Abstract

A novel laser sensor for position measurements of technical solid-state surfaces is proposed. An external Fabry–Perot laser cavity is assembled by use of an antireflection-coated laser diode together with the technical surface. Mode locking results from pumping the laser diode synchronously to the mode spacing of the cavity. The laser cavity length, i.e., the distance to the measurement object, is determined by evaluation of the modulation transfer function of the cavity by means of a phase-locked loop. The mode-locking external-cavity laser sensor incorporates a resonance effect that results in highly resolving position and displacement measurements. More than a factor-of-10 higher resolution than with conventional nonresonant sensing principles is achieved. Results of the displacement measurements of various technical surfaces are reported. Experimental and theoretical investigations are in good agreement.

© 2005 Optical Society of America

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References

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  1. H. K. Tönshoff, I. Inasaki, eds., Sensors in Manufacturing (Wiley-VCH, 2001).
    [CrossRef]
  2. W. Steen, Laser Material Processing (Springer-Verlag, 1998).
    [CrossRef]
  3. G. Dorsch, G. Häusler, J. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306–1314 (1994).
    [CrossRef] [PubMed]
  4. K. Wesolowicz, R. Sampson, “Laser radar range imaging sensor for commercial applications,” in Laser Radar II,R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 152–161 (1987).
    [CrossRef]
  5. S. Pellegrini, G. Buller, J. Smith, A. Wallace, S. Cova, “Laser-based distance measurement using picosecond resolution time-correlated single-photon counting,” Meas. Sci. Technol. 11, 712–716 (2000).
    [CrossRef]
  6. H. Höfler, G. Schmidtke, “Three dimensional contouring by an optical radar system,” in Laser Dimensional Metrology: Recent Advances for Industrial Application,M. J. Downs, ed., Proc. SPIE2088, 82–87 (1993).
    [CrossRef]
  7. R. Juskaitis, T. Wilson, “Imaging in reciprocal fiber-optic based confocal scanning microscopes,” Opt. Commun. 92, 315–325 (1992).
    [CrossRef]
  8. C.-J. Kim, M.-S. Kim, C.-M. Chung, “Demonstration of auto-focus control by chromatic filtering,” in Proceedings of ICALEO 1998 (Laser Institute of America, 1998), pp. C226–C235.
  9. E. Fischer, E. Dalhoff, S. Heim, U. Hofbauer, H. Tiziani, “Absolute interferometric distance measurement using a FM-demodulation technique,” Appl. Opt. 34, 5589–5594 (1995).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. J. Kato, J. Yamato, N. Kikuchi, R. Furutani, S. Ozono, “Non-contact optical probing sensor—applying optical feedback effects in laser diodes,” Measurement 9, 146–152 (1991).
    [CrossRef]
  15. C. Morgan, M. Bordovsky, I. White, R. Griffiths, “Noncontact vibration sensors based on current modulated external cavity semiconductor lasers,” in IEE Proc. Optoelectron. 147, 413–416 (2000).
    [CrossRef]
  16. P.-T. Ho, L. A. Glasser, E. P. Ippen, H. A. Haus, “Picosecond pulse generation with a cw GaAlAs laser diode,” Appl. Phys. Lett. 33, 241–242 (1978).
    [CrossRef]
  17. R. Nietzke, J. Sacher, W. Elsässer, E. O. Göbel, “Mode-locking of a semiconductor diode laser by self-synchronizing optoelectronic feedback of the longitudinal mode beats,” Electron. Lett. 26, 1016–1018 (1990).
    [CrossRef]
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    [CrossRef]
  21. J. van der Ziel, “Generation of short optical pulses in semiconductor lasers by combined dc microwave current injection,” IEEE J. Quantum Electron. QE-18, 1340–1350 (1982).
    [CrossRef]
  22. C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
    [CrossRef]
  23. D. Hjelme, A. Mickelson, “Gain nonlinearities due to carrier density dependent dispersion in semiconductor lasers,” IEEE J. Quantum Electron. 25, 1625–1631 (1989).
    [CrossRef]
  24. B. Tromborg, J. Mork, “Nonlinear injection locking dynamics and the onset of coherence collapse in external cavity lasers,” IEEE J. Quantum Electron. 26, 642–654 (1990).
    [CrossRef]

