Novel programmable microwave photonic filter with arbitrary filtering shape and linear phase

Xiaoqi Zhu; Feiya Chen; Huanfa Peng; Zhangyuan Chen

doi:10.1364/OE.25.009232

Novel programmable microwave photonic filter with arbitrary filtering shape and linear phase

Xiaoqi Zhu, Feiya Chen, Huanfa Peng, and Zhangyuan Chen

Optics Express
Vol. 25,
Issue 8,
pp. 9232-9243
(2017)
•https://doi.org/10.1364/OE.25.009232

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Xiaoqi Zhu, Feiya Chen, Huanfa Peng, and Zhangyuan Chen, "Novel programmable microwave photonic filter with arbitrary filtering shape and linear phase," Opt. Express 25, 9232-9243 (2017)

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Related Topics
Table of Contents Category
- RF and Microwave Photonics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Radio frequency photonics (060.5625)
- Frequency filtering (070.2615)

About this Article
History
- Original Manuscript: January 17, 2017
- Revised Manuscript: March 25, 2017
- Manuscript Accepted: April 10, 2017
- Published: April 12, 2017

Accessible

Open Access

Abstract

We propose and demonstrate a novel optical frequency comb (OFC) based microwave photonic filter which is able to realize arbitrary filtering shape with linear phase response. The shape of filter response is software programmable using finite impulse response (FIR) filter design method. By shaping the OFC spectrum using a programmable waveshaper, we can realize designed amplitude of FIR taps. Positive and negative sign of FIR taps are achieved by balanced photo-detection. The double sideband (DSB) modulation and symmetric distribution of filter taps are used to maintain the linear phase condition. In the experiment, we realize a fully programmable filter in the range from DC to 13.88 GHz. Four basic types of filters (lowpass, highpass, bandpass and bandstop) with different bandwidths, cut-off frequencies and central frequencies are generated. Also a triple-passband filter is realized in our experiment. To the best of our knowledge, it is the first demonstration of a programmable multiple passband MPF with linear phase response. The experiment shows good agreement with the theoretical result.

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References

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Publication

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F. Ohman, K. Yvind, and J. Mork, “Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics,” IEEE Photonics Technol. Lett. 19(15), 1145–1147 (2007).
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W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, “Enhancing light slow-down in semiconductor optical amplifiers by optical filtering,” Opt. Lett. 33(10), 1084–1086 (2008).
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W. Xue, S. Sales, J. Capmany, and J. Mørk, “Microwave phase shifter with controllable power response based on slow- and fast-light effects in semiconductor optical amplifiers,” Opt. Lett. 34(7), 929–931 (2009).
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[Crossref]
J. Liao, X. Xue, H. Wen, S. Li, X. Zheng, H. Zhang, and B. Zhou, “A spurious frequencies suppression method for optical frequency comb based microwave photonic filter,” Laser Photonics Rev. 7(4), 34–38 (2013).
[Crossref]
J. Mora, L. R. Chen, and J. Capmany, “Single-bandpass microwave photonic filter with tuning and reconfiguration capabilities,” J. Lightwave Technol. 26(15), 2663–2670 (2008).
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[Crossref]
D. Zou, X. Zheng, S. Li, H. Zhang, and B. Zhou, “High-Q microwave photonic filter with self-phase modulation spectrum broadening and third-order dispersion compensation,” Chin. Opt. Lett. 12(8), 080601 (2014).
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[Crossref]
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[Crossref]
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[Crossref]

2015 (2)

Y. Yu, S. Li, J. Liao, X. Zheng, H. Zhang, and B. Zhou, “Improving suppression ratio of microwave photonic filters using high-precision spectral shaping,” Opt. Eng. 54(5), 050501 (2015).
[Crossref]

C. Haffner, W. Heni, Y. Fedoryshyn, J. Niegemann, A. Melikyan, D. L. Elder, B. Baeuerle, Y. Salamin, A. Josten, U. Koch, C. Hoessbacher, F. Ducry, L. Juchli, A. Emboras, D. Hillerkuss, M. Kohl, L. R. Dalton, C. Hafner, and J. Leuthold,”All-plasmonic Mach-Zehnder modulator enabling optical high-speed communication at the microscale,” Nat. Photonics 9(8), 525–528 (2015).
[Crossref]

2014 (3)

