Abstract

We theoretically and experimentally evaluate the propagation, generation and amplification of signal, harmonic and intermodulation distortion terms inside a Semiconductor Optical Amplifier (SOA) under Coherent Population Oscillation (CPO) regime. For that purpose, we present a general optical field model, valid for any arbitrarily-spaced radiofrequency tones, which is necessary to correctly describe the operation of CPO based slow light Microwave Photonic phase shifters which comprise an electrooptic modulator and a SOA followed by an optical filter and supplements another recently published for true time delay operation based on the propagation of optical intensities. The phase shifter performance has been evaluated in terms of the nonlinear distortion up to 3rd order, for a modulating signal constituted of two tones, in function of the electrooptic modulator input RF power and the SOA input optical power, obtaining a very good agreement between theoretical and experimental results. A complete theoretical spectral analysis is also presented which shows that under small signal operation conditions, the 3rd order intermodulation products at 2Ω1 + Ω2 and 2Ω2 + Ω1 experience a power dip/phase transition characteristic of the fundamental tones phase shifting operation.

© 2010 OSA

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References

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  1. R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326(5956), 1074–1077 (2009).
    [Crossref] [PubMed]
  2. R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: Capabilities and fundamental limitations,” J. Lightwave Technol. 23(12), 4046–4066 (2005).
    [Crossref]
  3. G. M. Gehring, R. W. Boyd, A. L. Gaeta, D. J. Gauthier, and A. E. Willner, “Fiber based Slow-Light Technologies,” J. Lightwave Technol. 26(23), 3752–3762 (2008).
    [Crossref]
  4. T. Baba, “Slow Light in Photonic Crystals,” Nat. Photonics 2(8), 465–473 (2008).
    [Crossref]
  5. C. J. Chang Hasnain and S. L. Chuang, “Slow and Fast Light in Semiconductor Quantum Well and Quantum-Dot Devices,” J. Lightwave Technol. 24(12), 4642–4654 (2006).
    [Crossref]
  6. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
    [Crossref]
  7. J. P. Yao, “Microwave Photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
    [Crossref]
  8. S. Sales, W. Xue, J. Mork and I. Gasulla, “Slow and Fast Light Effects and their Applications to Microwave Photonics using Semiconductor Optical Amplifiers,” IEEE Trans. Microwave Theory Tech../J. Lightwave Technol., Joint Special issue on Microwave Photonics (to be published).
  9. G. P. Agrawal, “Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and amplifiers,” J. Opt. Soc. Am. B 5(1), 147–159 (1988).
    [Crossref]
  10. G. P. Agrawal and I. M. I. Habbab, “Effect of Four-Wave Mixing on Multichannel Amplification in Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 26(3), 501–505 (1990).
    [Crossref]
  11. T. Mukai and T. Saitoh, “Detuning Characteristics and Conversion Efficiency of Nearly Degenerate Four-Wave Mixing in a 1.5-μm Traveling-Wave Semiconductor Laser Amplifier,” IEEE J. Quantum Electron. 26(5), 865–875 (1990).
    [Crossref]
  12. H. Su, P. Kondratko, and S. L. Chuang, “Variable optical delay using population oscillation and four-wave-mixing in semiconductor optical amplifiers,” Opt. Express 14(11), 4800–4807 (2006).
    [Crossref] [PubMed]
  13. J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, “Slow light in a semiconductor waveguide at gigahertz frequencies,” Opt. Express 13(20), 8136–8145 (2005).
    [Crossref] [PubMed]
  14. W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, “Enhancing light slow-down in semiconductor optical amplifiers by optical filtering,” Opt. Lett. 33(10), 1084–1086 (2008).
    [Crossref] [PubMed]
  15. Y. Chen, W. Xue, F. Ohman, and J. Mork, “Theory of Optical Filtering Enhanced Slow and Fast Light Effects in Semiconductor Optical Waveguides,” J. Lightwave Technol. 26(23), 3734–3743 (2008).
    [Crossref]
  16. W. Xue, S. Sales, J. Capmany, and J. Mørk, “Microwave phase shifter with controllable power response based on slow- and fast-light effects in semiconductor optical amplifiers,” Opt. Lett. 34(7), 929–931 (2009).
    [Crossref] [PubMed]
  17. 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]
  18. F. Öhman, K. Yvind, and J. Mørk, “Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics,” IEEE Photon. Technol. Lett. 19(15), 1145–1147 (2007).
    [Crossref]
  19. W. Xue, S. Sales, J. Mork, and J. Capmany, “Widely tunable microwave photonic notch filter based on slow and fast light effects,” IEEE Photon. Technol. Lett. 21(3), 167–169 (2009).
    [Crossref]
  20. E. Shumakher, S. Á. O Duill, and G. Eisenstein, “Optoelectronic Oscillator Tunable by an SOA Based Slow Light Element,” J. Lightwave Technol. 27(18), 4063–4068 (2009).
    [Crossref]
  21. C. Cox, and W. S. C. Chang, “Figures of merit and performance analysis of photonic microwave links,” in RF-Photonic Technology in Optical Fiber Links, Cambridge University Press, UK, (2002).
  22. P. Berger, J. Bourderionnet, M. Alouini, F. Bretenaker, and D. Dolfi, “Theoretical study of the spurious-free dynamic range of a tunable delay line based on slow light in SOA,” Opt. Express 17(22), 20584–20597 (2009).
    [Crossref] [PubMed]
  23. E. Shumakher, S. Á. O Duill, and G. Eisenstein, “On the role of High-Order Coherent Population Oscillations in Slow and Light Propagation Using Semiconductor Optical Amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 578–584 (2009).
    [Crossref]
  24. I. Gasulla, J. Sancho, J. Lloret, S. Sales, and J. Capmany, “Harmonic Distortion in Microwave Photonic Phase Shifters Based on Coherent Population Oscillations in SOAs,” IEEE Photon. Technol. Lett. 22(12), 899–901 (2010).
    [Crossref]