2000

C. Morgan, M. Bordovsky, I. White, R. Griffiths, “Noncontact vibration sensors based on current modulated external cavity semiconductor lasers,” in IEE Proc. Optoelectron. 147, 413–416 (2000).
[CrossRef]

S. Pellegrini, G. Buller, J. Smith, A. Wallace, S. Cova, “Laser-based distance measurement using picosecond resolution time-correlated single-photon counting,” Meas. Sci. Technol. 11, 712–716 (2000).
[CrossRef]

1996

1995

1994

1993

T. Pfeifer, J. Thiel, “Absolutinterferometrie mit durchstimmbaren Halbleiterlasern,” Tech. Messen 60, 185–193 (1993).

1992

T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
[CrossRef] [PubMed]

R. Juskaitis, T. Wilson, “Imaging in reciprocal fiber-optic based confocal scanning microscopes,” Opt. Commun. 92, 315–325 (1992).
[CrossRef]

1991

J. Kato, J. Yamato, N. Kikuchi, R. Furutani, S. Ozono, “Non-contact optical probing sensor—applying optical feedback effects in laser diodes,” Measurement 9, 146–152 (1991).
[CrossRef]

1990

R. Nietzke, J. Sacher, W. Elsässer, E. O. Göbel, “Mode-locking of a semiconductor diode laser by self-synchronizing optoelectronic feedback of the longitudinal mode beats,” Electron. Lett. 26, 1016–1018 (1990).
[CrossRef]

B. Tromborg, J. Mork, “Nonlinear injection locking dynamics and the onset of coherence collapse in external cavity lasers,” IEEE J. Quantum Electron. 26, 642–654 (1990).
[CrossRef]

1989

D. Hjelme, A. Mickelson, “Gain nonlinearities due to carrier density dependent dispersion in semiconductor lasers,” IEEE J. Quantum Electron. 25, 1625–1631 (1989).
[CrossRef]

1982

J. van der Ziel, “Generation of short optical pulses in semiconductor lasers by combined dc microwave current injection,” IEEE J. Quantum Electron. QE-18, 1340–1350 (1982).
[CrossRef]

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

1981

J. van der Ziel, “Active mode locking of double heterostructure lasers in an external cavity,” J. Appl. Phys. 52, 4435–4446 (1981).
[CrossRef]

1980

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

1978

P.-T. Ho, L. A. Glasser, E. P. Ippen, H. A. Haus, “Picosecond pulse generation with a cw GaAlAs laser diode,” Appl. Phys. Lett. 33, 241–242 (1978).
[CrossRef]

Bordovsky, M.

C. Morgan, M. Bordovsky, I. White, R. Griffiths, “Noncontact vibration sensors based on current modulated external cavity semiconductor lasers,” in IEE Proc. Optoelectron. 147, 413–416 (2000).
[CrossRef]

Buller, G.

S. Pellegrini, G. Buller, J. Smith, A. Wallace, S. Cova, “Laser-based distance measurement using picosecond resolution time-correlated single-photon counting,” Meas. Sci. Technol. 11, 712–716 (2000).
[CrossRef]

Chung, C.-M.

C.-J. Kim, M.-S. Kim, C.-M. Chung, “Demonstration of auto-focus control by chromatic filtering,” in Proceedings of ICALEO 1998 (Laser Institute of America, 1998), pp. C226–C235.

Cova, S.

S. Pellegrini, G. Buller, J. Smith, A. Wallace, S. Cova, “Laser-based distance measurement using picosecond resolution time-correlated single-photon counting,” Meas. Sci. Technol. 11, 712–716 (2000).
[CrossRef]

Dalhoff, E.

Dorsch, G.

Dresel, T.

Elsässer, W.

R. Nietzke, J. Sacher, W. Elsässer, E. O. Göbel, “Mode-locking of a semiconductor diode laser by self-synchronizing optoelectronic feedback of the longitudinal mode beats,” Electron. Lett. 26, 1016–1018 (1990).
[CrossRef]

Fischer, E.

Furutani, R.

J. Kato, J. Yamato, N. Kikuchi, R. Furutani, S. Ozono, “Non-contact optical probing sensor—applying optical feedback effects in laser diodes,” Measurement 9, 146–152 (1991).
[CrossRef]

Glasser, L. A.

