Abstract

Self-mixing laser sensors require few components and can be used to measure velocity. The self-mixing laser sensor consists of a laser emitting a beam focused onto a rough target that scatters the beam with some of the emission re-entering the laser cavity. This ‘self-mixing’ causes measurable interferometric modulation of the laser output power that leads to a periodic Doppler signal spectrum with a peak at a frequency proportional to the velocity of the target. Scattering of the laser emission from a rough surface also leads to a speckle effect that modulates the Doppler signal causing broadening of the signal spectrum adding uncertainty to the velocity measurement. This article analyzes the speckle effect to provide an analytic equation to predict the spectral broadening of an acquired self-mixing signal and compares the predicted broadening to experimental results. To the best of our knowledge, the model proposed in this article is the first model that has successfully predicted speckle broadening in a self-mixing velocimetry sensor in a quantitative manner. It was found that the beam spot size on the target and the target speed affect the resulting spectral broadening caused by speckle. It was also found that the broadening is only weakly dependent on target angle. The experimental broadening was consistently greater than the theoretical speckle broadening due to other effects that also contribute to the total broadening.

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References

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  1. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron.QE-16, 347–355 (1980).
  2. Y. Mitsuhashi, J. Shimada, and S. Mitsutsuka, “Voltage change across the self-coupled semiconductor laser.” IEEE J. Quantum Electron.QE-17, 1216–1225 (1981).
  3. J. H. Churnside, “Laser Doppler velocimetry by modulating a CO2 laser with backscattered light,” Appl. Opt.23, 61–66 (1984).
    [PubMed]
  4. S. Shinohara, A. Mochizuki, H. Yoshida, and M. Sumi, “Laser Doppler velocimeter using the self-mixing effect of a semiconductor laser diode,” Appl. Opt.25, 1417–1419 (1986).
    [PubMed]
  5. M. Rudd, “A laser Doppler velocimeter employing the laser as a mixer-oscillator,” J. Phys. E1, 723–726 (1968).
  6. G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A Pure Appl. Opt.4, 283–294 (2002).
  7. L. E. Estes, L. M. Narducci, and R. A. Tuft, “Scattering of light from a rotating ground glass,” J. Opt. Soc. Am.61, 1301–1306 (1971).
  8. N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys.13, 2025–2032 (1974).
  9. E. Jakeman, “The effect of wavefront curvature on the coherence properties of laser light scattered by target centres in uniform motion,” J. Phys. A8, L23–L28 (1975).
  10. B. E. A. Saleh, “Speckle correlation measurement of the velocity of a small rotating rough object,” Appl. Opt.14, 2344–2346 (1975).
    [PubMed]
  11. P. N. Pusey, “Photon correlation study of laser speckle produced by a moving rough surface,” J. Phys. D9, 1399–1409 (1976).
  12. H. Jentink, F. de Mul, H. Suichies, J. Aarnoudse, and J. Greve, “Small laser Doppler velocimeter based on the self-mixing effect in a diode laser,” Appl. Opt.27, 379–385 (1988).
