Infrared imaging with nonlinear silicon resonator governed by high-Q quasi-BIC states

Nonlinear light-matter interactions have emerged as a promising platform for various applications, including imaging, nanolasing, background-free sensing, etc. Subwavelength dielectric resonators offer unique opportunities for manipulating light at the nanoscale and miniturising optical elements. Here, we explore the resonantly enhanced four-wave mixing (FWM) process from individual silicon resonators and propose an innovative FWM-enabled infrared imaging technique that leverages the capabilities of these subwavelength resonators. Specifically, we designed high-Q silicon resonators hosting dual quasi-bound states in the continuum at both the input pump and signal beams, enabling efficient conversion of infrared light to visible radiation. Moreover, by employing a point-scanning imaging technique, we achieve infrared imaging conversion while minimising the dependence on high-power input sources. This combination of resonant enhancement and point-scanning imaging opens up new possibilities for nonlinear imaging using individual resonators and shows potential in advancing infrared imaging techniques for high-resolution imaging, sensing, and optical communications.


Introduction
Frequency conversion in nonlinear optics, emanating from the nonlinear response of matter to light, has been extensively applied in different fields including imaging, sensing, holography, and quantum optics [1][2][3][4].Recently, dielectric nanoresonators, possessing the ability to support strong light-matter interactions utilising multipolar resonances, have been recognised as a super-compact and flexible platform to carry out frequency conversion [5][6][7][8][9][10].Via strong field enhancement, frequency conversion can be boosted several orders of magnitude stronger compared to unpatterned films of equal thickness [6,[11][12][13][14].Additionally, the amplitude, phase change, and directionality of converted frequency emissions can be precisely controlled through the careful design of resonator geometry and the manipulation of resonant responses [5,9,10].This ability to manipulate light at the sub-wavelength scale combined with minimal absorption allows dielectric resonators to overcome inherently low material nonlinear susceptibilities, and thus realise practical nonlinear applications [15][16][17][18][19].
To unlock the potential of nano-resonators, several innovative methods of boosting light-matter interactions have emerged, one of which is the use of bound states in the continuum (BICs).
BICs, originally predicted by von Neumann and Wigner in 1929 [20], have recently attracted attention in nanophotonics as a novel approach for enhancing light-matter interactions [21][22][23][24][25].To date, BICs have been widely demonstrated and exploited in photonic structures to procure controllable light confinement within nano-resonators [26][27][28][29][30][31][32][33][34].A widely explored type of BICs is the symmetry-protected BIC existing at the Γ point of a subdiffractive periodic structure.Due to the symmetry mismatch between the mode profile and external propagating modes, non-local resonant modes are formed and can be completely decoupled from the radiating waves [23,35].For the case of a single-particle structure, constructing an anti-crossing of pair modes can be used to form quasi-BICs based on the destructive interference of several far-field radiation channels [10,27,[36][37][38][39].
Infrared nonlinear imaging, utilising frequency conversion to capture images from infrared in the visible region, has gained growing attention due to its numerous applications in sensing, night vision, and spectroscopy [40].Relevant works of nonlinear imaging conversion have been demonstrated utilising arrays of nano-resonators, termed metasurfaces [4,18,40,41].These structures harbour non-local resonances demanding excitation using large irradiance areas needed to satisfy a periodic boundary condition [42].This mandates the use of high input powers for achieving sufficient nonlinear emission output.Moreover, because the resonators can be modelled as discretised point sources that combine to form the upconverted wavefront, the resolution of the upconverted image is limited by the periodicity of these arrays.By adopting a point-scanning imaging system, nonlinear imaging based on single particles can crucially reduce dependence on high power-inputs and improve attainable resolutions expanding the application range for nonlinear imaging further.Considering that identifying boundaries and extracting structural features from infrared images is becoming increasingly important for lots of applications, ranging from security to medical diagnostics, improving these imaging parameters is sought after.
In this paper, we explore resonantly enhanced nonlinear processes by employing an individual high-Q silicon resonator hosting quasi-BIC modes.The silicon resonator supports a quasi-BIC magnetic quadrupole (MQ) mode surrounding the signal beam which can be excited using a linearlypolarised (LP) beam, and a quasi-BIC magnetic hexadecapole (MH) mode surrounding the pump beam which is excitable by an azimuthally-polarised (AP) beam.Using a point-scanning imaging system, we obtain an optical image by converting an infrared signal to visible via the four-wave mixing (FWM) process enhanced by a single silicon resonator, as illustrated in figure 1.

