Abstract

An investigation of the minimum number of intersecting beams that is required for laser Doppler anemometry (LDA) incorporating only a single detector is presented. We aim to provide decisive arguments for using four beams as the minimum requirement for complete three-dimensional velocity reconstruction even though three beams supply three velocity components. We derive expressions for the detected signals of the most general LDA system. From a matrix analysis of these expressions, we conclude that there is no physically realizable arrangement of three beams that results in complete three-dimensional velocity reconstruction and that four beams is the minimum number of beams required. We also determine the optimal arrangement of the four incident beams for unambiguous LDA and for best signal separation and immunity to minor optical alignment errors. To ascertain the velocity components, we scan the specimen in a precise manner relative to the point of focus of the beams, whereas some other researchers alter the frequency of the incident beams. The results obtained with these two methods are equivalent. However, scanning is mechanically simpler than frequency shifting and also allows for the formation of velocity images—images of the flow velocity over a region in two- or three-dimensional space. In particular, we examine systems that are limited by the common practice of using only a single high-numerical-aperture objective for both focusing and detection. We show that using high-numerical-aperture objectives results in the best signal differentiation and immunity to minor alignment errors.

© 2000 Optical Society of America

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References

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  1. Y. Yeh, H. Z. Cummins, “Localized fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
    [CrossRef]
  2. J. W. Foreman, E. W. Georege, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flowmeter,” Appl. Phys. Lett. 7, 77–78 (1965).
    [CrossRef]
  3. D. C. Wisler, P. W. Mossey, “Gas velocity measurements within a compressor rotor passage using the laser Doppler velocimeter,” J. Eng. Power Trans. ASME 95, 91–96 (1973).
    [CrossRef]
  4. R. J. Baker, P. Hutchinson, J. H. Whitelaw, “Velocity measurements in the recirculation region of an industrial burner flame by laser anemometry with light frequency shifting,” Combust. Flame 23, 57–71 (1974).
    [CrossRef]
  5. S. Z. Kassab, A. E. Bakry, H. A. Warda, “Laser Doppler anemometry measurements in an axisymmetric turbulent jet,” Rev. Sci. Instrum. 67, 1842–1849 (1996).
    [CrossRef]
  6. T. Tanaka, C. Riva, I. Ben-Sira, “Blood velocity measurements in human retinal vessels,” Science 186(4166), 830–831 (1974).
    [PubMed]
  7. C. Tropea, “Laser Doppler anemometry: recent developments and future challenges,” Meas. Sci. Technol. 6, 605–619 (1995).
    [CrossRef]
  8. C. L. Dancey, “A review of three-component laser Doppler anemometry,” Int. J. Opt. Sens. 2, 5–6 (1987).
  9. S. Bopp, C. Tropea, L. Zhan, “The use of graded-index fibers in fiber-optic laser-Doppler anemometry probes,” Rev. Sci. Instrum. 60, 3195–3200 (1989).
    [CrossRef]
  10. O. Lanz, C. Johnson, S. Morikawa, “High-resolution laser Doppler velocity measurement of bidirectional pulsatile fluid flow,” Appl. Phys. Lett. 17, 523–525 (1970).
    [CrossRef]
  11. M. K. Mazumder, “Laser Doppler velocity measurement without directional ambiguity by using frequency shifted incident beams,” Appl. Phys. Lett. 16, 462–464 (1970).
    [CrossRef]
  12. C. J. Bates, “Results from a two-dimensional laser Doppler anemometer,” J. Phys. Sci. Instrum. 9, 616–618 (1976).
    [CrossRef]
  13. T. T. Nguyen, L. N. Binh, “A fiber-optic laser-Doppler anemometer,” Appl. Phys. Lett. 45, 1163–1165 (1984).
    [CrossRef]
  14. B. Lehmann, J. Mante, “On-axis velocity measurement by laser Doppler anemometry,” J. Phys. Sci. Instrum. 17, 455–457 (1984).
    [CrossRef]
  15. R. J. Hallermeier, “Design consideration for a 3-D laser Doppler velocimeter for studying gravity waves in shallow water,” Appl. Opt. 12, 294–300 (1973).
    [CrossRef] [PubMed]
  16. M. M. Antoine, R. L. Simpson, “A rapidly scanning three-velocity-component laser Doppler anemometer,” J. Phys. Sci. Instrum. 19, 853–858 (1986).
    [CrossRef]
  17. J. C. Owens, “Optical Doppler measurement of microscale wind velocity,” Proc. IEEE 57, 530–536 (1969).
    [CrossRef]
  18. R. M. Huffaker, “Laser Doppler detection systems for gas velocity measurement,” Appl. Opt. 9, 1026–1039 (1970).
    [CrossRef] [PubMed]
  19. R. J. Adrian, “A bipolar, two component laser-Doppler velocimeter,” J. Phys. Sci. Instrum. 8, 723–726 (1975).
    [CrossRef]
  20. W. M. Farmer, “Determination of a third orthogonal velocity component using two rotationally displaced laser Doppler velocimeter systems,” Appl. Opt. 11, 770–774 (1972).
    [CrossRef] [PubMed]
  21. K. A. Blake, “Simple two-dimensional laser velocimeter optics,” J. Phys. Sci. Instrum. 5, 623–624 (1972).
    [CrossRef]
  22. D. B. Brayton, H. T. Kalb, F. L. Crosswy, “Two-component dual-scatter laser Doppler velocimeter with frequency burst signal readout,” Appl. Opt. 12, 1145–1156 (1973).
    [CrossRef] [PubMed]
  23. M. Born, E. Wolf, Principles of Optics, 7th ed., expanded (Cambridge U. Press, Cambridge, 1999).
    [CrossRef]
  24. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
    [CrossRef]
  25. G. R. Grant, K. L. Orloff, “Two-color dual-beam backscatter laser Doppler velocimeter,” Appl. Opt. 12, 2913–2916 (1973).
    [CrossRef] [PubMed]

