## Abstract

In this paper, ultra-compact, broadband tunable optical bandstop filters (OBSFs) based on a multimode one-dimensional photonic crystal waveguide (PhCW) are proposed and systematically investigated. For the wavelengths in the mini-stopband, the input mode is coupled to a contra-propagating higher order mode by the PhCW and then radiates in a taper, resulting in a stopband at the output with low backreflection at the input. Three-dimensional finite-difference time-domain method is employed to study the OBSFs. The influence of main structural parameters is analyzed, and the design is optimized to reduce the back-reflection and band sidelobes. Using localized heating, we can shift the stopband and tune the bandwidth continuously by cascading the proposed structures. Due to the strong grating strength, our device provides a more compact footprint (40 μm × 1 μm) and much broader stopband (bandwidth of up to 84 nm), compared to the counterparts based on microrings, long-period waveguide gratings, and multimode two-dimensional PhCWs.

© 2016 Optical Society of America

## 1. Introduction

Photonic integrated circuits are regarded as a promising technique for on-chip optical interconnections, optical computing, and optical sensing. Optical bandstop filter (OBSF) is a basic component in photonic integrated circuits, which can attenuate a specific range of wavelengths to very low levels while passing the wavelengths at the two sides. OBSFs based on Bragg gratings and resonators have ever been reported [1–4]. However, the stopped light is mostly reflected to the input, causing detrimental effects when they are used in amplifiers or connected to reflectors in a photonic circuit. Therefore, OBSFs with low back-reflection is quite important in many cases. By far, low back-reflection OBSFs have already been proposed and demonstrated using microring resonators [5–7], long-period waveguide gratings [8–10], and multimode two-dimensional (2D) photonic crystal waveguides (PhCWs) [11–13].

For the microring based OBSFs, the low reflection is due to the fact that the input light in the stopband is consumed in the microring [5, 6] or tunneled into the drop port [7]. M. Popovic et al. reported an OBSF with an extinction exceeding 50 dB and a bandwidth of 32 GHz (~0.26 nm) using a multistage microring add-drop filter with a footprint of 200 μm × 70 μm [7]. In the long-period waveguide gratings, a stopband with low backreflection appears as the input mode is converted into a co-propagating higher order mode and then radiates into the cladding. A silicon OBSF with an extinction of 13 dB and a bandwidth of 15 nm was demonstrated using long-period waveguide gratings with a length of 500 μm [9]. Recently, another OBSF has been realized with a bandwidth of 4.3 nm using 1040 μm-long cladding modulated long-period gratings [10]. However, these structures are still limited by the large size and narrow stopband, because of the large microring/period and weak grating strength.

Mini-stopband (MSB) is prevalently existing in a multimode 2D PhCW [11–15], and the low order mode in the MSB can be strongly coupled to the contra-propagating higher order mode. Hence, a multimode 2D PhCW is suitable to realize OBSFs [14], and the backreflection would be greatly reduced if a tapered waveguide is added to get rid of the higher order mode. In recent years, one-dimensional (1D) PhCWs have attracted considerable attentions [16–21], since they are very compact, flexible, and compatible with the photonic devices in strip waveguides. Compared to the 2D PhCW with periodic hole arrays at the two sides, the 1D PhCW made of a line of holes in the center of a strip waveguide takes the advantages of even smaller size and stronger grating strength [20, 21]. Nevertheless, up to now, less attention has been paid to the multimode 1D PhCWs with a MSB and their applications. Then, owing to the high thermo-optic coefficient of silicon [22], silicon based optical filters can be significantly tuned by the localized heating [23–27]. Previously, thermal-tunable silicon OBSFs based on microring resonators and multimode 2D PhCWs have been proposed and analyzed in [6] and demonstrated in [15], respectively. However, OBSFs with tunable center wavelength and bandwidth are never considered using a multimode 1D PhCW.

