Low power reconfigurability and reduced crosstalk in integrated photonic circuits fabricated by femtosecond laser micromachining

Femtosecond laser writing is a powerful technique that allows rapid and cost-effective fabrication of photonic integrated circuits with unique 3D geometries, which today are playing upfront in quantum photonics applications. The possibility to reconfigure such devices by thermo-optic phase shifters represents a paramount feature, exploited to produce adaptive and programmable circuits. However, the scalability is strongly limited by the flaws of current thermal phase shifters, which require hundreds of milliwatts to operate and exhibit large thermal crosstalk. In this work, we exploit thermally-insulating three-dimensional microstructures to decrease the power needed to induce a 2{\pi} phase shift down to 37 mW and reduce the crosstalk to a few percent. Further improvement is demonstrated when operating in vacuum, with sub-milliwatt power dissipation and negligible crosstalk. These results pave the way towards a steep increase in the complexity of programmable integrated photonic circuits, opening exciting perspectives in integrated quantum photonics.


Introduction
Integrated photonics represents an enabling tool for several applications, providing a level of stability, miniaturization and complexity hardly attainable with bulk components and thus attracting a lot of attention from a large variety of areas, ranging from broadband optical communications [1] to photonic quantum information processing [2]. Among the qualities that photonic integrated circuits (PICs) can claim, it is worth mentioning the possibility to dynamically shift the relative phase between optical signals, a feature that provides unique abilities to the applications [3], [4]. Phase shifting can be implemented through different physical effects [5], [6], [7] but, since many applications in photonic quantum information require only a quasi-static reconfiguration (either to fine tune the circuit or to change the unitary transformation implemented), the most common approach in this field is to exploit the dependence of the refractive index on the local temperature: the thermo-optic effect. Thermal phase shifting has a straightforward implementation, because it requires only the integration of an electrical microheater (i.e. usually a resistor that dissipates electrical power by Joule effect), it does not introduce additional photon losses and, at the same time, it provides an excellent performance in terms of stability and accuracy.
Thermal shifters have been demonstrated for most of the integrated photonic platforms like silicon-on-insulator (SOI) [8], silica-on-silicon [9], UV laser written circuits [10] and femtosecond laser writing (FLW) [7]. In particular, the latter technology has recently attracted a lot of attention for quantum applications [11], thanks to the many advantages it provides [12]. Firstly, it does not require mask-based photolithography, thus allowing rapid and cost-effective fabrication of photonic circuits with arbitrary topology. Secondly, since the laser induces a modification confined in the focal volume of the beam, it allows the direct inscription of circuits featuring three-dimensional waveguide geometry that would be unfeasible with a planar process. Lastly, femtosecond laser written PICs (FLW-PICs) are able to provide low birefringence (down to 1.2E-6 [13]), which is mandatory for quantum applications in which the information is encoded in the polarization state of single photons, and low propagation losses (less than 0.3 dB/cm at 1550 nm [13]), which are essential to scale the number of single photons employed in quantum experiments.
FLW-PICs have already been exploited in several quantum applications and the introduction of thermal phase shifters has greatly extended both the quality and the applicability of these devices [14], [15], [16]. However, some design challenges prevent their deployment on a larger scale. Firstly, thermal shifters in FLW-PICs suffer from high power consumption and, up to now, a 2π phase modulation has been demonstrated only with a dissipation of hundreds of milliwatts [7], [17]. Since the total power dissipation that a photonic circuit can tolerate with no active cooling is limited to few watts, the total number of thermal shifters that can be integrated in the same chip is usually limited to no more than a dozen [17]. Another obstacle is represented by the thermal crosstalk: indeed, when a thermal shifter dissipates power, the heat can also reach waveguides different from the target one, inducing on them an undesired phase shift. To achieve an accurate phase control when multiple phases are tuned simultaneously, a massive calibration procedure is required including all possible combinations of dissipated powers in the different shifters, to also take into account that thermal shifters may have a nonlinear response with temperature [18]. This also means that all thermal shifters will have to be adjusted even if only one phase needs to be varied. Moreover, the additive nature of the crosstalk may require to set the waveguide temperature for a given phase shift at a much higher value than that required when operating the thermal shifters individually, thus jeopardizing both the stability of the phase response and the reliability of the microheater. Since the thermal crosstalk depends on the distance between thermal shifter and waveguides, a reduction of this phenomenon is necessary to increase the integration density of reconfigurable FLW-PICs that, up to now, have never been reported with an inter-waveguide pitch lower than 100 μm [17].
