## Abstract

In practical applications of the photonic nanojet (PNJ), a microscope objective can be used to illuminate a microsphere. Under such conditions, it is difficult to determine the absolute position of the sphere with respect to the focal point of the incident beam. A small change in the axial position of the sphere can change the form of the incident wavefront from diverging to converging and will significantly influence the evolution of PNJ. In our paper, we report a systematic study of the PNJ properties, including the effective focal length and the full-width at half maximum (FWHM) by changing the source curvature. We treat the cases of diverging and converging wavefronts with different NAs (from 0.1 to 0.8) and sphere diameters (from $1.6\lambda$ to $33\lambda$). We demonstrate that for a diverging source curvature, the effective focal length and FWHM of the PNJ increase as a function source NA for all the sphere diameters. By further increase of the source NA, for a NA of 0.8, the PNJ can be found only for sphere diameters less than $16\lambda$ (refractive index $n = 1.5$). At larger diameters, the microsphere behaves like a diffractive–refractive lens, collimating the light and showing aberrations. For the converging source, the effective focal length and FWHM tend to decrease as a function of source NA, the PNJ localizing inside the sphere for ${\rm NA}\gt 0.2$. Also, the PNJ shows similar behavior with respect to the NA of a converging source and sphere refractive index under plane wave illumination. Finally, we compare our theoretical 2D simulation results with 3D sphere geometries for diameters of $1.6\lambda$ and $8.3\lambda$; we find similar behavior for converging and diverging source wavefront curvatures.

© 2021 Optical Society of America

## 1. INTRODUCTION

A photonic nanojet (PNJ) is a high-intensity narrow propagating electromagnetic beam with a sub-wavelength dimension generated in the shadow side surface of a dielectric microparticle under the plane wave [1]. Furthermore, the PNJ is generally a non-evanescent propagating beam and a non-resonant phenomenon that can be applied to a wide range of sphere diameters until a certain limit [2]. Due to this property of the PNJ and also its small dimensions that can be smaller than the diffraction limit of $\lambda /2$, the PNJ is regarded as the foundation of a wide range of applications, including super-resolution optical imaging [3,4], nanoparticle optical detection [5,6], nanoparticle optical trapping [7,8], data storage [9], etc.

The PNJ can be characterized by key parameters including the full-width at half maximum (FWHM), effective focal length or distance of the hotspot from the structure, nanojet length, and field enhancement, which has been studied and manipulated in various works. The most important property influencing the PNJ characteristics is structure geometry (i.e., particle size and shape), which has been studied and manipulated in numerous works [10–15]. Also, the effect of structure optical properties, i.e., the contrast between the particle and the surrounding medium refractive index on PNJ parameters, has been investigated [1,16], especially multilayer graded-index structures, which have been applied in numerous studies [12,17–22]. Furthermore, we can manipulate the PNJ properties by modifying the source characteristics, including wavelength, polarization, intensity distribution, coherence, etc. [16,22]. In most papers, the configuration assumed is under plane wave illumination. One reason is that it is much easier to do the calculations. A few papers have reported non-planar and spherical wavefront sources. In Ref. [23], the PNJ is numerically reported for a spheroidal particle at the focus of a Gaussian beam. In Refs. [24–26], the PNJ is numerically or experimentally studied for a particle at the focus of a Gaussian beam or Bessel–Gaussian beam by changing the source polarization in comparison to plane wave illumination. In a recent paper [27], a numerical study is performed for sphere sizes of 2–90 µm, under a focused Gaussian beam by changing the source beam waist ${w_0}$, although no systematic study is presented considering the wavefront curvature of the incoming beam. Supposing the particle at the focal plane of the Gaussian beam, due to the flat wavefront curvature at the focal point, the source in practice can be considered a plane wave in these publications. In Ref. [26], the PNJ is experimentally observed for a sphere diameter of 12 µm under diverging and converging spherical wavefront curvatures, but no detailed study or guidelines are presented for different sphere diameters and source curvatures.

