Abstract

A triple band frequency generator based on the optoelectronic oscillator (OEO) with low phase noise has been proposed and experimentally demonstrated. In this novel scheme, the phase-coherent triple band frequency in C, Ku and K bands can be achieved simultaneously by biasing the first modulator in the oscillation loop at the minimum transmission point (MITP) and the second modulator near the MITP with a small deviation. In the proof-of-concept experiment, the triple band frequency signals at 5.36, 16.08 and 21.44 GHz are generated with the phase noise of −123.32, −113.68 and −111.19 dBc/Hz at 10 kHz offset frequency, respectively. The proposed scheme provides a novel strategy for phase-coherent multi-band frequency and low phase noise signals generation, which can be potentially used in the multi-function and multi-band frequency electrical systems.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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2016 (1)

2015 (2)

H. Peng, C. Zhang, and X. Xie, “Tunable DC-60 GHz RF generation utilizing a dual-loop optoelectronic oscillator based on stimulated Brillouin scattering,” J. Lightwave Technol. 33(13), 2707–2715 (2015).
[Crossref]

Y. Chen, Q. Zhang, N. Yuan, Y. Luo, and H. Lou, “An adaptive ISAR-Imaging-Considered task scheduling algorithm for multi-function phased array radars,” IEEE Trans. Signal Process. 63(19), 5096–5110 (2015).
[Crossref]

2014 (3)

J. Paziewski and P. Wielgosz, “Assessment of GPS+ Galileo and multi-frequency Galileo single-epoch precise positioning with network corrections,” GPS Solut. 18(4), 571–579 (2014).
[Crossref]

L. Jhe-Min, H. Wen-Jeng, C. Yu-Peng, P. Peng-Chun, and L. Hai-Han, “Demonstration of optical frequency quadrupling combined with direct/external signal double-sideband suppressed-carrier modulation,” Opt. Commun. 317(15), 34–39 (2014).

Y. Jiang, J. Liang, G. Bai, L. Hu, S. Cai, H. Li, Y. Shan, and C. Ma, “Multifrequency optoelectronic oscillator,” Opt. Eng. 53(11), 116106 (2014).
[Crossref]

2013 (3)

2012 (2)

W. Li and J. P. Yao, “Optically tunable frequency-multiplying optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(10), 812–814 (2012).
[Crossref]

D. Zhu, S. Pan, and D. Ben, “Tunable frequency-quadrupling dual-loop optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(3), 194–196 (2012).
[Crossref]

2011 (1)

L. Wang, N. Zhu, W. Li, and J. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

2009 (1)

2008 (2)

D. Eliyahu, D. Seidel, and L. Maleki, “RF amplitude and phase-noise reduction of an optical link and an opto-electronic oscillator,” IEEE Trans. Microw. Theory Tech. 56(2), 449–456 (2008).
[Crossref]

C. T. Lin, P. T. Shih, and J. Chen, “Optical millimeter wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photonics Technol. Lett. 20(12), 1027–1029 (2008).
[Crossref]

2007 (1)

J. Zhang, H. Chen, and M. Chen, “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression,” IEEE Photonics Technol. Lett. 19(14), 1057–1059 (2007).
[Crossref]

2000 (2)

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85(11), 2264–2267 (2000).
[Crossref] [PubMed]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–640 (2000).
[Crossref] [PubMed]

1996 (1)

Bai, G.

Y. Jiang, J. Liang, G. Bai, L. Hu, S. Cai, H. Li, Y. Shan, and C. Ma, “Multifrequency optoelectronic oscillator,” Opt. Eng. 53(11), 116106 (2014).
[Crossref]

Ben, D.

D. Zhu, S. Pan, and D. Ben, “Tunable frequency-quadrupling dual-loop optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(3), 194–196 (2012).
[Crossref]

Cai, S.

Y. Jiang, J. Liang, G. Bai, L. Hu, S. Cai, H. Li, Y. Shan, and C. Ma, “Multifrequency optoelectronic oscillator,” Opt. Eng. 53(11), 116106 (2014).
[Crossref]

Chen, H.

J. Zhang, H. Chen, and M. Chen, “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression,” IEEE Photonics Technol. Lett. 19(14), 1057–1059 (2007).
[Crossref]

Chen, J.

C. T. Lin, P. T. Shih, and J. Chen, “Optical millimeter wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photonics Technol. Lett. 20(12), 1027–1029 (2008).
[Crossref]

Chen, M.

