Abstract

An rf-modulated pulse train from a passively Q-switched Nd:YAG laser has been generated using an extra-cavity acousto-optic modulator. The rf modulation reproduces the spectral quality of the local oscillator. It leads to a high pulse-to-pulse phase coherence, i.e., phase memory, over thousands of pulses. The potentialities of this transmitter for lidar–radar are demonstrated by performing Doppler velocimetry on indoor moving targets. The experimental results are in good agreement with a model based on elementary signal processing theory. In particular, we show experimentally and theoretically that lidar–radar is a promising technique that allows discrimination between translation and rotation movements. Being independent of the laser internal dynamics, this scheme can be applied to any Q-switched laser.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. V. C. Chen, F. Li, S.-S. Ho, and H. Wechsler, “Analysis of micro-Doppler signatures,” IEE Proc. Radar Sonar Navig. 150, 271–276 (2003).
    [CrossRef]
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    [CrossRef]
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2012 (3)

2009 (2)

L. De Dominicis, M. Ferri de Collibus, G. Fornetti, M. Guarneri, M. Nuvoli, R. Ricci, and M. Francucci, “Improving underwater imaging in an amplitude modulated laser system with radio frequency control technique,” J. Eur. Opt. Soc. Rapid Pub. 5, 10004 (2009).
[CrossRef]

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for simultaneous velocity and high-resolution range profile measurements,” Proc. SPIE 7323, 73230Z (2009).
[CrossRef]

2008 (2)

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for high resolution ranging, profilometry and Doppler measurement,” Proc. SPIE 7114, 71140E (2008).
[CrossRef]

M. Brunel and M. Vallet, “Pulse-to-pulse coherent beat note generated by a passively Q-switched two-frequency laser,” Opt. Lett. 33, 2524–2526 (2008).
[CrossRef]

2007 (1)

2006 (1)

2004 (1)

2003 (1)

V. C. Chen, F. Li, S.-S. Ho, and H. Wechsler, “Analysis of micro-Doppler signatures,” IEE Proc. Radar Sonar Navig. 150, 271–276 (2003).
[CrossRef]

2002 (2)

N. D. Lai, M. Brunel, F. Bretenaker, and O. Emile, “Control of the pulse duration in one- and two-axis passively Q-switched solid-state lasers,” Eur. Phys. J. D 19, 403–410 (2002).

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712 (2002).
[CrossRef]

2001 (2)

M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and Ph. Thony, “Offset phase locking of Er:Yb:glass laser eigenstates for RF photonics applications,” IEEE Photon. Technol. Lett. 13, 367–369 (2001).
[CrossRef]

F. Pellen, P. Olivard, Y. Guern, J. Cariou, and J. Lotrian, “Radio frequency modulation on an optical carrier for target detection enhancement in sea-water,” J. Phys. D 34, 1122–1130 (2001).
[CrossRef]

1995 (1)

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of radar technology to aerial lidar systems for enhancement of shallow underwater target detection,” IEEE Trans. Microw. Theory Techn. 43, 2370–2377 (1995).
[CrossRef]

1994 (1)

1980 (1)

Alouini, M.

M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and Ph. Thony, “Offset phase locking of Er:Yb:glass laser eigenstates for RF photonics applications,” IEEE Photon. Technol. Lett. 13, 367–369 (2001).
[CrossRef]

Ando, T.

Asaka, K.

Bai, Y.

Benazet, B.

M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and Ph. Thony, “Offset phase locking of Er:Yb:glass laser eigenstates for RF photonics applications,” IEEE Photon. Technol. Lett. 13, 367–369 (2001).
[CrossRef]

Bretenaker, F.

N. D. Lai, M. Brunel, F. Bretenaker, and O. Emile, “Control of the pulse duration in one- and two-axis passively Q-switched solid-state lasers,” Eur. Phys. J. D 19, 403–410 (2002).

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712 (2002).
[CrossRef]

M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and Ph. Thony, “Offset phase locking of Er:Yb:glass laser eigenstates for RF photonics applications,” IEEE Photon. Technol. Lett. 13, 367–369 (2001).
[CrossRef]

Brunel, M.

M. Brunel and M. Vallet, “Pulse-to-pulse coherent beat note generated by a passively Q-switched two-frequency laser,” Opt. Lett. 33, 2524–2526 (2008).
[CrossRef]

N. D. Lai, M. Brunel, F. Bretenaker, and O. Emile, “Control of the pulse duration in one- and two-axis passively Q-switched solid-state lasers,” Eur. Phys. J. D 19, 403–410 (2002).

