Abstract

A single passband microwave photonic filter with ultrawide tunable range based on stimulated Brillouin scattering is theoretically analyzed. Combining the gain and loss spectrums, tuning range with 44GHz is obtained without crosstalk by introducing two pumps. Adding more pumps, Tuning range multiplying with the multiplication factor equaling to the total quantity of pump can be achieved, which has potential application in microwave and millimeter wave wireless communication systems.

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References

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  1. J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol.24, 201–229 (2006).
  2. R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE T Microw Theory54(2), 832–846 (2006).
    [CrossRef]
  3. J. Yao, “Photonics for microwave signal filtering,” in IEEE Sarnoff Symposium, 2009. SARNOFF '09(2009), 1–5.
  4. V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics6(3), 186–194 (2012).
    [CrossRef]
  5. R. Wu, C. M. Long, D. E. Leaird, and A. M. Weiner, “Directly generated Gaussian-shaped optical frequency comb for microwave photonic filtering and picosecond pulse generation,” IEEE Photon. Technol. Lett.24(17), 1484–1486 (2012).
    [CrossRef]
  6. T. X. H. Huang, X. Yi, and R. A. Minasian, “A high-order FIR microwave photonic filter,” in International Topical Meeting on Microwave Photonics, 2009. MWP '09(2009), 1–4.
  7. M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett.23(21), 1618–1620 (2011).
    [CrossRef]
  8. Y. Dai and J. Yao, “Nonuniformly-spaced photonic microwave delayline filter,” Opt. Express16(7), 4713–4718 (2008).
    [CrossRef] [PubMed]
  9. B. Vidal, J. L. Corral, and J. Martí, “All-optical WDM multi-tap microwave filter with flat bandpass,” Opt. Express14(2), 581–586 (2006).
    [CrossRef] [PubMed]
  10. J. Mora, B. Ortega, A. Díez, J. L. Cruz, M. V. Andrés, J. Capmany, and D. Pastor, “Photonic microwave tunable single-bandpass filter based on a Mach-Zehnder interferometer,” J. Lightwave Technol.24(7), 2500–2509 (2006).
    [CrossRef]
  11. X. Yi and R. A. Minasian, “Microwave photonic filter with single bandpass response,” Electron. Lett.45(7), 362–363 (2009).
    [CrossRef]
  12. W. Zhang and R. A. Minasian, “Widely Tunable Single-Passband Microwave Photonic Filter Based on Stimulated Brillouin Scattering,” IEEE Photon. Technol. Lett.23(23), 1775–1777 (2011).
    [CrossRef]
  13. W. Li, M. Li, and J. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted Fiber Bragg Grating,” IEEE T Microw Theory60(5), 1287–1296 (2012).
    [CrossRef]
  14. M. Bolea, J. Mora, B. Ortega, and J. Capmany, “Highly chirped single-bandpass microwave photonic filter with reconfiguration capabilities,” Opt. Express19(5), 4566–4576 (2011).
    [CrossRef] [PubMed]
  15. T. X. H. Huang, X. Yi, and R. A. Minasian, “Single passband microwave photonic filter using continuous-time impulse response,” Opt. Express19(7), 6231–6242 (2011).
    [CrossRef] [PubMed]
  16. Y. M. Chang and J. H. Lee, “Tunable, single passband photonic microwave filter based on stimulated Brillouin scattering in nonlinear fiber,” in IEEE LEOS Annual Meeting Conference Proceedings, 2009. LEOS '09(2009), 654–655.
  17. R. Pant, A. Byrnes, E. Li, D.-Y. Choi, C. G. Poulton, S. Fan, S. J. Madden, B. Luther-Davies, and B. J. Eggleton, “Photonic chip based tunable and dynamically reconfigurable microwave photonic filter using stimulated Brillouin scattering,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides(Optical Society of America, 2012).
  18. F. Zeng and J. Yao, “Investigation of phase-modulator-based all-optical bandpass microwave filter,” J. Lightwave Technol.23, 1113– 1117(2005).
  19. W. Zhang and R. A. Minasian, “Switchable and tunable microwave photonic Brillouin-based filter,” IEEE Photonics Journal4(5), 1443–1455 (2012).
    [CrossRef]
  20. P. D. Dragic, “Simplified model for effect of Ge doping on silica fibre acoustic properties,” Electron. Lett.45(5), 256–257 (2009).
    [CrossRef]
  21. N. Guo, R. C. Qiu, S. S. Mo, and K. Takahashi, “60-GHz millimeter-wave radio: principle, technology, and new Results,” EURASIP J. Wirel. Commun. Netw.2007, 1–48 (2007).
    [CrossRef]
  22. A. Loayssa, D. Benito, and M. José Garde, “Applications of optical carrier Brillouin processing to microwave photonics,” Opt. Fiber Technol.8(1), 24–42 (2002).
    [CrossRef]

