Muhammad Hamidullah
email: muhammad.hamidullah@femto-st.fr

Céline Élie-Caille

Thérèse Leblois

Higher-order Lamb waves with quasi-zero surface displacement components on a GaAs piezoelectric plate

Keywords: Lamb wave, zero surface displacement, sensor, acoustics, liquid media

Higher-order Lamb waves with quasi-zero surface displacement components are reported on (100)-cut GaAs propagating along the <110> direction where the total displacement at the surface of the plate is less than 10% of the maximum total displacement. The dispersion curves and the displacement component profiles show the reduction of total displacement at the surface of the plate starting when the phase velocities of the higher-order modes are crossing the shear bulk acoustic wave velocity to the value as low as 5%. Due to the concentration of acoustic energy inside the plate, the reported quasi-zero plate acoustic waves (QZ-PAW) further reduce the radiation of acoustic energy when the plate surface is in contact with liquid. The experimental results validate the occurrence of QZ-PAW with a reduction of viscous damping insertion loss compared to previously reported quasi-longitudinal Lamb waves (QL-LW). The results demonstrate the potential QZ-PAW mode for emerging applications such as dual-mode PAW sensors, PAW devices with integrated sensor and actuator, thin-film and ultra-high frequency (UHF) PAW sensors in highly viscous liquid media.

Introduction

Several types of acoustic waves (AW) devices have been extensively studied and reported for sensor application in liquid media [1]. The main aim is to prevent or minimise the radiation of acoustic energy into the liquid media. In general, AW can be classified into three different categories: bulk acoustic waves (BAW), surface acoustic waves (SAW) and plate acoustic waves (PAW). BAW-based devices with inplane components polarisation such as Quart Crystal Microbalance (QCM) [2], lateral field excitation BAW [3], and shear mode thin-film bulk acoustic resonator (TFBAR) [START_REF] Wingqvist | Yantchev[END_REF], for example, are suitable for a sensor in liquid media by having close-to-zero out-of-plane displacement component at the surface, thus preventing the conversion of acoustic energy into longitudinal pressure wave in the liquid. Similarly, several types of SAW are suitable for a sensor in liquid media, such as shear horizontal SAW (SH-SAW) [5] and Love mode (LM) [6] and high velocity pseudo-SAW (HVPSAW) [7].

The radiation of acoustic energy into the liquid can also be prevented by using the AW with phase velocity, vph, lower than the compressional sound velocity of liquid media (1500 m/s in water, for instance). For PAW-based sensors in liquid media, this condition can be achieved by using the fundamental antisymmetry Lamb wave (A0) mode [8]. However, it is limited only at a low plate thickness to wavelength ratio (h/λ), typically at an h/λ ratio lower than 0.1. Consequently, making the device very fragile or, if a thicker plate is used, a larger wavelength is necessary, thus lowering the operating frequency. Gao et al. reported the use of the mass-loading technique by using a high aspect ratio electrode to reduce the velocity of the waves lower than the velocity of the liquid with a trade-off in the reduction of electromechanical coupling coefficient and the complexity of the fabrication process to obtain high aspect ratio electrode [9].

PAW based devices have an advantage over SAW for sensor application in liquid media as they have two free surfaces that are both sensitive to surface perturbations. Thus when the interdigital transducer (IDT) is employed to convert the electrical energy into acoustic waves and vice versa in one of the surfaces, the opposite surface can be used as the sensing layer. Thus the electrical connection is separated from liquid contact without requiring an additional thin dielectric layer covering the IDT. Several types of PAW have in-plane polarisation, such as shear horizontal acoustic plate mode (SH-APM) [10] and fundamental symmetric Lamb waves (S0) [11]. For S0 mode, dominantly longitudinal displacement condition is also achieved at a low h/λ ratio. Therefore it has a similar limitation as A0 mode in terms of the robustness of the device.

While AW type with in-plane polarisation minimises the acoustic energy radiation, the loss of acoustic energy is still occurring due to the penetration of in-plane particle displacement and the wave-liquid interaction length at the solid/liquid interface. Due to non-zero in-plane particle displacement at the solid surface, or even in most cases are maximum at the surface, the in-plane particle displacement exponentially decreases to zero in the liquid. The thin liquid layer in the solid/liquid, where the liquid is coupled to the oscillating surface of the plate, is called the characteristic penetration depth or a decay length δ [12] and is given by :

δ = √ 2η 𝜌𝜔 Equation (1)
δ is dependent on the liquid viscosity (η), liquid mass density (𝜌) and the angular operating frequency (𝜔) of the AW. In water and for AW-based sensor with several MHz operating frequency, δ is around 50 nm. The viscous-damping insertion loss (IL) when the sensor is in contact with Newtonian liquid is given by :

IL = A√𝜌𝜔η Equation (2)
and

A ≈ 𝜔𝑙 𝜌 𝑠 𝑣 2 0
, where 𝑙 is the wave-liquid interaction length, 𝜌 𝑠 is the substrate mass density and the 𝑣 0 is the velocity of the wave.

As shown in equation ( 2), the insertion loss will increase as the liquid viscosity increases. Therefore the magnitude of insertion loss can be used to quantify the liquid viscosity. However, when the liquid viscosity is very high, the significant reduction in the signal strength affects the signal-to-noise ratio and the quality factor of the sensor, and consequently, the sensor's sensitivity. Additionally, the insertion loss will further increase when the sensor has a higher operating frequency, limiting the maximum operating frequency of the AW-based sensor application in liquid media.

In this paper, we report a type of higher-order plate acoustic wave with close-to-zero particle displacement at the surface of the plate, thus reducing the wave-liquid interaction length and the insertion loss. Finite element simulations were performed to obtain the dispersion curve, the displacement profile and the insertion loss of quasi-zero plate acoustic waves (QZ-PAW) due to viscous damping. The occurrence of QZ-PAW is validated experimentally. The performance of QZ-PAW is compared with inplane polarised quasi-longitudinal Lamb waves (QL-LW), followed by further discussion of the potential and future application of QZ-PAW for sensor application in highly viscous liquid media.

Lamb Waves Theory and Finite Element Simulation

Dispersion Curves and Displacement Profile

A general solution of Lamb's characteristic equation in an isotropic and homogenous plate results in Lamb wave (LW) dispersion curves with two sets of sinusoidal waves (symmetric S and antisymmetric A modes). Lamb waves have two-component displacements: longitudinal (in-plane, u1) and shear vertical (out-of-plane, u3) displacements [13]. The S0 and A0 modes converge into Rayleigh wave (RW) velocity in a thick plate, while the higher-order modes asymptotically reach the bulk shear wave (SBAW) velocity. Viktorov firstly reported the symmetric modes with dominantly u1 displacement component, when the phase velocities of S modes reach the value of bulk longitudinal wave phase velocity (vSBAW), and the u3 vanishes on the surface of the plate [START_REF] Viktorov | Rayleigh and Lamb Waves: Physical Theory and Application[END_REF]. These observations are further explained theoretically by Pilarsky et al. for the isotropic plate case [START_REF] Pilarski | [END_REF]. Furthermore, Veres et al. [16] discussed the maximum and minimum of the surface displacements of LW. They reported that the longitudinal displacement component vanishes at the surface with maximal shear vertical displacement component when the LW higher-order modes phase velocity (vph) at the crossing is vph = √2 vSBAW. Thus, it is possible to obtain LW with only in-plane (QL-LW) or out-of-plane (quasi-vertical LW, QV-LW) displacement components by choosing a specific plate thickness, frequency, and wavelength.

For anisotropic and piezoelectric material, there are two solutions of SBAW with two distinct vph (SBAWfast and SBAW-slow). Furthermore, when the plate is highly anisotropic (e.g. ST-cut Quartz), the solutions of the surface displacement component have a non-zero shear horizontal component (u2). The higher-order quasi longitudinal LW (QL-LW) and quasi shear horizontal LW (QSH-LW) are reported in ST-cut Quartz [17,18]. The higher-order modes LW with in-plane surface displacement expand the application of LW with higher frequency and higher plate thickness for a sensor in a liquid environment.

3D Comsol Multiphysics eigenfrequency simulation was used to obtain the Lamb waves resonant frequencies of GaAs plate at different h/λ ratios. The solutions of eigenfrequency for each mode are multiplied by the wavelength to obtain the phase velocities (vph). Figure 1a shows the dispersion curves of the vph vs h/λ ratio of (100)-cut GaAs Lamb waves propagating on <110>-direction. GaAs is a piezoelectric material and has a cubic crystal structure. There are three vph solutions of BAW that propagates at <110> direction: one longitudinal BAW (LBAW) and two SBAW. The BAW phase velocities are 5240 m/s, 3345 m/s and 2475 m/s for LBAW, SBAW-fast and SBAW-slow respectively [19]. The solution of SBAW-slow vph is lower than the RW vph. The particle displacement is out-of-plane for SBAW-fast (shear vertical) and in-plane for SBAW-slow (shear horizontal). As shown in figure 1a, the A0 and S0 modes vph converge into RW.

