Abstract

We demonstrate a tunable temporal photonic fractional-order differentiator based on a Mach–Zenhder interferometer (MZI) coupled microring. An MZI structure is designed to replace conventional coupling structure of straight waveguide and ring waveguide. When the refractive index of one arm of the MZI is adjusted by thermo-optic effect, the coupling coefficient between the straight waveguide and the microring can be changed and a tunable photonic differentiator is implemented. In the experiment, the device presents a tunable differentiation order range from 0.25 to 1.75, which is the largest tuning range of a tunable photonic differentiator using a single microring resonator to the best of our knowledge.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

With the rapid development of optical communication technology, all-optical signal processing technology has been implemented to improve the processing speed of the optical signal. The photonic temporal differentiator is one of the basic and critical devices for all-optical signal processing that provides temporal differentiation of the complex envelope of an arbitrary input optical pulse. The photonic temporal differentiator has received great attention due to its many advantages and wide applications in optical processing and computing [1, 2], pulse coding [3], optical metrology [4], pulse shaping [5], and optical sensing [6].

In recent years, lots of schemes have been proposed to perform photonic temporal differentiator, including the use of the long-period fiber grating [7], the phase-shifted fiber Bragg grating (FBG) [8], the semiconductor optical amplifier (SOA) [9], the Mach–Zehnder interferometer (MZI) [10], the silicon microring resonator (MRR) [11, 12], the directional coupler [13], and integrated Kerr optical combs [14]. In addition to regular integer-order differentiation, fractional-order differentiator is also implemented to approach complex mechanics and physics problems, which is the generalization of integer-order differentiator [15]. Cuadrado-Laborde et al. presented a fractional differentiator by a photonic MZI and an asymmetrical phase shifted-FBG in reflection, respectively [15, 16]. However, the differentiation order of these temporal differentiators is fixed and not tunable. Actually, the tunable range of the differentiation order is an important performance parameter for fractional-order differentiator. A titled FBG and optically pumping a tilted FBG were demonstrated further, in which the fractional order is tuned from 0.81 to 1.42 and from 1.25 to 1.72, respectively [17, 18]. Compared with fiber based schemes, silicon-based integrated waveguide is attracting more and more interests for its distinct advantages of small size, well integrated characteristic and compatibility with complementary metal-oxide semiconductor (CMOS) standard technology. Shahoei et al. proposed a tunable fractional-order differentiator based on a silicon ring resonator with a multimode interference (MMI) coupler for the first time [19]. The differentiation order ranges from 0.37 to 1.5, which, nevertheless, is highly polarization dependent. Then, we also demonstrated tunable fractional-order differentiators using electrically tuned MZI and MRR, where the tuning range was varied from 0.83 to 1.03 and 0.58 to 0.97 respectively [20, 21]. It was impractical if the input laser wavelength was fixed since the operating wavelength will be varied with the voltage applied on the MRR or MZI. Next, Zhang et al. presented a tunable multichannel fractional-order temporal differentiator based on a symmetric MZI incorporating cascaded MRRs [22]. And a photonic fractional-order temporal differentiator based on the inverse Raman scattering effect in an MRR is proposed by Jin et al. [23]. By changing the pump power, the tuning range of fractional order changes from 0.97 to 1.27 and 0.3 to 1.6, respectively. Obviously, both of these schemes required high power pump light to produce nonlinear effects. Furthermore, large-range tunable fractional-order differentiator based on cascaded MRRs was employed by Yang et al. with a fractional order ranging from 0.57 to 2 [24]. But alignment of the resonant wavelength of each MRR was very complicated. So far, a widely tunable fractional-order differentiator with robust and convenient operation has not been reported yet.

In this paper, we present a tunable fractional-order differentiator based on an MZI coupled microring resonator (MZI-MRR). An MZI structure replaces the coupling structure of straight waveguide and ring waveguide. When the arc-shaped waveguide in one arm of the MZI is heated, the effective coupling coefficient of the straight waveguide and ring waveguide can be changed. Thus, the coupling state of MRR can be varied and a tunable fractional-order differentiator can be realized. Compared to previous tuning schemes such as electrical tuning or optical pumping, thermal tuning provides enough phase shift for fully control of fractional order range and the fabrication process of thermal heaters is much easier and cheaper than ion implantation based electrical tuning schemes [25]. In the experiment, we demonstrate a tunable fractional-order differentiator with differentiation order ranging from 0.25 to 1.75, which is the largest tuning range of a tunable photonic differentiator using a single MRR to our knowledge. In addition, we designed two heaters for the arc-shaped waveguide and ring waveguide, so both the differentiation order and the operating wavelength are flexible to adjust.

2. Operation principle and device fabrication

In general, a temporal differentiator, which can be considered as an optical filter, performs the time derivate of the envelope of an optical signal [14]. The spectral transfer function of the temporal differentiator is given by

H(ω)=[j(ωω0)]N,
where N is the differentiation order, N can be either a fraction or an integer, ω is the optical frequency, ω0 is the optical carrier frequency. We can know from Eq. (1) that an Nth order temporal differentiator performed by an optical filter should have a magnitude response of |(ω-ω0)|Nand a phase shift of at ω0.

Figure 1(a) shows the structure of an add–drop MRR, whose frequency response can be expressed as

t=EoutEin=r1αr2ejϕ1αr1r2ejϕ,
where r1,κ1=1r12and r2, κ2=1r22are the transmission coefficient and the coupling coefficient of the through port and the drop port respectively, α is the round-trip amplitude transmission coefficient, and ϕ is the round-trip phase accumulation. By changing the coupling regime of MRR, the phase shift at the resonance frequency ranges from 0 to 2π and a fractional-order differentiator with 0<N<2 will be implemented. Figure 1(b) shows the schematic layout of our MZI-MRR. An arc-shaped arm has replaced the straight through port waveguide and is coupled to the MRR at two coupling regions. The upper half part of this structure can be considered as an asymmetrical MZI with arc-shape arm and ring arm. The arc-shaped arm has been reported to be effective in changing the coupling coefficient between the through port and MRR [25–27]. We can write the effective coupling coefficient κ1 of the through port as [28]
κ12=κ02(1κ02)×(tb+tr2tbtrcosφ),
whereκ0is the coupling coefficient of the coupling region, tb, tr and φb, φr are the transmission coefficient and phase of the arc-shape arm and ring arm respectively, φ=φbφr=2πλneff(LbLr)is the relative phase, λ is the wavelength of light,neffis the effective index,Lb,Lrare the lengths of the two interfering arms. Substituting Eq. (3) into Eq. (2), we can calculate the phase shift of MRR dependent on the relative phase φ. Besides, if the coupling coefficient of the drop port κ2 is designed to be tunable in our structure, it can serve as a differential-equation solver, which indicates the internal relations between the two functions [29].