    [PubMed]
  13. Şahin Kaya Özdemir, T. Takasu, S. Shinohara, H. Yoshida, and M. Sumi, “Simultaneous measurement of velocity and length of moving surfaces by a speckle velocimeter with two self-mixing laser diodes,” Appl. Opt.38, 1968–1974 (1999).
  14. X. Raoul, T. Bosch, G. Plantier, and N. Servagent, “A double-laser diode onboard sensor for velocity measurements,” IEEE Trans. Instr. Meas.53, 95–101 (2004).
  15. R.-H. Hage, T. Bosch, G. Plantier, and A. Sourice, “Modeling and analysis of speckle effects for velocity measurements with self-mixing laser diode sensors,” in Sensors, 2008 IEEE (IEEE, 2008), 953–956.
  16. D. Han, S. Chen, and L. Ma, “Autocorrelation of self-mixing speckle in an EDFR laser and velocity measurement,” Appl. Phys. B103, 695–700 (2011).
  17. H. Wang, J. Shen, B. Wang, B. Yu, and Y. Xu, “Laser diode feedback interferometry in flowing Brownian motion system: a novel theory,” Appl. Phys. B101, 173–183 (2010).
  18. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1975), chap. 2.
  19. J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).
  20. G. Giuliani and M. Norgia, “Laser diode linewidth measurement by means of self-mixing interferometry,” IEEE Photon. Technol. Lett.12, 1028–1030 (2000).
  21. A. Papoulis and S. U. Pillai, “Stochastic processes: General concepts,” in Probability, Random Variables, and Stochastic Processes (McGraw Hill, 2002), chap. 9, 4th ed.
  22. J. W. Goodman, “Fresnel and Fraunhofer diffraction,” in Introduction to Fourier Optics (Roberts & Company, 2005), chap. 4, 3rd ed.
  23. J. W. Goodman, “Effects of partial coherence on imaging systems,” in Statistical Optics (John Wiley & Sons, 1985), chap. 7.
  24. R. Juskaitis, N. Rea, and T. Wilson, “Semiconductor laser confocal microscopy,” Appl. Opt.33, 578–584 (1994).
    [PubMed]
  25. H. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer Verlag, 2003).
  26. R. S. Matharu, J. Perchoux, R. Kliese, Y. L. Lim, and A. D. Rakić, “Maintaining maximum signal-to-noise ratio in uncooled vertical-cavity surface-emitting laser-based self-mixing sensors,” Opt. Lett.36, 3690–3692 (2011).
    [PubMed]
  27. J. Scott, R. Geels, S. Corzine, and L. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron.29, 1295–1308 (1993).
  28. J. W. Goodman, “Random processes,” in Statistical Optics (John Wiley & Sons, 1985), chap. 3.
  29. D. Middleton, “Spectra, covariance, and correlation functions,” in An Introduction to Statistical Communication Theory (IEEE Press, 1996), chap. 3. Reprint.
  30. D. Middleton, “Statistical preliminaries,” in An Introduction to Statistical Communication Theory (IEEE Press, 1996), chap. 1. Reprint.
  31. Y. Suzaki and A. Tachibana, “Measurement of the μm sized radius of Gaussian laser beam using the scanning knife-edge,” Appl. Opt.14, 2809–2810 (1975).
    [PubMed]
  32. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt.22, 3406–3410 (1983).
    [PubMed]
  33. R. N. Bracewell, “The basic theorems,” in The Fourier Transform and Its Applications (McGraw Hill, 2000), chap. 6.
  34. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), 3rd ed.
  35. B. E. A. Saleh and M. C. Teich, “Beam optics,” in Fundamentals of Photonics (Wiley, 2007), chap. 3, 2nd ed.

