Abstract

I detail the results of adapting a rigorous algorithm, derived for multiple-scattering simulations in photon correlation spectroscopy, for modeling multiple-scattering suppression by a cross-correlation system that employs one laser beam and two slightly tilted detectors. The practical significance of the proposed numerical technique is shown for optimization of an arbitrary design configuration of cross correlation and for prediction of the ideal performance that is possible with that design. It is shown that the behavior of the coherent factor modeled versus the angle between detectors is in agreement with experimental data and analytical investigation. This factor permits mapping of the spatial extent of the single-scattering and the multiple-scattering speckles. The map holds important information about the optimal displacement of detectors for a given measurement setup, and it permits a comprehensive investigation of suppression of the scattering components, even when their magnitudes are small.

© 1998 Optical Society of America

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References

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  1. H. Wiese, D. Horn, “Single-mode fibers in fiber-optic quasielastic light scattering: a study of the dynamics of concentrated latex dispersions,” J. Chem. Phys. 94, 6429–6443 (1991).
    [CrossRef]
  2. R. G. W. Brown, A. E. Smart, “Practical consideration in photon correlation experiments,” Appl. Opt. 36, 7480–7492 (1997).
    [CrossRef]
  3. J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. 1. Theory,” J. Chem. Phys. 79, 1658–1663 (1983).
    [CrossRef]
  4. J. K. G. Dhont, “Multiple Rayleigh–Gans–Debye scattering in colloidal systems—dynamic light scattering,” Physica A 129, 374–394 (1985).
    [CrossRef]
  5. J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. II. Experimental,” J. Chem. Phys. 84, 45–49 (1986).
    [CrossRef]
  6. D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
    [CrossRef]
  7. R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorii-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
    [CrossRef]
  8. V. I. Ovod, D. W. Mackowski, R. Finsy, “Modeling of the effect of multiple scattering in photon correlation spectroscopy: plane-wave approach,” Langmuir 14, 2610–2618 (1998).
    [CrossRef]
  9. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configuration,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
    [CrossRef]
  10. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
    [CrossRef]
  11. W. V. Meyer, D. S. Cannell, A. E. Smart, T. W. Taylor, P. Tin, “Multiple scattering suppression by cross correlation,” Appl. Opt. 36, 7551–7558 (1997).
    [CrossRef]
  12. U. Nobbmann, S. W. Jones, B. J. Ackerson, “Multiple scattering suppression: cross correlation with tilted single-mode fibers,” Appl. Opt. 36, 7571–7576 (1997).
    [CrossRef]
  13. J. A. Lock, “The role of multiple scattering in cross-correlated light scattering employing a single laser beam,” Appl. Opt. 36, 7559–7570 (1997).
    [CrossRef]
  14. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  15. D. W. Mackowski, M. I. Mischenko, “Calculation of the T matrix and scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
    [CrossRef]
  16. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 36, 4573–4588 (1995).
    [CrossRef]
  17. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. 1. Linear chains,” Opt. Lett. 13, 90–92 (1988).
    [CrossRef] [PubMed]
  18. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. 1. Clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
    [CrossRef] [PubMed]
  19. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997).
    [CrossRef]
  20. G. Videen, R. G. Pinnick, D. Ngo, Q. Fu, P. Chýlek, “Asymmetric parameter and aggregate particles,” Appl. Opt. 37, 1104–1109 (1998).
    [CrossRef]
  21. V. I. Ovod, “Modeling of multiple scattering from ensemble of spheres in a laser beam,” in Proceedings of the Fifth International Congress on Optical Particle Sizing (Minneapolis, Minn., 1998).

