1. Introduction

Bragg gratings have been well known and attracted attention worldwide for decades [1,2]. They have been used for a wide range of applications such as optical filtering, dielectric mirrors, interferometers, spectrometers, lasers, wavelength-division multiplexing/ demultiplexing, differentiators, waveguiding, optical delay lines, dispersion compensators, and optical sensors. Bragg gratings have been implemented in different ways including free-space and waveguide embedded components. Examples of waveguide types are optical fibers, photonic crystals, and integrated optical waveguides. Examples of various materials are silica, silicon, polymers, electro-optic, acousto-optic, and composite materials.

Bragg gratings have been widely explored in numerous literatures work. To name some but a few, tens of picometers filter could be designed by Bragg grating loaded segments on a silicon-on-insulator (SOI) subwavelength grating metamaterial waveguide [3]. A silicon-on-insulator periodically-spaced sampled grating is designed to construct cascaded Fabry-Perot interferometers, which can produce a comb-like non-flat-top spectrum with Sinc-weighted amplitudes [4]. Four channels coarse wavelength demultiplexer is demonstrated using cascaded silicon multimode waveguide gratings [5]. A silicon-on-insulator circular Bragg grating mirror is demonstrated with high reflectivity over an ultra-broad bandwidth of 500 nm [6]. A silicon Bragg grating is formed by etching nano-holes within a waveguide core to achieve a 110 nm wide stop-band spectrum [7]. A thermally tunable band-stop filter is demonstrated using a multimode one-dimensional photonic crystal waveguide consisting of periodic holes [8]. However, its single stop-band shows a maximum bandwidth of just 84 nm and a footprint of 40 µm×1 µm. Two anti-symmetric π-shifted Bragg gratings were used to implement a single-wavelength resonator over the SOI platform [9]. Subwavelength gratings were used to implement SOI metamaterial waveguide that can be utilized to implement integrated optical components such as directional couplers and ring resonators [10]. An SOI long-period subwavelength metamaterial grating is used to have a diffraction-less homogeneous waveguide propagation [11]. An SOI electrically reconfigurable Bragg grating for programmable optical signal processing is demonstrated, where two sub-grating with a Fabry-Perot cavity was re-configured as uniform, phase-shifted, and chirped grating by programming the bias-voltages of PN junctions [12]. A circular diffraction grating consisting of an elliptical concave Bragg mirror is used to build an SOI micro-spectrometer, where it shows a 0.4 dB spectrum flatness over a 30 nm bandwidth [13]. An integrated apodized cladding-modulated Bragg grating on CMOS compatible ultra-silicon-rich nitride platform was demonstrated and utilized for Bragg soliton high compression [14].

In this work, a novel partial-width entrenched-core (PWEC) waveguide grating is numerically demonstrated using the finite-difference time-domain (FDTD) simulation method. A few-mode waveguide is chosen to be silicon-on-insulator (SOI) planar waveguide that is compatible with CMOS fabrication technology. The PWEC is simply a periodic nano-wide rectangular air-gaps that are entrenched in the middle of a core with a width less than that of the core. An input infrared TE fundamental mode couples within the PWEC region into guided zero and second-order even modes because of the periodic nano-focusing caused by air trenches. The PWEC acts as a stand-alone short (few-periods) partial-width uniform grating, which shows a unique double-hump (DH) spectrum that has two closely-separated Bragg wavelengths. Each Bragg wavelength corresponds to one of the waveguide even-modes. They can be tuned over a wide range (1 to 2 µm) of the shortwave infrared (SWIR) region. It is found that the air-trenches width and period are very crucial in the design of DH-spectrum characteristics. In general, the DH-spectrum characteristics can be engineered by the appropriate choice of the PWEC trenches width, period, as well as the number of periods. Thus, the double-hump reflection/ transmission spectra can be reshaped to obtain, for example, double stop-band, single wide/ narrow pass-band, notch-band, or ultra-wide super-band (e.g., full mirror). Such unique features of SOI-PWEC waveguide are attractive for various applications in planar lightwave integrated circuits such as optical signal processing, on-chip spectroscopy, optical filtering, optical sensing, and optical communications.