 

Fig. 1 Schematics of (a) an add–drop MRR and (b) an MZI coupled microring resonator.

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By changing the relative phase φ, a tunable fractional-order differentiator can be implemented. Assume thattb=0.98,tr=0.992,κ0=0.25andκ2=0.28, the intensity transmission and phase responses of the MZI-MRR are presented in Figs. 2(a) and 2(b). The blue solid lines in Figs. 2(a) and 2(b) represent the initial intensity transmission and phase responses of the simulated MZI-MRR. When the relative phase difference Δφ=φiφ1varies from 0.165 rad to 0.437 rad, the phase shift at the resonance wavelength is changed from less than π to greater than π, as shown in Fig. 2(b). These phase shift amounts represent a tunable fractional order range between 0 and 2. At the same time, the resonant wavelength is always fixed at about 1558.2nm but with a flexible resonant notch depth, as indicated in Fig. 2(a).

 

Fig. 2 (a) Intensity transmission and (b) phase response of the MZI-MRR with varying values of Δφ.

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When the amplitude and phase responses of the MZI-MRR is in the critical coupling regime, it can be regarded as a first-order differentiator, as shown in Figs. 3(a) and 3(b). From Fig. 3(a), the magnitude response of MZI-MRR near the resonant frequency is in good agreement with the ideal differentiator. This means the temporal differentiator using an MZI-MRR has a device-operation bandwidth (DOB). For the accuracy of a photonic differentiator, phase response is more sensitive than amplitude response [19]. Figure 3(b) shows the phase response of the MZI-MRR, which has an exact π phase shift at resonant frequency and an additional linear phase response. It means the critically-coupled MZI-MRR can perform as a first-order differentiator with a time delay [30]. Figure 3(c) shows the first order differentiator waveform of a 50 ps-pulsewidth Gaussian pulse employing MZI-MRR, which is consistent to the ideal differentiator. The deviation is defined as the mean absolute deviation of the simulated MZI-MRR differentiation results from the ideal results [31]. We calculate the computing deviation of the output waveform from the ideal differentiator when changing the input pulsewidth, as described in Fig. 3(d). One can see that the deviation decreases as the pulsewidth of the input waveform is increased. Figure 3(e) shows deviations of the simulated output response with different differentiation order of a 50 ps-pulsewidth. It can be found that the experimental results are in good agreement with the ideal differential results. Figure 4(a) illustrates the schematic concept diagram of the thermally tuned MZI-MRR. An arc-shaped waveguide, whose radius r1 is 32 μm, is designed to be a thermo-optical tunable region. So we can achieve the purpose of tuning the differentiation order by changing the refractive index of the arc-shaped waveguide. This waveguide is connected with the straight waveguides by bending waveguides with the radius r2 of 8 μm. There is a gap g1 of 550 nm between the arc-shape waveguide and the MRR with a radius r3 of 8 μm, and the coupling gap between the MRR and the drop waveguide is about 270 nm. The width and height of these waveguides are 500 nm and 220nm, respectively. We design and fabricate the device on a commercial silicon-on-insulator (SOI) wafer with 3 μm buried oxide and 220 nm top silicon. The waveguides and rings are first fabricated with electron beam lithography and inductively coupled plasma (ICP) etching, as shown in Fig. 4(b). Subsequently, 700nm thick silica is deposited as upper cladding using plasma-enhanced chemical vapor deposition (PECVD). Finally, 100 nm thick Ti heaters and Au electrical contact pads are deposited via electron beam evaporation and lift-off process. We designed two heaters on top of the arc-shaped waveguide and the ring waveguide, respectively. The design of heater for ring waveguide is to tune the operating wavelength of the MZI-MRR. Figures 4(c) and 4(d) show the microscope images of the fabricated MZI-MRR and the zoom-in ring region.

 

Fig. 3 (a) Magnitude response and (b) phase response of the MZI-MRR (dot lines) and the ideal first-order differentiator (solid lines). (c) Simulated the output response from the MZI-MRR (dot line) and the ideal first-order differentiator (solid line) of a 50 ps-pulsewidth Gaussian pulse. (d) Deviations of the simulated output response with different pulsewidth. (e) Deviations of the simulated output response with different differentiation order of a 50 ps-pulsewidth.

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Fig. 4 (a) Schematic concept diagram of the MZI-MRR. (b) Microscope image of the waveguides and rings after ICP etching. (c) Microscope image of fabricated MZI-MRR and (d) the zoom-in ring region.

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3. Experimental result

We first measure the transmission spectra of fabricated MZI-MRR at different heating powers applied to the heater of the arc-shaped waveguide, as shown in Fig. 5. The 3-dB bandwidth and resonant wavelength of the fabricated device is about 0.09 nm and about 1558.47 nm, respectively. The Q-factor and propagation loss of the microring we have calculated is about 17000 and 2.5 dB/cm, respectively. The heating power P applied to the heater is varied from 2 mW to 6 mW. The phase response is not measured due to our hardware restraint. In Fig. 5, we can see that the depth of the resonant notch is varied at different applied heating powers, which implies that the coupling regime of MZI-MRR is changed in this process. Since the refractive index of the MRR is not affected by the heater of the arc-shaped waveguide, the resonant wavelength is nearly constant when heating power is varied. In the measurement, the deepest resonant notch appears at the applied heating power of 3.92 mW (the red line), exhibiting a critical coupling regime. And a first-order differentiator can be implemented.

 

Fig. 5 Measured transmission spectra of the fabricated MZI-MRR at different heating powers applied to the heater above the arc-shaped waveguide.

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Figure 6 shows the experimental setup for the tunable fractional-order differentiator using an MZI-MRR. The continuous wave (CW) emitted by the tunable laser diode (TLD) is modulated by two cascaded Mach-Zehnder modulators (MZMs). The central wavelength of the CW is strictly aligned with the MRR resonant wavelength. MZMs are driven by a bit pattern generator (BPG). Because of the polarization sensitive characteristics of the MZM, a polarization controller (PC1) is placed before the MZM to control the polarization of input light. The output signal of MZM1 is amplified by an erbium doped fiber amplifier (EDFA1). An optical tunable delay line (OTDL) is used to align the two MZMs. The power of optical signal is adjusted by EDFA2 and an attenuator (ATT1). By adjusting PC3, the polarization state of optical signal can be TE mode. Then, the optical signal is coupled into the chip using a coupling grating. The total coupling loss of the input and the output coupling gratings is measured to be 11.7 dB. In this experiment, a direct-current heating power is applied to the heater of the arc-shaped waveguide to tune the differentiation order. The output temporal differential signal of the chip is finally analyzed by an oscilloscope and a spectrograph.

