Abstract

Within the accuracy of the first-order Born approximation, sufficient conditions are derived for the invariance of spectrum of an electromagnetic wave, which is generated by the scattering of an electromagnetic plane wave from an anisotropic random media. We show that the following restrictions on properties of incident fields and the anisotropic media must be simultaneously satisfied: 1) the elements of the dielectric susceptibility matrix of the media must obey the scaling law; 2) the spectral components of the incident field are proportional to each other; 3) the second moments of the elements of the dielectric susceptibility matrix of the media are inversely proportional to the frequency.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Far-zone spectral isotropy in weak scattering on spatially random media

Emil Wolf
J. Opt. Soc. Am. A 14(10) 2820-2823 (1997)

Spectrum of an electromagnetic light wave on scattering from an anisotropic semisoft boundary medium

Tao Wang, Zhenfei Jiang, Xiaoling Ji, and Daomu Zhao
J. Opt. Soc. Am. A 33(4) 625-629 (2016)

References

  • View by:
  • |
  • |
  • |

  1. D. F. V. James, “The Wolf effect and the redshift of quasars,” Pure Appl. Opt. 7(5), 959–970 (1998).
    [Crossref]
  2. M. Dashtdar and M. T. Tavassoly, “Redshift and blueshift in the spectra of lights coherently and diffusely scattered from random rough interfaces,” J. Opt. Soc. Am. A 26(10), 2134–2138 (2009).
    [Crossref] [PubMed]
  3. W. Gao, “Spectral changes of the light produced by scattering from tissue,” Opt. Lett. 35(6), 862–864 (2010).
    [Crossref] [PubMed]
  4. R. Zhu, S. Sridharan, K. Tangella, A. Balla, and G. Popescu, “Correlation-induced spectral changes in tissues,” Opt. Lett. 36(21), 4209–4211 (2011).
    [Crossref] [PubMed]
  5. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56(13), 1370–1372 (1986).
    [Crossref] [PubMed]
  6. H. Roychowdhury and E. Wolf, “Spectral invariance in fields generated by quasi-homogeneous scaling law sources,” Opt. Commun. 215(4-6), 199–203 (2003).
    [Crossref]
  7. H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241(1-3), 11–15 (2004).
    [Crossref]
  8. J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett. 31(14), 2097–2099 (2006).
    [Crossref] [PubMed]
  9. J. Pu, “Invariance of spectrum and polarization of electromagnetic Gaussian Schell-model beams propagating in free space,” Chin. Opt. Lett. 4, 196–198 (2006).
  10. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989).
    [Crossref]
  11. J. T. Foley and E. Wolf, “Frequency shifts of spectral lines generated by scattering from space-time fluctuations,” Phys. Rev. A 40(2), 588–598 (1989).
    [Crossref] [PubMed]
  12. D. F. V. James, M. P. Savedoff, and E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Phys. J. 359, 67–71 (1990).
  13. T. Shirai and T. Asakura, “Spectral changes of light induced by scattering from spatially random media under the Rytov approximation,” J. Opt. Soc. Am. A 12(6), 1354–1363 (1995).
    [Crossref]
  14. Y. Xin, Y. Chen, Q. Zhao, M. Zhou, and X. Yuan, “Changes of spectrum of light scattering on quasi-homogeneous random media,” Proc. SPIE 6786, 67864S (2007).
    [Crossref]
  15. X. Du and D. Zhao, “Spectral shifts produced by scattering from rotational quasi-homogeneous anisotropic media,” Opt. Lett. 36(24), 4749–4751 (2011).
    [Crossref] [PubMed]
  16. A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23(17), 1340–1342 (1998).
    [Crossref] [PubMed]
  17. X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun. 285(6), 934–936 (2012).
    [Crossref]
  18. M. Lahiri and E. Wolf, “Spectral changes of stochastic beams scattered on a deterministic medium,” Opt. Lett. 37(13), 2517–2519 (2012).
    [Crossref] [PubMed]
  19. E. Wolf, “Far-zone spectral isotropy in weak scattering on spatially random media,” J. Opt. Soc. Am. A 14(10), 2820–2823 (1997).
    [Crossref]
  20. T. Wang and D. Zhao, “Condition for far-zone spectral isotropy of an electromagnetic light wave on weak scattering,” Opt. Lett. 36(3), 328–330 (2011).
    [Crossref] [PubMed]
  21. J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
    [Crossref]
  22. L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley Press, 2000).
  23. T. Wang and D. Zhao, “Scattering theory of stochastic electromagnetic light waves,” Opt. Lett. 35(14), 2412–2414 (2010).
    [Crossref] [PubMed]
  24. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  25. L. Mandel, “Interference and the Alford and Gold effect,” J. Opt. Soc. Am. 52(12), 1335–1340 (1962).
    [Crossref]
  26. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
    [Crossref]
  27. T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. 34(24), 3866–3868 (2009).
    [Crossref] [PubMed]
  28. J. Chen, R. Lu, F. Chen, and J. Li, “Cross-spectrally pure light, cross-spectrally pure fields and statistical similarity in electromagnetic fields,” J. Mod. Opt. 61(14), 1164–1173 (2014).
    [Crossref]
  29. M. Lahiri and E. Wolf, “Effect of scattering on cross-spectral purity of light,” Opt. Commun. 330, 165–168 (2014).
    [Crossref]