1998

V. I. Ovod, D. W. Mackowski, R. Finsy, “Modeling of the effect of multiple scattering in photon correlation spectroscopy: plane-wave approach,” Langmuir 14, 2610–2618 (1998).
[CrossRef]

G. Videen, R. G. Pinnick, D. Ngo, Q. Fu, P. Chýlek, “Asymmetric parameter and aggregate particles,” Appl. Opt. 37, 1104–1109 (1998).
[CrossRef]

1997

1996

1995

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 36, 4573–4588 (1995).
[CrossRef]

1994

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
[CrossRef]

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorii-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

1991

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configuration,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

H. Wiese, D. Horn, “Single-mode fibers in fiber-optic quasielastic light scattering: a study of the dynamics of concentrated latex dispersions,” J. Chem. Phys. 94, 6429–6443 (1991).
[CrossRef]

1990

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

1988

1986

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. II. Experimental,” J. Chem. Phys. 84, 45–49 (1986).
[CrossRef]

1985

J. K. G. Dhont, “Multiple Rayleigh–Gans–Debye scattering in colloidal systems—dynamic light scattering,” Physica A 129, 374–394 (1985).
[CrossRef]

1983

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. 1. Theory,” J. Chem. Phys. 79, 1658–1663 (1983).
[CrossRef]

Ackerson, B. J.

U. Nobbmann, S. W. Jones, B. J. Ackerson, “Multiple scattering suppression: cross correlation with tilted single-mode fibers,” Appl. Opt. 36, 7571–7576 (1997).
[CrossRef]

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorii-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Brown, R. G. W.

Cannell, D. S.

Chýlek, P.

de Kruif, C. G.

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. II. Experimental,” J. Chem. Phys. 84, 45–49 (1986).
[CrossRef]

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. 1. Theory,” J. Chem. Phys. 79, 1658–1663 (1983).
[CrossRef]

Dhont, J. K. G.

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. II. Experimental,” J. Chem. Phys. 84, 45–49 (1986).
[CrossRef]

J. K. G. Dhont, “Multiple Rayleigh–Gans–Debye scattering in colloidal systems—dynamic light scattering,” Physica A 129, 374–394 (1985).
[CrossRef]

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. 1. Theory,” J. Chem. Phys. 79, 1658–1663 (1983).
[CrossRef]

Dorii-Nowkoorani, F.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorii-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Dougherty, R. L.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorii-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Finsy, R.

V. I. Ovod, D. W. Mackowski, R. Finsy, “Modeling of the effect of multiple scattering in photon correlation spectroscopy: plane-wave approach,” Langmuir 14, 2610–2618 (1998).
[CrossRef]

Fu, Q.

Fuller, K. A.

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

Horn, D.

H. Wiese, D. Horn, “Single-mode fibers in fiber-optic quasielastic light scattering: a study of the dynamics of concentrated latex dispersions,” J. Chem. Phys. 94, 6429–6443 (1991).
[CrossRef]

Jones, S. W.

Kattawar, G. W.

Lock, J. A.

Mackowski, D. W.

V. I. Ovod, D. W. Mackowski, R. Finsy, “Modeling of the effect of multiple scattering in photon correlation spectroscopy: plane-wave approach,” Langmuir 14, 2610–2618 (1998).
[CrossRef]

D. W. Mackowski, M. I. Mischenko, “Calculation of the T matrix and scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
[CrossRef]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configuration,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

Meyer, W. V.

Mischenko, M. I.

Ngo, D.

Nobbmann, U.

U. Nobbmann, S. W. Jones, B. J. Ackerson, “Multiple scattering suppression: cross correlation with tilted single-mode fibers,” Appl. Opt. 36, 7571–7576 (1997).
[CrossRef]

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorii-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Ovod, V. I.

V. I. Ovod, D. W. Mackowski, R. Finsy, “Modeling of the effect of multiple scattering in photon correlation spectroscopy: plane-wave approach,” Langmuir 14, 2610–2618 (1998).
[CrossRef]

V. I. Ovod, “Modeling of multiple scattering from ensemble of spheres in a laser beam,” in Proceedings of the Fifth International Congress on Optical Particle Sizing (Minneapolis, Minn., 1998).

Pine, D. J.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

Pinnick, R. G.

Reguigui, N. M.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorii-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Smart, A. E.

Taylor, T. W.