 

Fig. 6 Experimental setup for the fractional-order differentiator based on an MZI-MRR. TLD: Tunable laser diode. PC: Polarization controller. MZM: Mach-Zehnder modulator. EA: Electronic amplifier. BPG: Bit pattern generator. EDFA: Erbium doped fiber amplifier. OTDL: Optical tunable delay line. ATT: Attenuator. OC: Optical coupler.

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As shown in Fig. 7(a), the input Gaussian pulse with a pulsewidth of 56 ps is generated by two cascaded MZMs driven with a BPG. Since the fractional differentiation is indeed affected by the degree of chirp of the incoming light signal, we calculate the time-bandwidth product (TBWP) of the input Gaussian pulse, which is about 0.45 [32]. It validates that the pulse is unchirped. The experimental pulse waveform (blue dot line) agrees well with the theoretical Gaussian pulse (red solid line). Figures 7(b)-7(j) present the experimental differential signals (blue dot lines) at different heating powers and the ideal differential waveforms (red solid lines). When the applied heating power varies from 2.46 mW to 5.91 mW, the differentiation order N of experimental result ranges from 1.75 to 0.25. Although we are not able to measure the phase shift of this device, the experimental results show that the critical coupling MZI-MRR is performed when the applied heating power is 3.92 mW. The fabricated MZI-MRR works in the over-coupling regime with the heating power ranging from 2.46 mW to 3.47 mW. When we apply heating power from 4.24 mW to 5.91 mW, an under-coupling MZI-MRR is demonstrated. We also observe a distinct differentiator deviation when the applied heating power is less than 2.46 mW. The reason lies in that a shallow notch depth of MZI-MRR exists in this case, impairing the accuracy of the differentiator. Therefore, although a fractional-order differentiator with tunable differential order covering from 0 to 2 will be performed in theory, the MZI-MRR should work in the vicinity of critical coupling to ensure high accuracy of differential result. In fact, the device parameters, such as waveguide gap and length of arc waveguide, can be optimized carefully, so that the MZI-MRR can work initially near the critical coupling without heating power applied. As a result, the applied heating power can be greatly reduced. The averaged error is shown in Fig. 8, which is defined as the mean absolute deviation of experimental differentiation results from the simulated one. The average errors of these differential results fluctuate from 2% to 6% (2.03%, 2.73%, 2.93%, 3.46%, 3.53%, 3.75%, 3.73%, 3.43%, 5.72%). All the average errors of the differential result are less than 6%, which means the experimental results are in good agreement with the ideal differential results. Besides, we have experimentally examined the response speed of the device by measuring the 10%-to-90% rising and decaying times of output signal, and the calculated modulation rate based on this is about 2.7 kHz, which is limited by the intrinsic characteristic of thermo-optic effect. Since our device is not aimed for high-speed differentiation fraction tuning, this feature may be not considered as one of our main concerns [33].

 

Fig. 7 (a) The experimental input pulse (blue dot line) and the simulated Gaussian pulse with a pulsewidth of 56 ps (red solid line). The measured differentiated output pulses at applied heating power of (b) 2.46 mW, (c) 3.12 mW, (d) 3.47 mW, (e) 3.92 mW, (f) 4.24 mW, (g) 4.65 mW, (h) 5 mW, (i) 5.54 mW, (j) 5.91 mW. Sim: simulation, Exp: experiment.

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Fig. 8 Averaged error of the fabricated MZI-MRR as a function of differentiation order N.

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Figure 9 shows the spectral evolution at the critical coupling condition when the applied heating power is 3.92 mW. The input Gaussian pulse has a pulsewidth of 56 ps and the differentiation order is N = 1. The wavelength of TLD is set at 1558.47 nm, aligned to the resonant wavelength of MZI-MRR. We can see a notch at the central resonance wavelength in the output spectrum, which means the incident light frequency has been well aligned with the MRR resonant frequency. Because of the presence of the resonant notch, the pulse signal has energy loss after propagating in the MZI-MRR. Figure 3(d) shows that the input pulse signal with smaller bandwidth has lower differential error. However, smaller bandwidth also induces a low energy efficiency (EE), defined as the radio of the output power to the input power. So there is a tradeoff between higher energy efficiency and lower differential error [34].

 

Fig. 9 Measured spectra of the critical coupling MZI-MRR, the input Gaussian pulse with a pulsewidth of 56ps and the output pulse.

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4. Conclusions

In summary, we designed and fabricated a widely tunable fractional-order photonic differentiator using an MZI-MRR. The refractive index of arc-shape waveguide can be changed by thermo-optic effect. Thus, the fractional order of the differentiator will be tuned with the effective coupling coefficient of the through port. In order to ensure the accuracy of the fractional-order differentiator, the applied heating power is tuned from 2.46 mW to 5.91 mW to make the MZI-MRR work in the vicinity of the critical coupling condition. In the experiment, the operating wavelength of the differentiator is always aligned with the resonant wavelength of the MZI-MRR. The proposed differentiator presents a tunable differentiation order range of 0.25 to 1.75, and the average errors are less than 6%. Besides, the operating wavelength of the MZI-MRR can be tuned by the heater of ring waveguide, making the system robust.

Funding

The work is supported by the National Natural Science Foundation of China (No. 61475052, 61622502).

References and links

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2. Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express 14(26), 12670–12678 (2006). [CrossRef]   [PubMed]  

3. N. Ngo, S. Yu, S. Tjin, and C. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1–3), 115–129 (2004). [CrossRef]  

4. F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008). [CrossRef]   [PubMed]  

5. J. A. N. da Silva and M. L. R. de Campos, “Spectrally efficient UWB pulse shaping with application in orthogonal PSM,” IEEE Trans. Commun. 55(2), 313–322 (2007). [CrossRef]  

6. J. Xu, X. Zhang, J. Dong, D. Liu, and D. Huang, “High-speed all-optical differentiator based on a semiconductor optical amplifier and an optical filter,” Opt. Lett. 32(13), 1872–1874 (2007). [CrossRef]   [PubMed]  

7. J. Azaña and M. Kulishov, “All-fibre ultrafast optical differentiator based on π phase-shifted long-period grating,” Electron. Lett. 41(25), 1368–1369 (2005). [CrossRef]  

8. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Express 15(2), 371–381 (2007). [CrossRef]   [PubMed]  

9. Z. Li and C. Wu, “All-optical differentiator and high-speed pulse generation based on cross-polarization modulation in a semiconductor optical amplifier,” Opt. Lett. 34(6), 830–832 (2009). [CrossRef]   [PubMed]  

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11. Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012). [CrossRef]  

