Abstract

A continuous wave (CW) Lidar system for detection of scattering from atmospheric aerosol particles is presented which is useful in particular for remote sensing of wind velocities. It is based on a low-coherence interferometric setup powered by a synthetic broadband laser source with Gaussian power density spectrum. The laser bandwidth is electronically adjustable and determines the spatial resolution which is independent of range. The Lidar system has no moving parts. The location to be resolved can be shifted numerically after the measurement meaning that a single measurement already contains the full range information. The features of constant resolution and numerical range scanning are in sharp contrast to ordinary CW Lidar systems.

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. A. Kohlhaas, C. Frömchen, and E. Brinkmeyer, “High-resolution OCDR for testing integrated optical waveguide: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol.9, 1493–1502 (1991).
    [CrossRef]
  4. D. M. Baney and W. V. Sorin, “Extended-range optical low coherence reflectometry using a recirculation delay technique,” IEEE Photon. Technol. Lett.5, 1109–1102 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  14. E. Brinkmeyer and T. Waterholter, “Lidar-Messsystem,” pending European patent application P89429, November12, 2012.
  15. A. Yariv, Optical Electronics in Modern Communications (Oxford Univ. Press, 1997).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  21. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckles and Related Phenomena, J.C. Dainty, ed. (Springer, 1984).
  22. C. Weitkamp, Lidar Range-Resolved Optical Remote Sensing of the Atmosphere (Springer, 2005)
  23. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J.56, 703–718 (1977).
  24. F. L. Pedrotti, Introduction to Optics (Prentice-Hall, 1993).

2001

2000

1993

D. M. Baney and W. V. Sorin, “Extended-range optical low coherence reflectometry using a recirculation delay technique,” IEEE Photon. Technol. Lett.5, 1109–1102 (1993).
[CrossRef]

P. Lambelet, P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, C. Zimmer, and H. H. Gilgen, “Bragg grating characterization by optical low-coherence reflectometry,” IEEE Photon. Technol. Lett.5(5), 565–567 (1993).
[CrossRef]

U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” J. Lightwave Technol.11(8), 1377–1384 (1993).
[CrossRef]

K. Takada, T. Kitagawa, M. Shimizu, and M. Horiguchi, “High-sensitivity low coherence reflectometer using erbium-doped superfluorescent fiber source and erbium doped power amplifier,” Electron. Lett.29, 365–367 (1993).
[CrossRef]

1991

A. Kohlhaas, C. Frömchen, and E. Brinkmeyer, “High-resolution OCDR for testing integrated optical waveguide: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol.9, 1493–1502 (1991).
[CrossRef]

1988

1987

1980

T. Okoshi, K. Kikucho, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett.12, 630–631 (1980).
[CrossRef]

1977

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J.56, 703–718 (1977).

1971

Antoniou, I.

I. Antoniou and , “Remote sensing the wind using Lidars and Sodars,” Proceedings of the 2007 European Union Wind Energy Conference, Milan, Italy (2007).

Baney, D. M.

D. M. Baney and W. V. Sorin, “Extended-range optical low coherence reflectometry using a recirculation delay technique,” IEEE Photon. Technol. Lett.5, 1109–1102 (1993).
[CrossRef]

Brinkmeyer, E.

U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” J. Lightwave Technol.11(8), 1377–1384 (1993).
[CrossRef]

A. Kohlhaas, C. Frömchen, and E. Brinkmeyer, “High-resolution OCDR for testing integrated optical waveguide: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol.9, 1493–1502 (1991).
[CrossRef]

E. Brinkmeyer and T. Waterholter, “Lidar-Messsystem,” pending European patent application P89429, November12, 2012.

Carr, S.

Constant, G.

Davies, D.E.N.

Fonjallaz, P. Y.

P. Lambelet, P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, C. Zimmer, and H. H. Gilgen, “Bragg grating characterization by optical low-coherence reflectometry,” IEEE Photon. Technol. Lett.5(5), 565–567 (1993).
[CrossRef]

Frehlich, R.

E. Simley, L. Y. Pao, R. Frehlich, B. Jonkman, and N. Kelley, “Analysis of wind speed measurements using continuous wave Lidar for wind turbine control,” 49th AIAA Aerospace Sciences Meeting, Orlando, Florida (2011).

