Abstract

Whispering galleries propagating along large-radius concave meniscuses at the surfaces of rotating deionized water as well as hydrosol of ∼ 10 nm amorphous silica particles enriched by CsOH were probed by both x-ray reflectometry and x-ray fluorescence for the first time. The measurements were carried out at a wavelength of 1.5405 Å of Cu-Kα radiation by using a homemade diffractometer with a moving tube-sample-detector system. According to the experimental results, the x-ray beam deflection angle at the sol’s surface reaches 4°, which is roughly four times higher than that obtained on the water surface. The rigorous solution of the Helmholtz equation for the whispering gallery reflection mode at the concave liquid surface agrees well with experimental observations. We attribute the difference in the x-ray beam deflection angle for the studied liquids to the difference in their viscosity, which presumably is inversely proportional to the effective surface roughness.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Simple Device for Leveling Collimated Light Beams

Melvin D. Daybell
Appl. Opt. 4(7) 877_1-878 (1965)

The polarization patterns of skylight reflected off wave water surface

Guanhua Zhou, Wujian Xu, Chunyue Niu, and Huijie Zhao
Opt. Express 21(26) 32549-32565 (2013)

Simple Device for Leveling Collimated Light Beams

Roger H. Keith
Appl. Opt. 4(12) 1677-1677 (1965)

References

  • View by:
  • |
  • |
  • |

  1. J. W. Strutt(Lord Rayleigh), “The problem of the whispering gallery,” Philosophical magazine 20(120), 1001–1004 (1910).
  2. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
    [Crossref]
  3. M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
    [Crossref]
  4. N. V. Kryzhanovskaya, M. V. Maximov, and A. E. Zhukov, “Whispering-gallery mode microcavity quantum-dot lasers,” Quantum Electron. 44(3), 189–200 (2014).
    [Crossref]
  5. I. N. Bukreeva, I. V. Kozhevnikov, and A. V. Vinogradov, “Whispering gallery mirrors for the soft x-ray region: Properties and applications,” J. X-Ray Sci. Technol. 5(4), 397–419 (1995).
    [Crossref]
  6. E. Spiller, Soft x-ray Optics (SPIE, Bellingham, Washington, 1994).
  7. C. Liu and J. A. Golovchenko, “Surface Trapped X Rays: Whispering-Gallery Modes at $\lambda = 0.7$λ=0.7 Å,” Phys. Rev. Lett. 79(5), 788–791 (1997).
    [Crossref]
  8. I. V. Yakimchuk, B. S. Roshchin, I. V. Kozhevnikov, V. E. Asadchikov, and Zh. Wang, “Study of the whispering gallery effect on a spherical surface in the hard x-ray region,” Crystallogr. Rep. 53(6), 1054–1060 (2008).
    [Crossref]
  9. V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
    [Crossref]
  10. A. M. Tikhonov, “Ion-size effect at the surface of a silica hydrosol,” J. Chem. Phys. 130(2), 024512 (2009).
    [Crossref]
  11. L. I. Goray, N. I. Chkhalo, and G. E. Tsyrlin, “Determining angles of incidence and heights of quantum dot faces by analyzing x-ray diffuse and specular scattering,” Tech. Phys. 54(4), 561–568 (2009).
    [Crossref]
  12. L. I. Goray and G. Schmidt, “Boundary Integral Equation Methods for Conical Diffraction and Short Waves,” in Gratings: Theory and Numerical Applications, E. Popov, ed., 2nd rev. ed. (Presses Universitaires de Provence, AMU, 2014). Chap. 12.
  13. B. M. Ocko, in Spectroscopic and Diffraction Techniques in Interfacial Electrochemistry, C. Gutiérrez and C. Melendres, eds. (NATO ASI Series, 1990), p. 343.
  14. M. Fukuto, R. K. Heilmann, P. S. Pershan, J. A. Griffiths, S. M. Yu, and D. A. Tirrell, “X-Ray Measurements of Noncapillary Spatial Fluctuations from a Liquid Surface,” Phys. Rev. Lett. 81(16), 3455–3458 (1998).
    [Crossref]
  15. J. F. Seely, L. I. Goray, W. R. Hunter, and J. C. Rife, “Thin-film interference effects on the efficiency of a normal-incidence grating in the 100–350-Å wavelength region,” Appl. Opt. 38(7), 1251–1258 (1999).
    [Crossref]
  16. A. E. Muslimov, private communication.
  17. P. S. Pershan, “Effects of thermal roughness on x-ray studies of liquid surfaces,” Colloids Surf., A 171(1-3), 149–157 (2000).
    [Crossref]
  18. V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Theory (Alpha Science, Oxford, 2007).
  19. A. M. Tikhonov, “Compact Layer of Alkali Ions at the Surface of Colloidal Silica,” J. Phys. Chem. C 111(2), 930–937 (2007).
    [Crossref]
  20. A. Braslau, M. Deutsch, P. S. Pershan, A. H. Weiss, J. Als-Nielsen, and J. Bohr, “Surface Roughness of Water Measured by X-Ray Reflectivity,” Phys. Rev. Lett. 54(2), 114–117 (1985).
    [Crossref]
  21. V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
    [Crossref]
  22. A. M. Tikhonov, “Water Density in the Electric Double Layer at the Insulator/Electrolyte Solution Interface,” J. Phys. Chem. B 110(6), 2746–2750 (2006).
    [Crossref]
  23. The Center of x-ray Optics, http://henke.lbl.gov/ (2018).
  24. F. P. Buff, R. Lovett, and F. H. Stillinger, “Interfacial Density Profile for Fluids in the Critical Region,” Phys. Rev. Lett. 15(15), 621–623 (1965).
    [Crossref]
  25. A. Braslau, P. S. Pershan, G. Swislow, B. M. Ocko, and J. Als-Nielsen, “Capillary waves on the surface of simple liquids measured by x-ray reflectivity,” Phys. Rev. A 38(5), 2457–2470 (1988).
    [Crossref]
  26. D. K. Schwartz, M. L. Schlossman, E. H. Kawamoto, G. J. Kellogg, P. S. Pershan, and B. M. Ocko, “Thermal diffuse x-ray-scattering studies of the water-vapor interface,” Phys. Rev. A 41(10), 5687–5690 (1990).
    [Crossref]
  27. A. M. Tikhonov, V. E. Asadchikov, Yu. O. Volkov, B. S. Roshchin, V. Honkimäki, and M. V. Blanco, “Model-Independent X-Ray Scattering Study of a Silica Sol Surface,” JETP Lett. 107(6), 384–389 (2018).
    [Crossref]
  28. L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering on rough surfaces,” J. Appl. Phys. 108(3), 033516 (2010).
    [Crossref]
  29. L. I. Goray, A. Yu, and Egorov, “Breaking the efficiency limit for high-frequency blazed multilayer soft x-ray gratings: Conical vs classical diffraction,” Appl. Phys. Lett. 109(10), 103502 (2016).
    [Crossref]
  30. The PCGrate software, http://www.pcgrate.com/ (2018).
  31. The Grace products, https://grace.com/catalysts-and-fuels/en-us/Pages/ludox-emission-control.aspx (2018).
  32. E. D. Giuseppe, A. Davaille, E. Mittelstaedt, and M. Francois, “Rheological and mechanical properties of silica colloids: from newtonian liquid to brittle behaviour,” Rheol. Acta 51(5), 451–465 (2012).
    [Crossref]