2014 (2)

J. Chen, R. Lu, F. Chen, and J. Li, “Cross-spectrally pure light, cross-spectrally pure fields and statistical similarity in electromagnetic fields,” J. Mod. Opt. 61(14), 1164–1173 (2014).
[Crossref]

M. Lahiri and E. Wolf, “Effect of scattering on cross-spectral purity of light,” Opt. Commun. 330, 165–168 (2014).
[Crossref]

2012 (3)

X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun. 285(6), 934–936 (2012).
[Crossref]

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

M. Lahiri and E. Wolf, “Spectral changes of stochastic beams scattered on a deterministic medium,” Opt. Lett. 37(13), 2517–2519 (2012).
[Crossref] [PubMed]

2011 (3)

2010 (2)

2009 (2)

2007 (1)

Y. Xin, Y. Chen, Q. Zhao, M. Zhou, and X. Yuan, “Changes of spectrum of light scattering on quasi-homogeneous random media,” Proc. SPIE 6786, 67864S (2007).
[Crossref]

2006 (2)

2004 (1)

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241(1-3), 11–15 (2004).
[Crossref]

2003 (1)

H. Roychowdhury and E. Wolf, “Spectral invariance in fields generated by quasi-homogeneous scaling law sources,” Opt. Commun. 215(4-6), 199–203 (2003).
[Crossref]

1998 (2)

1997 (1)

1996 (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

1995 (1)

1990 (1)

D. F. V. James, M. P. Savedoff, and E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Phys. J. 359, 67–71 (1990).

1989 (2)

J. T. Foley and E. Wolf, “Frequency shifts of spectral lines generated by scattering from space-time fluctuations,” Phys. Rev. A 40(2), 588–598 (1989).
[Crossref] [PubMed]

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989).
[Crossref]

1986 (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56(13), 1370–1372 (1986).
[Crossref] [PubMed]

1962 (1)

Asakura, T.

Balla, A.

Chen, F.

J. Chen, R. Lu, F. Chen, and J. Li, “Cross-spectrally pure light, cross-spectrally pure fields and statistical similarity in electromagnetic fields,” J. Mod. Opt. 61(14), 1164–1173 (2014).
[Crossref]

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Chen, J.

J. Chen, R. Lu, F. Chen, and J. Li, “Cross-spectrally pure light, cross-spectrally pure fields and statistical similarity in electromagnetic fields,” J. Mod. Opt. 61(14), 1164–1173 (2014).
[Crossref]

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Chen, Y.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Y. Xin, Y. Chen, Q. Zhao, M. Zhou, and X. Yuan, “Changes of spectrum of light scattering on quasi-homogeneous random media,” Proc. SPIE 6786, 67864S (2007).
[Crossref]

Dashtdar, M.

Dogariu, A.

Du, X.

X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun. 285(6), 934–936 (2012).
[Crossref]

X. Du and D. Zhao, “Spectral shifts produced by scattering from rotational quasi-homogeneous anisotropic media,” Opt. Lett. 36(24), 4749–4751 (2011).
[Crossref] [PubMed]

Foley, J. T.

