1. Introduction

Targeting the future large-scale photonic integration, it is essential to develop a novel optical waveguide, which can simultaneously satisfy the following four requirements: (1) subwavelength mode confinement in the integrated plane of the circuit; (2) chip-scale propagation length; (3) highly efficient waveguide bends with compact footprints; (4) tiny center-to-center separation between adjacent waveguides with negligible crosstalk.

Silicon waveguides are of high mode confinement and long propagation range. However, the evanescent wave of its guiding mode still extends into air cladding over hundreds of nanometers at least, which leads to some coupling between two neighboring and parallel integrated silicon waveguides. In order to isolate the coupling, one has to enlarge the center-to-center separation of waveguides, which will decrease the on-chip integration density. The other drawback of silicon waveguide is large radius of bend to achieve high transmittance. Literature search shows that the bend radius of a silicon waveguide is at least one micron [1]. The common radius of bend is much larger than one micron [2].

Subsequently, the developments of surface plasmon (SP) waveguides [3] have aroused much attention because they could provide nano-scale waveguiding [4] and enhance light-matter interactions [5]. However, the tradeoff between the mode confinement and propagation length hinders their application in the large-scale photonic integration [6]. That means the propagation length can reach chip-scale at the expense of poor mode confinement, which gives rise to the large separation between adjacent parallel integrated waveguides with low crosstalk and the reduction of the on-chip integration density [7].

To overcome the above bottleneck and truly meet the four demands simultaneously, the long-range air-hole assisted subwavelength (LR-AHAS) waveguide was proposed [8]. The metallic sidewalls and two columns of periodic air-holes embedded in the silicon core can cooperate with each other in waveguiding to satisfy the above four requirements: (1) For TE (index-guided) mode transmitted in the LR-AHAS waveguide, the periodic air-holes in the waveguide can assist in tailoring the field profile and compressing the optical power as tightly as possible into a subwavelength scale (511.3-564.3 nm at 1.55 μm) as shown in the inset of Fig. 1(a); (2) While the propagation length of the waveguide can be significantly increased and reach chip-scale because of less contact between the optical field and the metal by introducing assisting air-holes. For example, the propagation length is 15.22 mm at wavelength of 1.55 μm. This phenomenon can be also explained by the effective index approach (detail in Appendix A); (3) The cooperation of the metallic sidewalls and assisting air-holes can provide a new path to realize the highly efficient PC-like direct bends with compact footprints; (4) The metallic sidewalls with skin-effect can enhance the isolation (50.3 dB) between adjacent integrated waveguides, and thus the on-chip integration density (877 mm⁻¹) can be boosted with a smaller value of the center-to-center separation (1.14 μm). In contrast, the separation and integration density are respectively 2.8 μm and 357 mm⁻¹ for silicon waveguides with a typical waveguide width of 500 nm at the same isolation level, based on our simulation results (detail in Appendix B). Here, we should emphasize that LR-AHAS waveguide performs broad optical bandwidth (1.47-1.80 μm) and almost 100% transmittances in the wavelength range from 1.48 to 1.65 μm in the transmission spectrum of the straight LR-AHAS waveguide, which is decided by the propagation loss of the fundamental mode transmitted in the LR-AHAS waveguide without back-refection. Besides that, it will be gradually influenced by the Bragg-reflection in the z-direction or multimode state with larger propagation loss of the higher-order modes for the wavelength larger than 1.65 μm or shorter than 1.48 μm, which give rise to slight drops in the transmission spectrum of the straight LR-AHAS waveguide. The calculated figure of merit (FOM) and corresponding integration density for the LR-AHAS waveguide are 6 × 10⁸ and 877 mm⁻¹, which achieved two orders of and a nearly five times improvement compared to the long-rang surface plasmon polariton (LR-SPP) waveguide with the best FOM [6], respectively. Therefore, the LR-AHAS waveguide is a promising candidate for the realization of large-scale photonic integration.

