Abstract

We introduce a numerical approach, the finite-difference time-domain (FDTD) method, to study the near-field effects on coherent anti-Stokes Raman scattering (CARS) microscopy on nanoparticles. Changes of the induced nonlinear polarization, scattering patterns, and polarization properties against different diameters of spherical nanoparticles are calculated and discussed in detail. The results show that due to near-field effects, the induced nonlinear polarization is significantly enhanced at the water-particle interface, with 1.5-fold increase in intensity compared to that inside the particles, and the near-field enhancement increases with decreasing diameters of nanoparticles. The enhanced scattering dominates over the scattering contribution from the particles when the nanoparticle size decreases down to the scale of less than a half wavelength of excitation light. Further studies show that near-field effects make the induced perpendicular polarization of CARS signals being strictly confined within the nanoparticles and the particle-water interface, and this perpendicular polarization component could contribute approximately 20% to the backward scattering. The ratio values of the perpendicular polarization component to the total CARS signals from nanoparticles sizing from 75 nm to 300 nm in backward scattering are approximately 3 to 5 times higher than those in forward scattering. Therefore, near-field effects can play an important role in CARS imaging. Employing the perpendicular polarization component of CARS signals can significantly improve the contrast of CARS images, and be particularly useful for revealing the fine structures of bio-materials with nano-scale resolutions.

© 2007 Optical Society of America

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References

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  1. R. J. H. Clark and R. E. Hester, Advances in Nonlinear Spectroscopy (Wiley, New York, 1988).
  2. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137,A801 (1965).
    [CrossRef]
  3. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  4. J. X. Cheng and X. S. Xie, “Coherent anti-Stokes scattering microscopy: instrumentation, theory, and application,” J. Phys. Chem. B 108,827 (2004).
    [CrossRef]
  5. J. X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19,1363 (2002).
    [CrossRef]
  6. H. Wang, Y. Fu, P. Zickmund, R. Shi, and X. J. Cheng, “Coherent anti-Stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys. J. 89,581 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. J. P. Ogilvie, E. Beaurepaire, A. Alexandrou, and M. Joffre, “Fourier-transform coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 31,480 (2006).
    [CrossRef] [PubMed]
  9. D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89,273001 (2002).
    [CrossRef]
  10. E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thϕgersen, and S. R. Keiding, “Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers,” J. Opt. Am. B 22,1934 (2005).
    [CrossRef]
  11. J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19,1604 (2002).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. K.S. Yee “Numerical solution of initial boundary value problem involving Maxwell equations in isotropic media,” IEEE Trans. Antennas Propagat. 14,302 (1966).
    [CrossRef]
  15. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).
  16. R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).
  17. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, 1995).
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    [CrossRef]
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    [CrossRef]
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    [PubMed]
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  22. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II: Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253,358 (1959).
    [CrossRef]
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  24. T. Wilson and J. B. Tan, “Finite sized coherent and incoherent detectors in confocal microscopy,” J. Microsc. 182,61 (1995).
    [CrossRef]

2006 (2)

2005 (3)

H. Wang, Y. Fu, P. Zickmund, R. Shi, and X. J. Cheng, “Coherent anti-Stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys. J. 89,581 (2005).
[CrossRef] [PubMed]

E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thϕgersen, and S. R. Keiding, “Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers,” J. Opt. Am. B 22,1934 (2005).
[CrossRef]

C. Liu and S. H. Park, “Anisotropy of near-field speckle patterns,” Opt. Lett. 30,1602 (2005).
[CrossRef] [PubMed]

2004 (2)

J. X. Cheng and X. S. Xie, “Coherent anti-Stokes scattering microscopy: instrumentation, theory, and application,” J. Phys. Chem. B 108,827 (2004).
[CrossRef]

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata, “Tip-enhanced coherent anti-Stokes Raman scattering for vibrational nano-imaging,” Phys. Rev. Lett. 92,220801 (2004).
[CrossRef] [PubMed]

2002 (3)

1999 (1)

1995 (1)

T. Wilson and J. B. Tan, “Finite sized coherent and incoherent detectors in confocal microscopy,” J. Microsc. 182,61 (1995).
[CrossRef]

1992 (1)

K. Takeda, Y. Ito, and C. Munakata, “Simultaneous measurement of size and refractive index of a fine particle in flowing liquid,” Meas. Sci. Technol. 3,27 (1992).
[CrossRef]

1966 (1)

K.S. Yee “Numerical solution of initial boundary value problem involving Maxwell equations in isotropic media,” IEEE Trans. Antennas Propagat. 14,302 (1966).
[CrossRef]

1965 (1)

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137,A801 (1965).
[CrossRef]

1962 (1)

H. Lotem, R. T. Lynch, and N. Blombergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. 126,1977 (1962).

1959 (1)

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

Alexandrou, A.

