different. The size is different and input source is different. I doubt this comparison and it is not scientific. In addition, "Input solar power" in horizontal axis is misleading. They should put real incident solar power on the collecting mirror. The incident solar power of two experiments is almost 4 times different.

With due respects, we think reviewer 1 has not been very careful in examining the manuscript. In Fig. 5, we are comparing ceramic Cr:Nd:YAG and Nd:YAG crystal data, the solar energy collection concentration system in PROMES-CNRS has the same size and reflectivity. The input solar irradiances are only slightly different for the two experiments in 2011 and 2012 respectively. The side-pumped Nd:YAG laser output performances, published by Optics and Laser Technology in 2012 (Ref. 10), are compared to the present results. Also from the comments of " the incident solar power of the two experiments are almost 4 times different", the reviewer might have mistakenly considered our Fresnel lens end-pumped solar laser published by Optics Express in 2011(Ref. 8) for his comparison. We did not compare this end-pumped result with the present results in Fig. 5. The collected solar power at the focus has been widely adopted as input solar power in many publications on solar-pumped lasers by Arashi (Ref 4), Weksler (Ref.5), Lando( Ref.7), Yabe ( Applied Physics Letters in 2007, Optics Letters in 2008), Liang (Ref. 8) etc, so we prefer not changing the rules during the game.

2. The authors mentioned 33.4W output but its collection efficiency is only 10W/m2 and a half of their previous result. This means the proposed scheme might have some difficulty in scaling up. This point should be discussed.

Once again, the reviewer must have mistakenly compared the 10W/m2 collection efficiency with the 19.3 W/m2 value in Reference 8, which matches well with "a half of their previous result". Instead, our previous result in Ref. 10 was only 9.6W/m2 with the same PROMES-CNRS system, so the collection efficiency is slightly enhanced by using Cr:Nd:YAG ceramic rod in 2012.

As mentioned in the manuscript, the total reflectivity of the whole solar energy collection and concentration system is only 59%. The back surface silver-coated plane mirror and parabolic mirror have more than 20 year´s service history. The collection efficiency can be largely improved by using front surface silver-coated mirrors.

3. On page 3, they claimed the demerit of using optical fiber. However, I did not find the superiority of their system for long distant transportation of many beams. This should be essential because we can not use huge mirror for obtaining MW class laser but we need to use 1000 bundles of kW laser, because no one show the feasibility of huge laser by single optical system.

We only mention some shortcomings of optical fibers for laser power transmission. We are admirers of the renewable large-scale Mg recovery scheme proposed by Professor T. Yabe with many optical fiber bundles for kW laser. Besides, we are also interested in the multi-rods solar-pumped laser scheme at the focus of the 1MW solar furnace as proposed by Uzbekistan scientists, without optical fibers in this case.

4. As for the beam quality, I would like to know the size of the beam spot which might be used to calculate the divergence of the beam. Far field laser beam divergence measurement was carried out again. We have measured the 1/e2 beam spot size of about 150 mm diameter, 10 meters away from the laser output coupler.

5. In the previous paper, they used CPC but now are using tapered cavity. I would like to know the reason of the change and how it changes the result.

The following text has been added to the revised manuscript: Since there is only less than 16% spectral overlap between the solar emission spectrum and Nd:YAG absorption spectrum, the absorbed solar pump power is limited. The compact CPC cavity is used to achieve efficient pumping along the 4 mm diameter Nd:YAG rod. ZEMAX© and LASCAD© numerical analysis has also indicated the effectiveness of the CPC scheme in attaining the maximum laser output power from the Nd:YAG rod. Since the overlap between the solar emission spectrum and Cr:Nd:YAG absorption spectrum can theoretically reach 40%, there are more absorbed solar pump power available, large diameter ceramic rod can then be used to achieve more laser output power. The compound cavity is adopted to pump the 7 mm diameter Cr:Nd:YAG ceramic rod. The rod diameter is also optimized by both ZEMAX© and LASCAD© analysis. The compound V-shaped pump cavity ensures the nearly top-hat absorbed pumped flux distribution, as shown in Fig. 4, which drastically reduces the laser beam divergence, when compared to other end-pumped configurations. The heat load of the 7 mm diameter Cr:Nd:YAG ceramic rod within the compound cavity is also lower than that of the 4 mm diameter single-crystal Nd:YAG rod within the CPC cavity. On the one hand, the Cr:Nd:YAG rod provides enhanced laser output performance within the large compound cavity, on the other hand, the Nd:YAG rod produce also its maximum laser output power within the compact CPC cavity, so the comparison between the ceramic and single-crystal laser rods are made within different pumping cavities, where the advantages of each laser rod, and therefore the laser output performances can be fully exploited.

