Abstract

Extracting precise target characteristics from microwave image is needed and calls for high-resolution microwave imaging radar systems. In this paper, a Ka-band ultra-wideband microwave photonic (MWP) imaging radar is developed and experimentally demonstrated. In the transmitter, continuous ultra-wideband linear frequency modulation (LFM) wave is generated based on optical frequency sextupling technique. In the receiver, a combination of optical frequency mixer with fiber delay lines and electric analog-to-digital converter (ADC) is capable of receiving target echoes and imaging targets with different distances. The maximum instantaneous bandwidth of the transmitted waveform is measured to be 10.02 GHz and corresponding range resolution is calibrated to be 1.68 cm. Out-field tests with demonstrator working at synthetic aperture radar (SAR) or inverse synthetic aperture radar (ISAR) mode are carried out. Different targets such as an unmanned aerial vehicle (UAV), airliner and Leifeng pagoda are imaged. Based on corresponding high-resolution microwave images, quantitative information of the targets can be identified, which shows the great potential of the radar demonstrator for various remote sensing applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (5)

2016 (1)

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

2015 (2)

F. Scotti, D. Onori, and F. Laghezza, “Fully Coherent S- and X-Band Photonics-Aided Radar System Demonstration,” IEEE Microw. Wirel. Co. 25(11), 757–759 (2015).
[Crossref]

J. Sheng, M. Xing, L. Zhang, M. Q. Mehmood, and L. Yang, “ISAR Cross-Range Scaling by Using Sharpness Maximization,” IEEE Geosci. Remote Sens. 12(1), 165–169 (2015).
[Crossref]

2014 (2)

J. D. McKinney, “Photonics illuminates the future of radar,” Nature 507(7492), 310–312 (2014).
[Crossref] [PubMed]

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]

2013 (1)

2011 (1)

L. Maleki, “The optoelectronic oscillator,” Nat. Photonics 5(12), 728–730 (2011).
[Crossref]

2010 (1)

J. Yao, “Arbitrary waveform generation,” Nat. Photonics 4(2), 79–80 (2010).
[Crossref]

2009 (1)

2007 (1)

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

1996 (1)

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]

Bogoni, A.

F. Scotti, F. Laghezza, D. Onori, and A. Bogoni, “Field trial of a photonics-based dual-band fully coherent radar system in a maritime scenario,” IET Radar Sonar Nav 11(3), 420–425 (2017).
[Crossref]

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]

F. Laghezza, F. Scotti, D. Onori, and A. Bogoni, “ISAR Imaging of Non-Cooperative Targets via Dual Band Photonics-Based Radar System,” Int Radar Symp Proc (2016).
[Crossref]

Capmany, J.

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, J.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Cui, Y.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Ding, M.

Du, P. F.

Gao, B.

Gasulla, I.

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]

Guo, P.

Guo, Q.

Guo, Q. S.

F. Z. Zhang, Q. S. Guo, Y. Zhang, Y. Yao, P. Zhou, D. Y. Zhu, and S. L. Pan, “Photonics-based real-time and high-resolution ISAR imaging of non-cooperative target,” Chin. Opt. Lett. 15(11), 95–98 (2017).

Laghezza, F.

F. Scotti, F. Laghezza, D. Onori, and A. Bogoni, “Field trial of a photonics-based dual-band fully coherent radar system in a maritime scenario,” IET Radar Sonar Nav 11(3), 420–425 (2017).
[Crossref]

F. Scotti, D. Onori, and F. Laghezza, “Fully Coherent S- and X-Band Photonics-Aided Radar System Demonstration,” IEEE Microw. Wirel. Co. 25(11), 757–759 (2015).
[Crossref]

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]

F. Laghezza, F. Scotti, D. Onori, and A. Bogoni, “ISAR Imaging of Non-Cooperative Targets via Dual Band Photonics-Based Radar System,” Int Radar Symp Proc (2016).
[Crossref]

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]

Li, R.

Li, S.

Li, W.

Li, Y.

Liang, X.

Lloret, J.

Long, X.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Luan, Y.

Luo, X.

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]

Maleki, L.

L. Maleki, “The optoelectronic oscillator,” Nat. Photonics 5(12), 728–730 (2011).
[Crossref]

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

McKinney, J. D.

