Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link

C. Sánchez; B. Ortega; J. L. Wei; J. Tang; J. Capmany

doi:10.1364/OE.21.007651

Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link

C. Sánchez, B. Ortega, J. L. Wei, J. Tang, and J. Capmany

Optics Express
Vol. 21,
Issue 6,
pp. 7651-7666
(2013)
•https://doi.org/10.1364/OE.21.007651

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C. Sánchez, B. Ortega, J. L. Wei, J. Tang, and J. Capmany, "Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link," Opt. Express 21, 7651-7666 (2013)

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Related Topics
Table of Contents Category
- Fiber Optics and Optical Communications
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fiber optics communications (060.2330)
- Modulation (060.4080)
- Lasers, fiber (060.3510)

About this Article
History
- Original Manuscript: November 26, 2012
- Revised Manuscript: January 22, 2013
- Manuscript Accepted: February 18, 2013
- Published: March 21, 2013

Accessible

Open Access

Abstract

We provide an analytical study on the propagation effects of a directly modulated OOFDM signal through a dispersive fiber and subsequent photo-detection. The analysis includes the effects of the laser operation point and the interplay between chromatic dispersion and laser chirp. The final expression allows to understand the physics behind the transmission of a multi-carrier signal in the presence of residual frequency modulation and the description of the induced intermodulation distortion gives us a detailed insight into the diferent intermodulation products which impair the recovered signal at the receiver-end side. Numerical comparisons between transmission simulations results and those provided by evaluating the expression obtained are carried out for different laser operation points. Results obtained by changing the fiber length, laser parameters and using single mode fiber with negative and positive dispersion are calculated in order to demonstrate the validity and versatility of the theory provided in this paper. Therefore, a novel analytical formulation is presented as a versatile tool for the description and study of IM/DD OOFDM systems with variable design parameters.

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References

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|
Author
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Publication

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Elsevier/Academic Press, 2009).
J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
[Crossref]
E. Hugues-Salas, R. P. Giddings, X. Q. Jin, J. L. Wei, X. Zheng, Y. Hong, C. Shu, and J. M. Tang, “Real-time experimental demonstration of low-cost VCSEL intensity-modulated 11.25Gb/s optical OFDM signal transmission over 25km PON systems,” Opt. Express 19(4), 2979–2988 (2011).
[Crossref] [PubMed]
D-Z. Hsu, C-C. Wei, H-Y. Chen, W-Y. Li, and J. Chen, “Cost-effective 33-Gbps intensity modulation direct detection multi-band OFDM LR-PON system employing a 10-GHz-based transceiver,” Opt. Express 19(18), 17546–17556 (2011).
[Crossref] [PubMed]
W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express 16(9), 6378–6386 (2008).
[Crossref] [PubMed]
D. Visani, C. Okonkwo, S. Loquai, H. Yang, Y. Shi, H. van de Boom, T. Ditewig, G. Tartarini, B. Schmauss, J. Lee, T. Koonen, and E. Tangdiongga, “Beyond 1Gbit/s transmission over 1 mm diameter plastic optical fiber employing DMT for in-home communication systems,” J. Lightwave Technol. 29(4), 622–628 (2011).
B. Franz, D. Suikat, R. Dischler, F. Buchali, and H. Buelow, “High speed OFDM data transmission over 5 km GI-multimode fiber using spatial multiplexing with 2×4 MIMO processing,” in ECOC2010, paper Tu3C4.
N. Cvijetic, D. Qian, and T. Wang, “10Gb/s free-space pptical transmission using OFDM,” in OFC/NFOEC2008, paper OThD2.
N. Cvijetic, D. Qian, J. Hu, and T. Wang, “Orthogonal frequency division multiple access PON (OFDMA-PON) for colorless upstream transmission beyond 10 Gb/s,” IEEE J. Sel. Areas Commun. 28(6), 781–790 (2010).
[Crossref]
X. Q. Jin, E. H-Salas, R. P. Giddings, J. L. Wei, J. Groenewald, and J. M. Tang, “First real-time experimental demonstrations of 11.25Gb/s optical OFDMA PONs with adaptive dynamic bandwidth allocation,” Opt. Express 19(21). 20557–20570 (2011).
[Crossref] [PubMed]
B. J. C. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental demonstrations of electronic dispersion compensation for long-haul transmission using direct-detection optical OFDM,” J. Lightwave Technol. 26(1), 196–203 (2008).
[Crossref]
Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelenth bandwidth access,” Opt. Express 17(11), 9421–9427 (2009).
[Crossref] [PubMed]
E. Vanin, “Performance evaluation of intensity modulated optical OFDM system with digital baseband distortion,” Opt. Express 19(5), 4280–4293 (2011).
[Crossref] [PubMed]
C-C. Wei, “Small-signal analysis of OOFDM signal transmission with directly modulated laser and direct detection,” Opt. Letters 36(2), 151–153 (2011).
[Crossref]
C-C. Wei, “Analysis and iterative equalization of transient and adiabatic chirp effects in DML-based OFDM transmission systems,” Opt. Express 20(23), 25774–25789 (2012).
[Crossref] [PubMed]
G. P. Agrawal and N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, 1993).
J. L. Wei, X. Q. Jin, and J. M. Tang, “The influence of directly modulated DFB lasers on the transmission performance of carrier-suppressed single-sideband optical OFDM signals over IMDD SMF systems,” J. Lightwave Technol. 27(13), 2412–2419 (2009).
[Crossref]
E. Peral, “Large-signal theory of the effect of dispersive propagation on the intensity modulation response of semiconductor lasers,” J. Lightwave Technol. 18(1), 84–89 (2000).
[Crossref]
J. Helms, “Intermodulation distortions of broad-band modulated laser diodes,” J. Lightwave Technol. 10(12), 1901–1906 (1992).
[Crossref]
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).
P. K. Vitthaladevuni, M-S. Alouini, and J. C. Kieffer, “Exact BER computation for cross QAM constellations,” IEEE Trans. Wireless Commun. 4(6), 3039–3050 (2005).
[Crossref]
X. Zheng, X. Q. Jin, R. P. Giddings, J. L. Wei, E. Hugues-Salas, Y. H. Hong, and J. M. Tang, “Negative power penalties of optical OFDM signal transmission in directly modulated DFB laser-based IMDD systems incorporating negative dispersion fibers,” IEEE Photon. J. 2(4), 532–542 (2010).
[Crossref]

