Abstract

A photonic approach for generating chirped microwave pulses with a flexible and fine parameter manipulation is proposed and experimentally demonstrated. In the proposed system, an intensity modulator (IM) biased at the minimum transmission point is used to generate two ± 1st-order optical sidebands which are then sent to a phase modulator (PM) for implementing large-signal phase modulations. A de-interleaver combined with an optical variable delay line (OVDL) is utilized to introduce a time delay between two phase-modulated optical signals. A second IM that acts as a time domain intensity switch (TDIS) is used to select different phase modulation ranges of the two phase-modulated optical signals. After the optical-electrical conversion in a photodetector (PD), chirped microwave pulses are generated. The key feature of this approach is that the parameters of the generated chirped microwave pulses including central frequency, pulse repetition frequency, and chirp rate can be flexibly and precisely manipulated by the radio frequency (RF) signals applied to modulators. A proof-of-principle experiment is carried out to verify the proposed approach. Consequently, positive or negative chirped microwave pulses with different central frequencies at 20, 22, 24 or 26 GHz and different pulse repetition frequencies at 1.5 or 2 GHz are generated, respectively.

© 2016 Optical Society of America

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References

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    [Crossref]
  9. M. Li and J. P. Yao, “Photonic generation of continuously tunable chirped microwave waveforms based on a temporal interferometer incorporating an optically pumped linearly chirped fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 59(12), 3531–3537 (2011).
    [Crossref]
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2015 (2)

M. Ma, M. Rochette, and L. R. Chen, “Generating chirped microwave pulses using an integrated distributed Fabry-Pérot cavity in silicon-on-insulator,” IEEE Photonics J. 7(2), 5500706 (2015).
[Crossref]

H. Zhang, W. Zou, and J. Chen, “Generation of a widely tunable linearly chirped microwave waveform based on spectral filtering and unbalanced dispersion,” Opt. Lett. 40(6), 1085–1088 (2015).
[Crossref] [PubMed]

2014 (3)

A. Rashidinejad and A. M. Weiner, “Photonic radio-frequency arbitrary waveform generation with maximal time-bandwidth product capability,” J. Lightwave Technol. 32(20), 3383–3393 (2014).
[Crossref]

Y. Mei, Y. Xu, H. Chi, X. Zhang, X. Jin, S. Zheng, and T. Jin, “Photonic generation of chirped microwave signals with high time-bandwidth product,” Opt. Commun. 316(1), 106–110 (2014).
[Crossref]

W. Li, W. T. Wang, and N. H. Zhu, “Photonic generation of radio-frequency waveforms based on dual-parallel Mach-Zehnder modulator,” IEEE Photonics J. 6(3), 5500608 (2014).
[Crossref]

2013 (4)

2011 (3)

M. Li and J. P. Yao, “Photonic generation of continuously tunable chirped microwave waveforms based on a temporal interferometer incorporating an optically pumped linearly chirped fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 59(12), 3531–3537 (2011).
[Crossref]

M. Li, L. Y. Shao, J. Albert, and J. P. Yao, “Tilted fiber Bragg grating for chirped microwave waveform generation,” IEEE Photonics Technol. Lett. 23(5), 314–316 (2011).
[Crossref]

J. P. Yao, “Photonic generation of microwave arbitrary waveforms,” Opt. Commun. 284(15), 3723–3736 (2011).
[Crossref]

2010 (3)

M. Li, C. Wang, W. Z. Li, and J. P. Yao, “An unbalanced temporal pulse-shaping system for chirped microwave waveform generation,” IEEE Trans. Microw. Theory Tech. 58(11), 2968–2975 (2010).
[Crossref]

R. Ashrafi, Y. Park, and J. Azana, “Fiber-based photonic generation of high-frequency microwave pulses with reconfigurable linear chirp control,” IEEE Trans. Microw. Theory Tech. 58(11), 3312–3319 (2010).
[Crossref]

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. H. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

2008 (2)

I. S. Lin and A. M. Weiner, “Selective correlation detection of photonically generated ultrawideband RF signals,” J. Lightwave Technol. 26(15), 2692–2699 (2008).
[Crossref]

C. Wang and J. P. Yao, “Photonic generation of chirped microwave pulses using superimposed chirped fiber Bragg gratings,” IEEE Photonics Technol. Lett. 20(11), 882–884 (2008).
[Crossref]

2007 (1)

H. Chi and J. P. Yao, “All-fiber chirped microwave pulses generation based on spectral shaping and wavelength-to-time conversion,” IEEE Trans. Microw. Theory Tech. 55(9), 1958–1963 (2007).
[Crossref]

2006 (1)

2005 (1)

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, “Optical generation of linearly chirped microwave pulses using fiber Bragg gratings,” IEEE Photonics Technol. Lett. 17(3), 660–662 (2005).
[Crossref]

2003 (1)

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 581–583 (2003).
[Crossref]

Albert, J.

