Abstract

A microwave photonic bandstop filter is proposed and experimentally demonstrated in this work. The filter exhibits promising performance combination of reconfigurability, frequency tunability, and bandwidth adjustment. The phase modulation on two orthogonal polarization states produces a bandpass and a lowpass MPF, respectively. The key concept of destructive interference between the bandpass and lowpass MPF enables the reconfiguration of MPF from bandpass to bandstop. By adjusting the wavelength of two orthogonally polarized optical carriers and the bandwidth of an optical bandpass filter, the bandstop filter is tunable in terms of center frequency and bandwidth.

© 2015 Optical Society of America

1. Introduction

Photonic processing of microwave signals using microwave photonic filter (MPF) has attracted considerable interest in the past decade because it is capable of overcoming the inherent bottlenecks of traditional electrical signal processing techniques whilst offering promising performance including reconfigurability, wideband, large tunability, and immunity to electromagnet interference [1–3 ]. MPF can be realized using multi-tap delay lines with a finite impulse response (FIR) [4–6 ]. The filter performance is related to the number of taps which is usually implemented using laser array or sliced broadband optical source. In order to design MPF with a high Q, delay-line filters with an infinite impulse response (IIR) is preferred [7,8 ]. However, either FIR or IIR MPF suffers from periodic magnitude response due to the discrete nature of the sampling process in time domain.

To overcome the periodic response nature of delay-line filters, many efforts have been made to design new MPFs [9–18 ]. MPFs based on stimulated Brillouin scattering (SBS) have been widely studied in the past few years [9–11 ]. It offers a single bandpass or bandstop response which is very attractive. Usually, SBS occurs in a spool of optical fiber that makes the system bulky and unstable [9–11 ]. Recently, chip-based SBS effect has been reported and has been applied to bandstop MPF [12]. However, it is hard to tailor the shape of the MPF due to the extreme narrow bandwidth of SBS interaction. Single bandpass or bandstop MPFs have also been demonstrated using a phase modulator (PM) and a fiber Bragg grating [13–15 ] or ring resonator [16]. However, the shape and bandwidth of the MPFs are the same as the optical filters which are not tunable. In order to adjust the bandwidth of the MPF, Chen et al. reported a method using PM and an optical bandpass filter (OBPF) [17]. However, only bandpass MPF can be implemented. Up to now, only a few schemes can realize bandstop MPF with wide tunability and adjustable bandwidth [18]. However, the bandwidth tuning range of MPF reported in [18] is relative small, ~1 GHz.

In this paper, we experimentally demonstrate a bandstop MPF using orthogonal phase modulation and optical filtering. Our MPF is tunable in terms of center frequency and bandwidth. In this scheme, the phase modulation to intensity modulation (PM-IM) conversion at two orthogonal polarization states generates a bandpass and a lowpass MPF, respectively. By tuning the optical wavelength, the bandpass and lowpass MPFs are perfectly out of phase. The destructive interference of the bandpass and lowpass MPFs results in a bandstop MPF. The gradual transform from bandpass to banstop MPF is controlled by the weighting of bandpass and lowpass MPFs. In addition, the center frequency and bandwidth of the bandstop MPF is tunable by adjusting the optical wavelength and the bandwidth of the OBPF.

2. Principle

The schematic diagram of the proposed bandstop MPF is illustrated in Fig. 1(a) . Two optical carriers from tunable laser sources (TLS1 and TLS2) are coupled by a polarization beam combiner (PBC). The state-of-polarizations (SOPs) of TLS1 and TLS2 are thus orthogonal, as shown in Fig. 1(b). The SOPs of TLS1 and TLS2 are aligned at x and y axis, respectively. Two polarization controllers (PC1 and PC2) are added to control the SOPs of TLS1 and TLS2, respectively, and consequently to adjust the power ratio between TLS1 and TLS2 at PBC output. The optical field at PBC output can be expressed as