2012 (1)

C-C. Wei, “Analysis and iterative equalization of transient and adiabatic chirp effects in DML-based OFDM transmission systems,” Opt. Express 20(23), 25774–25789 (2012).
[Crossref] [PubMed]

2011 (6)

E. Vanin, “Performance evaluation of intensity modulated optical OFDM system with digital baseband distortion,” Opt. Express 19(5), 4280–4293 (2011).
[Crossref] [PubMed]

C-C. Wei, “Small-signal analysis of OOFDM signal transmission with directly modulated laser and direct detection,” Opt. Letters 36(2), 151–153 (2011).
[Crossref]

E. Hugues-Salas, R. P. Giddings, X. Q. Jin, J. L. Wei, X. Zheng, Y. Hong, C. Shu, and J. M. Tang, “Real-time experimental demonstration of low-cost VCSEL intensity-modulated 11.25Gb/s optical OFDM signal transmission over 25km PON systems,” Opt. Express 19(4), 2979–2988 (2011).
[Crossref] [PubMed]

D-Z. Hsu, C-C. Wei, H-Y. Chen, W-Y. Li, and J. Chen, “Cost-effective 33-Gbps intensity modulation direct detection multi-band OFDM LR-PON system employing a 10-GHz-based transceiver,” Opt. Express 19(18), 17546–17556 (2011).
[Crossref] [PubMed]

D. Visani, C. Okonkwo, S. Loquai, H. Yang, Y. Shi, H. van de Boom, T. Ditewig, G. Tartarini, B. Schmauss, J. Lee, T. Koonen, and E. Tangdiongga, “Beyond 1Gbit/s transmission over 1 mm diameter plastic optical fiber employing DMT for in-home communication systems,” J. Lightwave Technol. 29(4), 622–628 (2011).

X. Q. Jin, E. H-Salas, R. P. Giddings, J. L. Wei, J. Groenewald, and J. M. Tang, “First real-time experimental demonstrations of 11.25Gb/s optical OFDMA PONs with adaptive dynamic bandwidth allocation,” Opt. Express 19(21). 20557–20570 (2011).
[Crossref] [PubMed]

2010 (2)

N. Cvijetic, D. Qian, J. Hu, and T. Wang, “Orthogonal frequency division multiple access PON (OFDMA-PON) for colorless upstream transmission beyond 10 Gb/s,” IEEE J. Sel. Areas Commun. 28(6), 781–790 (2010).
[Crossref]

X. Zheng, X. Q. Jin, R. P. Giddings, J. L. Wei, E. Hugues-Salas, Y. H. Hong, and J. M. Tang, “Negative power penalties of optical OFDM signal transmission in directly modulated DFB laser-based IMDD systems incorporating negative dispersion fibers,” IEEE Photon. J. 2(4), 532–542 (2010).
[Crossref]

2009 (3)

J. L. Wei, X. Q. Jin, and J. M. Tang, “The influence of directly modulated DFB lasers on the transmission performance of carrier-suppressed single-sideband optical OFDM signals over IMDD SMF systems,” J. Lightwave Technol. 27(13), 2412–2419 (2009).
[Crossref]

