Abstract

Piezoelectric transparent thin films are of great interest for use in tunable filters. We present experimental results on Ta2O5 single layers coated on fused-silica substrates with an electron-beam deposition process. Above 450 °C, coatings change from an amorphous to a polycrystallized structure. When this structure shows a preferred orientation matching the piezoelectric tensor of the Ta2O5 crystal and the external electric field, variation in the piezoelectric layer thickness is expected. We detail experimental results in terms of optical (spectrophotometric and scattering measurements) and nonoptical characterizations (x-ray diffraction and scanning electron microscopy). Then the resultant thickness variation under oscillating applied voltage is measured with an extrinsic Fabry-Perot interferometer setup.

© 2002 Optical Society of America

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  1. M. Lequime, R. Parmentier, F. Lemarchand, C. Amra, “Toward tunable thin-film filters for WDM applications,” Appl. Opt. 41, xxxx–xxxx (2002)
    [CrossRef]
  2. W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, 6th ed. (Van Nostrand, Princeton, N. J.1964), pp. 40–46.
  3. J. G. E. Gardeniers, Z. M. Rittersma, G. J. Burger, “Prefered orientation and piezoelectricity in sputtered ZnO films,” J. Appl. Phys. 83, 7844–7854 (1998).
    [CrossRef]
  4. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 233.
  5. C. Chaneliere, J. L. Autran, R. A. B. Devine, B. Balland, “Tantalum pentoxide thin films for advanced dielectric applications,” Mater. Sci. Eng. 22, 269–322 (1998).
    [CrossRef]
  6. D. R. Lide, Handbook of Chemistry and Physics, 76th ed. (CRC Press, Boca Raton, Fla., (1995).
  7. B. R. Jooste, H. J. Viljoen, “A study of piezoelectric orthorhombic Ta2O5,” J. Mater. Res. 13, 475–482 (1998).
    [CrossRef]
  8. H. K. Pulker, K. H. Guenther, “Reactive physical vapor deposition processes,” in Thin Films for Optical Systems, F. R. Flory, ed. (Marcel Dekker, New York, 1995), pp. 91–115.
  9. S. Boughaba, G. I. Sproule, J. P. McCaffrey, M. Islam, M. J. Graham, “Synthesis of tantalum pentoxide films by pulsed laser deposition: material characterization and scale-up,” Thin Solid Films 358, 104–113 (2000).
    [CrossRef]
  10. J. Y. Zhang, Q. Fang, I. W. Boyd, “Growth of tantalum pentoxide film by pulsed laser deposition,” Appl. Surf. Sci. 138–139, 320–324 (1999).
    [CrossRef]
  11. T. Dimitrova, K. Arshak, K. Atanassova, “Crystallization effects in oxygen annealed Ta2O5 thin films on Si,” Thin Solid Films 381, 31–38 (2001).
    [CrossRef]
  12. B. A. Movchan, A. V. Demchishin, “Investigations of the structure and properties of thick Ni, Ti, W, Al2O3 and ZrO2 vacuum condensates,” Fiz. Met. Metalloved. 28, 653–660 (1969).
  13. J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solid Films 102, 209–220 (1983).
    [CrossRef]
  14. C. Deumie, R. Richier, P. Dumas, C. Amra, “Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,” Appl. Opt. 35, 5583–5594 (1996).
    [CrossRef] [PubMed]
  15. International Centre for Diffraction Data, “Joint Committee on Powder Diffraction Standards (JCPDS) Card No. 25-0922” (ICDD, Newton Square, Pa. (1996).
  16. L. Burianová, M. Sule, M. Prokopová, “Determination of the piezoelectric coefficients dij of PZT ceramics and composites by laser interferometry,” J. Eur. Ceram. Soc. 21, 1397–1390 (2001).
    [CrossRef]
  17. M. Schmidt, B. Werther, N. Fuerstenau, M. Mathias, T. Melz, “Fiber-optic extrinsic Fabry-Perot interferometer strain sensor with 50 pm displacement resolution using three-wavelength digital phase demodulation,” Opt. Express 8, 475–480 (2001).
    [CrossRef] [PubMed]
  18. N. Felix, D. Certon, F. Patat, M. Lethiecq, “Piezoelectric materials, ultrasound transducers and arrays characterization by laser interferometry,” e-Journal Non-Destr. Test. Ultrasonics 5 (2000), www.ndt.net/article/v05n09/felix/felix.htm .
  19. M. Lequime, J. J. Guerin, “Large OPD extrinsic Fabry-Perot interferometers using thermally expanded core fiber,” in European Workshop on Optical Fibre Sensors, B. Culshaw, J. D. Jones, eds., Proc. SPIE3483, 179–183 (1998).