Figure 1b shows the inset where the higher-order modes are converging into the vph of SBAW-fast. As shown in figure 1b, at a higher h/λ ratio, the modes are crossing the vph of SBAW-fast and then stagnating with a slight fluctuation slightly below the SBAW-fast vph, which is different from the expected behaviour of higher-order modes based on Lamb waves characteristic equation in the case of isotropic plate material. In supplementary material 1 (S1), using the same COMSOL FEM method, the dispersion curves of A1 and S1 modes for (100)-cut GaAs propagation along <110> with an isotropic GaAs by assuming it has only two independent variables ((C11, C12, and C44 = (C11-C12)/2) are plotted to compare the different behaviour of isotropic and anisotropic plate. For the isotropic case, The A1 and S1 modes are asymptotically reaching the SBAW velocity at a higher h/λ ratio, which agrees with the solution of the LW characteristic equation. However, for anisotropic material, even for the simplest cubic crystal structure with three independent variables, the equations will become more complex, resulting in different behaviour of dispersion curves. In material with a higher degree of anisotropic such as ST-cut Quartz, the higher-order modes are lowly dispersive around the SBAW-FAST before the vph values drop asymptotically, reaching the value of SBAW-SLOW [18]. The values of SBAW-FAST (5100 m/s) and SBAW-SLOW ( 3298 m/s) are higher than the RW (3157 m/s) in ST-cut Quartz.

Furthermore, the piezoelectricity effect will also affect the behaviour of the dispersion curves In addition to the fluctuating value of vph slightly lower than the SBAW-FAST, the total displacement at the surface of the plate (usurf) of the higher-order modes is reducing relative to the maximum total displacement inside the plate (umax). The QZ-PAW for the first three higher-order modes (A1, S1 and A2) are shown in figure 1c. As shown in figure 1c, the particle displacement and the acoustic energy are concentrated inside the plate with almost zero total displacement values at the surfaces.

Figure 2a shows the total displacement on the plate surface normalised by the maximum total displacement inside the plate (usurf/umax) vs the h/λ ratio for the first four higher-order modes. As shown in figure 2a, the usurf/umax reduces to the value as low as 5% at h/λ ratio of 2.3 to 3.1, 3.5 to 4.8, and 3.8 to 4.7 for A1, S1 and A2 modes, respectively. Referencing the definition of QL-LW mode in [18], here we define the quasi-zero plate acoustic waves (QZ-PAW) as the LW with the usurf/umax equal to 10% or less. By this definition, the range of h/λ ratio for QZ-PAW is 2.1 to 3.3, 3.3 to 4.5 and 3.7 to 4.8 for A1, S1 and A2 modes, respectively. The total displacement is the sum of u1, u2, and u3 particle displacement components. The u1 and u2 are in-plane particle displacements, while u3 is out-of-plane displacement. In this case, due to the cubic crystal symmetry, u2 is zero. Thus only u1 and u3 are contributing to the total displacement. Figure 2b shows the

A 1 S 1 A 2 S 2 surface/maximum total displacement u 1 /u 3 h/l A 1 S 1 A 2 S 2
ratio of u1/ u3 at the surface of the plate. As shown in figure 2b, for A1 mode, the surface displacement is dominantly in-plane longitudinally polarised at the h/λ ratio of 2.7. Similarly, for S1 and A2 modes, the h/λ ratios with dominantly longitudinal displacement are 3.9 and 4.3. Thus, while the usurf/umax at different h/λ ratios are almost the same (5% for A1 mode at h/λ from 2.3 to 3.1, for instance), the u1/u3 are significantly different, which will affect the magnitude of insertion loss when the plate will be in contact with liquid.

In figure 2a, the usurf/umax of A1 mode is higher than 10% at the h/λ ratio of 3.4 to 4.2 and reducing again to lower than 10% at the h/λ ratio of 4.3 and higher, indicating a periodicity in usurf/umax with respect to the h/λ ratio. Similar periodic behaviour is also shown in the fluctuation of vph when the value is lower than vBAW-FAST. The usurf/umax is plotted with the vph/vBAW-FAST for A1 and S1 modes in figures 3a and 3b, respectively. As shown in figure 3, the lower value of usurf/umax correlates to the less dispersive region when the vph is almost constant to the change in the h/λ ratio. Conversely, when the wave is becoming more dispersive, the usurf/umax increases. This correlation shows that the condition of the QZ-PAW is achieved when the wave is less dispersive. Furthermore, the fluctuation in both vph/vBAW-FAST and usurf/umax ratio shows that higher-order modes remain dispersive at higher plate thickness. In contrast, the fundamental modes A0 and S0 are converted into a non-dispersive RW with a constant phase velocity value. Figure 4 shows the normalised displacement components profiles of A1, S1 and A2 modes at h/λ ratio of 2.7, 3.9 and 4.3, respectively. As shown in figure 4, the u3 for the three modes vanishes at the surface. The particle displacement at the surface is dominantly longitudinal with the normalised displacement amplitude of only 5% of the maximum displacement inside the plate. A small in-plane displacement on the surface of the plate in contact with the liquid will reduce the wave-liquid interaction length, thus reducing the insertion loss as given in equation ( 2). Therefore, they are expected to have the least acoustic energy radiation to the surrounding media and the minimum insertion loss due to viscous damping.

Electromechanical Coupling Coefficients

A general limitation of higher-order modes is the reduction of the electromechanical coupling coefficient (K 2 ) when a thicker plate or a higher-order number is used [20]. Furthermore, since the conversion of electrical to acoustic energy occurs at the surface where the IDT is located, it is expected that the QZ-PAW will have an even smaller K 2 . The deposition of the metal layer at the opposite surface of the IDT changes the electrical boundary condition and can potentially increase the value of K 2 [20,21]. Figures 5a andb show the K 2 of A1, S1, A2 and S2 modes with Plate-IDT (S-IDT) and Metal-Plate-IDT (M-P-IDT) coupling configuration, respectively, vs the h/λ ratio at the range where QZ-PAWs exist. As shown in figure 5a and5b, the K 2 value reduces with the minimum K 2 at h/λ ratio of 2.9, 4 and 4.5 for A1, S1 and A2 modes, respectively.

Comparing the lowest K 2 for each mode to the higher u1/ u3 at h/λ ratio, we can observe a shift of around 0.1 to 0.2 h/λ ratio. The h/λ ratios for the modes with the highest longitudinal component (u1/ u3 ratio) do not correspond to the h/λ ratios with the lowest K 2 . The correlation of K 2 to usurf/umax does not explain the lowest K 2 at a specific h/λ ratio. For A1 mode, for instance, the lowest K 2 is at h/λ ratio of 2.9, while the usurf/umax ratios are almost constant, around 5% for the h/λ ratio range from 2.3 to 3.1. One possible reason we cannot observe a direct correlation is that the K 2 value is affected by more complex interaction between both longitudinal and shear horizontal components to the piezoelectricity coupling coefficient. At h/λ ratio when the QZ-PAW is dominantly longitudinal, the K 2 is in the order of between 10 -5 to 10 -6 . Furthermore, adding a metal layer for M-P-IDT configuration, as shown in figure 5b, does not improve the K 2 . The K 2 is related to the conversion rate of electrical energy into acoustic energy and vice versa. For instance, for two-port delay line device implementation, the value of K 2 will affect the power received in the receiver IDT. Based on the K 2 values, we can expect the S21 transmission signal strength will be in the range of -50 dB to -60 dB. While the S21 signals will be relatively low, however level of the signal floor is more important in practical application. The signal floor is highly affected by the feedthrough of electromagnetic waves that instantaneously transmit the electrical power from the transmitter to receiver IDT. One of the methods to reduce this feedthrough is by electrically grounding the wafer's backside, thus preventing the feedthrough effect due to the substrate conductivity. This method can reduce the signal floor by 10 to -15 dB [START_REF] Visser | IEEE 1988 Ultrasonics Symposium Proceedings[END_REF]. While the addition of the metal at the back of the substrate does not improve the K 2 , it can reduce the signal floor in the S21 scattering transmission parameter by electrically grounding the metal layer.