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712 (2002).
[CrossRef]

M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and Ph. Thony, “Offset phase locking of Er:Yb:glass laser eigenstates for RF photonics applications,” IEEE Photon. Technol. Lett. 13, 367–369 (2001).
[CrossRef]

Cariou, J.

F. Pellen, P. Olivard, Y. Guern, J. Cariou, and J. Lotrian, “Radio frequency modulation on an optical carrier for target detection enhancement in sea-water,” J. Phys. D 34, 1122–1130 (2001).
[CrossRef]

Chan, S. C.

Chen, V. C.

V. C. Chen, F. Li, S.-S. Ho, and H. Wechsler, “Analysis of micro-Doppler signatures,” IEE Proc. Radar Sonar Navig. 150, 271–276 (2003).
[CrossRef]

Chen, Z.

Contarino, V. M.

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of radar technology to aerial lidar systems for enhancement of shallow underwater target detection,” IEEE Trans. Microw. Theory Techn. 43, 2370–2377 (1995).
[CrossRef]

De Dominicis, L.

L. De Dominicis, M. Ferri de Collibus, G. Fornetti, M. Guarneri, M. Nuvoli, R. Ricci, and M. Francucci, “Improving underwater imaging in an amplitude modulated laser system with radio frequency control technique,” J. Eur. Opt. Soc. Rapid Pub. 5, 10004 (2009).
[CrossRef]

Di Bin, P.

M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and Ph. Thony, “Offset phase locking of Er:Yb:glass laser eigenstates for RF photonics applications,” IEEE Photon. Technol. Lett. 13, 367–369 (2001).
[CrossRef]

Diaz, R.

Dolfi, D.

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for simultaneous velocity and high-resolution range profile measurements,” Proc. SPIE 7323, 73230Z (2009).
[CrossRef]

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for high resolution ranging, profilometry and Doppler measurement,” Proc. SPIE 7114, 71140E (2008).
[CrossRef]

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712 (2002).
[CrossRef]

Eberhard, W. L.

Emile, O.

N. D. Lai, M. Brunel, F. Bretenaker, and O. Emile, “Control of the pulse duration in one- and two-axis passively Q-switched solid-state lasers,” Eur. Phys. J. D 19, 403–410 (2002).

Ferri de Collibus, M.

L. De Dominicis, M. Ferri de Collibus, G. Fornetti, M. Guarneri, M. Nuvoli, R. Ricci, and M. Francucci, “Improving underwater imaging in an amplitude modulated laser system with radio frequency control technique,” J. Eur. Opt. Soc. Rapid Pub. 5, 10004 (2009).
[CrossRef]

Fornetti, G.

L. De Dominicis, M. Ferri de Collibus, G. Fornetti, M. Guarneri, M. Nuvoli, R. Ricci, and M. Francucci, “Improving underwater imaging in an amplitude modulated laser system with radio frequency control technique,” J. Eur. Opt. Soc. Rapid Pub. 5, 10004 (2009).
[CrossRef]

Francucci, M.

L. De Dominicis, M. Ferri de Collibus, G. Fornetti, M. Guarneri, M. Nuvoli, R. Ricci, and M. Francucci, “Improving underwater imaging in an amplitude modulated laser system with radio frequency control technique,” J. Eur. Opt. Soc. Rapid Pub. 5, 10004 (2009).
[CrossRef]

Guarneri, M.

L. De Dominicis, M. Ferri de Collibus, G. Fornetti, M. Guarneri, M. Nuvoli, R. Ricci, and M. Francucci, “Improving underwater imaging in an amplitude modulated laser system with radio frequency control technique,” J. Eur. Opt. Soc. Rapid Pub. 5, 10004 (2009).
[CrossRef]

Guern, Y.

F. Pellen, P. Olivard, Y. Guern, J. Cariou, and J. Lotrian, “Radio frequency modulation on an optical carrier for target detection enhancement in sea-water,” J. Phys. D 34, 1122–1130 (2001).
[CrossRef]

Herczfeld, P. R.

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of radar technology to aerial lidar systems for enhancement of shallow underwater target detection,” IEEE Trans. Microw. Theory Techn. 43, 2370–2377 (1995).
[CrossRef]

Hirano, Y.

Ho, S.-S.