2011 (2)

2010 (1)

H. Wang, J. Shen, B. Wang, B. Yu, and Y. Xu, “Laser diode feedback interferometry in flowing Brownian motion system: a novel theory,” Appl. Phys. B101, 173–183 (2010).

2004 (1)

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

2002 (1)

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A Pure Appl. Opt.4, 283–294 (2002).

2000 (1)

G. Giuliani and M. Norgia, “Laser diode linewidth measurement by means of self-mixing interferometry,” IEEE Photon. Technol. Lett.12, 1028–1030 (2000).

1999 (1)

1994 (1)

1993 (1)

J. Scott, R. Geels, S. Corzine, and L. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron.29, 1295–1308 (1993).

1988 (1)

1986 (1)

1984 (1)

1983 (1)

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).

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).

1976 (1)

P. N. Pusey, “Photon correlation study of laser speckle produced by a moving rough surface,” J. Phys. D9, 1399–1409 (1976).

1975 (3)

1974 (1)

N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys.13, 2025–2032 (1974).

1971 (1)

1968 (1)

M. Rudd, “A laser Doppler velocimeter employing the laser as a mixer-oscillator,” J. Phys. E1, 723–726 (1968).

Aarnoudse, J.

Albrecht, H.

H. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer Verlag, 2003).

Borys, M.

H. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer Verlag, 2003).

Bosch, T.

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

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A Pure Appl. Opt.4, 283–294 (2002).

R.-H. Hage, T. Bosch, G. Plantier, and A. Sourice, “Modeling and analysis of speckle effects for velocity measurements with self-mixing laser diode sensors,” in Sensors, 2008 IEEE (IEEE, 2008), 953–956.

Bracewell, R. N.

R. N. Bracewell, “The basic theorems,” in The Fourier Transform and Its Applications (McGraw Hill, 2000), chap. 6.

Chen, S.

D. Han, S. Chen, and L. Ma, “Autocorrelation of self-mixing speckle in an EDFR laser and velocity measurement,” Appl. Phys. B103, 695–700 (2011).

Churnside, J. H.

Coldren, L.

J. Scott, R. Geels, S. Corzine, and L. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron.29, 1295–1308 (1993).

Corzine, S.

J. Scott, R. Geels, S. Corzine, and L. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron.29, 1295–1308 (1993).

Damaschke, N.

H. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer Verlag, 2003).

de Mul, F.

Donati, S.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A Pure Appl. Opt.4, 283–294 (2002).

Estes, L. E.

Garetz, B. A.

Geels, R.

J. Scott, R. Geels, S. Corzine, and L. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron.29, 1295–1308 (1993).

Giuliani, G.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A Pure Appl. Opt.4, 283–294 (2002).

G. Giuliani and M. Norgia, “Laser diode linewidth measurement by means of self-mixing interferometry,” IEEE Photon. Technol. Lett.12, 1028–1030 (2000).

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1975), chap. 2.

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

J. W. Goodman, “Fresnel and Fraunhofer diffraction,” in Introduction to Fourier Optics (Roberts & Company, 2005), chap. 4, 3rd ed.

J. W. Goodman, “Effects of partial coherence on imaging systems,” in Statistical Optics (John Wiley & Sons, 1985), chap. 7.

J. W. Goodman, “Random processes,” in Statistical Optics (John Wiley & Sons, 1985), chap. 3.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), 3rd ed.

Greve, J.

Hage, R.-H.

R.-H. Hage, T. Bosch, G. Plantier, and A. Sourice, “Modeling and analysis of speckle effects for velocity measurements with self-mixing laser diode sensors,” in Sensors, 2008 IEEE (IEEE, 2008), 953–956.

Han, D.

D. Han, S. Chen, and L. Ma, “Autocorrelation of self-mixing speckle in an EDFR laser and velocity measurement,” Appl. Phys. B103, 695–700 (2011).

Jakeman, E.

E. Jakeman, “The effect of wavefront curvature on the coherence properties of laser light scattered by target centres in uniform motion,” J. Phys. A8, L23–L28 (1975).

Jentink, H.

Juskaitis, R.

Khosrofian, J. M.

Kliese, R.

Kobayashi, K.

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

Lang, R.

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

Lim, Y. L.

Ma, L.

D. Han, S. Chen, and L. Ma, “Autocorrelation of self-mixing speckle in an EDFR laser and velocity measurement,” Appl. Phys. B103, 695–700 (2011).

Matharu, R. S.

Middleton, D.

D. Middleton, “Spectra, covariance, and correlation functions,” in An Introduction to Statistical Communication Theory (IEEE Press, 1996), chap. 3. Reprint.

D. Middleton, “Statistical preliminaries,” in An Introduction to Statistical Communication Theory (IEEE Press, 1996), chap. 1. Reprint.

Mitsuhashi, Y.

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

Mitsutsuka, S.

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

Mochizuki, A.

Narducci, L. M.

Norgia, M.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A Pure Appl. Opt.4, 283–294 (2002).

G. Giuliani and M. Norgia, “Laser diode linewidth measurement by means of self-mixing interferometry,” IEEE Photon. Technol. Lett.12, 1028–1030 (2000).

Özdemir, Sahin Kaya

Papoulis, A.

