Abstract

A detailed theory of LDV in dense multiply scattering fluids such as blood has been developed. Using a Feynman diagram renormalization technique, we derived generalized transport equations for the Doppler spectral space irradiance in a moving medium. We developed a Monte Carlo algorithm for solution of these equations in realistic fiber LDV geometries. Numerical results indicate that hydrodynamic boundary layer effects on the performance of LDV systems could be substantially reduced by proper choice of fiber geometry and signal processing method. We give a theory of the sources of noise in LDV systems, which shows that such improvements are feasible.

© 1985 Optical Society of America

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References

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  1. C. W. White, C. B. Wright, D. B. Doty, L. F. Hiratza, C. L. Eastham, D. G. Harrison, M. L. Marcus, “Does Visual Interpretation of the Coronary Arteriogram Predict the Physiologic Importance of a Coronary Stenosis?” New Engl. J. Med. 310, 819 (1984).
    [CrossRef] [PubMed]
  2. C. Riva, B. Ross, G. Benedek, “Laser Doppler Measurements of Blood Flow in Capillary Tubes and Retinal Arteries,” Invest. Ophthalmol. 11, 936 (1972).
    [PubMed]
  3. T. Tanaka, G. B. Benedek, “Measurement of the Velocity of Blood Flow (in vivo) Using a Fiber Optic Catheter and Optical Mixing Spectroscopy,” Appl. Opt. 14, 189 (1975).
    [PubMed]
  4. M. D. Stern, “In Viυo Evaluation of Microcirculation by Coherent Light Scattering,” Nature London 254, 56 (1975).
    [CrossRef] [PubMed]
  5. M. D. Stern, D. L. Lappe, “Method and Apparatus for Measurement of Blood Flow Using Coherent Light,” U.S. Patent4,109,647 (1978).
  6. R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Continuous Real-time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
    [CrossRef]
  7. D. Watkins, G. A. Holloway, “An Instrument to Measure Cutaneous Blood Flow Using the Doppler Shift of Laser Light,” IEEE Trans. Biomed. Eng. 25, 28 (1978).
    [CrossRef] [PubMed]
  8. G. E. Nilsson, T. Tenland, P. A. Oberg, “A New Instrument for Continuous Measurement of Tissue Blood Flow by Light Beating Spectroscopy,” IEEE Trans. Biomed. Eng. 27, 12 (1980).
    [CrossRef] [PubMed]
  9. R. Bonner, R. Nossal, “Model for Laser Doppler Measurements of Blood Flow in Tissue,” Appl. Opt. 20, 2097 (1981).
    [CrossRef] [PubMed]
  10. D. Kilpatrick, T. Linderer, R. E. Sievers, J. V. Tyberg, “Measurement of Coronary Sinus Blood Flow by Fiber-Optic Laser Doppler Anemometry,” Am. J. Physiol. 242, H1114 (1982).
  11. H. Nishihara, J. Koyama, N. Hoki, F. Kajiya, M. Hironaga, M. Kano, “Optical-Fiber Laser Doppler Felocimeter for High-Resolution Measurement of Pulsatile Blood Flows,” Appl. Opt. 21, 1785 (1982).
    [CrossRef] [PubMed]
  12. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 271.
  13. G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. 16, 199 (1976).
    [CrossRef] [PubMed]
  14. L. G. Henyey, J. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 76 (1941).
    [CrossRef]
  15. V. Twersky, “Multiple Scattering by Biological Suspensions,” J. Opt. Soc. Am. 60, 1048 (1970).
    [CrossRef]
  16. H.C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), pp. 477–479.
  17. C. G. Caro, T. J. Pedley, R. C. Schroter, W. A. Seed, The Mechanics of the Circulation (Oxford U.P., Oxford, 1978).
  18. S. E. Miller, A. G. Chenoweth, Optical Fiber Telecommunications (Academic, New York, 1979).

1984 (1)

C. W. White, C. B. Wright, D. B. Doty, L. F. Hiratza, C. L. Eastham, D. G. Harrison, M. L. Marcus, “Does Visual Interpretation of the Coronary Arteriogram Predict the Physiologic Importance of a Coronary Stenosis?” New Engl. J. Med. 310, 819 (1984).
[CrossRef] [PubMed]

1982 (2)

D. Kilpatrick, T. Linderer, R. E. Sievers, J. V. Tyberg, “Measurement of Coronary Sinus Blood Flow by Fiber-Optic Laser Doppler Anemometry,” Am. J. Physiol. 242, H1114 (1982).

H. Nishihara, J. Koyama, N. Hoki, F. Kajiya, M. Hironaga, M. Kano, “Optical-Fiber Laser Doppler Felocimeter for High-Resolution Measurement of Pulsatile Blood Flows,” Appl. Opt. 21, 1785 (1982).
[CrossRef] [PubMed]

1981 (1)

1980 (1)

G. E. Nilsson, T. Tenland, P. A. Oberg, “A New Instrument for Continuous Measurement of Tissue Blood Flow by Light Beating Spectroscopy,” IEEE Trans. Biomed. Eng. 27, 12 (1980).
[CrossRef] [PubMed]

1978 (1)

D. Watkins, G. A. Holloway, “An Instrument to Measure Cutaneous Blood Flow Using the Doppler Shift of Laser Light,” IEEE Trans. Biomed. Eng. 25, 28 (1978).
[CrossRef] [PubMed]

1976 (1)

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. 16, 199 (1976).
[CrossRef] [PubMed]

1975 (2)

1972 (1)

C. Riva, B. Ross, G. Benedek, “Laser Doppler Measurements of Blood Flow in Capillary Tubes and Retinal Arteries,” Invest. Ophthalmol. 11, 936 (1972).
[PubMed]

1970 (1)

V. Twersky, “Multiple Scattering by Biological Suspensions,” J. Opt. Soc. Am. 60, 1048 (1970).
[CrossRef]

1941 (1)

L. G. Henyey, J. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 76 (1941).
[CrossRef]

Benedek, G.

