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Abstract
A compact, low-loss 2 × 1 angled-multi-mode-interference-based duplexer is proposed as an optical component for integrating several wavelengths with high coupling efficiency. The self-imaging principle in multimode waveguides is exploited to combine two target wavelengths, corresponding to distinctive absorption lines of important trace gases. The device performance has been numerically enhanced by engineering the geometrical parameters, offering trade-offs in coupling efficiency ratios. The proposed designs are used as versatile duplexers for detecting gas combinations such as ammonia-methane, ammonia-ethane, and ammonia-carbon dioxide, enabling customization for specific sensing applications. The duplexers designed are then fabricated and characterized, with a special focus on assessing the impact of the different target wavelengths on coupling efficiency.
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1. Introduction
The ever-increasing demand for passive optical components is critical in the interconnection of photonic integrated circuits (PICs) utilized in telecommunication and sensing applications [1–3]. The couplers stand as key enablers, and ongoing advances in waveguide-based integrated optical structures will facilitate future integration between optical sources and receivers. Distinctive functions of integrated optical couplers include modulation, power splitting and combining, signal routing and filtering, wavelength division multiplexing (WDM), etc [4–7]. Some of the passive optical components useful for realizing the above functions are directional and Y-junction couplers or multimode interference (MMI) couplers, also realized in angled configuration [8,9]. The angled MMI was first proposed by Y. Hu et al. and shows excellent characteristics in terms of low loss, compactness, superior fabrication tolerances and the capability to filter out higher order modes [10]. Furthermore, the angled design of angled MMI couplers can boost the coupling efficiency (CE), since it shortens the coupling region, resulting in lower optical loss. Due to its excellent compatibility with mature complementary metal-oxide-semiconductor (CMOS) technology, with low cost and high-volume production capability, silicon nitride (Si3N4) in silicon dioxide (SiO2) is an appealing platform for the realization of integrated devices [11,12].
The development of real-time, low-loss, portable multi-gas sensors is creating a need for high-performance duplexers. Trace gas sensing using technologies like laser absorption spectroscopy (LAS), photoacoustic spectroscopy (PAS), and quartz enhanced photoacoustic spectroscopy (QEPAS), compared to other techniques, are considerably faster, with response times >1 s, offer high gas specificity, are capable of part-per-quadrillion (ppq) detection sensitivity and unlock real time in-situ measurements. In LAS gas detection systems, increased radiation intensity is essential as it facilitates efficient coupling of the light source with the gas sample, resulting in heightened accuracy in measurements. In PAS and QEPAS, the measured signal is directly proposal to the radiation intensity. These techniques are typically limited to detecting a single gas [13,14]. High-coupling-efficiency-based duplexers not only are essential for sensing applications requiring high sensitivity and accuracy but can also enable multi-gas detection devices with a high temporal resolution by fast switching of two or more distinct optical sources, while alleviating the issues caused by the broad spectral separation of gases absorption lines. This is true, for example, in the case of industrial processes and combustion monitoring, toxic and flammable gas detection, agricultural fumes, hydrocarbon and explosives monitoring, as well as in breath analysis. Within the near infrared range (NIR), typical wavelengths of interest are λ1 = 1530 nm, λ2 = 1653.7 nm, λ3 = 1684 nm, and λ4 = 2003nm which correspond to the specific absorption characteristics of the following gases: ammonia (NH3), methane (CH4), ethane (C2H6) and carbon dioxide (CO2), respectively [15,16]. These absorption lines are spectrally very far apart from one another, posing a significant challenge for the combiner.
In this manuscript, we propose the design concept of a novel on-chip 2 × 1 angled MMI-based duplexer operating in the NIR region. This device is devoted to optical coupling of different spectrally separated laser sources. Specifically, the device is optimized for three independent cases, namely, to combine the following wavelength pairs: (i) λ1 and λ2, (ii) λ1 and λ3, (iii) λ1 and λ4. A rigorous evaluation of design parameters with trade-offs in CE ratios are numerically investigated for the fundamental, higher order mode profiles and using numerically calculated output beam profiles of the commercially available diode lasers. The proposed structure and design are used as broadband source for the realization of twofold-gas sensing applications.
