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Abstract
We present a method for evaluating the quality of optical glass using a high-resolution wavefront sensor, the wavefront phase imaging (WFPI) sensor. As shadowgraphy is a widely used method for inspecting striae in optical glass, it does not provide a quantitative metric that represents the potential optical quality of the glass and should be based on the operator’s experience. We compare the proposed method in two experiments. First, we compare it with the results obtained by shadowgraphy on a variety of samples. Second, we compare the results of a single-point chromatic confocal profilometer on a calibrated sample. The WFPI shows results comparable to the reference method in both cases but provides more information than shadowgraphy and avoids the human factor in the measurement.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. INTRODUCTION
An important property of optical glass is the homogeneity of the refractive index of the material. Lack of homogeneity can enlarge the circle of confusion in the image plane of an optical system and, thus, decrease definition. In highly demanding applications such as medicine, aerospace technology, or microscopy, small optical defects may seriously affect the optical performance while in other cases the effect may be unimportant, even when the defects are easily detectable. Therefore, inspection of glass quality is an important task for optical parts, as the error caused by inhomogeneity in the refractive index of a piece of glass can exceed that caused by surface polishing errors. However, the detection and quantitative determination of homogeneity is still an open problem. In accordance with existing regulations, the ISO 1101-4 standard [1] establishes two aspects as criteria for classifying the quality of glass. The first aspect is the maximum variation of the refractive index within the sample, which is divided into six classes ranging from $0.5\cdot {10^{- 6}}$ to $50\cdot {10^{- 6}}$. Since this criterion is not sufficient to entirely characterize the quality of a piece of glass because it does not consider the spatial scale, the standard also considers a second aspect: the density of striae that cause an optical path difference (OPD) of at least 30 nm. Striae are defined as spatially short-scale inhomogeneities [2] that originated during glass melting [3] that result in local differences from the refractive index of the surrounding glass, causing an angular deviation or refraction in the expected path of a light ray through the material [4]. Based on this criterion, the ISO standard classifies glass from class 5 (free of striae) to class 1 (with at least 10% density of striae). Striae affect not only the optical quality but also the mechanical strength of the glass [5].
There are basically three techniques to characterize the homogeneities in a glass: the Töpler–schlieren dark-field method (described in 1864 and based on the capture of schlieren images [6,7]), interferometry [8], and shadowgraphy [9]. With the first one being very little used, interferometry provides accurate quantitative results of phase delays resulting from internal striae. Generally, a collimated beam is chosen to illuminate the sample with a perpendicular incidence angle on the entrance and exit surfaces. An immersion fluid can be used to compensate for surface roughness. Otherwise, if the test sample is in air, the surfaces should be polished to optical quality to avoid surface artifacts [10].
The currently preferred method in the industry is shadowgraphy, mainly because of its simplicity. According to Gross et al. [4], the light of a mercury arc lamp with a small luminous area illuminates the sample without additional beam shaping components. The light cone that transmits the sample is nearly collimated due to a large distance (Fig. 1). The shadow image of the transmitted light is projected onto a distant screen. This image is recorded with a camera. The sample is arranged on a turntable in order to allow slight adjustments along the vertical axes and to choose the orientation of the striae relative to the illumination. One of the major problems in interpreting the shadow images is quantitative calibration and evaluation. For this purpose, a test plate can be used, which contains a matrix of small slits with well-defined widths and phase steps [11]. A human operator evaluates the contrast by comparing it with references.
In this work, we propose that, by measuring the phase of the wavefront in very high resolution, superior results can be obtained than those obtained with shadowgraphy (in the sense that they provide more information), in a quantitative way, and without the human factor, which provides a high degree of objectivity. To do this, we used the wavefront phase imaging (WFPI) technique to sense the OPD maps of the samples. This technique has been used with promising results in other fields, such as silicon metrology [12] or the characterization of ocular optics [13]. Glass samples provided by Schott AG (Mainz, Germany) have been used for the evaluation and have been fully characterized at their facilities using their standard methodology.
In Sections 2 and 3, the WFPI and the methodology are presented. In Section 3, we present results on real samples. First, we compare the raw OPD measurements obtained with WFPI with the characterization provided by Schott using shadowgraphy. In a second experiment, we compare the performance of WFPI in terms of absolute accuracy by comparing the measurements with a chromatic confocal (CC) sensor. Finally, in Section 4 the conclusions are shown.
2. METHODS
A. Shadowgraphy
As will be shown in Section 2.D, the results presented in this work are based on materials and classifications provided by Schott. Currently, Schott uses shadowgraphy as the method of choice for classifying a glass sample. The characterization is carried out in four types: A, B, C, and D; ranged from 10 to 60 nm in the variation of OPD [14]. This characterization also relates types A to D with the visibility of striae in the shadowgraphy images [15] (Table 1).
