Abstract
The optical allpass filter (APF), which exhibits a constant amplitude response and a variable phase response, is a key to manipulating the optical phase without inducing signal amplitude distortion. Highorder APFs are significantly demanded because they can afford large time delays and phase shifts. However, to date, only firstorder APFs based on lossy waveguides have been reported. Although highorder APFs can be simply obtained by cascading multiple firstorder APFs, the complexity and size are increased. To solve this problem, we propose and demonstrate a secondorder APF using Mach–Zehnder interferometerassisted microring resonators. The device is fabricated based on a silicononinsulator platform. Based on the secondorder APF, an adjustable time delay between 553 and 948 ps is obtained, and the corresponding amplitude variation is less than 1.7 dB. Meanwhile, a microwave photonic phase shifter is also obtained based on the APF. The microwave phase shift can be adjusted from 0 to 3.27π, with an RF power variation within 2.4 dB. Additionally, the secondorder APF can be reconfigured to a firstorder APF, which significantly enhances its flexibility. The reconfigured firstorder APF can realize an adjustable time delay between 257 and 429 ps, and the amplitude variation is less than 0.9 dB. The proposed highorder APF provides a novel approach to manipulating optical signals.
1 Introduction
Recently, silicononinsulator (SOI)based optical signal processing devices have attracted great interest because of their inherent advantages in size, weight, power, and cost (SWaPC) [1–3]. Optical filters are of fundamental importance in eliminating noise or extracting signals via amplitude manipulation [4–6]. In contrast to filters exploiting varied amplitude responses, allpass filters (APFs) have a constant amplitude response and a variable phase response, which is especially suitable for signal phase manipulation without introducing amplitude distortion [7], such as in adjustable delay lines [8, 9], microwave photonic phase shifters [10–12], optical adddrop multiplexers [13], optical dispersion compensators [14, 15], and Hilbert transformers [16].
To achieve an APF, the zero locations of the system function are mirror images about the unit circle from the pole locations [7]. In recent decades, APFs have been developed based on ideal lossless conditions, where the waveguide loss and coupling loss are omitted. However, lossless conditions cannot be obtained in practical passive waveguides, and there is always amplitude variation in the achieved APFs. To eliminate the amplitude variation induced by loss, several approaches have been proposed. First, a method called selfcompensation of loss was proposed, and an APF was obtained [17]. Then, obtaining an APF by using optical interference between the outputs of an allpass microring resonator and a straight waveguide was also proposed [18]. The obtained APF exhibits ultralow amplitude variation and the APFbased variable optical delay line and microwave photonic phase shifter also exhibit ultralow amplitude distortion. Notably, both achieved APFs are firstorder types. Hence, based on the firstorder APF, the achieved microwave photonic phase shift is less than 2π, and the achieved maximal time delay is also limited. To promote the system performance, highorder APFs are desired. Although highorder APFs have been proposed to achieve a large time delay by cascading multiple MRRs [7, 14, 19, 20], amplitude variation exists in the phase shift region of the APF and signal distortion is consequently induced. Because these high order APFs can only be obtained for lossless waveguides. Cascading firstorder APFs is a direct approach to acquire a highorder APF. However, the insertion loss, size and complexity are multiplied as the order of the APF increases.
In this paper, we propose and demonstrate a secondorder APF by simply cascading a microring resonator (MRR) and a structure of a firstorder APF. Compared with the secondorder APF obtained by simply cascading two firstorder APFs, the novel structure has superiorities in size, loss and simplicity, which is more obvious when extended to higherorder APFs. The device is fabricated based on an SOI wafer. Notably, the secondorder APF can also be reconfigured to a firstorder APF, which enhances the flexibility in applications. APFbased variable optical delay line and microwave photonic phase shifter are also demonstrated. The time delay and the microwave phase shift based on the secondorder APF can be continuously adjusted from 550 to 948 ps and from 0 to 3.27π, respectively. One advantage of the proposed approach is that the topology of the proposed secondorder APF can be easily extended to higherorder APFs (see Supplementary Material Section 1). The proposed highorder APF provides another dimension for manipulating optical
signals in addition to the amplitude response of optical filters, which means that pure phase manipulation can be achieved without resulting in amplitude distortion.