J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
[Crossref]

Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelenth bandwidth access,” Opt. Express 17(11), 9421–9427 (2009).
[Crossref] [PubMed]

2008 (2)

B. J. C. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental demonstrations of electronic dispersion compensation for long-haul transmission using direct-detection optical OFDM,” J. Lightwave Technol. 26(1), 196–203 (2008).
[Crossref]

W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express 16(9), 6378–6386 (2008).
[Crossref] [PubMed]

2005 (1)

P. K. Vitthaladevuni, M-S. Alouini, and J. C. Kieffer, “Exact BER computation for cross QAM constellations,” IEEE Trans. Wireless Commun. 4(6), 3039–3050 (2005).
[Crossref]

2000 (1)

E. Peral, “Large-signal theory of the effect of dispersive propagation on the intensity modulation response of semiconductor lasers,” J. Lightwave Technol. 18(1), 84–89 (2000).
[Crossref]

1992 (1)

J. Helms, “Intermodulation distortions of broad-band modulated laser diodes,” J. Lightwave Technol. 10(12), 1901–1906 (1992).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

Agrawal, G. P.

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, 1993).

Alouini, M-S.

P. K. Vitthaladevuni, M-S. Alouini, and J. C. Kieffer, “Exact BER computation for cross QAM constellations,” IEEE Trans. Wireless Commun. 4(6), 3039–3050 (2005).
[Crossref]

Armstrong, J.

J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
[Crossref]

Buchali, F.

B. Franz, D. Suikat, R. Dischler, F. Buchali, and H. Buelow, “High speed OFDM data transmission over 5 km GI-multimode fiber using spatial multiplexing with 2×4 MIMO processing,” in ECOC2010, paper Tu3C4.

Buelow, H.

Chen, H-Y.

Chen, J.

Chen, S.

Cvijetic, N.

N. Cvijetic, D. Qian, and T. Wang, “10Gb/s free-space pptical transmission using OFDM,” in OFC/NFOEC2008, paper OThD2.

Dischler, R.

Ditewig, T.

Djordjevic, I.

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Elsevier/Academic Press, 2009).

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, 1993).

Franz, B.

Giddings, R. P.

Groenewald, J.

Helms, J.

J. Helms, “Intermodulation distortions of broad-band modulated laser diodes,” J. Lightwave Technol. 10(12), 1901–1906 (1992).
[Crossref]

Hong, Y.

Hong, Y. H.

H-Salas, E.

Hsu, D-Z.

Hu, J.

Hugues-Salas, E.

Jin, X. Q.

Kieffer, J. C.

P. K. Vitthaladevuni, M-S. Alouini, and J. C. Kieffer, “Exact BER computation for cross QAM constellations,” IEEE Trans. Wireless Commun. 4(6), 3039–3050 (2005).
[Crossref]

Koonen, T.

Lee, J.

Li, W-Y.

Loquai, S.

Lowery, A. J.

Ma, Y.

W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express 16(9), 6378–6386 (2008).
[Crossref] [PubMed]

Okonkwo, C.

Peral, E.

E. Peral, “Large-signal theory of the effect of dispersive propagation on the intensity modulation response of semiconductor lasers,” J. Lightwave Technol. 18(1), 84–89 (2000).
[Crossref]

Qian, D.

N. Cvijetic, D. Qian, and T. Wang, “10Gb/s free-space pptical transmission using OFDM,” in OFC/NFOEC2008, paper OThD2.

Schmauss, B.

Schmidt, B. J. C.

Shi, Y.

Shieh, W.

W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express 16(9), 6378–6386 (2008).
[Crossref] [PubMed]

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Elsevier/Academic Press, 2009).

Shu, C.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

Suikat, D.

Tang, J. M.

Tang, Y.

Tangdiongga, E.

Tartarini, G.

van de Boom, H.

Vanin, E.

E. Vanin, “Performance evaluation of intensity modulated optical OFDM system with digital baseband distortion,” Opt. Express 19(5), 4280–4293 (2011).
[Crossref] [PubMed]

Visani, D.

Vitthaladevuni, P. K.

P. K. Vitthaladevuni, M-S. Alouini, and J. C. Kieffer, “Exact BER computation for cross QAM constellations,” IEEE Trans. Wireless Commun. 4(6), 3039–3050 (2005).
[Crossref]

Wang, T.

N. Cvijetic, D. Qian, and T. Wang, “10Gb/s free-space pptical transmission using OFDM,” in OFC/NFOEC2008, paper OThD2.

Wei, C-C.

C-C. Wei, “Analysis and iterative equalization of transient and adiabatic chirp effects in DML-based OFDM transmission systems,” Opt. Express 20(23), 25774–25789 (2012).
[Crossref] [PubMed]

C-C. Wei, “Small-signal analysis of OOFDM signal transmission with directly modulated laser and direct detection,” Opt. Letters 36(2), 151–153 (2011).
[Crossref]

Wei, J. L.