6. On page 10, "the temperature dependent reduction of the 1064 nm stimulated emission ... [9]" but there is no description on stimulated emission in ref. [9]. The authors should not write a misunderstanding sentence.

Yes, the sentence has been changed to: Side-pumping configuration provides the best heat load distribution along the rod and the temperature dependent reduction of the 1064 nm stimulated emission crosssection of Cr:Nd:YAG ceramic crystal [14] is not as significant as the case of end-pumped configurations.

7. The quality of ceramic laser depends on the production condition. Therefore the scattering loss changes depending on the lod. If the authors are careful enough, ref. [9] did not say the inferiority of the ceramic YAG but the ceramic they used has the high scattering coefficient. This is important because the quality changes one by one and should be improved for industrial application. Therefore they should remove the word "doubts" in abstract and introduction and so on.

Yes, the words doubt have been removed in the revised manuscript 8. Regarding the scattering coefficient, the authors mentioned on 9 that "0.004cm-1, as predicted in Ref. [9]". This is misleading because it was measured by integrated sphere but was not predicted. In addition, agreement with published value ref. [17] has no meaning because the ceramic quality is not so stable.

Yes, "0.004cm-1, as predicted in Ref. [9]" has been changed to, "0.004cm-1, as measured in Ref. [9]" The sentence"which agrees well with the published value of 0.002 cm-1 [17]" has been removed in the revised manuscript.

Dear Reviewer 2

Many thanks for your very insightful and helpful comments. We would like to answer your concerns one by one Reviewer #2: Solar laser attracts more attentions in research due to the potential applications in clean energy field. The authors present a comprehensive work on sidepumped CW Cr:Nd:YAG ceramic solar laser. The simulation as well as experimental results are very valuable reference for people working in this field. Nevertheless, a few issues are not clearly addressed in the manuscript.

1. Page 3, lines 39-40 describe the inner wall of the V-shaped pump cavity. What is the material of the V-shaped cavity? Why not have the pump cavity directly coated with gold or silver but bonded with a silver-coated aluminum foil?

The V-shaped cavity is bounded with silver-coated aluminum foil with 94% reflectivity by now. A gold-coated cavity will not reflect efficiently some useful solar pump power below 500 nm. Silver coating is not used due to the possible contamination with cooling water. Protected silver coating will be considered in the future version.

2. Page 6, line 44 to page 7, line 14 describe how the M2 parameter was measured, the method they used was too much simplified. Normally, the transverse beam profile at minimum five positions need to be recorded in order to fit the hyperbolic curve, so that the beam waist and far field divergence can be determined. The author only measured a very near field(5 mm) beam width and a far field beam width (1500 mm), the beam divergence calculated from these two values was not correct, therefore, the following up calculations of M2 and Figure of merit B were not reliable. In the same paragraph, the author mentioned that the diffraction-limited Gaussian beam of the same wavelength was 0.019 degree, how was this value calculated?

The following text has been added to replace the original text in the revised manuscript:

A linear fiber-optic array for measuring the one-dimensional laser beam intensity distribution in the near field is placed 5 mm away from the output coupler along the optical axis of the laser rod [8,10]. The 32 mm width, 128 optical fibers linear array is used to collect and transmit laser light to a Fairchild CCD 153A 512-element linear image sensor via a neutral density attenuator. This fiber-optic device has 0.25 mm core pitch resolution, so less than 2% laser beam diameter measurement error is found. This flexible fiber optic bundle has 2 m length. Outdoor solar laser beam diameter measurement is hence facilitated.

