Abstract

A newly reported method of making three-dimensional microstructures or photonic crystals by holographic lithography has some obvious advantages over other techniques with the same purpose. A systematic and comprehensive analysis of interference of four noncoplanar beams (IFNB) is provided. It shows that all 14 Bravais lattices can be formed by means of IFNB and gives explicit relationships between each lattice and the corresponding recording geometry. The concept of pattern contrast is extended to the case of IFNB, and it is indicated that a uniform contrast for each interference term can be obtained by properly choosing the beam ratio and polarization. A calculation algorithm is then developed to optimize the direction of polarization of each beam to ensure maximum uniform contrast. These results, verified by computer simulations, may lay a theoretical foundation for fabrication of photonic crystals with the approach of IFNB.

© 2002 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).
  2. C. M. West, Holographic Interferometry (Wiley, New York, 1979).
  3. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature (London) 404, 53–56 (2000).
    [CrossRef]
  4. J. D. Joannopoulos, P. R. Villeneuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
    [CrossRef]
  5. T. F. Krauss, R. M. De La Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
    [CrossRef]
  6. C. C. Cheng, A. Scherer, R. Tyan, Y. Fainman, G. Witzgall, E. Yablonovitch, “New fabrication techniques for high quality photonic crystals,” J. Vac. Sci. Technol. B 15, 2764–2767 (1997).
    [CrossRef]
  7. A. van Blaaderen, R. Ruel, P. Wiltzius, “Template-directed colloidal crystallization,” Nature (London) 385, 321–324 (1997).
    [CrossRef]
  8. M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, M. Stuke, “Laser rapid prototyping of photonic band-gap microstructures,” Science 275, 1284–1286 (1997).
    [CrossRef] [PubMed]
  9. B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
    [CrossRef]
  10. L. Z. Cai, X. L. Yang, Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. 26, 1858–1860 (2001).
    [CrossRef]

2001

2000

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature (London) 404, 53–56 (2000).
[CrossRef]

1999

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

T. F. Krauss, R. M. De La Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[CrossRef]

1997

C. C. Cheng, A. Scherer, R. Tyan, Y. Fainman, G. Witzgall, E. Yablonovitch, “New fabrication techniques for high quality photonic crystals,” J. Vac. Sci. Technol. B 15, 2764–2767 (1997).
[CrossRef]

A. van Blaaderen, R. Ruel, P. Wiltzius, “Template-directed colloidal crystallization,” Nature (London) 385, 321–324 (1997).
[CrossRef]

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, M. Stuke, “Laser rapid prototyping of photonic band-gap microstructures,” Science 275, 1284–1286 (1997).
[CrossRef] [PubMed]

J. D. Joannopoulos, P. R. Villeneuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Ananthavel, S. P.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Barlow, S.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Cai, L. Z.

Campbell, M.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature (London) 404, 53–56 (2000).
[CrossRef]

Cheng, C. C.

C. C. Cheng, A. Scherer, R. Tyan, Y. Fainman, G. Witzgall, E. Yablonovitch, “New fabrication techniques for high quality photonic crystals,” J. Vac. Sci. Technol. B 15, 2764–2767 (1997).
[CrossRef]

Cumpston, B. H.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

De La Rue, R. M.

T. F. Krauss, R. M. De La Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[CrossRef]

Denning, R. G.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature (London) 404, 53–56 (2000).
[CrossRef]

Dyer, D. L.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Ehrlich, J. E.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Erskine, L. L.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Fainman, Y.

C. C. Cheng, A. Scherer, R. Tyan, Y. Fainman, G. Witzgall, E. Yablonovitch, “New fabrication techniques for high quality photonic crystals,” J. Vac. Sci. Technol. B 15, 2764–2767 (1997).
[CrossRef]

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Harrison, M. T.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature (London) 404, 53–56 (2000).
[CrossRef]

Heikal, A. A.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Krauss, T. F.