2018 (1)

A. M. Tikhonov, V. E. Asadchikov, Yu. O. Volkov, B. S. Roshchin, V. Honkimäki, and M. V. Blanco, “Model-Independent X-Ray Scattering Study of a Silica Sol Surface,” JETP Lett. 107(6), 384–389 (2018).
[Crossref]

2016 (1)

L. I. Goray, A. Yu, and Egorov, “Breaking the efficiency limit for high-frequency blazed multilayer soft x-ray gratings: Conical vs classical diffraction,” Appl. Phys. Lett. 109(10), 103502 (2016).
[Crossref]

2015 (1)

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref]

2014 (1)

N. V. Kryzhanovskaya, M. V. Maximov, and A. E. Zhukov, “Whispering-gallery mode microcavity quantum-dot lasers,” Quantum Electron. 44(3), 189–200 (2014).
[Crossref]

2012 (1)

E. D. Giuseppe, A. Davaille, E. Mittelstaedt, and M. Francois, “Rheological and mechanical properties of silica colloids: from newtonian liquid to brittle behaviour,” Rheol. Acta 51(5), 451–465 (2012).
[Crossref]

2011 (1)

V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
[Crossref]

2010 (1)

L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering on rough surfaces,” J. Appl. Phys. 108(3), 033516 (2010).
[Crossref]

2009 (2)

A. M. Tikhonov, “Ion-size effect at the surface of a silica hydrosol,” J. Chem. Phys. 130(2), 024512 (2009).
[Crossref]

L. I. Goray, N. I. Chkhalo, and G. E. Tsyrlin, “Determining angles of incidence and heights of quantum dot faces by analyzing x-ray diffuse and specular scattering,” Tech. Phys. 54(4), 561–568 (2009).
[Crossref]

2008 (1)

I. V. Yakimchuk, B. S. Roshchin, I. V. Kozhevnikov, V. E. Asadchikov, and Zh. Wang, “Study of the whispering gallery effect on a spherical surface in the hard x-ray region,” Crystallogr. Rep. 53(6), 1054–1060 (2008).
[Crossref]

2007 (1)

A. M. Tikhonov, “Compact Layer of Alkali Ions at the Surface of Colloidal Silica,” J. Phys. Chem. C 111(2), 930–937 (2007).
[Crossref]

2006 (1)

A. M. Tikhonov, “Water Density in the Electric Double Layer at the Insulator/Electrolyte Solution Interface,” J. Phys. Chem. B 110(6), 2746–2750 (2006).
[Crossref]

2005 (1)

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref]

2000 (1)

P. S. Pershan, “Effects of thermal roughness on x-ray studies of liquid surfaces,” Colloids Surf., A 171(1-3), 149–157 (2000).
[Crossref]

1999 (1)

1998 (1)

M. Fukuto, R. K. Heilmann, P. S. Pershan, J. A. Griffiths, S. M. Yu, and D. A. Tirrell, “X-Ray Measurements of Noncapillary Spatial Fluctuations from a Liquid Surface,” Phys. Rev. Lett. 81(16), 3455–3458 (1998).
[Crossref]

1997 (1)