2010 (2)

I. Gasulla, J. Sancho, J. Lloret, S. Sales, and J. Capmany, “Harmonic Distortion in Microwave Photonic Phase Shifters Based on Coherent Population Oscillations in SOAs,” IEEE Photon. Technol. Lett. 22(12), 899–901 (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]

2009 (7)

W. Xue, S. Sales, J. Mork, and J. Capmany, “Widely tunable microwave photonic notch filter based on slow and fast light effects,” IEEE Photon. Technol. Lett. 21(3), 167–169 (2009).
[Crossref]

E. Shumakher, S. Á. O Duill, and G. Eisenstein, “On the role of High-Order Coherent Population Oscillations in Slow and Light Propagation Using Semiconductor Optical Amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 578–584 (2009).
[Crossref]

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326(5956), 1074–1077 (2009).
[Crossref] [PubMed]

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

W. Xue, S. Sales, J. Capmany, and J. Mørk, “Microwave phase shifter with controllable power response based on slow- and fast-light effects in semiconductor optical amplifiers,” Opt. Lett. 34(7), 929–931 (2009).
[Crossref] [PubMed]

E. Shumakher, S. Á. O Duill, and G. Eisenstein, “Optoelectronic Oscillator Tunable by an SOA Based Slow Light Element,” J. Lightwave Technol. 27(18), 4063–4068 (2009).
[Crossref]

P. Berger, J. Bourderionnet, M. Alouini, F. Bretenaker, and D. Dolfi, “Theoretical study of the spurious-free dynamic range of a tunable delay line based on slow light in SOA,” Opt. Express 17(22), 20584–20597 (2009).
[Crossref] [PubMed]

2008 (4)

2007 (2)

F. Öhman, K. Yvind, and J. Mørk, “Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics,” IEEE Photon. Technol. Lett. 19(15), 1145–1147 (2007).
[Crossref]

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

2006 (2)

2005 (2)

1990 (2)

G. P. Agrawal and I. M. I. Habbab, “Effect of Four-Wave Mixing on Multichannel Amplification in Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 26(3), 501–505 (1990).
[Crossref]

T. Mukai and T. Saitoh, “Detuning Characteristics and Conversion Efficiency of Nearly Degenerate Four-Wave Mixing in a 1.5-μm Traveling-Wave Semiconductor Laser Amplifier,” IEEE J. Quantum Electron. 26(5), 865–875 (1990).
[Crossref]

1988 (1)

Agrawal, G. P.

