Abstract

We demonstrate a silicon-based microwave photonic filter (MPF) with flattop passband and adjustable bandwidth. The proposed MPF is realized by using a 10th-order microring resonator (MRR) and a photodetector, both of which are integrated on a photonic chip. The full width at half-maximum (FWHM) bandwidth of the optical filter achieved at the drop port of the 10th-order MRR is 21.6 GHz. The ripple of the passband is less than 0.3 dB, while the rejection ratio is 32 dB. By adjusting the deviation of the optical carrier wavelength from the center wavelength of the optical bandpass filter, the bandwidth of the MPF can be greatly changed. In the experiment, the FWHM bandwidth of the proposed MPF is tuned from 5.3 to 19.5 GHz, and the rejection ratio is higher than 30 dB.

© 2019 Chinese Laser Press

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References

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    [Crossref]
  4. Y. Yu, E. Xu, J. Dong, L. Zhou, X. Li, and X. Zhang, “Switchable microwave photonic filter between high Q bandpass filter and notch filter with flat passband based on phase modulation,” Opt. Express 18, 25271–25282 (2010).
    [Crossref]
  5. F. Jiang, Y. Yu, H. Tang, L. Xu, and X. Zhang, “Tunable bandpass microwave photonic filter with ultrahigh stopband attenuation and skirt selectivity,” Opt. Express 24, 18655–18663 (2016).
    [Crossref]
  6. X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Widely tunable single-bandpass microwave photonic filter employing a non-sliced broadband optical source,” Opt. Express 19, 18423–18429 (2011).
    [Crossref]
  7. H. Tang, Y. Yu, C. Zhang, Z. Wang, L. Xu, and X. Zhang, “Analysis of performance optimization for a microwave photonic filter based on stimulated Brillouin scattering,” J. Lightwave Technol. 35, 4375–4383 (2017).
    [Crossref]
  8. M. Bolea, J. Mora, B. Ortega, and J. Capmany, “Highly chirped single-bandpass microwave photonic filter with reconfiguration capabilities,” Opt. Express 19, 4566–4576 (2011).
    [Crossref]
  9. L. Xu, X. Kong, Z. Wang, H. Tang, X. Liu, Y. Yu, J. Dong, and X. Zhang, “A tunable single passband microwave photonic filter of overcoming fiber dispersion induced amplitude fading,” IEEE Photon. J. 9, 5502008 (2017).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  17. A. Byrnes, R. Pant, E. Li, D.-Y. Choi, C. G. Poulton, S. Fan, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Photonic chip based tunable and reconfigurable narrowband microwave photonic filter using stimulated Brillouin scattering,” Opt. Express 20, 18836–18845 (2012).
    [Crossref]
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  20. L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (2014), paper Th2A–37.
  21. W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
    [Crossref]
  22. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
    [Crossref]
  23. P. Dong, W. Qian, H. Liang, R. Shafiiha, D. Feng, G. Li, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “Thermally tunable silicon racetrack resonators with ultralow tuning power,” Opt. Express 18, 20298–20304 (2010).
    [Crossref]
  24. P. Dong, W. Qian, H. Liang, R. Shafiiha, N.-N. Feng, D. Feng, X. Zheng, A. V. Krishnamoorthy, and M. Asghari, “Low power and compact reconfigurable multiplexing devices based on silicon microring resonators,” Opt. Express 18, 9852–9858 (2010).
    [Crossref]

2018 (1)

2017 (2)

H. Tang, Y. Yu, C. Zhang, Z. Wang, L. Xu, and X. Zhang, “Analysis of performance optimization for a microwave photonic filter based on stimulated Brillouin scattering,” J. Lightwave Technol. 35, 4375–4383 (2017).
[Crossref]

L. Xu, X. Kong, Z. Wang, H. Tang, X. Liu, Y. Yu, J. Dong, and X. Zhang, “A tunable single passband microwave photonic filter of overcoming fiber dispersion induced amplitude fading,” IEEE Photon. J. 9, 5502008 (2017).
[Crossref]

2016 (3)

2015 (1)

2014 (1)

2012 (3)

2011 (3)

2010 (3)

2009 (1)

2007 (3)

2002 (1)

A. J. Seeds, “Microwave photonics,” IEEE Trans. Microw. Theory Tech. 50, 877–887 (2002).
[Crossref]

Aryanfar, I.

Asghari, M.

Baets, R.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Ben-Ezra, Y.

Bienstman, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Blaize, S.