T. F. Krauss, R. M. De La Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[CrossRef]

Kuebler, S. M.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Lee, I. Y. S.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Lehmann, O.

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, M. Stuke, “Laser rapid prototyping of photonic band-gap microstructures,” Science 275, 1284–1286 (1997).
[CrossRef] [PubMed]

Marder, S. R.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

McCord-Maughon, D.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Müller, K.

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, M. Stuke, “Laser rapid prototyping of photonic band-gap microstructures,” Science 275, 1284–1286 (1997).
[CrossRef] [PubMed]

Perry, J. W.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Qin, J.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Röckel, H.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Ruel, R.

A. van Blaaderen, R. Ruel, P. Wiltzius, “Template-directed colloidal crystallization,” Nature (London) 385, 321–324 (1997).
[CrossRef]

Rumi, M.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Scherer, A.

C. C. Cheng, A. Scherer, R. Tyan, Y. Fainman, G. Witzgall, E. Yablonovitch, “New fabrication techniques for high quality photonic crystals,” J. Vac. Sci. Technol. B 15, 2764–2767 (1997).
[CrossRef]

Sharp, D. N.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature (London) 404, 53–56 (2000).
[CrossRef]

Stuke, M.

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, M. Stuke, “Laser rapid prototyping of photonic band-gap microstructures,” Science 275, 1284–1286 (1997).
[CrossRef] [PubMed]

Turberfield, A. J.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature (London) 404, 53–56 (2000).
[CrossRef]

Tyan, R.

C. C. Cheng, A. Scherer, R. Tyan, Y. Fainman, G. Witzgall, E. Yablonovitch, “New fabrication techniques for high quality photonic crystals,” J. Vac. Sci. Technol. B 15, 2764–2767 (1997).
[CrossRef]

van Blaaderen, A.

A. van Blaaderen, R. Ruel, P. Wiltzius, “Template-directed colloidal crystallization,” Nature (London) 385, 321–324 (1997).
[CrossRef]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Wang, Y. R.

Wanke, M. C.

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, M. Stuke, “Laser rapid prototyping of photonic band-gap microstructures,” Science 275, 1284–1286 (1997).
[CrossRef] [PubMed]

Wen, Q.

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, M. Stuke, “Laser rapid prototyping of photonic band-gap microstructures,” Science 275, 1284–1286 (1997).
[CrossRef] [PubMed]

West, C. M.

C. M. West, Holographic Interferometry (Wiley, New York, 1979).

Wiltzius, P.

A. van Blaaderen, R. Ruel, P. Wiltzius, “Template-directed colloidal crystallization,” Nature (London) 385, 321–324 (1997).
[CrossRef]

Witzgall, G.

C. C. Cheng, A. Scherer, R. Tyan, Y. Fainman, G. Witzgall, E. Yablonovitch, “New fabrication techniques for high quality photonic crystals,” J. Vac. Sci. Technol. B 15, 2764–2767 (1997).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Wu, X. L.

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

Yablonovitch, E.

C. C. Cheng, A. Scherer, R. Tyan, Y. Fainman, G. Witzgall, E. Yablonovitch, “New fabrication techniques for high quality photonic crystals,” J. Vac. Sci. Technol. B 15, 2764–2767 (1997).
[CrossRef]

Yang, X. L.

J. Vac. Sci. Technol. B

C. C. Cheng, A. Scherer, R. Tyan, Y. Fainman, G. Witzgall, E. Yablonovitch, “New fabrication techniques for high quality photonic crystals,” J. Vac. Sci. Technol. B 15, 2764–2767 (1997).
[CrossRef]

Nature (London)

A. van Blaaderen, R. Ruel, P. Wiltzius, “Template-directed colloidal crystallization,” Nature (London) 385, 321–324 (1997).
[CrossRef]

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X. L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature (London) 398, 51–54 (1999).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature (London) 404, 53–56 (2000).
[CrossRef]

J. D. Joannopoulos, P. R. Villeneuve, S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Opt. Lett.

Prog. Quantum Electron.