1996

S. Z. Kassab, A. E. Bakry, H. A. Warda, “Laser Doppler anemometry measurements in an axisymmetric turbulent jet,” Rev. Sci. Instrum. 67, 1842–1849 (1996).
[CrossRef]

1995

C. Tropea, “Laser Doppler anemometry: recent developments and future challenges,” Meas. Sci. Technol. 6, 605–619 (1995).
[CrossRef]

1989

S. Bopp, C. Tropea, L. Zhan, “The use of graded-index fibers in fiber-optic laser-Doppler anemometry probes,” Rev. Sci. Instrum. 60, 3195–3200 (1989).
[CrossRef]

1987

C. L. Dancey, “A review of three-component laser Doppler anemometry,” Int. J. Opt. Sens. 2, 5–6 (1987).

1986

M. M. Antoine, R. L. Simpson, “A rapidly scanning three-velocity-component laser Doppler anemometer,” J. Phys. Sci. Instrum. 19, 853–858 (1986).
[CrossRef]

1984

T. T. Nguyen, L. N. Binh, “A fiber-optic laser-Doppler anemometer,” Appl. Phys. Lett. 45, 1163–1165 (1984).
[CrossRef]

B. Lehmann, J. Mante, “On-axis velocity measurement by laser Doppler anemometry,” J. Phys. Sci. Instrum. 17, 455–457 (1984).
[CrossRef]

1976

C. J. Bates, “Results from a two-dimensional laser Doppler anemometer,” J. Phys. Sci. Instrum. 9, 616–618 (1976).
[CrossRef]

1975

R. J. Adrian, “A bipolar, two component laser-Doppler velocimeter,” J. Phys. Sci. Instrum. 8, 723–726 (1975).
[CrossRef]

1974

T. Tanaka, C. Riva, I. Ben-Sira, “Blood velocity measurements in human retinal vessels,” Science 186(4166), 830–831 (1974).
[PubMed]

R. J. Baker, P. Hutchinson, J. H. Whitelaw, “Velocity measurements in the recirculation region of an industrial burner flame by laser anemometry with light frequency shifting,” Combust. Flame 23, 57–71 (1974).
[CrossRef]

1973

1972

1970

O. Lanz, C. Johnson, S. Morikawa, “High-resolution laser Doppler velocity measurement of bidirectional pulsatile fluid flow,” Appl. Phys. Lett. 17, 523–525 (1970).
[CrossRef]

M. K. Mazumder, “Laser Doppler velocity measurement without directional ambiguity by using frequency shifted incident beams,” Appl. Phys. Lett. 16, 462–464 (1970).
[CrossRef]

R. M. Huffaker, “Laser Doppler detection systems for gas velocity measurement,” Appl. Opt. 9, 1026–1039 (1970).
[CrossRef] [PubMed]

1969

J. C. Owens, “Optical Doppler measurement of microscale wind velocity,” Proc. IEEE 57, 530–536 (1969).
[CrossRef]

1965

J. W. Foreman, E. W. Georege, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flowmeter,” Appl. Phys. Lett. 7, 77–78 (1965).
[CrossRef]

1964

Y. Yeh, H. Z. Cummins, “Localized fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
[CrossRef]

1959

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[CrossRef]

Adrian, R. J.