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989).
[Crossref]

J. T. Foley and E. Wolf, “Frequency shifts of spectral lines generated by scattering from space-time fluctuations,” Phys. Rev. A 40(2), 588–598 (1989).
[Crossref] [PubMed]

Friberg, A. T.

Gao, W.

Gori, F.

Hassinen, T.

James, D. F. V.

D. F. V. James, “The Wolf effect and the redshift of quasars,” Pure Appl. Opt. 7(5), 959–970 (1998).
[Crossref]

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

D. F. V. James, M. P. Savedoff, and E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Phys. J. 359, 67–71 (1990).

Korotkova, O.

Lahiri, M.

M. Lahiri and E. Wolf, “Effect of scattering on cross-spectral purity of light,” Opt. Commun. 330, 165–168 (2014).
[Crossref]

M. Lahiri and E. Wolf, “Spectral changes of stochastic beams scattered on a deterministic medium,” Opt. Lett. 37(13), 2517–2519 (2012).
[Crossref] [PubMed]

Li, J.

J. Chen, R. Lu, F. Chen, and J. Li, “Cross-spectrally pure light, cross-spectrally pure fields and statistical similarity in electromagnetic fields,” J. Mod. Opt. 61(14), 1164–1173 (2014).
[Crossref]

Lu, R.

J. Chen, R. Lu, F. Chen, and J. Li, “Cross-spectrally pure light, cross-spectrally pure fields and statistical similarity in electromagnetic fields,” J. Mod. Opt. 61(14), 1164–1173 (2014).
[Crossref]

Mandel, L.

Popescu, G.

Pu, J.

Roychowdhury, H.

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241(1-3), 11–15 (2004).
[Crossref]

H. Roychowdhury and E. Wolf, “Spectral invariance in fields generated by quasi-homogeneous scaling law sources,” Opt. Commun. 215(4-6), 199–203 (2003).
[Crossref]

Savedoff, M. P.

D. F. V. James, M. P. Savedoff, and E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Phys. J. 359, 67–71 (1990).

Shirai, T.

Sridharan, S.

Tangella, K.

Tavassoly, M. T.

Tervo, J.

Wang, T.

Wang, Y.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Wolf, E.

M. Lahiri and E. Wolf, “Effect of scattering on cross-spectral purity of light,” Opt. Commun. 330, 165–168 (2014).
[Crossref]

M. Lahiri and E. Wolf, “Spectral changes of stochastic beams scattered on a deterministic medium,” Opt. Lett. 37(13), 2517–2519 (2012).
[Crossref] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett. 31(14), 2097–2099 (2006).
[Crossref] [PubMed]

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241(1-3), 11–15 (2004).
[Crossref]

H. Roychowdhury and E. Wolf, “Spectral invariance in fields generated by quasi-homogeneous scaling law sources,” Opt. Commun. 215(4-6), 199–203 (2003).
[Crossref]

A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23(17), 1340–1342 (1998).
[Crossref] [PubMed]

E. Wolf, “Far-zone spectral isotropy in weak scattering on spatially random media,” J. Opt. Soc. Am. A 14(10), 2820–2823 (1997).
[Crossref]

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

D. F. V. James, M. P. Savedoff, and E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Phys. J. 359, 67–71 (1990).

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989).
[Crossref]

J. T. Foley and E. Wolf, “Frequency shifts of spectral lines generated by scattering from space-time fluctuations,” Phys. Rev. A 40(2), 588–598 (1989).
[Crossref] [PubMed]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56(13), 1370–1372 (1986).
[Crossref] [PubMed]

Xin, Y.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Y. Xin, Y. Chen, Q. Zhao, M. Zhou, and X. Yuan, “Changes of spectrum of light scattering on quasi-homogeneous random media,” Proc. SPIE 6786, 67864S (2007).
[Crossref]

Yuan, X.

Y. Xin, Y. Chen, Q. Zhao, M. Zhou, and X. Yuan, “Changes of spectrum of light scattering on quasi-homogeneous random media,” Proc. SPIE 6786, 67864S (2007).
[Crossref]

Zhao, D.

Zhao, Q.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Y. Xin, Y. Chen, Q. Zhao, M. Zhou, and X. Yuan, “Changes of spectrum of light scattering on quasi-homogeneous random media,” Proc. SPIE 6786, 67864S (2007).
[Crossref]

Zhou, M.