L. Alloatti, R. Palmer, S. Diebold, K. P. Pahl, B. Chen, R. Dinu, M. Fournier, J. Fedeli, T. Zwick, W. Freude, C. Koos, and J. Leuthold, “100 GHz silicon–organic hybrid modulator,” Light Sci. Appl. 3(5), e173 (2014).
[Crossref]

D. Zou, X. Zheng, S. Li, H. Zhang, and B. Zhou, “High-Q microwave photonic filter with self-phase modulation spectrum broadening and third-order dispersion compensation,” Chin. Opt. Lett. 12(8), 080601 (2014).
[Crossref]

L. Li, X. Yi, T. X. H. Huang, and R. Minasian, “High-resolution single bandpass microwave photonic filter with shape-invariant tunability,” IEEE Photonics Technol. Lett. 26(1), 82–85 (2014).
[Crossref]

2013 (7)

J. Liao, X. Xue, H. Wen, S. Li, X. Zheng, H. Zhang, and B. Zhou, “A spurious frequencies suppression method for optical frequency comb based microwave photonic filter,” Laser Photonics Rev. 7(4), 34–38 (2013).
[Crossref]

X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Analysis and compensation of third-order dispersion induced RF distortions in highly reconfigurable microwave photonic filters,” J. Lightwave Technol. 31(13), 2263–2270 (2013).
[Crossref]

J. Capmany, J. Mora, I. Gasulla, J. Sancho, J. Lloret, and S. Sales, “Microwave photonic signal processing,” J. Lightwave Technol. 31(4), 571–586 (2013).
[Crossref]

J. Liu, N. Guo, Z. Li, C. Yu, and C. Lu, “Ultrahigh-Q microwave photonic filter with tunable Q value utilizing cascaded optical-electrical feedback loops,” Opt. Lett. 38(21), 4304–4307 (2013).
[Crossref] [PubMed]

Y. Zhang and S. Pan, “Tunable multitap microwave photonic filter with all complex coefficients,” Opt. Lett. 38(5), 802–804 (2013).
[Crossref] [PubMed]

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

C. Wang and J. Yao, “A nonuniformly space microwave photonic filter using a spatially discrete chirped FBG,” IEEE Photonics Technol. Lett. 25(19), 1889–1892 (2013).
[Crossref]

2012 (1)

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radio frequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6(3), 186–194 (2012).
[Crossref]

2011 (6)

J. Sancho, J. Lloret, I. Gasulla, S. Sales, and J. Capmany, “Fully tunable 360° microwave photonic phase shifter based on a single semiconductor optical amplifier,” Opt. Express 19(18), 17421–17426 (2011).
[Crossref] [PubMed]

S. C. Chan, Q. Liu, Z. Wang, and K. S. Chiang, “Tunable negative-tap photonic microwave filter based on a cladding-mode coupler and an optically injected laser of large detuning,” Opt. Express 19(13), 12045–12052 (2011).
[Crossref] [PubMed]

E. J. Norberg, R. S. Guzzon, J. S. Parker, L. A. Johansson, and L. A. Coldren, “Programmable photonic microwave filters monolithically integrated in InP/InGaAsP,” J. Lightwave Technol. 29(11), 1611–1619 (2011).
[Crossref]

R. S. Guzzon, E. J. Norberg, J. S. Parker, L. A. Johansson, and L. A. Coldren, “Integrated InP-InGaAsP tunable coupled ring optical bandpass filters with zero insertion loss,” Opt. Express 19(8), 7816–7826 (2011).
[Crossref] [PubMed]

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express 19(15), 14233–14239 (2011).
[Crossref] [PubMed]

G. Venanzoni, A. Morini, and M. Farina, “Practical design of a high power L-band linear phase filter for radar applications,” Microw. Opt. Technol. Lett. 53(12), 2717–2721 (2011).
[Crossref]

2010 (7)

P. Dong, N. N. Feng, D. Feng, W. Qian, H. Liang, D. C. Lee, B. J. Luff, T. Banwell, A. Agarwal, P. Toliver, R. Menendez, T. K. Woodward, and M. Asghari, “GHz-bandwidth optical filters based on high-order silicon ring resonators,” Opt. Express 18(23), 23784–23789 (2010).
[Crossref] [PubMed]

N. N. Feng, P. Dong, D. Feng, W. Qian, H. Liang, D. C. Lee, J. B. Luff, A. Agarwal, T. Banwell, R. Menendez, P. Toliver, T. K. Woodward, and M. Asghari, “Thermally-efficient reconfigurable narrowband RF-photonic filter,” Opt. Express 18(24), 24648–24653 (2010).
[Crossref] [PubMed]