For far field laser beam profile measurement, a 10" x 10" industrial standard laser alignment thermal sensitive paper ZAP-IT@ is positioned 10 m away from the output coupler. Assuming 3 mm reading error in the thermal sensitive paper, less than 2% laser beam diameter measurement error is ensured.

The laser beam divergence θ is found by adopting the Eq. ( 1): see attachment (1) Where ϕ1 = 7.0 mm and ϕ2 = 150 mm are the measured laser beam diameters at 1/e2 width, 5 mm and points.

M2 factor is then calculated as: see attachment (2) where = 0.015º is the divergence of diffraction-limited Gaussian beam for 1.064 m and 1280 m, as calculated by LASCAD© laser beam propagation method for the 7mm diameter rod. For -2 m RoC output coupler, Mx2 = 27.5 ± 3% and My2 = 28.3 ± 3% are experimentally determined, indicating a near symmetric beam profile. Figure of merit B of 0.44 × 10-1 W is finally calculated. Even though this value is 6.5 times lower than our previous record of 2.9 × 10-1 W [10], it is still 6.7 times higher than that of the most recent end-pumped solar laser [9].

3. In section 4, the authors made some comparison between Nd:YAG and Cr, Nd: YAG ceramic, the experimental conditions were very different, e. g., the size of the crystal, mode volume of the resonator and pumping scheme were all different in two cases. But the authors put the results into one figure, (Fig. 5) and made comparison of the output and efficiency. So the conclusion based on this is not convincing, because other factors which could also influence the output power and efficiency were not excluded.

The following text has been added to the revised manuscript: Since there is only less than 16% spectral overlap between the solar emission spectrum and Nd:YAG absorption spectrum, the absorbed solar pump power is limited.

The compact CPC cavity is used to achieve efficient pumping along the 4 mm diameter Nd:YAG rod. ZEMAX© and LASCAD© numerical analysis has also indicated the effectiveness of the CPC scheme in attaining the maximum laser output power from the Nd:YAG rod. Since the overlap between the solar emission spectrum and Cr:Nd:YAG absorption spectrum can theoretically reach 40%, there are more absorbed solar pump power available, large diameter ceramic rod can then be used to achieve more laser output power. The compound cavity is adopted to pump the 7 mm diameter Cr:Nd:YAG ceramic rod. The rod diameter is also optimized by both ZEMAX© and LASCAD© analysis. The compound V-shaped pump cavity ensures the nearly top-hat absorbed pumped flux distribution, as shown in Fig. 4, which drastically reduces the laser beam divergence, when compared to other end-pumped configurations.

The heat load of the 7 mm diameter Cr:Nd:YAG ceramic rod within the compound cavity is also lower than that of the 4 mm diameter single-crystal Nd:YAG rod within the CPC cavity. On the one hand, the Cr:Nd:YAG rod provides enhanced laser output performance within the large compound cavity, on the other hand, the Nd:YAG rod produce also its maximum laser output power within the compact CPC cavity, so the comparison between the ceramic and single-crystal laser rods are made within different pumping cavities, where the advantages of each laser rod, and therefore the laser output performances can be fully exploited.

4. Page 10, line 2, the scattering loss of 0.018 cm-1, one zero was missing, the correct value should be 0.0018 cm-1.

Yes, the correct value of 0.0018cm-1 has been added to the revised manuscript. Abstract To clarify the advantages of Cr:Nd:YAG ceramics rods in solar-pumped lasers, a fused silica light guide with rectangular cross-section is coupled to a compound V-shaped cavity within which a 7 mm diameter 0.1 at% Cr: 1.0 at% Nd:YAG ceramic rod is uniformly pumped. The highly concentrated solar radiation at the focal spot of a 2 m diameter stationary parabolic mirror is transformed into a uniform pump radiation by the light guide. Efficient pump light absorption is achieved by pumping uniformly the ceramic rod within the V-shaped cavity. Optimum pumping parameters and solar laser output powers are found through ZEMAX © non-sequential ray-tracing and LASCAD © laser cavity analysis codes. 33.6 W continuous-wave laser power is measured, corresponding to 1.32 times enhancement over our previous results with a 4 mm diameter Nd:YAG single-crystal rod. High slope efficiency of 2.6% is also registered.