Frömchen, C.

A. Kohlhaas, C. Frömchen, and E. Brinkmeyer, “High-resolution OCDR for testing integrated optical waveguide: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol.9, 1493–1502 (1991).
[CrossRef]

Gilgen, H. H.

P. Lambelet, P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, C. Zimmer, and H. H. Gilgen, “Bragg grating characterization by optical low-coherence reflectometry,” IEEE Photon. Technol. Lett.5(5), 565–567 (1993).
[CrossRef]

Glombitza, U.

U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” J. Lightwave Technol.11(8), 1377–1384 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckles and Related Phenomena, J.C. Dainty, ed. (Springer, 1984).

Harris, M.

Horiguchi, M.

K. Takada, T. Kitagawa, M. Shimizu, and M. Horiguchi, “High-sensitivity low coherence reflectometer using erbium-doped superfluorescent fiber source and erbium doped power amplifier,” Electron. Lett.29, 365–367 (1993).
[CrossRef]

Horrigan, F. A.

Jonkman, B.

E. Simley, L. Y. Pao, R. Frehlich, B. Jonkman, and N. Kelley, “Analysis of wind speed measurements using continuous wave Lidar for wind turbine control,” 49th AIAA Aerospace Sciences Meeting, Orlando, Florida (2011).

Karlsson, C. J.

Kelley, N.

E. Simley, L. Y. Pao, R. Frehlich, B. Jonkman, and N. Kelley, “Analysis of wind speed measurements using continuous wave Lidar for wind turbine control,” 49th AIAA Aerospace Sciences Meeting, Orlando, Florida (2011).

Kikucho, K.

T. Okoshi, K. Kikucho, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett.12, 630–631 (1980).
[CrossRef]

Kitagawa, T.

K. Takada, T. Kitagawa, M. Shimizu, and M. Horiguchi, “High-sensitivity low coherence reflectometer using erbium-doped superfluorescent fiber source and erbium doped power amplifier,” Electron. Lett.29, 365–367 (1993).
[CrossRef]

Kohlhaas, A.

A. Kohlhaas, C. Frömchen, and E. Brinkmeyer, “High-resolution OCDR for testing integrated optical waveguide: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol.9, 1493–1502 (1991).
[CrossRef]

Lambelet, P.

P. Lambelet, P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, C. Zimmer, and H. H. Gilgen, “Bragg grating characterization by optical low-coherence reflectometry,” IEEE Photon. Technol. Lett.5(5), 565–567 (1993).
[CrossRef]

Letalick, D.

Limberger, H. G.

P. Lambelet, P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, C. Zimmer, and H. H. Gilgen, “Bragg grating characterization by optical low-coherence reflectometry,” IEEE Photon. Technol. Lett.5(5), 565–567 (1993).
[CrossRef]

Lindelöw, P.

P. Lindelöw and J. J. Mohr, “Coherent lidar modulated with frequency stepped pulse trains for unambiguous high duty cycle range and velocity sensing in the atmosphere,” International Geoscience and Remote Sensing Symposium (IGARSS), pages 2787–2790 (2008).

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J.56, 703–718 (1977).

Mohr, J. J.

P. Lindelöw and J. J. Mohr, “Coherent lidar modulated with frequency stepped pulse trains for unambiguous high duty cycle range and velocity sensing in the atmosphere,” International Geoscience and Remote Sensing Symposium (IGARSS), pages 2787–2790 (2008).

Nakayama, A.

T. Okoshi, K. Kikucho, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett.12, 630–631 (1980).
[CrossRef]

Okoshi, T.

T. Okoshi, K. Kikucho, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett.12, 630–631 (1980).
[CrossRef]

Olsson, F. A.

Pao, L. Y.

E. Simley, L. Y. Pao, R. Frehlich, B. Jonkman, and N. Kelley, “Analysis of wind speed measurements using continuous wave Lidar for wind turbine control,” 49th AIAA Aerospace Sciences Meeting, Orlando, Florida (2011).

Pedersen, A. T.

A. T. Pedersen, “Frequency swept fibre laser for wind speed measurements,” PhD thesis at the Technical University of Denmark, December 2011.

