Abstract

A dual-polarization coherent heterodyne optical communication system using a simplified and low-cost demodulation scheme, for high-capacity UDWDM-PON access networks, is proposed and demonstrated. In this scheme, the signal light and reference light occupying each of the polarization modes are emitted simultaneously from the transmitter. The random phase fluctuations between the signal light and reference light are obviated completely by means of the application of the phase-correlated orthogonal lights. When the signal light in the each polarization mode is modulated with M-amplitude-shift keying (M-ASK) or M2-quadrature amplitude modulation (M2-QAM), the phase-stable intermediate frequency (IF) signal with M-ASK or M2-QAM modulation in the corresponding polarization mode is available for conversion in the electrical domain by beating the modulated signal light with the un-modulated reference light. A new IF signal with M2 or M4-QAM can be synthesized by the IF signals in both modes as long as the power ratio and time delay between the two-modes optical signals are set at the proper values. This is achieved without using polarization demultiplexing and complicated algorithms and the synthesized IF signal can be received and demodulated directly. A proof-of-concept transmission link with dual-polarization 2-ASK is demonstrated. The experimental results are consistent with theoretical predictions.

© 2014 Optical Society of America

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References

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2014 (3)

M. Presi, F. Bottoni, R. Corsini, G. Cossu, and E. Ciaramella, “All DFB-based coherent UDWDM PON with 6.25 GHz spacing and a >40 dB power budget,” IEEE Photon. Technol. Lett. 26(2), 107–110 (2014).
[Crossref]

J. Zheng, J. Wang, J. Yu, M. Zhu, Z. Dong, X. Wang, T. Su, J. Liu, N. Zhu, and G. Chang, “Photonic microwave-signal-mixing technique using phase-coherent orthogonal optical carriers for radio-over-fiber application,” Opt. Lett. 39(18), 5263–5266 (2014).
[Crossref]

J. Zheng, L. Wang, Z. Dong, M. Xu, X. Wang, J. Liu, N. Zhu, S. LaRochelle, and G.-K. Chang, “Orthogonal Single-Sideband Signal Generation Using Improved Sagnac-Loop-Based Modulator,” IEEE Photon. Technol. Lett. 26(22), 2229–2231 (2014).
[Crossref]

2013 (2)

2012 (2)

M. Zhu, L. Zhang, S.-H. Fan, C. Su, G. Gu, and G.-K. Chang, “Efficient delivery of integrated wired and wireless services in UDWDM-RoF-PON coherent access network,” IEEE Photon. Technol. Lett. 24(13), 1127–1129 (2012).
[Crossref]

J. Prat, V. Polo, P. Zakynthinos, I. Cano, J. Tabares, J. M. Fàbrega, D. Klonidis, and I. Tomkos, “Simple intradyne PSK system for udWDM-PON,” Opt. Express 20(27), 28758–28763 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (2)

2009 (3)

2008 (1)

2007 (1)

1993 (1)

M. Lawrence, “Lithium niobate integrated optics,” Rep. Prog. Phys. 56(3), 363–429 (1993).
[Crossref]

Alfiad, M. S.

Avramopoulos, H.

S. Dris, P. Bakopoulos, I. Lazarou, B. Schrenk, and H. Avramopoulos, “Low-complexity DSP using undersampling for heterodyne receivers in coherent passive optical access networks,” European Conference and Exhibition on Optical Communication, Amsterdam, Netherlands, 2012, We.3.A.4.
[Crossref]

Awadalla, A.

Bakopoulos, P.

S. Dris, P. Bakopoulos, I. Lazarou, B. Schrenk, and H. Avramopoulos, “Low-complexity DSP using undersampling for heterodyne receivers in coherent passive optical access networks,” European Conference and Exhibition on Optical Communication, Amsterdam, Netherlands, 2012, We.3.A.4.
[Crossref]

Barros, D. J. F.

Bayvel, P.

Bottoni, F.

M. Presi, F. Bottoni, R. Corsini, G. Cossu, and E. Ciaramella, “All DFB-based coherent UDWDM PON with 6.25 GHz spacing and a >40 dB power budget,” IEEE Photon. Technol. Lett. 26(2), 107–110 (2014).
[Crossref]

Campillo, A. L.

Cano, I.