G. P. Agrawal and I. M. I. Habbab, “Effect of Four-Wave Mixing on Multichannel Amplification in Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 26(3), 501–505 (1990).
[Crossref]

G. P. Agrawal, “Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and amplifiers,” J. Opt. Soc. Am. B 5(1), 147–159 (1988).
[Crossref]

Alouini, M.

Baba, T.

T. Baba, “Slow Light in Photonic Crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

Berger, P.

Bourderionnet, J.

Boyd, R. W.

Bretenaker, F.

Capmany, J.

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]

I. Gasulla, J. Sancho, J. Lloret, S. Sales, and J. Capmany, “Harmonic Distortion in Microwave Photonic Phase Shifters Based on Coherent Population Oscillations in SOAs,” IEEE Photon. Technol. Lett. 22(12), 899–901 (2010).
[Crossref]

W. Xue, S. Sales, J. Mork, and J. Capmany, “Widely tunable microwave photonic notch filter based on slow and fast light effects,” IEEE Photon. Technol. Lett. 21(3), 167–169 (2009).
[Crossref]

W. Xue, S. Sales, J. Capmany, and J. Mørk, “Microwave phase shifter with controllable power response based on slow- and fast-light effects in semiconductor optical amplifiers,” Opt. Lett. 34(7), 929–931 (2009).
[Crossref] [PubMed]

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

Chang Hasnain, C. J.

Chang-Hasnain, C. J.

Chen, Y.

Chuang, S. L.

Dolfi, D.

Eisenstein, G.

E. Shumakher, S. Á. O Duill, and G. Eisenstein, “Optoelectronic Oscillator Tunable by an SOA Based Slow Light Element,” J. Lightwave Technol. 27(18), 4063–4068 (2009).
[Crossref]

E. Shumakher, S. Á. O Duill, and G. Eisenstein, “On the role of High-Order Coherent Population Oscillations in Slow and Light Propagation Using Semiconductor Optical Amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 578–584 (2009).
[Crossref]

Gaeta, A. L.

Gasulla, I.

I. Gasulla, J. Sancho, J. Lloret, S. Sales, and J. Capmany, “Harmonic Distortion in Microwave Photonic Phase Shifters Based on Coherent Population Oscillations in SOAs,” IEEE Photon. Technol. Lett. 22(12), 899–901 (2010).
[Crossref]

Gauthier, D. J.

Gehring, G. M.

Habbab, I. M. I.

G. P. Agrawal and I. M. I. Habbab, “Effect of Four-Wave Mixing on Multichannel Amplification in Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 26(3), 501–505 (1990).
[Crossref]

Kjær, R.

Kondratko, P.

Ku, P. C.

Lloret, J.

I. Gasulla, J. Sancho, J. Lloret, S. Sales, and J. Capmany, “Harmonic Distortion in Microwave Photonic Phase Shifters Based on Coherent Population Oscillations in SOAs,” IEEE Photon. Technol. Lett. 22(12), 899–901 (2010).
[Crossref]

Mork, J.

W. Xue, S. Sales, J. Mork, and J. Capmany, “Widely tunable microwave photonic notch filter based on slow and fast light effects,” IEEE Photon. Technol. Lett. 21(3), 167–169 (2009).
[Crossref]

Y. Chen, W. Xue, F. Ohman, and J. Mork, “Theory of Optical Filtering Enhanced Slow and Fast Light Effects in Semiconductor Optical Waveguides,” J. Lightwave Technol. 26(23), 3734–3743 (2008).
[Crossref]

Mørk, J.

Mukai, T.

T. Mukai and T. Saitoh, “Detuning Characteristics and Conversion Efficiency of Nearly Degenerate Four-Wave Mixing in a 1.5-μm Traveling-Wave Semiconductor Laser Amplifier,” IEEE J. Quantum Electron. 26(5), 865–875 (1990).
[Crossref]

Novak, D.

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

O Duill, S. Á.

E. Shumakher, S. Á. O Duill, and G. Eisenstein, “On the role of High-Order Coherent Population Oscillations in Slow and Light Propagation Using Semiconductor Optical Amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 578–584 (2009).
[Crossref]

E. Shumakher, S. Á. O Duill, and G. Eisenstein, “Optoelectronic Oscillator Tunable by an SOA Based Slow Light Element,” J. Lightwave Technol. 27(18), 4063–4068 (2009).
[Crossref]

Ohman, F.

