Abstract

Self-mixing laser sensors show promise for a wide range of sensing applications, including displacement, velocimetry, and fluid flow measurements. Several techniques have been developed to simulate self-mixing signals; however, a complete and succinct process for synthesizing self-mixing signals has so far been absent in the open literature. This article provides a systematic numerical approach for the analysis of self-mixing sensors using the steady-state solution to the Lang and Kobayashi model. Examples are given to show how this method can be used to synthesize self-mixing signals for arbitrary feedback levels and for displacement, distance, and velocity measurement. We examine these applications with a deterministic stimulus and discuss the velocity measurement of a rough surface, which necessitates the inclusion of a random stimulus.

© 2014 Optical Society of America

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2014 (1)

2013 (4)

U. Zabit, O. Bernal, and T. Bosch, “Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle,” IEEE Sensors J. 13, 824–831 (2013).
[CrossRef]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

A. Valavanis, P. Dean, Y. L. Lim, R. Alhathlool, M. Nikolić, R. Kliese, S. P. Khanna, D. Indjin, S. J. Wilson, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Self-mixing interferometry with terahertz quantum cascade lasers,” IEEE Sensors J. 13, 37–43 (2013).
[CrossRef]

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

2012 (3)

2011 (2)

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

S. Donati, “Responsivity and noise of self-mixing photodetection schemes,” IEEE J. Quantum Electron. 47, 1428–1433 (2011).
[CrossRef]

2010 (2)

U. Zabit, F. Bony, T. Bosch, and A. D. Rakić, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photon. Technol. Lett. 22, 410–412 (2010).
[CrossRef]

J. P. Toomey, D. M. Kane, M. W. Lee, and K. A. Shore, “Nonlinear dynamics of semiconductor lasers with feedback and modulation,” Opt. Express 18, 16955–16972 (2010).
[CrossRef]

2009 (4)

2007 (2)

D. Han, M. Wang, and J. Zhou, “Self-mixing speckle in an erbium-doped fiber ring laser and its application to velocity sensing,” IEEE Photon. Technol. Lett. 19, 1398–1400 (2007).
[CrossRef]

J. R. Tucker, A. D. Rakić, C. J. O’Brien, and A. V. Zvyagin, “Effect of multiple transverse modes in self-mixing sensors based on vertical-cavity surface-emitting lasers,” Appl. Opt. 46, 611–619 (2007).
[CrossRef]

2005 (2)

G. Plantier, C. Bes, and T. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005).
[CrossRef]

J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41, 1058–1064 (2005).
[CrossRef]

2004 (2)

K. Bertling, J. R. Tucker, and A. D. Rakić, “Optimum injection current waveform for a laser range finder based on the self-mixing effect,” Proc. SPIE 5277, 334–345 (2004).
[CrossRef]

X. Raoul, T. Bosch, G. Plantier, and N. Servagent, “A double-laser diode onboard sensor for velocity measurements,” IEEE Trans. Instrum. Meas. 53, 95–101 (2004).
[CrossRef]

2002 (1)

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A 4, S283–S294 (2002).
[CrossRef]

2001 (1)

T. Bosch, N. Servagent, and S. Donati, “Optical feedback interferometry for sensing application,” Opt. Eng. 40, 20–27 (2001).
[CrossRef]

2000 (1)

D. J. Young and N. C. Beaulieu, “The generation of correlated Rayleigh random variates by inverse discrete Fourier transform,” IEEE Trans. Commun. 48, 1114–1127 (2000).
[CrossRef]

1998 (2)

F. Gouaux, N. Servagent, and T. Bosch, “Absolute distance measurement with an optical feedback interferometer,” Appl. Opt. 37, 6684–6689 (1998).
[CrossRef]

S. Goeman, S. Boons, B. Dhoedt, K. Vandeputte, K. Caekebeke, P. Van Daele, and R. Baets, “First demonstration of highly reflective and highly polarization selective diffraction gratings (GIRO-gratings) for long-wavelength VCSELs,” IEEE Photon. Technol. Lett. 10, 1205–1207 (1998).
[CrossRef]

1997 (1)

P. Nerin, P. Puget, P. Besesty, and G. Chartier, “Self-mixing using a dual-polarisation Nd:YAG microchip laser,” Electron. Lett. 33, 491–492 (1997).
[CrossRef]

1995 (2)

