Abstract

A multi-octave highly-linear analog photonic link with simultaneous suppression of second-order intermodulation distortion (IMD2) and third-order intermodulation distortion (IMD3) is proposed and demonstrated based on a single integrated polarization-multiplexing dual-parallel Mach-Zehnder modulator (PM-DPMZM). The IMD2 is eliminated by biasing one sub-MZM in each sub-DPMZM at a point close to the maximum transmission point and the other sub-MZM at a point close to the minimum transmission point. The obtained fundamental frequency terms are in phase while the second-order harmonics are complementary when the two outputs of the two sub-MZMs are photodetected. The IMD3 is suppressed by adjusting the RF powers introduced to the two sub-DPMZMs, producing two complementary IMD3 terms when the modulated signals are photodetected. An experiment is carried out. Simultaneous suppression of IMD2 and IMD3 is achieved. The second-order spurious-free dynamic range (SFDR2) and third-order spurious-free dynamic range (SFDR3) are 82 dB·Hz1/2 and 110 dB·Hz2/3, respectively, indicating an improvement of 12 dB in SFDR2 and 13 dB in SFDR3 as compared with the low-biased MZM based analog photonic link, or an improvement of 3 dB in SFDR2 and 16 dB in SFDR3 as compared with the quadrature-biased MZM based photonic link.

© 2016 Optical Society of America

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References

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2015 (3)

D. Liang, Q. Tan, W. Jiang, Z. Zhu, X. Li, and Z. Yao, “Influence of Power Distribution on Performance of Intermodulation Distortion Suppression,” IEEE Photonics Technol. Lett. 27(15), 1639–1641 (2015).
[Crossref]

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[Crossref]

R. Waterhouse and D. Novack, “Realizing 5G: Microwave Photonics for 5G Mobile Wireless Systems,” IEEE Microw. Mag. 16(8), 84–92 (2015).
[Crossref]

2014 (2)

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

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[Crossref] [PubMed]

2013 (2)

2012 (1)

2010 (2)

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[Crossref]

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly linear radio-over-fiber system incorporating a single-drive dual-parallel Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 22(24), 1775–1777 (2010).
[Crossref]

2009 (2)

2007 (4)

2006 (1)

X. J. Meng and A. Karim, “Microwave photonic link with carrier suppression for increased dynamic range,” Fiber Integr. Opt. 25(3), 161–174 (2006).
[Crossref]

1996 (2)

W. Burns, G. Gopalakrishnan, and R. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photonics Technol. Lett. 8(1), 130–132 (1996).
[Crossref]

K. J. Williams, R. D. Esman, and M. Dagenais, “Nonlinearities in pin microwave photodetectors,” J. Lightwave Technol. 14(1), 84–96 (1996).
[Crossref]

Berizzi, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Bhatia, A.

Bogoni, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Burns, W.

W. Burns, G. Gopalakrishnan, and R. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photonics Technol. Lett. 8(1), 130–132 (1996).
[Crossref]

Capmany, J.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Capria, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Chen, Z.

Chizh, A. L.

Clark, T. R.

T. R. Clark and M. L. Dennis, “Coherent optical phase-modulation link,” IEEE Photonics Technol. Lett. 19(16), 1206–1208 (2007).
[Crossref]

Cowan, G. E.

Y. Shen, B. Hraimel, X. Zhang, G. E. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Cui, Y.

Y. Cui, K. Xu, Y. Dai, and J. Lin, “Suppression of second-order harmonic distortion in ROF links utilizing dual-output MZM and balanced detection,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2012), pp. 103–106.

Dagenais, M.

K. J. Williams, R. D. Esman, and M. Dagenais, “Nonlinearities in pin microwave photodetectors,” J. Lightwave Technol. 14(1), 84–96 (1996).
[Crossref]

Dai, Y.

Y. Cui, K. Xu, Y. Dai, and J. Lin, “Suppression of second-order harmonic distortion in ROF links utilizing dual-output MZM and balanced detection,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2012), pp. 103–106.

Dai, Y. T.

S. L. Pan, D. Zhu, S. F. Liu, K. Xu, Y. T. Dai, T. L. Wang, J. G. Liu, N. H. Zhu, Y. Xue, and N. J. Liu, “Satellite Payloads Pay Off,” IEEE Microw. Mag. 16(8), 61–73 (2015).
[Crossref]

Dennis, M. L.