2002 (1)

M. Lequime, R. Parmentier, F. Lemarchand, C. Amra, “Toward tunable thin-film filters for WDM applications,” Appl. Opt. 41, xxxx–xxxx (2002)
[CrossRef]

2001 (3)

T. Dimitrova, K. Arshak, K. Atanassova, “Crystallization effects in oxygen annealed Ta2O5 thin films on Si,” Thin Solid Films 381, 31–38 (2001).
[CrossRef]

L. Burianová, M. Sule, M. Prokopová, “Determination of the piezoelectric coefficients dij of PZT ceramics and composites by laser interferometry,” J. Eur. Ceram. Soc. 21, 1397–1390 (2001).
[CrossRef]

M. Schmidt, B. Werther, N. Fuerstenau, M. Mathias, T. Melz, “Fiber-optic extrinsic Fabry-Perot interferometer strain sensor with 50 pm displacement resolution using three-wavelength digital phase demodulation,” Opt. Express 8, 475–480 (2001).
[CrossRef] [PubMed]

2000 (2)

N. Felix, D. Certon, F. Patat, M. Lethiecq, “Piezoelectric materials, ultrasound transducers and arrays characterization by laser interferometry,” e-Journal Non-Destr. Test. Ultrasonics 5 (2000), www.ndt.net/article/v05n09/felix/felix.htm .

S. Boughaba, G. I. Sproule, J. P. McCaffrey, M. Islam, M. J. Graham, “Synthesis of tantalum pentoxide films by pulsed laser deposition: material characterization and scale-up,” Thin Solid Films 358, 104–113 (2000).
[CrossRef]

1999 (1)

J. Y. Zhang, Q. Fang, I. W. Boyd, “Growth of tantalum pentoxide film by pulsed laser deposition,” Appl. Surf. Sci. 138–139, 320–324 (1999).
[CrossRef]

1998 (3)

J. G. E. Gardeniers, Z. M. Rittersma, G. J. Burger, “Prefered orientation and piezoelectricity in sputtered ZnO films,” J. Appl. Phys. 83, 7844–7854 (1998).
[CrossRef]

C. Chaneliere, J. L. Autran, R. A. B. Devine, B. Balland, “Tantalum pentoxide thin films for advanced dielectric applications,” Mater. Sci. Eng. 22, 269–322 (1998).
[CrossRef]

B. R. Jooste, H. J. Viljoen, “A study of piezoelectric orthorhombic Ta2O5,” J. Mater. Res. 13, 475–482 (1998).
[CrossRef]

1996 (1)

1983 (1)

J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solid Films 102, 209–220 (1983).
[CrossRef]

1969 (1)

B. A. Movchan, A. V. Demchishin, “Investigations of the structure and properties of thick Ni, Ti, W, Al2O3 and ZrO2 vacuum condensates,” Fiz. Met. Metalloved. 28, 653–660 (1969).

Amra, C.

M. Lequime, R. Parmentier, F. Lemarchand, C. Amra, “Toward tunable thin-film filters for WDM applications,” Appl. Opt. 41, xxxx–xxxx (2002)
[CrossRef]

C. Deumie, R. Richier, P. Dumas, C. Amra, “Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,” Appl. Opt. 35, 5583–5594 (1996).
[CrossRef] [PubMed]

Arshak, K.

T. Dimitrova, K. Arshak, K. Atanassova, “Crystallization effects in oxygen annealed Ta2O5 thin films on Si,” Thin Solid Films 381, 31–38 (2001).
[CrossRef]

Atanassova, K.

