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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 5 — Mar. 1, 2010
  • pp: 5114–5123
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Long-term deformation at room temperature observed in fused silica

Maurizio Vannoni, Andrea Sordini, and Giuseppe Molesini  »View Author Affiliations


Optics Express, Vol. 18, Issue 5, pp. 5114-5123 (2010)
http://dx.doi.org/10.1364/OE.18.005114


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Abstract

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 OSA

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

1. B. Mysen, and P. Richet, Silicate Glasses and Melts (Elsevier, Amsterdam 2005).

3

3. R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt. 46(33), 8118–8133 (2007). [CrossRef] [PubMed]

]. In consideration of the above characteristics, fused silica was chosen as substrate of mirrors in interferometric gravitational wave detectors [4

4. M. Ando, K. Arai, R. Takahashi, G. Heinzel, S. Kawamura, D. Tatsumi, N. Kanda, H. Tagoshi, A. Araya, H. Asada, Y. Aso, M. A. Barton, M. K. Fujimoto, M. Fukushima, T. Futamase, K. Hayama, G. Horikoshi, H. Ishizuka, N. Kamikubota, K. Kawabe, N. Kawashima, Y. Kobayashi, Y. Kojima, K. Kondo, Y. Kozai, K. Kuroda, N. Matsuda, N. Mio, K. Miura, O. Miyakawa, S. M. Miyama, S. Miyoki, S. Moriwaki, M. Musha, S. Nagano, K. Nakagawa, T. Nakamura, K. Nakao, K. Numata, Y. Ogawa, M. Ohashi, N. Ohishi, S. Okutomi, K. Oohara, S. Otsuka, Y. Saito, M. Sasaki, S. Sato, A. Sekiya, M. Shibata, K. Somiya, T. Suzuki, A. Takamori, T. Tanaka, S. Taniguchi, S. Telada, K. Tochikubo, T. Tomaru, K. Tsubono, N. Tsuda, T. Uchiyama, A. Ueda, K. Ueda, K. Waseda, Y. Watanabe, H. Yakura, K. Yamamoto, and T. YamazakiM. AndoK. AraiR. TakahashiG. HeinzelS. KawamuraD. TatsumiN. KandaH. TagoshiA. ArayaH. AsadaY. AsoM. A. BartonM. K. FujimotoM. FukushimaT. FutamaseK. HayamaG. HorikoshiH. IshizukaN. KamikubotaK. KawabeN. KawashimaY. KobayashiY. KojimaK. KondoY. KozaiK. KurodaN. MatsudaN. MioK. MiuraO. MiyakawaS. M. MiyamaS. MiyokiS. MoriwakiM. MushaS. NaganoK. NakagawaT. NakamuraK. NakaoK. NumataY. OgawaM. OhashiN. OhishiS. OkutomiK. OoharaS. OtsukaY. SaitoM. SasakiS. SatoA. SekiyaM. ShibataK. SomiyaT. SuzukiA. TakamoriT. TanakaS. TaniguchiS. TeladaK. TochikuboT. TomaruK. TsubonoN. TsudaT. UchiyamaA. UedaK. UedaK. WasedaY. WatanabeH. YakuraK. YamamotoT. YamazakiTAMA Collaboration, “Stable operation of a 300-m laser interferometer with sufficient sensitivity to detect gravitational-wave events within our galaxy,” Phys. Rev. Lett. 86(18), 3950–3954 (2001). [CrossRef] [PubMed]

]. Optical components for satellite borne experiments are often made of fused silica [5

5. S. Jordan, “The GAIA project: Technique, performance and status,” Astron. Nachr. 329(9-10), 875–880 (2008). [CrossRef]

]. 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

1. B. Mysen, and P. Richet, Silicate Glasses and Melts (Elsevier, Amsterdam 2005).

], 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

6. G. W. Morey, “The flow of glass at room temperature,” J. Opt. Soc. Am. 42(11), 856–857 (1952). [CrossRef]

11

11. Y. M. Stokes, “Flowing windowpanes: a comparison of Newtonian and Maxwell fluid models,” Proc. R. Soc. Lond. A 456(2000), 1861–1864 (2000). [CrossRef]

]. No such doubts have been raised about fused silica, whose use for planarity standards has then been advocated whenever possible [12

12. G. D. Dew, “Some observations on the long-term stability of optical flat,” Opt. Acta (Lond.) 21, 609–614 (1974). [CrossRef]

].