2. Proposed structure and design
The proposed angled MMI-based duplexer, whose sketch is shown in Fig. 1, employs silicon nitride (Si3N4) guiding elements placed on a silicon dioxide (SiO2) layer.
The Si3N4 on SiO2 platform provides ultra-low loss optical propagation, relatively high refractive index contrast and a wide transparency window from 400 nm to 2350 nm [17]. The structure can be covered with a homogenous cladding material (the choice of which will be discussed further in this work) of thickness tc. The device is composed of three cascaded parts: an input section, a combiner (labeled as “core” in Fig. 1), and an output section. The input section consists of two waveguide ports WG1 and WG2 separated by a gap g, allowing the injection of two signals of different wavelengths, namely λi and λj, into the system. Specifically, λi will be fixed and equal to λ1, while λj will assume the value of wavelengths λ2, λ3, and λ4, depending on the duplexer configuration, with a different value of g for each, namely g1, g2 and g3, as depicted in Fig. 1. It is worth noting that the position of WG1 is fixed at the lower right corner of the combiner. Conversely, the position of WG2, uniquely identified by the gap g, is a function of the specific wavelength to be combined with λ1and will be a subject of optimization (for this reason, WG2 is depicted with evanescent colors in Fig. 1).
These waveguides are then prolonged and connected to the multimode waveguide region, where LC is the length and WC is the width. This multimode region is tilted on the substrate plane forming an angle β with respect to the direction of the input and output waveguides. Within this section, interference will result in a landscape of electromagnetic field maxima and minima that is determined by geometric parameters as well as spectral parameters and the guiding properties of the structures involved. The MMI structure allows the “long side” of the core region to be exploited as the insertion point of WG2. The asymmetry of the structure will depend on the choice of g, which, in turn, will affect the overall transmittance. Finally, an output section consists of a single waveguide having width Wo = Wi, whose position mirrors WG1 with respect to the center of the core region and to the x- and y- directions, where we expect the maximum light intensity according to the self-image principle [18].
When a single mode is injected into the multimodal region from an input port, the crossing of the discontinuity at the combiner inlet allows it to spatially disperse, resulting in the excitation of all available modes (while the total power is conserved). These modes then interfere with their own reflection from the multimodal region boundaries. According to the self-imaging principle in presence of modal dispersion, the injected mode is then regenerated along the propagation direction at specific positions, where the distance from the injection point is an integer multiple of the beating length. Here, the wavelength is inversely proportional to the length of the multimode region (Lc). Consequently, the required distance g increases with increasing wavelength separation between the two input sources [18].
The simulations were performed by the finite differences beam propagation method (FD-BPM) (BeamPROP simulation tool, provided by RSoft). Taking advantage of the monochromatic Helmholtz wave equation, FD-BPM provides an accurate yet simple computational framework for complex optical waveguide problems [19,20]. We use this method to optimize the several geometric parameters of the proposed device. After performing a convergence test, the mesh grid resolution was set to 50 nm, 100 nm, and 10 nm, along the x, y and z directions, respectively.
The material dispersion relation is introduced in simulations to account for the wavelength dependence of the refractive index and effectively manage loss and dispersion effects [21]. Wi is optimized to maximize the waveguide coupling to the mode profile of a typical diode laser, emphasizing butt coupling for efficient coupling of light [22,23]. To accomplish this, a width Wi = 3 µm and thickness t = 300 nm are used, with SiO2 cladding of tc = 1 µm thickness. The effective refractive index of fundamental mode (M00) for wavelength equal to 1530 nm, 1653.7 nm, 1684 and 2003nm are 1.649, 1.627, 1.621 and 1.535, respectively. It is important to point out that the input waveguide can also house higher order modes M01, which have effective refractive indexes of 1.499, 1.448 and 1.38, respectively. Implementing an adiabatic mode size converter to taper the input multi-mode waveguide to single-mode dimensions, aiming to avoid higher-order modes, could increase the gap between waveguides and improve overall device performance, however, this is counter-balanced either by a decrease of the device transmittance due to tapering losses or an increase in the device footprint.
