Abstract

We develop, analyze and apply a linearization technique based on dual parallel Mach-Zehnder modulator to self-beating microwave photonics systems. The approach enables broadband low-distortion transmission and reception at expense of a moderate electrical power penalty yielding a small optical power penalty (<1 dB).

© 2016 Optical Society of America

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References

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  1. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
    [Crossref]
  2. J. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
    [Crossref]
  3. J. E. Mitchell, “Integrated wireless backhaul over optical access networks,” J. Lightwave Technol. 32(20), 3373–3382 (2014).
    [Crossref]
  4. D. Pastor, B. Ortega, J. Capmany, P. Y. Fonjallaz, and M. Popov, “Tunable microwave photonic filter for noise and interference suppression in UMTS base stations,” Electron. Lett. 40(16), 997–999 (2004).
    [Crossref]
  5. A. J. Seeds and K. J. Williams, “Technology focus on microwave photonics,” Nat. Photonics 5, 723–736 (2011).
  6. A. L. Ricchiuti, J. Hervas, D. Barrera, S. Sales, and J. Capmany, “Microwave photonics filtering technique for interrogating a very-weak fiber bragg grating cascade sensor,” IEEE Photonics J. 6(6), 1–10 (2014).
    [Crossref]
  7. D. A. I. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
    [Crossref]
  8. S. Iezekiel, M. Burla, J. Klamkin, D. Marpaung, and J. Capmany, “RF engineering meets optoelectronics,” IEEE Microw. Mag. 16(8), 28–45 (2015).
    [Crossref]
  9. W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
    [Crossref]
  10. M. Li, Y. Deng, J. Tang, S. Sun, J. Yao, J. Azaña, and N. Zhu, “Reconfigurable optical signal processing based on a distributed feedback semiconductor optical amplifier,” Sci. Rep. 6, 19985 (2016).
    [Crossref] [PubMed]
  11. D. Pérez, I. Gasulla, J. Capmany, J. S. Fandiño, P. Muñoz, and H. Alavi, “Figures of merit for self-beating filtered microwave photonic systems,” Opt. Express 24(9), 10087–10102 (2016).
    [Crossref] [PubMed]
  12. C. H. Cox, III, Analog Optical Links: Theory and Practice (Cambridge University, 2004).
  13. S. K. Korotky and R. M. de Ridder, “Dual parallel modulation schemes for low-distortion analog optical transmission,” IEEE J. Sel. Areas Comm. 8(7), 1377–1381 (1990).
    [Crossref]
  14. H. Yamazaki, H. Takahashi, T. Goh, Y. Hashizume, T. Yamada, S. Mino, H. Kawakami, and Y. Miyamoto, “Optical modulator with a near-linear field response,” J. Lightwave Technol. 34 (2016, in press).
  15. M. Hochberg and L. Chrostowski, Silicon Photonics Design: From Devices to Systems (Cambridge University, 2015).

2016 (3)

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

M. Li, Y. Deng, J. Tang, S. Sun, J. Yao, J. Azaña, and N. Zhu, “Reconfigurable optical signal processing based on a distributed feedback semiconductor optical amplifier,” Sci. Rep. 6, 19985 (2016).
[Crossref] [PubMed]

D. Pérez, I. Gasulla, J. Capmany, J. S. Fandiño, P. Muñoz, and H. Alavi, “Figures of merit for self-beating filtered microwave photonic systems,” Opt. Express 24(9), 10087–10102 (2016).
[Crossref] [PubMed]

2015 (1)

S. Iezekiel, M. Burla, J. Klamkin, D. Marpaung, and J. Capmany, “RF engineering meets optoelectronics,” IEEE Microw. Mag. 16(8), 28–45 (2015).
[Crossref]

2014 (2)

A. L. Ricchiuti, J. Hervas, D. Barrera, S. Sales, and J. Capmany, “Microwave photonics filtering technique for interrogating a very-weak fiber bragg grating cascade sensor,” IEEE Photonics J. 6(6), 1–10 (2014).
[Crossref]

J. E. Mitchell, “Integrated wireless backhaul over optical access networks,” J. Lightwave Technol. 32(20), 3373–3382 (2014).
[Crossref]

2013 (1)

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

2011 (1)

A. J. Seeds and K. J. Williams, “Technology focus on microwave photonics,” Nat. Photonics 5, 723–736 (2011).

2009 (1)

2007 (1)

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

2004 (1)

D. Pastor, B. Ortega, J. Capmany, P. Y. Fonjallaz, and M. Popov, “Tunable microwave photonic filter for noise and interference suppression in UMTS base stations,” Electron. Lett. 40(16), 997–999 (2004).
[Crossref]

1990 (1)

S. K. Korotky and R. M. de Ridder, “Dual parallel modulation schemes for low-distortion analog optical transmission,” IEEE J. Sel. Areas Comm. 8(7), 1377–1381 (1990).
[Crossref]

Alavi, H.