Andresen, E. R.

E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thϕgersen, and S. R. Keiding, “Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers,” J. Opt. Am. B 22,1934 (2005).
[CrossRef]

Aoyama, N.

Beaurepaire, E.

Birkedal, V.

E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thϕgersen, and S. R. Keiding, “Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers,” J. Opt. Am. B 22,1934 (2005).
[CrossRef]

Blombergen, N.

H. Lotem, R. T. Lynch, and N. Blombergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. 126,1977 (1962).

Born, M.

M. Born and E Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th Edition) (Cambridge University Press, New York, 2002).
[PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).

Cheng, J. X.

Cheng, X. J.

H. Wang, Y. Fu, P. Zickmund, R. Shi, and X. J. Cheng, “Coherent anti-Stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys. J. 89,581 (2005).
[CrossRef] [PubMed]

Clark, R. J. H.

R. J. H. Clark and R. E. Hester, Advances in Nonlinear Spectroscopy (Wiley, New York, 1988).

Dudovich, N.

D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89,273001 (2002).
[CrossRef]

Evans, C. L.

Fu, Y.

H. Wang, Y. Fu, P. Zickmund, R. Shi, and X. J. Cheng, “Coherent anti-Stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys. J. 89,581 (2005).
[CrossRef] [PubMed]

Futamata, A.

Hasegawa, S. Y.

Hashimoto, M.

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata, “Tip-enhanced coherent anti-Stokes Raman scattering for vibrational nano-imaging,” Phys. Rev. Lett. 92,220801 (2004).
[CrossRef] [PubMed]

Hayazawa, N.

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata, “Tip-enhanced coherent anti-Stokes Raman scattering for vibrational nano-imaging,” Phys. Rev. Lett. 92,220801 (2004).
[CrossRef] [PubMed]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, New York, 2006).

Hester, R. E.

R. J. H. Clark and R. E. Hester, Advances in Nonlinear Spectroscopy (Wiley, New York, 1988).

Ichimura, T.

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata, “Tip-enhanced coherent anti-Stokes Raman scattering for vibrational nano-imaging,” Phys. Rev. Lett. 92,220801 (2004).
[CrossRef] [PubMed]

Inouye, Y.

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata, “Tip-enhanced coherent anti-Stokes Raman scattering for vibrational nano-imaging,” Phys. Rev. Lett. 92,220801 (2004).
[CrossRef] [PubMed]

Ito, Y.

K. Takeda, Y. Ito, and C. Munakata, “Simultaneous measurement of size and refractive index of a fine particle in flowing liquid,” Meas. Sci. Technol. 3,27 (1992).
[CrossRef]

Joffre, M.

Kawata, S.

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata, “Tip-enhanced coherent anti-Stokes Raman scattering for vibrational nano-imaging,” Phys. Rev. Lett. 92,220801 (2004).
[CrossRef] [PubMed]

Keiding, S. R.

E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thϕgersen, and S. R. Keiding, “Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers,” J. Opt. Am. B 22,1934 (2005).
[CrossRef]

Liu, C.

Lotem, H.

H. Lotem, R. T. Lynch, and N. Blombergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. 126,1977 (1962).

Lynch, R. T.

H. Lotem, R. T. Lynch, and N. Blombergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. 126,1977 (1962).

Maker, P. D.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137,A801 (1965).
[CrossRef]

Mukamel, S.

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, 1995).

Munakata, C.

K. Takeda, Y. Ito, and C. Munakata, “Simultaneous measurement of size and refractive index of a fine particle in flowing liquid,” Meas. Sci. Technol. 3,27 (1992).
[CrossRef]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, New York, 2006).

Ogilvie, J. P.

Oron, D.

D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89,273001 (2002).
[CrossRef]

Park, S. H.

Paulsen, H. N.

E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thϕgersen, and S. R. Keiding, “Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers,” J. Opt. Am. B 22,1934 (2005).
[CrossRef]

Potma, E. O.

Richards, B.

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

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Shi, R.

H. Wang, Y. Fu, P. Zickmund, R. Shi, and X. J. Cheng, “Coherent anti-Stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys. J. 89,581 (2005).
[CrossRef] [PubMed]

Silberberg, Y.

D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89,273001 (2002).
[CrossRef]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).

Takeda, K.