T. F. Krauss, R. M. De La Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[CrossRef]

Science

M. C. Wanke, O. Lehmann, K. Müller, Q. Wen, M. Stuke, “Laser rapid prototyping of photonic band-gap microstructures,” Science 275, 1284–1286 (1997).
[CrossRef] [PubMed]

Other

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

C. M. West, Holographic Interferometry (Wiley, New York, 1979).

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Figures (2)

Fig. 1
Fig. 1

Basis vectors of (a) triclinic, (b) body-centered orthorhombic, (c) face-centered orthorhombic, and (d) base-centered monoclinic lattices.

Fig. 2
Fig. 2

Computer simulations of four Bravais lattices formed by interference of four noncoplanar beams: (a) face-centered cubic lattice, (b) body-centered cubic lattice, (c) hexagonal lattice (c=3a/2), and (d) simple cubic lattice.

Tables (1)

Tables Icon

Table 1 Calculation Results of Polarization Optimization for Three Lattices

Equations (83)

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

Ej=Ejexp(iKjr)ej,j=1,, 4,
Kj=2πλKj0=2πλ (lj, mj, nj)
ej=(lj, mj, nj)
I=j=14Ej2+2i<jEiEjeijcos[(Ki-Kj)r+ij],
Vij=2EiEjeijj=14Ej2,i<j.
e12 e34=e13 e24=e14 e23,
E2E1=e13e23,E3E1=e12e23,E4E1=e12e24.
V=Vij=2e12 e13 e23e122+e132+e232+e132 e232 /e342
I=j=14Ej21+Vi<jcos[(Ki-Kj)r].
cos(K1-K2)r=cos(K1-K3)r=cos(K1-K4)  r=1.
l12 x+m12 y+n12 z=p1λ,
l13 x+m13 y+n13 z=p2λ,
l14 x+m14 y+n14 z=p3λ,
p1, p2, p3=0, ±1,±2,,
x=1Δp1m12n12p2m13n13p3m14n14,y=1Δl12p1n12l13p2n13l14p3n14,
z=1Δl12m12p1l13m13p2l14m14p3,
Δ=l12m12n12l13m13n13l14m14n14.
a1=λΔm13n13m14n14, n13l13n14l14, l13m13l14m14,
a2=λΔm14n14m12n12, n14l14n12l12, l14m14l12m12,
a3=λΔm12n12m13n13, n12l12n13l13, l12m12l13m13.
a1=(a, 0, 0),a2=(b cos γ, b sin γ, 0),
a3=(c cos β, c(cos α-cos β cos γ)/sin γ, cδ),
δ=[sin2 β-(cos α-cos β cos γ)2/sin2 γ]1/2.
Δ=λa-λ cos γa sin γλ(cos α cos γ-cos β)aδ sin2 γ0λb sin γλ(cos β cos γ-cos α)bδ sin2 γ00λcδ.
K1=K(l1, m1, n1),
K2=Kl1-λa, m1+λ cos γa sin γ, n1+λ(cos β-cos α cos γ)aδ sin2 γ,
K3=Kl1, m1-λb sin γ, n1+λ(cos α-cos β cos γ)bδ sin2 γ,
K4=Kl1, m1, n1-λcδ,
l1=λ(bc sin2 α+ac cos γ sin2 β+ab cos β sin2 γ)2abcδ2sin2 γ,