2 Device design and principle
In an optical filter, the optical interference can be used to adjust the zero locations without changing the pole locations. In our design, two cascaded Mach–Zehnder interferometers (MZIs) are used to arbitrarily adjust the zero locations of the device, and two MRRs provide the phase shift and time delay. Notably, to realize a broadband secondorder APF, the two MRRs should have the same free spectral range (FSR). Therefore, the two MRRs are designed with equal cavity lengths.
Figure 1(a) shows a schematic diagram of the proposed secondorder APF, which was designed based on an SOI wafer with a top layer of 220 nm (see in Supplementary Material Section 2). The optical signal, which is coupled into the device by a grating coupler (GC_{1}), is denoted as E _{1}. The propagation direction of the optical signal is denoted by the yellow arrow in Figure 1(a). Then, a multimode interferometer (MMI_{1}) equally divides the optical signal into the two arms of MZI_{1}. The phase difference between the two arms of MZI_{1} can be changed by adjusting the electric power applied to H_{1} or H_{2}. After phase adjustment, the two optical beams are combined and then redivided into two parts by MMI_{2}. The optical field of the upper (E _{2}) and lower (E _{3}) arms of MZI_{2} are described as
where ϕ _{1} is the phase difference between the two arms of MZI_{1}. The amplitude splitting ratio between the lower and upper arms of MZI_{2} x is expressed as
Then, the upper part of the light is coupled with an MRR (MRR_{1}), and the lower part is sent to the lower arm of MZI_{2}. To eliminate the coupling deviation from the designed value caused by fabrication error (see in Supplementary Material Section 3), the coupling between MRR_{1} and the upper arm of MZI_{2} is designed with a balanced MZI structure, and the coupling ratio can be changed by adjusting the electric power applied to microheater H_{6} or H_{7}. Microheaters H_{3} and H_{4} are used to adjust the phase difference between the two arms of MZI_{2}. Notably, MMI_{3} and MMI_{4} are designed to extract the transmissions of the two arms of MZI_{2}, and GC_{2} and GC_{3} are designed as the monitor ports. The optical signals in the two arms of MZI_{2} are combined in MMI_{5}. Omitting the optical loss caused by MMI_{3} and MMI_{4}, the optical field of the combined optical signal E _{4} is expressed as
where
where
where k is an arbitrary integer. Equation (7) indicates that MRR_{2} is overcoupled. Consequently, when Eqs. (5)–(7) are satisfied, the transmission of the secondorder APF can be expressed as
To make the principle understood intuitively, Figure 1(d) illustrates the principle of the secondorder APF. When the optical beams in the upper and lower arms of MZI_{2} interfere with each other, the bandstop response induced by MRR_{1} can be converted to a bandpass response after MMI_{5}. Then, the bandpass response is cascaded with MRR_{2}. When the notch response of MRR_{2} is complementary to the bandpass response, an allpass response is obtained. Additionally, the resonant wavelengths of MRR_{1} and MRR_{2} are aligned with each other to ensure that the amplitude response is constant. In the achieved APF, the phase variation in the FSR is twice that based on a single MRR [17, 18].
To validate our analysis, we carry out simulations. Figure 2 shows the simulation results when a and L are set as 2.5 dB/cm and 251.32 μm, respectively. When t _{1} is set as 0.99, the amplitude and phase responses of the secondorder APF are given by the red solid curve and the black short dashed curve in Figure 2(a), respectively. The amplitude response of the secondorder APF is constant, and the phase shift reaches 4.83π, from 1549.395 to 1551.685 nm, which corresponds to the FSR of the MRRs. The results show that a secondorder APF is obtained. The phase shift in the FSR is larger than 4π. This is caused by the MZI couplers for MRR_{1} and MRR_{2}. Compared with a directional coupler, an MZI coupler has a complex coupling coefficient, whose phase can increase the phase shift range of the APF. The corresponding time delay is shown in Figure 2(b), and the peak is 677 ps. Notably, the phase response can be changed by adjusting t _{1}. When t _{1} is set as 0.968, 0.978, 0.988 and 0.998, the phase responses of the second APF are given by the red dashed, green solid, black short dashed, and blue dasheddotted curves in Figure 2(c), respectively. The inset is a zoomedin view of the phase response from 1550.5 to 1550.6 nm. Figure 2(d) shows the variations in the insertion loss and the group delay of the secondorder APF versus selfcoupling coefficient t _{1}. When t _{1} is adjusted from 0.968 to 0.998, the insertion loss (blue solid curve) and delay (orange short dashed curve) of the secondorder APF increase from 3.3 to 29.4 dB and from 318 to 1219 ps, respectively.