C. Chauveau, P. Labeye, J.-M. Fedeli, S. Blaize, and G. Lerondel, “Study of the uniformity of 300  mm wafer through ring-resonator analysis,” in IEEE International Conference on Photonics in Switching (PS) (2012), pp. 1–3.

Bogaerts, W.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Bolea, M.

Byrnes, A.

Capmany, J.

Chauveau, C.

C. Chauveau, P. Labeye, J.-M. Fedeli, S. Blaize, and G. Lerondel, “Study of the uniformity of 300  mm wafer through ring-resonator analysis,” in IEEE International Conference on Photonics in Switching (PS) (2012), pp. 1–3.

Chen, T.

Choi, D.-Y.

Choudhary, A.

Chrostowski, L.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (2014), paper Th2A–37.

Claes, T.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Cunningham, J. E.

De Heyn, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

De Vos, K.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Dong, J.

X. Liu, Y. Yu, H. Tang, L. Xu, J. Dong, and X. Zhang, “Silicon-on-insulator-based microwave photonic filter with narrowband and ultrahigh peak rejection,” Opt. Lett. 43, 1359–1362 (2018).
[Crossref]

L. Xu, X. Kong, Z. Wang, H. Tang, X. Liu, Y. Yu, J. Dong, and X. Zhang, “A tunable single passband microwave photonic filter of overcoming fiber dispersion induced amplitude fading,” IEEE Photon. J. 9, 5502008 (2017).
[Crossref]

Y. Yu, H. Tang, L. Xu, X. Liu, F. Jiang, J. Dong, and X. Zhang, “Switchable microwave photonic filter between low-pass and high-pass responses,” IEEE Photon. J. 8, 5501408 (2016).
[Crossref]

Y. Yu, E. Xu, J. Dong, L. Zhou, X. Li, and X. Zhang, “Switchable microwave photonic filter between high Q bandpass filter and notch filter with flat passband based on phase modulation,” Opt. Express 18, 25271–25282 (2010).
[Crossref]

Dong, P.

Dumon, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Eggleton, B. J.

Fan, S.

Fard, S. T.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (2014), paper Th2A–37.

Fedeli, J.-M.

C. Chauveau, P. Labeye, J.-M. Fedeli, S. Blaize, and G. Lerondel, “Study of the uniformity of 300  mm wafer through ring-resonator analysis,” in IEEE International Conference on Photonics in Switching (PS) (2012), pp. 1–3.

Feng, D.

Feng, N.-N.

Flueckiger, J.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (2014), paper Th2A–37.

Jiang, F.

Y. Yu, H. Tang, L. Xu, X. Liu, F. Jiang, J. Dong, and X. Zhang, “Switchable microwave photonic filter between low-pass and high-pass responses,” IEEE Photon. J. 8, 5501408 (2016).
[Crossref]

F. Jiang, Y. Yu, H. Tang, L. Xu, and X. Zhang, “Tunable bandpass microwave photonic filter with ultrahigh stopband attenuation and skirt selectivity,” Opt. Express 24, 18655–18663 (2016).
[Crossref]

Kong, X.

L. Xu, X. Kong, Z. Wang, H. Tang, X. Liu, Y. Yu, J. Dong, and X. Zhang, “A tunable single passband microwave photonic filter of overcoming fiber dispersion induced amplitude fading,” IEEE Photon. J. 9, 5502008 (2017).
[Crossref]

Krishnamoorthy, A. V.

Labeye, P.

C. Chauveau, P. Labeye, J.-M. Fedeli, S. Blaize, and G. Lerondel, “Study of the uniformity of 300  mm wafer through ring-resonator analysis,” in IEEE International Conference on Photonics in Switching (PS) (2012), pp. 1–3.

Leaird, D. E.

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett. 23, 1618–1620 (2011).
[Crossref]

Lerondel, G.

C. Chauveau, P. Labeye, J.-M. Fedeli, S. Blaize, and G. Lerondel, “Study of the uniformity of 300  mm wafer through ring-resonator analysis,” in IEEE International Conference on Photonics in Switching (PS) (2012), pp. 1–3.

Li, E.

Li, G.

Li, L.

Li, M.

Li, W.

Li, X.

Liang, H.

Liu, J.

Liu, X.