EPBC(t)=[Ex(t)Ey(t)]=[E1exp(jω1t)E2exp(jω2t)]
where E 1 and E 2 is the amplitude of TLS1 and TLS2, respectively. ω 1 and ω 2 is the angular frequency of TLS1 and TLS2, respectively. A polarization modulator (PolM) is connected to realize phase modulation at two orthogonal polarization axes with opposite phase modulation indices [19]. By adjusting PC3, the SOPs of TLS1 and TLS2 are aligned with two principal axes of the PolM, respectively. The output field at x and y axes is given by
EPolM(t)=[Ex(t)Ey(t)]=[E1exp(jω1t+jβsin(ωmt))E2exp(jω2tjβsin(ωmt))]
where β = πV m/V π is the phase modulation index of the modulator. V m and ω m are the amplitude and angular frequency of the microwave signal to be processed. Under small-signal condition, only two first-order sidebands are considered. Equation (2) can be rewritten by Jacobi-Anger expansion as follows,
EPolM(t)=[Ex(t)Ey(t)]=[E1exp(jω1t)[J0+J1exp(jωmt+jπ)+J1exp(jωmt)]E2exp(jω2t)[J0+J1exp(jωmt)+J1exp(jωmt+jπ)]]
where J 0 and J 1 are the 0 and the first-order Bessel function of the first kind. The schematic optical spectra at x and y axes are separately illustrated in Fig. 1(c). Both TLS1 and TLS2 are phase modulated with opposite phase modulation indices. A polarization independent OBPF is attached after the PolM. Here we denote the OBPF bandwidth as f B and the frequency interval between TLS1 and the center of OBPF as f C.

 

Fig. 1 Schematic diagram of (a) the proposed MPF and (b)-(d) the principle behind the MPF.

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If we only consider the phase modulation at x axis, a bandpass MPF is realized after photodetection [17], as shown in Fig. 1(d) (see the red line in the top inset). The center frequency of the MPF is f B/2 and the bandwidth is 2f C. We give a brief explanation about the principle behind the bandpass MPF. Detail analysis can be find elsewhere [17,20 ]. For phase modulation, two optical sidebands with equal amplitudes are produced at two sides of TLS1. Moreover, they are out of phase, as shown in Fig. 1(c). When the microwave frequency to be processed, f m, is less than f B/2-f C, both the upper and lower sidebands fall into the passband of the OBPF. No microwave signal is recovered after photodetection. For f B/2-f C<f m<f B/2 + f C, the lower optical sideband is removed by the OBPF, the microwave signal at f m is recovered by beating between the upper sideband and the optical carrier. The beating signal has a phase of 0 since the optical carrier and upper sideband are in phase. For f m>f B/2 + f C, no microwave signal is detected because both optical sidebands are out of the passband of the OBPF. The center frequency of the bandpass MPF is thus tunable by adjusting the bandwidth of the OBPF, and the bandwidth of the MPF is tunable by adjusting the wavelength of the TLS1. After photodetection, the photocurrent generated at x axis is written as

ix(t)=REx(t)Ex*(t)={RE12[J02+J12+2J0J1cos(ωmt)],forfB2fC<ωm2π<fB2+fC0,else.
For TLS2 on y axis, we adjust the wavelength of TLS2 to the left edge of the OBPF to remove the lower optical sideband. The beating between the upper sideband and the optical carrier results in a lowpass MPF, as shown in Fig. 1(d) (see the blue line in the top inset). The photocurrent generated at y axis can be expressed as
iy(t)=REy(t)Ey*(t)={RE22[J02+J12+2J0J1cos(ωmt+π)],for0<ωm2π<fB0,else.
Equation (5) shows that the lowpass MPF has a phase of π because the optical carrier and upper sideband are out of phase, which is opposite to the case at x axis. The cut-off frequency of the lowpass MPF is f B. It is important to stress that there is no beating signal between TLS1 and TLS2 (including their sidebands) because they are orthogonally polarized. However, the generated bandpass and lowpass MPFs interfere with each other. The total photocurrent is given by
i(t)=ix(t)+iy(t)={RE22[J02+J12+2J0J1cos(ωmt+π)],for0<ωm2π<fB2fC,fB2+fC<ωm2π<fBRE12[J02+J12+2J0J1cos(ωmt)]RE22[J02+J12+2J0J1cos(ωmt)],forfB2fC<ωm2π<fB2+fC.
As can be seen from Eq. (6), the bandpass MPF generated at x axis and the lowpass MPF generated at y axis are out of phase, they cancel out each other perfectly at the overlapped frequencies if they have equal amplitudes (i.e. E 1 = E 2). This results in a bandstop MPF at the overlapped frequencies as shown in Fig. 1(d) (see the bottom inset). The bandstop MPF has a bandwidth of 2f C, a center frequency of f B/2, and a cut-off frequency of f B. By adjusting the power ratio between TLS1 and TLS2, bandpass, bandstop, and a transition MPF between bandpass and bandstop can be obtained. The center frequency and bandwidth of the stopband MPF is adjustable by tuning the OBPF bandwidth the wavelength of TLSs.

