Abstract

We describe a technology to mass-produce ultrathin mirror substrates for x-ray telescopes of near Wolter-I geometry. Thermal glass forming is a low-cost method to produce high-throughput, spaceborne x-ray mirrors for the 0.1–200-keV energy band. These substrates can provide the collecting area envisioned for future x-ray observatories. The glass microsheets are shaped into mirror segments at high temperature by use of a guiding mandrel, without polishing. We determine the physical properties and mechanisms that elucidate the formation process and that are crucial to improve surface quality. We develop a viscodynamic model for the glass strain as the forming proceeds to find the conditions for repeatability. Thermal forming preserves the x-ray reflectance and scattering properties of the raw glass. The imaging resolution is driven by a large wavelength figure. We discuss the sources of figure errors, and we calculate the relaxation time of surface ripples.

© 2003 Optical Society of America

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  1. M. C. Weisskopf, H. D. Tananbaum, L. P. Van Speybroeck, S. L. O’Dell, “Chandra X-ray Observatory (CXO): overview,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 2–16 (2000).
    [CrossRef]
  2. P. Gondoin, K. van Katwijk, B. R. Aschenbach, N. Schulz, R. Boerret, H. Glatzel, O. Citterio, “X-ray spectroscopy mission (XMM) telescope development,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 438–450 (1994).
    [CrossRef]
  3. F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
    [CrossRef]
  4. M. A. Jimenez-Garate, “The development of hard x-ray telescope optics and a theoretical model of x-ray emission from accretion disks,” Ph.D. dissertation (Columbia University, New York, 2001).
  5. W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
    [CrossRef]
  6. P. J. Serlemitsos, Y. Soong, “Foil X-ray mirrors,” Astrophys. Space. Sci. 239, 177–196 (1996).
    [CrossRef]
  7. H. Kunieda, M. Ishida, T. Endo, Y. Hidaka, H. Honda, K. Imamura, I. Ishida, M. Maeda, K. Misaki, R. Shibata, A. Furuzawa, K. Haga, Y. Ogasaka, T. Okajima, Y. Tawara, Y. Terashima, M. Watanabe, K. Yamashita, T. Yoshioka, P. J. Serlemitsos, Y. Soong, K. Chan, “X-ray telescope onboard Astro-E: optical design and fabrication of thin foil mirrors,” Appl. Opt. 40, 553–564 (2001).
    [CrossRef]
  8. H. D. Tananbaum, N. E. White, J. A. Bookbinder, F. E. Marshall, F. A. Cordova, “Constellation X-ray mission: implementation concept and science overview,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 62–72 (1999).
    [CrossRef]
  9. F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
    [CrossRef]
  10. L. Nevot, P. Croce, “Characterisation des surfaces par reflexion rasante de rayons X. Application a l’etude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
    [CrossRef]
  11. B. Aschenbach, H. Braeuninger, “Grazing incidence for ESA’s X-ray astronomy mission XMM,” in x-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. SPIE982, 10–21 (1988).
    [CrossRef]
  12. M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, F. E. Christensen, A. Hussain, “Fabrication, performance and figure metrology of epoxy replicated aluminum foils for hard x-ray focusing multilayer-coated segmented conical optics,” Opt. Eng. 39, 2982–2994 (2000).
    [CrossRef]
  13. C. J. Hailey, S. Abdali, F. E. Christensen, W. W. Craig, T. R. Decker, F. A. Harrison, M. A. Jiménez-Gárate, “Investigation of substrates and mounting techniques for the High-Energy Focusing Telescope (HEFT),” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 535–543 (1997).
    [CrossRef]
  14. W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
    [CrossRef]
  15. M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, “Fast optical metrology of the hard x-ray optics for the HEFT telescope,” in X-Ray Optics, Instruments, and Missions, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3444, 622–633 (1998).
    [CrossRef]
  16. D. R. Uhlmann, N. J. Kreidl, eds., Processing I, Vol. 2 of Glass Science and Technology (Academic, New York, 1984).
  17. I. Gutzow, J. Schmelzer, The Vitreous State: Thermodynamics, Structure, Rheology, and Crystallization (Springer, New York, 1995).
  18. J. C. Lapp, P. L. Bocko, J. W. Nelson, “Advanced glass substrates for flat panel displays,” in Advanced Flat Panel Display Technologies, P. S. Friedman, ed., Proc. SPIE2174, 129–138 (1994).
    [CrossRef]
  19. R. H. Doremus, Glass Science (Wiley, New York, 1994).
  20. H. Rawson, Glasses and their Applications (Institute of Metals, Brookfield, Vt., 1991).
  21. I. S. Grigoriev, E. Z. Meilikhov, Handbook of Physical Quantities (CRC Press, Boca Raton, Fla., 1997).
  22. H. Scholze, Glass: Nature, Structure, and Properties (Springer-Verlag, New York, 1991).
  23. N. P. Bansal, R. H. Doremus, Handbook of Glass Properties (Academic, Orlando, Fla., 1986).
  24. M. B. Volf, Mathematical Approach to Glass (Elsevier, New York, 1988).
  25. Y. S. Touloukian, ed., Thermophysical Properties of Matter, Vols. 4, 5, and 13 of the Purdue University Thermophysical Properties Research Center Data Series (IFI/Plenum, New York, 1970).
  26. J. H. Whitelaw, “Convective heat transfer,” in International Encyclopedia of Heat and Mass Transfer, G. F. Hewitt, G. L. Shires, Y. V. Polezhaev, eds. (CRC Press, Boca Raton, Fla., 1997), p. 237.
  27. J. Symonds, J. P. Vidosic, H. V. Hawkins, D. D. Dodge, “Strength of materials,” in Standard Handbook for Mechanical Engineers, E. A. Avallone, T. B. Baumeister, eds. (McGraw-Hill, New York, 1986), pp. 5–24.
  28. G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
    [CrossRef]
  29. A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
    [CrossRef]
  30. F. E. Christensen, Z. Shou-Hua, A. Hornstrup, H. W. Schnopper, P. Plag, J. Wood, “X-ray study of state-of-the-art small d-spacing W/B4C multilayers,” J. X-Ray Sci. Technology 3, 1–13 (1991).
    [CrossRef]
  31. A. M. Hussain, “Multilayer x-ray telescopes,” Ph.D. dissertation (University of Copenhagen, Copenhagen, Denmark, 1999).
  32. F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
    [CrossRef]
  33. P. H. Mao, F. A. Harrison, Y. Y. Platonov, D. Broadway, B. DeGroot, F. E. Christensen, W. W. Craig, C. J. Hailey, “Development of grazing incidence multilayer mirrors for hard x-ray focusing telescopes,” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 526–534 (1997).
    [CrossRef]
  34. D. L. Windt, F. E. Christensen, W. W. Craig, C. J. Hailey, F. A. Harrison, M. A. Jimenez-Garate, R. Kalyanaraman, P. H. Mao, “X-ray multilayer coatings for use at energies above 100 keV,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 442–447 (2000).
    [CrossRef]
  35. W. W. Zhang, K. Chan, D. A. Content, R. Petre, P. J. Serlemitsos, T. T. Saha, Y. Soong, “Development of mirror segments for the Constellation-X Observatory,” in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy, J. E. Truemper, H. D. Tananbaum, eds., Proc. SPIE4851, (2002).

