Abstract

We report the simultaneous implementation of the XOR and XNOR operations at two ports of a directed logic circuit based on two cascaded microring resonators (MRRs), which are both modulated through thermo-optic effect. Two electrical modulating signals applied to the MRRs represent the two operands of each logic operation. Simultaneous bitwise XOR and XNOR operations at 10 kbit/s are demonstrated in two different operating modes. We show that such a circuit can be readily realized using the plasma dispersion effect or the electric field effects, indicating its potential for high-speed operation. We further employ the scattering matrix method to analyze the spectral characteristics of the fabricated circuit, which can be regarded as a Mach-Zehnder interferometer (MZI) in whole. The two MRRs in the circuit act as wavelength-dependent splitting and combining units of the MZI. The degradation of the spectra observed in the experiment is found to be related to the length difference between the MZI’s two arms. The evolution of the spectra with this length difference is presented.

© 2011 OSA

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2011

J. B. Feng, Q. Q. Li, and Z. P. Zhou, “Single ring interferometer configuration with doubled free-spectral range,” IEEE Photon. Technol. Lett. 23(2), 79–81 (2011).
[CrossRef]

M. P. Fok and P. R. Prucnal, “All-optical XOR gate with optical feedback using highly Ge-doped nonlinear fiber and a terahertz optical asymmetric demultiplexer,” Appl. Opt. 50(2), 237–241 (2011).
[CrossRef] [PubMed]

2010

2009

2008

N. Sherwood-Droz, H. Wang, L. Chen, B. G. Lee, A. Biberman, K. Bergman, and M. Lipson, “Optical 4x4 hitless slicon router for optical networks-on-chip (NoC),” Opt. Express 16(20), 15915–15922 (2008).
[CrossRef] [PubMed]

J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[CrossRef]

2007

2006

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. 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]

2004

X. Zhang, Y. Wang, J. Q. Sun, D. M. Liu, and D. X. Huang, “All-optical AND gate at 10 Gbit/s based on cascaded single-port-couple SOAs,” Opt. Express 12(3), 361–366 (2004).
[CrossRef] [PubMed]

T. Fukazawa, T. Hirano, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
[CrossRef]

2002

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002).
[CrossRef]

A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14(4), 483–485 (2002).
[CrossRef]

2000

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[CrossRef]

1987

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

1961

U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124(6), 1866–1878 (1961).
[CrossRef]

Absil, P. P.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Asghari, M.

Baba, T.

T. Fukazawa, T. Hirano, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
[CrossRef]

Baehr-Jones, T.

J. Takayesu, M. Hochberg, T. Baehr-Jones, E. Chan, G. Wang, P. Sullivan, Y. Liao, J. Davies, L. Dalton, A. Scherer, and W. Krug, “A hybrid electrooptic microring resonator-based 1×4×1 ROADM for wafer scale optical interconnects,” J. Lightwave Technol. 27(4), 440–448 (2009).
[CrossRef]

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. 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.

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]

Beals, M.

J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[CrossRef]

Bennett, B. R.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Bergman, K.

Bernardis, S.

J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[CrossRef]

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]

Biberman, A.

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]

Caulfield, H. J.

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[CrossRef]

H. J. Caulfield, R. A. Soref, and C. S. Vikram, “Universal reconfigurable optical logic with silicon-on-insulator resonant structures,” Photonics Nanostruct. Fund. Appl. 5(1), 14–20 (2007).
[CrossRef]

Chan, E.

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. 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]

Chen, L.

Chen, P.

Cheng, J.

J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[CrossRef]

Chetrit, Y.

Ciftcioglu, B.

Dalton, L.

J. Takayesu, M. Hochberg, T. Baehr-Jones, E. Chan, G. Wang, P. Sullivan, Y. Liao, J. Davies, L. Dalton, A. Scherer, and W. Krug, “A hybrid electrooptic microring resonator-based 1×4×1 ROADM for wafer scale optical interconnects,” J. Lightwave Technol. 27(4), 440–448 (2009).
[CrossRef]

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. 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]

Davies, J.

Diederich, F.

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]

Dolev, S.

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[CrossRef]

Dong, P.

Dumon, P.

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]

Esembeson, B.

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]

Fan, S.

S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002).
[CrossRef]

Fang, Q.

Fano, U.

U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124(6), 1866–1878 (1961).
[CrossRef]

Feng, D.

