Scalable high Q-factor Fano resonance from air-mode photonic crystal nanobeam cavity

2.1 Structure design

Figure 1(a) shows the schematic of our proposed Fano resonance device, consisting of a side-coupled bus waveguide with PTE and an air-mode PCNC. Since the fundamental mode profile of the PCNC is highly concentrated in the central region of PCNC, an arc bus waveguide is designed to suppress the coupling of higher order modes. The radius of the arc feeding waveguide is 10 μm. The PTE is simply realized by etching two holes with radius r on the bus waveguide. The two holes are symmetric with respect to the red dashed line shown in Figure 1(a). Here, the center-to-center distance between the two air holes in x direction is labeled as d. For our fabricated devices, the distance d is set as 448 nm, which is the same as the lattice constant of the central tapered segment in PCNC. Note that to keep the fabrication uniformity, the holes’ radii are same for PCNC and PTE. As depicted schematically in Figure 1(a), Fano resonance occurs from optical interference between the discrete state (fundamental air-mode) of the PCNC and the continuum mode of the side-coupled line-defect waveguide with PTE. Part of the incident light launched at the input waveguide port with field amplitude S _i passes through the PTE directly, while the other part is coupled to the cavity mode in PCNC. A is the field amplitude of the fundamental mode in PCNC with a resonant frequency of ω ₀. The energy in PCNC can decay into input and output waveguides with decay rates γ ₁ and γ ₂, respectively, or decay as an intrinsic loss into the free space with intrinsic loss rate γ _i . t _B and r _B are the transmission and reflection coefficients of PTE, respectively, satisfying t B 2 + r B 2 = 1 . The value t _B can be tuned by controlling the distance between the two holes formed PTE, which can be used to engineer the parity and the shape of the Fano resonance spectrum [21–23, 40–42]. S _r is the reflected field amplitude at the input waveguide port. And the light is coupled out of the structure through outport of the waveguide with the transmitted field amplitude S _t.

Figure 1:

Structure design. (a) Schematics of the proposed air mode Fano resonance PTE-PCNC and side-coupling model. The structure is symmetric with respect to its center (red dashed line). w is the width of the waveguide. The air holes have radius r, and are kept constant. The lattice constant a(i) are quadratically modulated from center to both sides. (b) TE dispersion diagram of the PCNC with a = 448 nm (red line) and a = 400 nm (blue line). The red circle indicates the target resonant frequency. (c) Mirror strengths at different lattice constant from the 3D band diagram simulation.

The PCNC consists an array of circular holes etched into a Si strip waveguide with a width w of 450 nm and a thickness of 220 nm. The diameter of the holes is 2r = 180 nm, which conforms the minimum feature size. The structure is symmetric with respect to the red dashed line shown in Figure 1(a). The PCNC is optimized using the deterministic high-Q design method introduced by Quan et al. [43]. To create a Gaussian mirror reducing the intrinsic loss, for the tapered region, the hole lattice constant is quadratically tapered from a(1) = 448 nm in the center to a(N _taper) = 400 nm on both sides (a(i) = a(1) − (i − 1)²(a(1) − a(N _taper))/(N _taper − 1)², i increases from 1 to N _taper), while other parameters of the structure remain unchanged. The transverse-electric (TE) band diagram with lattice constant 448 nm and 400 nm simulated by a 3D-FDTD method with Bloch boundary conditions is given in Figure 1(b). Here, a(N _taper) is chosen to realize maximum mirror strength (γ), as shown in Figure 1(c). The mirror strength γ for different lattice constant is calculated by ( w 2 − w 1 ) 2 / ( w 2 + w 1 ) 2 − ( w 0 − w m ) 2 / w m 2 , where w ₀ is the proposed target resonance, and w ₂, w ₁, and w _m are the air band edge, dielectric band edge, and midgap frequency of each segment, respectively [43]. The air band frequency indicated by red circle in Figure 1(b) is the target resonant frequency. The decrease of lattice constant makes the effective refractive index of the guiding structure decreases, thus the cutoff frequency becomes higher correspondingly, as shown in Figure 1(b).