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

13. J. W. Berthold III, S. F. Jacobs, and M. A. Norton, “Dimensional stability of fused silica, Invar, and several ultralow thermal expansion materials,” Appl. Opt. 15(8), 1898–1899 (1976). [CrossRef]

, 14

14. J. W. Berthold III, S. F. Jacobs, and M. A. Norton, “Dimensional stability of fused silica, Invar, and several ultralow thermal expansion materials,” Metrologia 13(1), 9–16 (1977). [CrossRef]

]. 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

15. D. A. Ketelsen and D. S. Anderson, “Optical testing with large liquid flats,” Proc. Soc. Photo Opt. Instrum. Eng. 966, 365–371 (1988).

17

17. M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42(5), 389–393 (2005). [CrossRef]

]. 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

18. M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45(6), 1–3 (2006). [CrossRef]

21

21. R. D. Geckeler, “Optimal use of pentaprism in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18(1), 115–125 (2007). [CrossRef]

]. 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

22. G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6(6), 1077–1084 (1967). [CrossRef] [PubMed]

40

40. U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46(9), 093601 (2007). [CrossRef]

].

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

Table 1. Interferometric System Specifications

table-icon
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. The interferometer is placed on a granite table, passively isolated from the floor.

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
Fig. 1 Location of the pins and the pads in a plate mounting for interferometry.
; 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.

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.

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
Fig. 2 Shape change of flat L from 1998 to 2004 (Media 1).
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
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).
, along with a fitting to the equation
y=A[1e(xB)/C],
(1)
where A is the asymptotic excursion, B is the initial time, and C is the time constant.

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

44. M. Vannoni and G. Molesini, “Three-flat test with plates in horizontal posture,” Appl. Opt. 47(12), 2133–2145 (2008). [CrossRef] [PubMed]

]. As a major result, in our geometry the overall elastic deformation w is conveniently represented with a few terms of the Zernike polynomial representation:
w=c3Z3+c4Z4+c10Z10+c26Z26,
(2)
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

44. M. Vannoni and G. Molesini, “Three-flat test with plates in horizontal posture,” Appl. Opt. 47(12), 2133–2145 (2008). [CrossRef] [PubMed]

, 45

45. W. A. Bassali and H. G. Eggleston, “The transverse flexure of thin elastic plates supported at several points,” Proc. Camb. Philos. Soc. 53(03), 728–742 (1957). [CrossRef]

]. A reconstruction of the deformation by means of the terms in point is shown in Fig. 5
Fig. 5 Deformation of flat L as in Fig. 4, reconstructed with the Zernike terms of Table 2.
. 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.

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.

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
Fig. 7 Geometrical layout of a flat in horizontal posture to model laminar flow of fused silica.
, classical approaches provide the basic equation
dvdr=1ηFA,
(3)
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 F=ρgπr2h, A=2πrh, with ρ the density of fused silica, g the gravity, and h the thickness of the flat, so that

dv=ρg2ηrdr.
(4)

Integrating Eq. (4) and considering the boundary condition v(R)=0, with R the radius of the entire flat, one obtains
η=ρgR24v(0),
(5)
where v(0) 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

46. H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology (Elsevier, Amsterdam 1989), p. 11.

]; the range considered in Refs [7

7. E. D. Zanotto, “Do cathedral glasses flow?” Am. J. Phys. 66(5), 392–395 (1998). [CrossRef]

]. to [11

11. Y. M. Stokes, “Flowing windowpanes: a comparison of Newtonian and Maxwell fluid models,” Proc. R. Soc. Lond. A 456(2000), 1861–1864 (2000). [CrossRef]

