## Device design and operation principle

Figure 1a shows the proposed filter architecture consisting of a central grating-assisted F-P cavity and two symmetrically side-coupled bend waveguides. The F-P cavity consists of a central waveguide with length *L*c, two identical tapered gratings, and two Bragg gratings. The Bragg gratings are designed as two symmetrical mirrors to reflect the light to the central waveguide. Each grating is formed by side etching single-mode waveguide to get narrow transoms with width *d* periodically, as shown in Fig. 1b. The period is *Λ*, and the duty circle is 0.5. The tapered gratings connect the central waveguide and Bragg grating to reduce the scattering loss and satisfy the phase-matching condition between the fundamental Bloch and the waveguide modes. The curved waveguides are used to couple the light into and out of the central waveguide with the waveguide-input-cavity coupling gap *g*1 and the cavity-output-waveguide coupling gap *g*2, as shown in the cross-section view (Fig. 1c). The resonance tuning is actuated by a thermo-optical heater deposited over the F-P cavity. In addition, a scanning electron micrograph (SEM) was conducted to check the etching quality of the three coupled waveguides with the F-P cavity, as shown in Fig. 1d.

The proposed ADF is similar to a traditional one except for two symmetric Bragg gratings as the mirrors and band-stop filters. Instead of supporting the traveling wave in a conventional ADF’s cavity, such as a ring/disk resonator, the cavity of the proposed ADF supports a pure standing wave. The operating principle can be formulated by the coupled-mode theory[30]. The power \({P}_{R }\)reflected from the input port, the transmission power \({P}_{T1}\)through port T1, and both drop ports D1,2 can be expressed as follows:

$$\begin{array}{c}{P}_{R}=\frac{\frac{1}{{\tau }^{2}}}{{(\omega -{\omega }_{0})}^{2}+{(\frac{1}{\tau }+\frac{1}{{\tau }_{o}}+\frac{1}{{\tau }^{\text{'}}})}^{2}}\#\left(1.\right)\end{array}$$

$$\begin{array}{c}{P}_{T1}=\frac{{(\omega -{\omega }_{0})}^{2}+{(\frac{1}{{\tau }_{o}}+\frac{1}{{\tau }^{\text{'}}})}^{2}}{{(\omega -{\omega }_{0})}^{2}+{(\frac{1}{\tau }+\frac{1}{{\tau }_{o}}+\frac{1}{{\tau }^{\text{'}}})}^{2}}\#\left(2.\right)\end{array}$$

$$\begin{array}{c}{P}_{D\text{1,2}}=\frac{\frac{1}{\tau {\tau }^{\text{'}}}}{{(\omega -{\omega }_{0})}^{2}+{(\frac{1}{\tau }+\frac{1}{{\tau }_{o}}+\frac{1}{{\tau }^{\text{'}}})}^{2}}\#\left(3.\right)\end{array}$$

where \({\omega }_{0}\)is the resonant frequency, \(\frac{1}{{\tau }_{o}}\)is the decay rate due to loss, \(\frac{1}{\tau }\)and \(\frac{1}{{\tau }^{{\prime }}}\) are the rates of decay into the input and output curved waveguides. At the resonance of \(\omega ={\omega }_{0}\) and the condition of \(\frac{1}{\tau }=\frac{1}{{\tau }_{o}}+\frac{1}{{\tau }^{\text{'}}}\), the resonant mode decays equally into the forward and the backward propagating waveguide mode. The power transferred into the output waveguide at resonance is maximized; we can find \({P}_{R}={P}_{T1}=0.25\), \({P}_{D\text{1,2}}=0.25(1-\frac{\tau }{{\tau }_{o}})\). For the case of strong coupling between the input waveguide and the cavity with an ultralarge \(\frac{1}{\tau }\), the power is equally distributed into drop ports D1 and D2, near 0.25. The method to increase the drop efficiency will be discussed in detail later.