K. Petermann, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 480–489 (1995).
[CrossRef]

U. Fiedler and K. J. Ebeling, “Design of VCSEL’s for feedback insensitive data transmission and external cavity active mode-locking,” IEEE J. Sel. Top. Quantum Electron. 1, 442–450 (1995).
[CrossRef]

1993 (1)

1992 (2)

1990 (1)

K. Y. R. Billah and M. Shinozuka, “Numerical method for colored-noise generation and its application to a bistable system,” Phys. Rev. A 42, 7492–7495 (1990).
[CrossRef]

1989 (1)

P. J. de Groot and G. M. Gallatin, “Three-dimensional imaging coherent laser radar array,” Opt. Eng. 28, 284456 (1989).
[CrossRef]

1988 (1)

1986 (2)

1985 (1)

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[CrossRef]

1984 (2)

J. H. Churnside, “Laser Doppler velocimetry by modulating a CO2 laser with backscattered light,” Appl. Opt. 23, 61–66 (1984).
[CrossRef]

G. Acket, D. Lenstra, A. den Boef, and B. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. QE-20, 1163–1169 (1984).
[CrossRef]

1981 (1)

Y. Mitsuhashi, J. Shimada, and S. Mitsutsuka, “Voltage change across the self-coupled semiconductor laser,” IEEE J. Quantum Electron. QE-17, 1216–1225 (1981).
[CrossRef]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

1978 (1)

S. Donati, “Laser interferometry by induced modulation of cavity field,” J. Appl. Phys. 49, 495–497 (1978).
[CrossRef]

1967 (1)

P. Welch, “The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967).
[CrossRef]

1963 (1)

P. G. R. King and G. J. Steward, “Metrology with an optical maser,” New Scientist 17, 180 (1963).

Aarnoudse, J.

Aarnoudse, J. G.

Acket, G.

G. Acket, D. Lenstra, A. den Boef, and B. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. QE-20, 1163–1169 (1984).
[CrossRef]

Alhathlool, R.

A. Valavanis, P. Dean, Y. L. Lim, R. Alhathlool, M. Nikolić, R. Kliese, S. P. Khanna, D. Indjin, S. J. Wilson, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Self-mixing interferometry with terahertz quantum cascade lasers,” IEEE Sensors J. 13, 37–43 (2013).
[CrossRef]

Baets, R.

S. Goeman, S. Boons, B. Dhoedt, K. Vandeputte, K. Caekebeke, P. Van Daele, and R. Baets, “First demonstration of highly reflective and highly polarization selective diffraction gratings (GIRO-gratings) for long-wavelength VCSELs,” IEEE Photon. Technol. Lett. 10, 1205–1207 (1998).
[CrossRef]

Beaulieu, N. C.

D. J. Young and N. C. Beaulieu, “The generation of correlated Rayleigh random variates by inverse discrete Fourier transform,” IEEE Trans. Commun. 48, 1114–1127 (2000).
[CrossRef]

Beheim, G.

Bernal, O.

U. Zabit, O. Bernal, and T. Bosch, “Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle,” IEEE Sensors J. 13, 824–831 (2013).
[CrossRef]

Bertling, K.

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

Y. L. Lim, M. Nikolić, K. Bertling, R. Kliese, and A. D. Rakić, “Self-mixing imaging sensor using a monolithic VCSEL array with parallel readout,” Opt. Express 17, 5517–5525 (2009).
[CrossRef]

K. Bertling, J. R. Tucker, and A. D. Rakić, “Optimum injection current waveform for a laser range finder based on the self-mixing effect,” Proc. SPIE 5277, 334–345 (2004).
[CrossRef]

Bes, C.

G. Plantier, C. Bes, and T. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005).
[CrossRef]

Besesty, P.

P. Nerin, P. Puget, P. Besesty, and G. Chartier, “Self-mixing using a dual-polarisation Nd:YAG microchip laser,” Electron. Lett. 33, 491–492 (1997).
[CrossRef]

Billah, K. Y. R.

K. Y. R. Billah and M. Shinozuka, “Numerical method for colored-noise generation and its application to a bistable system,” Phys. Rev. A 42, 7492–7495 (1990).
[CrossRef]

Bony, F.