Here, we present and investigate the ultra-compact, broadband tunable OBSFs based on a multimode 1D PhCW for the first time. The proposed device consists of a multimode silicon waveguide with periodic holes in the center, input and output single-mode waveguides, and two linear tapers to connect them. In the MSB, the input fundamental mode is back-coupled to a higher order mode that ultimately eliminated by a taper waveguide. Three-dimensional finite-difference time-domain (3D-FDTD) simulations are performed to study the proposed structure systematically. The influence of main structure parameters are investigated and the design is optimized to suppress the backreflection and band sidelobes. The results indicate that OBSFs with low backreflection is achievable with an ultra-compact footprint of 40 μm × 1 μm and a broad bandwidth of up to 84 nm, showing advantages over the previous schemes. Furthermore, by means of localized heating on the 1D PhCW, the stopband of OBSFs can be shifted, and the bandwidth of stopband is continuously tunable when two such OBSFs are serially cascaded. The systematic investigation is valuable for the experimental realization and future applications of tunable silicon OBSFs based on a multimode 1D PhCW.

## 2. Basic device configuration

Figure 1(a) shows the basic configuration of the proposed OBSF based on a multimode 1D PhCW in silicon-on-insulator (SOI). The OBSF consists of a 1D PhCW embedded in a multimode waveguide that connected to the input and output single-mode waveguides through two identical tapered waveguides, as clearly seen in Fig. 1(b). The top silicon layer of SOI is 220 nm thick. The width of input and output waveguides is *W _{1}* = 450 nm. The 1D PhCW is formed by etching 1D periodic circular holes to a depth of 220 nm in the multimode waveguide with

*W*= 1 μm and

_{2}*L*= 30 μm. The length of linear taper is set as

_{2}*L*= 5 μm, which is a compromise between compactness and adiabatic propagation of the fundamental mode. The surface of the structure is covered by a SiO

_{1}_{2}layer, and then a micro-heater is placed on the top of the multimode waveguide and 1D PhCW for thermo-optic tuning. Throughout this work, 3D FDTD method is used for simulation, and the transverse-electric (TE) polarization is concerned. The refractive indices of Si and SiO

_{2}at room temperature are 3.478 and 1.444, respectively. As labeled in Figs. 1(b) and 1(c), the lattice constant, hole number, and hole diameter of the 1D PhCW are denoted by

*a*,

*N*, and

*d*, respectively.

## 3. Working principle of OBSF

To illustrate the working principle of the OBSF, we firstly simulate the photonic band structure of the 1D PhCW with *a* = 360 nm and *d* = 100 nm for example. Figure 2(a) shows the band structure of the TE-polarized Bloch modes in the 1D PhCW calculated using 3D FDTD method. For the unit cell, non-uniform meshing is used with *Δx _{min}* =

*Δy*= 10 nm, and

_{min}*Δz*= 20 nm. Bloch periodic boundary conditions are imposed on the two surfaces perpendicular to the x-axis, and the perfectly matched layer (PML) absorbing boundary conditions are imposed on the four surfaces parallel to the x-axis. The straight black line represents the light line in the SiO

_{min}_{2}cladding. Crossing and anti-crossing of the guided Bloch modes are observed. Note that the TE

_{0}and TE

_{2}modes of the same parity interact and anti-cross, resulting in a MSB at frequencies ranging from 187.7 THz to 193.2 THz (or wavelengths from 1.553 μm to 1.598 μm). The input TE

_{0}mode in the MSB will be coupled to the backward TE

_{2}mode in the 1D PhCW. The

*Ey*field distributions of TE

_{0}and TE

_{2}modes in the 1D PhCW are shown in Fig. 2(b). We can see in Fig. 2 that the dispersion curve of TE

_{0}(TE

_{2}) mode in the strip waveguide is close to that of the Bloch mode in the 1D PhCW, and their field distributions are also quite similar, indicating an efficient conversion between the strip waveguide mode and 1D PhCW mode of the same order. Therefore, optical filter with a stopband ranging from 1.553 μm to 1.598 μm is predictable if a higher-mode filter (i.e. linear taper) is added in front of the 1D PhCW. It is known that the conversion between the TE

_{0}and TE

_{2}modes in the 1D PhCW satisfies the phase matching condition expressed by

*λ/a*= (

*n*), where

_{1}+ n_{2}*n*and

_{1}*n*represent the effective refractive indices of the coupled two Bloch modes, and

_{2}*λ*is the wavelength. Hence, the lattice constant in our structure is much smaller than that of long-period grating with the phase matching condition

*λ/a*= (

*n*), which thereby permits the proposed OBSF with an ultra-compact footprint.

