effects of the introduction of the diffuser, we apply the technique on two coated samples, namely samples A and B. First, we examine scattering images from a sample with high defect density. Sample A is a super-polished silica substrate coated with a single $374$ nm GeO$_2$ layer. The sample is illuminated by our laser and imaged at $20^{\circ }$. Due to contaminants present during the coating of the substrate, the sample exhibits multiple point scatterers. This is less than ideal for a high optical quality mirror but useful for the purposes of our test. The effect of the low-speckle setup on the scattering images of this coated sample is seen in Fig. 6. All scattering defects seen were confirmed to be on the sample’s surface and not in the substrate bulk, using images from different angles.
Fig. 6. Images at $20^{\circ }$ viewing angle from the normal of sample A, a super-polished silica substrate coated with a single $374$ nm GeO$_2$ layer. a) The diffuser is not rotating, and the speckle is apparent. Many individual spots get distorted due to destructive interference from nearby points. b) Close-up of (a) in a region with high-defect density. c) The diffuser is rotating and speckle is minimal. All individual points appear as round spots even when overlapping with others. d) Close-up of (c), as in (b).
Figures 6(a) and 6(c) are both views of the entire beam spot on the same sample. In Fig. 6(a) the diffuser is not rotating, while in Fig. 6(c) the diffuser is rotating at $100^{\circ }/$s. The imaging conditions are otherwise identical. Figure 6(a) exhibits interference features between scatterers, which distort the image significantly in certain regions. Furthermore, depending on random phase relations, the scatterer might appear brighter or dimmer when the diffuser is not rotating. On the other hand, Fig. 6(c) presents clear airy disk scattering points, which can easily be identified using a point-detection algorithm. Figures 6(b) and 6d are zoomed-in versions of Figs. 6(a) and 6c, respectively, covering one of the high defect density areas. The region exhibits many more distinguishable defects in Fig. 6(d) than in Fig. 6(b). A clearer image helps with image analysis and accurate estimation of background noise.
All images of sample A have a diffuser in place, which is either in motion or stationary. Using a beam-profile camera, we have directly imaged the beam through the diffuser while rotating and while being stationary. In the former case, the profile is perfectly smooth, as can be seen in Fig. 3(b). In the latter, the profile is not as uniform, which might account for some of the distortions we see in Fig. 6.
Sample B is an HR mirror made of alternating layers of GeO$_2$ and SiO$_2$. Scattered light images of this sample are shown in Fig. 7. Figure 7(a) is an image of the sample without a diffuser in place, while Fig. 7(b) is with the diffuser rotating at optimum speed. Illumination of the sample in the two cases is not identical due to the changes in the optics of the set-up, but reasonably close. Unlike sample A, sample B has relatively low density of large defects, and surface scattering is significant, as is evident by the images. By surface scattering here we mean scattering that follows the profile of the beam rather than individual point scatterers. This surface scattering may be the result of surface roughness, or of very high density sub-micron size defects. In this case, it is most likely the latter. For the purposes of our tests however, this is not relevant.
Fig. 7. Images at $20^{\circ }$ viewing angle from the normal of sample B, a super-polished silica substrate coated with multiple pairs of GeO$_2$/SiO$_2$ layers. a) The image was taken without a diffuser. A small number of large defects can be identified, as well as a broad speckled pattern from surface scattering. b) Image of the same region as in (a) but with the diffuser on. The same defects can be identified in both images, with a few exceptions which are likely dust particles. The beam profile on the surface is smooth and doesn’t interfere with the larger defects. The labels Defect 1 and Defect 2 depict the two defects for which we present the BRDF results in Fig. 8.
Most defects that appear in Fig. 7 are isolated and easily identifiable in both images. Defects that overlap with the surface scattering profile are more distinguishable when the diffuser is on. The speckle pattern makes it nearly impossible to identify whether there are any dimmer defects in the area. Using a defect identification algorithm on the speckled image, returns a large number of false positives, while it works as expected for image 7b.
In order to evaluate the advantages of the diffuser as they relate directly to the BRDF, we analyzed scattered light from two point defects found on sample B. These defects are depicted as Defect 1 and Defect 2 in Fig. 7. Measurements were performed with the diffuser both on and out. The measured BRDF of the individual defects are shown in Fig. 8. Defect 1 is the weaker of the two, and its analysis is largely hindered by the speckle, when the diffuser is out. The BRDF of Defect 1 (Fig. 8(a)) as measured by the set-up with the diffuser on (red circles) is a smooth function of angle, with relatively small error bars. Without the diffuser (blue squares), the BRDF is unreliable and unusable for further analysis.
Fig. 8. BRDF measurements of Defect 1 and Defect 2 shown in Fig. 8. a) Defect 1 BRDF data taken with the diffuser on and the diffuser out. b) BRDF data of Defect 2. The error bars only represent random uncertainty in determining the image counts under the area of interest.
On the other hand, Defect 2 is stronger, and is less affected by the speckle. The BRDF data shown in 8 match very well at all angles, though the data without the diffuser carry higher uncertainty.
6. Conclusion
We present a novel experimental setup for performing angle-resolved scattering measurements. An important feature of this scattero-meter is its low-speckle imaging capability, which helps with the identification and study of individual defects present in coatings. We believe that this setup will allow measurements that will aid with the development of new low-loss coatings as well as the understanding and mitigation of defects.
Funding
Bard College; Bard Summer Research Institute, Bard College.
Acknowledgments
The authors would like to thank Joshua R. Smith from California State University, Fullerton, for useful discussions regarding the design of the setup, and Carmen Menoni, Colorado State University, Fort Collins, for supplying the coated substrates.
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.
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