M. Li, L. Y. Shao, J. Albert, and J. P. Yao, “Tilted fiber Bragg grating for chirped microwave waveform generation,” IEEE Photonics Technol. Lett. 23(5), 314–316 (2011).
[Crossref]

Ashrafi, R.

R. Ashrafi, Y. Park, and J. Azana, “Fiber-based photonic generation of high-frequency microwave pulses with reconfigurable linear chirp control,” IEEE Trans. Microw. Theory Tech. 58(11), 3312–3319 (2010).
[Crossref]

Azana, J.

R. Ashrafi, Y. Park, and J. Azana, “Fiber-based photonic generation of high-frequency microwave pulses with reconfigurable linear chirp control,” IEEE Trans. Microw. Theory Tech. 58(11), 3312–3319 (2010).
[Crossref]

J. Azana, N. K. Berger, B. Levit, and B. Fischer, “Broadband arbitrary waveform generation based on microwave frequency upshifting in optical fibers,” J. Lightwave Technol. 24(7), 2663–2675 (2006).
[Crossref]

Berger, N. K.

Bolea, M.

M. Bolea, J. Mora, B. Ortega, and J. Capmany, “Reconfigurability and tunability of a chirped microwave photonic pulse generator,” in International Topical Meeting on Microwave Photonics (MWP), 167–170 (2010).
[Crossref]

Capmany, J.

M. Bolea, J. Mora, B. Ortega, and J. Capmany, “Reconfigurability and tunability of a chirped microwave photonic pulse generator,” in International Topical Meeting on Microwave Photonics (MWP), 167–170 (2010).
[Crossref]

Chen, H.

Chen, J.

Chen, L. R.

M. Ma, M. Rochette, and L. R. Chen, “Generating chirped microwave pulses using an integrated distributed Fabry-Pérot cavity in silicon-on-insulator,” IEEE Photonics J. 7(2), 5500706 (2015).
[Crossref]

Chen, M.

Chi, H.

Y. Mei, Y. Xu, H. Chi, X. Zhang, X. Jin, S. Zheng, and T. Jin, “Photonic generation of chirped microwave signals with high time-bandwidth product,” Opt. Commun. 316(1), 106–110 (2014).
[Crossref]

H. Chi and J. P. Yao, “All-fiber chirped microwave pulses generation based on spectral shaping and wavelength-to-time conversion,” IEEE Trans. Microw. Theory Tech. 55(9), 1958–1963 (2007).
[Crossref]

Chou, J.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 581–583 (2003).
[Crossref]

Dezfooliyan, A.

Fischer, B.

Gao, H.

Goh, C. S.

Han, Y.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 581–583 (2003).
[Crossref]

Horowitz, M.

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, “Optical generation of linearly chirped microwave pulses using fiber Bragg gratings,” IEEE Photonics Technol. Lett. 17(3), 660–662 (2005).
[Crossref]

Jalali, B.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 581–583 (2003).
[Crossref]

Jin, T.

Y. Mei, Y. Xu, H. Chi, X. Zhang, X. Jin, S. Zheng, and T. Jin, “Photonic generation of chirped microwave signals with high time-bandwidth product,” Opt. Commun. 316(1), 106–110 (2014).
[Crossref]

Jin, X.

Y. Mei, Y. Xu, H. Chi, X. Zhang, X. Jin, S. Zheng, and T. Jin, “Photonic generation of chirped microwave signals with high time-bandwidth product,” Opt. Commun. 316(1), 106–110 (2014).
[Crossref]

Khan, M. H.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. H. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Kong, F. Q.

Leaird, D. E.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. H. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Lei, C.

Levinson, O.

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, “Optical generation of linearly chirped microwave pulses using fiber Bragg gratings,” IEEE Photonics Technol. Lett. 17(3), 660–662 (2005).
[Crossref]

Levit, B.