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Y. Xin, Y. Chen, Q. Zhao, M. Zhou, and X. Yuan, “Changes of spectrum of light scattering on quasi-homogeneous random media,” Proc. SPIE 6786, 67864S (2007).
[Crossref]

Zhu, R.

Chin. Opt. Lett. (1)

J. Mod. Opt. (1)

J. Chen, R. Lu, F. Chen, and J. Li, “Cross-spectrally pure light, cross-spectrally pure fields and statistical similarity in electromagnetic fields,” J. Mod. Opt. 61(14), 1164–1173 (2014).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (5)

M. Lahiri and E. Wolf, “Effect of scattering on cross-spectral purity of light,” Opt. Commun. 330, 165–168 (2014).
[Crossref]

H. Roychowdhury and E. Wolf, “Spectral invariance in fields generated by quasi-homogeneous scaling law sources,” Opt. Commun. 215(4-6), 199–203 (2003).
[Crossref]

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241(1-3), 11–15 (2004).
[Crossref]

X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun. 285(6), 934–936 (2012).
[Crossref]

J. Chen, F. Chen, Y. Chen, Y. Xin, Y. Wang, Q. Zhao, and M. Zhou, “Coherence properties of the scattered field generated by anisotropic quasi-homogeneous media,” Opt. Commun. 285(19), 3955–3960 (2012).
[Crossref]

Opt. Lett. (9)

Phys. J. (1)

D. F. V. James, M. P. Savedoff, and E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Phys. J. 359, 67–71 (1990).

Phys. Rev. A (1)

J. T. Foley and E. Wolf, “Frequency shifts of spectral lines generated by scattering from space-time fluctuations,” Phys. Rev. A 40(2), 588–598 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56(13), 1370–1372 (1986).
[Crossref] [PubMed]

Proc. SPIE (1)

Y. Xin, Y. Chen, Q. Zhao, M. Zhou, and X. Yuan, “Changes of spectrum of light scattering on quasi-homogeneous random media,” Proc. SPIE 6786, 67864S (2007).
[Crossref]

Pure Appl. Opt. (1)

D. F. V. James, “The Wolf effect and the redshift of quasars,” Pure Appl. Opt. 7(5), 959–970 (1998).
[Crossref]

Rep. Prog. Phys. (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

Other (2)

L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley Press, 2000).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Schematic diagram for illustrating the weak scattering theory of a plane wave from an anisotropic random media.

Fig. 2
Fig. 2

Normalized spectrum of scattered field produced by considering two types of anisotropic media, of which the second moments of the elements of the dielectric susceptibility suffice Eq. (39) and (40), respectively.

Equations (45)