Yang, H.

Yang, Q.

W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express 16(9), 6378–6386 (2008).
[Crossref] [PubMed]

Zheng, X.

IEEE J. Sel. Areas Commun. (1)

IEEE Photon. J. (1)

IEEE Trans. Wireless Commun. (1)

P. K. Vitthaladevuni, M-S. Alouini, and J. C. Kieffer, “Exact BER computation for cross QAM constellations,” IEEE Trans. Wireless Commun. 4(6), 3039–3050 (2005).
[Crossref]

J. Lightwave Technol. (6)

J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
[Crossref]

E. Peral, “Large-signal theory of the effect of dispersive propagation on the intensity modulation response of semiconductor lasers,” J. Lightwave Technol. 18(1), 84–89 (2000).
[Crossref]

J. Helms, “Intermodulation distortions of broad-band modulated laser diodes,” J. Lightwave Technol. 10(12), 1901–1906 (1992).
[Crossref]

Opt. Express (7)

E. Vanin, “Performance evaluation of intensity modulated optical OFDM system with digital baseband distortion,” Opt. Express 19(5), 4280–4293 (2011).
[Crossref] [PubMed]

W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express 16(9), 6378–6386 (2008).
[Crossref] [PubMed]

C-C. Wei, “Analysis and iterative equalization of transient and adiabatic chirp effects in DML-based OFDM transmission systems,” Opt. Express 20(23), 25774–25789 (2012).
[Crossref] [PubMed]

Opt. Letters (1)

C-C. Wei, “Small-signal analysis of OOFDM signal transmission with directly modulated laser and direct detection,” Opt. Letters 36(2), 151–153 (2011).
[Crossref]

Other (5)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

N. Cvijetic, D. Qian, and T. Wang, “10Gb/s free-space pptical transmission using OFDM,” in OFC/NFOEC2008, paper OThD2.

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, 1993).

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Elsevier/Academic Press, 2009).

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Fig. 1

Schematic illustration of the simulated OOFDM system.

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Fig. 2

Constellations obtained by simulation of the 4-QAM IM/DD OOFDM system and evaluation of Eq. (14). (a) BW =2.5GHz, FS=32, N=14, T_pre=T_pos=0.125, L=0km, i₀=40mA, Δi=0.005, α=4; (b) same as (a), but L=60km; (c) same as (b), but L=80km and i₀=60mA; (d) same as (c) but BW =5.5GHz; (e) same as (d), but Δ i=0.01mA; (f) same as (e), but α=10; (g) same as (f), but FS=128, N=55 and α=4; (h) same as (g), but the nonlinear gain coefficient of the laser is equal to 3×10⁻²³ m³, i₀=80mA, Δi=0.015 and L=100km.

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Fig. 3

Constellations obtained by simulation of the 32-QAM IM/DD OOFDM system and evaluation of Eq. (14). BW =5.5GHz, FS=128, N=55, T_pre=T_pos=0.125, L=60km, α=4, i₀=60mA and Δi=0.01mA. (a) the nonlinear gain coefficient is equal to 3×10⁻²⁴ m³; (b) the nonlinear gain coefficient is equal to 3×10⁻²³ m³; (c) same as (b) but the fiber dispersion is D=−7ps/(nm·km).

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Fig. 4

Comparison of the carrier to interference power ratio obtained through simulation of an IM/DD OOFDM system and that obtained through numerical computation of Eq. (14), and carrier to different interfering terms power ratios for a dispersive fiber link of length (a) 20 km, (b) 40 km, (c) 60 km, and (d) 80 km.

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Fig. 5

Comparison of the BER obtained through numerical simulations and calculated by the obtained expression Eq. (14) for standard and negative dispersion fibers.

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Tables (1)

Table 1 Laser Parameters

View Table

Equations (21)

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s (t) \propto \sum_{k = 1}^{N} | X_{k} | \cdot \cos (Ω_{k} t + φ x_{k})

\begin{array}{l} \frac{d p (t)}{d t} = [Γ \cdot v_{g} a_{g} \frac{n (t) - n_{t}}{1 + ε_{n l} \cdot p (t)} - \frac{1}{τ_{p}}] p (t) + ζ Γ B \cdot n^{2} (t) \\ \frac{d n (t)}{d t} = \frac{i (t)}{e \cdot V} - A n - B n^{2} - C n^{3} - v_{g} a_{g} \frac{n (t) - n_{t}}{1 + ε_{n l} \cdot p (t)} \cdot p (t) \\ \frac{d ϕ}{d t} = \frac{1}{2} α Γ v_{g} a_{g} (n (t) - n_{t}) \end{array}