3. Experiment

We carried out experiments to confirm the proposed MPF. The key parameters of devices used in our experiments are as follows: two TLSs which are tunable in C-band with a minimum step of 0.1 pm; a 40 GHz PolM with a half-wave voltage of 3.5 V; a photodetector (PD) with 3-dB bandwidth of ~38 GHz; an OBPF which has a tunable bandwidth from 32 to 650 pm, a tunable center wavelength from 1480 to 1620 nm, a polarization dependent loss of ± 0.2 dB, and a typical edge roll-off of 800 dB/nm. We first set the center wavelength and bandwidth of the OBPF at 1549.616 nm and 263 pm, respectively. Consequently, we adjusted the wavelength of TLS1 and TLS2 to be 1549.571 and 1549.513 nm, respectively. They were combined by a PBC. The measured optical spectrum at the output of the PBC is shown in Fig. 2(a) as black line. In order to show the polarization extinction ratio between the two orthogonally polarized optical carriers, we added a PC and a polarization beam splitter (PBS) after the PBC. The measured optical spectra at two ports of the PBS are shown in Fig. 2(a) as red and blue line, respectively. The polarization extinction ratio is better than 30 dB. We applied a sinusoidal microwave signal of 7 GHz to the PolM. The optical spectra at the output of the PolM when TLS1 and TLS2 were turned on in turn are shown in Fig. 2(b). For microwave frequency of 7 GHz, the power imbalance between the two optical sidebands of TLS1 and TLS2 is 0.7 and 25 dB, respectively. It means that the microwave signal at 7 GHz can be recovered at y axis (TLS2) but nearly eliminated at x axis (TLS1). The transmission response of the OBPF is also shown in Fig. 2(b). The wavelength interval between TLS1 and the center of OBPF is 45 pm, corresponding to a frequency interval of 5.625 GHz. According to previous analysis, the bandpass MPF should have a bandwidth of 11.25 GHz and a center frequency of 16.43 GHz. The cut-off frequency of the lowpass MPF should be 32.875 GHz.

 

Fig. 2 Measured optical spectra of (a) orthogonally polarized optical carriers and (b) the modulated optical carriers.

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The measured magnitude responses of the bandpass and bandstop MPFs when TLS1 and TLS2 were turned on in turn are depicted in Fig. 3(a) . The bandpass MPF generated by TLS1 has a center frequency of 16.45 GHz and a bandwidth of 11 GHz, which agrees with the calculated ones. The cut-off frequency is around 30 GHz. The phase responses of the bandpass MPF generated by TLS1 and lowpass MPF generated by TLS2 are shown in Fig. 3(b). A phase difference of 180° between the bandpass and lowpass MPFs is obtained (see the green line in Fig. 3(b)). By adjusting the power of TLS1 and TLS2, the bandpass and lowpass MPFs have unique amplitude but a 180° phase difference, leading to power cancellation. A bandstop MPF, as shown in Fig. 3(a), is thus obtained. The bandstop MPF has a center frequency of 16.5 GHz and a suppression ratio of 30 dB inside the notch. The insertion loss of the bandstop MPF is measured to be ~50 dB. In our proof-of-concept experiment, no optical or electrical amplifiers were involved. Moreover, the optical signal set to the PD is around 0 dBm, which can be further improved to generate strong microwave signal since the saturation power of the PD is around 13 dBm. In our condition, the expected insertion loss is around 45 dB. The extra insertion loss is mainly due to the loss of the cable and the insertion loss of the PD. To reduce the insertion loss of the MPF, optical and electrical amplifiers can be used to boost the optical and electrical signals. As can be seen from Fig. 3(a), the passband response generally follows the response of the lowpass filter. The destructive interference in the bandstop will not affect the loss of passband because the bandpass response is significantly suppressed around this frequency range.

