Abstract

Junction structures provide the foundation of digital electronics and spintronics today. An equivalent, a photonic junction to achieve systematic and drastic control of photon flow is currently missing, but is mandatory for serious all-optical signal processing. Here we propose a photonic junction built upon mode-orthogonal hetero-structures, as a fundamental structural unit for photonic integrated circuits. Controlling the optical potential of mode-orthogonal junctions, the flow of photons can be dynamically manipulated, to complete the correspondence to the electronic junction structures. Of the possible applications, we provide examples of a photonic junction diode and a multi-junction half-adder, with exceptional performance metrics. Highly directional (41dB), nearly unity throughput, ultra-low threshold-power, high quality signal regeneration at 200Gb/s, and all-optic logic operations are successfully derived with the self-induced, bi-level dynamic mode-conversion process across the junction.

© 2011 OSA

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2011 (1)

2010 (5)

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4(7), 477–483 (2010).
[CrossRef]

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[CrossRef] [PubMed]

F. Leo, S. Coen, P. Kockaert, S. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics 4(7), 471–476 (2010).
[CrossRef]

R. Slavík, F. Parmigiani, J. Kakande, C. Lundstro¨m, M. Sjo¨din, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Gru¨ner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[CrossRef]

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[CrossRef]

2009 (3)

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457(7228), 455–458 (2009).
[CrossRef] [PubMed]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
[CrossRef]

2008 (4)

X. Hu, P. Jiang, C. Ding, H. Yang, and Q. Gong, “Picosecond and low-power all-optical switching based on an organic photonic-bandgap microcavity,” Nat. Photonics 2(3), 185–189 (2008).
[CrossRef]

Q. Liu, Z. Ouyang, C. J. Wu, C. P. Liu, and J. C. Wang, “All-optical half adder based on cross structures in two-dimensional photonic crystals,” Opt. Express 16(23), 18992–19000 (2008).
[CrossRef] [PubMed]

S. Yu, S. Koo, and N. Park, “Coded output photonic A/D converter based on photonic crystal slow-light structures,” Opt. Express 16(18), 13752–13757 (2008).
[CrossRef] [PubMed]

Y. J. Jung, C. W. Son, Y. M. Jhon, S. Lee, and N. Park, “One-level simplification method for all-optical combinational logic circuits,” IEEE Photon. Technol. Lett. 20(10), 800–802 (2008).
[CrossRef]

2007 (1)

R. Philip, M. Anija, C. S. Yelleswarapu, and D. V. G. L. N. Rao, “Passive all-optical diode using asymmetric nonlinear absorption,” Appl. Phys. Lett. 91(14), 141118 (2007).
[CrossRef]

2006 (6)

2005 (3)

J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005).
[CrossRef]

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4(5), 383–387 (2005).
[CrossRef] [PubMed]

A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and S. Yuasa, “Spin-torque diode effect in magnetic tunnel junctions,” Nature 438(7066), 339–342 (2005).
[CrossRef] [PubMed]

2004 (2)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[CrossRef] [PubMed]

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004).
[CrossRef] [PubMed]

2003 (3)

S. Tatsuura, M. Furuki, Y. Sato, I. Iwasa, M. Tian, and H. Mitsu, “Semiconductor carbon nanotubes as ultrafast switching materials for optical telecommunications,” Adv. Mater. (Deerfield Beach Fla.) 15(6), 534–537 (2003).
[CrossRef]

M. F. Yanik, S. Fan, M. Soljacić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28(24), 2506–2508 (2003).
[CrossRef] [PubMed]

X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83(13), 2518–2520 (2003).
[CrossRef]

2002 (2)

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601 (2002).
[CrossRef] [PubMed]

S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B 19(9), 2241–2249 (2002).
[CrossRef]

2001 (2)

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294(5546), 1488–1495 (2001).
[CrossRef] [PubMed]

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79(3), 314–316 (2001).
[CrossRef]

2000 (1)

1999 (1)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[CrossRef]

1997 (1)

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The Nonlinear Optical Properties of AlGaAs at the Half Band Gap,” IEEE J. Quantum Electron. 33(3), 341–348 (1997).
[CrossRef]

1947 (1)

J. H. Scaff and R. S. Ohl, “Development of silicon crystal rectifiers for microwave radar receivers,” Bell Syst. Tech. J. 26, 1–30 (1947).

