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Abstract
Frequency conversion based on χ(2) nonlinear optical interactions can be made very efficient in waveguide structures. Fabrication imperfections remain very often a limiting factor. They can induce strong distortions in the spectral shape and lower the efficiency. To overcome these imperfections a post-trimming method based on a cascade of ten micro-heaters on top of the waveguide along the interaction length is demonstrated. As an example, the second harmonic generation in a Ti-indiffused waveguide in periodically poled LiNbO3 is investigated. A tailoring of the spectral characteristics of the phase-matching curves could be successfully demonstrated.
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1. Introduction
Frequency converters based on nonlinear optical $\chi ^{(2)}$ processes play a key role for many photonic applications ranging e.g. from low-cost intra-cavity frequency doublers in green laser pointers to nonlinear mixers for the generation of ultra-broad frequency combs. Many of such devices are realized as integrated converters in guided-wave structures [1]. Using optical waveguides is very beneficial because it can provide higher efficiency due to a tight confinement of the interacting waves over a long interaction length without beam divergence due to diffraction spreading. On the other hand, in many practical devices the efficiency is limited by fabrication imperfections of the waveguide [2–4]. Efficient nonlinear interaction requires phase-matching (or quasi-phase-matching in periodically poled structures). This requirement of momentum conservation relates the wave vectors of the interacting modes, which propagate collinearly in the waveguide. These wave vectors are very sensitive to variations of fabrication parameters of the waveguide. Thus, even small waveguide imperfections can significantly impact the nonlinear process and degrade the maximum efficiency [5].
To mitigate the influence of imperfections, fabrication parameters may be optimized to minimize the variation of phase-matching along the interaction length [7,8]. Although this so-called non-critical phase-matching provides an improvement, the useful overall length is typically still limited. According to a detailed theoretical study [6] there exists a critical device length for such phase-matched processes.
Another approach to improve the performance of such nonlinear integrated devices is based on a post-fabrication trimming of the local phase-matching. The basic idea is to use a specifically tailored temperature profile to counteract the local variations of the phase-matching function. A coarse version of this concept has already been demonstrated for a second-harmonic process in a Ti-indiffused LiNbO$_3$ waveguide [9]. The authors placed five individually controllable thermo-electric heaters (TEHs) below a waveguide sample and showed that they could manipulate the spectral characteristics of the phase-matching curve. The generation of arbitrary temperature profiles with such TEHs is limited to long-ranging variations. As these heaters are placed relatively far away from the waveguide, heating is not really local and cross-talk between adjacent sections must be expected.
In this work we present a completely refined version of such a post-trimming. We use micro-heaters directly deposited on the sample to generate a temperature profile along the PPLN waveguide. As a result, we can generate really local profiles with steeper temperature gradients. In this work we use a cascade comprising of ten micro-heaters to generate the temperature profiles. The generation of temperature profiles with steeper gradients is also important in view of a higher integration density. When more nonlinear elements are integrated on a single chip, e.g. two parallel waveguides for second harmonic generation, it may be necessary to generated different temperature profiles for their optimum performance, which is hardly possible with TEHs below the chip.
Microheaters on top of the waveguide are frequently used as phase-shifters in photonic circuits realized in passive materials (see e.g. [10,11]. In integrated optical circuits in LiNbO$_3$ such micro-heaters are rarely applied because phase-shifting is usually accomplished via the electro-optic effect. Recently, some applications of micro-heaters in thin film LiNbO$_3$ (TFLN) devices were demonstrated. The heaters acted as phase-shifters to tune micro-ring filters [12] and to tune SHG [13,14]. In this work we use micro-heaters in combination with a conventional Ti-indiffused periodically poled LiNbO$_3$ waveguide.
The paper is organized as follows. In the next section we present a theoretical model which describes how a temperature profile impacts the spectral shape of the nonlinear conversion. We apply this model especially for the second harmonic generation (SHG) in a waveguide with the fundamental wavelength in the 1500 nm range. In Sec. 3 we describe details of the sample design and fabrication. Results on detailed investigations on the performance of the micro-heaters are given in Sec. 4. In Sec. 5 we discuss experiments and their results on tailoring the spectral characteristics of an SHG-process.