In order to cope with these challenges, some solutions have been already proposed in the literature. On the one hand, in a previous work [18] we have proved that the power dissipation can be reduced with a compact design of the microheater and, thanks to this approach, we demonstrated a 2π phase shift for light at 800 nm by dissipating only 200 mW. On the other hand, Chaboyer et al. [19] have achieved a similar level of power dissipation, along with a reduced thermal crosstalk by introducing isolation trenches between the waveguides. In this work, we aim at expanding the above concepts by 3D structuring the glass around the waveguide to significantly concentrate the heat diffusion. This yields a dramatic reduction of the dissipated power, as well as of the crosstalk. By combining waveguide writing and water-assisted laser ablation, we present a new generation of thermal phase shifters in glass photonic circuits with great potential for scalability and compactness. Power dissipation, dynamic response, thermal crosstalk and stability of the phase response are thoroughly characterized both in air and in vacuum, where we achieved an unprecedented level of performance.

Device structure and operation
To show the capabilities of the new technology, we report here the results of the experimental characterization performed on reconfigurable Mach-Zehnder interferometers (MZIs), which represent the basic building block for a universal multiport device able to implement any unitary transformation of a photonic quantum state [20]. The interferometers are inscribed in a boro-aluminosilicate glass (Corning EAGLE XG [21], 1.1 mm thick) and optimized for single-mode operation at 1550 nm wavelength. The basic structure of the MZI is depicted in Figure 1a: the optical waveguides are inscribed at 30 μm below the chip surface, forming an optical circuit composed of two 3 dB directional couplers (interaction length 1.2 mm, coupling distance 9 μm), that are connected by sinusoidal S-bend waveguides (minimum radius of curvature 45 mm) to the central arms of the interferometer (two straight waveguides having length L and distance p). Light propagation is characterized by losses of 0.29 dB/cm and a mode diameter (1/e 2 ) of 9 μm. Such a circuit enables the coupling to standard single-mode optical fibers (SMF-28) with losses as low as 0.27 dB/facet.
Given coherent light injected in one of the two input ports, the optical power Iout measured at one of the two outputs can be modulated by acting on the phase difference φ that is present between the two optical paths. Mathematically speaking where Imax is the sum of the two output optical powers and ν is the visibility of the interference fringe. In order to tune the phase difference φ, we exploit a Cr-Au resistive microheater (length Lr = L and width Wr) fabricated on top of one of the two central arms (see Figure 1a). Given the linearity of the heat equation and assuming a linear relation between refractive index and temperature change of the substrate, we can model the phase difference φ between the two arms as [7] = + , where Φ is the phase difference when the microheater is not biased, α is the tuning coefficient of the interferometer and P is the electrical power dissipated by the microheater. In order to induce a given phase difference with minimal power dissipation, it is necessary to maximize the tuning coefficient α. To this aim, it is useful to derive an analytical expression for this parameter. Considering an infinitesimal waveguide segment having length dl, the corresponding phase dφ induced by the microheater is where λ is the wavelength, Δn(l) and ΔT(l) are, respectively, the refractive index and the temperature difference between the two waveguides at a given coordinate l and nt is the thermo-optic coefficient of the substrate. By integrating Equation 3 over the entire optical path γ and substituting Δφ = φ -Φ into Equation 2, one reads where we have introduced the linear power dissipation density P = P/L, and the thermal efficiency factor R = ΔT/P. The approximation in Equation 4 is valid whenever L is much greater than the transverse dimensions of the device. Indeed, in this condition the thermal problem can be studied with a 2D geometry, in which the heat flux is always orthogonal to the light propagation and in which there exists a temperature difference ΔT that is constant and different from zero only for a length L, as long as the waveguide is beneath the microheater. The quantity R has the dimensions of a 2D thermal resistance and depends only on the transverse geometry of the device and on the thermal conductivity k of the substrate. For given