Tin, P.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Videen, G.

Weitz, D. A.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

Wiese, H.

H. Wiese, D. Horn, “Single-mode fibers in fiber-optic quasielastic light scattering: a study of the dynamics of concentrated latex dispersions,” J. Chem. Phys. 94, 6429–6443 (1991).
[CrossRef]

Xu, Y.-L.

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997).
[CrossRef]

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 36, 4573–4588 (1995).
[CrossRef]

Zhu, J. X.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

Appl. Opt.

J. Chem. Phys.

H. Wiese, D. Horn, “Single-mode fibers in fiber-optic quasielastic light scattering: a study of the dynamics of concentrated latex dispersions,” J. Chem. Phys. 94, 6429–6443 (1991).
[CrossRef]

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. 1. Theory,” J. Chem. Phys. 79, 1658–1663 (1983).
[CrossRef]

J. K. G. Dhont, C. G. de Kruif, “Scattered light intensity cross correlation. II. Experimental,” J. Chem. Phys. 84, 45–49 (1986).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. (Paris)

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorii-Nowkoorani, U. Nobbmann, “Correlation transfer: development and application,” J. Quant. Spectrosc. Radiat. Transfer 52, 713–727 (1994).
[CrossRef]

Langmuir

V. I. Ovod, D. W. Mackowski, R. Finsy, “Modeling of the effect of multiple scattering in photon correlation spectroscopy: plane-wave approach,” Langmuir 14, 2610–2618 (1998).
[CrossRef]

Opt. Lett.

Physica A

J. K. G. Dhont, “Multiple Rayleigh–Gans–Debye scattering in colloidal systems—dynamic light scattering,” Physica A 129, 374–394 (1985).
[CrossRef]

Proc. R. Soc. London Ser. A

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configuration,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

Other

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

V. I. Ovod, “Modeling of multiple scattering from ensemble of spheres in a laser beam,” in Proceedings of the Fifth International Congress on Optical Particle Sizing (Minneapolis, Minn., 1998).

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Figures (7)

Fig. 1
Fig. 1

Schematic representation of the cross-correlation setup at θ = 90°. Hidden detector B is located in the same xy plane as is shown for the detector A. A laser beam propagates through a scattering cell of volume V, illuminating sample volume V 1. Both detectors view the same sample volume, V 2. The intersection of V 1 and V 2 volumes is denoted V o . The average numbers of particles in these volumes are N, N 1, N 2, and N o , respectively.

Fig. 2
Fig. 2

Flow diagram of the simulations: CCF, cross-correlation function.

Fig. 3
Fig. 3

Scattering schematic illustrating by numerical modeling the multiple-scattering suppression in a particle suspension of volume fraction φ = 0.005. Only volume V o (2R x = 2R y = 2R z = 0.683 μm) contributes to single scattering. The incident field is polarized in the x direction [e x = 1 and e y = 0; Eqs. (23) and (24)] or in the y direction (e x = 0, e y = 1). The orthonormal vectors, θ̂ and φ̂, denote two polarization states of a receiver polarizer. Two photodetectors view all N = 9000 particles contained in the cell of volume V (2l x = 2l z = 45.8 μm and 2l y = 0.683 μm).

Fig. 4
Fig. 4

Suppression of the single-scattering cross term C 11 2 (curves 1 and 4), the second-order scattering cross term C 22 2 (curves 3 and 6), and the modeled cross-correlation function C 2 (curves 2 and 5) versus detector displacement. The receiver polarizer is absent. The numbers of particles in the corresponding volumes are N o = 3, N 1 = 3, and N 2 = 9000. Two sets of curves (curves 1–3 and 4–6) correspond to the x and y polarization, respectively, of the incident field. The detector displacement range of 0.05–0.07 rad can be used as the optimal one for a given scattering geometry. S, D, and S + D denote single, double, and combined (single and double) scattering, respectively. The subscripts x and y correspond to x and y polarization, respectively, of the incident field. Hence curves 1–3 and 4–6 correspond to the so-called (VV + VH) and (HV + HH) input–output polarization states, respectively.