However, for the higher wavelength 2003nm, the confinement factor of the higher order mode is negligible. The corresponding mode profiles have been used to investigate the device performance. The presence of higher order modes in the system has a negative impact on the performance of the duplexer-based sensor. Higher order modes lead to increased spatial mode dispersion, decreased self-imaging fidelity, and higher crosstalk [24]. Multimode interference couplers excel in suppressing higher-order modes, thereby alleviating the performance limitations seen in comparison to alternative coupling methods. We have exclusively considered TE mode throughout our simulations, as we envisage the future integration of diode lasers operating in TE.
3. Simulation analysis
When β = 0 (i.e., “straight” MMI case), no common self-focusing positions of the two waveguides can be found for each target pairs of wavelengths. Therefore, from this point on, we consider β > 0, and commence the optimization of the duplexer for λ1 and λ2, at g1 configuration using input ports WG1 and WG2. Having fixed Wi, Wo, t and tc, the performance of the proposed duplexer was optimized by investigating the effects of varying Lc, Wc, β, and g on the total transmittance To. [25]. Ideally the duplexer must maximize output power To for both the target wavelengths, while keeping the device footprint as small as possible. A figure of merit (FOM) is defined as To (λ1) × To (λ2)/(100WC) to distinguish the optimal duplexer configuration. This formula helps to identify configurations that offer superior coupling efficiencies while minimizing the footprint (Lc), which is proportional to the square of WC. Table 1 summarises some of the best configurations in terms of FOM for the proposed duplexer, obtained in the investigated parametric space. The best configuration obtained in terms of FOM maximization is the one labelled “Case 1”. For the latter, a value of g1 equal to 1.56 µm is deemed sufficient to prevent evanescent mode coupling between the two input waveguides [25]. The components of the output transmittance To obtained for λ1 = 1530 nm and λ2 = 1653.7 nm, are equal to 86% and 88%, respectively.
Table 1. Optimized output transmittance for various duplexer configurations
Figure 2 shows the optimization process performed for Case 1 as outlined in Table 1, where To as a function of Lc, with Wc equal to 10 µm, for different values of g1. When WG1 and WG2 are individually excited at wavelengths λi = λ1 = 1530 nm (dotted lines) and λj = λ2 = 1653.7 nm (solid lines), respectively - this allows us to identify To depending on the excited waveguide. In particular, To is calculated for different values of g1 (0.5 µm, 1 µm, 1.5 µm and 2 µm).
Fig. 2. Transmittance To as a function of LC for different values of g1at fixed WC = 10 µm when WG1 and WG2 are individually excited at wavelengths 1530 nm (dotted lines) and 1653.7 nm (solid lines), respectively.
In general, it is possible to observe a bell-shaped trend of To with varying Lc. When WG1 is solely excited at λ1, the change in gap value modulates the transmittance intensity, although the absolute maximum is obtained for almost the same values of LC (i.e., in Fig. 2 the distribution of the dashed curves does not shift significantly along Lc axis), indicating that the beating length remains relatively stable with respect to g1. Conversely, when WG2 is solely excited at λ2, changing g has only a minor effect in on the maximum transmittance intensity. Nonetheless, the beating length exhibits significant variation, spanning a range of approximately 10 microns (i.e., in Fig. 2 the distribution of the solid curves shifts toward longer Lc for increasing g). It is worth mentioning that, for a fixed value of the gap and for each different value of Lc, it is possible to identify a distinct combination of wavelength coupling ratios (i.e., To for pairs of the same color).
