Abstract

Quasi-static interferometric signals in lasers under feedback arise from slowly varying perturbations of the intracavity electric field resulting from the reinjection of a portion of the emitted field into the cavity. Such interferometric signals are well described by the steady-state solution to the Lang–Kobayashi rate equation model. We give an exact series expansion for this steady-state solution that shows precisely how Acket’s characteristic parameter C and Henry’s linewidth enhancement factor α influence such signals. We show how the series coefficients can be extracted easily and explain how to determine C and α directly from them. Moreover, we draw a precise analogy between self-mixing and FM signals, showing that C plays exactly the same role in self-mixing as the modulation index does in FM.

© 2014 Optical Society of America

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References

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  1. G. Plantier, C. Bès, and T. M. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005).
    [CrossRef]
  2. J. Xi, Y. Yu, J. F. Chicharo, and T. M. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41, 1058–1064 (2005).
    [CrossRef]
  3. C. Bès, G. Plantier, and T. M. Bosch, “Displacement measurements using a self-mixing laser diode under moderate feedback,” IEEE Trans. Instrum. Meas. 55, 1101–1105 (2006).
    [CrossRef]
  4. G. Giuliani, S. Donati, and W. Elsässer, “Measurement of linewidth enhancement factor of different semiconductor lasers in operating conditions,” Proc. SPIE 6184, 61841D (2006).
    [CrossRef]
  5. Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43, 527–534 (2007).
    [CrossRef]
  6. R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
    [CrossRef]
  7. L. Wei, J. T. Xi, Y. G. Yu, and J. F. Chicharo, “Linewidth enhancement factor measurement based on optical feedback self-mixing effect: a genetic algorithm approach,” J. Opt. A 11, 045505 (2009).
    [CrossRef]
  8. Y. Yu, J. Xi, and J. F. Chicharo, “Measuring the feedback parameter of a semiconductor laser with external optical feedback,” Opt. Express 19, 9582–9593 (2011).
    [CrossRef]
  9. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
    [CrossRef]
  10. K. Petermann, Laser Diode Modulation and Noise (Kluwer, 1991).
  11. G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
    [CrossRef]
  12. 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), pp. 23–54.
  13. G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. 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]
  14. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
    [CrossRef]
  15. M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
    [CrossRef]
  16. F. Bessel, “Untersuchung des theils der planetarischen störungen, welcher aus der bewegung der sonne entsteht,” in Abhandlungen der mathematischen Klasse der Königlichen Akademie der Wissenschaften zu Berlin. Aus dem Jahre 1824 (1826), pp. 1–52.
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  19. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  20. A. B. Carlson and P. B. Crilly, Communication Systems: An Introduction to Signals and Noise in Electrical Communication, 5th ed. (McGraw-Hill, 2009).
  21. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  22. C. F. Gauss, Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium (Frid. Perthes et I.H. Besser, 1809).
  23. Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” Proc. SPIE 6038, 60381O (2006).
    [CrossRef]
  24. C. G. J. Jacobi, “Formula transformationis integralium definitorum,” Crelle Journal für die Reine und Angewandte Mathematik 15, 1–26 (1836).
  25. C. T. Anger, Untersuchungen über die Function Ikh mit Anwendungen auf das Kepler’sche Problem (Neueste Schrift. d. Naturforch. Ges. in Danzig, 1855).
  26. 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]

2013 (1)

2011 (1)

2009 (1)

L. Wei, J. T. Xi, Y. G. Yu, and J. F. Chicharo, “Linewidth enhancement factor measurement based on optical feedback self-mixing effect: a genetic algorithm approach,” J. Opt. A 11, 045505 (2009).
[CrossRef]

2008 (1)

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

2007 (1)

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43, 527–534 (2007).
[CrossRef]

2006 (3)

C. Bès, G. Plantier, and T. M. Bosch, “Displacement measurements using a self-mixing laser diode under moderate feedback,” IEEE Trans. Instrum. Meas. 55, 1101–1105 (2006).
[CrossRef]

G. Giuliani, S. Donati, and W. Elsässer, “Measurement of linewidth enhancement factor of different semiconductor lasers in operating conditions,” Proc. SPIE 6184, 61841D (2006).
[CrossRef]

Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” Proc. SPIE 6038, 60381O (2006).
[CrossRef]

2005 (2)

G. Plantier, C. Bès, and T. M. 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. M. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41, 1058–1064 (2005).
[CrossRef]

1995 (1)

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

1987 (1)

M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[CrossRef]

1984 (1)

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. 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]

1982 (1)

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

1980 (1)

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

1854 (1)

P. L. Tchebychef, “Théorie des mécanismes connus sous le nom de parallélogrammes,” Mémoires présentés à l’Académie Impériale des Sciences de St.-Pétersbourg par Divers Savants 7, 539–568 (1854).

1836 (1)

C. G. J. Jacobi, “Formula transformationis integralium definitorum,” Crelle Journal für die Reine und Angewandte Mathematik 15, 1–26 (1836).

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Acket, G. A.

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. 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]

Anger, C. T.

C. T. Anger, Untersuchungen über die Function Ikh mit Anwendungen auf das Kepler’sche Problem (Neueste Schrift. d. Naturforch. Ges. in Danzig, 1855).

Beere, H. E.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Beltram, F.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Bertling, K.

Bès, C.

C. Bès, G. Plantier, and T. M. Bosch, “Displacement measurements using a self-mixing laser diode under moderate feedback,” IEEE Trans. Instrum. Meas. 55, 1101–1105 (2006).
[CrossRef]

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

Bessel, F.

F. Bessel, “Untersuchung des theils der planetarischen störungen, welcher aus der bewegung der sonne entsteht,” in Abhandlungen der mathematischen Klasse der Königlichen Akademie der Wissenschaften zu Berlin. Aus dem Jahre 1824 (1826), pp. 1–52.

Bosch, T. M.

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43, 527–534 (2007).
[CrossRef]

C. Bès, G. Plantier, and T. M. Bosch, “Displacement measurements using a self-mixing laser diode under moderate feedback,” IEEE Trans. Instrum. Meas. 55, 1101–1105 (2006).
[CrossRef]

G. Plantier, C. Bès, and T. M. 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. M. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41, 1058–1064 (2005).
[CrossRef]

Buus, J.

M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[CrossRef]

Carlson, A. B.

A. B. Carlson and P. B. Crilly, Communication Systems: An Introduction to Signals and Noise in Electrical Communication, 5th ed. (McGraw-Hill, 2009).

Chicharo, J. F.

Y. Yu, J. Xi, and J. F. Chicharo, “Measuring the feedback parameter of a semiconductor laser with external optical feedback,” Opt. Express 19, 9582–9593 (2011).
[CrossRef]

L. Wei, J. T. Xi, Y. G. Yu, and J. F. Chicharo, “Linewidth enhancement factor measurement based on optical feedback self-mixing effect: a genetic algorithm approach,” J. Opt. A 11, 045505 (2009).
[CrossRef]

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43, 527–534 (2007).
[CrossRef]

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

Crilly, P. B.

A. B. Carlson and P. B. Crilly, Communication Systems: An Introduction to Signals and Noise in Electrical Communication, 5th ed. (McGraw-Hill, 2009).

Davies, A. G.

Dean, P.

den Boef, A. J.

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. 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]

Donati, S.

G. Giuliani, S. Donati, and W. Elsässer, “Measurement of linewidth enhancement factor of different semiconductor lasers in operating conditions,” Proc. SPIE 6184, 61841D (2006).
[CrossRef]

Elsässer, W.

G. Giuliani, S. Donati, and W. Elsässer, “Measurement of linewidth enhancement factor of different semiconductor lasers in operating conditions,” Proc. SPIE 6184, 61841D (2006).
[CrossRef]

Gauss, C. F.

C. F. Gauss, Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium (Frid. Perthes et I.H. Besser, 1809).

Giuliani, G.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

G. Giuliani, S. Donati, and W. Elsässer, “Measurement of linewidth enhancement factor of different semiconductor lasers in operating conditions,” Proc. SPIE 6184, 61841D (2006).
[CrossRef]

Green, R. P.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Harrison, P.