 

Fig. 1 (a) A drawing of the LR-AHAS waveguide array with five channels. Inset at the left top is the contour profile of the field Ey (TE-mode) transmitted in the LR-AHAS waveguide at λ = 1.55 μm. The first and fourth channels counted from the left are wave guiding ones. The second, third, and fifth channels are embedded with proposed reflective filters (indicated by dashed boxes), in which the corresponding resonant light with tunable wavelength can be highly reflected back. (b) Schematic of the in-line stub-like (ILSL) filter (region) corresponding to dashed part in the fifth channel. (c) Transmission spectra of the ILSL filters with silver sidewalls (solid line) and without them (dashed line). Inset is the structure of the ILSL filter without silver sidewalls, where r_e = 0.38•Period, and r_s = 0. (d) The contour profiles of field Ey of the ILSL filter at the wavelengths of 1.624 μm and 1.550 μm corresponding to the solid line in the spectra.

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Naturally, a new and intriguing topic emerges, that is, whether various kinds of functional components for light manipulating and signal processing can be developed based on the LR-AHAS waveguides, instead of only for the subwavelength guiding of light. If successful, it will boost the functionalization and practical applicability of the LR-AHAS waveguide. Among various kinds of functional waveguide components, optical filter is an important one, which can be utilized as the wavelength-selection, wavelength-dropping or multiplexing elements. The Bragg gratings fabricated on the platforms of a silicon waveguide [9] and a LR-SPP waveguide [10] have been demonstrated. However, due to the principle of periodically modulating the effective index along the waveguide and large period numbers, the total length of the Bragg grating must be at least over dozens of micrometers. Another common pathway to achieve reflective wavelength-selection relies on the side-coupled ring resonator. Normally, the radius of the ring in the practical fabrication with low bending loss is around 5 μm [11,12]. These components far exceed the subwavelength-scale and the on-chip integration density will be largely decreased when adding them into the system. As a result, in this paper, we focused our attention on designing a compact optical filter based on a LR-AHAS waveguide with narrow bandwidth, high extinction ratio, and ultrahigh on-chip integration density. What's more, the proposed optical filter can also act as the demultiplexing element when coupling with a dropping waveguide. These schemes could promote the development of LR-AHAS waveguides and make a step further towards large-scale photonic integration.

2. Characteristics and principles of in-line stub-like (ILSL) filters based on LR-AHAS waveguides

The LR-AHAS waveguide array embedded with three different reflective filters is vividly depicted in Fig. 1(a). The yellow, gray and black sections stand for the metal, silicon and air, respectively. The first and fourth LR-AHAS waveguides are utilized as the subwavelength-scale signal transmission lines.

To constitute a filtering function component in the LR-AHAS waveguide, we removed or decreased the radius of one assisting air-hole in the waveguide to form the asymmetric configuration. The shape of the configuration is like a stub junction as depicted in each dashed box in Fig. 1(a). Thus, it is called the in-line stub-like (ILSL) filter in the following. Because of the symmetry breaking, the k-vector of the guided mode will be deviated from the z-axis, which is imposed by the scattering of the air-holes in the ILSL region and the reflection of the metal-dielectric interface with high reflectivity. To be noted that two enlarged adjacent air-holes (r_e ≥ 0.35•Period) in the same column can constitute a Bragg mirror with high reflectivity, which means that a pair of Bragg mirrors (four enlarged air-holes in the same column) at left or right side of silicon core can confine the local resonant light in the z direction. To provide strong localization in the z-direction, we tried to enlarge eight air-holes at top and bottom sides of the centre row at the same time, however, the radius of right centered air-hole between a pair of Bragg mirrors should be very close to or larger than that of the enlarged air-holes in the Bragg mirror in case of the splitting (two) resonant dips emerged in the spectrum instead of only a single dip due to the formation of double stubs with different cavity sizes in the same row. To exclude the splitting resonant dips, we enlarged nine air-holes in the ILSL region at the same time, and the standing wave pattern will obviously emerge at the stub junction under the resonant condition, and the incident light will be highly reflected back as shown in the upper contour profile of Fig. 1(d). The directions of the k-vectors of resonant light and incident light are almost orthogonal. As a result, the filtering phenomenon of the ILSL region can be comprehended with the coherent interference between the scattering light denoted by the red arrow, which is partly splitting (or escaping) from the resonant light caused by the T-shaped splitting structure, and the z-direction transmitted light, which is represented by the blue arrows as shown in Fig. 1(b). More interestingly, the reflective peak wavelength can be easily tuned and blue shifted by compressing the duty circle of the silicon core in the ILSL region. Later, we will discuss the above phenomena based on the simple and robust model.