Azaña, J.

M. Li, Y. Deng, J. Tang, S. Sun, J. Yao, J. Azaña, and N. Zhu, “Reconfigurable optical signal processing based on a distributed feedback semiconductor optical amplifier,” Sci. Rep. 6, 19985 (2016).
[Crossref] [PubMed]

Barrera, D.

A. L. Ricchiuti, J. Hervas, D. Barrera, S. Sales, and J. Capmany, “Microwave photonics filtering technique for interrogating a very-weak fiber bragg grating cascade sensor,” IEEE Photonics J. 6(6), 1–10 (2014).
[Crossref]

Burla, M.

S. Iezekiel, M. Burla, J. Klamkin, D. Marpaung, and J. Capmany, “RF engineering meets optoelectronics,” IEEE Microw. Mag. 16(8), 28–45 (2015).
[Crossref]

Capmany, J.

D. Pérez, I. Gasulla, J. Capmany, J. S. Fandiño, P. Muñoz, and H. Alavi, “Figures of merit for self-beating filtered microwave photonic systems,” Opt. Express 24(9), 10087–10102 (2016).
[Crossref] [PubMed]

S. Iezekiel, M. Burla, J. Klamkin, D. Marpaung, and J. Capmany, “RF engineering meets optoelectronics,” IEEE Microw. Mag. 16(8), 28–45 (2015).
[Crossref]

A. L. Ricchiuti, J. Hervas, D. Barrera, S. Sales, and J. Capmany, “Microwave photonics filtering technique for interrogating a very-weak fiber bragg grating cascade sensor,” IEEE Photonics J. 6(6), 1–10 (2014).
[Crossref]

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

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

D. Pastor, B. Ortega, J. Capmany, P. Y. Fonjallaz, and M. Popov, “Tunable microwave photonic filter for noise and interference suppression in UMTS base stations,” Electron. Lett. 40(16), 997–999 (2004).
[Crossref]

Coldren, L. A.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

de Ridder, R. M.

S. K. Korotky and R. M. de Ridder, “Dual parallel modulation schemes for low-distortion analog optical transmission,” IEEE J. Sel. Areas Comm. 8(7), 1377–1381 (1990).
[Crossref]

Deng, Y.

M. Li, Y. Deng, J. Tang, S. Sun, J. Yao, J. Azaña, and N. Zhu, “Reconfigurable optical signal processing based on a distributed feedback semiconductor optical amplifier,” Sci. Rep. 6, 19985 (2016).
[Crossref] [PubMed]

Fandiño, J. S.

Fonjallaz, P. Y.

D. Pastor, B. Ortega, J. Capmany, P. Y. Fonjallaz, and M. Popov, “Tunable microwave photonic filter for noise and interference suppression in UMTS base stations,” Electron. Lett. 40(16), 997–999 (2004).
[Crossref]

Gasulla, I.

Guzzon, R. S.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Heideman, R.

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

Hervas, J.

A. L. Ricchiuti, J. Hervas, D. Barrera, S. Sales, and J. Capmany, “Microwave photonics filtering technique for interrogating a very-weak fiber bragg grating cascade sensor,” IEEE Photonics J. 6(6), 1–10 (2014).
[Crossref]

Iezekiel, S.

S. Iezekiel, M. Burla, J. Klamkin, D. Marpaung, and J. Capmany, “RF engineering meets optoelectronics,” IEEE Microw. Mag. 16(8), 28–45 (2015).
[Crossref]

Klamkin, J.

S. Iezekiel, M. Burla, J. Klamkin, D. Marpaung, and J. Capmany, “RF engineering meets optoelectronics,” IEEE Microw. Mag. 16(8), 28–45 (2015).
[Crossref]

Korotky, S. K.