E (t, z = 0) = \sqrt{P (t)} \cdot \exp (j \cdot ϕ_{1} (t)) \cdot \exp (j \cdot ω_{0} t)

ϕ_{1} (t) = \sum_{k = 1}^{N} m_{k} \cdot \sin (Ω_{k} t + φ_{m_{k}})

H (ω) = \exp (j \cdot β (ω) \cdot L) = \exp (j \cdot (β_{0} + β_{1} \cdot (ω - ω_{0}) + \frac{1}{2} β_{2} \cdot {(ω - ω_{0})}^{2}) \cdot L)

E (t, z = L) = F T^{- 1} {F T {E (t, z = 0)} \cdot H (ω)}

I_{p h} (t) = 𝔕 \cdot {| E (t, z = L) |}^{2}

i (t) = i_{0} + \sum_{k = 1}^{N} 2 i_{k} \cdot \cos (Ω_{k} t + φ_{i_{k}})

\begin{array}{l} P (t) \approx P_{0} + P_{1} (t) + P_{2} (t) + P_{11} (t) = P_{0} + \sum_{k = 1}^{N} 2 p_{k} \cdot \cos (Ω_{k} t + φ_{p_{k}}) + \sum_{k = 1}^{N} 2 p_{2 k} \cdot \cos (2 Ω_{k} t + φ_{p_{2 k}}) + \\ \sum_{k = 1}^{N} \sum_{l = 1}^{k - 1} 2 p_{k l} \cdot \cos ((Ω_{k} + Ω_{t}) t + φ_{p_{k l}}) + \sum_{k = 1}^{N} \sum_{l = 1}^{k - 1} 2 p_{k_l} \cdot \cos (((Ω_{k} - Ω_{l}) + φ_{p_{k_l}})) \end{array}

\begin{matrix} p_{k} \cdot \exp (j \cdot φ_{p_{k}}) = H_{p 1} (Ω_{k}) \cdot i_{k} \cdot \exp (j \cdot φ_{i_{k}}) \\ p_{2 k} \cdot \exp (φ_{p_{2 k}}) = H_{p 2} (Ω_{k}) \cdot i_{k}^{2} \cdot \exp (j \cdot 2 \cdot φ_{i_{k}}) \\ p_{k l} \cdot \exp (j \cdot φ_{p_{k l}}) = H_{p 11} ((Ω_{k}, Ω_{l})) \cdot i_{k} \cdot i_{l} \cdot \exp (j \cdot (φ_{i_{k}} + φ_{i_{l}})) \\ p_{k_l} \cdot \exp (j \cdot φ_{p_{k_l}}) = H_{p 11} ((Ω_{k}, - Ω_{l})) \cdot i_{k} \cdot i_{l} \cdot \exp (j \cdot (φ_{i_{k}} - φ_{i_{l}})) \end{matrix}

\begin{array}{l} | E (t) | = a + b \cdot P_{1} (t) + c \cdot P_{2} (t) + c \cdot P_{11} (t) + d \cdot \sum_{k = 1}^{N} \sum_{\begin{array}{l} l = 1 \\ l \neq k \end{array}}^{N} 2 p_{k} \cdot p_{l} \cdot \cos ((Ω_{k} + Ω_{l}) t + φ_{p_{k}} + φ_{p_{l}}) \\ + d \cdot \sum_{k = 1}^{N} \sum_{\begin{array}{l} l = 1 \\ l \neq k \end{array}}^{N} 2 p_{k} \cdot p_{l} \cdot \cos ((Ω_{k} - Ω_{l}) t + φ_{p_{k}} - φ_{p_{l}}) + d \sum_{k = 1}^{N} 2 p_{k}^{2} \cdot \cos (2 (Ω_{k} t + φ_{p_{k}})) \end{array}

a = P_{0}^{\frac{1}{2}}, b = c = \frac{1}{2 P_{0}^{\frac{1}{2}}}, d = - \frac{1}{8 P_{0}^{\frac{3}{2}}}

E (t) = | E (t) | \cdot \exp (j \cdot \sum_{k = 1}^{N} m_{k} \cdot \sin (Ω_{k} t + φ_{m_{k}}))