The schematic of the ILSL filter is shown in Fig. 1(b), where r_s, r_e and r are respectively the radii of the defective air-hole (the smallest one in ILSL region), the rest nine air-holes in the ILSL region and the assisting air-holes in the LR-AHAS waveguides. l_r denotes the distance between the centers of two adjacent air-holes in the same row. l_s is the distance between the centers of the defective air-hole and its adjacent ones in the same column, while the distance of all the rest adjacent air-holes in the same column is defined as the symbol Period because of the periodic distribution in the z direction. w_s and w_m denote the width of the silicon core and the metallic sidewalls, respectively. The symbols of ε_a, ε_s, and ε_m are the relative permittivity of air, silicon, and metal, respectively.

In the following, a Finite-Difference Time-Domain (FDTD) Maxwell equation solver was used for the numerical simulations. The parameters of the structure were chosen as follows: l_r = 745 nm, l_s = 410 nm, r = 130 nm, Period = 430 nm, w_s = 1.04 μm, w_m = 100 nm, ε_a = 1, and ε_s = 12.25. The metal is silver, whose relative permittivity ε_m can be described by the well-known Drude model with (ε_∞, ω_p, γ) = (3.7, 9.1eV, 0.018eV) [13], where ε_∞ is the dielectric constant at the infinite frequency, γ and ω_p represent the electron collision frequency and bulk plasma frequency, respectively. The grid size is 5nm for both x and z directions and perfectly matched layers (PML) conditions are applied to all the boundaries. In addition, we should emphasize that the value of w_s should be kept in a proper range, such as 0.86 tο 1.16 μm for the ILSL filter with r_e = 0.38•Period, to observe obvious wavelength filtering phenomena in the spectra. On the other hand, the propagation length will decrease if w_s becomes smaller because more portion of light will contact with lossy metal. w_s was chosen as 1.04 μm to make sure that the LR-AHAS waveguide has both the merits of compact configuration and optimal propagation length.

The metallic sidewalls are necessary elements for the proposed filters. Without them, the highly efficient wavelength-selection filters are impossible to realize and the spectrum is meaningless in practical application as shown in Fig. 1(c), in which the dashed curve represents the corresponding situation. Without metal mirrors, there is only weak resonance effect for the silicon waveguide with the same air-holes embedded because reflectivity of air-silicon interface is much smaller than that of silver-silicon interface. Meanwhile, there is a large portion of light will radiate to air, escape from the resonator and become the radiation mode instead of resonant mode, which will decrease the Q-factor of the stub resonator and form the spectrum with wider FWHM and smaller transmission contrast in Fig. 1(c). Thus, metal mirror with high reflectivity and skin-effect, which can prevent radiation from the cavity, accomplishes the crucial role in the narrow spectral response.

From the solid curve in Fig. 1(c), one can see that the spectrum of the proposed ILSL filter is more favorable. The full width at half-maximum (FWHM) and the extinction ratio are 11.0 nm and −17.8 dB, respectively. Thus, it is a new pathway to realize the compact wavelength filters and can be developed as the signal processing components in the LR-AHAS waveguides. When the Gaussian-type continuous wave (CW) incidents from the left, the standing wave pattern will be formed at the resonant wavelength of 1.624 μm, while the perturbation imposed by the ILSL region upon the guided mode can be neglected in terms of the off-resonant state as shown in Fig. 1(d).