To further shed light on the behavior of the device, the transmittance has been studied by varying the angle of the core region. In particular, Fig. 3 shows To as a function of Lc for increasing discrete values of β (equal to 3, 4, 5, and 6, degrees respectively), when Wc, Lc and g are set as in “case 1” (see Table 1) and when WG1 and WG2 are individually excited at wavelengths λi = λ1 = 1530 nm (dotted lines) and λj = λ2 = 1653.7 nm (solid lines), respectively. The vertical line in Fig. 3 represents the Lc = 458 µm at which maximum To obtained by means of figure of merit i.e., at β = 4, when wavelengths launched in their respective waveguides. The configuration referred to as “case 1” presents an excellent compromise in terms of both transmittance and compactness. For this configuration, using the mode profiles of commercially available diode laser as the input excitation leads to To values of 71% and 77% for wavelengths of 1530 nm and 1653.7 nm, respectively. In addition, when launching higher order modes (M01), To reduces to 0.5% and 9% for the same wavelengths, respectively, demonstrating a good high-order mode rejection behavior. Therefore, we set the values of Wc, Lc and β to 10 µm, 458 µm and 4°, respectively.
Fig. 3. Transmittance To as a function of LC, for different values of β, when WG1 and WG2 are individually excited at wavelengths 1530 nm (dotted lines) and 1653.7 nm (solid lines), respectively.
These geometric parameters of the core region are then used to test the operation of the duplexer in the presence of other desired wavelength combinations (i.e., λ1-λ3 and λ1-λ4). For each of these combinations, we optimized the corresponding gap value that maximizes the output transmittance. Table 2 summarizes the numerical results obtained for the three different wavelength pairs of interest, where λ3 and λ4 are launched at WG2.
Table 2. Values of transmittances obtained for three different pairs of excitation wavelengths when the distance between the input waveguides g is optimized
Figure 4 illustrates the amplitude of electric field (TE mode) as function of x and y axis. The distribution of the propagating light in the proposed device is at z = tc/2 = 0.15 µm. These maps provide valuable insights into the self-focusing principle and demonstrate how the gap between the two waveguides influences the focusing effect for different input waveguides targeting specific wavelengths.
Fig. 4. Electric field distributions as a function of the x and y coordinates at a z = 0.15 µm. The sole WG1 is fed with the fundamental mode at λ1 = 1530 nm (a) at g1 = 1.56 µm (c) g2 = 2.03 µm and (e) at g3 = 9.8 µm. When WG2 is fed with the fundamental mode at (b) λ2 = 1653.7 nm, at g1 = 1.56 µm (d) λ3 = 1684 nm at g2 = 2.03 µm and (f) λ4 = 2003nm at g3 = 9.8 µm.
The To of the duplexer with g3 configuration for λ1 is noticeably lower (<50%), due to modal dispersion, which is a consequence of the asymmetry in MMI width caused by the presence of WG2. A portion of the field experiences leakage at the terminus of the MMI structure, primarily due to the presence of the dielectric interface between Si3N4 and SiO2 slab region.
Figure 5 represents the transmittance as a function of the wavelength within the spectra range between 1.450 µm and 2.150 µm, for the three configurations reported in Table 2. In particular, the blue color corresponds the To for λ1 injected in WG1 and red, gray, and green color for wavelengths λ2, λ3, λ4 respectively injected in WG2 for corresponding duplexers. The vertical line corresponds to the To at target wavelengths.
4. Experimental results
The angled multimode interference duplexers, discussed in previous section, have been fabricated using electron beam lithography and dry etching. In order to optimize the fabrication process and transmission capabilities, various device configurations were utilized.
The fabrication process starts with the spin-coating of a 450 nm thick layer of ZEP 520A resist onto a 300 nm thick Si3N4 layer, which had been deposited using plasma-enhanced chemical vapor deposition (PECVD) onto a 4-inch bulk silicon wafer. This silicon wafer had a thermally oxidized (buried oxide layer) with a thickness of 2.2 µm, needed for the confinement of the optical mode in the thin Si3N4 layer. The devices were then patterned using electron beam lithography (EBL) at 100 kV before being developed in n-amyl acetate solution for 90 seconds and rinsed with IPA. Then, the patterns were transferred onto the PECVD Si3N4 layer via an inductively coupled plasma (ICP) etch step in O2:CHF3 chemistry in a 4:21 ratio (etch rate of ∼90 nm/min). Any residual resist was removed using a bath of 1165 remover for 30 minutes and an O2 plasma ashing step. Following SEM inspection, a 1 µm-thick SiO2 layer was deposited using PECVD. The details of the fabricated angled MMI duplexer samples can be seen in Fig. 6. A trench with a width of 2 µm is sufficient to prevent the waveguide mode from leaking into the neighboring region. From measurements of micro-ring resonators Q-factors, fabricated using the same process, we estimate the propagation loss to be 4 dB/cm [26].