] is from a minimum of 1019 (taken as an “ultra-conservative” order of magnitude [11

11. Y. M. Stokes, “Flowing windowpanes: a comparison of Newtonian and Maxwell fluid models,” Proc. R. Soc. Lond. A 456(2000), 1861–1864 (2000). [CrossRef]

]) up to 1041 Pa s (a “more realistic” estimate [11

11. Y. M. Stokes, “Flowing windowpanes: a comparison of Newtonian and Maxwell fluid models,” Proc. R. Soc. Lond. A 456(2000), 1861–1864 (2000). [CrossRef]

]). 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

dρρ=dhh.
(6)

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

References and links

1.

B. Mysen, and P. Richet, Silicate Glasses and Melts (Elsevier, Amsterdam 2005).

2.

K. Numata, K. Yamamoto, H. Ishimoto, S. Otsuka, K. Kawabe, M. Ando, and K. Tsubono, “Systematic measurement of the intrinsic losses in various kinds of bulk fused silica,” Phys. Lett. A 327(4), 263–271 (2004). [CrossRef]

3.

R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt. 46(33), 8118–8133 (2007). [CrossRef] [PubMed]

4.

M. Ando, K. Arai, R. Takahashi, G. Heinzel, S. Kawamura, D. Tatsumi, N. Kanda, H. Tagoshi, A. Araya, H. Asada, Y. Aso, M. A. Barton, M. K. Fujimoto, M. Fukushima, T. Futamase, K. Hayama, G. Horikoshi, H. Ishizuka, N. Kamikubota, K. Kawabe, N. Kawashima, Y. Kobayashi, Y. Kojima, K. Kondo, Y. Kozai, K. Kuroda, N. Matsuda, N. Mio, K. Miura, O. Miyakawa, S. M. Miyama, S. Miyoki, S. Moriwaki, M. Musha, S. Nagano, K. Nakagawa, T. Nakamura, K. Nakao, K. Numata, Y. Ogawa, M. Ohashi, N. Ohishi, S. Okutomi, K. Oohara, S. Otsuka, Y. Saito, M. Sasaki, S. Sato, A. Sekiya, M. Shibata, K. Somiya, T. Suzuki, A. Takamori, T. Tanaka, S. Taniguchi, S. Telada, K. Tochikubo, T. Tomaru, K. Tsubono, N. Tsuda, T. Uchiyama, A. Ueda, K. Ueda, K. Waseda, Y. Watanabe, H. Yakura, K. Yamamoto, and T. YamazakiM. AndoK. AraiR. TakahashiG. HeinzelS. KawamuraD. TatsumiN. KandaH. TagoshiA. ArayaH. AsadaY. AsoM. A. BartonM. K. FujimotoM. FukushimaT. FutamaseK. HayamaG. HorikoshiH. IshizukaN. KamikubotaK. KawabeN. KawashimaY. KobayashiY. KojimaK. KondoY. KozaiK. KurodaN. MatsudaN. MioK. MiuraO. MiyakawaS. M. MiyamaS. MiyokiS. MoriwakiM. MushaS. NaganoK. NakagawaT. NakamuraK. NakaoK. NumataY. OgawaM. OhashiN. OhishiS. OkutomiK. OoharaS. OtsukaY. SaitoM. SasakiS. SatoA. SekiyaM. ShibataK. SomiyaT. SuzukiA. TakamoriT. TanakaS. TaniguchiS. TeladaK. TochikuboT. TomaruK. TsubonoN. TsudaT. UchiyamaA. UedaK. UedaK. WasedaY. WatanabeH. YakuraK. YamamotoT. YamazakiTAMA Collaboration, “Stable operation of a 300-m laser interferometer with sufficient sensitivity to detect gravitational-wave events within our galaxy,” Phys. Rev. Lett. 86(18), 3950–3954 (2001). [CrossRef] [PubMed]

5.

