Abstract

We theoretically study a 1D elastic photonic crystal containing air voids as an opto-pneumatic medium. This medium is sensitive to weak deviations of the external pressure and, owing to its elasticity, can vary its geometry depending on the external conditions. We show that the reflectivity can be drastically changed at a chosen working frequency near the photonic band-gap edge or the reflection window. The resonance properties of such pneumatic photonic crystals made of glass, silicon, and mica with directly excited eigenmodes in the infrared region of frequencies are analyzed. The ways to determine small changes in the pressure on the micro- and nanobar scale are discussed.

©2010 Optical Society of America

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References

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  1. A. Werber and H. Zappe, “Tunable, membrane-based, pneumatic micro-mirrors,” J. Opt. A, Pure Appl. Opt. 8(7), S313–S317 (2006).
    [Crossref]
  2. V. Pervak, I. Ahmad, M. K. Trubetskov, A. V. Tikhonravov, and F. Krausz, “Double-angle multilayer mirrors with smooth dispersion characteristics,” Opt. Express 17(10), 7943–7951 (2009).
    [Crossref] [PubMed]
  3. N. Tokranova, B. Xu, and J. Castracane, “Fabrication of flexible one-dimensional porous silicon photonic band-gap structures,” MRS Proceedings 797, 1 (2004).
  4. V. I. Beloglazov, N. Langhoff, V. V. Tuchin, A. Bjeoumikhov, Z. Bjeoumikhova, R. Wedel, N. B. Skibina, Yu. S. Skibina, and M. V. Chainikov, “Technologies of manufacturing polycapillary optics for x-ray engineering,” J. of X-Ray Sci. Techn., 13, 179 (2005), http://spie.org/x35457.xml?ArticleID=x35457 (2009).
  5. G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, and A. M. Kamchatnov, “Theory of optical dispersive shock waves,” Phys. Rev. A 76(5), 053813 (2007).
    [Crossref]
  6. S. Darmanyan, A. Kamchatnov, and F. Lederer, “Optical shock waves in media with quadratic nonlinearity,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(4), R4120–R4123 (1998).
    [Crossref]
  7. E. J. Reed, M. Soljacić, and J. D. Joannopoulos, “Color of shock waves in photonic crystals,” Phys. Rev. Lett. 90(20), 203904 (2003).
    [Crossref] [PubMed]
  8. K. Hosokawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12(1), 301 (2002).
    [Crossref]
  9. B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators 86(1-2), 81–85 (2000).
    [Crossref]
  10. A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
    [Crossref]
  11. E. Ya. Glushko, “Analytical solution for the field in photonic structures containing cubic nonlinearity,” Opt. Commun. 259(1), 342–349 (2006).
    [Crossref]
  12. E. Ya. Glushko, “All-optical signal processing in photonic structures with nonlinearity,” Opt. Commun. 247(4-6), 275–280 (2005).
    [Crossref]
  13. L. D. Landau, and E. M. Lifshitz, Theory of Elasticity, Pergamon Press, New York, 1970.
  14. Tables of Physical Values (in Russian). Ed. by I.K. Kikoin, Atomizdat, Moscow, 1976; L.B. Freund and S. Suresh, Thin film materials. Cambridge University Press, Cambridge, 2003.
  15. J. L. Caslaevsky and K. Vedam, “Muscovites with isotropic and anisotropic elasticity in the basal plane,” Amer. Miner. 55, 1633 (1970).

2009 (1)

2007 (1)

G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, and A. M. Kamchatnov, “Theory of optical dispersive shock waves,” Phys. Rev. A 76(5), 053813 (2007).
[Crossref]

2006 (3)

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

E. Ya. Glushko, “Analytical solution for the field in photonic structures containing cubic nonlinearity,” Opt. Commun. 259(1), 342–349 (2006).
[Crossref]

A. Werber and H. Zappe, “Tunable, membrane-based, pneumatic micro-mirrors,” J. Opt. A, Pure Appl. Opt. 8(7), S313–S317 (2006).
[Crossref]

2005 (1)

E. Ya. Glushko, “All-optical signal processing in photonic structures with nonlinearity,” Opt. Commun. 247(4-6), 275–280 (2005).
[Crossref]

2004 (1)

N. Tokranova, B. Xu, and J. Castracane, “Fabrication of flexible one-dimensional porous silicon photonic band-gap structures,” MRS Proceedings 797, 1 (2004).

