Abstract

We introduce a formalism based on complex unitary vectors for the direction of propagation and for the polarization in order to describe in detail the propagation of inhomogeneous plane waves in absorbing isotropic media. We obtain analytic expressions for the displacement vector, the electric field, the magnetic field, and the Poynting vector, and we study their geometry in terms of the geometrical interpretation of the complex directions of propagation inside the material. We introduce a complex coordinate system based on complex unitary vectors, where the description of the polarization states of the field vectors and the Poynting vector becomes simpler. The physical meaning and the interpretation of the mathematical operations involving these complex unitary vectors is provided.

© 2004 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  2. M. Berek, “Optische Messmethoden im polarisierten Auflicht in sonderheit zur Bestimmung der erzmineralien, mit einer theorie der Optick absorbierender Kristalle,” Fortschr. Mineral. 22, 1–104 (1937).
  3. B. D. Cervelle, R. Caye, J. Billard, “Determination de l’ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale,” Bull. Soc. Fr. Mineral. Cristallogr. 93, 72–82 (1970).
  4. G. N. Borzdov, “Waves with quadratic amplitude dependence on coordinates in uniaxial crystals,” J. Mod. Opt. 37, 281–284 (1990).
    [Crossref]
  5. C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
    [Crossref]
  6. J. M. Cabrera, F. Agulló, F. J. López, Optica Electromagnética. Vol. I: Materials y Aplicaciones, 1st ed. (Addison-Wesley Iberoamericana Española, Madrid, 1998).
  7. R. M. A. Azzam, E. Ericsson, “Angular range for reflection of p-polarized light at the surface of an absorbing medium with reflectance below that at normal incidence,” J. Opt. Soc. Am. A 19, 112–115 (2002).
    [Crossref]
  8. R. Echarri, M. T. Garea, “Behavior of the Poynting vector in uniaxial absorbant media,” Pure Appl. Opt. 3, 931–941 (1994).
    [Crossref]
  9. M. V. Berry, M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London Ser. A 459, 1261–1292 (2003).
    [Crossref]
  10. T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [Crossref]
  11. T. Saastamoinen, J. Tervo, “Geometrical interpretation of the degree of polarization for arbitrary electromagnetic fields,” presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, June 30–July 3, 2003.
  12. Y. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 1–62.
  13. R. A. Egorchenkov, Y. A. Kravtsov, “Complex ray-tracingalgorithms with application to optical problems,” J. Opt. Soc. Am. A 18, 650–656 (2001).
    [Crossref]
  14. M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, E. Poli, “Exact, beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams,” in 28th EPS Conference on Controlled Fusion and Plasma Physics, ECA Vol. 25A (ECA, Arlington, Va., 2001), pp. 293–300.
  15. P. Halevi, A. Mendoza, “Temporal and spatial behavior of the Poynting vector in dissipative media: refraction from vacuum into a medium,” J. Opt. Soc. Am. A 71, 1238–1242 (1981).
    [Crossref]

2003 (1)

M. V. Berry, M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London Ser. A 459, 1261–1292 (2003).
[Crossref]

2002 (3)

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
[Crossref]

R. M. A. Azzam, E. Ericsson, “Angular range for reflection of p-polarized light at the surface of an absorbing medium with reflectance below that at normal incidence,” J. Opt. Soc. Am. A 19, 112–115 (2002).
[Crossref]

2001 (1)

1994 (1)

R. Echarri, M. T. Garea, “Behavior of the Poynting vector in uniaxial absorbant media,” Pure Appl. Opt. 3, 931–941 (1994).
[Crossref]

1990 (1)

G. N. Borzdov, “Waves with quadratic amplitude dependence on coordinates in uniaxial crystals,” J. Mod. Opt. 37, 281–284 (1990).
[Crossref]

1981 (1)

P. Halevi, A. Mendoza, “Temporal and spatial behavior of the Poynting vector in dissipative media: refraction from vacuum into a medium,” J. Opt. Soc. Am. A 71, 1238–1242 (1981).
[Crossref]

1970 (1)

B. D. Cervelle, R. Caye, J. Billard, “Determination de l’ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale,” Bull. Soc. Fr. Mineral. Cristallogr. 93, 72–82 (1970).

1937 (1)

M. Berek, “Optische Messmethoden im polarisierten Auflicht in sonderheit zur Bestimmung der erzmineralien, mit einer theorie der Optick absorbierender Kristalle,” Fortschr. Mineral. 22, 1–104 (1937).