2012 (4)

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics6(3), 186–194 (2012).
[CrossRef]

R. Wu, C. M. Long, D. E. Leaird, and A. M. Weiner, “Directly generated Gaussian-shaped optical frequency comb for microwave photonic filtering and picosecond pulse generation,” IEEE Photon. Technol. Lett.24(17), 1484–1486 (2012).
[CrossRef]

W. Li, M. Li, and J. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted Fiber Bragg Grating,” IEEE T Microw Theory60(5), 1287–1296 (2012).
[CrossRef]

W. Zhang and R. A. Minasian, “Switchable and tunable microwave photonic Brillouin-based filter,” IEEE Photonics Journal4(5), 1443–1455 (2012).
[CrossRef]

2011 (4)

M. Bolea, J. Mora, B. Ortega, and J. Capmany, “Highly chirped single-bandpass microwave photonic filter with reconfiguration capabilities,” Opt. Express19(5), 4566–4576 (2011).
[CrossRef] [PubMed]

T. X. H. Huang, X. Yi, and R. A. Minasian, “Single passband microwave photonic filter using continuous-time impulse response,” Opt. Express19(7), 6231–6242 (2011).
[CrossRef] [PubMed]

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett.23(21), 1618–1620 (2011).
[CrossRef]

W. Zhang and R. A. Minasian, “Widely Tunable Single-Passband Microwave Photonic Filter Based on Stimulated Brillouin Scattering,” IEEE Photon. Technol. Lett.23(23), 1775–1777 (2011).
[CrossRef]

2009 (2)

X. Yi and R. A. Minasian, “Microwave photonic filter with single bandpass response,” Electron. Lett.45(7), 362–363 (2009).
[CrossRef]

P. D. Dragic, “Simplified model for effect of Ge doping on silica fibre acoustic properties,” Electron. Lett.45(5), 256–257 (2009).
[CrossRef]

2008 (1)

2007 (1)

N. Guo, R. C. Qiu, S. S. Mo, and K. Takahashi, “60-GHz millimeter-wave radio: principle, technology, and new Results,” EURASIP J. Wirel. Commun. Netw.2007, 1–48 (2007).
[CrossRef]

2006 (4)

2005 (1)

F. Zeng and J. Yao, “Investigation of phase-modulator-based all-optical bandpass microwave filter,” J. Lightwave Technol.23, 1113– 1117(2005).

2002 (1)

A. Loayssa, D. Benito, and M. José Garde, “Applications of optical carrier Brillouin processing to microwave photonics,” Opt. Fiber Technol.8(1), 24–42 (2002).
[CrossRef]

Andrés, M. V.

Benito, D.

A. Loayssa, D. Benito, and M. José Garde, “Applications of optical carrier Brillouin processing to microwave photonics,” Opt. Fiber Technol.8(1), 24–42 (2002).
[CrossRef]

Bolea, M.

Capmany, J.

Corral, J. L.

Cruz, J. L.

Dai, Y.

Díez, A.

Dragic, P. D.

