1. Introduction

Nanobeam photonic crystal cavities (NPCs) are important building blocks in many on-chip photonic applications. From its first proposition [1,2] to recently demonstrated high-Q design methodology [3–5], NPCs are widely employed in functional devices including lasers [6, 7], modulators [8, 9], switches, sensors [10–12] and optomechanics [13, 14]. NPCs feature ultra-compact waveguide structure, ultrahigh Q/V (ratio of quality factor to modal value) thus lower power consumption and higher speed. These unique characteristics indicate their great potentials in future very large scale integrated (VLSI) photonic circuits.

So far, most of the applications of NPCs rely on Bloch states near the boundary of first Brillouin zone [15,16]. For instances, dielectric-mode and air-mode NPCs were demonstrated by engineering the band edge defect modes of the fundamental dielectric band and the lowest air band. Slow light dispersion near the band edge was also reported [17], although in which flat band structure is a little bit difficult to design and achieve. In addition, in NPCs-based applications such as filters, the reflection from the involved direct-coupled or side-coupled NPCs need careful treatment, otherwise would influence the stability of the system. The reflection of side-coupled NPCs is well eliminated by using modes destruction interference effect [18–20]. However, the reflection of a direct-coupled NPC is intractable to deal with since intrinsically lack of mode cancellation in such configurations.

Recently, antisymmetric multimode Bragg waveguide gratings and components attracted lots of interest due to the promising prospect of on-chip Mode-division-Multiplexing (MDM) and polarization handling. Huang et al. reported a design of bandstop filter based on a multimode one-dimensional photonic crystal waveguide [21]. By using the mode coupling between the input fundamental transverse electric modes (TE₀) and the contra-propagating second order transverse electric mode (TE₂) in the mini-stopband, they obtained a reflection-free bandstop filter. We also demonstrated experimentally a series of photonic components based on antisymmetric multimode Bragg gratings [22, 23], where the injected fundamental modes are designed to couple to the contra-propagating first order modes. Up to now, reflection-free high-Q bandpass filters based on NPCs have not been demonstrated experimentally, although those are crucial for implementing compact functional devices.

In this work, we propose and demonstrate antisymmetric multimode nanobeam photonic crystal cavities (AM-NPCs) [Fig. 1(b)]. Unlike the conventional symmetric NPCs [Fig. 1(a)], the AM-NPCs are implemented on antisymmetric multimode periodic waveguides (AMPW) with two rows of staggered (a/2-shifted) periodic holes [Fig. 1(b)]. Due to transverse symmetry-breaking of the antisymmetric periodic waveguide, anti-crossing of fundamental mode and 1st-order mode is achieved and confirmed by plane wave extension method calculation. By utilizing coupling mechanism between these two modes, two-mode filtering and reflection-free cavity filters are designed and demonstrated on silicon-on-insulator (SOI) platform. Benefited from the higher index contrast of SOI, the total waveguide length is smaller than 20 μm and consequently leads to an ultra-compact footprint. And as far as we know, this is the first demonstration of reflection-free AM-NPCs with Q-factors higher than 7 × 10⁴.

 

Fig. 1 SEM images (only the left half parts shown) of (a) a traditional symmetric NPC and (b) an antisymmetric AM-NPC.

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2. Analysis and design

The dispersion relation of the antisymmetric multimode periodic waveguide are calculated by the MIT MPB packages [24]. In the calculation, all parameters are normalized to the period a of the unit cell (4 periods are shown in Fig. 2(a) i–vi) and the following parameters are used: waveguide width w = 2.424a, waveguide height h = 0.667a, and the AM-NPC is envisioned to be fabricated in a silica (n_sio2=1.447) cladded 200-nm-thick silicon (n_si=3.47) layer. As shown in Fig. 2(a), the band structure of the antisymmetric multimode periodic waveguide looks very different from that of the traditional symmetric periodic waveguide. The solid curves show the bands of the antisymmetric periodic waveguide with r = 0.15a, which supports two guided Bloch modes and opens a bandgap between the second band and the third band. For comparison, the dot-dash curves reveal the crossing of fundamental mode and 1st-order mode of a multi-mode channel waveguide with the equivalent refractive index, which is calculated from the unit cell of the antisymmetric periodic waveguide with r = 0.15a. When periodic refractive index modulation is introduced in this waveguide, the bands repel one another (anti-crossing) and a big bandgap of the AM-NPC appears [25]. The Bloch mode profiles in Fig. 2(a) i–vi prove that modes with various order exist in these same bands.

 

Fig. 2 (a) Band structure (TE-polarized modes) of the antisymmetric multimode periodic waveguide, (b) and (c) Mode profiles (E_y @z = 0) of the degenerate resonant states of the AM-NPC. (The blue and red solid curves correspond to the TE₀ and TE₁ modes of the AMPW (r = 0.15a) respectively. The dotted curves are the bands near the bandgap of r = 0.31a AMPW. The dot-dash curves are the bands of the channel waveguide with equivalent refractive index of the r = 0.15a AMPW. i–vi correspond to the profiles of Bloch modes in the bands where the arrows indicate.)