C. Riva, B. Ross, G. Benedek, “Laser Doppler Measurements of Blood Flow in Capillary Tubes and Retinal Arteries,” Invest. Ophthalmol. 11, 936 (1972).
[PubMed]

Benedek, G. B.

Bonner, R.

R. Bonner, R. Nossal, “Model for Laser Doppler Measurements of Blood Flow in Tissue,” Appl. Opt. 20, 2097 (1981).
[CrossRef] [PubMed]

R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Continuous Real-time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
[CrossRef]

Bowen, P. D.

R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Continuous Real-time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
[CrossRef]

Bowman, R. L.

R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Continuous Real-time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
[CrossRef]

Caro, C. G.

C. G. Caro, T. J. Pedley, R. C. Schroter, W. A. Seed, The Mechanics of the Circulation (Oxford U.P., Oxford, 1978).

Chenoweth, A. G.

S. E. Miller, A. G. Chenoweth, Optical Fiber Telecommunications (Academic, New York, 1979).

Clem, T. R.

R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Continuous Real-time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
[CrossRef]

Doty, D. B.

C. W. White, C. B. Wright, D. B. Doty, L. F. Hiratza, C. L. Eastham, D. G. Harrison, M. L. Marcus, “Does Visual Interpretation of the Coronary Arteriogram Predict the Physiologic Importance of a Coronary Stenosis?” New Engl. J. Med. 310, 819 (1984).
[CrossRef] [PubMed]

Eastham, C. L.

C. W. White, C. B. Wright, D. B. Doty, L. F. Hiratza, C. L. Eastham, D. G. Harrison, M. L. Marcus, “Does Visual Interpretation of the Coronary Arteriogram Predict the Physiologic Importance of a Coronary Stenosis?” New Engl. J. Med. 310, 819 (1984).
[CrossRef] [PubMed]

Greenstein, J.

L. G. Henyey, J. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 76 (1941).
[CrossRef]

Harrison, D. G.

C. W. White, C. B. Wright, D. B. Doty, L. F. Hiratza, C. L. Eastham, D. G. Harrison, M. L. Marcus, “Does Visual Interpretation of the Coronary Arteriogram Predict the Physiologic Importance of a Coronary Stenosis?” New Engl. J. Med. 310, 819 (1984).
[CrossRef] [PubMed]

Henyey, L. G.

L. G. Henyey, J. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 76 (1941).
[CrossRef]

Hiratza, L. F.

C. W. White, C. B. Wright, D. B. Doty, L. F. Hiratza, C. L. Eastham, D. G. Harrison, M. L. Marcus, “Does Visual Interpretation of the Coronary Arteriogram Predict the Physiologic Importance of a Coronary Stenosis?” New Engl. J. Med. 310, 819 (1984).
[CrossRef] [PubMed]

Hironaga, M.

Hoki, N.

Holloway, G. A.

D. Watkins, G. A. Holloway, “An Instrument to Measure Cutaneous Blood Flow Using the Doppler Shift of Laser Light,” IEEE Trans. Biomed. Eng. 25, 28 (1978).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 271.

Kajiya, F.

Kano, M.

Kilpatrick, D.

D. Kilpatrick, T. Linderer, R. E. Sievers, J. V. Tyberg, “Measurement of Coronary Sinus Blood Flow by Fiber-Optic Laser Doppler Anemometry,” Am. J. Physiol. 242, H1114 (1982).

Koyama, J.

Lappe, D. L.

M. D. Stern, D. L. Lappe, “Method and Apparatus for Measurement of Blood Flow Using Coherent Light,” U.S. Patent4,109,647 (1978).

Linderer, T.

D. Kilpatrick, T. Linderer, R. E. Sievers, J. V. Tyberg, “Measurement of Coronary Sinus Blood Flow by Fiber-Optic Laser Doppler Anemometry,” Am. J. Physiol. 242, H1114 (1982).

Marcus, M. L.

C. W. White, C. B. Wright, D. B. Doty, L. F. Hiratza, C. L. Eastham, D. G. Harrison, M. L. Marcus, “Does Visual Interpretation of the Coronary Arteriogram Predict the Physiologic Importance of a Coronary Stenosis?” New Engl. J. Med. 310, 819 (1984).
[CrossRef] [PubMed]

McCormick, N. J.

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. 16, 199 (1976).
[CrossRef] [PubMed]

Miller, S. E.

S. E. Miller, A. G. Chenoweth, Optical Fiber Telecommunications (Academic, New York, 1979).

Nilsson, G. E.

G. E. Nilsson, T. Tenland, P. A. Oberg, “A New Instrument for Continuous Measurement of Tissue Blood Flow by Light Beating Spectroscopy,” IEEE Trans. Biomed. Eng. 27, 12 (1980).
[CrossRef] [PubMed]

Nishihara, H.

Nossal, R.

Oberg, P. A.

G. E. Nilsson, T. Tenland, P. A. Oberg, “A New Instrument for Continuous Measurement of Tissue Blood Flow by Light Beating Spectroscopy,” IEEE Trans. Biomed. Eng. 27, 12 (1980).
[CrossRef] [PubMed]

Pedersen, G. D.