material and wavelength, R represents the only degree of freedom for the optimization of α. In particular, this factor can be increased by avoiding the heat diffusion all over the chip and, thus, by thermally isolating the target waveguide from the rest of the circuit. Of course, this approach is beneficial also for the thermal crosstalk between different integrated interferometers in the same substrate. Two different isolating structures are investigated in this work: deep isolation trenches ( Figure 1b) and the bridge waveguide ( Figure 1c). On the one hand, the isolation trenches ( Figure 1b) are fabricated by removing glass boxes (dimensions Lt × Wt × Dt) from both sides of the microheater. They are removed as close as possible to the waveguide, in order to limit the portion of glass subject to the heating and to reduce the width W of the slab that thermally connects the waveguide to the rest of the circuit. A nominal width W = 20 μm represents a value that, given the dimensions of the optical mode, does not affect the insertion losses of the circuit and, at the same time, is compatible with the fabrication of a microheater that occupies all the area between the two trenches (i.e. Wr = W). The dimensions of the boxes are chosen to match the length of the microheater (i.e. Lt = Lr = L) and to allow the reduction of the inter-waveguide pitch p down to 80 μm (i.e. Wt = p -W = 60 μm). A photomicrograph of deep isolation trenches having Dt = 150, 300, 450 μm is reported in Figure 1d. On the other hand, the bridge waveguide ( Figure 1c) is a structure based on the former one, but in which the glass is removed also under the optical path, thus leaving the waveguide inside a suspended bridge whose section has nominal dimensions W = 20 μm and D = 60 μm. The dimensions of the lateral boxes are Lt = Lr = L, Wt = 60 μm and Dt = 90 μm. A photomicrograph of a bridge waveguide is reported in Figure 1e.

Power dissipation
The electrical power P2π that a microheater must dissipate to induce the maximum useful phase shift (i.e. Δφ = 2π) is directly related to the tuning coefficient α through the relation This quantity has been experimentally characterized on reconfigurable MZIs (L = 3 mm, p = 127 μm) featuring isolation trenches with depth Dt ranging from 0 (no trenches) to 450 μm. The power dissipation P2π is reported in Figure 2a as a function of the depth Dt: the use of isolation trenches allows a reduction of the power dissipation of more than an order of magnitude, from 611 mW for a MZI with no isolation, to 57 mW for a MZI with trenches 450 μm deep. It is clear from Figure  2a that, for Dt > 300 μm, the power dissipation has already saturated to the minimum value and a further increase of the thermal isolation produces no effect on the MZI performance. However, modelling such phenomenon by considering heat dissipation only through the glass slab underneath the microheater does not predict this saturation trend. Indeed, by considering a trench depth Dt sufficiently large, the 2D thermal resistance Rslab (see the inset of Figure 2a) between the microheater and the heat sink at the bottom of the substrate (fixed at room temperature Troom) can be calculated as This quantity determines the temperature of the target waveguide, that is T1 = Troom + Rslab P. By assuming negligible heating on the second arm of the MZI (i.e. T2 = Troom), the temperature difference between the waveguides is ΔT = Rslab P and, therefore, the thermal efficiency factor is R = Rslab. In conclusion, Equation 5 becomes It is clear from Equation 7 that, for an increasing trench depth Dt, the power dissipation P2π should approach zero. On the contrary, the saturation observed during the experimental characterization is consistent with the presence of a thermal leakage that breaks the isolation achieved with the trenches. Indeed, finite-element simulations demonstrate that, by taking into account the thermal conduction through the air gaps, the experimental behavior can be predicted with an error that is lower than 12% over the entire dataset (see Figure 2a). Even though the presence of air limits the effectiveness of the isolation structures, a further reduction of the power dissipation P2π can be achieved with the bridge waveguide. Figure 2b reports the optical power Iout measured at one of the output ports of a MZI as a function of the electrical power P dissipated by a microheater on a bridge waveguide: a complete reconfiguration is achieved with a power dissipation as low as 37 mW. The experimental dataset is reported along with its best sinusoidal fit, obtained by exploiting the mathematical model described by Equations 1 and 2.