Agulló, F.

J. M. Cabrera, F. Agulló, F. J. López, Optica Electromagnética. Vol. I: Materials y Aplicaciones, 1st ed. (Addison-Wesley Iberoamericana Española, Madrid, 1998).

Alberdi, C.

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
[Crossref]

Alfonso, S.

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
[Crossref]

Asatryan, A. A.

Y. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 1–62.

Azzam, R. M. A.

Berek, M.

M. Berek, “Optische Messmethoden im polarisierten Auflicht in sonderheit zur Bestimmung der erzmineralien, mit einer theorie der Optick absorbierender Kristalle,” Fortschr. Mineral. 22, 1–104 (1937).

Berrogui, M.

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
[Crossref]

Berry, M. V.

M. V. Berry, M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London Ser. A 459, 1261–1292 (2003).
[Crossref]

Billard, J.

B. D. Cervelle, R. Caye, J. Billard, “Determination de l’ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale,” Bull. Soc. Fr. Mineral. Cristallogr. 93, 72–82 (1970).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Bornatici, M.

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, E. Poli, “Exact, beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams,” in 28th EPS Conference on Controlled Fusion and Plasma Physics, ECA Vol. 25A (ECA, Arlington, Va., 2001), pp. 293–300.

Borzdov, G. N.

G. N. Borzdov, “Waves with quadratic amplitude dependence on coordinates in uniaxial crystals,” J. Mod. Opt. 37, 281–284 (1990).
[Crossref]

Cabrera, J. M.

J. M. Cabrera, F. Agulló, F. J. López, Optica Electromagnética. Vol. I: Materials y Aplicaciones, 1st ed. (Addison-Wesley Iberoamericana Española, Madrid, 1998).

Caye, R.

B. D. Cervelle, R. Caye, J. Billard, “Determination de l’ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale,” Bull. Soc. Fr. Mineral. Cristallogr. 93, 72–82 (1970).

Cervelle, B. D.

B. D. Cervelle, R. Caye, J. Billard, “Determination de l’ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale,” Bull. Soc. Fr. Mineral. Cristallogr. 93, 72–82 (1970).

Dennis, M. R.

M. V. Berry, M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London Ser. A 459, 1261–1292 (2003).
[Crossref]

Diñeiro, J. M.

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
[Crossref]

Echarri, R.

R. Echarri, M. T. Garea, “Behavior of the Poynting vector in uniaxial absorbant media,” Pure Appl. Opt. 3, 931–941 (1994).
[Crossref]

Egorchenkov, R. A.

R. A. Egorchenkov, Y. A. Kravtsov, “Complex ray-tracingalgorithms with application to optical problems,” J. Opt. Soc. Am. A 18, 650–656 (2001).
[Crossref]

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, E. Poli, “Exact, beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams,” in 28th EPS Conference on Controlled Fusion and Plasma Physics, ECA Vol. 25A (ECA, Arlington, Va., 2001), pp. 293–300.

Ericsson, E.

Forbes, G. W.

Y. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 1–62.

Friberg, A. T.

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Garea, M. T.

R. Echarri, M. T. Garea, “Behavior of the Poynting vector in uniaxial absorbant media,” Pure Appl. Opt. 3, 931–941 (1994).
[Crossref]

Halevi, P.

P. Halevi, A. Mendoza, “Temporal and spatial behavior of the Poynting vector in dissipative media: refraction from vacuum into a medium,” J. Opt. Soc. Am. A 71, 1238–1242 (1981).
[Crossref]

Hernandez, B.

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
[Crossref]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Kravtsov, Y. A.

R. A. Egorchenkov, Y. A. Kravtsov, “Complex ray-tracingalgorithms with application to optical problems,” J. Opt. Soc. Am. A 18, 650–656 (2001).
[Crossref]

Y. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 1–62.

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, E. Poli, “Exact, beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams,” in 28th EPS Conference on Controlled Fusion and Plasma Physics, ECA Vol. 25A (ECA, Arlington, Va., 2001), pp. 293–300.

López, F. J.

J. M. Cabrera, F. Agulló, F. J. López, Optica Electromagnética. Vol. I: Materials y Aplicaciones, 1st ed. (Addison-Wesley Iberoamericana Española, Madrid, 1998).

Maj, O.

