Abstract

This paper combines and extends two optical models based on a two-collimated-flux approach that we previously proposed for the reflectance and transmittance of nonscattering elements, i.e., stacked nonscattering plastic films on the one hand, and films printed in halftone on the other hand. Those two models are revisited and combined by introducing different reflectances and transmittances on the two sides of a printed film, a common situation in practice. We then address the special case of stacks of identical films for which we obtain closed-form expressions for the reflectance and transmittance of the stacks as functions of the number of films. Experimental testing has been carried out on several different films printed with an inkjet printer. The accuracy of the model is good up to 16 films in most cases, despite a slight decrease in the case of yellow ink, which is more scattering than the other inks. By transposing the model to thin diffusing layers and considering diffuse fluxes instead of collimated ones, the closed-form expressions yield the well-known Kubelka–Munk reflectance and transmittance formulas. When these stacks of films are backed by a colored specular reflector, the reflectance is in certain conditions independent of the number of films.

© 2012 Optical Society of America

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  1. P. Kubelka, “New contributions to the optics of intensely light-scattering material. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef]
  2. S. Mourad, P. Emmel, K. Simon, and R. D. Hersch, “Extending Kubelka-Munk’s theory with lateral light scattering,” in Proceedings of IS&T NIP17: International Conference on Digital Printing Technologies, Fort Lauderdale, Fla., USA (2001), pp. 469–473.
  3. L. Yang and B. Kruse, “Revised Kubelka–Munk theory. I. Theory and application,” J. Opt. Soc. Am. A 21, 1933–1941 (2004).
    [CrossRef]
  4. L. Yang, B. Kruse, and S. J. Miklavcic, “Revised Kubelka–Munk theory. II. Unified framework for homogeneous and inhomogeneous optical media,” J. Opt. Soc. Am. A 21, 1942–1952 (2004).
    [CrossRef]
  5. L. Yang and S. J. Miklavcic, “Revised Kubelka-Munk theory. III. A general theory of light propagation in scattering and absorptive media,” J. Opt. Soc. Am. A 22, 1866–1873 (2005).
    [CrossRef]
  6. F. Rousselle, M. Hébert, and R. D. Hersch, “Predicting the reflectance of paper dyed with ink mixtures by describing light scattering as a function of ink absorbance,” J. Imaging Sci. Technol. 54, 050501 (2010).
    [CrossRef]
  7. M. Hébert, R. D. Hersch, and L. Simonot, “Spectral prediction model for piles of nonscattering sheets,” J. Opt. Soc. Am. A 25, 2066–2077 (2008).
    [CrossRef]
  8. P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behavior of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96 Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.
  9. F. R. Ruckdeschel and O. G. Hauser, “Yule-Nielsen effect in printing: a physical analysis,” Appl. Opt. 17, 3376–3383 (1978).
    [CrossRef]
  10. J. Machizaud and M. Hébert, “Spectral transmittance model for stacks of transparencies printed with halftone colors,” in SPIE/IS&T Color Imaging XVII: Displaying, Processing, Hardcopy, and Applications (Society for Imaging Science and Technology, 2012).
  11. M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
  12. M. E. Demichel, Procédés 26, 17–21 (1924).
  13. J. A. S. Viggiano, “The color of halftone tints,” Proc. TAGA 37, 647–661 (1985).
  14. J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proc. TAGA 3, 65–76 (1951).
  15. M. Hébert and R. D. Hersch, “Yule–Nielsen based recto—verso color halftone transmittance prediction model,” Appl. Opt. 50, 519–525 (2011).
    [CrossRef]
  16. R. D. Hersch and F. Crété, “Improving the Yule-Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions,” Proc. SPIE 5667, 434–445 (2005).
    [CrossRef]
  17. J. Machizaud and M. Hébert, “Spectral reflectance and transmittance prediction model for stacked transparency and paper both printed with halftone colors,” J. Opt. Soc. Am. A, 29, 1537–1548 (2012).
    [CrossRef]
  18. G. Sharma, Digital Color Imaging Handbook (CRC, 2003).
  19. P. Kubelka, “New contributions to the optics of intensely light-scattering materials, part II: Non homogeneous layers,” J. Opt. Soc. Am. 44, 330–335 (1954).
    [CrossRef]
  20. V. Ostromoukhov and R. D. Hersch, “Stochastic clustered-dot dithering,” J. Electron. Imaging 8, 439–445 (1999).
    [CrossRef]
  21. B. Maheu, J. N. Letouzan, and G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorentz-Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
    [CrossRef]
  22. P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
    [CrossRef]
  23. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  24. K. Klier, “Absorption and scattering in plane parallel turbid media,” J. Opt. Soc. Am. 62, 882–885 (1971).
    [CrossRef]
  25. M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
    [CrossRef]
  26. C. K. Yap, Fundamental Problems of Algorithmic Algebra(Oxford University, 2000).
  27. G. Strang, Applied Mathematics (MIT, 1986).