 

Fig. 3 Measured (a) magnitude and (b) phase responses of the bandpass, lowpass, and bandstop filters.

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It is worth noting that a suppression on the low frequency side is observed from both lowpass and bandstop MPFs. This is attributed to the non-ideal rectangular response of the OBPF whose edge is not steep enough to fully remove the lower sideband of TLS2. The phase-to-intensity modulation conversion generates a notch on the low frequency side. The MPF has slow roll-off edge and a bandwidth of ~6 GHz. The roll-off property and the bandwidth of the MPF is mainly determined by the performance of the OBPF. In order to realize a bandstop MPF with sharp edge, it is better to choose an OBPF with sharp roll-off edge. In addition, the real operation bandwidth of the MPF is limited by the bandwidth of the OBPF. The measured phase response of the bandstop MPF is shown in Fig. 3(b). The bandstop MPF generally follows the phase response of lowpass MPF except for the notch. Inside the notch, the phase response is distorted. For the undesired signal inside the notch, the phase distortion is acceptable. Figure 4 shows the transform of MPF from bandpass to bandstop when TLS2 was disconnected and was attenuated by 5.0, 3.0, 1.5, 1.0, and 0 dB, respectively.

 

Fig. 4 The transform from bandpass to bandstop MPF when the power of TLS2 was tuned.

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We also experimentally confirmed the frequency tunability of the bandstop MPF. In this case, TLS1 was fixed at 1549.581 nm. The bandwidth of OBPF was tuned from 250 to 350 pm and the TLS2 was tuned from 1549.521 to 1549.465 nm to match the OBPF. The center frequency of the bandpass and bandstop MPFs was tuned from 15.6 to 21.5 GHz whilst keeping the bandwidth almost unchanged, as shown in Figs. 5(a) and 5(b) , respectively. For the bandwidth adjustment of the MPF, the OBPF was set at a fixed bandwidth of 263 pm. TLS2 was fixed at 1549.513 nm, while TLS1 was tuned. When the TLS1 was detuned from 1549.571 nm for 0 to 24 pm, the bandwidth of the bandpass MPF was altered from 11 to 5 GHz, as shown in Fig. 6(a) . Meanwhile, the bandwidth of bandstop MPF was changed from ~6 to 1.5 GHz. The bandwidth of the bandstop MPF is determined by the frequency difference between the TLS1 and the center frequency of the OBPF. To create a bandstop MPF with very narrow bandwidth, the frequency interval between the TLS1 and the center of the OBPF should be as small as possible. Moreover, it is better to choose an OBPF with steep roll-off edge.

 

Fig. 5 The center frequency tunability of (a) the bandpass and (b) the bandstop MPF by adjusting the bandwidth of the OBPF and the wavelength of TLS2.

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Fig. 6 The bandwidth tunability of (a) the bandpass and (b) the bandstop MPF by adjusting the wavelength of the TLS1.

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For traditional bandstop MPF based on single-sideband (SSB) modulation and bandstop optical filter, the bandwidth and suppression of the MPF is determined by the shape of the bandstop optical filter. However, the bandwidth of the proposed bandstop MPF is mainly determined by the frequency interval between the TLS and the center frequency of the OBPF. In principle, the proposed MPF could have ultra-narrow bandwidth by adjusting the TLS. Moreover, the suppression of the proposed MPF is related to the power cancellation between the bandpass and lowpass MPFs, which can be extremely deep if a perfect 180° phase difference between the two MPFs can be ensured.

The stability of the MPF is affected by the relative frequency-stability between the TLS and the OBPF. The relative drift between TLS and the OBPF results in variation of the MPF bandwidth. In addition, the suppression performance of the MPF is sensitive to the polarization change of the laser sources. The variation of the SOP of the TLSs causes magnitude unbalance between the generated lowpass and bandpass MPFs, and finally degrades the suppression of the bandstop MPF. In our lab, the suppression of the bandstop MPF degraded around 5 dB after running for 2 hours. In our scheme, a bias-free PolM, which can also be considered as a special PM, is used to generate the MPF, we believe that the performance degradation due to bias-drift of conventional modulator has been avoided [21]. Thus, the suppression degradation of our filter can be mainly attributed to the polarization instability of the laser sources. We believe the use of polarization maintaining components is helpful to enhance the stability of the MPF.