Fig. 6. Fabricated angled MMI duplexer sample under (a) microscopic imaging. SEM image (zoom) (b) between input and multimode waveguide (c) between multimode waveguide and output.
Considering optimum numerical result of case 1 from Table 1, we fabricated an array of samples having three fixed Lc for varying g1 (1 µm, 1.5 µm and 2 µm) values, with the aim to explore the optimum fabrication tolerances and device performance. A straight waveguide with width Wi = 3 µm and thickness tc = 300 nm, having the same footprint of the designed duplexer (4 mm), is included in the fabricated chip for the normalization of the device performance.
The optical characterization of the fabricated samples was performed using an end-fire measurement setup similar to that shown in [27]. The input waveguides of each device are separated by 100 µm using S-bend waveguides, ultimately allowing sufficient space for bonding the diode laser to the chip. Individual alignment procedures for each wavelength are carried, considering that the optics utilized during characterization are wavelength dependent; specifically, a 60X lens to focus light into the waveguide and a 10X lens to collimate the light into the detectors. Additionally, the measurement has been performed using a TE polarizer.
The alignment process is performed using single wavelength laser (Yenista Optics TLS) for WG1 (1530 nm) and diode laser (Eblana Photonics) for WG2 (1653.7 nm) by maximizing the output power obtained from Wo using Thorlabs power meter. After the alignment process, the TLS was replaced with an Amonics ALS broadband light source and the diode laser with an Amonics ASLD superluminescent source. The output transmittance spectrum at Wo is collected using an Yenista Optics optical spectrum analyzer (OSA).
We employed two normalization processes for our measurements. One using the transmittance collected from a blank waveguide integrated into the fabricated chip, while the other uses the output spectra of the sources under the same conditions as those applied to the chip. Table 3 shows the calculated CE for each sample obtained by normalizing the duplexer transmittance spectrum with respect to the transmittance spectrum obtained from the reference straight waveguide.
Table 3. The normalized transmittance of the duplexer configurations
Results exhibit a favorable trade-off in CE concerning variations in the geometric parameters Lc and g. Samples configurations 6 and 8 (Table 3) are specifically highlighted due to their notable performance in terms of the figure of merit (FOM). However, other samples do exhibit good performance, and their CE trends align with the simulation data concerning variations in Lc and g (see Table 3). The drift from optimized Lc and g values causes a decrease in the CE values for λ1 and λ2. Sample 6 has adopted the same device configuration of as the optimized angled MMI duplexer with Lc value of 458 µm and g1 = 1.56 µm (case 1 from Table 1). Considering the FOM in CE, Sample 6 shows promising CE values of 62%: 67%. Nevertheless, Sample 8 shows the most promising performance, where Lc = 460 µm and g1 = 1 µm having CE 59.5% and 71.5%, for λ1 and λ2, respectively.
The normalized output transmittance spectra of the duplexer collected from Wo with respect to the source are shown in Fig. 7(a) and (b) for the selected samples (6 and 8). Figure 7(a) corresponds to the wavelength launched at WG1 and Fig. 7(b) at WG2. The vertical line in the figures indicates the target wavelengths λ1 and λ2.
