Abstract

Dynamically reconfigurable and transparent signal spectral conversion is expected to play a vital role in seamlessly integrating traditional metropolitan optical networks and mobile fronthaul/backhaul networks. In this paper, a simple digital signal processing (DSP)-enabled spectral converter is proposed and extensively investigated, for the first time, which just utilizes a single standard dual-parallel Mach-Zehnder modulator (DP-MZM) driven by SDN-controllable RF signals and DC bias currents. As an important thrust of the paper, optimum operating conditions of the proposed converter are analytically identified, statistically examined and experimentally verified. Optimum operating condition-supported spectral converter performances in IMDD-based network nodes are explored both theoretically and experimentally in terms of frequency detuning range-dependent conversion efficiency, spectral conversion-induced OSNR/power penalty and transparency to input signal characteristics. The proposed spectral converter has unique advantages including low configuration complexity, strict transparency, SDN-controllable performance reconfigurability and flexibility, as well as negligible spectral conversion-induced latency.

© 2017 Optical Society of America

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References

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  1. J. E. Mitchell, “Integrated wireless backhaul over optical access networks,” J. Lightwave Technol. 32(20), 3373–3382 (2014).
    [Crossref]
  2. P. Chanclou, A. Cui, F. Geilhardt, J. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
    [Crossref]
  3. Y. Okumura and J. Terada, “Optical network technologies and architectures for backhaul/fronthaul of future radio access supporting big mobile data,” in Optical Fiber Communication Conference (Optical Society of America, 2014), Paper Tu3F.1.
    [Crossref]
  4. M. Channegowda, R. Nejabati, and D. Simeonidou, “Software-defined optical networks technology and infrastructure: enabling software-defined optical network operations [Invited],” J. Opt. Netw. 5(10), A274–A282 (2013).
    [Crossref]
  5. A. E. Willner, S. Khaleghi, M. R. Chitgarha, and O. F. Yilmaz, “All-optical signal processing,” J. Lightwave Technol. 32(4), 660–680 (2014).
    [Crossref]
  6. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14(6), 955–966 (1996).
    [Crossref]
  7. A. Nguyen, C. Porzi, G. Serafino, F. Fresi, G. Contestabile, and A. Bogoni, “All-optical gated wavelength converter-eraser using a single SOA-MZI,” IEEE Photonics Technol. Lett. 23(21), 1621–1623 (2011).
    [Crossref]
  8. G. Contestabile, Y. Yoshida, A. Maruta, and K. Kitayama, “Ultra-broad band, low power, highly efficient coherent wavelength conversion in quantum dot SOA,” Opt. Express 20(25), 27902–27907 (2012).
    [Crossref] [PubMed]
  9. E. Temprana, V. Ataie, A. Peric, N. Alic, and S. Radic, “Wavelength conversion of QPSK signals in single-pump FOPA with 20 dB conversion efficiency,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), Paper Th1H.2.
    [Crossref]
  10. H. Murai, Y. Kanda, M. Kagawa, and S. Arahira, “Regenerative SPM-based wavelength conversion and field demonstration of 160-Gb/s all-optical 3R operation,” J. Lightwave Technol. 28(6), 910–921 (2009).
    [Crossref]
  11. D. Zhu and J. Yao, “Dual-chirp microwave waveform generation using a dual-parallel Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 27(13), 1410–1413 (2015).
    [Crossref]
  12. Z. Cao, J. Yu, F. Li, L. Chen, Q. Shu, Q. Tang, and L. Chen, “Energy efficient and transparent platform for optical wireless networks based on reverse modulation,” IEEE J. Sel. Areas Comm. 31(12), 804–814 (2013).
    [Crossref]
  13. W. Jin, X. Duan, Y. Dong, B. Cao, R. P. Giddings, C. F. Zhang, K. Qiu, and J. M. Tang, “DSP-enabled flexible ROADMs without optical filters and O-E-O conversions,” J. Lightwave Technol. 33(19), 4124–4131 (2015).
    [Crossref]
  14. W. Jin, C. F. Zhang, X. Duan, M. R. Kadhum, Y. X. Dong, R. P. Giddings, N. Jiang, K. Qiu, and J. M. Tang, “Improved performance robustness of DSP-enabled flexible ROADMs free from optical filters and O-E-O conversions,” J. Opt. Commun. Netw. 8(8), 521–529 (2016).
    [Crossref]
  15. X. Duan, M. L. Deng, W. Jin, R. P. Giddings, S. Mansoor, and J. M. Tang, “Experimental demonstration of DSP-enabled drop operations of flexible ROADMs excluding optical filters and O-E-O conversions,” in Optical Fiber Communication Conference, (Optical Society of America, 2016), Paper M3E.4.
  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications Inc. 1968).