The calculated transmission spectra of the designed filter at drop ports (D1 and D2) and through port (T1) are attained by the three-dimensional finite-difference time-domain (3D FDTD) method, as shown in Fig. 2a. These simulation parameters are *R*r=20 µm, *w* = 500 nm, *d* = 300 nm, Λ = 316 nm, *L*c=0 µm, *g*1 = 200 nm, *g*2 = 250 nm, *N*1 = 5 and *N*2 = 150, respectively. It can be observed that only one single resonant mode is excited at D1 and D2 ports from 1400 to 1630 nm. Across the entire wavelength range, the in-band insertion loss at drop ports is about − 6.5 dB (drop efficiency ~ 0.22), which agrees well with the theoretical analysis. The OBR at the drop port is about 16 dB. Figure 2b exhibits the electric field distribution of the whole structure at the wavelengths of 1400, 1516, and 1630 nm. At the resonant wavelength of 1516 nm, the light is almost equally divided into four parts. At non-resonant wavelengths, the filter is considered a tri-waveguide directional coupler. The light propagates to the T1 port directly, while very little light is transmitted to the D2 port. Thus, the optical power at the D2 port is larger than that at the D1 port.

The spectral response of our proposed filter depends mainly on the FSRFP of the F-P cavity with an ideal reflector and the stopband of the Bragg grating Δλsb. The strict FSR-free condition is Δλsb < FSRFP, only one resonant mode of the F-P cavity in the stopband of Bragg grating is excited. The FSR-free condition can also be satisfied by FSRFP < Δλsb < 2FSRFP only when the resonant wavelength is around the central wavelength of the stopband λc; otherwise, there exist two resonant modes. Considering that 3D FDTD simulation for the entire device is time-consuming, it is challenging to study the relationship between the device response and the key parameters profoundly and determine the device geometry with the FSR-free operation capability at the desired wavelength range by 3D FDTD.

In this case, the investigation of the coupled-mode theory seems particularly important. For the Bragg grating, it is known that the optical rejection *R*e and Δλsb of the Bragg grating are decided by the grating geometry[31]

$$\begin{array}{c}{R}_{\text{e}}={\text{t}\text{a}\text{n}\text{h}}^{2}\left(\kappa {L}_{\text{g}}\right)\#\left(4.\right)\end{array}$$

$$\begin{array}{c}\varDelta {{\lambda }}_{\text{s}\text{b}}=\frac{{\lambda }_{0}^{2}}{\pi {n}_{\text{g}}}\sqrt{{\kappa }^{2}+\frac{{\pi }^{2}}{{L}_{\text{g}}^{2}}}\#\left(5.\right)\end{array}$$

where \({{\lambda }}_{0}\), \({n}_{\text{g}}\), κ, and *L*g are the central wavelength of stopband, the group index, the coupling coefficient, and the grating length. \({{\lambda }}_{0}\) is decided by the Bragg phase-matching condition \({{\lambda }}_{0}=2{\Lambda }{n}_{\text{e}\text{f}\text{f}}\), \({n}_{\text{e}\text{f}\text{f}}\)is the effective index of the mode propagating through the grating. Once the geometry of the grating is given, \(\varDelta {{\lambda }}_{\text{s}\text{b}}\), \(\kappa ,\) and \({n}_{\text{g}}\) can be acquired by the 3D eigenmode expansion (3D EME) simulation algorithm according to Equations (4) and (5). The 3D EME algorithm is an exact and efficient simulation tool. Specially, it is much more time-saving than 3D FDTD for periodic structures. The FSR is given by

$$\begin{array}{c}{FSR}_{\text{F}\text{P}}=\frac{{\lambda }_{0}^{2}}{2\left({L}_{\text{c}}{n}_{\text{g}1}{+2L}_{\text{p}\text{d}}{n}_{\text{g}}+{2L}_{\text{t}}{n}_{\text{g}2}\right)}\#\left(6.\right)\end{array}$$

where \({L}_{\text{p}\text{d}}\) and \({L}_{\text{t}}\) are the penetration depth of the Bragg grating and the length of the tapered grating. \({n}_{\text{g}1}\), \({n}_{\text{g}}\), and \({n}_{\text{g}2}\) are the group refractive indices of central waveguide, Bragg grating, and the tapered grating, respectively. \({L}_{\text{p}\text{d}}\)can be given by\(\)effective refraction index of wide and narrow parts of the Bragg grating \({n}_{\text{n}}\) and \({n}_{\text{w}}\) [32]

$$\begin{array}{c}{L}_{\text{p}\text{d}}=\frac{1}{2}\left(\frac{{{\lambda }}_{0}}{4{n}_{\text{n}}}+\frac{{{\lambda }}_{0}}{4{n}_{\text{w}}}\right)/\text{ln}\left(\frac{{n}_{\text{w}}}{{n}_{\text{n}}}\right)\#\left(7.\right)\end{array}$$