A. Papoulis and S. U. Pillai, “Stochastic processes: General concepts,” in Probability, Random Variables, and Stochastic Processes (McGraw Hill, 2002), chap. 9, 4th ed.

Perchoux, J.

Pillai, S. U.

A. Papoulis and S. U. Pillai, “Stochastic processes: General concepts,” in Probability, Random Variables, and Stochastic Processes (McGraw Hill, 2002), chap. 9, 4th ed.

Plantier, G.

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

R.-H. Hage, T. Bosch, G. Plantier, and A. Sourice, “Modeling and analysis of speckle effects for velocity measurements with self-mixing laser diode sensors,” in Sensors, 2008 IEEE (IEEE, 2008), 953–956.

Pusey, P. N.

P. N. Pusey, “Photon correlation study of laser speckle produced by a moving rough surface,” J. Phys. D9, 1399–1409 (1976).

Rakic, A. D.

Raoul, X.

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

Rea, N.

Rudd, M.

M. Rudd, “A laser Doppler velocimeter employing the laser as a mixer-oscillator,” J. Phys. E1, 723–726 (1968).

Saleh, B. E. A.

B. E. A. Saleh, “Speckle correlation measurement of the velocity of a small rotating rough object,” Appl. Opt.14, 2344–2346 (1975).
[PubMed]

B. E. A. Saleh and M. C. Teich, “Beam optics,” in Fundamentals of Photonics (Wiley, 2007), chap. 3, 2nd ed.

Scott, J.

J. Scott, R. Geels, S. Corzine, and L. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron.29, 1295–1308 (1993).

Servagent, N.

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

Shen, J.

H. Wang, J. Shen, B. Wang, B. Yu, and Y. Xu, “Laser diode feedback interferometry in flowing Brownian motion system: a novel theory,” Appl. Phys. B101, 173–183 (2010).

Shimada, J.

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

Shinohara, S.

Sourice, A.

R.-H. Hage, T. Bosch, G. Plantier, and A. Sourice, “Modeling and analysis of speckle effects for velocity measurements with self-mixing laser diode sensors,” in Sensors, 2008 IEEE (IEEE, 2008), 953–956.

Suichies, H.

Sumi, M.

Suzaki, Y.

Tachibana, A.

Takai, N.

N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys.13, 2025–2032 (1974).

Takasu, T.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, “Beam optics,” in Fundamentals of Photonics (Wiley, 2007), chap. 3, 2nd ed.

Tropea, C.

H. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer Verlag, 2003).

Tuft, R. A.

Wang, B.

H. Wang, J. Shen, B. Wang, B. Yu, and Y. Xu, “Laser diode feedback interferometry in flowing Brownian motion system: a novel theory,” Appl. Phys. B101, 173–183 (2010).

Wang, H.

H. Wang, J. Shen, B. Wang, B. Yu, and Y. Xu, “Laser diode feedback interferometry in flowing Brownian motion system: a novel theory,” Appl. Phys. B101, 173–183 (2010).

Wilson, T.

Xu, Y.

H. Wang, J. Shen, B. Wang, B. Yu, and Y. Xu, “Laser diode feedback interferometry in flowing Brownian motion system: a novel theory,” Appl. Phys. B101, 173–183 (2010).

Yoshida, H.

Yu, B.

H. Wang, J. Shen, B. Wang, B. Yu, and Y. Xu, “Laser diode feedback interferometry in flowing Brownian motion system: a novel theory,” Appl. Phys. B101, 173–183 (2010).

Appl. Opt. (8)

Appl. Phys. B (2)

D. Han, S. Chen, and L. Ma, “Autocorrelation of self-mixing speckle in an EDFR laser and velocity measurement,” Appl. Phys. B103, 695–700 (2011).

H. Wang, J. Shen, B. Wang, B. Yu, and Y. Xu, “Laser diode feedback interferometry in flowing Brownian motion system: a novel theory,” Appl. Phys. B101, 173–183 (2010).

