Abstract

Eikonal analyses are applied to a hybrid micro/macro-optical shuffle interconnection approach that minimizes distortion in a multichip smart-pixel shuffle interconnection system. The optical system uses off-axis imaging elements to link clusters of dense arrays of vertical-cavity surface-emitting laser (VCSEL) sources to matching clusters within arrays of detectors. A critical requirement for such a system is that the images of the two-dimensional arrays of the VCSELs must be registered on their associated detector arrays with a precision of the order of 10 μm across the entire multichip array. The hybrid approach exploits the typical narrow-beam cone angles of VCSELs by use of beam-deflecting micro-optics to create a distortion-canceling symmetry about a central aperture in the optical system for each VCSEL–detector link. The second- and third-order aberrations of the plane-symmetric system created by the global off-axis imaging system are analyzed. The results prove that the hybrid concept cancels distortion and minimizes the spot size at the detector array plane.

© 2002 Optical Society of America

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References

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  1. A. W. Lohmann, “What classical optics can do for the digital optical computer,” Appl. Opt. 25, 1543–1549 (1986).
    [CrossRef] [PubMed]
  2. G. Eichmann, Y. Li, “Compact optical generalized perfect shuffle,” Appl. Opt. 26, 1167–1169 (1987).
    [CrossRef]
  3. S.-H. Lin, T. F. Krile, J. F. Walkup, “2-D optical multistage interconnection networks,” in Digital Optical Computing, R. Arrathoon, ed., Proc. SPIE752, 209–216 (1987).
    [CrossRef]
  4. K.-H. Brenner, A. Huang, “Optical implementations of the perfect shuffle interconnection,” Appl. Opt. 27, 135–137 (1988).
    [CrossRef] [PubMed]
  5. C. W. Stirk, R. A. Athale, M. W. Haney, “Folded perfect shuffle optical processor,” Appl. Opt. 27, 202–203 (1988).
    [CrossRef] [PubMed]
  6. A. A. Sawchuk, I. Glaser, “Geometries for optical implementations of the perfect shuffle,” in Optical Computing ’88, P. H. Chavel, J. W. Goodman, G. Roblin, eds., Proc. SPIE963, 270–282 (1988).
    [CrossRef]
  7. M. W. Haney, J. J. Levy, “Optically efficient free-space folded perfect shuffle network,” Appl. Opt. 30, 2833–2840 (1991).
    [CrossRef] [PubMed]
  8. G. C. Marsden, P. J. Marchand, P. Harvey, S. C. Esener, “Optical transpose interconnection system architecture,” Opt. Lett. 18, 1083–1085 (1993).
    [CrossRef]
  9. M. W. Haney, “Pipelined optoelectronic free-space permutation network,” Opt. Lett. 17, 283–285 (1992).
    [CrossRef]
  10. M. W. Haney, M. P. Christensen, “Performance scaling comparison for free-space optical and electrical interconnection approaches,” special issue on Optics in Computing, Appl. Opt. 37, 2886–2894 (1998).
    [CrossRef]
  11. M. W. Haney, M. P. Christensen, “Optical free-space sliding tandem Banyan architecture for self-routing switching networks,” in Digest of the International Conference on Optical Computing (Heriot-Watt University, Edinburgh, UK, 1994), pp. 249–250.
  12. R. R. Michael, M. P. Christensen, Michael W. Haney, “Experimental evaluation of the 3-D optical shuffle module of the sliding Banyan architecture,” J. Lightwave Technol. 14, 1970–1978 (1996).
    [CrossRef]
  13. M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
    [CrossRef]
  14. T. Smith, “The changes in aberrations when the object and stop are moved,” Trans. Opt. Soc. 23, 139–153 (1921/1922).
  15. G. C. Steward, The Symmetrical Optical System (Cambridge U. Press, Cambridge, UK, 1928).
  16. R. Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978).
  17. M. P. Christensen, P. Milojkovic, M. W. Haney, “Low-distortion hybrid optical shuffle concept,” Opt. Lett. 24, 169–171 (1999).
    [CrossRef]
  18. M. P. Christensen, P. Milojkovic, M. W. Haney, “Aberration analysis of beam-steering in the FAST-Net global interconnection system,” in Proceedings of the Lasers and Electro-Optics Society 2000 Annual Meeting (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2000), pp. 236–237.
  19. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, UK, 1970).
  20. C. H. F. Velzel, “Image formation by a general optical system, using Hamilton’s method,” J. Opt. Soc. Am. A 4, 1342–1348 (1987).
    [CrossRef]
  21. P. J. Sands, “Aberration coefficients of plane symmetric system,” J. Opt. Soc. Am. 62, 1211–1220 (1972).
    [CrossRef]
  22. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).
  23. A. Walther, “Some unsolved problems in image formation,” in Optics in Four Dimensions, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981), pp. 271–275.

