Abstract

Four distinct conditions for primitive-lattice-vector-direction equal contrasts in four-beam interference are introduced and described. By maximizing the absolute contrast subject to an equal contrast condition, lithographically useful interference patterns are found. Each condition is described in terms of the corresponding constraints on the plane wave wave vectors, polarizations, and intensities. The resulting locations of global intensity maxima, minima, and saddle points are presented. Subordinate conditions for unity absolute contrast are also developed. Three lattices are treated for each condition: simple cubic, face-centered cubic, and body-centered cubic.

© 2009 Optical Society of America

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2008

A. J. Danner, B. Wang, S.-J. Chua, and J.-K. Hwang, “Fabrication of efficient light-emitting diodes with a self-assembled photonic crystal array of polystyrene nanoparticles,” IEEE Photon. Technol. Lett. 20, 48-50 (2008).
[CrossRef]

J. L. Stay and T. K. Gaylord, “Three-beam-interference lithography: contrast and crystallography,” Appl. Opt. 47, 3221-3230 (2008).
[CrossRef] [PubMed]

J. L. Stay and T. K. Gaylord, “Contrast in four-beam-interference lithography,” Opt. Lett. 33, 1434-1436(2008).
[CrossRef] [PubMed]

2007

A. P. Hynninen, J. H. J. Thijssen, E. C. M. Vermolen, M. Dijkstra, and A. Van Blaaderen, “Self-assembly route for photonic crystals with a bandgap in the visible region,” Nat. Mater. 6, 202-205 (2007).
[CrossRef] [PubMed]

R. C. Rumpf, A. Mehta, P. Srinivasan, and E. G. Johnson, “Design and optimization of space-variant photonic crystal filters,” Appl. Opt. 46, 5755-5761 (2007).
[CrossRef] [PubMed]

A. Mehta, R. Rumpf, Z. Roth, and E. G. Johnson, “Simplified fabrication process of 3-D photonic crystal optical transmission filter,” Proc. SPIE 6462, 64621D (2007).
[CrossRef]

2006

A. Locatelli, M. Conforti, D. Modotto, and C. De Angelis, “Discrete negative refraction in photonic crystal waveguide arrays,” Opt. Lett. 31, 1343-1345 (2006).
[CrossRef] [PubMed]

B. Momeni, J. Huang, M. Soltani, M. Askari, S. Mohammadi, M. Rakhshandehroo, and A. Adibi, “Compact wavelength demultiplexing using focusing negative index photonic crystal superprisms,” Opt. Express 14, 2413-2422(2006).
[CrossRef] [PubMed]

S. Cabrini, L. Businaro, M. Prasciolu, A. Carpentiro, D. Gerace, M. Galli, L. C. Andreani, F. Riboli, L. Pavesi, and E. Di Fabrizio, “Focused ion beam fabrication of one-dimensional photonic crystals on Si3N4/SiO2 channel waveguides,” J. Opt. A: Pure Appl. Opt. 8, 550-553 (2006).
[CrossRef]

B. Momeni and A. Adibi, “Demultiplexers harness photonic-crystal dispersion properties,” Laser Focus World 42, 125-128 (2006).

2005

T. Prasad, R. Rengarajan, D. M. Mittleman, and V. L. Colvin, “Advanced photonic crystal architectures from colloidal self-assembly techniques,” Opt. Mater. 27, 1250-1254 (2005).
[CrossRef]

T. Kamalakis and T. Sphicopoulos, “Numerical study of the implications of size nonuniformities in the performance of photonic crystal couplers using coupled mode theory,” IEEE J. Quantum Electron. 41, 863-871 (2005).
[CrossRef]

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76-84 (2005).
[CrossRef]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

S. Chakravarty, J. Topol'ancik, P. Bhattacharya, S. Chakrabarti, Y. Kang, and M. E. Meyerhoff, “Ion detection with photonic crystal microcavities,” Opt. Lett. 30, 2578-2580(2005).
[CrossRef] [PubMed]