Solid/Liquid Contact Insertion Loss

3D COMSOL Multiphysics eigenfrequency simulations were performed to calculate insertion loss when one of the plate surfaces is in contact with liquid. We used the isotropic viscoelastic material model for the liquid domain to be able to consider both shear (ηv) and bulk viscosity (ηb) of the liquid [START_REF] Mirea | [END_REF], with more detail in the simulation model described in supplementary material 2 (S2). The value of ηv and ηb are 0.00093 Pa.s and 0.0023 Pa.s. Figure 6a shows the total displacement profile of QZ-PAW A1, S1 and A2 modes at h/λ ratios of 2.7, 3.9 and 4.3, respectively. As shown in figure 6a, the total displacement and acoustic energy are concentrated inside the plate and close to zero at the plate/liquid interface. The complex resonant frequencies were obtained from the simulation, and the complex phase velocities (vph = vr + jvi) were obtained by multiplying the resonance frequency with the wavelength. The imaginary part of the phase velocities indicates a leaky Lamb wave where some of the acoustic energies are radiated into the liquid. Higher radiation of acoustic energy will be translated into a higher insertion loss (IL) value in dB/λ, which can be calculated by the equation below [24] 𝐼𝐿 = -40π(𝑙𝑜𝑔 10 𝑒)

𝑣 𝑖 𝑣 𝑟 = -54.6 𝑣 𝑖 𝑣 𝑟 (Equation 3)
The theoretical insertion losses vs the h/λ ratios are plotted in figure 6b for the first four higher-order LW modes, in contact with DI water and propagating at a distance of 250 λ. The wavelength of the LW devices in this simulation is 168 μm, so the propagation distance is equal to 42mm. As shown in figure 6b, the ILs is less than 1 dB for the h/λ range of QZ-PAW when the surface displacement is less than 10% of the maximum displacement inside the plate. The lowest losses, as expected, are achieved at the h/λ ratios of 2.7, 3.9 and 4.3 for A1, S1, and A2 modes, respectively, when the surface displacement components are dominantly longitudinal and usurf/umax of around 5%.

The lowest theoretical ILs are 0.00894 dB for A1 mode, 0.00453 for S1 mode and 0.00274 for A2 mode. However, at the h/λ ratios with the lowest IL, several other modes have a very close phase velocity and resonant frequency. For example, at h/λ of 2.7, the A1 and S1 modes have resonant frequencies of 19.786 and 19.78 MHz, respectively. Thus the difference between the resonance peak is merely 6kHz. While A1 is a QZ-PAW with the lowest theoretical IL, S1 mode is not a QZ-PAW with usurf/umax of 22% and the theoretical IL of 3.45. Therefore, the resonant bandwidth must be less than 6 kHz to separate the resonant peaks for A1 and S1 modes in the frequency response signal. This bandwidth value will be challenging to achieve, especially in the two-port delay line device implementation. The summary of the non-QZ modes merged into the QZ-PAW modes is summarised in Table 1. As shown in Table 1, a challenge of the multiple modes merging into a single peak also similarly occurs in h/λ ratio of 3.9 and 4.3, limiting the minimum insertion loss for the practical applications of the device.

Time Domain Simulation

Two-port delay line QZ-PAW devices were simulated using COMSOL Multiphysics time-domain study to obtain the S21 frequency response of the devices. From the 3D model, the u2 is always zero across the plate. Therefore to simplify the simulation, a 2D model of a two-port delay line will be sufficient to obtain a relatively accurate simulation result. An impulse response of 1 ns is applied in the transmitter IDT to excite the Lamb wave modes. The waves are propagating across the plate, and the output voltage signals at the receiver IDT were recorded. The IDTs wavelength is 168 μm, and the IDTs have 50 finger pairs with a delay length of 200λ, thus equal to the IDT centre to centre distance of 250λ. The signal's bandwidth can be approximated by dividing the resonance frequency by the number of IDT finger pairs. Thus for the QZ-PAW resonance peak of around 19.7 MHz, the bandwidth will be around 394 kHz. The output signals are converted into the frequency domain by Fourier transform, and the S21 transmission scattering parameter is calculated from the ratio of output and input voltage with the equation below : 4)

S21 (dB) = 20 log (Vout/Vin) (Equation
The simulated S21 scattering parameter frequency response at the h/λ ratios of 2.7, 3.9, and 4.3 is shown in figures 7a, 7b and 7c, respectively. As shown in figure 7a, the A1 and S1 modes are merged into a single resonant peak. Similar conditions are also shown in figures 7b and 7c for h/λ 3.9 and 4.3, respectively. In figure 7b, at 3.9 h/λ ratio, the A1 mode is no longer classified as QZ-PAW with usurf/umax value of 13%. However, since the total displacement at the surface is still relatively low, the theoretical insertion loss in DI water liquid contact is reasonably low at 1.27 dB. The main challenge is to separate the S2 mode with usurf/umax value of 33% and the insertion loss of 5.78 dB. The resonant frequency difference between A2 and S2 modes is 170 kHz. In figure 7c, at h/λ ratio of 4.3, both S2 and A3 modes are non-QZ-PAW modes with resonant frequency differences from A2 mode of 12 kHz and 421 kHz, respectively. A2 and S2 modes resonant peaks distance can be increased by using h/λ ratio lower than 3.9 to increase the S2 vph. For example, as shown in Table 2, when the h/λ ratio is reduced to 3.7, the difference between A2 and S2 modes is increased from 170 kHz to 319 kHz to separate the resonance peaks, with a trade-off in the increase of with usurf/umax and insertion loss of A1 mode to 14% and 1.79 dB. Another advantage of using the h/λ ratio of 3.7 is to obtain the QL S6 mode with a dominantly longitudinal displacement [20], to compare the insertion loss between QZ-PAW and QL-LW, which will be demonstrated in the experimental results.

Experimental Validation and Discussion

To validate the occurrence of QZ-PAW experimentally, we designed a two-port delay line devices using a standard 3-inch (100)-cut GaAs wafers with thicknesses of 625 µm and 650 µm, with the main hypothesis in the reduction of insertion loss when the surface of the plate is in contact with liquid. The IDT designs are identical for both wafers with a wavelength of 168 µm. Thus the h/λ ratios are 3.7 and 3.9 for the 625 µm and 650 µm wafers, respectively. The IDTs have 50 finger pairs with an aperture of 50λ. The delay line length is 200λ so that the centre to centre distance between the transmitter and receiver IDTs is 250λ or equal to 42 mm. The direction of propagation is parallel to the wafer flat, which corresponds to <110> direction. The Piranha process cleaned the GaAs to remove organic contamination, followed by dehydration baking. For IDTs patterning, the wafer was coated by Ti prime as a photoresist adhesion layer and negative photoresist of nLOF AZ2000. The thickness of the photoresist is 1.3 µm. The wafer was exposed using EVG 6200 photolithography tools followed by photoresist hardbake and development process. 100 nm of aluminium is used as an electrode with 10 nm Cr as an adhesion layer, and they were deposited by the electron beam evaporator. After metal deposition, the photoresist was stripped using photoresist remover for the lift-off process. 50nm Au was deposited at the back of the wafer to obtain an M-P-IDT coupling configuration. Even though the M-P-IDT configuration does not improve the K 2 , the electrically short boundary condition at the sensing layer will reduce the electromagnetic feedthrough [START_REF] Visser | IEEE 1988 Ultrasonics Symposium Proceedings[END_REF] and decouple liquid conductivity [25]. The wafers were attached to PCB with a centre hole so that the IDTs and the delay line are on the wafer that is suspended to obtain free surfaces mechanical boundary condition. The backside of the wafer is electrically connected to the ground of the PCB to obtain an electrically grounded boundary condition. The image of the fabricated devices on the PCB and the experimental set-up can be found in supplementary material 3 (S3). Figure 8 shows the characterisation results of two two-port delay line devices having an h/λ ratio of 3.9 and 3.7 in figures 8a and 8b, respectively, with and without liquid contact. DI water and 100% glycerol were used as liquid contact. Despite the low K 2 value, the signal resonant peaks of around -56 dB are observable from the signal floor of -68 dB. As shown in figure 8a, at 3.9 h/λ, A1, S1, A2 and S2 modes are combined into a single peak, as previously predicted in the time domain simulation results. The first resonant peak in figure 8a has a large bandwidth due to a merge of four waves with theoretical frequency ranges from 19.692 MHz to 20.033 MHz. However, since the S2 mode has a total surface displacement of 33 % and theoretical IL of 5.78 dB, the S2 mode will have a much higher insertion loss the plate is in contact with liquid. As shown in figure 8a, the reduction in S21 is asymmetrical with around 6 dB reduction in the frequency area higher than 20 MHz, which corresponds to the S2 mode. Thus, the significant reduction in the S2 mode separates the S2 mode from the other three modes with lower resonance frequencies.

