Abstract

We propose and experimentally demonstrate a novel and practical microwave photonic system that is capable of executing cascaded signal processing functions comprising a microwave photonic bandpass filter and a phase shifter, while providing separate and independent control for each function. The experimental results demonstrate a single bandpass microwave photonic filter with a 3-dB bandwidth of 15 MHz and an out-of-band ratio of over 40 dB, together with a simultaneous RF phase tuning control of 0-215° with less than ± 3 dB filter shape variance.

© 2016 Optical Society of America

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References

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  6. J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24(1), 201–229 (2006).
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  7. W. Xue, S. Sales, J. Capmany, and J. Mørk, “Wideband 360° microwave photonic phase shifter based on slow light in semiconductor optical amplifiers,” Opt. Express 18(6), 6156–6163 (2010).
    [Crossref] [PubMed]
  8. A. Loayssa and F. J. Lahoz, “Broadband RF photonic phase shifter based on stimulated Brillouin scattering and single side-band modulation,” IEEE Photonics Technol. Lett. 18(1), 208–210 (2006).
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  22. S. Hu, L. Li, X. Yi, and C. Yu, “Ultraflat widely tuned single bandpass filter based on stimulated Brillouin Scattering,” IEEE Photonics Technol. Lett. 26(14), 1466–1469 (2014).
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    [Crossref]
  24. M. Nikles, L. Thévenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
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  28. G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, and J. Bowers, “III-V/silicon photonics for on-chip and intra-chip optical interconnects,” Laser Photonics Rev. 4(6), 751–779 (2010).
    [Crossref]
  29. J. K. Doylend and A. P. Knights, “The evolution of silicon photonics as an enabling technology for optical interconnection,” Laser Photonics Rev. 6(4), 504–525 (2012).
    [Crossref]
  30. M. Pu, L. Liu, W. Xue, Y. Ding, H. Ou, K. Yvind, and J. M. Hvam, “Widely tunable microwave phase shifter based on silicon-on-insulator dual-microring resonator,” Opt. Express 18(6), 6172–6182 (2010).
    [Crossref] [PubMed]
  31. M. Pu, L. Liu, W. Xue, Y. Ding, H. Ou, K. Yvind, J. M. Hvam, and J. M. Hvam, “Widely tunable microwave phase shifter based on silicon-on-insulator dual-microring resonator,” Opt. Express 18(6), 6172–6182 (2010).
    [Crossref] [PubMed]
  32. B. J. Eggleton, C. G. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photonics 5(4), 536–587 (2013).
    [Crossref]
  33. M. Merklein, I. V. Kabakova, T. F. Büttner, D. Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun. 6, 6396 (2015).
    [Crossref] [PubMed]

2015 (2)

M. Merklein, I. V. Kabakova, T. F. Büttner, D. Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun. 6, 6396 (2015).
[Crossref] [PubMed]

J. F. Diehl, J. M. Singley, C. E. Sunderman, and V. J. Urick, “Microwave photonic delay line signal processing,” Appl. Opt. 54(31), F35–F41 (2015).
[Crossref] [PubMed]

2014 (2)

W. Wei, L. Yi, Y. Jaouën, and W. Hu, “Bandwidth-tunable narrowband rectangular optical filter based on stimulated Brillouin scattering in optical fiber,” Opt. Express 22(19), 23249–23260 (2014).
[Crossref] [PubMed]

S. Hu, L. Li, X. Yi, and C. Yu, “Ultraflat widely tuned single bandpass filter based on stimulated Brillouin Scattering,” IEEE Photonics Technol. Lett. 26(14), 1466–1469 (2014).
[Crossref]

2013 (3)

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

B. J. Eggleton, C. G. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photonics 5(4), 536–587 (2013).
[Crossref]

R. A. Minasian, E. H. W. Chan, and X. Yi, “Microwave photonic signal processing,” Opt. Express 21(19), 22918–22936 (2013).
[Crossref] [PubMed]

2012 (5)

2011 (1)

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(23), 1775–1777 (2011).
[Crossref]

2010 (5)

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, and J. Bowers, “III-V/silicon photonics for on-chip and intra-chip optical interconnects,” Laser Photonics Rev. 4(6), 751–779 (2010).
[Crossref]

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O’Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay lines in silicon photonics: coupled resonators and photonic crystals, a comparison,” IEEE Photonics J. 2(2), 181–194 (2010).
[Crossref]

W. Xue, S. Sales, J. Capmany, and J. Mørk, “Wideband 360° microwave photonic phase shifter based on slow light in semiconductor optical amplifiers,” Opt. Express 18(6), 6156–6163 (2010).
[Crossref] [PubMed]