In addition, the APF can be reconfigured to change the order. By adjusting the selfcoupling coefficient of MRR_{2} to 0 or 1, which indicates that no resonance exists in MRR_{2}, the APF can be reconfigured from second order to first order. Therefore, the flexibility of the APF is enhanced to adapt to different scenarios. Based on the theory in [18], a firstorder APF can be realized when the parameters satisfy
Substituting Eqs. (9) and (10) into Eq. (4) and assuming t _{2} = 1, we can derive the transmission of the firstorder APF, which can be expressed as
The simulation results of the firstorder APF are shown in Figure 3. When the selfcoupling coefficient of MRR_{1} is set as 0.99, which is the same as that in Figure 2(a), the amplitude and phase frequency responses are as shown in Figure 3(a). The amplitude response is constant and the phase shift, from 1549.395 to 1551.685 nm, is 2.42π. The phase shift is larger than 2π, which is also caused by the MZI coupler. The corresponding group delay is shown in Figure 3(b), and the peak is 446 ps. The phase response can also be adjusted by changing t _{1}, which is the selfcoupling coefficient of MRR_{1}. When t _{1} is set as 0.968, 0.978, 0.988, and 0.998, which are the same as those in Figure 2(c), the phase responses of the firstorder APF are given by the red dashed, green solid, black short dashed and blue dasheddotted curves in Figure 3(c). Figure 3(d) shows the insertion loss and group delay of the firstorder APF versus t _{1}, and the trend is similar to that for the secondorder APF. When the coupling coefficient of MRR_{1} remains the same, both the insertion loss and delay of the firstorder APF are less than those of the corresponding secondorder APF.
3 Results
Based on the theoretical analysis, our proposed device is fabricated based on an SOI wafer. A micrograph of the fabricated device is shown in Figure 4(a). To measure the amplitude and frequency responses of the fabricated device, the experimental setup illustrated in Figure 4(b) is adopted. Continuouswave (CW) light at 1550.00 nm emitted from a laser source (LS, Koheras BasiK E15) is launched into a phase modulator (PM, Covega Mach40). A polarization controller (PC_{1}) is used to align the state of polarization (SOP) of the CW light with the polarization axis of the PM. Then, an optical bandpass filter (OBPF, Alnair BVF300CL) is used to eliminate the lower sideband of the phasemodulated signal, and a singlesideband (SSB) signal is obtained. An erbiumdoped fibre amplifier (EDFA) and a variable optical attenuator (VOA) are used to adjust the optical power launched into the device. Then, the optical signal is coupled into the chip via GC_{1}. After processing by the device, the output signal is coupled out of the chip via GC_{4} and launched into a photodetector (PD, SHF AG Berlin). Then, the converted electric signals are received and analysed by a vector network analyser (VNA, Anritsu, MS4647B). Figure 4(c) shows the experimental setup used to measure the optical transmission spectrum of the fabricated device. The light emitted from a broadband optical source (BOS) is polarized by a polarization beam splitter (PBS). Then, the SOP of the linearly polarized optical beam is adjusted to be aligned with the polarization axis of the optical waveguide by a PC. The broadband light is coupled into and out of the chip via GC_{1} and GC_{4}, respectively. Finally, the optical transmission spectrum of the device is measured by an optical spectrum analyser (OSA, YOKOGAWA, AQ6370C).