J. D. McKinney, “Photonics illuminates the future of radar,” Nature 507(7492), 310–312 (2014).
[Crossref] [PubMed]

Mehmood, M. Q.

J. Sheng, M. Xing, L. Zhang, M. Q. Mehmood, and L. Yang, “ISAR Cross-Range Scaling by Using Sharpness Maximization,” IEEE Geosci. Remote Sens. 12(1), 165–169 (2015).
[Crossref]

Mora, J.

Novak, D.

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

Onori, D.

F. Scotti, F. Laghezza, D. Onori, and A. Bogoni, “Field trial of a photonics-based dual-band fully coherent radar system in a maritime scenario,” IET Radar Sonar Nav 11(3), 420–425 (2017).
[Crossref]

F. Scotti, D. Onori, and F. Laghezza, “Fully Coherent S- and X-Band Photonics-Aided Radar System Demonstration,” IEEE Microw. Wirel. Co. 25(11), 757–759 (2015).
[Crossref]

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]

F. Laghezza, F. Scotti, D. Onori, and A. Bogoni, “ISAR Imaging of Non-Cooperative Targets via Dual Band Photonics-Based Radar System,” Int Radar Symp Proc (2016).
[Crossref]

Pan, S.

Pan, S. L.

F. Z. Zhang, Q. S. Guo, Y. Zhang, Y. Yao, P. Zhou, D. Y. Zhu, and S. L. Pan, “Photonics-based real-time and high-resolution ISAR imaging of non-cooperative target,” Chin. Opt. Lett. 15(11), 95–98 (2017).

Peng, S.

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]

Sales, S.

Sancho, J.

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.

F. Scotti, F. Laghezza, D. Onori, and A. Bogoni, “Field trial of a photonics-based dual-band fully coherent radar system in a maritime scenario,” IET Radar Sonar Nav 11(3), 420–425 (2017).
[Crossref]

F. Scotti, D. Onori, and F. Laghezza, “Fully Coherent S- and X-Band Photonics-Aided Radar System Demonstration,” IEEE Microw. Wirel. Co. 25(11), 757–759 (2015).
[Crossref]

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]

F. Laghezza, F. Scotti, D. Onori, and A. Bogoni, “ISAR Imaging of Non-Cooperative Targets via Dual Band Photonics-Based Radar System,” Int Radar Symp Proc (2016).
[Crossref]

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]

Sheng, J.

J. Sheng, M. Xing, L. Zhang, M. Q. Mehmood, and L. Yang, “ISAR Cross-Range Scaling by Using Sharpness Maximization,” IEEE Geosci. Remote Sens. 12(1), 165–169 (2015).
[Crossref]

Sun, J.

Tian, Y.

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, A.

Wang, Z.

Wen, Z.

Wo, J. H.

Wu, D.

Xiao, X.

Xing, M.

J. Sheng, M. Xing, L. Zhang, M. Q. Mehmood, and L. Yang, “ISAR Cross-Range Scaling by Using Sharpness Maximization,” IEEE Geosci. Remote Sens. 12(1), 165–169 (2015).
[Crossref]

Xing, T.

Xu, X.

Xue, X.

Yang, L.

J. Sheng, M. Xing, L. Zhang, M. Q. Mehmood, and L. Yang, “ISAR Cross-Range Scaling by Using Sharpness Maximization,” IEEE Geosci. Remote Sens. 12(1), 165–169 (2015).
[Crossref]

Yao, J.

J. Yao, “Arbitrary waveform generation,” Nat. Photonics 4(2), 79–80 (2010).
[Crossref]

J. Yao, “Microwave Photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
[Crossref]

Yao, X. S.

Yao, Y.

F. Z. Zhang, Q. S. Guo, Y. Zhang, Y. Yao, P. Zhou, D. Y. Zhu, and S. L. Pan, “Photonics-based real-time and high-resolution ISAR imaging of non-cooperative target,” Chin. Opt. Lett. 15(11), 95–98 (2017).

Yu, L.

Yu, S.

Zhang, D. M.

Zhang, F.

Zhang, F. Z.

F. Z. Zhang, Q. S. Guo, Y. Zhang, Y. Yao, P. Zhou, D. Y. Zhu, and S. L. Pan, “Photonics-based real-time and high-resolution ISAR imaging of non-cooperative target,” Chin. Opt. Lett. 15(11), 95–98 (2017).