Öhman, F.

W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, “Enhancing light slow-down in semiconductor optical amplifiers by optical filtering,” Opt. Lett. 33(10), 1084–1086 (2008).
[Crossref] [PubMed]

F. Öhman, K. Yvind, and J. Mørk, “Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics,” IEEE Photon. Technol. Lett. 19(15), 1145–1147 (2007).
[Crossref]

Saitoh, T.

T. Mukai and T. Saitoh, “Detuning Characteristics and Conversion Efficiency of Nearly Degenerate Four-Wave Mixing in a 1.5-μm Traveling-Wave Semiconductor Laser Amplifier,” IEEE J. Quantum Electron. 26(5), 865–875 (1990).
[Crossref]

Sales, S.

Sancho, J.

I. Gasulla, J. Sancho, J. Lloret, S. Sales, and J. Capmany, “Harmonic Distortion in Microwave Photonic Phase Shifters Based on Coherent Population Oscillations in SOAs,” IEEE Photon. Technol. Lett. 22(12), 899–901 (2010).
[Crossref]

Shumakher, E.

E. Shumakher, S. Á. O Duill, and G. Eisenstein, “On the role of High-Order Coherent Population Oscillations in Slow and Light Propagation Using Semiconductor Optical Amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 578–584 (2009).
[Crossref]

E. Shumakher, S. Á. O Duill, and G. Eisenstein, “Optoelectronic Oscillator Tunable by an SOA Based Slow Light Element,” J. Lightwave Technol. 27(18), 4063–4068 (2009).
[Crossref]

Su, H.

Tucker, R. S.

van der Poel, M.

Willner, A. E.

Xue, W.

Yao, J. P.

Yvind, K.

F. Öhman, K. Yvind, and J. Mørk, “Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics,” IEEE Photon. Technol. Lett. 19(15), 1145–1147 (2007).
[Crossref]

J. Mørk, R. Kjær, M. van der Poel, and K. Yvind, “Slow light in a semiconductor waveguide at gigahertz frequencies,” Opt. Express 13(20), 8136–8145 (2005).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (2)

G. P. Agrawal and I. M. I. Habbab, “Effect of Four-Wave Mixing on Multichannel Amplification in Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 26(3), 501–505 (1990).
[Crossref]

T. Mukai and T. Saitoh, “Detuning Characteristics and Conversion Efficiency of Nearly Degenerate Four-Wave Mixing in a 1.5-μm Traveling-Wave Semiconductor Laser Amplifier,” IEEE J. Quantum Electron. 26(5), 865–875 (1990).
[Crossref]

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

E. Shumakher, S. Á. O Duill, and G. Eisenstein, “On the role of High-Order Coherent Population Oscillations in Slow and Light Propagation Using Semiconductor Optical Amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(3), 578–584 (2009).
[Crossref]

IEEE Photon. Technol. Lett. (3)

I. Gasulla, J. Sancho, J. Lloret, S. Sales, and J. Capmany, “Harmonic Distortion in Microwave Photonic Phase Shifters Based on Coherent Population Oscillations in SOAs,” IEEE Photon. Technol. Lett. 22(12), 899–901 (2010).
[Crossref]

F. Öhman, K. Yvind, and J. Mørk, “Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics,” IEEE Photon. Technol. Lett. 19(15), 1145–1147 (2007).
[Crossref]

W. Xue, S. Sales, J. Mork, and J. Capmany, “Widely tunable microwave photonic notch filter based on slow and fast light effects,” IEEE Photon. Technol. Lett. 21(3), 167–169 (2009).
[Crossref]

J. Lightwave Technol. (6)

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

Nat. Photonics (2)

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

T. Baba, “Slow Light in Photonic Crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Science (1)

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326(5956), 1074–1077 (2009).
[Crossref] [PubMed]

Other (2)

S. Sales, W. Xue, J. Mork and I. Gasulla, “Slow and Fast Light Effects and their Applications to Microwave Photonics using Semiconductor Optical Amplifiers,” IEEE Trans. Microwave Theory Tech../J. Lightwave Technol., Joint Special issue on Microwave Photonics (to be published).

C. Cox, and W. S. C. Chang, “Figures of merit and performance analysis of photonic microwave links,” in RF-Photonic Technology in Optical Fiber Links, Cambridge University Press, UK, (2002).