T. Dimitrova, K. Arshak, K. Atanassova, “Crystallization effects in oxygen annealed Ta2O5 thin films on Si,” Thin Solid Films 381, 31–38 (2001).
[CrossRef]

Autran, J. L.

C. Chaneliere, J. L. Autran, R. A. B. Devine, B. Balland, “Tantalum pentoxide thin films for advanced dielectric applications,” Mater. Sci. Eng. 22, 269–322 (1998).
[CrossRef]

Balland, B.

C. Chaneliere, J. L. Autran, R. A. B. Devine, B. Balland, “Tantalum pentoxide thin films for advanced dielectric applications,” Mater. Sci. Eng. 22, 269–322 (1998).
[CrossRef]

Borgogno, J. P.

J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solid Films 102, 209–220 (1983).
[CrossRef]

Boughaba, S.

S. Boughaba, G. I. Sproule, J. P. McCaffrey, M. Islam, M. J. Graham, “Synthesis of tantalum pentoxide films by pulsed laser deposition: material characterization and scale-up,” Thin Solid Films 358, 104–113 (2000).
[CrossRef]

Boyd, I. W.

J. Y. Zhang, Q. Fang, I. W. Boyd, “Growth of tantalum pentoxide film by pulsed laser deposition,” Appl. Surf. Sci. 138–139, 320–324 (1999).
[CrossRef]

Burger, G. J.

J. G. E. Gardeniers, Z. M. Rittersma, G. J. Burger, “Prefered orientation and piezoelectricity in sputtered ZnO films,” J. Appl. Phys. 83, 7844–7854 (1998).
[CrossRef]

Burianová, L.

L. Burianová, M. Sule, M. Prokopová, “Determination of the piezoelectric coefficients dij of PZT ceramics and composites by laser interferometry,” J. Eur. Ceram. Soc. 21, 1397–1390 (2001).
[CrossRef]

Certon, D.

N. Felix, D. Certon, F. Patat, M. Lethiecq, “Piezoelectric materials, ultrasound transducers and arrays characterization by laser interferometry,” e-Journal Non-Destr. Test. Ultrasonics 5 (2000), www.ndt.net/article/v05n09/felix/felix.htm .

Chaneliere, C.

C. Chaneliere, J. L. Autran, R. A. B. Devine, B. Balland, “Tantalum pentoxide thin films for advanced dielectric applications,” Mater. Sci. Eng. 22, 269–322 (1998).
[CrossRef]

Demchishin, A. V.

B. A. Movchan, A. V. Demchishin, “Investigations of the structure and properties of thick Ni, Ti, W, Al2O3 and ZrO2 vacuum condensates,” Fiz. Met. Metalloved. 28, 653–660 (1969).

Deumie, C.

Devine, R. A. B.

C. Chaneliere, J. L. Autran, R. A. B. Devine, B. Balland, “Tantalum pentoxide thin films for advanced dielectric applications,” Mater. Sci. Eng. 22, 269–322 (1998).
[CrossRef]

Dimitrova, T.

T. Dimitrova, K. Arshak, K. Atanassova, “Crystallization effects in oxygen annealed Ta2O5 thin films on Si,” Thin Solid Films 381, 31–38 (2001).
[CrossRef]

Dumas, P.

Fang, Q.

J. Y. Zhang, Q. Fang, I. W. Boyd, “Growth of tantalum pentoxide film by pulsed laser deposition,” Appl. Surf. Sci. 138–139, 320–324 (1999).
[CrossRef]

Felix, N.

N. Felix, D. Certon, F. Patat, M. Lethiecq, “Piezoelectric materials, ultrasound transducers and arrays characterization by laser interferometry,” e-Journal Non-Destr. Test. Ultrasonics 5 (2000), www.ndt.net/article/v05n09/felix/felix.htm .

Fuerstenau, N.

Gardeniers, J. G. E.

J. G. E. Gardeniers, Z. M. Rittersma, G. J. Burger, “Prefered orientation and piezoelectricity in sputtered ZnO films,” J. Appl. Phys. 83, 7844–7854 (1998).
[CrossRef]

Graham, M. J.