12. J. Dong, A. Zheng, D. Gao, S. Liao, L. Lei, D. Huang, and X. Zhang, “High-order photonic differentiator employing on-chip cascaded microring resonators,” Opt. Lett. 38(5), 628–630 (2013). [CrossRef]   [PubMed]  

13. T. L. Huang, A. L. Zheng, J. J. Dong, D. S. Gao, and X. L. Zhang, “Terahertz-bandwidth photonic temporal differentiator based on a silicon-on-isolator directional coupler,” Opt. Lett. 40(23), 5614–5617 (2015). [CrossRef]   [PubMed]  

14. X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017). [CrossRef]  

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32. L. Poveda-Wong, A. Carrascosa, C. Cuadrado-Laborde, J. L. Cruz, and M. V. Andrés, “Experimental demonstration of fractional order differentiation using a long-period grating-based in-fiber modal interferometer,” Opt. Commun. 380, 35–40 (2016). [CrossRef]  

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References

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  1. F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. 32(22), 3364–3366 (2007).
    [Crossref] [PubMed]
  2. Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express 14(26), 12670–12678 (2006).
    [Crossref] [PubMed]
  3. N. Ngo, S. Yu, S. Tjin, and C. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1–3), 115–129 (2004).
    [Crossref]
  4. F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008).
    [Crossref] [PubMed]
  5. J. A. N. da Silva and M. L. R. de Campos, “Spectrally efficient UWB pulse shaping with application in orthogonal PSM,” IEEE Trans. Commun. 55(2), 313–322 (2007).
    [Crossref]
  6. J. Xu, X. Zhang, J. Dong, D. Liu, and D. Huang, “High-speed all-optical differentiator based on a semiconductor optical amplifier and an optical filter,” Opt. Lett. 32(13), 1872–1874 (2007).
    [Crossref] [PubMed]
  7. J. Azaña and M. Kulishov, “All-fibre ultrafast optical differentiator based on π phase-shifted long-period grating,” Electron. Lett. 41(25), 1368–1369 (2005).
    [Crossref]
  8. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Express 15(2), 371–381 (2007).
    [Crossref] [PubMed]
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  10. J. Dong, A. Zheng, D. Gao, L. Lei, D. Huang, and X. Zhang, “Compact, flexible and versatile photonic differentiator using silicon Mach-Zehnder interferometers,” Opt. Express 21(6), 7014–7024 (2013).
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  11. Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012).
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  12. J. Dong, A. Zheng, D. Gao, S. Liao, L. Lei, D. Huang, and X. Zhang, “High-order photonic differentiator employing on-chip cascaded microring resonators,” Opt. Lett. 38(5), 628–630 (2013).
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  18. H. Shahoei, J. Albert, and J. Yao, “Fractional Order Temporal Differentiator by Optically Pumping a Tilted Fiber Bragg Grating,” IEEE Photonics Technol. Lett. 24(9), 730–732 (2012).
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  19. H. Shahoei, D. Xu, J. H. Schmid, and J. Yao, “Photonic Fractional-Order Differentiator Using an SOI Microring Resonator With an MMI Coupler,” IEEE Photonics Technol. Lett. 25(15), 1408–1411 (2013).
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  31. J. Dong, B. Luo, Y. Zhang, D. Huang, and X. Zhang, “Reconfigurable photonic differentiators based on all-optical phase modulation and linear filtering,” Opt. Commun. 284(24), 5792–5797 (2011).
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  34. S. Yan, Y. Zhang, J. Dong, A. Zheng, S. Liao, H. Zhou, Z. Wu, J. Xia, and X. Zhang, “Operation bandwidth optimization of photonic differentiators,” Opt. Express 23(15), 18925–18936 (2015).
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2017 (2)

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

W. Liu, W. Zhang, and J. Yao, “Silicon-Based Integrated Tunable Fractional Order Photonic Temporal Differentiators,” J. Lightwave Technol. 35(12), 2487– 2493 (2017).

2016 (2)

L. Poveda-Wong, A. Carrascosa, C. Cuadrado-Laborde, J. L. Cruz, and M. V. Andrés, “Experimental demonstration of fractional order differentiation using a long-period grating-based in-fiber modal interferometer,” Opt. Commun. 380, 35–40 (2016).
[Crossref]

T. Yang, S. Liao, L. Liu, and J. Dong, “Large-range tunable fractional-order differentiator based on cascaded microring resonators,” Front. Optoelectron. 9(3), 1–7 (2016).
[Crossref]

2015 (5)

2014 (3)

2013 (3)

2012 (2)

Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012).
[Crossref]

H. Shahoei, J. Albert, and J. Yao, “Fractional Order Temporal Differentiator by Optically Pumping a Tilted Fiber Bragg Grating,” IEEE Photonics Technol. Lett. 24(9), 730–732 (2012).
[Crossref]

2011 (2)

J. Dong, B. Luo, Y. Zhang, D. Huang, and X. Zhang, “Reconfigurable photonic differentiators based on all-optical phase modulation and linear filtering,” Opt. Commun. 284(24), 5792–5797 (2011).
[Crossref]

M. Li, L. Shao, J. Albert, and J. Yao, “Continuously tunable photonic fractional temporal differentiator based on tilted fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(4), 251–253 (2011).
[Crossref]

2010 (2)

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[Crossref] [PubMed]

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

2009 (2)

2008 (2)

2007 (5)

2006 (1)

2005 (1)

J. Azaña and M. Kulishov, “All-fibre ultrafast optical differentiator based on π phase-shifted long-period grating,” Electron. Lett. 41(25), 1368–1369 (2005).
[Crossref]

2004 (1)

N. Ngo, S. Yu, S. Tjin, and C. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1–3), 115–129 (2004).
[Crossref]

Albert, J.

H. Shahoei, J. Albert, and J. Yao, “Fractional Order Temporal Differentiator by Optically Pumping a Tilted Fiber Bragg Grating,” IEEE Photonics Technol. Lett. 24(9), 730–732 (2012).
[Crossref]

M. Li, L. Shao, J. Albert, and J. Yao, “Continuously tunable photonic fractional temporal differentiator based on tilted fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(4), 251–253 (2011).
[Crossref]

Andrés, M. V.

L. Poveda-Wong, A. Carrascosa, C. Cuadrado-Laborde, J. L. Cruz, and M. V. Andrés, “Experimental demonstration of fractional order differentiation using a long-period grating-based in-fiber modal interferometer,” Opt. Commun. 380, 35–40 (2016).
[Crossref]

C. Cuadrado-Laborde and M. V. Andrés, “In-fiber all-optical fractional differentiator,” Opt. Lett. 34(6), 833–835 (2009).
[Crossref] [PubMed]

Azaña, J.

Berger, N. K.

Cao, P.

Carrascosa, A.