In our contribution, we propose a configuration in which we can systematically investigate the effect of curved wavefront sources (from low to high NAs of 0.8) apart from the effect of beam diameter for sphere sizes of 1–20 µm. To our knowledge, such a situation is neglected by other papers but leads to a fundamentally different PNJ behavior for converging and diverging source curvatures. Our study has the following major differences from the previous works: first, the papers study mainly the PNJ under a Gaussian beam at the focal point where the wavefront curvature is flat. Thus, for a larger beam waist ${w_0}$ of the source at its focus, there exists more interference between the incoming beam and the sphere, resulting in a tighter PNJ [25,27]. However, we study the effect of source curvature and not source diameter. Second, in many works, the focused beam is modeled by introducing a single-mode Gaussian beam under the paraxial approximation, which cannot be implemented to investigate high NA sources [25,27].

From the practical point of view, the PNJ is often generated using a microscope objective to illuminate a microsphere, as seen in Fig. 1(a). In such a setting, it is very tricky to determine the absolute position of the sphere with respect to the incoming beam. This is because an active alignment is used in the experiment, and one usually searches for the best spot, which might not be at the plane wave illumination distance [3]. For heavily focused beams using a large NA objective, a small displacement of the sphere along the axis around the incoming beam focal point can significantly influence the generation of PNJ. Our theoretical findings show that the PNJ has completely different behaviors for different positions of the particle around the microscope objective focal point, and our investigation is helpful to understand the observations in the experiment.

## 2. CONFIGURATION

Our proposed configurations are shown in Figs. 1(b) and 1(c). The illuminating beam that in practice is focused by employing a microscope objective in Fig. 1(a) is modeled by making use of a lens under plane wave illumination (beam propagation along ${+}x$). The focusing beam wavefront curvature changes by changing the lens focal distance, and in such a way that sources of different NAs from low to high are generated. Then, the beam out of the lens illuminates a microsphere of radius $R$ to form PNJ. Assuming the sphere position to be before the lens focal point along $x$ axis, the converging beams hit the sphere, as shown in Fig. 1(b), and considering the sphere to be located after the lens focal point, the diverging beams cover the sphere, according to Fig. 1(c). A similar configuration is also proposed in Ref. [28] for investigating the effect of high NA sources on PNJ for a fused silica particle with a diameter of 2 µm. To do a systematic investigation by changing the source NA for all sphere sizes, we choose the sphere position such that it is fully covered by the illuminating beam but the beam does not transmit through the surrounding medium in the air. To design such a configuration, assuming the lens NA to be ${\rm NA} = {\sin}$($\theta$), the sphere position with respect to the lens focal point can be geometrically calculated, which is ${+}\frac{R}{{\rm NA}}$ for a converging area and ${-}\frac{R}{{\rm NA}}$ for a diverging area. The advantage of implementing such a configuration is that the whole sphere is fully illuminated by the beam for all sphere radii of $R$, and we can consistently compare the effect of source wavefront curvature for different sphere sizes.

Several techniques have been used to solve the problem, such as analytical Mie theory [26], finite difference time domain (FDTD), and finite element method (FEM) [20,24–26]. We perform 2D FDTD simulations using the software package of Lumerical FDTD [29] in which the focused beam out of the lens is modeled using a thin element approximation [30] to generate a high NA source input for rigorous simulations. The rigorous 2D simulation area is shown in Figs. 1(b) and 1(c), and the perfectly matched layers (PMLs) boundary condition is implemented along $x$ and $y$ axes. The source wavelength is 600 nm and polarized along $y$ axis. The sphere has a refractive index of $n = 1.5$, and the surrounding medium is air with a refractive index of one. In this work, we investigate two main parameters of a PNJ: effective focal length ($f$) and FWHM. As seen in Fig. 1(c), the effective focal length $f$ is the distance between the center of the sphere and the point in which the PNJ intensity is maximum [24].

We start by comparing PNJ behavior under a converging, diverging, and plane wave illumination for a sphere of 5 µm diameter to illustrate the effect of source curvature. The plane wave illumination case is considered as a reference for comparison. Later, we compare a small (2 µm diameter) and a large microsphere (15 µm diameter). In the rest of the paper, we analyze the PNJ in more detail for different source NAs of 0.1, 0.2, 0.4, 0.6, and 0.8 and for various sphere diameters of 1, 2, 5, 10, 15, and 20 µm. The simulation results show that the PNJ has a completely different behavior under converging and diverging illuminations by changing the sphere size. For diverging beam, the PNJ is localized outside the sphere, and its FWHM increases for large source NAs. For the very high NA of 0.8 and large sphere sizes of 15 and 20 µm, no PNJ is generated, as the sphere behaves like a diffractive–refractive ball lens. In contrast, under the converging beam, the FWHM in most cases is smaller than the plane wave, and the PNJ moves toward the center of the sphere by increasing the source NA. For the high NA converging beam, the PNJ is located inside the sphere. Finally, we find an analogy between PNJ behaviors by changing the source NA for the sphere refractive index of $n$ in comparison to changing the sphere refractive index of $n$ under plane wave illumination.