P. D. Dragic, “Simplified model for effect of Ge doping on silica fibre acoustic properties,” Electron. Lett.45(5), 256–257 (2009).
[CrossRef]

Ferdous, F.

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics6(3), 186–194 (2012).
[CrossRef]

Guo, N.

N. Guo, R. C. Qiu, S. S. Mo, and K. Takahashi, “60-GHz millimeter-wave radio: principle, technology, and new Results,” EURASIP J. Wirel. Commun. Netw.2007, 1–48 (2007).
[CrossRef]

Hamidi, E.

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics6(3), 186–194 (2012).
[CrossRef]

Huang, T. X. H.

José Garde, M.

A. Loayssa, D. Benito, and M. José Garde, “Applications of optical carrier Brillouin processing to microwave photonics,” Opt. Fiber Technol.8(1), 24–42 (2002).
[CrossRef]

Leaird, D. E.

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics6(3), 186–194 (2012).
[CrossRef]

R. Wu, C. M. Long, D. E. Leaird, and A. M. Weiner, “Directly generated Gaussian-shaped optical frequency comb for microwave photonic filtering and picosecond pulse generation,” IEEE Photon. Technol. Lett.24(17), 1484–1486 (2012).
[CrossRef]

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett.23(21), 1618–1620 (2011).
[CrossRef]

Li, M.

W. Li, M. Li, and J. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted Fiber Bragg Grating,” IEEE T Microw Theory60(5), 1287–1296 (2012).
[CrossRef]

Li, W.

W. Li, M. Li, and J. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted Fiber Bragg Grating,” IEEE T Microw Theory60(5), 1287–1296 (2012).
[CrossRef]

Loayssa, A.

A. Loayssa, D. Benito, and M. José Garde, “Applications of optical carrier Brillouin processing to microwave photonics,” Opt. Fiber Technol.8(1), 24–42 (2002).
[CrossRef]

Long, C. M.

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics6(3), 186–194 (2012).
[CrossRef]

R. Wu, C. M. Long, D. E. Leaird, and A. M. Weiner, “Directly generated Gaussian-shaped optical frequency comb for microwave photonic filtering and picosecond pulse generation,” IEEE Photon. Technol. Lett.24(17), 1484–1486 (2012).
[CrossRef]

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett.23(21), 1618–1620 (2011).
[CrossRef]

Martí, J.

Minasian, R. A.

W. Zhang and R. A. Minasian, “Switchable and tunable microwave photonic Brillouin-based filter,” IEEE Photonics Journal4(5), 1443–1455 (2012).
[CrossRef]

W. Zhang and R. A. Minasian, “Widely Tunable Single-Passband Microwave Photonic Filter Based on Stimulated Brillouin Scattering,” IEEE Photon. Technol. Lett.23(23), 1775–1777 (2011).
[CrossRef]

T. X. H. Huang, X. Yi, and R. A. Minasian, “Single passband microwave photonic filter using continuous-time impulse response,” Opt. Express19(7), 6231–6242 (2011).
[CrossRef] [PubMed]

X. Yi and R. A. Minasian, “Microwave photonic filter with single bandpass response,” Electron. Lett.45(7), 362–363 (2009).
[CrossRef]

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE T Microw Theory54(2), 832–846 (2006).
[CrossRef]

Mo, S. S.

N. Guo, R. C. Qiu, S. S. Mo, and K. Takahashi, “60-GHz millimeter-wave radio: principle, technology, and new Results,” EURASIP J. Wirel. Commun. Netw.2007, 1–48 (2007).
[CrossRef]

Mora, J.

Ortega, B.

Pastor, D.

Qiu, R. C.

N. Guo, R. C. Qiu, S. S. Mo, and K. Takahashi, “60-GHz millimeter-wave radio: principle, technology, and new Results,” EURASIP J. Wirel. Commun. Netw.2007, 1–48 (2007).
[CrossRef]

Seo, D.

