Abstract

We propose and experimentally demonstrate a simple and cost-effective bidirectional radio-over-fiber (RoF) system for transmission of downstream multiband signals and upstream data. At the central station (CS), the multiband data consisting of baseband, micro-wave (MW) and millimeter-wave (MMW) signals are generated using only one single-drive x-cut Mach-Zehnder modulator (MZM), which is driven by a clock signal at radio frequency (RF) port and an electrical data at bias port. Upstream data transmission is realized by re-modulation of filtered frequency-shift-keying (FSK) signal, selected from the multiband signals.

© 2011 OSA

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References

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  1. Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
    [CrossRef]
  2. J. Yao, “Microwave Photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
    [CrossRef]
  3. A. Martinez, V. Polo, and J. Marti, “Simultaneous baseband and RF optical modulation scheme for feeding wireless and wireline heterogeneous access network,” IEEE Trans. Microw. Theory Tech. 49(10), 2018–2024 (2001).
    [CrossRef]
  4. G. Yoffe, R. Waterhouse, D. Novak, A. Nirmalathas, and C. Lim, “Millimeter-Wave broad-band fiber-wireless system incorporating baseband data transmission over fiber and remote LO delivery,” J. Lightwave Technol. 18(10), 1355–1363 (2000).
    [CrossRef]
  5. C. Lin, J. Chen, P. Peng, C. Peng, W. Peng, B. Chiou, and S. Chi, “Hybrid optical access network integrating fiber-to-the-home and radio-over-fiber systems,” IEEE Photon. Technol. Lett. 19(8), 610–612 (2007).
    [CrossRef]
  6. M. Bakaul, A. Nirmalathas, C. Lim, D. Novak, and R. Waterhouse, “Hybrid multiplexing of multi-band optical access technologies towards an integrated DWDM network,” IEEE Photon. Technol. Lett. 18(21), 2311–2313 (2006).
    [CrossRef]
  7. Q. Chang, H. Fu, and Y. Su, “Simultaneous Generation and Transmission of Downstream Multi-band Signals and Upstream Data in a Bidirectional Radio over Fiber System,” IEEE Photon. Technol. Lett. 20(3), 181–183 (2008).
    [CrossRef]
  8. Y. Hsueh, Z. Jia, H. Chien, J. Yu, and G. Chang, “A Novel Bidirectional 60-GHz Radio-Over-Fiber Scheme With Multiband Signal Generation Using a Single Intensity Modulator,” IEEE Photon. Technol. Lett. 21(18), 1338–1340 (2009).
    [CrossRef]
  9. K. Ikeda, T. Kuri, and K. Kitayama, “Simultaneous three-band modulation and fiber-optic transmission of 2.5-Gb/s baseband, microwave-, and 60-GHz-band signals on a single wavelength,” J. Lightwave Technol. 21(12), 3194–3202 (2003).
    [CrossRef]
  10. L. Zhang, X. Hu, P. Cao, T. Wang, and Y. Su, “A Multiband Radio over Fiber System Using a Single-drive Mach-Zehder Modulator,” in Proceeding of ACP2010, FP5.
  11. Q. Chang, Y. Tian, J. Gao, T. Ye, Q. Li, and Y. Su, “Generation and Transmission of Optical Carrier Suppressed-Optical Differential (Qudrature) Phase-Shift Keying (OCS-OD(Q)PSK) Signals in Radio Over Fiber Systems,” J. Lightwave Technol. 26(15), 2611–2618 (2008).
    [CrossRef]
  12. J. Yu, Z. Jia, L. Yi, Y. Su, G. K. Chang, and T. Wang, “Optical millimetre wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18(1), 265–267 (2006).
    [CrossRef]

2009 (2)

Y. Hsueh, Z. Jia, H. Chien, J. Yu, and G. Chang, “A Novel Bidirectional 60-GHz Radio-Over-Fiber Scheme With Multiband Signal Generation Using a Single Intensity Modulator,” IEEE Photon. Technol. Lett. 21(18), 1338–1340 (2009).
[CrossRef]

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

2008 (3)

Q. Chang, Y. Tian, J. Gao, T. Ye, Q. Li, and Y. Su, “Generation and Transmission of Optical Carrier Suppressed-Optical Differential (Qudrature) Phase-Shift Keying (OCS-OD(Q)PSK) Signals in Radio Over Fiber Systems,” J. Lightwave Technol. 26(15), 2611–2618 (2008).
[CrossRef]