_{1}- n_{2}The bandwidth *Δf* of the MSB can be evaluated by the following expression [12],

*f*is the center frequency of MSB,

_{c}*κ*is the coupling constant describing the grating strength,

*n*

_{g1}and

*n*

_{g2}are the group indices of the two coupled modes. From Fig. 2(a), we obtain

*Δf/f*= 0.029,

_{c}*n*

_{g1}= 3.466 and

*n*

_{g2}= 4.150, and thus extract the coupling constant

*κ =*0.346

*a*

^{−1}. The light is coupled more strongly and more light is reflected in the PhCW with periodic holes in the center, as compared to that with periodic perturbation at the sidewalls [20, 21]. Owing to the stronger grating strength, the bandwidth of MSB in our 1D PhCW can be much larger than that in the line-defect waveguides in 2D photonic crystal slab.

Then, the spectral responses of the OBSF based on a 1D PhCW with *N* = 60 are examined using 3D FDTD method. For the whole structure, we use non-uniform meshing with *Δx _{min}* =

*Δy*= 10 nm, and

_{min}*Δz*= 20 nm. The calculation domain is surrounded by PML absorbing boundaries. Figure 3(a) shows the transmission and reflection spectra of this structure as the TE

_{min}_{0}mode is injected. It is found that a stopband is generated with nearly zero transmission at wavelengths ranging from 1.553 μm to 1.598 μm, just the same as the wavelength range of MSB. The reflection decreases from the maximum to near zero with the wavelength in the MSB, due to the fact that the mismatch between the TE

_{0}modes in strip waveguide and 1D PhCW is continuously enlarged as the frequency increases, as seen in Fig. 2(a). The center reflection in the stopband is as low as 0.03. The mode reflections shown in Fig. 3(a) illustrate that the major component of the backward fields in the 1D PhCW is the TE

_{2}mode that filtered by the taper ultimately, while the minor component is the TE

_{0}mode that is left and contributes to the reflection of the OBSF. Moreover, the spectral responses show that the TE

_{0}mode is mostly reflected at wavelengths larger than 1.77 μm in the bandgap of 1D PhCW, while it is transmitted with certain loss at wavelengths smaller than 1.49 μm, as the guided light in the 1D PhCW is above the light cone and lossy. The |

*Ey|*field distributions in the device at the typical wavelengths of 1.42 μm, 1.58 μm, 1.67 μm, and 1.79 μm are presented in Figs. 3(b)-3(e), respectively. The field distributions is consistent with the responses in Fig. 3(a) and provide more insight in the operation principle of our OBSF. It is observed in Fig. 3(c) that the input TE

_{0}mode at 1.58 μm is all reflected and converted into a higher-order mode by the PhCW, which is then radiated into the cladding in the tapered waveguide.

## 4. The influence of the main structural parameters

In this section, we study the influence of the main structural parameters including *N*, *d*, and *a* of the 1D PhCW on the optical filtering properties, such as the minimum transmission, 3-dB bandwidth, center backreflection, and center wavelength of the stopband. Firstly, to reveal the effect of hole number, we simulate the spectral responses of the OBSFs by varying *N* when keeping *a* = 360 nm, *d* = 100 nm, as shown in Fig. 4(a). It is seen that as *N* increases, the band edge becomes much steeper, and the stopband transmission decreases significantly. Meanwhile the 3-dB bandwidth of stopband decreases slightly, but the reflection in the stopband does not change notably. Figure 4(b) plots the minimum transmission and center backreflection in the stopband as a function of *N.* As seen, the minimum transmission (*T _{min}*) in the stopband decreases with

*N*monotonously, which is −7.2 dB for

*N*= 20 and −44.2 dB for

*N*= 80, while the center backreflection keeps nearly constant around −16 dB. Thus, the filtering efficiencies ( = 1 -

*T*) [5] can be estimated to be 80.945% and 99.996% for

_{min}*N*= 20 and 80, respectively. Therefore, we find that the increase of

*N*is very effective to increase the stopband extinction (or filtering efficiency), whereas its impact on the backreflection is not remarkable.