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

Ikonic, Z.

Indjin, D.

Jacobi, C. G. J.

C. G. J. Jacobi, “Formula transformationis integralium definitorum,” Crelle Journal für die Reine und Angewandte Mathematik 15, 1–26 (1836).

Khanna, S. P.

Kobayashi, K.

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

Lachab, M.

Lang, R.

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

Lenstra, D.

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. 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]

Lim, Y. L.

Linfield, E. H.

Mahler, L.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Osinski, M.

M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[CrossRef]

Petermann, K.

K. Petermann, Laser Diode Modulation and Noise (Kluwer, 1991).

Pierce, I.

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), pp. 23–54.

Plantier, G.

C. Bès, G. Plantier, and T. M. Bosch, “Displacement measurements using a self-mixing laser diode under moderate feedback,” IEEE Trans. Instrum. Meas. 55, 1101–1105 (2006).
[CrossRef]

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

Rakic, A. D.

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), pp. 23–54.

Rio, P.

Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” Proc. SPIE 6038, 60381O (2006).
[CrossRef]

Ritchie, D. A.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Spencer, 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), pp. 23–54.

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Taimre, T.

Tchebychef, P. L.

P. L. Tchebychef, “Théorie des mécanismes connus sous le nom de parallélogrammes,” Mémoires présentés à l’Académie Impériale des Sciences de St.-Pétersbourg par Divers Savants 7, 539–568 (1854).

Tredicucci, A.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Tucker, J.

Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” Proc. SPIE 6038, 60381O (2006).
[CrossRef]

Valavanis, A.

van Tartwijk, G. H. M.

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

Verbeek, B. H.

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. 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]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1966).

Wei, L.

L. Wei, J. T. Xi, Y. G. Yu, and J. F. Chicharo, “Linewidth enhancement factor measurement based on optical feedback self-mixing effect: a genetic algorithm approach,” J. Opt. A 11, 045505 (2009).
[CrossRef]

Wilson, S. J.

Xi, J.

Y. Yu, J. Xi, and J. F. Chicharo, “Measuring the feedback parameter of a semiconductor laser with external optical feedback,” Opt. Express 19, 9582–9593 (2011).
[CrossRef]

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43, 527–534 (2007).
[CrossRef]

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

Xi, J. T.

L. Wei, J. T. Xi, Y. G. Yu, and J. F. Chicharo, “Linewidth enhancement factor measurement based on optical feedback self-mixing effect: a genetic algorithm approach,” J. Opt. A 11, 045505 (2009).
[CrossRef]

Xu, J.-H.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Yu, Y.

Y. Yu, J. Xi, and J. F. Chicharo, “Measuring the feedback parameter of a semiconductor laser with external optical feedback,” Opt. Express 19, 9582–9593 (2011).
[CrossRef]

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43, 527–534 (2007).
[CrossRef]

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

Yu, Y. G.

L. Wei, J. T. Xi, Y. G. Yu, and J. F. Chicharo, “Linewidth enhancement factor measurement based on optical feedback self-mixing effect: a genetic algorithm approach,” J. Opt. A 11, 045505 (2009).
[CrossRef]

Appl. Phys. Lett. (1)

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Crelle Journal für die Reine und Angewandte Mathematik (1)

C. G. J. Jacobi, “Formula transformationis integralium definitorum,” Crelle Journal für die Reine und Angewandte Mathematik 15, 1–26 (1836).

IEEE J. Quantum Electron. (7)

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43, 527–534 (2007).
[CrossRef]

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

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. 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]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[CrossRef]

G. Plantier, C. Bès, and T. M. 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. M. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41, 1058–1064 (2005).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

C. Bès, G. Plantier, and T. M. Bosch, “Displacement measurements using a self-mixing laser diode under moderate feedback,” IEEE Trans. Instrum. Meas. 55, 1101–1105 (2006).
[CrossRef]

J. Opt. A (1)

L. Wei, J. T. Xi, Y. G. Yu, and J. F. Chicharo, “Linewidth enhancement factor measurement based on optical feedback self-mixing effect: a genetic algorithm approach,” J. Opt. A 11, 045505 (2009).
[CrossRef]

Mémoires présentés à l’Académie Impériale des Sciences de St.-Pétersbourg par Divers Savants (1)

P. L. Tchebychef, “Théorie des mécanismes connus sous le nom de parallélogrammes,” Mémoires présentés à l’Académie Impériale des Sciences de St.-Pétersbourg par Divers Savants 7, 539–568 (1854).

