2.1.

Device Design

The design parameters were optimized with a SOI device layer thickness (H) of 220 nm and box layer thickness of $2\mu \mathrm{m}$ [see Fig. 1(b)]. An unperturbed waveguide width ($W_g$) of 400 nm and a slab height ($h$) of 160 nm were chosen with a sidewall corrugation ($\mathrm{\Delta }W$) of 80 nm at both sides of the waveguide to operate within the single mode regime supporting only a transverse electric (TE)-like guided mode. The waveguide width of 400 nm is chosen to ensure tighter mode confinement and enhanced evanescent field, which can lead to a higher coupling coefficient as compared with a typical waveguide width of 500 nm or similar. A grating period ($\mathrm{\Lambda }$) of 296 nm was considered to obtain the filter passband centered around ${\lambda }_c\sim 1570\mathrm{nm}$ as per Bragg condition. As we know from the theory of DBR, the number of gratings (N) is an important design aspect of the filter, which greatly influences the resonance bandwidth as well as the rejection bandwidth, extinction ratio, and shape factor. In the n’th-order filter, the N of different grating stages (denoted as $N_i,i=\mathrm{1,2},3,\dots ,n,n+1$) can be varied to obtain the desired figures of merit as listed previously. The cavity length $C_i,i=\mathrm{1,2},3,\dots ,n$ decides the number of longitudinal modes as per the standard Fabry–Pérot resonator concept. It has been fixed at $C_i=\mathrm{\Lambda }/2$ to obtain a single coupled-cavity resonance, which can be used as a flat-top rectangular bandpass filter.

The transmission spectrum corresponding to the filters has been simulated using the transfer matrix method, as discussed by Chrostowski and Hochberg²⁵ The simulation method uses the Fresnel equations to calculate the grating coupling coefficient using plane-wave approximations. Then, the coupling coefficient is further used to calculate the reflection/transmission coefficient matrix for each period of the grating. The propagation coefficient matrix is also calculated using the waveguide loss and the phase constant. Therefore, the overall transmission/reflection of the entire grating structure is the accumulation of the individual grating transmission/reflection and calculated by the matrix multiplication of individual gratings. In simulations, the waveguide propagation loss has been considered 5 dB/cm (typical value as per our in-house fabrication facility). Figures 2(a) and 2(c) show the simulated transmission spectrum of a fourth- and fifth-order DBR resonant filter, respectively. The number of gratings used for the beginning and end grating stages is kept fixed at $N_1=N_5=100$ and $N_1=N_6=100$ for the fourth- and fifth-order resonance filters, respectively. The central grating stages ($N_3$ for the fourth order and $N_3,N_4$ for the fifth order) consist of $N=200$, which is twice that of $N_1$, whereas the rest of the grating stages have $N=190$ number of gratings.

Fig. 2

(a) Simulated transmission characteristics of a fourth-order DBR resonator for $N_1=N_5=100$, $N_2=N_4=190$, and $N_3=200$; (b) zoomed view of the passband transmission in the cavity; (c) Simulated transmission characteristics of a fifth-order DBR resonator for $N_1=N_6=100$, $N_2=N_5=190$, and $N_3=N_4=200$; and (d) zoomed view of the passband transmission in the cavity.

As the overall device length for the fifth-order filter is higher at $\sim 290\mu \mathrm{m}$ as compared with that of the fourth-order filter ($\sim 231.5\mu \mathrm{m}$), the extinction ratio of the stopband is also higher at $\sim 120\mathrm{dB}$ for the former and $\sim 75\mathrm{dB}$ for the latter case, in accordance with the theory. Figure 2(b) shows a zoomed transmission of the passband of the fourth-order filter with a 3-dB bandwidth of 0.79 nm and 10-dB bandwidth of 0.9 nm, resulting in a shape factor of 0.87. Similarly, the zoomed transmission of the passband of the fifth-order filter in Fig. 2(d) shows a 3-dB bandwidth of 0.59 nm and a 10-dB bandwidth of 0.65 nm, which gives a shape factor of 0.91. We observe the shape factor of the fifth-order filter design to be higher along with fewer in-band ripples as compared with the fourth-order design. The insertion loss in the passband was found to be 0.62 and 1.6 dB for the fourth- and fifth-order filter, respectively, which implies that the passband extinction is almost equivalent to the stopband extinction ratio. With careful selection of the grating parameters ($\mathrm{\Delta }W,\mathrm{\Lambda },N_i,C_i$) and order of the filter ($n$), a lossless and high extinction filter can be designed with a high shape factor.