Next, we change the hole diameter while keeping *a* = 360 nm and *N* = 60. The spectral responses of the OBSFs in Fig. 5(a) illustrate that as *d* increases, the short-wavelength band edge shifts leftwards, resulting in the stopband extension, and the sidelobes of the stopband become larger because of the stronger grating strength. At the same time, the left edge of the reflection band also shifts to the short wavelength, and the reflection continues increasing as the wavelength gets shorter. The minimum transmission in the stopband decreases with *d*, and when *d >* 100 nm, it is lower than −30 dB, indicating a filtering efficiency larger than 99.9%. From Fig. 5(b) we can find that the 3-dB bandwidth of the stopband increases approximately linearly from 22 nm to 84 nm, and the center backreflection increases monotonously from −19.2 dB to −10.4 dB, when *d* increases from 60 nm to 160 nm. Hence, the increase of *d* is a highly efficient way to increase the bandwidth of stopband, and it will leads to a higher center backreflection at the same time.

Then, the effect of lattice constant on the OBSF is studied when *N* = 60 and *d* = 100 nm are maintained. The transmission and reflection spectra of the OBSFs with different *a* shown in Fig. 6(a) indicate that as *a* increases, the stopband and reflection band both are red-shifted significantly, while the 3-dB bandwidth, sidelobes, and center backreflection become a little smaller. Figure 6(a) present the dependence of center wavelength *λ _{c}* and 3-dB bandwidth of the stopband on the lattice constant quantitatively. As seen, when

*a*increases from 0.30 μm to 0.42 μm,

*λ*increases with

_{c}*a*linearly from 1.42 μm to 1.72 μm with slope = 2.48, and the 3-dB bandwidth decreases monotonously from 52.2 nm to 37.7 nm. As a result, we know that the change of

*a*is very useful to tune the center wavelength of the stopband, and it also influences the 3-dB bandwidth to some extent.

## 5. Suppressing the backreflection and sidelobes

For an ideal bandstop filter, low backreflection and weak sidelobes are of great importance. In this section, we aim to optimize the OBSF design to suppress the backreflection and sidelobes. It is pointed out previously that the reflected TE_{0} mode is mainly caused by the mismatch between waveguide mode and Bloch mode at the interface between multimode strip waveguide and 1D PhCW. Therefore, we add three tapered holes with *d _{1} = d - 3∙δd, d_{2} = d - 2∙δd,* and

*d*at each side of the uniform 1D PhCW to reduce the mismatch, as plotted in Fig. 7(a). Figure 7(b) gives the transmission and reflection of the improved OBSFs that have additional tapered holes, in comparison with that of the original OBSF. It is shown that the reflection is significantly lowered by using smaller holes at the interface. The reflection level is lower than −20 dB throughout the stopband when

_{3}= d - δd*δd*= 20 nm. However, the stopband does not changes greatly meanwhile, as shown in the inset of Fig. 7(b). That is to say, adding tapered holes at the interface is useful to suppress the backreflection, but its impact on the stopband is weak.

To suppress the sidelobes of stopband, Gaussian apodization is applied to the periodic holes, as seen in Fig. 8(a). Here, we use a 1D PhCW with *N* = 60, *a* = 360 nm, and *d _{0}* = 100 nm for example, and the diameter of the

*i-*th hole is described by

*i*is the sequence number of the holes and

*G*denotes the apodization coefficient. Figures 8(b) and (c) show the spectral responses of the apodized OBSFs. It is found that the transmission sidebands become flat and the sidelobes are greatly suppressed as the apodization is used. The sidelobe level is reduced from - 3.3 dB to - 0.5 dB and - 0.1 dB for the apodized OBSFs with

*G*= 0.1 and

*G*= 0.2, respectively. We can see that it is a trade-off between the sidelobe level and stopband extinction. In addition, the maximum loss of the passband on the left side of stopband is also reduced as

*G*increases. Note that the maximum backreflection in the stopband is significantly reduced from −12 dB to −20 dB by the apodization as well. We can attribute it to the fact that the mismatch becomes weaker between modes in the strip waveguide and 1D PhCW, as the hole diameter is smaller at the interface and changes gradually under the apodization. Therefore, it is concluded that apodization is an effective method to suppress the sidelobes and backreflection simultaneously, though it will deteriorate the stopband extinction to some degree.