M. Pu, L. Liu, W. Xue, Y. Ding, H. Ou, K. Yvind, and J. M. Hvam, “Widely tunable microwave phase shifter based on silicon-on-insulator dual-microring resonator,” Opt. Express 18(6), 6172–6182 (2010).
[Crossref] [PubMed]

M. Pu, L. Liu, W. Xue, Y. Ding, H. Ou, K. Yvind, J. M. Hvam, and J. M. Hvam, “Widely tunable microwave phase shifter based on silicon-on-insulator dual-microring resonator,” Opt. Express 18(6), 6172–6182 (2010).
[Crossref] [PubMed]

2009 (2)

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

Q. Chang, Q. Li, Z. Zhang, M. Qiu, T. Ye, and Y. Su, “A tunable broadband photonic RF phase shifter based on a silicon microring resonator,” IEEE Photonics Technol. Lett. 21(1), 60–62 (2009).
[Crossref]

2008 (1)

2007 (1)

2006 (4)

J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24(1), 201–229 (2006).
[Crossref]

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006).
[Crossref]

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006).
[Crossref]

A. Loayssa and F. J. Lahoz, “Broadband RF photonic phase shifter based on stimulated Brillouin scattering and single side-band modulation,” IEEE Photonics Technol. Lett. 18(1), 208–210 (2006).
[Crossref]

2005 (2)

J. Wang, F. Zeng, and J. P. Yao, “All-optical microwave bandpass filters implemented in a radio-over-fiber link,” IEEE Photonics Technol. Lett. 17(8), 1737–1739 (2005).
[Crossref]

J. P. Yao, G. Maury, Y. L. Guennec, and B. Cabon, “All-optical subcarrier frequency conversion using an electrooptic phase modulator,” IEEE Photonics Technol. Lett. 17(11), 2427–2429 (2005).
[Crossref]

2002 (1)

A. J. Seeds, “Microwave photonics,” IEEE Trans. Microw. Theory Tech. 50(3), 877–887 (2002).
[Crossref]

1997 (1)

M. Nikles, L. Thévenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

Adams, D. B.

Ben, D.

Bowers, J.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, and J. Bowers, “III-V/silicon photonics for on-chip and intra-chip optical interconnects,” Laser Photonics Rev. 4(6), 751–779 (2010).
[Crossref]

Büttner, T. F.

M. Merklein, I. V. Kabakova, T. F. Büttner, D. Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun. 6, 6396 (2015).
[Crossref] [PubMed]

Cabon, B.

J. P. Yao, G. Maury, Y. L. Guennec, and B. Cabon, “All-optical subcarrier frequency conversion using an electrooptic phase modulator,” IEEE Photonics Technol. Lett. 17(11), 2427–2429 (2005).
[Crossref]

Cai, S.

Canciamilla, A.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O’Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay lines in silicon photonics: coupled resonators and photonic crystals, a comparison,” IEEE Photonics J. 2(2), 181–194 (2010).
[Crossref]

Capmany, J.

Chan, E. H. W.

Chang, Q.

Q. Chang, Q. Li, Z. Zhang, M. Qiu, T. Ye, and Y. Su, “A tunable broadband photonic RF phase shifter based on a silicon microring resonator,” IEEE Photonics Technol. Lett. 21(1), 60–62 (2009).
[Crossref]

Chen, T.

Choi, D. Y.

M. Merklein, I. V. Kabakova, T. F. Büttner, D. Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun. 6, 6396 (2015).
[Crossref] [PubMed]

De La Rue, R.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O’Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay lines in silicon photonics: coupled resonators and photonic crystals, a comparison,” IEEE Photonics J. 2(2), 181–194 (2010).
[Crossref]

Diehl, J. F.

Ding, Y.

Doylend, J. K.

J. K. Doylend and A. P. Knights, “The evolution of silicon photonics as an enabling technology for optical interconnection,” Laser Photonics Rev. 6(4), 504–525 (2012).
[Crossref]

Eggleton, B. J.

M. Merklein, I. V. Kabakova, T. F. Büttner, D. Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun. 6, 6396 (2015).
[Crossref] [PubMed]

B. J. Eggleton, C. G. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photonics 5(4), 536–587 (2013).
[Crossref]

Eyal, A.

Fang, A.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, and J. Bowers, “III-V/silicon photonics for on-chip and intra-chip optical interconnects,” Laser Photonics Rev. 4(6), 751–779 (2010).
[Crossref]

Ferrari, C.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O’Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay lines in silicon photonics: coupled resonators and photonic crystals, a comparison,” IEEE Photonics J. 2(2), 181–194 (2010).
[Crossref]

Guennec, Y. L.