The contour plots of the calculated Δλsb, FSRFP, and ratio *R* = Δλsb/ FSRFP are shown in Figs. 3a-3c, respectively, as functions of Λ and *d.* Note that the dispersion of effective refraction index and group index has been considered for the whole calculation process. The waveguide width *w* and *L*c are 0.5 µm and 0 µm, respectively. As shown in Fig. 3c, Λ is almost independent of the ratio *R*, which is more sensitive to the variation of *d*. Figure 3d exhibits a contour map of the calculated *R* as a function of *L*c and *d* for Λ = 316 nm. The ratio *R* has a strong correlation with *L*c and *d*. The white and blue contour lines of *R* = 1 and *R* = 2 in Figs. 3c and 3d divide the figures into three areas, corresponding to the three cases discussed in our previous work[23]. Therefore, the contour maps provide useful guidance for designing an FSR-free filter.

We further simulated temperature field distributions at different bias voltages using the finite-element solver to confirm the filter’s external wavelength tuning performance. The waveguide was embedded between the 3-µm-thick buried oxide as the bottom layer and the 1-µm-thick oxide as the top cladding layer. 100-nm-thick titanium (Ti) and 100-nm-thick gold (Au) serve as microheaters and contact pads. The device was simulated with the bias voltage set from 2 to 5 V. We can observe the obvious temperature rise in the waveguide, as shown in Fig. 4a. Figure 4b shows the transmission spectra with the bias voltage set from 0 to 6 V by 3D FDTD simulations. The resonant wavelength redshifts with the increasing bias voltage and the FSR-free operation feature has been observed for all the tuning configurations.

For device characterization, a broadband tunable laser system (Santec full-band TSL, 1260–1630 nm) is employed to characterize the fabricated devices. We also fabricated and characterized a reference structure near the proposed filter on the same chip. The reference structure comprises two grating couplers and a straight waveguide of 200 microns in length. Our proposed filter is connected by the grating couplers with the same parameters at both ends to facilitate characterization. Considering that the propagation loss of the straight waveguide can be ignored, the response spectra of the filter can be obtained by subtracting the insertion loss of the reference structure. The grating coupler with a 150-nm etching depth has a peak coupling efficiency of -7.5 dB and a 10-dB bandwidth of ~ 200 nm, which is much broader than that of the shallow-etched grating couplers with a 70-nm etching depth.

Figure 5 shows the normalized spectral response at D1 and T1 ports for the fabricated optical filter associated with various pitches. The critical parameters of *R*r=20 µm, *w* = 500 nm, *d* = 300 nm, *L*c=0 µm, *g*1 = 350 nm, *g*2 = 300 nm, *N*1 = 5, and *N*2 = 150 are adopted. As the pitch Λ deviates from 292 to 316 nm with a 6 nm interval, the resonant wavelength has a redshift from 1431.3 to 1505.9 nm. We can only observe a single sharp peak with an OBR of ~ 21.5 dB in a 220-nm wavelength span from 1360 to 1580 nm at the D1 port. The strong noises from 1360 to 1420 nm result from the limited bandwidth of the grating coupler. The insertion loss is lower than 8.4 dB, and the 3-dB bandwidth is smaller than 0.5 nm. Figure 6a shows the measured spectral responses at the D1 and T1 ports of the fabricated optical filter with Λ = 298 nm, which demonstrates the tunability and provides a guide to accurate wavelength routing. When the applied bias voltage for heating varied from 0 to 8 V in the experiments, the wavelength can be tuned with a wavelength shift of ~ 12.3 nm. The inset shows the zoom-in view of the resonant peaks. There is a linear relationship between the resonant wavelength shift and the heating power, as shown in Fig. 6b. The thermo-tuning efficiency, namely, the slope of the fitted curve, is 97 pm/mW, which can be further increased by improving the alignment between the waveguide and the microheater. In addition, the tuning range can be increased to about 30 nm by utilizing isolation trenches surrounding the filter[33]. Compared to the microheater without trenches, the one with trenches can avoid heat dissipation from the side to enhance the heating efficiency. However, the maximum resonance wavelength shift will be limited by the highest power the microheater could withstand. As a result, it is difficult to cover the entire FSR-free range via heating due to the small thermo-optic coefficient of silicon (1.86×10− 4 at 1550 nm). However, tens of nanometers of tuning range are also meaningful and enable a lot of applications in sensing, communication, and spectroscopy. The 3-dB bandwidth and the insertion loss variation are smaller than 0.17 nm and 1.1 dB, respectively, as shown in Fig. 6c.