## 3. RESULTS AND DISCUSSION

To introduce the subject, we choose an example with fixed parameters. We first investigate the PNJ of a microsphere under diverging and converging sources with NA of 0.4 and compare it with the plane wave illumination for a sphere diameter of 5 µm at 600 nm wavelengths (sphere diameter $8.3\lambda$). Results are shown in Fig. 2 in three representations: amplitude, phase, and Poynting vector streamlines. From the amplitude distributions in Fig. 2, one observes that the hot spot is formed inside the sphere for the converging source, while it moves to the outside of the sphere for plane wave and diverging beams, i.e., providing a bigger effective focal length $f$. It demonstrates that the sphere behaves like a ball lens, and for this reason, the hotspot moves along the propagation direction by going from the converging to the diverging source.

The FWHM under a plane wave is $0.53\lambda$. Compared to plane wave illumination, the FWHM is increased by 92% from $0.53\lambda$ to $1.02\lambda$, for the diverging beam, while it is decreased by 25% from $0.53\lambda$ to $0.4\lambda$ for the converging beam. Rather than source wavefront curvature, the FWHM can also be explained by the refractive index of the medium in which the hot spot is formed. In the case of a converging beam, as the hot spot is localized inside the sphere of refractive index $n$, the FWHM is $n$ times smaller than the hot spot in the air, meaning that the FWHM is $0.53\lambda /1.2 \approx 0.44\lambda$. In the following sections, we will demonstrate that the FWHM can go down to even smaller dimensions than ${\rm FWHM}/n$ for high NA converging beams.

As seen from Poynting vector streamlines, the energy flow around the hot spot is not the same for all illuminating beam curvatures. For the converging beam, the Poynting vectors around the hot spot are mostly scattered outside, although for the diverging beam, the hot spot is elongated along the propagation direction since the Poynting vectors around the focal point are almost parallel. Furthermore, the Poynting vector streamlines confirm the propagating character of the PNJ, as the vector’s intensity is higher at the hot spot position for all illuminating beam conditions. Also, the phase distributions in Fig. 2 compare the incoming beam wavefront curvatures as well as the wavefront modulations around the hot spot. As seen, a hot spot with a higher NA is generated for a converging beam in comparison to a diverging beam and plane wave illumination.

#### A. Comparison of the PNJ for Sphere Diameters of $3.3\lambda$ and $25\lambda$

Here, we demonstrate that even for small diameters of the spheres, the PNJ still strongly depends on the incident wavefront of the illumination and hence cannot be seen as a scattering phenomenon only. Figure 3 shows the evolution of PNJ for the small sphere diameter of 2 µm (or $3.3\lambda$) in comparison to a much larger diameter of 15 µm (or $25\lambda$) under diverging and converging beams of NA = 0.4 and plane wave illumination. PNJs for sphere diameters of 2 and 15 µm have the following common aspects that are also observed in Fig. 2 for the sphere diameter of 5 µm: from the field distributions, one observes that for the converging beam, the hot spot is formed inside the particle with a smaller FWHM in comparison to diverging and plane wave illumination in which the hot spot is localized outside the sphere and elongated along the propagation direction with a bigger FWHM. Also as seen from the phase modulations for both sphere dimensions, a hot spot with a higher NA is obtained for the converging beam compared to the plane wave and diverging beam.

On the other side, the PNJ for small and big diameters of 2 and 15 µm has the following different aspects: according to the field distributions, the PNJ behaves similarly to a ball lens for the sphere diameter of 15 µm, meaning that the side lobes around the hot spot are pronounced, especially under the plane wave and diverging beam. It can be explained by the lens aberration for large sphere sizes, although for the small sphere diameter of 2 µm, the side lobes are diminished around the hot spot. Figures 3(b) and 3(d) compare the transverse intensity profiles at the hot spot plane for 2 and 15 µm diameters, clarifying the intensity distribution in the side lobes. Furthermore, the effective focal length $f$ increases by 83% for the diameter of 15 µm by going from a converging to a diverging source, although $f$ increases by 40% only for the 2 µm diameter (dimensions normalized to the sphere size). This comparison confirms that the lens behavior of the sphere is dominant for a large sphere diameter of 15 µm. For this reason, we will more deeply investigate the PNJ behavior for sphere diameters of 1–20 µm (or $1.6\lambda$ to $33\lambda$).