C. Liu and J. A. Golovchenko, “Surface Trapped X Rays: Whispering-Gallery Modes at $\lambda = 0.7$λ=0.7 Å,” Phys. Rev. Lett. 79(5), 788–791 (1997).
[Crossref]

1995 (1)

I. N. Bukreeva, I. V. Kozhevnikov, and A. V. Vinogradov, “Whispering gallery mirrors for the soft x-ray region: Properties and applications,” J. X-Ray Sci. Technol. 5(4), 397–419 (1995).
[Crossref]

1990 (1)

D. K. Schwartz, M. L. Schlossman, E. H. Kawamoto, G. J. Kellogg, P. S. Pershan, and B. M. Ocko, “Thermal diffuse x-ray-scattering studies of the water-vapor interface,” Phys. Rev. A 41(10), 5687–5690 (1990).
[Crossref]

1988 (1)

A. Braslau, P. S. Pershan, G. Swislow, B. M. Ocko, and J. Als-Nielsen, “Capillary waves on the surface of simple liquids measured by x-ray reflectivity,” Phys. Rev. A 38(5), 2457–2470 (1988).
[Crossref]

1985 (1)

A. Braslau, M. Deutsch, P. S. Pershan, A. H. Weiss, J. Als-Nielsen, and J. Bohr, “Surface Roughness of Water Measured by X-Ray Reflectivity,” Phys. Rev. Lett. 54(2), 114–117 (1985).
[Crossref]

1965 (1)

F. P. Buff, R. Lovett, and F. H. Stillinger, “Interfacial Density Profile for Fluids in the Critical Region,” Phys. Rev. Lett. 15(15), 621–623 (1965).
[Crossref]

1910 (1)

J. W. Strutt(Lord Rayleigh), “The problem of the whispering gallery,” Philosophical magazine 20(120), 1001–1004 (1910).

Als-Nielsen, J.

A. Braslau, P. S. Pershan, G. Swislow, B. M. Ocko, and J. Als-Nielsen, “Capillary waves on the surface of simple liquids measured by x-ray reflectivity,” Phys. Rev. A 38(5), 2457–2470 (1988).
[Crossref]

A. Braslau, M. Deutsch, P. S. Pershan, A. H. Weiss, J. Als-Nielsen, and J. Bohr, “Surface Roughness of Water Measured by X-Ray Reflectivity,” Phys. Rev. Lett. 54(2), 114–117 (1985).
[Crossref]

Asadchikov, V. E.

A. M. Tikhonov, V. E. Asadchikov, Yu. O. Volkov, B. S. Roshchin, V. Honkimäki, and M. V. Blanco, “Model-Independent X-Ray Scattering Study of a Silica Sol Surface,” JETP Lett. 107(6), 384–389 (2018).
[Crossref]

V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
[Crossref]

I. V. Yakimchuk, B. S. Roshchin, I. V. Kozhevnikov, V. E. Asadchikov, and Zh. Wang, “Study of the whispering gallery effect on a spherical surface in the hard x-ray region,” Crystallogr. Rep. 53(6), 1054–1060 (2008).
[Crossref]

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Babak, V. G.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Babich, V. M.

V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Theory (Alpha Science, Oxford, 2007).

Blanco, M. V.

A. M. Tikhonov, V. E. Asadchikov, Yu. O. Volkov, B. S. Roshchin, V. Honkimäki, and M. V. Blanco, “Model-Independent X-Ray Scattering Study of a Silica Sol Surface,” JETP Lett. 107(6), 384–389 (2018).
[Crossref]

Bohr, J.

A. Braslau, M. Deutsch, P. S. Pershan, A. H. Weiss, J. Als-Nielsen, and J. Bohr, “Surface Roughness of Water Measured by X-Ray Reflectivity,” Phys. Rev. Lett. 54(2), 114–117 (1985).
[Crossref]

Braslau, A.

A. Braslau, P. S. Pershan, G. Swislow, B. M. Ocko, and J. Als-Nielsen, “Capillary waves on the surface of simple liquids measured by x-ray reflectivity,” Phys. Rev. A 38(5), 2457–2470 (1988).
[Crossref]

A. Braslau, M. Deutsch, P. S. Pershan, A. H. Weiss, J. Als-Nielsen, and J. Bohr, “Surface Roughness of Water Measured by X-Ray Reflectivity,” Phys. Rev. Lett. 54(2), 114–117 (1985).
[Crossref]

Buff, F. P.

F. P. Buff, R. Lovett, and F. H. Stillinger, “Interfacial Density Profile for Fluids in the Critical Region,” Phys. Rev. Lett. 15(15), 621–623 (1965).
[Crossref]

Bukreeva, I. N.

I. N. Bukreeva, I. V. Kozhevnikov, and A. V. Vinogradov, “Whispering gallery mirrors for the soft x-ray region: Properties and applications,” J. X-Ray Sci. Technol. 5(4), 397–419 (1995).
[Crossref]

Buldyrev, V. S.

V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Theory (Alpha Science, Oxford, 2007).

Buzmakov, A. V.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Cheremukhin, E. A.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Cheremukhina, G. A.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Chernenko, S. P.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Chkhalo, N. I.

L. I. Goray, N. I. Chkhalo, and G. E. Tsyrlin, “Determining angles of incidence and heights of quantum dot faces by analyzing x-ray diffuse and specular scattering,” Tech. Phys. 54(4), 561–568 (2009).
[Crossref]

Chulichkov, A. I.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Davaille, A.