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To achieve a high-Q factor, we design the AM-NPC by applying the mode-gap concept that is well-established basing on Fourier analysis of resonant mode profiles of incomplete bandgap photonic crystal cavities [5]. As illustrated in Fig. 1(b) and Figs. 2(b)–2(c), the AM-NPC is constructed by linearly and slowly decreasing the radii of a/2-shifted two rows of periodic air holes, from the center (x = 0) to the both ends (x = ±Na) of the AM-NPC. The radii of the holes in the center r₀ and the ends r_N are 0.31a and 0.15a, respectively. Fig. 2 also shows the band structure of the periodic waveguide with r = 0.31a (dotted lines). It’s clear that dielectric defect modes are engineered in the band gap and the expected resonant frequency is around 0.217, the normalized bandedge frequency of the periodic waveguide with r = 0.31a. Taking this bandedge frequency as the initial seed for the resonant frequency calculations, we can easily investigate the resonant mode profiles and frequencies using the open-source 3D-FDTD package MEEP [26]. The calculation results for N = 16 AM-NPC are reported in Figs. 2(b) and 2(c). Due to the multimode feature, the AM-NPC supports two degenerate transverse electric (TE) modes with different symmetries, as a result of superposition of the fundamental mode and 1st-order mode. When N > 30, the calculated Q-factors of fundamental modes of these AM-NPCs are higher than 10⁶.

We also conducted transmission simulations to explore the characteristics of wavelength filtering. Both TE-polarized fundamental (TE₀) and 1st-order (TE₁) guided modes are used as incident mode sources. The separate simulation results are illustrated in Figs. 3(a) and 3(b) respectively. It is evident that when a TE₀ mode source is launched, the transmission wave of on-resonance wavelength is a TE₀ mode and the residual reflection wave is a TE₁ mode, and vice versa. Similar TE₀/TE₁ mode conversion phenomena were observed in reflection spectra of antisymmetric Bragg gratings [22, 23, 27], and here it further shows that the transmission wave has the same mode-order as the incident wave. For off-resonance light, the launched mode source (TE₀/TE₁) is mostly converted into a reflection wave of different mode order (TE₁/TE₀). Such scenarios are graphically illustrated in Figs. 3(c) and 3(d). It is important to note that the radii-linearly-tapered holes are crucial for efficient reflection mode conversion [28] and high transmission [3]. In our 40 tapered holes (N = 40) case, the calculated TE₀/TE₁ conversion efficiency (reflection) is higher than 98% over the whole band gap [Fig. 4].

 

Fig. 3 Mode profiles of (a,b) on-resonance transmission and (c,d) off-resonance reflection waves of the AM-NPC at z = 0 plane.

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Fig. 4 Reflection of the TE₀−TE₁ and TE₀−TE₀ of the tapered antisymmetric multi-mode periodic waveguide mirror (N=40).

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Basing on this TE₀−TE₁ conversion character of the antisymmetric periodic waveguide, we design a reflection-free AM-NPC which relies on an taper waveguide to eliminate the unwanted TE₁ reflection wave. The input TE₀ wave could evolve adiabatically to the fundamental mode of the wider multimode section whereas the reflection high-order mode radiate out at the single mode section [21]. Such a configuration is shown in Fig. 5(a), in which a Y splitter is also added for measuring the reflection spectra [29]. In order to investigate the characteristics of two-mode filtering as those shown in Fig. 3, a pair of well-verified adiabatic taper couplers [27] are used to couple TE₀ and TE₁ wave in and out from the AM-NPCs, respectively. In our design, a narrow tapered waveguide is parallel to a wide tapered waveguide with coupling length 230 μm and gap 200 nm. When an adiabatic coupler works as an input port, it converts the injected TE₀ mode in the narrow branch waveguide to the TE₁ mode of the wide waveguide [TE₁ in in Fig. 5(b)]. Similarly, if it acts as an output-port, the TE₁ mode in the wide waveguide is converted to the TE₀ mode of the narrow branch waveguide [TE₁ out in Fig. 5(b)]. A snapshot of the layout of the adiabatic couplers and cavity is shown in Fig. 5(b).

 

Fig. 5 Layouts of the designed (a) reflection-free AM-NPC and (b) two-mode filtering AM-NPC.

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3. Experimental results and discussion

The devices were defined by single-step E-beam lithography using negative-tone resist (HSQ) and subsequent inductively coupled plasma reactive ion etching (ICP RIE). To share the same etching depth as the strip waveguide, we employ the fully-etched focusing sub-wavelength grating couplers [30]. After fabrication, more than hundreds of devices were measured by an automated probe station under the control of pyOptomip [29].One typical SEM image of the fabricated AM-NPC is shown in Fig. 1(b). As a control, we also designed and fabricated a conventional symmetric single-mode NPC (w = 500 nm, a = 350 nm), which is shown in Fig. 1(a). Unless otherwise indicated, the following experimental spectra data (reflection, transmission) are normalized to those of the similar configurations but without the NPCs respectively.