Results and discussion
We first consider a silicon disk-type resonator capable of supporting resonant leaky modes that exhibit strong interactions with the surrounding environment due to its open boundaries.The aspect ratio is defined as the radius to the height of the disk r/h.The size parameter is defined as the ratio of the disk radius to the mode wavelength.For computational simplicity, we assume the silicon resonator is surrounded by homogeneous medium air.Based on our previous work [8], the presence of silica substrate did not lead to qualitative changes in the mode structures.Our focus is on the formation of quasi-BICs at the coupling region for two distinct modes within the NIR and SWIR spectral range for our FWM enhancement and imaging application.Specifically, we focus on a pump wavelength in the region of 800-1200 nm, and a signal beam wavelength in the region of 1400-1800 nm.In the designed coupling region A as shown in figure 2(a), we have modes A1 and A2, which are rotationally symmetric TE-polarised modes, meaning their electric field does not depend on the azimuthal angle ϕ.The designed coupling region B, figure 2(b), is formed by modes B1 and B2, existing as TE-polarised modes whose field is mirror symmetric with respect to xz and yz planes.Based on the mode properties, modes A1 and A2 can be excited under AP light, and modes B1 and B2 can be excited using a conventional LP light.Figure 2(a) gives the calculated dispersion of the A-mode.Two dispersion curves depicted as blue-coloured and green-coloured dots exhibit characteristic anti-crossing around r/h = 0.55.The size of the dots indicates the inverse of the mode's Q-factor at that point.As anticipated, in the vicinity of the anti-crossing regime, mode A1's Q-factor is maximised and mode A2's Q-factor is minimised leading to the formation of a quasi-BIC state.Similarly, for modes B1 and B2, we observe the same trend at r/h = 0.55.This formation of quasi-BICs can be explained through each modes multipolar transformations, in figure 3.
The scattering properties of the resonator can be described by the supported leaky eigenmodes and their interference in the system.Each mode can be viewed as a superposition of the spherical multipoles with different orbital index n and  azimuthal index m.Spherical multipolar analysis was performed for the modes of our nanodisks.For the multipolar analysis of the resonant modes, first, electromagnetic field distributions of the resonant mode were calculated using the finite element method solver through eigenfrequency analysis in COMSOL Multiphysics.We then employ the polarisation currents induced inside the nanostructure to obtain the contributions associated with multipoles with different orbital index n and azimuthal index m [43].The calculated Q-factor trends are plotted for these four modes at various aspect ratio values in figures 3(a) and (b).To gain a better understanding of the multipolar transformation, we have performed the multipolar analysis of the eigenmode under the spherical coordinate system, as presented in figures 3(c)-(f).For multipoles, it is important to note that with the increase of the orbital index n, the higher-order multipoles become less leaky than lower-order multipoles.For modes A1 and A2, as can be seen from figures 3(c) and (e), they are constructed by MQ with n = 2, m = 0 and MH with n = 4, m = 0.When the frequencies of these two modes approach each other at r/h = 0.55, this leads to a suppression of MQ m=0 n=2 and an enhancement of MH m=0 n=4 for mode A1.Conversely, for mode A2, when the aspect ratio approaches the anti-crossing region r/h = 0.55, MH m=0 n=4 is suppressed, and as a result, the more leaky multipole MQ m=0 n=2 dominates the multipolar structure of mode A2.Consequently, the radiative leakage of mode A1 is significantly suppressed with the presence of less leaky multipole MH and the absence of more radiative multipole MQ in the structure, while mode A2 becomes leakier due to the absence of less radiative multipole MH and the presence of more radiative multipole MQ in the structure.Importantly, both of these two modes Mode A1 and A2 can be efficiently excited under the AP vector beam since the AP beam can be decomposed solely to magnetic multipoles with m = 0 that matches the multipolar composition of these two modes.For details about the multipolar content of an AP beam, see section I in the supplementary material.For Mode B1 and B2, they are primarily dominated by an electric dipole (ED) with n = 1, m ± 1 and MQ with n = 2, m ± 1.Both of these two modes can be excited directly under LP beam incidence as they share the same multipole contents m = ±1 (see section I in the supplementary material).Due to the less radiative nature of MQ as compared to ED, similar to modes A1 and A2, we observe an increase of the Q-factor for Mode B1 when both modes' frequencies are close to each other at the aspect ratio r/h = 0.55 where ED is gradually suppressed in the multipole content of Mode B1 meanwhile, a decrease of Q-factor for Mode B2 in the vicinity of r/h = 0.55 when the MQ is suppressed in the multipole content of Mode B2.Although with a relatively low Q-factor for mode B1 and B2, due to the same formation mechanism of the multipolar transformation model for quasi-BIC formation in individual nanostructure [10,38], we still apply the term quasi-BICs to this region.
We expect that by exciting the modes at such anti-crossing region, the electromagnetic field can be strongly enhanced, and the nonlinear light-matter interactions can be boosted significantly.In the following, we simulate the third-order nonlinear processes: third-harmonic generation (THG) and FWM for different aspect ratios with the input pump and signal light at the corresponding frequencies of coupling region A and region B, for each respective r/h as shown in figure 4. To ensure the efficient coupling of our light into the designed modes, at region A, we model our simulation sources as an AP vector beam focused by an objective with numerical aperture NA = 0.7, and an LP plane-wave input frequency at region B. The power-independent FWM conversion efficiency is defined as η FWM = P FWM /(P s P 2 p ), where P FWM is the total generated nonlinear emission power, and P s and P p are the total input powers from the fundamental signal beam and pump beam, respectively.For a special case when the pump and signal beams are the same, it gives the powerindependent THG conversion efficiency η THG = P THG /(P 3 FW ) with P THG being the total generated THG power, and P FW being the total input power for the fundamental wave.The linear and nonlinear optical responses from our silicon resonators are simulated using the finite element method solver in COMSOL Multiphysics in the frequency domain [44][45][46][47][48].For details of the calculation, see section II in the supplementary materials.