2.2.

Device Fabrication

The devices were fabricated using our SOI (device layer $H=220\mathrm{nm}$, box layer = $2\mu \mathrm{m}$, and handle layer = $500\mu \mathrm{m}$)-based in-house waveguide process technology at the CNNP, IIT Madras. The devices were patterned using single-step e-beam lithography using negative tone resist followed by inductively coupled plasma reactive-ion etching. The devices were terminated with grating couplers centered around 1550 nm (period = 620 nm) for fiber-optic couplings at the input/output access waveguides. Reference waveguides of length the same as the corresponding devices were also fabricated to help normalize the wavelength-dependent grating coupler response of device transmission characteristics. The list of fourth-order DBR cavity devices with the corresponding grating parameters is provided in Table 1.

Table 1

Fabricated device parameters for fourth-order DBR cavity.

The SEM image of one of the fabricated fourth-order filters is shown in Fig. 3. The measured device parameters are close to the design values except for the perturbation width, which is slightly higher than designed. However, it may be observed that the grating modulations are rounded around the edges, which can have an impact on the coupling coefficient.

Fig. 3

SEM images of the fourth-order filter with $N_1=N_5=100$ and $N_2=N_4=N_3=180$: (a) DBR region and (b) cavity region.

2.3.

Device Characterization

Figure 4 shows the schematic block diagram of the experimental set-up for device characterization. The transmission characteristics of the fabricated devices were measured using an optical source-cum-spectrum analyzer (APEX AP2043B, France) with a resolution bandwidth of 0.8 pm and sensitivity of $-75\mathrm{dBm}$ for a wavelength range of $1520\mathrm{nm}\le \lambda \le 1625\mathrm{nm}$. All the measured device transmission characteristics have been normalized with respect to the reference waveguide transmission to eliminate the wavelength-dependent grating coupler response.

Fig. 4

Schematic block diagram of characterization set-up.

The normalized transmission characteristics of DBR cavities are shown in Fig. 5. The observed Bragg stopband of the devices is found to be $\sim 20\mathrm{nm}$ red-shifted in comparison with that of the simulation results, as shown in Fig. 2. Also, the central passband is slightly blue-shifted with respect to the phase-matched Bragg wavelength. The deviation in the position of the stopband may be attributed to the deviation in grating modulation, slab height thickness variation, and slight phase errors introduced in the cavity after fabrication during fabrication. For devices $D_1,D_2$, and $D_3$, the number of gratings in the central DBRs ($N_2,N_3,N_4$) is slightly less than twice that of $N_1,N_5$. The 3-dB bandwidth in the case of $D_1$ is the lowest, whereas the 10-dB shape factor for $D_3$ is the highest among the three devices. Thus, we observe that a grating ratio $N_3/N_1$ of 1.95 and $N_2/N_1$ of 1.9 (both slightly lesser than the ratio 2) gives a better shape factor in exchange for a slightly higher 3-dB bandwidth. But, the 3-dB bandwidth mainly depends on the grating coupling coefficient because the overall grating length here does not change significantly from device to device. For device $D_4$, with $\frac{N_3}{N_1}=2$ and $\frac{N_2}{N_1}=1.95$, the shape factor is similar to that of $D_1$ but with a higher 3-dB bandwidth. As the ratio becomes exactly 2 for both $\frac{N_3}{N_1}$ and $\frac{N_2}{N_1}$ in $D_5$, the shape factor further reduces to 0.8 with a broader passband. In device $D_6$ with $\frac{N_3}{N_1}=2.05$ and $\frac{N_2}{N_1}=2$, the shape factor is the lowest at 0.74 with the highest 3-dB bandwidth. Although the overall insertion loss in the passband for all the devices is well within 1 dB with the in-band ripple of $\sim 1\mathrm{dB}$, some of them show slight distortion in the flat-top envelope. Another observation can be made in terms of the asymmetry of the stopbands around the passband in all the devices, which may be attributed to fabrication-induced phase errors in the grating structures around the cavities and can be mitigated with better process control or active tuning. The extinction ratio of the passband in devices $D_1,D_2$, and $D_3$ is $\sim 30\mathrm{dB}$, whereas it reduces to $\sim 10\mathrm{dB}$ for device $D_6$. The sideband ripples in all cases are within $\sim 10\mathrm{dB}$, which can be further reduced with apodization techniques, as discussed in Sec. 3.

