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Cases of long-term deformation of fused silica flats are reported. The phenomenon is detected at the scale of the nanometer, and exhibits a time constant of the order of 9 years. The observed deformation appears related to gravity and constraints, but a change of physical properties locally resulting in non-homothetic behavior is also hypothesized.
©2010 Optical Society of America
1. Introduction
When reliable performance and mechanical stability of optical components are of concern, fused silica often is the material of choice. Its optical, chemical, mechanical and electrical properties have been studied in detail, and are available in the literature [1–3]. In consideration of the above characteristics, fused silica was chosen as substrate of mirrors in interferometric gravitational wave detectors [4]. Optical components for satellite borne experiments are often made of fused silica [5]. Due to its internal transmittance, extending in the ultraviolet region of the light spectrum, it is also used to fabricate microlithography optical systems. As to laboratory environment, type III fused silica [1], a synthetic silica glass obtained by flame hydrolysis, is generally used in the form of reference and transmission flats for interferometry applications such as optical testing.
A particular problem that is faced in optical testing is the measurement of absolute planarity. A method that is used in most laboratories to work out the absolute height map of reference flats is by the so-called three flat test. According to this approach, three flats are interferometrically compared two by two in the course of four or more measurements. The interferograms are then processed, and the absolute shape of each flat is recovered. Afterwards, such flats are used as reference surfaces or primary standards in metrology laboratories to calibrate other plates subtracting the shape of the reference as a systematic error. Naturally, mechanical stability is a key feature required from the flats; also, low thermal expansion coefficient is helpful to reduce the measurement uncertainty. In fact, when the standard has to work in reflection only, glass ceramics such as Zerodur is often used for reference flats. When the standard is also working in transmission, high quality optical glass or fused silica are generally used. As to glass, however, doubts concerning its stability at room temperature have been expressed, although the conclusion seems to be that no observable effect is occurring over human lifetime [6–11]. No such doubts have been raised about fused silica, whose use for planarity standards has then been advocated whenever possible [12].
As regards the dimensional stability of fused silica, however, most accurate measurements report daily changes ΔL over a length L of relative amount ΔL/L = −0.5 parts per billion [13, 14]. Considering a plate of thickness L = 20 mm, this means a change ΔL = −3.7 nm per year; as to the time constant of the phenomenon, no data is available. If such a dimensional change occurs homothetically in the bulk material of the fused silica, it does not affect the planarity features of the flat. In case though the change somehow departs from such a behavior, surface deformations can be produced, and observed in the long term if sufficient sensitivity and measurement accuracy are available.
Interferometry is an effective technique to detect surface deformations; however, in its simplest form, it is a differential approach requiring in turn a reference standard whose shape is known a priori. As to absolute measurements, liquid flats have been used as reference surfaces; the accuracy so far achieved with such measurements is though limited [15–17]. A promising technique still being studied is deflectometric pentaprism scanning; such an approach does not need a further reference to compare with, but yet requires full metrologic validation as to achievable accuracy [18–21]. The three-flat test is instead a well assessed method that provides the absolute shape of the three primary flats, and has been established to a high degree of accuracy [22–40].
In the framework of legal metrology, the three-flat test resulting in the primary standard is routinely repeated every year, so that a significant number of data on the shape history of the reference flats is now becoming available. This paper is about the behavior of a set of three fused silica flats whose shape has been monitored during a period of ten years. Being used as planarity standards, such flats are individually identified as to provenance and age, and have been maintained all the time in a clean room under carefully controlled environmental conditions. In the case of one of such flats, a deformation behavior above the measurement uncertainty is reported, and the time constant of the process is estimated. Two more flats were examined ten years ago, to serve as replacements in case of need; one of these flats is now found to also exhibit a deformation of significance.
2. Experimental apparatus and measuring procedure
The optical metrology laboratory where the primary standards and the measuring equipment are housed consists of an ISO class 7 clean room with environmental conditions of (20 ± 1) °C temperature and (45 ± 5) % relative humidity. We use a Fizeau-type phase-shift interferometer whose main characteristics are given in Table 1 . The interferometer is placed on a granite table, passively isolated from the floor.
Table 1. Interferometric System Specifications
The three fused silica flats making up the primary standard have been purchased in different times from the same manufacturer. The flats are mounted on aluminum frames with two pins for bayonet insertion into the interferometer’s flange. The flats are suspended in their frame with three elastomer pads at unevenly spaced locations, as schematically shown in Fig. 1 ; as to front and back, the flats are confined between a raised edge and a retaining ring.
For identification purposes, the flats are labeled K, L, M. The measuring operations are carried out according to written procedures, implementing well established laboratory practices. The flats are interferometrically compared two by two in the sequence K-M, L-M, L-MR, L-K, where MR represents the flat M azimuthally rotated through a fixed angle (54°); for each pair, 100 acquisitions are averaged. Executing the sequence takes approximately one hour. The sequence is repeated 40 times, distributed in a period of three consecutive weeks, for a total of 16000 acquisitions. The data collected are then processed with analysis programs.