S. Jordan, “The GAIA project: Technique, performance and status,” Astron. Nachr. 329(9-10), 875–880 (2008). [CrossRef]

6.

G. W. Morey, “The flow of glass at room temperature,” J. Opt. Soc. Am. 42(11), 856–857 (1952). [CrossRef]

7.

E. D. Zanotto, “Do cathedral glasses flow?” Am. J. Phys. 66(5), 392–395 (1998). [CrossRef]

8.

M. Pasachoff, “Comment on ‘Do cathedral glasses flow?’,” Am. J. Phys. 66(11), 1021 (1998). [CrossRef]

9.

E. D. Zanotto and P. K. Gupta, “Do cathedral glasses flow? – Additional remarks,” Am. J. Phys. 67(3), 260–262 (1999). [CrossRef]

10.

Y. M. Stokes, “Flowing windowpanes: fact or fiction?” Proc. R. Soc. Lond. A 455(1987), 2751–2756 (1999). [CrossRef]

11.

Y. M. Stokes, “Flowing windowpanes: a comparison of Newtonian and Maxwell fluid models,” Proc. R. Soc. Lond. A 456(2000), 1861–1864 (2000). [CrossRef]

12.

G. D. Dew, “Some observations on the long-term stability of optical flat,” Opt. Acta (Lond.) 21, 609–614 (1974). [CrossRef]

13.

J. W. Berthold III, S. F. Jacobs, and M. A. Norton, “Dimensional stability of fused silica, Invar, and several ultralow thermal expansion materials,” Appl. Opt. 15(8), 1898–1899 (1976). [CrossRef]

14.

J. W. Berthold III, S. F. Jacobs, and M. A. Norton, “Dimensional stability of fused silica, Invar, and several ultralow thermal expansion materials,” Metrologia 13(1), 9–16 (1977). [CrossRef]

15.

D. A. Ketelsen and D. S. Anderson, “Optical testing with large liquid flats,” Proc. Soc. Photo Opt. Instrum. Eng. 966, 365–371 (1988).

16.

I. Powell and E. Goulet, “Absolute figure measurements with a liquid-flat reference,” Appl. Opt. 37(13), 2579–2588 (1998). [CrossRef]

17.

M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42(5), 389–393 (2005). [CrossRef]

18.

M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45(6), 1–3 (2006). [CrossRef]

19.

P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46(2), 1–9 (2007). [CrossRef]

20.

J. Yellowhair and J. H. Burge, “Analysis of a scanning pentaprism system for measurements of large flat mirrors,” Appl. Opt. 46(35), 8466–8474 (2007). [CrossRef] [PubMed]

21.

R. D. Geckeler, “Optimal use of pentaprism in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18(1), 115–125 (2007). [CrossRef]

22.

G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6(6), 1077–1084 (1967). [CrossRef] [PubMed]

23.

G. Schulz, J. Schwider, C. Hiller, and B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10(4), 929–934 (1971). [CrossRef] [PubMed]

24.

J. Grzanna and G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77(2-3), 107–112 (1990). [CrossRef]

25.

G. Schulz and J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31(19), 3767–3780 (1992). [CrossRef] [PubMed]

26.

G. Schulz, “Absolute flatness testing by an extended rotation method using two angles of rotation,” Appl. Opt. 32(7), 1055–1059 (1993). [CrossRef] [PubMed]

27.

J. Grzanna, “Absolute testing of optical flats at points on a square grid: error propagation,” Appl. Opt. 33(28), 6654–6661 (1994). [CrossRef] [PubMed]

28.

B. B. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39(28), 5161–5171 (2000). [CrossRef]

29.

S. Sonozaki, K. Iwata, and Y. Iwahashi, “Measurement of profiles along a circle on two flat surfaces by use of a Fizeau interferometer with no standard,” Appl. Opt. 42(34), 6853–6858 (2003). [CrossRef] [PubMed]

30.

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 33, 379–383 (1984).

31.