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

η( r',ω )=( η x ( r',ω ) 0 0 0 η y ( r',ω ) 0 0 0 η z ( r',ω ) ),
F j ( r',ω )= ( ω/c ) 2 η j ( r',ω ),(j=x,y,z),
U j ( i ) ( r',ω )= a j ( ω )exp( ik s 0 r' ),(j=x,y),
U x ( s ) ( rs,ω )= D F x ( r',ω ) U x ( i ) ( r',ω )G( rs,r',ω ) d 3 r',
U y ( s ) ( rs,ω )= cos 2 θ D F y ( r',ω ) U y ( i ) ( r',ω )G( rs,r',ω ) d 3 r',
U z ( s ) ( rs,ω )=sinθcosθ D F z ( r',ω ) U y ( i ) ( r',ω ) G( rs,r',ω ) d 3 r',
G( rs,r',ω ) exp( ikr ) r exp( iksr' ).
S ( s ) ( rs,ω )=Tr{ W ij ( s ) ( rs,rs,ω ) },(i.j=x,y,z),
W ij ( s ) ( r 1 s 1 , r 2 s 2 ,ω )=[ U i ( s ) ( r 1 s 1 ,ω ) U j ( s ) ( r 2 s 2 ,ω ) ],
S ( s ) ( rs,ω )= 1 r 2 ( ω c ) 4 { C ˜ xx ( η ) ( K,K,ω ) S x ( i ) ( ω )+ cos 4 θ C ˜ yy ( η ) ( K,K,ω ) S y ( i ) ( ω ) + sin 2 θ cos 2 θ C ˜ zz ( η ) ( K,K,ω ) S y ( i ) ( ω ) },
C ˜ ij ( η ) ( K,K,ω )= D D C ij ( η ) ( r 1 ', r 2 ',ω )exp[ iK( r 2 ' r 1 ' ) ] d 3 r 1 ' d 3 r 2 ',(i,j=x,y,z),
C ij ( η ) ( r 1 ', r 2 ',ω )=[ η i * ( r 1 ',ω ) η j ( r 2 ',ω ) ]=[ C ij ( η ) ( r 2 ' r 1 ',ω ) ],( r 1 ', r 2 ' )V,
C ˜ ij ( η ) ( K,K,ω )V C ˜ ij ( η ) ( K,ω ),
S ( s ) ( rs,ω )= V r 2 ( ω c ) 4 { C ˜ xx ( η ) ( K,ω ) S x ( i ) ( ω )+ cos 4 θ C ˜ yy ( η ) ( K,ω ) S y ( i ) ( ω )+ sin 2 θ cos 2 θ C ˜ zz ( η ) ( K,ω ) S y ( i ) ( ω ) }.
S N ( s ) ( rs,ω )= S ( s ) ( rs,ω ) 0 S ( s ) ( rs,ω' )dω' .
S N ( s ) ( rs,ω )= ω 4 S x ( i ) ( ω ) C ˜ xx ( η ) ( K,ω )+ cos 4 θ ω 4 S y ( i ) ( ω ) C ˜ yy ( η ) ( K,ω )+ sin 2 θ cos 2 θ ω 4 S y ( i ) ( ω ) C ˜ zz ( η ) ( K,ω ) 0 ω ' 4 S x ( i ) ( ω' ) C ˜ xx ( η ) ( K,ω' )dω'+ cos 4 θ 0 ω ' 4 S y ( i ) ( ω' ) C ˜ yy ( η ) ( K,ω' )dω' + sin 2 θ cos 2 θ 0 ω ' 4 S y ( i ) ( ω' ) C ˜ zz ( η ) ( K,ω' )dω' .
S N ( i ) ( ω )= S ( i ) ( ω ) 0 S ( i ) ( ω' )dω' = S x ( i ) ( ω )+ S y ( i ) ( ω ) 0 S x ( i ) ( ω' )dω' + 0 S y ( i ) ( ω' )dω' .
C ˜ jj ( η ) ( K,ω )= F jj ( ω ) H jj ( K/k ),(j=x,y,z),
C ˜ jj ( η ) ( 0,ω )= F jj ( ω ) H jj ( 0 ).
C ˜ jj ( η ) ( K,ω )= C ˜ jj ( η ) ( 0,ω ) H jj 1 ( 0 ) H jj ( K/k ).
C jj ( η ) ( r',ω )= 1 ( 2π ) 3 C ˜ jj ( η ) ( 0,ω ) H jj 1 ( 0 ) H jj ( K/k )exp( iKr' ) d 3 K.
C jj ( η ) ( r',ω )= k 3 C ˜ jj ( η ) ( 0,ω ) H jj 1 ( 0 ) H ˜ jj ( kr' ),
H ˜ jj ( K )= H jj ( r' )exp(iKr') d 3 r' ,
μ jj ( η ) ( r',ω )= C jj ( η ) ( r',ω ) C jj ( η ) ( 0,ω ) .