Li, M.

M. Li and J. P. Yao, “Photonic generation of continuously tunable chirped microwave waveforms based on a temporal interferometer incorporating an optically pumped linearly chirped fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 59(12), 3531–3537 (2011).
[Crossref]

M. Li, L. Y. Shao, J. Albert, and J. P. Yao, “Tilted fiber Bragg grating for chirped microwave waveform generation,” IEEE Photonics Technol. Lett. 23(5), 314–316 (2011).
[Crossref]

M. Li, C. Wang, W. Z. Li, and J. P. Yao, “An unbalanced temporal pulse-shaping system for chirped microwave waveform generation,” IEEE Trans. Microw. Theory Tech. 58(11), 2968–2975 (2010).
[Crossref]

Li, W.

W. Li, W. T. Wang, and N. H. Zhu, “Photonic generation of radio-frequency waveforms based on dual-parallel Mach-Zehnder modulator,” IEEE Photonics J. 6(3), 5500608 (2014).
[Crossref]

Li, W. Z.

W. Z. Li, F. Q. Kong, and J. P. Yao, “Arbitrary microwave waveform generation based on a tunable optoelectronic oscillator,” J. Lightwave Technol. 31(23), 3780–3786 (2013).
[Crossref]

M. Li, C. Wang, W. Z. Li, and J. P. Yao, “An unbalanced temporal pulse-shaping system for chirped microwave waveform generation,” IEEE Trans. Microw. Theory Tech. 58(11), 2968–2975 (2010).
[Crossref]

Lin, I. S.

Ma, M.

M. Ma, M. Rochette, and L. R. Chen, “Generating chirped microwave pulses using an integrated distributed Fabry-Pérot cavity in silicon-on-insulator,” IEEE Photonics J. 7(2), 5500706 (2015).
[Crossref]

Mei, Y.

Y. Mei, Y. Xu, H. Chi, X. Zhang, X. Jin, S. Zheng, and T. Jin, “Photonic generation of chirped microwave signals with high time-bandwidth product,” Opt. Commun. 316(1), 106–110 (2014).
[Crossref]

Mora, J.

M. Bolea, J. Mora, B. Ortega, and J. Capmany, “Reconfigurability and tunability of a chirped microwave photonic pulse generator,” in International Topical Meeting on Microwave Photonics (MWP), 167–170 (2010).
[Crossref]

Ortega, B.

M. Bolea, J. Mora, B. Ortega, and J. Capmany, “Reconfigurability and tunability of a chirped microwave photonic pulse generator,” in International Topical Meeting on Microwave Photonics (MWP), 167–170 (2010).
[Crossref]

Pan, S. L.

Y. M. Zhang, X. W. Ye, and S. L. Pan, “Photonic generation of linear frequency-modulated waveform with improved time-bandwidth product,” in International Topical Meeting on Microwave Photonics (MWP), 1–4 (2015).
[Crossref]

Park, Y.

R. Ashrafi, Y. Park, and J. Azana, “Fiber-based photonic generation of high-frequency microwave pulses with reconfigurable linear chirp control,” IEEE Trans. Microw. Theory Tech. 58(11), 3312–3319 (2010).
[Crossref]

Qi, M. H.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. H. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Rashidinejad, A.

Rochette, M.

M. Ma, M. Rochette, and L. R. Chen, “Generating chirped microwave pulses using an integrated distributed Fabry-Pérot cavity in silicon-on-insulator,” IEEE Photonics J. 7(2), 5500706 (2015).
[Crossref]

Set, S. Y.

Shao, L. Y.

M. Li, L. Y. Shao, J. Albert, and J. P. Yao, “Tilted fiber Bragg grating for chirped microwave waveform generation,” IEEE Photonics Technol. Lett. 23(5), 314–316 (2011).
[Crossref]

Shen, H.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. H. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Shi, J. W.

Stepanov, S.

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, “Optical generation of linearly chirped microwave pulses using fiber Bragg gratings,” IEEE Photonics Technol. Lett. 17(3), 660–662 (2005).
[Crossref]

Wang, C.

M. Li, C. Wang, W. Z. Li, and J. P. Yao, “An unbalanced temporal pulse-shaping system for chirped microwave waveform generation,” IEEE Trans. Microw. Theory Tech. 58(11), 2968–2975 (2010).
[Crossref]

C. Wang and J. P. Yao, “Photonic generation of chirped microwave pulses using superimposed chirped fiber Bragg gratings,” IEEE Photonics Technol. Lett. 20(11), 882–884 (2008).
[Crossref]

Wang, W. T.