As mentioned above, the directions of k-vectors of scattered light and transmitted light are all along the z-axis. If the phase displacement between them satisfies (2m + 1)π, the z-propagation light will be highly reflected. The phase delay of the scattering light can be expressed as follows:

Obviously, two different silver-dielectric interfaces in the ILSL region act as two reflection walls of the resonator. To further explain it, we only enlarge the radius of the one air-hole embedded in the cavity, as shown in the black dotted circle in the inset of Fig. 2(a), and the radii of the rest air-holes are kept in the same as 0.38•Period. The length L of the silicon material in the cavity is decreased with the increase in the radius r of the one air-hole, with the expression of ΔL = −Δr. As a result, the effective length, and thus the resonant wavelength are decreased. From the blue line in Fig. 2(a), the original resonant wavelength is 1.624 μm. When ΔL is −30 nm, the resonant wavelength is shifted to 1.569 μm. The other factor to influence the resonant wavelength is the width of the stub. A pair of Bragg mirrors were moved closer, as shown in the red dotted circles in the inset of Fig. 2(a). When ΔW equals −30 nm, the resonant wavelength is shifted from 1.624 to 1.621 μm (Δλ = 3 nm). The slope of the red line is much smaller than the blue one. This can be comprehended that the effective refractive index is decreased a little as a pair of Bragg mirrors get closer, which derives small decrease in the integral of n_eff (x, λ) and thus small decrease in the effective length. Both the major and the minor variation factors prove that the resonant cavity is between two metal-dielectric interfaces, and it is in accordance with field evolution of the standing wave patterns shown in Fig. 2(b).

 

Fig. 2 (a) The resonant wavelength versus the variation of radius of one air-hole in the cavity (ΔL) and versus the width of the stub (ΔW). (b) The evolution of the standing wave pattern in the ILSL filter, which is the direct proof that two silver-dielectric interfaces act as two reflection walls of the cavity and the directions of k-vectors of resonant light and incident light are almost orthogonal. (c) Transmission spectra of the ILSL filters with three different r_s (0, 0.1•Period and 0.15•Period) while r_e is fixed at 0.38•Period and 0.41•Period for the six curves, respectively. (d) Transmission spectra of the ILSL filters with five different r_e while removing the defective air-hole from the ILSL region (r_s = 0).

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In Fig. 2(c), we obtained two sets of spectra by fixing r_e as 0.38•Period and 0.41•Period, while varying the radius of the defective air-hole. Obviously, by decreasing the size of the defective air-hole, the resonant wavelength can be red-shifted. The minimum transmittances of the red and cyan curves are smaller than the rest ones. Therefore, the defective air-hole acts as the scattering or obstructive object for the resonant light. To obtain a high extinction ratio, the defective air-hole should be removed from the ILSL filter. And under this condition, the spectra of the proposed filters with five different values of r_e were investigated as shown in Fig. 2(d). The resonant wavelength is blue shifted while increasing the value of r_e. It can be comprehended that the effective length of the resonator is decreased. For the above five structures with r_e from 0.38 to 0.42•Period, the Q-factors are 143.2, 147.0, 149.6, 167.5, and 184.3, respectively. The average values of FWHM and minimum transmittance at the spectral dips are 10.1 nm and −17.5 dB. Because the performances of plasmonic single stub filters [13–15] are hindered by the large dumping of SPP mode, the average value of FWHM is larger than 100 nm. Therefore, by utilizing the proposed filter, a specific wavelength with a narrow bandwidth can be highly reflected back, and it can be used as the signal processing component in the LR-AHAS waveguides. The on-chip integration density of the LR-AHAS waveguide array (877 mm⁻¹) can be maintained because of the in-line configuration without the lateral extension or opening the silver sidewalls.

More interestingly, the strong Fano-type asymmetric resonances were observed in Figs. 2(c) and 2(d), which can increase the sensitivity of the filter. Next, we demonstrated how to control the asymmetry factor or Fano-type asymmetry of spectra. According to the previous article based on couple-mode-theory (CMT) [16], Fig. 3(a) illustrates equivalent model for CMT analysis, assuming an effective low-Q junction resonator in between the stub and main waveguide. The spectra of stub filter will be affected by the presence of junction resonator and its resonance characteristics. The transmittance of the stub filter can be derived as:

 

Fig. 3 (a) Equivalent model based on Fano resonance from local mode of an effective low Q-factor resonator in stub junction. (b) In-line stub-like (ILSL) filter with refractive index n_f marked with the yellow square in junction resonator region. The transmission spectra of ILSL filters with different refractive index (c) n_f = 2, (d) n_f = 3.5 and (e) n_f = 4.5.