2003 (1)

E. J. Reed, M. Soljacić, and J. D. Joannopoulos, “Color of shock waves in photonic crystals,” Phys. Rev. Lett. 90(20), 203904 (2003).
[Crossref] [PubMed]

2002 (1)

K. Hosokawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12(1), 301 (2002).
[Crossref]

2000 (1)

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators 86(1-2), 81–85 (2000).
[Crossref]

1998 (1)

S. Darmanyan, A. Kamchatnov, and F. Lederer, “Optical shock waves in media with quadratic nonlinearity,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(4), R4120–R4123 (1998).
[Crossref]

1970 (1)

J. L. Caslaevsky and K. Vedam, “Muscovites with isotropic and anisotropic elasticity in the basal plane,” Amer. Miner. 55, 1633 (1970).

Ahmad, I.

Arsenaul, A. C.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Bertolotti, J.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Cademartiri, L.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Caslaevsky, J. L.

J. L. Caslaevsky and K. Vedam, “Muscovites with isotropic and anisotropic elasticity in the basal plane,” Amer. Miner. 55, 1633 (1970).

Castracane, J.

N. Tokranova, B. Xu, and J. Castracane, “Fabrication of flexible one-dimensional porous silicon photonic band-gap structures,” MRS Proceedings 797, 1 (2004).

Clark, T. J.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Darmanyan, S.

S. Darmanyan, A. Kamchatnov, and F. Lederer, “Optical shock waves in media with quadratic nonlinearity,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(4), R4120–R4123 (1998).
[Crossref]

El, G. A.

G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, and A. M. Kamchatnov, “Theory of optical dispersive shock waves,” Phys. Rev. A 76(5), 053813 (2007).
[Crossref]

Gammal, A.

G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, and A. M. Kamchatnov, “Theory of optical dispersive shock waves,” Phys. Rev. A 76(5), 053813 (2007).
[Crossref]

Glushko, E. Ya.

E. Ya. Glushko, “Analytical solution for the field in photonic structures containing cubic nonlinearity,” Opt. Commun. 259(1), 342–349 (2006).
[Crossref]

E. Ya. Glushko, “All-optical signal processing in photonic structures with nonlinearity,” Opt. Commun. 247(4-6), 275–280 (2005).
[Crossref]

Grzybowski, B.

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators 86(1-2), 81–85 (2000).
[Crossref]

Haag, R.

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators 86(1-2), 81–85 (2000).
[Crossref]

Hanada, K.

K. Hosokawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12(1), 301 (2002).
[Crossref]

Hosokawa, K.

K. Hosokawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12(1), 301 (2002).
[Crossref]

Joannopoulos, J. D.

E. J. Reed, M. Soljacić, and J. D. Joannopoulos, “Color of shock waves in photonic crystals,” Phys. Rev. Lett. 90(20), 203904 (2003).
[Crossref] [PubMed]

John, S.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Kamchatnov, A.

S. Darmanyan, A. Kamchatnov, and F. Lederer, “Optical shock waves in media with quadratic nonlinearity,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(4), R4120–R4123 (1998).
[Crossref]

Kamchatnov, A. M.

G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, and A. M. Kamchatnov, “Theory of optical dispersive shock waves,” Phys. Rev. A 76(5), 053813 (2007).
[Crossref]

Khamis, E. G.

G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, and A. M. Kamchatnov, “Theory of optical dispersive shock waves,” Phys. Rev. A 76(5), 053813 (2007).
[Crossref]

Kitaev, V.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Kraenkel, R. A.

G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, and A. M. Kamchatnov, “Theory of optical dispersive shock waves,” Phys. Rev. A 76(5), 053813 (2007).
[Crossref]

Krausz, F.

Lederer, F.

S. Darmanyan, A. Kamchatnov, and F. Lederer, “Optical shock waves in media with quadratic nonlinearity,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(4), R4120–R4123 (1998).
[Crossref]

Maeda, R.

K. Hosokawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12(1), 301 (2002).
[Crossref]

Mannersi, I.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Ozin, G. A.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Pervak, V.

Qin, D.

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators 86(1-2), 81–85 (2000).
[Crossref]

Reed, E. J.

E. J. Reed, M. Soljacić, and J. D. Joannopoulos, “Color of shock waves in photonic crystals,” Phys. Rev. Lett. 90(20), 203904 (2003).
[Crossref] [PubMed]

Sapienza, R.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Soljacic, M.

E. J. Reed, M. Soljacić, and J. D. Joannopoulos, “Color of shock waves in photonic crystals,” Phys. Rev. Lett. 90(20), 203904 (2003).
[Crossref] [PubMed]

Tikhonravov, A. V.

Tokranova, N.