Pedrotti, F. L.

F. L. Pedrotti, Introduction to Optics (Prentice-Hall, 1993).

Salathé, R. P.

P. Lambelet, P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, C. Zimmer, and H. H. Gilgen, “Bragg grating characterization by optical low-coherence reflectometry,” IEEE Photon. Technol. Lett.5(5), 565–567 (1993).
[CrossRef]

Shimizu, M.

K. Takada, T. Kitagawa, M. Shimizu, and M. Horiguchi, “High-sensitivity low coherence reflectometer using erbium-doped superfluorescent fiber source and erbium doped power amplifier,” Electron. Lett.29, 365–367 (1993).
[CrossRef]

Simley, E.

E. Simley, L. Y. Pao, R. Frehlich, B. Jonkman, and N. Kelley, “Analysis of wind speed measurements using continuous wave Lidar for wind turbine control,” 49th AIAA Aerospace Sciences Meeting, Orlando, Florida (2011).

Sonnenschein, C. M.

Sorin, W. V.

D. M. Baney and W. V. Sorin, “Extended-range optical low coherence reflectometry using a recirculation delay technique,” IEEE Photon. Technol. Lett.5, 1109–1102 (1993).
[CrossRef]

W. V. Sorin, “Optical reflectometry for component characterization,” in Fiber Optic Test and Measurement, D. Derrickson, ed. (Prentice-Hall, 1998).

W. V. Sorin, “Noise sources in optical measurements,” in Fiber Optic Test and Measurement, D. Derrickson, ed. (Prentice-Hall, 1998).

Takada, K.

K. Takada, T. Kitagawa, M. Shimizu, and M. Horiguchi, “High-sensitivity low coherence reflectometer using erbium-doped superfluorescent fiber source and erbium doped power amplifier,” Electron. Lett.29, 365–367 (1993).
[CrossRef]

Wang, J. Y.

Ward, C.

Waterholter, T.

E. Brinkmeyer and T. Waterholter, “Lidar-Messsystem,” pending European patent application P89429, November12, 2012.

Weitkamp, C.

C. Weitkamp, Lidar Range-Resolved Optical Remote Sensing of the Atmosphere (Springer, 2005)

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications (Oxford Univ. Press, 1997).

Youngquist, R. C.

Zimmer, C.

P. Lambelet, P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, C. Zimmer, and H. H. Gilgen, “Bragg grating characterization by optical low-coherence reflectometry,” IEEE Photon. Technol. Lett.5(5), 565–567 (1993).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J.56, 703–718 (1977).

Electron. Lett.

T. Okoshi, K. Kikucho, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett.12, 630–631 (1980).
[CrossRef]

K. Takada, T. Kitagawa, M. Shimizu, and M. Horiguchi, “High-sensitivity low coherence reflectometer using erbium-doped superfluorescent fiber source and erbium doped power amplifier,” Electron. Lett.29, 365–367 (1993).
[CrossRef]

IEEE Photon. Technol. Lett.

D. M. Baney and W. V. Sorin, “Extended-range optical low coherence reflectometry using a recirculation delay technique,” IEEE Photon. Technol. Lett.5, 1109–1102 (1993).
[CrossRef]

P. Lambelet, P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, C. Zimmer, and H. H. Gilgen, “Bragg grating characterization by optical low-coherence reflectometry,” IEEE Photon. Technol. Lett.5(5), 565–567 (1993).
[CrossRef]

J. Lightwave Technol.

U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” J. Lightwave Technol.11(8), 1377–1384 (1993).
[CrossRef]

A. Kohlhaas, C. Frömchen, and E. Brinkmeyer, “High-resolution OCDR for testing integrated optical waveguide: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol.9, 1493–1502 (1991).
[CrossRef]

Opt. Lett.

Other

W. V. Sorin, “Noise sources in optical measurements,” in Fiber Optic Test and Measurement, D. Derrickson, ed. (Prentice-Hall, 1998).

P. Lindelöw and J. J. Mohr, “Coherent lidar modulated with frequency stepped pulse trains for unambiguous high duty cycle range and velocity sensing in the atmosphere,” International Geoscience and Remote Sensing Symposium (IGARSS), pages 2787–2790 (2008).