Table 1. Definition of Shadowgraph Grades Used by Schotta
In terms of physical optics, the method of shadowgraphy can be analyzed using the transport of intensity equation (TIE) [17]. The TIE equation can be written as
where ${{\Phi}}$ is the wavefront phase aberration produced by the sample, $\lambda$ is the wavelength, and $\frac{d}{{dz}}I$ is the intensity derivative along propagation axis. Equation (1) relates the intensity observed at the screen in a shadowgraphy setup to the Laplacian of the wavefront phase aberration and, thus, to the OPD.If we consider the Mahajan approximation [18] of the Strehl ratio (SR) [Eq. (2)], it shows that the maximum SR attainable by any part of the glass is proportional to the root mean square (RMS) of its wavefront phase (${\sigma _\varphi}^2$) and is not related to the Laplacian of it,
B. Wavefront Phase Imaging Technique
In this work, we propose the use of WFPI to objectively quantify the OPD maps of the samples. WFPI is based on capturing two intensity images around a conjugated sample plane using conventional image sensors. The sensor output is the OPD gradients along two orthogonal directions. Finally, a numerical integration is performed to estimate the value of the OPD.
In an exemplary implementation, a collimated light beam propagates through the sample, whereby the output beam is distorted according to the OPD map of the sample. A 4f relay translates the sample plane to the mid-distance between the planes where the intensity images should be captured.
In Fig. 2. The schematic drawing can be found to characterize a transparent sample using WFPI, where a collimated monochromatic light beam is refracted by the sample. Then, two images, ${I_1}$ and ${I_2}$, are captured around a conjugated plane of the sample, with symmetric positions before and after it.
Fig. 2. WFPI measurement schematic drawing. ${L_1}$ collimates the light source, and ${L_2}$ and ${L_3}$ form a 4f relay.
To better describe the method, we first must define $H({x,y})$ and $D({f,g})$ functions:
Let $H({x,y})$ be a continuous two-dimensional function defined for positive values of $x,y$ and taking only positive numbers as values, and let $V$ be an auxiliary transformation function acting on $H$ as follows:
Let $D({f,g})$ be another auxiliary function that operates over $f(x)$ and $g(x)$ defined as
To obtain the phase derivatives ${\varphi _x}$ and ${\varphi _y}$ from the input images ${I_1}({x,y})$ and ${I_2}({x,y})$, the operation is as follows:
C. Measurement of Striae in Glass Blocks
The samples provided by Schott consist of a set of three glass blocks marked with two square areas each. Each area will be denoted with numbers from 1 to 6 for simplicity (Fig. 3).
For the measurement of these samples, we have assembled a setup as described in Fig. 2. As light source, we used a 650 nm LED (MTPS9067, Marktech Optoelectronics, New York, USA), collimated using a lens (L1 in Fig. 2). In this implementation, the two images (${I_1}$ and ${I_2}$) used by the WFPI method are obtained by a single imaging sensor (Trius SX-36, Starlight Xpress, Berks, UK) coupled to a translation stage. This allows images to be captured equally spaced with respect to the conjugate plane of the sample, with a separation between the two of 75 mm. The field of view for each measurement was a circle with a diameter of 33 mm.
Each measurement describes the OPD map in the measurement plane, with the spatial sampling defined by the image sensor used. Accordingly, the lateral sampling of the measurements is 19 µm, producing measurements with 2.2 million data points.
D. Measurement of Artificial Striae Plate
To deepen the results of the technique and have an estimate of the quantitative quality of the results, we have carried out measurements on an artificial striae plate also provided by Schott. The sample provided consists of a ${{68}} \times {{68}}\;{\rm{mm}}$ glass plate (5 mm thick) with rectangular areas with variable separation and variable thickness (Fig. 4).
The plate is made of Schott N-BK7 glass. Although the WFPI output is an OPD map, knowing the value of the refractive index of the material at the wavelength used allows us to know the variations in the thickness of the plate.
For the measurement of this sample, we developed a setup based on stitching of several smaller areas to cover a larger area (Fig. 5). In this configuration, the sample is placed on a ${{XY}}$ translation stage. It has been configured to capture 676 OPD maps (${{26}} \times {{26}}$), with some areas of overlap, so that the entire sample is covered. Each one subtends a 10 mm diameter circle with a distance between them of 2 mm. The pixel size in these measurements was 8.9 µm, and the full extent after stitching and cropping borders of all measurements in this setup is ${{50}} \times {{50}}\;{\rm{mm}}$.
The stitching method is simple and consists of placing each measurement on a large grid of the estimated sample size of the camera sample in relation to the pixel size in the sample plane (${{6337}} \times {{6337}}\;{\rm{pixels}}$), subsequently translating each measurement to the actual position within the field of view. The stitched OPD map is finally obtained by averaging all the images.
3. RESULTS
A. Results on the Measurement of Striae in Glass Blocks
In Fig. 6, the two leftmost columns show each image ${I_2}$ for every sample area as captured by the WFPI setup. The contrast has been enhanced for better visualization. Each image ${I_2}$ produces a result similar to what a standard shadowgraphy method would create. They have been incorporated to help the reader understand the major differences between shadowgraphy and WFPI. The two rightmost columns in Fig. 6 show the false-color OPD for each sample area.