2012 (1)

2011 (1)

2010 (1)

F. Rousselle, M. Hébert, and R. D. Hersch, “Predicting the reflectance of paper dyed with ink mixtures by describing light scattering as a function of ink absorbance,” J. Imaging Sci. Technol. 54, 050501 (2010).
[CrossRef]

2008 (2)

M. Hébert, R. D. Hersch, and L. Simonot, “Spectral prediction model for piles of nonscattering sheets,” J. Opt. Soc. Am. A 25, 2066–2077 (2008).
[CrossRef]

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[CrossRef]

2005 (2)

L. Yang and S. J. Miklavcic, “Revised Kubelka-Munk theory. III. A general theory of light propagation in scattering and absorptive media,” J. Opt. Soc. Am. A 22, 1866–1873 (2005).
[CrossRef]

R. D. Hersch and F. Crété, “Improving the Yule-Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions,” Proc. SPIE 5667, 434–445 (2005).
[CrossRef]

2004 (2)

1999 (1)

V. Ostromoukhov and R. D. Hersch, “Stochastic clustered-dot dithering,” J. Electron. Imaging 8, 439–445 (1999).
[CrossRef]

1985 (1)

J. A. S. Viggiano, “The color of halftone tints,” Proc. TAGA 37, 647–661 (1985).

1984 (1)

1978 (1)

1971 (2)

1954 (1)

1951 (1)

J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proc. TAGA 3, 65–76 (1951).

1948 (1)

1924 (1)

M. E. Demichel, Procédés 26, 17–21 (1924).

Amidror, I.

P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behavior of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96 Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.

Becker, J.-M.

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[CrossRef]

Bhatia, A.

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Born, M.

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Crété, F.

R. D. Hersch and F. Crété, “Improving the Yule-Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions,” Proc. SPIE 5667, 434–445 (2005).
[CrossRef]

Demichel, M. E.

M. E. Demichel, Procédés 26, 17–21 (1924).

Emmel, P.

P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behavior of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96 Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.

S. Mourad, P. Emmel, K. Simon, and R. D. Hersch, “Extending Kubelka-Munk’s theory with lateral light scattering,” in Proceedings of IS&T NIP17: International Conference on Digital Printing Technologies, Fort Lauderdale, Fla., USA (2001), pp. 469–473.

Gouesbet, G.

Hauser, O. G.

Hébert, M.

J. Machizaud and M. Hébert, “Spectral reflectance and transmittance prediction model for stacked transparency and paper both printed with halftone colors,” J. Opt. Soc. Am. A, 29, 1537–1548 (2012).
[CrossRef]

M. Hébert and R. D. Hersch, “Yule–Nielsen based recto—verso color halftone transmittance prediction model,” Appl. Opt. 50, 519–525 (2011).
[CrossRef]

F. Rousselle, M. Hébert, and R. D. Hersch, “Predicting the reflectance of paper dyed with ink mixtures by describing light scattering as a function of ink absorbance,” J. Imaging Sci. Technol. 54, 050501 (2010).
[CrossRef]

M. Hébert, R. D. Hersch, and L. Simonot, “Spectral prediction model for piles of nonscattering sheets,” J. Opt. Soc. Am. A 25, 2066–2077 (2008).
[CrossRef]

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[CrossRef]

J. Machizaud and M. Hébert, “Spectral transmittance model for stacks of transparencies printed with halftone colors,” in SPIE/IS&T Color Imaging XVII: Displaying, Processing, Hardcopy, and Applications (Society for Imaging Science and Technology, 2012).

Hersch, R. D.

