Abstract

The presented new type of interferometer combines the principle of two-beam interferometry and the technique of phase-shift keying of holographic gratings. On the basis of the phase-shift keying technique, the interferometer employs two different geometries for the recording and the readout process. Two holographic Bragg gratings are recorded in transmission geometry and simultaneously read out in reflection geometry using a tunable IR laser. Both gratings have the same grating period but a relative phase shift. The wavelength of the readout beam is fitted to the Bragg condition for the gratings. Using a tunable IR laser for the readout process, we can measure the spectral transfer function of both combined gratings. The shape of the measured transfer function is extremely sensitive to the phase shift between the two gratings. We demonstrate an application of this method by the measurement of refractive-index variations of gases due to pressure changes of the gases. The achieved resolution with respect to the measurement of phase shifts is approximately 1/40π. We present experimental investigations on two kinds of gas (an inert gas and a gas composition) as well as an efficient numerical approach to simulate the transfer function for Bragg gratings with a phase shift. Furthermore, we present a method to increase the resolution based on the controlled manipulation of the transfer function.

© 2005 Optical Society of America

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References

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  1. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
    [CrossRef]
  2. V. M. Petrov, C. Denz, J. Petter, T. Tschudi, “Enhancing the sensitivity of an adaptive holographic interferometer using non-Bragg diffraction orders,” Opt. Lett. 22, 1902–1904 (1997).
    [CrossRef]
  3. G. Boensch, E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlén’s formulae,” Meas. Sci. Technol. 4, 133–139 (1993).
  4. B. Culshaw, D. Davies, S. Kingsley, “Fibre optic strain, pressure and temperature sensors,” in Proceedings of the 4th European Conference on Optical Communication (4th ECOC) (Institution of Electrical Engineers, 1978), Vol. 229, pp. 115–126.
  5. G. Cedilnik, M. Esselbach, A. Kiessling, R. Kowarschik, “Real-time holographic interferometry with double two-wave mixing in photorefractive crystals,” Opt. Commun. 138, 2091–2100 (1997).
  6. V. Petrov, S. Lichtenberg, J. Petter, T. Tschudi, “Control of the optical transfer function by phase-shift keying of a holographic Bragg grating,” Opt. Commun. 229, 131–139 (2004).
    [CrossRef]
  7. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  8. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, 1993).
  9. P. W. Atkins, Physical Chemistry (Oxford U. Press, 1998).
  10. J. Bartels, H. Borchers, H. Hausen, K. Hellwege, K. Schaefer, E. Schmidt, Landolt-Boernstein: Zahlenwerte aus Physik, Chemie, Astronomie, Geophysik und Technik (Springer-Verlag, 1962).
  11. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 2003).

2004 (1)

V. Petrov, S. Lichtenberg, J. Petter, T. Tschudi, “Control of the optical transfer function by phase-shift keying of a holographic Bragg grating,” Opt. Commun. 229, 131–139 (2004).
[CrossRef]

1997 (2)

V. M. Petrov, C. Denz, J. Petter, T. Tschudi, “Enhancing the sensitivity of an adaptive holographic interferometer using non-Bragg diffraction orders,” Opt. Lett. 22, 1902–1904 (1997).
[CrossRef]

G. Cedilnik, M. Esselbach, A. Kiessling, R. Kowarschik, “Real-time holographic interferometry with double two-wave mixing in photorefractive crystals,” Opt. Commun. 138, 2091–2100 (1997).

1993 (2)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

G. Boensch, E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlén’s formulae,” Meas. Sci. Technol. 4, 133–139 (1993).

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Atkins, P. W.

P. W. Atkins, Physical Chemistry (Oxford U. Press, 1998).

Bartels, J.

J. Bartels, H. Borchers, H. Hausen, K. Hellwege, K. Schaefer, E. Schmidt, Landolt-Boernstein: Zahlenwerte aus Physik, Chemie, Astronomie, Geophysik und Technik (Springer-Verlag, 1962).

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

Boensch, G.

G. Boensch, E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlén’s formulae,” Meas. Sci. Technol. 4, 133–139 (1993).

Borchers, H.

J. Bartels, H. Borchers, H. Hausen, K. Hellwege, K. Schaefer, E. Schmidt, Landolt-Boernstein: Zahlenwerte aus Physik, Chemie, Astronomie, Geophysik und Technik (Springer-Verlag, 1962).

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 2003).

Cedilnik, G.

G. Cedilnik, M. Esselbach, A. Kiessling, R. Kowarschik, “Real-time holographic interferometry with double two-wave mixing in photorefractive crystals,” Opt. Commun. 138, 2091–2100 (1997).