2001 (1)

2000 (3)

W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
[CrossRef]

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, F. E. Christensen, A. Hussain, “Fabrication, performance and figure metrology of epoxy replicated aluminum foils for hard x-ray focusing multilayer-coated segmented conical optics,” Opt. Eng. 39, 2982–2994 (2000).
[CrossRef]

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

1996 (1)

P. J. Serlemitsos, Y. Soong, “Foil X-ray mirrors,” Astrophys. Space. Sci. 239, 177–196 (1996).
[CrossRef]

1991 (1)

F. E. Christensen, Z. Shou-Hua, A. Hornstrup, H. W. Schnopper, P. Plag, J. Wood, “X-ray study of state-of-the-art small d-spacing W/B4C multilayers,” J. X-Ray Sci. Technology 3, 1–13 (1991).
[CrossRef]

1980 (1)

L. Nevot, P. Croce, “Characterisation des surfaces par reflexion rasante de rayons X. Application a l’etude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
[CrossRef]

Abdali, S.

C. J. Hailey, S. Abdali, F. E. Christensen, W. W. Craig, T. R. Decker, F. A. Harrison, M. A. Jiménez-Gárate, “Investigation of substrates and mounting techniques for the High-Energy Focusing Telescope (HEFT),” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 535–543 (1997).
[CrossRef]

Aschenbach, B.

B. Aschenbach, H. Braeuninger, “Grazing incidence for ESA’s X-ray astronomy mission XMM,” in x-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. SPIE982, 10–21 (1988).
[CrossRef]

Aschenbach, B. R.

P. Gondoin, K. van Katwijk, B. R. Aschenbach, N. Schulz, R. Boerret, H. Glatzel, O. Citterio, “X-ray spectroscopy mission (XMM) telescope development,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 438–450 (1994).
[CrossRef]

Bansal, N. P.

N. P. Bansal, R. H. Doremus, Handbook of Glass Properties (Academic, Orlando, Fla., 1986).

Bocko, P. L.

J. C. Lapp, P. L. Bocko, J. W. Nelson, “Advanced glass substrates for flat panel displays,” in Advanced Flat Panel Display Technologies, P. S. Friedman, ed., Proc. SPIE2174, 129–138 (1994).
[CrossRef]

Boerret, R.

P. Gondoin, K. van Katwijk, B. R. Aschenbach, N. Schulz, R. Boerret, H. Glatzel, O. Citterio, “X-ray spectroscopy mission (XMM) telescope development,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 438–450 (1994).
[CrossRef]

Boggs, S. E.

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

Bolotnikov, A. E.

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

Bookbinder, J. A.

H. D. Tananbaum, N. E. White, J. A. Bookbinder, F. E. Marshall, F. A. Cordova, “Constellation X-ray mission: implementation concept and science overview,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 62–72 (1999).
[CrossRef]

Braeuninger, H.

B. Aschenbach, H. Braeuninger, “Grazing incidence for ESA’s X-ray astronomy mission XMM,” in x-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. SPIE982, 10–21 (1988).
[CrossRef]

Broadway, D.

P. H. Mao, F. A. Harrison, Y. Y. Platonov, D. Broadway, B. DeGroot, F. E. Christensen, W. W. Craig, C. J. Hailey, “Development of grazing incidence multilayer mirrors for hard x-ray focusing telescopes,” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 526–534 (1997).
[CrossRef]

Butler, N.

G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
[CrossRef]

Chakan, J. M.

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

Chan, K.

H. Kunieda, M. Ishida, T. Endo, Y. Hidaka, H. Honda, K. Imamura, I. Ishida, M. Maeda, K. Misaki, R. Shibata, A. Furuzawa, K. Haga, Y. Ogasaka, T. Okajima, Y. Tawara, Y. Terashima, M. Watanabe, K. Yamashita, T. Yoshioka, P. J. Serlemitsos, Y. Soong, K. Chan, “X-ray telescope onboard Astro-E: optical design and fabrication of thin foil mirrors,” Appl. Opt. 40, 553–564 (2001).
[CrossRef]

W. W. Zhang, K. Chan, D. A. Content, R. Petre, P. J. Serlemitsos, T. T. Saha, Y. Soong, “Development of mirror segments for the Constellation-X Observatory,” in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy, J. E. Truemper, H. D. Tananbaum, eds., Proc. SPIE4851, (2002).

Chen, C. G.