We have three independent analysis programs. The first one performs the data reduction with Fritz’s method [30], modified to take into account the data decorrelation and to compute the measurement uncertainty [36]. The second program works on a pixel-based iterative algorithm that provides the shape of the three flats with high spatial resolution [41]. The third program uses the Zernike representation as Fritz’s method, but proceeds to the result by a numerical iterative procedure [42]. The compatibility of the programs on various data sets has been verified within the measurement uncertainty. The latter has been evaluated according to the directions of legal metrology [43]. The expanded uncertainty U, expressed with a coverage factor of 2, i.e. as twice the standard deviation σ, is found to range from 0.4 nm in the center of the flats to 1.0 nm at the pupil’s edge, with a mean value of U = 0.5 nm (2σ); under the hypothesis of normal distribution of the error contributions, such a value corresponds to an interval having a level of confidence of approximately 95% [36].
The fused silica flats have a thickness of 19 mm, and a full diameter of 114 mm. The data we collect and process are taken within a software mask that selects a central portion of the plates, with a diameter of 94 mm. The corresponding diameter of the detected image is 169 pixels, for a total of 22333 pixels over the entire mask. Each time we run the three-flat test, the data of all the 40 sequences are saved and permanently stored in the archive of the laboratory. As to the flats themselves, L and M are accommodated in horizontal posture in their boxes, and maintained on a shelf at a reserved location within the clean room; they are only moved and placed in upright posture during the three weeks of the annual run of the three-flat test, or in case of extraordinary measurements. The flat K is used for the calibration of other plates submitted to the laboratory, and rests horizontally in its box when not in use.
We started to use the three-flat test in its fixed form in 1998, at the occasion of an international interlaboratory comparison (“round robin”). The flats taking part into the exercise were L and M, although in a different order with respect to the sequence given above; the third flat was the one circulating in the round robin. In 2000 our laboratory was acknowledged as a planarity calibration center for legal metrology within the European Co-operation for Accreditation (EA), and since then the three-flat test was run every year according to the established sequence, in compliance of legal metrology policies.
3. Measurement results
To show the results of the annual measurements we refer to the analysis output in the form of 37-terms Zernike polynomials. The surface quality of the three flats is better than 10 nm Peak-to-Valley (P-V). The results of the measurements made over the years are given as differences between the actual shape and the first-measured one. As to flats K and M, considering the measurement uncertainty no significant shape change is noticed over the interval of observation. As to flat L, its behavior from 1998 to 2004 is shown in Fig. 2 and associated movie.
In this latter case there is evidence of a significant progressive deformation of the surface. The basic form of the deformation can be referred to the shape assumed by a plate laying on a circular support; in addition, prominences at the locations of the pads in Fig. 1 are clearly identified. Referring to the power term of the deformation, the time evolution is presented in Fig. 3 , along with a fitting to the equation
where A is the asymptotic excursion, B is the initial time, and C is the time constant.
Fig. 3 Power term of flat L in the period 1998-2004. The error bar about the data is ± 2σ. Full line: curve fitting to Eq. (1).
The deformation observed exceeds the uncertainty level of the measurements; best fit is obtained with a time constant C = 9.1 years. Having monitored such a behavior year after year, in 2004 after the annual run of the three-flat test we decided to store the flat L in reverse posture. In Fig. 4 and associated movie we show the observed evolution of the flat L from 2004 to 2009, given as a difference with respect to the shape of 2004. It is found that the behavior of the flat is apparently different from the previous case, being more closely related to the deformation of a plate laying on three supporting points. It is then inferred that in the reverse posture the flat is missing true contact with the retaining ring, and is only suspended at the pads.
Fig. 4 Deformation of flat L intervened in the period 2004-2009, after 2004 upside down reversal (Media 2).
The elastic deformation of a flat in horizontal posture under its own weight was recently studied analytically, with finite element analysis, and in experiments [44]. As a major result, in our geometry the overall elastic deformation w is conveniently represented with a few terms of the Zernike polynomial representation:
where Zj are the polynomials, and cj the coefficients. The above terms correspond to power (Z 3), x astigmatism (Z 4), y trefoil (Z 10), and y pentafoil (Z 26).To account for the actual deformation we observed (Fig. 4), more terms of the Zernike representation need to be taken into account, namely, y coma (Z 7), primary spherical (Z 8), x secondary astigmatism (Z 11), and x tetrafoil (Z 16). Although the deformation here of concern is not elastic, it is noticed that all such terms are also compliant with the biharmonic equation that is at the basis of the theory of elastic deformation of plates [44, 45]. A reconstruction of the deformation by means of the terms in point is shown in Fig. 5 . The used coefficients cj are given in Table 2 .
Table 2. Zernike coefficients used to represent the deformation of plate L, expressed in wave units x 10−6.