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, E. Poli, “Exact, beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams,” in 28th EPS Conference on Controlled Fusion and Plasma Physics, ECA Vol. 25A (ECA, Arlington, Va., 2001), pp. 293–300.

Mendoza, A.

P. Halevi, A. Mendoza, “Temporal and spatial behavior of the Poynting vector in dissipative media: refraction from vacuum into a medium,” J. Opt. Soc. Am. A 71, 1238–1242 (1981).
[Crossref]

Poli, E.

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, E. Poli, “Exact, beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams,” in 28th EPS Conference on Controlled Fusion and Plasma Physics, ECA Vol. 25A (ECA, Arlington, Va., 2001), pp. 293–300.

Saastamoinen, T.

T. Saastamoinen, J. Tervo, “Geometrical interpretation of the degree of polarization for arbitrary electromagnetic fields,” presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, June 30–July 3, 2003.

Sáenz, C.

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
[Crossref]

Setälä, T.

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Tervo, J.

T. Saastamoinen, J. Tervo, “Geometrical interpretation of the degree of polarization for arbitrary electromagnetic fields,” presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, June 30–July 3, 2003.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Bull. Soc. Fr. Mineral. Cristallogr. (1)

B. D. Cervelle, R. Caye, J. Billard, “Determination de l’ellipsoïde complexe des indices de cristaux uniaxes fortement absorbants. Application à la pyrrhotite hexagonale,” Bull. Soc. Fr. Mineral. Cristallogr. 93, 72–82 (1970).

Fortschr. Mineral. (1)

M. Berek, “Optische Messmethoden im polarisierten Auflicht in sonderheit zur Bestimmung der erzmineralien, mit einer theorie der Optick absorbierender Kristalle,” Fortschr. Mineral. 22, 1–104 (1937).

J. Mod. Opt. (2)

G. N. Borzdov, “Waves with quadratic amplitude dependence on coordinates in uniaxial crystals,” J. Mod. Opt. 37, 281–284 (1990).
[Crossref]

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
[Crossref]

J. Opt. Soc. Am. A (3)

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Proc. R. Soc. London Ser. A (1)

M. V. Berry, M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London Ser. A 459, 1261–1292 (2003).
[Crossref]

Pure Appl. Opt. (1)

R. Echarri, M. T. Garea, “Behavior of the Poynting vector in uniaxial absorbant media,” Pure Appl. Opt. 3, 931–941 (1994).
[Crossref]

Other (5)

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

J. M. Cabrera, F. Agulló, F. J. López, Optica Electromagnética. Vol. I: Materials y Aplicaciones, 1st ed. (Addison-Wesley Iberoamericana Española, Madrid, 1998).

T. Saastamoinen, J. Tervo, “Geometrical interpretation of the degree of polarization for arbitrary electromagnetic fields,” presented at the ICO Topical Meeting on Polarization Optics, Polvijärvi, Finland, June 30–July 3, 2003.

Y. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 1–62.

M. Bornatici, R. A. Egorchenkov, Y. A. Kravtsov, O. Maj, E. Poli, “Exact, beam tracing and complex geometrical optics solutions for the propagation of Gaussian electromagnetic beams,” in 28th EPS Conference on Controlled Fusion and Plasma Physics, ECA Vol. 25A (ECA, Arlington, Va., 2001), pp. 293–300.

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Figures (5)

Fig. 1
Fig. 1

Possible orientations of the vectors uRDxy and uIDxy in the XY plane. Options (1) and (2) in uIDxy correspond to the minus and plus signs in Eqs. (23)–(24).

Fig. 2
Fig. 2

Polarization ellipse defined by u˜D, the polarization ellipse corresponding to u˜Dxy and the real vectors sR and sI.

Fig. 3
Fig. 3

Unitary vectors u˜D, u˜H, and u˜HD (projection of u˜H in the plane defined by ).

Fig. 4
Fig. 4

Ellipses of vibration corresponding to the unitary vectors u˜D, u˜Dxy, and u˜H, real vectors sR and sI, and the directions of propagation of the wave front spr, attenuation sat, and propagation of the energy 〈S〉.

Fig. 5
Fig. 5

Real and imaginary components of , , and 〈S〉 in the OXYZ system for a given direction.