G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
[CrossRef]

Christensen, F. E.

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, F. E. Christensen, A. Hussain, “Fabrication, performance and figure metrology of epoxy replicated aluminum foils for hard x-ray focusing multilayer-coated segmented conical optics,” Opt. Eng. 39, 2982–2994 (2000).
[CrossRef]

W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
[CrossRef]

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

F. E. Christensen, Z. Shou-Hua, A. Hornstrup, H. W. Schnopper, P. Plag, J. Wood, “X-ray study of state-of-the-art small d-spacing W/B4C multilayers,” J. X-Ray Sci. Technology 3, 1–13 (1991).
[CrossRef]

D. L. Windt, F. E. Christensen, W. W. Craig, C. J. Hailey, F. A. Harrison, M. A. Jimenez-Garate, R. Kalyanaraman, P. H. Mao, “X-ray multilayer coatings for use at energies above 100 keV,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 442–447 (2000).
[CrossRef]

P. H. Mao, F. A. Harrison, Y. Y. Platonov, D. Broadway, B. DeGroot, F. E. Christensen, W. W. Craig, C. J. Hailey, “Development of grazing incidence multilayer mirrors for hard x-ray focusing telescopes,” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 526–534 (1997).
[CrossRef]

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

C. J. Hailey, S. Abdali, F. E. Christensen, W. W. Craig, T. R. Decker, F. A. Harrison, M. A. Jiménez-Gárate, “Investigation of substrates and mounting techniques for the High-Energy Focusing Telescope (HEFT),” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 535–543 (1997).
[CrossRef]

W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

Citterio, O.

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

P. Gondoin, K. van Katwijk, B. R. Aschenbach, N. Schulz, R. Boerret, H. Glatzel, O. Citterio, “X-ray spectroscopy mission (XMM) telescope development,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 438–450 (1994).
[CrossRef]

Content, D. A.

W. W. Zhang, K. Chan, D. A. Content, R. Petre, P. J. Serlemitsos, T. T. Saha, Y. Soong, “Development of mirror segments for the Constellation-X Observatory,” in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy, J. E. Truemper, H. D. Tananbaum, eds., Proc. SPIE4851, (2002).

Cook, W. R.

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

Cordova, F. A.

H. D. Tananbaum, N. E. White, J. A. Bookbinder, F. E. Marshall, F. A. Cordova, “Constellation X-ray mission: implementation concept and science overview,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 62–72 (1999).
[CrossRef]

Craig, W. W.

W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
[CrossRef]

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, F. E. Christensen, A. Hussain, “Fabrication, performance and figure metrology of epoxy replicated aluminum foils for hard x-ray focusing multilayer-coated segmented conical optics,” Opt. Eng. 39, 2982–2994 (2000).
[CrossRef]

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

D. L. Windt, F. E. Christensen, W. W. Craig, C. J. Hailey, F. A. Harrison, M. A. Jimenez-Garate, R. Kalyanaraman, P. H. Mao, “X-ray multilayer coatings for use at energies above 100 keV,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 442–447 (2000).
[CrossRef]

P. H. Mao, F. A. Harrison, Y. Y. Platonov, D. Broadway, B. DeGroot, F. E. Christensen, W. W. Craig, C. J. Hailey, “Development of grazing incidence multilayer mirrors for hard x-ray focusing telescopes,” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 526–534 (1997).
[CrossRef]

C. J. Hailey, S. Abdali, F. E. Christensen, W. W. Craig, T. R. Decker, F. A. Harrison, M. A. Jiménez-Gárate, “Investigation of substrates and mounting techniques for the High-Energy Focusing Telescope (HEFT),” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 535–543 (1997).
[CrossRef]

W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
[CrossRef]

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, “Fast optical metrology of the hard x-ray optics for the HEFT telescope,” in X-Ray Optics, Instruments, and Missions, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3444, 622–633 (1998).
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

Croce, P.

L. Nevot, P. Croce, “Characterisation des surfaces par reflexion rasante de rayons X. Application a l’etude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
[CrossRef]

Decker, T. A.

W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
[CrossRef]

Decker, T. R.

C. J. Hailey, S. Abdali, F. E. Christensen, W. W. Craig, T. R. Decker, F. A. Harrison, M. A. Jiménez-Gárate, “Investigation of substrates and mounting techniques for the High-Energy Focusing Telescope (HEFT),” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 535–543 (1997).
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

DeGroot, B.

P. H. Mao, F. A. Harrison, Y. Y. Platonov, D. Broadway, B. DeGroot, F. E. Christensen, W. W. Craig, C. J. Hailey, “Development of grazing incidence multilayer mirrors for hard x-ray focusing telescopes,” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 526–534 (1997).
[CrossRef]

del Rio, M. S.

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

Dodge, D. D.

J. Symonds, J. P. Vidosic, H. V. Hawkins, D. D. Dodge, “Strength of materials,” in Standard Handbook for Mechanical Engineers, E. A. Avallone, T. B. Baumeister, eds. (McGraw-Hill, New York, 1986), pp. 5–24.

Doremus, R. H.

N. P. Bansal, R. H. Doremus, Handbook of Glass Properties (Academic, Orlando, Fla., 1986).

R. H. Doremus, Glass Science (Wiley, New York, 1994).

Endo, T.

Freund, A. K.

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

Furuzawa, A.

Gehrels, N. A.

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

Glatzel, H.

P. Gondoin, K. van Katwijk, B. R. Aschenbach, N. Schulz, R. Boerret, H. Glatzel, O. Citterio, “X-ray spectroscopy mission (XMM) telescope development,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 438–450 (1994).
[CrossRef]

Gondoin, P.

P. Gondoin, K. van Katwijk, B. R. Aschenbach, N. Schulz, R. Boerret, H. Glatzel, O. Citterio, “X-ray spectroscopy mission (XMM) telescope development,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 438–450 (1994).
[CrossRef]

Gorenstein, P.