J. Witzens, T. Baehr-Jones, and A. Scherer, “Hybrid superprism with low insertion losses and suppressed cross-talk,” Phys. Rev. E 71, 026604 (2005).
[CrossRef]

T. Matsumoto, S. Fujita, and T. Baba, “Wavelength demultiplexer consisting of photonic crystal superprism and superlens,” Opt. Express 13, 10768-10783 (2005).
[CrossRef] [PubMed]

2004

2003

2002

C. Caloz, A. K. Skrivervik, and F. E. Gardiol, “An efficient method to determine Green's functions of a two-dimensional photonic crystal excited by a line source--the phased-array method,” IEEE Trans. Microwave Theory Tech. 50, 1380-1391(2002).
[CrossRef]

A. Chelnokov, S. David, K. Wang, F. Marty, and J. M. Lourtioz, “Fabrication of 2-D and 3-D silicon photonic crystals by deep etching,” IEEE J. Sel. Top. Quantum Electron. 8, 919-927(2002).
[CrossRef]

L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. 27, 900-902 (2002).
[CrossRef]

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. A 19, 2238-2244(2002).
[CrossRef]

2001

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. 26, 1858-1860 (2001).
[CrossRef]

A. Mekis and J. D. Joannopoulos, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,” J. Lightwave Technol. 19, 861-865 (2001).
[CrossRef]

S. Kawata, H.-B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412, 697-698(2001).
[CrossRef] [PubMed]

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Muller, U. Gosele, J. Mlynek, and V. Sandoghdar, “Direct spectroscopy of a deep two-dimensional photonic crystal microresonator,” Phys. Rev. B 64, 233102-1-233102-4 (2001).
[CrossRef]

2000

T. Kim and C. Seo, “A novel photonic bandgap structure for low-pass filter of wide stopband,” IEEE Microwave Guid. Wave Lett. 10, 13-15 (2000).
[CrossRef]

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Properties of the slab modes in photonic crystal optical waveguides,” J. Lightwave Technol. 18, 1554-1564 (2000).
[CrossRef]

A. Birner, A. P. Li, F. Mueller, U. Goesele, P. Kramper, V. Sandoghdar, J. Mlynek, K. Busch, and V. Lehmann, “Transmission of a microcavity structure in a two-dimensional photonic crystal based on macroporous silicon,” Mater. Sci. Semicond. Process. 3, 487-491 (2000).
[CrossRef]

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

M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, “Design and fabrication of silicon photonic crystal optical waveguides,” J. Lightwave Technol. 18, 1402-1411 (2000).
[CrossRef]

1999

S. Rowson, A. Chelnokov, and J. M. Lourtioz, “Two-dimensional photonic crystals in macroporous silicon: from mid-infrared (10 μm) to telecommunication wavelengths (1.3-1.5 μm),” J. Lightwave Technol. 17, 1989-1995 (1999).
[CrossRef]

T. Zijlstra, E. Van Der Drift, M. J. A. De Dood, E. Snoeks, and A. Polman, “Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,” J. Vac. Sci. Technol. B 17, 2734-2739 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals: toward microscale lightwave circuits,” J. Lightwave Technol. 17, 2032-2038 (1999).
[CrossRef]

M. M. Beaky, J. B. Burk, H. O. Everitt, M. A. Haider, and S. Venakides, “Two-dimensional photonic crystal Fabry-Perot resonators with lossy dielectrics,” IEEE Trans. Microwave Theory Tech. 47, 2085-2091 (1999).
[CrossRef]

1997

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Fabrication of a 2D photonic bandgap by a holographic method,” Electron. Lett. 33, 425-426 (1997).
[CrossRef]

1995

U. Gruning, V. Lehmann, and C. M. Engelhardt, “Two-dimensional infrared photonic band gap structure based on porous silicon,” Appl. Phys. Lett. 66, 3254-3256 (1995).
[CrossRef]

Adibi, A.