K. Takeda, Y. Ito, and C. Munakata, “Simultaneous measurement of size and refractive index of a fine particle in flowing liquid,” Meas. Sci. Technol. 3,27 (1992).
[CrossRef]

Tan, J. B.

T. Wilson and J. B. Tan, “Finite sized coherent and incoherent detectors in confocal microscopy,” J. Microsc. 182,61 (1995).
[CrossRef]

Terhune, R. W.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137,A801 (1965).
[CrossRef]

Th?gersen, J.

E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thϕgersen, and S. R. Keiding, “Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers,” J. Opt. Am. B 22,1934 (2005).
[CrossRef]

Uchiyama, T.

Volkmer, A.

Wang, H.

H. Wang, Y. Fu, P. Zickmund, R. Shi, and X. J. Cheng, “Coherent anti-Stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys. J. 89,581 (2005).
[CrossRef] [PubMed]

Wilson, T.

T. Wilson and J. B. Tan, “Finite sized coherent and incoherent detectors in confocal microscopy,” J. Microsc. 182,61 (1995).
[CrossRef]

Wolf, E

M. Born and E Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th Edition) (Cambridge University Press, New York, 2002).
[PubMed]

Wolf, E.

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

Xie, X. S.

Yee, K.S.

K.S. Yee “Numerical solution of initial boundary value problem involving Maxwell equations in isotropic media,” IEEE Trans. Antennas Propagat. 14,302 (1966).
[CrossRef]

Zickmund, P.

H. Wang, Y. Fu, P. Zickmund, R. Shi, and X. J. Cheng, “Coherent anti-Stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys. J. 89,581 (2005).
[CrossRef] [PubMed]

Appl. Opt. (1)

Biophys. J. (1)

H. Wang, Y. Fu, P. Zickmund, R. Shi, and X. J. Cheng, “Coherent anti-Stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys. J. 89,581 (2005).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propagat. (1)

K.S. Yee “Numerical solution of initial boundary value problem involving Maxwell equations in isotropic media,” IEEE Trans. Antennas Propagat. 14,302 (1966).
[CrossRef]

J. Microsc. (1)

T. Wilson and J. B. Tan, “Finite sized coherent and incoherent detectors in confocal microscopy,” J. Microsc. 182,61 (1995).
[CrossRef]

J. Opt. Am. B (1)

E. R. Andresen, H. N. Paulsen, V. Birkedal, J. Thϕgersen, and S. R. Keiding, “Broadband multiplex coherent anti-Stokes Raman scattering microscopy employing photonic-crystal fibers,” J. Opt. Am. B 22,1934 (2005).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. Chem. B (1)

J. X. Cheng and X. S. Xie, “Coherent anti-Stokes scattering microscopy: instrumentation, theory, and application,” J. Phys. Chem. B 108,827 (2004).
[CrossRef]

Meas. Sci. Technol. (1)

K. Takeda, Y. Ito, and C. Munakata, “Simultaneous measurement of size and refractive index of a fine particle in flowing liquid,” Meas. Sci. Technol. 3,27 (1992).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. (2)

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137,A801 (1965).
[CrossRef]

H. Lotem, R. T. Lynch, and N. Blombergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. 126,1977 (1962).

Phys. Rev. Lett. (2)

D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89,273001 (2002).
[CrossRef]

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata, “Tip-enhanced coherent anti-Stokes Raman scattering for vibrational nano-imaging,” Phys. Rev. Lett. 92,220801 (2004).
[CrossRef] [PubMed]

Proc. Roy. Soc. A (1)

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

Other (7)

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, New York, 2006).

M. Born and E Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th Edition) (Cambridge University Press, New York, 2002).
[PubMed]

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, 1995).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

R. J. H. Clark and R. E. Hester, Advances in Nonlinear Spectroscopy (Wiley, New York, 1988).

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Figures (7)

Fig. 1.
Fig. 1.

(a). Schematic diagram of CARS experiments. The collinear pump and Stokes light beams with parallel polarization in x-direction are tightly focused onto the sample located in the focal volume of a high numerical aperture, water-immersion microscope objective; (b) amplitude distribution of the focused light field in water (without nanoparticles) in the x-z plane, and (c) the corresponding phase distribution of the focused light field in the x-z plane. Note that in Fig. 1(a) the red and blue arrows represent the pump and the Stokes light beams, respectively, and the rectangle in dotted lines stands for the excitation volume of light fields for FDTD simulation.

Fig. 2.
Fig. 2.