V. C. Chen, F. Li, S.-S. Ho, and H. Wechsler, “Analysis of micro-Doppler signatures,” IEE Proc. Radar Sonar Navig. 150, 271–276 (2003).
[CrossRef]

Huignard, J.-P.

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for simultaneous velocity and high-resolution range profile measurements,” Proc. SPIE 7323, 73230Z (2009).
[CrossRef]

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for high resolution ranging, profilometry and Doppler measurement,” Proc. SPIE 7114, 71140E (2008).
[CrossRef]

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712 (2002).
[CrossRef]

Jezequel, V.

Kameyama, S.

Kane, T. J.

Kao, D. C.

Kliese, R.

Lai, N. D.

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712 (2002).
[CrossRef]

N. D. Lai, M. Brunel, F. Bretenaker, and O. Emile, “Control of the pulse duration in one- and two-axis passively Q-switched solid-state lasers,” Eur. Phys. J. D 19, 403–410 (2002).

Le Floch, A.

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712 (2002).
[CrossRef]

M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and Ph. Thony, “Offset phase locking of Er:Yb:glass laser eigenstates for RF photonics applications,” IEEE Photon. Technol. Lett. 13, 367–369 (2001).
[CrossRef]

Le Jeune, B.

Li, F.

V. C. Chen, F. Li, S.-S. Ho, and H. Wechsler, “Analysis of micro-Doppler signatures,” IEE Proc. Radar Sonar Navig. 150, 271–276 (2003).
[CrossRef]

Liu, J. M.

Lotrian, J.

F. Pellen, P. Olivard, Y. Guern, J. Cariou, and J. Lotrian, “Radio frequency modulation on an optical carrier for target detection enhancement in sea-water,” J. Phys. D 34, 1122–1130 (2001).
[CrossRef]

Morvan, L.

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for simultaneous velocity and high-resolution range profile measurements,” Proc. SPIE 7323, 73230Z (2009).
[CrossRef]

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for high resolution ranging, profilometry and Doppler measurement,” Proc. SPIE 7114, 71140E (2008).
[CrossRef]

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712 (2002).
[CrossRef]

Mullen, L. J.

D. C. Kao, T. J. Kane, and L. J. Mullen, “Development of an amplitude-modulated Nd:YAG pulsed laser with modulation frequency tunability up to 60 GHz by dual seed injection,” Opt. Lett. 29, 1203–1205 (2004).
[CrossRef]

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of radar technology to aerial lidar systems for enhancement of shallow underwater target detection,” IEEE Trans. Microw. Theory Techn. 43, 2370–2377 (1995).
[CrossRef]

Nuvoli, M.

L. De Dominicis, M. Ferri de Collibus, G. Fornetti, M. Guarneri, M. Nuvoli, R. Ricci, and M. Francucci, “Improving underwater imaging in an amplitude modulated laser system with radio frequency control technique,” J. Eur. Opt. Soc. Rapid Pub. 5, 10004 (2009).
[CrossRef]

Olivard, P.

F. Pellen, P. Olivard, Y. Guern, J. Cariou, and J. Lotrian, “Radio frequency modulation on an optical carrier for target detection enhancement in sea-water,” J. Phys. D 34, 1122–1130 (2001).
[CrossRef]

Otsuka, K.

Pellen, F.

F. Pellen, V. Jezequel, G. Zion, and B. Le Jeune, “Detection of an underwater target through modulated lidar experiments at grazing incidence in a deep wave basin,” Appl. Opt. 51, 7690–7700 (2012).
[CrossRef]

F. Pellen, P. Olivard, Y. Guern, J. Cariou, and J. Lotrian, “Radio frequency modulation on an optical carrier for target detection enhancement in sea-water,” J. Phys. D 34, 1122–1130 (2001).
[CrossRef]

Pillet, G.

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for simultaneous velocity and high-resolution range profile measurements,” Proc. SPIE 7323, 73230Z (2009).
[CrossRef]

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for high resolution ranging, profilometry and Doppler measurement,” Proc. SPIE 7114, 71140E (2008).
[CrossRef]

Qian, L.

Qu, Y.

Rakic, A. D.

Ren, D.

Ricci, R.

L. De Dominicis, M. Ferri de Collibus, G. Fornetti, M. Guarneri, M. Nuvoli, R. Ricci, and M. Francucci, “Improving underwater imaging in an amplitude modulated laser system with radio frequency control technique,” J. Eur. Opt. Soc. Rapid Pub. 5, 10004 (2009).
[CrossRef]

Schotland, R. M.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Thony, Ph.