The solar laser output performances of both the ceramic and the single-crystal rods are finally compared, revealing the relative advantage of the Cr:Nd:YAG rod in conversion efficiency. Low scattering coefficient of 0.0018 cm -1 is deduced for the ceramic rod.

Heat load is considered as a key factor affecting the ceramic laser output performance.

Introduction

Solar-pumped lasers have gained an ever-increasing importance in recent years [1].

Compared to electrically powered lasers, solar laser is much simpler and more reliable due to the complete elimination of the electrical power generation and conditioning equipments. This technology has a large potential for many applications, e.g. hightemperature materials processing, free space laser communications, space to earth power transmission, and so on. The renewable recovery of Mg from MgO is another very interesting topic for solar-pumped lasers [2]. Ultra-high brightness renewable solar-pumped laser beams can be very conveniently focused to heat the magnesium oxide to more than 4000 K and thus create pure magnesium. Magnesium can be easily stored and transported in the form of "pellets" and, when necessary, reacts with water to produce both hydrogen and thermal energy for fuel cell vehicles and other applications.

The first solar-pumped laser was reported by Young in 1966 [3]. Since then, researchers have been exploiting both parabolic mirrors and Fresnel lenses to attain enough concentrated solar radiation at focal point and several pumping schemes have been constructed for enhancing solar laser output performances [1][2][3][4][5][START_REF] Krupkin | Proc. SPIE[END_REF][START_REF] Lando | [END_REF][8][9][10][11][12]. To improve the efficiency of Nd 3+ -doped YAG laser, cross-pumped Cr 3+ and Nd 3+ co-doped YAG ceramic material has attracted more attentions in recent years [1,2,[11][12][13]. The sensitizer Cr 3+ ions have broad absorption bands in the visible region. By the 4 T 2 to 4 A 2 transition of Cr 3+ ions, energy is transferred from Cr 3+ to Nd 3+ ions. For single-shot laser operation with a 0.1 at% Cr 3+ and 1.0 at% Nd 3+ co-doped YAG ceramic rod, the laser efficiency is found to be more than twice that of a 1.0 at% Nd 3+ :YAG ceramic rod. At low repetition rates, the average output power of Cr:Nd:YAG rod is higher than that of Nd:YAG. However, this tendency gradually decreases with increasing repetition rates [13].

Despite the interests in Cr:Nd:YAG ceramic medium, researchers have achieved significant laser efficiencies with different Nd:YAG single-crystal rods. 19.3 W/m 2 collection efficiency has been reported by us last year [8] with a 4 mm diameter Nd:YAG single-crystal rod pumped through a 0.9 m diameter Fresnel lens. The most recent solar-pumped laser with a liquid light guide lens and also a 6 mm diameter Nd:YAG rod has produced 30.0 W/m 2 collection efficiency, despite its very low laser beam brightness figure of merit B = 6.6×10 -3 W. The collection efficiency with the Nd:YAG rod is unexpectedly better than that with Cr:Nd:YAG ceramic rods [9]. Large scattering loss of 0.004 cm -1 for Cr: Nd;YAG ceramics is considered as the main reason for this unfavorable surprise. While it is clear about the effectiveness of Nd:YAG single-crystal rods for solar laser operation, there still exist, in our opinion, some concerns about the advantages of Cr:Nd:YAG ceramic medium in solar-pumped lasers.

Although the most efficient laser systems have end-pumping approaches, sidepumping is an effective configuration for power scaling as it gives uniform absorption along the rod axis and spreads the absorbed power within the laser medium, reducing hence the associated thermal loading problems. The solar laser beam brightness from a side-pumping configuration can be higher than those by end-pumping configurations with Fresnel lenses. Indeed, significant improvement in solar-pumped laser beam brightness has been achieved by us in 2011 with the same PROMES-CNRS medium size heliostat-parabolic mirror system. By side-pumping the 4 mm diameter 30mm

length Nd:YAG single-crystal rod through a light guide / modified 2D-CPC cavity, record-high brightness figure of merit of 2.9×10 -1 W is registered [10]. Therefore, light guide side-pumping configuration is chosen here for comparing the laser performances of both the Cr:Nd:YAG ceramic and the Nd:YAG single-crystal rod.