The A1 mode has a relatively higher theoretical insertion loss of 1.27 dB than the S1 and A2 modes, with theoretical insertion losses of 0.005 and 0.1 dB. As the reduction in S21 is around 1 dB in the frequency area lower than 20 MHz, it could be expected that the A1 mode contributes to the reduction in the S21 signal and the remaining S21 peak observed when the surface in contact with DI water consists of the QZ-PAW modes of S1 and A2. Hence, while the four modes merged into a single peak in a dry environment, higher S21 signal reduction in the liquid media will filter out the non-QZ mode from the QZ-PAW resonant peak. Furthermore, when the liquid contact is replaced from DI water to glycerol with much higher viscosity, there is a further reduction of around 0.8 dB in the resonance peak of the QZ mode. Additionally, as shown in figure 8a, the A3 and S3 modes are fully damped with liquid contact. The signal is fully damped because both modes have a maximum displacement at the surface with out-of-plane u3 components. Consequently, acoustic energy is fully radiated into the liquid and is not received in the receiving IDT.

For practical applications of sensor devices in quantitatively monitoring physical parameters of the liquids, it is necessary to separate the S2 mode without the liquid contact so that the QZ-mode signal peak and the bandwidth can be identified and calculated. For separating the S2 mode peak from the lower frequency peak, a thinner GaAs wafer with a thickness of 625 μm was used to lower the h/λ. At 3.7 h/λ, the vph is higher, thus increasing the difference in the resonant frequency peak between S2 and A2 modes. As shown in Figure 8b, the S2 mode is further separated from the other three modes as the theoretical resonant frequency difference increases from 170 kHz to 319 kHz.

However, the resonance frequency difference of 319 kHz is still lower than the signal bandwidth of 394 kHz, thus meaning that the S2 mode is not completely separated, which can be the reason for the appearance of two peaks below the S2 mode label in figure 8b. The frequency difference can be further increased by a lower h/λ ratio of 3.6, with the trade-off in higher surface displacement and higher insertion loss. Despite this limitation, the reduction in IL when the surface is in contact with liquid can be qualitatively calculated in QZ-PAW with 3.7 h/λ. Furthermore, at the h/λ ratio of 3.7, the GaAs device will also excite the S6 mode with dominantly longitudinal displacement at the surface (QL-S6 mode) with the vph close to LBAW vph [20]. The detail on the QL-6 mode is described in supplementary material 4 (S4). Thus it will be possible to compare the performance of QZ-PAW mode with QL-LW mode when the surface is in contact with highly viscous liquid. Figures 9a and9b compare the QZ-PAW mode and QL-LW mode in dry and liquid contact. The QL-LW has a higher vph, thus a higher resonant frequency. As shown in figure 9a, the QZ-PAW mode has 1.8 dB reduction in S21 from dry to glycerol contact, about one-third of the S21 reduction of QL-LW mode. The loss of 5.1 dB of the S6 QL-LW mode, as shown in figure 9b, is comparable with the previously reported S1 QL-LW mode in ST-cut quartz with a similar operating frequency [26].

Figure 9c shows the change in the insertion loss when the plate surface is in contact with different DI water/glycerol mixtures, with the insertion loss in DI water as the reference value. The concentration of glycerol is 25%, 50%, 75% and 100%. The viscosity and mass density increase as the increase of glycerol concentrations and the change in insertion loss is plotted against the square root of the mass density and viscosity product. As shown in figure 8c, using the DI water as a reference, the change in the insertion loss is 0.95 dB when 100% glycerol is used for QZ-PAW and 3.9 dB for QL-LW. Thus the QL-LW has almost four times higher insertion loss. The saturation of insertion loss at higher viscosity is similar to the previously reported viscosity sensor based on SH-APM mode [12]. The result shows the advantage of the QZ-PAW mode in further reducing acoustic energy radiation due to lower total displacement at the surface, which reduces the wave-liquid interaction length, and consequently, the insertion loss due to viscous damping. The sensor signals (phase velocity, frequency, or attenuation) are highly affected by the surface condition. Since the QZ-PAW modes are not sensitive to surface perturbation, thus they are not suitable for the sensing mechanism based on surface perturbation principle as liquid viscosity or mass sensors, which is the main disadvantage of QZ-PAW for sensor application. For example, QZ-PAW will have lower sensitivity but a wider dynamic range for highly viscous media than QL-LW for viscosity sensors. Furthermore, QZ-PAW can be used for a physical sensor that affects the elasticity constant of the whole plate, such as temperature, strain and pressure sensor [27,28].

This advantage is especially beneficial for sensor application in highly viscous media as it will prevent a significant reduction in signal strength due to viscous damping. Additionally, for guided acoustic waves with maximum displacement at the surface, the surface roughness is one of the leading causes of signal attenuation in [29], especially for the modes with out-of-plane displacement, due to acoustic energy scattering at the surface as the wave propagating. With QZ-PAW, this problem can be prevented and will be beneficial when the plate is thinned down by a micromachining process such as dry etching, which will introduce roughness at the surface of the plate.

Despite several advantages of QZ-PAW, there are still several challenges to be addressed to utilise the QZ-PAW in practical application. First, while the signal resonant peaks of around -56 dB are observable from the signal floor of -68 dB, it is important to improve the signal to noise ratio by increasing the signal strength. In this case, the weak signal is directly related to the low value of K 2 . There are two possible methods in improving the K 2 related to the piezoelectric materials. First, due to weak piezoelectricity in a cubic GaAs plate, a thin-film piezoelectric layer such as ZnO can be deposited on GaAs for piezoelectric field enhancement and signal strength improvement in RW [30] and LW [31] devices. However, the deposition of the thin film will also affect the displacement component profile across the plate, which may increase the displacement on the surface of the plate. Thus, it is necessary to find the optimum thin-film thickness to improve the signal strength while keeping the surface displacement lower than 10% of maximum displacement inside the plate. Another possible improvement is replacing the GaAs plate with another piezoelectric material with a more complex crystal structure and higher piezoelectricity. Similarly, future study is required to verify the occurrence of QZ-PAW in piezoelectric material with different crystal structures, crystal cut and direction of propagation.

Furthermore, as different modes types with different displacement profiles can be excited in the same device with the same IDT, dual-mode PAW sensors can be implemented to decouple physical parameters such as density and viscosity [32] or pressure and temperature [33], utilising different behaviours of QL-LW and QZ-PAW in response to different physical parameters. Furthermore, the out-of-plane mode can be utilised for liquid actuation, thus integrating the liquid sensor and actuator functions within the same device [34]. Finally, the sensor sensitivity is proportional to the operating frequency; a sensor with GHz operating frequency is needed to obtain a high precision sensor and reduce the limit of detection [35]. However, for current state-of-the-art AW-based sensors, the GHz operating frequency in liquid media is impractical due to too high insertion loss.

QZ-PAW modes can potentially solve the limitation of AW-based GHz sensor devices in a liquid environment, especially when individual QZ-PAW mode resonant peaks can be separated to achieve the lowest theoretical insertion loss. However, as previously shown in the experimental result, separation of an individual peak in multi-mode devices remains a challenge, which may require the trade-off between the minimum practical insertion loss and the separation of individual modes. To solve this challenge, the improvement could be achieved by designing QZ-PAW devices with lower bandwidth using a higher number of IDTs or by exploring different piezoelectric materials, crystal cuts, and propagation directions.

Conclusion

We presented a higher-order Lamb wave type with a quasi-zero surface displacement component on a GaAs piezoelectric plate. We defined the quasi-zero plate acoustic waves (QZ-PAW) modes as the higherorder Lamb waves modes with the concentration of acoustic energy inside the plate and the total surface displacement of less than 10% relative to the maximum total displacement inside the plate. We performed finite element simulation using COMSOL Multiphysics software to calculate the h/λ range and the theoretical insertion loss of QZ-A1, QZ-S2, and QZ-A3 modes when the surface of the plate is in contact with liquid. Despite several challenges in the practical implementation of QZ-PAW devices, we demonstrate experimentally the two-port delay line devices utilising QZ-PAW modes. Furthermore, the improvement in the insertion loss in highly viscous liquid is shown compared to QL-LW modes. Finally, we discussed the potential application of QZ-PAW to answer some challenges in the future emerging application of AW-based sensors such as dual-mode sensors and UHF sensors in highly viscous media.

Figure 1 :

 1 Figure 1: (a) Lamb wave dispersion curve of (100)-cut GaAs propagating along the <110>direction (b) the conversion of Lamb waves into QZ-PAW when the phase velocities of higher-order modes are lower than the fast shear BAW phase velocity, and (c) the QZ-PAW displacements of the first three higher-order modes.