Aggarwal, I. D.

Aitchison, J. S.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The Nonlinear Optical Properties of AlGaAs at the Half Band Gap,” IEEE J. Quantum Electron. 33(3), 341–348 (1997).
[CrossRef]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[CrossRef] [PubMed]

Andrekson, P. A.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundstro¨m, M. Sjo¨din, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Gru¨ner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[CrossRef]

Anija, M.

R. Philip, M. Anija, C. S. Yelleswarapu, and D. V. G. L. N. Rao, “Passive all-optical diode using asymmetric nonlinear absorption,” Appl. Phys. Lett. 91(14), 141118 (2007).
[CrossRef]

Asano, T.

Assanto, G.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79(3), 314–316 (2001).
[CrossRef]

Awschalom, D. D.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294(5546), 1488–1495 (2001).
[CrossRef] [PubMed]

Baehr-Jones, T.

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. Y. Jen, L. Dalton, and A. Scherer, “Terahertz all-optical modulation in a silicon-polymer hybrid system,” Nat. Mater. 5(9), 703–709 (2006).
[CrossRef] [PubMed]

Baets, R.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4(3), 182–187 (2010).
[CrossRef]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[CrossRef] [PubMed]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[CrossRef] [PubMed]

Biaggio, I.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Biancalana, F.

Bogaerts, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Bogris, A.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundstro¨m, M. Sjo¨din, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Gru¨ner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[CrossRef]

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[CrossRef] [PubMed]

Buhrman, R. A.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294(5546), 1488–1495 (2001).
[CrossRef] [PubMed]

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[CrossRef] [PubMed]

Chen, B.

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. Y. Jen, L. Dalton, and A. Scherer, “Terahertz all-optical modulation in a silicon-polymer hybrid system,” Nat. Mater. 5(9), 703–709 (2006).
[CrossRef] [PubMed]

Cheng, B.

X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83(13), 2518–2520 (2003).
[CrossRef]

Cheong, S. W.

Chtchelkanova, A. Y.

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: a spin-based electronics vision for the future,” Science 294(5546), 1488–1495 (2001).
[CrossRef] [PubMed]

Coen, S.

F. Leo, S. Coen, P. Kockaert, S. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics 4(7), 471–476 (2010).
[CrossRef]

Dalton, L.

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. Y. Jen, L. Dalton, and A. Scherer, “Terahertz all-optical modulation in a silicon-polymer hybrid system,” Nat. Mater. 5(9), 703–709 (2006).
[CrossRef] [PubMed]

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Adv. Mater. (Deerfield Beach Fla.) (1)

S. Tatsuura, M. Furuki, Y. Sato, I. Iwasa, M. Tian, and H. Mitsu, “Semiconductor carbon nanotubes as ultrafast switching materials for optical telecommunications,” Adv. Mater. (Deerfield Beach Fla.) 15(6), 534–537 (2003).
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Appl. Phys. Lett. (4)

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Figures (7)

Fig. 1
Fig. 1

(a) Examples of orthogonal mode junctions constructed between two orthogonal modes, providing different frequency separation and modal overlap properties. (b) Examples of molecular modes, which could be used to construct a variety of molecular mode junction (S, D, T: Single-, Di-, Tri- atomic molecular modes). A T- / T0 mode junction, for example, can be constructed between two structures providing orthogonal modes of T- and T0 (composed of optical atoms in single mode, sharing an identical polarization). Dashed circles in Fig. 1(a) and 1(b) represent optical atoms. (c) Operation principles of the mode junction. Excited modes at the operation frequency are marked with filled curves. NT-(E) and NT0(E) axis represents the density of states for T- and T0 mode. Adjusting the optical potential, the dominant mode of the controlled region can be switched between T- (non-shifted, yellow) and T0 (potential ΔE-shifted, pink) modes, to give a junction throughput of either a ‘0’ (yellow) or ‘1’ (pink). The coupling across the junction is determined by the inner product between eigenmodes of the left/right structures (equivalently, multiplying spatial mode profiles at the same frequency along the identical mode axis (T- or T0)).

