Abstract

A compact frequency-modulated, continuous wave (FM-cw) lidar system for measurement of distributed aerosol plumes and hard targets is presented. The system is based on intensity modulation of a laser diode and quadrature detection of the return signals. The advantages of using laser diode amplitude modulation and quadrature detection is a large reduction in the hardware required for processing and storing return signals as well as the availability of off-the-shelf integrated electronic components from the wireless and telecommunication communities. Equations to invert the quadrature signal components and determine spatial distributions of multiple targets are derived. Spatial scattering intensities are used to extract aerosol backscatter coefficients, which can then be directly compared to microphysics aerosol models for environmental measurements. Finally, results from laboratory measurements with a monostatic FM-cw lidar system with both hard targets and aerosols are discussed.

© 2005 Optical Society of America

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  1. K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.
  2. J. E. Nettleton, B. W. Schilling, D. N. Barr, J. S. Lei, “Monoblock laser for a low-cost, eyesafe, microlaser range finder,” Appl. Opt. 39, 2428–2432 (2000).
    [CrossRef]
  3. J. Massa, G. Buller, A. Walker, G. Smith, S. Cova, M. Umasuthan, A. Wallace, “Optical design of three-dimensional imaging and ranging system based on time-correlated, single-photon counting,” Appl. Opt. 41, 1063–1070 (2002).
    [CrossRef] [PubMed]
  4. M. Bashkansky, H. R. Burris, E. E. Funk, R. Mahon, C. I. Moore, “RF phase-coded random-modulation LIDAR,” Opt. Commun. 231, 93–98 (2004).
    [CrossRef]
  5. R. B. Chadwick, R. G. Strauch, “Processingof FM-cw Doppler radar signals from distributed targets,” IEEE Trans. Aerosp. Electron. Syst. AES-15, 185–188 (1979).
    [CrossRef]
  6. M. L. Simpson, C. A. Bennett, M. S. Emery, D. P. Hutchinson, G. H. Miller, R. K. Richards, D. N. Sitter, “Coherent imaging with two-dimensional focal-plane arrays: design and application,” Appl. Opt. 36, 6913–6920 (1997).
    [CrossRef]
  7. B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270– 3278, (1996).
    [CrossRef]
  8. B. Stann, A. Abou-Auf, K. Aliberti, M. Giza, B. Ovrebo, W. Ruff, D. Simon, M. Stead, “Research progress on scannerless ladar systems using a laser diode transmitter and FM/cw radar principles,” in Aerosense 2002 Laser Radar Technology and Applications VII, G. W. Kamerman, ed., Proc. SPIE47234723–4733, (2002).
  9. R. L. Schmitt, R. J. Williams, J. D. Matthews, “High-frequency scannerless imaging laser radar for industrial inspection and measurement applications,” (Sandia National Laboratory, 1996).
    [CrossRef]
  10. E. Hecht, Optics (Addison-Wesley, 1987), pp. 12–21 and 246–247.
  11. A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1989), pp. 514–561.
  12. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), p. 181.
  13. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  14. W. K. Pratt, Laser Communications Systems (Wiley, 1969), pp. 178–183.
  15. J. Rothermel, D. M. Chambers, M. A. Jarzembski, V. Srivastava, D. A. Bowdle, W. D. Jones, “Signal processing and calibration of continuous-wave focused CO2 Doppler lidars for atmospheric backscatter measurement,” Appl. Opt. 35, 2083–2095 (1996).
    [CrossRef] [PubMed]
  16. J. H. Seinfeld, S. N. Pandis, Atmospheric Chemistry and Physics (Wiley, 1998), p. 1118.
  17. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), p. 589.

2004 (1)

M. Bashkansky, H. R. Burris, E. E. Funk, R. Mahon, C. I. Moore, “RF phase-coded random-modulation LIDAR,” Opt. Commun. 231, 93–98 (2004).
[CrossRef]

2002 (1)

2000 (1)

1997 (1)

1996 (2)

1979 (1)

R. B. Chadwick, R. G. Strauch, “Processingof FM-cw Doppler radar signals from distributed targets,” IEEE Trans. Aerosp. Electron. Syst. AES-15, 185–188 (1979).
[CrossRef]

Abou-Auf, A.

B. Stann, A. Abou-Auf, K. Aliberti, M. Giza, B. Ovrebo, W. Ruff, D. Simon, M. Stead, “Research progress on scannerless ladar systems using a laser diode transmitter and FM/cw radar principles,” in Aerosense 2002 Laser Radar Technology and Applications VII, G. W. Kamerman, ed., Proc. SPIE47234723–4733, (2002).

Akerman, A.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Aliberti, K.

