Abstract

A direct numerical inversion method for the determination of the refractive index and the thickness of the outermost layer of a thin transparent film on top of a multilayer has been developed. This method is based on a second-order Taylor decomposition of the coefficients of the Abelès matrices of the newly grown layer. The variations of the real-time spectroscopic ellipsometry data are expressed as polynomial functions depending on the dielectric constant and the thickness of the newly grown layer. The method is fast, capable of single-wavelength and multiwavelength inversion of continuous as well as discontinuous-index profiles, and can be adapted to many different polarimetric instruments.

© 2002 Optical Society of America

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References

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  1. D. E. Aspnes, N. Dietz, “ Optical approaches for controlling epitaxial growth,” Appl. Surf. Sci. 130–132, 367–376 (1998).
    [CrossRef]
  2. W. M. Duncan, S. A. Henck, “In situ ellipsometry for real-time measurement and control,” Appl. Surf. Sci. 63, 9–16 (1993).
    [CrossRef]
  3. M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
    [CrossRef]
  4. T. Heitz, A. Hofrichter, P. Bulkin, B. Drévillon, “Real time control of plasma deposited optical filters by multiwavelength ellipsometry,” J. Vac. Sci. Technol. A 18, 1303–1307 (2000).
    [CrossRef]
  5. A. Hofrichter, T. Heitz, P. Bulkin, B. Drévillon, “An ellipsometric method for real time control of thin film deposition on imperfect substrates” (to be published).
  6. S. Callard, A. Gagnaire, J. Joseph, “New method for in situ control of Bragg reflector fabrication,” Appl. Phys. Lett. 68, 2335–2336 (1996).
    [CrossRef]
  7. D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
    [CrossRef]
  8. D. E. Aspnes, W. E. Quinn, S. Gregory, “Application of ellipsometry to crystal growth by organometallic molecular beam epitaxy,” Appl. Phys. Lett. 56, 2569–2571 (1990).
    [CrossRef]
  9. M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
    [CrossRef]
  10. M. Kildemo, R. Brenot, B. Drévillon, “Spectroellipsometric method for process monitoring semiconductor thin films and interfaces,” Appl. Opt. 37, 5145–5149 (1998).
    [CrossRef]
  11. A. V. Hofrichter, D. Kouznetsov, P. Bulkin, B. Drévillon, “Direct numerical inversion method for kinetic ellipsometric data. II. Implementation and experimental verification,” Appl. Opt. 41, 4519–4525 (2002).
    [CrossRef] [PubMed]
  12. F. Abelès, “ Recherches sur la propagation des ondes electromagnetiques sinusoïdales dans les millieux stratifies. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782(1950).
  13. B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mater. 27, 1–87 (1993).
    [CrossRef]
  14. Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism of data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995).
    [CrossRef]
  15. M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent refelction model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313, 108–113 (1998).
    [CrossRef]
  16. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University, Cambridge, UK, 1989).
  17. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1987).
  18. M. Kildemo, “Real-time monitoring and growth control of Si-gradient-index structures by multiwavelength ellipsometry,” Appl. Opt. 37, 113–124 (1998).
    [CrossRef]
  19. J. Lekner, “Inversion of reflection ellipsometric data,” Appl. Opt. 33, 5159–5165 (1994).
    [CrossRef] [PubMed]
  20. J. P. Drolet, S. C. Russev, M. I. Boyanov, R. M. Leblanc, “Polynomial inversion of the single transparent layer problem in ellipsometry,” J. Opt. Soc. Am. A 11, 3284–3291 (1994).
    [CrossRef]
  21. D. Charlot, A. Maruani, “Ellipsometric data processing: an efficient method and an analysis of the relative errors,” Appl. Opt. 24, 3368–3373 (1985).
    [CrossRef] [PubMed]

2002 (1)

2000 (1)

T. Heitz, A. Hofrichter, P. Bulkin, B. Drévillon, “Real time control of plasma deposited optical filters by multiwavelength ellipsometry,” J. Vac. Sci. Technol. A 18, 1303–1307 (2000).
[CrossRef]

1998 (4)

D. E. Aspnes, N. Dietz, “ Optical approaches for controlling epitaxial growth,” Appl. Surf. Sci. 130–132, 367–376 (1998).
[CrossRef]

M. Kildemo, R. Brenot, B. Drévillon, “Spectroellipsometric method for process monitoring semiconductor thin films and interfaces,” Appl. Opt. 37, 5145–5149 (1998).
[CrossRef]

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent refelction model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313, 108–113 (1998).
[CrossRef]

