Abstract

A novel signal processing technique using sinusoidal optical frequency modulation of an inexpensive continuous-wave laser diode source is proposed that allows highly linear interferometric phase measurements in a simple, self-referencing setup. Here, the use of a smooth window function is key to suppress unwanted signal components in the demodulation process. Signals from several interferometers with unequal optical path differences can be multiplexed, and, in contrast to prior work, the optical path differences are continuously variable, greatly increasing the practicality of the scheme. In this paper, the theory of the technique is presented, an experimental implementation using three multiplexed interferometers is demonstrated, and detailed investigations quantifying issues such as linearity and robustness against instrument drift are performed.

© 2015 Optical Society of America

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References

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  1. F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48(3 pt 1), 603–609 (1970).
    [Crossref]
  2. E. Voges, O. Ostwald, B. Schiek, and A. Neyer, “Optical phase and amplitude measurement by single sideband homodyne detection,” IEEE J. Quantum Electron. 18(1), 124–129 (1982).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  5. R. P. Tatam, J. D. C. Jones, and D. A. Jackson, “Opto-electronic processing schemes for the measurement of circular birefringence,” Optica Acta 33(12), 1519–1528 (1986).
    [Crossref]
  6. A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
    [Crossref]
  7. A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fibre gyroscope,” Electron. Lett. 20(9), 368–370 (1984).
    [Crossref]
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    [Crossref]
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2013 (3)

2012 (2)

2009 (1)

2007 (1)

2006 (1)

2004 (1)

2002 (1)

M. Bauer, F. Ritter, and G. Siegmund, “High-precision laser vibrometers based on digital Doppler signal processing,” Proc. SPIE 4827, 50–61 (2002).
[Crossref]

2001 (1)

1996 (1)

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

1987 (1)

I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5(7), 932–940 (1987).
[Crossref]

1986 (1)

R. P. Tatam, J. D. C. Jones, and D. A. Jackson, “Opto-electronic processing schemes for the measurement of circular birefringence,” Optica Acta 33(12), 1519–1528 (1986).
[Crossref]

1984 (1)

A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fibre gyroscope,” Electron. Lett. 20(9), 368–370 (1984).
[Crossref]

1982 (4)

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18(25), 1081–1083 (1982).
[Crossref]

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21(14), 2470 (1982).
[Crossref] [PubMed]

E. Voges, O. Ostwald, B. Schiek, and A. Neyer, “Optical phase and amplitude measurement by single sideband homodyne detection,” IEEE J. Quantum Electron. 18(1), 124–129 (1982).
[Crossref]

1970 (1)

F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48(3 pt 1), 603–609 (1970).
[Crossref]

Andrews, F. A.

F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48(3 pt 1), 603–609 (1970).
[Crossref]

Austin, E.

Bauer, M.

M. Bauer, F. Ritter, and G. Siegmund, “High-precision laser vibrometers based on digital Doppler signal processing,” Proc. SPIE 4827, 50–61 (2002).
[Crossref]

Baumann, E.

Charrett, T. O. H.

T. Kissinger, T. O. H. Charrett, and R. P. Tatam, “Fibre segment interferometry using code-division multiplexed optical signal processing for strain sensing applications,” Meas. Sci. Technol. 24, 094011 (2013).
[Crossref]

Chua, S.

Coddington, I.

Corke, M.

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18(25), 1081–1083 (1982).
[Crossref]

Cranch, G. A.

Dandridge, A.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

C. K. Kirkendall, A. D. Kersey, A. Dandridge, M. J. Marrone, and A. R. Davis, “Sensitivity limitations due to aliased high frequency phase noise in high channel-count TDM interferometric arrays,” in Optical Fiber Sensors (Optical Society of America, 1996), paper Fr1-4.
[Crossref]

Davis, A. R.

C. K. Kirkendall, A. D. Kersey, A. Dandridge, M. J. Marrone, and A. R. Davis, “Sensitivity limitations due to aliased high frequency phase noise in high channel-count TDM interferometric arrays,” in Optical Fiber Sensors (Optical Society of America, 1996), paper Fr1-4.
[Crossref]

De Vine, G.

Eberhardt, F. J.

F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48(3 pt 1), 603–609 (1970).
[Crossref]

Freund, C. H.

Gerberding, O.

Giallorenzi, T. G.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

Giorgetta, F. R.

Gray, M. B.

Hanson, R. K.

Heinze, G.

Herrmann, J.

Hsu, M. T. L.

Itoh, K.

Jackson, D. A.