Fig. 5
Fig. 5

Suppression of the single-scattering cross term C 11 2 (curves 1 and 4), the second-order scattering cross term C 22 2 (curves 3 and 6), and the modeled cross-correlation function C 2, (curves 2 and 5) versus detector displacement. Only the P θ component of the scattered light is transmitted through the detector polarizer. The numbers of particles in the corresponding volumes are N o = 3, N 1 = 3, and N 2 = 9000. Two sets of curves (curves 1–3 and 4–6) correspond to x and y polarization, respectively, of the incident field. The single-scattering speckle (curve 1) disappears for the x-polarized incident field, as was shown first experimentally.12 S, D, and S + D denote single, double, and combined (single and double) scattering, respectively. The subscripts x and y correspond to x and y polarization, respectively, of the incident field. Hence curves 1–3 and 4–6 correspond to the VH and HH input–output polarization states, respectively.

Fig. 6
Fig. 6

Relative suppression coefficient RC2(δ) versus the detector displacement for three scattering components: the second-order scattering component from the x-polarized incident field (curve 1) and second-order scattering (curve 2) and single-scattering (curve 3) from a y-polarized incident field. The detector displacement range of 0.05–0.07 rad can be used as the optimal one for a given scattering geometry. S and D denote single and double scattering, respectively. The subscripts x and y correspond to x and y polarization, respectively, of the incident field.

Fig. 7
Fig. 7

Square root of the intensity cross-correlation function (CCF) modeled for the single-scattering term, g 11 ( 2 ) (τ) (curves 2 and 3), and for the second-order scattering cross term, g 22 ( 2 ) (τ) (curves 4 and 5), and at δ = 0.05 rad and for a system of Brownian particles (curve 1). Curves 2 and 4 are for an x-polarized incident field; curves 3 and 5 are for a y-polarized incident field. S, D, and S + D denote single, double, and combined (single and double) scattering, respectively. The subscripts x and y correspond to x and y polarization, respectively, of the incident field.

Equations (50)