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. 16, 199 (1976).
[CrossRef] [PubMed]

Pedley, T. J.

C. G. Caro, T. J. Pedley, R. C. Schroter, W. A. Seed, The Mechanics of the Circulation (Oxford U.P., Oxford, 1978).

Reynolds, L. O.

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. 16, 199 (1976).
[CrossRef] [PubMed]

Riva, C.

C. Riva, B. Ross, G. Benedek, “Laser Doppler Measurements of Blood Flow in Capillary Tubes and Retinal Arteries,” Invest. Ophthalmol. 11, 936 (1972).
[PubMed]

Ross, B.

C. Riva, B. Ross, G. Benedek, “Laser Doppler Measurements of Blood Flow in Capillary Tubes and Retinal Arteries,” Invest. Ophthalmol. 11, 936 (1972).
[PubMed]

Schroter, R. C.

C. G. Caro, T. J. Pedley, R. C. Schroter, W. A. Seed, The Mechanics of the Circulation (Oxford U.P., Oxford, 1978).

Seed, W. A.

C. G. Caro, T. J. Pedley, R. C. Schroter, W. A. Seed, The Mechanics of the Circulation (Oxford U.P., Oxford, 1978).

Sievers, R. E.

D. Kilpatrick, T. Linderer, R. E. Sievers, J. V. Tyberg, “Measurement of Coronary Sinus Blood Flow by Fiber-Optic Laser Doppler Anemometry,” Am. J. Physiol. 242, H1114 (1982).

Stern, M. D.

M. D. Stern, “In Viυo Evaluation of Microcirculation by Coherent Light Scattering,” Nature London 254, 56 (1975).
[CrossRef] [PubMed]

M. D. Stern, D. L. Lappe, “Method and Apparatus for Measurement of Blood Flow Using Coherent Light,” U.S. Patent4,109,647 (1978).

Tanaka, T.

Tenland, T.

G. E. Nilsson, T. Tenland, P. A. Oberg, “A New Instrument for Continuous Measurement of Tissue Blood Flow by Light Beating Spectroscopy,” IEEE Trans. Biomed. Eng. 27, 12 (1980).
[CrossRef] [PubMed]

Twersky, V.

V. Twersky, “Multiple Scattering by Biological Suspensions,” J. Opt. Soc. Am. 60, 1048 (1970).
[CrossRef]

Tyberg, J. V.

D. Kilpatrick, T. Linderer, R. E. Sievers, J. V. Tyberg, “Measurement of Coronary Sinus Blood Flow by Fiber-Optic Laser Doppler Anemometry,” Am. J. Physiol. 242, H1114 (1982).

van de Hulst, H.C.

H.C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), pp. 477–479.

Watkins, D.

D. Watkins, G. A. Holloway, “An Instrument to Measure Cutaneous Blood Flow Using the Doppler Shift of Laser Light,” IEEE Trans. Biomed. Eng. 25, 28 (1978).
[CrossRef] [PubMed]

White, C. W.

C. W. White, C. B. Wright, D. B. Doty, L. F. Hiratza, C. L. Eastham, D. G. Harrison, M. L. Marcus, “Does Visual Interpretation of the Coronary Arteriogram Predict the Physiologic Importance of a Coronary Stenosis?” New Engl. J. Med. 310, 819 (1984).
[CrossRef] [PubMed]

Wright, C. B.

C. W. White, C. B. Wright, D. B. Doty, L. F. Hiratza, C. L. Eastham, D. G. Harrison, M. L. Marcus, “Does Visual Interpretation of the Coronary Arteriogram Predict the Physiologic Importance of a Coronary Stenosis?” New Engl. J. Med. 310, 819 (1984).
[CrossRef] [PubMed]

Am. J. Physiol. (1)

D. Kilpatrick, T. Linderer, R. E. Sievers, J. V. Tyberg, “Measurement of Coronary Sinus Blood Flow by Fiber-Optic Laser Doppler Anemometry,” Am. J. Physiol. 242, H1114 (1982).

Appl. Opt. (3)

Astrophys. J. (1)

L. G. Henyey, J. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 76 (1941).
[CrossRef]

Biophys. J. (1)

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport Calculations for Light Scattering in Blood,” Biophys. J. 16, 199 (1976).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (2)

D. Watkins, G. A. Holloway, “An Instrument to Measure Cutaneous Blood Flow Using the Doppler Shift of Laser Light,” IEEE Trans. Biomed. Eng. 25, 28 (1978).
[CrossRef] [PubMed]

G. E. Nilsson, T. Tenland, P. A. Oberg, “A New Instrument for Continuous Measurement of Tissue Blood Flow by Light Beating Spectroscopy,” IEEE Trans. Biomed. Eng. 27, 12 (1980).
[CrossRef] [PubMed]

Invest. Ophthalmol. (1)

C. Riva, B. Ross, G. Benedek, “Laser Doppler Measurements of Blood Flow in Capillary Tubes and Retinal Arteries,” Invest. Ophthalmol. 11, 936 (1972).
[PubMed]

J. Opt. Soc. Am. (1)

V. Twersky, “Multiple Scattering by Biological Suspensions,” J. Opt. Soc. Am. 60, 1048 (1970).
[CrossRef]

Nature London (1)

M. D. Stern, “In Viυo Evaluation of Microcirculation by Coherent Light Scattering,” Nature London 254, 56 (1975).
[CrossRef] [PubMed]

New Engl. J. Med. (1)

C. W. White, C. B. Wright, D. B. Doty, L. F. Hiratza, C. L. Eastham, D. G. Harrison, M. L. Marcus, “Does Visual Interpretation of the Coronary Arteriogram Predict the Physiologic Importance of a Coronary Stenosis?” New Engl. J. Med. 310, 819 (1984).
[CrossRef] [PubMed]

Other (6)

M. D. Stern, D. L. Lappe, “Method and Apparatus for Measurement of Blood Flow Using Coherent Light,” U.S. Patent4,109,647 (1978).