Zhang, G.

Zhang, H.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Zhang, J.

Zhang, L.

J. Sheng, M. Xing, L. Zhang, M. Q. Mehmood, and L. Yang, “ISAR Cross-Range Scaling by Using Sharpness Maximization,” IEEE Geosci. Remote Sens. 12(1), 165–169 (2015).
[Crossref]

Zhang, S.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Zhang, Y.

F. Z. Zhang, Q. S. Guo, Y. Zhang, Y. Yao, P. Zhou, D. Y. Zhu, and S. L. Pan, “Photonics-based real-time and high-resolution ISAR imaging of non-cooperative target,” Chin. Opt. Lett. 15(11), 95–98 (2017).

Zheng, X.

Zhou, B.

Zhou, L.

Zhou, P.

F. Z. Zhang, Q. S. Guo, Y. Zhang, Y. Yao, P. Zhou, D. Y. Zhu, and S. L. Pan, “Photonics-based real-time and high-resolution ISAR imaging of non-cooperative target,” Chin. Opt. Lett. 15(11), 95–98 (2017).

F. Zhang, Q. Guo, Z. Wang, P. Zhou, G. Zhang, J. Sun, and S. Pan, “Photonics-based broadband radar for high-resolution and real-time inverse synthetic aperture imaging,” Opt. Express 25(14), 16274–16281 (2017).
[Crossref] [PubMed]

Zhu, D. Y.

F. Z. Zhang, Q. S. Guo, Y. Zhang, Y. Yao, P. Zhou, D. Y. Zhu, and S. L. Pan, “Photonics-based real-time and high-resolution ISAR imaging of non-cooperative target,” Chin. Opt. Lett. 15(11), 95–98 (2017).

Zhu, Y.

Zou, W.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Chin. Opt. Lett. (2)

A. Wang, J. H. Wo, J. Zhang, X. Luo, X. Xu, D. M. Zhang, P. F. Du, and L. Yu, “Radio-frequency arbitrary waveform generation based on dispersion compensated tunable optoelectronic oscillator with ultra-wide tunability,” Chin. Opt. Lett. 15(10), 100603 (2017).
[Crossref]

F. Z. Zhang, Q. S. Guo, Y. Zhang, Y. Yao, P. Zhou, D. Y. Zhu, and S. L. Pan, “Photonics-based real-time and high-resolution ISAR imaging of non-cooperative target,” Chin. Opt. Lett. 15(11), 95–98 (2017).

IEEE Geosci. Remote Sens. (1)

J. Sheng, M. Xing, L. Zhang, M. Q. Mehmood, and L. Yang, “ISAR Cross-Range Scaling by Using Sharpness Maximization,” IEEE Geosci. Remote Sens. 12(1), 165–169 (2015).
[Crossref]

IEEE Microw. Wirel. Co. (1)

F. Scotti, D. Onori, and F. Laghezza, “Fully Coherent S- and X-Band Photonics-Aided Radar System Demonstration,” IEEE Microw. Wirel. Co. 25(11), 757–759 (2015).
[Crossref]

IET Radar Sonar Nav (1)

F. Scotti, F. Laghezza, D. Onori, and A. Bogoni, “Field trial of a photonics-based dual-band fully coherent radar system in a maritime scenario,” IET Radar Sonar Nav 11(3), 420–425 (2017).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (1)

Nat. Photonics (3)

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

L. Maleki, “The optoelectronic oscillator,” Nat. Photonics 5(12), 728–730 (2011).
[Crossref]

J. Yao, “Arbitrary waveform generation,” Nat. Photonics 4(2), 79–80 (2010).
[Crossref]

Nature (2)

J. D. McKinney, “Photonics illuminates the future of radar,” Nature 507(7492), 310–312 (2014).
[Crossref] [PubMed]

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]

Opt. Express (3)

Sci. Rep. (1)

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Other (4)

F. Laghezza, F. Scotti, D. Onori, and A. Bogoni, “ISAR Imaging of Non-Cooperative Targets via Dual Band Photonics-Based Radar System,” Int Radar Symp Proc (2016).
[Crossref]

V. Chen and M. Martorella, Inverse Synthetic Aperture Radar Imaging: Principles, Algorithms, and Applications (SciTech Publishing, 2014)

W. L. Melvin and J. A. Scheer, Principles of Modern Radar: Vol. II Advanced Techniques (SciTech Publishing, 2012).

https://www.ll.mit.edu//publications/technotes/TechNote_HUSIR.pdf .