L. Poveda-Wong, A. Carrascosa, C. Cuadrado-Laborde, J. L. Cruz, and M. V. Andrés, “Experimental demonstration of fractional order differentiation using a long-period grating-based in-fiber modal interferometer,” Opt. Commun. 380, 35–40 (2016).
[Crossref]

Chen, J.

Chen, L.

Chu, S. T.

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

Chu, T.

Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012).
[Crossref]

Cruz, J. L.

L. Poveda-Wong, A. Carrascosa, C. Cuadrado-Laborde, J. L. Cruz, and M. V. Andrés, “Experimental demonstration of fractional order differentiation using a long-period grating-based in-fiber modal interferometer,” Opt. Commun. 380, 35–40 (2016).
[Crossref]

Cuadrado-Laborde, C.

L. Poveda-Wong, A. Carrascosa, C. Cuadrado-Laborde, J. L. Cruz, and M. V. Andrés, “Experimental demonstration of fractional order differentiation using a long-period grating-based in-fiber modal interferometer,” Opt. Commun. 380, 35–40 (2016).
[Crossref]

C. Cuadrado-Laborde and M. V. Andrés, “In-fiber all-optical fractional differentiator,” Opt. Lett. 34(6), 833–835 (2009).
[Crossref] [PubMed]

C. Cuadrado-Laborde, “All-optical ultrafast fractional differentiator,” Opt. Quantum Electron. 40(13), 983–990 (2008).
[Crossref]

da Silva, J. A. N.

J. A. N. da Silva and M. L. R. de Campos, “Spectrally efficient UWB pulse shaping with application in orthogonal PSM,” IEEE Trans. Commun. 55(2), 313–322 (2007).
[Crossref]

de Campos, M. L. R.

J. A. N. da Silva and M. L. R. de Campos, “Spectrally efficient UWB pulse shaping with application in orthogonal PSM,” IEEE Trans. Commun. 55(2), 313–322 (2007).
[Crossref]

Dong, J.

T. Yang, S. Liao, L. Liu, and J. Dong, “Large-range tunable fractional-order differentiator based on cascaded microring resonators,” Front. Optoelectron. 9(3), 1–7 (2016).
[Crossref]

S. Yan, Y. Zhang, J. Dong, A. Zheng, S. Liao, H. Zhou, Z. Wu, J. Xia, and X. Zhang, “Operation bandwidth optimization of photonic differentiators,” Opt. Express 23(15), 18925–18936 (2015).
[Crossref] [PubMed]

A. Zheng, J. Dong, L. Zhou, X. Xiao, Q. Yang, X. Zhang, and J. Chen, “Fractional-order photonic differentiator using an on-chip microring resonator,” Opt. Lett. 39(21), 6355–6358 (2014).
[Crossref] [PubMed]

A. Zheng, T. Yang, X. Xiao, Q. Yang, X. Zhang, and J. Dong, “Tunable fractional-order differentiator using an electrically tuned silicon-on-isolator Mach-Zehnder interferometer,” Opt. Express 22(15), 18232–18237 (2014).
[Crossref] [PubMed]

J. Dong, A. Zheng, D. Gao, L. Lei, D. Huang, and X. Zhang, “Compact, flexible and versatile photonic differentiator using silicon Mach-Zehnder interferometers,” Opt. Express 21(6), 7014–7024 (2013).
[Crossref] [PubMed]

J. Dong, A. Zheng, D. Gao, S. Liao, L. Lei, D. Huang, and X. Zhang, “High-order photonic differentiator employing on-chip cascaded microring resonators,” Opt. Lett. 38(5), 628–630 (2013).
[Crossref] [PubMed]

J. Dong, B. Luo, Y. Zhang, D. Huang, and X. Zhang, “Reconfigurable photonic differentiators based on all-optical phase modulation and linear filtering,” Opt. Commun. 284(24), 5792–5797 (2011).
[Crossref]

J. Xu, X. Zhang, J. Dong, D. Liu, and D. Huang, “High-speed all-optical differentiator based on a semiconductor optical amplifier and an optical filter,” Opt. Lett. 32(13), 1872–1874 (2007).
[Crossref] [PubMed]

Dong, J. J.

Fan, L.

Farrell, G.

Fischer, B.

Gao, D.

Gao, D. S.

Hu, X.

Hu, Y.

Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012).
[Crossref]

Huang, D.

Huang, T. L.

Jiang, X.

Jin, B.

B. Jin, J. Yuan, C. Yu, X. Sang, Q. Wu, F. Li, K. Wang, B. Yan, G. Farrell, and P. K. A. Wai, “Tunable fractional-order photonic differentiator based on the inverse Raman scattering in a silicon microring resonator,” Opt. Express 23(9), 11141–11151 (2015).
[Crossref] [PubMed]

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Kam, C.

N. Ngo, S. Yu, S. Tjin, and C. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1–3), 115–129 (2004).
[Crossref]

Khan, M. H.

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[Crossref] [PubMed]

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Kulishov, M.

Leaird, D. E.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Lei, L.

Levit, B.

Li, F.

Li, M.

M. Li, L. Shao, J. Albert, and J. Yao, “Continuously tunable photonic fractional temporal differentiator based on tilted fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(4), 251–253 (2011).
[Crossref]

Li, W.

Li, Y.

Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012).
[Crossref]

Li, Z.

Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012).
[Crossref]

Z. Li and C. Wu, “All-optical differentiator and high-speed pulse generation based on cross-polarization modulation in a semiconductor optical amplifier,” Opt. Lett. 34(6), 830–832 (2009).
[Crossref] [PubMed]

Liao, S.

Lipson, M.

Little, B. E.

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

Liu, D.

Liu, F.

Liu, L.

T. Yang, S. Liao, L. Liu, and J. Dong, “Large-range tunable fractional-order differentiator based on cascaded microring resonators,” Front. Optoelectron. 9(3), 1–7 (2016).
[Crossref]

Liu, W.

Lu, C.

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Luo, B.

J. Dong, B. Luo, Y. Zhang, D. Huang, and X. Zhang, “Reconfigurable photonic differentiators based on all-optical phase modulation and linear filtering,” Opt. Commun. 284(24), 5792–5797 (2011).
[Crossref]

Mitchell, A.

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

Morandotti, R.

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

Moss, D. J.

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

Ngo, N.

N. Ngo, S. Yu, S. Tjin, and C. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1–3), 115–129 (2004).
[Crossref]

Nguyen, T. G.

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

Ouyang, J.

Pan, T.

Park, Y.

Plant, D. V.

Poveda-Wong, L.

L. Poveda-Wong, A. Carrascosa, C. Cuadrado-Laborde, J. L. Cruz, and M. V. Andrés, “Experimental demonstration of fractional order differentiation using a long-period grating-based in-fiber modal interferometer,” Opt. Commun. 380, 35–40 (2016).
[Crossref]

Qi, M.