S. K. Korotky and R. M. de Ridder, “Dual parallel modulation schemes for low-distortion analog optical transmission,” IEEE J. Sel. Areas Comm. 8(7), 1377–1381 (1990).
[Crossref]

Leinse, A.

D. A. I. 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, M.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

M. Li, Y. Deng, J. Tang, S. Sun, J. Yao, J. Azaña, and N. Zhu, “Reconfigurable optical signal processing based on a distributed feedback semiconductor optical amplifier,” Sci. Rep. 6, 19985 (2016).
[Crossref] [PubMed]

Liu, W.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Lu, M.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Marpaung, D.

S. Iezekiel, M. Burla, J. Klamkin, D. Marpaung, and J. Capmany, “RF engineering meets optoelectronics,” IEEE Microw. Mag. 16(8), 28–45 (2015).
[Crossref]

Marpaung, D. A. I.

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

Mitchell, J. E.

Muñoz, P.

Norberg, E. J.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Novak, D.

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

Ortega, B.

D. Pastor, B. Ortega, J. Capmany, P. Y. Fonjallaz, and M. Popov, “Tunable microwave photonic filter for noise and interference suppression in UMTS base stations,” Electron. Lett. 40(16), 997–999 (2004).
[Crossref]

Parker, J. S.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Pastor, D.

D. Pastor, B. Ortega, J. Capmany, P. Y. Fonjallaz, and M. Popov, “Tunable microwave photonic filter for noise and interference suppression in UMTS base stations,” Electron. Lett. 40(16), 997–999 (2004).
[Crossref]

Pérez, D.

Popov, M.

D. Pastor, B. Ortega, J. Capmany, P. Y. Fonjallaz, and M. Popov, “Tunable microwave photonic filter for noise and interference suppression in UMTS base stations,” Electron. Lett. 40(16), 997–999 (2004).
[Crossref]

Ricchiuti, A. L.

A. L. Ricchiuti, J. Hervas, D. Barrera, S. Sales, and J. Capmany, “Microwave photonics filtering technique for interrogating a very-weak fiber bragg grating cascade sensor,” IEEE Photonics J. 6(6), 1–10 (2014).
[Crossref]

Roeloffzen, C.

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

Sales, S.

A. L. Ricchiuti, J. Hervas, D. Barrera, S. Sales, and J. Capmany, “Microwave photonics filtering technique for interrogating a very-weak fiber bragg grating cascade sensor,” IEEE Photonics J. 6(6), 1–10 (2014).
[Crossref]

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

Seeds, A. J.

A. J. Seeds and K. J. Williams, “Technology focus on microwave photonics,” Nat. Photonics 5, 723–736 (2011).

Sun, S.

M. Li, Y. Deng, J. Tang, S. Sun, J. Yao, J. Azaña, and N. Zhu, “Reconfigurable optical signal processing based on a distributed feedback semiconductor optical amplifier,” Sci. Rep. 6, 19985 (2016).
[Crossref] [PubMed]

Tang, J.

M. Li, Y. Deng, J. Tang, S. Sun, J. Yao, J. Azaña, and N. Zhu, “Reconfigurable optical signal processing based on a distributed feedback semiconductor optical amplifier,” Sci. Rep. 6, 19985 (2016).
[Crossref] [PubMed]

Williams, K. J.

A. J. Seeds and K. J. Williams, “Technology focus on microwave photonics,” Nat. Photonics 5, 723–736 (2011).

Yao, J.

M. Li, Y. Deng, J. Tang, S. Sun, J. Yao, J. Azaña, and N. Zhu, “Reconfigurable optical signal processing based on a distributed feedback semiconductor optical amplifier,” Sci. Rep. 6, 19985 (2016).
[Crossref] [PubMed]

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

Yao, J. P.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Zhu, N.