Figure 2 :

 2 Figure 2: The ratio between (a) total displacement at the surface of the plate and the maximum displacement inside the plate (usurf/umax) and (b) Longitudinal and shear vertical (u1/ u3) at the surface of the plate

Figure 3 :

 3 Figure 3: The correlation in the fluctuation of vph/vBAW-FAST and usurf/umax of (a) A1 mode and (b) S1 mode

Figure 4 :

 4 Figure 4: Particle displacement components profile of the first three QZ-PAW at h/λ ratio of 2.7 (A1), 3.9 (S1) and 4.3 (A2)

Figure 5 :

 5 Figure 5: K 2 for (a) P-IDT and (b) M-P-IDT configuration

Figure 6 :

 6 Figure 6: (a) Total displacement profile of QZ-PAW in contact with liquid and (b) Insertion loss of the A1, S1, A2 and S2 modes in dB for the waves travelling at the distance of 250λ

Figure 7

 7 Figure 7 Simulation results of S21 transmission scattering parameter for GaAs two-port delay line device with h/λ of (a) 2.7, (b) 3.9 and (c) 4.3

Figure 8 .

 8 Figure 8. S21 scattering parameter vs frequency experimental results of Lamb waves two-port delay line device with h/λ of (a) 3.9 and (b) 3.7

Figure 9

 9 Figure 9 Insertion loss comparison (a) QZ mode (b) QL mode in contact with DI water and glycerol, and (c) the change in S21 insertion loss at different concentrations of DI Water/glycerol mixture

Table 1 :

 1 List of the merged LW higher-order modes with QZ-PAW resonant peak h/λ of 2.7, 3.9 and 4.3

	h/λ	Modes	Freq (MHz)	usurf/umax (%)	IL (dB)	Types
	2.7	A1 S1	19.786 19.78	5.6 22	0.00894 3.45	QZ non-QZ
		A1	19.692	13	1.27	non-QZ
	3.9	S1 A2	19.695 19.865	5.8 5.9	0.00453 0.106	QZ QZ
		S2	20.033	33	5.78	non-QZ
		A1	19.661	8.5	0.495	QZ
		S1	19.702	6.4	0.228	QZ
	4.3	A2	19.871	5.3	0.00274	QZ
		S2	19.859	22	2.85	non-QZ
		A3	20.292	45	7.80	non-QZ

Table 2

 2 List of the merged modes with QZ-PAW resonant peaks at h/λ = 3.7

	h/λ	Modes	Freq (MHz)	usurf/umax (%)	IL (dB)	Types
		A1	19.739	14	1.79	non-QZ
	3.7	S1	19.704	5.6	0.0510	QZ
		A2	19.878	8.3	0.385	QZ

Acknowledgements

This journal publication is part of the SmOoC project that has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 844135. The work was partly supported by the French Renatech network and its FEMTO-ST technological facilityReferences


Jean-Loup Skora

Olivier Gaiffe

Sylwester Bargiel

Jean-Marc Cote

Laurent Tavernier

Michel De Labachelerie

Nicolas Passilly
email: nicolas.passilly@femto-st.fr

High-fidelity glass micro-axicons fabricated by laser-assisted wet etching

We report on the fabrication of micro-axicons made of glass by laser-assisted wet etching (LAE) and laser polishing. The employed technique, relying on a direct-writing process using a femtosecond laser, allows revealing high fidelity profiles when the exposed glass samples are etched in a heated potassium hydroxide (KOH) solution. The remaining surface roughness is then decreased by carbon dioxide (CO 2 ) laser polishing. Such polishing is limited to the superficial layer of the component so that the tip is only slightly rounded, with a radius of curvature of nearly 200 µm. It is then shown with 500 µm-diameter axicons that a quasi-Bessel beam is generated closely after the tip, and features a 5.3 µm diameter maintained over a propagation distance of almost 3.5 mm.

Introduction

Refractive axicons, first proposed in 1954 [START_REF] Mcleod | The axicon: a new type of optical element[END_REF], are conical-shaped continuous profile transmissive components. They are known to produce quasi-Bessel beams (QBB) from an incident collimated beam (Fig. 1). Such beams are characterized by an electric-field profile following a Bessel function generated from the interference of all deviated beams and diffraction on the conical vertex. This shaping results in a long and narrow focal line along the optical axis ("diffraction-free beam"), rather than a point as in the case of convex lenses. Such non-diffracting unique property makes them useful in applications like alignment and metrology [START_REF] Häusler | Light sectioning with large depth and high resolution[END_REF][START_REF] Davis | Range-finding by triangulation with nondiffracting beams[END_REF], second harmonic generation [START_REF] Piskarskas | Noncollinear second-harmonic generation in periodically poled KTiOPO 4 excited by the bessel beam[END_REF], 2D micro-fabrication [START_REF] Li | Application of bessel beams to 2D microfabrication[END_REF], waveguide writing [START_REF] Xin | Using bessel beams to induce optical waveguides[END_REF], optical trapping [START_REF] Garces-Chavez | Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam[END_REF][START_REF] Suarez | Experimental optical trapping with frozen waves[END_REF], or optical coherence tomography [START_REF] Lee | Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range[END_REF][START_REF] Lorenser | Energy-efficient low-fresnel-number bessel beams and their application in optical coherence tomography[END_REF] where the associated improved depth of field is an asset. To do so, the axicon deviates the incident collimated beam propagating along the z-axis with an angle 𝛽 determined, in the paraxial approximation, by the opening angle (also called "base" or "wedge" angle) 𝛼 of the cone and the refractive index 𝑛 of the employed material [START_REF] Mcgloin | Bessel beams: diffraction in a new light[END_REF]:

𝛽 = (𝑛 -1)𝛼. ( 1
)
If 𝑊 0 is the beam-waist of the incident beam (Fig. 1), the length of the focal line, 𝑍 𝑚𝑎𝑥 , can be approximated to:

𝑍 𝑚𝑎𝑥 = 𝑊 0 /tan 𝛽, (2)
whereas the radius 𝑟 𝐵 of the central lobe (first zero) of the QBB is described by [START_REF] Brzobohatý | High quality quasi-bessel beam generated by round-tip axicon[END_REF]:

𝑟 𝐵 = 2.4048/(𝑘 sin 𝛽), (3)
𝑟 𝐵 can then be related to the full width at half maximum (FWHM) of the central lobe FWHM = 1.0532 × 𝑟 𝐵 , which is easier to derive experimentally in the ranges where the QBB is not properly shaped. Refractive axicons are usually fabricated by diamond turning [START_REF] Bass | Handbook of Optics[END_REF] or grinding and polishing [START_REF] Van Le | [END_REF] of glass or epoxy. In the case of micro-fabrication, most of the employed techniques lead to microstructures made of resist, whose shape can be generated by electron beam lithography [START_REF] Cheong | Fabrication of efficient microaxicon by direct electron-beam lithography for long nondiffracting distance of bessel beams for optical manipulation[END_REF][START_REF] Ahluwalia | Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids[END_REF], electric field shaping of SU8 droplets [START_REF] Cao | Fabrication of refractive axicons utilizing electrostatic force[END_REF], interference lithography [START_REF] Stankevičius | Bessel-like beam array formation by periodical arrangement of the polymeric round-tip microstructures[END_REF], or 2-photon laser writing [START_REF] Schmid | Three-dimensional direct laser written achromatic axicons and multi-component microlenses[END_REF]. Other means as direct-laser writing of UV-curable epoxy [START_REF] Huang | Fabrication of micro-axicons using direct-laser writing[END_REF] or chemical etching (positive or negative) to produce axicons at the tip of optical fibers [START_REF] Philip | Fabrication of negative micro axicons in optical fibers via chemical etching[END_REF] have also been proposed. Diamond tools are also used to generate molds that are subsequently employed for replication [START_REF] Weber | Highly compact imaging using bessel beams generated by ultraminiaturized multi-micro-axicon systems[END_REF]. Still, such approaches lead to components in polymer, which are not as robust as if they were made in glass. This can be critical for applications where high laser power has to be handled, e.g. for ablation (microcapillarities, etc.) [START_REF] Dudutis | Non-ideal axicon-generated bessel beam application for intra-volume glass modification[END_REF], or when the components should be used in harsh conditions.

Previously, we developed a wafer-level technique based on glass-blowing from multiple siliconglass cavities to generate micro-axicons [START_REF] Carrión | Microfabrication of axicons by glass blowing at a wafer-level[END_REF]. Nevertheless, although the wafer-level character along with the high level of rotational symmetry are undeniable advantages, the technique intrinsically leads to blunt the tip with a radius of curvature reaching a few millimeters. Recently, axicons made of glass with shorter radii of curvature have been fabricated by femtosecond laser ablation followed by CO 2 laser polishing [START_REF] Schwarz | Fabrication of a high-quality axicon by femtosecond laser ablation and CO 2 laser polishing for quasi-bessel beam generation[END_REF][START_REF] Schwarz | Fabrication and evaluation of negative axicons for ultrashort pulsed laser applications[END_REF][START_REF] Schwarz | Rapid fabrication of precise glass axicon arrays by an all laser-based manufacturing technology[END_REF][START_REF] Dudutis | Laser-fabricated axicons challenging the conventional optics in glass processing applications[END_REF]. Despite the fact that it is a direct writing method, more adapted to prototyping than large-scale production, it leads to a more conical shape allowing the generation of Bessel beams closer from the tip. However, the significant roughness resulting from the laser ablation requires to perform a strong thermal surface polishing, eventually rounding the tip as well.