R. Teysseyre, F. Bony, J. Perchoux, and T. Bosch, “Laser dynamics in sawtooth-like self-mixing signals,” Opt. Lett. 37, 3771–3773 (2012).
[CrossRef]

U. Zabit, F. Bony, T. Bosch, and A. D. Rakić, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photon. Technol. Lett. 22, 410–412 (2010).
[CrossRef]

Boons, S.

S. Goeman, S. Boons, B. Dhoedt, K. Vandeputte, K. Caekebeke, P. Van Daele, and R. Baets, “First demonstration of highly reflective and highly polarization selective diffraction gratings (GIRO-gratings) for long-wavelength VCSELs,” IEEE Photon. Technol. Lett. 10, 1205–1207 (1998).
[CrossRef]

Bosch, T.

U. Zabit, O. Bernal, and T. Bosch, “Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle,” IEEE Sensors J. 13, 824–831 (2013).
[CrossRef]

R. Teysseyre, F. Bony, J. Perchoux, and T. Bosch, “Laser dynamics in sawtooth-like self-mixing signals,” Opt. Lett. 37, 3771–3773 (2012).
[CrossRef]

U. Zabit, F. Bony, T. Bosch, and A. D. Rakić, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photon. Technol. Lett. 22, 410–412 (2010).
[CrossRef]

J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41, 1058–1064 (2005).
[CrossRef]

G. Plantier, C. Bes, and T. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005).
[CrossRef]

X. Raoul, T. Bosch, G. Plantier, and N. Servagent, “A double-laser diode onboard sensor for velocity measurements,” IEEE Trans. Instrum. Meas. 53, 95–101 (2004).
[CrossRef]

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A 4, S283–S294 (2002).
[CrossRef]

T. Bosch, N. Servagent, and S. Donati, “Optical feedback interferometry for sensing application,” Opt. Eng. 40, 20–27 (2001).
[CrossRef]

F. Gouaux, N. Servagent, and T. Bosch, “Absolute distance measurement with an optical feedback interferometer,” Appl. Opt. 37, 6684–6689 (1998).
[CrossRef]

J. Perchoux and T. Bosch, “Multimode VCSELs for self-mixing velocity measurements,” in IEEE Sensors (IEEE, 2007), pp. 419–422.

Botev, Z. I.

D. P. Kroese, T. Taimre, and Z. I. Botev, “Random process generation,” in Handbook of Monte Carlo Methods (Wiley, 2011), Chap. 5.

Boyle, W. J. O.

Caekebeke, K.

S. Goeman, S. Boons, B. Dhoedt, K. Vandeputte, K. Caekebeke, P. Van Daele, and R. Baets, “First demonstration of highly reflective and highly polarization selective diffraction gratings (GIRO-gratings) for long-wavelength VCSELs,” IEEE Photon. Technol. Lett. 10, 1205–1207 (1998).
[CrossRef]

Chartier, G.

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Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

U. Zabit, F. Bony, T. Bosch, and A. D. Rakić, “A self-mixing displacement sensor with fringe-loss compensation for harmonic vibrations,” IEEE Photon. Technol. Lett. 22, 410–412 (2010).
[CrossRef]

Y. L. Lim, M. Nikolić, K. Bertling, R. Kliese, and A. D. Rakić, “Self-mixing imaging sensor using a monolithic VCSEL array with parallel readout,” Opt. Express 17, 5517–5525 (2009).
[CrossRef]

J. R. Tucker, A. D. Rakić, C. J. O’Brien, and A. V. Zvyagin, “Effect of multiple transverse modes in self-mixing sensors based on vertical-cavity surface-emitting lasers,” Appl. Opt. 46, 611–619 (2007).
[CrossRef]

K. Bertling, J. R. Tucker, and A. D. Rakić, “Optimum injection current waveform for a laser range finder based on the self-mixing effect,” Proc. SPIE 5277, 334–345 (2004).
[CrossRef]

Raoul, X.

X. Raoul, T. Bosch, G. Plantier, and N. Servagent, “A double-laser diode onboard sensor for velocity measurements,” IEEE Trans. Instrum. Meas. 53, 95–101 (2004).
[CrossRef]

Rees, P.

P. Spencer, P. Rees, and I. Pierce, “Theoretical analysis,” in Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, D. M. Kane and K. A. Shore, eds. (Wiley, 2005), Chap. 2.

Servagent, N.