E. Xu, X. Zhang, L. Zhou, Y. Zhang, Y. Yu, X. Li, and D. Huang, “Ultrahigh-Q microwave photonic filter with Vernier effect and wavelength conversion in a cascaded pair of active loops,” Opt. Lett. 35(8), 1242–1244 (2010).
[Crossref] [PubMed]

E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable programmable microwave photonic fitlers based on an optical frequency comb,” IEEE Trans. Microw. Theory Tech. 58(11), 3269–3278 (2010).
[Crossref]

W. Xue, S. Sales, J. Capmany, and J. Mørk, “Wideband 360 degrees microwave photonic phase shifter based on slow light in semiconductor optical amplifiers,” Opt. Express 18(6), 6156–6163 (2010).
[Crossref] [PubMed]

S. Sales, W. Xue, J. Mork, and I. Gasulla, “Slow and fast light effects and their applications tomicrowave photonics using semiconductor optical amplifiers,” IEEE Trans. Microw. Theory Tech. 58(11), 3022–3038 (2010).
[Crossref]

X. Yi, T. X. H. Huang, and R. A. Minasian, “Tunable and reconfigurable photonic signal processor with programmable all-optical complex coefficients,” IEEE Trans. Microw. Theory Tech. 58(11), 3088–3093 (2010).
[Crossref]

2009 (1)

W. Xue, S. Sales, J. Capmany, and J. Mørk, “Microwave phase shifter with controllable power response based on slow- and fast-light effects in semiconductor optical amplifiers,” Opt. Lett. 34(7), 929–931 (2009).
[Crossref] [PubMed]

2008 (5)

W. Xue, Y. Chen, F. Öhman, S. Sales, and J. Mørk, “Enhancing light slow-down in semiconductor optical amplifiers by optical filtering,” Opt. Lett. 33(10), 1084–1086 (2008).
[Crossref] [PubMed]

Y. Dai and J. Yao, “Nonuniformly-spaced photonic microwave delayline filter,” Opt. Express 16(7), 4713–4718 (2008).
[Crossref] [PubMed]

J. Mora, L. R. Chen, and J. Capmany, “Single-bandpass microwave photonic filter with tuning and reconfiguration capabilities,” J. Lightwave Technol. 26(15), 2663–2670 (2008).
[Crossref]

M. D. Manzanedo, J. Mora, and J. Capmany, “Continuously tunable microwave photonic filter with negative coefficients using cross-phase modulation in an SOA-MZ interferometer,” IEEE Photonics Technol. Lett. 20(7), 526–528 (2008).
[Crossref]

Q. Wang and J. P. Yao, “Multitap photonic microwave filters with arbitrary positive and negative coefficients using a polarization modulator and an optical polarizer,” IEEE Photonics Technol. Lett. 20(2), 78–80 (2008).
[Crossref]

2007 (6)

J. P. Yao and Q. Wang, “Photonic microwave bandpass filter with negative coefficients using a polarization modulator,” IEEE Photonics Technol. Lett. 19(9), 644–646 (2007).
[Crossref]

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

F. Li, X. Zhang, Q. Meng, L. Sun, Q. Zhang, C. Li, S. Li, A. He, H. Li, and Y. He, “Superconducting filter with a linear phase for third-generation mobile communications,” Supercond. Sci. Technol. 20(7), 611–615 (2007).
[Crossref]

Y. Yan and J. Yao, “A tunable photonic microwave filter with a complex coefficient using an optical RF phase shifter,” IEEE Photonics Technol. Lett. 19(19), 1472–1474 (2007).
[Crossref]

F. Ohman, K. Yvind, and J. Mork, “Slow light in a semiconductor waveguide for true-time delay applications in microwave photonics,” IEEE Photonics Technol. Lett. 19(15), 1145–1147 (2007).
[Crossref]

J. H. Lee, Y. M. Chang, Y.-G. Han, S. B. Lee, and H. Y. Chung, “Fully reconfigurable photonic microwave transversal filter based on digital micromirror device and continuous-wave, incoherent supercontinuum source,” Appl. Opt. 46(22), 5158–5167 (2007).
[Crossref] [PubMed]

2006 (2)

R. A. Minasian, “Photonic signal processing of microwave signal,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006).
[Crossref]

J. Mora, J. Capmany, A. Loayssa, and D. Pastor, “Novel technique for implementing incoherent microwave photonic filters with negative coefficients using phase modulation and single sideband selection,” IEEE Photonics Technol. Lett. 18(18), 1943–1945 (2006).
[Crossref]

