Abstract

Multiplexed volume Bragg gratings can be applied to many types of broad- and narrowband spectral systems. However, there are often deleterious side effects to combining several gratings into a single holographic optical element, including loss of efficiency in diffracted waves of interest and the introduction of spurious waves. Design of these spectral systems requires analysis methods that are flexible and efficient and that take these side effects into account. We present a matrix-based algorithm for determining diffraction efficiencies of significant coupled waves in these multiplexed grating Holographic optical elements (HOEs). Several carefully constructed experiments with spectrally multiplexed gratings in dichromated gelatin verify our conclusions.

© 2014 Optical Society of America

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References

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  1. I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100  W Yb–fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
    [Crossref]
  2. B. Chann, A. K. Goyal, T. Y. Fan, A. Sanchez-Rubio, B. L. Volodin, and V. S. Ban, “Efficient, high-brightness wavelength-beam-combined commercial off-the-shelf diode stacks achieved by use of a wavelength-chirped volume Bragg grating,” Opt. Lett. 31, 1253–1255 (2006).
    [Crossref]
  3. A. Sevian, O. Andrusyak, I. V. Ciapurin, V. I. Smirnov, G. B. Venus, and L. B. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33, 384–386 (2008).
    [Crossref]
  4. P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, and M. Martinelli, “1550  nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
    [Crossref]
  5. S. F. Chen, C. S. Wu, and C. C. Sun, “Design for a high dense wavelength division multiplexer based on volume holographic gratings,” Opt. Eng. 43, 2028–2033 (2004).
    [Crossref]
  6. S. Datta and S. R. Forrest, “Low through-channel loss wavelength multiplexer using multiple transmission volume Bragg gratings,” J. Opt. Soc. Am. A 22, 1624–1629 (2005).
    [Crossref]
  7. D. Lin, E. Torrey, J. Leger, and P. Cohen, “Lossless holographic spectrum splitter in lateral photovoltaic devices,” in 37th IEEE Photovoltaic Specialists Conference (IEEE, 2011), pp. 894–898.
  8. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  9. R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
    [Crossref]
  10. R. Alferness and S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
    [Crossref]
  11. S. K. Case, “Coupled wave theory for multiply exposed thick holographic gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
    [Crossref]
  12. K. Tu, T. Tamir, and H. Lee, “Multiple-scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. A 7, 1421–1435 (1990).
    [Crossref]
  13. J. H. Zhao, X. N. Shen, and X. Y. Xia, “Beam splitting, combining, and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
    [Crossref]
  14. R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
    [Crossref]
  15. R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in reflection volume holograms,” Opt. Quantum Electron. 10, 171–178 (1978).
    [Crossref]
  16. V. Minier, A. Kevorkian, and J. M. Xu, “Superimposed phase gratings in planar optical waveguides for demultiplexing applications,” IEEE Photon. Technol. Lett. 5, 330–333 (1993).
    [Crossref]
  17. V. Minier and J. M. Xu, “Coupled-mode analysis of superimposed phase grating guided-wave structures and integrating coupling effects,” Opt. Eng. 32, 2054–2063 (1993).
    [Crossref]
  18. M. G. Moharam and T. K. Gaylord, “Rigorous coupled wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [Crossref]
  19. T. Rowland, “Orthonormal basis,” 2014, http://mathworld.wolfram.com/OrthonormalBasis.html .
  20. A. Villamarin, J. Atencia, M. V. Collados, and M. Quintanilla, “Characterization of transmission volume holographic gratings recorded in Slavich PFG04 dichromated gelatin plates,” Appl. Opt. 48, 4348–4353 (2009).
    [Crossref]
  21. A. Barnett, “Very high efficiency solar cell modules,” Prog. Photovoltaics 17, 75–83 (2009).
    [Crossref]
  22. A. Othonos, J. Bismuth, M. Sweeny, A. Kevorkian, and J. M. Xu, “Superimposed grating wavelength division multiplexing in Ge-doped SiO2/Si planar waveguides,” Opt. Eng. 37, 717–720 (1998).
    [Crossref]
  23. X. Fu, M. Fay, and T. M. Xu, “18 supergrating wavelength-division demultiplexer in a silica planar waveguide,” Opt. Lett. 22, 1627–1629 (1997).
    [Crossref]

2009 (2)

2008 (1)

2006 (1)

2005 (1)

2004 (1)

S. F. Chen, C. S. Wu, and C. C. Sun, “Design for a high dense wavelength division multiplexer based on volume holographic gratings,” Opt. Eng. 43, 2028–2033 (2004).
[Crossref]

2003 (1)

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100  W Yb–fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[Crossref]

2001 (1)

J. H. Zhao, X. N. Shen, and X. Y. Xia, “Beam splitting, combining, and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
[Crossref]