Opt. Express (2)

Proc. SPIE (2)

Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” Proc. SPIE 6038, 60381O (2006).
[CrossRef]

G. Giuliani, S. Donati, and W. Elsässer, “Measurement of linewidth enhancement factor of different semiconductor lasers in operating conditions,” Proc. SPIE 6184, 61841D (2006).
[CrossRef]

Quantum Semiclass. Opt. (1)

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

Other (9)

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), pp. 23–54.

K. Petermann, Laser Diode Modulation and Noise (Kluwer, 1991).

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Figures (2)

Fig. 1.
Fig. 1.

Relationship of series coefficients for a self-mixing signal with C=1.2 and α=5. Horizontal lines are a guide for the eye. (a) Exact (plus) and extracted (circles and crosses) series coefficients. Five chain lines: ratio of second and third coefficients giving α=5. (b) First 13 Bessel Jn curves as a function of C. Black: n even. Red: n odd. Vertical chain line: C=1.2.

Fig. 2.
Fig. 2.

Linear-displacement self-mixing signal with C=1.2 and α=5. Shaded blocks indicate the signal portion whose sign must differ from the remainder. Inset in (a): Linear-displacement self-mixing signal viewed as a function of time over five periods. The shaded central portion is a representative single period, elaborated on in (a)–(c). (a) Exact (thick red) and reconstructed (solid black) cosine signal. Reconstruction with alternate sign set (dotted black). (b) Exact (thick red) and reconstructed (solid black) sine signal. Reconstruction with alternate sign set (dotted black). Reconstruction with same positive sign (thick cyan). (c) Thirteen calculated basis functions used to reconstruct the cosine signal in (a).

Tables (1)

Tables Icon

Table 1. Exact and Extracted Parameter Pairs C and α for Both Weak (left) and Moderate (right) Feedback Regimes

Equations (13)

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φSφFB=Csin(φFB+arctanα),
V=V0cos(φFB)=V0cos(φSCsin(φFB+arctanα))=V0cos(φS)cos(Csin(φFB+arctanα))+V0sin(φS)sin(Csin(φFB+arctanα)).
cos(xsin(θ))=J0(x)+2n=1J2n(x)cos(2nθ),sin(xsin(θ))=2n=1J2n1(x)sin((2n1)θ),
V=V0cos(φFB)=V0cos(φS){J0(C)+2n=1J2n(C)cos(2n(φFB+arctanα))}+V0sin(φS){2n=1J2n1(C)sin((2n1)(φFB+arctanα))}
V=V0n=Jn(C)cos(φSnϑFB),
xc(t)=Acn=Jn(β)cos(ωc+nωm)t
V˜=cos(φFB)=J0(C)cos(φS)+n=12J2n1(C)α1+α2U2(n1)(11+α2)sin(φS)T2n1(cosφFB)+n=12J2n1(C)T2n1(11+α2)sin(φS)sin(φFB)U2(n1)(cosφFB)n=12J2n(C)α1+α2U2n1(11+α2)cos(φS)sin(φFB)U2n1(cosφFB)+n=12J2n(C)T2n(11+α2)cos(φS)T2n(cosφFB),
T0(x)=1,T1(x)=x,Tn(x)=2xTn1(x)Tn2(x),U0(x)=1,U1(x)=2x,Un(x)=2xUn1(x)Un2(x),
Csin(φFB+arctanα)=C1+α2(sin(φFB)+αcos(φFB)).
2J1(C)α1+α2U0(11+α2)2J1(C)T1(11+α2)=α1+α211+α2=α,
φS(t)=φ0+ΦΔtθR,
sin(φFB)=±1cos2(φFB)=±sin(arccos(cosφFB)),
(ATA)β=ATy.

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