This normalization process facilitates accurate and meaningful comparisons across different samples and experimental conditions. These findings indicate that the designed device exhibits an optimal performance within a spectral range of approximately 10 nm centered around these specific wavelengths. Notably, these results demonstrate a satisfactory agreement with the corresponding simulation outcomes. A duplexer at 1684 nm was also tested, as this wavelength is at the limit of that provided by the super luminescent diode, the signal to noise was low and we estimated the CE to be 32% and 8% for samples 6 and sample 8, respectively. Optical losses in the measurement are attributed to several practical factors, including challenges in properly coupling the light to the chip's input port (free-space end-fire setup using lenses to resize the optical fiber mode to match the waveguide mode), fabrication imperfections, parasitic reflections within the waveguide structure due to cleaved facets, or the filtering of higher-order modes within the MMI region. These factors collectively contribute to the reported losses measured.
5. Conclusion
We have presented results for a high-performance duplexer based on, on-chip angled MMI configuration, for wavelengths 1530 nm - 1653.7 nm, 1530 nm - 1684 nm, 1530 nm - 2003nm. The coupling efficiencies are numerically verified for various modes of operation, including the fundamental, higher-order, and realistic diode lasers mode profiles, for all target wavelengths. The experimental results show significant enhancement in measured coupling efficiency of the duplexer with 57% and 68% for target wavelengths of 1530 nm and 1653.7 nm, respectively. Following similar design and fabrication principles, duplexers can be designed for operation at other wavelengths such as 1684 nm 2003nm amongst others and be utilized for customized applications in multi-gas sensing and telecommunication fields.
Funding
H2020 Marie Skłodowska-Curie Actions (860808); Horizon 2020 Framework Programme (101016956).
Acknowledgment
We gratefully acknowledge useful discussions with Vincenzo Spagnolo and Pietro Patimisco, PolysenSe, University of Bari, (UNIBA).
GM is supported by a grant from Region Puglia “Research for Innovation” (REFIN). REFIN is an intervention co-financed by the European Union under the POR Puglia 2014-2020, Priority Axis OT X “Investing in education, training and professional training for skills and lifelong learning” - Action 10.4 - DGR 1991/2018 - Notice 2/FSE/2020 n. 57 of 13/05/2019 (BURP n. 52 of 16/06/2019).
Disclosures
The authors declare that there is no conflict of interest.
Data availability
Data implicit in the results submitted in this report are not publicly accessible at this time but may be obtained from the authors upon reasonable request.
References
1. S. Iadanza, J. H. Mendoza-Castro, T. Oliveira, et al., “High-Q asymmetrically cladded silicon nitride 1D photonic crystals cavities and hybrid external cavity lasers for sensing in air and liquids,” Nanophotonics 11(18), 4183–4196 (2022). [CrossRef]
2. A. Shakoor, J. Grant, M. Grande, et al., “Towards portable nanophotonic sensors,” Sensors 19(7), 1715 (2019). [CrossRef]
3. A. Annunziato, F. Anelli, P. Teilleul, et al., “Fused optical fiber combiner based on indium fluoride glass: new perspectives for mid-IR applications,” Opt. Express 30(24), 44160 (2022). [CrossRef]
4. C. D. Truong, D. N. Thi Hang, H. Chandrahalim, et al., “On-chip silicon photonic controllable 2 × 2 four-mode waveguide switch,” Sci. Rep. 11(1), 897 (2021). [CrossRef]
5. Y. Akihama and K. Hane, “Single and multiple optical switches that use freestanding silicon nanowire waveguide couplers,” Light: Sci. Appl. 1(6), e16 (2012). [CrossRef]
6. S. Hassan and D. Chack, “Design and analysis of polarization independent MMI based power splitter for PICs,” Microelectron. J. 104, 104887 (2020). [CrossRef]
7. S. Iadanza, G. C. R. Devarapu, A. Blake, et al., “Polycrystalline silicon PhC cavities for CMOS on-chip integration,” Sci. Rep. 12(1), 17097 (2022). [CrossRef]
8. Y. Luo, Y. Yu, M. Ye, et al., “Integrated dual-mode 3 dB power coupler based on tapered directional coupler,” Sci. Rep. 6(1), 23516 (2016). [CrossRef]
9. J. Mu, S. A. Vázquez-Córdova, M. A. Sefunc, et al., “A low-loss and broadband MMI-based multi/demultiplexer in Si3N4/SiO2 technology,” J. Lightwave Technol. 34(15), 3603–3609 (2016). [CrossRef]
10. Y. Hu, R. M. Jenkins, F. Y. Gardes, et al., “Wavelength division (de)multiplexing based on dispersive self-imaging,” Opt. Lett. 36(23), 4488–4490 (2011). [CrossRef]
11. J. Zhang, L. Han, B. P.-P. Kuo, et al., “Broadband angled arbitrary ratio SOI MMI couplers with enhanced fabrication tolerance,” J. Lightwave Technol. 38(20), 5748–5755 (2020). [CrossRef]
12. L. O’Faolain, S. Iadanza, A. P. Bakoz, et al., “Hybrid lasers using CMOS compatible nanostructures,” in 2019 IEEE 8th International Conference on Advanced Optoelectronics and Lasers (CAOL)1–4 (2019).