2000 (1)

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, and M. Martinelli, “1550  nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[Crossref]

1998 (1)

A. Othonos, J. Bismuth, M. Sweeny, A. Kevorkian, and J. M. Xu, “Superimposed grating wavelength division multiplexing in Ge-doped SiO2/Si planar waveguides,” Opt. Eng. 37, 717–720 (1998).
[Crossref]

1997 (1)

1993 (2)

V. Minier, A. Kevorkian, and J. M. Xu, “Superimposed phase gratings in planar optical waveguides for demultiplexing applications,” IEEE Photon. Technol. Lett. 5, 330–333 (1993).
[Crossref]

V. Minier and J. M. Xu, “Coupled-mode analysis of superimposed phase grating guided-wave structures and integrating coupling effects,” Opt. Eng. 32, 2054–2063 (1993).
[Crossref]

1990 (1)

1981 (1)

1978 (2)

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
[Crossref]

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in reflection volume holograms,” Opt. Quantum Electron. 10, 171–178 (1978).
[Crossref]

1975 (3)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Alferness, R.

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[Crossref]

R. Alferness and S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
[Crossref]

Andrusyak, O.

Atencia, J.

Ban, V. S.

Barnett, A.

A. Barnett, “Very high efficiency solar cell modules,” Prog. Photovoltaics 17, 75–83 (2009).
[Crossref]

Bismuth, J.

A. Othonos, J. Bismuth, M. Sweeny, A. Kevorkian, and J. M. Xu, “Superimposed grating wavelength division multiplexing in Ge-doped SiO2/Si planar waveguides,” Opt. Eng. 37, 717–720 (1998).
[Crossref]

Boffi, P.

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, and M. Martinelli, “1550  nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[Crossref]

Case, S. K.

Chann, B.

Chen, S. F.

S. F. Chen, C. S. Wu, and C. C. Sun, “Design for a high dense wavelength division multiplexer based on volume holographic gratings,” Opt. Eng. 43, 2028–2033 (2004).
[Crossref]

Ciapurin, I. V.

A. Sevian, O. Andrusyak, I. V. Ciapurin, V. I. Smirnov, G. B. Venus, and L. B. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33, 384–386 (2008).
[Crossref]

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100  W Yb–fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[Crossref]

Cohen, P.

D. Lin, E. Torrey, J. Leger, and P. Cohen, “Lossless holographic spectrum splitter in lateral photovoltaic devices,” in 37th IEEE Photovoltaic Specialists Conference (IEEE, 2011), pp. 894–898.

Collados, M. V.

Datta, S.

Fan, T. Y.

Fay, M.

Forrest, S. R.

Frascolla, C.

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, and M. Martinelli, “1550  nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[Crossref]

Fu, X.

Gaylord, T. K.

Glebov, L. B.

A. Sevian, O. Andrusyak, I. V. Ciapurin, V. I. Smirnov, G. B. Venus, and L. B. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33, 384–386 (2008).
[Crossref]

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100  W Yb–fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[Crossref]

Glebova, L. N.

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100  W Yb–fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[Crossref]

Goyal, A. K.

Kevorkian, A.

A. Othonos, J. Bismuth, M. Sweeny, A. Kevorkian, and J. M. Xu, “Superimposed grating wavelength division multiplexing in Ge-doped SiO2/Si planar waveguides,” Opt. Eng. 37, 717–720 (1998).
[Crossref]

V. Minier, A. Kevorkian, and J. M. Xu, “Superimposed phase gratings in planar optical waveguides for demultiplexing applications,” IEEE Photon. Technol. Lett. 5, 330–333 (1993).
[Crossref]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kowarschik, R.

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
[Crossref]

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in reflection volume holograms,” Opt. Quantum Electron. 10, 171–178 (1978).
[Crossref]

Lee, H.

Leger, J.

D. Lin, E. Torrey, J. Leger, and P. Cohen, “Lossless holographic spectrum splitter in lateral photovoltaic devices,” in 37th IEEE Photovoltaic Specialists Conference (IEEE, 2011), pp. 894–898.

Lin, D.

D. Lin, E. Torrey, J. Leger, and P. Cohen, “Lossless holographic spectrum splitter in lateral photovoltaic devices,” in 37th IEEE Photovoltaic Specialists Conference (IEEE, 2011), pp. 894–898.

Martinelli, M.

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, and M. Martinelli, “1550  nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[Crossref]

Minier, V.

V. Minier, A. Kevorkian, and J. M. Xu, “Superimposed phase gratings in planar optical waveguides for demultiplexing applications,” IEEE Photon. Technol. Lett. 5, 330–333 (1993).
[Crossref]

V. Minier and J. M. Xu, “Coupled-mode analysis of superimposed phase grating guided-wave structures and integrating coupling effects,” Opt. Eng. 32, 2054–2063 (1993).
[Crossref]

Moharam, M. G.