A. T. Pedersen, “Frequency swept fibre laser for wind speed measurements,” PhD thesis at the Technical University of Denmark, December 2011.

M. Harris, “Introduction to continuous-wave Doppler lidar,” in Remote Sensing for Wind Energy, A. Peña and C. B. Hasager, eds. (RisøDTU, 2011), pp. 41–64.

I. Antoniou and , “Remote sensing the wind using Lidars and Sodars,” Proceedings of the 2007 European Union Wind Energy Conference, Milan, Italy (2007).

E. Simley, L. Y. Pao, R. Frehlich, B. Jonkman, and N. Kelley, “Analysis of wind speed measurements using continuous wave Lidar for wind turbine control,” 49th AIAA Aerospace Sciences Meeting, Orlando, Florida (2011).

E. Brinkmeyer and T. Waterholter, “Lidar-Messsystem,” pending European patent application P89429, November12, 2012.

A. Yariv, Optical Electronics in Modern Communications (Oxford Univ. Press, 1997).

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckles and Related Phenomena, J.C. Dainty, ed. (Springer, 1984).

C. Weitkamp, Lidar Range-Resolved Optical Remote Sensing of the Atmosphere (Springer, 2005)

W. V. Sorin, “Optical reflectometry for component characterization,” in Fiber Optic Test and Measurement, D. Derrickson, ed. (Prentice-Hall, 1998).

F. L. Pedrotti, Introduction to Optics (Prentice-Hall, 1993).

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Figures (20)

Fig. 1
Fig. 1

Low coherence CW-Lidar setup

Fig. 2
Fig. 2

Division of measurement range into zones of length Δzc

Fig. 3
Fig. 3

Simulated normalized electrical power density spectrum for power ratios 〈Ptot〉 / 〈Pcoh〉 = [1, 10, 100]

Fig. 4
Fig. 4

Setup for source with adjustable spectral width

Fig. 5
Fig. 5

Normalized optical power density spectrum for ΔνL = 10 MHz (a) and ΔνL = 100 MHz (b), green curve: Gaussian target spectrum, blue curve: iteration result; bottom: cutout of phase Θ(t)

Fig. 6
Fig. 6

Result from iteration and Gaussian target function for ΔνL = 38.2 MHz: a) Normalized optical power density spectrum; b) temporal degree of coherence |γ11|

Fig. 7
Fig. 7

Normalized power spectral density for uniformly weighted zones w(mc) ≡ 1; inset: central peak

Fig. 8
Fig. 8

Normalized power spectral density for increased scattered power from outside the matched zone: w̱ = [1 1 1 1 10  1 1 1 1 1 1]

Fig. 9
Fig. 9

Normalized power spectral density for increased scattered power from outside the matched zone: w̱ = [1 1 10 10 10  1  1 1 1 1 1]

Fig. 10
Fig. 10

SNR for uniform weighting w(mc) ≡ 1 as function of averages Mav

Fig. 11
Fig. 11

Normalized power spectral density for uniformly weighted zones w(mc) ≡ 1 with 〈Pcoh〉 = 250 fW

Fig. 12
Fig. 12

Normalized power spectral density for uniformly weighted zones w(mc) ≡ 1 with 〈Pcoh〉 = 1 pW

Fig. 13
Fig. 13

Simulation of post-hoc shifting the matched zone; a) shift by +4Δzc = +20 m; b) shift by −3Δzc = −15 m

Fig. 14
Fig. 14

Normalized measured electrical power spectral density of the synthetic broad-band source with delayed-self-heterodyne-method, Δfel = 55.4 MHz, Δ ν L = Δ f el / 2 = 39.2 MHz, N = 16384 data points, δf = 1/(N * δt), Mav = 1, 16384 data points

Fig. 15
Fig. 15

Lidar Measuremtent with static target; (a) Normalized power density spectrum with target at z = 10 m, inside matched zone mc = 0, inset: closeup of central peak of; (b) Normalized power density spectrum with target at z = 5 m inside zone mc = −1

Fig. 16
Fig. 16

Lidar Measuremtent with moving target; (a) Normalized power density spectrum with target at z = 10 m, inside matched zone mc = 0, inset: closeup of central peak of; (b) Normalized power density spectrum with target at z = 5 m inside zone mc = −1