4. Conclusion

We have reported a bandstop MPF using orthogonal phase modulation and optical filtering. The PM-IM conversion at two orthogonal modulation axes produced a bandpass and a lowpass MPF, respectively. The destructive interference between the two MPFs generated a bandstop MPF. The transform from bandpass to bandstop MPF has been clearly observed. The center frequency and bandwidth tunability of the MPF has been confirmed by tuning the wavelength of TLSs and the bandwidth of the OBPF. The proposed scheme has potential for integration since all the key components in our scheme, i.e. modulator, optical filter, and PD, can be integrated. We believe that the integrated MPF is very promising in practical applications.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (NSFC) under 61377069 and 61335005, in part by the National High Technology Research and Development Program (863 Program) under 2015AA017002, in part by the National Basic Research Program of China (973 Program) under 2012CB315703, and in part by Beijing Nova program under xxjh2015B076.

References and links

1. J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24(1), 201–229 (2006). [CrossRef]  

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

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

4. 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]  

5. L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011). [CrossRef]  

6. Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013). [CrossRef]  

7. E. H. W. Chan and R. A. Minasian, “High-resolution photonics-based interference suppression filter with wide passband,” J. Lightwave Technol. 21(12), 3144–3149 (2003). [CrossRef]  

8. 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(24), 25271–25282 (2010). [CrossRef]   [PubMed]  

9. X. S. Yao, “Brillouin selective sideband amplification of microwave photonic signals,” IEEE Photonics Technol. Lett. 10(1), 138–140 (1998). [CrossRef]  

10. W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(23), 1775–1777 (2011). [CrossRef]  

11. W. Li, L. X. Wang, and N. H. Zhu, “All-optical microwave photonic single-passband filter based on polarization control through stimulated Brillouin scattering,” IEEE Photonics J. 5(4), 5501411 (2013). [CrossRef]  

12. D. Marpaung, B. Morrison, M. Pagani, R. Pant, D. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Low-power, chip-based stimulated Brillouin scattering microwave photonic filter with ultrahigh selectivity,” Optica 2(2), 76–83 (2015). [CrossRef]  

13. X. Yi and R. A. Minasian, “Microwave photonic filter with single bandpass response,” Electron. Lett. 45(7), 362–363 (2009). [CrossRef]  

14. W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012). [CrossRef]  

15. E. Xu and J. Yao, “Frequency-and notch-depth-tunable single-notch microwave photonic filter,” IEEE Photonics Technol. Lett. 27(19), 2063–2066 (2015). [CrossRef]  

16. J. Palací, G. E. Villanueva, J. V. Galán, J. Martí, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010). [CrossRef]  

17. T. Chen, X. Yi, L. Li, and R. Minasian, “Single passband microwave photonic filter with wideband tunability and adjustable bandwidth,” Opt. Lett. 37(22), 4699–4701 (2012). [CrossRef]   [PubMed]  

18. J. Ge, H. Feng, G. Scott, and M. P. Fok, “High-speed tunable microwave photonic notch filter based on phase modulator incorporated Lyot filter,” Opt. Lett. 40(1), 48–51 (2015). [CrossRef]   [PubMed]  

19. J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004). [CrossRef]  

20. W. Li, J. G. Liu, and N. H. Zhu, “A widely and continuously tunable frequency doubling optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(13), 1461–1464 (2015). [CrossRef]  

21. S. Shahnia, M. Pagani, B. Morrison, B. J. Eggleton, and D. Marpaung, “Independent manipulation of the phase and amplitude of optical sidebands in a highly-stable RF photonic filter,” Opt. Express 23(18), 23278–23286 (2015). [CrossRef]   [PubMed]  