2. Modelling
In this section we derive a model to simulate the impact of various temperature profiles on phase-matching for a second harmonic generation process. We consider a Ti-indiffused optical waveguide in a periodically poled LiNbO$_3$ (PPLN) substrate. The poling period $\Lambda$ is chosen to provide phase-matching for the SHG process. As Ti-diffused PPLN waveguides can be fabricated with low losses, we neglect any loss terms in the following calculations.
The evolution of the amplitudes $A_\text {f}$ and $A_\text {SH}$ of the fields of the fundamental and the SH waves, respectively, along the propagation direction $z$ can be described using the coupled mode theory [16]. We used the notation given in [15] but adapted to the propagation of guided modes in a waveguide. This yields the following differential equation system:
The coefficients $\gamma _j$ are given by
In the ideal case of a waveguide, which is homogeneous along the interaction length, the coefficients $\gamma _j$’s are independent of $z$ and the phase-factor $\phi (z)=\Delta \beta \,z$ with
The shape of the spectral characteristics of practical systems, however, differs very often from the theoretically predicted one. The origin of this deviation is mostly due to inhomogeneities along the interaction region. Such inhomogeneities may arise during the fabrication process of the waveguides or even be already present in the wafer material due to stoichiometry fluctuations. To simulate the impact of such inhomogeneities one can use a refined version of the above model. In the presence of inhomogeneities the waveguide properties, i.e. the mode distributions and the propagation constants, can vary along the interaction lengths. As long as the variations are relative small, it is reasonable to neglect variations of modal distributions and we can assume that the coefficients $\gamma _j$ remain constant along the interaction length. The phase factor $\phi (z)$, however, has to be modified according to:
i.e. the phase factor at $z$ is given by the phase difference which is accumulated along the interaction region from 0 to $z$.Apart from local variations of the propagation constants due to fabrication related inhomogeneities, temperature gradients can also modify the propagation constants. This enables to counteract fabrication-induced inhomogeneities and allows a post-fabrication trimming of the device performance with a specifically adopted temperature profile. The effective index of a waveguide mode can be approximated by $n_\text {eff}=n_\text {bulk}(\lambda,T)+\delta n_\text {WG}(\lambda )$ with $n_\text {bulk}(\lambda,T)$ being the wavelength and temperature dependent refractive index of the substrate material and $\delta n_\text {WG}(\lambda )$ the increase due to the waveguiding which depends on the (local) fabrication parameters and the wavelength. The temperature dependence of the latter term can be neglected, but the substrate index may be strongly temperature dependent.
To study the impact of temperature profiles on SHG we calculated the spectral characteristics for different temperature gradients. Examples are shown in Fig. 1. All calculations were performed under the assumption of a low conversion efficiency. Thus, depletion of the fundamental wave due to the conversion process is neglectable. We modelled a type 0 phase-matched SHG-process in a 42 mm long periodically poled waveguide (poling period 17.8 $\mu$m). For this process the change of the phase-mismatch with varying temperature is given by $\frac {\partial \,\Delta \beta }{\partial T}\approx -60\, \textrm{m}^{-1}{}^{o}\textrm{C}^{-1}$. This corresponds to a shift of the phase-matching wavelength $\frac {\partial \,\lambda _f}{\partial T}\approx 0.1~$nm$\,^o$C$^{-1}$ at fixed poling period. The modelling parameters are adapted to the experimental scenarios discussed in the following sections.
Fig. 1. Modelled SH-power versus wavelength of the fundamental wave of type 0 phase-matched SHG in a 42 mm long Ti-indiffused waveguide in PPLN for various temperature profiles.
Phase-matching for the chosen parameter set occurs at $\lambda =1505\,$nm for a homogeneous profile at room temperature (T=21.5 $^o$C) and the spectral characteristics is the expected sinc$^2$-function with a half-width $0.25$ nm. A parabolic variation of the temperature along the interaction length results in an asymmetric increase of the sidelobes, i.e. one can observe that stronger sidelobes occur on one side of the main lobe. In the case of a linear temperature profile we observe a symmetric broadening to the phase-matching curve. A fourth example shows in Fig. 1 the spectral characteristics belonging to a strongly asymmetric temperature profile.