To realize a secondorder APF, Eqs. (5)–(7) must be satisfied, which can be realized by adjusting the electric power applied to the microheaters on MZI_{1}, MZI_{2}, MRR_{1}, and MRR_{2} (see in Supplementary Material Section 7). The measured results are shown in Figure 5. When the electric power applied to H_{2}, H_{4}, H_{5}, H_{7}, and H_{8} is 58.1, 63.3, 9.4, 23.4, and 28.9 mW, respectively, the achieved amplitude and phase responses are given by the blue solid curves in Figure 5(a) and (b), respectively. Obtainment of a secondorder APF can be observed. The amplitude variation is within 1.4 dB, and the phase shift is 3.89π, from 5 to 40 GHz. A maximal time delay of 553 ps is achieved, shown by the blue solid curve in Figure 5(c). The corresponding insertion loss is 7.2 dB (see in Supplementary Material Sections 8 and 9), shown by the blue solid curve in Figure 5(d). By adjusting the electric power applied to H_{2}, H_{4}, H_{5}, H_{6}, H_{7}, H_{8}, and H_{10}, the phase response and time delay are correspondingly changed (see in Supplementary Material Section 10). When the maximal time delay is adjusted to 643, 805, and 948 ps, shown by the black, red, and green solid curves in Figure 5(c), respectively, the corresponding amplitude responses, phase responses and transmission spectrum are given by the black, red, and green solid curves in Figure 5(a), (b) and (d), respectively. The inset of Figure 5(b) is a zoomedin view of the phase response from 22 to 24 GHz. In Figure 5(a) and (d), we can observe amplitude variations in the measured results. This phenomenon is caused by the Fabry–Pérot (FP) cavity, environmental fluctuations and the amplitude variation around 22.75 GHz. The FP cavity results from the pair of input and output GCs. These amplitude variations cause fluctuations in the amplitude response of the APF and consequently induce signal power variation. In addition, the amplitude variation around 22.75 GHz shown in Figure 5(a) is mainly caused by the issue that the bandpass and bandstop responses are not totally complementary. This may be caused by misaligned resonant wavelengths of the two MRRs and insufficient adjustment resolution of the power supply to microheaters. In the secondorder APF, MRR_{2} is overcoupled, and the bandpass response of MRR_{1} and the bandstop response of MRR_{2} are approximately complementary. Therefore, the extinction ratios of both the bandpass and the bandstop are enlarged as t _{2} increases. When the resonant wavelengths of MRR_{1} and MRR_{2} are misaligned, a high extinction ratio will lead to a large amplitude variation. Therefore, the measured magnitude of the amplitude variation in the phase shift region is enlarged as the group delay increases, as shown in Figure 5(a).
The phase response of the secondorder APF can also be tuned. When the electric powers applied to microheaters H_{1}, H_{4}, H_{5}, H_{6}, H_{7}, H_{8}, and H_{10} are 42.9, 14.2, 58.9, 9.1, 56.6, 30.4, and 0 mW, respectively, the device works as a secondorder APF, and the achieved time delay is 630 ps. Then, we adjust the electric powers applied to H_{5} and H_{8} to tune the resonant wavelengths of MRR_{1} and MRR_{2}, respectively. To maintain allpass transmission, both resonant wavelengths are adjusted to be aligned with each other. Based on the experimental setup shown in Figure 4(b), we adjust the centre of the phase shift area of the secondorder APF from 5.7 to 38.6 GHz, as shown in Figure 6. Figure 6(a) shows that the transmission variation is less than 1.9 dB during the tuning process. The power variation is mainly caused by environmental fluctuations. For example, the mechanical vibration causes the SOP of the optical signal in fiber link varied. Thus, the transmitted power of the APF is changed because the fabricated GCs in the chip is polarization dependent. Figure 6(b) and (c) show the measured phase responses and time delay of the secondorder APF when the electric powers applied to H_{5} and H_{8} are adjusted from 58.1 to 62.1 mW and from 25.6 to 31.2 mW, respectively.
The secondorder APF is an ideal candidate to realize a broadband tuneable microwave photonic phase shifter. The optical carrier is placed within the phase shift area of the secondorder APF, and the sideband is placed outside of the phase shift area [22]. By tuning the phase shift region of the secondorder APF and adjusting the electric powers applied to H_{5} and H_{8}, the phase difference between the optical carrier and the sideband is accordingly changed, and a tuneable microwave photonic phase shifter is obtained. Figure 7(a) and (b) show the amplitude and phase responses of the microwave photonic phase shifter. The microwave phase can be shifted from 0 to 3.27π, while the RF power variation is 2.4 dB, which is larger than the predicted result. The RF power variation is mainly caused by three factors. The first is environmental fluctuations. The second is the relative wavelength drift between the resonant wavelengths of MRR_{1} and MRR_{2}. Additionally, the FP effect caused by reflections of the GC pair also contributes to the power variation. Notably, in the previously reported microwave photonic phase shifter based on the firstorder APF [17, 18], the microwave phase shift is less than 2π. In contrast, the microwave photonic phase shifter based on the secondorder APF has a much larger phase shift range, and the phase shift exceeds 2π, which is significantly important for phased array antennas.