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To investigate Fano-resonance in the spectral responses, a effective material with refractive index n_f can be embedded in the centre of the ILSL filter. Through modifying the n_f, one can adjust the resonance frequency (ω_R) or FWHM (Γ₀) of the junction resonator. As a result, the asymmetry factor (A_R) as well as Fano asymmetry of spectrum can be controlled.

In the numerical simulations, for n_f = 2, the lineshape of spectrum is nearly symmetrical (A_R~0) at the wavelength near resonant dip (λ = 1.546 μm), and the corresponding FWHM is 4.8 nm (Q-factor = 322). Increasing n_f to larger value, the Fano-type asymmetry will gradually occur. Especially, the strong Fano-type resonance can be seen in Fig. 3(e), whose resonant wavelength is 1.555 μm and FWHM is 15.9 nm. Strong Fano-asymmetry in the spectrum accompanied by the emergence of local mode in the junction resonator shown as the inset in Fig. 3(e). As a result, the asymmetry factor (A_R) is tunable by employing different refractive index material in the junction resonator, which can increase the sensitivity of the filter, compress FWHM at a symmetrical point and may find potential application in the low-power optical switching devices [17].

3. Characteristics and dropping efficiencies enhancement of wavelength demultiplexers based on LR-AHAS waveguides

Because directions of k-vectors of resonant light and incident light are almost perpendicular in the ILSL region as shown in Fig. 1(d), ILSL filters can also act as the wavelength-dropping/demultiplexing components, if the dropping channels are assigned nearby. To vertically drop the resonant light out of the LR-AHAS waveguides, one can build the in-line stub-like channel drop filter (ILSL CDF) shown as the upper structure in Fig. 4(a). The configuration of the dropping waveguide is simplified and a TE-mode MDM waveguide is utilized to side couple with the ILSL filter. Noted that there is a silver gap between one end of the MDM waveguide and the top of stub-like region. The thickness of silver gap will influence the transmission (dropping) efficiency for the ILSL CDF. Most of parameters of the ILSL CDF are maintained the same as the ones of the above ILSL filter. Here, the widths of the silicon core and silver sidewalls for the TE-mode MDM waveguide are respectively denoted as w_dc and w_dm, and are chosen to be w_dc = 0.3 μm and w_dm = 0.25 μm. The relation between the transmission efficiency of the ILSL CDF (r_e = 0.38•Period and r_s = 0) and the thickness of the silver gap is investigated, and is shown as the black curve with triangle dots in Fig. 4(c). When the thickness of the silver gap equals 20 nm, the transmittance is 0.396. It follows exponential decay as increasing the thickness of the silver gap. The transmittance is nearly zero when the thickness equals 50 nm. As a result, the thickness of 20 nm is chosen for ILSL CDFs in the following.

 

Fig. 4 (a) Schematics of a single in-line stub-like channel drop filter (ILSL CDF) and a combined unit consisted of an ILSL CDF and an ILSL filter, which are indicated by the black and blue dashed boxes in the lower structure. The directions of drop and incident light are perpendicular. Based on the resonant tunneling effect, the combined unit can enhance the drop efficiency compared to the single ILSL CDF. (b) The dropping wavelength (black curve with circles) and the corresponding value of FWHM (blue circles) versus the radius of nine enlarged air-holes in the black dashed box of a single ILSL CDF. (c) The transmittance of a single ILSL CDF and a combined unit with r_e = 0.38•Period and r_s = 0 versus the thickness of the gap and versus the center-to-center separation S, respectively.

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As shown in Fig. 4(b), the dropping wavelength can be easily tuned by only changing the radius of nine enlarged air-holes in the black dashed box of a single ILSL CDF shown in Fig. 4(a). The larger the radius, the shorter the dropping wavelength because of the smaller value of the effective length of the resonator as discussed above. All dropping wavelengths are in the range of telecommunication band. Especially, the dropping wavelength is 1.554 μm when r_e = 0.42•Period. The corresponding values of the FWHM are also plotted as the blue circles in Fig. 4(b), all of which are between 10 nm and 20 nm. The average value of the FWHM is calculated as 16.2 nm, which is a little larger than that of the ILSL filter because a portion of light is coupled to the dropping waveguide as well as the additional coupling loss, which reduce the Q-factor of the stub-like resonator.