P.-T. Ho, L. A. Glasser, E. P. Ippen, H. A. Haus, “Picosecond pulse generation with a cw GaAlAs laser diode,” Appl. Phys. Lett. 33, 241–242 (1978).
[CrossRef]

Göbel, E. O.

R. Nietzke, J. Sacher, W. Elsässer, E. O. Göbel, “Mode-locking of a semiconductor diode laser by self-synchronizing optoelectronic feedback of the longitudinal mode beats,” Electron. Lett. 26, 1016–1018 (1990).
[CrossRef]

Griffiths, R.

C. Morgan, M. Bordovsky, I. White, R. Griffiths, “Noncontact vibration sensors based on current modulated external cavity semiconductor lasers,” in IEE Proc. Optoelectron. 147, 413–416 (2000).
[CrossRef]

Haus, H. A.

P.-T. Ho, L. A. Glasser, E. P. Ippen, H. A. Haus, “Picosecond pulse generation with a cw GaAlAs laser diode,” Appl. Phys. Lett. 33, 241–242 (1978).
[CrossRef]

Häusler, G.

Heim, S.

Henry, C.

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

Herrmann, J.

Hjelme, D.

D. Hjelme, A. Mickelson, “Gain nonlinearities due to carrier density dependent dispersion in semiconductor lasers,” IEEE J. Quantum Electron. 25, 1625–1631 (1989).
[CrossRef]

Ho, P.-T.

P.-T. Ho, L. A. Glasser, E. P. Ippen, H. A. Haus, “Picosecond pulse generation with a cw GaAlAs laser diode,” Appl. Phys. Lett. 33, 241–242 (1978).
[CrossRef]

Hofbauer, U.

Höfler, H.

H. Höfler, G. Schmidtke, “Three dimensional contouring by an optical radar system,” in Laser Dimensional Metrology: Recent Advances for Industrial Application,M. J. Downs, ed., Proc. SPIE2088, 82–87 (1993).
[CrossRef]

Ippen, E. P.

P.-T. Ho, L. A. Glasser, E. P. Ippen, H. A. Haus, “Picosecond pulse generation with a cw GaAlAs laser diode,” Appl. Phys. Lett. 33, 241–242 (1978).
[CrossRef]

Juskaitis, R.

R. Juskaitis, T. Wilson, “Imaging in reciprocal fiber-optic based confocal scanning microscopes,” Opt. Commun. 92, 315–325 (1992).
[CrossRef]

Kato, J.

J. Kato, J. Yamato, N. Kikuchi, R. Furutani, S. Ozono, “Non-contact optical probing sensor—applying optical feedback effects in laser diodes,” Measurement 9, 146–152 (1991).
[CrossRef]

Kikuchi, N.

J. Kato, J. Yamato, N. Kikuchi, R. Furutani, S. Ozono, “Non-contact optical probing sensor—applying optical feedback effects in laser diodes,” Measurement 9, 146–152 (1991).
[CrossRef]

Kim, C.-J.

C.-J. Kim, M.-S. Kim, C.-M. Chung, “Demonstration of auto-focus control by chromatic filtering,” in Proceedings of ICALEO 1998 (Laser Institute of America, 1998), pp. C226–C235.

Kim, M.-S.

C.-J. Kim, M.-S. Kim, C.-M. Chung, “Demonstration of auto-focus control by chromatic filtering,” in Proceedings of ICALEO 1998 (Laser Institute of America, 1998), pp. C226–C235.

Kobayashi, K.

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

Lang, R.

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

Liu, X.

X. Liu, Untersuchung eines neuartigen Abstandsmessverfahrens, basierend auf der Bestimmung von Laserpulswiederholfrequenzen (Fortschrittberichte VDI, 1996). Reihe 8: Meß-, Steuerungs- und Regelungstechnik, Nr. 566, ISBN 3-18-356608-7.

Maier, N.

Mickelson, A.

D. Hjelme, A. Mickelson, “Gain nonlinearities due to carrier density dependent dispersion in semiconductor lasers,” IEEE J. Quantum Electron. 25, 1625–1631 (1989).
[CrossRef]

Morgan, C.