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[Crossref] [PubMed]

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Qiang, L.

Qiu, C.

Qiu, M.

Sang, X.

B. Jin, J. Yuan, C. Yu, X. Sang, Q. Wu, F. Li, K. Wang, B. Yan, G. Farrell, and P. K. A. Wai, “Tunable fractional-order photonic differentiator based on the inverse Raman scattering in a silicon microring resonator,” Opt. Express 23(9), 11141–11151 (2015).
[Crossref] [PubMed]

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Schmid, J. H.

H. Shahoei, D. Xu, J. H. Schmid, and J. Yao, “Photonic Fractional-Order Differentiator Using an SOI Microring Resonator With an MMI Coupler,” IEEE Photonics Technol. Lett. 25(15), 1408–1411 (2013).
[Crossref]

Shahoei, H.

W. Zhang, W. Liu, W. Li, H. Shahoei, and J. Yao, “Independently Tunable Multichannel Fractional-Order Temporal Differentiator Based on a Silicon-Photonic Symmetric Mach–Zehnder Interferometer Incorporating Cascaded Microring Resonators,” J. Lightwave Technol. 33(2), 361–367 (2015).
[Crossref]

H. Shahoei, D. Xu, J. H. Schmid, and J. Yao, “Photonic Fractional-Order Differentiator Using an SOI Microring Resonator With an MMI Coupler,” IEEE Photonics Technol. Lett. 25(15), 1408–1411 (2013).
[Crossref]

H. Shahoei, J. Albert, and J. Yao, “Fractional Order Temporal Differentiator by Optically Pumping a Tilted Fiber Bragg Grating,” IEEE Photonics Technol. Lett. 24(9), 730–732 (2012).
[Crossref]

Shao, L.

M. Li, L. Shao, J. Albert, and J. Yao, “Continuously tunable photonic fractional temporal differentiator based on tilted fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(4), 251–253 (2011).
[Crossref]

Shen, H.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[Crossref] [PubMed]

Sherwood-Droz, N.

Shoeiby, M.

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

Slavík, R.

Su, Y.

Tjin, S.

N. Ngo, S. Yu, S. Tjin, and C. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1–3), 115–129 (2004).
[Crossref]

Tremblay, C.

Varghese, L. T.

Wai, P. K. A.

B. Jin, J. Yuan, C. Yu, X. Sang, Q. Wu, F. Li, K. Wang, B. Yan, G. Farrell, and P. K. A. Wai, “Tunable fractional-order photonic differentiator based on the inverse Raman scattering in a silicon microring resonator,” Opt. Express 23(9), 11141–11151 (2015).
[Crossref] [PubMed]

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Wang, K.

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

B. Jin, J. Yuan, C. Yu, X. Sang, Q. Wu, F. Li, K. Wang, B. Yan, G. Farrell, and P. K. A. Wai, “Tunable fractional-order photonic differentiator based on the inverse Raman scattering in a silicon microring resonator,” Opt. Express 23(9), 11141–11151 (2015).
[Crossref] [PubMed]

Wang, T.

Weiner, A. M.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Wu, C.

Wu, J.

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

J. Wu, P. Cao, X. Hu, X. Jiang, T. Pan, Y. Yang, C. Qiu, C. Tremblay, and Y. Su, “Compact tunable silicon photonic differential-equation solver for general linear time-invariant systems,” Opt. Express 22(21), 26254–26264 (2014).
[Crossref] [PubMed]

Wu, Q.

B. Jin, J. Yuan, C. Yu, X. Sang, Q. Wu, F. Li, K. Wang, B. Yan, G. Farrell, and P. K. A. Wai, “Tunable fractional-order photonic differentiator based on the inverse Raman scattering in a silicon microring resonator,” Opt. Express 23(9), 11141–11151 (2015).
[Crossref] [PubMed]

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Wu, Z.

Xia, J.

Xiao, S.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Xiao, X.

Xu, D.

H. Shahoei, D. Xu, J. H. Schmid, and J. Yao, “Photonic Fractional-Order Differentiator Using an SOI Microring Resonator With an MMI Coupler,” IEEE Photonics Technol. Lett. 25(15), 1408–1411 (2013).
[Crossref]

Xu, J.

Xu, X.

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

Xuan, Y.

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[Crossref] [PubMed]

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Yan, B.

B. Jin, J. Yuan, C. Yu, X. Sang, Q. Wu, F. Li, K. Wang, B. Yan, G. Farrell, and P. K. A. Wai, “Tunable fractional-order photonic differentiator based on the inverse Raman scattering in a silicon microring resonator,” Opt. Express 23(9), 11141–11151 (2015).
[Crossref] [PubMed]

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Yan, S.

Yang, Q.

Yang, T.

T. Yang, S. Liao, L. Liu, and J. Dong, “Large-range tunable fractional-order differentiator based on cascaded microring resonators,” Front. Optoelectron. 9(3), 1–7 (2016).
[Crossref]

A. Zheng, T. Yang, X. Xiao, Q. Yang, X. Zhang, and J. Dong, “Tunable fractional-order differentiator using an electrically tuned silicon-on-isolator Mach-Zehnder interferometer,” Opt. Express 22(15), 18232–18237 (2014).
[Crossref] [PubMed]

Yang, Y.

Yao, J.

W. Liu, W. Zhang, and J. Yao, “Silicon-Based Integrated Tunable Fractional Order Photonic Temporal Differentiators,” J. Lightwave Technol. 35(12), 2487– 2493 (2017).

W. Zhang, W. Liu, W. Li, H. Shahoei, and J. Yao, “Independently Tunable Multichannel Fractional-Order Temporal Differentiator Based on a Silicon-Photonic Symmetric Mach–Zehnder Interferometer Incorporating Cascaded Microring Resonators,” J. Lightwave Technol. 33(2), 361–367 (2015).
[Crossref]

H. Shahoei, D. Xu, J. H. Schmid, and J. Yao, “Photonic Fractional-Order Differentiator Using an SOI Microring Resonator With an MMI Coupler,” IEEE Photonics Technol. Lett. 25(15), 1408–1411 (2013).
[Crossref]

H. Shahoei, J. Albert, and J. Yao, “Fractional Order Temporal Differentiator by Optically Pumping a Tilted Fiber Bragg Grating,” IEEE Photonics Technol. Lett. 24(9), 730–732 (2012).
[Crossref]

M. Li, L. Shao, J. Albert, and J. Yao, “Continuously tunable photonic fractional temporal differentiator based on tilted fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(4), 251–253 (2011).
[Crossref]

Yaw Tam, H.

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Ye, T.

Yu, C.