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

Grigoriev, I. S.

I. S. Grigoriev, E. Z. Meilikhov, Handbook of Physical Quantities (CRC Press, Boca Raton, Fla., 1997).

Grindlay, J. E.

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

Gutzow, I.

I. Gutzow, J. Schmelzer, The Vitreous State: Thermodynamics, Structure, Rheology, and Crystallization (Springer, New York, 1995).

Haga, K.

Hailey, C. J.

W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
[CrossRef]

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, F. E. Christensen, A. Hussain, “Fabrication, performance and figure metrology of epoxy replicated aluminum foils for hard x-ray focusing multilayer-coated segmented conical optics,” Opt. Eng. 39, 2982–2994 (2000).
[CrossRef]

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

D. L. Windt, F. E. Christensen, W. W. Craig, C. J. Hailey, F. A. Harrison, M. A. Jimenez-Garate, R. Kalyanaraman, P. H. Mao, “X-ray multilayer coatings for use at energies above 100 keV,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 442–447 (2000).
[CrossRef]

P. H. Mao, F. A. Harrison, Y. Y. Platonov, D. Broadway, B. DeGroot, F. E. Christensen, W. W. Craig, C. J. Hailey, “Development of grazing incidence multilayer mirrors for hard x-ray focusing telescopes,” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 526–534 (1997).
[CrossRef]

C. J. Hailey, S. Abdali, F. E. Christensen, W. W. Craig, T. R. Decker, F. A. Harrison, M. A. Jiménez-Gárate, “Investigation of substrates and mounting techniques for the High-Energy Focusing Telescope (HEFT),” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 535–543 (1997).
[CrossRef]

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, “Fast optical metrology of the hard x-ray optics for the HEFT telescope,” in X-Ray Optics, Instruments, and Missions, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3444, 622–633 (1998).
[CrossRef]

W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

Harrison, F. A.

W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
[CrossRef]

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

D. L. Windt, F. E. Christensen, W. W. Craig, C. J. Hailey, F. A. Harrison, M. A. Jimenez-Garate, R. Kalyanaraman, P. H. Mao, “X-ray multilayer coatings for use at energies above 100 keV,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 442–447 (2000).
[CrossRef]

P. H. Mao, F. A. Harrison, Y. Y. Platonov, D. Broadway, B. DeGroot, F. E. Christensen, W. W. Craig, C. J. Hailey, “Development of grazing incidence multilayer mirrors for hard x-ray focusing telescopes,” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 526–534 (1997).
[CrossRef]

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
[CrossRef]

C. J. Hailey, S. Abdali, F. E. Christensen, W. W. Craig, T. R. Decker, F. A. Harrison, M. A. Jiménez-Gárate, “Investigation of substrates and mounting techniques for the High-Energy Focusing Telescope (HEFT),” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 535–543 (1997).
[CrossRef]

Hawkins, H. V.

J. Symonds, J. P. Vidosic, H. V. Hawkins, D. D. Dodge, “Strength of materials,” in Standard Handbook for Mechanical Engineers, E. A. Avallone, T. B. Baumeister, eds. (McGraw-Hill, New York, 1986), pp. 5–24.

Heilmann, R. K.

G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
[CrossRef]

Hidaka, Y.

Hill, R. M.

W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
[CrossRef]

Honda, H.

Honkimaki, V.

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

Hornstrup, A.

F. E. Christensen, Z. Shou-Hua, A. Hornstrup, H. W. Schnopper, P. Plag, J. Wood, “X-ray study of state-of-the-art small d-spacing W/B4C multilayers,” J. X-Ray Sci. Technology 3, 1–13 (1991).
[CrossRef]

Hussain, A.

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, F. E. Christensen, A. Hussain, “Fabrication, performance and figure metrology of epoxy replicated aluminum foils for hard x-ray focusing multilayer-coated segmented conical optics,” Opt. Eng. 39, 2982–2994 (2000).
[CrossRef]

Hussain, A. M.

W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

A. M. Hussain, “Multilayer x-ray telescopes,” Ph.D. dissertation (University of Copenhagen, Copenhagen, Denmark, 1999).

Imamura, K.

Ishida, I.

Ishida, M.

J. Serlemitsos, P.

W. W. Zhang, K. Chan, D. A. Content, R. Petre, P. J. Serlemitsos, T. T. Saha, Y. Soong, “Development of mirror segments for the Constellation-X Observatory,” in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy, J. E. Truemper, H. D. Tananbaum, eds., Proc. SPIE4851, (2002).

Jimenez-Garate, M. A.

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, F. E. Christensen, A. Hussain, “Fabrication, performance and figure metrology of epoxy replicated aluminum foils for hard x-ray focusing multilayer-coated segmented conical optics,” Opt. Eng. 39, 2982–2994 (2000).
[CrossRef]

W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
[CrossRef]

M. A. Jimenez-Garate, “The development of hard x-ray telescope optics and a theoretical model of x-ray emission from accretion disks,” Ph.D. dissertation (Columbia University, New York, 2001).

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
[CrossRef]

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, “Fast optical metrology of the hard x-ray optics for the HEFT telescope,” in X-Ray Optics, Instruments, and Missions, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3444, 622–633 (1998).
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

D. L. Windt, F. E. Christensen, W. W. Craig, C. J. Hailey, F. A. Harrison, M. A. Jimenez-Garate, R. Kalyanaraman, P. H. Mao, “X-ray multilayer coatings for use at energies above 100 keV,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 442–447 (2000).
[CrossRef]

Jiménez-Gárate, M. A.

C. J. Hailey, S. Abdali, F. E. Christensen, W. W. Craig, T. R. Decker, F. A. Harrison, M. A. Jiménez-Gárate, “Investigation of substrates and mounting techniques for the High-Energy Focusing Telescope (HEFT),” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 535–543 (1997).
[CrossRef]

Kalyanaraman, R.