1999 (1)

1998 (1)

1996 (1)

R. R. Michael, M. P. Christensen, Michael W. Haney, “Experimental evaluation of the 3-D optical shuffle module of the sliding Banyan architecture,” J. Lightwave Technol. 14, 1970–1978 (1996).
[CrossRef]

1993 (1)

1992 (1)

M. W. Haney, “Pipelined optoelectronic free-space permutation network,” Opt. Lett. 17, 283–285 (1992).
[CrossRef]

1991 (1)

1988 (2)

1987 (2)

1986 (1)

1972 (1)

Athale, R. A.

Bounnak, S.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Brenner, K.-H.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, UK, 1970).

Chandramani, P.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Christensen, M. P.

M. P. Christensen, P. Milojkovic, M. W. Haney, “Low-distortion hybrid optical shuffle concept,” Opt. Lett. 24, 169–171 (1999).
[CrossRef]

M. W. Haney, M. P. Christensen, “Performance scaling comparison for free-space optical and electrical interconnection approaches,” special issue on Optics in Computing, Appl. Opt. 37, 2886–2894 (1998).
[CrossRef]

R. R. Michael, M. P. Christensen, Michael W. Haney, “Experimental evaluation of the 3-D optical shuffle module of the sliding Banyan architecture,” J. Lightwave Technol. 14, 1970–1978 (1996).
[CrossRef]

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

M. W. Haney, M. P. Christensen, “Optical free-space sliding tandem Banyan architecture for self-routing switching networks,” in Digest of the International Conference on Optical Computing (Heriot-Watt University, Edinburgh, UK, 1994), pp. 249–250.

M. P. Christensen, P. Milojkovic, M. W. Haney, “Aberration analysis of beam-steering in the FAST-Net global interconnection system,” in Proceedings of the Lasers and Electro-Optics Society 2000 Annual Meeting (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2000), pp. 236–237.

Eichmann, G.

Ekman, J.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Esener, S. C.

Glaser, I.

A. A. Sawchuk, I. Glaser, “Geometries for optical implementations of the perfect shuffle,” in Optical Computing ’88, P. H. Chavel, J. W. Goodman, G. Roblin, eds., Proc. SPIE963, 270–282 (1988).
[CrossRef]

Haney, M. W.

M. P. Christensen, P. Milojkovic, M. W. Haney, “Low-distortion hybrid optical shuffle concept,” Opt. Lett. 24, 169–171 (1999).
[CrossRef]

M. W. Haney, M. P. Christensen, “Performance scaling comparison for free-space optical and electrical interconnection approaches,” special issue on Optics in Computing, Appl. Opt. 37, 2886–2894 (1998).
[CrossRef]

M. W. Haney, “Pipelined optoelectronic free-space permutation network,” Opt. Lett. 17, 283–285 (1992).
[CrossRef]

M. W. Haney, J. J. Levy, “Optically efficient free-space folded perfect shuffle network,” Appl. Opt. 30, 2833–2840 (1991).
[CrossRef] [PubMed]

C. W. Stirk, R. A. Athale, M. W. Haney, “Folded perfect shuffle optical processor,” Appl. Opt. 27, 202–203 (1988).
[CrossRef] [PubMed]

M. P. Christensen, P. Milojkovic, M. W. Haney, “Aberration analysis of beam-steering in the FAST-Net global interconnection system,” in Proceedings of the Lasers and Electro-Optics Society 2000 Annual Meeting (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2000), pp. 236–237.

M. W. Haney, M. P. Christensen, “Optical free-space sliding tandem Banyan architecture for self-routing switching networks,” in Digest of the International Conference on Optical Computing (Heriot-Watt University, Edinburgh, UK, 1994), pp. 249–250.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Haney, Michael W.

R. R. Michael, M. P. Christensen, Michael W. Haney, “Experimental evaluation of the 3-D optical shuffle module of the sliding Banyan architecture,” J. Lightwave Technol. 14, 1970–1978 (1996).
[CrossRef]

Harvey, P.

Hibbs-Brenner, M.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Huang, A.

Kalweit, E.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Kiamilev, F.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978).

Krile, T. F.

S.-H. Lin, T. F. Krile, J. F. Walkup, “2-D optical multistage interconnection networks,” in Digital Optical Computing, R. Arrathoon, ed., Proc. SPIE752, 209–216 (1987).
[CrossRef]

Levy, J. J.