R. P. Tatam, J. D. C. Jones, and D. A. Jackson, “Opto-electronic processing schemes for the measurement of circular birefringence,” Optica Acta 33(12), 1519–1528 (1986).
[Crossref]

A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fibre gyroscope,” Electron. Lett. 20(9), 368–370 (1984).
[Crossref]

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18(25), 1081–1083 (1982).
[Crossref]

Jeffries, J. B.

Jones, J. D. C.

R. P. Tatam, J. D. C. Jones, and D. A. Jackson, “Opto-electronic processing schemes for the measurement of circular birefringence,” Optica Acta 33(12), 1519–1528 (1986).
[Crossref]

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18(25), 1081–1083 (1982).
[Crossref]

Kersey, A. D.

A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fibre gyroscope,” Electron. Lett. 20(9), 368–370 (1984).
[Crossref]

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18(25), 1081–1083 (1982).
[Crossref]

C. K. Kirkendall, A. D. Kersey, A. Dandridge, M. J. Marrone, and A. R. Davis, “Sensitivity limitations due to aliased high frequency phase noise in high channel-count TDM interferometric arrays,” in Optical Fiber Sensors (Optical Society of America, 1996), paper Fr1-4.
[Crossref]

Kingsley, S. A.

Kirkendall, C. K.

C. K. Kirkendall, A. D. Kersey, A. Dandridge, M. J. Marrone, and A. R. Davis, “Sensitivity limitations due to aliased high frequency phase noise in high channel-count TDM interferometric arrays,” in Optical Fiber Sensors (Optical Society of America, 1996), paper Fr1-4.
[Crossref]

Kissinger, T.

T. Kissinger, T. O. H. Charrett, and R. P. Tatam, “Fibre segment interferometry using code-division multiplexed optical signal processing for strain sensing applications,” Meas. Sci. Technol. 24, 094011 (2013).
[Crossref]

Knabe, K.

Lam, T. T.

Lewin, A. C.

A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fibre gyroscope,” Electron. Lett. 20(9), 368–370 (1984).
[Crossref]

Li, H.

Liao, Y.

Liu, X.

Marrone, M. J.

C. K. Kirkendall, A. D. Kersey, A. Dandridge, M. J. Marrone, and A. R. Davis, “Sensitivity limitations due to aliased high frequency phase noise in high channel-count TDM interferometric arrays,” in Optical Fiber Sensors (Optical Society of America, 1996), paper Fr1-4.
[Crossref]

McClelland, D. E.

McRae, T. G.

Nash, P. J.

Newbury, N. R.

Neyer, A.

E. Voges, O. Ostwald, B. Schiek, and A. Neyer, “Optical phase and amplitude measurement by single sideband homodyne detection,” IEEE J. Quantum Electron. 18(1), 124–129 (1982).
[Crossref]

Ostwald, O.

E. Voges, O. Ostwald, B. Schiek, and A. Neyer, “Optical phase and amplitude measurement by single sideband homodyne detection,” IEEE J. Quantum Electron. 18(1), 124–129 (1982).
[Crossref]

Parry, G.

I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5(7), 932–940 (1987).
[Crossref]

Rabeling, D. S.

Richardson, D. J.

Rieker, G. B.

Ritter, F.

M. Bauer, F. Ritter, and G. Siegmund, “High-precision laser vibrometers based on digital Doppler signal processing,” Proc. SPIE 4827, 50–61 (2002).
[Crossref]

Sakai, I.

I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5(7), 932–940 (1987).
[Crossref]

Schiek, B.

E. Voges, O. Ostwald, B. Schiek, and A. Neyer, “Optical phase and amplitude measurement by single sideband homodyne detection,” IEEE J. Quantum Electron. 18(1), 124–129 (1982).
[Crossref]

Shaddock, D. A.

Siegmund, G.

M. Bauer, F. Ritter, and G. Siegmund, “High-precision laser vibrometers based on digital Doppler signal processing,” Proc. SPIE 4827, 50–61 (2002).
[Crossref]

Sinclair, L. C.

Slagmolen, B. J.

Su, C. S.

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

Sutton, A. J.

Swann, W. C.

Tatam, R. P.

T. Kissinger, T. O. H. Charrett, and R. P. Tatam, “Fibre segment interferometry using code-division multiplexed optical signal processing for strain sensing applications,” Meas. Sci. Technol. 24, 094011 (2013).
[Crossref]

R. P. Tatam, J. D. C. Jones, and D. A. Jackson, “Opto-electronic processing schemes for the measurement of circular birefringence,” Optica Acta 33(12), 1519–1528 (1986).
[Crossref]

Tveten, A. B.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

Voges, E.