2. PWEC operation principles

Figure 1(a) shows a three-dimensional schematic diagram of a partial-width entrenched-core waveguide. The silicon waveguide core has a width of 850 nm to ensure a few-mode operation (almost 4 modes), a height of 220 nm to be compatible with standard CMOS fabrication technology. The silicon core (refractive index: n=3.4) is surrounded by air-cladding (n=1) everywhere except for underneath silicon-dioxide (SiO₂) substrate (n=1.5). A partial-width (W) of silicon core is entrenched to form periodic rectangular air-gaps with a depth equal to the core height (220 nm). Thus, a stand-alone uniform Bragg grating consisting of alternating air-trenches and silicon-stripes is partially formed within the core with a total length of ‘L’. Each pair of silicon-stripe and air-gap constitutes one Bragg-period ‘Λ’. The Bragg duty-cycle is 50%. The rest of the core portion before the grating region is considered an input/ reflection port of PWEC, whereas the rest of the core portion after the grating region is considered a transmission port.

 

Fig. 1. The PWEC waveguide structure and its principle of operation (Λ=0.32 µm, N=30, and W=50 nm). (a) The three-dimensional schematic diagram of silicon-on-insulator partial-width entrenched-core waveguide (PWEC), the duty-cycle is 50%. (b, c) The two-dimensional electric-field magnitudes over PWEC-region for the excited second-order mode (short Bragg-wavelength, λ_B1=1.386 µm), and the excited zero-order (fundamental) mode (longer Bragg-wavelength, λ_B2=1.605 µm), respectively. (d) A typical example of double-hump reflection and transmission spectra of PWEC. (Check “Visualization 1” for an animation of evolving electric-fields magnitudes along PWEC).

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The input port is excited with a linearly-polarized TE zero-order (fundamental) mode. This mode suffers a strong perturbation during propagation through the PWEC region. It mainly experiences a periodic focusing inside nano-wide air-trenches followed by expansion across core total width within silicon-stripes. This focusing takes place at the middle of the core, and thus stimulates and excites the second-order even-mode within the PWEC region. Therefore, both zero and second-order modes couple to multiple reflections and transmissions along the PWEC grating region, resulting in presence of two different Bragg-wavelengths that are closely separated. The coupled modes also experience periodic concentration and expansion along the Bragg region.

Figures 1(b) and 1(c) show the XY cross-sections of second-order and zero-order electric-fields magnitude across the grating region, respectively. Check also “Visualization 1” for an animation of the evolving normalized electric-fields magnitude along PWEC length. The incident fundamental mode couples during reflections and transmissions to the second-order waveguide mode, resulting in short Bragg-wavelength ‘λ_B1’ with low effective index ‘n_eff1’. Also, it couples to reflections and transmissions of zero-order mode, resulting in long Bragg-wavelength ‘λ_B2’with high effective index ‘n_eff2’. The result is double-hump (DH) reflectance and transmittance (P_out/ P_in) spectra of PWEC waveguide. Each hump surrounds one of the Bragg wavelengths, as shown in Fig. 1(d). On the left side, hump #1 corresponds to the second-order mode, with λ_B1=1.386 µm, peak-reflectance (R_pk1) = 98% to 100%, bandwidth BW₁=66 nm, and extinction-ratio ≅ -31.5dB. On the right side, hump #2 corresponds to the zero-order mode, with λ_B2=1.605 µm, R_pk2 = 97% to 100%, BW₂=50.5 nm, and extinction ratio ≅ -27dB.

By following a simple mathematical analysis, the two Bragg-wavelengths can be derived by applying the conservation of momentum condition (i.e. phase-matching) to the PWEC region [1,2]. Let’s assume a broadband input fundamental mode that has a one wavelength at λ_B1 with propagation constant ‘β_i1’, and another wavelength at λ_B2 with propagation constant ‘β_i2’. Where β_{i 1,2} = 2πn_{eff 1,2} /λ_o, and n_{eff 1, 2} is the effective index corresponding to each mode, in addition λ_o is the free-space wavelength. The input modes couple to reflected second-order mode at λ_B1 and zero-order mode at λ_B2 with propagation constants β_r1 = -2πn_eff-r1 /λ_o, and β_r2 = -2πn_{eff 2} /λ_o, respectively. Where n_eff-r1 is the effective index of second-order reflected mode. The negative-sign indicates counter-propagation direction with respect to incident mode. The effective index of zero-order mode (λ_B2) remains the same because the reflected wave couples to the same mode order. The Bragg-grating wave-vector can be defined for counter-propagation direction as K = -2π /Λ . Thus, the conservation of momentum condition mandates:

By substituting with propagation constants, the two Bragg resonance wavelengths becomes:

The PWEC waveguide is numerically simulated using variational finite-difference time-domain FDTD (2.5D-varFDTD) method [15]. This method is adequately comparable in accuracy to the conventional three-dimensional FDTD, especially when evaluating planar integrated lightwave waveguides. It has also the advantages of less simulation time and memory size requirements (i.e. less computational cost). When compared to conventional coupled-mode theory, the FDTD has the advantages of more accurate three-dimensional device designs, precise excitation and propagation of modes inside multimode waveguides, in addition to consideration of all waveguide effects such as radiation modes, dispersion, and nonlinearities. The simulations are performed here using perfectly-matched boundary conditions on all sides of the PWEC planar structure. The minimum x-boundary condition is set to anti-symmetric to reduce the simulation run-time. A broadband TE-polarized (along x-direction) fundamental-mode is utilized as an input source to the silicon waveguide. The simulation temperature is set to room-temperature at 300°K. The silicon and silicon-dioxide materials models are obtained from the Ansys-Lumerical default library of Palik [16]. A multi-coefficient model is selected to accurately fit the materials constants.

3. PWEC characterization

 

Fig. 2. The Characterization of double-hump spectral properties as a function of Bragg-period detuning (W=50 nm, N=30): (a) The resonance Bragg-wavelengths tuning. The dashed-lines outline the boundaries of each hump, (b) The effective indices of the two modes, (c) The curve-fitted maximum-reflectance, (d) The humps bandwidths, (e) The coupling overlap-factor between modes, (f) The curve-fitted average dispersion across each reflected hump.

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The PWEC spectral properties are examined and characterized by varying different grating parameters such as the trench-width (W), Bragg-period (Λ), and the number of Bragg-periods (N). The examined spectral properties are Bragg wavelengths (λ_B), humps bandwidths (BW), peaks reflectance (R_Pk), coupling overlap-factors (η), effective indices (n_B), and dispersion.

Figure 2 illustrates the effect of detuning Bragg-grating period (Λ) on the double-hump spectral properties. In Fig. 2(a), by varying the Bragg-period, both Bragg resonances (λ_B1 and λ_B2) can be tuned over a wide wavelength range covering a large part of SWIR (from 1 to 2 µm). The dashed lines in the figure indicate the minimum and maximum wavelength boundaries for each hump. The separation between each set of dashed-lines is equal to the hump bandwidth. In Fig. 2(b), the effective index of each Bragg wavelength (i.e., each mode) is shown as a function of Bragg-period detuning. The indices reduce as longer ‘Λ’ increases, besides ‘n_B1’ is always less than ‘n_B2’. Figure 2(c) shows that the peak reflectances of humps are always close to unity with a possible minimum value of ≅ 93%. In Fig. 2(d), the bandwidth of the first hump stays around a value of 65 nm, whereas the bandwidth of the second hump increases up to 110 nm. The BW becomes almost constant, according to Eq. (5) , when ‘η’ starts to reduce, as shown in Fig. 2(e), while ‘λ_B’ increases. In Fig. 2(f), the estimated average dispersion across each hump ranges between ± 0.2 fs/nm, which is considered very small.

 

Fig. 3. The Characterization of double-hump spectral properties as a function of trench-width (Λ=0.32 µm, N=30): (a) The resonance Bragg-wavelengths tuning. The dashed-lines outline the boundaries of each hump, (b) The effective indices of each mode, (c) The curve-fitted maximum-reflectance, (d) The humps bandwidth, (e) The coupling overlap-factor between modes, (f) The curve-fitted average dispersion across each reflected hump.