Feng, J. B.

J. B. Feng, Q. Q. Li, and Z. P. Zhou, “Single ring interferometer configuration with doubled free-spectral range,” IEEE Photon. Technol. Lett. 23(2), 79–81 (2011).
[CrossRef]

Feng, N.-N.

Fok, M. P.

Freude, 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]

Fukazawa, T.

T. Fukazawa, T. Hirano, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
[CrossRef]

Gardes, F. Y.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Geng, M. M.

Grover, R.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Hardy, J.

Harvard, K.

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. 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]

Hirano, T.

T. Fukazawa, T. Hirano, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
[CrossRef]

Ho, P.-T.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Hochberg, M.

J. Takayesu, M. Hochberg, T. Baehr-Jones, E. Chan, G. Wang, P. Sullivan, Y. Liao, J. Davies, L. Dalton, A. Scherer, and W. Krug, “A hybrid electrooptic microring resonator-based 1×4×1 ROADM for wafer scale optical interconnects,” J. Lightwave Technol. 27(4), 440–448 (2009).
[CrossRef]

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. 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]

Huang, D. X.

Ibrahim, T. A.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Izhaky, N.

Jen, A. K.

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. 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]

Ji, R. Q.

Jia, L. X.

Jiang, Z. Y.

Johnson, F. G.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Kimerling, L. C.

J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[CrossRef]

Koos, C.

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]

Krishnamoorthy, A. V.

Krug, W.

Lawson, R.

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. 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]

Lee, B. G.

Leuthold, J.

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]

Li, G.

Li, Q. Q.

J. B. Feng, Q. Q. Li, and Z. P. Zhou, “Single ring interferometer configuration with doubled free-spectral range,” IEEE Photon. Technol. Lett. 23(2), 79–81 (2011).
[CrossRef]

Liang, H.

Liao, L.

Liao, S.

Liao, Y.

Lipson, M.

Liu, A. S.

Liu, D. M.

Liu, J. F.

J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[CrossRef]

Liu, Y. L.

Lu, Y. Y.

Luo, J.

M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. 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]

Madsen, C. K.

Manipatruni, S.

Mashanovich, G.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Michel, J.

J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[CrossRef]

Michinobu, T.

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]

Nguyen, H.

Ohno, F.

T. Fukazawa, T. Hirano, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. 43(2), 646–647 (2004).
[CrossRef]

Paniccia, M.

Pomerene, A.

J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[CrossRef]

Prucnal, P. R.

Reed, G. T.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Rubin, D.

Scherer, A.

J. Takayesu, M. Hochberg, T. Baehr-Jones, E. Chan, G. Wang, P. Sullivan, Y. Liao, J. Davies, L. Dalton, A. Scherer, and W. Krug, “A hybrid electrooptic microring resonator-based 1×4×1 ROADM for wafer scale optical interconnects,” J. Lightwave Technol. 27(4), 440–448 (2009).
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Figures (11)

Fig. 1
Fig. 1

(a) Schematic and (b) micrograph of the XOR/XNOR directed logic circuit based on two cascaded microring resonators (CW: continuous wave, EPT: electrical pulse train, OPT: optical pulse train, MRR: microring resonator).

Fig. 2
Fig. 2

Response spectra obtained at (a-d) the drop port and (e-h) the through port of the device. Voltages applied to MRR1 and MRR2 are both 0 V in (a) and (e), 3.13 V and 0 V in (b) and (f), 0 V and 2.15 V in (c) and (g), and 3.13 V and 2.15 V in (d) and (h). The dashed arrow indicates the location of the working wavelength λ w1.

Fig. 3
Fig. 3

Signals applied to two MRRs and detected at the drop and through ports in the first operating mode. Signals applied to (a) MRR1 and (b) MRR2. Results of (c) XNOR operation result at the drop port and (d) XOR operation result at the through port.

Fig. 4
Fig. 4

Response spectra obtained at (a-d) the drop port and (e-h) the through port of the device. Voltages applied to MRR1 and MRR2 are 0.8 V and 0 V in (a) and (e), 3.07 V and 0 V in (b) and (f), 0.8 V and 2.15 V in (c) and (g), and 3.07 V and 2.15 V in (d) and (h). The dashed arrow indicates the location of the working wavelength λ w2.