2.2 Temporal coupled-mode theory

Here, we analyze the coupled fundamental air-mode of PTE-PCNC according to temporal coupled-mode theory (TCMT). For an input light with field amplitude S _i = e^−iωt , therefore, in the steady state, the resonant field amplitude of fundamental mode dA/dt = −iωA. The equations for the evolution of the PTE-PCNC mode amplitude in time are given by

Where e i θ 1 and e i θ 2 are the phase coupling coefficients of the coupling between the input/out ports and the PCNC, respectively. Accordingly, the power transmission of the PTE-PCNC coupling system can be calculated as follows:

The phase term e i θ 1 and e i θ 2 can be calculated from the values of t _B , γ _1, and γ ₂ using the requirements of energy conservation and time-reversal symmetry as

Here, the factor P takes the value +1(−1) for the red (blue) parity Fano resonances. The parity of a Fano line is defined by whether the transmission minimum is red- or blue-shifted relative to the maximum [23]. For our proposed symmetric PTE-PCNC design, the fundamental air-mode is odd with respect to the mirror symmetry plane. Therefore, γ ₁ = γ ₂, θ ₁ = θ ₂, and cos ( 2 θ 1 ) = ( γ 2 − γ 1 ) t B 2 − 2 γ 1 γ B 2 / 2 γ 1 γ B = − γ B [29]. Correspondingly, for P = ±1, Eq. (3) can be deduced as:

Apparently, the cavity loss is composed of the intrinsic loss (characterized by Q _i = ω ₀/2γ _i ) and the coupling loss (characterized by Q _c = ω ₀/2(γ ₁ + γ ₂) = ω ₀/4γ ₁) to the coupling waveguide. Thus, the net dimensionless total decay rate 1/Q _t can be written as the sum of two decay rates: 1/Q _t = 1/Q _i + 1/Q _c . In practice, the intrinsic loss of the cavity includes the radiation loss into the free space (characterized by Q _r ) and scattering loss induced by fabrication imperfection (characterized by Q _s ). While, in simulation, only the Q _r is considered. And the Q _c and Q _r are evaluated by Q = w ₀ U/P using 3D-FDTD simulation, where ω ₀ is the resonance frequency, U is the electromagnetic energy in the cavity and P is the rate of energy loss. Hence, according to Eq. (5), for given t _B , ω ₀, Q _c and Q _r , the transmission spectrum can be calculated. Similarly, for given measured transmission spectrum, Q _c and Q _i can be extracted by fitting to Eq. (5).

4.1 Fabrication

Our devices were fabricated on an 8-inch SOI wafer with a 220 nm-thick top-silicon layer and a 3 µm-thick BOX in a CMOS pilot line. To realize optical characterization, the waveguides were integrated with TE grating couplers. The DUV photolithography was employed to form the devices patterns on photoresist as a soft mask. Double Inductively coupled plasma (ICP) etching processes were applied to transfer the patterns from the photoresist layer to silicon to form waveguides and devices. 220 nm full etching was defined for waveguides and circle holes, and 70 nm shallow etching was defined for grating lines. The devices were oxide-embedded through oxide deposition by plasma-enhanced chemical vapor deposition (PECVD) and planarization down to the top of the silicon layer.

Figure 4(a) shows the finished 8-inch wafer fabricated by MPW run in a CMOS pilot line at an ∼180 nm technology node. Figure 4(b) shows the single die from the 8-inch wafer. And the inset microscope image shows the fabricated devices with different N _taper and g. Figure 4 (c) shows the scanning electron microscope (SEM) image (xy plane) of the fabricated device, in which light is coupled into the bus waveguide by the focusing grating coupler, then passes through the PTE-PCNC, and finally gets directed out by the other grating coupler. The SEM images of the fabricated grating coupler and PTE-PCNC are shown in Figure 4(d) and (e), respectively. A close-up SEM image of the evanescent coupling region between PCNC and coupling defect waveguide is shown in the zoom-in picture of Figure 4(e), indicating well-defined circular holes and the coupling region. The fabricated devices were characterized by measuring the transmission, using a tunable laser (from 1500 to 1620 nm), and cleaved single-mode fibers for input/output coupling. In the following experimental results, the shown measured transmission spectra were obtained by normalizing the spectra to a reference waveguide.