E. D. Giuseppe, A. Davaille, E. Mittelstaedt, and M. Francois, “Rheological and mechanical properties of silica colloids: from newtonian liquid to brittle behaviour,” Rheol. Acta 51(5), 451–465 (2012).
[Crossref]

Dembo, A.

V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
[Crossref]

Deutsch, M.

A. Braslau, M. Deutsch, P. S. Pershan, A. H. Weiss, J. Als-Nielsen, and J. Bohr, “Surface Roughness of Water Measured by X-Ray Reflectivity,” Phys. Rev. Lett. 54(2), 114–117 (1985).
[Crossref]

Dorokhin, Yu. P.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Egorov,

L. I. Goray, A. Yu, and Egorov, “Breaking the efficiency limit for high-frequency blazed multilayer soft x-ray gratings: Conical vs classical diffraction,” Appl. Phys. Lett. 109(10), 103502 (2016).
[Crossref]

Fateev, V. D.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Foreman, M. R.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref]

Francois, M.

E. D. Giuseppe, A. Davaille, E. Mittelstaedt, and M. Francois, “Rheological and mechanical properties of silica colloids: from newtonian liquid to brittle behaviour,” Rheol. Acta 51(5), 451–465 (2012).
[Crossref]

Frolov, D. A.

V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
[Crossref]

Fukuto, M.

M. Fukuto, R. K. Heilmann, P. S. Pershan, J. A. Griffiths, S. M. Yu, and D. A. Tirrell, “X-Ray Measurements of Noncapillary Spatial Fluctuations from a Liquid Surface,” Phys. Rev. Lett. 81(16), 3455–3458 (1998).
[Crossref]

Giuseppe, E. D.

E. D. Giuseppe, A. Davaille, E. Mittelstaedt, and M. Francois, “Rheological and mechanical properties of silica colloids: from newtonian liquid to brittle behaviour,” Rheol. Acta 51(5), 451–465 (2012).
[Crossref]

Glagolev, I. P.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Golovchenko, J. A.

C. Liu and J. A. Golovchenko, “Surface Trapped X Rays: Whispering-Gallery Modes at $\lambda = 0.7$λ=0.7 Å,” Phys. Rev. Lett. 79(5), 788–791 (1997).
[Crossref]

Goray, L. I.

L. I. Goray, A. Yu, and Egorov, “Breaking the efficiency limit for high-frequency blazed multilayer soft x-ray gratings: Conical vs classical diffraction,” Appl. Phys. Lett. 109(10), 103502 (2016).
[Crossref]

L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering on rough surfaces,” J. Appl. Phys. 108(3), 033516 (2010).
[Crossref]

L. I. Goray, N. I. Chkhalo, and G. E. Tsyrlin, “Determining angles of incidence and heights of quantum dot faces by analyzing x-ray diffuse and specular scattering,” Tech. Phys. 54(4), 561–568 (2009).
[Crossref]

J. F. Seely, L. I. Goray, W. R. Hunter, and J. C. Rife, “Thin-film interference effects on the efficiency of a normal-incidence grating in the 100–350-Å wavelength region,” Appl. Opt. 38(7), 1251–1258 (1999).
[Crossref]

L. I. Goray and G. Schmidt, “Boundary Integral Equation Methods for Conical Diffraction and Short Waves,” in Gratings: Theory and Numerical Applications, E. Popov, ed., 2nd rev. ed. (Presses Universitaires de Provence, AMU, 2014). Chap. 12.

Griffiths, J. A.

M. Fukuto, R. K. Heilmann, P. S. Pershan, J. A. Griffiths, S. M. Yu, and D. A. Tirrell, “X-Ray Measurements of Noncapillary Spatial Fluctuations from a Liquid Surface,” Phys. Rev. Lett. 81(16), 3455–3458 (1998).
[Crossref]

Heilmann, R. K.

M. Fukuto, R. K. Heilmann, P. S. Pershan, J. A. Griffiths, S. M. Yu, and D. A. Tirrell, “X-Ray Measurements of Noncapillary Spatial Fluctuations from a Liquid Surface,” Phys. Rev. Lett. 81(16), 3455–3458 (1998).
[Crossref]

Honkimäki, V.

A. M. Tikhonov, V. E. Asadchikov, Yu. O. Volkov, B. S. Roshchin, V. Honkimäki, and M. V. Blanco, “Model-Independent X-Ray Scattering Study of a Silica Sol Surface,” JETP Lett. 107(6), 384–389 (2018).
[Crossref]

Hunter, W. R.

Kawamoto, E. H.

D. K. Schwartz, M. L. Schlossman, E. H. Kawamoto, G. J. Kellogg, P. S. Pershan, and B. M. Ocko, “Thermal diffuse x-ray-scattering studies of the water-vapor interface,” Phys. Rev. A 41(10), 5687–5690 (1990).
[Crossref]

Kellogg, G. J.

D. K. Schwartz, M. L. Schlossman, E. H. Kawamoto, G. J. Kellogg, P. S. Pershan, and B. M. Ocko, “Thermal diffuse x-ray-scattering studies of the water-vapor interface,” Phys. Rev. A 41(10), 5687–5690 (1990).
[Crossref]

Kozhevnikov, I. V.