R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Continuous Real-time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 271.

H.C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), pp. 477–479.

C. G. Caro, T. J. Pedley, R. C. Schroter, W. A. Seed, The Mechanics of the Circulation (Oxford U.P., Oxford, 1978).

S. E. Miller, A. G. Chenoweth, Optical Fiber Telecommunications (Academic, New York, 1979).

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Figures (9)

Fig. 1
Fig. 1

Feynman diagram expansion of the two-time coherence tensor to fourth order in the dielectric fluctuations of the scattering medium. The large circles represent the coherence tensor of the sources of the field, the wavy lines represent cumulant autocorrelation moments of the dielectric fluctuations, and the straight lines represent free-field propagators of the electric field. The rules for translation of the diagrams are in the text.

Fig. 2
Fig. 2

Diagramatic representation of the series for the dynamic polarizability tensor to fifth order. The diagrams are computed by the same rules as in Fig. 1, but external lines are omitted, and only diagrams which cannot be split into two parts by cutting a propagator line are included.

Fig. 3
Fig. 3

Diagram series for the scattering kernel K to fifth order. The heavy lines represent renormalized propagators Jlm; the thin external lines are not included in the computation. All diagrams are irreducible; i.e., they cannot be divided by cutting two propagator lines.

Fig. 4
Fig. 4

One-dimensional Doppler sensitivity field for configurations of one or two 50 μm core optical fibers at 45° to a plane laminar flow. The algebraic mean Doppler shift of light collected from the receiving fiber is obtained by integrating the sensitivity field multiplied by the velocity as a function of distance from the plane of the fiber tips. For uniform flow velocity (block flow), the mean Doppler shift equals the area under the curve, which is the same for the two configurations.

Fig. 5
Fig. 5

Comparison of the algebraic mean Doppler shift and Doppler bandwidth for a single-fiber LDV as a function of thickness of a stagnant boundary layer adjacent to the plane containing the fiber tip. The flow velocity is assumed homogeneous except in the boundary layer where it is zero. The Doppler frequencies are expressed per unit free stream flow velocity.

Fig. 6
Fig. 6

Comparison of the boundary layer effect on the output of two LDV systems, one using a single fiber with the algebraic mean Doppler shift as a velocity estimator, the other using two fibers separated by 0.5 mm with Doppler bandwidth as output. The two-fiber system is relatively insensitive to boundary layer thickness over a much wider range of thicknesses than the single-fiber system; the rising limb of the Doppler bandwidth curve for thin boundary layers can be eliminated by prediffusing the light at the fiber tip, as discussed in the text.

Fig. 7
Fig. 7

Computed Doppler spectra of light backscattered through a single optical fiber from a plane laminar flow for two thicknesses of hydrodynamic boundary layer. The velocity profiles (left panel) are represented by the same function (a ratio of two second-degree polynomials), but the horizontal scale is 66% greater in profile B. The corresponding spectra (right panel) differ markedly, but both show steep dropoff above the maximum Doppler frequency of 1.41, which would be found for single scattering from a particle moving at the free stream velocity.

Fig. 8
Fig. 8

Input SNR of hypothetical LDV systems computed from Eqs. (68) and (69). (A) Input SNR as a function of the number of modes of light collected on the detector assuming 0.1 nW/mode, a PIN diode with a quantum efficiency of 0.7, laser amplitude noise of 0.2%, and either homodyne detection or heterodyne detection with optimum reference beam intensity. (B) The same conditions as A except for use of an avalanche photodiode with gain of 25 and excess noise factor 2. (C) Input SNR as a function of fiber core diameter assuming quadratic index fiber with uniform excitation of modes and complete collection of output light on the detector. The optimum core diameter is smaller for higher gain detectors or heterodyne detection. In practice, a variety of communications fibers are suitable because output noise will often be dominated by intrinsic signal fluctuations when the input SNR is >1.

Fig. 9
Fig. 9

Output noise level of a hypothetical heterodyne LDV system using the algebraic mean Doppler shift to estimate velocity with an output bandwidth of 10 Hz and an input Doppler center frequency and bandwidth of 250 kHz. For sufficiently high collected power, the output noise is constant, limited by stochastic fluctuations of the signal. For received power <1 μW, the input SNR becomes important, and a photodetector with gain shows superior performance.