\begin{array}{l} I_{p h} (t) \approx 𝔕 \cdot \sum_{n_{1} \dots n_{N} = - \infty}^{\infty} J_{n_{1}} (μ_{1}) \cdot \cdot J_{n_{N}} (μ_{N}) \cdot (\overset{T 0}{\overset{︷}{a^{2} + 2 \cdot b^{2} \sum_{k = 1}^{N} p_{k}^{2} \cdot \cos (2 θ_{k}) +}} \\ \overset{T 1}{\overset{︷}{2 a b \sum_{k = 1}^{N} \frac{p_{k}}{J_{n_{k}} (μ_{k})} \cdot \cos (θ_{k}) (J_{n_{k} + 1} (μ_{k}) \cdot \exp (j (Δ φ_{k} + \frac{π}{2})) + J_{n_{k} - 1} (μ_{k}) \cdot \exp (- j (Δ φ_{k} + \frac{π}{2})))}} + \\ \overset{T 2}{\overset{︷}{2 a c \sum_{k = 1}^{N} \frac{p_{2 k}}{J_{n_{k}} (μ_{k})} \cos (2 θ_{k}) (J_{n_{k} + 2} (μ_{k}) \cdot \exp (j (2 φ_{m_{k}} - φ_{p_{2 k}} + π)) + J_{n_{k} - 2} (μ_{k}) \cdot \exp (- j (2 φ_{m_{k}} - φ_{p_{2 k}} + π)))}} \\ \overset{T 3}{\overset{︷}{+ 2 a c \sum_{k = 1}^{N} \sum_{l = 1}^{k - 1} \frac{p_{k l}}{J_{n_{k}} (μ_{k}) J_{n_{l}} (μ_{l})} \cos (θ_{k} + θ_{l}) (J_{n_{k} + 1} (μ_{k}) \cdot J_{n_{l} + 1} (μ_{l}) \cdot \exp (j (φ_{m_{k}} + φ_{m_{l}} - φ_{p_{k l}} + π))) +}} \\ \overset{T 3}{\overset{︷}{J_{n_{k} - 1} (μ_{k}) \cdot J_{n_{l} - 1} (μ_{l}) \cdot \exp (- j (φ_{m_{k}} + φ_{m_{l}} - φ_{p_{k l}} + π)) + 2 a c \sum_{k = 1}^{N} \sum_{l = 1}^{k - 1} \frac{p_{k_l}}{J_{n_{k}} (μ_{k}) J_{n_{l}} (μ_{l})} \cos (θ_{k} - θ_{l})}} \\ \overset{T 3}{\overset{︷}{(J_{n_{k} + 1} (μ_{k}) \cdot J_{n_{l} - 1} (μ_{l}) \cdot \exp (j (φ_{m_{k}} - φ_{m_{l}} - φ_{p_{k_l}})) + J_{n_{k} - 1} (μ_{k}) \cdot J_{n_{l} + 1} (μ_{l}) \cdot \exp (- j (φ_{m_{k}} - φ_{m_{l}} - φ_{p_{k_l}})))}} \\ \overset{T 4}{\overset{︷}{+ \sum_{k = 1}^{N} \sum_{\begin{array}{l} l = 1 \\ l \neq k \end{array}}^{N} \frac{p_{k} \cdot p_{l}}{J_{n_{k}} (μ_{k}) J_{n_{l}} (μ_{l})} (J_{n_{k} + 1} (μ_{k}) \cdot J_{n_{l} + 1} (μ_{l}) \cdot \exp (j (Δ φ_{k} + Δ φ_{l} + π)) + J_{n_{k} - 1} (μ_{k}) \cdot J_{n_{l} - 1} (μ_{l})) \cdot}} \\ \overset{T 4}{\overset{︷}{\exp (- j (Δ φ_{k} + Δ φ_{l} + π))) (2 a d \cdot \cos (θ_{k} + θ_{l}) + b^{2}) + \sum_{k = 1}^{N} \sum_{\begin{array}{l} l = 1 \\ l \neq k \end{array}}^{N} \frac{p_{k} \cdot p_{l}}{J_{n_{k}} (μ_{k}) J_{n_{l}} (μ_{l})} (J_{n_{k} + 1} (μ_{k}) \cdot J_{n_{l} - 1} (μ_{l}) \cdot}} \\ \overset{T 4}{\overset{︷}{\exp (j (Δ φ_{k} - Δ φ_{l})) + J_{n_{k} - 1} (μ_{k}) \cdot J_{n_{l} + 1} (μ_{l}) \cdot \exp (- j (Δ φ_{k} - Δ φ_{l}))) (2 a d \cdot \cos (θ_{k} - θ_{l}) + b^{2})}} + \\ \overset{T 5}{\overset{︷}{\sum_{k = 1}^{N} p_{k}^{2} (\frac{J_{n_{k} + 2} (μ_{k}) \cdot \exp (j 2 (Δ φ_{k} + \frac{π}{2})) + J_{n_{k} - 2} (μ_{k}) \cdot \exp (- j 2 (Δ φ_{k} + \frac{π}{2}))}{J_{n_{k}} (μ_{k})} (2 a d \cdot \cos (2 θ_{k} + b^{2}))}}) \\ \cdot \exp (j (Ω_{imp} t + \sum_{k = 1}^{N} n_{k} (φ_{m_{k}} + \frac{π}{2}))) \end{array}