V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
[Crossref]

I. V. Yakimchuk, B. S. Roshchin, I. V. Kozhevnikov, V. E. Asadchikov, and Zh. Wang, “Study of the whispering gallery effect on a spherical surface in the hard x-ray region,” Crystallogr. Rep. 53(6), 1054–1060 (2008).
[Crossref]

I. N. Bukreeva, I. V. Kozhevnikov, and A. V. Vinogradov, “Whispering gallery mirrors for the soft x-ray region: Properties and applications,” J. X-Ray Sci. Technol. 5(4), 397–419 (1995).
[Crossref]

Krivonosov, Yu. S.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Kryzhanovskaya, N. V.

N. V. Kryzhanovskaya, M. V. Maximov, and A. E. Zhukov, “Whispering-gallery mode microcavity quantum-dot lasers,” Quantum Electron. 44(3), 189–200 (2014).
[Crossref]

Liu, C.

C. Liu and J. A. Golovchenko, “Surface Trapped X Rays: Whispering-Gallery Modes at $\lambda = 0.7$λ=0.7 Å,” Phys. Rev. Lett. 79(5), 788–791 (1997).
[Crossref]

Lovett, R.

F. P. Buff, R. Lovett, and F. H. Stillinger, “Interfacial Density Profile for Fluids in the Critical Region,” Phys. Rev. Lett. 15(15), 621–623 (1965).
[Crossref]

Mamich, V. F.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Maximov, M. V.

N. V. Kryzhanovskaya, M. V. Maximov, and A. E. Zhukov, “Whispering-gallery mode microcavity quantum-dot lasers,” Quantum Electron. 44(3), 189–200 (2014).
[Crossref]

McHedlishvili, B. V.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Mittelstaedt, E.

E. D. Giuseppe, A. Davaille, E. Mittelstaedt, and M. Francois, “Rheological and mechanical properties of silica colloids: from newtonian liquid to brittle behaviour,” Rheol. Acta 51(5), 451–465 (2012).
[Crossref]

Moseiko, L. A.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Moseiko, N. I.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Muslimov, A. E.

A. E. Muslimov, private communication.

Ocko, B. M.

D. K. Schwartz, M. L. Schlossman, E. H. Kawamoto, G. J. Kellogg, P. S. Pershan, and B. M. Ocko, “Thermal diffuse x-ray-scattering studies of the water-vapor interface,” Phys. Rev. A 41(10), 5687–5690 (1990).
[Crossref]

A. Braslau, P. S. Pershan, G. Swislow, B. M. Ocko, and J. Als-Nielsen, “Capillary waves on the surface of simple liquids measured by x-ray reflectivity,” Phys. Rev. A 38(5), 2457–2470 (1988).
[Crossref]

B. M. Ocko, in Spectroscopic and Diffraction Techniques in Interfacial Electrochemistry, C. Gutiérrez and C. Melendres, eds. (NATO ASI Series, 1990), p. 343.

Pershan, P. S.

P. S. Pershan, “Effects of thermal roughness on x-ray studies of liquid surfaces,” Colloids Surf., A 171(1-3), 149–157 (2000).
[Crossref]

M. Fukuto, R. K. Heilmann, P. S. Pershan, J. A. Griffiths, S. M. Yu, and D. A. Tirrell, “X-Ray Measurements of Noncapillary Spatial Fluctuations from a Liquid Surface,” Phys. Rev. Lett. 81(16), 3455–3458 (1998).
[Crossref]

D. K. Schwartz, M. L. Schlossman, E. H. Kawamoto, G. J. Kellogg, P. S. Pershan, and B. M. Ocko, “Thermal diffuse x-ray-scattering studies of the water-vapor interface,” Phys. Rev. A 41(10), 5687–5690 (1990).
[Crossref]

A. Braslau, P. S. Pershan, G. Swislow, B. M. Ocko, and J. Als-Nielsen, “Capillary waves on the surface of simple liquids measured by x-ray reflectivity,” Phys. Rev. A 38(5), 2457–2470 (1988).
[Crossref]

A. Braslau, M. Deutsch, P. S. Pershan, A. H. Weiss, J. Als-Nielsen, and J. Bohr, “Surface Roughness of Water Measured by X-Ray Reflectivity,” Phys. Rev. Lett. 54(2), 114–117 (1985).
[Crossref]

Rife, J. C.

Roshchin, B. S.

A. M. Tikhonov, V. E. Asadchikov, Yu. O. Volkov, B. S. Roshchin, V. Honkimäki, and M. V. Blanco, “Model-Independent X-Ray Scattering Study of a Silica Sol Surface,” JETP Lett. 107(6), 384–389 (2018).
[Crossref]

V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
[Crossref]

I. V. Yakimchuk, B. S. Roshchin, I. V. Kozhevnikov, V. E. Asadchikov, and Zh. Wang, “Study of the whispering gallery effect on a spherical surface in the hard x-ray region,” Crystallogr. Rep. 53(6), 1054–1060 (2008).
[Crossref]

Savel’ev, S. V.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Schlossman, M. L.

D. K. Schwartz, M. L. Schlossman, E. H. Kawamoto, G. J. Kellogg, P. S. Pershan, and B. M. Ocko, “Thermal diffuse x-ray-scattering studies of the water-vapor interface,” Phys. Rev. A 41(10), 5687–5690 (1990).
[Crossref]

Schmidt, G.