T. R. Clark and M. L. Dennis, “Coherent optical phase-modulation link,” IEEE Photonics Technol. Lett. 19(16), 1206–1208 (2007).
[Crossref]

Devenport, J.

A. Karim and J. Devenport, “Low noise figure microwave photonic link,” in Proceedings of IEEE/MTT-S International Microwave Symposium (IEEE, 2007), pp. 1519–1522.

Devgan, P. S.

Diehl, J. F.

Esman, R. D.

K. J. Williams, R. D. Esman, and M. Dagenais, “Nonlinearities in pin microwave photodetectors,” J. Lightwave Technol. 14(1), 84–96 (1996).
[Crossref]

Foster, M. A.

Fu, J.

Ghelfi, P.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Gopalakrishnan, G.

W. Burns, G. Gopalakrishnan, and R. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photonics Technol. Lett. 8(1), 130–132 (1996).
[Crossref]

Hraimel, B.

Y. Shen, B. Hraimel, X. Zhang, G. E. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Huang, M.

Jiang, W.

D. Liang, Q. Tan, W. Jiang, Z. Zhu, X. Li, and Z. Yao, “Influence of Power Distribution on Performance of Intermodulation Distortion Suppression,” IEEE Photonics Technol. Lett. 27(15), 1639–1641 (2015).
[Crossref]

Karim, A.

X. J. Meng and A. Karim, “Microwave photonic link with carrier suppression for increased dynamic range,” Fiber Integr. Opt. 25(3), 161–174 (2006).
[Crossref]

A. Karim and J. Devenport, “Low noise figure microwave photonic link,” in Proceedings of IEEE/MTT-S International Microwave Symposium (IEEE, 2007), pp. 1519–1522.

Laghezza, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Lazzeri, E.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Lee, K.-L.

Li, P.

Li, S.

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly linear radio-over-fiber system incorporating a single-drive dual-parallel Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 22(24), 1775–1777 (2010).
[Crossref]

Li, W.

Li, X.

D. Liang, Q. Tan, W. Jiang, Z. Zhu, X. Li, and Z. Yao, “Influence of Power Distribution on Performance of Intermodulation Distortion Suppression,” IEEE Photonics Technol. Lett. 27(15), 1639–1641 (2015).
[Crossref]

Liang, D.

D. Liang, Q. Tan, W. Jiang, Z. Zhu, X. Li, and Z. Yao, “Influence of Power Distribution on Performance of Intermodulation Distortion Suppression,” IEEE Photonics Technol. Lett. 27(15), 1639–1641 (2015).
[Crossref]

Lim, C.

Lin, J.

Y. Cui, K. Xu, Y. Dai, and J. Lin, “Suppression of second-order harmonic distortion in ROF links utilizing dual-output MZM and balanced detection,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2012), pp. 103–106.

Liu, J. G.

S. L. Pan, D. Zhu, S. F. Liu, K. Xu, Y. T. Dai, T. L. Wang, J. G. Liu, N. H. Zhu, Y. Xue, and N. J. Liu, “Satellite Payloads Pay Off,” IEEE Microw. Mag. 16(8), 61–73 (2015).
[Crossref]

Liu, N. J.

S. L. Pan, D. Zhu, S. F. Liu, K. Xu, Y. T. Dai, T. L. Wang, J. G. Liu, N. H. Zhu, Y. Xue, and N. J. Liu, “Satellite Payloads Pay Off,” IEEE Microw. Mag. 16(8), 61–73 (2015).
[Crossref]

Liu, S. F.

S. L. Pan, D. Zhu, S. F. Liu, K. Xu, Y. T. Dai, T. L. Wang, J. G. Liu, N. H. Zhu, Y. Xue, and N. J. Liu, “Satellite Payloads Pay Off,” IEEE Microw. Mag. 16(8), 61–73 (2015).
[Crossref]

Liu, T.

Y. Shen, B. Hraimel, X. Zhang, G. E. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Luo, B.

Malacarne, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Malyshev, S. A.

Meng, X. J.

X. J. Meng and A. Karim, “Microwave photonic link with carrier suppression for increased dynamic range,” Fiber Integr. Opt. 25(3), 161–174 (2006).
[Crossref]

Moeller, R.