2016 (1)

2015 (2)

W. Jin, X. Duan, Y. Dong, B. Cao, R. P. Giddings, C. F. Zhang, K. Qiu, and J. M. Tang, “DSP-enabled flexible ROADMs without optical filters and O-E-O conversions,” J. Lightwave Technol. 33(19), 4124–4131 (2015).
[Crossref]

D. Zhu and J. Yao, “Dual-chirp microwave waveform generation using a dual-parallel Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 27(13), 1410–1413 (2015).
[Crossref]

2014 (2)

2013 (2)

M. Channegowda, R. Nejabati, and D. Simeonidou, “Software-defined optical networks technology and infrastructure: enabling software-defined optical network operations [Invited],” J. Opt. Netw. 5(10), A274–A282 (2013).
[Crossref]

Z. Cao, J. Yu, F. Li, L. Chen, Q. Shu, Q. Tang, and L. Chen, “Energy efficient and transparent platform for optical wireless networks based on reverse modulation,” IEEE J. Sel. Areas Comm. 31(12), 804–814 (2013).
[Crossref]

2012 (2)

G. Contestabile, Y. Yoshida, A. Maruta, and K. Kitayama, “Ultra-broad band, low power, highly efficient coherent wavelength conversion in quantum dot SOA,” Opt. Express 20(25), 27902–27907 (2012).
[Crossref] [PubMed]

P. Chanclou, A. Cui, F. Geilhardt, J. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[Crossref]

2011 (1)

A. Nguyen, C. Porzi, G. Serafino, F. Fresi, G. Contestabile, and A. Bogoni, “All-optical gated wavelength converter-eraser using a single SOA-MZI,” IEEE Photonics Technol. Lett. 23(21), 1621–1623 (2011).
[Crossref]

2009 (1)

1996 (1)

S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14(6), 955–966 (1996).
[Crossref]

Arahira, S.

Bogoni, A.

A. Nguyen, C. Porzi, G. Serafino, F. Fresi, G. Contestabile, and A. Bogoni, “All-optical gated wavelength converter-eraser using a single SOA-MZI,” IEEE Photonics Technol. Lett. 23(21), 1621–1623 (2011).
[Crossref]

Cao, B.

Cao, Z.

Z. Cao, J. Yu, F. Li, L. Chen, Q. Shu, Q. Tang, and L. Chen, “Energy efficient and transparent platform for optical wireless networks based on reverse modulation,” IEEE J. Sel. Areas Comm. 31(12), 804–814 (2013).
[Crossref]

Chanclou, P.

P. Chanclou, A. Cui, F. Geilhardt, J. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[Crossref]

Channegowda, M.

M. Channegowda, R. Nejabati, and D. Simeonidou, “Software-defined optical networks technology and infrastructure: enabling software-defined optical network operations [Invited],” J. Opt. Netw. 5(10), A274–A282 (2013).
[Crossref]

Chen, L.

Z. Cao, J. Yu, F. Li, L. Chen, Q. Shu, Q. Tang, and L. Chen, “Energy efficient and transparent platform for optical wireless networks based on reverse modulation,” IEEE J. Sel. Areas Comm. 31(12), 804–814 (2013).
[Crossref]

Z. Cao, J. Yu, F. Li, L. Chen, Q. Shu, Q. Tang, and L. Chen, “Energy efficient and transparent platform for optical wireless networks based on reverse modulation,” IEEE J. Sel. Areas Comm. 31(12), 804–814 (2013).
[Crossref]

Chitgarha, M. R.