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Figure 3 illustrates the effect of detuning the PWEC trench-width (W) on double-hump spectral properties. As explained earlier, the first spectral hump is mainly associated with the guided second-order mode, whereas the second spectral hump is associate with the zero-order (fundamental) mode. Therefore, varying ‘W’ affects the effective index of each coupled mode, and also affect the spectral properties of DH-spectrum. This is a unique property of PWEC waveguide that has proven to be crucial in the control of DH spectral characteristics, as discussed later. Figure 3(a) shows a wide tuning range of Bragg resonance wavelengths when varying the trench-width, which is a consequence of large changes of effective indices, as shown in Fig. 3(b). For example, by varying ‘W’ from 50 nm up to 600 nm, the effective indices of zero-order and second-order modes reduce by ≅ 0.5, which is a large difference. The dashed lines, in Fig. 3(a), indicate the minimum and maximum wavelength boundaries of each hump. Unlike ‘Λ’ detuning in Fig. 2(a), the Bragg-resonance wavelengths, in Fig. 3-(a), reduce with wider trenches, and the humps’ inner boundaries approach each other to become closer. In Fig. 3(d), the first hump has a large bandwidth variation. It can reach a bandwidth up to ≅ 450 nm, while the second hump bandwidth stays around ≅ 100 nm. That is because the hump bandwidths mainly follow the coupling overlap-factor in Fig. 3(e), which agrees with Eq. (5) , taking into consideration that Λ_{1, 2} = λ_{B1, 2} /2n_{B1, 2}. In Fig. 3(c), the peak reflectances of DH-spectrum are still close to unity with a possible minimum value of ≅ 92%. In Fig. 4(f), the estimated average dispersion across each hump ranges between ± 0.3 fs/nm, which is still very small. It is worth mentioning that beyond W = 600 nm (i.e., when trench-width becomes comparable to the core-width of 850 nm) the second-order mode seizes to exist (i.e., cut-off) and we are left with only one single hump corresponding to the zero-order fundamental mode.

4. PWEC spectral reshaping

 

Fig. 4. The reshaped double-hump PWEC spectra: (a) The double stop-band spectra covering two of the E, S, C-bands in addition to a part of the SWIR band. (b) A wide pass-band transmission around 1556 nm with a wide bandwidth of 200 nm. (c) A narrow pass-band transmission around 1536 nm with a narrow bandwidth of 10 nm, (d) the notch-reflection spectrum corresponding to pass-band of part (c). (e) An ultra-wide stop-band reflection spectrum around 1565 nm with a bandwidth of 530 nm.

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From the characterization curves in Figs. 2 and 3, it is obvious that PWEC width (W) and period (Λ) are crucial design parameters for the DH-spectral characteristics. The interaction between the zero and second-order modes, and in turn the shape of the two humps, can be controlled by the appropriate choice of ‘Λ’ and ‘W’, in addition to ‘N’ (i.e., PWEC length) as well. In other words, the ‘Λ’ and ‘W’ values of the short, stand-alone, and uniform PWEC grating can be engineered together to reshape the DH-spectrum. This in turn allows for the following different spectral shapes such as wide, narrow, closely-separated, far-separated, pass, or stop-bands spectra. All of them can be suitable for different applications.

4.1. Double stop-band reflection spectrum

The double-humps in the reflection spectrum of PWEC correspond, to double stop-bands in its transmission spectrum [check Fig. 1(d)]. Figure 4(a) shows the ability to tune the DH-spectrum to selectively filter different bands within the shortwave infrared (SWIR) region. For the solid red curve, the PWEC parameters are adjusted to Λ = 0.41 µm, W=150 nm, and N=25. That gives a total PWEC length of L = 25 × 0.41 = 10.25 µm. The first red-hump reflects both the optical communications S-band (1460-1530 nm) and C-band (1530 -1565 nm), whereas the second red-hump reflects a wide range of SWIR band starting from 1700 to 1910 nm (i.e., BW₂ ≅ 210 nm). For the dashed-blue curve, the PWEC parameters are adjusted to Λ = 0.41 µm, W=250 nm, and N=20. The first blue-hump reflects both the optical communications E-band (1380 - 1460 nm) and S-band (1460-1530 nm), whereas the second blue-hump reflects a wide range of the SWIR-band starting from 1625 to 1895 nm (i.e., BW₂ ≅ 270 nm).