X. Raoul, T. Bosch, G. Plantier, and N. Servagent, “A double-laser diode onboard sensor for velocity measurements,” IEEE Trans. Instrum. Meas. 53, 95–101 (2004).
[CrossRef]

T. Bosch, N. Servagent, and S. Donati, “Optical feedback interferometry for sensing application,” Opt. Eng. 40, 20–27 (2001).
[CrossRef]

F. Gouaux, N. Servagent, and T. Bosch, “Absolute distance measurement with an optical feedback interferometer,” Appl. Opt. 37, 6684–6689 (1998).
[CrossRef]

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[CrossRef]

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J. Ohtsubo, H. Kumagai, and R. Shogenji, “Numerical study of doppler dynamics in self-mixing semiconductor lasers,” IEEE Photon. Technol. Lett. 21, 742–744 (2009).
[CrossRef]

Shore, K. A.

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P. Spencer, P. Rees, and I. Pierce, “Theoretical analysis,” in Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, D. M. Kane and K. A. Shore, eds. (Wiley, 2005), Chap. 2.

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T. Taimre and A. D. Rakić, “On the nature of Acket’s characteristic parameter C in semiconductor lasers,” Appl. Opt. 53, 1001–1006 (2014).
[CrossRef]

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
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A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
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J. R. Tucker, A. D. Rakić, C. J. O’Brien, and A. V. Zvyagin, “Effect of multiple transverse modes in self-mixing sensors based on vertical-cavity surface-emitting lasers,” Appl. Opt. 46, 611–619 (2007).
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[CrossRef]

Valavanis, A.

A. Valavanis, P. Dean, Y. L. Lim, R. Alhathlool, M. Nikolić, R. Kliese, S. P. Khanna, D. Indjin, S. J. Wilson, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Self-mixing interferometry with terahertz quantum cascade lasers,” IEEE Sensors J. 13, 37–43 (2013).
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A. Valavanis, P. Dean, Y. L. Lim, R. Alhathlool, M. Nikolić, R. Kliese, S. P. Khanna, D. Indjin, S. J. Wilson, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Self-mixing interferometry with terahertz quantum cascade lasers,” IEEE Sensors J. 13, 37–43 (2013).
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Figures (11)

Fig. 1.
Fig. 1.

Schematic of the three-mirror configuration. Solid lines and arrows show the beam directions, the reflections from the mirrors, and the transmissions through the partially transparent laser mirrors that are relevant to the model.

Fig. 2.
Fig. 2.

Plot of the r.h.s. of Eq. (6) for C=0.8. The bounds, ϕmin and ϕmax, are indicated and supplied to the root finding algorithm to determine the zero crossing (circled).

Fig. 3.
Fig. 3.

Plot of the r.h.s. of Eq. (6) for C=8. The solid and broken lines indicate the regions where stable and unstable solutions are found, respectively. Circles indicate the stable solutions, and crosses the unstable solutions. Here mlower and mupper indicate the regions of the lowest and highest values of m where a solution exists. The bounds, ϕmin and ϕmax, have been labeled for the lower region.

Fig. 4.
Fig. 4.

Surface plot of the r.h.s. of Eq. (6) for a range of values of ϕ and ϕ0 (the external phase) with λ0=800nm, C=8, and α=5. The locus of points where the function is zero is indicated by the plotted line and corresponds to the solutions of Eq. (6).

Fig. 5.
Fig. 5.

Plot (a) shows the sinusoidal external phase function with a period T, and plot (b) is the resulting values of ϕ for λ0=800nm, C=8, and α=5. The thin solid and dotted lines show the possible solutions to the phase equations; the thick solid and thick broken lines trace the locus of solutions in plot (b) to the external phase function in plot (a).

Fig. 6.
Fig. 6.

Plot (a) shows the target displacement motion used to generate the synthetic self-mixing signal in plots (b)–(e) using the provided MATLAB code with a laser wavelength of 850 nm and a target moving in harmonic motion with an amplitude of 2.5 μm at a frequency of 100 Hz. Plots (b)–(e) show the evolution of the self-mixing signals as the feedback parameter, C, increases through 0.5, 5, 20, and 60.

Fig. 7.
Fig. 7.