Contestabile, G.

G. Contestabile, Y. Yoshida, A. Maruta, and K. Kitayama, “Ultra-broad band, low power, highly efficient coherent wavelength conversion in quantum dot SOA,” Opt. Express 20(25), 27902–27907 (2012).
[Crossref] [PubMed]

A. Nguyen, C. Porzi, G. Serafino, F. Fresi, G. Contestabile, and A. Bogoni, “All-optical gated wavelength converter-eraser using a single SOA-MZI,” IEEE Photonics Technol. Lett. 23(21), 1621–1623 (2011).
[Crossref]

Cui, A.

P. Chanclou, A. Cui, F. Geilhardt, J. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[Crossref]

Dong, Y.

Dong, Y. X.

Duan, X.

Fresi, F.

A. Nguyen, C. Porzi, G. Serafino, F. Fresi, G. Contestabile, and A. Bogoni, “All-optical gated wavelength converter-eraser using a single SOA-MZI,” IEEE Photonics Technol. Lett. 23(21), 1621–1623 (2011).
[Crossref]

Geilhardt, F.

P. Chanclou, A. Cui, F. Geilhardt, J. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[Crossref]

Giddings, R. P.

Jiang, N.

Jin, W.

Kadhum, M. R.

Kagawa, M.

Kanda, Y.

Khaleghi, S.

Kitayama, K.

Li, F.

Z. Cao, J. Yu, F. Li, L. Chen, Q. Shu, Q. Tang, and L. Chen, “Energy efficient and transparent platform for optical wireless networks based on reverse modulation,” IEEE J. Sel. Areas Comm. 31(12), 804–814 (2013).
[Crossref]

Maruta, A.

Mitchell, J. E.

Murai, H.

Nakamura, J.

P. Chanclou, A. Cui, F. Geilhardt, J. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[Crossref]

Nejabati, R.

M. Channegowda, R. Nejabati, and D. Simeonidou, “Software-defined optical networks technology and infrastructure: enabling software-defined optical network operations [Invited],” J. Opt. Netw. 5(10), A274–A282 (2013).
[Crossref]

Nesset, D.

P. Chanclou, A. Cui, F. Geilhardt, J. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[Crossref]

Nguyen, A.

A. Nguyen, C. Porzi, G. Serafino, F. Fresi, G. Contestabile, and A. Bogoni, “All-optical gated wavelength converter-eraser using a single SOA-MZI,” IEEE Photonics Technol. Lett. 23(21), 1621–1623 (2011).
[Crossref]

Porzi, C.

A. Nguyen, C. Porzi, G. Serafino, F. Fresi, G. Contestabile, and A. Bogoni, “All-optical gated wavelength converter-eraser using a single SOA-MZI,” IEEE Photonics Technol. Lett. 23(21), 1621–1623 (2011).
[Crossref]

Qiu, K.

Serafino, G.

A. Nguyen, C. Porzi, G. Serafino, F. Fresi, G. Contestabile, and A. Bogoni, “All-optical gated wavelength converter-eraser using a single SOA-MZI,” IEEE Photonics Technol. Lett. 23(21), 1621–1623 (2011).
[Crossref]

Shu, Q.

Z. Cao, J. Yu, F. Li, L. Chen, Q. Shu, Q. Tang, and L. Chen, “Energy efficient and transparent platform for optical wireless networks based on reverse modulation,” IEEE J. Sel. Areas Comm. 31(12), 804–814 (2013).
[Crossref]

Simeonidou, D.

M. Channegowda, R. Nejabati, and D. Simeonidou, “Software-defined optical networks technology and infrastructure: enabling software-defined optical network operations [Invited],” J. Opt. Netw. 5(10), A274–A282 (2013).
[Crossref]

Tang, J. M.