M. Kildemo, “Real-time monitoring and growth control of Si-gradient-index structures by multiwavelength ellipsometry,” Appl. Opt. 37, 113–124 (1998).
[CrossRef]

1996 (3)

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

S. Callard, A. Gagnaire, J. Joseph, “New method for in situ control of Bragg reflector fabrication,” Appl. Phys. Lett. 68, 2335–2336 (1996).
[CrossRef]

1995 (1)

Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism of data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995).
[CrossRef]

1994 (2)

1993 (3)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mater. 27, 1–87 (1993).
[CrossRef]

D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
[CrossRef]

W. M. Duncan, S. A. Henck, “In situ ellipsometry for real-time measurement and control,” Appl. Surf. Sci. 63, 9–16 (1993).
[CrossRef]

1990 (1)

D. E. Aspnes, W. E. Quinn, S. Gregory, “Application of ellipsometry to crystal growth by organometallic molecular beam epitaxy,” Appl. Phys. Lett. 56, 2569–2571 (1990).
[CrossRef]

1985 (1)

1950 (1)

F. Abelès, “ Recherches sur la propagation des ondes electromagnetiques sinusoïdales dans les millieux stratifies. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782(1950).

Abelès, F.

F. Abelès, “ Recherches sur la propagation des ondes electromagnetiques sinusoïdales dans les millieux stratifies. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782(1950).

Abelson, J. R.

Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism of data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, N. Dietz, “ Optical approaches for controlling epitaxial growth,” Appl. Surf. Sci. 130–132, 367–376 (1998).
[CrossRef]

D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
[CrossRef]

D. E. Aspnes, W. E. Quinn, S. Gregory, “Application of ellipsometry to crystal growth by organometallic molecular beam epitaxy,” Appl. Phys. Lett. 56, 2569–2571 (1990).
[CrossRef]

Boyanov, M. I.

Brenot, R.

Bulkin, P.

A. V. Hofrichter, D. Kouznetsov, P. Bulkin, B. Drévillon, “Direct numerical inversion method for kinetic ellipsometric data. II. Implementation and experimental verification,” Appl. Opt. 41, 4519–4525 (2002).
[CrossRef] [PubMed]

T. Heitz, A. Hofrichter, P. Bulkin, B. Drévillon, “Real time control of plasma deposited optical filters by multiwavelength ellipsometry,” J. Vac. Sci. Technol. A 18, 1303–1307 (2000).
[CrossRef]

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

A. Hofrichter, T. Heitz, P. Bulkin, B. Drévillon, “An ellipsometric method for real time control of thin film deposition on imperfect substrates” (to be published).

Callard, S.

S. Callard, A. Gagnaire, J. Joseph, “New method for in situ control of Bragg reflector fabrication,” Appl. Phys. Lett. 68, 2335–2336 (1996).
[CrossRef]

Charlot, D.

Deniau, S.

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

Dietz, N.

D. E. Aspnes, N. Dietz, “ Optical approaches for controlling epitaxial growth,” Appl. Surf. Sci. 130–132, 367–376 (1998).
[CrossRef]

Drévillon, B.

A. V. Hofrichter, D. Kouznetsov, P. Bulkin, B. Drévillon, “Direct numerical inversion method for kinetic ellipsometric data. II. Implementation and experimental verification,” Appl. Opt. 41, 4519–4525 (2002).
[CrossRef] [PubMed]

T. Heitz, A. Hofrichter, P. Bulkin, B. Drévillon, “Real time control of plasma deposited optical filters by multiwavelength ellipsometry,” J. Vac. Sci. Technol. A 18, 1303–1307 (2000).
[CrossRef]

M. Kildemo, R. Brenot, B. Drévillon, “Spectroellipsometric method for process monitoring semiconductor thin films and interfaces,” Appl. Opt. 37, 5145–5149 (1998).
[CrossRef]

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mater. 27, 1–87 (1993).
[CrossRef]

A. Hofrichter, T. Heitz, P. Bulkin, B. Drévillon, “An ellipsometric method for real time control of thin film deposition on imperfect substrates” (to be published).

Drolet, J. P.

Duncan, W. M.

W. M. Duncan, S. A. Henck, “In situ ellipsometry for real-time measurement and control,” Appl. Surf. Sci. 63, 9–16 (1993).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University, Cambridge, UK, 1989).

Gagnaire, A.

S. Callard, A. Gagnaire, J. Joseph, “New method for in situ control of Bragg reflector fabrication,” Appl. Phys. Lett. 68, 2335–2336 (1996).
[CrossRef]

Gregory, S.