2005 (3)

F. Zeng, J. Wang, and J. Yao, “All-optical microwave bandpass filter with negative coefficients based on a phase modulator and linearly chirped fiber Bragg gratings,” Opt. Lett. 30(17), 2203–2205 (2005).
[Crossref] [PubMed]

J. Wang, F. Zeng, and J. P. Yao, “All-optical microwave bandpass filter with negative coefficients based on PM-IM conversion,” IEEE Photonics Technol. Lett. 17(10), 2176–2178 (2005).
[Crossref]

B. Ortega, J. Mora, J. Capmany, D. Pastor, and R. Garcia-Olcina, “Highly selective microwave photonic filters based on active optical recirculating cavity and tuned modulator hybrid structure,” Electron. Lett. 41(20), 1113–1135 (2005).
[Crossref]

1999 (1)

J. Capmany, D. Pastor, and B. Ortega, “New and flexible fiber-optic delay-line filters using chirped Bragg grattings and laser arrays,” IEEE Trans. Microw. Theory Tech. 47(7), 1321–1326 (1999).
[Crossref]

Agarwal, A.

Alloatti, L.

Asghari, M.

Baeuerle, B.

Banwell, T.

Borja, V.

V. Borja, L. C. Juan, and M. Javier, “Multi-tap all-optical microwave filter with negative coefficients based on multiple optical carriers and dispersive media,” in IEEE International Topical Meeting on Microwave Photonics (2005).

Capmany, J.

J. Capmany, J. Mora, I. Gasulla, J. Sancho, J. Lloret, and S. Sales, “Microwave photonic signal processing,” J. Lightwave Technol. 31(4), 571–586 (2013).
[Crossref]

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

J. Mora, L. R. Chen, and J. Capmany, “Single-bandpass microwave photonic filter with tuning and reconfiguration capabilities,” J. Lightwave Technol. 26(15), 2663–2670 (2008).
[Crossref]

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Chan, S. C.

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Chen, Y.

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Y. Dai and J. Yao, “Nonuniformly-spaced photonic microwave delayline filter,” Opt. Express 16(7), 4713–4718 (2008).
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[Crossref]

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M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express 19(15), 14233–14239 (2011).
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M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express 19(15), 14233–14239 (2011).
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M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express 19(15), 14233–14239 (2011).
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D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
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M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express 19(15), 14233–14239 (2011).
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M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express 19(15), 14233–14239 (2011).
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R. A. Minasian, “Photonic signal processing of microwave signal,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006).
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J. Capmany, J. Mora, I. Gasulla, J. Sancho, J. Lloret, and S. Sales, “Microwave photonic signal processing,” J. Lightwave Technol. 31(4), 571–586 (2013).
[Crossref]

J. Mora, L. R. Chen, and J. Capmany, “Single-bandpass microwave photonic filter with tuning and reconfiguration capabilities,” J. Lightwave Technol. 26(15), 2663–2670 (2008).
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G. Venanzoni, A. Morini, and M. Farina, “Practical design of a high power L-band linear phase filter for radar applications,” Microw. Opt. Technol. Lett. 53(12), 2717–2721 (2011).
[Crossref]

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M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express 19(15), 14233–14239 (2011).
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J. P. Yao and Q. Wang, “Photonic microwave bandpass filter with negative coefficients using a polarization modulator,” IEEE Photonics Technol. Lett. 19(9), 644–646 (2007).
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J. Wang, F. Zeng, and J. P. Yao, “All-optical microwave bandpass filter with negative coefficients based on PM-IM conversion,” IEEE Photonics Technol. Lett. 17(10), 2176–2178 (2005).
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J. Wang, F. Zeng, and J. P. Yao, “All-optical microwave bandpass filter with negative coefficients based on PM-IM conversion,” IEEE Photonics Technol. Lett. 17(10), 2176–2178 (2005).
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Zhang, H.

Zhang, Q.