Dynamic response
As shown in the previous section, thermal isolation has a beneficial effect on the static power dissipation P2π. However, a higher isolation can slow down the dynamic response of the device and, thus, increase the switching time τ (10 to 90%) necessary for the reconfiguration. In order to take into account both the effects, silicon PICs are usually compared with the aid of a figure of merit (FOM) based on the product of these two quantities [22]. However, in order to compare devices operating at different wavelengths, it is possible to introduce a more general FOM, based on the former one and defined as Step response and FOM have been assessed by using the thermal shifter to induce a phase change Δφ = π in a MZI, corresponding to a complete switch of the optical power from an output port to the other. The switching time τ is reported in Figure 3a for the same MZIs presented in the previous section: isolation trenches slow down the step response, with a switching time that goes from 12 ms, for a MZI with no trenches, to 46 ms, for a MZI with a trench depth Dt = 450 μm. However, if this worsening is considered along with the strong improvement in terms of static power dissipation, it is possible to conclude that the overall performance of the device benefits from the use of isolation trenches. This fact is quantified by the FOM, that is reported in Figure 3a along with the switching time: indeed, by introducing the isolation trenches the FOM improves (i.e. decreases) of about a factor of 3, starting from 4730 Ws/m and saturating down to about 1500 Ws/m. Again, it is worth noting that isolation trenches deeper than 300 μm do not provide any improvement.
Finally, Figure 3b reports a comparison between the step responses, normalized to the final steady state value, for a MZI with no isolation, with isolation trenches (Dt = 300 μm) and with a bridge waveguide. The latter features a switching time τ = 33 ms, that is comparable to the one achieved with the trenches. Given the lower dissipated power, this result leads to a further improvement of the FOM that reaches a value as low as 788 Ws/m.

Thermal crosstalk
Up to now, we have considered a single MZI and the effects of a thermal shifter fabricated right upon it. However, due to thermal crosstalk, an interferometer can be affected also by the power dissipated in close-by devices, resulting in an undesired contribution Δφct to the actual phase φ. Under the linearity hypothesis stated in the former sections, the total phase φ induced on a given MZI can be modeled by generalizing Equation 2 as where N is the total number of thermal shifters integrated in the device and the subscript 0 refers to the microheater right upon the MZI. In order to guarantee an effective control on the phase φ, the phase contribution Δφct induced by the other microheaters has to be as low as possible. Thermal crosstalk has been measured on three groups of four MZIs each (L = 3 mm, p = 127 μm). In all the three groups there is one MZI with a microheater on top, each characterized by a different isolation approach (no isolation, isolation trenches having depth Dt = 300 μm and the bridge waveguide). As depicted in Figure 4a, the four interferometers in each group have a different distance x from the microheater, that ranges from x = 0 μm (MZI fabricated beneath the thermal shifter) to x = 6p = 762 μm (separation step Δx = 2p = 254 μm). A comparison among the three isolation approaches is reported in Figure 4b in terms of the phase shift Δφ as a function of the distance x. When the thermal shifter induces a 2π phase shift on the target interferometer (i.e. at x = 0 μm), the phase shift on the next-neighbor MZI (i.e. x = 254 μm) is Δφct = 3.64 rad when no isolation is used. On the contrary, by exploiting one of the two isolation strategies, this value reduces to 0.23 and 0.27 rad, respectively. For MZIs that are further apart, the slope of the three curves is very similar, consistently with the fact that no isolation is replicated on these interferometers and, thus, all the improvement is gained in the isolation structure produced around the microheater.

Miniaturization
The reduction of the optical circuit dimensions would be advantageous not only to increase the integration density attainable with this platform, but also to reduce the insertion losses of the device as the propagation distance would be diminished. Firstly, it is interesting to investigate the scaling of the inter-waveguide distance p. A few problems can arise when reducing this parameter: indeed, by adopting p = 80 μm in interferometers that are not isolated, the power P2π dissipated by a thermal shifter set at Δφ = 2π increases from 611 to 776 mW, while the phase Δφct induced by the same thermal shifter on an adjacent MZI (i.e. x = 160 μm) increases from 3.64 to 4.48 rad. On the contrary, the detrimental effects of the scaling are not observed on MZIs isolated with trenches (Dt = 300 μm): indeed, by adopting p = 80 μm on such interferometers, the power dissipation P2π remains as low as 57 mW, while the phase shift Δφct induced on the adjacent MZIs remains as low as 0.22 rad.