B. Momeni, J. Huang, M. Soltani, M. Askari, S. Mohammadi, M. Rakhshandehroo, and A. Adibi, “Compact wavelength demultiplexing using focusing negative index photonic crystal superprisms,” Opt. Express 14, 2413-2422(2006).
[CrossRef] [PubMed]

B. Momeni and A. Adibi, “Demultiplexers harness photonic-crystal dispersion properties,” Laser Focus World 42, 125-128 (2006).

A. Jafarpour, E. Chow, C. M. Reinke, J. Huang, A. Adibi, A. Grot, L. W. Mirkarimi, G. Girolami, R. K. Lee, and Y. Xu, “Large-bandwidth ultra-low-loss guiding in bi-periodic photonic crystal waveguides,” Appl. Phys. B 79, 409-414(2004).
[CrossRef]

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Properties of the slab modes in photonic crystal optical waveguides,” J. Lightwave Technol. 18, 1554-1564 (2000).
[CrossRef]

Agio, M.

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Muller, U. Gosele, J. Mlynek, and V. Sandoghdar, “Direct spectroscopy of a deep two-dimensional photonic crystal microresonator,” Phys. Rev. B 64, 233102-1-233102-4 (2001).
[CrossRef]

Andreani, L. C.

S. Cabrini, L. Businaro, M. Prasciolu, A. Carpentiro, D. Gerace, M. Galli, L. C. Andreani, F. Riboli, L. Pavesi, and E. Di Fabrizio, “Focused ion beam fabrication of one-dimensional photonic crystals on Si3N4/SiO2 channel waveguides,” J. Opt. A: Pure Appl. Opt. 8, 550-553 (2006).
[CrossRef]

Asakawa, K.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE J. Quantum Electron. 41, 76-84 (2005).
[CrossRef]

Askari, M.

Assanto, G.

Baba, T.

Baehr-Jones, T.

J. Witzens, T. Baehr-Jones, and A. Scherer, “Hybrid superprism with low insertion losses and suppressed cross-talk,” Phys. Rev. E 71, 026604 (2005).
[CrossRef]

Beaky, M. M.

M. M. Beaky, J. B. Burk, H. O. Everitt, M. A. Haider, and S. Venakides, “Two-dimensional photonic crystal Fabry-Perot resonators with lossy dielectrics,” IEEE Trans. Microwave Theory Tech. 47, 2085-2091 (1999).
[CrossRef]

Berger, V.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Fabrication of a 2D photonic bandgap by a holographic method,” Electron. Lett. 33, 425-426 (1997).
[CrossRef]

Bhattacharya, P.

Birner, A.

P. Kramper, M. Kafesaki, C. M. Soukoulis, A. Birner, F. Muller, U. Gosele, R. B. Wehrspohn, J. Mlynek, and V. Sandoghdar, “Near-field visualization of light confinement in a photonic crystal microresonator,” Opt. Lett. 29, 174-176 (2004).
[CrossRef] [PubMed]

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Muller, U. Gosele, J. Mlynek, and V. Sandoghdar, “Direct spectroscopy of a deep two-dimensional photonic crystal microresonator,” Phys. Rev. B 64, 233102-1-233102-4 (2001).
[CrossRef]

A. Birner, A. P. Li, F. Mueller, U. Goesele, P. Kramper, V. Sandoghdar, J. Mlynek, K. Busch, and V. Lehmann, “Transmission of a microcavity structure in a two-dimensional photonic crystal based on macroporous silicon,” Mater. Sci. Semicond. Process. 3, 487-491 (2000).
[CrossRef]

Boag, A.

Borel, P. I.

Burk, J. B.

M. M. Beaky, J. B. Burk, H. O. Everitt, M. A. Haider, and S. Venakides, “Two-dimensional photonic crystal Fabry-Perot resonators with lossy dielectrics,” IEEE Trans. Microwave Theory Tech. 47, 2085-2091 (1999).
[CrossRef]

Busch, K.

A. Birner, A. P. Li, F. Mueller, U. Goesele, P. Kramper, V. Sandoghdar, J. Mlynek, K. Busch, and V. Lehmann, “Transmission of a microcavity structure in a two-dimensional photonic crystal based on macroporous silicon,” Mater. Sci. Semicond. Process. 3, 487-491 (2000).
[CrossRef]

Businaro, L.