Li, Y.

Lin, S.-H.

S.-H. Lin, T. F. Krile, J. F. Walkup, “2-D optical multistage interconnection networks,” in Digital Optical Computing, R. Arrathoon, ed., Proc. SPIE752, 209–216 (1987).
[CrossRef]

Liu, Y.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Lohmann, A. W.

Marchand, P. J.

Marsden, G. C.

Marta, T.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Michael, R. R.

R. R. Michael, M. P. Christensen, Michael W. Haney, “Experimental evaluation of the 3-D optical shuffle module of the sliding Banyan architecture,” J. Lightwave Technol. 14, 1970–1978 (1996).
[CrossRef]

Milojkovic, P.

M. P. Christensen, P. Milojkovic, M. W. Haney, “Low-distortion hybrid optical shuffle concept,” Opt. Lett. 24, 169–171 (1999).
[CrossRef]

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

M. P. Christensen, P. Milojkovic, M. W. Haney, “Aberration analysis of beam-steering in the FAST-Net global interconnection system,” in Proceedings of the Lasers and Electro-Optics Society 2000 Annual Meeting (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2000), pp. 236–237.

Nohava, J.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Rozier, R.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Sands, P. J.

Sawchuk, A. A.

A. A. Sawchuk, I. Glaser, “Geometries for optical implementations of the perfect shuffle,” in Optical Computing ’88, P. H. Chavel, J. W. Goodman, G. Roblin, eds., Proc. SPIE963, 270–282 (1988).
[CrossRef]

Smith, T.

T. Smith, “The changes in aberrations when the object and stop are moved,” Trans. Opt. Soc. 23, 139–153 (1921/1922).

Steward, G. C.

G. C. Steward, The Symmetrical Optical System (Cambridge U. Press, Cambridge, UK, 1928).

Stirk, C. W.

Velzel, C. H. F.

Walkup, J. F.

S.-H. Lin, T. F. Krile, J. F. Walkup, “2-D optical multistage interconnection networks,” in Digital Optical Computing, R. Arrathoon, ed., Proc. SPIE752, 209–216 (1987).
[CrossRef]

Walterson, B.

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

Walther, A.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).

A. Walther, “Some unsolved problems in image formation,” in Optics in Four Dimensions, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981), pp. 271–275.

Appl. Opt. (6)

J. Lightwave Technol. (1)

R. R. Michael, M. P. Christensen, Michael W. Haney, “Experimental evaluation of the 3-D optical shuffle module of the sliding Banyan architecture,” J. Lightwave Technol. 14, 1970–1978 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Lett. (3)

Trans. Opt. Soc. (1)

T. Smith, “The changes in aberrations when the object and stop are moved,” Trans. Opt. Soc. 23, 139–153 (1921/1922).

Other (10)

G. C. Steward, The Symmetrical Optical System (Cambridge U. Press, Cambridge, UK, 1928).

R. Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978).

M. W. Haney, M. P. Christensen, P. Milojkovic, J. Ekman, P. Chandramani, R. Rozier, F. Kiamilev, Y. Liu, M. Hibbs-Brenner, J. Nohava, E. Kalweit, S. Bounnak, T. Marta, B. Walterson, “ FAST-Net optical interconnection prototype demonstration program,” in Optoelectronic Interconnects V, R. T. Chen, J. P. Bristow, eds., Proc. SPIE3288, 194–203 (1998).
[CrossRef]

M. P. Christensen, P. Milojkovic, M. W. Haney, “Aberration analysis of beam-steering in the FAST-Net global interconnection system,” in Proceedings of the Lasers and Electro-Optics Society 2000 Annual Meeting (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2000), pp. 236–237.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, UK, 1970).

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).

A. Walther, “Some unsolved problems in image formation,” in Optics in Four Dimensions, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981), pp. 271–275.

M. W. Haney, M. P. Christensen, “Optical free-space sliding tandem Banyan architecture for self-routing switching networks,” in Digest of the International Conference on Optical Computing (Heriot-Watt University, Edinburgh, UK, 1994), pp. 249–250.