J. Zhang, H. Chen, and M. Chen, “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression,” IEEE Photonics Technol. Lett. 19(14), 1057–1059 (2007).
[Crossref]

Chen, Y.

Y. Chen, Q. Zhang, N. Yuan, Y. Luo, and H. Lou, “An adaptive ISAR-Imaging-Considered task scheduling algorithm for multi-function phased array radars,” IEEE Trans. Signal Process. 63(19), 5096–5110 (2015).
[Crossref]

Y. Chen, W. Li, A. Wen, and J. Yao, “Frequency-multiplying optoelectronic oscillator with a tunable multiplication factor,” IEEE Trans. Microw. Theory Tech. 61(9), 3479–3485 (2013).
[Crossref]

Chen, Z.

Cundiff, S. T.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–640 (2000).
[Crossref] [PubMed]

Diddams, S. A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–640 (2000).
[Crossref] [PubMed]

Eliyahu, D.

D. Eliyahu, D. Seidel, and L. Maleki, “RF amplitude and phase-noise reduction of an optical link and an opto-electronic oscillator,” IEEE Trans. Microw. Theory Tech. 56(2), 449–456 (2008).
[Crossref]

Guo, P.

Hai-Han, L.

L. Jhe-Min, H. Wen-Jeng, C. Yu-Peng, P. Peng-Chun, and L. Hai-Han, “Demonstration of optical frequency quadrupling combined with direct/external signal double-sideband suppressed-carrier modulation,” Opt. Commun. 317(15), 34–39 (2014).

Hall, J. L.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–640 (2000).
[Crossref] [PubMed]

Hänsch, T. W.

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85(11), 2264–2267 (2000).
[Crossref] [PubMed]

Holzwarth, R.

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85(11), 2264–2267 (2000).
[Crossref] [PubMed]

Hu, L.

Y. Jiang, J. Liang, G. Bai, L. Hu, S. Cai, H. Li, Y. Shan, and C. Ma, “Multifrequency optoelectronic oscillator,” Opt. Eng. 53(11), 116106 (2014).
[Crossref]

Hu, W.

Jhe-Min, L.

L. Jhe-Min, H. Wen-Jeng, C. Yu-Peng, P. Peng-Chun, and L. Hai-Han, “Demonstration of optical frequency quadrupling combined with direct/external signal double-sideband suppressed-carrier modulation,” Opt. Commun. 317(15), 34–39 (2014).

Jiang, Y.

Y. Jiang, J. Liang, G. Bai, L. Hu, S. Cai, H. Li, Y. Shan, and C. Ma, “Multifrequency optoelectronic oscillator,” Opt. Eng. 53(11), 116106 (2014).
[Crossref]

Jones, D. J.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–640 (2000).
[Crossref] [PubMed]

Knight, J. C.

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85(11), 2264–2267 (2000).
[Crossref] [PubMed]

Kong, F.

Li, H.

Y. Jiang, J. Liang, G. Bai, L. Hu, S. Cai, H. Li, Y. Shan, and C. Ma, “Multifrequency optoelectronic oscillator,” Opt. Eng. 53(11), 116106 (2014).
[Crossref]

Li, S.

Li, W.

Y. Chen, W. Li, A. Wen, and J. Yao, “Frequency-multiplying optoelectronic oscillator with a tunable multiplication factor,” IEEE Trans. Microw. Theory Tech. 61(9), 3479–3485 (2013).
[Crossref]

F. Kong, W. Li, and J. Yao, “Transverse load sensing based on a dual-frequency optoelectronic oscillator,” Opt. Lett. 38(14), 2611–2613 (2013).
[Crossref] [PubMed]

W. Li and J. P. Yao, “Optically tunable frequency-multiplying optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(10), 812–814 (2012).
[Crossref]

L. Wang, N. Zhu, W. Li, and J. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

Liang, J.

Y. Jiang, J. Liang, G. Bai, L. Hu, S. Cai, H. Li, Y. Shan, and C. Ma, “Multifrequency optoelectronic oscillator,” Opt. Eng. 53(11), 116106 (2014).
[Crossref]

Lin, C. T.

C. T. Lin, P. T. Shih, and J. Chen, “Optical millimeter wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photonics Technol. Lett. 20(12), 1027–1029 (2008).
[Crossref]

Liu, J.

L. Wang, N. Zhu, W. Li, and J. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

Lou, H.