N. Tokranova, B. Xu, and J. Castracane, “Fabrication of flexible one-dimensional porous silicon photonic band-gap structures,” MRS Proceedings 797, 1 (2004).

Trubetskov, M. K.

Vedam, K.

J. L. Caslaevsky and K. Vedam, “Muscovites with isotropic and anisotropic elasticity in the basal plane,” Amer. Miner. 55, 1633 (1970).

Vekrisi, E.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

von Freymann, G.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Wang, R. Z.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Werber, A.

A. Werber and H. Zappe, “Tunable, membrane-based, pneumatic micro-mirrors,” J. Opt. A, Pure Appl. Opt. 8(7), S313–S317 (2006).
[Crossref]

Whitesides, G. M.

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators 86(1-2), 81–85 (2000).
[Crossref]

Wiersma, D.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Wong, S.

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Xu, B.

N. Tokranova, B. Xu, and J. Castracane, “Fabrication of flexible one-dimensional porous silicon photonic band-gap structures,” MRS Proceedings 797, 1 (2004).

Zappe, H.

A. Werber and H. Zappe, “Tunable, membrane-based, pneumatic micro-mirrors,” J. Opt. A, Pure Appl. Opt. 8(7), S313–S317 (2006).
[Crossref]

Amer. Miner. (1)

J. L. Caslaevsky and K. Vedam, “Muscovites with isotropic and anisotropic elasticity in the basal plane,” Amer. Miner. 55, 1633 (1970).

J. Micromech. Microeng. (1)

K. Hosokawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices,” J. Micromech. Microeng. 12(1), 301 (2002).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

A. Werber and H. Zappe, “Tunable, membrane-based, pneumatic micro-mirrors,” J. Opt. A, Pure Appl. Opt. 8(7), S313–S317 (2006).
[Crossref]

MRS Proceedings (1)

N. Tokranova, B. Xu, and J. Castracane, “Fabrication of flexible one-dimensional porous silicon photonic band-gap structures,” MRS Proceedings 797, 1 (2004).

Nat. Mater. (1)

A. C. Arsenaul, T. J. Clark, G. von Freymann, L. Cademartiri, R. Sapienza, J. Bertolotti, E. Vekrisi, S. Wong, V. Kitaev, I. Mannersi, R. Z. Wang, S. John, D. Wiersma, and G. A. Ozin, “From colour fingerprinting to the control of photoluminescence in elastic photonic crystals,” Nat. Mater. 5(3), 179–184 (2006).
[Crossref]

Opt. Commun. (2)

E. Ya. Glushko, “Analytical solution for the field in photonic structures containing cubic nonlinearity,” Opt. Commun. 259(1), 342–349 (2006).
[Crossref]

E. Ya. Glushko, “All-optical signal processing in photonic structures with nonlinearity,” Opt. Commun. 247(4-6), 275–280 (2005).
[Crossref]

Opt. Express (1)

Phys. Rev. A (1)

G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, and A. M. Kamchatnov, “Theory of optical dispersive shock waves,” Phys. Rev. A 76(5), 053813 (2007).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

S. Darmanyan, A. Kamchatnov, and F. Lederer, “Optical shock waves in media with quadratic nonlinearity,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(4), R4120–R4123 (1998).
[Crossref]

Phys. Rev. Lett. (1)

E. J. Reed, M. Soljacić, and J. D. Joannopoulos, “Color of shock waves in photonic crystals,” Phys. Rev. Lett. 90(20), 203904 (2003).
[Crossref] [PubMed]

Sens. Actuators (1)

B. Grzybowski, D. Qin, R. Haag, and G. M. Whitesides, “Elastomeric optical elements with deformable surface topographies: applications to force measurements, tunable light transmission and light focusing,” Sens. Actuators 86(1-2), 81–85 (2000).
[Crossref]

Other (3)

V. I. Beloglazov, N. Langhoff, V. V. Tuchin, A. Bjeoumikhov, Z. Bjeoumikhova, R. Wedel, N. B. Skibina, Yu. S. Skibina, and M. V. Chainikov, “Technologies of manufacturing polycapillary optics for x-ray engineering,” J. of X-Ray Sci. Techn., 13, 179 (2005), http://spie.org/x35457.xml?ArticleID=x35457 (2009).

L. D. Landau, and E. M. Lifshitz, Theory of Elasticity, Pergamon Press, New York, 1970.

Tables of Physical Values (in Russian). Ed. by I.K. Kikoin, Atomizdat, Moscow, 1976; L.B. Freund and S. Suresh, Thin film materials. Cambridge University Press, Cambridge, 2003.