Fig. 2
Fig. 2

Operation of a mode junction diode: under (a) Forward bias below threshold, (b) Forward bias above threshold, and (c) Reverse bias. Corresponding field patterns in the photonic crystal realization are shown in (d) ~(f) (details of numerical analysis in Appendix-A). Unidirectional transmission of the signal is evident only for the state 2(e), confirming the diode operation above threshold. For other states of operation, the wave propagation is inhibited; (d), at the odd-mode coupler for forward bias. (f), at the right end barrier of the di-atomic resonator, for reverse bias.

Fig. 3
Fig. 3

(a) Impedance matched (1/τL = 1/τR1 + 1/τR2) low reflection design is achieved with a junction diode, by adjusting τL. (e) Illustration of impedance imbalance, for the case of single band photonic diode (case of τR = 4τL). Mode-dependent field intensity inside resonators (even: |a1 + a2|2, odd: |a1-a2|2) for structures in (a) and (e) are shown for; (b) and (f) - forward bias before threshold, (c) and (g) - forward bias after threshold, (d) and (h) - under reverse bias. For two-band operation (b) at the operation frequency ωop (dashed line), on resonance feeding and low power excitation of the resonator is achieved. κ = 0.003ωop, and Q = 200. Coupled-mode-theory was used for the calculation.

Fig. 4
Fig. 4

(a) Numerically (FDTD, blue lines, Appendix A), and analytically (CMT, red lines, Appendix B) obtained response curve of the Ψe-oo junction diode. (b) Temporal-CMT calculated threshold power (blue circle), breakdown power (green circle), and operation bandwidth (red circle) as a function of the loaded Q-factor of a di-atomic resonator. Solid triangles overlaid to the plot are results of the FDTD, obtained with photonic diode realizations of different loaded quality factors Q1 = 1094, Q2 = 10945, and Q3 = 74895 (achieved by adjusting the number of dielectric rods around the resonator). (d) and (f) show the regenerated optical eyes at the output of the junction diode, at 100Gbit/s and 200Gbit/s respectively for the input signals (c) and (e). Input signals were FDTD generated to include amplitude noise (Gaussian random distribution. for level-0, between point p2 and p3, and for level-1 between p4 and p5). For output signals from the regenerator, optical Butterworth filters were assumed to conform to the signal bandwidth.

Fig. 5
Fig. 5

Multi-junction realization of monolithic half-adder; (a) coupling to S (XOR) port with a single logic input (IA or IB) power below the threshold. (b) coupling to C (AND) port under two input signals (IA · IB) for their total power above the threshold. Even (state 1)- / Odd (state 2)- mode excitation for the central di-atomic resonator and couplings to the even- / odd- mode coupler at the S / C port (left / right) of the half-adder is evident from the FDTD generated field amplitude plot. Figure (c) shows the logic operations of AND & XOR, under the two input signals at 50Gbps (de-correlated, PRBS). Figure (d) and (e) show the optical eye patterns for AND & XOR outputs. To note, for the generation of phase / time synchronized two input signals (IA and IB) for the proper logic operation, a single source was assumed, which are power divided and then separately modulated [14].

Fig. 6
Fig. 6

Layout of the photonic junction diode (case of Q = 1094). Red ellipsoids are dielectric defects composing di-atomic nonlinear resonators. Blue marked rods are position shifted (parameters in Table 1) in order; to introduce phase shift of π between the upper and lower waveguides (Box C, D), to fine-tune the resonance of the upper resonator (Box A), and to adjust couplings into the waveguide (Box B).

Fig. 7
Fig. 7

Analytical model of the di-atomic mode junction diode, used in the temporal CMT analysis.