Equations (97)

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D˜=˜E˜,
n˜=(˜/0)1/2=n-ik=n(1-iχ)=n(1+χ2)1/2exp(-iδ),
F˜=F˜0expiωt-n˜crs˜,
H˜×s˜=cn˜D˜,
s˜×E˜=cμon˜H˜,
D˜s˜=0,
H˜s˜=0,
F˜=F0u˜Fexp(iθF)expiωt-n˜crs˜,
sR2-sI2=1,sRsI.
uRF2-uIF2=1,uRF  uIF.
sR=sRux,sI=sIuy,
F˜=F0u˜Fexp(iθF)exp-ωc nr(χsRux-sIuy)expiωt-ncr(sRux+χsIuy).
spr=(sR2+χ2sI2)-1/2(sR+χsI),
sat=(χ2sR2+sI2)-1/2(χsR-sI).
exp-ωc nr(χsR-sI),expiωt-ncr(sR+χsI)
At=exp-ωc nr(χsR-sI)(attenuationfactor),
Pr=expiωt-ncr(sR+χsI)
(propagationfactor).
F˜=F0u˜Fexp(iθF)AtPr.
D˜=D0u˜DAtPr.
D˜xy=D0xyu˜Dxyexp(iθDxy)AtPr,
D˜z=D0zuzexp(iθDz)AtPr.
(uRDxy+iuIDxy)(sR+isI)=0,
sRuRDxyux-sIuIDxyuy=0,
sIuRDxyuy+sRuIDxyux=0.
sRuRDxycos φsIuIDxycos φ=0,
sIuRDxysin φsRuIDxysin φ=0,
sI2uRDxy2=sR2uIDxy2(sR2-1)uRDxy2=sR2(uRDxy2-1),
uRDxy=sR,
uIDxy=sI.
φ=π/2uRDxy=sRuy,uIDxy=-sIux;
φ=-π/2uRDxy=-sRuy,uIDxy=sIux.
D˜xy=D0[u˜D(uz×s˜)](uz×s˜)AtPr.
D0xy2=D02[(sRuRDy+sIuIDx)2+(sRuIDy-sIuRDx)2],
D0xyexp(iθDxy)=D0u˜D(u˜z×s˜)
 tan θDxy
=sRuIDy-sIuRDxsRuRDy+sIuIDx=Im(u˜Du˜Dxy)Re(u˜Du˜Dxy).
D˜z=D0(u˜Duz)u˜zAtPr.
D0z2=D02[(uIDuz)2+(uRDuz)2],
D0zexp(iθDz)=D0[u˜Duz]
tan θDz=uIDzuRDz=Im(u˜Du˜z)Re(u˜Du˜z).
E˜=E0u˜Eexp(iθE)AtPr
=D˜˜=exp(i2δ)0n2(1+χ2) D0u˜DAtPr.
E0=D0on2(1+χ2),θE=2δ,
u˜E=uRE+iuIE=uRD+iuID=u˜D,
E˜xy=E0xy(uRExy+iuIExy)exp(iθExy)AtPr,
E˜z=E0zexp(iθEz)uzAtPr,
E0xy=D0xyon2(1+χ2),θExy=2δ+θDxy,
E0z=D0zon2(1+χ2),θEz=2δ+θDz,
uRExy=uRDxy=sRuy,uIExy=uIDxy=-sIux.
H˜=H0u˜Hexp(iθH)AtPr=cD0n(1+χ2)1/2 (s˜×u˜D)exp(iδ)AtPr,
H0=cD0n(1+χ2)1/2,θH=δ,
u˜H=uRH+iuIH=s˜×u˜D,
uRH=sR×uRD-sI×uID,
uIH=sR×uID+sI×uRD.
H˜xy=H0xy(uRHxy+iuIHxy)exp(iθHxy)AtPr,
H˜z=H0zexp(iθHz)uzAtPr,
H0z=cD0xyn(1+χ2)1/2,θHz=δ+θDxy.
H0xy=cD0zn(1+χ2)1/2,θHxy=δ+θDz,
uRHxy=sR×uz=-sRuy,
uIHxy=sI×uz=sIux.
S=Re(E˜)×Re(H˜)=12Re(E˜×H˜)+12Re(E˜*×H˜).
12Re(E˜×H˜)=M Re[u˜D×(s˜×u˜D)]expi2ωt-ncr(sR+χsI)+3iδ=M Res˜expi2ωt-ncr(sR+χsI)+3iδ=MsRcos2ωt-ncr(sR+χsI)+3δ-sIsin2ωt-ncr(sR+χsI)+3δ,