Q. Chang, H. Fu, and Y. Su, “Simultaneous Generation and Transmission of Downstream Multi-band Signals and Upstream Data in a Bidirectional Radio over Fiber System,” IEEE Photon. Technol. Lett. 20(3), 181–183 (2008).
[CrossRef]

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

2007 (1)

C. Lin, J. Chen, P. Peng, C. Peng, W. Peng, B. Chiou, and S. Chi, “Hybrid optical access network integrating fiber-to-the-home and radio-over-fiber systems,” IEEE Photon. Technol. Lett. 19(8), 610–612 (2007).
[CrossRef]

2006 (2)

M. Bakaul, A. Nirmalathas, C. Lim, D. Novak, and R. Waterhouse, “Hybrid multiplexing of multi-band optical access technologies towards an integrated DWDM network,” IEEE Photon. Technol. Lett. 18(21), 2311–2313 (2006).
[CrossRef]

J. Yu, Z. Jia, L. Yi, Y. Su, G. K. Chang, and T. Wang, “Optical millimetre wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18(1), 265–267 (2006).
[CrossRef]

2003 (1)

2001 (1)

A. Martinez, V. Polo, and J. Marti, “Simultaneous baseband and RF optical modulation scheme for feeding wireless and wireline heterogeneous access network,” IEEE Trans. Microw. Theory Tech. 49(10), 2018–2024 (2001).
[CrossRef]

2000 (1)

Bakaul, M.

M. Bakaul, A. Nirmalathas, C. Lim, D. Novak, and R. Waterhouse, “Hybrid multiplexing of multi-band optical access technologies towards an integrated DWDM network,” IEEE Photon. Technol. Lett. 18(21), 2311–2313 (2006).
[CrossRef]

Buck, J. A.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Chang, G.

Y. Hsueh, Z. Jia, H. Chien, J. Yu, and G. Chang, “A Novel Bidirectional 60-GHz Radio-Over-Fiber Scheme With Multiband Signal Generation Using a Single Intensity Modulator,” IEEE Photon. Technol. Lett. 21(18), 1338–1340 (2009).
[CrossRef]

Chang, G. K.

J. Yu, Z. Jia, L. Yi, Y. Su, G. K. Chang, and T. Wang, “Optical millimetre wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18(1), 265–267 (2006).
[CrossRef]

Chang, G.-K.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Chang, Q.

Q. Chang, H. Fu, and Y. Su, “Simultaneous Generation and Transmission of Downstream Multi-band Signals and Upstream Data in a Bidirectional Radio over Fiber System,” IEEE Photon. Technol. Lett. 20(3), 181–183 (2008).
[CrossRef]

Q. Chang, Y. Tian, J. Gao, T. Ye, Q. Li, and Y. Su, “Generation and Transmission of Optical Carrier Suppressed-Optical Differential (Qudrature) Phase-Shift Keying (OCS-OD(Q)PSK) Signals in Radio Over Fiber Systems,” J. Lightwave Technol. 26(15), 2611–2618 (2008).
[CrossRef]

Chen, J.

C. Lin, J. Chen, P. Peng, C. Peng, W. Peng, B. Chiou, and S. Chi, “Hybrid optical access network integrating fiber-to-the-home and radio-over-fiber systems,” IEEE Photon. Technol. Lett. 19(8), 610–612 (2007).
[CrossRef]

Chi, S.

C. Lin, J. Chen, P. Peng, C. Peng, W. Peng, B. Chiou, and S. Chi, “Hybrid optical access network integrating fiber-to-the-home and radio-over-fiber systems,” IEEE Photon. Technol. Lett. 19(8), 610–612 (2007).
[CrossRef]

Chien, H.

Y. Hsueh, Z. Jia, H. Chien, J. Yu, and G. Chang, “A Novel Bidirectional 60-GHz Radio-Over-Fiber Scheme With Multiband Signal Generation Using a Single Intensity Modulator,” IEEE Photon. Technol. Lett. 21(18), 1338–1340 (2009).
[CrossRef]

Chien, H.-C.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Chiou, B.