B. Stann, A. Abou-Auf, K. Aliberti, M. Giza, B. Ovrebo, W. Ruff, D. Simon, M. Stead, “Research progress on scannerless ladar systems using a laser diode transmitter and FM/cw radar principles,” in Aerosense 2002 Laser Radar Technology and Applications VII, G. W. Kamerman, ed., Proc. SPIE47234723–4733, (2002).

Ayers, C. W.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Barr, D. N.

Bashkansky, M.

M. Bashkansky, H. R. Burris, E. E. Funk, R. Mahon, C. I. Moore, “RF phase-coded random-modulation LIDAR,” Opt. Commun. 231, 93–98 (2004).
[CrossRef]

Bennett, C. A.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), p. 589.

Bowdle, D. A.

Buller, G.

Burris, H. R.

M. Bashkansky, H. R. Burris, E. E. Funk, R. Mahon, C. I. Moore, “RF phase-coded random-modulation LIDAR,” Opt. Commun. 231, 93–98 (2004).
[CrossRef]

Chadwick, R. B.

R. B. Chadwick, R. G. Strauch, “Processingof FM-cw Doppler radar signals from distributed targets,” IEEE Trans. Aerosp. Electron. Syst. AES-15, 185–188 (1979).
[CrossRef]

Chambers, D. M.

Cheng, M.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Cova, S.

Dam, T. Q.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Emery, M. S.

Fisher, W. G.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Funk, E. E.

M. Bashkansky, H. R. Burris, E. E. Funk, R. Mahon, C. I. Moore, “RF phase-coded random-modulation LIDAR,” Opt. Commun. 231, 93–98 (2004).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), p. 181.

Giza, M.

B. Stann, A. Abou-Auf, K. Aliberti, M. Giza, B. Ovrebo, W. Ruff, D. Simon, M. Stead, “Research progress on scannerless ladar systems using a laser diode transmitter and FM/cw radar principles,” in Aerosense 2002 Laser Radar Technology and Applications VII, G. W. Kamerman, ed., Proc. SPIE47234723–4733, (2002).

Goedeke, S.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1987), pp. 12–21 and 246–247.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), p. 589.

Hutchinson, D. P.

Jarzembski, M. A.

Jones, W. D.

Lei, J. S.

Lenox, K. E.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Mahon, R.

M. Bashkansky, H. R. Burris, E. E. Funk, R. Mahon, C. I. Moore, “RF phase-coded random-modulation LIDAR,” Opt. Commun. 231, 93–98 (2004).
[CrossRef]

Massa, J.

Matthews, J. D.

R. L. Schmitt, R. J. Williams, J. D. Matthews, “High-frequency scannerless imaging laser radar for industrial inspection and measurement applications,” (Sandia National Laboratory, 1996).
[CrossRef]

McGill, R. N.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Miller, G. H.

Moore, C. I.

M. Bashkansky, H. R. Burris, E. E. Funk, R. Mahon, C. I. Moore, “RF phase-coded random-modulation LIDAR,” Opt. Commun. 231, 93–98 (2004).
[CrossRef]

Nettleton, J. E.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1989), pp. 514–561.

Ovrebo, B.

B. Stann, A. Abou-Auf, K. Aliberti, M. Giza, B. Ovrebo, W. Ruff, D. Simon, M. Stead, “Research progress on scannerless ladar systems using a laser diode transmitter and FM/cw radar principles,” in Aerosense 2002 Laser Radar Technology and Applications VII, G. W. Kamerman, ed., Proc. SPIE47234723–4733, (2002).

Pandis, S. N.

J. H. Seinfeld, S. N. Pandis, Atmospheric Chemistry and Physics (Wiley, 1998), p. 1118.

Pratt, W. K.

W. K. Pratt, Laser Communications Systems (Wiley, 1969), pp. 178–183.

Richards, R. K.

M. L. Simpson, C. A. Bennett, M. S. Emery, D. P. Hutchinson, G. H. Miller, R. K. Richards, D. N. Sitter, “Coherent imaging with two-dimensional focal-plane arrays: design and application,” Appl. Opt. 36, 6913–6920 (1997).
[CrossRef]

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Rothermel, J.

Ruff, W.

B. Stann, A. Abou-Auf, K. Aliberti, M. Giza, B. Ovrebo, W. Ruff, D. Simon, M. Stead, “Research progress on scannerless ladar systems using a laser diode transmitter and FM/cw radar principles,” in Aerosense 2002 Laser Radar Technology and Applications VII, G. W. Kamerman, ed., Proc. SPIE47234723–4733, (2002).

Ruff, W. C.

B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270– 3278, (1996).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1989), pp. 514–561.

Schilling, B. W.