Othonos, A.

A. Othonos, J. Bismuth, M. Sweeny, A. Kevorkian, and J. M. Xu, “Superimposed grating wavelength division multiplexing in Ge-doped SiO2/Si planar waveguides,” Opt. Eng. 37, 717–720 (1998).
[Crossref]

Piccinin, D.

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, and M. Martinelli, “1550  nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[Crossref]

Quintanilla, M.

Rotari, E. V.

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100  W Yb–fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[Crossref]

Sanchez-Rubio, A.

Sevian, A.

Shen, X. N.

J. H. Zhao, X. N. Shen, and X. Y. Xia, “Beam splitting, combining, and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
[Crossref]

Smirnov, V. I.

A. Sevian, O. Andrusyak, I. V. Ciapurin, V. I. Smirnov, G. B. Venus, and L. B. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33, 384–386 (2008).
[Crossref]

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100  W Yb–fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[Crossref]

Sun, C. C.

S. F. Chen, C. S. Wu, and C. C. Sun, “Design for a high dense wavelength division multiplexer based on volume holographic gratings,” Opt. Eng. 43, 2028–2033 (2004).
[Crossref]

Sweeny, M.

A. Othonos, J. Bismuth, M. Sweeny, A. Kevorkian, and J. M. Xu, “Superimposed grating wavelength division multiplexing in Ge-doped SiO2/Si planar waveguides,” Opt. Eng. 37, 717–720 (1998).
[Crossref]

Tamir, T.

Torrey, E.

D. Lin, E. Torrey, J. Leger, and P. Cohen, “Lossless holographic spectrum splitter in lateral photovoltaic devices,” in 37th IEEE Photovoltaic Specialists Conference (IEEE, 2011), pp. 894–898.

Tu, K.

Ubaldi, M. C.

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, and M. Martinelli, “1550  nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[Crossref]

Venus, G. B.

Villamarin, A.

Volodin, B. L.

Wu, C. S.

S. F. Chen, C. S. Wu, and C. C. Sun, “Design for a high dense wavelength division multiplexer based on volume holographic gratings,” Opt. Eng. 43, 2028–2033 (2004).
[Crossref]

Xia, X. Y.

J. H. Zhao, X. N. Shen, and X. Y. Xia, “Beam splitting, combining, and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
[Crossref]

Xu, J. M.

A. Othonos, J. Bismuth, M. Sweeny, A. Kevorkian, and J. M. Xu, “Superimposed grating wavelength division multiplexing in Ge-doped SiO2/Si planar waveguides,” Opt. Eng. 37, 717–720 (1998).
[Crossref]

V. Minier and J. M. Xu, “Coupled-mode analysis of superimposed phase grating guided-wave structures and integrating coupling effects,” Opt. Eng. 32, 2054–2063 (1993).
[Crossref]

V. Minier, A. Kevorkian, and J. M. Xu, “Superimposed phase gratings in planar optical waveguides for demultiplexing applications,” IEEE Photon. Technol. Lett. 5, 330–333 (1993).
[Crossref]

Xu, T. M.

Zhao, J. H.

J. H. Zhao, X. N. Shen, and X. Y. Xia, “Beam splitting, combining, and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
[Crossref]

Appl. Opt. (1)

Appl. Phys. (1)

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

IEEE Photon. Technol. Lett. (2)

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, and M. Martinelli, “1550  nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[Crossref]

V. Minier, A. Kevorkian, and J. M. Xu, “Superimposed phase gratings in planar optical waveguides for demultiplexing applications,” IEEE Photon. Technol. Lett. 5, 330–333 (1993).
[Crossref]

J. Opt. Soc. Am. (3)

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

Opt. Acta (1)

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
[Crossref]

Opt. Eng. (3)

V. Minier and J. M. Xu, “Coupled-mode analysis of superimposed phase grating guided-wave structures and integrating coupling effects,” Opt. Eng. 32, 2054–2063 (1993).
[Crossref]

S. F. Chen, C. S. Wu, and C. C. Sun, “Design for a high dense wavelength division multiplexer based on volume holographic gratings,” Opt. Eng. 43, 2028–2033 (2004).
[Crossref]

A. Othonos, J. Bismuth, M. Sweeny, A. Kevorkian, and J. M. Xu, “Superimposed grating wavelength division multiplexing in Ge-doped SiO2/Si planar waveguides,” Opt. Eng. 37, 717–720 (1998).
[Crossref]

Opt. Laser Technol. (1)

J. H. Zhao, X. N. Shen, and X. Y. Xia, “Beam splitting, combining, and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in reflection volume holograms,” Opt. Quantum Electron. 10, 171–178 (1978).
[Crossref]