W. Burns, G. Gopalakrishnan, and R. Moeller, “Multi-octave operation of low-biased modulators by balanced detection,” IEEE Photonics Technol. Lett. 8(1), 130–132 (1996).
[Crossref]

Nirmalathas, A. T.

Novack, D.

R. Waterhouse and D. Novack, “Realizing 5G: Microwave Photonics for 5G Mobile Wireless Systems,” IEEE Microw. Mag. 16(8), 84–92 (2015).
[Crossref]

Novak, D.

Onori, D.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Pan, S.

Pan, S. L.

S. L. Pan, D. Zhu, S. F. Liu, K. Xu, Y. T. Dai, T. L. Wang, J. G. Liu, N. H. Zhu, Y. Xue, and N. J. Liu, “Satellite Payloads Pay Off,” IEEE Microw. Mag. 16(8), 61–73 (2015).
[Crossref]

Pan, W.

Pinna, S.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Porzi, C.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Scaffardi, M.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Scotti, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Serafino, G.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Shen, Y.

Y. Shen, B. Hraimel, X. Zhang, G. E. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Sunderman, C. E.

Tan, Q.

D. Liang, Q. Tan, W. Jiang, Z. Zhu, X. Li, and Z. Yao, “Influence of Power Distribution on Performance of Intermodulation Distortion Suppression,” IEEE Photonics Technol. Lett. 27(15), 1639–1641 (2015).
[Crossref]

Ting, H.-F.

Urick, V. J.

Vercesi, V.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Wang, T. L.

S. L. Pan, D. Zhu, S. F. Liu, K. Xu, Y. T. Dai, T. L. Wang, J. G. Liu, N. H. Zhu, Y. Xue, and N. J. Liu, “Satellite Payloads Pay Off,” IEEE Microw. Mag. 16(8), 61–73 (2015).
[Crossref]

Waterhouse, R.

Williams, K. J.

Wu, K.

Y. Shen, B. Hraimel, X. Zhang, G. E. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Xu, K.

S. L. Pan, D. Zhu, S. F. Liu, K. Xu, Y. T. Dai, T. L. Wang, J. G. Liu, N. H. Zhu, Y. Xue, and N. J. Liu, “Satellite Payloads Pay Off,” IEEE Microw. Mag. 16(8), 61–73 (2015).
[Crossref]

Y. Cui, K. Xu, Y. Dai, and J. Lin, “Suppression of second-order harmonic distortion in ROF links utilizing dual-output MZM and balanced detection,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2012), pp. 103–106.

Xue, Y.

S. L. Pan, D. Zhu, S. F. Liu, K. Xu, Y. T. Dai, T. L. Wang, J. G. Liu, N. H. Zhu, Y. Xue, and N. J. Liu, “Satellite Payloads Pay Off,” IEEE Microw. Mag. 16(8), 61–73 (2015).
[Crossref]

Yan, L.

Yao, J.

Yao, Z.

D. Liang, Q. Tan, W. Jiang, Z. Zhu, X. Li, and Z. Yao, “Influence of Power Distribution on Performance of Intermodulation Distortion Suppression,” IEEE Photonics Technol. Lett. 27(15), 1639–1641 (2015).
[Crossref]

Zhang, H.

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly linear radio-over-fiber system incorporating a single-drive dual-parallel Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 22(24), 1775–1777 (2010).
[Crossref]

Zhang, X.

Y. Shen, B. Hraimel, X. Zhang, G. E. Cowan, K. Wu, and T. Liu, “A novel analog broadband RF predistortion circuit to linearize electro-absorption modulators in multiband OFDM radio-over-fiber systems,” IEEE Trans. Microw. Theory Tech. 58(11), 3327–3335 (2010).
[Crossref]

Zheng, X.

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly linear radio-over-fiber system incorporating a single-drive dual-parallel Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 22(24), 1775–1777 (2010).
[Crossref]

Zhou, B.

S. Li, X. Zheng, H. Zhang, and B. Zhou, “Highly linear radio-over-fiber system incorporating a single-drive dual-parallel Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 22(24), 1775–1777 (2010).
[Crossref]

Zhou, T.

Zhu, D.

S. L. Pan, D. Zhu, S. F. Liu, K. Xu, Y. T. Dai, T. L. Wang, J. G. Liu, N. H. Zhu, Y. Xue, and N. J. Liu, “Satellite Payloads Pay Off,” IEEE Microw. Mag. 16(8), 61–73 (2015).
[Crossref]

Zhu, N. H.