Chang, G.

Chang, G.-K.

J. Zheng, L. Wang, Z. Dong, M. Xu, X. Wang, J. Liu, N. Zhu, S. LaRochelle, and G.-K. Chang, “Orthogonal Single-Sideband Signal Generation Using Improved Sagnac-Loop-Based Modulator,” IEEE Photon. Technol. Lett. 26(22), 2229–2231 (2014).
[Crossref]

M. Zhu, L. Zhang, S.-H. Fan, C. Su, G. Gu, and G.-K. Chang, “Efficient delivery of integrated wired and wireless services in UDWDM-RoF-PON coherent access network,” IEEE Photon. Technol. Lett. 24(13), 1127–1129 (2012).
[Crossref]

Z. Dong, J. Yu, H.-C. Chien, N. Chi, L. Chen, and G.-K. Chang, “Ultra-dense WDM-PON Delivering Carrier-Centralized Nyquist-WDM Uplink with Digital Coherent Detection,” Opt. Express 19(12), 11100–11105 (2011).
[Crossref] [PubMed]

Chen, L.

Chi, N.

Chien, H.-C.

Ciaramella, E.

M. Presi, F. Bottoni, R. Corsini, G. Cossu, and E. Ciaramella, “All DFB-based coherent UDWDM PON with 6.25 GHz spacing and a >40 dB power budget,” IEEE Photon. Technol. Lett. 26(2), 107–110 (2014).
[Crossref]

Corsini, R.

M. Presi, F. Bottoni, R. Corsini, G. Cossu, and E. Ciaramella, “All DFB-based coherent UDWDM PON with 6.25 GHz spacing and a >40 dB power budget,” IEEE Photon. Technol. Lett. 26(2), 107–110 (2014).
[Crossref]

Cossu, G.

M. Presi, F. Bottoni, R. Corsini, G. Cossu, and E. Ciaramella, “All DFB-based coherent UDWDM PON with 6.25 GHz spacing and a >40 dB power budget,” IEEE Photon. Technol. Lett. 26(2), 107–110 (2014).
[Crossref]

Dong, Z.

Dris, S.

S. Dris, P. Bakopoulos, I. Lazarou, B. Schrenk, and H. Avramopoulos, “Low-complexity DSP using undersampling for heterodyne receivers in coherent passive optical access networks,” European Conference and Exhibition on Optical Communication, Amsterdam, Netherlands, 2012, We.3.A.4.
[Crossref]

Fàbrega, J. M.

Fan, S.-H.

M. Zhu, L. Zhang, S.-H. Fan, C. Su, G. Gu, and G.-K. Chang, “Efficient delivery of integrated wired and wireless services in UDWDM-RoF-PON coherent access network,” IEEE Photon. Technol. Lett. 24(13), 1127–1129 (2012).
[Crossref]

Ferreira, R.

A. Shahpari, R. S. Luís, J. D. Reis, R. Ferreira, Z. Vujicic, J. M. D. Mendinueta, M. Lima, N. Wada, and A. Teixeira, “Fully coherent self- homodyne bi-directional enhanced performance PON,” in Optical Fiber Communication Conference, San Francisco, 2014, W4G.1.
[Crossref]

Gottwald, E.

H. Rohde, S. Smolorz, J. S. Wey, and E. Gottwald, “Coherent optical access networks,” in Optical Fiber Communication Conference, Los Angeles, 2011, OTuB1.
[Crossref]

Gu, G.

M. Zhu, L. Zhang, S.-H. Fan, C. Su, G. Gu, and G.-K. Chang, “Efficient delivery of integrated wired and wireless services in UDWDM-RoF-PON coherent access network,” IEEE Photon. Technol. Lett. 24(13), 1127–1129 (2012).
[Crossref]

Hauske, F. N.

Ionescu, M.

Ip, E.

Kahn, J. M.

Klonidis, D.

Krause, D. J.

Kuschnerov, M.

Lankl, B.

Laperle, C.

LaRochelle, S.

J. Zheng, L. Wang, Z. Dong, M. Xu, X. Wang, J. Liu, N. Zhu, S. LaRochelle, and G.-K. Chang, “Orthogonal Single-Sideband Signal Generation Using Improved Sagnac-Loop-Based Modulator,” IEEE Photon. Technol. Lett. 26(22), 2229–2231 (2014).
[Crossref]

Lau, A. P. T.