D. L. Windt, F. E. Christensen, W. W. Craig, C. J. Hailey, F. A. Harrison, M. A. Jimenez-Garate, R. Kalyanaraman, P. H. Mao, “X-ray multilayer coatings for use at energies above 100 keV,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 442–447 (2000).
[CrossRef]

Konkola, P. T.

G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
[CrossRef]

Kroeger, R. A.

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

Kuneida, H.

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

Kunieda, H.

Lapp, J. C.

J. C. Lapp, P. L. Bocko, J. W. Nelson, “Advanced glass substrates for flat panel displays,” in Advanced Flat Panel Display Technologies, P. S. Friedman, ed., Proc. SPIE2174, 129–138 (1994).
[CrossRef]

Madsen, A.

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

Maeda, M.

Mammen, C.

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

Mao, P. H.

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
[CrossRef]

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

D. L. Windt, F. E. Christensen, W. W. Craig, C. J. Hailey, F. A. Harrison, M. A. Jimenez-Garate, R. Kalyanaraman, P. H. Mao, “X-ray multilayer coatings for use at energies above 100 keV,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 442–447 (2000).
[CrossRef]

P. H. Mao, F. A. Harrison, Y. Y. Platonov, D. Broadway, B. DeGroot, F. E. Christensen, W. W. Craig, C. J. Hailey, “Development of grazing incidence multilayer mirrors for hard x-ray focusing telescopes,” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 526–534 (1997).
[CrossRef]

Marshall, F. E.

H. D. Tananbaum, N. E. White, J. A. Bookbinder, F. E. Marshall, F. A. Cordova, “Constellation X-ray mission: implementation concept and science overview,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 62–72 (1999).
[CrossRef]

McGuirk, M.

G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
[CrossRef]

Meilikhov, E. Z.

I. S. Grigoriev, E. Z. Meilikhov, Handbook of Physical Quantities (CRC Press, Boca Raton, Fla., 1997).

Misaki, K.

Mongrard, O.

G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
[CrossRef]

Monnelly, G. P.

G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
[CrossRef]

Nelson, J. W.

J. C. Lapp, P. L. Bocko, J. W. Nelson, “Advanced glass substrates for flat panel displays,” in Advanced Flat Panel Display Technologies, P. S. Friedman, ed., Proc. SPIE2174, 129–138 (1994).
[CrossRef]

Nevot, L.

L. Nevot, P. Croce, “Characterisation des surfaces par reflexion rasante de rayons X. Application a l’etude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
[CrossRef]

O’Dell, S. L.

M. C. Weisskopf, H. D. Tananbaum, L. P. Van Speybroeck, S. L. O’Dell, “Chandra X-ray Observatory (CXO): overview,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 2–16 (2000).
[CrossRef]

Ogasaka, Y.

Okajima, T.

Pareschi, G.

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

Parsons, A. M.

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

Petre, R.

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

W. W. Zhang, K. Chan, D. A. Content, R. Petre, P. J. Serlemitsos, T. T. Saha, Y. Soong, “Development of mirror segments for the Constellation-X Observatory,” in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy, J. E. Truemper, H. D. Tananbaum, eds., Proc. SPIE4851, (2002).

Plag, P.

F. E. Christensen, Z. Shou-Hua, A. Hornstrup, H. W. Schnopper, P. Plag, J. Wood, “X-ray study of state-of-the-art small d-spacing W/B4C multilayers,” J. X-Ray Sci. Technology 3, 1–13 (1991).
[CrossRef]

Platonov, Y. Y.

P. H. Mao, F. A. Harrison, Y. Y. Platonov, D. Broadway, B. DeGroot, F. E. Christensen, W. W. Craig, C. J. Hailey, “Development of grazing incidence multilayer mirrors for hard x-ray focusing telescopes,” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 526–534 (1997).
[CrossRef]

Rawson, H.

H. Rawson, Glasses and their Applications (Institute of Metals, Brookfield, Vt., 1991).

Ricker, G. R.

G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
[CrossRef]

Romaine, S. E.

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

Saha, T. T.

W. W. Zhang, K. Chan, D. A. Content, R. Petre, P. J. Serlemitsos, T. T. Saha, Y. Soong, “Development of mirror segments for the Constellation-X Observatory,” in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy, J. E. Truemper, H. D. Tananbaum, eds., Proc. SPIE4851, (2002).

Schattenburg, M. L.

G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
[CrossRef]

Schindler, S. M.

W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
[CrossRef]

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

Schmelzer, J.

I. Gutzow, J. Schmelzer, The Vitreous State: Thermodynamics, Structure, Rheology, and Crystallization (Springer, New York, 1995).

Schnopper, H. W.

F. E. Christensen, Z. Shou-Hua, A. Hornstrup, H. W. Schnopper, P. Plag, J. Wood, “X-ray study of state-of-the-art small d-spacing W/B4C multilayers,” J. X-Ray Sci. Technology 3, 1–13 (1991).
[CrossRef]

Scholze, H.

H. Scholze, Glass: Nature, Structure, and Properties (Springer-Verlag, New York, 1991).

Schulz, N.

P. Gondoin, K. van Katwijk, B. R. Aschenbach, N. Schulz, R. Boerret, H. Glatzel, O. Citterio, “X-ray spectroscopy mission (XMM) telescope development,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 438–450 (1994).
[CrossRef]

Serlemitsos, P. J.

Shibata, R.

Shou-Hua, Z.

F. E. Christensen, Z. Shou-Hua, A. Hornstrup, H. W. Schnopper, P. Plag, J. Wood, “X-ray study of state-of-the-art small d-spacing W/B4C multilayers,” J. X-Ray Sci. Technology 3, 1–13 (1991).
[CrossRef]

Soong, Y.