Figure 4:

Fabrication results. (a) Photography of the finished 8-inch wafer fabricated by MPW run in a CMOS pilot line. (b) Image of a single die from the 8-inch wafer. The inset microscope image shows the fabricated devices. The SEM images of fabricated (c) device, (d) grating coupler, and (e) side-coupled PCNC with PTE fabricated at ∼180 nm technology node.

4.2 Measurement and analysis

To demonstrate the characteristics of the proposed Fano resonance from air-mode PTE-PCNC, the normal side-coupled air-mode PCNC with N _taper = 6 (g = 200 nm), the PTE-PCNCs with g = 200 nm (N _taper = 6, 18 and 30) and N _taper = 30 (g = 400, 600 nm) are fabricated and measured. The normalized experimental transmission spectra are shown in Figure 5. And the extracted resonance frequency ω ₀, asymmetry parameter q, total Q-factor Q _Γ , intrinsic Q-factor Q _i, and coupling Q-factor Q _c are marked in Figure 5 correspondingly. For the fabricated device, Q _i is composed of Q _r and Q _s .

Figure 5:

Measured and fitted transmittance spectra for the fundamental modes of the normal side-coupled PCNC and PTE-PCNCs with w = 450 nm, r = 90 nm, a _center = 448 nm, and a _end = 480 nm at various g and N _taper.

As shown, for normal side-coupled PCNC with N _taper = 6 and g = 200 nm, the measured transmission has Lorentzian resonance line-shape (q = 0) at resonance frequency of 193.0921 THz. The small frequency mismatch between the simulation and the experiment is possibly caused by fabrication imperfections or simulation parameter settings, such as mesh accuracy. The insertion loss is ∼0.5 dB at the non-resonance frequency. A Lorentzian fit to the measured transmission yields a total Q-factor of 805, which for the measured ER of 1.2 dB corresponds to an intrinsic Q-factor of Q _i = 902 and a coupling Q-factor of Q _c = 7422. Compared with the simulation results shown in Figure 2(a), the smaller measured Q _c may be caused by the shallow etching depth in the coupling region due to the RIE-lag effect when g = 200 nm.

As expected, for PTE-PCNC with N _taper = 6 and g = 200 nm, a sharper and higher-ER asymmetric Fano line-shape appears in the measured transmission spectrum, though the total Q-factor Q _Γ is 323 achieved by fitting the transmission spectra using the Fano line-shape formula. The asymmetric parameter q is estimated as q = 1.62, proving the consistency of the simulation result. The insertion loss is ∼2.5 dB at the non-resonance frequency. Inserting the data extracted from the spectra into Eq. (5) and 1/Q _Γ = 1/Q _i + 1/Q _c , the experimental Q _i = 850 and Q _c = 521 are derived. As N _taper grows from 6 to 30, the resonance frequency of the fundamental mode decreases from 193.6433 THz to 191.2015 THz, and the corresponding total Q-factor Q _Γ increases from 323 to 541. Though both derived Q _c and Q _i increase with the growth of N _taper, the results are much smaller than the theoretical ones shown in Figure 3(b). When N _taper = 30 and g = 200 nm, the derived experimental Q _i and Q _c are 3880 and 629, respectively, resulting a high ER ∼21.5 dB and SR ∼7.3 dB/nm. Since the simulated Q _r is larger than 2 × 10⁴, the Q _i is mainly limited by fabrication imperfection induced scattering loss Q _s .