S. Cabrini, L. Businaro, M. Prasciolu, A. Carpentiro, D. Gerace, M. Galli, L. C. Andreani, F. Riboli, L. Pavesi, and E. Di Fabrizio, “Focused ion beam fabrication of one-dimensional photonic crystals on Si3N4/SiO2 channel waveguides,” J. Opt. A: Pure Appl. Opt. 8, 550-553 (2006).
[CrossRef]

Cabrini, S.

S. Cabrini, L. Businaro, M. Prasciolu, A. Carpentiro, D. Gerace, M. Galli, L. C. Andreani, F. Riboli, L. Pavesi, and E. Di Fabrizio, “Focused ion beam fabrication of one-dimensional photonic crystals on Si3N4/SiO2 channel waveguides,” J. Opt. A: Pure Appl. Opt. 8, 550-553 (2006).
[CrossRef]

Cai, L. Z.

Caloz, C.

C. Caloz, A. K. Skrivervik, and F. E. Gardiol, “An efficient method to determine Green's functions of a two-dimensional photonic crystal excited by a line source--the phased-array method,” IEEE Trans. Microwave Theory Tech. 50, 1380-1391(2002).
[CrossRef]

Campbell, M.

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

Carpentiro, A.

S. Cabrini, L. Businaro, M. Prasciolu, A. Carpentiro, D. Gerace, M. Galli, L. C. Andreani, F. Riboli, L. Pavesi, and E. Di Fabrizio, “Focused ion beam fabrication of one-dimensional photonic crystals on Si3N4/SiO2 channel waveguides,” J. Opt. A: Pure Appl. Opt. 8, 550-553 (2006).
[CrossRef]

Chakrabarti, S.

Chakravarty, S.

Chelnokov, A.

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S. Cabrini, L. Businaro, M. Prasciolu, A. Carpentiro, D. Gerace, M. Galli, L. C. Andreani, F. Riboli, L. Pavesi, and E. Di Fabrizio, “Focused ion beam fabrication of one-dimensional photonic crystals on Si3N4/SiO2 channel waveguides,” J. Opt. A: Pure Appl. Opt. 8, 550-553 (2006).
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Reinke, C. M.

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A. Mehta, R. Rumpf, Z. Roth, and E. G. Johnson, “Simplified fabrication process of 3-D photonic crystal optical transmission filter,” Proc. SPIE 6462, 64621D (2007).
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[CrossRef] [PubMed]

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Muller, U. Gosele, J. Mlynek, and V. Sandoghdar, “Direct spectroscopy of a deep two-dimensional photonic crystal microresonator,” Phys. Rev. B 64, 233102-1-233102-4 (2001).
[CrossRef]

A. Birner, A. P. Li, F. Mueller, U. Goesele, P. Kramper, V. Sandoghdar, J. Mlynek, K. Busch, and V. Lehmann, “Transmission of a microcavity structure in a two-dimensional photonic crystal based on macroporous silicon,” Mater. Sci. Semicond. Process. 3, 487-491 (2000).
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Scherer, A.

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T. Kim and C. Seo, “A novel photonic bandgap structure for low-pass filter of wide stopband,” IEEE Microwave Guid. Wave Lett. 10, 13-15 (2000).
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M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53-56 (2000).
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T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. 87, 151112 (2005).
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Figures (8)

Fig. 1
Fig. 1

Orientation of polarizations for the subordinate condition for unity absolute contrast [or any combination of the polarization vectors ( e ^ i ), where one, multiple, or all are inverted ( e ^ i )] for C 4 ( 6 ) , where V 4 ( 6 ) = 1 / 3 . α = cos 1 ( 1 / 3 ) 109.47 ° .

Fig. 2
Fig. 2

Intensity contours for interference patterns with body-centered cubic (BCC), face-centered cubic (FCC), and simple cubic periodicity that satisfy the C + 4 ( 6 ) (upper) and C 4 ( 6 ) (lower) conditions for primitive-lattice-vector-direction equal contrasts.