Please note, we kept the coupling coefficients $\gamma _j$ and the incoupled power of the fundamental wave constant in all shown examples. The SH power generated in the maximum and, thus, the efficiency of the SHG process drops with increasing inhomogeneities. As already pointed out in [17], the area below such phase-matching curves is kept constant to a first approximation. This emphasizes again how important a homogeneous interaction is to obtain high conversion efficiencies.
3. PPLN waveguide sample
For the experimental investigations we fabricated samples comprising straight Ti-indiffused waveguides in z-cut LiNbO$_3$. Single-mode guiding in the telecom range is obtained by an indiffusion of 7 $\mu$m wide Ti-stripes for 9 h into the LiNbO$_3$-substrate at 1060 $^o$C. Typical waveguide losses are in the range of 0.1 dB/cm. Subsequent to waveguide fabrication electric field assisted periodic domain inversion with 16.9 $\mu$m poling period was performed to obtain type 0 phase-matching for SHG.
On top of the 42 mm long sample micro-heaters were placed above the waveguide separated by a planar 400 nm thick SiO$_2$-layer which is necessary to avoid optical excess loss due to the metallic heater electrodes. A cascade of 10 heaters (each 2.5 mm long) was fabricated within the central 30 mm of the sample (Fig. 2 (a)). Details on position and heater dimensions are shown in Fig. 2 (b). Each heater comprises a 15 $\mu$m wide heating Au electrode on top of the waveguide and bond pads adjacent to it. The targeted thickness of the sputter deposited gold electrodes was 100 nm (with an additional a few nanometer thick Cr adhesion layer). However, due to shadowing during the sputter deposition the electrode thickness varied along the sample resulting in a variation of the ohmic resistance for the heaters between 55 $\Omega$ and 90 $\Omega$. We repeated the characterization of the waveguide losses and could not observe any increase of these losses after the electrode deposition.
Fig. 2. Details of the PPLN waveguide sample with micro-heaters placed on top of the waveguide. (a) Schematics of the sample with the $42\,$mm long waveguide and the 10 micro-heaters placed within a long $30\,$mm region. (b) The micro-heaters are composed of two 300 $\mu$m $\times$ 360 $\mu$m large bond pads and a 2.5 mm long and 15 $\mu$m wide heating wire. (c) Photograph of the bonded waveguide sample.
A double-band antireflective-coating on both polished waveguide facets minimizes reflective Fresnel-losses to < 2% for the input of the fundamental wave and the output SH-wave.
The sample was mounted on a copper holder, which could be temperature controlled with a stability of about 0.1$^{\circ }$C, i.e. the bottom of the 0.5 mm thick sample is stabilized to a fixed temperature. The electrical contacts to the micro-heaters were realized with wire bonding between the bond pads of the electrodes and a printed circuit board directly adjacent to the sample, which is shown in Fig. 2 (c). To operate the device the individual currents flowing through the different heaters could be set via a personal computer and a USB-controlled multichannel D/A converter (Meilhaus Redlab USB-3114).
4. Micro-heater performance
We performed the initial characterization of the micro-heaters prior to the antireflection coating of the end-faces. This enabled us to benefit from the low-finesse resonator formed by the waveguide and its uncoated end-facets, which have a reflectivity of about 14 %. We used the fringing of the resonator transmission to determine the phase-shift induced by the micro-heaters. Experimentally, we coupled a TM-polarized narrowband cw laser beam ($\lambda \approx 1550$ nm) into the waveguide and monitored the output with a photodiode. When heating the waveguide region the round-trip phase in the waveguide resonator changes and the output signals varies accordingly. To characterize the impact of the micro-heaters we slowly ramped up the heater current and monitored the transmission. A low ramping slope was chosen to ensure that we could monitor all oscillations in the transmission. Typical measurement curves are shown in Fig. 3. A complete oscillation period corresponds to a single-pass phase-shift of $\pi$ (A phase-shift of $2\pi$ for the complete round-trip in the waveguide resonator). Please note, the oscillations speeds up with increasing voltage because the heating power scales proportional to the square of this voltage.