In numerical simulations, it was found that if an ILSL CDF is cascaded with an ILSL filter as the combined unit, shown as the lower structure in Fig. 4(a), the transmittance of the dropping waveguide can be largely enhanced because of the resonant tunneling effect [18]. The ILSL CDF and the ILSL filter have the same configuration except the latter one without the dropping MDM waveguide, to make sure that their resonant wavelengths are almost same. The center-to-center separation between two filter cavities is denoted as S. Based on the resonant tunneling effect, the dropping efficiency can be maximally enhanced if the phase shift φ ( = 2πS•n_eff /λ) corresponding to the center-to-center separation satisfies (2m’ + 1)π/2, where m’ is an integer. It gives S = (2m’ + 1)λ/(4n_eff). Here, the radius of each air-hole in the ILSL CDF and the ILSL filter is chosen as 0.38•Period, one can adjust the center-to-center separation to reinforce the dropping efficiencies. The effective index of the TE-eigenmode in the LR-AHAS waveguide with r_e of 0.38•Period is 3.24, by utilizing the supercell technique as described in the previous article [8]. For m' = 8 and λ = 1.628 μm, S is calculated to be 2.14 μm. According to the simulation results shown as the blue curve with triangular dots in Fig. 4(c), the dropping efficiencies can be significantly enhanced from 0.396 to the maximum value of 0.63, when the center-to-center separation equals 2.16 μm, which is close to the theoretical value.

Based on the above discussions, we cascaded three different ILSL CDFs acting as the demultiplexing units to form a 1 × 3 wavelength demultiplexer as shown in Fig. 5(a). The radii of the enlarged air holes in three units are denoted as r₁, r₂, and r₃, respectively. Here, we choose r₁ = 0.35•Period, r₂ = 0.39•Period, and r₃ = 0.42•Period, while the radius of each assisting air-hole in the LR-AHAS waveguide maintains the same value. The corresponding dropping wavelengths for three channels are 1.675 μm, 1.614 μm and 1.554 μm. When the light with one of the resonant wavelengths incidents from the bottom of the demultiplexer, a standing wave will be formed at the corresponding demultiplexing unit, and the energy is partially coupled to its dropping channel. As shown in Fig. 5(b), the peak transmittances of three dropping channels are 34.8%, 38.5% and 35.5%, respectively. The FWHMs of the first channel to the third one are 18.9 nm, 17.5 nm, and 16.9 nm. While the average value of the FWHM is around 120 nm for the plasmonic wavelength demultiplexers [19–21].

 

Fig. 5 (a) Schematic of a 1 × 3 wavelength demultiplexer consisted of three different ILSL CDFs acting as demultiplexing units indicated by the black dashed boxes in the LR-AHAS waveguide. Each demultiplexing unit can be replaced with the corresponding combined unit to enhance the drop efficiency based on the resonant tunneling effect. (b) Transmission spectra of three dropping channels with and without the ILSL filters (or enhancement). (c) The contour profiles of field Ey of the 1 × 3 wavelength demultiplexer at the corresponding dropping wavelengths from the first channel to the third one: 1.675 μm, 1.614 μm, and 1.554 μm.

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The drop efficiencies of three channels are all below 40%. To overcome this drawback, one can utilize the above discussed method based on the resonant tunneling effect, that is, the combined unit with an ILSL filter and a CDF in a proper separation. The original demultiplexing unit with low transmission efficiency in Fig. 5(a) can be replaced with the corresponding combined unit with the same value of r_i. One can adjust the center-to-center separation S_i to enhance the drop efficiency. When S₁ ~S₃ are chosen as 2.21 μm, 2.11 μm, and 2.31 μm, the drop efficiencies from the first channel to the third one are respectively increased to be 61.4%, 64.3% and 65.0%, as shown in Fig. 5(b). The drop efficiencies are improved by more than 65% compared to the case without ILSL filters. The FWHMs of the drop efficiency enhanced structure are 16.2 nm, 18.87 nm, and 16.8 nm, respectively. Thus, the proposed 1 × 3 wavelength demultiplexer has important potential for the design of narrow bandwidth and highly efficient WDM systems in large-scale photonic integrated circuits.