Fresnel lenses have attracted much more attentions of solar laser researchers. The advantage of heliostat-parabolic mirror system for solar laser research is, however, comparatively neglected in recent years. We have, instead, insisted on using the heliostat-parabolic mirror system. The advantage of having a fixed laser head at the focus of a stationary parabolic mirror becomes much more pronounced when an Mg reduction vacuum chamber is to be installed nearby. The solar laser head pumped by a

Fresnel lens usually moves together with the whole tracking structure, an optical fiber thus becomes inevitable for the transportation of solar laser power from the laser head to the reduction chamber. Beside a lot of practical inconveniences, fiber optic transmission loss will not be avoided, which will influence negatively the final collection efficiency of the whole laser system. It is therefore very meaningful to improve the performance of the solar laser pumped through a heliostat-parabolic mirror solar energy collection and concentration system. The ultra-high power heliostat-parabolic mirror system, such as the 1 MW solar furnace of PROMES-CNRS in France, might well become a super solar laser power station in the future. homogenizer by transforming the near-Gaussian profile of the concentrated light spot at its input end into a uniform pump distribution at its output end [10]. Uniform absorbed pump distribution along the ceramic rod is achieved.

Side-pumped

Fig.2

The tracking error of the heliostat shifts the focal spot at the input face of the light guide, resulting only in a uniform reduction in power intensity at its output end. The absorbed pump power profile within the laser rod, and hence the laser power, is not significantly affected.

Compound V-shaped pump cavity

The compound V-shaped pump cavity plays an important role by coupling the pump radiation from the output end face of the light guide into the 7 mm diameter 30 mm length 0.1 at% Cr:1.0 at% Nd:YAG ceramic rod. To improve the absorbed pump distribution, the compound V-shaped pump cavity is optimized by ZEMAX © nonsequential ray-tracing code. With an entrance aperture of 21 mm × 23 mm, 24 mm depth and 9 mm separation between the output end of the guide and the rod optical axis, as shown in Fig. 3, this cavity is effective in coupling the light rays with different incidence angles into the laser rod. For example, ray 1 pass through the rod once and is bounced back by the lower plane section A of the cavity, so that double-pass absorption of the pump radiation is obtained. While ray 2 pass through the rod only once, ray 3 passes through the rod twice, due to the successive reflections from the two symmetric plane sections B. The upper plane section C of the cavity redirects the ray 4 to the laser rod so that at least one passage can be accomplished. A relatively large pump cavity is designed to allow a simultaneous lateral access to cooling water, ensuring hence a uniform thermal load along the laser rod. Both the ceramic rod and the cavity are actively cooled by water with 7 L/min flow rate. The inner wall of the whole pumping cavity is bonded with a protected silver-coated aluminum foil with 94% reflectivity.

Fig. 3

The standard solar spectra [15] for one-and-a-half air mass (AM1.5) are used as the reference data for consulting the spectral irradiance (W/m 2 /nm) at each wavelength. overlap between the absorption spectrum of the 0.1 at% Cr: 1.0 at% Nd:YAG medium [13] and the solar spectrum [15]. ZEMAX © ray-tracing code is used to both maximize the absorbed pump power and optimize the absorption profile within the rod, as shown in Fig. 4. The absorbed pump flux data from the ZEMAX © analysis is then processed by LASCAD © software. The stimulated emission cross-section of 2.38 × 10 -19 cm -1 , the fluorescence life time of 220 s [14] and a typical scattering loss of 0.002 cm -1 for the 0.1 at% Cr: 1.0 at% Nd:YAG medium are adopted in LASCAD © analysis. A concaveconcave stable laser resonator of 300 mm length, an averaged solar pump wavelength of 560 nm [13] are used in the LASCAD © analysis. Output couplers of different reflectivity, ranging from 85% to 99%, are tested individually to maximize the multimode laser power. According to different resonant cavity parameters, various input solar power/output laser power characteristics are numerically analyzed. For example, the maximum solar laser power of 34.0 W can be achieved by adopting the 98% output coupler with -2 m radius of curvature (RoC) in LASCAD © analysis.