Lavery, D.

Lawrence, M.

M. Lawrence, “Lithium niobate integrated optics,” Rep. Prog. Phys. 56(3), 363–429 (1993).
[Crossref]

Lazarou, I.

S. Dris, P. Bakopoulos, I. Lazarou, B. Schrenk, and H. Avramopoulos, “Low-complexity DSP using undersampling for heterodyne receivers in coherent passive optical access networks,” European Conference and Exhibition on Optical Communication, Amsterdam, Netherlands, 2012, We.3.A.4.
[Crossref]

Li, G.

Li, W.

Lima, M.

A. Shahpari, R. S. Luís, J. D. Reis, R. Ferreira, Z. Vujicic, J. M. D. Mendinueta, M. Lima, N. Wada, and A. Teixeira, “Fully coherent self- homodyne bi-directional enhanced performance PON,” in Optical Fiber Communication Conference, San Francisco, 2014, W4G.1.
[Crossref]

Liu, J.

J. Zheng, J. Wang, J. Yu, M. Zhu, Z. Dong, X. Wang, T. Su, J. Liu, N. Zhu, and G. Chang, “Photonic microwave-signal-mixing technique using phase-coherent orthogonal optical carriers for radio-over-fiber application,” Opt. Lett. 39(18), 5263–5266 (2014).
[Crossref]

J. Zheng, L. Wang, Z. Dong, M. Xu, X. Wang, J. Liu, N. Zhu, S. LaRochelle, and G.-K. Chang, “Orthogonal Single-Sideband Signal Generation Using Improved Sagnac-Loop-Based Modulator,” IEEE Photon. Technol. Lett. 26(22), 2229–2231 (2014).
[Crossref]

Liu, J. G.

Luís, R. S.

A. Shahpari, R. S. Luís, J. D. Reis, R. Ferreira, Z. Vujicic, J. M. D. Mendinueta, M. Lima, N. Wada, and A. Teixeira, “Fully coherent self- homodyne bi-directional enhanced performance PON,” in Optical Fiber Communication Conference, San Francisco, 2014, W4G.1.
[Crossref]

Maher, R.

Makovejs, S.

Mendinueta, J. M. D.

A. Shahpari, R. S. Luís, J. D. Reis, R. Ferreira, Z. Vujicic, J. M. D. Mendinueta, M. Lima, N. Wada, and A. Teixeira, “Fully coherent self- homodyne bi-directional enhanced performance PON,” in Optical Fiber Communication Conference, San Francisco, 2014, W4G.1.
[Crossref]

Millar, D. S.

Napoli, A.

O’Sullivan, M.

Piyawanno, K.

Polo, V.

Prat, J.

Presi, M.

M. Presi, F. Bottoni, R. Corsini, G. Cossu, and E. Ciaramella, “All DFB-based coherent UDWDM PON with 6.25 GHz spacing and a >40 dB power budget,” IEEE Photon. Technol. Lett. 26(2), 107–110 (2014).
[Crossref]

Reis, J. D.

A. Shahpari, R. S. Luís, J. D. Reis, R. Ferreira, Z. Vujicic, J. M. D. Mendinueta, M. Lima, N. Wada, and A. Teixeira, “Fully coherent self- homodyne bi-directional enhanced performance PON,” in Optical Fiber Communication Conference, San Francisco, 2014, W4G.1.
[Crossref]

Roberts, K.

Rohde, H.

H. Rohde, S. Smolorz, J. S. Wey, and E. Gottwald, “Coherent optical access networks,” in Optical Fiber Communication Conference, Los Angeles, 2011, OTuB1.
[Crossref]

Savory, S. J.

Schrenk, B.

S. Dris, P. Bakopoulos, I. Lazarou, B. Schrenk, and H. Avramopoulos, “Low-complexity DSP using undersampling for heterodyne receivers in coherent passive optical access networks,” European Conference and Exhibition on Optical Communication, Amsterdam, Netherlands, 2012, We.3.A.4.
[Crossref]

Shahpari, A.