Tang, Q.

Z. Cao, J. Yu, F. Li, L. Chen, Q. Shu, Q. Tang, and L. Chen, “Energy efficient and transparent platform for optical wireless networks based on reverse modulation,” IEEE J. Sel. Areas Comm. 31(12), 804–814 (2013).
[Crossref]

Willner, A. E.

Yao, J.

D. Zhu and J. Yao, “Dual-chirp microwave waveform generation using a dual-parallel Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 27(13), 1410–1413 (2015).
[Crossref]

Yilmaz, O. F.

Yoo, S. J. B.

S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14(6), 955–966 (1996).
[Crossref]

Yoshida, Y.

Yu, J.

Z. Cao, J. Yu, F. Li, L. Chen, Q. Shu, Q. Tang, and L. Chen, “Energy efficient and transparent platform for optical wireless networks based on reverse modulation,” IEEE J. Sel. Areas Comm. 31(12), 804–814 (2013).
[Crossref]

Zhang, C. F.

Zhu, D.

D. Zhu and J. Yao, “Dual-chirp microwave waveform generation using a dual-parallel Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 27(13), 1410–1413 (2015).
[Crossref]

IEEE J. Sel. Areas Comm. (1)

Z. Cao, J. Yu, F. Li, L. Chen, Q. Shu, Q. Tang, and L. Chen, “Energy efficient and transparent platform for optical wireless networks based on reverse modulation,” IEEE J. Sel. Areas Comm. 31(12), 804–814 (2013).
[Crossref]

IEEE Netw. (1)

P. Chanclou, A. Cui, F. Geilhardt, J. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[Crossref]

IEEE Photonics Technol. Lett. (2)

A. Nguyen, C. Porzi, G. Serafino, F. Fresi, G. Contestabile, and A. Bogoni, “All-optical gated wavelength converter-eraser using a single SOA-MZI,” IEEE Photonics Technol. Lett. 23(21), 1621–1623 (2011).
[Crossref]

D. Zhu and J. Yao, “Dual-chirp microwave waveform generation using a dual-parallel Mach–Zehnder modulator,” IEEE Photonics Technol. Lett. 27(13), 1410–1413 (2015).
[Crossref]

J. Lightwave Technol. (5)

J. Opt. Commun. Netw. (1)

J. Opt. Netw. (1)

M. Channegowda, R. Nejabati, and D. Simeonidou, “Software-defined optical networks technology and infrastructure: enabling software-defined optical network operations [Invited],” J. Opt. Netw. 5(10), A274–A282 (2013).
[Crossref]

Opt. Express (1)

Other (4)

E. Temprana, V. Ataie, A. Peric, N. Alic, and S. Radic, “Wavelength conversion of QPSK signals in single-pump FOPA with 20 dB conversion efficiency,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), Paper Th1H.2.
[Crossref]

Y. Okumura and J. Terada, “Optical network technologies and architectures for backhaul/fronthaul of future radio access supporting big mobile data,” in Optical Fiber Communication Conference (Optical Society of America, 2014), Paper Tu3F.1.
[Crossref]

X. Duan, M. L. Deng, W. Jin, R. P. Giddings, S. Mansoor, and J. M. Tang, “Experimental demonstration of DSP-enabled drop operations of flexible ROADMs excluding optical filters and O-E-O conversions,” in Optical Fiber Communication Conference, (Optical Society of America, 2016), Paper M3E.4.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications Inc. 1968).