M=cD02At22on3(1+χ2)3/2,
u˜Ds˜=0,
u˜D*×(s˜×u˜D)=s˜(uRD2+uID2)-u˜D(u˜D*s˜)=s˜(uRD2+uID2)+s˜×(u˜D*×u˜D),
12Re(E˜*×H˜)=cD02At22on3(1+χ2)3/2Re{[u˜D*×(s˜×u˜D)]exp(-iδ)}=cD02At22on3(1+χ2)2 [(uRD2+uID2)×(sR+χsI)+2(χsR-sI)×(uRD×uID)],
S=cD02At22on3(1+χ2)2 {(uRD2+uID2)(sR+χsI)-2uRD[(sR+χsI)uRD]-2uID[(sR+χsI)uID]}.
S=cD02At22on3(1+χ2)2 [(sR+χsI)].
S=cD02At22on3(1+χ2)2 (uRD2+uID2)sR.
u˜x×u˜y=(sRux+isIuy)×(-isIux+sRuy)=uz,
u˜y×uz=(-isIux+sRuy)×uz=u˜x,
uz×u˜x=uz×(sRux+isIuy)=u˜y.
u˜Du˜x=cos α˜D=u˜Ds˜=0α˜D=±π/2,
u˜Du˜y=cos β˜D=u˜D(sRuy-isIux),
u˜Duz=cos γ˜D=uRDz+iuIDz.
cos2 β˜D+cos2 γ˜D=1,
cos β˜D=u˜Du˜y=[1-(u˜Duz)2]1/2=sin γ˜D.
u˜D=sin γ˜Du˜y+cos γ˜Duz=1D0 (D0xycos θDxyu˜y+D0zcos θDzuz)+i 1D0 (D0xysin θDxyu˜y+D0zsin θDzuz)=uRD+iuID,
uRDx=0,uIDx=0,
uRDy=1D0 D0xycos θDxy=Re(sin γ˜D),
uIDy=1D0 D0xysin θDxy=Im(sin γ˜D),
uRDz=1D0 D0zcos θDz=uRDz=Re(cos γ˜D),
uIDz=1D0 D0zsin θDz=uIDz=Im(cos γ˜D).
u˜Hu˜x=cos α˜H=u˜Hs˜=0α˜H=±π/2,
u˜Hu˜y=cos β˜H=(u˜x×u˜D)u˜y=(u˜y×u˜x)u˜D=-u˜Duz=-cos γ˜D,
u˜Huz=cos γ˜H=(u˜x×u˜D)uz=(uz×u˜x)u˜D=u˜Du˜y=sin γ˜D.
u˜H=-cos γ˜Du˜y+sin γ˜Duz=1D0 (-D0zcos θDzu˜y+D0xycos θDxyuz)+i 1D0 (-D0zsin θDzu˜y+D0xysin θDxyuz)=uRH+iuIH,
uRHx=0,uIHx=0,
uRHy=-1D0 D0zcos γDz=-uRDz=-Re(cos γ˜D),
uIHy=-1D0 D0zsin γDz=-uIDz=-Im(cos γ˜D),
uRHz=1D0 D0xycos γDxy=Re(sin γ˜D),
uIHz=1D0 D0xysin γDxy=Im(sin γ˜D).
Re{[u˜D*×(s˜×u˜D)]exp(-iδ)}=sRcos(γ˜D-γ˜D*)cos δux-sIcos(γ˜D+γ˜D*)sin δuy+sRsI[sin(γ˜D+γ˜D*)sin δ-i sin(γ˜D-γ˜D*)cos δ]uz.
u˜RS=M{sR2cos(γ˜D-γ˜D*)cos δu˜x-sRsIcos(γ˜D+γ˜D*)sin δu˜y+sRsI[sin(γ˜D+γ˜D*)sin δ-i sin(γ˜D-γ˜D*)cos δ]uz}
u˜IS=M[sI2cos(γ˜D+γ˜D*)sin δu˜x-sRsIcos(γ˜D-γ˜D*)cos δu˜y],
M=12 {sR2cos2(γ˜D-γ˜D*)cos2 δ+sI2cos2(γ˜D+γ˜D*)sin2 δ+[sRsIsin(γ˜D+γ˜D*)sin δ-sRsIi sin(γ˜D-γ˜D*)cos δ]2}-1/2.

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