Looking for further cases of deformation, two extra flats (labeled A and B) have been examined. Such flats were calibrated in 2000, and then remained stored in the clean room to serve as replacements in case of need. Calibrated again in 2009, flat B is found unchanged within the measurement uncertainty; conversely, flat A exhibits a significant depression about the center (Fig. 6 ). Unfortunately, intermediate measurements in the period 2000-2009 were not taken, so that the time constant cannot be estimated. The deformation does not appear related to the way how the flat is suspended in its frame; it rather occurs as the fused silica had different physical properties in the central area with respect to the outer annular region. The amount of the deformation is 7.6 nm P-V, which is of the same order of magnitude as that of flat L.
4. Discussion
The surface deformations observed in flats L and A might have several origins. A first possibility is plastic deformation of fused silica under the action of gravity (although the hypothesis of constant volume, implied by such type of a process, could not be verified). In particular, fused silica can be modeled as a liquid of very high viscosity, flowing at very slow rate under laminar regime. Since the plates are maintained in horizontal position, the surface deformation occurring over a period of time can be regarded as a map of the downward velocity. Considering cylindrical symmetry and referring to Fig. 7 , classical approaches provide the basic equation
where v is the velocity at which fused silica is flowing, r is the radius, F is the shear force, A is the area where the force is applied, and η is the viscosity. In our case we have , , with ρ the density of fused silica, g the gravity, and h the thickness of the flat, so thatIntegrating Eq. (4) and considering the boundary condition , with R the radius of the entire flat, one obtains
where is the velocity at the center of the plate. The latter velocity can be obtained from the measured surface deformation of the flat over the years, either using the raw power data or their fitting to Eq. (1). In particular, considering that the power deformation of flat L in the period 1998-2000 is 1.3 nm, a velocity v 1998-2000(0) = 2.1 10−17 m s−1 is computed; referring instead to the period 2003-2004 the power deformation of the same flat is 0.4 nm, and the velocity is v 2003-2004(0) = 1.3 10−17 m s−1. With g = 9.8 m s−2, ρ = 2.2 103 Kg m−3, R = 4.7 10−2 m, viscosity values η 1998-2000 = 5.7 1017 Pa s, η 2003-2004 = 9.2 1017 Pa s are correspondingly computed; naturally, due to uncertainty, such values are here intended as orders of magnitude only. For comparison, however, in the case of soda-lime-silica common glass at room temperature the approximate viscosity quoted in rheology is 1040 Pa s [46]; the range considered in Refs [7]. to [11] is from a minimum of 1019 (taken as an “ultra-conservative” order of magnitude [11]) up to 1041 Pa s (a “more realistic” estimate [11]). The viscosity values computed in our case for fused silica under the hypothesis of plastic deformation are then significantly smaller than those generally attributed to common glass at room temperature.A second possibility to explain the surface deformation observed in experiments is densification of fused silica, assuming that dimensional changes are local. In particular, still considering cylindrical symmetry as above, to a simple approach the flat can be modeled as a disk whose thickness h is varied as a function of the radius r. With the further hypothesis of mirror deformation of the opposite surface, and also knowing the mean thickness of the flat, one can then estimate the density change from the center to the edge. Simple considerations lead to the relationship
The quantity dh can be taken as twice the power deformation observed; over the period 1998-2004 we have dh = 6.7 nm. Since h = 19 mm, the result is dρ/ρ = 3.5 10−7; unfortunately, however, no such data could be found in the literature for comparison. Besides, dρ/ρ is very small, and can hardly be related to changes of other independently measurable quantities such as the refractive index (the nominal value of the refractive index of fused silica at room temperature, 632.8 nm wavelength, is 1.45705).
More possibilities include surface leaking, caused by localized dissolution of a thin layer of silica due to residual humidity. In any case, to identify the actual process taking place, more information would be required, and specific experiments should be planned in the future.
5. Concluding remarks
Two cases of deformation of fused silica flats have been identified, out of five flats that were examined; the observation period has been of approximately 10 years. In one case, the time constant of the phenomenon could be estimated, the result being of the order of 9 years. The entity of the deformation is very small but it is significant, and is almost compatible with data on dimensional stability that are available in the literature [13, 14]. The deformation appears to occur in a way related to gravity and constraints, but might also be locally dependent on physical properties of the bulk material departing from homothetic behavior. While phenomena such as viscoelasticity of Newtonian and Maxwell fluids [9, 11, 47], nonequilibrium statistical mechanics [48] and shear thinning or thickening [49] among others might be invoked to explain the cases of deformation observed, it is understood that the statistics presently available is not sufficient for a thorough model to be developed. Also, the information presently available does not allow indicating possible reasons why some flats are less stable than others. On the other hand, the measurements are difficult to be made, and require a long time to come out with data of significance. The cases here reported are, to the best of our knowledge, the first ones quantitatively documenting deformation of fused silica flats at room temperature over the years. As interferometers of improved sensitivity and better accuracy are becoming available, and absolute form metrology is becoming more developed, it should be possible to collect an appropriate base of data in shorter times, to be used for a thorough and detailed study of the stability properties of vitreous materials at room temperature.
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