C. Morgan, M. Bordovsky, I. White, R. Griffiths, “Noncontact vibration sensors based on current modulated external cavity semiconductor lasers,” in IEE Proc. Optoelectron. 147, 413–416 (2000).
[CrossRef]

Mork, J.

B. Tromborg, J. Mork, “Nonlinear injection locking dynamics and the onset of coherence collapse in external cavity lasers,” IEEE J. Quantum Electron. 26, 642–654 (1990).
[CrossRef]

Nietzke, R.

R. Nietzke, J. Sacher, W. Elsässer, E. O. Göbel, “Mode-locking of a semiconductor diode laser by self-synchronizing optoelectronic feedback of the longitudinal mode beats,” Electron. Lett. 26, 1016–1018 (1990).
[CrossRef]

Ozono, S.

J. Kato, J. Yamato, N. Kikuchi, R. Furutani, S. Ozono, “Non-contact optical probing sensor—applying optical feedback effects in laser diodes,” Measurement 9, 146–152 (1991).
[CrossRef]

Pellegrini, S.

S. Pellegrini, G. Buller, J. Smith, A. Wallace, S. Cova, “Laser-based distance measurement using picosecond resolution time-correlated single-photon counting,” Meas. Sci. Technol. 11, 712–716 (2000).
[CrossRef]

Pfeifer, T.

T. Pfeifer, J. Thiel, “Absolutinterferometrie mit durchstimmbaren Halbleiterlasern,” Tech. Messen 60, 185–193 (1993).

Rothe, A.

Sacher, J.

R. Nietzke, J. Sacher, W. Elsässer, E. O. Göbel, “Mode-locking of a semiconductor diode laser by self-synchronizing optoelectronic feedback of the longitudinal mode beats,” Electron. Lett. 26, 1016–1018 (1990).
[CrossRef]

Sampson, R.

K. Wesolowicz, R. Sampson, “Laser radar range imaging sensor for commercial applications,” in Laser Radar II,R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 152–161 (1987).
[CrossRef]

Schmidtke, G.

H. Höfler, G. Schmidtke, “Three dimensional contouring by an optical radar system,” in Laser Dimensional Metrology: Recent Advances for Industrial Application,M. J. Downs, ed., Proc. SPIE2088, 82–87 (1993).
[CrossRef]

Smith, J.

S. Pellegrini, G. Buller, J. Smith, A. Wallace, S. Cova, “Laser-based distance measurement using picosecond resolution time-correlated single-photon counting,” Meas. Sci. Technol. 11, 712–716 (2000).
[CrossRef]

Steen, W.

W. Steen, Laser Material Processing (Springer-Verlag, 1998).
[CrossRef]

Thiel, J.

T. Pfeifer, J. Thiel, “Absolutinterferometrie mit durchstimmbaren Halbleiterlasern,” Tech. Messen 60, 185–193 (1993).

Tiziani, H.

Tromborg, B.

B. Tromborg, J. Mork, “Nonlinear injection locking dynamics and the onset of coherence collapse in external cavity lasers,” IEEE J. Quantum Electron. 26, 642–654 (1990).
[CrossRef]

van der Ziel, J.

J. van der Ziel, “Generation of short optical pulses in semiconductor lasers by combined dc microwave current injection,” IEEE J. Quantum Electron. QE-18, 1340–1350 (1982).
[CrossRef]

J. van der Ziel, “Active mode locking of double heterostructure lasers in an external cavity,” J. Appl. Phys. 52, 4435–4446 (1981).
[CrossRef]

Venzke, H.

Voges, E.

E. Voges, Bauelemente und Schaltungen, Vol. 1 of Hochfrequenztechnik (Hüthig, Heidelberg, 1991).

Wallace, A.

S. Pellegrini, G. Buller, J. Smith, A. Wallace, S. Cova, “Laser-based distance measurement using picosecond resolution time-correlated single-photon counting,” Meas. Sci. Technol. 11, 712–716 (2000).
[CrossRef]

Wesolowicz, K.

K. Wesolowicz, R. Sampson, “Laser radar range imaging sensor for commercial applications,” in Laser Radar II,R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 152–161 (1987).
[CrossRef]

White, I.

C. Morgan, M. Bordovsky, I. White, R. Griffiths, “Noncontact vibration sensors based on current modulated external cavity semiconductor lasers,” in IEE Proc. Optoelectron. 147, 413–416 (2000).
[CrossRef]

Wilson, T.