By increasing the coupling gap from 200 nm to 600 nm with a step of 200 nm, the experimental coupling condition changes from initial over-coupling (Q _c < Q _i ) to under-coupling (Q _c > Q _i ). The increase tendency of the experimental results (Q _Γ, Q _c and Q _i ) are in agreement with the simulation results shown in Figure 3. And Q _c grows more rapidly than Q _i , resulting the monotonically decreasing of ER. When g = 400 nm, the fundamental mode is achieved at resonance frequency of 195.5281 THz. The corresponding Q _Γ, ER, and SR are 6.52 × 10³, 10.8 dB, and 53 dB/nm, respectively. The derived Q _i and Q _c are 1.52 × 10⁴ and 1.34 × 10⁴, respectively. Increasing the gap distance to 600 nm, the derived Q _i grows weakly to 2.03 × 10⁴, while Q _c increases rapidly to 7.62 × 10⁴, resulting in a lower ER ∼ 2.91 dB and SR ∼ 49 dB/nm. The phenomenon demonstrates that the measured total Q-factor 1.60 × 10⁴ is mainly limited by the fabrication induced scattering loss since the calculated Q _r is ∼2 × 10⁵. Therefore, in further studies, how to decrease the fabrication induced imperfection in CMOS line should be studied to achieve higher Q and ER.

In addition, as observed, higher-order resonances modes can be observed when N _taper = 30. Compared with the fundamental mode, higher-order modes are significantly suppressed, due to the spatial separation of the cavity and waveguide modes. For g = 400 nm, higher-order resonance mode at frequency of 197. 4576 THz is present with higher total Q-factors of 6.57 × 10³. However, the ER is only ∼3.9 dB, and it disappears at larger gaps. This feature enables single-frequency operation over a very broad frequency range, compared to ring resonators.

To achieve more insightful information than single device, a sufficiently large amount (30 dies) of PTE-PCNCs with N _taper = 30 is tested. The variations of the measured resonance wavelengths, Q-factors, and ERs derived from the measured transmission spectra for g = 400 nm and 600 nm are shown in Figure 6(a) and (b), respectively. Fluctuations in the measured values are due to intrinsic disorder introduced by the fabrication process. The average resonance wavelength values for g = 400 nm and 600 nm are 1537.5 nm and 1532.5 nm, respectively. The result reveals red-shift of resonance due to increased effective refractive index as the coupling gap g narrows. The standard deviations are 5.9 nm and 8.1 nm, respectively, which should be decreased in further study by improving the structure fabrication tolerance. For g = 400 nm and 600 nm, the highest Q-factors are 1.29 × 10⁴ and 2.41 × 10⁴, which correspond to ERs of 6.3 dB and 3.1 dB, respectively. In average, the Q-factors of the fundamental mode for g = 400 nm and 600 nm are about 7.58 × 10³ and 1.58 × 10⁴, respectively, for 30 measured resonances in total. Thus, here a large Q factor has been achieved systematically. These results indicate that the mass manufacture of high-Q Fano resonance from air-mode PTE-PCNC is feasible by using CMOS-compatible technologies.

Figure 6:

Experimental resonant wavelengths, Q and ER distributions of 30 measured dies for PTE-PCNCs with N _taper = 30, (a) g = 400 nm and (b) g = 600 nm, respectively. The solid line indicates the average values. The dashed line indicates the standard deviation values.