m1=λ[b(cos α-cos β cos γ)+c sin2 β]2bcδ2sin γ,
n1=λ2cδ,
λ=2abcδ2[a2b2+b2c2sin4 α csc4 γ+a2c2sin4 β csc4 γ+2abc csc2 γ(a cos α sin2 β+b cos β sin2 α+c cos γ sin2 α sin2 β csc2 γ)]-1/2.
K1=πc+a cos βac sin2 β, 1b, 1c sin β,
K2=πa cos β+c cos 2βac sin2 β, 1b, a+2c cos βac sin β,
K3=πc+a cos βac sin2 β, -1b, 1c sin β,
K4=πc+a cos βac sin2 β, 1b, -1c sin β;
λ=2abc(sin2 β)(a2b2+b2c2+a2c2sin4 β+2ab2c cos β)-1/2.
K1=π1a, 1b, 1c,K2=π-1a, 1b, 1c,
K3=π1a, -1b, 1c,K4=π1a, 1b, -1c;
λ=2abc(a2b2+b2c2+a2c2)-1/2.
K1=πa (1, 1, 1),K2=πa (-1, 1, 1),
K3=πa (1, -1, 1),K4=πa (1, 1, -1);
λ=2a/3.
K1=πacot2α2, cotα2, cotα2(1+2 cos α)-1/2,
K2=πa-2+cot2α2, cotα2+2 cot α,cotα2+2 cot α(1+2 cos α)-1/2,
K3=πacot2α2, -tanα2,cotα2+2 cot α×(1+2 cos α)-1/2,
K4=πacot2α2, cotα2,-cotα2(1+2 cos α)-1/2;
λ=2(3)-1/2a(1+2 cos α)1/2cot-2α2.
K1=2π13a, 13a, 12c,K2=2π-23a, 0, 12c,
K3=2π13a, -13a, 12c,
K4=2π13a, 13a, -12c;
λ=6ac(9a2+16c2)-1/2.
a1=12(-a,b, c),a2=12(a, -b, c),a3=12(a, b, -c).
K1=π1a, 1b, 1c, K2=π1a, -1b, -1c,
K3=π-1a, 1b, -1c,K4=π-1a, -1b, 1c;
λ=2abc(a2b2+b2c2+c2a2)-1/2.
K1=πa (1, 1, 1),K2=πa (1, -1, -1),
K3=πa (-1, 1, -1),K4=πa (-1, -1, 1),
a1=12 (a, b, 0),a2=12 (0, b, c),a3=12 (a, 0, c).
K1=πa1a2+1b2+1c2,b1a2+1b2+1c2,c1a2+1b2+1c2,
K2=πa-1a2+1b2+1c2,b1a2-1b2+1c2,c1a2+1b2+3c2,
K3=πa3a2+1b2+1c2,b1a2-1b2+1c2,c1a2+1b2-1c2,
K4=πa-1a2+1b2+1c2,b1a2+3b2+1c2,c1a2+1b2-1c2;
λ=21a2+1b2+1c2-1(a2+b2+c2)-1/2.
K1=πa (3, 3, 3),K2=πa (1, 1, 5),
K3=πa (5, 1, 1), K=πa (1, 5, 1);
λ=2a33.
a1=(a, 0, 0),a2=a2, b2, 0,
a3=(c cos β, 0, c sin β),
K1=π1a sin2 β+3ab2+cos βc sin2 β, 2b, 1c sin β,
K2=πcos 2βa sin2 β+3ab2+cos βc sin2 β, 4b, a+2c cos βac sin β,
K3=π1a sin2 β+3ab2+cos βc sin2 β, -2b, 1c sin β,
K4=π1a sin2 β+3ab2+cos βc sin2 β, 2b, -1c sin β;
 
λ=24b2+1c2sin2 β+1a sin2 β+3ab2+cos βc sin2 β2-1/2.
K1=π1a+3ab2, 2b, 1c,K2=π-1a+3ab2, 4b, 1c,
K3=π1a+3ab2, -2b, 1c,
K4=π1a+3ab2, 2b, -1c;
λ=21a2+10b2+1c2+9a2b4-1/2.
f(l1, m1, n1, l2, m2, n2, l3, m3, n3, l4, m4, n4, λ1, , λ10)
=V+j=14λj(lj2+mj2+nj2-1)+j=14λj+4(ljlj+mjmj+njnj)+λ9(e12 e34-e13 e24)+λ10(e12 e34-e14 e23),
Vabs=IM-ImIM+Im,
Vabs=4V-1+2,

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