Culshaw, B.

B. Culshaw, D. Davies, S. Kingsley, “Fibre optic strain, pressure and temperature sensors,” in Proceedings of the 4th European Conference on Optical Communication (4th ECOC) (Institution of Electrical Engineers, 1978), Vol. 229, pp. 115–126.

Davies, D.

B. Culshaw, D. Davies, S. Kingsley, “Fibre optic strain, pressure and temperature sensors,” in Proceedings of the 4th European Conference on Optical Communication (4th ECOC) (Institution of Electrical Engineers, 1978), Vol. 229, pp. 115–126.

Denz, C.

Esselbach, M.

G. Cedilnik, M. Esselbach, A. Kiessling, R. Kowarschik, “Real-time holographic interferometry with double two-wave mixing in photorefractive crystals,” Opt. Commun. 138, 2091–2100 (1997).

Hausen, H.

J. Bartels, H. Borchers, H. Hausen, K. Hellwege, K. Schaefer, E. Schmidt, Landolt-Boernstein: Zahlenwerte aus Physik, Chemie, Astronomie, Geophysik und Technik (Springer-Verlag, 1962).

Hellwege, K.

J. Bartels, H. Borchers, H. Hausen, K. Hellwege, K. Schaefer, E. Schmidt, Landolt-Boernstein: Zahlenwerte aus Physik, Chemie, Astronomie, Geophysik und Technik (Springer-Verlag, 1962).

Kiessling, A.

G. Cedilnik, M. Esselbach, A. Kiessling, R. Kowarschik, “Real-time holographic interferometry with double two-wave mixing in photorefractive crystals,” Opt. Commun. 138, 2091–2100 (1997).

Kingsley, S.

B. Culshaw, D. Davies, S. Kingsley, “Fibre optic strain, pressure and temperature sensors,” in Proceedings of the 4th European Conference on Optical Communication (4th ECOC) (Institution of Electrical Engineers, 1978), Vol. 229, pp. 115–126.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kowarschik, R.

G. Cedilnik, M. Esselbach, A. Kiessling, R. Kowarschik, “Real-time holographic interferometry with double two-wave mixing in photorefractive crystals,” Opt. Commun. 138, 2091–2100 (1997).

Lichtenberg, S.

V. Petrov, S. Lichtenberg, J. Petter, T. Tschudi, “Control of the optical transfer function by phase-shift keying of a holographic Bragg grating,” Opt. Commun. 229, 131–139 (2004).
[CrossRef]

Petrov, V.

V. Petrov, S. Lichtenberg, J. Petter, T. Tschudi, “Control of the optical transfer function by phase-shift keying of a holographic Bragg grating,” Opt. Commun. 229, 131–139 (2004).
[CrossRef]

Petrov, V. M.

Petter, J.

V. Petrov, S. Lichtenberg, J. Petter, T. Tschudi, “Control of the optical transfer function by phase-shift keying of a holographic Bragg grating,” Opt. Commun. 229, 131–139 (2004).
[CrossRef]

V. M. Petrov, C. Denz, J. Petter, T. Tschudi, “Enhancing the sensitivity of an adaptive holographic interferometer using non-Bragg diffraction orders,” Opt. Lett. 22, 1902–1904 (1997).
[CrossRef]

Potulski, E.

G. Boensch, E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlén’s formulae,” Meas. Sci. Technol. 4, 133–139 (1993).

Schaefer, K.

J. Bartels, H. Borchers, H. Hausen, K. Hellwege, K. Schaefer, E. Schmidt, Landolt-Boernstein: Zahlenwerte aus Physik, Chemie, Astronomie, Geophysik und Technik (Springer-Verlag, 1962).

Schmidt, E.

J. Bartels, H. Borchers, H. Hausen, K. Hellwege, K. Schaefer, E. Schmidt, Landolt-Boernstein: Zahlenwerte aus Physik, Chemie, Astronomie, Geophysik und Technik (Springer-Verlag, 1962).

Tschudi, T.

V. Petrov, S. Lichtenberg, J. Petter, T. Tschudi, “Control of the optical transfer function by phase-shift keying of a holographic Bragg grating,” Opt. Commun. 229, 131–139 (2004).
[CrossRef]

V. M. Petrov, C. Denz, J. Petter, T. Tschudi, “Enhancing the sensitivity of an adaptive holographic interferometer using non-Bragg diffraction orders,” Opt. Lett. 22, 1902–1904 (1997).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 2003).

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, 1993).