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Y. Zhang and S. Pan, “Tunable multitap microwave photonic filter with all complex coefficients,” Opt. Lett. 38(5), 802–804 (2013).
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Appl. Opt. (1)

Chin. Opt. Lett. (1)

Electron. Lett. (1)

IEEE Photonics Technol. Lett. (9)

J. P. Yao and Q. Wang, “Photonic microwave bandpass filter with negative coefficients using a polarization modulator,” IEEE Photonics Technol. Lett. 19(9), 644–646 (2007).
[Crossref]

J. Wang, F. Zeng, and J. P. Yao, “All-optical microwave bandpass filter with negative coefficients based on PM-IM conversion,” IEEE Photonics Technol. Lett. 17(10), 2176–2178 (2005).
[Crossref]

Y. Yan and J. Yao, “A tunable photonic microwave filter with a complex coefficient using an optical RF phase shifter,” IEEE Photonics Technol. Lett. 19(19), 1472–1474 (2007).
[Crossref]

C. Wang and J. Yao, “A nonuniformly space microwave photonic filter using a spatially discrete chirped FBG,” IEEE Photonics Technol. Lett. 25(19), 1889–1892 (2013).
[Crossref]

IEEE Trans. Microw. Theory Tech. (5)

R. A. Minasian, “Photonic signal processing of microwave signal,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006).
[Crossref]

J. Lightwave Technol. (4)

J. Capmany, J. Mora, I. Gasulla, J. Sancho, J. Lloret, and S. Sales, “Microwave photonic signal processing,” J. Lightwave Technol. 31(4), 571–586 (2013).
[Crossref]

J. Mora, L. R. Chen, and J. Capmany, “Single-bandpass microwave photonic filter with tuning and reconfiguration capabilities,” J. Lightwave Technol. 26(15), 2663–2670 (2008).
[Crossref]

Laser Photonics Rev. (2)

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

Light Sci. Appl. (1)

Microw. Opt. Technol. Lett. (1)

G. Venanzoni, A. Morini, and M. Farina, “Practical design of a high power L-band linear phase filter for radar applications,” Microw. Opt. Technol. Lett. 53(12), 2717–2721 (2011).
[Crossref]

Nat. Photonics (3)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Opt. Eng. (1)

Opt. Express (8)

Y. Dai and J. Yao, “Nonuniformly-spaced photonic microwave delayline filter,” Opt. Express 16(7), 4713–4718 (2008).
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Opt. Lett. (6)

Y. Zhang and S. Pan, “Tunable multitap microwave photonic filter with all complex coefficients,” Opt. Lett. 38(5), 802–804 (2013).
[Crossref] [PubMed]

Supercond. Sci. Technol. (1)

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J. Wang, Y. Xuan, A. J. Metcalf, P. Wang, X. Xue, D. E. Leaird, A. M. Weiner, and M. Qi, A multi-channel thermally reconfigurable SiN spectral shaper,” in CLEO:2014, OSA Technical Digest (online) (Optical Society of America, 2014), paper SM3G.4.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. Tom, and A. M. Weiner, Spectral line-by-line pulse shaping of an on-chip microresonator frequency comb,” in CLEO: 2011 - Laser Science to Photonic Applications (OSA, 2011).

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Fig. 1 Structure of the proposed filter.

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Fig. 2 Amplitude response G(ω) of DSB modulation caused by the fiber dispersion.

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Fig. 3 Experimental setup of the proposed filter.

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Fig. 4 Optical spectra of (a) the optical frequency comb, and the waveshaper outputs for (b) positive taps, and (c) negative taps.

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Fig. 5 Normalized S21 response of (a) lowpass filters; (b) highpass filters; (c) bandpass filters; (d) bandstop filters. (Solid lines: experimental results. Dashed lines: theoretical curves).

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Fig. 6 (a) Normalized amplitude response and (b) phase response of a triple passband filter (Blue line: Response with only 2nd order dispersion; Yellow line: Response with 2nd and 3rd order dispersion; Red line: Experimental result)

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Equations (17)

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s_{o} = \sum_{n = 0}^{N - 1} h (n) s_{i} (t - n T)

H (ω) = \sum_{n = 0}^{N - 1} h (n) \exp (- j n ω T)

H (ω) {=e}^{j φ (ω)} | H (ω) | = {\begin{matrix} e^{- j ω T (N - 1) / 2} \cdot \sum_{n = 0}^{N - 1} h (n) \cos [(\frac{N - 1}{2} - n) ω T)], when h (n) = h (N - 1 - n) \\ e^{j π / 2 - j ω T (N - 1) / 2} \cdot \sum_{n = 0}^{N - 1} h (n) \sin [(\frac{N - 1}{2} - n) ω T)], when h (n) = - h (N - 1 - n) \end{matrix}