#### B. FWHM and Effective Focal Length $f$ versus Sphere Diameter *D* for Diverging Beam

In this section, we study more systematically the PNJ for a sphere of diameters between 1 and 20 µm and with changing the source NA from 0.1 to 0.8. We concentrate on the diverging source and compare it to plane wave illumination as a reference. As seen in Fig. 4(a), the effective focal length $f$ for all NAs and also for plane wave illumination is greater than $f = D/2$, i.e., the hot spot is generated in air, outside the sphere in all cases. Also, for every source NA, the effective focal length $f$ tends to increase for a large diameter $D$. For plane wave and source NAs of 0.1 and 0.2, respectively, $f$ linearly increases as a function of $D$. For ${\rm NA}\; \gt \;{0.2}$, the linear behavior of the curve vanishes, and $f$ significantly increases for a large diameter $D$. For a high NA of 0.8, no hot spot is generated in practice for large diameters of 15 µm ($25\lambda$) and 20 µm ($33.3\lambda$) because the sphere behaves like a lens that collimates the beams.

As seen in Fig. 4(b), the smallest FWHM is achieved under plane wave illumination for all sphere diameters $D$. Also, for all source NAs, the FWHM generally increases as the particle diameter $D$ increases, providing a lager focus. For source NAs below 0.4, the FWHM with respect to $D$ is almost a stable function. By further increase of NA, the FWHM tends to increase steeply as a function of $D$, and for high NAs of 0.6 and 0.8, the FWHM is much larger than the plane wave illumination. For a high NA of 0.8, no PNJ is formed for large diameters of 15 µm ($25\lambda$) and 20 µm ($33.3\lambda$), as we explained earlier. Finally, Fig. 4 emphasizes that the generation of PNJ as a function of the sphere diameter $D$ is much more sensitive for a high NA diverging source than plane wave illumination. This point should be carefully considered when employing a high NA objective to generate the source for optical trapping applications [7], etc.

To explore the lens behavior of spheres for high NA diverging sources as a function of sphere diameter $D$, we show the field distributions in Fig. 5 for the source ${\rm NA} = {0.8}$ and various sphere diameters of 1, 2, 5, 10, 15, and 20 µm. As seen in Figs. 5(a) and 5(b), for small sphere diameters of 1 µm ($1.6\lambda$) and 2 µm ($3.3\lambda$), a PNJ is formed outside the sphere. For diameters of 5 µm ($8.3\lambda$) and 10 µm ($16.6\lambda$), the hot spot is elongated along the propagation direction having side lobes around the central lobe [Figs. 5(c) and 5(d)]. The side lobes around the hot spot for this range of diameters are the effect of aberration, which demonstrates a transition from the PNJ scattering origin to the behavior of a ball lens having aberrations. According to Figs. 5(e) and 5(f), for 15 µm ($25\lambda$) and 20 µm ($33.3\lambda$) diameter, the sphere behaves like a lens and collimates the beam out of the sphere, although the diffracted beams are also visible in the collimated beam, i.e., the sphere in practice is a diffracting lens. In this case, no PNJ is generated.

#### C. FWHM and Effective Focal Length $f$ versus Sphere Diameter $D$ for Converging Beam

Here, we study the evolution of PNJ as a function of sphere diameter $D$ for converging beams with different NAs in comparison to plane wave illumination. As seen in Fig. 6(a), the effective focal length $f$ linearly increases by increasing the particle diameter $D$ for all NAs of the converging beam and plane wave illumination. For larger NAs, the slope of the effective focal length curve decreases, the PNJ immersing inside the sphere for NAs of 0.4, 0.6, and 0.8 (curves are below $f = D/2$). In contrast, the PNJ is localized in the air, outside the sphere for the plane wave and low NAs of 0.1 and 0.2.