Y. Chen, Q. Zhang, N. Yuan, Y. Luo, and H. Lou, “An adaptive ISAR-Imaging-Considered task scheduling algorithm for multi-function phased array radars,” IEEE Trans. Signal Process. 63(19), 5096–5110 (2015).
[Crossref]

Luo, Y.

Y. Chen, Q. Zhang, N. Yuan, Y. Luo, and H. Lou, “An adaptive ISAR-Imaging-Considered task scheduling algorithm for multi-function phased array radars,” IEEE Trans. Signal Process. 63(19), 5096–5110 (2015).
[Crossref]

Ma, C.

Y. Jiang, J. Liang, G. Bai, L. Hu, S. Cai, H. Li, Y. Shan, and C. Ma, “Multifrequency optoelectronic oscillator,” Opt. Eng. 53(11), 116106 (2014).
[Crossref]

Main, R.

L. Naidoo, R. Mathieu, and R. Main, “The assessment of data mining algorithms for modelling Savannah Woody cover using multi-frequency (X-, C-, L-band) synthetic aperture radar (SAR) datasets,” in Proc. of the IEEE Int. Geosci. and Remote Sensing Symp. (IEEE, 2014), pp. 1049–1052.
[Crossref]

Maleki, L.

D. Eliyahu, D. Seidel, and L. Maleki, “RF amplitude and phase-noise reduction of an optical link and an opto-electronic oscillator,” IEEE Trans. Microw. Theory Tech. 56(2), 449–456 (2008).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13(8), 1725–1735 (1996).
[Crossref]

Mathieu, R.

L. Naidoo, R. Mathieu, and R. Main, “The assessment of data mining algorithms for modelling Savannah Woody cover using multi-frequency (X-, C-, L-band) synthetic aperture radar (SAR) datasets,” in Proc. of the IEEE Int. Geosci. and Remote Sensing Symp. (IEEE, 2014), pp. 1049–1052.
[Crossref]

Naidoo, L.

L. Naidoo, R. Mathieu, and R. Main, “The assessment of data mining algorithms for modelling Savannah Woody cover using multi-frequency (X-, C-, L-band) synthetic aperture radar (SAR) datasets,” in Proc. of the IEEE Int. Geosci. and Remote Sensing Symp. (IEEE, 2014), pp. 1049–1052.
[Crossref]

Pan, S.

D. Zhu, S. Pan, and D. Ben, “Tunable frequency-quadrupling dual-loop optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(3), 194–196 (2012).
[Crossref]

Paziewski, J.

J. Paziewski and P. Wielgosz, “Assessment of GPS+ Galileo and multi-frequency Galileo single-epoch precise positioning with network corrections,” GPS Solut. 18(4), 571–579 (2014).
[Crossref]

Peng, H.

Peng-Chun, P.

L. Jhe-Min, H. Wen-Jeng, C. Yu-Peng, P. Peng-Chun, and L. Hai-Han, “Demonstration of optical frequency quadrupling combined with direct/external signal double-sideband suppressed-carrier modulation,” Opt. Commun. 317(15), 34–39 (2014).

Ranka, J. K.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–640 (2000).
[Crossref] [PubMed]

Russell, P. S.

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85(11), 2264–2267 (2000).
[Crossref] [PubMed]

Seidel, D.

D. Eliyahu, D. Seidel, and L. Maleki, “RF amplitude and phase-noise reduction of an optical link and an opto-electronic oscillator,” IEEE Trans. Microw. Theory Tech. 56(2), 449–456 (2008).
[Crossref]

Shan, Y.

Y. Jiang, J. Liang, G. Bai, L. Hu, S. Cai, H. Li, Y. Shan, and C. Ma, “Multifrequency optoelectronic oscillator,” Opt. Eng. 53(11), 116106 (2014).
[Crossref]

Shih, P. T.

C. T. Lin, P. T. Shih, and J. Chen, “Optical millimeter wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photonics Technol. Lett. 20(12), 1027–1029 (2008).
[Crossref]

Stentz, A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–640 (2000).
[Crossref] [PubMed]

Sun, T.

Udem, T.

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85(11), 2264–2267 (2000).
[Crossref] [PubMed]

Wadsworth, W. J.

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85(11), 2264–2267 (2000).
[Crossref] [PubMed]

Wang, L.

L. Wang, N. Zhu, W. Li, and J. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

Wen, A.