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Figures (13)

Fig. 1
Fig. 1

Schematic of the microwave photonics phase shifter and representative diagram of the spectral complex amplitude components at the SOA input.

Fig. 2
Fig. 2

Schematic of the experimental setup.

Fig. 3
Fig. 3

Measured FBG magnitude and phase shift responses.

Fig. 4
Fig. 4

Theoretical (lines) and experimental (markers) photodetected RF power (solid lines and circles) from the fundamental [(Pd (1,0) and Pd (0,1))] and IMD3 [Pd (2,-1) and Pd (−1,2)] terms as function of the input RF power, including 3 rd order SFDR determination (SFDR3). The dashed lines and the points correspond to the RF power at the SOA input.

Fig. 5
Fig. 5

Theoretical (lines) and experimental (markers) photodetected RF power (solid lines and circles) from the fundamental [(Pd (1,0) and Pd (0,1))], HD2 [Pd (2,0) and Pd (0,2)] and IMD2 [Pd (1,1) and Pd (−1,1)] terms as function of the input RF power, including 2 nd order SFDR determination (SFDR2). The dashed lines and the points correspond to the RF power at the SOA input.

Fig. 6
Fig. 6

Theoretical (lines) and experimental (markers) photodetected RF power (a) from the fundamental [(Pd (1,0), Pd (0,1))] and the IMD3 [Pd (2,-1), Pd (−1,2)] terms versus the SOA input optical power for f 1 = 10.5GHz and f 2 = 11GHz. (b) Microwave phase shift [( ϕ ( 1 , 0 ) , ϕ ( 0 , 1 ) )] and [ ϕ ( 2 , 1 ) , ϕ ( 1 , 2 ) ].

Fig. 7
Fig. 7

Theoretical (lines) and experimental (markers) photodetected RF power (a) from the IMD3 [Pd (2,1) and Pd (1,2)] terms versus the SOA input optical power for f 1 = 10.5GHz and f 2 = 11GHz. (b) Microwave phase shift [ ϕ ( 2 , 1 ) and ϕ ( 1 , 2 ) ].

Fig. 8
Fig. 8

Theoretical (lines) and experimental (markers) photodetected RF power (a) from the HD2 [Pd (2,0), Pd (0,2)] and IMD2 [Pd (1,1), Pd (−1,1)] terms versus the SOA input optical power for f 1 = 10.5GHz and f 2 = 11 GHz. (b) Microwave phase shift [ ϕ ( 2 , 0 ) , ϕ ( 0 , 2 ) ] and [ ϕ ( 1 , 1 ) , ϕ ( 1 , 1 ) ].

Fig. 9
Fig. 9

Theoretical (lines) and experimental (markers) photodetected RF power (a) from the fundamental [(Pd (1,0), Pd (0,1))] and the IMD3 [Pd (2,-1), Pd (−1,2)] terms versus the SOA input optical power for f 1 = 9 GHz and f 2 = 12 GHz. (b) Microwave phase shift [( ϕ ( 1 , 0 ) , ϕ ( 0 , 1 ) )] and [ ϕ ( 2 , 1 ) , ϕ ( 1 , 2 ) ].

Fig. 11
Fig. 11

Theoretical (lines) and experimental (markers) photodetected RF power (a) from the HD2 [Pd (2,0), Pd (0,2)] and IMD2 [Pd (1,1), Pd (−1,1)] terms versus the SOA input optical power for f 1 = 9 GHz and f 2 = 12 GHz. (b) Microwave phase shift [ ϕ ( 2 , 0 ) , ϕ ( 0 , 2 ) ] and [ ϕ ( 1 , 1 ) , ϕ ( 1 , 1 ) ].

Fig. 10
Fig. 10

Theoretical (lines) and experimental (markers) photodetected RF power (a) from the IMD3 [Pd (2,1) and Pd (1,2)] terms versus the SOA input optical power for f 1 = 9 GHz and f 2 = 12 GHz. (b) Microwave phase shift [ ϕ ( 2 , 1 ) , ϕ ( 1 , 2 ) ].