As seen in Fig. 6(b) for low NAs of 0.1 and 0.2, the FWHM variation follows more closely the plane wave illumination, cases in which the PNJ is located outside the sphere in the air, but for high NAs of 0.4, 0.6, and 0.8, the FWHM curve is slowly varying as a function of the sphere diameter $D$. A similar trend between these curves also comes from the fact that the PNJ is localized inside the sphere for all these cases. Also, the FWHM can decrease to $0.33\lambda$ for converging sources of high NAs 0.6 and 0.8, even for large sphere diameters $D$, although for plane wave illumination, the FWHM increases for large sphere diameters and reaches $0.7\lambda$. The achieved ultra-narrow FWHM width for high NA sources is not only because of the medium refractive index $n$ but also depends highly on the source NA. In summary, using a high NA microscope objective as the source in the experiment, we have to carefully consider that there is a possibility that the PNJ is generated inside the particle, and no field enhancement is achieved outside the sphere to apply for optical trapping [7] and other applications.

#### D. Analogy Between Converging Source NA and Sphere Refractive Index $n$

As we discussed, the PNJ for a glass sphere with a refractive index of 1.5 moves in the opposite direction of field propagation, toward the sphere surface, by increasing the converging source NA, and the PNJ is immersed inside the sphere for NA = 0.4, 0.6, and 0.8. On the other hand, it is known that for plane wave illumination, according to geometrical optics, the focal length of the ball lens with a radius of $R$ and refractive index of $n$ is given by [31]

According to this equation, the focal point is outside the sphere for $n \lt 2$. For $n = 2$, the focal point is at the sphere surface. And for $n \gt 2$, the focal point is inside the sphere and moves toward the center of the sphere by a further increase in refractive index $n$ [31]. This geometrical optics approximation for the focal length of the ball lens under plane wave follows behavior similar to increasing the converging source NA for a glass sphere, as discussed above. To provide more insight, we show in Fig. 7 the field distribution for a glass sphere of 15 µm diameter under a converging beam in comparison to a sphere with a refractive index of 2.5 under plane wave illumination. As seen, the PNJ is localized inside the glass sphere under the converging beam and also for the high refractive index sphere under plane wave illumination [1,18]. It means that the effect of a large source NA is equivalent to a high refractive index sphere under a plane wave, although, a PNJ with a higher NA is generated for a converging source in comparison to the high refractive index sphere under plane wave illumination, as seen in Fig. 7.

Next, we compare small and large diameters of 2 and 15 µm. The curves in Fig. 8, show the effective focal length $f$ and FWHM for $n = 1.5$, by changing the source NA from zero to 0.8, in comparison to plane wave illumination by changing the refractive index $n$ between 1.5 and 2.5. When comparing Figs. 8(a) and 8(b), we find a similar trend between the effective focal length curves versus the NA and refractive index $n$. As seen in Fig. 8(a), the PNJ is localized inside the sphere for ${\rm NA} \gt 0.25$ for sphere diameters of both 2 and 15 µm. According to Fig. 8(b), the PNJ for a sphere diameter of 15 µm, is localized inside the sphere for $n \gt 1.8$, although for the small diameter of 2 µm, the PNJ is immersed inside the sphere for $n \gt 1.6$, i.e., $f$ as a function of refractive index $n$ is more sensitive than $f$ versus the source NA for small and large sphere diameters. It can also be seen in Fig. 8(b) that using the ray-tracing equation, the beams are focused inside the sphere for $n \gt 2$ for sphere diameters of both 2 and 15 µm, showing that the ray tracing does not include the effect of particle size, although $f$ depends on both the refractive index $n$ and size of the sphere, for this diffractive lens [16,17].

In Fig. 8(c), the FWHM for a sphere diameter of 2 µm is first increased and then tends to decrease by increasing the source NA. For a sphere diameter of 5 µm, the FWHM is decreased by the source NA. As seen in Fig. 8(d), the FWHM tends to increase for a large refractive index $n$ [16]. For the large diameter of 15 µm, both curves in Figs. 8(c) and 8(d) follow similar trends, although for a small diameter of 2 µm, the curves behave differently. On the other side, the minimum FWHM obtained value as a function of NA is $0.33\lambda$, although the FWHM can decrease to $0.25\lambda$ for a large refractive index of 2.5.