{Err}_{y} [k] = \sqrt{\frac{〈 {| Y_{{rec}_{sim}} [k] - Y_{{rec}_{theo}} [k] |}^{2} 〉}{〈 {| Y_{{rec}_{sim}} [k] |}^{2} 〉}} \times 100

{CIPR}_{Ω_{k}} = \frac{Information Signal Power at Ω_{k}}{Interference Power at Ω_{k}}

\begin{array}{l} H_{p_{1}} (Ω_{k}) & = C_{p} \frac{1}{e \cdot V} \frac{ψ}{X M - E Ψ - Ω_{k}^{2} + j Ω_{k} (X + M)} \\ H_{p_{2}} (Ω_{k}) & = C_{p} \frac{v_{g} a_{g}}{1 + ε_{n l} p_{0}} \frac{(ψ - Γ (2 j Ω_{k} + X)) (P - j \frac{Ω_{k}}{ψ})}{X M - E Ψ - 4 Ω_{k}^{2} + 2 j Ω_{k} (X + M)} \cdot {(\frac{H_{p_{1}} (Ω_{k})}{C_{p}})}^{2} \\ H_{p_{1, 1}} (Ω_{k}, Ω_{l}) & = C_{p} \frac{v_{g} a_{g}}{1 + ε_{n l} p_{0}} \frac{(ψ - Γ (j (Ω_{k} + Ω_{l}) + X)) (2 P - j \frac{Ω_{k} + Ω_{l}}{ψ})}{X M - E ψ - {(Ω_{k} + Ω_{l})}^{2} + j (Ω_{k} + Ω_{l}) (X + M)} \cdot \frac{H_{p_{1}} (Ω_{k})}{C_{p}} \cdot \frac{H_{p_{1}} (Ω_{l})}{C_{p}} \end{array}

\begin{array}{l} ψ & = Γ \cdot v_{g} a_{g} \frac{p_{0}}{1 + ε_{n l} \cdot p_{0}} + 2 β B Γ \cdot n_{0} \\ M & = Γ \cdot v_{g} a_{g} \frac{n_{0} - n_{t}}{1 + ε_{n l} \cdot p_{0}} (\frac{ε_{n l} \cdot p_{0}}{1 + ε_{n l} \cdot p_{0}} - 1) + \frac{1}{τ_{p}} \\ X & = A + 2 B \cdot n_{0} + 3 C \cdot n_{0}^{2} + v_{g} a_{g} \frac{p_{0}}{1 + ε_{n l} p_{0}} \\ E & = v_{g} a_{g} \frac{n_{0} - n_{t}}{1 + ε_{n l} \cdot p_{0}} (\frac{p_{0}}{1 + ε_{n l} \cdot p_{0}} - 1) \\ P & = \frac{n_{0} - n_{t}}{1 + ε_{n l} p_{0}} ε_{n l} - \frac{M}{ψ} \end{array}

s_{c} (t) = \sum_{k = - N / 2}^{N / 2 - 1} X_{k} \cdot \exp (Ω_{k} \cdot t)

s (t) = Re {s_{c} (t) \cdot \exp (Ω_{r f} \cdot t)} = \sum_{k = - N / 2}^{N / 2 - 1} | X_{k} | \cdot \cos ((Ω_{r f} + Ω_{k}) \cdot t + φ_{X_{k}})