## 6. Thermal-optic tuning of the stopband filters

Due to the high thermo-optic coefficient of silicon, the OBSF can be tuned easily with the stopband shifted by changing the temperature through the micro-heater. For simplicity, as seen in Fig. 9(a), we assume that when the micro-heater is working, the covered region experiences uniform temperature change [23], which causes the refractive-index of the local Si and SiO_{2} to change according to the thermo-optic coefficients of 1.86 × 10^{−4} /K and 1.00 × 10^{−5} /K [22], respectively. The studied OBSF here consists of a Gaussian apodized 1D PhCW with *N* = 80, *a* = 0.34 μm, *d _{0}* = 80 nm, and

*G*= 0.1. Figure 9(b) shows us that high-performance stopband filtering is obtained at room temperature with the minimum transmission of −15 dB (or filtering efficiency of 96.7%), the maximum backreflection < −20 dB, 3-dB bandwidth of 33 nm,

*λ*near 1.55 μm, sidelobe levels > −0.3 dB, and a broad passband at each side of the stopband with loss < 1.0 dB. Then, as the temperature increases by 150 K and 300 K, the stopband is red-shifted significantly by 11 nm and 23 nm without deformation of the spectral lineshape, respectively, as indicated in Fig. 9(c). Note that the stopband shift against temperature change is determined to be 0.08 nm/K, a little smaller than the previous reported results [26, 27], because the core material of 1D PhCW is the hybrid of Si and SiO

_{c}_{2}in holes and the effective thermo-optic coefficient is thereby smaller than that of pure Si.

Now we will show that the bandwidth of stopband is also continuously tunable. Figure 10(a) presents the schematics of series-cascaded two such OBSF units that can be thermally adjusted separately. Here, the right unit is tuned, while the left one is fixed, and the simulation results are given in Fig. 10(b). The cascade of two OBSFs is simulated as a whole configuration, and the results are given in red and blue curves (solid: transmission; dot: reflection). The dash dot curves represent the multiplication of the transmission spectra of two independent OBSF units (i.e. *T _{1}* ×

*T*). As seen, the 3-dB bandwidth of stopband is expanded from 37.2 nm to 53.5 nm, while the stopband transmission increases slightly and the maximum backreflection is kept less than 0.01, as the temperature of the right unit is raised by 300 K. It is noteworthy that the transmission of the whole structure is almost equal to the multiplication of the transmissions of two independent OBSF units, which is attributed to the low backreflection of OBSF unit. Therefore, when the temperature of one unit is raised continuously, it is expected that the bandwidth of stopband will be enlarged continuously without any resonant peaks in the stopband, which is different from the transmission of cascaded bandstop filters with considerable backreflection. After all, those results validate that the proposed OBSF provides highly flexible tunability in filtering.

_{2}## 7. Conclusions

In summary, we have proposed an ultra-compact tunable OBSF with a broad stopband and low backreflection based on a multimode 1D PhCW. For wavelengths in the MSB, the input TE_{0} mode will be converted to the contra-propagating TE_{2} mode in the 1D PhCW that leaks out in a linear taper, leading to the desired bandstop filtering with low backreflection. We have studied the proposed device systematically using the 3D-FDTD simulations. The numerical results show that OBSFs are achievable with a footprint of 40 μm × 1 μm, bandwidth of up to 84 nm, and filtering efficiency larger than 99.9%, which are superior to the reported schemes in terms of compactness and bandwidth. We can reduce the maximum backreflection to lower than −20 dB using Gaussian apodization. Moreover, by localized heating, the stopband can be shifted with 0.08 nm/K, and its bandwidth is continuously tunable as two OBSF units are cascaded. The research is useful for the future experiment and applications of the proposed OBSFs.

## Funding

National Natural Science Foundation of China (NSFC) (61675084, 61376055, 61335002, 61435004); Major State Basic Research Development Program of China (2013CB933303, 2013CB632104); National High Technology Research and Development Program of China (2015AA016904).

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