M. Hébert and R. D. Hersch, “Yule–Nielsen based recto—verso color halftone transmittance prediction model,” Appl. Opt. 50, 519–525 (2011).
[CrossRef]

F. Rousselle, M. Hébert, and R. D. Hersch, “Predicting the reflectance of paper dyed with ink mixtures by describing light scattering as a function of ink absorbance,” J. Imaging Sci. Technol. 54, 050501 (2010).
[CrossRef]

M. Hébert, R. D. Hersch, and L. Simonot, “Spectral prediction model for piles of nonscattering sheets,” J. Opt. Soc. Am. A 25, 2066–2077 (2008).
[CrossRef]

R. D. Hersch and F. Crété, “Improving the Yule-Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions,” Proc. SPIE 5667, 434–445 (2005).
[CrossRef]

V. Ostromoukhov and R. D. Hersch, “Stochastic clustered-dot dithering,” J. Electron. Imaging 8, 439–445 (1999).
[CrossRef]

P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behavior of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96 Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.

S. Mourad, P. Emmel, K. Simon, and R. D. Hersch, “Extending Kubelka-Munk’s theory with lateral light scattering,” in Proceedings of IS&T NIP17: International Conference on Digital Printing Technologies, Fort Lauderdale, Fla., USA (2001), pp. 469–473.

Klier, K.

Kruse, B.

Kubelka, P.

Letouzan, J. N.

Machizaud, J.

J. Machizaud and M. Hébert, “Spectral reflectance and transmittance prediction model for stacked transparency and paper both printed with halftone colors,” J. Opt. Soc. Am. A, 29, 1537–1548 (2012).
[CrossRef]

J. Machizaud and M. Hébert, “Spectral transmittance model for stacks of transparencies printed with halftone colors,” in SPIE/IS&T Color Imaging XVII: Displaying, Processing, Hardcopy, and Applications (Society for Imaging Science and Technology, 2012).

Maheu, B.

Miklavcic, S. J.

Mourad, S.

S. Mourad, P. Emmel, K. Simon, and R. D. Hersch, “Extending Kubelka-Munk’s theory with lateral light scattering,” in Proceedings of IS&T NIP17: International Conference on Digital Printing Technologies, Fort Lauderdale, Fla., USA (2001), pp. 469–473.

Mudgett, P. S.

Nielsen, W. J.

J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proc. TAGA 3, 65–76 (1951).

Ostromoukhov, V.

V. Ostromoukhov and R. D. Hersch, “Stochastic clustered-dot dithering,” J. Electron. Imaging 8, 439–445 (1999).
[CrossRef]

P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behavior of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96 Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.

Richards, L. W.

Rousselle, F.

F. Rousselle, M. Hébert, and R. D. Hersch, “Predicting the reflectance of paper dyed with ink mixtures by describing light scattering as a function of ink absorbance,” J. Imaging Sci. Technol. 54, 050501 (2010).
[CrossRef]

Ruckdeschel, F. R.

Sharma, G.

G. Sharma, Digital Color Imaging Handbook (CRC, 2003).

Simon, K.

S. Mourad, P. Emmel, K. Simon, and R. D. Hersch, “Extending Kubelka-Munk’s theory with lateral light scattering,” in Proceedings of IS&T NIP17: International Conference on Digital Printing Technologies, Fort Lauderdale, Fla., USA (2001), pp. 469–473.

Simonot, L.

Strang, G.

G. Strang, Applied Mathematics (MIT, 1986).

Viggiano, J. A. S.

J. A. S. Viggiano, “The color of halftone tints,” Proc. TAGA 37, 647–661 (1985).

Wolf, E.

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Yang, L.

Yap, C. K.

C. K. Yap, Fundamental Problems of Algorithmic Algebra(Oxford University, 2000).

Yule, J. A. C.

J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proc. TAGA 3, 65–76 (1951).