Tables (1)

Tables Icon

Table 1 List of Structural Parameters for the Photonic Junction Diode Layout in Fig. 6

Equations (20)

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d dt ( a 1 a 2 a 3 )=( i ω 1 ( 1 τ 1 + 1 τ 2 ) i κ 12 0 i κ 21 i ω 2 1 τ 3 0 0 0 i ω 3 ( 1 τ 4 + 1 τ 5 + 1 τ 6 ) )( a 1 a 2 a 3 )+( 2 τ 1 S +1 + 2 τ 2 S +2 2 τ 3 S +3 2 τ 4 S +4 + 2 τ 5 S +5 + 2 τ 6 S +6 )
( S 1 S 2 )=( S +1 S +2 )+( 2 τ 1 2 τ 2 ) a 1
S 3 = S +3 + 2 τ 3 a 2
( S 4 S 5 S 6 )=( S +4 S +5 S +6 )+( 2 τ 4 2 τ 5 2 τ 6 ) a 3
( S +2 S +4 )=( S 4 S 2 ) e j θ 1
( S +3 S +5 )=( S 5 S 3 ) e j θ 2
S +2 = 2 τ 2 e i θ 1 2isin θ 1 a 1 + 2 τ 4 2isin θ 1 a 3 α 21 a 1 + α 23 a 3
S +3 = 2 τ 3 e i θ 2 2isin θ 2 a 2 + 2 τ 5 2isin θ 2 a 3 α 32 a 2 + α 33 a 3
S +4 = 2 τ 2 2isin θ 1 a 1 + 2 τ 4 e i θ 1 2isin θ 1 a 3 α 41 a 1 + α 43 a 3
S +5 = 2 τ 3 2isin θ 2 a 2 + 2 τ 5 e i θ 2 2isin θ 2 a 3 α 52 a 2 + α 53 a 3
a 1 = ( c 5 c 6 M 2 c 3 ) 2 τ 1 S +1 +( c 12 c 2 M 2 c 3 ) 2 τ 6 S +6 ( M 1 c 21 c 2 c 3 )( c 5 c 6 M 2 c 3 )( c 4 c 6 c 21 c 3 )( c 12 c 2 M 2 c 3 )
a 2 = ( c 4 c 6 c 21 c 3 ) 2 τ 1 S +1 +( M 1 c 21 c 2 c 3 ) 2 τ 6 S +6 ( M 1 c 21 c 2 c 3 )( c 5 c 6 M 2 c 3 )( c 4 c 6 c 21 c 3 )( c 12 c 2 M 2 c 3 )
a 3 = c 4 M 2 c 21 c 5 c 3 c 5 c 6 M 2 ( c 5 c 6 M 2 c 3 ) 2 τ 1 S +1 +( c 12 c 2 M 2 c 3 ) 2 τ 6 S +6 ( M 1 c 21 c 2 c 3 )( c 5 c 6 M 2 c 3 )( c 4 c 6 c 21 c 3 )( c 12 c 2 M 2 c 3 )
M 1 =i(ω ω 1 )( 1 τ 1 + 1 τ 2 )+ 2 τ 2 α 21
M 2 =i(ω ω 2 ) 1 τ 3 + 2 τ 3 α 32
| a 3 | 2 = | c 1 c 5 c 4 M 2 | 2 | c 3 c 5 c 6 M 2 | 2 | a 1 | 2
| a 1 | 2 = | c 5 c 6 M 2 c 3 | 2 2 τ 1 P I | ( M 1 c 21 c 2 c 3 )( c 5 c 6 M 2 c 3 )( c 4 c 6 c 21 c 3 )( c 12 c 2 M 2 c 3 ) | 2
| a 2 | 2 | a 1 | 2 = | c 3 c 4 c 6 c 21 | 2 | c 3 c 5 c 6 M 2 | 2 .
| a 1 | 2 = | c 12 c 2 M 2 c 3 | 2 2 τ 6 P I | ( M 1 c 21 c 2 c 3 )( c 5 c 6 M 2 c 3 )( c 4 c 6 c 21 c 3 )( c 12 c 2 M 2 c 3 ) | 2
| a 2 | 2 | a 1 | 2 = | c 3 M 1 c 2 c 21 | 2 | c 3 c 12 c 2 M 2 | 2 .

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