Equations (86)

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D = 4 π ρ , × E + 1 c B t = 0 , B = 0 , j + ρ t = 0 . × H = 4 π j c + 1 c D t
ϵ E = ( χ E + S ) × E i ω 0 c H = 0 H = 0 × H + i ω 0 ϵ c E = i ω 0 c [ χ E + S ] .
E l ( x ) = G lm ( x x ) [ χ E m ( x ) S m ( x ) ] d 3 x .
G lm ( r ) = k 0 2 ϵ exp ( ik 0 r ) r ( δ lm r l r m r 2 ) + exp ( ik 0 r ) r ϵ ( 1 r 2 ik 0 r ) ( 3 r l r m r 2 δ lm ) , k 0 ω 0 ϵ c .
E l ( x , t ) = G lm ( x x ) S m ( x ) d 3 x 1 + n = 1 G lm 1 ( x x 1 ) G m n m ( x n x ) χ ( x 11 , t ) χ ( x n , t ) S m ( x ) d 3 x 1 d 3 x n d 3 x
Γ lm ( x 1 , x 2 , τ ) E l ( x 1 , t ) E m * ( x 2 , t + τ ) , S lm ( x 1 , x 2 ) S l ( x 1 ) S m * ( x 2 )
Γ lm ( x 1 , x 2 , τ ) = n 1 = 0 n 2 = 0 G lm 1 ( x 1 y 1 ) G m n 1 p ( y n 1 y ) G m q 1 * ( x 2 y 1 ) G q n 2 u * ( y n 2 y ) × χ ( y 1 ) χ ( y n 1 ) χ * ( y 1 τ ) χ * ( y n 2 , τ ) S pu ( y , y ) , × d 3 y 1 d 3 y n 1 d 3 y 1 d 3 y n 2 d 3 y d 3 y ,
C ( z 1 z 2 ) = z 1 z n c ( π 1 ) c ( π m ) all partitions π 1 | π 2 | π m ,
z 1 z n = c ( z 1 z n ) + all π c ( π 1 ) c ( π m ) .
J lm ( x ) = G lm ( x ) + G ln ( x x 1 ) × Π np ( x 1 x 2 ) J pm ( x 2 ) d 3 x 1 d 3 x 2 .
Γ lm ( x 1 , x 2 , τ ) = J ln ( x 1 x 1 ) J mp * ( x 2 x 2 ) S np ( x 1 , x 2 ) d 3 x 1 d 3 x 2 + J ln ( x 1 y 1 ) J mp * ( x 2 y 2 ) K npqr ( y 1 , y 2 , x 1 , x 2 , τ ) Γ qr ( x 1 , x 2 , τ ) d 3 x 1 d 3 x 2 d 3 y 1 d 3 y 2 .
W lm ( x , k , τ ) Γ lm ( x + z 2 , x z 2 , τ ) exp ( ik z ) d 3 z , T lmnp ( r , k ) J ln ( r + z 2 ) J mp * ( r z 2 ) exp ( ik z ) d 3 z , Q npqr ( x , x 2 , k 1 , k 2 , τ ) 1 ( 2 π ) 3 K npqr ( x 1 + z 2 , x 1 z 2 , x 2 + z 2 , x 2 z 2 , τ ) exp [ i ( k 1 z k 2 z ) ] d 3 d 3 z , S np ( x , k ) S np ( x + z 2 , x z 2 ) exp ( i k z ) d 3 z ,
W lm ( x , k , τ ) = T lnmp ( x x , k ) S np ( x , k ) d 3 x + T lnmp ( x x , k ) Q npqr ( x 1 x , k , k , τ ) × W qr ( x , k , τ ) d 3 x d 3 x d 3 k .
W lm ( x , k , w ) = 1 2 π W lm ( x , k , τ ) exp ( iw τ ) d τ , Q npqr ( x 1 , x 2 , k 1 , k 2 , w ) = 1 2 π Q npqr ( x 1 , x 2 , k 1 , k 2 , τ ) × exp ( iw τ ) d τ ,
W lm ( x , k , w ) = U lm ( x , k ) δ ( w ) + P lnqr ( x , x , k , k , w w ) × W qr ( x , k , w ) d 3 x d 3 k d 3 w ,
U lm ( x , k ) T lnmp ( x x , k ) S np ( x , k ) d 3 x , P lnqr ( x 1 , x 2 , k 1 , k 2 , Δ w ) T lnmp ( x 1 x , k 1 ) Q npqr ( x , x 2 , k 1 , k 2 , Δ w ) d 3 x .
J ̂ lm ( k ) = G ̂ lm ( k ) + G ̂ ln ( k ) π ̂ np ( k ) J ̂ pm ( k ) .
G ̂ lm ( k ) = k 0 2 δ lm k l k m ϵ ( k 2 k 0 2 ) .
J ̂ lm ( k ) = J 1 ( k 2 ) δ lm J 2 ( k 2 ) k l k m , π ̂ lm ( k ) = π 1 ( k 2 ) δ lm π 2 ( k 2 ) k l k m , π 1 ( k 2 ) = 1 2 [ π ̂ ll ( k ) π ̂ lm k l k m k 2 ] , π 2 ( k 2 ) = 1 2 k 2 [ π ̂ u 3 π lm k l k m k 2 ] ,