#### E. Comparison between 2D and 3D Simulations

In a real, practical scenario, the microstructure is a 3D sphere illuminated by a microscope objective; however, our simulations are 2D. In Ref. [32], the behavior of PNJ is studied for 3D spheres in comparison to the 2D cylinder geometries for diameters of 1–5 µm under plane wave illumination. The optical characteristics of PNJs for 2D and 3D geometries are different; however, they follow a similar trend as a function of structure diameter. In this section, we compare the optical characteristics of PNJ for cylinders and spheres of 1 and 5 µm diameters under different source wavefront curvatures. For sphere diameters larger than 5 µm, a 3D simulation is computationally extensive and not practical.

In Fig. 9(a), the evolution of PNJ is shown for a sphere diameter of 1 µm under converging and diverging sources (NA = 0.4, 0.8) in comparison to plane wave illumination. For both 2D and 3D geometries, by going from a high-converging to a high-diverging source, the formed PNJ moves along the direction of propagation (${+}x$). However, for all the illumination conditions, a longer PNJ is realized for a cylinder compared to 3D sphere geometry [32].

In Fig. 9(b), the FWHM and effective focal length $f$ of PNJs are shown. As seen, the optical characteristics of PNJs are not equal for the cylinder and sphere; however, both configurations show a similar trend under different wavefront curvatures. Going from a high-converging to a high-diverging source, the following tendencies are observed in the curves for both cylinder and sphere: the FWHM is increased, except that a dip is realized in the FWHM curves in the case of plane wave illumination. Also, the effective focal length of $f$ tends to increase.

In Fig. 10(a), the field distributions are demonstrated for a larger sphere diameter of 5 µm. As observed, the PNJ moves along the direction of propagation by going from a high-converging to a high-diverging source wavefront. Moving the hot spot along the direction of propagation is more pronounced for a diameter of 5 µm, compared to the smaller diameter of 1 µm. Also, the PNJ is elongated along the propagation direction with a longer length for the cylinder compared to a sphere geometry. In Fig. 10(b), the FWHM and effective focal length ($f$) of PNJ are compared for cylinder and sphere geometries. The FWHM tends to increase by the evolution of source curvature from converging to diverging, although a minimum is visible in the FWHM curve for plane wave illumination. Also, the effective focal length of $f$ increases by changing the source curvature from converging to diverging.

Here, we observed similar PNJ behavior for 2D and 3D scenarios according to the diameters of microstructures for different source wavefront curvatures. It demonstrates that our theoretical findings in this paper for a 2D cylinder can be useful and informative for understanding the behavior of real 3D spheres in practical applications.

## 4. CONCLUSION

In conclusion, we systematically study the PNJ formed by a microsphere under diverging and converging sources of low and high wavefront curvatures in comparison to plane wave illumination, by conducting 2D FDTD simulations. We study the characteristic parameters of PNJ, including the effective focal length and FWHM. The PNJ shows a completely different behavior under converging and diverging illuminations with NAs of 0.1 to 0.8 for sphere diameters of $1.6\lambda$ to $33\lambda$. For a diverging source curvature, the PNJ is found to move along the propagation direction by increasing the source NA, forming a hot spot with a larger effective focal length and FWHM for all sphere diameters. By further increase of NA to 0.8, PNJ is generated only for sphere diameters less than $16\lambda$, and for larger diameters, the microsphere is a diffractive–refractive ball lens that collimates the beams. In contrast, under the converging beam, the effective focal length and FWHM tend to decrease as a function of source NA, and for ${\rm NA} \gt 0.2$, the PNJ is immersed inside the sphere with a FWHM that can be as small as $0.33\lambda$. We also found similar PNJ behavior for two configurations of a glass sphere (i) under a converging beam by increasing the source NA from 0.1 to 0.8 and (ii) increasing the sphere refractive index from 1.5 to 2.5 under plane wave illumination.

Finally, we compare our theoretical 2D simulation results with 3D sphere geometries and find similar behavior for converging and diverging source wavefront curvatures. Thus, our theoretical findings are helpful for those practical applications that implement a high NA objective to illuminate 3D real particle geometries for optical trapping, etc.

## Funding

Horizon 2020 Framework Programme (675745).

## Acknowledgment

The authors thank Prof. O. Martin (NAM-EPFL) and D. Ray (NAM-EPFL) for useful discussions.

## Disclosures

The authors declare no conflicts of interest.

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