Fig. 3
Fig. 3

Orientation of polarizations for the subordinate conditions for unity absolute contrast [or any combination of the polarization vectors ( e ^ i ), where one, multiple, or all are inverted ( e ^ i )] for (a)  C + 4 ( 5 ) , where V 4 ( 5 ) = 1 / 3 and (b)  C 4 ( 5 ) , where V 4 ( 5 ) = 1 / 3 and α = cos 1 ( 1 / 3 ) 109.47 ° .

Fig. 4
Fig. 4

Intensity contours for interference patterns with body-centered cubic (BCC), face-centered cubic (FCC), and simple cubic periodicity that satisfy the C + 4 ( 5 ) (upper) and C 4 ( 5 ) (lower) conditions for primitive-lattice-vector-direction equal contrasts.

Fig. 5
Fig. 5

Orientation of polarizations for the subordinate conditions for unity absolute contrast [or any combination of the polarization vectors ( e ^ i ), where one, multiple, or all are inverted ( e ^ i )] for (a)  C + 4 ( 4 ) , where V 4 ( 4 ) = 2 / 5 and (b)  C 4 ( 4 ) where V 4 ( 4 ) = 1 / 4 and α = cos 1 ( 1 / 3 ) 125.26 ° .

Fig. 6
Fig. 6

Intensity contours for interference patterns with body-centered cubic (BCC), face-centered cubic (FCC), and simple cubic periodicity that satisfy the C + 4 ( 4 ) (upper) and C 4 ( 4 ) (lower) conditions for primitive-lattice-vector-direction equal contrasts.

Fig. 7
Fig. 7

Orientation of polarizations for the subordinate conditions for unity absolute contrast [or any combination of the polarization vectors ( e ^ i ), where one, multiple, or all are inverted ( e ^ i )] for C 4 ( 3 ) , where V 4 ( 3 ) = ± 1 / 3 and α = cos 1 ( 3 / 3 ) 54.74 ° .

Fig. 8
Fig. 8

Intensity contours for interference patterns with body-centered cubic (BCC), face-centered cubic (FCC), and simple cubic periodicity that satisfy the C 4 ( 3 ) condition for primitive-lattice- vector-direction equal contrasts.

Tables (8)

Tables Icon

Table 1 Primitive Basis Vectors and their Corresponding Recording Wave Vectors

Tables Icon

Table 2 Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for C 4 ( 6 )

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Table 3 Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for C + 4 ( 6 )

Tables Icon

Table 4 Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for C 4 ( 5 )

Tables Icon

Table 5 Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for C + 4 ( 5 )

Tables Icon

Table 6 Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for C 4 ( 4 )

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Table 7 Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for C + 4 ( 4 )

Tables Icon

Table 8 Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for C + 4 ( 3 )

Equations (47)