C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32(25), 4698–4705 (1993). [CrossRef] [PubMed]

32.

C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35(7), 1015–1021 (1996). [CrossRef] [PubMed]

33.

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 36(9), 2478–2481 (1997). [CrossRef]

34.

C. J. Evans, “Comment on the paper ‘Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 37(6), 1880–1882 (1998). [CrossRef]

35.

R. E. Parks, L.-Z. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37(25), 5951–5956 (1998). [CrossRef]

36.

V. Greco, R. Tronconi, C. D. Vecchio, M. Trivi, and G. Molesini, “Absolute measurement of planarity with Fritz’s method: uncertainty evaluation,” Appl. Opt. 38(10), 2018–2027 (1999). [CrossRef]

37.

K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40(10), 1637–1648 (2001). [CrossRef]

38.

M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Stuttg.) 112(9), 381–391 (2001). [CrossRef]

39.

U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45(23), 5856–5865 (2006). [CrossRef] [PubMed]

40.

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46(9), 093601 (2007). [CrossRef]

41.

M. Vannoni and G. Molesini, “Iterative algorithm for three flat test,” Opt. Express 15(11), 6809–6816 (2007). [CrossRef] [PubMed]

42.

M. Vannoni and G. Molesini, “Absolute planarity with three-flat test: an iterative approach with Zernike polynomials,” Opt. Express 16(1), 340–354 (2008). [CrossRef] [PubMed]

43.

International Bureau of Weights and Measures, International Electrotechnical Commission, International Federation of Clinical Chemistry, International Organization for Standardization, International Union of Pure and Applied Chemistry, International Union of Pure and Applied Physics, and International Organization of Legal Metrology, Guide to the Expression of Uncertainty in Measurements (International Organization for Standardization, Geneva, 1993).

44.

M. Vannoni and G. Molesini, “Three-flat test with plates in horizontal posture,” Appl. Opt. 47(12), 2133–2145 (2008). [CrossRef] [PubMed]

45.

W. A. Bassali and H. G. Eggleston, “The transverse flexure of thin elastic plates supported at several points,” Proc. Camb. Philos. Soc. 53(03), 728–742 (1957). [CrossRef]

46.

H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology (Elsevier, Amsterdam 1989), p. 11.

47.

L. Landau, and E. Lifchitz, Théorie de l’Élasticité (MIR, Moscow 1967), p. 201.

48.

J. Langer, “The mysterious glass transition,” Phys. Today 60(2), 8–9 (2007). [CrossRef]

49.

N. J. Wagner and J. F. Brady, “Shear thickening in colloidal dispersions,” Phys. Today 62(10), 27–32 (2009). [CrossRef]

OCIS Codes
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3940) Instrumentation, measurement, and metrology : Metrology
(120.4800) Instrumentation, measurement, and metrology : Optical standards and testing
(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: November 25, 2009
Revised Manuscript: January 11, 2010
Manuscript Accepted: January 22, 2010
Published: February 25, 2010

Citation
Maurizio Vannoni, Andrea Sordini, and Giuseppe Molesini, "Long-term deformation at room temperature observed in fused silica," Opt. Express 18, 5114-5123 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-5114