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Figures (3)

Fig. 1
Fig. 1 (a) Pressure-sensitive pneumatic resonator with the normal incidence of light beam (arrow); in-plane geometry. (b) P(z) dependence calculated by (7) for a pneumooptic SiO2/air system with d1=4.0 μm, d2=6.0 μm, R=800 μm, P(z)Є(999–1001) mbar.

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Fig. 2
Fig. 2 Vertical panel: the frequency dependence of field modes inside the total reflection area for a 10-period 1D (SiO2/Air)15SiO2 structure; the angles are (43.6–50°). The SiO2-layer width d1=4.0 µm, the size of air voids d2=6.0 µm. Horizontal panel: the frequency diagram for the reflection of external incident light; the angles are (0–90°). Photon energies are (0–0.5) eV.

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Fig. 3
Fig. 3 Glass/air system with d1=4 μm, d2=6 μm, N=100, under the normal incidence. (a) Reflection vs the photon energy. Curves 1-10 correspond to pressures of 1000-1000.1 mbar. The arrow indicates the chosen working photon energy of 0.31660 eV. Changed colour in the upper part corresponds to the reflection inside the gap. (b) Reflection vs the pressure. Curves 1-6 cover R=550-800 μm. (с) shows a multiscale pressure indicator. The stack of plates with sensibilities η1=102 , η2=101 , ..., η7=10–4 mbar−1 cover several orders of magnitude of the measured pressure.

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Tables (1)

Tables Icon

Table 1 * Inside the TIR area, ** negative magnitude; values of the Young modulus and the Poisson’s ratio were taken from [14, 15].

View Table

Equations (15)

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

D Δ 2 ξ = δ P ,
( ∂ ξ ∂ r ) r = R = ( ξ ) r = R = 0
ξ ( r ) = β ( R 2 − r 2 ) 2 , β = δ P / 64 D ,
{ P s = Q P 0 Q + P s + 1 − P s − 1 , s = 1 , 2.. [ ( N − 2 ) / 2 ] ,
P N − 1 2 = Q P 0 Q + P N − 1 2 − P N − 3 2 , odd N ; P N 2 = Q P 0 Q + 2 ( P N 2 − P N − 2 2 ) , even N .
∂ P ∂ z = Q ˜ P 0 − P P ,
P − P ( 0 ) + P 0 L n | P 0 − P P 0 − P ( 0 ) | = − Q ˜ z .
∂ ( δ P ) ∂ z = − q δ P .
E j → = ( sin θ j , cos θ j ) A j e i k j x x + ( sin θ j , − cos θ j ) B j e − i k j x x ,
{ M ⌢ s − 1 ( A l B l ) = L ⌢ s ( A 1 B 1 ) M ⌢ s ( A 1 B 1 ) = L ⌢ s + 1 ( A 2 B 2 )   , s = 1..2 N + 1 ,
L ⌢ s = ( cos θ s - cos θ s ε s sin θ s ε s sin θ s ) ; M ⌢ s = ( e i k z z cos θ s - e − i k z z cos θ s e i k z z ε s sin θ s e − i k z z ε s sin θ s ) , s = 0 … 2 N + 1.
Λ ⌢ = ( μ 1 , ν 1 λ 1 , μ 1 ) ( μ 2 , ν 2 λ 2 , μ 2 ) = ( μ , ν λ , μ ¯ ) μ j = sin 2 θ j ⋅ cos k j z d j / Λ ; λ j = − 2 i sin 2 θ j ⋅ sin k j z d j / Λ ν j = − 2 i cos 2 θ j ⋅ sin k j z d j / Λ ,
( cos θ l c o s θ r ⋅ I 1 1 + ε r sin θ r ⋅ I 1 2 − ε l sin θ l cos θ r ⋅ I 2 1 + ε r sin θ r ⋅ I 2 2 ) ( B l A r ) = A l ( cos θ l ε l sin θ l )   ,
f 1 N ( − x 12 cos θ l + i ν ε l sin θ l ) ( x ˜ 12 cos θ r + i ν ˜ ε r sin θ r ) − f 2 N ( − x 22 cos θ l + i ν ε l sin θ l ) ( x ˜ 22 cos θ r + i ν ˜ ˜ ε r sin θ r ) = 0 ,
x 11 = x 21 = ν ; x j 2 = f j − μ ; ν ˜ = ν μ 1 − ν 1 x 22 ; ν ˜ ˜ = ν μ 1 − ν 1 x 12 ; x ˜ j 2 = x s ( j ) 2 μ 1 − λ 1 ν ;

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