Y. Chen, W. Li, A. Wen, and J. Yao, “Frequency-multiplying optoelectronic oscillator with a tunable multiplication factor,” IEEE Trans. Microw. Theory Tech. 61(9), 3479–3485 (2013).
[Crossref]

Wen, H.

Wen-Jeng, H.

L. Jhe-Min, H. Wen-Jeng, C. Yu-Peng, P. Peng-Chun, and L. Hai-Han, “Demonstration of optical frequency quadrupling combined with direct/external signal double-sideband suppressed-carrier modulation,” Opt. Commun. 317(15), 34–39 (2014).

Wielgosz, P.

J. Paziewski and P. Wielgosz, “Assessment of GPS+ Galileo and multi-frequency Galileo single-epoch precise positioning with network corrections,” GPS Solut. 18(4), 571–579 (2014).
[Crossref]

Windeler, R. S.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–640 (2000).
[Crossref] [PubMed]

Xiao, X.

Xie, X.

Xie, Z.

Yan, H.

Yao, J.

F. Kong, W. Li, and J. Yao, “Transverse load sensing based on a dual-frequency optoelectronic oscillator,” Opt. Lett. 38(14), 2611–2613 (2013).
[Crossref] [PubMed]

Y. Chen, W. Li, A. Wen, and J. Yao, “Frequency-multiplying optoelectronic oscillator with a tunable multiplication factor,” IEEE Trans. Microw. Theory Tech. 61(9), 3479–3485 (2013).
[Crossref]

Yao, J. P.

W. Li and J. P. Yao, “Optically tunable frequency-multiplying optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(10), 812–814 (2012).
[Crossref]

Yao, X. S.

Yuan, N.

Y. Chen, Q. Zhang, N. Yuan, Y. Luo, and H. Lou, “An adaptive ISAR-Imaging-Considered task scheduling algorithm for multi-function phased array radars,” IEEE Trans. Signal Process. 63(19), 5096–5110 (2015).
[Crossref]

Yu-Peng, C.

L. Jhe-Min, H. Wen-Jeng, C. Yu-Peng, P. Peng-Chun, and L. Hai-Han, “Demonstration of optical frequency quadrupling combined with direct/external signal double-sideband suppressed-carrier modulation,” Opt. Commun. 317(15), 34–39 (2014).

Zhang, C.

Zhang, H.

Zhang, J.

J. Zhang, H. Chen, and M. Chen, “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression,” IEEE Photonics Technol. Lett. 19(14), 1057–1059 (2007).
[Crossref]

Zhang, Q.

Y. Chen, Q. Zhang, N. Yuan, Y. Luo, and H. Lou, “An adaptive ISAR-Imaging-Considered task scheduling algorithm for multi-function phased array radars,” IEEE Trans. Signal Process. 63(19), 5096–5110 (2015).
[Crossref]

Zhao, Y.

Zheng, X.

Zhou, B.

Zhu, D.

D. Zhu, S. Pan, and D. Ben, “Tunable frequency-quadrupling dual-loop optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(3), 194–196 (2012).
[Crossref]

Zhu, L.

Zhu, N.

L. Wang, N. Zhu, W. Li, and J. Liu, “A frequency-doubling optoelectronic oscillator based on a dual-parallel Mach-Zehnder modulator and a chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(22), 1688–1690 (2011).
[Crossref]

Zhu, X.

GPS Solut. (1)

J. Paziewski and P. Wielgosz, “Assessment of GPS+ Galileo and multi-frequency Galileo single-epoch precise positioning with network corrections,” GPS Solut. 18(4), 571–579 (2014).
[Crossref]

IEEE Photonics Technol. Lett. (5)

C. T. Lin, P. T. Shih, and J. Chen, “Optical millimeter wave signal generation using frequency quadrupling technique and no optical filtering,” IEEE Photonics Technol. Lett. 20(12), 1027–1029 (2008).
[Crossref]

J. Zhang, H. Chen, and M. Chen, “A photonic microwave frequency quadrupler using two cascaded intensity modulators with repetitious optical carrier suppression,” IEEE Photonics Technol. Lett. 19(14), 1057–1059 (2007).
[Crossref]

D. Zhu, S. Pan, and D. Ben, “Tunable frequency-quadrupling dual-loop optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(3), 194–196 (2012).
[Crossref]