Plots showing the agreement between experimental self-mixing displacement signals and the corresponding fits to synthetic self-mixing signals generated using the algorithm described in this article. The target displacement is plotted in (a) and is relative to a nominal position of 100 mm from the laser. Plots (b) and (c) show the experimental signals (broken lines) and the corresponding synthetic self-mixing signals (solid lines). A moderate feedback signal was obtained in plot (b) (C=6.8) while a weak feedback signal was obtained in plot (c) (C=0.51) by introducing a neutral density filter into the beam path. The signal was acquired from the terminal voltage variations of an 850 nm wavelength VCSEL (Litrax LX-VCS-850-T101).

Fig. 8.
Fig. 8.

Plot (a) shows synthetic (solid line) and experimental (broken line) absolute distance self-mixing sensor signals. Plot (b) shows the result of numerically differentiating the synthetic signal in (a). The experimental signal was extracted from [6, Fig. 2(a)]. The synthetic signal target distance was 24 mm plus a small offset of 0.5 μm to align the synthetic signal fringe positions with the experimental fringe positions. The laser frequency sweep range is 46 GHz. The modulation period of the triangle waveform is 29.3 μs.

Fig. 9.
Fig. 9.

Plot (a) shows the synthetic absolute distance and velocity self-mixing sensor signal, which is numerically differentiated in plot (b) for a target with a velocity of 0.3mm/s at a distance of 24 mm and a laser frequency sweep over a range of 46 GHz. The modulation period of the triangle waveform is 29.3 μs.

Fig. 10.
Fig. 10.

Plots from various stages in the generation of a synthetic self-mixing velocimetry signal. The plots in (a) show the amplitude (solid line) and unwrapped phase (broken line) of a realization of a random process that models the beam scattered from a rough target. (The linear component of the unwrapped phase has been removed to make the phase variations visible.) Plot (b) is the self-mixing signal obtained prior to adding noise. Plot (c) is the signal after white noise has been added to model the laser intensity noise.

Fig. 11.
Fig. 11.

PSD of the self-mixing power variations obtained from the velocimetry simulation (solid line) along with an experimentally obtained signal for comparison (broken line).

Tables (7)

Tables Icon

Algorithm 1. Self-mixing power algorithm

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Algorithm 2. Algorithm for generating synthetic self-mixing signals for a given time-series of round-trip phase values

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Table 3 Listing 1. selmixpower.m

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Table 4 Listing 2. harmonic_motion.m

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Table 5 Listing 3. absolute_distance.m

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Table 6 Listing 4. absolute_distance+velocity.m

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Table 7 Listing 5. velocimetry.m

Equations (29)

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0=2πτext(νν0)+Csin(2πντext+arctanα),
C=τextτκext1+α2,
κext=εRextRs(1Rs),
ϕ=2πντext,
ϕ0=2πν0τext,
0=ϕϕ0+Csin(ϕ+arctanα).
Δgth=κextLcosϕ,
ΔP=βcos(ϕ),
0=ϕ[ϕϕ0+Csin(ϕ+arctanα)]=Ccos(ϕ+arctanα)+1.
ϕmin=(2m+1)π+arccos1Carctanα,
ϕmax=(2m+3)πarccos1Carctanα,
mlower=ϕ0+arctanα+arccos1C2πCsin[arccos(1C)]2π32.
mlower=ϕ0+arctanα+arccos1C2πC212π32.
mupper=ϕ0+arctanαarccos1C2π+C212π12.
ϕ0(tn)=4πν0c[L0+d(tn)],
d(tn)=Acos(2πftn).
ϕ0(tn)=4πλ0[L0+Acos(2πftn)],
Δν(tn)=ΔFTri(tn),
ν0(tn)=ν0+Δν(tn)=ν0+ΔFTri(tn).
ϕ0(tn)=4π[ν0+ΔFTri(tn)]Lc.
ϕ0(tn)=4πL[1λ0+ΔFTri(tn)c].
p(tn)=p(tn)+ρTri(tn).
L(tn)=L0+vtn.
ϕ0(tn)=4π(L0+vtn)[1λ0+ΔFTri(tn)c].
S(f)=exp[4loge2(ffD)2FWHM2],
p(tn)βSELMIXPOWER(C,ϕ0(tn),α)
p(tn)β|ψ(tn)|SELMIXPOWER(C0|ψ(tn)|,ϕ0(tn),α),
ϕ0(tn)=4πL0λ+arg[ψ(tn)],
ψ(tn)=IFFT[S(fn)exp(iϑ(fn))],

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