H. Kunieda, M. Ishida, T. Endo, Y. Hidaka, H. Honda, K. Imamura, I. Ishida, M. Maeda, K. Misaki, R. Shibata, A. Furuzawa, K. Haga, Y. Ogasaka, T. Okajima, Y. Tawara, Y. Terashima, M. Watanabe, K. Yamashita, T. Yoshioka, P. J. Serlemitsos, Y. Soong, K. Chan, “X-ray telescope onboard Astro-E: optical design and fabrication of thin foil mirrors,” Appl. Opt. 40, 553–564 (2001).
[CrossRef]

P. J. Serlemitsos, Y. Soong, “Foil X-ray mirrors,” Astrophys. Space. Sci. 239, 177–196 (1996).
[CrossRef]

W. W. Zhang, K. Chan, D. A. Content, R. Petre, P. J. Serlemitsos, T. T. Saha, Y. Soong, “Development of mirror segments for the Constellation-X Observatory,” in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy, J. E. Truemper, H. D. Tananbaum, eds., Proc. SPIE4851, (2002).

Souvorov, A.

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

Stern, M.

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

Symonds, J.

J. Symonds, J. P. Vidosic, H. V. Hawkins, D. D. Dodge, “Strength of materials,” in Standard Handbook for Mechanical Engineers, E. A. Avallone, T. B. Baumeister, eds. (McGraw-Hill, New York, 1986), pp. 5–24.

Tananbaum, H. D.

M. C. Weisskopf, H. D. Tananbaum, L. P. Van Speybroeck, S. L. O’Dell, “Chandra X-ray Observatory (CXO): overview,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 2–16 (2000).
[CrossRef]

H. D. Tananbaum, N. E. White, J. A. Bookbinder, F. E. Marshall, F. A. Cordova, “Constellation X-ray mission: implementation concept and science overview,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 62–72 (1999).
[CrossRef]

Tawara, Y.

Terashima, Y.

Tucoulou, R.

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

van Katwijk, K.

P. Gondoin, K. van Katwijk, B. R. Aschenbach, N. Schulz, R. Boerret, H. Glatzel, O. Citterio, “X-ray spectroscopy mission (XMM) telescope development,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 438–450 (1994).
[CrossRef]

Van Speybroeck, L. P.

M. C. Weisskopf, H. D. Tananbaum, L. P. Van Speybroeck, S. L. O’Dell, “Chandra X-ray Observatory (CXO): overview,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 2–16 (2000).
[CrossRef]

Vidosic, J. P.

J. Symonds, J. P. Vidosic, H. V. Hawkins, D. D. Dodge, “Strength of materials,” in Standard Handbook for Mechanical Engineers, E. A. Avallone, T. B. Baumeister, eds. (McGraw-Hill, New York, 1986), pp. 5–24.

Volf, M. B.

M. B. Volf, Mathematical Approach to Glass (Elsevier, New York, 1988).

Watanabe, M.

Weisskopf, M. C.

M. C. Weisskopf, H. D. Tananbaum, L. P. Van Speybroeck, S. L. O’Dell, “Chandra X-ray Observatory (CXO): overview,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 2–16 (2000).
[CrossRef]

White, N. E.

H. D. Tananbaum, N. E. White, J. A. Bookbinder, F. E. Marshall, F. A. Cordova, “Constellation X-ray mission: implementation concept and science overview,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 62–72 (1999).
[CrossRef]

Whitelaw, J. H.

J. H. Whitelaw, “Convective heat transfer,” in International Encyclopedia of Heat and Mass Transfer, G. F. Hewitt, G. L. Shires, Y. V. Polezhaev, eds. (CRC Press, Boca Raton, Fla., 1997), p. 237.

Windt, D. L.

W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
[CrossRef]

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

D. L. Windt, F. E. Christensen, W. W. Craig, C. J. Hailey, F. A. Harrison, M. A. Jimenez-Garate, R. Kalyanaraman, P. H. Mao, “X-ray multilayer coatings for use at energies above 100 keV,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 442–447 (2000).
[CrossRef]

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

Wood, J.

F. E. Christensen, Z. Shou-Hua, A. Hornstrup, H. W. Schnopper, P. Plag, J. Wood, “X-ray study of state-of-the-art small d-spacing W/B4C multilayers,” J. X-Ray Sci. Technology 3, 1–13 (1991).
[CrossRef]

Yamashita, K.

Yoshioka, T.

Zhang, W. W.

W. W. Zhang, K. Chan, D. A. Content, R. Petre, P. J. Serlemitsos, T. T. Saha, Y. Soong, “Development of mirror segments for the Constellation-X Observatory,” in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy, J. E. Truemper, H. D. Tananbaum, eds., Proc. SPIE4851, (2002).

Ziegler, E.

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

Appl. Opt. (1)

Astrophys. Space. Sci. (1)

P. J. Serlemitsos, Y. Soong, “Foil X-ray mirrors,” Astrophys. Space. Sci. 239, 177–196 (1996).
[CrossRef]

J. X-Ray Sci. Technology (1)

F. E. Christensen, Z. Shou-Hua, A. Hornstrup, H. W. Schnopper, P. Plag, J. Wood, “X-ray study of state-of-the-art small d-spacing W/B4C multilayers,” J. X-Ray Sci. Technology 3, 1–13 (1991).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (1)

F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, C. J. Hailey, F. A. Harrison, P. H. Mao, J. M. Chakan, E. Ziegler, V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical X-ray telescopes,” Nucl. Instrum. Methods Phys. Res. A 415, 572–581 (2000).
[CrossRef]

Opt. Eng. (1)

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, F. E. Christensen, A. Hussain, “Fabrication, performance and figure metrology of epoxy replicated aluminum foils for hard x-ray focusing multilayer-coated segmented conical optics,” Opt. Eng. 39, 2982–2994 (2000).
[CrossRef]

Opt. Exp. (1)