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δ coh x = 1 kR x ,
δ coh z = 1 kR z
k | k o | = 2 π n L λ
E s , ε i = E o i D X i n = 1 m = - n n a mn 1 , ε i N mn 3 k R d i + a mn 2 , ε i M mn 3 k R d i ,
E o i = E o   exp i kL e exp - k m im L bi 2 ,
a ± 1 np , ε i , s = 1 = - α np i U X i p ± 1 np , ε   exp i k o X i ,
p 1 n 1 , x = - i n + 1 2 2 n + 1 n n + 1 , p - 1 n 1 , x = i n + 1 2 2 n + 1 , p mnp , ε = 0 ,     | m | 1 ,
p mn 3 - p , ε = mp mnp , ε ,     p mnp , y = 1 i p mn 3 - p , x .
a mnp , ε i , s = 2 = - α np , ε i j = 1 j i N 3 l = 1 N o j k = - l l A klmn 3 kR ij ,   Θ ij ,   Φ ij a mnp , ε j + B klmn 3 kR ij ,   Θ ij ,   Φ ij a mn 3 - p , ε j ,
N o j ρ j + 4 ρ j 1 / 3 + 2 ,
a mnp , ε i = s = 1 2   a mnp , ε i , s ,
a mnp , ε j = s = 1 2   a mnp , ε j , s ,
a ± 1 np , ε j , s = 1 = - α np j U X j p ± 1 np , ε   exp i k o X j ,
a mnp , ε j , s = 2 = - α np , ε j i = 1 i j N 3 l = 1 N o i k = - l l × A klmn 3 kR ji ,   Θ ji ,   Φ ji a mnp , ε i + B klmn 3 kR ji ,   Θ ji ,   Φ ji a mn 3 - p , ε i
E s , ε = i = 1 N 2 E s , ε i .
M mn 3 k R d i = - i n + 1 exp ik s i R d i kR d i exp - k m im L id 2 × i τ mn 2 θ i θ ˆ - τ mn 1 θ i φ ˆ exp i m φ i ,
N mn 3 k R d i = - i n exp i k s i R d i kR d i exp - k m im L id 2 × τ mn 1 θ i θ ˆ + i τ mn 2 θ i φ ˆ exp i m φ i ,
τ mn 1 θ = d d θ   P n m cos θ ,     τ mn 2 θ = m sin θ   P n m cos θ
k s i = k R d i R d i = k sin   θ i   cos   φ i x ˆ + sin   θ i   sin   φ i y ˆ + cos   θ i z ˆ ,
R d i = R d - X i .
E s i = iT P s i ,
T = E o D X i exp i k s i R d kR d exp ikL e exp - k m im L bi + L id 2
P θ i = exp - i k s i X i S 2 i θ i ,   φ i e x i + S 3 i θ i ,   φ i e y i ,
P φ i = exp - i k s i X i S 4 i θ i ,   φ i e x i + S 1 i θ i ,   φ i e y i ,
S 1 i θ i ,   φ i = n = 1 m = - n n p = 1 2 - i n a mnp , ε = y i τ mn 3 - p θ i × exp i m   φ i ,
S 2 i θ i ,   φ i = n = 1 m = - n n p = 1 2 - i n + 1 a mnp , ε = x i τ mnp θ i × exp i m   φ i ,
S 3 i θ i ,   φ i = n = 1 m = - n n p = 1 2 - i n + 1 a mnp , ε = y i τ mnp θ i ×   exp i m   φ i ,
S 4 i θ i ,   φ i = n = 1 m = - n n p = 1 2 - i n a mnp , ε = x i τ mn 3 - p θ i × exp i m   φ i ,
e x i = E x E o ,
e y i = E y E o ,
a mnp , ε i , s = 2 = - α np , ε i j = 1 j i N 1   F   l = 1 N o j k = - l l A klmn 3 kR ij ,   Θ ij ,   Φ ij a mnp , ε j + B klmn 3 kR ij ,   Θ ij ,   Φ ij a mn 3 - p , ε j ,
F = 1 multiple-order scattering exp - k m im R ij 2 second-order scattering .
G 2 τ ,   δ = -   I A t I B t + τ d t ,
G 2 τ ,   δ = BL + | Y | 2 + | W | 2 ,
Y = - E A t E B t + τ d t , W = - E A * t E B t + τ d t ;
BL = - E A * t E A t d t   - E B * t E B t d t
g 2 τ ,   δ = G 2 τ ,   δ - BL BL = γ 2 δ | g 1 τ | 2 ,
Y τ ,   δ = f = 1 2 g = 1 2   Y fg τ ,   δ ,     W τ ,   δ = f = 1 2 g = 1 2   W fg τ ,   δ ,
Y fg τ ,   δ = - E Af t ,   δ E Bg t + τ ,   δ d t ,
W τ ,   δ = - E Af * t ,   δ E Bg t + τ ,   δ d t .
g fg 2 τ ,   δ = | Y fg τ ,   δ | 2 + | W fg τ ,   δ | 2 BL ,
C fg , pol , ε 2 δ = γ fg , pol , ε 2 δ γ fg , pol , ε 2 δ = 0 ,
γ fg , pol , ε 2 δ = g fg , pol , ε 2 τ = 0 ,   δ .
RC fg , pol , ε 2 δ = C fg , pol , ε 2 δ C base 2 δ ,
C fg , pol , ε 2 δ = γ fg , pol , ε 2 δ γ pol , ε 2 δ = 0 .
M : S pol , ε = BL 22 , pol , ε BL 11 , pol , ε 1 / 2 .
V = 2 l x 2 l y 2 l z = d p 3 π N p 6 φ .
C 22 , φ , x δ coh x 4 16 π R x 15 R ij ,
2 R x 15 8 π   C 22 , φ , x δ coh x 4 R ij 0.06 R ij .
2 R x 0.06 l z / 2 .

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