J. P. Yao, G. Maury, Y. L. Guennec, and B. Cabon, “All-optical subcarrier frequency conversion using an electrooptic phase modulator,” IEEE Photonics Technol. Lett. 17(11), 2427–2429 (2005).
[Crossref]

Heideman, R.

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

Hu, S.

S. Hu, L. Li, X. Yi, and C. Yu, “Ultraflat widely tuned single bandpass filter based on stimulated Brillouin Scattering,” IEEE Photonics Technol. Lett. 26(14), 1466–1469 (2014).
[Crossref]

Hu, W.

Hvam, J. M.

Jaouën, Y.

Jones, R.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, and J. Bowers, “III-V/silicon photonics for on-chip and intra-chip optical interconnects,” Laser Photonics Rev. 4(6), 751–779 (2010).
[Crossref]

Kabakova, I. V.

M. Merklein, I. V. Kabakova, T. F. Büttner, D. Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun. 6, 6396 (2015).
[Crossref] [PubMed]

Knights, A. P.

J. K. Doylend and A. P. Knights, “The evolution of silicon photonics as an enabling technology for optical interconnection,” Laser Photonics Rev. 6(4), 504–525 (2012).
[Crossref]

Koch, B.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, and J. Bowers, “III-V/silicon photonics for on-chip and intra-chip optical interconnects,” Laser Photonics Rev. 4(6), 751–779 (2010).
[Crossref]

Krauss, T. F.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O’Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay lines in silicon photonics: coupled resonators and photonic crystals, a comparison,” IEEE Photonics J. 2(2), 181–194 (2010).
[Crossref]

Lahoz, F. J.

A. Loayssa and F. J. Lahoz, “Broadband RF photonic phase shifter based on stimulated Brillouin scattering and single side-band modulation,” IEEE Photonics Technol. Lett. 18(1), 208–210 (2006).
[Crossref]

Leinse, A.

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

Li, L.

S. Hu, L. Li, X. Yi, and C. Yu, “Ultraflat widely tuned single bandpass filter based on stimulated Brillouin Scattering,” IEEE Photonics Technol. Lett. 26(14), 1466–1469 (2014).
[Crossref]

T. Chen, X. Yi, L. Li, and R. Minasian, “Single passband microwave photonic filter with wideband tunability and adjustable bandwidth,” Opt. Lett. 37(22), 4699–4701 (2012).
[Crossref] [PubMed]

Li, Q.

Q. Chang, Q. Li, Z. Zhang, M. Qiu, T. Ye, and Y. Su, “A tunable broadband photonic RF phase shifter based on a silicon microring resonator,” IEEE Photonics Technol. Lett. 21(1), 60–62 (2009).
[Crossref]

Liang, D.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, and J. Bowers, “III-V/silicon photonics for on-chip and intra-chip optical interconnects,” Laser Photonics Rev. 4(6), 751–779 (2010).
[Crossref]

Liu, L.

Loayssa, A.

A. Loayssa and F. J. Lahoz, “Broadband RF photonic phase shifter based on stimulated Brillouin scattering and single side-band modulation,” IEEE Photonics Technol. Lett. 18(1), 208–210 (2006).
[Crossref]

Luther-Davies, B.

M. Merklein, I. V. Kabakova, T. F. Büttner, D. Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun. 6, 6396 (2015).
[Crossref] [PubMed]

Madden, S. J.

M. Merklein, I. V. Kabakova, T. F. Büttner, D. Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun. 6, 6396 (2015).
[Crossref] [PubMed]

Madsen, C. K.

Marpaung, D.

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

Marti, J.

B. Vidal, T. Menguak, and J. Marti, “Photonic microwave filter with single bandpass response based on Brillouin processing and SSB-SC,” in Proc. IEEE. Int. Meeting on Microwave Photonics (MWP, 2009), pp. 1–4.

Maury, G.

J. P. Yao, G. Maury, Y. L. Guennec, and B. Cabon, “All-optical subcarrier frequency conversion using an electrooptic phase modulator,” IEEE Photonics Technol. Lett. 17(11), 2427–2429 (2005).
[Crossref]

Melloni, A.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O’Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay lines in silicon photonics: coupled resonators and photonic crystals, a comparison,” IEEE Photonics J. 2(2), 181–194 (2010).
[Crossref]

Menguak, T.

B. Vidal, T. Menguak, and J. Marti, “Photonic microwave filter with single bandpass response based on Brillouin processing and SSB-SC,” in Proc. IEEE. Int. Meeting on Microwave Photonics (MWP, 2009), pp. 1–4.

Merklein, M.

M. Merklein, I. V. Kabakova, T. F. Büttner, D. Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun. 6, 6396 (2015).
[Crossref] [PubMed]

Minasian, R.

Minasian, R. A.