13. I. Galli, S. Bartalini, S. Borri, et al., “Molecular gas sensing below parts per trillion: radiocarbon-dioxide optical detection,” Phys. Rev. Lett. 107(27), 270802 (2011). [CrossRef]
14. P. Patimisco, G. Scamarcio, F. K. Tittel, et al., “Quartz-enhanced photoacoustic spectroscopy: a review,” Sensors 14(4), 6165–6206 (2014). [CrossRef]
15. A. Sampaolo, P. Patimisco, M. Giglio, et al., “Quartz-enhanced photoacoustic spectroscopy for multi-gas detection: A review,” Anal. Chim. Acta 1202, 338894 (2022). [CrossRef]
16. Z. Shang, S. Li, B. Li, et al., “Quartz-enhanced photoacoustic NH3 sensor exploiting a large-prong-spacing quartz tuning fork and an optical fiber amplifier for biomedical applications,” Photoacoustics 26, 100363 (2022). [CrossRef]
17. H. Lin, Z. Luo, T. Gu, et al., “Mid-infrared integrated photonics on silicon: a perspective,” Nanophotonics 7(2), 393–420 (2017). [CrossRef]
18. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995). [CrossRef]
19. K. Okamoto, “Chapter 7 - Beam propagation method,” in Fundamentals of Optical Waveguides (Second Edition), (Academic, 2006), pp. 329–397.
20. G. R. Hadley, “Wide-angle beam propagation using Pad approximant operators,” Opt. Lett. 17(20), 1426 (1992). [CrossRef]
21. K. Luke, Y. Okawachi, M. R. E. Lamont, et al., “Broadband mid-infrared frequency comb generation in a Si3N4 microresonator,” Opt. Lett. 40(21), 4823–4826 (2015). [CrossRef]
22. M. Paparella, L. M. Bartolome, J.-B. Rodriguez, et al., “Analysis of the optical coupling between gaSb diode lasers and passive waveguides: a step toward monolithic integration on Si platforms,” IEEE Photonics J. 14(5), 1–6 (2022). [CrossRef]
23. R. Marchetti, C. Lacava, L. Carroll, et al., “Coupling strategies for silicon photonics integrated chips [Invited],” Photon. Res. 7(2), 201 (2019). [CrossRef]
24. X. Jiang, H. Wu, and D. Dai, “Low-loss and low-crosstalk multimode waveguide bend on silicon,” Opt. Express 26(13), 17680 (2018). [CrossRef]
25. M. Pollock Clifford and R. Lipson, “Introduction and overview,” in Integrated Photonics (Springer, 2003), pp. 1–7.
26. J. Li, L. O’Faolain, S. A. Schulz, et al., “Low loss propagation in slow light photonic crystal waveguides at group indices up to 60,” Photonics Nanostruct 10(4), 589–593 (2012). [CrossRef]
27. S. Iadanza, C. Devarapu, A. Liles, et al., “hybrid external cavity laser with an amorphous silicon-based photonic crystal cavity mirror,” Appl. Sci. 10(1), 240 (2019). [CrossRef]