IEEE J. Quantum Electron. (3)

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

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

J. Scott, R. Geels, S. Corzine, and L. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron.29, 1295–1308 (1993).

IEEE Photon. Technol. Lett. (1)

G. Giuliani and M. Norgia, “Laser diode linewidth measurement by means of self-mixing interferometry,” IEEE Photon. Technol. Lett.12, 1028–1030 (2000).

IEEE Trans. Instr. Meas. (1)

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

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

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A Pure Appl. Opt.4, 283–294 (2002).

J. Opt. Soc. Am. (1)

J. Phys. A (1)

E. Jakeman, “The effect of wavefront curvature on the coherence properties of laser light scattered by target centres in uniform motion,” J. Phys. A8, L23–L28 (1975).

J. Phys. D (1)

P. N. Pusey, “Photon correlation study of laser speckle produced by a moving rough surface,” J. Phys. D9, 1399–1409 (1976).

J. Phys. E (1)

M. Rudd, “A laser Doppler velocimeter employing the laser as a mixer-oscillator,” J. Phys. E1, 723–726 (1968).

Jpn. J. Appl. Phys. (1)

N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys.13, 2025–2032 (1974).

Opt. Lett. (1)

Other (13)

H. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer Verlag, 2003).

A. Papoulis and S. U. Pillai, “Stochastic processes: General concepts,” in Probability, Random Variables, and Stochastic Processes (McGraw Hill, 2002), chap. 9, 4th ed.

J. W. Goodman, “Fresnel and Fraunhofer diffraction,” in Introduction to Fourier Optics (Roberts & Company, 2005), chap. 4, 3rd ed.

J. W. Goodman, “Effects of partial coherence on imaging systems,” in Statistical Optics (John Wiley & Sons, 1985), chap. 7.

J. W. Goodman, “Random processes,” in Statistical Optics (John Wiley & Sons, 1985), chap. 3.

D. Middleton, “Spectra, covariance, and correlation functions,” in An Introduction to Statistical Communication Theory (IEEE Press, 1996), chap. 3. Reprint.

D. Middleton, “Statistical preliminaries,” in An Introduction to Statistical Communication Theory (IEEE Press, 1996), chap. 1. Reprint.

R. N. Bracewell, “The basic theorems,” in The Fourier Transform and Its Applications (McGraw Hill, 2000), chap. 6.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), 3rd ed.

B. E. A. Saleh and M. C. Teich, “Beam optics,” in Fundamentals of Photonics (Wiley, 2007), chap. 3, 2nd ed.

R.-H. Hage, T. Bosch, G. Plantier, and A. Sourice, “Modeling and analysis of speckle effects for velocity measurements with self-mixing laser diode sensors,” in Sensors, 2008 IEEE (IEEE, 2008), 953–956.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1975), chap. 2.

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

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Figures (8)

Fig. 1
Fig. 1

Geometry used for the derivation of the dynamic speckle statistics at point P. A field which has the x, y coordinate system is incident on the rough surface moving with velocity v which is inclined at an angle θ from the x axis. This field is scattered by the rough surface and the resulting time autocorrelation function of the field at P with coordinates ξ, η is derived.

Fig. 2
Fig. 2

A plot of the normalised power spectral density with the FWHM and peak frequency indicated.

Fig. 3
Fig. 3

Light-current plots for the Firecomms laser (a) and the Litrax laser (b) used in the experiments showing threshold currents measured at 25°C. The power rollover is typical of VCSELs [27]. The insets show the emission spectra for the Firecomms and Litrax lasers at 1.00 mA and 4.08 mA respectively; both lase with a single mode.

Fig. 4
Fig. 4

Plots of the laser far-field profiles obtained from the CCD (circles) and the Gaussian fits (solid lines) used to infer the laser beam waist radii for the Firecomms laser (a) with an extracted beam waist diameter of 2.22 μm and the Litrax laser (b) with an extracted beam waist diameter of 2.48 μm.