Fig. 5
Fig. 5

Signals applied to two MRRs and detected at the drop and through ports in the second operating mode. Signals applied to (a) MRR1 and (b) MRR2. Result of (c) XNOR operation result at the drop port and (d) XOR operation result at the through port.

Fig. 6
Fig. 6

Response spectra obtained at (a-d) the drop port and (e-h) the through port when the device operates in the first mode. No voltages are applied in (a) and (e), MRR1 is tuned to λw1 in (b) and (f), MRR2 is tuned to λw1 in (c) and (g), and both MRRs are tuned to λw1 in (d) and (h).

Fig. 7
Fig. 7

Scattering matrix model of the circuit (Z1~Z4: coupling regions, S1~S2: straight waveguides, R1~R2: ring waveguides).

Fig. 8
Fig. 8

Response spectra obtained in simulations at (a-d) the drop port and (e-h) the through port when the device operates in the first mode. No voltages are applied in (a) and (e), MRR1 is tuned to λw1 in (b) and (f), MRR2 is tuned to λw1 in (c) and (g), and both MRRs are tuned to λw1 in (d) and (h).

Fig. 9
Fig. 9

Amplitudes of the two constituent parts of E v3 (Ev 3- v 2 and Ev 3- p 2) as functions of the wavelength.

Fig. 10
Fig. 10

Amplitude of the electric field E v3 and the phase difference of its two constituent parts (Ev 3- v 2 and Ev 3- p 2) when the length difference of S1 and S2 is (a) odd multiples of half the perimeter of the ring waveguides (p = 1), (b) even multiples of half the perimeter of the ring waveguides (p = 0).

Fig. 11
Fig. 11

The spectra obtained at (a-c) the drop port and (d-f) the through port when the length differences between S1 and S2 are C/8, C/4, and 3C/8. C represents the ring waveguide’s perimeter.

Tables (1)

Tables Icon

Table 1 Parameters adopted in the numerical simulation

Equations (11)

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( E p 1 E r 2 ) = ( t j k j k t ) ( E in E r 1 ) ,
E p 1 = t ( 1 α exp( j θ )) 1- α t 2 exp ( j θ ) × E i n , E v 1 = k 2 α 1 / 4 exp( j θ / 4 ) 1- α t 2 exp ( j θ ) × E i n .
θ = β 2 π R = 4 π 2 n e f f R λ ,
ϕ p 1 = arctan [ α sin ( θ ) 1 α cos ( θ ) ] arctan [ α t 2 sin ( θ ) 1 α t 2 cos ( θ ) ] , ϕ v 1 = π + θ 4 arctan [ α t 2 sin ( θ ) 1 α t 2 cos ( θ ) ] ,
| E p 1 | = [ t 2 ( 1 + α 2 2 α cos ( θ ) ) 1 + α 2 t 4 2 α t 2 cos ( θ ) ] 1 / 2 , | E v 1 | = [ k 4 α 1 / 2 1 + α 2 t 4 2 α t 2 cos ( θ ) ] 1 / 2 .
| E v 3 v 2 | = t 2 ( 1 + α 2 2 α cos ( θ ) ) 1 + α 2 t 4 2 α t 2 cos ( θ ) × α 1 , | E v 3 p 2 | = k 4 α 1 / 2 1 + α 2 t 4 2 α t 2 cos ( θ ) × α 2 ,
φ v 3 v 2 = 2 arctan [ α sin ( θ ) 1 α cos ( θ ) ] 2 arctan [ α t 2 sin ( θ ) 1 α t 2 cos ( θ ) ] + θ 1 , φ v 3 p 2 = θ 2 2 arctan [ α t 2 sin ( θ ) 1 α t 2 cos ( θ ) ] + θ 2 ,
Δ φ v 3 = 2 arctan [ α sin ( θ ) 1 α cos ( θ ) ] θ 2 + ( θ 1 θ 2 ) .
Δ φ v 3 = 2 arctan [ 1 ε 2 + ( 1 / α ) 1 ε ] + ( p 1 ) m π { π + ( p 1 ) m π , w h e n ε = 2 [ ( 1 / α ) 1 ] , π + ( p 1 ) m π , w h e n ε = 2 [ ( 1 / α ) 1 ] .
| E v 3 | = | | E v 3 v 2 | | E v 3 p 2 | |
| E v 3 | = | | E v 3 v 2 | + | E v 3 p 2 | |

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