Schmitt, R. L.

R. L. Schmitt, R. J. Williams, J. D. Matthews, “High-frequency scannerless imaging laser radar for industrial inspection and measurement applications,” (Sandia National Laboratory, 1996).
[CrossRef]

Seinfeld, J. H.

J. H. Seinfeld, S. N. Pandis, Atmospheric Chemistry and Physics (Wiley, 1998), p. 1118.

Simon, D.

B. Stann, A. Abou-Auf, K. Aliberti, M. Giza, B. Ovrebo, W. Ruff, D. Simon, M. Stead, “Research progress on scannerless ladar systems using a laser diode transmitter and FM/cw radar principles,” in Aerosense 2002 Laser Radar Technology and Applications VII, G. W. Kamerman, ed., Proc. SPIE47234723–4733, (2002).

Simpson, M. L.

M. L. Simpson, C. A. Bennett, M. S. Emery, D. P. Hutchinson, G. H. Miller, R. K. Richards, D. N. Sitter, “Coherent imaging with two-dimensional focal-plane arrays: design and application,” Appl. Opt. 36, 6913–6920 (1997).
[CrossRef]

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Sitter, D. N.

Smith, G.

Srivastava, V.

Stann, B.

B. Stann, A. Abou-Auf, K. Aliberti, M. Giza, B. Ovrebo, W. Ruff, D. Simon, M. Stead, “Research progress on scannerless ladar systems using a laser diode transmitter and FM/cw radar principles,” in Aerosense 2002 Laser Radar Technology and Applications VII, G. W. Kamerman, ed., Proc. SPIE47234723–4733, (2002).

Stann, B. L.

B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270– 3278, (1996).
[CrossRef]

Stead, M.

B. Stann, A. Abou-Auf, K. Aliberti, M. Giza, B. Ovrebo, W. Ruff, D. Simon, M. Stead, “Research progress on scannerless ladar systems using a laser diode transmitter and FM/cw radar principles,” in Aerosense 2002 Laser Radar Technology and Applications VII, G. W. Kamerman, ed., Proc. SPIE47234723–4733, (2002).

Storey, J. M.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Strauch, R. G.

R. B. Chadwick, R. G. Strauch, “Processingof FM-cw Doppler radar signals from distributed targets,” IEEE Trans. Aerosp. Electron. Syst. AES-15, 185–188 (1979).
[CrossRef]

Sztankay, Z. G.

B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270– 3278, (1996).
[CrossRef]

Umasuthan, M.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

Wachter, E. A.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

Walker, A.

Wallace, A.

Williams, R. J.

R. L. Schmitt, R. J. Williams, J. D. Matthews, “High-frequency scannerless imaging laser radar for industrial inspection and measurement applications,” (Sandia National Laboratory, 1996).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Aerosp. Electron. Syst. (1)

R. B. Chadwick, R. G. Strauch, “Processingof FM-cw Doppler radar signals from distributed targets,” IEEE Trans. Aerosp. Electron. Syst. AES-15, 185–188 (1979).
[CrossRef]

Opt. Commun. (1)

M. Bashkansky, H. R. Burris, E. E. Funk, R. Mahon, C. I. Moore, “RF phase-coded random-modulation LIDAR,” Opt. Commun. 231, 93–98 (2004).
[CrossRef]

Opt. Eng. (1)

B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270– 3278, (1996).
[CrossRef]

Other (10)

B. Stann, A. Abou-Auf, K. Aliberti, M. Giza, B. Ovrebo, W. Ruff, D. Simon, M. Stead, “Research progress on scannerless ladar systems using a laser diode transmitter and FM/cw radar principles,” in Aerosense 2002 Laser Radar Technology and Applications VII, G. W. Kamerman, ed., Proc. SPIE47234723–4733, (2002).

R. L. Schmitt, R. J. Williams, J. D. Matthews, “High-frequency scannerless imaging laser radar for industrial inspection and measurement applications,” (Sandia National Laboratory, 1996).
[CrossRef]

E. Hecht, Optics (Addison-Wesley, 1987), pp. 12–21 and 246–247.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1989), pp. 514–561.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), p. 181.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

W. K. Pratt, Laser Communications Systems (Wiley, 1969), pp. 178–183.

K. E. Lenox, A. Akerman, C. W. Ayers, T. Q. Dam, S. Goedeke, R. N. McGill, J. M. Storey, M. Cheng, R. K. Richards, M. L. Simpson, W. G. Fisher, E. A. Wachter, “Further development of remote sensing instrumentation for NOx and PM emissions from heavy duty trucks,” in Proceedings Air and Waste Management 96th Annual Conference, A. Knopes, ed. (Air and Waste Management Association, 2003), paper 69731.