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Figures (15)

Fig. 1
Fig. 1 (a) Schematic illustration of the proposed spectral converter utilizing a single standard DP-MZM. (b) Representative converted optical components and frequency detuning range definition.
Fig. 2
Fig. 2 An illustration of utilizing spectral converters with a soft-ROADM to dynamically integrate mobile fronthaul and backhaul networks.
Fig. 3
Fig. 3 Spectral conversion efficiencies of various converted optical components under optimum and randomly selected operating condition parameters.
Fig. 4
Fig. 4 Converted optical component index-dependent total number of parasitic optical components in the converted optical signals obtained under optimum and randomly selected operating condition parameters.
Fig. 5
Fig. 5 BERs of the 5-th converted optical component for optimum and randomly selected operating condition parameters. The OSNR of input signal is fixed at 28dB.
Fig. 6
Fig. 6 BERs of the 5-th converted optical component for optimum and randomly selected operating condition parameters. The OSNR of the 5-th converted optical component is fixed at 20dB.
Fig. 7
Fig. 7 Spectral conversion efficiency versus frequency detuning range for different input optical signal powers. The identified optimum operating conditions are adopted.
Fig. 8
Fig. 8 BER versus signal OSNR for the 1-st, 5-th and 9-th converted optical components, as well as a spectral conversion-free input signal. (a) BER versus input signal OSNR; (b) BER versus converted component OSNR. The identified optimum operating conditions are adopted for all these two cases.
Fig. 9
Fig. 9 Normalized signal waveforms of the 1-st, 5-th and 9-th converted optical components as well as the conversion-free input optical signal. The identified optimum operating conditions are adopted.
Fig. 10
Fig. 10 5-th converted optical component BER performances versus OSNR of input optical signals of various characteristics.
Fig. 11
Fig. 11 5-th converted optical component BER performances versus OSNR of input signal for different widths of optical filters.
Fig. 12
Fig. 12 Impacts of DP-MZM extinction ratio on the spectral converter performance for the 1-, 5- and 9-th converted optical components. (a) conversion efficiency versus extinction ratio, (b) total number of parasitic optical components versus extinction ratio, and (c) BER performance versus extinction ratio. In calculating these three figures, the OSNR of each converted optical component is fixed at 20dB.
Fig. 13
Fig. 13 Representative optical spectrum (a) emerging from the DP-MZM subject to the optimum operating conditions for the 1-st converted optical component; (b) emerging from the DP-MZM subject to non-optimized operating conditions.
Fig. 14
Fig. 14 BERs versus optical launch power for the 1-st and 3-rd converted optical components each obtained under its optimum operating conditions. A BER curve for the spectral conversion-free input optical signal is also plotted for comparison.
Fig. 15
Fig. 15 Normalized signal waveforms of the 1-st and 3-rd converted optical component as well as the conversion-free input optical signal. In measuring the waveforms of the converted optical components, the identified optimum operating conditions are adopted.

Tables (1)

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Table 1 Simulation Parameters

Equations (18)