Notably, we observe strong THG emission when exciting the quasi-BIC Mode B1 around r/h = 0.55 as shown in figure 4(a).As expected, when pumping light at the frequency of the more leaky mode B2 near the anti-crossing region, the THG signal generally remains significantly lower when compared to the case with mode B1 excitation.This also agrees well with the calculated Q-factors of modes B1 and B2 for different aspect ratios.Interestingly, when the silicon resonator is illuminated in the vicinity of the higher-Q mode B1, the generated far-field pattern of the TH emission exhibits a directionality in the forward and backward directions across different aspect ratios.When the silicon resonator is illuminated in the vicinity of the lower-Q mode B2, the generated far-field pattern of the TH emission shows a transverse directionality for different aspect ratios.This nonlinear-type transverse Kerker effect can be interpreted by the nonlinearly generated multipolar interference effect [6,15], similar to the linear case for generalised Kerker effect which has been widely studied in the past [49][50][51][52][53].For the A-mode region, when the silicon resonator is pumped at the resonance position using a AP beam, a substantial enhancement of the electric near field has been observed numerically.Consequently, this enhancement leads to intense nonlinear light-matter interactions, resulting in a sizeable nonlinear electric field and strong conversion at the short wavelength region 250 nm to 400 nm (see figure S3i in supplementary material).this numerical prediction suggests that using a quasi-BIC-type silicon resonator offers the potential for the development of nanoscale ultraviolet light sources.For details of THG from A-mode region, see sections II and III in the supplementary material.
Next, we consider inputting two beams to investigate the FWM process.Excitation conditions were chosen fixing the signal to excited mode B1 and calculating FWM efficiency when the pump beam was coupled to mode A1 and A2, respectively.Figure 4(b) presents the calculated FWM efficiency for different mode combinations (A1B1 and A2B1) at various aspect ratios.Profound enhancement of FWM emission is numerically predicted when the pump frequency approaches the quasi-BIC point r/h = 0.55.Here, in contrast to the THG case, we observe a strong normal emission of our FWM signal.In the case of A2B1, it is noteworthy that, by varying the aspect ratio, not only does the multipolar structure of the pump and signal beams change (as shown in figure 3), but the nonlinear emission also transitions from forward emission (r/h = 0.45) to backward emission (r/h = 0.55) and even to entirely transverse emission (r/h = 0.65).This intriguing behaviour can be elucidated by considering the effects of nonlinear multipolar interference [6,15].More details about the evolution of the nonlinear far-field pattern with different aspect ratios r/h can be found in section III of the supplementary material.In our experiments, we fabricated a set of individual silicon nanodisks with radii ranging from 205 nm to 500 nm and a constant height of 550 nm (r/h = 0.37 : 0.01 : 0.91) on a quartz substrate.Each two neighbouring nanodisks were positioned 10 µm apart to isolate local resonances, as shown in figure 5(a).To investigate the designed resonant features we focused exclusively on the nonlinear response of each nanodisk under different excitation conditions.To elicit nonlinear responses we employed two femotsecond laser sources independently tuned from 1425 to 1575 nm and 900 to 1000 nm for signal, λ s , and pump, λ p , beams respectively.The pump wavefront was structured into an AP beam using a vortex half-wave plate, VWP in figure 5(e).Figures 5(b)-(d) show the beam intensity profile with and without subsequent linear polarisation, confirming AP beam generation.The signal beam was left unaltered maintaining a LP beam profile.To maximise particle irradiance both beams were tightly focused using a 100x objective with high N.A. = 0.7 incident through the substrate.Any resulting nonlinear emission was collected in the forward direction by a 50x objective N.A. = 0.42.For FWM processes, to ensure optimal λ s -λ p pulse overlap, the pump beam path length was carefully adjusted using precise on-axis movement of prism P1 constituting an optical delay line, as shown in figure 5(e).
Initially, nanodisks were excited solely by λ s with a fixed time averaged input power of 10 mW to generate TH radiation.Note any reference to average input power is measured preceeding objective L1 in figure 5(e).Broadly scanning both aspect ratio and λ s exposed two resonant modes: one centred at r/h = 0.61 providing high THG enhancement and another offering low THG enhancement at r/h = 0.49 (see figure S5i  With the goal of achieving strong FWM emission, pump resonances were experimentally probed by evaluating FWM emission enhancement for nanodisks excited simultaneously by λ s and λ p with optimised spatial and temporal coincidence.The highest Q-factor for both modes A1 and B1 are numerically shown to converge at the same aspect ratio, r/h = 0.55 in figures 3(a) and (b).Assuming equivalent experimental behaviour, λ s was fixed at 1500 nm to ensure any maximum FWM enhancement that emerges should correspond to λ p and r/h values that approach the experimental ideal.Measurements were performed varying λ p and r/h with each nanodisk positioned at points of maximum FWM (2ω p − ω s ) enhancement, implying central AP excitation for aspect ratios possessing predominantly AP resonant modes.Results in figure 7(a) reveal significant FWM enhancement at r/h = 0.61 for λ p = 920 nm.For both the THG and FWM emission cases the  r/h value for maximum enhancement is identical indicating successful overlap of high-Q regions for modes A1 and B1 at this aspect ratio.The two peaks observed at the anticipated position of mode A1 near r/h = 0.61 were interpreted not as two separate modes but instead as the consequence of the pump laser linewidth exceeding the linewidth of the designed high-Q mode A1 preventing efficient coupling at this r/h region [4,54,55].Both peaks are therefore attributed to mode A1.Spectra in figure 7(b) provide an unobstructed view of FWM enhancement at λ s = 920 nm highlighting this behaviour.Distinguishing mode A2 amongst non-designed neighbouring low-Q resonances with different multipolar origins was unfeasible with our setup.Characterisation of mode A2 was excluded for this reason.As with figure 6(a), the inset in figure 7(a) shows nonlinear emission is strong enough to be imaged by a handheld camera producing a white colour due to a combination of RGB components: 2ω p − ω s , 3ω s , and 2ω s + ω p quantified in figure 7(b).
Experimental conversion efficiency was estimated by measuring FWM power in the forward direction at 10 mW input power from both the signal and pump to get a value of 0.5nW.This represents the portion of nonlinear emission in 0.67/4π of the total spherical emission area.Based on the far field emission pattern simulated for excitation parameters at high-Q positions of modes B1 and A1, the collected area is assumed to contain 7% of nonlinear emission power estimated to be 7nW in total.We define FWM conversion efficiency as the ratio of FWM power to input signal, P FWM /P s , which equates to 7.14 × 10 −7 and normalised FWM efficiency as P FWM /(P s P 2 p ) which gives 0.7%W −2 for our input laser specifications.
To demonstrate the dependency of THG and FWM strength on the input power of the pump and signal, nonlinear emission strengths were measured over increasing average input power.Figures 8(a) and (b) demonstrate expected linear dependency of FWM(2ω p − ω s ) with respect to signal power, quadratic dependency of FWM(2ω p − ω s ) with respect to pump power, and cubic dependency of THG with respect to signal power.Deviations from nonlinear theory, noticed especially for the quadratic FWM pump dependence, arise from changes in refractive index caused by the Kerr effect and thermal effects which shift the resonance position, in this case slightly in favour of FWM ehancement [44].Notably, demonstrated in figure 8(b), owing to the quadratic dependence of the FWM emission intensity on the pump power, even with comparatively low signal power where the THG emission is undetectable with the hardware configuration, sizeable FWM emission intensity can still be observed.This capability to efficiently convert low-power infrared light into visible radiation with the help of a pump beam suggests a promising route to realise FWM-based infrared imaging techniques.
Utilising automated stage movement and high-speed spectral data acquisition to realise a point scanning system, third order nonlinear imaging of a chrome on fused silica imaging target was achieved via frequency conversion enhancement within a single nanodisk.λ p , λ s , and r/h were chosen based on optimal values established during nonlinear characterisation, at 920 nm, 1500 nm, and 0.61 respectively.The target was positioned between two lenses to obstruct the signal beam, AT in figure 5(e), and the delay line was tuned to overlap λ s -λ p pulses when the signal passed through the fused silica substrate.A custom .NET application was programmed to automate motorised stage movement and trigger spectrometer captures.Images were formed by raster scanning the glass slide 3 mm at 1 mm s −1 horizontally whilst sequentially shifting downwards by 10 µm after each scan.Spectra were captured at 100 Hz requiring high FWM/THG strength for adequate emission detection.Prompting both the stage and spectrometer concurrently, spatially encoded nonlinear spectral data was reconstructed to produce infrared images of an arbitrary imaging target with 10 micron resolution.Resolution is dictated by the ratio of scanning speed to spectral acquisition rate and limited by the beam waist size incident on the target.As shown in figure 9, images obtained show clear features of the imaging target even resolving intricate details.To emphasise the significance of utilising a FWM process, figure 9(b) shows images captured at a low signal input power of 1 mW.Consequently, a undetectable THG signal is unable to resolve encoded image data but by boosting nonlinear conversion via the designed high-Q quasi-BIC resonances the pump degenerate FWM process produces an image with good clarity nonetheless.This convincingly demonstrates the potential of FWM to provide an additional degree of freedom for improving nonlinear conversion efficiency compared to harmonic nonlinear processes, such as SHG and THG, where emission strengths rely exclusively on the light intensity from an imaging target.This is crucial in situations where the use of low signal intensities is necessary such as when avoiding photodamage in bioimaging and sensing applications or for achieving nonlinear imaging in ambient environments.The point-scanning techniqe evidently sacrifices temporal resolution for improvements in sensitivity and spatial resolution and in this respect metasurfaces retain an advantage for dynamic imaging situations.The horizontal artefacts in FWM images appear due to highly sensitive positional dependency of nanodisk with respect to the AP beam centre as well as signal and pump overlap.Hence, gradual deviation in nanodisk position over the scan duration causes fluctuations in FWM emission intensity.Interestingly, additional images show that taking into account the stronger light scattering at target edges, we can highlight the edges of a target by mapping the ratio of FWM strength to THG strength (see figure S5iii in supplementary material).