J. H. Seinfeld, S. N. Pandis, Atmospheric Chemistry and Physics (Wiley, 1998), p. 1118.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), p. 589.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the stepped frequency lidar system. The laser is intensity modulated with a rf signal in 10 MHz steps from 10 to 200 MHz. The amplitude and the phase of the return lidar signal are detected at each frequency by use of a lock-in amplifier, and the resultant vector is then postprocessed to yield range information.

Fig. 2
Fig. 2

Complex representation of the modulated lidar signal. Scattering is depicted from two regions in space, resulting in a phase difference in the two return signals. The observer sees a superposition of the two return signal waveforms.

Fig. 3
Fig. 3

Schematic of the lock-in amplifier. The signal used to drive the laser provides a reference signal from which sine and cosine waveforms are generated. These sine and cosine reference waveforms are then mixed with the return lidar signal and filtered. The outputs of the lock-in amplifier are then the quadrature components of the return lidar signal from which amplitude and phase can be extracted.

Fig. 4
Fig. 4

Expected return signal waveform (postprocessing) for a single target from a stepped frequency lidar is a sinc function with amplitude proportional to the backscatter signal amplitude and width inversely proportional to twice the modulation bandwidth. The FWHM of the sinc function determines the range resolution of the lidar system.

Fig. 5
Fig. 5

Processed scattering waveform from a single translucent target located 2.5 m from the stepped frequency lidar system. The return signal is a sinc function as expected and has a spatial resolution of 0.75 m (FWHM), corresponding to a modulation bandwidth of 190 MHz.

Fig. 6
Fig. 6

Normalized scattering waveform (postprocessing) from multiple translucent targets, demonstrating the capability of stepped frequency lidar to measure distributed targets. The actual target locations are provided above the peaks.

Fig. 7
Fig. 7

Results from a laboratory diesel engine test, demonstrating the correlation between calculated beta values from sampled SMPS data and lidar SNR measurements, shown on a dual-axis plot. The lidar results are from measurements made on two separate days. The larger value of SNR at 0 km/h on day 2 is due to a cold engine start.

Equations (15)

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E ( x , t ) = E 0 cos ( k x ± ω t + ɛ ) .
E obs = E obs exp [ i ( k x ± ω t + ɛ ) ] = E 1 exp [ i ( 2 k x 1 ± ω t + ɛ ) ] + E 2 exp [ i ( 2 k x 2 ± ω t + ɛ ) ] .
E meas = E ω 0 exp [ i ( Θ ω 0 ) ] = E 1 exp [ i ( 2 k x 1 ) ] + E 2 exp [ i ( 2 k x 2 ) ] ,
E ω j exp [ i ( Θ ω j ) ] = l = 0 M 1 E l exp [ i ( 2 ω j c x l ) ] , j = 0 N 1 .
E norm ( ω j ) = E norm ω j exp [ i ( Θ ω j Θ ref ω j ) ] = l = 0 M 1 E norm l exp { i [ 2 ω j c ( x l x ref ) ] } , j = 0 N 1 ,
f ( x ) = F ( β ) exp [ i 2 π β x ] d β .
E norm ( x ) = E norm ( ω ) exp [ i 2 ω c ( x x ref ) ] d ω .
E norm ( x l ) = 1 N j = 0 N 1 Re { E norm ω j exp [ i ( Θ ω j Θ ref ω j ) ] × exp [ i 2 ω j c ( x l x ref ) ] } , l = 0 M 1 .
G ( f ) = A rect ( f f 0 b ) = { 0 | f f 0 b | > 1 2 A 2 | f f 0 b | = 1 2 A | f f 0 b | < 1 2 ,
g ( x ) = G ( f ) exp ( i 2 π x f ) d f = A b sinc ( b x ) exp ( i 2 π x f 0 ) .
SNR receiver = [ G ( η q / h f ) ] 2 P sig 2 R L 16 G 2 q { ( η q / h f ) [ ( P sig / 2 ) + P bkg ] + I drk } B R L + 32 kTB ,
η P sig = η sys P laser β Ω opt .
SNR system = [ G ( η sys q / h f ) ] 2 P laser 2 β 2 Ω opt 2 R L 16 G 2 q { ( q / h f ) [ ( η sys P laser β Ω opt / 2 ) + η P bkg ] + I drk } B R L + 32 kTB .
SNR system = η sys P laser β Ω opt 8 hfB .
SNR meas ( x k ) = [ E norm meas ( x k ) E norm bkg 1 ( x k ) ] 2 1 M k [ E norm bkg 2 ( x k ) E norm bkg 1 ( x k ) ] 2 ,

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