E. Voges, O. Ostwald, B. Schiek, and A. Neyer, “Optical phase and amplitude measurement by single sideband homodyne detection,” IEEE J. Quantum Electron. 18(1), 124–129 (1982).
[Crossref]

Wu, C. M.

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

Wuchenich, D. M.

Youngquist, R.

I. Sakai, R. Youngquist, and G. Parry, “Multiplexing of optical fiber sensors using a frequency-modulated source and gated output,” J. Lightwave Technol. 5(7), 932–940 (1987).
[Crossref]

Zheng, J.

Appl. Opt. (3)

Electron. Lett. (2)

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18(25), 1081–1083 (1982).
[Crossref]

A. D. Kersey, A. C. Lewin, and D. A. Jackson, “Pseudo-heterodyne detection scheme for the fibre gyroscope,” Electron. Lett. 20(9), 368–370 (1984).
[Crossref]

IEEE J. Quantum Electron. (2)

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18(10), 1647–1653 (1982).
[Crossref]

E. Voges, O. Ostwald, B. Schiek, and A. Neyer, “Optical phase and amplitude measurement by single sideband homodyne detection,” IEEE J. Quantum Electron. 18(1), 124–129 (1982).
[Crossref]

J. Acoust. Soc. Am. (1)

F. J. Eberhardt and F. A. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoust. Soc. Am. 48(3 pt 1), 603–609 (1970).
[Crossref]

J. Lightwave Technol. (3)

Meas. Sci. Technol. (2)

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[Crossref]

T. Kissinger, T. O. H. Charrett, and R. P. Tatam, “Fibre segment interferometry using code-division multiplexed optical signal processing for strain sensing applications,” Meas. Sci. Technol. 24, 094011 (2013).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Optica Acta (1)

R. P. Tatam, J. D. C. Jones, and D. A. Jackson, “Opto-electronic processing schemes for the measurement of circular birefringence,” Optica Acta 33(12), 1519–1528 (1986).
[Crossref]

Proc. SPIE (1)

M. Bauer, F. Ritter, and G. Siegmund, “High-precision laser vibrometers based on digital Doppler signal processing,” Proc. SPIE 4827, 50–61 (2002).
[Crossref]

Other (1)

C. K. Kirkendall, A. D. Kersey, A. Dandridge, M. J. Marrone, and A. R. Davis, “Sensitivity limitations due to aliased high frequency phase noise in high channel-count TDM interferometric arrays,” in Optical Fiber Sensors (Optical Society of America, 1996), paper Fr1-4.
[Crossref]

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Figures (12)