A. Shahpari, R. S. Luís, J. D. Reis, R. Ferreira, Z. Vujicic, J. M. D. Mendinueta, M. Lima, N. Wada, and A. Teixeira, “Fully coherent self- homodyne bi-directional enhanced performance PON,” in Optical Fiber Communication Conference, San Francisco, 2014, W4G.1.
[Crossref]

Smolorz, S.

H. Rohde, S. Smolorz, J. S. Wey, and E. Gottwald, “Coherent optical access networks,” in Optical Fiber Communication Conference, Los Angeles, 2011, OTuB1.
[Crossref]

Spinnler, B.

Su, C.

M. Zhu, L. Zhang, S.-H. Fan, C. Su, G. Gu, and G.-K. Chang, “Efficient delivery of integrated wired and wireless services in UDWDM-RoF-PON coherent access network,” IEEE Photon. Technol. Lett. 24(13), 1127–1129 (2012).
[Crossref]

Su, T.

Sun, H.

Tabares, J.

Teixeira, A.

A. Shahpari, R. S. Luís, J. D. Reis, R. Ferreira, Z. Vujicic, J. M. D. Mendinueta, M. Lima, N. Wada, and A. Teixeira, “Fully coherent self- homodyne bi-directional enhanced performance PON,” in Optical Fiber Communication Conference, San Francisco, 2014, W4G.1.
[Crossref]

Thomsen, B. C.

Tomkos, I.

Torrengo, E.

Vujicic, Z.

A. Shahpari, R. S. Luís, J. D. Reis, R. Ferreira, Z. Vujicic, J. M. D. Mendinueta, M. Lima, N. Wada, and A. Teixeira, “Fully coherent self- homodyne bi-directional enhanced performance PON,” in Optical Fiber Communication Conference, San Francisco, 2014, W4G.1.
[Crossref]

Wada, N.

A. Shahpari, R. S. Luís, J. D. Reis, R. Ferreira, Z. Vujicic, J. M. D. Mendinueta, M. Lima, N. Wada, and A. Teixeira, “Fully coherent self- homodyne bi-directional enhanced performance PON,” in Optical Fiber Communication Conference, San Francisco, 2014, W4G.1.
[Crossref]

Wang, H.

Wang, J.

Wang, L.

J. Zheng, L. Wang, Z. Dong, M. Xu, X. Wang, J. Liu, N. Zhu, S. LaRochelle, and G.-K. Chang, “Orthogonal Single-Sideband Signal Generation Using Improved Sagnac-Loop-Based Modulator,” IEEE Photon. Technol. Lett. 26(22), 2229–2231 (2014).
[Crossref]

Wang, L. X.

Wang, X.

J. Zheng, J. Wang, J. Yu, M. Zhu, Z. Dong, X. Wang, T. Su, J. Liu, N. Zhu, and G. Chang, “Photonic microwave-signal-mixing technique using phase-coherent orthogonal optical carriers for radio-over-fiber application,” Opt. Lett. 39(18), 5263–5266 (2014).
[Crossref]

J. Zheng, L. Wang, Z. Dong, M. Xu, X. Wang, J. Liu, N. Zhu, S. LaRochelle, and G.-K. Chang, “Orthogonal Single-Sideband Signal Generation Using Improved Sagnac-Loop-Based Modulator,” IEEE Photon. Technol. Lett. 26(22), 2229–2231 (2014).
[Crossref]

Wey, J. S.

H. Rohde, S. Smolorz, J. S. Wey, and E. Gottwald, “Coherent optical access networks,” in Optical Fiber Communication Conference, Los Angeles, 2011, OTuB1.
[Crossref]

Wu, K.-T.

Xu, M.

J. Zheng, L. Wang, Z. Dong, M. Xu, X. Wang, J. Liu, N. Zhu, S. LaRochelle, and G.-K. Chang, “Orthogonal Single-Sideband Signal Generation Using Improved Sagnac-Loop-Based Modulator,” IEEE Photon. Technol. Lett. 26(22), 2229–2231 (2014).
[Crossref]

Yu, J.

Zakynthinos, P.

Zhang, L.

M. Zhu, L. Zhang, S.-H. Fan, C. Su, G. Gu, and G.-K. Chang, “Efficient delivery of integrated wired and wireless services in UDWDM-RoF-PON coherent access network,” IEEE Photon. Technol. Lett. 24(13), 1127–1129 (2012).
[Crossref]

Zheng, J.