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

I T ( r ) = I 0 ( 1 + i = 1 N j > i N V i j cos ( G j i · r + ϕ i ϕ j ) ) ,
I 0 = 1 2 k = 1 N E k 2 , V i j = E i E j e i j I 0 , e i j = e ^ i · e ^ j , G j i = k j k i ,
A = 2 π b × c a · b × c , B = 2 π c × a a · b × c , C = 2 π a × b a · b × c .
P = 1 2 | A | 2 ( B × C ) + | B | 2 ( C × A ) + | C | 2 ( A × B ) A · B × C .
k 1 = P , k 2 = P A , k 3 = P B , k 4 = P C .
I T = I 0 [ 1 + V 12 cos ( G 21 · r ) + V 13 cos ( G 31 · r ) + V 14 cos ( G 41 · r ) + V 23 cos ( G 32 · r ) + V 24 cos ( G 42 · r ) + V 34 cos ( G 43 · r ) ] .
V abs = I max I min I max + I min .
V 4 ( 6 ) = V 12 = V 13 = V 14 = V 23 = V 24 = V 34 [ Condition     C 4 ( 6 ) ] .
e 12 e 34 = e 13 e 24 = e 14 e 23 ,
E 2 E 1 = e 13 e 23 , E 3 E 1 = e 12 e 23 , E 4 E 1 = e 12 e 24 .
V 4 ( 6 ) = 2 e 12 e 13 e 23 e 12 2 + e 13 2 + e 23 2 + e 13 2 e 23 2 / e 34 2 .
V abs = | 4 1 / V 4 ( 6 ) + 2 | .
e i j = e 12 = e 13 = e 14 = e 23 = e 24 = e 34 = 1 / 3
e i j = e 12 = e 13 = e 14 = e 23 = e 24 = e 34 = 1
V 4 ( 5 ) = V 12 = V 13 = V 14 = V 23 = V 24 , V 34 = 0 [ Condition     C 4 ( 5 ) ] .
e 13 e 24 = e 14 e 23 , e 34 = 0
E 2 E 1 = e 13 e 23 , E 3 E 1 = e 12 e 23 , E 4 E 1 = e 12 e 24 .
V 4 ( 5 ) = 2 e 12 e 13 e 23 e 12 2 + e 13 2 + e 23 2 + e 12 2 e 23 2 / e 24 2 .
V abs = | 4 1 / V 4 ( 5 ) + 1 | .
e 12 = 1 / 3 , e 13 = e 14 = e 23 = e 24 = 1 / 6
e 12 = 1 , e 13 = e 14 = e 23 = e 24 = 1 / 2
V 4 ( 4 ) = V 12 = V 13 = V 14 = V 23 , V 24 = V 34 = 0 [ Condition C 4 ( 4 ) ] .
e 24 = e 34 = 0
E 2 E 1 = e 13 e 23 , E 3 E 1 = e 12 e 23 , E 4 E 1 = e 12 e 13 e 14 e 23 .
V 4 ( 4 ) = 2 e 12 e 13 e 23 e 12 2 + e 13 2 + e 23 2 + e 12 2 e 13 2 / e 14 2 .
V abs = | 13 4 / V 4 ( 4 ) + 3 | .
e 12 = e 13 = 1 / 6 , e 14 = 1 / 3 , e 23 = 1 / 2
e 12 = e 13 = e 14 = 1 / 2 , e 23 = 1
V 4 ( 3 ) = V 12 = V 13 = V 14 , V 23 = V 24 = V 34 = 0 [ Condition C 4 ( 3 ) ] .
e 23 = e 24 = e 34 = 0
E 3 E 2 = e 12 e 13 , E 4 E 2 = e 12 e 14 .
V 4 ( 3 ) = 2 E 1 E 2 e 12 E 1 2 + E 2 2 ( 1 + e 12 2 / e 13 2 + e 12 2 / e 14 2 ) .
V abs = | 3 V 4 ( 3 ) | .
E 2 E 1 = 1 1 + e 12 2 / e 13 2 + e 12 2 / e 14 2 .
V 4 ( 3 ) = e 12 e 13 e 14 e 13 2 e 14 2 + e 12 2 e 14 2 + e 12 2 e 13 2 .
I max I b = I ( r ) , I min + I b = I ( r + a / 2 + b / 2 + c / 2 ) .
e 12 = e 13 = e 14 = 1 / 3
e ^ i = ( e i , x , e i , y , e i , z ) .
| e ^ i | = e i , x 2 + e i , y 2 + e i , z 2 = 1
e ^ i · k i = e i , x k i , x + e i , y k i , y + e i , z k i , z = 0 ,
k i = ( k i , x , k i , y , k i , z ) .
f ( x 1 , , x n ) ,
g 1 ( x 1 , , x n ) = c 1 , , g m ( x 1 , , x n ) = c m ,
Λ ( x 1 , , x n , λ 1 , , λ m ) = f ( x 1 , , x n ) + λ 1 ( g 1 ( x 1 , , x n ) c 1 ) + + λ m ( g m ( x 1 , , x n ) c m )
Λ = 0 ,
e ^ i , 0 = z ^ × k i
e ^ i = R z ( ϕ ) R y ( θ ) R z ( ψ ) R y ( θ ) R z ( ϕ ) e ^ i , 0 ,

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