R. J. Adrian, “A bipolar, two component laser-Doppler velocimeter,” J. Phys. Sci. Instrum. 8, 723–726 (1975).
[CrossRef]

Antoine, M. M.

M. M. Antoine, R. L. Simpson, “A rapidly scanning three-velocity-component laser Doppler anemometer,” J. Phys. Sci. Instrum. 19, 853–858 (1986).
[CrossRef]

Baker, R. J.

R. J. Baker, P. Hutchinson, J. H. Whitelaw, “Velocity measurements in the recirculation region of an industrial burner flame by laser anemometry with light frequency shifting,” Combust. Flame 23, 57–71 (1974).
[CrossRef]

Bakry, A. E.

S. Z. Kassab, A. E. Bakry, H. A. Warda, “Laser Doppler anemometry measurements in an axisymmetric turbulent jet,” Rev. Sci. Instrum. 67, 1842–1849 (1996).
[CrossRef]

Bates, C. J.

C. J. Bates, “Results from a two-dimensional laser Doppler anemometer,” J. Phys. Sci. Instrum. 9, 616–618 (1976).
[CrossRef]

Ben-Sira, I.

T. Tanaka, C. Riva, I. Ben-Sira, “Blood velocity measurements in human retinal vessels,” Science 186(4166), 830–831 (1974).
[PubMed]

Binh, L. N.

T. T. Nguyen, L. N. Binh, “A fiber-optic laser-Doppler anemometer,” Appl. Phys. Lett. 45, 1163–1165 (1984).
[CrossRef]

Blake, K. A.

K. A. Blake, “Simple two-dimensional laser velocimeter optics,” J. Phys. Sci. Instrum. 5, 623–624 (1972).
[CrossRef]

Bopp, S.

S. Bopp, C. Tropea, L. Zhan, “The use of graded-index fibers in fiber-optic laser-Doppler anemometry probes,” Rev. Sci. Instrum. 60, 3195–3200 (1989).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed., expanded (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Brayton, D. B.

Crosswy, F. L.

Cummins, H. Z.

Y. Yeh, H. Z. Cummins, “Localized fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
[CrossRef]

Dancey, C. L.

C. L. Dancey, “A review of three-component laser Doppler anemometry,” Int. J. Opt. Sens. 2, 5–6 (1987).

Farmer, W. M.

Foreman, J. W.

J. W. Foreman, E. W. Georege, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flowmeter,” Appl. Phys. Lett. 7, 77–78 (1965).
[CrossRef]

Georege, E. W.

J. W. Foreman, E. W. Georege, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flowmeter,” Appl. Phys. Lett. 7, 77–78 (1965).
[CrossRef]

Grant, G. R.

Hallermeier, R. J.

Huffaker, R. M.

Hutchinson, P.

R. J. Baker, P. Hutchinson, J. H. Whitelaw, “Velocity measurements in the recirculation region of an industrial burner flame by laser anemometry with light frequency shifting,” Combust. Flame 23, 57–71 (1974).
[CrossRef]

Johnson, C.

O. Lanz, C. Johnson, S. Morikawa, “High-resolution laser Doppler velocity measurement of bidirectional pulsatile fluid flow,” Appl. Phys. Lett. 17, 523–525 (1970).
[CrossRef]

Kalb, H. T.

Kassab, S. Z.

S. Z. Kassab, A. E. Bakry, H. A. Warda, “Laser Doppler anemometry measurements in an axisymmetric turbulent jet,” Rev. Sci. Instrum. 67, 1842–1849 (1996).
[CrossRef]

Lanz, O.

O. Lanz, C. Johnson, S. Morikawa, “High-resolution laser Doppler velocity measurement of bidirectional pulsatile fluid flow,” Appl. Phys. Lett. 17, 523–525 (1970).
[CrossRef]

Lehmann, B.