Fig. 5

Normalized transmission of fourth-order DBR cavity: (a) $D_1$: FWHM = 1.74 nm, shape factor = 0.81; (b) $D_2$: FWHM = 1.96 nm, shape factor = 0.83; (c) $D_3$: FWHM = 2.02 nm, shape factor = 0.88; (d) $D_4$: FWHM = 2.24 nm, shape factor = 0.81; (e) $D_5$: FWHM = 2.66 nm, shape factor = 0.8; and (f) $D_6$: FWHM = 3.11 nm, shape factor = 0.74.

The normalized transmission spectrum for a fifth-order DBR cavity is shown in Fig. 6(a) along with the zoomed passband in Fig. 6(b). A lossless flat-top spectrum is observed with in-band ripples within $\sim 2\mathrm{dB}$. The 3- and 10-dB bandwidths are measured to be 2.13 and 2.34 nm, respectively, which result in a 10-dB shape factor of 0.91. The combination of $\frac{N_3}{N_1}=2.0$ and $\frac{N_2}{N_1}=1.9$ is chosen after the trends observed in the fourth-order filter characteristics, which can be further explored for enhanced shape factor. As observed in the simulations, with more coupled cavities, the rectangular shape of the passband can be engineered to provide high roll-off filter characteristics. The behavioral trend of higher-order filters can be considered analogous to a higher-order Butterworth filter where the passband roll-off gets sharper with increasing filter order. Haus and Schmidt²⁰ discussed the equivalent inductor-capacitor-based model in detail and provided a framework for modeling the higher-order Bragg resonators in terms of filter response.

Fig. 6

Normalized transmission of fifth-order DBR cavity is shown in (a) $N_1=N_6=100$, $N_2=N_5=190$, and $N_3=N_4=200$; (b) zoomed spectrum of the flat-top passband, shape factor = 0.91, $\mathrm{\Delta }{\lambda }_{3\mathrm{dB}}=2.13\mathrm{nm}$.

3.1.

Device Design

The top and cross-sectional views of the proposed device have been shown schematically in Fig. 7, with important design parameters annotated in the schematic. The grating width is apodized from $W_0$ to $W_m$ in the middle along with linearly tapering the input/output single-mode waveguide width from $W_0$ down to $W_0^{\prime }$. The values of grating parameters $L_g,W_m,W_0^{\prime }$ and the grating period $\mathrm{\Lambda }$ (50% duty cycle), critical to obtaining ultra-broad stopband and high Q without introducing significant insertion loss in the passband, have been carefully chosen. The grating periodicity ($\mathrm{\Lambda }$) has been chosen to be 292 nm to fix the Bragg wavelength ${\lambda }_B\sim 1550\mathrm{nm}$ with the desired ultra-broad stopband ($\mathrm{\Delta }\lambda >60\mathrm{nm}$). The length of the DBR mirrors denoted as $L_g$ is stretched from $-\frac{L_g}{2}$ to $+\frac{L_g}{2}$ along the X-axis, as shown in Fig. 7(b). The fundamental guided mode (TE-like) launched from the input single-mode waveguide expands adiabatically to have maximum overlap with the grating structure at the center of the DBR resulting in very strong coupling strength at the phase-matched Bragg wavelength ${\lambda }_B=2n_{\mathrm{eff}}\mathrm{\Lambda }$ and then symmetrically tapered down into the output single mode waveguide, where $n_{\mathrm{eff}}$ is the average effective index of the grating region at ${\lambda }_B$. This apodization technique ensures that the effective indices of the forward and backward coupled fundamental modes along the device length can be maintained nearly uniformly and the same as that of the input/output waveguide.

Fig. 7

(a) Schematic top-view of the proposed device, (b) adiabatically apodized DBR (used as grating mirror) integrated into a single-mode rib waveguide, and (c) cross-section view of the apodized DBR at the center: $W_0$, input/output waveguide width for fundamental TE mode guidance; $W_m$, maximum width of the grating; $W_0^{\prime }$, minimum width of the linear taper; $\mathrm{\Lambda }$, grating period; H, device layer thickness; h, rib height; $L_g$, length of Bragg grating mirror; $L_c$, length of the cavity.