\begin{array}{l} I_{p h} (t) \approx 𝔕 \cdot \sum_{n_{- N / 2} \dots n_{N / 2 - 1} = - \infty}^{\infty} J_{n_{- N / 2}} (μ_{- N / 2}) \cdot \cdot J_{n_{N / 2 - 1}} (μ_{N / 2 - 1}) \cdot (a^{2} + 2 \cdot b^{2} \sum_{k = - N / 2}^{N / 2 - 1} p_{k}^{2} \cdot \cos (2 θ_{k}) + \\ 2 a b \sum_{k = - N / 2}^{N / 2 - 1} \frac{p_{k}}{J_{n_{k}} (μ_{k})} \cdot \cos (θ_{k}) (J_{n_{k} + 1} (μ_{k}) \cdot \exp (j (Δ φ_{k} + \frac{π}{2})) + J_{n_{k} - 1} (μ_{k}) \cdot \exp (- j (Δ φ_{k} + \frac{π}{2}))) + \\ 2 a c \sum_{k = - N / 2}^{N / 2 - 1} \frac{p_{2 k}}{J_{n_{k}} (μ_{k})} \cos (2 θ_{k}) (J_{n_{k} + 2} (μ_{k}) \cdot \exp (j (2 φ_{m_{k}} - φ_{p_{2 k}} + π)) + J_{n_{k} - 2} (μ_{k}) \cdot \exp (- j (2 φ_{m_{k}} - φ_{p_{2 k}} + π))) \\ + 2 a c \sum_{k = - N / 2}^{N / 2 - 1} \sum_{l = - N / 2}^{k - 1} \frac{p_{k l}}{J_{n_{k}} (μ_{k}) J_{n_{l}} (μ_{l})} \cos (θ_{k} + θ_{l}) (J_{n_{k} + 1} (μ_{k}) \cdot J_{n_{l} + 1} (μ_{l}) \cdot \exp (j (φ_{m_{k}} + φ_{m_{l}} - φ_{p_{k l}} + π)) + \\ J_{n_{k} - 1} (μ_{k}) \cdot J_{n_{l} - 1} (μ_{l}) \cdot \exp (- j (φ_{m_{k}} + φ_{m_{l}} - φ_{p_{k l}} + π))) + 2 a c \sum_{k = - N / 2}^{N} \sum_{l = - N / 2}^{k - 1} \frac{p_{k_l}}{J_{n_{k}} (μ_{k}) J_{n_{l}} (μ_{l})} cos (θ_{k} - θ_{l}) \\ (J_{n_{k} + 1} (μ_{k}) \cdot J_{n_{l} - 1} (μ_{l}) \cdot \exp (j (φ_{m_{k}} - φ_{m_{l}} - φ_{p_{k_l}})) + J_{n_{k} - 1} (μ_{k}) \cdot J_{n_{l} + 1} (μ_{l}) \cdot \exp (- j (φ_{m_{k}} - φ_{m_{l}} - φ_{p_{k_l}}))) \\ + \sum_{k = - N / 2}^{N / 2 - 1} \sum_{\begin{array}{l} l = - N / 2 \\ l \neq k \end{array}}^{N / 2 - 1} \frac{p_{k} \cdot p_{l}}{J_{n_{k}} (μ_{k}) J_{n_{l}} (μ_{l})} (J_{n_{k} + 1} (μ_{k}) \cdot J_{n_{l} + 1} (μ_{l}) \cdot \exp (j (Δ φ_{k} + Δ φ_{l} + π)) + J_{n_{k} - 1} (μ_{k}) \cdot J_{n_{l} - 1} (μ_{l}) \cdot \\ \exp (- j (Δ φ_{k} + Δ φ_{l} + π))) (2 a d \cdot \cos (θ_{k} + θ_{l}) + b^{2}) + \sum_{k = - N / 2}^{N / 2 - 1} \sum_{\begin{array}{l} l = - N / 2 \\ l \neq k \end{array}}^{N / 2 - 1} \frac{p_{k} \cdot p_{l}}{J_{n_{k}} (μ_{k}) J_{n_{l}} (μ_{l})} (J_{n_{k} + 1} (μ_{k}) \cdot J_{n_{l} - 1} (μ_{l}) \cdot \\ \exp (j (Δ φ_{k} - Δ φ_{l})) + J_{n_{k} - 1} (μ_{k}) \cdot J_{n_{l} + 1} (μ_{l}) \cdot \exp (- j (Δ φ_{k} - Δ φ_{l}))) (2 a d \cdot \cos (θ_{k} - θ_{l}) + b^{2}) + \\ \sum_{k = - N / 2}^{N / 2 - 1} p_{k}^{2} (\frac{J_{n_{k} + 2} (μ_{k}) \cdot \exp (j 2 (Δ φ_{k} + \frac{π}{2})) + J_{n_{k} - 2} (μ_{k}) \cdot \exp (- j 2 (Δ φ_{k} + \frac{π}{2}))}{J_{n_{k}} (μ_{k})} (2 a d \cdot \cos (2 θ_{k}) + b^{2}))) \\ \cdot \exp (j (Ω_{imp} t + \sum_{k = - N / 2}^{N / 2 - 1} n_{k} (φ_{m_{k}} + \frac{π}{2}))) \end{array}

Parameter	Symbol	Value

Cavity length, width and thickness	l, w, d	300, 2 and 0.2 μm
Confinement factor	Γ	0.3
Group refractive index	n_g	4
Linear gain coefficient	a_g	4 × 10⁻²⁰ m²
Nonlinear gain coefficient	ε_nl	3 × 10⁻²⁴ m³
Transparency carrier density	n_t	1.5 × 10²⁴ m⁻³
Non radiative-recombination coefficient	A	0.1 × 10⁸s⁻¹
Biomolecular recombination coefficient	B	1.0 × 10⁻¹⁶ m³/s
Auger recombination coefficient	C	3 × 10⁻⁴¹ m⁶/s
Internal loss	α_int	2873 m⁻¹
Frac. Spont. Emission into the fund. Mode	ζ	10⁻⁵
Optical efficiency	η	1
Left facet reflectivity	R_l	0.3
Right facet reflectivity	R_r	0.3

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