Distributions of the focal light field surrounding the nanoparticle in the x-z plane (a), and in the x-y plane (b), respectively. (c) Comparison of the intensity profiles of the focal light field in x-direction (red curve) and in y-direction (green curve) with nanoparticles, and the intensity profile of the focal light field distribution in pure water (blue curve).

Fig. 3.
Fig. 3.

The induced nonlinear polarizations (x-component) for nanoparticles with diameters of: (a) 75 nm; (b) 125 nm; (c) 200 nm; (d) 250 nm; (e) 300 nm, and (f) 325 nm, respectively. The top two panels represent the amplitude distribution of focal light field in the x-z plane and the x-y plane, respectively. The third panel is the corresponding intensity profiles across the particles along the x-direction as indicated in second panel. The bottom panel represents the corresponding phase distributions of light fields in the x-z plane.

Fig. 4.
Fig. 4.

The total scattering patterns from both the surrounding water and the nanoparticle with diameters of (a) 75 nm; (b) 125 nm; (c) 250 nm, and (d) 325 nm, respectively. (e), (f), (g), and (h) represent the scattering patterns from the nanoparticle alone with diameters of 75 nm; 125 nm; 250 nm, and 325 nm, respectively. Note that in z-axis, the negative values stand for CARS intensities in the forwarding scattering, whereas the postive values correspond to CARS intensities in the backward scattering.

Fig. 5.
Fig. 5.

Changes of scattering amplitudes as a function of particle diameters: (a) in F-CARS; and (b) in E-CARS. Note that the curve oe-15-7-4118-i001 corresponds to the total scattering from inside the particle and the surrounding water outside the particle, the curve oe-15-7-4118-i002 stands for the scattering from inside the particle, and the curve oe-15-7-4118-i003 stands for the scattering from the surrounding water outside the particle.

Fig. 6.
Fig. 6.

Distribution of the induced polarization of the focal light fields on the x-y plane. (a), (b) and (c) represent the distribution for x-component, z-component, and y-component of pure water, respectively; (d), (e), and (f) correspond to the related distributions with 140 nm particles placed in the focal volume of a 0.9 N.A., water objective.

Fig. 7.
Fig. 7.

Comparison of scattering amplitude of the induced perpendicular polarization component from nanoparticles with the total scattering and the scattering from surrounding water outside the nanoparticle: (a) in forward direction, and (b) in backward direction; (c) ratio of the perpendicular polarization component (|Ey|) to the total scattering (|E|) from nanoparticles in both forward and backward directions.

Equations (6)

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× × E r t + n 2 c 2 2 E r t 2 t = 4 π c 2 2 P ( 3 ) r t 2 t
P i ( 3 ) r ω as t = 3 jkl χ ijkl ( 3 ) E j P r ω p t E k P r ω p t E l S * r ω s t
{ × H = D t × E = B t
{ E x n + 1 ( i + 0.5 , j , k ) = CA ( m ) E x n ( i + 0.5 , j , k ) + CB ( m ) [ ( H z n + 0.5 ( i + 0.5 , j + 0.5 , k ) H z n + 0.5 ( i + 0.5 , j 0.5 , k ) ) Δy ( H y n + 0.5 ( i + 0.5 , j , k + 0.5 ) H y n + 0.5 ( i + 0.5 , j , k 0.5 ) ) Δz E y n + 1 ( i , j + 0.5 , k ) = CA ( m ) E x n ( i , j + 0.5 , k ) + CB ( m ) [ ( H x n + 0.5 ( i , j + 0.5 , k + 0.5 ) H x n + 0.5 ( i , j + 0.5 , k 0.5 ) ) Δz ( H y n + 0.5 ( i + 0.5 , j + 0.5 , k ) H z n + 0.5 ( i 5 , j 0.5 k ) ) Δ x E z n + 1 ( i , j , k + 0.5 ) = CA ( m ) E x n ( i , j , k + 0.5 ) + CB ( m ) [ ( H y n + 0.5 ( i + 0.5 , j , k + 0.5 ) H y n + 0.5 ( i 0 . 5 , j , k + 0.5 ) ) Δ x ( H x n + 0.5 ( i , j + 0.5 , k + 0.5 ) H x n + 0.5 ( i , j 5 , k 0.5 ) ) Δ y
χ sca ( 3 ) R ( ω as ) = χ 1111 ( 3 ) R ( ω as ) = A R ω R ( ω p ω s ) i Γ R
P i = 3 j , k , 1 [ ( χ 1122 ( 3 ) NR + χ 1122 ( 3 ) R ) ( δ ij + δ kl ) + ( χ 1221 ( 3 ) NR + χ 1221 ( 3 ) R ) δ il δ jk ] E j p E k p E 1 s *

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