B. Lehmann, J. Mante, “On-axis velocity measurement by laser Doppler anemometry,” J. Phys. Sci. Instrum. 17, 455–457 (1984).
[CrossRef]

Lewis, R. D.

J. W. Foreman, E. W. Georege, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flowmeter,” Appl. Phys. Lett. 7, 77–78 (1965).
[CrossRef]

Mante, J.

B. Lehmann, J. Mante, “On-axis velocity measurement by laser Doppler anemometry,” J. Phys. Sci. Instrum. 17, 455–457 (1984).
[CrossRef]

Mazumder, M. K.

M. K. Mazumder, “Laser Doppler velocity measurement without directional ambiguity by using frequency shifted incident beams,” Appl. Phys. Lett. 16, 462–464 (1970).
[CrossRef]

Morikawa, S.

O. Lanz, C. Johnson, S. Morikawa, “High-resolution laser Doppler velocity measurement of bidirectional pulsatile fluid flow,” Appl. Phys. Lett. 17, 523–525 (1970).
[CrossRef]

Mossey, P. W.

D. C. Wisler, P. W. Mossey, “Gas velocity measurements within a compressor rotor passage using the laser Doppler velocimeter,” J. Eng. Power Trans. ASME 95, 91–96 (1973).
[CrossRef]

Nguyen, T. T.

T. T. Nguyen, L. N. Binh, “A fiber-optic laser-Doppler anemometer,” Appl. Phys. Lett. 45, 1163–1165 (1984).
[CrossRef]

Orloff, K. L.

Owens, J. C.

J. C. Owens, “Optical Doppler measurement of microscale wind velocity,” Proc. IEEE 57, 530–536 (1969).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[CrossRef]

Riva, C.

T. Tanaka, C. Riva, I. Ben-Sira, “Blood velocity measurements in human retinal vessels,” Science 186(4166), 830–831 (1974).
[PubMed]

Simpson, R. L.

M. M. Antoine, R. L. Simpson, “A rapidly scanning three-velocity-component laser Doppler anemometer,” J. Phys. Sci. Instrum. 19, 853–858 (1986).
[CrossRef]

Tanaka, T.

T. Tanaka, C. Riva, I. Ben-Sira, “Blood velocity measurements in human retinal vessels,” Science 186(4166), 830–831 (1974).
[PubMed]

Tropea, C.

C. Tropea, “Laser Doppler anemometry: recent developments and future challenges,” Meas. Sci. Technol. 6, 605–619 (1995).
[CrossRef]

S. Bopp, C. Tropea, L. Zhan, “The use of graded-index fibers in fiber-optic laser-Doppler anemometry probes,” Rev. Sci. Instrum. 60, 3195–3200 (1989).
[CrossRef]

Warda, H. A.

S. Z. Kassab, A. E. Bakry, H. A. Warda, “Laser Doppler anemometry measurements in an axisymmetric turbulent jet,” Rev. Sci. Instrum. 67, 1842–1849 (1996).
[CrossRef]

Whitelaw, J. H.

R. J. Baker, P. Hutchinson, J. H. Whitelaw, “Velocity measurements in the recirculation region of an industrial burner flame by laser anemometry with light frequency shifting,” Combust. Flame 23, 57–71 (1974).
[CrossRef]

Wisler, D. C.

D. C. Wisler, P. W. Mossey, “Gas velocity measurements within a compressor rotor passage using the laser Doppler velocimeter,” J. Eng. Power Trans. ASME 95, 91–96 (1973).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th ed., expanded (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Yeh, Y.

Y. Yeh, H. Z. Cummins, “Localized fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
[CrossRef]

Zhan, L.