M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and Ph. Thony, “Offset phase locking of Er:Yb:glass laser eigenstates for RF photonics applications,” IEEE Photon. Technol. Lett. 13, 367–369 (2001).
[CrossRef]

Vallet, M.

M. Brunel and M. Vallet, “Pulse-to-pulse coherent beat note generated by a passively Q-switched two-frequency laser,” Opt. Lett. 33, 2524–2526 (2008).
[CrossRef]

M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and Ph. Thony, “Offset phase locking of Er:Yb:glass laser eigenstates for RF photonics applications,” IEEE Photon. Technol. Lett. 13, 367–369 (2001).
[CrossRef]

Vieira, A. J. C.

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of radar technology to aerial lidar systems for enhancement of shallow underwater target detection,” IEEE Trans. Microw. Theory Techn. 43, 2370–2377 (1995).
[CrossRef]

Wadaka, S.

Wechsler, H.

V. C. Chen, F. Li, S.-S. Ho, and H. Wechsler, “Analysis of micro-Doppler signatures,” IEE Proc. Radar Sonar Navig. 150, 271–276 (2003).
[CrossRef]

Zhao, W.

Zion, G.

Appl. Opt. (5)

Eur. Phys. J. D (1)

N. D. Lai, M. Brunel, F. Bretenaker, and O. Emile, “Control of the pulse duration in one- and two-axis passively Q-switched solid-state lasers,” Eur. Phys. J. D 19, 403–410 (2002).

IEE Proc. Radar Sonar Navig. (1)

V. C. Chen, F. Li, S.-S. Ho, and H. Wechsler, “Analysis of micro-Doppler signatures,” IEE Proc. Radar Sonar Navig. 150, 271–276 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. Alouini, B. Benazet, M. Vallet, M. Brunel, P. Di Bin, F. Bretenaker, A. Le Floch, and Ph. Thony, “Offset phase locking of Er:Yb:glass laser eigenstates for RF photonics applications,” IEEE Photon. Technol. Lett. 13, 367–369 (2001).
[CrossRef]

IEEE Trans. Microw. Theory Techn. (1)

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of radar technology to aerial lidar systems for enhancement of shallow underwater target detection,” IEEE Trans. Microw. Theory Techn. 43, 2370–2377 (1995).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

L. De Dominicis, M. Ferri de Collibus, G. Fornetti, M. Guarneri, M. Nuvoli, R. Ricci, and M. Francucci, “Improving underwater imaging in an amplitude modulated laser system with radio frequency control technique,” J. Eur. Opt. Soc. Rapid Pub. 5, 10004 (2009).
[CrossRef]

J. Phys. D (1)

F. Pellen, P. Olivard, Y. Guern, J. Cariou, and J. Lotrian, “Radio frequency modulation on an optical carrier for target detection enhancement in sea-water,” J. Phys. D 34, 1122–1130 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Proc. SPIE (2)

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for high resolution ranging, profilometry and Doppler measurement,” Proc. SPIE 7114, 71140E (2008).
[CrossRef]

G. Pillet, L. Morvan, D. Dolfi, and J.-P. Huignard, “Wideband dual-frequency lidar-radar for simultaneous velocity and high-resolution range profile measurements,” Proc. SPIE 7323, 73230Z (2009).
[CrossRef]

Other (1)

A. E. Siegman, Lasers (University Science, 1986).

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Figures (8)

Fig. 1.
Fig. 1.

Experimental setup of the transmitter. PQSL: passively Q-switched Nd:YAG laser. M1,2: cavity mirrors. LD: laser diode. AO: acousto-optic frequency shifter driven by a local oscillator LO. M3: feedback mirror. L: matching lens.

Fig. 2.
Fig. 2.

(a) Experimental output pulse power versus time. Inset: 25 ns time window centered at the pulse peak power. (b) Output pulse obtained from a rate equation model.

Fig. 3.
Fig. 3.

Electrical spectral analysis of the output power. (a) Span 500 MHz; RBW 30 kHz. (b) Span 30 kHz; RBW 30 Hz; the horizontal axis corresponds to the spectrum analyzer floor (100dBm).

Fig. 4.
Fig. 4.

Schematic of the experimental setup for measuring the linear velocity v of an indoor target. PBS: polarization beamsplitter. QWP: quarter wave plate.

Fig. 5.
Fig. 5.