Secondly, another parameter that is desirable to scale is the thermal shifter length L. Since the thermal efficiency factor R does not depend on L, by considering Equation 4 one may think that this parameter can be reduced as much as we want, with no effect on the tuning performance of the circuit. Actually, this is not true: while the power dissipation P needed to induce a given phase φ is not dependent on L, the corresponding temperature difference ΔT induced between the two arms is inversely proportional to this parameter. Therefore, the microheater miniaturization leads to higher operating temperatures that, in turn, can degrade the resistive materials during the operation of the device. This phenomenon can lead to long-term drifts of the phase φ or, in the worst case, to the breakdown of the thermal shifter. Mathematically speaking, Equation 5 can be exploited to estimate the maximum temperature difference ΔT2π (i.e. the temperature difference corresponding to a 2π phase shift) as For the microheaters considered so far (L = 3 mm) and a thermo-optic coefficient nt = 6.8E-6 K -1 , the temperature difference is ΔT2π = 76 °C. The compatibility of this value with a stable operation of the thermal shifter has been already demonstrated [18], but it is interesting to investigate a further reduction of the length L. Therefore, we have characterized the phase stability of MZIs featuring 1.5-mm-long microheaters on bridge waveguides, corresponding to an estimated temperature difference ΔT2π = 152 °C. Figure 5a reports the phase φ monitored from the output power distribution of such an interferometer, continuously operating for almost 13 h. The thermal shifter induces a constant phase φ = 3 2 π, a value that guarantees the maximum sensitivity of the measurement (see Equation 1) and, at the same time, a temperature difference close to the maximum one. No evidence of long-term drifts is present, and the short-term phase fluctuations are characterized by a small standard deviation of 23.5 mrad. Such fluctuation includes the effects strictly due to the thermal shifter, but also those due to the experimental setup (e.g. laser power fluctuations, alignment instabilities, room temperature changes, etc.). In order to isolate the contribution of the thermal shifter, the electrical resistance of the microheater has been monitored over the whole period of the measurement. Figure 5b reports this quantity as a function of the time: the standard deviation is as low as 59 mΩ, corresponding to a phase variation of only 5.3 mrad. This is the value that could be achieved by the thermal tuning process if no other sources of fluctuations were present.
Finally, we investigated the electrical reliability of these devices on longer periods. To this aim, three microheaters have been continuously operated for over 2 weeks, with no evidence of damages or drifts of the electrical properties of the microheater.

Performance in vacuum
The experimental characterization presented so far demonstrates that the performance of these 3D-structured reconfigurable circuits is limited by the presence of air. Therefore, in order to investigate the possibility of further improving both the power dissipation and the thermal crosstalk, we have repeated in a vacuum chamber the main steps of the experimental characterization for a MZI (L = 3 mm, p = 127 μm) featuring a bridge waveguide. Thanks to a two stage pumping system, the experimental setup allows measurements at two different vacuum levels: a medium vacuum, corresponding to an absolute pressure = 1.1E-4 bar, and a high vacuum, corresponding to = 2.7E-7 bar. However, thanks to the fact that the pressure decreases with a slow transient, additional measurements can be collected at intermediate pressure values. In this way, the performance of the MZI can be characterized on an extended pressure range.
This modus operandi is at the basis of Figure 6a, that reports the power dissipation P2π as a function of the pressure : a value of 1E-3 bar is already enough to achieve P2π = 3.49 mW, an improvement in terms of power dissipation of more than an order of magnitude with respect to the value measured in ambient conditions (i.e. = 1 bar). By waiting for the pressure to reach the medium vacuum level, the value of P2π drops down to 1.04 mW. However, the power dissipation has not reached its minimum value yet: indeed, when the second pumping stage is turned on, this quantity further decreases and enters the sub-milliwatt range, with a power dissipation as low as 0.72 mW. Finally, it is worth noting that, close to the high vacuum level, the power dissipation P2π seems to become independent on the pressure and, thus, it is possible to conclude that the minimum power dissipation achievable with the bridge waveguide is reached.
The excellent performance in terms of power dissipation is again counterbalanced by the slowing down of the step response. Figure 6b reports the normalized output power Iout' as a function of the time t, when the thermal shifter is operated to induce a phase step having amplitude Δφ = π: the switching time increases from 33 ms to 0.96 s and 1.35 s for the medium and the high vacuum level. However, the overall performance of the device benefits from the operation in vacuum: indeed, the corresponding FOM drops to 644 and 604 Ws/m, respectively.