R. A. Minasian, E. H. W. Chan, and X. Yi, “Microwave photonic signal processing,” Opt. Express 21(19), 22918–22936 (2013).
[Crossref] [PubMed]

W. Zhang and R. A. Minasian, “Switchable and tunable microwave photonic Brillouin-based filter,” IEEE Photonics J. 4(5), 1443–1455 (2012).
[Crossref]

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(23), 1775–1777 (2011).
[Crossref]

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006).
[Crossref]

Morichetti, F.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O’Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay lines in silicon photonics: coupled resonators and photonic crystals, a comparison,” IEEE Photonics J. 2(2), 181–194 (2010).
[Crossref]

Mørk, J.

Nikles, M.

M. Nikles, L. Thévenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

O’Faolain, L.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. O’Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “Tunable delay lines in silicon photonics: coupled resonators and photonic crystals, a comparison,” IEEE Photonics J. 2(2), 181–194 (2010).
[Crossref]

Ortega, B.

Ou, H.

Pan, S.

Pant, R.

B. J. Eggleton, C. G. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photonics 5(4), 536–587 (2013).
[Crossref]

Pastor, D.

Poulton, C. G.

B. J. Eggleton, C. G. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photonics 5(4), 536–587 (2013).
[Crossref]

Pu, M.

Qiu, M.

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Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

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IEEE Photonics J. (2)

W. Zhang and R. A. Minasian, “Switchable and tunable microwave photonic Brillouin-based filter,” IEEE Photonics J. 4(5), 1443–1455 (2012).
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IEEE Photonics Technol. Lett. (6)

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[Crossref]

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(23), 1775–1777 (2011).
[Crossref]

S. Hu, L. Li, X. Yi, and C. Yu, “Ultraflat widely tuned single bandpass filter based on stimulated Brillouin Scattering,” IEEE Photonics Technol. Lett. 26(14), 1466–1469 (2014).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the distributed optical signal processing MWP subsystem with cascaded functionalities.

Fig. 2
Fig. 2

Experimental setup of the proposed system. Inset: Top-view scanning electron microscope (SEM) image of the fabricated on-chip microring resonator

Fig. 3
Fig. 3

Measured RF response of the continuously tunable single passband MWP filter

Fig. 4
Fig. 4

Measured responses of the MPPS (a) continuous RF phase tuning (b) superimposed RF power variations at various RF phase shifts. Inset: Optical response of the phase shifter.

Fig. 5
Fig. 5

Measured RF responses (a) Superimposed RF spectra at various RF phase shifts of the bandpass filter response at 20.828 GHz (b) Continuous RF phase tuning at passband

Fig. 6
Fig. 6

Measured RF responses (a) Superimposed RF spectra at various RF phase shifts of the bandpass filter response at 30.828 GHz (b) Continuous RF phase tuning at passband

Equations (8)

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E PM (t)= E 0 [ J 0 ( m 1 ) e j2π f c t J 1 ( m 1 ) e j2π( f c f RF )t + J 1 ( m 1 ) e j2π( f c + f RF )t ]
E IM = E 0 [ J 0 ( m 1 )( e j2π f c t + m 2 2 e j2π( f c + f m )t + m 2 2 e j2π( f c f m )t ) J 1 ( m 1 )( e j2π( f c f RF )t + m 2 2 e j2π( f c f RF + f m )t + m 2 2 e j2π( f c f RF f m )t ) + J 1 ( m 1 )( e j2π( f c + f RF )t + m 2 2 e j2π( f c + f RF + f m )t + m 2 2 e j2π( f c + f RF f m )t ) ]
E P1 E 0 [ J 0 ( m 1 ) e j2π f c t J 1 ( m 1 ) e j2π( f c f RF )t + J 1 ( m 1 ) e j2π( f c + f RF )t e G( f c + f RF ) ]
G( f c + f RF )= g 0 I p 2 ( Γ B /2) 2 (f) 2 + ( Γ B /2) 2 +j g 0 I p 4 f Γ B (f) 2 + ( Γ B /2) 2
I PD 2 E 0 2 J 0 ( m 1 ) J 1 ( m 1 )| H( f c ) |[ G B cos( 2π f RF t ϕ c + Φ B )cos( 2π f RF t+ ϕ c ) ]
G B =exp[ g 0 2 P p L A eff ( Γ B /2) 2 ( f m f B f RF ) 2 + ( Γ B /2) 2 ]
Φ B = g 0 4 P p L A eff ( f m f B f RF ) Γ B ( f m f B f RF ) 2 + ( Γ B /2) 2
P PD ( f RF )=2R 2 E 0 4 J 0 2 ( m 1 ) J 1 2 ( m 1 ) | H( f c ) | 2 [ G B 2 +12 G B cos( 2 ϕ c Φ B ) ]

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