S. Bopp, C. Tropea, L. Zhan, “The use of graded-index fibers in fiber-optic laser-Doppler anemometry probes,” Rev. Sci. Instrum. 60, 3195–3200 (1989).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

Y. Yeh, H. Z. Cummins, “Localized fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
[CrossRef]

J. W. Foreman, E. W. Georege, R. D. Lewis, “Measurement of localized flow velocities in gases with a laser Doppler flowmeter,” Appl. Phys. Lett. 7, 77–78 (1965).
[CrossRef]

O. Lanz, C. Johnson, S. Morikawa, “High-resolution laser Doppler velocity measurement of bidirectional pulsatile fluid flow,” Appl. Phys. Lett. 17, 523–525 (1970).
[CrossRef]

M. K. Mazumder, “Laser Doppler velocity measurement without directional ambiguity by using frequency shifted incident beams,” Appl. Phys. Lett. 16, 462–464 (1970).
[CrossRef]

T. T. Nguyen, L. N. Binh, “A fiber-optic laser-Doppler anemometer,” Appl. Phys. Lett. 45, 1163–1165 (1984).
[CrossRef]

Combust. Flame

R. J. Baker, P. Hutchinson, J. H. Whitelaw, “Velocity measurements in the recirculation region of an industrial burner flame by laser anemometry with light frequency shifting,” Combust. Flame 23, 57–71 (1974).
[CrossRef]

Int. J. Opt. Sens.

C. L. Dancey, “A review of three-component laser Doppler anemometry,” Int. J. Opt. Sens. 2, 5–6 (1987).

J. Eng. Power Trans. ASME

D. C. Wisler, P. W. Mossey, “Gas velocity measurements within a compressor rotor passage using the laser Doppler velocimeter,” J. Eng. Power Trans. ASME 95, 91–96 (1973).
[CrossRef]

J. Phys. Sci. Instrum.

C. J. Bates, “Results from a two-dimensional laser Doppler anemometer,” J. Phys. Sci. Instrum. 9, 616–618 (1976).
[CrossRef]

B. Lehmann, J. Mante, “On-axis velocity measurement by laser Doppler anemometry,” J. Phys. Sci. Instrum. 17, 455–457 (1984).
[CrossRef]

M. M. Antoine, R. L. Simpson, “A rapidly scanning three-velocity-component laser Doppler anemometer,” J. Phys. Sci. Instrum. 19, 853–858 (1986).
[CrossRef]

R. J. Adrian, “A bipolar, two component laser-Doppler velocimeter,” J. Phys. Sci. Instrum. 8, 723–726 (1975).
[CrossRef]

K. A. Blake, “Simple two-dimensional laser velocimeter optics,” J. Phys. Sci. Instrum. 5, 623–624 (1972).
[CrossRef]

Meas. Sci. Technol.

C. Tropea, “Laser Doppler anemometry: recent developments and future challenges,” Meas. Sci. Technol. 6, 605–619 (1995).
[CrossRef]

Proc. IEEE

J. C. Owens, “Optical Doppler measurement of microscale wind velocity,” Proc. IEEE 57, 530–536 (1969).
[CrossRef]

Proc. R. Soc. London A

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[CrossRef]

Rev. Sci. Instrum.

S. Bopp, C. Tropea, L. Zhan, “The use of graded-index fibers in fiber-optic laser-Doppler anemometry probes,” Rev. Sci. Instrum. 60, 3195–3200 (1989).
[CrossRef]

S. Z. Kassab, A. E. Bakry, H. A. Warda, “Laser Doppler anemometry measurements in an axisymmetric turbulent jet,” Rev. Sci. Instrum. 67, 1842–1849 (1996).
[CrossRef]

Science

T. Tanaka, C. Riva, I. Ben-Sira, “Blood velocity measurements in human retinal vessels,” Science 186(4166), 830–831 (1974).
[PubMed]

Other

M. Born, E. Wolf, Principles of Optics, 7th ed., expanded (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram representing the photomultiplier signal with (a) three incident beams and (b) four incident beams. In both (a) and (b) there is a dc pedestal. It is vital to separate the signals by as much as possible to avoid any confusion in the detected beat frequencies.

Fig. 2
Fig. 2

Optical arrangement of the LDA system showing one (of possibly many) incident beams i. Note that the use of a single objective limits the possible incident-wave vectors because of the NA of the element.

Fig. 3
Fig. 3

Shown is the collapse of a set of randomly chosen values for the different parameters of the LDA system as they are altered by iteration to a solution that ensures that all the beat frequencies are uniformly separated.

Fig. 4
Fig. 4

Data from which the relations for α > π/2 were taken: Examination of many sets of variables that maintain the maximum beat-frequency separation reveals the relations that hold between different parameters. In particular, we see that θ w is always equal to θ4 (curve with squares), that θ w always sums with θ1 (curve with crosses) to equal π, and that θ w is independent of both ϕ1 (curve with circles) and ϕ4 (curve with triangles).