Proc. SPIE (1)

I. V. Ciapurin, L. B. Glebov, L. N. Glebova, V. I. Smirnov, and E. V. Rotari, “Incoherent combining of 100  W Yb–fiber laser beams by PTR Bragg grating,” Proc. SPIE 4974, 209–219 (2003).
[Crossref]

Prog. Photovoltaics (1)

A. Barnett, “Very high efficiency solar cell modules,” Prog. Photovoltaics 17, 75–83 (2009).
[Crossref]

Other (2)

T. Rowland, “Orthonormal basis,” 2014, http://mathworld.wolfram.com/OrthonormalBasis.html .

D. Lin, E. Torrey, J. Leger, and P. Cohen, “Lossless holographic spectrum splitter in lateral photovoltaic devices,” in 37th IEEE Photovoltaic Specialists Conference (IEEE, 2011), pp. 894–898.

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Figures (7)

Fig. 1.
Fig. 1.

Momentum or k-space diagram for a single grating. Inset: the physical model of a single grating defining plane wave angles θ1 and θ2, and the grating thickness d.

Fig. 2.
Fig. 2.

k-space diagram for defining two multiplexed gratings. These gratings share one of their defining angles and have different central operating wavelengths.

Fig. 3.
Fig. 3.

k-space diagram for reconstructing two multiplexed gratings. From the diagram, waves associated with σ1 and σ2 are qualitatively expected to be significant to the solution. However, σ3 is not qualitatively expected to be significant because |σ3| differs so greatly from β.

Fig. 4.
Fig. 4.

Diffraction efficiency of Grating 1 as a function of input angle for an input wavelength of 632.8 nm. Solid lines show the theoretical efficiency after adjusting the model to fit the measured data.

Fig. 5.
Fig. 5.

Diffraction efficiency of the significant output waves of the interfering grating pair as a function of input wavelength for an input angle (in air) of 18.5°. Measured data is superimposed on theoretical data. The lighter dashed curves indicate what the S-wave diffraction efficiency of each grating would be if the other grating was not present in the holographic element.

Fig. 6.
Fig. 6.

Diffraction efficiency of the significant output waves of the noninterfering grating pair as a function of input wavelength for an input angle (in air) of 18.5°. Measured data is superimposed on theoretical data.

Fig. 7.
Fig. 7.

Diffraction efficiencies versus wavelength of various diffraction orders for an example multiplexed transmission grating pair compared with an ideal bandpass response. Note how the combination of the two directly coupled waves (S1 and S2) approaches the ideal response, but multiplexing the gratings also gives rise to stray light in the system in the form of cross coupled waves T12 and T21.

Tables (2)

Tables Icon

Table 1. Parameters of Interfering Grating Pair as Determined by Fitting Measured Diffraction Efficiency Data to the Theoretical Model

Tables Icon

Table 2. Parameters of NonInterfering Grating Pair Determined by Fitting Measured Diffraction Efficiency Data to the Theoretical Model

Equations (27)

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

E(x,y,z,t)=R(z)ej(ρ·xωt)+S(z)ej(σ·xωt).
ϵ(x,y,z)=ϵ0+ϵ1cosK·x,
cRR=jκS,
cSSjϑS=jκR.
n1=ϵ12ϵ0.
cR=cosθ1=ρzβ,
cS=cosθ1|K|βcosϕ=σzβ,
ϑ=β2σ22β=|K|cos(ϕθ)|K|24πnλ.
Mx=γx,
M=[0jκcRjκcSjϑcS],
x=[RS].
x˜[R˜S˜][cR00cRcS]x.
M˜=[0jκcRcSjκcRcSjϑcS].
|γjκcRcSjκcRcSγjϑcS|=0,
γ±=j2(ϑcS±ϑ2cS2+4κ2cRcS).
G=[1cR001cRcS][ξ+ξ][eγ+d00eγd][ξ+ξ]T[cR00cRcS],
xd=Gx0.
η=cScRSdSd*,
M˜mm=jϑmcSm,m0.
M˜mn=M˜nm=jκpcSmcSn,mn.
M˜13=M˜31=jκ2cS1cS3.
M˜01=M˜10=jκ1cRcS1.
M˜=[0··0·00··0·00·0·].
M˜mn=M˜nm=jκpsm·sncSmcSn,mn.
G=[1cR00001cRcS100001cRcS200001cRcS3][ξ1ξ2ξ3ξ4][eγ1d0000eγ2d0000eγ3d0000eγ4d]×[ξ1ξ2ξ3ξ4]T[cR0000cRcS10000cRcS20000cRcS3],
xd=[RdS1dS2dS3d]=G[1000].
ηi=cSicRSidSid*.

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