The performance of the 1 × 3 wavelength demultiplexer at three dropping wavelengths is visually illustrated in Fig. 5(c) that shows the contour profiles of the TE-mode electric field. One can see that the electromagnetic wave at the wavelength of 1.675 μm is coupled out from the first dropping channel, while a portion of light passes through the whole system. In the same way, the incident lights at the wavelengths of 1.614 μm and 1.554 μm are coupled out from the second and the third dropping channels. This is in good agreement with the transmission spectra shown in Fig. 5(b).

4. Conclusion

We proposed and numerically investigated the ILSL filters and a 1 × 3 wavelength demultiplexer based on LR-AHAS waveguides. The structures are simple and compact. The proposed components have distinguished wavelength-filtering/dropping characters in terms of the narrow bandwidth. The peak filtering/dropping wavelength can be easily tuned by changing the radius of air-holes in the ILSL region. The dropping efficiencies of the wavelength demultiplexer can be significantly enhanced by adding the ILSL filters into the structure. These schemes could be useful in developing the fundamental signal processing devices and WDM systems for the LR-AHAS waveguides, and will meet the demand of large-scale photonic integrations.

Appendix A

The strong Bragg reflection in the lateral x-direction of LR-AHAS waveguide is not supported because there is only one row of air-holes in each side of the waveguide. However, the Bragg reflection can be supported in the propagation direction because of the periodic distribution of air-holes. In Fig. 6(a), an equivalent waveguide (right one) is used to highlight this effect. Based on the effective index approach, regions I and II modulated with periodic air-holes can be equivalent to a special dielectric material with the periodic distribution of the average refractive index (averaged in the x-direction) of N(z)=n_SiW_Si(z)/W_total+n_airW_air(z)/W_total. And W_total= W_Si(z) + W_air(z), where W_Si(z) and W_air(z) are respectively local widths of silicon and air-hole in region I or II at the coordinate of z.

 

Fig. 6 (a) The equivalent waveguide (right one), in which all air-holes are replaced with the square-shape material with gradual index change from 1.3 to 3.5, considering the Bragg-effect along the propagation direction. (b) A further approximation that defines homogeneous and effective relative permittivity ε_eff of regions I and II to the left equivalent waveguide, for operational wavelengths far away from the resonant wavelengths of Bragg-effect.

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For a LR-AHAS waveguide, the effective index approach in z-direction can be used again conditionally to obtain approximately characteristic equation and n_eff(λ). When the wavelength is far away from the Bragg wavelengths or transmission dips of the LR-AHAS waveguide, one can make further approximation to neglect the Bragg-effect in propagation direction. Based on negligible Bragg reflection and the effective index approximation, the right structure in Fig. 6(b) gives the equivalent treatment for the left equivalent waveguide of a period of a LR-AHAS waveguide. ε_eff is the effective relative permittivity for regions I and II. Here, we define n_eff =p·n_air+(1-p)·n_Si, where p=πR²/(Period·d₂), d₂=0.295 μm, d₁=0.45 μm, R=0.13 μm and Period=0.43 μm. Thus ε_eff= n_eff² = 6.02. The transverse electric-field of TE₀ mode in the equivalent waveguide can be generally written a

$E(x)=\{\begin{array}{l}A\exp [-r_m(x-d_1/2-d_2)]\left(\text{x}>d_{\text{1}}/\text{2}+d_{\text{2}}\right)\\ B\exp [r_{eff}(x-d_1/2-d_2)]+C\exp [-r_{eff}(x-d_1/2)]\left(d_{\text{1}}/\text{2}<\text{x}<d_{\text{1}}/\text{2}+d_{\text{2}}\right)\\ D\exp [r_{Si}(x-d_1/2)]+E\exp [-r_{Si}(x+d_1/2)]\left(-d_{\text{1}}/\text{2}<\text{x}<d_{\text{1}}/\text{2}\right)\\ F\exp [r_{eff}(x+d_1/2)]+G\exp [-r_{eff}(x+d_1/2+d_2)]\left(-d_{\text{1}}/\text{2}-d_{\text{2}}<\text{x}<-d_{\text{1}}/\text{2}\right)\\ H\exp [r_m(x+d_1/2+d_2)]\left(\text{x}<-d_{\text{1}}/\text{2}-d_{\text{2}}\right)\end{array}$