References

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  • |

  1. J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24(1), 201–229 (2006).
    [Crossref]
  2. V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6(3), 186–194 (2012).
    [Crossref]
  3. J. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
    [Crossref]
  4. 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]
  5. L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
    [Crossref]
  6. Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013).
    [Crossref]
  7. E. H. W. Chan and R. A. Minasian, “High-resolution photonics-based interference suppression filter with wide passband,” J. Lightwave Technol. 21(12), 3144–3149 (2003).
    [Crossref]
  8. 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(24), 25271–25282 (2010).
    [Crossref] [PubMed]
  9. X. S. Yao, “Brillouin selective sideband amplification of microwave photonic signals,” IEEE Photonics Technol. Lett. 10(1), 138–140 (1998).
    [Crossref]
  10. W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(23), 1775–1777 (2011).
    [Crossref]
  11. W. Li, L. X. Wang, and N. H. Zhu, “All-optical microwave photonic single-passband filter based on polarization control through stimulated Brillouin scattering,” IEEE Photonics J. 5(4), 5501411 (2013).
    [Crossref]
  12. D. Marpaung, B. Morrison, M. Pagani, R. Pant, D. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Low-power, chip-based stimulated Brillouin scattering microwave photonic filter with ultrahigh selectivity,” Optica 2(2), 76–83 (2015).
    [Crossref]
  13. X. Yi and R. A. Minasian, “Microwave photonic filter with single bandpass response,” Electron. Lett. 45(7), 362–363 (2009).
    [Crossref]
  14. W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
    [Crossref]
  15. E. Xu and J. Yao, “Frequency-and notch-depth-tunable single-notch microwave photonic filter,” IEEE Photonics Technol. Lett. 27(19), 2063–2066 (2015).
    [Crossref]
  16. J. Palací, G. E. Villanueva, J. V. Galán, J. Martí, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
    [Crossref]
  17. T. Chen, X. Yi, L. Li, and R. Minasian, “Single passband microwave photonic filter with wideband tunability and adjustable bandwidth,” Opt. Lett. 37(22), 4699–4701 (2012).
    [Crossref] [PubMed]
  18. J. Ge, H. Feng, G. Scott, and M. P. Fok, “High-speed tunable microwave photonic notch filter based on phase modulator incorporated Lyot filter,” Opt. Lett. 40(1), 48–51 (2015).
    [Crossref] [PubMed]
  19. J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
    [Crossref]
  20. W. Li, J. G. Liu, and N. H. Zhu, “A widely and continuously tunable frequency doubling optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(13), 1461–1464 (2015).
    [Crossref]
  21. S. Shahnia, M. Pagani, B. Morrison, B. J. Eggleton, and D. Marpaung, “Independent manipulation of the phase and amplitude of optical sidebands in a highly-stable RF photonic filter,” Opt. Express 23(18), 23278–23286 (2015).
    [Crossref] [PubMed]

2015 (5)

2013 (2)

W. Li, L. X. Wang, and N. H. Zhu, “All-optical microwave photonic single-passband filter based on polarization control through stimulated Brillouin scattering,” IEEE Photonics J. 5(4), 5501411 (2013).
[Crossref]

Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013).
[Crossref]

2012 (3)

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

W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
[Crossref]

T. Chen, X. Yi, L. Li, and R. Minasian, “Single passband microwave photonic filter with wideband tunability and adjustable bandwidth,” Opt. Lett. 37(22), 4699–4701 (2012).
[Crossref] [PubMed]

2011 (2)

L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
[Crossref]

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(23), 1775–1777 (2011).
[Crossref]

2010 (3)

J. Palací, G. E. Villanueva, J. V. Galán, J. Martí, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (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]

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(24), 25271–25282 (2010).
[Crossref] [PubMed]

2009 (2)

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

X. Yi and R. A. Minasian, “Microwave photonic filter with single bandpass response,” Electron. Lett. 45(7), 362–363 (2009).
[Crossref]

2006 (1)

2004 (1)

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

2003 (1)

1998 (1)

X. S. Yao, “Brillouin selective sideband amplification of microwave photonic signals,” IEEE Photonics Technol. Lett. 10(1), 138–140 (1998).
[Crossref]

Bull, J. D.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Capmany, J.

Chan, E. H. W.

Chen, T.

Choi, D.

Dong, J.

Eggleton, B. J.

Fairburn, M.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Feng, H.

Ferdous, F.

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

Fok, M. P.

Galán, J. V.

J. Palací, G. E. Villanueva, J. V. Galán, J. Martí, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Ge, J.

Ghanipour, P.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Guo, Y.-H.

L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
[Crossref]

Hamidi, E.

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

Huang, T. X. H.

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]

Jaeger, N. A. F.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Jiang, H.-Y.

L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
[Crossref]

Kato, H.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Leaird, D. E.