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Figures (4)

Fig. 1
Fig. 1 Schematic diagram of the proposed linearized analog photonic link based on a polarization-multiplexing dual-parallel Mach-Zehnder modulator. LD: laser diode; DPMZM; dual parallel Mach-Zehnder modulator. PR: 90°polarization rotator; SMF: single mode fiber; PD: photodetector; ATT: microwave attenuator.
Fig. 2
Fig. 2 Electrical spectra of the output fundamental signal and their IMD3 for (a) the quadrature-biased MZM, (b) the low-biased MZM and (c) the proposed PM-DPMZM based photonic links; Electrical spectra of the output SHD and IMD2 for (d) the quadrature-biased MZM, (e) the low-biased MZM and (f) the proposed PM-DPMZM based photonic links; (g) electrical spectrum with a span of 24 GHz for the proposed linearized photonic link based on the PM-DPMZM.
Fig. 3
Fig. 3 SFDR performance of the (a) quadrature-biased and (b) low-biased MZM based photonic links.
Fig. 4
Fig. 4 SFDR performance of the proposed multi-octave linearized link based on the PM-DPMZM.

Equations (7)

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[ E 01 E 02 ] = [ E I e j ω c t cos ( ϕ 11 + ϕ 1 m ( t ) 2 ) + E I e j ( ω c t + ϕ 13 cos ( ϕ 12 + ϕ 1 m ( t ) 2 ) E I j ω c t cos ( ϕ 11 + ϕ 1 m ( t ) 2 ) + E I e j ( ω c t + ϕ 23 cos ( ϕ 22 + ϕ 1 m ( t ) 2 ) ]
I ( t ) = | E O 1 | 2 + | E O 2 | 2 = i = 1 2 [ E I 2 cos 2 ( ϕ i 1 + ϕ i m ( t ) 2 ) + E I 2 cos 2 ( ϕ i 2 + ϕ i m ( t ) 2 ) + 2 E I 2 cos ϕ i 3 cos ( ϕ i 1 + ϕ i m ( t ) 2 ) cos ( ϕ i 2 + ϕ i m ( t ) 2 ) ]
I ( t ) = E I 2 sin ϕ 11 sin ϕ 1 m E I 2 sin ϕ 21 sin ϕ 2 m
I ( t ) = E 1 2 sin ϕ 11 sin [ β 1 cos ( ω 1 t ) + β 1 cos ( ω 2 t ) ] E 2 2 sin ϕ 21 sin [ β 2 cos ( ω 1 t ) + β 2 cos ( ω 2 t ) ] = E 1 2 sin ϕ 11 { p = p = + q = q = + J p ( β 1 ) J q ( β 1 ) sin [ p ω 1 + q ω 2 + ( p + q ) π / 2 ] } E 2 2 sin ϕ 21 { p = p = + q = q = + J p ( β 2 ) J q ( β 2 ) sin [ p ω 1 + q ω 2 + ( p + q ) π / 2 ] } = i = 1 2 { Γ i 0 + Γ i 1 [ cos ( ω 1 t ) + cos ( ω 2 t ) ] + Γ i 3 [ cos ( 2 ω 1 ω 2 ) + cos ( 2 ω 2 ω 1 ) ] + }
I ( t ) E I 2 { [ sin ϕ 11 J 0 ( β 1 ) J 1 ( β 1 ) sin ϕ 21 J 0 ( β 2 ) J 1 ( β 2 ) ] [ cos ( ω 1 t ) + cos ( ω 2 t ) ] + [ sin ϕ 11 J 1 ( β 1 ) J 2 ( β 1 ) + sin ϕ 21 J 1 ( β 2 ) J 2 ( β 2 ) ] [ cos ( 2 ω 1 ω 2 ) + cos ( 2 ω 2 ω 1 ) ] + }
{ sin ϕ 11 J 1 ( β 1 ) J 2 ( β 1 ) = sin ϕ 21 J 1 ( β 2 ) J 2 ( β 2 ) sin ϕ 11 J 0 ( β 1 ) J 1 ( β 1 ) sin ϕ 21 J 0 ( β 2 ) J 1 ( β 2 )
{ sin ϕ 11 / sin ϕ 21 = β 2 3 / β 1 3 sin ϕ 11 / sin ϕ 21 β 2 / β 1

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