C. Lin, J. Chen, P. Peng, C. Peng, W. Peng, B. Chiou, and S. Chi, “Hybrid optical access network integrating fiber-to-the-home and radio-over-fiber systems,” IEEE Photon. Technol. Lett. 19(8), 610–612 (2007).
[CrossRef]

Chowdhury, A.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Fu, H.

Q. Chang, H. Fu, and Y. Su, “Simultaneous Generation and Transmission of Downstream Multi-band Signals and Upstream Data in a Bidirectional Radio over Fiber System,” IEEE Photon. Technol. Lett. 20(3), 181–183 (2008).
[CrossRef]

Gao, J.

Hsueh, Y.

Y. Hsueh, Z. Jia, H. Chien, J. Yu, and G. Chang, “A Novel Bidirectional 60-GHz Radio-Over-Fiber Scheme With Multiband Signal Generation Using a Single Intensity Modulator,” IEEE Photon. Technol. Lett. 21(18), 1338–1340 (2009).
[CrossRef]

Hsueh, Y.-T.

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

Ikeda, K.

Jia, Z.

Y. Hsueh, Z. Jia, H. Chien, J. Yu, and G. Chang, “A Novel Bidirectional 60-GHz Radio-Over-Fiber Scheme With Multiband Signal Generation Using a Single Intensity Modulator,” IEEE Photon. Technol. Lett. 21(18), 1338–1340 (2009).
[CrossRef]

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

J. Yu, Z. Jia, L. Yi, Y. Su, G. K. Chang, and T. Wang, “Optical millimetre wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18(1), 265–267 (2006).
[CrossRef]

Kitayama, K.

Kuri, T.

Li, Q.

Lim, C.

M. Bakaul, A. Nirmalathas, C. Lim, D. Novak, and R. Waterhouse, “Hybrid multiplexing of multi-band optical access technologies towards an integrated DWDM network,” IEEE Photon. Technol. Lett. 18(21), 2311–2313 (2006).
[CrossRef]

G. Yoffe, R. Waterhouse, D. Novak, A. Nirmalathas, and C. Lim, “Millimeter-Wave broad-band fiber-wireless system incorporating baseband data transmission over fiber and remote LO delivery,” J. Lightwave Technol. 18(10), 1355–1363 (2000).
[CrossRef]

Lin, C.

C. Lin, J. Chen, P. Peng, C. Peng, W. Peng, B. Chiou, and S. Chi, “Hybrid optical access network integrating fiber-to-the-home and radio-over-fiber systems,” IEEE Photon. Technol. Lett. 19(8), 610–612 (2007).
[CrossRef]

Marti, J.

A. Martinez, V. Polo, and J. Marti, “Simultaneous baseband and RF optical modulation scheme for feeding wireless and wireline heterogeneous access network,” IEEE Trans. Microw. Theory Tech. 49(10), 2018–2024 (2001).
[CrossRef]

Martinez, A.

A. Martinez, V. Polo, and J. Marti, “Simultaneous baseband and RF optical modulation scheme for feeding wireless and wireline heterogeneous access network,” IEEE Trans. Microw. Theory Tech. 49(10), 2018–2024 (2001).
[CrossRef]

Nirmalathas, A.

M. Bakaul, A. Nirmalathas, C. Lim, D. Novak, and R. Waterhouse, “Hybrid multiplexing of multi-band optical access technologies towards an integrated DWDM network,” IEEE Photon. Technol. Lett. 18(21), 2311–2313 (2006).
[CrossRef]

G. Yoffe, R. Waterhouse, D. Novak, A. Nirmalathas, and C. Lim, “Millimeter-Wave broad-band fiber-wireless system incorporating baseband data transmission over fiber and remote LO delivery,” J. Lightwave Technol. 18(10), 1355–1363 (2000).
[CrossRef]

Novak, D.

M. Bakaul, A. Nirmalathas, C. Lim, D. Novak, and R. Waterhouse, “Hybrid multiplexing of multi-band optical access technologies towards an integrated DWDM network,” IEEE Photon. Technol. Lett. 18(21), 2311–2313 (2006).
[CrossRef]

G. Yoffe, R. Waterhouse, D. Novak, A. Nirmalathas, and C. Lim, “Millimeter-Wave broad-band fiber-wireless system incorporating baseband data transmission over fiber and remote LO delivery,” J. Lightwave Technol. 18(10), 1355–1363 (2000).
[CrossRef]

Peng, C.