Fig. 3. Examples of measurement curves to characterize the micro-heaters. The current flowing through the heaters was slowly ramped up (starting at about $t=40$ s) to a fixed value and the transmission through the low-finesse waveguide resonator was monitored. Left diagram: The single heater #2 was driven with 60 mW heating power resulting in an overall (single-pass) phase-shift of $\Delta \phi =(2.5 \pm 0.1) \pi$, right: All 10 heaters are driven with 300 mW electric power resulting in a phase-shift of $\Delta \phi =(13.8\pm 0.1) \pi$.
We performed measurements where we applied the same heating power simultaneously to all 10 micro-heaters and measured the overall phase-shift. As shown in Fig. 4 (left) we could observe an almost perfect linear dependence between the overall heating power and the related phase-shift. When driving the heaters separately, we discovered that they perform slightly different. In the right diagram in Fig. 4 we have plotted the obtained phase-shift, when only a single micro-heater is driven with 60 mW heating power. We observe that e.g. heater #5 induces less phase-shift than the other heaters. The origin of this different is not finally proven but we attribute it to a non-homogeneous heat contact mediated by the thermal grease between waveguide sample and sample holder.
Fig. 4. Left: Phase-shift as function of the overall heating power. The heating was homogeneously distributed over the 10 micro-heaters. Right: Phase-shift obtained when driving a single heater with 60 mW power.
Next we linked the observed phase-shift with the local temperature. We define a local average temperature difference $\Delta T_\text {ave}$, i.e. we approximate the temperature profile from a single micro-heater by an average temperature, which is homogeneous along the $L=2.5$ mm long heater and also homogeneous over the waveguide cross-section. The observed phase-shift arises due to the temperature dependent refractive index and the thermal expansion of the crystal. Taking these two contributions into account, the measured phase-shift as function of $\Delta T_\text {ave}$ is given by
5. Tailored SHG characteristics
We performed experimental investigations on tailoring the shape of the spectral phase-matching characteristics. Light from a tunable laser in the telecom band was coupled into the waveguide and the generated SH power was measured as function of the wavelength of the fundamental wave ( see Fig. 5). To minimize spectral distortions of the phase-matching characteristics due to photo-refraction we kept the coupled power into the waveguide low ($< 0.7$ mW) and used a lock-in technique to measure the SH-signal.
Fig. 5. Experimental setup to investigate the SHG spectral characteristics. Linear polarized light from a tunable laser is coupled into the waveguide using an aspheric lenses with 8 mm focal length. A similar lens is used at the rear facet for out-coupling. The generated SHG power is measured with a silicon photo-diode and a lock-in amplifier. Power of the fundamental wave is measured in front and behind the waveguide using a power meter (not shown in the diagram).
In Fig. 6 some measured spectral characteristics of the SH phase-matching are shown. At a constant temperature we observed in the investigated waveguide a very broad SH characteristics covering about 1 nm around 1505 nm. According to our simulation results as discussed in Sec.2, we attribute the observed broadening mainly to a linear shift of the phase-matching along the interaction length. To counteract the broadening we applied linear temperature profiles. The predicted change of the spectral shape could clearly be observed (middle and right curves in Fig. 6).
Fig. 6. Data of manipulated SHG spectra. The manipulation was performed by micro-heater induced temperature profiles shown below. The ideal sinc$^2$ spectrum is normalized to the integral of the measured data. Spectra applying three linear temperature profiles with different gradients are shown.
In general, arbitrary spectral shapes can be generated with a proper temperature profiles. Some examples of profiles including their resulting SH spectra are shown in Fig. 7. The shown curves are just a few examples to demonstrate the capabilities of spectral tailoring and reflect well the expected qualitative behavior according to our simulations.
Fig. 7. Data of manipulated SHG spectra. The manipulation was performed by micro-heater induced temperature profiles shown below. The ideal sinc$^2$ spectrum is normalized to the integral of the measured data.
However when only considering high SH efficiency, the sinc$^2$ shape outputs the highest maximum SH power and therefore the highest efficiency when used with a narrow excitation spectrum. In the present example the best approximation to a sinc$^2$-function could be achieved with a linear temperature ramp (the spectrum in the right of Fig. 6). A direct comparison of the optimized and the original spectra at constant temperature is given in Fig. 8. With the linear temperature profile with a temperature difference of approximately $4.3$ $^{\circ }$C a significant narrowing of the spectral peak can be observed. To quantify the improvement we can compare the SH efficiency defined by
where $P_{\text {SHG}}$ is the power of the SH signal, $P_{\text {fund}}$ the power of the fundamental wave coupled out of the waveguide and $L$ the length of the waveguide. From our experimental results we observed an increase from $28\,\frac {{\% }}{\text {W}\cdot \text {cm}^2}$ to $43\,\frac {{\% }}{\text {W}\cdot \text {cm}^2}$ with the optimized temperature profile.Fig. 8. Comparison of the phase-matching spectrum with homogeneous temperature profile and with optimized T-profile to maximize the SH-efficiency.