Finally, the experimental characterization in vacuum has been concluded with the assessment of the thermal crosstalk performance: when the microheater is biased in order to induce a 2π phase shift on the target MZI, the phase variation measured on the adjacent interferometer (i.e. x = 254 μm) is as low as Δφct = 6 mrad.

Discussion
The technological platform that we have presented here demonstrates that it is possible to integrate state-of-theart waveguide circuits inscribed by FLW with isolation structures that allow an efficient reconfiguration of the device. Indeed, the microstructures result in no compromise on the optical performance of the circuit, whose losses are comparable to the best results reported in the literature for FLW integrated circuits at telecom C-band wavelength [13], [23]. Two different isolation structures have been considered throughout the article: deep trenches and the bridge waveguide. Both provide a dramatic decrease in the dissipated power required to induce a 2π phase shift in a MZI, with a better performance of the bridge waveguide. The improvement is also evident when both the static and the dynamic responses are assessed with the FOM that we have defined.
A comparison among the best results achieved with FLW is reported in Table 1. The bridge waveguide compares favorably with the other FLW platforms, resulting the best solution both in terms of power dissipation and in terms of the overall performance, assessed by means of the FOM. Furthermore, with this work we have also demonstrated that, by operating the device in vacuum, the power dissipation needed to induce a complete phase shift by the thermo-optic effect drops to the best value ever reported for an integrated MZI. Indeed, even considering the most efficient integrated circuits based on microstructured SOI substrates [24], [25], our MZI is the first device able to fully reconfigure its behavior with a power dissipation as low as 0.72 mW. This value is even more remarkable if we think that it can be achieved along with negligible thermal crosstalk (less than 0.1%) on the adjacent interferometers. Such a performance can be exploited also in standard conditions, by suitably encapsulating the device, or alternatively in extreme environment conditions, like in outer space or in a cryostat, as required by some applications in photonic quantum information science [23].
Concerning the size of the FLW circuits, most of the devices presented in this work have been designed to achieve an integration scale that is comparable with the state of the art. However, it is worth remarking that, with our 3D structuring method, we have demonstrated short microheaters, with length L = 1.5 mm, providing an operation stability comparable to that achieved in longer FLW devices [18], [19]. In addition, we have also demonstrated that a reduction of the inter-waveguide pitch, down to p = 80 μm, can be achieved with negligible effect on the thermal crosstalk; in fact, even without vacuum operation, only 3.5% of the phase induced on one MZI is transferred on the next-neighbor device. Such a low value makes the performance of our FLW circuits comparable to the one attained with devices based on the SOI technology, that are able to reach a thermal crosstalk as low as 1% [27].
In conclusion, all these results pave the way towards a level of control, complexity and integration density never achieved before in a FLW device, opening exciting scenarios in photonic quantum information processing.

Fabrication process
The fabrication process starts with the waveguide inscription by means of femtosecond laser writing in Corning EAGLE XG glass slides (1.1 mm thick), followed by the micromachining of the isolation structures by water-assisted laser ablation (WALA) [28]. To allow the realization of both these steps with a single alignment, the two processes share the same fabrication setup. The laser is a Yb-based commercial fiber laser (Amplitude Satsuma HP), which delivers pulses with a duration of 230 fs at 1030 nm wavelength. Since the WALA works on the rear surface of the substrate, thus producing structures that are flipped with respect to the typical usage, the bottom of the sample must be in contact with the water. Therefore, two edges of the sample are secured on top of a pair of 1 mm thick spare glasses, allowing the chip to be suspended at the cost of a negligible reduction in the workable area. The system is then glued to a 60 mm diameter Petri dish, mounted on a XY air-bearing stage (Aerotech ANT series) to allow a full coverage of the sample with deionized water. In order to reduce aberrations, a water-immersion objective (Zeiss 20X -0.5 NA) is employed in the fabrication process. The objective is held on a Z-axis translation stage, thus ensuring the full 3D capability of the system. A schematic representation of the writing configuration is reported in Figure 7a. For the waveguide inscription, the laser is configured in order to provide pulses of 580 nJ, at a repetition rate of 1 MHz. The substrate is translated at a speed of 20 mm/s for a total number of 6 successive scans.