J ̂ lm ( k ) = k 0 2 δ lm [ ϵ π 2 ( k 2 ) k 0 2 ϵ + π 1 ( k 2 ) k 2 π 2 ( k 2 ) ] k l k m ϵ k 2 [ ϵ + π 1 ( k 2 ) ] k 0 2 .
T lnmp ( r , k ) = { k 0 2 δ ln [ ϵ π 2 ( k 2 + q 2 4 + k q ) ϵ + π 1 ( k 2 + q 2 4 + k q ) ( k 2 + q 2 4 + k q ) π 2 ( k 2 + q 2 4 + k q ) ] ( k l + q l 2 ) ( k n + q n 2 ) } { ϵ ( k 2 + q 2 4 + k q ) [ ϵ + π 1 ( k 2 + q 2 4 + k q ) ] k 0 2 } × { k 0 2 * δ m p [ ϵ * π 2 * ( k 2 + q 2 4 k q ) ϵ * + π 1 * ( k 2 + q 2 4 k q ) ( k 2 + q 2 4 k q ) π 2 * ( k 2 + q 2 4 l R c q ) ] ( k m q n 2 ) ( k p q p 2 ) } { ϵ * ( k 2 + q 2 4 k q ) [ ϵ * + π 1 * ( k 2 + q 2 4 k q ) ] k 0 2 * } × exp ( iq r ) d 3 q .
ϵ k 1 2 = [ ϵ + π 1 ( k 1 2 ) ] k 0 2
J ̂ lm ( k ) 1 [ ϵ 1 k 1 2 π 1 ( k 1 2 ) ] k 1 2 δ lm k l k m k 2 k 1 2 ,
J lm ( r ) ω 0 2 / c 2 1 ω 0 2 c 2 π 1 ( k 1 2 ) exp ( ik 0 r ) r × exp ( σ r 2 ) ( δ lm r l r m r 2 ) + local field ,
J lm ( r ± z 2 ) ω 0 2 / c 2 1 ω 0 2 c 2 π 1 ( k 1 2 ) exp [ i ( k 0 r ± k 0 r z 2 r ) ] r × exp [ σ 2 ( r ± r z 2 r ) ] ( δ lm r l r m r 2 ) ,
T lnmp ( r , k ) [ ω 0 2 / c 2 1 ω 0 2 c 2 π 1 1 ( k 1 2 ) ] 2 exp ( σ r ) r 2 ( δ lm r l r n r 2 ) × ( δ mp r m r p r 2 ) exp [ i ( k 0 r r k ) z d 3 z = ( 2 π ) 3 [ ω 0 2 / c 2 1 ω 0 2 c 2 π 1 ( k 1 2 ) ] 2 exp ( σ r ) r 2 ( δ l n k l k n k 2 ) × ( δ mp k m k p k 2 ) δ ( k k 0 r r ) .
K npqr ( x ̂ 1 , x 2 , y 1 , y 1 , y 2 , τ ) = k npqr ( x 1 , x 2 υ τ , y 1 , y 2 υ τ , 0 ) .
Q ( x , x , k , k , τ ) = Q ( x x , k , k , 0 ) exp [ i τ ( k k ) v ] ,
k ̂ k / | k | , D lm ( x , k ̂ , τ ) 0 W lm ( x , k k ̂ , τ ) k 2 dk , S ̂ np ( x , k ̂ ) ( 2 π ) 3 [ ω 0 2 / c 2 1 ω 0 2 c 2 π 1 ( k 1 2 ) ] 2 S np ( x , k 0 k ̂ ) npqr ( k ̂ 1 , k ̂ 2 ) ( 2 π ) 3 [ ω 0 2 / c 2 1 ω 0 2 c 2 π 1 ( k 1 2 ) ] 2 × Q npqr ( x x , k 0 k ̂ 1 , k 0 k ̂ 2 , 0 ) d 3 x ,
D lm ( x , k ̂ , τ ) = exp ( σ r ) ( δ ln k ̂ l k ̂ n ) ( δ mp k ̂ m k ̂ p ) S ̂ np ( x r k ̂ , k ̂ ) dr + exp ( σ r ) ( δ ln k ̂ l k ̂ n ) ( δ mp k ̂ m k p ) npqr ( k ̂ , k ̂ ) D qr ( x rk , k , τ ) exp [ i τ k 0 ( k ̂ k ̂ ) v ( x r k ̂ ) ] d Ω ( k ̂ ) dr .
D lm ( x , k ̂ , w ) 1 2 π D lm ( x , k ̂ , τ ) exp ( iw τ ) d τ , a 1 2 σ ( δ np k ̂ n k ̂ p ) ( δ q r k ̂ q k ̂ r ) × npqr ( k ̂ , k ̂ ) d Ω ( k ̂ ) , Φ lmqr ( k ̂ , k ̂ ) 4 π a σ ( δ ln k ̂ l k ̂ n ) ( δ mp k ̂ m k ̂ p ) × npqr ( k ̂ , k ̂ ) ,
D lm ( x , k ̂ , w ) = δ ( w ) exp ( σ r ) ( δ ln k ̂ l k ̂ n ) × ( δ mp k ̂ m k ̂ p ) S np ( x r k ̂ , k ̂ ) dr + a σ 4 π exp ( σ r ) Φ lmqr ( k ̂ , k ̂ ) × D qr [ x r k ̂ , k ̂ , w k 0 v ( x r k ) ( k ̂ k ̂ ) ] d Ω ( k ̂ ) dr .
k ̂ x f ( x k ̂ r ) = r f ( x k ̂ r )
k ̂ x D lm ( x , k ̂ , w ) = σ D lm ( x , k ̂ , w ) + a σ 4 π Φ lmqr ( k ̂ , k ̂ ) × D qr [ x , k ̂ , w k 0 v ( x ) ( k ̂ k ̂ ) ] d Ω ( k ̂ ) ,