It is beyond the scope of this work to analyze in detail the sources for the distorted phase-matching characteristics, which we observed with the homogeneous temperature profile. The distortions are caused by inhomogeneities along waveguides. Most probably, these inhomogeneities arise during the fabrication of the waveguide. It is well-known that fabrication parameters like width and thickness of the lithographically processed Ti-stripes (before indiffusion) strongly impact the propagation constants of the waveguide modes. For instance, we estimated that the phase-match wavelength shifts with a slope of $\frac {\partial \, \lambda _f}{\partial W}\approx 1.8$ nm/$\mu$m with changing waveguide width $W$. Similarly, we calculate for the dependence of the phase-match wavelength on the poling period $\frac {\partial \, \lambda _f}{\partial \Lambda }\approx 42$ nm/$\mu$m. These values emphasize that it is difficult to fabricate long periodically poled waveguides with good homogeneity and a post-trimming method can significantly improve the performance.
6. Conclusions
Based on theoretical investigations on temperature dependence of quasi-phase-matching, we demonstrated an efficient method to tailor the spectral profile of nonlinear optical processes in PPLN waveguides. With a cascade of ten micro-heaters on top of a waveguide sample a post-trimming of the spectral shape could be experimentally demonstrated for a type 0 phase-matched SHG process. The SH efficiency could be increased from $28\,\frac {{\% }}{\text {W}\cdot \text {cm}^2}$ to $43\,\frac {{\% }}{\text {W}\cdot \text {cm}^2}$ by using a linear temperature profile with a difference of $4.3$ $^{\circ }$C along a $30$ mm section of a $42$ mm long waveguide. Furthermore the temperature change induced by the $2.5$ mm long micro-heaters was measured to be at around $11$ $^{\circ }$C for the maximum tested power of $60$ mW. A proportionality between micro-heater power and induced temperature change could be demonstrated. The temperature changes were calculated from the phase-shifts which we determined from transmission fringes of the waveguide resonator.
In this proof-of-principle demonstration we developed a SHG-device with 10 micro-heaters. The length of a micro-heater section and the overall number of heaters along the interaction region was not yet optimized. More heaters might allow to generate more sophisticated temperature profiles. But, ultimately, thermal cross-talk between adjacent heater sections will limit the useful density of individual heaters. Further studies are necessary to elaborate the optimum trade-off for best performance.
The application of this trimming method is not limited to optimize the efficiency of SHG processes in the waveguide. It should be applicable to many other quasi-phasematched nonlinear processes as well. Our method provides a promising tool to fine-tune the spectral characteristics. For instance, this trimming method may be combined with engineering of the poling pattern (e.g. variation of the poling period and/or the duty cycle of the poling along the structure) for the realization of e.g. specifically engineered quantum light sources [20,21] or ultrabroadband or apodized frequency converters [22,23].
Funding
Ministerium für Kultur und Wissenschaft des Landes Nordrhein-Westfalen (Project PhoQC).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. T. Suhara and M. Fujimura, Waveguide nonlinear-optic devices, Vol. 11, (Springer Science & Business Media, 2003).