A. A. Sawchuk, I. Glaser, “Geometries for optical implementations of the perfect shuffle,” in Optical Computing ’88, P. H. Chavel, J. W. Goodman, G. Roblin, eds., Proc. SPIE963, 270–282 (1988).
[CrossRef]

S.-H. Lin, T. F. Krile, J. F. Walkup, “2-D optical multistage interconnection networks,” in Digital Optical Computing, R. Arrathoon, ed., Proc. SPIE752, 209–216 (1987).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic side view of a multichip global optical shuffle interconnection pattern. There is one lens over each chip. Each chip communicates with every chip in the receiving array. The system is “folded” along the dotted line by use of a mirror to facilitate packaging and alignment.11,12

Fig. 2
Fig. 2

(From Ref. 17.) Depiction of VCSEL beams as they pass through the on-axis interconnection system. The VCSEL planes are on the left, and the detector planes are on the right. (a) Telecentric interconnection system, (b) symmetric interconnection system, (c) symmetric interconnect system with auxiliary microbeam-deflection elements.

Fig. 3
Fig. 3

Off-axis imaging creates a plane of symmetry containing the axes of symmetry of the transmitting and receiving lenses.

Fig. 4
Fig. 4

Schematic depiction of a steered VCSEL cone describing parameters Δ (angle between the ray and the chief ray) and θ (angle of rotation about the chief ray).

Equations (54)

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

x=-γxWL-τxWM-WL,
y=-γyWM-τyWL-WM,
x=WL-WL,
y=WM-WM,
W(3)(L, M, L, M)=i+j+k+l=3αijklLiMjLkMl,
W(L, M, L, M)=W(L, M, L, M).
α1110=α1011,
α0102=α0201,
α0300=α0003,
α2001=α0120,
α3000=α0030,
α1020=α2010,
α2100=α0021,
α1002=α0210,
α0111=α1101,
α1200=α0012.
α1110=α2001=α2100.
α0102=α0300.
x=3α3000L2+3α1002M2+3α1200M2+3α1020L2-3α3000L2-3α1020L2-3α1002M2-3α1200M2,
y=L[6M(α1200-α0111)+6M(α0111-α1002)]+L[6M(α1002-α0111)+6M(α0111-α1200)].
LL0+N0Δ cos θ,
LL0-N0Δ cos θ,
MM0-N0Δ sin θ,
MM0+N0Δ sin θ,
x=N0Δ(12α3000L0cos θ-12α1002M0sin θ+12α1200M0sin θ-12α1020L0cos θ),
y=N0Δ(12α1002L0sin θ-12α1002M0cos θ-24α0111L0sin θ+12α1200L0sin θ+12α1200M0cos θ).
W(4)(L, M, L, M)=i+j+k+l=4αijklLiMjLkMl,
W(L, M, L, M)=W(L, M, L, M).
α0121=α2101,
α1210=α1012,
α0112=α1201,
α0211=α1102,
α2011=α1120,
α2110=α1021,
α1030=α3010,
α0130=α3001,
α0022=α2200,
α0220=α2002,
α0310=α1003,
α0013=α1300,
α0400=α0004,
α0040=α4000.
α2011=α2110=α0130=α0031,
α0211=α0112,
α0013=α0310.
α0211=α0013.
x=4(L-L)(α0040L2-α1030L2-3α2020LL+2α1030LL+α0040LL-α1030L2+α0040L2)+12M2(-α0022L-α1210L+α0220L+α1210L)+24MM(α1111L-α1111L+α0121L-α0121L)+12M2(-α1210L-α0220L+α0022L+α1210L),
y=4(M-M)(α0400 M2-α0103 M2-3α0202 MM+2α0103MM+α0400MM-α0103M2+α0400M2)+12L2(-α0022M-α0121M+α0220M+α0121M)+24LL(α1111M-α1111M+α1210M-α1210M)+12L2(-α0121M-α0220M+α0022M+α0121M).
LL0+N0Δ cos θ,
LL0-N0Δ cos θ,
MM0-N0Δ sin θ,
MM0+N0Δ sin θ,
x=8N03Δ3cos θ(-3α0220sin2 θ-3α0022sin2 θ+6α1210sin2 θ-6α1111sin2 θ+6α0121sin2 θ+3α2020cos2 θ-4α1030cos2 θ+α0040cos2 θ)+N0Δ[24M02cos θ(α0220+2α0120-2α1111+α0022-2α1210)+48L0M0sin θ(α0022-α0220)+24L02cos θ(α0040-α2020)],
y=8N03Δ3sin θ(-α0400sin2 θ-3α0202sin2 θ+4α0103sin2 θ+3α0220cos2 θ-6α0121cos2 θ-6α1210cos2 θ+6α1111cos2 θ+3α0022cos2 θ)+N0Δ[24M02sin θ(α0400-α0202)+48L0M0cos θ(α0022+α0220)+24L02sin θ(α0022+α0220-2α1111-2α0121+2α1210)].

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