I lm ( x , k ̂ ) D lm ( x , k ̂ , w ) dw , F lm ( x , k ̂ ) w D lm ( x , k ̂ , w ) dw , G lm ( x , k ̂ ) w 2 D lm ( x , k ̂ , w ) dw . ( 39 )
k ̂ x I lm ( x , k ̂ ) = σ I lm ( x , k ̂ ) + a σ 4 π Φ lmqr ( k ̂ , k ̂ ) I qr ( x , k ̂ ) d Ω ( k ̂ ) ,
k ̂ x F l m ( x , k ̂ ) = σ F lm ( x , k ̂ ) + a σ 4 π Φ lmqr ( k ̂ , k ̂ ) [ F qr ( x , k ̂ ) + k 0 v ( x ) ( k ̂ k ̂ ) I qr ( x , k ̂ ) ] d Ω ( k ̂ ) ,
k ̂ x G lm ( x , k ̂ ) = σ G lm ( x , k ̂ ) + a σ 4 π Φ lmqr ( k ̂ , k ̂ ) { G qr ( x , k ̂ ) + 2 k 0 υ ( x ) ( k ̂ k ̂ ) F qr ( x , k ̂ ) + [ k 0 υ ( x ) ( k ̂ k ̂ ) ] 2 I qr ( x , k ̂ ) } d Ω ( k ̂ ) .
Δ w ¯ = R F ( x , k ̂ ) da ( x ) d Ω ( k ̂ ) R I ( x , k ̂ ) da ( x ) d Ω ( k ̂ ) = R F ( x , k ̂ ) da ( x ) d Ω ( k ̂ ) E 12 ,
Δ w ¯ = P Δ ( x ) v ( x ) d 3 x .
Δ w ¯ = k 0 a σ 4 π E 12 𝒢 R ( x , k ̂ ) [ Φ ( k ̂ k ̂ ) I ( x , k ̂ ) ( k ̂ k ̂ ) d Ω ( k ̂ ) ] v ( x ) d 3 xd Ω ( k ̂ ) .
𝒢 R ( x , k ̂ ) = A R E 2 I + ( x , k ̂ ) ,
P Δ ( x ) = k 0 a σ 4 π E 12 E 2 I ( x , k ̂ ) I + ( x , k ̂ ) Φ ( k ̂ k ̂ ) d Ω ( k ̂ ) d Ω ( k ̂ ) = k 0 A R E 2 E 12 k ̂ [ I ( x , k ̂ ) k ̂ x I + ( x , k ̂ ) + I + ( x , k ̂ ) k ̂ x I ( x , k ̂ ) ] d Ω ( k ̂ ) = div T
T ij ( x ) k 0 A R E 2 E 12 k ̂ i k ̂ j I ( x , k ̂ ) I + ( x , k ̂ ) d Ω ( k ̂ )
P Δ ( x ) = k 0 a σ 4 π E 12 A R E 2 [ ψ ( x ) J + ( x ) + ψ + ( x ) J ( x ) ] ,
ψ ( x ) I ( x , k ̂ ) d Ω ( k ̂ ) , ψ + ( x ) I + ( x , k ̂ ) d Ω ( k ̂ ) , J ( x ) k ̂ I ( x , k ̂ ) d Ω ( k ̂ ) , J + ( x ) k ̂ I + ( x , k ̂ ) d Ω ( k ̂ ) .
J = D ψ ,
P Δ ( x ) = k 0 a σ D 4 π E 12 A R E 2 [ ψ ( x ) ψ + ( x ) ] .
Δ w = P Δ ( x ) v ( x ) d 3 x = k 0 a σ D A R 4 π E 12 E 2 [ B ψ ( x ) ψ + ( x ) v n d a ( x ) ψ ( x ) ψ + ( x ) v d 3 x ] = 0 ,
E 12 = R I ( x , k ̂ ) R ( x , k ̂ ) da ( x ) d Ω ( k ̂ ) ,
E 12 = n = 1 ( a σ 4 π ) n { exp [ σ ( r s 1 + r 12 + + r nR ) ] r s 1 2 + r 12 2 r nR 2 Φ ( k ̂ s 1 k ̂ 12 ) Φ ( k ̂ n 1 , n k ̂ n , R ) × S ( x s , k ̂ s 1 ) R ( x R , k ̂ nR ) } d 3 x 1 d 3 x n da ( x s ) da ( x R ) ,
E 12 = ( a σ 4 π ) n K n ( x 1 x n , x S , x R ) x R , p n P ( x 1 x n , x S , x R ) ,
dp ( x ) = [ σ exp ( σ r ) dr ] [ ½ Φ ( cos θ ) d θ ] ( d ϕ 2 π ) sin θ = σ exp ( r σ ) Φ 4 π r 2 d 3 x .
Φ 0 ( x S , k ̂ s 1 ) 4 π S ( x , k ̂ ) S ( x , k ̂ ) d Ω ( k ̂ ) , P S ( x S ) S ( x , k ̂ ) d Ω ( k ̂ ) S S ( x , k ̂ ) d Ω ( k ̂ ) da ( x ) = S ( x , k ̂ ) d Ω ( k ̂ ) E 1 , Φ n + 1 ( x R , k nR ) 4 π R ( x , k ̂ ) R ( x , k ̂ ) d Ω ( k ̂ ) P R ( x R ) R ( x , k ̂ ) d Ω ( k ̂ ) R R ( x , k ̂ ) d Ω ( k ̂ ) da ( x ) = R ( x , k ̂ ) d Ω ( k ̂ ) A R , Φ k Φ ( k ̂ k 1 , k k ̂ k , k + 1 ) ,
P ( x 1 x n , x S , x R ) = P S ( x S ) P R ( x R ) n + 1 k = 0 n ( σ 4 π ) n Φ 0 Φ k 1 Φ k + 2 Φ n + 1 exp ( σ i k r i , i + 1 ) r 2 s 1 r 2 k 1 , k r 2 k + 1 , k + 2 r 2 n R .