W. Li, W. T. Wang, and N. H. Zhu, “Photonic generation of radio-frequency waveforms based on dual-parallel Mach-Zehnder modulator,” IEEE Photonics J. 6(3), 5500608 (2014).
[Crossref]

Wei, C. C.

Weiner, A. M.

Wun, J. M.

Xiao, S. J.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. H. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Xie, S.

Xing, F.

Xu, Y.

Y. Mei, Y. Xu, H. Chi, X. Zhang, X. Jin, S. Zheng, and T. Jin, “Photonic generation of chirped microwave signals with high time-bandwidth product,” Opt. Commun. 316(1), 106–110 (2014).
[Crossref]

Xuan, Y.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. H. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Yao, J. P.

W. Z. Li, F. Q. Kong, and J. P. Yao, “Arbitrary microwave waveform generation based on a tunable optoelectronic oscillator,” J. Lightwave Technol. 31(23), 3780–3786 (2013).
[Crossref]

M. Li, L. Y. Shao, J. Albert, and J. P. Yao, “Tilted fiber Bragg grating for chirped microwave waveform generation,” IEEE Photonics Technol. Lett. 23(5), 314–316 (2011).
[Crossref]

M. Li and J. P. Yao, “Photonic generation of continuously tunable chirped microwave waveforms based on a temporal interferometer incorporating an optically pumped linearly chirped fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 59(12), 3531–3537 (2011).
[Crossref]

J. P. Yao, “Photonic generation of microwave arbitrary waveforms,” Opt. Commun. 284(15), 3723–3736 (2011).
[Crossref]

M. Li, C. Wang, W. Z. Li, and J. P. Yao, “An unbalanced temporal pulse-shaping system for chirped microwave waveform generation,” IEEE Trans. Microw. Theory Tech. 58(11), 2968–2975 (2010).
[Crossref]

C. Wang and J. P. Yao, “Photonic generation of chirped microwave pulses using superimposed chirped fiber Bragg gratings,” IEEE Photonics Technol. Lett. 20(11), 882–884 (2008).
[Crossref]

H. Chi and J. P. Yao, “All-fiber chirped microwave pulses generation based on spectral shaping and wavelength-to-time conversion,” IEEE Trans. Microw. Theory Tech. 55(9), 1958–1963 (2007).
[Crossref]

Ye, X. W.

Y. M. Zhang, X. W. Ye, and S. L. Pan, “Photonic generation of linear frequency-modulated waveform with improved time-bandwidth product,” in International Topical Meeting on Microwave Photonics (MWP), 1–4 (2015).
[Crossref]

Zeitouny, A.

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, “Optical generation of linearly chirped microwave pulses using fiber Bragg gratings,” IEEE Photonics Technol. Lett. 17(3), 660–662 (2005).
[Crossref]

Zhang, H.

Zhang, X.

Y. Mei, Y. Xu, H. Chi, X. Zhang, X. Jin, S. Zheng, and T. Jin, “Photonic generation of chirped microwave signals with high time-bandwidth product,” Opt. Commun. 316(1), 106–110 (2014).
[Crossref]

Zhang, Y. M.

Y. M. Zhang, X. W. Ye, and S. L. Pan, “Photonic generation of linear frequency-modulated waveform with improved time-bandwidth product,” in International Topical Meeting on Microwave Photonics (MWP), 1–4 (2015).
[Crossref]

Zhao, L.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. H. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Zheng, S.

Y. Mei, Y. Xu, H. Chi, X. Zhang, X. Jin, S. Zheng, and T. Jin, “Photonic generation of chirped microwave signals with high time-bandwidth product,” Opt. Commun. 316(1), 106–110 (2014).
[Crossref]

Zhu, N. H.

W. Li, W. T. Wang, and N. H. Zhu, “Photonic generation of radio-frequency waveforms based on dual-parallel Mach-Zehnder modulator,” IEEE Photonics J. 6(3), 5500608 (2014).
[Crossref]

Zou, W.