M. Li, Y. Deng, J. Tang, S. Sun, J. Yao, J. Azaña, and N. Zhu, “Reconfigurable optical signal processing based on a distributed feedback semiconductor optical amplifier,” Sci. Rep. 6, 19985 (2016).
[Crossref] [PubMed]

Electron. Lett. (1)

D. Pastor, B. Ortega, J. Capmany, P. Y. Fonjallaz, and M. Popov, “Tunable microwave photonic filter for noise and interference suppression in UMTS base stations,” Electron. Lett. 40(16), 997–999 (2004).
[Crossref]

IEEE J. Sel. Areas Comm. (1)

S. K. Korotky and R. M. de Ridder, “Dual parallel modulation schemes for low-distortion analog optical transmission,” IEEE J. Sel. Areas Comm. 8(7), 1377–1381 (1990).
[Crossref]

IEEE Microw. Mag. (1)

S. Iezekiel, M. Burla, J. Klamkin, D. Marpaung, and J. Capmany, “RF engineering meets optoelectronics,” IEEE Microw. Mag. 16(8), 28–45 (2015).
[Crossref]

IEEE Photonics J. (1)

A. L. Ricchiuti, J. Hervas, D. Barrera, S. Sales, and J. Capmany, “Microwave photonics filtering technique for interrogating a very-weak fiber bragg grating cascade sensor,” IEEE Photonics J. 6(6), 1–10 (2014).
[Crossref]

J. Lightwave Technol. (2)

Laser Photonics Rev. (1)

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

Nat. Photonics (3)

A. J. Seeds and K. J. Williams, “Technology focus on microwave photonics,” Nat. Photonics 5, 723–736 (2011).

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. P. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

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

Opt. Express (1)

Sci. Rep. (1)

M. Li, Y. Deng, J. Tang, S. Sun, J. Yao, J. Azaña, and N. Zhu, “Reconfigurable optical signal processing based on a distributed feedback semiconductor optical amplifier,” Sci. Rep. 6, 19985 (2016).
[Crossref] [PubMed]

Other (3)

C. H. Cox, III, Analog Optical Links: Theory and Practice (Cambridge University, 2004).

H. Yamazaki, H. Takahashi, T. Goh, Y. Hashizume, T. Yamada, S. Mino, H. Kawakami, and Y. Miyamoto, “Optical modulator with a near-linear field response,” J. Lightwave Technol. 34 (2016, in press).

M. Hochberg and L. Chrostowski, Silicon Photonics Design: From Devices to Systems (Cambridge University, 2015).

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Figures (4)

Fig. 1
Fig. 1 (Left) DPMZM scheme and (Right) self-homodyne MWP system architecture.
Fig. 2
Fig. 2 Photodetected spectrum without (Left) and with the applied linearization (Right).
Fig. 3
Fig. 3 Case 1. (Upper) Optical penalties vs Electrical Penalties, (Lower) Coupler coefficient vs Electrical Penalty for Case 1 (left) and Case 2 (right).
Fig. 4
Fig. 4 Configuration for a Near-linear Field Response modulator reported in [12].

Equations (30)