L. I. Goray and G. Schmidt, “Boundary Integral Equation Methods for Conical Diffraction and Short Waves,” in Gratings: Theory and Numerical Applications, E. Popov, ed., 2nd rev. ed. (Presses Universitaires de Provence, AMU, 2014). Chap. 12.

Schwartz, D. K.

D. K. Schwartz, M. L. Schlossman, E. H. Kawamoto, G. J. Kellogg, P. S. Pershan, and B. M. Ocko, “Thermal diffuse x-ray-scattering studies of the water-vapor interface,” Phys. Rev. A 41(10), 5687–5690 (1990).
[Crossref]

Seely, J. F.

Senin, R. A.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Shilin, Yu. N.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Shishkov, V. A.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Smykov, L. P.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Spiller, E.

E. Spiller, Soft x-ray Optics (SPIE, Bellingham, Washington, 1994).

Stillinger, F. H.

F. P. Buff, R. Lovett, and F. H. Stillinger, “Interfacial Density Profile for Fluids in the Critical Region,” Phys. Rev. Lett. 15(15), 621–623 (1965).
[Crossref]

Strutt, J. W.

J. W. Strutt(Lord Rayleigh), “The problem of the whispering gallery,” Philosophical magazine 20(120), 1001–1004 (1910).

Swaim, J. D.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref]

Swislow, G.

A. Braslau, P. S. Pershan, G. Swislow, B. M. Ocko, and J. Als-Nielsen, “Capillary waves on the surface of simple liquids measured by x-ray reflectivity,” Phys. Rev. A 38(5), 2457–2470 (1988).
[Crossref]

Tikhonov, A. M.

A. M. Tikhonov, V. E. Asadchikov, Yu. O. Volkov, B. S. Roshchin, V. Honkimäki, and M. V. Blanco, “Model-Independent X-Ray Scattering Study of a Silica Sol Surface,” JETP Lett. 107(6), 384–389 (2018).
[Crossref]

V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
[Crossref]

A. M. Tikhonov, “Ion-size effect at the surface of a silica hydrosol,” J. Chem. Phys. 130(2), 024512 (2009).
[Crossref]

A. M. Tikhonov, “Compact Layer of Alkali Ions at the Surface of Colloidal Silica,” J. Phys. Chem. C 111(2), 930–937 (2007).
[Crossref]

A. M. Tikhonov, “Water Density in the Electric Double Layer at the Insulator/Electrolyte Solution Interface,” J. Phys. Chem. B 110(6), 2746–2750 (2006).
[Crossref]

Tirrell, D. A.

M. Fukuto, R. K. Heilmann, P. S. Pershan, J. A. Griffiths, S. M. Yu, and D. A. Tirrell, “X-Ray Measurements of Noncapillary Spatial Fluctuations from a Liquid Surface,” Phys. Rev. Lett. 81(16), 3455–3458 (1998).
[Crossref]

Tsyrlin, G. E.

L. I. Goray, N. I. Chkhalo, and G. E. Tsyrlin, “Determining angles of incidence and heights of quantum dot faces by analyzing x-ray diffuse and specular scattering,” Tech. Phys. 54(4), 561–568 (2009).
[Crossref]

Tudosi, G. A.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref]

Vinogradov, A. V.

I. N. Bukreeva, I. V. Kozhevnikov, and A. V. Vinogradov, “Whispering gallery mirrors for the soft x-ray region: Properties and applications,” J. X-Ray Sci. Technol. 5(4), 397–419 (1995).
[Crossref]

Volkov, V. V.

V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
[Crossref]

Volkov, Yu. O.

A. M. Tikhonov, V. E. Asadchikov, Yu. O. Volkov, B. S. Roshchin, V. Honkimäki, and M. V. Blanco, “Model-Independent X-Ray Scattering Study of a Silica Sol Surface,” JETP Lett. 107(6), 384–389 (2018).
[Crossref]

V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
[Crossref]

Vollmer, F.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref]

Wang, Zh.

I. V. Yakimchuk, B. S. Roshchin, I. V. Kozhevnikov, V. E. Asadchikov, and Zh. Wang, “Study of the whispering gallery effect on a spherical surface in the hard x-ray region,” Crystallogr. Rep. 53(6), 1054–1060 (2008).
[Crossref]

Weiss, A. H.

A. Braslau, M. Deutsch, P. S. Pershan, A. H. Weiss, J. Als-Nielsen, and J. Bohr, “Surface Roughness of Water Measured by X-Ray Reflectivity,” Phys. Rev. Lett. 54(2), 114–117 (1985).
[Crossref]

Yakimchuk, I. V.

I. V. Yakimchuk, B. S. Roshchin, I. V. Kozhevnikov, V. E. Asadchikov, and Zh. Wang, “Study of the whispering gallery effect on a spherical surface in the hard x-ray region,” Crystallogr. Rep. 53(6), 1054–1060 (2008).
[Crossref]

Yu, A.

L. I. Goray, A. Yu, and Egorov, “Breaking the efficiency limit for high-frequency blazed multilayer soft x-ray gratings: Conical vs classical diffraction,” Appl. Phys. Lett. 109(10), 103502 (2016).
[Crossref]

Yu, S. M.