S. Boughaba, G. I. Sproule, J. P. McCaffrey, M. Islam, M. J. Graham, “Synthesis of tantalum pentoxide films by pulsed laser deposition: material characterization and scale-up,” Thin Solid Films 358, 104–113 (2000).
[CrossRef]

Guenther, K. H.

H. K. Pulker, K. H. Guenther, “Reactive physical vapor deposition processes,” in Thin Films for Optical Systems, F. R. Flory, ed. (Marcel Dekker, New York, 1995), pp. 91–115.

Guerin, J. J.

M. Lequime, J. J. Guerin, “Large OPD extrinsic Fabry-Perot interferometers using thermally expanded core fiber,” in European Workshop on Optical Fibre Sensors, B. Culshaw, J. D. Jones, eds., Proc. SPIE3483, 179–183 (1998).

Islam, M.

S. Boughaba, G. I. Sproule, J. P. McCaffrey, M. Islam, M. J. Graham, “Synthesis of tantalum pentoxide films by pulsed laser deposition: material characterization and scale-up,” Thin Solid Films 358, 104–113 (2000).
[CrossRef]

Jooste, B. R.

B. R. Jooste, H. J. Viljoen, “A study of piezoelectric orthorhombic Ta2O5,” J. Mater. Res. 13, 475–482 (1998).
[CrossRef]

Lazarides, B.

J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solid Films 102, 209–220 (1983).
[CrossRef]

Lemarchand, F.

M. Lequime, R. Parmentier, F. Lemarchand, C. Amra, “Toward tunable thin-film filters for WDM applications,” Appl. Opt. 41, xxxx–xxxx (2002)
[CrossRef]

Lequime, M.

M. Lequime, R. Parmentier, F. Lemarchand, C. Amra, “Toward tunable thin-film filters for WDM applications,” Appl. Opt. 41, xxxx–xxxx (2002)
[CrossRef]

M. Lequime, J. J. Guerin, “Large OPD extrinsic Fabry-Perot interferometers using thermally expanded core fiber,” in European Workshop on Optical Fibre Sensors, B. Culshaw, J. D. Jones, eds., Proc. SPIE3483, 179–183 (1998).

Lethiecq, M.

N. Felix, D. Certon, F. Patat, M. Lethiecq, “Piezoelectric materials, ultrasound transducers and arrays characterization by laser interferometry,” e-Journal Non-Destr. Test. Ultrasonics 5 (2000), www.ndt.net/article/v05n09/felix/felix.htm .

Lide, D. R.

D. R. Lide, Handbook of Chemistry and Physics, 76th ed. (CRC Press, Boca Raton, Fla., (1995).

Mason, W. P.

W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, 6th ed. (Van Nostrand, Princeton, N. J.1964), pp. 40–46.

Mathias, M.

McCaffrey, J. P.

S. Boughaba, G. I. Sproule, J. P. McCaffrey, M. Islam, M. J. Graham, “Synthesis of tantalum pentoxide films by pulsed laser deposition: material characterization and scale-up,” Thin Solid Films 358, 104–113 (2000).
[CrossRef]

Melz, T.

Movchan, B. A.

B. A. Movchan, A. V. Demchishin, “Investigations of the structure and properties of thick Ni, Ti, W, Al2O3 and ZrO2 vacuum condensates,” Fiz. Met. Metalloved. 28, 653–660 (1969).

Parmentier, R.

M. Lequime, R. Parmentier, F. Lemarchand, C. Amra, “Toward tunable thin-film filters for WDM applications,” Appl. Opt. 41, xxxx–xxxx (2002)
[CrossRef]

Patat, F.

N. Felix, D. Certon, F. Patat, M. Lethiecq, “Piezoelectric materials, ultrasound transducers and arrays characterization by laser interferometry,” e-Journal Non-Destr. Test. Ultrasonics 5 (2000), www.ndt.net/article/v05n09/felix/felix.htm .

Prokopová, M.