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{ E out_up ( t )= A( t ) e j( ω c t+ θ c ) t ff 4 { e j( Δ φ up + β 1 cos( ω rf t+ θ 1 ) ) + e j( β 2 cos( ω rf t+ θ 2 ) ) } E out_down ( t )= A( t ) e j( ω c t+ θ c ) t ff 4 { e j( Δ φ down + β 3 cos( ω rf t+ θ 3 ) ) + e j( β 4 cos( ω rf t+ θ 4 ) ) }
E out ( t )= E out_up ( t )+ e jϕ E out_down ( t ) = A( t ) e j( ω c t+ θ c ) t ff 4 { + k= J k ( β 1 ) e j[ Δ φ up + kπ 2 +k( ω rf t+ θ 1 ) ] + k= J k ( β 2 ) e j[ kπ 2 +k( ω rf t+ θ 2 ) ] + e jϕ k= J k ( β 3 ) e j[ Δ φ down + kπ 2 +k( ω rf t+ θ 3 ) ] + e jϕ k= J k ( β 4 ) e j[ kπ 2 +k( ω rf t+ θ 4 ) ] }
E out ( ω c t+p( ω rf t ) )= A( t ) t ff 4 K p e j( ω c t+ θ c +p ω rf t+ pπ 2 )
K p = J p ( β 1 ) e j( Δ φ up +p θ 1 ) + J p ( β 2 ) e j( p θ 2 ) + J p ( β 3 ) e j( Δ φ down +p θ 3 +ϕ ) + J p ( β 4 ) e j( p θ 4 +ϕ )
η p = t ff | K p | 2 16
| K p | 2 ={ J p 2 ( β 1 )+ J p 2 ( β 2 )+ J p 2 ( β 3 )+ J p 2 ( β 4 ) +2 J p ( β 1 ) J p ( β 2 )cos( p θ 1 p θ 2 +Δ φ up ) +2 J p ( β 1 ) J p ( β 3 )cos( p θ 1 p θ 3 +Δ φ up Δ φ down ϕ ) +2 J p ( β 1 ) J p ( β 4 )cos( p θ 1 p θ 4 +Δ φ up ϕ ) +2 J p ( β 2 ) J p ( β 3 )cos( p θ 2 p θ 3 Δ φ down ϕ ) +2 J p ( β 2 ) J p ( β 4 )cos( p θ 2 p θ 4 ϕ ) +2 J p ( β 3 ) J p ( β 4 )cos( p θ 3 p θ 4 +Δ φ down )
{ p θ 1 p θ 2 +Δ φ up =2mπ p θ 1 p θ 4 +Δ φ up ϕ=2mπ p θ 2 p θ 3 Δ φ down ϕ=2mπ p θ 3 p θ 4 +Δ φ down =2mπ β p_opt = β 1 = β 2 = β 3 = β 4
K 0 =2 J o ( β p_opt )[ cos Δ φ up 2 e j Δ φ up 2 + e jϕ cos Δ φ down 2 e j Δ φ down 2 ]
{ Δ φ up =Δ φ down &ϕ=( 2m1 )π,forcaseI Δ φ up =Δ φ down =π+2mπ,forcaseII
K k = J k ( β p_opt )[ 2cos( k θ 1 θ 3 2 ϕ 2 ) e j( Δ φ up +Δ φ down +k θ 1 +k θ 3 +ϕ 2 ) +2cos( k θ 2 θ 4 2 ϕ 2 ) e j( k θ 2 +k θ 4 +ϕ 2 ) ]
{ ϕ=( 2m1 )π θ 1 θ 3 =( 2m1 )π θ 2 θ 4 =( 2m1 )π
E out ( t )= E out_up ( t )+ e jϕ E out_down ( t ) = A( t ) t ff J p_max ( β p_opt ) e j[ ω c t+p ω rf t+ θ c + Ψ p ] Convertedopticalcomponent +A( t ) t ff k=,kp J k ( β p_opt ) ×[ cos( ( pk ) 2 ( θ 1 θ 2 + θ 3 θ 4 ) 2 ) e j[ ω c t+k ω rf t+ θ c + Ψ k ] ]
Ψ p = Δ φ up +Δ φ down +2pπ+p( θ 1 + θ 2 + θ 3 + θ 4 ) 4
( θ 1 θ 2 + θ 3 θ 4 ) 2 =( π 2 +mπ)
E out = E out_up + e jϕ E out_down = A( t ) t ff J p_max ( β p_opt ) e j( ω c t+p ω rf t+ θ c + Ψ p ) Convertedopticalcomponent + A( t ) t ff J p ( β p_opt )[ +... + J p8 ( β p_opt ) e j( ω c t+( p8 ) ω rf t+ θ c + Ψ ( p8 ) ) J p4 ( β p_opt ) e j( ω c t+( p4 ) ω rf t+ θ c + Ψ ( p4 ) ) J p+4 ( β p_opt ) e j( ω c t+( p+4 ) ω rf t+ θ c + Ψ ( p+4 ) ) + J p+8 ( β p_opt ) e j( ω c t+( p+8 ) ω rf t+ θ c + Ψ ( p+8 ) ) +... ] Parasiticopticalcomponents
η p = J p_max 2 ( β p_opt ) t ff
{ ϕ=π+2mπ Δθ=Δ θ 12 =Δ θ 23 =Δ θ 34 =Δ θ 41 = π 2 +mπ Δ φ up =Δ φ down =pΔθ β p_opt = β 1 = β 2 = β 3 = β 4
f d =p f RF

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