For the ablation of the isolation structures, we designed the WALA irradiation pattern by coping with the idea of speeding up the process. Indeed, the ablation is not performed on the full volume but, following the method proposed in [28], only on the external surfaces of rectangular boxes. First, the irradiation is performed on the sidewalls of the box (Figure 7b), patterning multiple rectangular perimeters, made of straight lines connected with arcs of circumference of 5 µm radius, and spaced in depth by 1 µm. After the desired height of the box is achieved, the top part is irradiated ( Figure  7b) by describing smaller and smaller concentric rectangular paths, until the glass piece is fully detached. For the case of the isolation trenches, the full structure is sliced in 150 µm deep building blocks and, then, these blocks are ablated one after the other in order to remove deep portion of the material and to allow a higher yield with respect to the direct ablation of an entire big box. For the case of the bridge waveguides (Figure 7c), the two lateral trenches are connected by ablating small rectangular boxes from the edges towards the center of the structure. Such pieces of glass are sufficiently small to fall out through the lateral trenches.
The ablation process exploits laser pulses of 4 µJ, delivered at a repetition rate of 50 kHz. This choice allows the definition of the isolation trenches at a writing speed of 4 mm/s, a value that is at least one order of magnitude higher than that used in previous works [28], [29] and that allows a significant speed up of the ablation process. By exploiting these irradiation parameters and by stacking up 150 µm boxes, isolation trenches can reach an overall depth up to Dt = 450 µm. On the other hand, the bridge waveguide requires a less demanding fabrication process, due to its limited depth Dt of only 90 µm. For this last structure, the laser pulse energy required for an optimal ablation is only 2.4 µJ. As a matter of fact, the adopted irradiation pattern, repetition rate and writing speed allow the fabrication of a pair of isolation trenches (L = 3 mm, Dt = 300 μm) in around 20 minutes. A bridge waveguide (L = 3 mm) requires the same amount of time for its realization.
Following the method proposed by Arriola and co-workers [30] to obtain low losses and low birefringence waveguides, the fabrication continues with a thermal annealing process [13] and, finally, it is concluded by the Cr-Au deposition and by the microheaters patterning through a subsequent step of femtosecond laser ablation [18]. By exploiting reference markers dug into the device, a 2 µm resolution re-alignment is achieved during the metal ablation process for the definition of microheaters and contact pads.

Experimental characterization
The characterization of the static performance (i.e. power dissipation, thermal crosstalk and phase stability) is carried out by coupling light at 1550 nm wavelength from a laser diode (Thorlabs L1550P5DFB) into the devices by means of a standard single-mode fiber (Thorlabs SMF-28-J9). The two output ports of the MZI are simultaneously collected by an aspheric lens (Thorlabs C230TMD-C) with 0.55 NA and directed towards two photodiode heads (Ophir PD300-IR) connected to a computer-controlled power meter (Ophir Nova II). The driving currents to the thermal shifters are provided by a DC power supply (Keithley 2231A). All the measurements are performed by exploiting a MATLAB program for controlling the current scan and the acquisition of the optical signals measured by the photodiodes.
For the case of the step response, the optical power is measured only at one output of the MZI by using a largebandwidth photodiode (Thorlabs PDA20CS-EC) and by controlling the thermal shifters with a function generator (Tektronix AFG3011C). Waveforms are recorded with a digital oscilloscope (Tektronix DPO2024B) and then analyzed with a MATLAB software.

Simulation
The simulation of the electrical power P2π (see Figure 2a) is carried out with the heat transfer module provided by COMSOL Multiphysics. The MZI with deep isolation trenches is represented with a 2D geometry composed of three parts: the glass substrate, the gold microheater and the air that surrounds the entire device and completely fills the trenches. The thermal conductivity employed for the substrate is reported in the datasheet [21], while, for the other materials, the thermal properties are the ones provided by the COMSOL material library. All the boundaries of the simulation domain are subject to a no flux condition, with the exception of the bottom of the device, whose temperature is fixed at Troom = 20 °C.
The entire curve is obtained by measuring the power P2π uniformly generated in the thermal shifter volume and necessary to induce a temperature difference ΔT2π = 76 °C between the MZI arms. According to Equation 10, this is the required temperature difference for a 2π phase shift considering a thermo-optic coefficient nt = 6.8E-6 K -1 . Although this information is not reported in the datasheet of the substrate [21], it can be retrieved from other works using the Corning EAGLE glasses [18], [19].