E_{s} (t) = \sum_{n = 0}^{N - 1} \sqrt{P_{0} (n)} \exp (- j ω_{n} t) = \sum_{n = 0}^{N - 1} \sqrt{P_{0} (n)} \exp [- j (ω_{0} + n ω_{r}) t]

\begin{array}{l} E_{m} (t) = E_{s} (t) \cos (\frac{π}{4} + \frac{π V_{R F} (t)}{2 V_{π}}) \\ \approx \frac{\sqrt{2}}{2} \sum_{n = 0}^{N - 1} \sqrt{P_{0} (n)} [\frac{1}{2} J_{1} (\frac{π A_{R F}}{2 V_{π}}) e^{- j (ω_{n} t - ω_{R F} t) - j \frac{π}{2}} + J_{0} (\frac{π A_{R F}}{2 V_{π}}) e^{- j ω_{n} t} + \frac{1}{2} J_{1} (\frac{π A_{R F}}{2 V_{π}}) e^{- j (ω_{n} t + ω_{R F} t) + j \frac{π}{2}}] \end{array}

θ (ω) = \frac{β_{2} L}{2} {(ω - ω_{0})}^{2}

\begin{array}{l} E_{D} (t) = \frac{\sqrt{2}}{2} \sum_{n = 0}^{N - 1} \sqrt{P_{0} (n)} [\frac{- 1}{2} J_{1} (\frac{π A_{R F}}{2 V_{π}}) e^{- j [ω_{n} t - ω_{R F} t + θ (ω_{n} - ω_{R F}) + j \frac{π}{2}]} \\ + J_{0} (\frac{π A_{R F}}{2 V_{π}}) e^{- j (ω_{n} t + θ (ω_{n}))} + \frac{1}{2} J_{1} (\frac{π A_{R F}}{2 V_{π}}) e^{- j [ω_{n} t + ω_{R F} t + θ (ω_{n} + ω_{R F}) - j \frac{π}{2}]}] \end{array}

\begin{array}{l} I_{o} (t) = A_{D} \cos (\frac{β_{2} L ω_{R F}^{2}}{2}) [\sum_{n, h (n) > 0} P_{s} (n) \cos (ω_{R F} t + n β_{2} L ω_{R F} ω_{r}) - \sum_{n, h (n) < 0} P_{s} (n) \cos (ω_{R F} t + n β_{2} L ω_{R F} ω_{r})] \\ = A_{D} \cos (\frac{β_{2} L ω_{R F}^{2}}{2}) \sum_{n = 0}^{N - 1} sign [h (n) \cdot P_{s} (n)] \cos {ω_{R F} [t -(- n β_{2} L ω_{r})]} \end{array}

A_{D} = α R Z J_{0} (\frac{π A_{R F}}{2 V_{π}}) J_{1} (\frac{π A_{R F}}{2 V_{π}})

H (ω) = G (ω) \sum_{n = 0}^{N - 1} [sign (h (n)) \cdot P_{s} (n)] \exp (- j n ω T)

T =- β_{2} ω_{r} L

G (ω) = A_{D} \cos (β_{2} L ω^{2} / 2)

H (ω) =G(ω) \cdot e^{- j ω T (N - 1) / 2} \cdot \sum_{n = 0}^{N - 1} [si g n (h (n)) \cdot P_{s} (n)] \cos [(\frac{N - 1}{2} - n) ω T)]

G (ω) = {\begin{matrix} A_{S} \exp (j β_{2} L ω^{2} / 2), Single-sideband modulation \\ A_{P} \sin (β_{2} L ω^{2} / 2), Phase-modulation \\ A_{D} \cos (β_{2} L ω^{2} / 2), Double-sideband modulation \end{matrix}

H (ω) = A_{D} \sum_{n = 0}^{N - 1} [si g n (h (n)) \cdot P_{s} (n)] \cos (\frac{β_{2} L ω^{2}}{2} + \frac{n β_{3} L ω^{2} ω_{r}}{2}) \exp (- j (n ω β_{2} L ω_{r} + \frac{1}{6} β_{3} L ω^{3} + \frac{n^{2}}{2} β_{3} L ω_{r}^{2} ω))

H (ω) \approx A_{D} \cos (\frac{β_{2} L ω^{2}}{2}) \sum_{n = 0}^{N - 1} P_{s} (n) \exp (- j (n ω β_{2} L ω_{r} + \frac{n^{2}}{2} β_{3} L ω_{r}^{2} ω))

T (n) = β_{2} L ω_{r} + \frac{1}{2} β_{3} L ω_{r}^{2} + n β_{3} L ω_{r}^{2}