L. Burianová, M. Sule, M. Prokopová, “Determination of the piezoelectric coefficients dij of PZT ceramics and composites by laser interferometry,” J. Eur. Ceram. Soc. 21, 1397–1390 (2001).
[CrossRef]

Pulker, H. K.

H. K. Pulker, K. H. Guenther, “Reactive physical vapor deposition processes,” in Thin Films for Optical Systems, F. R. Flory, ed. (Marcel Dekker, New York, 1995), pp. 91–115.

Richier, R.

Rittersma, Z. M.

J. G. E. Gardeniers, Z. M. Rittersma, G. J. Burger, “Prefered orientation and piezoelectricity in sputtered ZnO films,” J. Appl. Phys. 83, 7844–7854 (1998).
[CrossRef]

Roche, P.

J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solid Films 102, 209–220 (1983).
[CrossRef]

Schmidt, M.

Sproule, G. I.

S. Boughaba, G. I. Sproule, J. P. McCaffrey, M. Islam, M. J. Graham, “Synthesis of tantalum pentoxide films by pulsed laser deposition: material characterization and scale-up,” Thin Solid Films 358, 104–113 (2000).
[CrossRef]

Sule, M.

L. Burianová, M. Sule, M. Prokopová, “Determination of the piezoelectric coefficients dij of PZT ceramics and composites by laser interferometry,” J. Eur. Ceram. Soc. 21, 1397–1390 (2001).
[CrossRef]

Viljoen, H. J.

B. R. Jooste, H. J. Viljoen, “A study of piezoelectric orthorhombic Ta2O5,” J. Mater. Res. 13, 475–482 (1998).
[CrossRef]

Werther, B.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 233.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 233.

Zhang, J. Y.

J. Y. Zhang, Q. Fang, I. W. Boyd, “Growth of tantalum pentoxide film by pulsed laser deposition,” Appl. Surf. Sci. 138–139, 320–324 (1999).
[CrossRef]

Appl. Opt. (2)

M. Lequime, R. Parmentier, F. Lemarchand, C. Amra, “Toward tunable thin-film filters for WDM applications,” Appl. Opt. 41, xxxx–xxxx (2002)
[CrossRef]

C. Deumie, R. Richier, P. Dumas, C. Amra, “Multiscale roughness in optical multilayers: atomic force microscopy and light scattering,” Appl. Opt. 35, 5583–5594 (1996).
[CrossRef] [PubMed]

Appl. Surf. Sci. (1)

J. Y. Zhang, Q. Fang, I. W. Boyd, “Growth of tantalum pentoxide film by pulsed laser deposition,” Appl. Surf. Sci. 138–139, 320–324 (1999).
[CrossRef]

e-Journal Non-Destr. Test. Ultrasonics (1)

N. Felix, D. Certon, F. Patat, M. Lethiecq, “Piezoelectric materials, ultrasound transducers and arrays characterization by laser interferometry,” e-Journal Non-Destr. Test. Ultrasonics 5 (2000), www.ndt.net/article/v05n09/felix/felix.htm .

Fiz. Met. Metalloved. (1)

B. A. Movchan, A. V. Demchishin, “Investigations of the structure and properties of thick Ni, Ti, W, Al2O3 and ZrO2 vacuum condensates,” Fiz. Met. Metalloved. 28, 653–660 (1969).

J. Appl. Phys. (1)

J. G. E. Gardeniers, Z. M. Rittersma, G. J. Burger, “Prefered orientation and piezoelectricity in sputtered ZnO films,” J. Appl. Phys. 83, 7844–7854 (1998).
[CrossRef]

J. Eur. Ceram. Soc. (1)

L. Burianová, M. Sule, M. Prokopová, “Determination of the piezoelectric coefficients dij of PZT ceramics and composites by laser interferometry,” J. Eur. Ceram. Soc. 21, 1397–1390 (2001).
[CrossRef]

J. Mater. Res. (1)

B. R. Jooste, H. J. Viljoen, “A study of piezoelectric orthorhombic Ta2O5,” J. Mater. Res. 13, 475–482 (1998).
[CrossRef]

Mater. Sci. Eng. (1)