Fig. 5
Fig. 5

Superposition of the Monte Carlo simulation results with the theoretical results of the maximum beat-frequency separation as functions of the half-angle of the NA.

Fig. 6
Fig. 6

Superposition of the Monte Carlo simulation results with the theoretical results for the determinant of the arrangement matrix [Eq. (15)] as a function of the half-angle of the NA. The determinant is a measure of the optical arrangement—a zero determinant means that no optical arrangement is possible. From this figure, we can see that a fairly large NA is required for strong stability of the optical LDA arrangement.

Tables (2)

Tables Icon

Table 1 Maximum Number of Velocity Components Obtained with Multiple Incident Beams

Tables Icon

Table 2 Results for the Six Beam Combinations for Substitution in Expression (A2) to Determine the Minimum Scanning-Stage Speed by Case Analysisa

Equations (25)

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

Ei=Ai cos 2πνit,
Vn  i=1n Eij=1m Ej=i=1n Ai cos 2πνit2=12i=1n Ai2+i=1nj=1i AiAj cos 2πtνi+νj+i=1nj=1i1-δijAiAj cos 2πtνi-νj,
δij=1if i=j0if ij.
i=1nj=1i1-δij=nC2,
ki=2πλsin θi cos ϕi i+sin θi sin ϕi j+cos θik
ks=2πλsin θs cos ϕs isin θs sin ϕsj+cos θsk.
W=W0sin θw cos ϕwi+sin θw sin ϕwj+cos θwk,  U=Uxi+Uyj+Uzk.
νi=ν0+12πks-kiU+W,
Δνij=|νj-νi|=ν0+12πks-kjU+W-ν0+12πks-kiU+W=12π |ki-kjU+W|,
Δνij=ν0c |sin θi cos ϕi-sin θj cos ϕji+sin θi sin ϕi-sin θj sin ϕjj+cos θi-cos θjkU+W|=ν0c |Ciji+Sijj+RijkU+W|,
Cij=sin θi cos ϕi-sin θj cos ϕj,  Sij=sin θi sin ϕi-sin θj sin ϕj,  Cij=cos ϕi-cos ϕj.
Ciji+Sijj+RijkW0.
Δνij=ν0cCiji+Sijj+RijkU+W.
ν1=ν0+12πks-k1U+W,  ν2=ν0+12πks-k2U+W,  ν3=ν0+12πks-k3U+W.
Δν12Δν23Δν13=ν0cC12S12R12C23S23R23C13S13R13UxUyUz+WxWyWz,
ν2-ν1=Δν,  ν3-ν1=4Δν,  ν4-ν1=6Δν,
Δν12Δν34Δν23=ν0cC12S12R12C34S34R34C23S23R23UxUyUz+WxWyWz.
ϕ1=π+ϕw,  ϕ4=ϕw,  θ1=α,  θ4=α,
ϕ2=cos-1-23+ϕw,  θ2=α,  ϕ3=cos-11+ϕw,  θ3=sin-113 sin α.
ϕ1=π+ϕw,  θ1=α,  ϕ2=cos-1-23+ϕw,  θ2=α,  ϕ3=cos-11+ϕw,  θ3=sin-113 sin α,  ϕ4=ϕw,  θ4=α,  ϕw=ϕw,  θw=π2.
ϕ1=π+ϕw,  ϕ4=ϕw,  θ1=π-θw,   θ4=θw,
ϕ2=cos-1-2-3 cos θ2 cos θw3 sin θ2 sin θw+ϕw,  ϕ3=cos-11-3 cos θ3 cos θw3 sin θ3 sin θw+ϕw.
ϕ1=π+ϕw,  ϕ4=ϕw,  θ1=π-θw,  θ4=θw,  ϕ2=cos-1-2-3 cos θ2 cos θw3 sin θ2 sin θw+ϕw,  ϕ3=cos-11-3 cos θ3 cos θw3 sin θ3 sin θw+ϕw.
Δνij=12π |ki-kjU+W|,
Wmin>Umaxcos θi-cos θjcos θu+cosϕu-ϕisin θi sin θu-cosϕu-ϕjsin θj sin θucosϕj-ϕwsin θj-cosϕi-ϕwsin θi

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