X. Liu, Y. Yu, H. Tang, L. Xu, J. Dong, and X. Zhang, “Silicon-on-insulator-based microwave photonic filter with narrowband and ultrahigh peak rejection,” Opt. Lett. 43, 1359–1362 (2018).
[Crossref]

L. Xu, X. Kong, Z. Wang, H. Tang, X. Liu, Y. Yu, J. Dong, and X. Zhang, “A tunable single passband microwave photonic filter of overcoming fiber dispersion induced amplitude fading,” IEEE Photon. J. 9, 5502008 (2017).
[Crossref]

Y. Yu, H. Tang, L. Xu, X. Liu, F. Jiang, J. Dong, and X. Zhang, “Switchable microwave photonic filter between low-pass and high-pass responses,” IEEE Photon. J. 8, 5501408 (2016).
[Crossref]

Long, C. M.

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett. 23, 1618–1620 (2011).
[Crossref]

Luther-Davies, B.

Madden, S.

Marpaung, D.

Martí, J.

Minasian, R.

Mora, J.

Morrison, B.

Novak, D.

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

Ortega, B.

Pant, R.

Piqueras, M.

Poulton, C. G.

Qian, W.

Rooks, M.

Schneider, T.

Seeds, A. J.

A. J. Seeds, “Microwave photonics,” IEEE Trans. Microw. Theory Tech. 50, 877–887 (2002).
[Crossref]

Sekaric, L.

Selvaraja, S. K.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Seo, D.

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett. 23, 1618–1620 (2011).
[Crossref]

Shafiiha, R.

Shahnia, S.

Song, M.

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett. 23, 1618–1620 (2011).
[Crossref]

Stern, Y.

Tang, H.

Tur, M.

Van Thourhout, D.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Van Vaerenbergh, T.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Vidal, B.

Vlasov, Y.

Vu, K.

Wang, L.

Wang, X.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (2014), paper Th2A–37.

Wang, Y.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (2014), paper Th2A–37.

Wang, Z.

L. Xu, X. Kong, Z. Wang, H. Tang, X. Liu, Y. Yu, J. Dong, and X. Zhang, “A tunable single passband microwave photonic filter of overcoming fiber dispersion induced amplitude fading,” IEEE Photon. J. 9, 5502008 (2017).
[Crossref]

H. Tang, Y. Yu, C. Zhang, Z. Wang, L. Xu, and X. Zhang, “Analysis of performance optimization for a microwave photonic filter based on stimulated Brillouin scattering,” J. Lightwave Technol. 35, 4375–4383 (2017).
[Crossref]

Weiner, A. M.

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett. 23, 1618–1620 (2011).
[Crossref]

Wu, R.

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett. 23, 1618–1620 (2011).
[Crossref]

Wu, Y.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (2014), paper Th2A–37.

Xia, F.

Xu, E.

Xu, L.

Xue, X.

Yang, C.

Yao, J.

Yi, X.

Yu, Y.

Yuan, Z.

Zadok, A.

Zhang, C.

Zhang, H.

Zhang, R.

Zhang, X.

Zheng, X.

Zhong, K.

Zhou, B.

Zhou, L.

Zhu, N.

IEEE Photon. J. (2)

L. Xu, X. Kong, Z. Wang, H. Tang, X. Liu, Y. Yu, J. Dong, and X. Zhang, “A tunable single passband microwave photonic filter of overcoming fiber dispersion induced amplitude fading,” IEEE Photon. J. 9, 5502008 (2017).
[Crossref]

Y. Yu, H. Tang, L. Xu, X. Liu, F. Jiang, J. Dong, and X. Zhang, “Switchable microwave photonic filter between low-pass and high-pass responses,” IEEE Photon. J. 8, 5501408 (2016).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. Song, C. M. Long, R. Wu, D. Seo, D. E. Leaird, and A. M. Weiner, “Reconfigurable and tunable flat-top microwave photonic filters utilizing optical frequency combs,” IEEE Photon. Technol. Lett. 23, 1618–1620 (2011).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

A. J. Seeds, “Microwave photonics,” IEEE Trans. Microw. Theory Tech. 50, 877–887 (2002).
[Crossref]

J. Lightwave Technol. (2)

Laser Photon. Rev. (1)

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Nat. Photonics (1)

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

Opt. Express (9)

M. Bolea, J. Mora, B. Ortega, and J. Capmany, “Highly chirped single-bandpass microwave photonic filter with reconfiguration capabilities,” Opt. Express 19, 4566–4576 (2011).
[Crossref]