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Meas. Sci. Technol. (2)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

G. Boensch, E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlén’s formulae,” Meas. Sci. Technol. 4, 133–139 (1993).

Opt. Commun. (2)

G. Cedilnik, M. Esselbach, A. Kiessling, R. Kowarschik, “Real-time holographic interferometry with double two-wave mixing in photorefractive crystals,” Opt. Commun. 138, 2091–2100 (1997).

V. Petrov, S. Lichtenberg, J. Petter, T. Tschudi, “Control of the optical transfer function by phase-shift keying of a holographic Bragg grating,” Opt. Commun. 229, 131–139 (2004).
[CrossRef]

Opt. Lett. (1)

Other (5)

B. Culshaw, D. Davies, S. Kingsley, “Fibre optic strain, pressure and temperature sensors,” in Proceedings of the 4th European Conference on Optical Communication (4th ECOC) (Institution of Electrical Engineers, 1978), Vol. 229, pp. 115–126.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, 1993).

P. W. Atkins, Physical Chemistry (Oxford U. Press, 1998).

J. Bartels, H. Borchers, H. Hausen, K. Hellwege, K. Schaefer, E. Schmidt, Landolt-Boernstein: Zahlenwerte aus Physik, Chemie, Astronomie, Geophysik und Technik (Springer-Verlag, 1962).

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 2003).

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Figures (7)

Fig. 1
Fig. 1

(a) Experimental setup: 1, Nd:YAG laser; 2, beam splitter; 3, mirror; 4, aperture; 5, cuvette; 6, BaTiO3 crystal; 6a and 6b, gratings; 7, IR laser; 8, photodiode; 9a and 9b, reference and signal beams; 9c recording beam. The labeled areas mark the two different interferometer types. Note that both interferometers merge at beam splitter 2b and the photorefractive crystal 6. (b) The recording process of the phase-shifted Bragg holograms. Depicted are the holographic recording beams, the phase shift in the wave fronts of the signal and reference beams, and the resulting Bragg gratings.

Fig. 2
Fig. 2

Phase-shifted grating consisting of two parts recorded by a signal beam (ϕ1) and a reference beam (ϕ2) (each of them overlapping with a recording beam). I1 and I0 are the reflected and the transmitted portion of the incoming beam Iin, respectively. (a) Illustration of the geometry and the resulting phase shift; (b) illustration of the resulting grating discontinuity due to the phase shift.

Fig. 3
Fig. 3

Calculated (solid curves) and measured transfer function for a (a) 60, (b) 147, and (c) 200 deg phase shift in the Bragg grating. The investigated gas was krypton. In addition to the given phase shift, an amplitude mismatch ratio of 4:1 between the two gratings was used for the theoretical calculation.

Fig. 4
Fig. 4

Calculated (solid curves) and measured transfer function for a (a) 50, (b) 100, and (c) 135 deg phase shift in the Bragg grating. The investigated gas was an atmospheric composition, i.e., approximately 78% N2, 21% O2. In addition to the given phase shift, an amplitude mismatch ratio of 2:1 between the two gratings was used for the theoretical calculation.

Fig. 5
Fig. 5

Refractive-index change δn versus pressure change δp (in millibars) for krypton gas. Crosses mark experimental data points; the solid line was calculated using the ideal gas equation. The marked data points a, b, and c correspond to the transfer functions depicted in Fig. 3.

Fig. 6
Fig. 6

Refractive-index change δn versus pressure change δp (in millibars) for regular atmospheric gas composition. Crosses mark experimental data points; the solid line was calculated using the ideal gas equation. The marked data points a, b, and c correspond to the transfer functions depicted in Fig. 4.

Fig. 7
Fig. 7

Difference between two transfer functions: (a) calculated transfer function without phase shift (solid curve) and with a phase shift of 10° (dotted curve), (b) calculated transfer function with a 180° (solid curve) and 190° (dotted curve) phase shift, respectively.

Equations (7)

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n ( z ) = { n 0 + Δ n 1 cos ( K z z + ϕ 1 ) 0 z L 1 n 0 + Δ n 2 cos ( K z z + ϕ 2 ) L 1 z L 2 ,
2 Λ = λ rec sin Θ = λ read n ( λ read ) ,
D ( λ ) = A out ( λ ) A in ( λ ) ,
η ( λ ) = I 1 I in = D 2 .
D c ( λ ) = k F { Δ n ( z ) exp [ i Φ ( z ) ] } .
Δ n = Δ ϕ 2 π λ l ,
Δ n = ( n 0 - 1 ) ( p / T ) ( T 0 / p 0 ) ,

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