R. Juskaitis, T. Wilson, “Imaging in reciprocal fiber-optic based confocal scanning microscopes,” Opt. Commun. 92, 315–325 (1992).
[CrossRef]

Yamato, J.

J. Kato, J. Yamato, N. Kikuchi, R. Furutani, S. Ozono, “Non-contact optical probing sensor—applying optical feedback effects in laser diodes,” Measurement 9, 146–152 (1991).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

P.-T. Ho, L. A. Glasser, E. P. Ippen, H. A. Haus, “Picosecond pulse generation with a cw GaAlAs laser diode,” Appl. Phys. Lett. 33, 241–242 (1978).
[CrossRef]

Electron. Lett.

R. Nietzke, J. Sacher, W. Elsässer, E. O. Göbel, “Mode-locking of a semiconductor diode laser by self-synchronizing optoelectronic feedback of the longitudinal mode beats,” Electron. Lett. 26, 1016–1018 (1990).
[CrossRef]

IEEE J. Quantum Electron.

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

J. van der Ziel, “Generation of short optical pulses in semiconductor lasers by combined dc microwave current injection,” IEEE J. Quantum Electron. QE-18, 1340–1350 (1982).
[CrossRef]

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

D. Hjelme, A. Mickelson, “Gain nonlinearities due to carrier density dependent dispersion in semiconductor lasers,” IEEE J. Quantum Electron. 25, 1625–1631 (1989).
[CrossRef]

B. Tromborg, J. Mork, “Nonlinear injection locking dynamics and the onset of coherence collapse in external cavity lasers,” IEEE J. Quantum Electron. 26, 642–654 (1990).
[CrossRef]

in IEE Proc. Optoelectron.

C. Morgan, M. Bordovsky, I. White, R. Griffiths, “Noncontact vibration sensors based on current modulated external cavity semiconductor lasers,” in IEE Proc. Optoelectron. 147, 413–416 (2000).
[CrossRef]

J. Appl. Phys.

J. van der Ziel, “Active mode locking of double heterostructure lasers in an external cavity,” J. Appl. Phys. 52, 4435–4446 (1981).
[CrossRef]

Meas. Sci. Technol.

S. Pellegrini, G. Buller, J. Smith, A. Wallace, S. Cova, “Laser-based distance measurement using picosecond resolution time-correlated single-photon counting,” Meas. Sci. Technol. 11, 712–716 (2000).
[CrossRef]

Measurement

J. Kato, J. Yamato, N. Kikuchi, R. Furutani, S. Ozono, “Non-contact optical probing sensor—applying optical feedback effects in laser diodes,” Measurement 9, 146–152 (1991).
[CrossRef]

Opt. Commun.

R. Juskaitis, T. Wilson, “Imaging in reciprocal fiber-optic based confocal scanning microscopes,” Opt. Commun. 92, 315–325 (1992).
[CrossRef]

Tech. Messen

T. Pfeifer, J. Thiel, “Absolutinterferometrie mit durchstimmbaren Halbleiterlasern,” Tech. Messen 60, 185–193 (1993).

Other

K. Wesolowicz, R. Sampson, “Laser radar range imaging sensor for commercial applications,” in Laser Radar II,R. J. Becherer, R. C. Harney, eds., Proc. SPIE783, 152–161 (1987).
[CrossRef]

H. Höfler, G. Schmidtke, “Three dimensional contouring by an optical radar system,” in Laser Dimensional Metrology: Recent Advances for Industrial Application,M. J. Downs, ed., Proc. SPIE2088, 82–87 (1993).
[CrossRef]

C.-J. Kim, M.-S. Kim, C.-M. Chung, “Demonstration of auto-focus control by chromatic filtering,” in Proceedings of ICALEO 1998 (Laser Institute of America, 1998), pp. C226–C235.

H. K. Tönshoff, I. Inasaki, eds., Sensors in Manufacturing (Wiley-VCH, 2001).
[CrossRef]

W. Steen, Laser Material Processing (Springer-Verlag, 1998).
[CrossRef]

E. Voges, Bauelemente und Schaltungen, Vol. 1 of Hochfrequenztechnik (Hüthig, Heidelberg, 1991).

X. Liu, Untersuchung eines neuartigen Abstandsmessverfahrens, basierend auf der Bestimmung von Laserpulswiederholfrequenzen (Fortschrittberichte VDI, 1996). Reihe 8: Meß-, Steuerungs- und Regelungstechnik, Nr. 566, ISBN 3-18-356608-7.