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett.23(21), 1618–1620 (2011).
[CrossRef]

Song, M.

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett.23(21), 1618–1620 (2011).
[CrossRef]

Supradeepa, V. R.

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics6(3), 186–194 (2012).
[CrossRef]

Takahashi, K.

N. Guo, R. C. Qiu, S. S. Mo, and K. Takahashi, “60-GHz millimeter-wave radio: principle, technology, and new Results,” EURASIP J. Wirel. Commun. Netw.2007, 1–48 (2007).
[CrossRef]

Vidal, B.

Weiner, A. M.

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics6(3), 186–194 (2012).
[CrossRef]

R. Wu, C. M. Long, D. E. Leaird, and A. M. Weiner, “Directly generated Gaussian-shaped optical frequency comb for microwave photonic filtering and picosecond pulse generation,” IEEE Photon. Technol. Lett.24(17), 1484–1486 (2012).
[CrossRef]

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett.23(21), 1618–1620 (2011).
[CrossRef]

Wu, R.

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics6(3), 186–194 (2012).
[CrossRef]

R. Wu, C. M. Long, D. E. Leaird, and A. M. Weiner, “Directly generated Gaussian-shaped optical frequency comb for microwave photonic filtering and picosecond pulse generation,” IEEE Photon. Technol. Lett.24(17), 1484–1486 (2012).
[CrossRef]

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett.23(21), 1618–1620 (2011).
[CrossRef]

Yao, J.

W. Li, M. Li, and J. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted Fiber Bragg Grating,” IEEE T Microw Theory60(5), 1287–1296 (2012).
[CrossRef]

Y. Dai and J. Yao, “Nonuniformly-spaced photonic microwave delayline filter,” Opt. Express16(7), 4713–4718 (2008).
[CrossRef] [PubMed]

F. Zeng and J. Yao, “Investigation of phase-modulator-based all-optical bandpass microwave filter,” J. Lightwave Technol.23, 1113– 1117(2005).

Yi, X.

Zeng, F.

F. Zeng and J. Yao, “Investigation of phase-modulator-based all-optical bandpass microwave filter,” J. Lightwave Technol.23, 1113– 1117(2005).

Zhang, W.

W. Zhang and R. A. Minasian, “Switchable and tunable microwave photonic Brillouin-based filter,” IEEE Photonics Journal4(5), 1443–1455 (2012).
[CrossRef]

W. Zhang and R. A. Minasian, “Widely Tunable Single-Passband Microwave Photonic Filter Based on Stimulated Brillouin Scattering,” IEEE Photon. Technol. Lett.23(23), 1775–1777 (2011).
[CrossRef]

Electron. Lett. (2)

X. Yi and R. A. Minasian, “Microwave photonic filter with single bandpass response,” Electron. Lett.45(7), 362–363 (2009).
[CrossRef]

P. D. Dragic, “Simplified model for effect of Ge doping on silica fibre acoustic properties,” Electron. Lett.45(5), 256–257 (2009).
[CrossRef]

EURASIP J. Wirel. Commun. Netw. (1)

N. Guo, R. C. Qiu, S. S. Mo, and K. Takahashi, “60-GHz millimeter-wave radio: principle, technology, and new Results,” EURASIP J. Wirel. Commun. Netw.2007, 1–48 (2007).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

W. Zhang and R. A. Minasian, “Widely Tunable Single-Passband Microwave Photonic Filter Based on Stimulated Brillouin Scattering,” IEEE Photon. Technol. Lett.23(23), 1775–1777 (2011).
[CrossRef]

R. Wu, C. M. Long, D. E. Leaird, and A. M. Weiner, “Directly generated Gaussian-shaped optical frequency comb for microwave photonic filtering and picosecond pulse generation,” IEEE Photon. Technol. Lett.24(17), 1484–1486 (2012).
[CrossRef]

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett.23(21), 1618–1620 (2011).
[CrossRef]

IEEE Photonics Journal (1)