Additionally, the secondorder APF can be reconfigured to a firstorder APF, which significantly enhances the flexibility of the APF. The firstorder APF can be obtained when the selfcoupling coefficient between the waveguide and MRR_{2} is 0 or 1. In the experiment, when the electric power applied to H_{10} is 25.1 mW, the selfcoupling coefficient between the waveguide and MRR_{2} is almost 1. Therefore, the optical signal is not resonant in MRR_{2}. When the electric powers applied to microheaters H_{2}, H_{4}, and H_{5} are adjusted to 58.8, 68.3, and 26.6 mW, respectively, a firstorder APF with a 257 ps delay can be realized, as shown by the blue solid curve in Figure 8. By adjusting the electric powers applied to H_{2}, H_{4}, H_{5}, and H_{6} (see in Supplementary Material Section 10), the phase response of the firstorder APF can be changed, and the corresponding time delay is correspondingly adjusted. When the electric powers applied to H_{2}, H_{4}, and H_{6} are adjusted from 58.8 to 62.5 mW, 68.3–76.0 mW, and 0–12.4 mW, respectively, the amplitude and phase responses are as shown in Figure 8(a) and (b), respectively. Figure 8(a) shows that the maximum amplitude variation is 0.9 dB during the adjustment process. Two factors contribute to the amplitude variation occurred at 22 GHz. The first is the notch existing in the transmission spectrum of MRR_{2} because the crosscoupling coefficient between the waveguide and MRR_{2} cannot be exactly adjusted to 0. Therefore, slight optical signal is coupled into MRR_{2} and a transmission notch is generated. The second factor is the limited resolution of adjusting the power applied to the microheaters, which caused the corresponding parameters slightly deviate from the desired values. Figure 8(b) shows that the rolloff rate of the phase response is consequently changed. As a result, the corresponding time delay is adjusted from 247 to 429 ps, as shown in Figure 8(c). Figure 8(d) shows the corresponding optical transmission spectrum of the firstorder APF measured by the OSA. The FP effect is also observed in the transmission spectrum. The insertion loss also increases as the time delay increases. Notably, if we simply cascade two firstorder APFs which contains the part from GC_{1} to MMI_{5}, the device will occupy much more size and more heaters will be required than those of the structure displayed in Figure 1(a). Therefore, our proposed device shows advantages of more compact size, simpler structure, and less power consumption.
4 Conclusions
In conclusion, we have realized a tuneable and reconfigurable optical secondorder APF based on SOI. Both simulations and experiments are carried out to demonstrate the APF. Compared with the secondorder APF achieved by simply cascading two firstorder APFs, our proposed secondorder APF is much simpler and more compact. The secondorder APFbased adjustable delay line and microwave photonic phase shifter are also investigated. The results show that when the time delay is adjusted from 553 to 948 ps, the amplitude variation is within 1.7 dB. Based on the secondorder APF, a microwave photonic phase shifter with a phase shift from 0 to 3.27π is achieved, and the corresponding RF power variation is less than 2.4 dB. The secondorder APF can also be reconfigured to a firstorder APF when the selfcoupling coefficient of MRR_{2} is set as 0 or 1. In the firstorder APF, the amplitude variation is less than 0.9 dB, and the group delay can be adjusted from 247 to 429 ps. Another advantage is that the proposed approach can be easily extended to realize APFs with even higher orders. Compared with highorder APFs obtained by simply cascading firstorder APFs, our proposed approach is much simpler and more compact since fewer optical elements are used.
Funding source: the National Key R&D Program of China
Award Identifier / Grant number: 2018YFB2201700
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 61975249
Funding source: the Program for HUST Academic Frontier Youth Team
Award Identifier / Grant number: 2018QYTD08

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Research funding: This research was supported by the National Key R&D Program of China (2018YFB2201700), the National Natural Science Foundation of China (61975249), the Program for HUST Academic Frontier Youth Team (2018QYTD08).

Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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Supplementary Material
The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph20220140).
© 2022 Yu Chen et al., published by De Gruyter, Berlin/Boston
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