Conclusion
In conclusion, we have studied resonantly enhanced FWM process from individual nonlinear silicon resonators featuring double BICs.Our designed silicon resonator exhibits strong quasi-BICs around both the signal beam and pump beams, enabling efficient FWM and THG processes.Based on these third-order nonlinear processes, we develop a novel pointscanning imaging platform for infrared-to-visible image conversion.Notably, the quadratic dependence of FWM emission power on pump beam power opens up possibilities for efficient infrared-to-visible conversion through pump power control, significantly reducing the need for high input signal beam power.The ability to control and manipulate nonlinear optical interactions in silicon resonators with double BICs not only enhances nonlinear optical interactions but also paves the way for the development of ultra-thin nonlinear photonic chips for advanced infrared imaging technologies and signal processing applications.

Figure 1 .
Figure 1.Schematic representation of the degenerate four-wave mixing process from our proposed silicon resonator: a signal beam at shortwave infrared (SWIR) frequency ωs is mixed with a pump structured beam at near-infrared (NIR) frequency ωp on the silicon resonator, to excite quasi-BICs and generate an idler output at visible ω i = 2ωp − ωs, to convert the optical image from infrared to visible.

Figure 2 .
Figure 2. Dispersion of the eigenmodes A1 & A2, and B1 & B2.(a) Dispersion of A-modes.(b) Dispersion of B-modes.The size of the circles indicates the inverse of quality factor of the mode at each aspect ratio.For both figures, the inset gives the xz-view of near-field distributions of the electric field magnitude: the top row corresponds to the higher-Q modes A1/B1 at r/h = 0.45, 0.55, 0.65; The bottom row corresponds to the lower-Q modes A2/B2 at r/h = 0.45, 0.55, 0.65.