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C( Ω 1 )= I b ( Ω 1 )= I dc α U α L CD( ϕ RF1 4 )[ ( 1a )( 1b ) sin( ϕ DC1 ) A 1, Ω 1 I,B +γ ab sin( ϕ DC2 ) A 2, Ω 1 I,B ],
A i, Ω 1 I,B =cosφ [ H i ( ω o + Ω 1 ) H i * ( ω o Ω 1 ) ] sin( ϕ DCi 2 ) , H i ( ω )=H( ω ) e j ψ 1 .
IMD2( Ω 1 Ω 2 )= I b ( Ω 1 Ω 2 )=j I dc α U α L CD ( ϕ RF 4 ) 2 [ ( 1a )( 1b ) A 1, Ω 1 Ω 2 I,B + γ 2 ab A 2, Ω 1 Ω 2 I,B ],
A i, Ω 1 Ω 2 I,B =2[ H i ( ω o + Ω 1 Ω 2 ) H i * ( ω o Ω 1 + Ω 2 ) ]sin( ϕ DCi /2 )cosφ,
IMD3( 2 Ω 1 Ω 2 )= I b ( 2 Ω 1 Ω 2 )= I dc CD α U α L ( ϕ RF 4 ) 3 [ sin( ϕ dc1 ) ( 1a )( 1b ) A 1,2 Ω 1 Ω 2 I,B + γ 3 sin( ϕ dc2 ) ab A 2,2 Ω 1 Ω 2 I,B ],
A i,2 Ω 1 Ω 2 I,B = [ H i ( ω o +2 Ω 1 Ω 2 ) H i * ( ω o 2 Ω 1 + Ω 2 ) ] 2sin( ϕ dci /2 ) cosφ.
γ= ( 1a )( 1b ) ab [ H( ω o +2 Ω 1 Ω 2 ) e j ψ 1 H * ( ω o 2 Ω 1 + Ω 2 ) e j ψ 1 H( ω o +2 Ω 1 Ω 2 ) e j ψ 2 H * ( ω o 2 Ω 1 + Ω 2 ) e j ψ 2 ] cos( ϕ dc1 /2 ) cos( ϕ dc2 /2 ) 3 .
γ= ( 1a )( 1b ) ab cos( ϕ dc1 /2 ) cos( ϕ dc2 /2 ) 3 .
CIMD 2 L CIMD2 = C L / IMD 2 L C/ IMD2 = I b,L ( Ω 1 )/ I b,L ( Ω 1 Ω 2 ) I b ( Ω 1 )/ I b ( Ω 1 Ω 2 ) .
( 1a )( 1b ) γ 3 ab .=0
C L C = ( 1a )( 1b ) γ ab
a=b= 1 1+ γ 3 .
C L C | MAX = γ( γ 2 1 ) 1+ γ 3 ;
IMD 2 L IMD2 = I L ( Ω 1 Ω 2 ) I ( Ω 1 Ω 2 ) = γ 2 ( γ1 ) 1+ γ 3 .
CIMD 2 L CIMD2 =1+ 1 γ .
a= 1 1+ γ 6 ;
C L C | max = γ( γ 2 1 ) 2( 1+ γ 6 ) .
IMD 2 L IMD2 = γ 2 ( γ1 ) 2( 1+ γ 6 ) .
E out,modulator = E in [ 1r sin( ϕ RF 2 ) r 2 sin( ϕ RF ) ]= = E in,modulator 2j { 1r [ e j ϕ RF 2 e j ϕ RF 2 ] r 2 [ e j ϕ RF e j ϕ RF ] },
  I bL ( Ω 1 )j2 I dc α U α L CD[ 1r ( ϕ RF 4 ) r 2 ( ϕ RF 2 ) ] A Ω 1 I,B , A Ω 1 I,B ={ H[ ω o + Ω 1 ]+ H * [ ω o Ω 1 ] }cosφ.
  I ( Ω 1 Ω 2 )=j2 I dc α U α L CD{ B 1,1 H[ ω o +n Ω 1 +k Ω 2 ] e j( ω o +n Ω 1 +k Ω 2 )t B 1,1 * H * [ ω o +n Ω 1 +k Ω 2 ] e j( ω o +n Ω 1 +k Ω 2 )t }cosφ= ={ B 1,1 = B 1,1 =0 }=0,
I ( 2 Ω 1 Ω 2 )=j I dc CD α U α L CD[ 1r ( ϕ RF 4 ) 3 r 2 ( ϕ RF 2 ) 3 ] A 2 Ω 1 Ω 1 I,B , A 2 Ω 1 Ω 1 I,B ={ H[ ω o +2 Ω 1 Ω 2 ]+ H * [ ω o 2 Ω 1 + Ω 2 ] }cosφ.
I ( 2 Ω 1 Ω 2 )=0 1r ( ϕ RF 4 ) 3 r 2 ( ϕ RF 2 ) 3 =0, 1r 64 = r 16 1r=16rr= 1 17 0.06
I bL ( Ω 1 )j I dc α U α L CD( ϕ RF 2 )[ 16 17 1 17 ] A Ω 1 I,B = =j0.364 I dc α U α L CD ϕ RF A Ω 1 I,B , A Ω 1 I,B ={ H[ ω o + Ω 1 ]+ H * [ ω o Ω 1 ] }cosφ.
I bL ( Ω 1 ) I b ( Ω 1 ) =0.728.
L( a,b )= ( 1a )( 1b ) γ ab +λ[ ( 1a )( 1b ) γ 3 ab ],
L a = ( 1b ) 2 ( 1a ) + γ b 2 a +λ[ ( 1b ) 2 ( 1a ) + γ 3 b 2 a ]=0, L b = ( 1a ) 2 ( 1b ) + γ a 2 b +λ[ ( 1a ) 2 ( 1b ) + γ 3 a 2 b ]=0, L λ = ( 1a )( 1b ) γ 3 ab =0.
L a =0λ= a( 1b ) +γ ( 1a )b a( 1b ) γ 3 ( 1a )b .
L b =0 1a a = 1b b a=b.
L λ =0( 1a ) γ 3 a=0a=b= 1 1+ γ 3 .

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