D. E. Aspnes, W. E. Quinn, S. Gregory, “Application of ellipsometry to crystal growth by organometallic molecular beam epitaxy,” Appl. Phys. Lett. 56, 2569–2571 (1990).
[CrossRef]

Heitz, T.

T. Heitz, A. Hofrichter, P. Bulkin, B. Drévillon, “Real time control of plasma deposited optical filters by multiwavelength ellipsometry,” J. Vac. Sci. Technol. A 18, 1303–1307 (2000).
[CrossRef]

A. Hofrichter, T. Heitz, P. Bulkin, B. Drévillon, “An ellipsometric method for real time control of thin film deposition on imperfect substrates” (to be published).

Henck, S. A.

W. M. Duncan, S. A. Henck, “In situ ellipsometry for real-time measurement and control,” Appl. Surf. Sci. 63, 9–16 (1993).
[CrossRef]

Hofrichter, A.

T. Heitz, A. Hofrichter, P. Bulkin, B. Drévillon, “Real time control of plasma deposited optical filters by multiwavelength ellipsometry,” J. Vac. Sci. Technol. A 18, 1303–1307 (2000).
[CrossRef]

A. Hofrichter, T. Heitz, P. Bulkin, B. Drévillon, “An ellipsometric method for real time control of thin film deposition on imperfect substrates” (to be published).

Hofrichter, A. V.

Joseph, J.

S. Callard, A. Gagnaire, J. Joseph, “New method for in situ control of Bragg reflector fabrication,” Appl. Phys. Lett. 68, 2335–2336 (1996).
[CrossRef]

Kildemo, M.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent refelction model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313, 108–113 (1998).
[CrossRef]

M. Kildemo, “Real-time monitoring and growth control of Si-gradient-index structures by multiwavelength ellipsometry,” Appl. Opt. 37, 113–124 (1998).
[CrossRef]

M. Kildemo, R. Brenot, B. Drévillon, “Spectroellipsometric method for process monitoring semiconductor thin films and interfaces,” Appl. Opt. 37, 5145–5149 (1998).
[CrossRef]

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

Kouznetsov, D.

Leblanc, R. M.

Lekner, J.

Maruani, A.

Ossikovski, R.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent refelction model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313, 108–113 (1998).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University, Cambridge, UK, 1989).

Quinn, W. E.

D. E. Aspnes, W. E. Quinn, S. Gregory, “Application of ellipsometry to crystal growth by organometallic molecular beam epitaxy,” Appl. Phys. Lett. 56, 2569–2571 (1990).
[CrossRef]

Russev, S. C.

Stchakovsky, M.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent refelction model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313, 108–113 (1998).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University, Cambridge, UK, 1989).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University, Cambridge, UK, 1989).

Yang, Y. H.

Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism of data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995).
[CrossRef]

Ann. Phys. (Paris) (1)

F. Abelès, “ Recherches sur la propagation des ondes electromagnetiques sinusoïdales dans les millieux stratifies. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782(1950).

Appl. Opt. (5)

Appl. Phys. Lett. (3)

S. Callard, A. Gagnaire, J. Joseph, “New method for in situ control of Bragg reflector fabrication,” Appl. Phys. Lett. 68, 2335–2336 (1996).
[CrossRef]

D. E. Aspnes, W. E. Quinn, S. Gregory, “Application of ellipsometry to crystal growth by organometallic molecular beam epitaxy,” Appl. Phys. Lett. 56, 2569–2571 (1990).
[CrossRef]

M. Kildemo, P. Bulkin, S. Deniau, B. Drévillon, “Real time control of plasma deposited multilayers by multiwavelength ellipsometry,” Appl. Phys. Lett. 68, 3395–3397 (1996).
[CrossRef]

Appl. Surf. Sci. (2)

D. E. Aspnes, N. Dietz, “ Optical approaches for controlling epitaxial growth,” Appl. Surf. Sci. 130–132, 367–376 (1998).
[CrossRef]

W. M. Duncan, S. A. Henck, “In situ ellipsometry for real-time measurement and control,” Appl. Surf. Sci. 63, 9–16 (1993).
[CrossRef]

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

J. Vac. Sci. Technol. A (2)

T. Heitz, A. Hofrichter, P. Bulkin, B. Drévillon, “Real time control of plasma deposited optical filters by multiwavelength ellipsometry,” J. Vac. Sci. Technol. A 18, 1303–1307 (2000).
[CrossRef]

Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism of data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995).
[CrossRef]

Prog. Cryst. Growth Charact. Mater. (1)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Mater. 27, 1–87 (1993).
[CrossRef]

Rev. Sci. Instrum. (1)

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

Thin Solid Films (1)

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent refelction model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313, 108–113 (1998).
[CrossRef]

Other (3)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University, Cambridge, UK, 1989).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1987).