Fig. 5
Fig. 5

Schematic of the experimental setup used to obtain Doppler velocimetry signals along with a typical averaged spectrum. The setup is shown looking from above with the laser beam waist imaged onto the disc disc below the axis or rotation such that there is a component of velocity in the direction of the optical axis. The attenuator is used to keep the sensor operating in the weak feedback regime. The laser is driven by a constant current source. The laser terminal voltage variations are amplified, then sampled to produce a spectrum via the FFT.

Fig. 6
Fig. 6

Doppler peak broadening for a range of target speeds measured using a 2.5× imaging lens configuration (a), and for a range of spot sizes on the target with at a fixed target velocity of 111 mm/s (b). The circles and triangles show the experimental broadening measurements acquired from the Firecomms and Litrax lasers respectively. The solid and broken lines indicate the theoretical broadening for the Firecomms and Litrax lasers respectively. Plots (c) and (d) show the differences between the experimental broadening (which includes all broadening effects) and the theoretical values that only include speckle broadening.

Fig. 7
Fig. 7

Doppler peak frequency (a) and FWHM broadening (b) with a fixed target speed of 111 mm/s over a range of target angles. The circles and triangles show the experimental broadening measurements acquired from the Firecomms and Litrax lasers respectively. The solid and broken lines indicate the theoretical broadening for the Firecomms and Litrax laser’s respectively.

Fig. 8
Fig. 8

(a) Near-field image of Firecomms laser below threshold (0.21 mA) suggestive of non-circular carrier injection. The disc where emission is present is defined by the metalization on the front facet of the laser. Non-uniform proton implantation may be responsible for the irregular emission shape. (b) Diagram of the setup used to infer the laser near-field spot size from a far-field image. (c) A sample far-field image obtained from the CCD array. The speckle caused by the rough surface of the screen is evident in the CCD image.

Tables (2)

Tables Icon

Table 1 Parameters of the two lasers used for experimental validation of the dynamic speckle theory.

Tables Icon

Table 2 The focal lengths and resulting magnifications for lens combinations used to obtain a range of spot sizes on the target.

Equations (29)