Appl. Opt. (4)

J. Electron. Imaging (1)

V. Ostromoukhov and R. D. Hersch, “Stochastic clustered-dot dithering,” J. Electron. Imaging 8, 439–445 (1999).
[CrossRef]

J. Imaging Sci. Technol. (1)

F. Rousselle, M. Hébert, and R. D. Hersch, “Predicting the reflectance of paper dyed with ink mixtures by describing light scattering as a function of ink absorbance,” J. Imaging Sci. Technol. 54, 050501 (2010).
[CrossRef]

J. Opt. A (1)

M. Hébert and J.-M. Becker, “Correspondence between continuous and discrete two-flux models for reflectance and transmittance of diffusing layers,” J. Opt. A 10, 035006 (2008).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Proc. SPIE (1)

R. D. Hersch and F. Crété, “Improving the Yule-Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions,” Proc. SPIE 5667, 434–445 (2005).
[CrossRef]

Proc. TAGA (2)

J. A. S. Viggiano, “The color of halftone tints,” Proc. TAGA 37, 647–661 (1985).

J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proc. TAGA 3, 65–76 (1951).

Procédés (1)

M. E. Demichel, Procédés 26, 17–21 (1924).

Other (8)

G. Sharma, Digital Color Imaging Handbook (CRC, 2003).

P. Emmel, I. Amidror, V. Ostromoukhov, and R. D. Hersch, “Predicting the spectral behavior of colour printers for transparent inks on transparent support,” in Proceedings of IS&T/SID 96 Color Imaging Conference (Society for Imaging Science, 1996), pp. 86–91.

J. Machizaud and M. Hébert, “Spectral transmittance model for stacks of transparencies printed with halftone colors,” in SPIE/IS&T Color Imaging XVII: Displaying, Processing, Hardcopy, and Applications (Society for Imaging Science and Technology, 2012).

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

S. Mourad, P. Emmel, K. Simon, and R. D. Hersch, “Extending Kubelka-Munk’s theory with lateral light scattering,” in Proceedings of IS&T NIP17: International Conference on Digital Printing Technologies, Fort Lauderdale, Fla., USA (2001), pp. 469–473.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

C. K. Yap, Fundamental Problems of Algorithmic Algebra(Oxford University, 2000).

G. Strang, Applied Mathematics (MIT, 1986).

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Figures (7)

Fig. 1.
Fig. 1.

Trajectories of collimated light within a transparency.

Fig. 2.
Fig. 2.

Spectral reflectance measured at normal incidence on the inked face (dotted curve) and the opposite face (solid curve) of a film printed with cyan and yellow inks at surface coverages 0.57, respectively 0.12.

Fig. 3.
Fig. 3.

Multiple reflections of collimated flux between two nonscattering films.

Fig. 4.
Fig. 4.

Spectral front reflectances (top graph) and front-to-back transmittances (bottom graph) of single green film (dashed curves) and of stacks of 2–16 green films (solid curves) measured at normal incidence. The numbers of films in the stack appear as circled. Infinite stack reflectance denoted by symbol is predicted according to formula (31).

Fig. 5.
Fig. 5.

Variation of the ΔE94 value as a function of the number of films for the front reflectance, the back reflectance, and the front-to-back transmittance of green (g), blue (b), magenta (m), and yellow (y) films when the single film transmittance T is measured (solid curves) or deduced from Eq. (33) (dotted curves). Horizontal dashed lines indicate the visibility threshold of color differences (ΔE94=1).

Fig. 6.
Fig. 6.

Evolution of the spectral reflectance of (a) blue films in front of a red reflector and (b) green films in front of a magenta reflector. The circled numbers denote numbers of films and line colors roughly reproduce the colors associated to the plotted spectra. Spectral reflectances of the backing alone and of an infinite stack of films are in dotted and dashed curves, respectively.

Fig. 7.
Fig. 7.

Variation of the spectral reflectance of a red paper background covered by yellow films; the numbers attached to the spectra indicate the number of added films.

Tables (3)

Tables Icon

Table 1. Prediction Accuracy of the Model for Spectral Reflectance and Transmittance of Printed Films

Tables Icon

Table 2. Prediction Accuracy for the Spectral Transmittance of Stacks of Printed Films

Tables Icon

Table 3. Average (max) ΔE94 Values Obtained for the Different Films and Geometries

Equations (62)