W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, D. L. Windt, F. A. Harrison, P. H. Mao, F. E. Christensen, A. M. Hussain, “Development of thermally formed glass optics for astronomical hard x-ray telescopes,” Opt. Exp. 7, 178–185 (2000); http://www.opticsexpress.org .
[CrossRef]

Rev. Phys. Appl. (1)

L. Nevot, P. Croce, “Characterisation des surfaces par reflexion rasante de rayons X. Application a l’etude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
[CrossRef]

Other (28)

B. Aschenbach, H. Braeuninger, “Grazing incidence for ESA’s X-ray astronomy mission XMM,” in x-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. SPIE982, 10–21 (1988).
[CrossRef]

A. M. Hussain, “Multilayer x-ray telescopes,” Ph.D. dissertation (University of Copenhagen, Copenhagen, Denmark, 1999).

P. H. Mao, F. A. Harrison, Y. Y. Platonov, D. Broadway, B. DeGroot, F. E. Christensen, W. W. Craig, C. J. Hailey, “Development of grazing incidence multilayer mirrors for hard x-ray focusing telescopes,” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 526–534 (1997).
[CrossRef]

D. L. Windt, F. E. Christensen, W. W. Craig, C. J. Hailey, F. A. Harrison, M. A. Jimenez-Garate, R. Kalyanaraman, P. H. Mao, “X-ray multilayer coatings for use at energies above 100 keV,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 442–447 (2000).
[CrossRef]

W. W. Zhang, K. Chan, D. A. Content, R. Petre, P. J. Serlemitsos, T. T. Saha, Y. Soong, “Development of mirror segments for the Constellation-X Observatory,” in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy, J. E. Truemper, H. D. Tananbaum, eds., Proc. SPIE4851, (2002).

H. D. Tananbaum, N. E. White, J. A. Bookbinder, F. E. Marshall, F. A. Cordova, “Constellation X-ray mission: implementation concept and science overview,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 62–72 (1999).
[CrossRef]

F. A. Harrison, W. R. Cook, F. E. Christensen, O. Citterio, W. W. Craig, N. A. Gehrels, P. Gorenstein, J. E. Grindlay, C. J. Hailey, R. A. Kroeger, H. Kuneida, G. Pareschi, A. M. Parsons, R. Petre, S. E. Romaine, “Technology development for the Constellation-X hard x-ray telescope,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy X, O. H. Siegmund, K. A. Flanagan, eds., Proc. SPIE3765, 104–111 (1999).
[CrossRef]

M. C. Weisskopf, H. D. Tananbaum, L. P. Van Speybroeck, S. L. O’Dell, “Chandra X-ray Observatory (CXO): overview,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 2–16 (2000).
[CrossRef]

P. Gondoin, K. van Katwijk, B. R. Aschenbach, N. Schulz, R. Boerret, H. Glatzel, O. Citterio, “X-ray spectroscopy mission (XMM) telescope development,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 438–450 (1994).
[CrossRef]

F. A. Harrison, S. E. Boggs, A. E. Bolotnikov, F. E. Christensen, W. R. Cook, W. W. Craig, C. J. Hailey, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, D. L. Windt, “Development of the High-Energy Focusing Telescope (HEFT) balloon experiment,” in X-Ray Optics, Instruments, and Missions III, J. Truemper, B. Aschenbach, eds., Proc. SPIE4012, 693–699 (2000).
[CrossRef]

M. A. Jimenez-Garate, “The development of hard x-ray telescope optics and a theoretical model of x-ray emission from accretion disks,” Ph.D. dissertation (Columbia University, New York, 2001).

C. J. Hailey, S. Abdali, F. E. Christensen, W. W. Craig, T. R. Decker, F. A. Harrison, M. A. Jiménez-Gárate, “Investigation of substrates and mounting techniques for the High-Energy Focusing Telescope (HEFT),” in EUV, X-Ray and Gamma-Ray Instrumentation for Astronomy VIII, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3114, 535–543 (1997).
[CrossRef]

W. W. Craig, F. E. Christensen, T. A. Decker, C. J. Hailey, F. A. Harrison, R. M. Hill, M. A. Jimenez-Garate, P. H. Mao, S. M. Schindler, “Hard x-ray optics for the HEFT balloon-borne payload: prototype design and performance,” in EUV, X-ray, and Gamma-Ray Instrumentation for Astronomy IX, O. H. Siegmund, M. A. Gummin, eds., Proc. SPIE3445, 112–120 (1998).
[CrossRef]

M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, “Fast optical metrology of the hard x-ray optics for the HEFT telescope,” in X-Ray Optics, Instruments, and Missions, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3444, 622–633 (1998).
[CrossRef]

D. R. Uhlmann, N. J. Kreidl, eds., Processing I, Vol. 2 of Glass Science and Technology (Academic, New York, 1984).

I. Gutzow, J. Schmelzer, The Vitreous State: Thermodynamics, Structure, Rheology, and Crystallization (Springer, New York, 1995).

J. C. Lapp, P. L. Bocko, J. W. Nelson, “Advanced glass substrates for flat panel displays,” in Advanced Flat Panel Display Technologies, P. S. Friedman, ed., Proc. SPIE2174, 129–138 (1994).
[CrossRef]

R. H. Doremus, Glass Science (Wiley, New York, 1994).

H. Rawson, Glasses and their Applications (Institute of Metals, Brookfield, Vt., 1991).

I. S. Grigoriev, E. Z. Meilikhov, Handbook of Physical Quantities (CRC Press, Boca Raton, Fla., 1997).

H. Scholze, Glass: Nature, Structure, and Properties (Springer-Verlag, New York, 1991).

N. P. Bansal, R. H. Doremus, Handbook of Glass Properties (Academic, Orlando, Fla., 1986).

M. B. Volf, Mathematical Approach to Glass (Elsevier, New York, 1988).

Y. S. Touloukian, ed., Thermophysical Properties of Matter, Vols. 4, 5, and 13 of the Purdue University Thermophysical Properties Research Center Data Series (IFI/Plenum, New York, 1970).