W. Zhang and R. A. Minasian, “Switchable and tunable microwave photonic Brillouin-based filter,” IEEE Photonics Journal4(5), 1443–1455 (2012).
[CrossRef]

IEEE T Microw Theory (2)

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE T Microw Theory54(2), 832–846 (2006).
[CrossRef]

W. Li, M. Li, and J. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted Fiber Bragg Grating,” IEEE T Microw Theory60(5), 1287–1296 (2012).
[CrossRef]

J. Lightwave Technol. (3)

Nat. Photonics (1)

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics6(3), 186–194 (2012).
[CrossRef]

Opt. Express (4)

Opt. Fiber Technol. (1)

A. Loayssa, D. Benito, and M. José Garde, “Applications of optical carrier Brillouin processing to microwave photonics,” Opt. Fiber Technol.8(1), 24–42 (2002).
[CrossRef]

Other (4)

Y. M. Chang and J. H. Lee, “Tunable, single passband photonic microwave filter based on stimulated Brillouin scattering in nonlinear fiber,” in IEEE LEOS Annual Meeting Conference Proceedings, 2009. LEOS '09(2009), 654–655.

R. Pant, A. Byrnes, E. Li, D.-Y. Choi, C. G. Poulton, S. Fan, S. J. Madden, B. Luther-Davies, and B. J. Eggleton, “Photonic chip based tunable and dynamically reconfigurable microwave photonic filter using stimulated Brillouin scattering,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides(Optical Society of America, 2012).

T. X. H. Huang, X. Yi, and R. A. Minasian, “A high-order FIR microwave photonic filter,” in International Topical Meeting on Microwave Photonics, 2009. MWP '09(2009), 1–4.

J. Yao, “Photonics for microwave signal filtering,” in IEEE Sarnoff Symposium, 2009. SARNOFF '09(2009), 1–5.

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Figures (9)

Fig. 1
Fig. 1

Conventional configuration of the tunable single passband MPF. CW: Continuous Wave, PM: Phase Modulation, EDFA: Erbium-Doped Fiber Amplifier, PC: Polarization Controller, DSF: Dispersion-Shifted Fiber, VNA: Vector Network Analyzer, PD: Photodetector.

Fig. 2
Fig. 2

Phase modulation spectrum.

Fig. 3
Fig. 3

(a) RF passband generation with the SBS process for a single pump (b) RF passband broadened generation with the SBS process for two pumps.

Fig. 4
Fig. 4

(a)SBS gain spectrum and (b) gain related nonlinear phase shift (c) SBS loss spectrum and (d) loss related nonlinear phase shift.

Fig. 5
Fig. 5

Comparison of frequency response in different number of pump.

Fig. 6
Fig. 6

Filter continuous tuning within 44GHz range.

Fig. 7
Fig. 7

Frequency response for two pumps with 1MHz offset.

Fig. 8
Fig. 8

Amplitude response for undesired passband at different frequency detuning.

Fig. 9
Fig. 9

Configuration of the proposed microwave photonic filter. LD: Laser Diode, CS-SSB: Carrier suppressed Single Sideband Modulation, CS-DSB: Carrier Suppressed Double Sideband Modulation.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