C. Lin, J. Chen, P. Peng, C. Peng, W. Peng, B. Chiou, and S. Chi, “Hybrid optical access network integrating fiber-to-the-home and radio-over-fiber systems,” IEEE Photon. Technol. Lett. 19(8), 610–612 (2007).
[CrossRef]

Peng, P.

C. Lin, J. Chen, P. Peng, C. Peng, W. Peng, B. Chiou, and S. Chi, “Hybrid optical access network integrating fiber-to-the-home and radio-over-fiber systems,” IEEE Photon. Technol. Lett. 19(8), 610–612 (2007).
[CrossRef]

Peng, W.

C. Lin, J. Chen, P. Peng, C. Peng, W. Peng, B. Chiou, and S. Chi, “Hybrid optical access network integrating fiber-to-the-home and radio-over-fiber systems,” IEEE Photon. Technol. Lett. 19(8), 610–612 (2007).
[CrossRef]

Polo, V.

A. Martinez, V. Polo, and J. Marti, “Simultaneous baseband and RF optical modulation scheme for feeding wireless and wireline heterogeneous access network,” IEEE Trans. Microw. Theory Tech. 49(10), 2018–2024 (2001).
[CrossRef]

Su, Y.

Q. Chang, H. Fu, and Y. Su, “Simultaneous Generation and Transmission of Downstream Multi-band Signals and Upstream Data in a Bidirectional Radio over Fiber System,” IEEE Photon. Technol. Lett. 20(3), 181–183 (2008).
[CrossRef]

Q. Chang, Y. Tian, J. Gao, T. Ye, Q. Li, and Y. Su, “Generation and Transmission of Optical Carrier Suppressed-Optical Differential (Qudrature) Phase-Shift Keying (OCS-OD(Q)PSK) Signals in Radio Over Fiber Systems,” J. Lightwave Technol. 26(15), 2611–2618 (2008).
[CrossRef]

J. Yu, Z. Jia, L. Yi, Y. Su, G. K. Chang, and T. Wang, “Optical millimetre wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18(1), 265–267 (2006).
[CrossRef]

Tian, Y.

Wang, T.

J. Yu, Z. Jia, L. Yi, Y. Su, G. K. Chang, and T. Wang, “Optical millimetre wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18(1), 265–267 (2006).
[CrossRef]

Waterhouse, R.

M. Bakaul, A. Nirmalathas, C. Lim, D. Novak, and R. Waterhouse, “Hybrid multiplexing of multi-band optical access technologies towards an integrated DWDM network,” IEEE Photon. Technol. Lett. 18(21), 2311–2313 (2006).
[CrossRef]

G. Yoffe, R. Waterhouse, D. Novak, A. Nirmalathas, and C. Lim, “Millimeter-Wave broad-band fiber-wireless system incorporating baseband data transmission over fiber and remote LO delivery,” J. Lightwave Technol. 18(10), 1355–1363 (2000).
[CrossRef]

Yao, J.

Ye, T.

Yi, L.

J. Yu, Z. Jia, L. Yi, Y. Su, G. K. Chang, and T. Wang, “Optical millimetre wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18(1), 265–267 (2006).
[CrossRef]

Yoffe, G.

Yu, J.

Y. Hsueh, Z. Jia, H. Chien, J. Yu, and G. Chang, “A Novel Bidirectional 60-GHz Radio-Over-Fiber Scheme With Multiband Signal Generation Using a Single Intensity Modulator,” IEEE Photon. Technol. Lett. 21(18), 1338–1340 (2009).
[CrossRef]

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

J. Yu, Z. Jia, L. Yi, Y. Su, G. K. Chang, and T. Wang, “Optical millimetre wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18(1), 265–267 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (6)

C. Lin, J. Chen, P. Peng, C. Peng, W. Peng, B. Chiou, and S. Chi, “Hybrid optical access network integrating fiber-to-the-home and radio-over-fiber systems,” IEEE Photon. Technol. Lett. 19(8), 610–612 (2007).
[CrossRef]