4.2. Single pass-band transmission spectrum

The separation between the double stop-bands in the PWEC transmission spectrum can be used to define a certain pass-band at the transmission port. Figure 4(b) shows one example of such a wide band-pass spectrum region around a center wavelength of ≅ 1556 nm with a full-width half-maximum (FWHM) ≅ 200 nm, extinction ratio ≅ -17dB, and a semi-unity flat-top with ≅ 10% average ripples. To achieve that, the PWEC parameters are adjusted to Λ = 0.34 µm, W=20 nm, and N=30, which in turn gives λ_B1 = 1.44 µm and λ_B2 = 1.700 µm, as shown at spectrum minima. In Figs. 4(c) and 4(d), the two humps are brought very close to each other so that the transmission region forms a narrow band-pass region [Fig. 4(c)] corresponding to a notch within the reflection spectrum in Fig. 4(d). In this case, the center wavelength is 1536 nm with FWHM ≅ 10 nm and extinction ratio ≅ -30dB. It is worth mentioning that the 10 nm pass-band falls within an ultra-wide 500 nm stop-band. The PWEC parameters in this case are adjusted to Λ = 0.38 µm, W=400 nm, and N=10, which in turn gives λ_B1 = 1443.5 nm and λ_B2 = 1700 nm.

4.3. Ultra-wide stop-band reflection spectrum

Figure 4(e) shows an ultra-wide bandwidth reflection spectrum acting like a full mirror. The center wavelength is equal to 1565 nm with FWHM ≅ 530 nm, extinction ratio ≅ -17.5dB, and semi-unity flat-top having ≅ 5% average ripples with reflectance more than 90%. To achieve this super-hump spectrum, the PWEC parameters are adjusted to Λ = 0.39 µm, W= 0.7 µm, and N=20. Which in turn gives λ_B2 = 1565 nm. The first hump seizes to exist, as discussed earlier because the second-order mode is cut-off beyond W=0.6 µm. The ultra-wide bandwidth of this super-hump (full mirror) is achieved, according to Eq. (5) , because the effective index reduces below 2 [Fig. 3(b)], and in the meanwhile, the coupling overlap-factor increases beyond 30% [Fig. 3(e)], thus the bandwidth could exceed 450 nm [Fig. 3(d)].

5. Conclusions

A novel partial-width entrenched core (PWEC) waveguide grating is numerically demonstrated. The PWEC shows a closely-separated double-hump Bragg spectrum with each hump supporting one of the waveguide even-modes. The double-hump spectrum characteristics are highly dependent on the air-trenches width, period, and numbers. The two resonance Bragg wavelengths can be tuned over a wide wavelength range from 1 to 2 µm. The spectrum of this stand-alone few-periods (maximum length here is 10.25 µm) uniform PWEC grating can be reshaped by mainly engineering the air-trenches width and period. The overall estimated dispersion of PWEC is found to be very small (less than 0.3 fs/nm). The double stop-band spectrum could be re-shaped to cover two standard optical communication bands in addition to a wide range of SWIR region. Also, a wide pass-band transmission spectrum with a bandwidth of 200 nm could be obtained. Besides, a narrow transmission pass-band with 10nm bandwidth that falls within 500nm ultra-wide transmission stop-band could be obtained (corresponding to a notch in reflection-spectrum). Finally, an ultra-wideband full-mirror (single super-hump) could be obtained with 530 nm bandwidth. The PWEC grating has a compact size of less than 20 µm×0.85 µm. The ultra-wide bandwidth and compact size are considered advantageous when compared to some previous literature, e.g. [8]. The PWEC can be simply fabricated by entrenching periodic air-gaps with partial core-width inside most of the few-mode waveguide types and materials. The chosen waveguide here was a silicon-on-insulator planar type that is compatible with CMOS fabrication technology. The SOI waveguide can be fabricated using one of the well-known conventional methods. While the air-trenches could be simply entrenched on top of silicon planar waveguide core using electron-beam lithography method.