E(t)= J 0 (m)exp( j2π f c t ) + J 1 (m)exp{ j[ 2π( f c + f m )t+ π 2 ] } J 1 (m)exp{ j[ 2π( f c f m )t π 2 ] }
g(f)= g 0 I p 2 ( Γ B /2) 2 f 2 + ( Γ B /2) 2 +j g 0 I p 4 Γ B f f 2 + ( Γ B /2) 2
α(f)= g 0 I p 2 ( Γ B /2) 2 f 2 + ( Γ B /2) 2 j g 0 I p 4 Γ B f f 2 + ( Γ B /2) 2
E(t)=exp( j2π f c t ){ J 0 (m) + J 1 (m)exp[ { g[( f p ν B )( f c + f m )]+α[( f p + ν B )( f c + f m )] }L +j(2π f m t+ π 2 ) ] J 1 (m)exp[ j(2π f m t π 2 ) ] }
P2 J 0 (m) J 1 (m){ G( f m )A( f m )cos[ 2π f m t+ π 2 + ϕ g ( f m )+ ϕ α ( f m ) ]cos(2π f m t+ π 2 ) }
G( f m )=exp( Re[ g( f p ν B ( f c + f m ) ) ]L ) =exp{ g 0 I p L 2 ( Γ B /2 ) 2 [ f p ν B ( f c + f m ) ] 2 + ( Γ B /2 ) 2 }
A( f m )=exp( Re[ α( f p + ν B ( f c + f m ) ) ]L ) =exp{ g 0 I p L 2 ( Γ B /2) 2 [ f p + ν B ( f c + f m ) ] 2 + ( Γ B /2) 2 }
ϕ g ( f m )=Im[ g( f p ν B ( f c + f m ) ) ]L = g 0 I p L 4 Γ B [ f p ν B ( f c + f m ) ] [ f p ν B ( f c + f m ) ] 2 + ( Γ B /2) 2
ϕ α ( f m )=Im[ α( f p + ν B ( f c + f m ) ) ]L = g 0 I p L 4 Γ B [ f p + ν B ( f c + f m ) ] [ f p + ν B ( f c + f m ) ] 2 + ( Γ B /2) 2
E RF (t)=<P> G( f m )A( f m )cos[ 2π f m t+ π 2 + ϕ g ( f m )+ ϕ α ( f m ) ]cos(2π f m t+ π 2 )
| H(f) | 2 = P RF out P RF in 1+G( f m ) A 2 ( f m ) 2 2G( f m )A( f m )cos[ ϕ g ( f m )+ ϕ α ( f m ) ]
P2 J 0 (m) J 1 (m){ G 1 ( f m ) G 2 ( f m ) A 1 ( f m ) A 2 ( f m )cos[ 2π f m t+ π 2 + ϕ g1 ( f m ) + ϕ α1 ( f m )+ ϕ g 2 ( f m )+ ϕ α2 ( f m ) ] cos(2π f m t+ π 2 ) }
G k ( f m )=exp( Re{ g( f pk ν B ( f c + f m ) ) }L ) =exp{ g 0 I pk L 2 ( Γ B /2) 2 [ f pk ν B ( f c + f m ) ] 2 + ( Γ B /2) 2 }
A k ( f m )=exp( Re{ α( f pk + ν B ( f c + f m ) ) }L ) =exp{ g 0 I pk L 2 ( Γ B /2) 2 [ f pk + ν B ( f c + f m ) ] 2 + ( Γ B /2) 2 }
ϕ gk ( f m )=Im[ g( f pk ν B ( f c + f m ) ) ]L = g 0 I pk L 4 Γ B [ f pk ν B ( f c + f m ) ] [ f pk ν B ( f c + f m ) ] 2 + ( Γ B /2) 2
ϕ αk ( f m )=Im[ α( f pk + ν B ( f c + f m ) ) ]L = g 0 I pk L 4 Γ B [ f pk + ν B ( f c + f m ) ] [ f pk + ν B ( f c + f m ) ] 2 + ( Γ B /2) 2
| H(f) | 2 =1+ G 1 ( f m ) G 2 2 ( f m ) A 2 1 ( f m ) 2 A 2 ( f m ) 2 2 G 1 ( f m ) G 2 ( f m ) A 1 ( f m ) A 2 ( f m )cos[ ϕ g1 ( f m )+ ϕ α1 ( f m ) + ϕ g2 ( f m )+ ϕ α2 ( f m ) ]
| H(f) | 2 =1+ k=1 N G k ( f m ) A 2 k ( f m ) 2 2[ k=1 N G k ( f m ) A k ( f m ) ]cos{ k=1 N [ ϕ gk ( f m )+ ϕ αk ( f m ) ] }

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