E 12 = E 1 A R ( 4 π ) 2 ( n + 1 ) a n p n 1 k = 0 n r k , k + 1 2 exp ( σ r k , k + 1 ) Φ k Φ k + 1 .
σ k diff ( a , g ) = σ k diff ( a , g ) , 1 ag 1 a = 1 a g 1 a .
P = i | E R + z i | 2 A i = P S + P R ,
P S = i A i ( E R z i * + E R * z i ) + i z i z i * A i ; P R = | E R | 2 A .
P ¯ S = i z i z i * A i = n P 0
i ¯ = η q h υ ( P ¯ S + P R ) , i p = η q h υ i ( E R z * i + E * R z i ) A i ,
i p 2 = 2 η 2 q 2 h 2 ν 2 P R P ¯ S n .
( SNR ) heterodyne = 2 η 2 q 2 h 2 ν 2 P R P ¯ S n M 2 2 q [ η q h ν ( P ¯ S + P R ) + I d ] M 2 F ( M ) B I + i 2 amp + α 2 η 2 q 2 h 2 ν 2 M 2 ( P ¯ S + P R ) 2 ,
P R = P ¯ S 2 + B P ¯ S + C A ,
( SNR ) heterodyne = 2 η 2 q 2 h 2 ν 2 P 0 M 2 ( n 2 P 2 0 + B P 0 n + C A ) 1 / 2 A [ n P 0 + ( n 2 P 2 0 + B P 0 n + C A ) 1 / 2 ] 2 + B [ n P 0 + ( n 2 P 2 0 + B P 0 n + C A ) 1 / 2 ] + C , A α 2 η 2 q 2 M 2 h 2 ν 2 , B 2 M 2 F ( M ) B I η q 2 h v , C i 2 amp + 2 q M 2 F ( M ) B I I d .
( SNR ) hemodyne = M 2 η 2 q 2 h 2 ν 2 n P 0 2 A n 2 P 0 2 + B n P 0 + C ,
n = ½ ( n 1 kr ) 2 Δ ,
i s = a exp ( iw 0 t ) + a * exp ( iw 0 t ) ,
g ( τ ) = a ( t + τ ) a * ( t ) = P ( w ) exp ( iw τ ) dw .
Δ w ¯ = wP ( w ) dw P ( w ) dw = i ( dg d τ ) τ = 0 g ( 0 ) = i a ˙ ( t ) a * ( t ) a ( t ) a * ( t ) .
Δ w ( t ) = i a ˙ ( t ) a * ( t ) a ( t ) a * ( t ) ,
Δ w B ( t ) = F [ a ˙ ( t ) a * ( t ) ] a ( t ) a * ( t ) ,
v = Δ w B 2 ¯ Δ w 2 ¯ Δ w 2 ¯ 2 B 1 P Δ w ( 0 ) Δ w 2 ¯ = B 1 π G ( τ ) d τ Δ w 2 ¯ , G ( τ ) Δ w ( t + τ ) Δ w ( t ) Δ w 2 ¯ .
v = B 1 π | a ˙ ( t + τ ) a * ( t + τ ) a ˙ ( t ) a * ( t ) a ˙ ( t ) a * ( t ) 2 | d τ [ w P ( w ) dw ] 2 = B 1 π | dg d τ | 2 d τ [ wP ( w ) dw ] 2 = 2 B 1 w 2 P 2 ( w ) dw [ wP ( w ) dw ] 2 .
P ( w ) = S B 0 2 π exp [ ( w Δ w ¯ ) 2 / 2 B 0 2 ] + N 2 B I
δ rms = v = B 1 Δ w ¯ { [ B 0 2 Δ w ¯ + Δ w ¯ B 0 ] 1 π + 2 ( N S ) [ B 0 2 B 1 Δ w ¯ + Δ w ¯ B 1 ] + ( N S ) 2 B I 3 Δ w ¯ } .
δ rms = B 1 Δ w ¯ { 4 3 + 8 Δ w ¯ 3 B I ( N S ) + B I 3 Δ w ¯ ( N S ) 2 } .
i s ( t ) i s ( t + τ ) g 2 ( τ ) = P 2 ( w ) exp ( iw τ ) dw .
B ¯ = | w | P 2 ( w ) dw P 2 ( w ) dw ,
B ¯ = | f [ i s ( t ) ] | 2 i s 2 y 2 ( t ) / i s 2 ,
y ( t ) y ( t + τ ) = q ( τ ) = Q ( w ) exp ( iwt ) dw ; Q ( w ) | w | P 2 ( w ) .
v = 1 i s 2 2 B 1 π [ y 2 ( t ) y 2 ( t + τ ) y 2 ( t ) 2 ] d τ = 2 B 1 π i s 2 2 q 2 ( τ ) d τ = 4 B 1 i s 2 2 Q 2 ( w ) dw = 4 B 1 w 2 P 2 2 ( w ) dw [ α P 2 ( w ) dw ] 2 .
δ rms = v B ¯ B N = 2 α w 2 P 2 2 ( w ) dw α | w | [ P 2 ( w ) P N ( w ) ] dw .
P 2 ( w ) = P ( w ) P ( w + w ) dw + noise .
P 2 ( w ) = S 2 B 0 π exp ( w 2 / 4 B 0 2 ) + N 2 B I ,
δ rms = π B 1 2 B 0 [ I 2 π + 4 ( N S ) B 0 B I + ( N S ) 2 B I 3 B 0 ]

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