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References

  1. B. Mysen, and P. Richet, Silicate Glasses and Melts (Elsevier, Amsterdam 2005).
  2. K. Numata, K. Yamamoto, H. Ishimoto, S. Otsuka, K. Kawabe, M. Ando, and K. Tsubono, “Systematic measurement of the intrinsic losses in various kinds of bulk fused silica,” Phys. Lett. A 327(4), 263–271 (2004). [CrossRef]
  3. R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt. 46(33), 8118–8133 (2007). [CrossRef] [PubMed]
  4. M. Ando, K. Arai, R. Takahashi, G. Heinzel, S. Kawamura, D. Tatsumi, N. Kanda, H. Tagoshi, A. Araya, H. Asada, Y. Aso, M. A. Barton, M. K. Fujimoto, M. Fukushima, T. Futamase, K. Hayama, G. Horikoshi, H. Ishizuka, N. Kamikubota, K. Kawabe, N. Kawashima, Y. Kobayashi, Y. Kojima, K. Kondo, Y. Kozai, K. Kuroda, N. Matsuda, N. Mio, K. Miura, O. Miyakawa, S. M. Miyama, S. Miyoki, S. Moriwaki, M. Musha, S. Nagano, K. Nakagawa, T. Nakamura, K. Nakao, K. Numata, Y. Ogawa, M. Ohashi, N. Ohishi, S. Okutomi, K. Oohara, S. Otsuka, Y. Saito, M. Sasaki, S. Sato, A. Sekiya, M. Shibata, K. Somiya, T. Suzuki, A. Takamori, T. Tanaka, S. Taniguchi, S. Telada, K. Tochikubo, T. Tomaru, K. Tsubono, N. Tsuda, T. Uchiyama, A. Ueda, K. Ueda, K. Waseda, Y. Watanabe, H. Yakura, K. Yamamoto, T. Yamazaki, and TAMA Collaboration, “Stable operation of a 300-m laser interferometer with sufficient sensitivity to detect gravitational-wave events within our galaxy,” Phys. Rev. Lett. 86(18), 3950–3954 (2001). [CrossRef] [PubMed]
  5. S. Jordan, “The GAIA project: Technique, performance and status,” Astron. Nachr. 329(9-10), 875–880 (2008). [CrossRef]
  6. G. W. Morey, “The flow of glass at room temperature,” J. Opt. Soc. Am. 42(11), 856–857 (1952). [CrossRef]
  7. E. D. Zanotto, “Do cathedral glasses flow?” Am. J. Phys. 66(5), 392–395 (1998). [CrossRef]
  8. M. Pasachoff, “Comment on ‘Do cathedral glasses flow?’,” Am. J. Phys. 66(11), 1021 (1998). [CrossRef]
  9. E. D. Zanotto and P. K. Gupta, “Do cathedral glasses flow? – Additional remarks,” Am. J. Phys. 67(3), 260–262 (1999). [CrossRef]
  10. Y. M. Stokes, “Flowing windowpanes: fact or fiction?” Proc. R. Soc. Lond. A 455(1987), 2751–2756 (1999). [CrossRef]
  11. Y. M. Stokes, “Flowing windowpanes: a comparison of Newtonian and Maxwell fluid models,” Proc. R. Soc. Lond. A 456(2000), 1861–1864 (2000). [CrossRef]
  12. G. D. Dew, “Some observations on the long-term stability of optical flat,” Opt. Acta (Lond.) 21, 609–614 (1974). [CrossRef]
  13. J. W. Berthold, S. F. Jacobs, and M. A. Norton, “Dimensional stability of fused silica, Invar, and several ultralow thermal expansion materials,” Appl. Opt. 15(8), 1898–1899 (1976). [CrossRef]
  14. J. W. Berthold, S. F. Jacobs, and M. A. Norton, “Dimensional stability of fused silica, Invar, and several ultralow thermal expansion materials,” Metrologia 13(1), 9–16 (1977). [CrossRef]
  15. D. A. Ketelsen and D. S. Anderson, “Optical testing with large liquid flats,” Proc. Soc. Photo Opt. Instrum. Eng. 966, 365–371 (1988).
  16. I. Powell and E. Goulet, “Absolute figure measurements with a liquid-flat reference,” Appl. Opt. 37(13), 2579–2588 (1998). [CrossRef]
  17. M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42(5), 389–393 (2005). [CrossRef]
  18. M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45(6), 1–3 (2006). [CrossRef]
  19. P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46(2), 1–9 (2007). [CrossRef]
  20. J. Yellowhair and J. H. Burge, “Analysis of a scanning pentaprism system for measurements of large flat mirrors,” Appl. Opt. 46(35), 8466–8474 (2007). [CrossRef] [PubMed]
  21. R. D. Geckeler, “Optimal use of pentaprism in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18(1), 115–125 (2007). [CrossRef]
  22. G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6(6), 1077–1084 (1967). [CrossRef] [PubMed]