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

t(λ)=eα(λ)h.
[t(λ)]1/cosψ.
θ1=arcsin(sinθ/n1).
t(θ,λ)=[t(λ)]1/cosθ1=[t(λ)]n1/n12sin2θ.
R(θ,λ)=rθ+(1rθ)2rθt2(θ,λ)1rθ2t2(θ,λ),
T(θ,λ)=(1rθ)2t(θ,λ)1rθ2t2(θ,λ).
T(0,λ)=16n12t(λ)(n1+1)4(n11)4t2(λ).
t(λ)=64n14+(n121)4T2(0,λ)8n12(n11)4T(0,λ).
aw=(1xc)(1xm)(1xy),ac=xc(1xm)(1xy),am=(1xc)xm(1xy),ay=(1xc)(1xm)xy,am+y=(1xc)xmxy,ac+y=xc(1xm)xy,ac+m=xcxm(1xy),ac+m+y=xcxmxy.
R(θ,λ)=k=18akRk(θ,λ).
R(θ,λ)=[k=18akRk1/n(θ,λ)]n,
T(θ,λ)=[k=18akTk1/n(θ,λ)]n.
Ri/u(a,λ)=[(1a)Ru1/n(0,λ)+aRi+u1/n(0,λ)]n.
ai/u=argmina[λ(Ri/u(a,λ)Rm(λ))2].
x c = ( 1 x m ) ( 1 x y ) f c / w ( c ) + x m ( 1 x y ) f c / m ( c ) + ( 1 x m ) x y f c / y ( c ) + x m x y f c / m + y ( c ) , x m = ( 1 x c ) ( 1 x y ) f m / w ( m ) + x c ( 1 x y ) f m / c ( m ) + ( 1 x c ) x y f m / y ( m ) + x c x y f m / c + y ( m ) , x y = ( 1 x c ) ( 1 x m ) f y / w ( y ) + x c ( 1 x m ) f y / c ( y ) + ( 1 x c ) x m f y / m ( y ) + x c x m f y / c + m ( y ) .
R21=R2+T2T2R11R1R2.
T21=T2T11R1R2.
R21=R1+T1T1R21R1R2,
T21=T1T21R1R2.
Rqp1=Rq+TqTqRp11Rp1Rq,
Tqp1=TqTp11Rp1Rq,
Rqp1=Rp1+Tp1Tp1Rq1Rp1Rq,
Tqp1=TqTp11Rp1Rq.
RN=1αβ(121[1(α+β)R1(αβ)R]N),
TN=2bTN(α+β)[1(αβ)R]N(αβ)[1(α+β)R]N,
α=1+RRTT2R,
β=α2RR.
RN=RNRR,
TN=TN(TT)N.
R=R+TTR1RR.
R=(αβ)RR=1α+β.
R=αβ=RR·1α+β.
T=TkTk1(1Rk1R).
Pp1=Rp1+Tp1Tp1P01P0Rp1.
PN+1=R+TTPN1PNR.
PN+1PN=R1PNR[PNRR(αβ)]·[PNRR(α+β)].
PN+1PN=R1PNR(PNR)(PN1/R).
R+TT1PNRR+TT1RR;
α=limN1+(Sh/N)2[1(K+S)h/N]22Sh/N=K+SS=a.
β=α21=b.
limN(1xN)N=ex,
Rkm(h)=limNRN=1ab(121e2bSh)=sinh(bSh)asinh(bSh)+bcosh(bSh).
Tkm(h)=limnTN=2beaSh(a+b)e(ab)Sh(ab)e(a+b)Sh=2basinh(bSh)+bcosh(bSh).
RN=R+TTR+1RN1.
v0+u1v1+u2v2+ukvk
(UV)=(1v001)·(0u11v1)·(0u21v2)(0uk1vk).
(UV)=(1R01)·[(0TT1R)·(011R)]N1
(UV)=(0110)·(1RRTTRR)N1·(011R).
M=E·(e100e2)·E1
E=(αβα+β11).
MN1=E·(e1N100e2N1)·E1.
(UV)=12b(e1Ne2N(αβ)e1N(α+β)e2N).
RN=e1Ne2N(αβ)e1N(α+β)e2N,
RN=1αβ(121(e1/e2)N).
α=1RR+TT2R=αRR,
β=α2RR=βRR,
e1,2=1(α±β)R=e1,2.
TNTN1=T1RN1R.
TNTN1=TuN1uN,
uk=(αβ)e1k(α+β)e2k.
TN=T(TNTN1·TN1TN2T2T)=TN(uN1uN·uN2uN1u1u2)=TN(2buN),
TN=2bTN(α+β)e2N(αβ)e1N.

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