IEEE Photonics J. (2)

W. Li, W. T. Wang, and N. H. Zhu, “Photonic generation of radio-frequency waveforms based on dual-parallel Mach-Zehnder modulator,” IEEE Photonics J. 6(3), 5500608 (2014).
[Crossref]

M. Ma, M. Rochette, and L. R. Chen, “Generating chirped microwave pulses using an integrated distributed Fabry-Pérot cavity in silicon-on-insulator,” IEEE Photonics J. 7(2), 5500706 (2015).
[Crossref]

IEEE Photonics Technol. Lett. (4)

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 581–583 (2003).
[Crossref]

M. Li, L. Y. Shao, J. Albert, and J. P. Yao, “Tilted fiber Bragg grating for chirped microwave waveform generation,” IEEE Photonics Technol. Lett. 23(5), 314–316 (2011).
[Crossref]

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, “Optical generation of linearly chirped microwave pulses using fiber Bragg gratings,” IEEE Photonics Technol. Lett. 17(3), 660–662 (2005).
[Crossref]

C. Wang and J. P. Yao, “Photonic generation of chirped microwave pulses using superimposed chirped fiber Bragg gratings,” IEEE Photonics Technol. Lett. 20(11), 882–884 (2008).
[Crossref]

IEEE Trans. Microw. Theory Tech. (4)

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Y. M. Zhang, X. W. Ye, and S. L. Pan, “Photonic generation of linear frequency-modulated waveform with improved time-bandwidth product,” in International Topical Meeting on Microwave Photonics (MWP), 1–4 (2015).
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Figures (7)

Fig. 1
Fig. 1 Conceptual diagram of the proposed chirped microwave pulses generation scheme. LD: laser diode; PC: polarization controller; IM: intensity modulator; PM: phase modulator; OVDL: optical variable delay line; OC: optical coupler; PD: photodetector; PS: phase shifter; Div: divider; EA: electrical amplifier; MG: microwave generator; RF: radio frequency signal.
Fig. 2
Fig. 2 Principle of the proposed approach for generating chirped microwave pulses. (a) Frequency modulations of two phase-modulated optical signals; (b) frequency modulation of the generated microwave signal; (c) temporal waveform in the linear chirp range.
Fig. 3
Fig. 3 Optical spectra at (a) the output port of IM1, (b) the output port of PM, (c) the port 1 and (d) the port 2 of the de-interleaver
Fig. 4
Fig. 4 Generated chirped microwave pulse. (a) Single-period waveform, (b) instantaneous frequency (solid line: theoretical value, circles: measured value).
Fig. 5
Fig. 5 Generated chirped microwave pulses with different central frequencies at (a) 20GHz, (b) 22GHz, (c) 24GHz, and (d) 26GHz.
Fig. 6
Fig. 6 Generated chirped microwave pulses with (a) negative chirp and (b) positive chirp.
Fig. 7
Fig. 7 Generated chirped microwave pulses with different pulse repetition frequencies at (a) 2 GHz and (d) 1.5 GHz.

Equations (7)

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E ( t ) { exp [ j ω c t + j β 1 cos ( ω R F 1 t ) ] exp [ j ω c t j β 1 cos ( ω R F 1 t ) ] }
E ( t ) = E 1 exp j ( ω c ω R F 1 ) t + E + 1 exp j ( ω c + ω R F 1 ) t
E ( t ) = E 1 exp [ j ( ω c ω R F 1 ) t + j β 2 cos ( ω R F 2 t ) ] + E + 1 exp [ j ( ω c + ω R F 1 ) t + j β 2 cos ( ω R F 2 t ) ]
E ( t ) E 1 exp { j ( ω c ω R F 1 ) ( t + τ ) + j β 2 cos [ ω R F 2 ( t + τ ) ] } + E + 1 exp [ j ( ω c + ω R F 1 ) t + j β 2 cos ( ω R F 2 t ) ]
H ( t ) = 1 2 cos [ π 2 cos ( ω R F 3 t + ϕ ) + π 2 ] + 1 2
I ( t ) E 1 E + 1 { cos [ π 2 cos ( ω R F 3 t + ϕ ) + π 2 ] + 1 } { cos [ 2 ω R F 1 t + 2 β 2 sin ω R F 2 τ 2 sin ( ω R F 2 t + ω R F 2 τ 2 ) ] + A }
f ( t ) = 1 2 π d θ d t = 1 π { ω R F 1 + β 2 ω R F 2 sin ω R F 2 τ 2 cos [ ω R F 2 ( t + τ 2 ) ] }

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