Here, the proposed LAE technique is, similarly to laser ablation, a direct writing method, but only for laser-exposing the component's surface so that it is subsequently primarily revealed by wet-etching, especially in potassium hydroxide (KOH) aqueous solution. The etch-rate of KOH, better known for anisotropic etching of silicon, rather than glass, is substantially increased for laser-exposed areas, allowing glass 3-D structuration. Proposed in the early 2000s, such a technique of femtosecond laser pre-irradiation was first employed to enhance etching based on aqueous solution of Hydrofluoric (HF) acid [START_REF] Marcinkevičius | Femtosecond laser-assisted three-dimensional microfabrication in silica[END_REF], but with a much lower selectivity than the one obtained further in KOH [START_REF] Matsuo | Femtosecond laser-assisted etching of pyrex glass with aqueous solution of KOH[END_REF]. In addition, it has been more focused on microfluidics circuits generation [START_REF] Bellouard | Fabrication of high-aspect ratio, micro-fluidic channels and tunnels using femtosecond laser pulses and chemical etching[END_REF][START_REF] Sima | Three-dimensional femtosecond laser processing for lab-on-a-chip applications[END_REF][START_REF] Liu | Hybrid laser precision engineering of transparent hard materials: challenges, solutions and applications[END_REF] or microfluidics combined with waveguides for optical sensing [START_REF] Osellame | Integration of optical waveguides and microfluidic channels both fabricated by femtosecond laser irradiation[END_REF][START_REF] Bellini | Femtosecond laser fabricated monolithic chip for optical trapping and stretching of single cells[END_REF], rather than on micro-optical component fabrication. Moreover, when it is employed for this purpose, it is often used to carve a basic geometry which is subsequently shaped by reflow, such as a cube toward a sphere [START_REF] Drs | Laser-assisted morphing of complex three dimensional objects[END_REF] or a disk toward a toroid [START_REF] Lin | On-chip three-dimensional high-Q microcavities fabricated by femtosecond laser direct writing[END_REF] to generate microresonators, or pillars reflowed in an array of microlenses [START_REF] Yang | CO 2 laser thermal reflow shaped convex glass microlens array after bessel picosecond laser inscribing and hydrofluoric acid processing[END_REF]. Even when the primary shape of the lens is drawn from LAE, the final sought state, e.g. achieved by flame polishing [START_REF] Qiao | Fabrication of a micro-optical lens using femtosecond laser 3D micromachining for two-photon imaging of bio-tissues[END_REF][START_REF] Ross | Towards optical quality micro-optic fabrication by direct laser writing and chemical etching[END_REF][START_REF] Ross | A miniature fibre-optic raman probe fabricated by ultrafast laser-assisted etching[END_REF], is close to the one minimizing the surface tensions and only slightly modified.

In the case of an axicon, featuring a much higher spatial frequency at its tip, the laser surface polishing should minimize surface re-shaping. In this framework, we consequently show that the surface roughness resulting from LAE is lower than for laser ablation, and then requires lighter thermal surface polishing, eventually better preserving the profile singularities of the component such as the axicon tip. This feature is particularly interesting for micro-axicons since it favors the quasi-Bessel generation in its nearest neighborhood and gives it more potential to be used, e.g., in compact lab-on-chip devices.

Micro-axicon generation

Laser assisted etching

High-fidelity micro-axicons with 500 µm diameter and apex angle of 170 • (𝛼= 5 • ) are fabricated in fused silica through a LAE manufacturing process [START_REF] Marcinkevičius | Femtosecond laser-assisted three-dimensional microfabrication in silica[END_REF] whose direct laser writing step is achieved with a f100 aHead Enhanced system from FEMTOprint (Muzzano, Switzerland). The system consists of a horizontal XY-moving hollow stage holding the glass wafer to be irradiated. The stage stands above a 20× microscope objective, mounted on a vertical Z-translation stage and used to focus inside the glass bulk the beam emitted by a 1030 nm Yb:YAG femtosecond laser (spot size specified at 1.5 µm). The beam pulse energy is typically set to 230nJ and produced at a repetition rate of 1 MHz. The system enables to run programmed sequences synchronizing the laser spot XYZ-displacements with the laser output power (Fig. 2(a)). The LAE process relies on the structural modification of the glass material resulting from the intense irradiation undergone at the focal point of the femtosecond laser. Since the efficiency of the laser inscription is polarization-sensitive, at least in the femtosecond regime [START_REF] Li | Polarization-insensitive space-selective etching in fused silica induced by picosecond laser irradiation[END_REF], the incident linear polarization is aligned perpendicularly to the writing direction [START_REF] Hnatovsky | Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica[END_REF], in order to optimize the material modification.

Here, we use 500 µm-thick 20× 20 mm polished fused-silica samples (Corning 7980), although the system can process wafers up to 1 mm-thick and 100 mm diameter. The 3D surface of the refractive micro-optical component is raster-drawn with 1 µm line spacing along Y and located few tens of microns below the sample surface. Just before the component's surface, vertical accesses are first laser-exposed. They are aimed to provide evenly distributed access of the etching solution through the sacrificial volume to the 3D laser-exposed surface. By doing so, the sacrificial material does not require to be entirely laser-irradiated but is eventually detached during wet etching [START_REF] Butkutė | Optimization of selective laser etching (SLE) for glass micromechanical structure fabrication[END_REF]. In addition, this scanning strategy prevents from excessive internal stress due to overexposure. The laser irradiation step requires typically 60 minutes for a 500 µm-diameter component. This duration is not proportional to the volume but to the surface of the contour and then is little dependent on the superfluous material, unlike laser ablation for which it has to be entirely laser-treated.

The next step (Fig. 2(b)) consists in a liquid phase etching of the irradiated glass wafer. For this purpose, the glass wafer is immersed into a 10 mol/L KOH aqueous solution, maintained at a constant temperature of 85 • C under sporadic ultrasonic agitation. It takes approximately 60min for the KOH solution to react and dissolve the considered areas, i.e. surface and accesses, of the laser-modified glass whatever their number on the wafer. Hence, this etching sub-step allows fast and uniform batch processing. Since these areas were defined as the lens outlines, their dissolution leads to the separation of the undesirable glass parts located above the axicon profile. The laser-modified areas can be etched nearly 900times faster than the unmodified glass [START_REF] Ross | Optimisation of ultrafast laser assisted etching in fused silica[END_REF], allowing them to be removed with a negligible alteration of the surrounding glass, hence resulting in good machining accuracy.

Nevertheless, the surface of the released micro-axicons shows right after etching a nonnegligible roughness that avoids their qualification as optical components, at least for visible wavelengths. Therefore, an additional surface-polishing step is required.

𝐶𝑂 2 laser polishing

The last fabrication step (Fig. 2(c)) consists in polishing the component's surface through local heating by a CO 2 laser [START_REF] Nowak | Efficient laser polishing of silica micro-optic components[END_REF][START_REF] Cao | Ground fused silica processed by combined chemical etching and CO 2 laser polishing with super-smooth surface and high damage resistance[END_REF]. The Diamond C-30 laser (from Coherent, Santa Clara, CA, USA) emits an infrared beam focused and raster scanned on the surface of the component produced by LAE. This scanning is done by moving the component with a motorized XY-stage. For each axicon, 13 scans are performed on the X and the Y directions at a speed of 2 mm/s with an overlap of 100 µm. The laser power is set at 5.3W and the CO 2 laser energy is deposited on a circular Gaussian beam with a 230 µm beam waist (1/e 2 spot radius).

Since fused silica is highly absorbing at 10.6 µm, the laser light is absorbed in the vicinity of the lens surface. Therefore, this absorption leads to a significant increase of the glass temperature, limited to a thin layer below the lens surface. As this thin layer temperature increases, its viscosity decreases and eventually reaches a level where the glass is soft enough for surface roughness relaxation effects to occur, which results in a smoothing effect. Indeed, multiple phenomena involving surface tension lead, at first, to the relaxation of high spatial frequencies corresponding to roughness. Since the glass heating is limited to the glass surface, the shape of the lens is rather well-preserved during this process.