To support fundamental TE-like mode, waveguides with $W_0=450\mathrm{nm}$ and $h=150\mathrm{nm}$ have been chosen for input/output access waveguides. An optimized maximum grating width $W_m=1.5\mu \mathrm{m}$ and a minimum taper width $W_0^{\prime }=100\mathrm{nm}$ are used for further device simulation. The coupled mode theory-aided-transfer matrix method (CMT-aided-TMM)–based semi-analytical simulation approach is discussed in detail elsewhere.²⁷ To briefly summarize the approach for CMT-aided-TMM model, we have used the solutions of the coupled mode equations used to model the Bragg structures to generate the transmission coefficient matrix of each grating. This is different than the approach described in Sec. 2 where the Fresnel equations using plane wave approximation are used to obtain the same matrix for each grating. Our proposed method is helpful in providing the flexibility to model apodized grating structures while remaining within the confines of the standard coupled mode theory. The overall transmission is calculated similar to the previous transfer matrix method by sequentially multiplying the grating transmission matrices.

Figure 8 shows the simulated transmission spectra of the DBR resonators for a cavity length of $L_c=146\mathrm{nm}$ and other optimized grating parameters as mentioned above. For a fixed $W_m=1.5\mu \mathrm{m}$, the Q value of the resonance and stopband extinction ratio is observed to increase with $L_g$. The inset figure shows the blue shift in the resonance wavelength due to a reduction in the average effective index as the $W_m$ and the apodization function are maintained, whereas $L_g$ increases. The insertion loss at resonant wavelengths increases with $L_g$ because of the higher reflectivity of the DBR mirrors (lower extracted power from the cavity in the transmission characteristics, even though the Q value increases).

Fig. 8

Calculated transmission for DBR resonator for a cavity length $L_c=146\mathrm{nm}$: (a) with $W_m=1.5\mu \mathrm{m}$ for N = 61, 71, and 81. (The waveguide propagation loss $\alpha$ is assumed as 1 dB/cm.)

3.2.

Device Fabrication

The fabrication steps for this device are the same as those described in Sec. 2.2. Subsequently, the SEM image of one of the smallest fabricated apodized DBR cavity devices is shown in Fig. 9.

Fig. 9

SEM image of a DBR resonator with $N=61$, $W_m=1.5\mu \mathrm{m}$, and cavity length of 146 nm (total device length $35.76\mu \mathrm{m}$).

3.3.

Device Characterization

The characterization setup shown in Sec. 2.3 is used for measuring the transmission of this device. Figure 10 shows the spectra of a set of singly resonant apodized DBR cavity filters for three different $L_g$ with $L_c=146\mathrm{nm}$ and $W_m=1.5\mu \mathrm{m}$. The observed stopband $\mathrm{\Delta }{\lambda }_{\mathrm{stop}}$ was found to be $\sim 60\mathrm{nm}$, which shows that the apodized DBR designs can result in an ultra-broad stopband. The stopband of the DBR cavity decreases as the DBR length increases following theoretical prediction.²⁸ In addition, the observed blueshift in resonances as a function of $L_g$ is found to be in accordance with the simulation result discussed in Fig. 8. Although the stopband extinction is not seen to be significantly improving as a function of grating length as predicted by the theoretical simulation, the observed discrepancy can be attributed to the sensitivity of the optical spectrum analyzer used in our experimental setup. One could evaluate the DBR length-dependent extinction by increasing the input power level.¹⁶ Nevertheless, the resonator Q values are increased with grating length from $\sim 17,000$ to $\sim 51,000$ as expected. As a consequence, the insertion loss at the resonance transmission peak also increases as discussed earlier.

Fig. 10

Normalized transmission characteristics of singly resonant devices with cavity length $L_c=0.146\mu \mathrm{m}$, $W_m=1.5\mu \mathrm{m}$ for $L_g=17.81\mu \mathrm{m}$, 20.73 $\mu \mathrm{m}$, and $23.65\mu \mathrm{m}$; inset shows the zoomed resonance peaks.