To better understand the origin of the statistical variations of the device characteristics, we numerically estimate the effects of fabrication errors including fluctuation of air–hole radii, positions, waveguide width and coupling gap. Taking PTE-PCNC with N _taper = 30, g = 400 nm for example, to study the fabrication error tolerance of the cavity, firstly, the influence of fabrication offsets in the air hole radius with uniform offset (∆r) is studied. We set the radius offset varying from −10 nm to 10 nm with a step of 5 nm. As shown in Figure 7(a), with the increase of holes’ radius, the resonance wavelength shifts to shorter wavelength range due to the decrease of effective refractive index. And the total Q-factor decreases with the increase of offset. When the offset is ±10 nm, the total Q-factor of above 6332 is achievable. However, in practice, the radius offset for each holes on the device is not uniform. To study the fabrication error in more detail, from SEM images of the devices on the die from center of the wafer, the distribution of air hole diameters can be described by a Gaussian distribution with mean value being 180.2 nm and standard deviation of 9.7 nm, as shown in Figure 7(b). Since the positions of the air holes are hard to determine accurately due to the quadratically tapered lattice constant structure, we also assume the maximum deviation of positions to be same as the maximum deviation of radii. Therefore, 3D-FDTD calculations taking the randomness of the hole radii and positions into account are performed. As shown in Figure 7(c), the holes’ radii and positions are perturbed by random deviations δr, δ _x, and δ _y varying from −5 nm to 5 nm. Figure 7(d) shows 20 different random results. The average resonant wavelength, Q-factor, and ER are 1579.7 nm, 7215, and 14.1 dB, respectively. And the corresponding standard deviations are 2.1 nm, 1080, and 1.7 dB, respectively. The average Q-factor corresponds well with the measured characteristics of the devices shown in Figure 7(a). Note that for different dies the mean value of the holes’ diameters has slightly shift, therefore the simulated standard deviations here are smaller than the measured results.

Figure 7:

Fabrication tolerance analysis. (a) The variation of resonance wavelength and total Q-factor versus uniform hole radius offset ∆r. (b) The statistical distribution of the air holes’ diameters exhibiting Gaussian distributions. (c) Holes’ radii and positions perturbed by random deviations δ _r , δ _x, and δ _y . (d) The corresponding fluctuation of resonance wavelength, total Q-factor and ER for 20 different random patterns.

Since the measured average resonance wavelength of the devices shift to 1537.5 nm, we study the effect of the fabrication variation of waveguide width and coupling gap of the PTE-PCNC structure on the performance of the resonance mode, as presented in Figure 8. Because the fabrication variation of waveguide width generally fluctuates within 10 %, therefore, we set the waveguide width offset (∆w) varying from −40 nm to 40 nm with a step of 10 nm. As shown in Figure 8(a), with the increase of the waveguide width, both the resonance wavelength and Q-factor increase due to the increased effective refractive index and stronger light confinement. Keeping the waveguide width 450 nm unchanged, setting the coupling offset (∆g) varying from −40 nm to 40 nm as shown in Figure 8(b), the resonance wavelength almost keep unchanged, while the Q-factor increases with the decreased coupling gap due to the decreased coupling loss. In practice, as the waveguide width decreased, the coupling gap correspondingly became larger. Therefore, as shown in Figure 8(c), when ∆w and ∆g working together, the total Q-factor of 6258–8935 is achievable. When waveguide width and coupling gap equal to 410 nm and 440 nm, respectively, the corresponding resonant wavelength, Q-factor are 1539 nm and 8115, respectively. This, together with the results shown in Figure 7 for tuning of the holes’ size and position, demonstrates the good correspondence between the measured and simulated operating wavelength, Q-factor and ER taking into account a reduction of 40 nm in width, increase of 40 nm in coupling gap, 5 nm random fluctuation in holes’ radii and position. This in turn demonstrates the accuracy and reliability of CMOS technology for the fabrication of the proposed PTE-PCNC structure. Therefore, the structure retains good performance even after several nanometer fabrication fluctuations. Note that for PTE-PCNC structure with larger coupling gap, the waveguide and gap offsets caused by fabrication are larger, resulting in the red-shift of the measurement average resonance wavelength when g = 600 nm as shown in Figure 6. Though the value of random deviations δr, δ _x, and δ _y are a few times larger than the value obtained with EBL where standard deviation of hole diameters less than 1 nm has been reported [46, 47], the performance of our device exceeds previous reported air-mode PCNC [48] and Fano resonance from PCNC [17, 21, 28] fabricated by EBL.

Figure 8:

The variation of resonance wavelength and total Q-factor versus (a) waveguide width offset ∆w, (b) coupling gap offset ∆g, and (c) ∆w and ∆g changing together.