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Figures (13)

Fig. 1
Fig. 1

Schematic of the MoLECL sensor. A laser diode with partially reflecting (PR) and antireflection- (AR-) coated facets is arranged in an external cavity, with the surface of the measurement object used as an external reflector. Laser light detected with a photodiode is used as the control circuit to generate the modulation signal of the laser diode.

Fig. 2
Fig. 2

Typical transfer function of the optical cavity measured with a vector network analyzer: (a) measured amplitude response, (b) measured phase response.

Fig. 3
Fig. 3

Schematic of the MoLECL sensor. The beat signal generated by the photoreceiver is electrically mixed with the signal of the VCO. A 90° phase shifter and a delay line are employed as the working point of the phase detector and to produce a path length difference of zero, respectively. The VCO signal is superposed with a bias current by a bias-T circuit, resulting in the modulation signal of the laser diode. AR, antireflection coating; PR, partial reflector; LD, laser diode; PD, phase detector.

Fig. 4
Fig. 4

Changes of the laser power at tilting of technical surfaces (solid curve, steel; dashed curve, paper).

Fig. 5
Fig. 5

Phase-response function for three measurement objects: mirror, steel, and paper.

Fig. 6
Fig. 6

Phase-transfer function for a nonfocal setup. The technical surface is moved out of the focal plane of the imaging optics by distance d (solid curve, d = 0 mm; dashed curve, d = 10 mm; dotted curve, d = 20 mm).

Fig. 7
Fig. 7

Transient behavior of the distance measurement when changing the finesse of the cavity.

Fig. 8
Fig. 8

Phase-response function for a Fabry–Perot cavity with four optical attenuations.

Fig. 9
Fig. 9

Measured phase-response function for five laser-diode bias currents Idc.

Fig. 10
Fig. 10

Resonance frequency of an external-cavity laser determined by zero crossing of the phase-transfer function with and without an AGC. The reference value corresponds to the mode spacing of the cavity.

Fig. 11
Fig. 11

Resonance frequency fR determined by zero crossing of the phase-transfer function for sinusoidal (dashed curve) and pulsed (solid curve) modulation of laser-diode current.

Fig. 12
Fig. 12

Simulation results: time functions of the sinusoidal pump current signal and photon density for different signal frequencies.

Fig. 13
Fig. 13

Calculated phase-response function for five laser-diode bias currents Idc; cf. Fig. 9.

Tables (1)

Tables Icon

Table 1 Laser-Diode Parameters Used for Numerical Simulations

Equations (12)

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

Δ f = c / ( 2 n L ) ,
d f d Φ | f = f R ,
Δ f R = Δ Φ d f d Φ | f = f R .
H = 1 ( 1 + Q 2 v 2 ) 1 / 2 ,             Φ = arctan ( Q v ) .
d Φ d f | f = f R = 2 Q f R .
d S d t = Γ G n ( N - N th ) ( 1 - ɛ g S ) S + 2 ln ( R e ) t l S + γ N t s ,
d Φ d t = 1 2 R Γ G n ( N - N th ) ( 1 - ɛ g S ) + arg ( R e ) t l ,
d N d t = η I e V - Γ G n ( N - N 0 ) ( 1 - ɛ g S ) S - N t s
ln ( R e ) = 1 2 ln × { [ 1 + ( q 00 + q 11 ) cos ϕ r + q 00 q 11 ] 2 + [ ( q 00 - q 11 ) sin ϕ r ] 2 ( 1 + 2 q 11 cos ϕ r + q 11 2 ) 2 } ,
arg ( R e ) = arctan [ ( q 00 - q 11 ) sin ϕ r 1 + ( q 00 + q 11 ) cos ϕ r + q 00 q 11 ] ,
q 00 = R 3 R 2 [ S ( t - t e ) S ( t ) ] 1 / 2 ,
q 11 = R 3 R 2 [ S ( t - t e ) S ( t ) ] 1 / 2 .

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