Component's profile and surface characterization

The surfaces of the fabricated axicons were characterized by two different systems. First, an optical profilometer based on white light interferometry (MSA 500 from Polytec, Waldbronn, Germany) was employed to determine surface topography and roughness on a 886 x 662 µm 2 area. Second, a mechanical stylus profilometer (Dektak XT from Brüker) was used to measure the profiles of non-polished components, whose roughness is too large for optical profilometer to collect the signal from all the surface points.

Two batches of components have been fabricated, with a slightly different meshing of the laser-insulated paths. The first tested micro-axicon (MA1), from the first batch, is characterized by a coarser meshing than the second tested one (MA2), taken from the second batch. In addition, those components are compared to a precision axicon (AX125 from Thorlabs) specified at a similar angle (𝛼 = 5 • ) than the fabricated micro-axicons. The various acquired profiles are fitted with Eq. 4 [START_REF] Dépret | Characterization and modelling of the hollow beam produced by a real conical lens[END_REF] in order to derive average angles as well as radii of curvature of the axicon tips. The roughness of the polished component, extracted by considering a cut-off length of 10 µm, drops significantly. The former rough surface from (c) is now polished and shown in (g) with a 125x magnification of the ordinate (vertical scale). The calculated roughness is then equal to 𝑅 𝑞 = 0.9 nm. Note that cross sections are mostly derived from white-light profilometry (MSA-500 from Polytec) whereas the rough profiles (b), (c) result from stylus profilometry (Bruker Dektak XT).

𝑒(𝑟) = 𝑒 0 -𝑅 𝑐 tan 2 (𝛼) √ 1 + 𝑟 2 𝑅 2 𝑐 tan 2 (𝛼) . ( 4
Figure 3(a) reports the topography of one of the 500 µm diameter conical lenses, namely MA1, buried 63 µm under the wafer surface and measured by white light interferometry. One of the cross-sections of MA1 is shown in Fig. 3(b) before the polishing step. This rough profile is acquired with the mechanical profilometer equipped with a 2 µm probe. It remains close to the ideal conical shape, with an associated radius of curvature measured around 28 µm. The initial roughness displayed on Fig. 3(c) is at the level of 𝑅 𝑞 = 94 nm evaluated when applying a 10 µm cut-off filter. It can be noted that the roughness observed before polishing varies according to the different axes passing through the axicon tip. Indeed, since the 3D surface of the axicon is generated by writing parallel lines with 1 µm overlap, the KOH solution preferentially dissolves and diffuses along the writing lines rather than along the perpendicular direction, resulting in an anisotropic roughness pattern (𝑅 𝑞 = 45 nm RMS along the writing lines versus 𝑅 𝑞 = 90 nm RMS perpendicularly to them).

The subsequent laser polishing tends to round the tip and thus increases the curvature as shown in Fig. 3(d) where one of MA1's cross-sectional profiles is displayed (𝑅 𝑐 = 174 µm). The tip rounding is highlighted as well when displaying the shape error (Fig. 3(e), resulting from subtraction between both curves of Fig. 3(d)). Here, it reaches almost 1.4 µm for MA1. Nevertheless, this maximum profile deviation remains almost twice lower than the one measured on the AX125 axicon (3 µm maximum shape error) and 2 to 4 times lower than the one reported in [START_REF] Dudutis | Laser-fabricated axicons challenging the conventional optics in glass processing applications[END_REF] measured on laser-ablated components (maximum shape deviation of 3 to 6 µm and associated 𝑅 𝑐 = 600 µm). This higher profile fidelity obtained for the fabricated axicons can be attributed to a lighter surface polishing and an associated higher cut-off of spatial frequencies due to lower initial roughness produced by the LAE process.

From the difference between the shape error profile and the primary profile (dashed black line derived from Eq. 4 with a radius set at 𝑅 𝑐 = 207 µm), both displayed on Fig. 3(e), we can subsequently derive the waviness profile (Fig. 3(f)) along with the roughness profile (Fig. 3

(g)).

The waviness shown here is measured at 𝑊 𝑞 = 54 nm. CO 2 laser processing leads to a significant decrease of the surface roughness, at the level of 𝑅 𝑞 = 0.9 nm (Fig. 3(g)). The profile parameters concerning the three considered components are gathered in Tab. 1. According to the reported values, MA2 appears to be more obtuse than MA1 (by ≈ 0.25 • ) and differs stronger from the specified 5 • than MA1. It is also slightly more asymmetric. Indeed, whereas MA1 wedge angle ranges from 5.08 • (y-axis) to 5.22 • (x-axis), it is spread between 5.20 • (y-axis) to 5.45 • (x-axis) in the case of MA2. This asymmetry is attributed to a higher applied irradiation dose, probably consecutive to the finer meshing, resulting also in uneven etch rate. The AX125 mean angle is measured similarly and appears to be more uniform at 4.67 • (standard deviation (𝑆𝐷) = 0.01), value from which a slightly smaller focusing power can however be expected. Note that the angle of the AX125 is evaluated from topography measurements applied only on the 500 µm-diameter disk centered on the tip, despite its 6.35 mm radius. This can explain the discrepancy with the specified angle of 5 • .

As reported in Tab. 1, MA1 shows a higher fidelity around the tip (𝑅 mean 𝑐 = 191 µm) than MA2 (𝑅 mean 𝑐 = 228 µm) but with a larger 𝑆𝐷 (37 µm against 19 µm) revealing a more asymmetrical tip. In addition, although MA1 appears to be more asymmetric than MA2 around the tip, it is the opposite concerning their lower part as underlined earlier. Such asymmetries are not experienced with commercial AX125 axicon fabricated by standard grinding and polishing, however, the latter leads to a larger radius of curvature (𝑅 mean 𝑐 = 344 µm). The surface of the AX125 is also more stretched with nearly twice lower 𝑊 mean 𝑞 = 34 nm than the one of MA1 (𝑊 mean 𝑞 = 68 nm). In addition, the measured waviness is 25% larger for MA1 than for MA2, as it could be expected from the different meshing of exposed surfaces. Note that, as previously, the wavelength of the applied cut-off filter is set at 10 µm. Finally, the measured surface roughness of the commercial AX125 (𝑅 mean 𝑞 = 0.7 nm) is close to its specification given at 0.6 nm and on the same order as the mean surface roughness of MA1 and MA2 measured at 0.9 nm and 0.8 nm, respectively.

Bessel beam generation

Such components are then employed to generate QBB. The resulting intensity distributions are studied with a 3D intensity point spread function (IPSF) measurement system [START_REF] Baranski | A simple method for quality evaluation of micro-optical components based on 3D IPSF measurement[END_REF] where each transverse plane is imaged onto a movable camera so that the whole distribution of the focal volume can be reconstructed. The axicons under test are illuminated with a He-Ne laser (𝜆 = 632.8 nm), whose beam is collimated by a beam expander (at 𝑊 0 ≈ 240 µm), whereas the imaging part uses a 10x plan achromat microscope objective from Olympus (NA = 0.25) associated with a tube lens of 150 mm focal length to ensure a magnification of 8.33. Consequently, each pixel of the CMOS camera (µEye UI-1545LE-M-GL from IDS having 1280 × 1024 pixels of 5.2 µm size) corresponds to 0.62 µm in the focal plane. Transverse planes are acquired every 25 µm along the propagation direction thanks to a translation achieved by a servo motor (Ealing 37-1104).

Figure 4 displays transverse (XY intensity distributions -Fig. 4(a) (b) and (c)) and longitudinal (XZ intensity distributions -Fig. 4(d) (e) and (f)) sections of the recorded focal volumes. The origin of the longitudinal distance is set at the position of the axicon tip. The Bessel-like intensity distributions, characterized by a sharp and intense central peak surrounded by concentric and evenly separated rings (square of the zeroth-order Bessel function profile) in the transverse planes, and by a non-diffracting feature in the longitudinal plane, are visible confirming the generation of a QBB by the three considered components. The first transverse distributions (the three most to the left) of the beams are noteworthy since they display an earlier shaped QBB for the fabricated micro-axicons MA1 and MA2 than for the commercial AX125, as well as a smaller beam radius, also clearly visible on the longitudinal planes. It can moreover be noticed that such beam shaping is shorter than the one achieved by laser-ablated axicons for which it typically starts at 𝑧 ≈ 3 mm [START_REF] Schwarz | Fabrication of a high-quality axicon by femtosecond laser ablation and CO 2 laser polishing for quasi-bessel beam generation[END_REF][START_REF] Dudutis | Laser-fabricated axicons challenging the conventional optics in glass processing applications[END_REF]. The beam central core early ellipticity of MA1 (Fig. 4(a): 𝑧 = 0.5 mm, 𝜖 = 1.66) is attributed to the asymmetry of the component's surface profile around the tip reported in the previous section. On the other side, i.e. at 𝑧 > 4 mm, the beam shaped by MA1 (Fig. 4(a)) does not loose it symmetry as much as the one shaped by MA2 (Fig. 4(b)), for which asymmetrical side lobes materialize from 𝑧 > 5 mm, in good agreement with the earlier observations of an elliptical contour of the surface profile.