B. Jin, J. Yuan, C. Yu, X. Sang, Q. Wu, F. Li, K. Wang, B. Yan, G. Farrell, and P. K. A. Wai, “Tunable fractional-order photonic differentiator based on the inverse Raman scattering in a silicon microring resonator,” Opt. Express 23(9), 11141–11151 (2015).
[Crossref] [PubMed]

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Yu, J.

Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012).
[Crossref]

Yu, S.

N. Ngo, S. Yu, S. Tjin, and C. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1–3), 115–129 (2004).
[Crossref]

Yu, Y.

Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012).
[Crossref]

Yuan, J.

B. Jin, J. Yuan, C. Yu, X. Sang, Q. Wu, F. Li, K. Wang, B. Yan, G. Farrell, and P. K. A. Wai, “Tunable fractional-order photonic differentiator based on the inverse Raman scattering in a silicon microring resonator,” Opt. Express 23(9), 11141–11151 (2015).
[Crossref] [PubMed]

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Zhang, L.

Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012).
[Crossref]

Zhang, W.

Zhang, X.

S. Yan, Y. Zhang, J. Dong, A. Zheng, S. Liao, H. Zhou, Z. Wu, J. Xia, and X. Zhang, “Operation bandwidth optimization of photonic differentiators,” Opt. Express 23(15), 18925–18936 (2015).
[Crossref] [PubMed]

A. Zheng, T. Yang, X. Xiao, Q. Yang, X. Zhang, and J. Dong, “Tunable fractional-order differentiator using an electrically tuned silicon-on-isolator Mach-Zehnder interferometer,” Opt. Express 22(15), 18232–18237 (2014).
[Crossref] [PubMed]

A. Zheng, J. Dong, L. Zhou, X. Xiao, Q. Yang, X. Zhang, and J. Chen, “Fractional-order photonic differentiator using an on-chip microring resonator,” Opt. Lett. 39(21), 6355–6358 (2014).
[Crossref] [PubMed]

J. Dong, A. Zheng, D. Gao, L. Lei, D. Huang, and X. Zhang, “Compact, flexible and versatile photonic differentiator using silicon Mach-Zehnder interferometers,” Opt. Express 21(6), 7014–7024 (2013).
[Crossref] [PubMed]

J. Dong, A. Zheng, D. Gao, S. Liao, L. Lei, D. Huang, and X. Zhang, “High-order photonic differentiator employing on-chip cascaded microring resonators,” Opt. Lett. 38(5), 628–630 (2013).
[Crossref] [PubMed]

J. Dong, B. Luo, Y. Zhang, D. Huang, and X. Zhang, “Reconfigurable photonic differentiators based on all-optical phase modulation and linear filtering,” Opt. Commun. 284(24), 5792–5797 (2011).
[Crossref]

J. Xu, X. Zhang, J. Dong, D. Liu, and D. Huang, “High-speed all-optical differentiator based on a semiconductor optical amplifier and an optical filter,” Opt. Lett. 32(13), 1872–1874 (2007).
[Crossref] [PubMed]

Zhang, X. L.

Zhang, Y.

S. Yan, Y. Zhang, J. Dong, A. Zheng, S. Liao, H. Zhou, Z. Wu, J. Xia, and X. Zhang, “Operation bandwidth optimization of photonic differentiators,” Opt. Express 23(15), 18925–18936 (2015).
[Crossref] [PubMed]

J. Dong, B. Luo, Y. Zhang, D. Huang, and X. Zhang, “Reconfigurable photonic differentiators based on all-optical phase modulation and linear filtering,” Opt. Commun. 284(24), 5792–5797 (2011).
[Crossref]

Zhang, Z.

Zhao, L.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[Crossref] [PubMed]

Zheng, A.

Zheng, A. L.

Zhou, G.

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Zhou, H.

Zhou, L.

Zhou, X.

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

APL Photonics (1)

X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photonics 2(9), 096104 (2017).
[Crossref]

Electron. Lett. (1)

J. Azaña and M. Kulishov, “All-fibre ultrafast optical differentiator based on π phase-shifted long-period grating,” Electron. Lett. 41(25), 1368–1369 (2005).
[Crossref]

Front. Optoelectron. (1)

T. Yang, S. Liao, L. Liu, and J. Dong, “Large-range tunable fractional-order differentiator based on cascaded microring resonators,” Front. Optoelectron. 9(3), 1–7 (2016).
[Crossref]

IEEE Photonics Technol. Lett. (3)

M. Li, L. Shao, J. Albert, and J. Yao, “Continuously tunable photonic fractional temporal differentiator based on tilted fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(4), 251–253 (2011).
[Crossref]

H. Shahoei, J. Albert, and J. Yao, “Fractional Order Temporal Differentiator by Optically Pumping a Tilted Fiber Bragg Grating,” IEEE Photonics Technol. Lett. 24(9), 730–732 (2012).
[Crossref]

H. Shahoei, D. Xu, J. H. Schmid, and J. Yao, “Photonic Fractional-Order Differentiator Using an SOI Microring Resonator With an MMI Coupler,” IEEE Photonics Technol. Lett. 25(15), 1408–1411 (2013).
[Crossref]

IEEE Trans. Commun. (1)

J. A. N. da Silva and M. L. R. de Campos, “Spectrally efficient UWB pulse shaping with application in orthogonal PSM,” IEEE Trans. Commun. 55(2), 313–322 (2007).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. (1)

Y. Hu, L. Zhang, X. Xiao, Z. Li, Y. Li, T. Chu, Y. Su, Y. Yu, and J. Yu, “An ultra-high-speed photonic temporal differentiator using cascaded SOI microring resonators,” J. Opt. 14(6), 065501 (2012).
[Crossref]

Nat. Photonics (1)

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Opt. Commun. (3)

J. Dong, B. Luo, Y. Zhang, D. Huang, and X. Zhang, “Reconfigurable photonic differentiators based on all-optical phase modulation and linear filtering,” Opt. Commun. 284(24), 5792–5797 (2011).
[Crossref]

L. Poveda-Wong, A. Carrascosa, C. Cuadrado-Laborde, J. L. Cruz, and M. V. Andrés, “Experimental demonstration of fractional order differentiation using a long-period grating-based in-fiber modal interferometer,” Opt. Commun. 380, 35–40 (2016).
[Crossref]

N. Ngo, S. Yu, S. Tjin, and C. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1–3), 115–129 (2004).
[Crossref]

Opt. Express (9)

F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008).
[Crossref] [PubMed]

Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express 14(26), 12670–12678 (2006).
[Crossref] [PubMed]

N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Express 15(2), 371–381 (2007).
[Crossref] [PubMed]