Y. Yu, E. Xu, J. Dong, L. Zhou, X. Li, and X. Zhang, “Switchable microwave photonic filter between high Q bandpass filter and notch filter with flat passband based on phase modulation,” Opt. Express 18, 25271–25282 (2010).
[Crossref]

F. Jiang, Y. Yu, H. Tang, L. Xu, and X. Zhang, “Tunable bandpass microwave photonic filter with ultrahigh stopband attenuation and skirt selectivity,” Opt. Express 24, 18655–18663 (2016).
[Crossref]

X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Widely tunable single-bandpass microwave photonic filter employing a non-sliced broadband optical source,” Opt. Express 19, 18423–18429 (2011).
[Crossref]

W. Li, C. Yang, L. Wang, Z. Yuan, J. Liu, M. Li, and N. Zhu, “Microwave photonic bandstop filter with wide tunability and adjustable bandwidth,” Opt. Express 23, 33579–33586 (2015).
[Crossref]

A. Byrnes, R. Pant, E. Li, D.-Y. Choi, C. G. Poulton, S. Fan, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Photonic chip based tunable and reconfigurable narrowband microwave photonic filter using stimulated Brillouin scattering,” Opt. Express 20, 18836–18845 (2012).
[Crossref]

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
[Crossref]

P. Dong, W. Qian, H. Liang, R. Shafiiha, D. Feng, G. Li, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “Thermally tunable silicon racetrack resonators with ultralow tuning power,” Opt. Express 18, 20298–20304 (2010).
[Crossref]

P. Dong, W. Qian, H. Liang, R. Shafiiha, N.-N. Feng, D. Feng, X. Zheng, A. V. Krishnamoorthy, and M. Asghari, “Low power and compact reconfigurable multiplexing devices based on silicon microring resonators,” Opt. Express 18, 9852–9858 (2010).
[Crossref]

Opt. Lett. (4)

Photon. Res. (1)

Other (2)

C. Chauveau, P. Labeye, J.-M. Fedeli, S. Blaize, and G. Lerondel, “Study of the uniformity of 300  mm wafer through ring-resonator analysis,” in IEEE International Conference on Photonics in Switching (PS) (2012), pp. 1–3.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (2014), paper Th2A–37.

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Figures (10)

Fig. 1.
Fig. 1. Optical spectra of the phase-modulated signal and the flattop optical filter when the frequency of the optical carrier is tuned (a) aligned with the center of the optical filter and (b) away from the center of the optical filter.
Fig. 2.
Fig. 2. Structure of the photonic chip.
Fig. 3.
Fig. 3. Simulated results of (a) output at the drop port of the 10th-order MRR and (b) proposed MPF with adjustable bandwidths.
Fig. 4.
Fig. 4. Optical micrograph of the fabricated device.
Fig. 5.
Fig. 5. Experimental setup of tuning the 10th-order MRR. BOS, broadband optical source; PBS, polarization beam splitter; PC, polarization controller; OSA, optical spectrum analyzer.
Fig. 6.
Fig. 6. Optimized optical filter at the drop port.
Fig. 7.
Fig. 7. Experimental setup of the proposed MPF. LD, laser diode; PC, polarization controller; PM, phase modulator; EA, electronic amplifier; VNA, vector network analyzer.
Fig. 8.
Fig. 8. (a) Measured dark current of the Ge PD; (b) measured response of the Ge PD.
Fig. 9.
Fig. 9. Optical spectra of the bandpass filter and the phase-modulated signals after the filter.
Fig. 10.
Fig. 10. Measured frequency responses of the proposed MPF with different bandwidths.

Equations (5)

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E ( t ) = E 0 { J 0 ( m ) cos ( ω 0 t ) + J 1 ( m ) cos [ ( ω 0 + ω m ) t + π 2 ] + J 1 ( m ) cos [ ( ω 0 ω m ) t π 2 ] } ,
E ( t ) = E 0 { α 0 J 0 ( m ) cos ( ω 0 t ) + α 1 J 1 ( m ) cos [ ( ω 0 + ω m ) t + π 2 ] + α 1 J 1 ( m ) cos [ ( ω 0 ω m ) t π 2 ] } ,
i = R · | E 0 | 2 [ α 0 α 1 J 0 J 1 cos ( ω m t + π 2 ) + α 0 α 1 J 0 J 1 cos ( ω m t + π 2 ) ] ,
i = R · | E 0 | 2 α 0 ( α 1 α 1 ) J 0 J 1 cos ( ω m t + π 2 ) .
FSR = c / ( n g · L ) ,

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