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

Li, L.

Li, M.

W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
[Crossref]

Li, W.

W. Li, J. G. Liu, and N. H. Zhu, “A widely and continuously tunable frequency doubling optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(13), 1461–1464 (2015).
[Crossref]

W. Li, L. X. Wang, and N. H. Zhu, “All-optical microwave photonic single-passband filter based on polarization control through stimulated Brillouin scattering,” IEEE Photonics J. 5(4), 5501411 (2013).
[Crossref]

W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
[Crossref]

Li, X.

Liu, J. G.

W. Li, J. G. Liu, and N. H. Zhu, “A widely and continuously tunable frequency doubling optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(13), 1461–1464 (2015).
[Crossref]

Long, C. M.

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

Luo, B.

L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
[Crossref]

Luther-Davies, B.

Madden, S. J.

Marpaung, D.

Martí, J.

J. Palací, G. E. Villanueva, J. V. Galán, J. Martí, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Minasian, R.

Minasian, R. A.

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(23), 1775–1777 (2011).
[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]

X. Yi and R. A. Minasian, “Microwave photonic filter with single bandpass response,” Electron. Lett. 45(7), 362–363 (2009).
[Crossref]

E. H. W. Chan and R. A. Minasian, “High-resolution photonics-based interference suppression filter with wide passband,” J. Lightwave Technol. 21(12), 3144–3149 (2003).
[Crossref]

Morrison, B.

Ortega, B.

Pagani, M.

Palací, J.

J. Palací, G. E. Villanueva, J. V. Galán, J. Martí, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Pan, S.

Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013).
[Crossref]

Pan, W.

L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
[Crossref]

Pant, R.

Pastor, D.

Reid, A.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Scott, G.

Shahnia, S.

Supradeepa, V. R.

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

Vidal, B.

J. Palací, G. E. Villanueva, J. V. Galán, J. Martí, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Villanueva, G. E.

J. Palací, G. E. Villanueva, J. V. Galán, J. Martí, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Wang, L. X.

W. Li, L. X. Wang, and N. H. Zhu, “All-optical microwave photonic single-passband filter based on polarization control through stimulated Brillouin scattering,” IEEE Photonics J. 5(4), 5501411 (2013).
[Crossref]

Weiner, A. M.

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

Wu, R.

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

Xu, E.

Yan, L.-S.

L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
[Crossref]

Yao, J.

E. Xu and J. Yao, “Frequency-and notch-depth-tunable single-notch microwave photonic filter,” IEEE Photonics Technol. Lett. 27(19), 2063–2066 (2015).
[Crossref]

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

Yao, J. P.

W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
[Crossref]

Yao, X. S.

L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
[Crossref]

X. S. Yao, “Brillouin selective sideband amplification of microwave photonic signals,” IEEE Photonics Technol. Lett. 10(1), 138–140 (1998).
[Crossref]

Ye, J.

L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
[Crossref]

Yi, A.-L.

L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
[Crossref]

Yi, X.

T. Chen, X. Yi, L. Li, and R. Minasian, “Single passband microwave photonic filter with wideband tunability and adjustable bandwidth,” Opt. Lett. 37(22), 4699–4701 (2012).
[Crossref] [PubMed]

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]

X. Yi and R. A. Minasian, “Microwave photonic filter with single bandpass response,” Electron. Lett. 45(7), 362–363 (2009).
[Crossref]

Yu, Y.

Zhang, W.

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(23), 1775–1777 (2011).
[Crossref]

Zhang, X.

Zhang, Y.

Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013).
[Crossref]

Zhou, L.

Zhu, N. H.

W. Li, J. G. Liu, and N. H. Zhu, “A widely and continuously tunable frequency doubling optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(13), 1461–1464 (2015).
[Crossref]

W. Li, L. X. Wang, and N. H. Zhu, “All-optical microwave photonic single-passband filter based on polarization control through stimulated Brillouin scattering,” IEEE Photonics J. 5(4), 5501411 (2013).
[Crossref]

Electron. Lett. (1)

X. Yi and R. A. Minasian, “Microwave photonic filter with single bandpass response,” Electron. Lett. 45(7), 362–363 (2009).
[Crossref]

IEEE Photonics J. (1)