  23. G. Schulz, J. Schwider, C. Hiller, and B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10(4), 929–934 (1971). [CrossRef] [PubMed]
  24. J. Grzanna and G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77(2-3), 107–112 (1990). [CrossRef]
  25. G. Schulz and J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31(19), 3767–3780 (1992). [CrossRef] [PubMed]
  26. G. Schulz, “Absolute flatness testing by an extended rotation method using two angles of rotation,” Appl. Opt. 32(7), 1055–1059 (1993). [CrossRef] [PubMed]
  27. J. Grzanna, “Absolute testing of optical flats at points on a square grid: error propagation,” Appl. Opt. 33(28), 6654–6661 (1994). [CrossRef] [PubMed]
  28. B. B. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39(28), 5161–5171 (2000). [CrossRef]
  29. S. Sonozaki, K. Iwata, and Y. Iwahashi, “Measurement of profiles along a circle on two flat surfaces by use of a Fizeau interferometer with no standard,” Appl. Opt. 42(34), 6853–6858 (2003). [CrossRef] [PubMed]
  30. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 33, 379–383 (1984).
  31. C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32(25), 4698–4705 (1993). [CrossRef] [PubMed]
  32. C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35(7), 1015–1021 (1996). [CrossRef] [PubMed]
  33. P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 36(9), 2478–2481 (1997). [CrossRef]
  34. C. J. Evans, “Comment on the paper ‘Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 37(6), 1880–1882 (1998). [CrossRef]
  35. R. E. Parks, L.-Z. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37(25), 5951–5956 (1998). [CrossRef]
  36. V. Greco, R. Tronconi, C. D. Vecchio, M. Trivi, and G. Molesini, “Absolute measurement of planarity with Fritz’s method: uncertainty evaluation,” Appl. Opt. 38(10), 2018–2027 (1999). [CrossRef]
  37. K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40(10), 1637–1648 (2001). [CrossRef]
  38. M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Stuttg.) 112(9), 381–391 (2001). [CrossRef]
  39. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45(23), 5856–5865 (2006). [CrossRef] [PubMed]
  40. U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46(9), 093601 (2007). [CrossRef]
  41. M. Vannoni and G. Molesini, “Iterative algorithm for three flat test,” Opt. Express 15(11), 6809–6816 (2007). [CrossRef] [PubMed]
  42. M. Vannoni and G. Molesini, “Absolute planarity with three-flat test: an iterative approach with Zernike polynomials,” Opt. Express 16(1), 340–354 (2008). [CrossRef] [PubMed]
  43. International Bureau of Weights and Measures, International Electrotechnical Commission, International Federation of Clinical Chemistry, International Organization for Standardization, International Union of Pure and Applied Chemistry, International Union of Pure and Applied Physics, and International Organization of Legal Metrology, Guide to the Expression of Uncertainty in Measurements (International Organization for Standardization, Geneva, 1993).
  44. M. Vannoni and G. Molesini, “Three-flat test with plates in horizontal posture,” Appl. Opt. 47(12), 2133–2145 (2008). [CrossRef] [PubMed]
  45. W. A. Bassali and H. G. Eggleston, “The transverse flexure of thin elastic plates supported at several points,” Proc. Camb. Philos. Soc. 53(03), 728–742 (1957). [CrossRef]
  46. H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology (Elsevier, Amsterdam 1989), p. 11.
  47. L. Landau, and E. Lifchitz, Théorie de l’Élasticité (MIR, Moscow 1967), p. 201.
  48. J. Langer, “The mysterious glass transition,” Phys. Today 60(2), 8–9 (2007). [CrossRef]
  49. N. J. Wagner and J. F. Brady, “Shear thickening in colloidal dispersions,” Phys. Today 62(10), 27–32 (2009). [CrossRef]

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