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Figures (5)

Fig. 1
Fig. 1 Schematic of the MWP ultra-wideband imaging radar. MZM: Mach-Zehnder modulator; DDS: direct digital synthesizer; OF: optical filter; PD: photodetector; PA: power amplifier; TA: transmitting antenna; OTD: optical time delayer; EDFA: erbium doped fiber amplifier; LPF: low-pass filter; ADC: analog-digital converter; LNA: low-noise amplifier; RA: receiving antenna.
Fig. 2
Fig. 2 The photograph of the radar demonstrator (a) and its performance test results: (b) is the optical spectrum of light output from MZM1, (c) is the spectrum of the generated LFWM signal and (d) is the resolution calibration result, in which P1 and P2 represent the reflectors.
Fig. 3
Fig. 3 The diagram of the UAV ISAR experiment in overhead view (a), the photograph of the six-rotor UAV used in the experiment (b) and its ISAR image (c) obtained by the demonstrator.
Fig. 4
Fig. 4 Experimental imaging results of Boeing 737. (a) ISAR imaging result of the whole plane, the inset is photograph of a Boeing 737; (b) enlarged view of the down wing part in (a), the inset is the enlarged view of the rectangle in white dashed line; (c) the photograph of the wing part, the inset corresponds to the enlarged view in (b).
Fig. 5
Fig. 5 The photograph of the Leifeng pagoda and the analysis of its SAR image. (a) the photograph of the Leifeng pagoda; (a) SAR imaging results of the pagoda; (c) enlarged view of the spire part in (b); (d) photograph of the spire part corresponding to (c).

Equations (8)

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E o1 ( t )=rect( t T )* n=1 + ( 1 ) n J 2n1 ( β 1 ) E l { cos[ ω l t+( 2n1 )( ω 0 t+π K 0 t 2 ) ]+ cos[ ω l t( 2n1 )( ω 0 t+π K 0 t 2 ) ] }
E o1 ( t )=rect( t T ) J 3 ( β 1 ) E l { cos[ ω l t+3( ω 0 t+π K 0 t 2 ) ] +cos[ ω l t3( ω 0 t+π K 0 t 2 ) ] }.
I 1 ( t )rect( t T )cos[ 6( ω 0 t+π K 0 t 2 ) ].
E l ( t )=rect( tτ T ) E l { cos[ ω l ( tτ )+3( ω 0 ( tτ )+π K 0 ( tτ ) 2 ) ] +cos[ ω l ( tτ )3( ω 0 ( tτ )+π K 0 ( tτ ) 2 ) ] }.
V eco ( t )=rect( t t R T ) V eco cos[ 6 ω 0 ( t t R )+6π K 0 ( t t R ) 2 ],
E o2 ( t )= E l ( t )cos[ Δϕ+π V eco ( t ) V π2 ],
E o2 ( t )= 2 2 rect( t 1 T )rect( t 2 T )* E l { { J 0 ( β 2 )cos[ ω l t 1 +3( ω 0 t 1 +π K 0 t 1 2 ) ] J 1 ( β 2 )cos[ ω l t 1 +3( ω 0 t 1 +π K 0 t 1 2 )+6( ω 0 t 2 +π K 0 t 2 2 ) ] J 1 ( β 2 )cos[ ω l t 1 +3( ω 0 t 1 +π K 0 t 1 2 )6( ω 0 t 2 +π K 0 t 2 2 ) ] } +{ J 0 ( β 2 )cos[ ω l t 1 3( ω 0 t 1 +π K 0 t 1 2 ) ] J 1 ( β 2 )cos[ ω l t 1 3( ω 0 t 1 +π K 0 t 1 2 )+6( ω 0 t 2 +π K 0 t 2 2 ) ] J 1 ( β 2 )cos[ ω l t 1 3( ω 0 t 1 +π K 0 t 1 2 )6( ω 0 t 2 +π K 0 t 2 2 ) ] } }.
I( t )rect( tτ T )rect( t t R T )*cos[ 12π K 0 ( τ t R )t+6 ω 0 ( τ t R )+6π K 0 ( t R 2 τ 2 ) ].

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