$r_x=\sqrt{{\beta }^2-k_0^2{\epsilon }_x},(x=m,Si,\text{or}eff)$

As a result, the characteristic equation can be obtained from the above expressions:

(5)$$\exp (r_{Si}{d}_1)=\frac{(r_{eff}-r_m)(r_{Si}+r_{eff})+(r_{eff}+r_m)(r_{Si}-r_{eff})\exp (2r_{eff}{d}_2)}{(r_{eff}-r_m)(r_{Si}-r_{eff})+(r_{eff}+r_m)(r_{Si}+r_{eff})\exp (2r_{eff}{d}_2)}$$

By solving characteristic equations of the equivalent waveguide (Eq. (5)) and the corresponding MDM waveguide (Eq. (1) in Ref [8]), one can obtain the direct comparison of effective index between these two kinds of waveguides as shown in Fig. 7.

 

Fig. 7 (a) Real and (b) imaginary parts of the effective index for the corresponding MDM waveguide (blue line with triangular dots) and the equivalent waveguide corresponding to the right one in Fig. 6(b) (red line with round dots).

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From Fig. 7(b), one can note that the imaginary part of the equivalent waveguide is much smaller than that of the corresponding MDM waveguide in the wavelength range from 1.45 to 1.70 μm, which is away from the Bragg wavelength range (1.85-2.10 μm). It means that the equivalent waveguide has much smaller propagation loss. Similarly, direct numerical simulation on the LR-AHAS waveguide has proved that it has longer propagation length than that of the corresponding MDM waveguide by introducing two rows of periodic air-holes in the silicon core of the waveguide, which is shown in Fig. 2(a) of Ref [8].

Appendix B

To prove the merit of high isolation of LR-AHAS waveguides, we provide the isolation (in absolute value) between two parallel integrated silicon waveguides versus the separation as shown in Fig. 8. The Width and the Length of silicon waveguides are respectively chosen as a typical number of 500 nm and 100 μm. For the isolation of 50.3 dB, the separation between two silicon waveguides is 2.8 μm, while the separation between two LR-AHAS waveguides is 1.14 μm. If the integration density is defined to be the number of parallelly-integrated waveguides with the high isolation over 50 dB in one millimeter in the x direction, the integration densities of LR-AHAS waveguides and silicon waveguides are 877 mm⁻¹ and 357 mm⁻¹, respectively. Inset of Fig. 8 shows a vivid comparison between LR-AHAS waveguide array and silicon waveguide array to highlight the difference in terms of on-chip integration density.

 

Fig. 8 Silicon waveguide isolation versus the separation. Inset is the comparison of integration density between silicon and LR-AHAS waveguide arrays with the same waveguide isolation of 50.3 dB under the same plotting scale.

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Finally, we would emphasize that the silver sidewall is an indispensable part for the LR-AHAS waveguides and the proposed filters. It has three important functions as follows: (1) The silver sidewalls can improve the isolation between adjacent integrated waveguides. The on-chip integration density can be boosted with a very small center-to-center separation of waveguides. (2) The cooperation of the silver sidewalls with assisting air-holes can provide a new path to improve filed confinement and to realize highly efficient direct bends with compact footprints built with LR-AHAS waveguides. (3) The filters without silver sidewalls are meaningless for wavelength-selection function. On the contrary, the proposed filters are of the characters of high transmission contrasts and narrow bandwidths by employing the silver sidewalls and air-holes. Additionally, the on-chip integration density (877 mm⁻¹) is maintained when the filters are integrated into the LR-AHAS waveguides.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61077038) and Guangdong Province High-Tech Zone Development Program (No. 2012B010900022).

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