Fig. 1
Fig. 1 Illustration of typical signal shapes occurring for the case of two constituent interferometers at A1 = 40 rad and A2 = 80 rad. (a) shows the applied sinusoidal optical frequency modulation, (b) plots the resultant photo detector signal U(t), (c) shows the complex carrier C(t) set for demodulation of the second constituent interferometer by letting Ad = A2 along with the window function W(t) for width parameter σ = 0.0225. Finally, (d) plots the resulting complex quadrature signal Q(t) = W(t)C(t)U(t).
Fig. 2
Fig. 2 Illustration of the effect of windowing on the baseband component of a single complex exponential term E(t, 40, 0). (a) plots the real and imaginary part of E(t, 40, 0) over one modulation period Tm as well as W(t) for σ = 0.0225 and, additionally, a rectangular window function Π(t) for comparison with prior work. (b) compares the corresponding amplitude spectra of E(t, 40, 0) with and without application of W(t) or Π(t).
Fig. 3
Fig. 3 The baseband crosstalk suppression that results from applying W(t) of Eq. (8) to a complex exponential term of Eq. (6) is plotted as a function of phase carrier amplitude A for a range of window width parameters σ and without windowing, i.e. σ → ∞
Fig. 4
Fig. 4 Illustration of the effect of the application of the window function W(t). Here the complex quadrature signal Q(t) for the example configuration previously used for Fig. 1 is plotted in the frequency domain, comparing the cases with (σ = 0.0225) and without (σ → ∞) windowing. Here, for the case with windowing, the spectrum separates into distinct peak regions and the inset reveals the resulting comb-like spectrum in the baseband.
Fig. 5
Fig. 5 The experimental setup used in this paper is a nested MZ interferometer with arms A, B and C, where the three constituent interferometers I1, I2 and I3 correspond to the interference between arms A&B, B&C and A&C, respectively. PZTs are integrated in arms A and C to induce suitable test signals. Both modulation and signal processing are controlled by an Field Programmable Gate Array (FPGA) via digital-to-analogue (DAC) and analogue-to-digital (ADC) converters, respectively. The FPGA performs time-critical demodulation steps and sends the data to a personal computer for final processing.
Fig. 6
Fig. 6 (a) plots the photo detector signal U(t), sampled by the ADC, over one modulation period. Analogous to Fig. 4, (b) shows the Fourier spectrum after demodulating at Ad = 103 rad with (σ = 0.0225) and without (σ → ∞) application of the window function W(t). Here the signal processing delay tsp has been compensated as described later in Sec. 3.3.
Fig. 7
Fig. 7 Determination of demodulation parameters using demodulation phase carrier amplitude Ad versus signal processing delay tsp maps with σ = 0.0225 for the cases without corrections (a) and with corrections (b) according to Sec. 3.2. The maps plot the normalized baseband signal amplitude as a function of Ad and tsp with a common colorbar shown on the right. The white, vertical line is the chosen evaluation location and the signal along this line is also plotted next to each map on a logarithmic scale.
Fig. 8
Fig. 8 The low-pass filtered, complex quadrature signal Q(t) is shown in (a) as a time series over 0.2s for I1 without any corrections discussed previously in Sec. 3.2, while (b) shows the polar plots of all three constituent interferometers in the complex plane.
Fig. 9
Fig. 9 Time traces of the phase signals over 0.2s in the main plots and over 5s in the insets are shown without any corrections according to Sec. 3.2. (a) plots the signals from the three constituent interferometers I1, I2 and I3, exhibiting excitations at 10Hz from PZT A (in I1 and I3) and at 180Hz from PZT C (in I2 and I3). (b) plots the offset signals (induced by PZT C) obtained directly (I2) or indirectly (I3 - I1) along with their difference, the residual signal. (c) shows the differential signal (induced by PZT A) obtained directly (I1) or indirectly (I3 - I2), again with the residual signal as their difference. (d) plots only the residual signal (I3 - I2 - I1), mathematically identical for both (b) and (c). Here a low-pass filtered signal (cut-off at 600Hz) is also plotted and the inset only shows the filtered signal.
Fig. 10
Fig. 10 The Fourier spectra (over 1 s) of the direct and indirect versions of the differential signal from PZT A as well as their residual signal are shown on a double logarithmic scale up to a bandwidth of 48kHz, without (a) and with (b) corrections according to Sec. 3.2.
Fig. 11
Fig. 11 Residual maps for the cases of (a) σ = 0.0075 and (b) σ = 0.0225, note the different colorbar scaling, are shown both with and without corrections according to Sec. 3.2. (c) plots the maximum residual absolute values over a wide range of width parameters σ.
Fig. 12
Fig. 12 The positions of the 9 parameter combinations (a,...,i) used are shown on the demodulation parameter map in (a), drawn analogously to Fig. 7(a). The resultant maximum residual absolute values are compared in (b), where uncorrected and corrected results are represented by the blue and green bars, respectively, with values in units of mrad also given.

Equations (9)

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ω ( t ) = ω 0 + Δ ω sin [ ω m t ]
U ( t ) = RP avg + n = 1 N RP n V n cos [ A n sin [ ω m ( t t sp 0.5 τ n ) ] + φ n ( t ) ]
A n = 2 Δ ω ω m sin [ ω m τ n 2 ] Δ ω τ n for α n 1 ω m
C ( t ) = exp [ j A d sin [ ω m ( t t sp 0.5 τ d ) ] ]
Q ( t ) = W ( t ) C ( t ) U ( t ) = W ( t ) RP avg exp [ j A d sin [ ω m ( t t sp 0.5 τ d ) ] ] + W ( t ) R n = 1 N 0.5 P n V n { exp [ j ( A d sin [ ω m ( t t sp 0.5 τ d ) ] A n sin [ ω m ( t t sp 0.5 τ n ) ] φ n ( t ) ) ] + exp [ j ( A d sin [ ω m ( t t sp + 0.5 τ d ) ] + A n sin [ ω m ( t t sp 0.5 τ n ) ] + φ n ( t ) ) ] }
E ( t , A , φ ( t ) ) = exp [ j ( A sin [ ω m ( t t sp ) ] + φ ( t ) ) ]
Q ( t ) W ( t ) RP avg E ( t , A d , 0 ) + W ( t ) R n = 1 N 0.5 P n V n { E ( t , ( A d A n ) , φ n ( t ) ) + E ( t , ( A d + A n ) , φ n ( t ) ) }
W ( t ) = n = { exp [ 1 2 ( ( t t sp ) n T m T m σ ) 2 ] + exp [ 1 2 ( ( t t sp ) ( n + 0.5 ) T m T m σ ) 2 ] }
OPD min c A min Δ ω

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