M. Bakaul, A. Nirmalathas, C. Lim, D. Novak, and R. Waterhouse, “Hybrid multiplexing of multi-band optical access technologies towards an integrated DWDM network,” IEEE Photon. Technol. Lett. 18(21), 2311–2313 (2006).
[CrossRef]

Q. Chang, H. Fu, and Y. Su, “Simultaneous Generation and Transmission of Downstream Multi-band Signals and Upstream Data in a Bidirectional Radio over Fiber System,” IEEE Photon. Technol. Lett. 20(3), 181–183 (2008).
[CrossRef]

Y. Hsueh, Z. Jia, H. Chien, J. Yu, and G. Chang, “A Novel Bidirectional 60-GHz Radio-Over-Fiber Scheme With Multiband Signal Generation Using a Single Intensity Modulator,” IEEE Photon. Technol. Lett. 21(18), 1338–1340 (2009).
[CrossRef]

J. Yu, Z. Jia, L. Yi, Y. Su, G. K. Chang, and T. Wang, “Optical millimetre wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett. 18(1), 265–267 (2006).
[CrossRef]

Z. Jia, J. Yu, Y.-T. Hsueh, A. Chowdhury, H.-C. Chien, J. A. Buck, and G.-K. Chang, “Multiband signal generation and dispersion-tolerant transmission based on photonic frequency tripling technology for 60-GHz radio-over-fiber systems,” IEEE Photon. Technol. Lett. 20(17), 1470–1472 (2008).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

A. Martinez, V. Polo, and J. Marti, “Simultaneous baseband and RF optical modulation scheme for feeding wireless and wireline heterogeneous access network,” IEEE Trans. Microw. Theory Tech. 49(10), 2018–2024 (2001).
[CrossRef]

J. Lightwave Technol. (4)

Other (1)

L. Zhang, X. Hu, P. Cao, T. Wang, and Y. Su, “A Multiband Radio over Fiber System Using a Single-drive Mach-Zehder Modulator,” in Proceeding of ACP2010, FP5.

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Figures (5)

Fig. 1
Fig. 1

Principle of the generation of multiband signals.

Fig. 2
Fig. 2

Schematic diagram of the proposed RoF system.

Fig. 3
Fig. 3

Experimental setup of the proposed bidirectional RoF system. (i) Optical eye diagram of baseband signal, (ii) electrical eye diagram of MW signal, (iii) electrical eye diagram of MMW signal, (iv) optical eye diagram of FSK signal, (v) optical eye diagram of re-modulated ASK signal. (a) - (f) correspond to the optical spectra shown in the Fig. 4.

Fig. 4
Fig. 4

Optical spectra taken at different positions as indicated in Fig. 3. Spectral resolution: 0.07 nm. X-axis: 0.2 nm/div; Y-axis: 5 dB/div. (a) Multiband signals, (b) reflected baseband signal, (c) passing signal from the first FBG, (d) reflected MW signal, (e) passing MMW signal, (f) filtered FSK signal.

Fig. 5
Fig. 5

BER curves and electrical eye diagrams after 25-km SMF transmission. (a) Downstream baseband signal, (b) downstream MW signal, (c) downstream MMW signal, (d) upstream re-modulation signal.

Equations (3)

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E o u t = E 0 J 0 ( π 2 α ) cos [ π 2 ε ( t ) ] cos ( ω c t + θ 0 ) + E 0 J 1 ( π 2 α ) sin [ π 2 ε ( t ) ] cos [ ( ω c + ¯ ω R F ) t + θ 0 + ¯ θ ) ] + E 0 J 2 ( π 2 α ) cos [ π 2 ε ( t ) ] cos [ ( ω c + ¯ 2 ω R F ) t + ϕ 0 + ¯ 2 θ ) ] ,
E o u t 1 = E 0 J 1 ( π 2 α ) sin [ π 2 ε ( t ) ] cos [ ( ω c + ¯ ω R F ) t + θ 0 + ¯ θ ) ] .
E o u t 2 = E 0 J 0 ( π 2 α ) cos [ π 2 ε ( t ) ] cos ( ω c t + θ 0 ) + E 0 J 2 ( π 2 α ) cos [ π 2 ε ( t ) ] cos [ ( ω c + ¯ 2 ω R F ) t + θ 0 + ¯ 2 θ ) ] .

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