Fig. 6. Left, image ${I_2}$ used by WFPI method for every measurement (contrast enhanced). Right, OPD maps for every measurement area.
Table 2 shows the classification provided by Schott for each area using its standard procedure (last column). In addition, the results are provided in terms of RMS (second column) and peak to valley (third column) obtained from the OPD maps in Fig. 6. The metrics used RMS, and peak to valley has been chosen as they describe different characteristics of the sample. The RMS is related to the SR by Eq. (2), which provides an indication of the maximum optical quality attainable by some optical system. In the other hand, the peak to valley is the metric used in ISO 1101-4 [1] standard to quantify refractive index variations.
Table 2. Summary of OPD RMS and Peak to Valley Values for Each Measurement Area
Sample number 1 was given a B grade since its shadowgraph does not reveal any major stria. However, its OPD reveals low spatial frequency variations that are not detectable by shadowgraphy. Another notable difference is in samples 2 and 4, as they are given a C grade based on shadowgraphy, but looking at the RMS, the sample number 4 is the best, so it should have been classified as A grade.
From Fig. 6, it can be extracted that the data from the shadowgraphy is related to the OPD. In fact, shadowgraphy produces intensity variations that are proportional to the second derivative of the OPD map as stated in Eq. (1) and, therefore, is much more sensitive to high spatial frequency variations than to low spatial frequencies. From the inspection of Fig. 6, this effect can be observed: in sample 3, the characterization according to shadowgraphy is of the worst degree; nevertheless, its OPD RMS value is better than other samples that were given a better grade.
B. Results on the Measurement of an Artificial Striae Plate
Figure 7 shows the final result of the sample measurement. Note that this map includes information regarding not only the rectangular patterns in the striae plate but also information about the thickness and/or refractive index variations of the sample.
Figure 8 shows the horizontal OPD profiles of each row of patterns, obtained from Fig. 7. It can be seen that the measured height of each slit does not depend on its width, which means that WFPI is able to resolve the smallest feature of the plate, which is 0.125 mm without loss of contrast. The measured OPD of each slit for each row is summarized in Table 3.
Table 3. Summary of the OPD Height in Nanometers of the Sample in Every Slit for Each Row of Patterns
To put our results in context, the artificial striae plate has also been measured using a commercial CC-based optical profiler S-Neox (Sensofar Medical, Catalonia, Spain). Table 4 summarizes the data that Schott obtained with the CC versus the data obtained using the WFPI, and both were compared with the nominal value.
Table 4. Comparison of Data Obtained by a Chromatic Confocal and WFPIa
From the analysis in Table 4, it can be seen that the data obtained by WFPI slightly differ from the CC measurement and from the specified values.
It is important to note that a CC profiler measures the height of the top layer of the sample on a single point, while the WFPI measures the OPD, so each point on the OPD map represents the product of the refractive index and the thickness of the sample of the entire pattern. As seen in Fig. 8, the OPD map of the sample contains inhomogeneities that can be explained as variations in the refractive index within the sample, variations in thickness, or both. For these reasons, comparisons of measurements between sensors that measure different magnitudes should be carefully examined. Even so, the results provided by WFPI are certainly similar to those provided by CC.
4. CONCLUSION
Shadowgraphy is the current method of choice for some of the largest optical glass manufacturers. However, this method is based on inspection and, therefore, is not quantitative. In addition, since the projection on the screen is proportional to the Laplacian operator of the OPD, the sensitivity to low-order aberrations is reduced after the original OPD is differentiated twice, making all low spatial frequency variations appear closer to the noise level. It is also important to note that the result obtained by shadowgraphy, although it is possible for a trained operator to obtain a classification of the quality of the glass, cannot determine its potential optical quality. The best potential optical quality of a glass sample can only be known if the wavefront aberrations are effectively measured with a wavefront phase-specific measure.
In this work, we show a method based on the characterization in very high resolution of the wavefront phase for estimating the overall quality of optical glass. As in the case of shadowgraphy, the necessary optical assembly is simple and does not require specialized hardware. However, it can produce an objective characterization of the quality of the glass and eliminate the need for a trained operator to obtain a reliable characterization as occurs in shadowgraphy. In addition to information about the striae present in the material, it provides a measure of potential optical quality.
The results derived from the measurement of six samples provided a certain degree of coincidence between the measurements carried out with shadowgraphy by an expert technician and those carried out with the proposed methodology. The causes of the discrepancy could be explained by using information about the optical quality of the samples. Regarding its absolute accuracy, the method solves different spatial scale features without loss of contrast, which is a proof of its effective high resolution.
Acknowledgment
The authors thank Dr. Fabian Rupp from Schott AG for lending us the samples to perform the experiments in this work.
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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