Fig. 17
Fig. 17

(a) Evaluation of data for moving target at z = 10 m ; (b) Reevaluation of data: numerical shift of matched zone to location of static target at z = 5 m

Fig. 18
Fig. 18

Gaussian beam of the Lidar setup

Fig. 19
Fig. 19

Factors considered for calculation of η(x,y,z)

Fig. 20
Fig. 20

Received power per length element p(z) and launching efficiency η̄(z)

Equations (49)

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S L ( ν ) = S ^ L exp [ ( ν ν 0 Δ ν L / 2 ) 2 ]
E ( t ) = E ^ exp ( j ω 0 t ) exp [ j Θ ( t ) ]
| γ 11 ( τ ) | = exp [ ( π Δ ν L 2 τ ) 2 ] = exp [ ( τ τ c ) 2 ]
δ i a , b ( t ) = { δ P 2 + P LO 2 ± 2 δ P 2 P LO 2 cos [ ω AOM t + Θ ( t + t d ) Θ ( t ) + Ψ ( z ) ] }
δ i ( t ) = δ i a ( t ) δ i b ( t ) = δ i ^ 2 exp { j [ ω AOM t + ϕ ( t ; t d ) + ψ ( z ) ] } + c . c .
δ S ( f ) = ( δ i ^ ) 2 T 2 π { 1 Δ f rec { sin [ π ( f f AOM ) T ] π ( f f AOM ) T } 2 for t d τ c 1 Δ f exp [ ( f f AOM Δ f / 2 ) 2 ] for t d τ c .
δ S ( f ; t d ) = ( δ i ^ ) 2 T 2 π ( 1 Δ f rec exp [ 2 ( t d τ c ) 2 ] { sin [ π ( f f AOM ) T ] π ( f f AOM ) T } 2 + 1 Δ f { 1 exp ( 2 ( t d τ c ) 2 ) } exp [ ( f f AOM Δ f / 2 ) 2 ] ) .
q narrow ( m c ) = 1 2 τ c ( 2 m c 1 ) τ c ( 2 m c + 1 ) τ c exp [ 2 ( t d τ c ) 2 ] d t d = π 32 { erf [ 2 ( 2 m c + 1 ) ] erf [ 2 ( 2 m c 1 ) ] } = { 0.6 for m c = 0 0.014 for m c = ± 1 < 6 10 10 for | m c | 2
q broad ( m c ) = 1 q narrow ( m c ) = { 0.4 for m c = 0 0.986 for m c = ± 1 1 for | m c | 2
Δ S narrow ( p ) ( m c ) = A LO 1 Δ f rec Δ P ( m c ) q narrow ( m c )
Δ S broad ( p ) ( m c ) = A LO 1 Δ f Δ P ( m c ) q broad ( m c )
S narrow ( p ) A LO 1 Δ f rec 0.6 P coh
S broad ( p ) A LO 1 Δ f P coh ( μ 0.6 )
σ narrow = S narrow
σ broad = S broad
S ( f ) = S narrow ( p ) exp [ ( f f AOM Δ f sp / 2 ) 2 ] + S broad ( p ) exp [ ( f f AOM Δ f / 2 ) 2 ]
SNR = ( S up S low ) 2 σ up 2 + σ low 2
SNR = S narrow ( p ) 2 S up 2 + S low 2 = 1 1 + 2 ξ + 2 ξ 2
ξ = Δ f sp ( μ 0.6 ) Δ f 0.6
SNR sp = M av 1 + 2 ξ + 2 ξ 2
SNR SN = M av S narrow ( p ) 2 2 σ SN 2 = M av S narrow ( p ) 2 2 ( 2 e P LO T ) 2 = M av S narrow 2 2 ( A LO π e / ) 2
SNR 1 = SNR sp 1 + SNR SN 1
E ( t ) = E ^ exp ( j ω 0 t ) exp [ j Θ ( t ) ]