Figures 6(a) and 6(b) show the measured spectra of the control NPC and the reflection-free AM-NPC, respectively. It is evident that the back reflection of the AM-NPC is well suppressed below −10dB, over a broad band from 1540–1640 nm. The measured resonant wavelength of fundamental mode of the control NPC is around 1615.8 nm, which is red-shifted relative to the designed value 1593 nm. This discrepancy mostly comes from the fabrication errors owing to incomplete ICP etching of holes and variation of width of waveguide [31]. By extracting the resonant wavelength λ_res and 3dB width Δλ_3dB [Fig. 6(b)], we can calculate the quality factors of the NPC using the equation Q = λ_res/Δλ_3dB. The calculated Q factor for the fundamental mode of the control NPC is around 1 × 10⁵. Although this Q factor is not as high as the record high-Q value of air-cladded SOI NPCs [32], here the loaded (directly coupled) NPCs are silica encapsulated and feature higher compatibility and robustness.

 

Fig. 6 Measured spectra and enlarged views for fundamental modes of the fabricated (a,b) control NPC (a = 350 nm, w = 500 nm) and (c,d) AM-NPC (a = 350 nm, w = 800 nm).

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From the enlarged view of spectra of the fundamental mode as shown in Fig. 6(d), it can be observed that the resonance of the AM-NPC splits into double peaks which have close Q-factors around 7 × 10⁴. Because the AM-NPC supports a pair of degenerate fundamental modes with distinct mode profiles, this resonance splitting can be attributed to the intercoupling of these two modes. The structure variation (loss of adiabatic-taper and rigorous anti-symmetry structure) due to fabrication error would decrease the reflection mode-conversion efficiency, as a result, strong coupling between the degenerate resonant modes lead to apparent resonance splitting. This behaviour is further confirmed by inspecting the transmission spectra of the two-mode filtering AM-NPCs (w = 800 nm, a = 330 nm) as shown in Fig. 7. One may find that the cross transmission has comparable power level to bar transmission in the bandgap whereas −10 dB lower than that in the dielectric bands. Since only one TE-polarized mode (TE₀ or TE₁) is injected, these results mean that strong mode conversion of transmission occurs in the band gap especially on the resonant wavelengths. By normalizing the cross transmission (TE₁−TE₀, TE₀−TE₁) to the bar transmission(TE₁−TE₁, TE₀−TE₀) respectively, we can extract the mode extinction ratio as a function of wavelength and the results are shown in Figs. 7 (b) and 7(d). Higher mode conversion is observed in the wavelength range of band gap, which corresponds well to the anti-crossing range of the air band and dielectric band near the band gap [Fig. 2(a)]. It is important to note the resonance splitting of high-order modes is not as severe as the low-order modes. This phenomena can be well explained by comparing the mode conversion abilities at these wavelengths in Figs. 7 (b) and 7(d). For higher-order modes which are far away from the point of anti-crossing, the mode extinction ratio is lower than −10 dB. However, the transmission mode conversion is quite strong in the band gap where lower-order modes exist, therefore apparent resonance splitting happens.

 

Fig. 7 Measured transmission and mode extinction ratio of the fabricated AM-NPCs (a = 330 nm, w = 800 nm) as the input light is (a,b) TE₁ and (c,d) TE₀ mode.

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Figures 7(b) and 7(d) also show that the transmission mode conversion spectra are single-peaked, which indicates that the transmission mode conversion was enhanced greatly by resonance. Theoretically, the resonant light transmit through the AM-NPC without mode-conversion if the mode-conversion reflection mirrors are perfect, although the resonant light endures multiple reflections from the tapered antisymmetric photonic crystal mirrors. However, due to the actual incomplete reflection mode-conversion, the transmission mode conversion builds up for these high-Q modes and results in resonance enhanced transmission mode conversion and resonance splitting.

4. Conclusion

We have proposed and demonstrated antisymmetric multimode nanobeam photonic crystal cavities (AM-NPCs). By using reflection mode conversion characteristic due to anti-crossing of bands, we have successfully achieved low-reflection AM-NPCs with reflection suppression lower than −10 dB. The measured Q-factors of these AM-NPCs are higher than 7 × 10⁴, which is applicable to most functional devices such as filters, modulators and switches. We have also demonstrated two-mode filtering of AM-NPCs and observed the transmission resonance splitting. It is found that this resonances splittings are caused by the inter-coupling between the degenerate resonant modes, owing to the incomplete mode conversion of the anti-symmetric photonic crystal mirrors. The results presented here have directive significance for AM-NPCs based band-stop and add-drop filters design.