Nevertheless, as seen on the longitudinal (XZ plane) evolutions of beams shaped by MA1 and MA2 between 𝑧 = 0.75 mm and 𝑧 = 4.75 mm (Fig. 4(d) and(e)), the central peak remains stable without significant modulation. Note on these graphs that images are normalized (from transverse plane to plane) to better visualize the entire propagation, but the QBB quality can be better appreciated with the on-axis intensity distributions (Fig. 4(g)(h)(i)) where its early generation with respect to the beam shaped by the AX125 is also manifest. According to Eq. 2, the length of the focal line is influenced by the axicon angle. Taking into account the average wedge angles reported in the Tab. 1, 𝑍 𝑚𝑎𝑥 is equal to 5.92 mm, 5.66 mm and 6.44 mm for MA1, MA2 and the AX125, respectively. The higher angles of the fabricated axicons lead to shortening of their generated focal line length as it can be seen in Fig. 4(g) and (h) compared to Fig. 4(i). Additionally, the decline of the on-axis intensity for MA2 (Fig. 4(h)) is accentuated by the loss of symmetry at the component's contour. Finally, Fig. 5 displays other features of the shaped beams, in the form of the experimental central lobe's FWHM (Fig. 5(a)(b)) and ellipticity (Fig. 5(c)(d)) as a function of the longitudinal distance. As expected from Eq. 3, larger angles, such as for MA1 and especially MA2, lead to smaller central lobes. Indeed, whereas the beam generated by the AX125 is characterized by FWHM ≈ 6 µm, the ones generated by MA1 and MA2 are closer to 5 µm. Note that dash-dotted lines display the approximated FWHM of an ideal axicon, i.e. fully conical, evaluated according to Eq. 3 and to data from Tab. 1. These lines are plotted at FWHM = 6.2 µm, 5.5 µm and 5.3 µm for AX125, MA1 and MA2, respectively. As already shown on the on-axis intensities on Figs. 4(g)-(h), the QBB is generated closer from the tip with MA1 and MA2, i.e. around 𝑧 = 1 mm, than with the AX125 for which the QBB appears right after 𝑧 = 2 mm. This feature, achieved thanks to the reduced radius of curvature of the tip, is particularly interesting for micro-axicons intended to be used within integrated optical microsystems.

From 𝑧 = 1 mm to 𝑧 ≈ 4.5 mm, the ellipticity of the central lobes of MA1 and MA2 remains reasonable (1.00 < 𝜖 < 1.15) (Fig. 5(d)-(e)). Taking also into account the on-axis intensities and FWHM, this propagation range can be defined as the operational length of the generated QBB. It can be noticed that despite a slightly shorter length for the beam shaped by MA2 (𝑧 ≈ 4.50 mm instead of 𝑧 ≈ 4.75 mm), the ellipticity remains close to unity all along its working range. This behavior is in good agreement with the previous observations, such as a lower waviness and more symmetrical tip attributed to the finer meshing of the exposed surface of MA2. The latter is then capable of e.g. shaping closely from the axicon tip (𝑧 = 1 mm) until 𝑧 = 4.5 mm, a tiny visible QBB of FWHM = 5.3 µm, featuring an aspect ratio of 660.

Conclusion

This paper describes the use of LAE, a rapid prototyping method, for the fabrication of microoptical components made of glass, and in particular micro-axicons employed for QBB generation in the visible range. It is shown that LAE, based on femtosecond laser pre-inscription of glass wafer, followed by wet KOH etching, allows building high fidelity refractive surfaces characterized by a reasonable roughness (𝑅 𝑞 < 100 nm). Hence, the required polishing step performed by heating the glass surface with a CO 2 laser faithfully preserves the designed shape of the component whereas decreasing the roughness until less than 1 nm RMS. Thereby, micro-axicons featuring tips with a low radius of curvature (𝑅 𝑐 < 200 µm) are fabricated, leading to the generation of QBB closely from the component's surface (1 mm). This is particularly interesting for micro-optical components embedded in microsystems that cannot tolerate delayed beam shaping. The QBB are then characterized by 5.3 µm diameter central core over an effective length of 3.5 mm.

Fig. 1 .

 1 Fig. 1. Sketch of Bessel beam formation behind an axicon.

Fig. 2 .

 2 Fig. 2. Flowchart of the complete fabrication process of a single micro-axicon in a glass wafer : (a) the axicon outline areas to be subtracted are irradiated by scanning the femtosecond laser-focused spot inside the wafer bulk from its backside, along a defined path and starting from the top surface. Here, the irradiated glass is structurally modified and only the refractive index change is visible. (b) These outline areas are then etched in a 85 • C KOH solution leading to the release of the axicon shape. (c) This surface is then smoothed by raster scanning a 150 µm diameter CO 2 laser spot, (d) the resulting axicon surface is optically polished.

Fig. 3 .

 3 Fig.3. Surface profile characterization of the 500 µm -diameter MA1 : (a) 3-D surface topography, (b) Cross-section profile measured right after wet etching. The fitted profile features 𝑅 𝑐 = 28.2 µm and 𝛼 = 5.12 • . (c) Roughness profile from which RMS roughness is derived at the level of 𝑅 𝑞 = 94 nm. (d) Cross-section profile after laser polishing compared to the ideal conical shape (dashed black line). The polished profile now features a more rounded tip with an associated 𝑅 𝑐 = 174 µm. (e) Resulting shape error derived from comparing the two curves of (d). The dashed black line shows the primary profile and allows deriving the RMS waviness in (f) which reaches 𝑊 𝑞 = 54 nm. The roughness of the polished component, extracted by considering a cut-off length of 10 µm, drops significantly. The former rough surface from (c) is now polished and shown in (g) with a 125x magnification of the ordinate (vertical scale). The calculated roughness is then equal to 𝑅 𝑞 = 0.9 nm. Note that cross sections are mostly derived from white-light profilometry (MSA-500 from Polytec) whereas the rough profiles (b), (c) result from stylus profilometry (Bruker Dektak XT).

Fig. 4 .

 4 Fig. 4. Transverse intensity distributions recorded onto the camera at different picked longitudinal distances: Beam shaped by (a) MA1 (b) MA2 (c) commercial AX125 axicon. The white bar is 25 µm long; Associated longitudinally normalized intensity distributions: (d) MA1 (e) MA2 (f) commercial AX125 axicon; On-axis intensity: (g) MA1 (h) MA2 (i) commercial AX125 axicon.

Fig. 5 .

 5 Fig. 5. (a)-(b) MA1 and MA2 beam central lobe's FWHM derived from experimental IPSF volumes, each being compared to the beam FWHM obtained after the AX125. Dash-dotted lines display the corresponding expected FWHM calculated from approximated Eq. 3 of an equivalent ideal (conical) axicon; (c)-(d) Ellipticity of the beam central lobe as a function of the propagation distance for MA1 and MA2.

Table 1 .

 1 Representative parameters of axicons' surfaces, derived from 8 radial crosssections (each 22.5 • ) collected from surfaces measured with the optical profilometer. Given 𝛼, 𝑅 𝑐 , 𝑊 𝑞 and 𝑅 𝑞 are mean values, averaged from the 8 considered crosssections.

	Element	𝜶 (𝑺𝑫 𝒂 )	Range	𝑹 𝒄 (𝑺𝑫)	Range	𝑾 𝒒 (𝑺𝑫) Range 𝑹 𝒒 (𝑺𝑫) Range
		( • )	( • )	(µm)	(µm)	(nm)	(nm)	(nm)	(nm)
	MA1	5.08 (0.07) 5.02 -5.22 191 (37) 157 -246 68 (13) 47 -85 0.9 (0.1) 0.7 -1
	MA2	5.32 (0.10) 5.20 -5.45 228 (19) 208 -258	55 (9)	40 -71 0.8 (0.2) 0.6 -1.1
	AX125 4.67 (0.01) 4.66 -4.67 344 (2) 340 -346	34 (7)	23 -43 0.7 (0.1) 0.6 -0.8

𝑎 𝑆𝐷 stands for standard deviation

Acknowledgments

The authors thank the french RENATECH network and its FEMTO-ST technological facility, Luca Furfaro for fruitfull discussions as well as Sarah Benchabane for discussions and careful reading of the manuscript.

Funding

French National Research Agency (ANR) DOCT-VCSEL and µROCS projects (ANR-15-CE19-0012 and ANR-17-CE19-0005), INSERM Robot project (OPE-2017-0123), SMYLE Collegium.

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.

Disclosures

The authors declare no conflicts of interest.