W. Li, L. X. Wang, and N. H. Zhu, “All-optical microwave photonic single-passband filter based on polarization control through stimulated Brillouin scattering,” IEEE Photonics J. 5(4), 5501411 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (7)

X. S. Yao, “Brillouin selective sideband amplification of microwave photonic signals,” IEEE Photonics Technol. Lett. 10(1), 138–140 (1998).
[Crossref]

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(23), 1775–1777 (2011).
[Crossref]

E. Xu and J. Yao, “Frequency-and notch-depth-tunable single-notch microwave photonic filter,” IEEE Photonics Technol. Lett. 27(19), 2063–2066 (2015).
[Crossref]

J. Palací, G. E. Villanueva, J. V. Galán, J. Martí, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

L.-S. Yan, J. Ye, H.-Y. Jiang, W. Pan, B. Luo, A.-L. Yi, Y.-H. Guo, and X. S. Yao, “A photonic comb filter with independently and digitally tunable bandwidth and frequency spacing,” IEEE Photonics Technol. Lett. 23(13), 857–859 (2011).
[Crossref]

Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013).
[Crossref]

W. Li, J. G. Liu, and N. H. Zhu, “A widely and continuously tunable frequency doubling optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(13), 1461–1464 (2015).
[Crossref]

IEEE Trans. Microw. Theory Tech. (2)

W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
[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]

J. Lightwave Technol. (3)

Nat. Photonics (1)

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

Opt. Express (2)

Opt. Lett. (2)

Optica (1)

Proc. SPIE (1)

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Schematic diagram of (a) the proposed MPF and (b)-(d) the principle behind the MPF.
Fig. 2
Fig. 2 Measured optical spectra of (a) orthogonally polarized optical carriers and (b) the modulated optical carriers.
Fig. 3
Fig. 3 Measured (a) magnitude and (b) phase responses of the bandpass, lowpass, and bandstop filters.
Fig. 4
Fig. 4 The transform from bandpass to bandstop MPF when the power of TLS2 was tuned.
Fig. 5
Fig. 5 The center frequency tunability of (a) the bandpass and (b) the bandstop MPF by adjusting the bandwidth of the OBPF and the wavelength of TLS2.
Fig. 6
Fig. 6 The bandwidth tunability of (a) the bandpass and (b) the bandstop MPF by adjusting the wavelength of the TLS1.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

E P B C ( t ) = [ E x ( t ) E y ( t ) ] = [ E 1 exp ( j ω 1 t ) E 2 exp ( j ω 2 t ) ]
E P o l M ( t ) = [ E x ( t ) E y ( t ) ] = [ E 1 exp ( j ω 1 t + j β sin ( ω m t ) ) E 2 exp ( j ω 2 t j β sin ( ω m t ) ) ]
E P o l M ( t ) = [ E x ( t ) E y ( t ) ] = [ E 1 exp ( j ω 1 t ) [ J 0 + J 1 exp ( j ω m t + j π ) + J 1 exp ( j ω m t ) ] E 2 exp ( j ω 2 t ) [ J 0 + J 1 exp ( j ω m t ) + J 1 exp ( j ω m t + j π ) ] ]
i x ( t ) = R E x ( t ) E x * ( t ) = { R E 1 2 [ J 0 2 + J 1 2 + 2 J 0 J 1 cos ( ω m t ) ] , f o r f B 2 f C < ω m 2 π < f B 2 + f C 0 , e l s e .
i y ( t ) = R E y ( t ) E y * ( t ) = { R E 2 2 [ J 0 2 + J 1 2 + 2 J 0 J 1 cos ( ω m t + π ) ] , f o r 0 < ω m 2 π < f B 0 , e l s e .
i ( t ) = i x ( t ) + i y ( t ) = { R E 2 2 [ J 0 2 + J 1 2 + 2 J 0 J 1 cos ( ω m t + π ) ] , f o r 0 < ω m 2 π < f B 2 f C , f B 2 + f C < ω m 2 π < f B R E 1 2 [ J 0 2 + J 1 2 + 2 J 0 J 1 cos ( ω m t ) ] R E 2 2 [ J 0 2 + J 1 2 + 2 J 0 J 1 cos ( ω m t ) ] , f o r f B 2 f C < ω m 2 π < f B 2 + f C .

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