A. Hofrichter, T. Heitz, P. Bulkin, B. Drévillon, “An ellipsometric method for real time control of thin film deposition on imperfect substrates” (to be published).

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Figures (7)

Fig. 1
Fig. 1

Error induced on the matrix elements cos q i dx (circles) and sin q i dx (squares) by the polynomial approximation expressed by an equivalent error on the refractive index Δn (see text for details) as a function of the thickness dx of the approximated layer for the case (open symbols) of low index (1.455) long wavelength (700 nm) and for the case (filled symbols) of high refractive index (2.17) and short wavelength (300 nm).

Fig. 2
Fig. 2

Averaged refractive index (symbols) and total reconstructed thickness (line) for 1.8 and 3.8 eV as a function of the step size for simulated data without noise of the growth of a 2500-Å-thick silicon nitride layer on Corning glass at a 58° angle of incidence.

Fig. 3
Fig. 3

Target (curve) and reconstruction (symbols) for 1.8- (squares) and 3.8-(circles) eV single-wavelength reconstruction a 58° angle of incidence from simulated data without noise of the growth of a 2500-Å-thick silicon nitride layer on Corning glass.

Fig. 4
Fig. 4

Reconstructed refractive index at 1.8 (squares) and 3.8 (circles) eV versus thickness for single-wavelength reconstruction (filled symbols) and multiwavelength reconstruction (open symbols) from simulated data without noise of the growth of a 2500-Å-thick silicon nitride layer on Corning glass at a 58° angle of incidence at a step size of 50 Å.

Fig. 5
Fig. 5

Averaged refractive index for 1.8 (squares) and 3.8 eV (circles) as a function of the step size for simulated data with simulated experimental noise of the growth of a 2500-Å-thick silicon nitride layer on Corning glass at a 58° angle of incidence for the inversion with a single wavelength (filled symbols) and for inversion with 11 wavelengths equally spaced between 1.8 and 3.8 eV (open symbols).

Fig. 6
Fig. 6

Reconstructed refractive index versus thickness for a 2000-Å-thick linear gradient layer between silicon oxide and silicon nitride reconstructed with 16 wavelengths equally spaced between 1.5 and 5 eV at a 58° angle of incidence.

Fig. 7
Fig. 7

Reconstructed refractive index versus thickness for a 1-µm-thick alternating index layer, reconstructed with 16 wavelengths at a 58° angle of incidence.

Equations (24)

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

Mi=cos ϕiiqisin ϕiiqi sin ϕicos ϕi,
α=asin γa=isin γi=ssin γs,
M=m11m12m21m22=i=1nMi.
r=qam11-qsm22+qaqsm12-m21qam11+qsm22+qaqsm12+m21.
ρ=rprs=tan Ψ expiΔ
Is=sin 2Ψ sin Δ=2 Imrs*rprsrs*+rprp*, Ic=sin 2Ψ cos Δ=2 Rers*rprsrs*+rprp*,
rs,prs,p*=rs,prs,p*+ts,pts,p*ts,pts,p*rbs,prbs,p*exp-4 Im βs1-rbs,prbs,p*rbs,prbs,p*exp-4 Im βs,
dMs1+-12 k2-α2dx2ikdxi-α2kdx1+-12 k2-α2dx2,
dMp1+-12 k2-α2dx2i1-1 α2kdxikdx1+-12 k2-α2dx2.
A±2dx2+B±1dx+C,
B±1=C1B+C0B+C-1B,
A±2=C2A2+C1A+C0A+C-1A+C-2A2,
rs,pprod=A±2dx2+B±1dx+CA±2dx2+B±1dx+C.
rs,pprodrs,p+drs,p=C+A±2dx2+B±1dx,
dIc=Ac±2dx2+Bc±1dx,
dIs=As±2dx2+Bs±2dx.
dx1=dIcBs-dIsBcdIsAc-dIcAs=T1±1T±2
dx2=dIsAc-dIcAsBsAc-BcAs=T±2T2±3
P±4=T±22-T1±1T2±3=0.
σ2=dIc*-dIc2ΔIc2+dIs*-dIs2ΔIs2,
dx=i=1Ndxiσi2i=1N1σi2,
χ2=1ΔIC2 |dIc*, dx-dIc|2+1ΔIs2 |dIs*, dx-dIs|2,
ΔE1=sin2πλn2-α21/2dx-2πλn2-α21/2dx.
ΔE2=sin2πλn2-α21/2dx-sin2πλn+Δn2-α21/2dx.

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