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

R A ( τ ) = exp ( i 4 π v τ sin θ λ ) λ 2 z 2 U ( x + v τ cos θ , y ) U * ( x . y ) × exp { i π [ ( x + v τ cos θ ) 2 x 2 ] λ z } d x d y .
U G ( x , y ) = 2 π w t 2 exp ( x 2 + y 2 w t 2 ) ,
S A ( f ) = exp [ 2 π 2 w t 2 ( f 2 v sin θ λ ) 2 v 2 ] .
FWHM = v cos θ 2 log e 2 π w t .
f D = 2 v sin θ λ ,
w t = α w 0 .
R A ( t 1 , t 2 ) = E [ A ( t 1 ) A * ( t 2 ) ] ,
R A ( τ ) = E [ A ( t τ ) A * ( t ) ] .
V ( x , y ) = exp ( i 4 π x tan θ λ ) .
α ( x , y ) = U ( x , y ) V ( x , y ) Ψ ( x , y ) ,
A ( ξ , η ) = exp ( i 2 π z λ ) i λ z exp [ i π λ z ( ξ 2 + η 2 ) ] × α ( x , y ) exp [ i π λ z ( x 2 + y 2 ) ] exp [ i π λ z ( ξ x + η y ) ] d x d y .
A ( t ) = 1 λ z α ( x , y , t ) exp [ i π λ z ( x 2 + y 2 ) ] d x d y .
α ( x , y , t ) = Ψ ( x + v t cos θ , y ) U ( x , y ) V ( x , y ) .
R A ( t 1 , t 2 ) = E [ 1 λ z Ψ ( x + v t 1 cos θ , y ) U ( x , y ) V ( x , y ) exp [ i π ( x 2 + y 2 ) λ z ] d x d y × conj { 1 λ z Ψ ( x + v t 2 cos θ , y ) U ( x , y ) V ( x , y ) exp [ i π ( x 2 + y 2 ) λ z ] d x d y } ]
R A ( t 1 , t 2 ) = 1 λ 2 z 2 E [ Ψ ( x 1 + v t 1 cos θ , y 1 ) Ψ * ( x 2 + v t 2 cos θ , y 2 ) ] × U ( x 1 , y 1 ) V ( x 1 , y 1 ) U * ( x 2 , y 2 ) V * ( x 2 , y 2 ) exp [ i π ( x 1 2 + y 1 2 x 2 2 y 2 2 ) λ z ] d x 1 d y 1 d x 2 d y 2 .
E [ Ψ ( x 1 + v t 1 cos θ , y 1 ) Ψ * ( x 2 + v t 2 cos θ , y 2 ) ] = δ [ x 2 + v t 2 cos θ x 1 v t 1 cos θ , y 2 y 1 ] ,
δ ( x 2 x 1 + v τ cos θ , y 2 y 1 ) .
R A ( τ ) = 1 λ 2 z 2 U ( x 2 + v τ cos θ , v 2 ) V ( x 2 + v τ cos θ , y 1 ) U * ( x 2 , y 2 ) V * ( x 2 , y 2 ) exp { i π [ ( x 2 + v τ cos θ ) 2 x 2 2 ] λ z } d x 2 d y 2 .
V ( x 2 + v τ cos θ , y 1 ) V * ( x 2 , y 2 ) = exp [ i 4 π ( x 2 + v τ cos θ ) tan θ λ ] exp [ i 4 π x 2 tan θ λ ] = exp ( i 4 π v τ cos θ tan θ λ ) = exp ( i 4 π v τ sin θ λ ) .
R A ( τ ) = exp ( i 4 π v τ sin θ λ ) λ 2 z 2 U ( x 2 + v τ cos θ , y 2 ) U * ( x 2 , y 2 ) exp { i π [ ( x 2 + v τ cos θ ) 2 x 2 2 ] λ z } d x 2 d y 2 .
R A ( τ ) = exp ( i 4 π v τ sin θ λ ) ( λ z ) 2 4 ( π w t 2 ) 2 exp [ ( x + v τ cos θ ) 2 + y 2 + x 2 + y 2 w t 2 ] × exp { i π [ ( x + v τ cos θ ) 2 x 2 ] λ z } d x d y = 4 exp ( i 4 π v τ sin θ λ ) ( λ z π w t 2 ) 2 exp ( 2 y 2 w t 2 ) × exp { i π λ z [ ( x + v τ ) 2 x 2 ] 1 w t 2 [ ( x + v τ ) 2 + x 2 ] } d x d y = 4 exp ( i 4 π v τ sin θ λ ) ( λ z π w t 2 ) 2 exp ( 2 y 2 w t 2 ) π w t 2 exp [ v 2 τ 2 2 ( π 2 w t 2 λ 2 z 2 + 1 w t 2 ) ] d y = 4 ( λ z π w t 2 ) 2 π w t π w t 2 exp [ v 2 τ 2 2 ( π 2 w t 2 λ 2 z 2 + 1 w t 2 ) + i 4 π v τ sin θ λ ] = 2 2 π ( λ z w t ) 2 exp [ v 2 τ 2 2 ( π 2 w t 2 λ 2 z 2 + 1 w t 2 ) + i 4 π v τ sin θ λ ] .
π 2 w t 4 λ 2 z 2 1 .
R A ( τ ) = exp ( v 2 τ 2 2 w t 2 + i 4 π v τ sin θ λ ) .
S A ( f ) = { R A ( τ ) } / max ( { R A ( τ ) } ) = exp [ 2 π 2 w t 2 ( f 2 v sin θ λ ) 2 v 2 ] .
f ( u ) f * ( u + x ) d u = | { f } | 2 ,
I ( x , y ) exp [ 2 ( x 2 + y 2 ) w 2 ( z ) ] ,
w ( z ) = w 0 1 + ( z z R ) 2 .
I ( x ) exp [ 2 x 2 w 2 ( z ) ] .
I ( θ ) exp [ 2 ( π θ w 0 λ ) 2 . ]

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