μ jj ( η ) ( r',ω )= h jj ( kr' ),(j=x,y,z),
h jj ( kr' )= H jj ( kr' ) H jj 1 ( 0 ).
S N ( s ) ( rs,ω )= [ H xx ( s s 0 ) H xx 1 ( 0 ) ω 4 S x ( i ) ( ω ) C ˜ xx ( η ) ( 0,ω )+ cos 4 θ H yy ( s s 0 ) H yy 1 ( 0 ) ω 4 S y ( i ) ( ω ) C ˜ yy ( η ) ( 0,ω ) + sin 2 θ cos 2 θ H zz ( s s 0 ) H zz 1 ( 0 ) ω 4 S y ( i ) ( ω ) C ˜ zz ( η ) ( 0,ω ) ]/ [ H xx ( s s 0 ) H xx 1 ( 0 ) 0 ω ' 4 S x ( i ) ( ω' ) C ˜ xx ( η ) ( 0,ω )dω' + cos 4 θ H yy ( s s 0 ) H yy 1 ( 0 ) 0 ω ' 4 S y ( i ) ( ω' ) C ˜ yy ( η ) ( 0,ω' )dω' + sin 2 θ cos 2 θ H zz ( s s 0 ) H zz 1 ( 0 ) × 0 ω ' 4 S y ( i ) ( ω' ) C ˜ zz ( η ) ( 0,ω' )dω' ].
ω 4 S x ( i ) ( ω ) C ˜ xx ( η ) ( 0,ω )= S ( i ) ( ω ),
ω 4 S y ( i ) ( ω ) C ˜ yy ( η ) ( 0,ω )= S ( i ) ( ω ),
ω 4 S y ( i ) ( ω ) C ˜ zz ( η ) ( 0,ω )= S ( i ) ( ω ).
S y ( i ) ( ω )=γ S x ( i ) ( ω ),
C ˜ xx ( η ) ( 0,ω )= γ+1 ω 4 , C ˜ yy ( η ) ( 0,ω )= C ˜ zz ( η ) ( 0,ω )= γ+1 γ ω 4 .
C ˜ jj ( η ) ( 0,ω )= C jj ( η ) ( r',ω ) d 3 r' ,(i=x,y,z).
C ˜ jj ( η ) ( 0,ω )= C jj ( η ) ( 0,ω ) μ jj ( η ) ( r',ω ) d 3 r'.
C ˜ jj ( η ) ( 0,ω )= C jj ( η ) ( 0,ω ) h ˜ jj ( 0 ) k 3 ,
h ˜ jj ( 0 )= 1 k 3 h jj ( kr' ) d 3 r'
C xx ( η ) ( 0,ω )= γ+1 ω c 3 h ˜ xx ( 0 ) , C yy ( η ) ( 0,ω )= γ+1 ω c 3 γ h ˜ yy ( 0 ) , C zz ( η ) ( 0,ω )= γ+1 ω c 3 γ h ˜ zz ( 0 ) .
η j * ( r',ω ) η j ( r',ω ) 1 ω ,(j=x,y,z).
C xx1 ( η ) ( 0,ω )= γ+1 ω c 3 h ˜ xx ( 0 ) , C yy1 ( η ) ( 0,ω )= γ+1 ω c 3 γ h ˜ yy ( 0 ) , C zz1 ( η ) ( 0,ω )= γ+1 ω c 3 γ h ˜ zz ( 0 ) .
C xx2 ( η ) ( 0,ω )= γ+1 ω c 3 h ˜ xx ( 0 ) , C yy2 ( η ) ( 0,ω )= γ+1 c 3 γ h ˜ yy ( 0 ) ω, C zz2 ( η ) ( 0,ω )= γ+1 c 3 γ h ˜ zz ( 0 ) ω,
S N1 ( s ) ( rs,ω )= S ( i ) ( ω ) 0 S ( i ) ( ω )dω .
S N2 ( s ) ( rs,ω )= [ H xx ( s s 0 )/ H xx ( 0 ) S ( i ) ( ω ) +γ cos 4 θ H yy ( s s 0 )/ H yy ( 0 ) ω 2 S ( i ) ( ω ) +γ sin 2 θ cos 2 θ H zz ( s s 0 )/ H zz ( 0 ) ω 2 S ( i ) ( ω ) ]/ [ H xx ( s s 0 )/ H xx ( 0 ) 0 S ( i ) ( ω )dω +γ cos 4 θ H yy ( s s 0 )/ H yy ( 0 ) 0 ω 2 S ( i ) ( ω )dω +γ sin 2 θ cos 2 θ H zz ( s s 0 )/ H zz ( 0 ) 0 ω 2 S ( i ) ( ω )dω ].
η x ( r',ω )= η y ( r',ω )= η z ( r',ω )=η( r',ω ),
| C ˜ xx ( η ) ( K 1 , K 2 ,ω ) |=| C ˜ yy ( η ) ( K 1 , K 2 ,ω ) |=| C ˜ zz ( η ) ( K 1 , K 2 ,ω ) |=| C ˜ ( η ) ( K 1 , K 2 ,ω ) |,
| C ˜ ( η ) ( K 1 , K 2 ,ω ) | C( ω ) ω 4 ,

Metrics