J. H. Whitelaw, “Convective heat transfer,” in International Encyclopedia of Heat and Mass Transfer, G. F. Hewitt, G. L. Shires, Y. V. Polezhaev, eds. (CRC Press, Boca Raton, Fla., 1997), p. 237.

J. Symonds, J. P. Vidosic, H. V. Hawkins, D. D. Dodge, “Strength of materials,” in Standard Handbook for Mechanical Engineers, E. A. Avallone, T. B. Baumeister, eds. (McGraw-Hill, New York, 1986), pp. 5–24.

G. P. Monnelly, N. Butler, C. G. Chen, R. K. Heilmann, P. T. Konkola, M. McGuirk, O. Mongrard, G. R. Ricker, M. L. Schattenburg, “Novel methods for shaping thin-foil optics for x-ray astronomy,” in X-Ray Optics for Astronomy: Telescopes, Multilayers, Spectrometers, and Missions, P. Gorenstein, R. B. Hoover, eds., Proc. SPIE4496, 62–72 (2002).
[CrossRef]

A. M. Hussain, F. E. Christensen, M. A. Jimenez-Garate, W. W. Craig, C. J. Hailey, T. R. Decker, M. Stern, D. L. Windt, P. H. Mao, F. A. Harrison, G. Pareschi, M. S. del Rio, A. Souvorov, A. K. Freund, R. Tucoulou, A. Madsen, C. Mammen, “X-ray scatter measurements from thermally slumped thin glass substrates for the HEFT hard x-ray telescopes,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 184–196 (1999).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Cross section of an x-ray telescope with a conical approximation to the Wolter-I geometry. X-rays from a celestial source reflect twice and come into focus.

Fig. 2
Fig. 2

Wire diagram of the setup and cross-sectional schematic of four time steps in the TF process. A semicylindrical quartz mandrel serves as a guide for the glass microsheet. The resulting TFG shell is cut axially to reduce edge distortion.

Fig. 3
Fig. 3

Thermal processing curve for glass forming. Glass forming occurs when TT soak. The other parts of the thermal processing curve are meant to either reduce thermal gradients (thermalization) or to anneal the glass after forming (cool down).

Fig. 4
Fig. 4

Axial laser scan of a free-standing TFG substrate of 200-mm length. This piece would make a telescope with a HPD of 46 arc sec. Mounted performance can be better. Some of the edge region is visible at 60 mm.4,12

Fig. 5
Fig. 5

Cross sections of some mandrel geometries for the TF of Wolter-I or conical mirror segments. The dark lines delineate the mandrel, whereas the lighter lines delineate several forming stages of a sheet. (a) Convex mandrel as master; (b) the concave mandrel we use for the High-Energy Focusing Telescope, with the glass on the cylinder edges; (c) like (b) but with a smaller glass sheet inside the cylinder; (d) concave mandrel with ramps; (e) and (f) like (d) but with smaller mandrels; (g) two perpendicular surfaces; (h) two curved surfaces.

Fig. 6
Fig. 6

Viscosity versus temperature for various glass microsheets. The data points are from Table 5, and the curves are fits with the VFT Eq. (3). In the TF region, glass is formed in an ∼5–60-min soak.

Fig. 7
Fig. 7

Thermal equilibration time for various heat transfer mechanisms.

Fig. 8
Fig. 8

Successful TF parameters in separate trials for a fixed glass sheet size (0.3 mm × 200 mm × 168 mm) and a mandrel radius of 84 mm (Corning 1737 glass). The slope of the line is fit by the viscodynamic model of Eq. (16).

Fig. 9
Fig. 9

Viscosity to soak time ratio versus TFG arc length. The data are shown in symbols and the curve is the viscodynamic model with one free normalization parameter. Errors were estimated as in Fig. 8.

Fig. 10
Fig. 10

Predicted self-relaxation time of surface ripples versus ripple wavelength.

Fig. 11
Fig. 11

Reflectance at 8 keV of a typical glass microsheet before (Raw) and after TF (TFG). The glass is uncoated. For both data sets, the best fit is σ = 3.3 Å.

Tables (9)

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Table 1 Mass and Performance of X-Ray Telescope Mirrorsa

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Table 2 Mirror Figure Requirements for Future Missionsa

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Table 3 Dominant Figure and Assembly HPD Termsa

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Table 4 Mirror Substrate Parameters for X-Ray Telescopesa

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Table 5 Viscosity Data and Fit Parameters from Eq. (3)a

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Table 6 Thermophysical Properties of TFG Substrate and Mandrel Materials

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Table 7 Convective Heat Transfer Coefficient Θ for Typical Conditionsa

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Table 8 Linear Coefficients of Thermal Expansiona

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Table 9 Measurements of Glass Surface before and after Forminga

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

FyxA=η dvydx, or Π=η dγdt,
γt= Πηdt.
ηT=η0 expT0T-T,
P=Sr,
S=S0-δT-T0,
τDρcpd2κd2D,
dQdAdt=ΘΔT,
τΘ=ρcpdΘ.
dQdAdtσT04-Tg4,
dQdAdt4σT03ΔT.
τρcpd4σT03.
smax=5Wl3384YI,
I=12/Lw3,
Geffηsoaktsoak,
Yeff=2Geff1+ν,
ηsoaktsoak=5Cρgl464smax1+νw2  ρgl3w2,
FyxA=η dvydx,
WV=FyxdyAdx=η dy2dtdx2.
WV=Cρgh=η w2tr2,
ηsoaktsoak=Cρghr2w2  ρgl3w2
UAρgh24+8Sh2λ2,
WA=ηh2τλ,
τηλρg/4+8S/λ,
42Δnz
42ΔrropL
42ΔrrapL
42Δrabsf
22Δrripf
Lrmax+rmin16f2

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