C. Chaneliere, J. L. Autran, R. A. B. Devine, B. Balland, “Tantalum pentoxide thin films for advanced dielectric applications,” Mater. Sci. Eng. 22, 269–322 (1998).
[CrossRef]

Opt. Express (1)

Thin Solid Films (3)

J. P. Borgogno, B. Lazarides, P. Roche, “An improved method for the determination of the extinction coefficient of thin film materials,” Thin Solid Films 102, 209–220 (1983).
[CrossRef]

T. Dimitrova, K. Arshak, K. Atanassova, “Crystallization effects in oxygen annealed Ta2O5 thin films on Si,” Thin Solid Films 381, 31–38 (2001).
[CrossRef]

S. Boughaba, G. I. Sproule, J. P. McCaffrey, M. Islam, M. J. Graham, “Synthesis of tantalum pentoxide films by pulsed laser deposition: material characterization and scale-up,” Thin Solid Films 358, 104–113 (2000).
[CrossRef]

Other (6)

M. Lequime, J. J. Guerin, “Large OPD extrinsic Fabry-Perot interferometers using thermally expanded core fiber,” in European Workshop on Optical Fibre Sensors, B. Culshaw, J. D. Jones, eds., Proc. SPIE3483, 179–183 (1998).

International Centre for Diffraction Data, “Joint Committee on Powder Diffraction Standards (JCPDS) Card No. 25-0922” (ICDD, Newton Square, Pa. (1996).

D. R. Lide, Handbook of Chemistry and Physics, 76th ed. (CRC Press, Boca Raton, Fla., (1995).

H. K. Pulker, K. H. Guenther, “Reactive physical vapor deposition processes,” in Thin Films for Optical Systems, F. R. Flory, ed. (Marcel Dekker, New York, 1995), pp. 91–115.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 233.

W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, 6th ed. (Van Nostrand, Princeton, N. J.1964), pp. 40–46.

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Figures (11)

Fig. 1
Fig. 1

(a) Schematic drawing of a randomly oriented crystallite structure and (b) an organized structure with a preferred orientation in the vertical direction If the piezoelectric tensor matches the vertical axis and the electric field direction, a piezoelectric displacement equivalent to the single-crystal structure displacement can occur in the vertical direction.

Fig. 2
Fig. 2

Evaporation chamber that uses the electron-beam deposition process. An optical monitoring and a quartz crystal measurement is performed on the reference substrate. The temperature of the heated substrate can reach 950 °C on the rear side.

Fig. 3
Fig. 3

Calculated real parts of the refractive indices N for S1, S2, and S3 as a function of the wavelength.

Fig. 4
Fig. 4

Bidirectional reflectance distribution function (BRDF) as a function of the measurement angle. Samples are illuminated under normal incidence.

Fig. 5
Fig. 5

X-ray diffraction spectra of Ta2O5 powder. Theta is the angle between the incident light and the sample. 2theta is the angle between the incident light and the reflected beam in the incident plane. The theoretical diffraction efficiency is also shown.

Fig. 6
Fig. 6

X-ray diffraction spectra of samples: (a) S4, (b) S5, (c) S6, and (d) S7.

Fig. 7
Fig. 7

Scanning electron microscopy of samples: (a) S4, (b) S5, (c) S6, and (d) S7.

Fig. 8
Fig. 8

Extrinsic Fabry-Perot interferometer experimental setup. The photoreceptor measures the interference fringes between the sample and the glass-air interface of the fiber.

Fig. 9
Fig. 9

Theoretical intensity detected by a photoreceptor as a function of the distance z between the fiber exit and the sample surface.

Fig. 10
Fig. 10

Principle of the measurement of a small f 0 displacement from the intensity detected for two working points.

Fig. 11
Fig. 11

Interferometric intensity measured as a function of the incident wavelength, the distance between the fiber and the sample being constant (black curve). Under a 400-V voltage at f 0 = 105 Hz, we show the f 0-filtered signal for sample 7 (dark gray curve) and for sample 4 (light gray curve).

Equations (4)

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Δtj/tj=dijEi,
I=I01+m cosϕ,
ϕ=4πz/λ.
ΔIω=-I0m sinϕ4πλ Δzω.

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