Figure 4 .
Figure 4. (a) Normalised THG efficiency when exciting modes B1 and B2 at different aspect ratios.(b) Normalised FWM efficiency when exciting different modes A and B at different aspect ratios.For both figures, the inset gives the far-field patterns of the THG/FWM emissions: the top row corresponds to the high-quality modes at r/h =0.45, 0.55, 0.65; The bottom row corresponds to the low-quality mode at r/h =0.45, 0.55, 0.65.
in supplementary material).For both modes, peak enhancement occurred when excited close to λ s = 1500 nm and measured spectra are shown in figure 6(b).Each peaks relative position and degree of enhancement agrees with the numerically simulated Q-factor trend in figure 3(b) and suggests the existence and successful coupling to designed modes B1 and B2.Mode B1 was further experimentally explored by precisely scanning aspect ratio and λ s , confirming peak Qfactor at 1500 nm which corroborates size parameter r/λ 0 ≈ 0.2 in figure 2(b).In direct comparison with figure 2(b) at λ s ≈ 1500 nm, simulated maximum Q-factors appear for modes B1 and B2 respectively at r/h = 0.55 and r/h = 0.45 indicating potential nanodisk fabrication errors such as a consistent geometric shift and/or difference in experimental and simulated refractive indices.Inset in figure 6(a) demonstrates noticeable THG emission enhancement enough for capture with a handheld VIS camera under ambient lighting.

Figure 6 .
Figure 6.(a) Normalised THG emission for varying nanodisk aspect ratio and λs wavelength with a time averaged input power of 10 mW.The inset shows a photographed image of the THG emission with handheld VIS camera under ambient lighting.(b) Experimentally measured spectra showing THG emission for different nanodisks excited by 1500 nm signal with time averaged input power of 10 mW.Extrusion magnifies THG enhancement arising from mode B2.

Figure 7 .
Figure 7. (a) Normalised FWM emission for fixed λs = 1500 nm with varying nanodisk aspect ratio and λp wavelength.Both λs and λp have time averaged output power of 10 mW.The inset shows a photographic image of the FWM emission with handheld VIS camera under ambient lighting.(b) Experimentally measured visible spectrum for nanodisk emission when different nanodisks are excited by λs = 1500 nm and λp = 920 nm both at time averaged power of 10 mW.

Figure 8 .
Figure 8.(a) Experimentally measured nonlinear emission with increasing pump and signal input powers.Points denote individual data points with corresponding linear regression line.(b) THG/FWM emission spectra with different time averaged signal input power.The pump power here is fixed as 10 mW.

Figure 9 .
Figure 9. Experimentally measured and reconstructed nonlinear images using a point scanning system at (a) 10 mW and (b) 1 mW signal powers.Target pattern depicts George Green's Windmill, a historical scientific landmark in Nottingham, UK.First column: the images of targets under white light illumination.Second column: reconstructed image from FWM process.Third column: reconstructed images from THG process.Fourth column: Averaged spectra over entire image acquisition duration quantifying THG and FWM signals for each signal power case.