J. Zheng, J. Wang, J. Yu, M. Zhu, Z. Dong, X. Wang, T. Su, J. Liu, N. Zhu, and G. Chang, “Photonic microwave-signal-mixing technique using phase-coherent orthogonal optical carriers for radio-over-fiber application,” Opt. Lett. 39(18), 5263–5266 (2014).
[Crossref]

J. Zheng, L. Wang, Z. Dong, M. Xu, X. Wang, J. Liu, N. Zhu, S. LaRochelle, and G.-K. Chang, “Orthogonal Single-Sideband Signal Generation Using Improved Sagnac-Loop-Based Modulator,” IEEE Photon. Technol. Lett. 26(22), 2229–2231 (2014).
[Crossref]

Zheng, J. Y.

Zhu, M.

J. Zheng, J. Wang, J. Yu, M. Zhu, Z. Dong, X. Wang, T. Su, J. Liu, N. Zhu, and G. Chang, “Photonic microwave-signal-mixing technique using phase-coherent orthogonal optical carriers for radio-over-fiber application,” Opt. Lett. 39(18), 5263–5266 (2014).
[Crossref]

M. Zhu, L. Zhang, S.-H. Fan, C. Su, G. Gu, and G.-K. Chang, “Efficient delivery of integrated wired and wireless services in UDWDM-RoF-PON coherent access network,” IEEE Photon. Technol. Lett. 24(13), 1127–1129 (2012).
[Crossref]

Zhu, N.

J. Zheng, J. Wang, J. Yu, M. Zhu, Z. Dong, X. Wang, T. Su, J. Liu, N. Zhu, and G. Chang, “Photonic microwave-signal-mixing technique using phase-coherent orthogonal optical carriers for radio-over-fiber application,” Opt. Lett. 39(18), 5263–5266 (2014).
[Crossref]

J. Zheng, L. Wang, Z. Dong, M. Xu, X. Wang, J. Liu, N. Zhu, S. LaRochelle, and G.-K. Chang, “Orthogonal Single-Sideband Signal Generation Using Improved Sagnac-Loop-Based Modulator,” IEEE Photon. Technol. Lett. 26(22), 2229–2231 (2014).
[Crossref]

Zhu, N. H.

Adv. Opt. Photon. (1)

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

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (3)

M. Zhu, L. Zhang, S.-H. Fan, C. Su, G. Gu, and G.-K. Chang, “Efficient delivery of integrated wired and wireless services in UDWDM-RoF-PON coherent access network,” IEEE Photon. Technol. Lett. 24(13), 1127–1129 (2012).
[Crossref]

M. Presi, F. Bottoni, R. Corsini, G. Cossu, and E. Ciaramella, “All DFB-based coherent UDWDM PON with 6.25 GHz spacing and a >40 dB power budget,” IEEE Photon. Technol. Lett. 26(2), 107–110 (2014).
[Crossref]

J. Zheng, L. Wang, Z. Dong, M. Xu, X. Wang, J. Liu, N. Zhu, S. LaRochelle, and G.-K. Chang, “Orthogonal Single-Sideband Signal Generation Using Improved Sagnac-Loop-Based Modulator,” IEEE Photon. Technol. Lett. 26(22), 2229–2231 (2014).
[Crossref]

J. Lightwave Technol. (3)

Opt. Express (4)

Opt. Lett. (3)

Rep. Prog. Phys. (1)

M. Lawrence, “Lithium niobate integrated optics,” Rep. Prog. Phys. 56(3), 363–429 (1993).
[Crossref]

Other (3)

A. Shahpari, R. S. Luís, J. D. Reis, R. Ferreira, Z. Vujicic, J. M. D. Mendinueta, M. Lima, N. Wada, and A. Teixeira, “Fully coherent self- homodyne bi-directional enhanced performance PON,” in Optical Fiber Communication Conference, San Francisco, 2014, W4G.1.
[Crossref]

S. Dris, P. Bakopoulos, I. Lazarou, B. Schrenk, and H. Avramopoulos, “Low-complexity DSP using undersampling for heterodyne receivers in coherent passive optical access networks,” European Conference and Exhibition on Optical Communication, Amsterdam, Netherlands, 2012, We.3.A.4.
[Crossref]