S. Yan, Y. Zhang, J. Dong, A. Zheng, S. Liao, H. Zhou, Z. Wu, J. Xia, and X. Zhang, “Operation bandwidth optimization of photonic differentiators,” Opt. Express 23(15), 18925–18936 (2015).
[Crossref] [PubMed]

A. Zheng, T. Yang, X. Xiao, Q. Yang, X. Zhang, and J. Dong, “Tunable fractional-order differentiator using an electrically tuned silicon-on-isolator Mach-Zehnder interferometer,” Opt. Express 22(15), 18232–18237 (2014).
[Crossref] [PubMed]

J. Wu, P. Cao, X. Hu, X. Jiang, T. Pan, Y. Yang, C. Qiu, C. Tremblay, and Y. Su, “Compact tunable silicon photonic differential-equation solver for general linear time-invariant systems,” Opt. Express 22(21), 26254–26264 (2014).
[Crossref] [PubMed]

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[Crossref] [PubMed]

B. Jin, J. Yuan, C. Yu, X. Sang, Q. Wu, F. Li, K. Wang, B. Yan, G. Farrell, and P. K. A. Wai, “Tunable fractional-order photonic differentiator based on the inverse Raman scattering in a silicon microring resonator,” Opt. Express 23(9), 11141–11151 (2015).
[Crossref] [PubMed]

J. Dong, A. Zheng, D. Gao, L. Lei, D. Huang, and X. Zhang, “Compact, flexible and versatile photonic differentiator using silicon Mach-Zehnder interferometers,” Opt. Express 21(6), 7014–7024 (2013).
[Crossref] [PubMed]

Opt. Lett. (8)

C. Cuadrado-Laborde and M. V. Andrés, “In-fiber all-optical fractional differentiator,” Opt. Lett. 34(6), 833–835 (2009).
[Crossref] [PubMed]

L. Chen, N. Sherwood-Droz, and M. Lipson, “Compact bandwidth-tunable microring resonators,” Opt. Lett. 32(22), 3361–3363 (2007).
[Crossref] [PubMed]

Z. Li and C. Wu, “All-optical differentiator and high-speed pulse generation based on cross-polarization modulation in a semiconductor optical amplifier,” Opt. Lett. 34(6), 830–832 (2009).
[Crossref] [PubMed]

J. Xu, X. Zhang, J. Dong, D. Liu, and D. Huang, “High-speed all-optical differentiator based on a semiconductor optical amplifier and an optical filter,” Opt. Lett. 32(13), 1872–1874 (2007).
[Crossref] [PubMed]

F. Li, Y. Park, and J. Azaña, “Complete temporal pulse characterization based on phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. 32(22), 3364–3366 (2007).
[Crossref] [PubMed]

J. Dong, A. Zheng, D. Gao, S. Liao, L. Lei, D. Huang, and X. Zhang, “High-order photonic differentiator employing on-chip cascaded microring resonators,” Opt. Lett. 38(5), 628–630 (2013).
[Crossref] [PubMed]

T. L. Huang, A. L. Zheng, J. J. Dong, D. S. Gao, and X. L. Zhang, “Terahertz-bandwidth photonic temporal differentiator based on a silicon-on-isolator directional coupler,” Opt. Lett. 40(23), 5614–5617 (2015).
[Crossref] [PubMed]

A. Zheng, J. Dong, L. Zhou, X. Xiao, Q. Yang, X. Zhang, and J. Chen, “Fractional-order photonic differentiator using an on-chip microring resonator,” Opt. Lett. 39(21), 6355–6358 (2014).
[Crossref] [PubMed]

Opt. Quantum Electron. (1)

C. Cuadrado-Laborde, “All-optical ultrafast fractional differentiator,” Opt. Quantum Electron. 40(13), 983–990 (2008).
[Crossref]

Sci. Rep. (1)

B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216 (2015).
[Crossref] [PubMed]

Other (1)

M. A. Popovic, T. Barwicz, F. Gan, M. S. Dahlem, C. W. Holzwarth, P. T. Rakich, H. I. Smith, E. P. Ippen, and F. X. Kärtner, “Transparent Wavelength Switching of Resonant Filters,” in Conference on Lasers and Electro-Optics (2007), paper CPDA2.

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Figures (9)

Fig. 1
Fig. 1 Schematics of (a) an add–drop MRR and (b) an MZI coupled microring resonator.
Fig. 2
Fig. 2 (a) Intensity transmission and (b) phase response of the MZI-MRR with varying values of Δφ.
Fig. 3
Fig. 3 (a) Magnitude response and (b) phase response of the MZI-MRR (dot lines) and the ideal first-order differentiator (solid lines). (c) Simulated the output response from the MZI-MRR (dot line) and the ideal first-order differentiator (solid line) of a 50 ps-pulsewidth Gaussian pulse. (d) Deviations of the simulated output response with different pulsewidth. (e) Deviations of the simulated output response with different differentiation order of a 50 ps-pulsewidth.
Fig. 4
Fig. 4 (a) Schematic concept diagram of the MZI-MRR. (b) Microscope image of the waveguides and rings after ICP etching. (c) Microscope image of fabricated MZI-MRR and (d) the zoom-in ring region.
Fig. 5
Fig. 5 Measured transmission spectra of the fabricated MZI-MRR at different heating powers applied to the heater above the arc-shaped waveguide.
Fig. 6
Fig. 6 Experimental setup for the fractional-order differentiator based on an MZI-MRR. TLD: Tunable laser diode. PC: Polarization controller. MZM: Mach-Zehnder modulator. EA: Electronic amplifier. BPG: Bit pattern generator. EDFA: Erbium doped fiber amplifier. OTDL: Optical tunable delay line. ATT: Attenuator. OC: Optical coupler.
Fig. 7
Fig. 7 (a) The experimental input pulse (blue dot line) and the simulated Gaussian pulse with a pulsewidth of 56 ps (red solid line). The measured differentiated output pulses at applied heating power of (b) 2.46 mW, (c) 3.12 mW, (d) 3.47 mW, (e) 3.92 mW, (f) 4.24 mW, (g) 4.65 mW, (h) 5 mW, (i) 5.54 mW, (j) 5.91 mW. Sim: simulation, Exp: experiment.
Fig. 8
Fig. 8 Averaged error of the fabricated MZI-MRR as a function of differentiation order N.
Fig. 9
Fig. 9 Measured spectra of the critical coupling MZI-MRR, the input Gaussian pulse with a pulsewidth of 56ps and the output pulse.

Equations (3)

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H( ω )= [ j( ω ω 0 ) ] N ,
t= E out E in = r 1 α r 2 e jϕ 1α r 1 r 2 e jϕ ,
κ 1 2 = κ 0 2 ( 1 κ 0 2 )×( t b + t r 2 t b t r cosφ ),

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