M. Fukuto, R. K. Heilmann, P. S. Pershan, J. A. Griffiths, S. M. Yu, and D. A. Tirrell, “X-Ray Measurements of Noncapillary Spatial Fluctuations from a Liquid Surface,” Phys. Rev. Lett. 81(16), 3455–3458 (1998).
[Crossref]

Zanevskii, Yu. V.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Zhukov, A. E.

N. V. Kryzhanovskaya, M. V. Maximov, and A. E. Zhukov, “Whispering-gallery mode microcavity quantum-dot lasers,” Quantum Electron. 44(3), 189–200 (2014).
[Crossref]

Zryuev, V. N.

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

Adv. Opt. Photonics (1)

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

L. I. Goray, A. Yu, and Egorov, “Breaking the efficiency limit for high-frequency blazed multilayer soft x-ray gratings: Conical vs classical diffraction,” Appl. Phys. Lett. 109(10), 103502 (2016).
[Crossref]

Colloids Surf., A (1)

P. S. Pershan, “Effects of thermal roughness on x-ray studies of liquid surfaces,” Colloids Surf., A 171(1-3), 149–157 (2000).
[Crossref]

Crystallogr. Rep. (1)

I. V. Yakimchuk, B. S. Roshchin, I. V. Kozhevnikov, V. E. Asadchikov, and Zh. Wang, “Study of the whispering gallery effect on a spherical surface in the hard x-ray region,” Crystallogr. Rep. 53(6), 1054–1060 (2008).
[Crossref]

Instrum. Exp. Tech. (1)

V. E. Asadchikov, A. V. Buzmakov, V. N. Zryuev, Yu. S. Krivonosov, B. V. McHedlishvili, R. A. Senin, Yu. N. Shilin, V. A. Shishkov, V. G. Babak, Yu. P. Dorokhin, I. P. Glagolev, V. F. Mamich, V. D. Fateev, Yu. V. Zanevskii, L. P. Smykov, S. P. Chernenko, G. A. Cheremukhina, L. A. Moseiko, N. I. Moseiko, S. V. Savel’ev, G. A. Tudosi, E. A. Cheremukhin, and A. I. Chulichkov, “An x-ray diffractometer with a mobile emitter-detector system,” Instrum. Exp. Tech. 48(3), 364–372 (2005).
[Crossref]

J. Appl. Phys. (1)

L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering on rough surfaces,” J. Appl. Phys. 108(3), 033516 (2010).
[Crossref]

J. Chem. Phys. (1)

A. M. Tikhonov, “Ion-size effect at the surface of a silica hydrosol,” J. Chem. Phys. 130(2), 024512 (2009).
[Crossref]

J. Phys. Chem. B (1)

A. M. Tikhonov, “Water Density in the Electric Double Layer at the Insulator/Electrolyte Solution Interface,” J. Phys. Chem. B 110(6), 2746–2750 (2006).
[Crossref]

J. Phys. Chem. C (1)

A. M. Tikhonov, “Compact Layer of Alkali Ions at the Surface of Colloidal Silica,” J. Phys. Chem. C 111(2), 930–937 (2007).
[Crossref]

J. X-Ray Sci. Technol. (1)

I. N. Bukreeva, I. V. Kozhevnikov, and A. V. Vinogradov, “Whispering gallery mirrors for the soft x-ray region: Properties and applications,” J. X-Ray Sci. Technol. 5(4), 397–419 (1995).
[Crossref]

JETP Lett. (2)

V. E. Asadchikov, V. V. Volkov, Yu. O. Volkov, A. Dembo, I. V. Kozhevnikov, B. S. Roshchin, D. A. Frolov, and A. M. Tikhonov, “Condensation of silica nanoparticles on a phospholipid membrane,” JETP Lett. 94(7), 585–587 (2011).
[Crossref]

A. M. Tikhonov, V. E. Asadchikov, Yu. O. Volkov, B. S. Roshchin, V. Honkimäki, and M. V. Blanco, “Model-Independent X-Ray Scattering Study of a Silica Sol Surface,” JETP Lett. 107(6), 384–389 (2018).
[Crossref]

Nature (1)

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref]

Philosophical magazine (1)

J. W. Strutt(Lord Rayleigh), “The problem of the whispering gallery,” Philosophical magazine 20(120), 1001–1004 (1910).

Phys. Rev. A (2)

A. Braslau, P. S. Pershan, G. Swislow, B. M. Ocko, and J. Als-Nielsen, “Capillary waves on the surface of simple liquids measured by x-ray reflectivity,” Phys. Rev. A 38(5), 2457–2470 (1988).
[Crossref]

D. K. Schwartz, M. L. Schlossman, E. H. Kawamoto, G. J. Kellogg, P. S. Pershan, and B. M. Ocko, “Thermal diffuse x-ray-scattering studies of the water-vapor interface,” Phys. Rev. A 41(10), 5687–5690 (1990).
[Crossref]

Phys. Rev. Lett. (4)

F. P. Buff, R. Lovett, and F. H. Stillinger, “Interfacial Density Profile for Fluids in the Critical Region,” Phys. Rev. Lett. 15(15), 621–623 (1965).
[Crossref]