S L ( f opt ) = | E ^ exp ( j 2 π f opt t ) exp [ j Θ ( t ) ] d t | 2
{ exp [ j Θ ( t ) ] } = exp [ 1 2 ( f opt Δ ν L / 2 ) 2 ] exp [ j ψ ( f opt ) ]
Θ ( t ) = arg { 1 { exp [ 1 2 ( f opt Δ ν L / 2 ) 2 ] exp [ j ψ ( f opt ) ] } }
ψ ( f opt ) = arg { { exp [ j Θ ( t ) ] } exp [ + 1 2 ( f opt Δ ν L / 2 ) 2 ] } = arg { { exp [ j Θ ( t ) ] } }
h shift ( t ; Δ t shift ) = exp { j [ Θ ( t + Δ t shift ) Θ ( t ) ] }
h shift ( t ; Δ t shift ) exp { j [ ω AOM t + Θ ( t + t d ) Θ ( t ) ] } = exp { j [ ω AOM t + Θ ( t + t d ) Θ ( t + Δ t shift ) ] }
f D = 2 v LOS λ
w _ = [ w ( M c 2 ) w ( M c 2 + 1 ) w ( 1 ) w ( 0 ) = 1 w ( 1 ) w ( M c 2 1 ) w ( M c 2 ) ]
h shift ( t ; Δ t shift ) = exp [ j ( Θ ( t + Δ t shift ) Θ ( t ) ) ]
E f ( x f , y f ) = E ^ f exp ( x f 2 + x f 2 w 0 2 )
z 1 = f + f 2 ( z 0 f ) ( z 0 f ) 2 + ( π w 0 2 λ ) 2
1 w 1 2 = 1 w 0 2 ( 1 z 0 f ) 2 + 1 f 2 ( π w 0 λ ) 2
w ( z ) = w 1 1 + ( z z R 1 ) 2
δ P ( 1 ) = η A lens 4 π z 2 δ P sc ( 1 )
η = | E sc ( 1 ) ( x f , y f ) E f * ( x f , y f ) d x f d y f | 2 | E sc ( 1 ) ( x f , y f ) | 2 d x f d y f | E f ( x f , y f ) | 2 d x f d y f
E sc ( 1 ) E ^ sc ( 1 ) exp ( x f 2 + y f 2 w 0 2 )
r lens = 1.29 λ f π w 0
η ( x , y , z ) = η long ( z ) exp [ ( δ x f ) 2 + ( δ y f ) 2 w η 2 ]
η long ( z ) = 4 / { [ w ˜ 0 ( z ) w 0 + w 0 w ˜ 0 ( z ) ] 2 + [ k w 0 w ˜ 0 ( z ) 2 R curv ] 2 } w ˜ 0 2 ( z ) = w 0 2 [ 1 + ( δ z f δ z 0 z R 0 ) 2 ] R curv = ( δ z f δ z 0 ) [ 1 + z R 0 2 ( δ z f δ z 0 ) 2 ] 1 w η 2 = 2 w ˜ 0 2 + Re { [ k 2 2 ( 1 f 1 R curv + j 2 k w ˜ 0 2 ) 2 ] / ( 1 w 0 2 + 1 w ˜ 0 2 + j k 2 R curv ) } k = 2 π / λ ; δ x f x f z ; δ y f x f z
δ P sc ( z ) = δ z I ( x , y , z ) ρ ( x , y , z ) d x d y
I ( x , y , z ) = 2 π P 0 exp [ a ( z ) ] w 2 ( z ) exp [ 2 x 2 + y 2 w 2 ( z ) ]
δ P = δ z I ( x , y , z ) ρ ¯ ( z ) η ( x , y , z ) d x d y A lens 4 π z 2 exp [ a ( z ) ]
δ P ( z ) = η ¯ ( z ) A lens 4 π z 2 δ P sc ( z ) exp [ a ( z ) ]
η ¯ ( z ) = exp [ 2 x 2 + y 2 w 2 ( z ) ] η ( x , y , z ) d x d y exp [ 2 x 2 + y 2 w 2 ( z ) ] d x d y = η long ( z ) / [ 1 + 1 2 ( f z ) 2 w 2 w η 2 ]
p ( z ) = δ P ( z ) δ z = η ¯ ( z ) A lens 4 π z 2 ρ ¯ ( z ) P 0 exp [ 2 a ( z ) ]
Δ z focus = p ( z ) d z p max

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