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[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Schematic of the proposed triple band frequency generation based on the OEO. LD: laser diode; PC: polarization controller; MZM: Mach-Zehnder modulator; OC: optical coupler; SMF: single-mode fiber; PD: photo detector; EA: electrical amplifier; BPF: bandpass filter; EC: electrical coupler; EDFA: erbium-doped fiber amplifier; PS: phase shifter; ESA: electrical spectrum analyzer; OSA: optical spectrum analyzer.
Fig. 2
Fig. 2 Fundamental microwave signal generated by the OEO. (a) Electrical spectrum of the generated signal at 5.36 GHz. (b) Zoom-in view of the spectrum.
Fig. 3
Fig. 3 Optical spectra for generating triple band frequency. (a) Optical spectrum at the output of OC1. (b) Optical spectrum at the output of the MZM2. (c) Optical spectrum at the output of EDFA.
Fig. 4
Fig. 4 Electrical spectrum of the generated triple band frequency signals at 5.36, 16.08 and 21.44 GHz, respectively.
Fig. 5
Fig. 5 Zoom-in view of the electrical spectra (a-c) and phase noise measurements (d-f) of the generated 5.36 GHz, 16.08 GHz and 21.44 GHz signals, respectively.

Equations (8)

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E i n ( t ) = E 0 cos ( ω 0 t )
V 1 ( t ) = V e 1 cos ( ω e t + ϕ 1 )
E o u t 1 ( t ) = E 0 cos ( ω 0 t ) cos [ β 1 cos ( ω e t + ϕ 1 ) + π 2 ] E 0 J 1 ( β 1 ) 2 { cos [ ( ω 0 + ω e ) t + ϕ 1 ] + cos [ ( ω 0 ω e ) t ϕ 1 ] }
V 2 ( t ) = V e 2 cos ( ω e t + ϕ 2 )
E o u t 2 ( t ) = E o u t 1 ( t ) cos [ β 2 cos ( ω e t + ϕ 2 ) + π 2 + δ ] E 0 J 1 ( β 1 ) J 0 ( β 2 ) sin δ 2 { cos [ ( ω 0 + ω e ) t + ϕ 1 ] + cos [ ( ω 0 ω e ) t ϕ 1 ] } + E 0 J 1 ( β 1 ) J 1 ( β 2 ) 4 { cos [ ( ω 0 + 2 ω e ) t + ϕ 1 + ϕ 2 + δ ] + cos [ ( ω 0 2 ω e ) t ϕ 1 ϕ 2 δ ] + cos ( ω 0 t ) cos ( ϕ 1 ϕ 2 δ ) }
E o u t 2 ( t ) E 0 J 1 ( β 1 ) J 0 ( β 2 ) sin δ 2 { cos [ ( ω 0 + ω e ) t + ϕ 1 ] + cos [ ( ω 0 ω e ) t ϕ 1 ] } + E 0 J 1 ( β 1 ) J 1 ( β 2 ) 4 { cos [ ( ω 0 + 2 ω e ) t + ϕ 1 + ϕ 2 + δ ] + cos [ ( ω 0 2 ω e ) t ϕ 1 ϕ 2 δ ] } .
V o u t ( t ) = E 0 2 J 1 2 ( β 1 ) 2 { J 0 ( β 2 ) J 1 ( β 2 ) sin δ cos ( ω e t + ϕ 2 + δ ) + J 0 2 ( β 2 ) sin 2 δ cos ( 2 ω e t + ϕ 1 ) + J 0 ( β 2 ) J 1 ( β 2 ) sin δ cos ( 3 ω e t + 2 ϕ 1 + ϕ 2 + δ ) + J 1 2 ( β 2 ) 4 cos ( 4 ω e t + 2 ϕ 1 + 2 ϕ 2 + δ ) } .
A 1 = 1 2 E 0 2 J 1 2 ( β 1 ) J 0 ( β 2 ) J 1 ( β 2 ) sin δ = R J 0 ( β 2 ) J 1 ( β 2 ) sin δ ; A 2 = 1 2 E 0 2 J 1 2 ( β 1 ) J 0 2 ( β 2 ) sin 2 δ = R J 0 2 ( β 2 ) sin 2 δ ; A 3 = 1 2 E 0 2 J 1 2 ( β 1 ) J 0 ( β 2 ) J 1 ( β 2 ) sin δ = R J 0 ( β 2 ) J 1 ( β 2 ) sin δ ; A 4 = 1 8 E 0 2 J 1 2 ( β 1 ) J 1 2 ( β 2 ) = R J 1 2 ( β 2 ) / 4

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