A. Braslau, M. Deutsch, P. S. Pershan, A. H. Weiss, J. Als-Nielsen, and J. Bohr, “Surface Roughness of Water Measured by X-Ray Reflectivity,” Phys. Rev. Lett. 54(2), 114–117 (1985).
[Crossref]

C. Liu and J. A. Golovchenko, “Surface Trapped X Rays: Whispering-Gallery Modes at $\lambda = 0.7$λ=0.7 Å,” Phys. Rev. Lett. 79(5), 788–791 (1997).
[Crossref]

M. Fukuto, R. K. Heilmann, P. S. Pershan, J. A. Griffiths, S. M. Yu, and D. A. Tirrell, “X-Ray Measurements of Noncapillary Spatial Fluctuations from a Liquid Surface,” Phys. Rev. Lett. 81(16), 3455–3458 (1998).
[Crossref]

Quantum Electron. (1)

N. V. Kryzhanovskaya, M. V. Maximov, and A. E. Zhukov, “Whispering-gallery mode microcavity quantum-dot lasers,” Quantum Electron. 44(3), 189–200 (2014).
[Crossref]

Rheol. Acta (1)

E. D. Giuseppe, A. Davaille, E. Mittelstaedt, and M. Francois, “Rheological and mechanical properties of silica colloids: from newtonian liquid to brittle behaviour,” Rheol. Acta 51(5), 451–465 (2012).
[Crossref]

Tech. Phys. (1)

L. I. Goray, N. I. Chkhalo, and G. E. Tsyrlin, “Determining angles of incidence and heights of quantum dot faces by analyzing x-ray diffuse and specular scattering,” Tech. Phys. 54(4), 561–568 (2009).
[Crossref]

Other (8)

L. I. Goray and G. Schmidt, “Boundary Integral Equation Methods for Conical Diffraction and Short Waves,” in Gratings: Theory and Numerical Applications, E. Popov, ed., 2nd rev. ed. (Presses Universitaires de Provence, AMU, 2014). Chap. 12.

B. M. Ocko, in Spectroscopic and Diffraction Techniques in Interfacial Electrochemistry, C. Gutiérrez and C. Melendres, eds. (NATO ASI Series, 1990), p. 343.

E. Spiller, Soft x-ray Optics (SPIE, Bellingham, Washington, 1994).

V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Theory (Alpha Science, Oxford, 2007).

A. E. Muslimov, private communication.

The PCGrate software, http://www.pcgrate.com/ (2018).

The Grace products, https://grace.com/catalysts-and-fuels/en-us/Pages/ludox-emission-control.aspx (2018).

The Center of x-ray Optics, http://henke.lbl.gov/ (2018).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1. Schematic the set up for probing x-ray whispering gallery modes propagating along the concave surface of the rotating liquid. The shape of the concave meniscus at the surface of the rotating liquid $z(x)= ({{\omega^2}/2g} ){x^2}$ z ( x ) = ( ω 2 / 2 g ) x 2 is defined by the angular velocity, ω, of the rotation around Oz-axis and acceleration of gravity, g. The center of the rotation is at x, z = 0. Inset: integral count-rate with a noise reduction of the detector along Oy-axis as a function of the scattering angle ψ / 2, (1) for the direct beam; (2) for the specular reflectance from the water meniscus; (3,4) for the specular reflectances from the concave silica sol surface.
Fig. 2.
Fig. 2. Distribution of x-ray fluorescence counts at λ ≈ 2.892 Ǻ along the concave surface of the cesium-enriched hydrosol. The experimental data (squares) are approximated by two Gaussian functions (dashed lines) centered at x ≈ –30 mm and x ≈ 0. The solid line is the sum of the two dashed curves. Inset: scheme of reflection from the concave surface.
Fig. 3.
Fig. 3. X-ray reflectivity, R, as a function of the wave vector component, ${q_z} = ({4\pi /\lambda } )sin \theta $ q z = ( 4 π / λ ) s i n θ , normal to the surface of the cesium-enriched colloidal solution: dots represent the solution Ludox SM of ∼ 10-nm particles; circles are for the solution Ludox FM of ∼ 7-nm particles. Cesium bulk concentration in the sols ∼ 0.7 mol / L. Solid lines correspond to the hybrid model discussed in the text. Dotted lines indicate calculation results using the model-independent approach [27].
Fig. 4.
Fig. 4. Specular reflectance, R, for the meniscus of the cesium-enriched hydrosol Ludox SM as a function of the grazing incidence angle φ between a tangential plane to the hydrosol surface and the direction of incidence. Solid line represents calculations for the rotation angle Ψ = 2°. Dashed line corresponds to calculations for Ψ = 4°. Dots depict experimental data, which correspond to the insert in Fig. 1 with a detector noise reduction.
Fig. 5.
Fig. 5. Integral count-rate accounting a noise of the detector along Oy-axis as a function of the scattering angle ψ / 2, (highest black curve) for the direct beam; (various color shapes) for the multiple successive reflections from the concave silica sol surface.

Equations (3)

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

R ( θ , r , ψ ) = I 0 ( θ , r , 0 ) e x p { 2 ψ R e [ η ε 1 θ η 2 ε 1 ] } ,
R ( θ , r , ψ ) = I 0 ( θ , r , 0 ) exp { 2 ψ I m [ 1 1 ϵ ] } .
σ eff 2 = σ capil 2 + σ fract 2 .