Spectral analysis of the optical power backscattered by a target having a linear motion with velocity v. As schematized, the PSD is shown around 2fAO. Left peak: v=1.8ms1. Central peak: v=0ms1. Right peak: v=1.8ms1. Span: 50 Hz. RBW: 1 Hz.

Fig. 6.
Fig. 6.

Experimental Doppler shift fD versus target velocity v.

Fig. 7.
Fig. 7.

Schematic of the experimental setups to study rotation of indoor targets. (a) Rotating diffusing panel. (b) Rotating corner cube. See text for details.

Fig. 8.
Fig. 8.

Experimental PSD around 2fAO of the signal reflected by a rotating corner cube. Span 100 Hz; RBW 1 Hz. Dashed line: Gaussian fit of the envelope.

Equations (31)

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

dEidt=(Γ+κna)Ei/2,
dndt=γ((a0+Γ)ρ/κn)ξ|Ei|2n,
dadt=γa(a0a)μ|Ei|2a,
Eout(t)=T2Ei(t)+η1T2Eout(tτ)e4πjfAOt.
Pout(t)|Ei(t)+ηEi(tτ)e4πjfAOt|2.
Pout(t)=[1+mcos(4πfAOt+φ0(t))][A(t)ΔTrep(t)].
P˜out(f)=1TrepnA˜(n/Trep){δ(fn/Trep)+m2[Fφ0(f2fAOn/Trep)+Fφ0(f+2fAOn/Trep)]}.
|P˜out(f)|2=1Trep2n|A˜(n/Trep)|2{δ(fn/Trep)+m24δ(f2fAOn/Trep)+m24δ(f+2fAOn/Trep)}.
SP(f)m2Trep2n|A˜(n/Trep)|2δ(f2fAOn/Trep).
SR|Π˜1/τ(f)|2δ(f2fAOfD),
PR(t)=p=0q1{(ΠT/qΔT)(tpT/q)×exp(jφ(t+D((tpT/q)[T])))×exp(jφp,0)},
SR(f)=nq2T2|Π˜q/T(nq/TfD)|2×δ(f2fAOn/T).
SR(f)=q2T2|Π˜q/T(fD)|2×δ(f2fAO)δ(f2fAO).
PR(t)=(ΠτΔT)(t)×exp(jφ(t+D(t[T]))).
SR(f)=1T2n|Π˜1/τ(n/TfD)|2δ(f2fAOn/T).
SR(f)=1T2n|Π˜1/τ(n/TfDR)|2δ(f2fAOfDTn/T),
SR(f)=δ(f2fAOfDT).
PR(t)=p=0q1{(ΠT/qΔT)(tpT/q)×exp(4πjfA0t+2πjfD((tpT/q)[T]))}.
exp(2πjfD(t[T]))=(ΠT(t)exp(2πjfDt))ΔT(t),
PR(t)=e4πjfA0tp=0q1{(ΠT/q(tpT/q)ΠT(tpT/q)exp(2πjfD(tpT/q)))ΔT(tpT/q)}.
PR(t)=e4πjfA0tp=0q1{(ΠT/q(tpT/q)exp(2πjfD(tpT/q)))ΔT(tpT/q)}.
P˜R(f)=δ(f2fAO)(1Tp=0q1e2πj(pT/q)fΠ˜q/T(ffD)·Δ1/T(f)),
P˜R(f)=δ(f2fAO)(1Tp=0q1e2πjpn/qnΠ˜qT(n/TfD)δ(fn/T)),
P˜R(f)=1Tn{(p=0q1e2πjpn/q)Π˜q/T(n/TfD)δ(f2fAOn/T)}.
P˜R(f)=qTn{Π˜q/T(n/TfD)δ(f2fAOn/T)}.
PR(t)=e4πjfA0t((Πτ(t)exp(2πjfDt))ΔT(t)),
P˜R(f)=1Tn{Π˜1/τ(n/TfD)δ(f2fAOn/T)}.
PR(t)=p=0q1{(ΠT/qΔT)(tpT/q)×exp(jφ(t+DT(tpT/q)+DR((tpT/q)[T])))},
PR(t)=e2πj(2fA0+fDT)tp=0q1{(ΠT/q(tpT/q)exp(2πjfDR(tpT/q)))ΔT(tpT/q)}.
PR(t)=(ΠτΔT)(t)×exp(jφ(t+DT(t)+DR(t[T]))).
PR(t)=e2πj(2fAO+fDT)t(Πτ(t)exp(2πjfDRt[T]))ΔT(t).

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