H. Rohde, S. Smolorz, J. S. Wey, and E. Gottwald, “Coherent optical access networks,” in Optical Fiber Communication Conference, Los Angeles, 2011, OTuB1.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Schematic diagram of the proposed coherent dual-polarization optical communication system; insets (1)-(4) correspond to schematic representations of the modulation optical spectra at different positions of the link. OLT: optical line terminal; TE: transverse electric; TM: transverse magnetic; POLG: phase-correlated orthogonal lights generator; VOA: variable optical attenuator; ODL: optical delay line; EOM: electrical optical modulator; PBC: polarization beam combiner; SMF: single mode fiber; PD: photodetector; ELO: electrical local oscillator; ONU: optical network unit. SL: signal light, RL: reference light, POLs: phase-correlated orthogonal lights.
Fig. 2
Fig. 2 Transfer relationship between the TE/TM mode optical coherent signals and the synthesized IF signal
Fig. 3
Fig. 3 Experiment setup of the dual-polarization 2-ASK coherent communication system; insets (1)-(5) are schematic representations of the optical spectra at different positions in the link. LD: laser diode; PR: polarization rotator; MZM: Mach-Zehnder modulator, LPF: low pass filter.
Fig. 4
Fig. 4 Optical spectra of the POLG output measured before (a) or after the polarizer is set at-β = 0° (b), 90° (c), and 45° (d), electrical spectra of the POLG before (e) and after (f) the LPF with β = 45°. β is the angle between the polarization direction of the OC ν0 and the principal axis of the polarizer.
Fig. 5
Fig. 5 Eye diagrams of the 5-GHz IF 2-ASK signal generated by beating between the phased-correlated signal light and reference light in TE mode (a), TM mode (b), and TE + TM modes (c)-(d). (a)-(c) were recorded as back-to-back (B-to-B) transmission, (d) was recorded after 25-km SMF transmission.
Fig. 6
Fig. 6 EVM versus received optical power for synthesized IF signal as 2-ASK modulation at the TE and TM mode. Inset: the constellations after 25-km SMF transmission.

Equations (10)

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[ E 0,y (t) E 1,x (t) ]=[ e j(2π ν 0 t) e j(2π ν 1 + φ 1 ) ]
[ E 0,y (t) E 1,x (t) ]=[ ( A m,I,TE +j A m,Q,TE ) e j(2π ν 0 t) e j(2π ν 1 + φ 1 ) ] (m=1,2,,M)
A m =(2m1M)d,
E 45 ,TE (t)= 2 2 (( A m,I,TE +j A m,Q,TE ) e j(2π ν 0 t) + e j(2π ν 1 + φ 1 ) ) (m=1,2,,M),
E 45 ,TM (t)= 2 2 k(( A m,I,TM +j A m,Q,TM ) e j(2π ν 0 t) + e j(2π ν 1 + φ 1 ) ) (m=1,2,,M)
I(t)= I TE (t)+ I TM (t) E 45°,TE E 45°,TE + E 45°,TM E 45°,TM =1+ 1 2 ( A m,I,TE 2 + A m,Q,TE 2 + k 2 A m,I,TM 2 + k 2 A m,Q,TM 2 ) + A m,I,TE cos(2πft+ φ 1 ) A m,Q,TE sin(2πft+ φ 1 ) +k A m,I,TM cos(2πft+ φ 1 +Φ)k A m,Q,TM sin(2πft+ φ 1 +Φ)
S(t) A m,I,TE cos(2πft+ φ 1 ) A m,Q,TE sin(2πft+ φ 1 ) +k A m,I,TM cos(2πft+ φ 1 +Φ)k A m,Q,TM sin(2πft+ φ 1 +Φ)
S(t) A m,I,TE cos(2πft+ φ 1 )+k A m,I,TM cos(2πft+ φ 1 +Φ)
S(t) A m,I,TE cos(2πft+ φ 1 ) A m,I,TM sin(2πft+ φ 1 )
S(t)( A m,I,TE +M A m,I,TM )cos(2πft+ φ 1 ) ( A m,Q,TE +M A m,Q,TM )sin(2πft+ φ 1 )

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