2. S. Helmfrid, G. Arvidsson, and J. Webjörn, “Influence of various imperfections on the conversion efficiency of second-harmonic generation in quasi-phase-matching lithium niobate waveguides,” J. Opt. Soc. Am. B 10(2), 222–229 (1993). [CrossRef]
3. D. Jedrzejczyk, R. Güther, K. Paschke, et al., “Diode laser frequency doubling in a ppMgO:LN ridge waveguide: influence of structural imperfection, optical absorption and heat generation,” Appl. Phys. B 109(1), 33–42 (2012). [CrossRef]
4. A. C. Gray, S. A. Berry, L. G. Carpenter, et al., “Investigation of PPLN Waveguide Uniformity via Second Harmonic Generation Spectra,” IEEE Photonics Technol. Lett. 32(1), 63–66 (2020). [CrossRef]
5. M. Santandrea, M. Stefszky, R. Ganael, et al., “Characterisation of fabrication inhomogeneities in Ti:LiNbO3 waveguides,” New J. Phys. 21(12), 123005 (2019). [CrossRef]
6. M. Santandrea, M. Stefszky, and C. Silberhorn, “General framework for the analysis of imperfections in nonlinear systems,” Opt. Lett. 44(22), 5398–5401 (2019). [CrossRef]
7. E. J. Lim, S. Matsumoto, and M. M. Fejer, “Noncritical phase matching for guided–wave frequency conversion,” Appl. Phys. Lett. 57(22), 2294–2296 (1990). [CrossRef]
8. M. L. Bortz, S. J. Field, M. M. Fejer, et al., “Noncritical quasi-phase-matched second harmonic generation in an annealed proton-exchanged LiNbO3 waveguide,” IEEE J. Quantum Electron. 30(12), 2953–2960 (1994). [CrossRef]
9. Y. L. Lee, Y.-C. Noh, C. Jung, et al., “Reshaping of a second-harmonic curve in periodically poled Ti:LiNbO3 channel waveguide by a local-temperature-control technique,” Appl. Phys. Lett. 86(1), 011104 (2005). [CrossRef]
10. F. Ceccarelli, S. Atzeni, A. Prencipe, et al., “Thermal Phase Shifters for Femtosecond Laser Written Photonic Integrated Circuits,” J. Lightwave Technol. 37(17), 4275–4281 (2019). [CrossRef]
11. N. C. Harris, Y. Ma, J. Mower, et al., “Efficient, compact and low loss thermo-optic phase shifter in silicon,” Opt. Express 22(9), 10487–10493 (2014). [CrossRef]
12. Y. Ding, S. Tao, X. Wang, et al., “Thermo-optic tunable optical filters with GHz-bandwidth and flat-top passband on thin film lithium niobate platform,” Opt. Express 30(12), 22135–22142 (2022). [CrossRef]
13. X. Liu, C. Zhang, Y. Pan, et al., “Thermally tunable and efficient second-harmonic generation on thin-film lithium niobate with integrated micro-heater,” Opt. Lett. 47(19), 4921–4924 (2022). [CrossRef]
14. X. Li, H. Li, Z. Wang, et al., “Advancing large-scale thin-film PPLN nonlinear photonics with segmented tunable micro-heaters,” arXiv, arXiv:2312.09568 (2023). [CrossRef]
15. R. W. Boyd, Nonlinear Optics, Fourth Edition, (Academic Press, 2020).
16. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973). [CrossRef]
17. F. R. Nash, G. D. Boyd, M. Sargent III, et al., “Effect of optical inhomogeneities on phase matching in nonlinear crystals,” J. Appl. Phys. 41(6), 2564–2576 (1970). [CrossRef]
18. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. 22(20), 1553–1555 (1997). [CrossRef]
19. Y. S. Kim and R. T. Smith, “Thermal expansion of lithium tantalate and lithium niobate single crystals,” J. Appl. Phys. 40(11), 4637–4641 (1969). [CrossRef]
20. A. Branczyk, A. Fedrizzi, T. M. Stace, et al., “Engineered optical nonlinearity for quantum light sources,” Opt. Express 19(1), 55–65 (2011). [CrossRef]
21. A. Thomas, M. van Camp, O. Minaeva, et al., “Spectrally engineered broadband photon source for two-photon quantum interferometry,” Opt. Express 24(22), 24947–24958 (2016). [CrossRef]
22. C. R. Phillips, C. Langrock, D. Chang, et al., “Apodization of chirped quasi-phasematching devices,” J. Opt. Soc. Am. B 30(6), 1551–1568 (2013). [CrossRef]
23. B. Cao, M. Hisamitsu, K. Tokuda, et al., “Efficient generation of ultra-broadband parametric fluorescence using chirped quasi-phase-matched waveguide devices,” Opt. Express 29(14), 21615–21628 (2021). [CrossRef]