Fig. 12
Fig. 12

Photodetected RF power from the fundamental tones [(Pd (1,0) and Pd (0,1))], the HD2 [Pd (2,0) and Pd (0,2)], IMD2 [Pd (1,1) and Pd (−1,1)] and IMD3 [Pd (2,-1), Pd (−1,2), Pd (1,2) and Pd (2,1)] terms versus the modulating frequency f 1 for different modulation indices m.

Fig. 13
Fig. 13

Microwave phase shift from the fundamental tones [( ϕ ( 1 , 0 ) and ϕ ( 0 , 1 ) )], the HD2 [ ϕ ( 2 , 0 ) and ϕ ( 0 , 2 ) ], IMD2 [ ϕ ( 1 , 1 ) and ϕ ( 1 , 1 ) ] and IMD3 [ ϕ ( 2 , 1 ) , ϕ ( 1 , 2 ) , ϕ ( 1 , 2 ) and ϕ ( 2 , 1 ) ] versus the modulating frequency f 1 for different modulation indices m.

Equations (13)

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d N d t = I e V N τ s Γ a ( N N t r ) | E | 2
E ( t , z ) = k 1 = M M ... k N = M | i = 1 N k i | M M E ( k 1 ... ,   k N ) e j   [ ( ω 0 + i = 1 N k i Ω i ) t z i = 1 N β k i Ω i ]
d E ( k 1 ... ,   k N ) d z = γ int 2 E ( k 1 ... ,   k N ) + ( 1 j α ) 2 m 1 = M + k 1 M + k 1 ... m N = M + k N   M + k N g ( m 1 ... ,   m N ) E ( k 1 m 1 ... ,   k N m N ) for  | i = 1 N k i | M  and  | i = 1 N m i | 2 M
g ( m 1 , ...   m N ) = g ( 0 , ...0 ) S ( m 1 , ...   m N ) / P s [ 1 + S ( 0 , ...0 ) / P S j ( i = 1 N m i Ω i ) τ s ]
S ( m 1 , ... m N ) = k 1 = M + m 1 M ... k N = M + m N | i = 1 N k i | M  and  | i = 1 N k i m i | M   M E ( k 1 ... ,   k N ) E ( k 1 m 1 ... ,   k N m N ) * .
V a ( t ) = V D C a + V R F 1 cos ( Ω 1 t + ϕ 1 a ) + V R F 2 cos ( Ω 2 t + ϕ 2 a ) V b ( t ) = V D C b + V R F 2 cos ( Ω 1 t + ϕ 1 b ) + V R F 2 cos ( Ω 2 t + ϕ 2 b ) .
E o u t ( t ) | E O M = E s 2 e j π V π [ V a ( t ) + V b ( t ) 2 ] cos  [ π V π ( V a ( t ) V b ( t ) 2 ) ]
E o u t ( t ) | E O M = E e 2 e j ϕ s k = l = ( j ) k + l J k ( m 1 ) J l ( m 2 ) e j ( k Ω 1 + l Ω 2 ) t cos  [ φ + k ( ϕ 1 b ϕ 1 a ) + l ( ϕ 2 b ϕ 2 a ) 2 ] e j [ k ( ϕ 1 a + ϕ 1 b ) + l ( ϕ 2 a + ϕ 2 b ) 2 ]
E ( k 1 , k 2 ) ( t , 0 ) = E e 2 cos [ φ / 2 + ( | k 1 | + | k 2 | ) π / 2 ] J k 1 ( m 1 ) J k 2 ( m 2 ) e j ( k 1 Ω 1 + k 2 Ω 2 )   t
P R F | i n = V R F 2 Z o = ( m V π / π ) 2 Z o
P n o i s e = 10  log  ( σ R I N 2 + σ A S E 2 + σ T h e r m a l 2 + σ S h o t 2 )    ( d B m / H z )
S D F R 2 = 1 / 2 ( I P 2 P n o i s e )    ( d B / H z 1 / 2 ) S D F R 3 = 2 / 3 ( I P 3 P n o i s e )    ( d B / H z 2 / 3 )
S F D R 2 = { P d ( 1 , 1 ) i , P d ( 1 , 1 ) i }   P d ( 1 , 0 ) i    ( d B / H z 1 / 2 ) S F D R 3 = { P d ( 2 , 1 ) i , P d ( 1 , 2 ) i }   P d ( 1 , 0 ) i    ( d B / H z 2 / 3 )

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