Fig. 4

At high average laser power, even a nearly uniform gain distribution in a watercooled laser rod, as given in Fig. 4, has been shown to induce a non-parabolic heat distribution as a result of the temperature dependence of the thermal conductivity. This results in a radially dependent refractive power of the thermal lens, which has a maximum along the rod axis [16]. Nevertheless, comparing to other end-pumping schemes, the laser beam divergence of our side-pumped Cr:Nd:YAG ceramic laser is found to be significantly reduced in LASCAD © analysis, due principally to the uniform absorbed pump distributions shown both in Fig. 2 and Fig. 4.

Experimental results of the side-pumped Cr:Nd:YAG ceramic solar laser

The 7 mm diameter, 30 mm length 0.1 at% Cr: 1.0 at% Nd:YAG ceramic rod is supplied by Konoshima Chemical Co. Ltd. Japan. The optical resonator with 300 mm length is comprised of two opposing concave-concave mirrors at right angles to the optical axis of the rod, as shown in Fig. 1. The rear mirror is high reflection coated (HR, 99.98%), while the output mirror is partial reflection coated (PR, 98%). Output mirrors with RoC varying between -0.5 m and -5 m are used to test the solar laser performances.

The -2 m RoC output couplers offer the best solution. Four -2 m RoC output mirrors with 90%, 94%, 98% and 99% reflectivity are therefore chosen to study the solar input / laser output performances. Fig. 5. shows the laser output power as a function of the input solar power at the focus. We define the slope efficiency at the focus as  focus, since only the solar power at the focus of the parabolic mirror is considered in calculation.  laser is then defined as the laser slope efficiency when the combined reflection losses of the heliostat-parabolic mirror system are taken into account. Direct solar irradiance is measured simultaneously during lasing with a Kipp & Zonen CH1 pyrheliometer on a Kipp & Zonen 2AP solar tracker. It varies between 930 and 1030 W/m 2 during the period of measurement in July, 2012. Laser output power is detected by a Molectron PowerMax 500D with less than 3% measurement uncertainty. Two sliding doors and a shutter with motorized blades are used to regulate the incoming solar power from the heliostat. To achieve the maximum laser power, the shutter is totally removed. The maximum laser output power of 33.6 W is measured for -2.0 m RoC output coupler with 98% reflectivity. The slope efficiency of  focus = 2.6% is finally determined.

A linear fiber-optic array for measuring the one-dimensional laser beam intensity distribution in the near field is placed 5 mm away from the output coupler along the optical axis of the laser rod [8,10]. The 32 mm width, 128 optical fibers linear array is used to collect and transmit laser light to a Fairchild CCD 153A 512-element linear image sensor via a neutral density attenuator. This fiber-optic device has 0.25 mm core pitch resolution, so less than 2% laser beam diameter measurement error is found. This flexible fiber optic bundle has 2 m length. Outdoor solar laser beam diameter measurement is hence facilitated.

For far field laser beam profile measurement, a 10" x 10" industrial standard laser alignment thermal sensitive paper ZAP-IT @ is positioned 10 m away from the output coupler. Assuming 3 mm reading error in the thermal sensitive paper, less than 2% laser beam diameter measurement error is ensured.

The laser beam divergence θ is found by adopting the Eq. ( 1):

(1)

Where ϕ 1 = 7.0 mm and ϕ 2 = 150 mm are the measured laser beam diameters at 1/e 2 width, 5 mm and 10 m away from the output mirror respectively and L is the distance between these two points.

M 2 factor is then calculated as:

(2) where = 0.015º is the divergence of diffraction-limited Gaussian beam for =1.064 µm and ω 0 =1280 µm, as calculated by LASCAD © laser beam propagation method for the 7 mm diameter rod. For -2 m RoC output coupler, M x 2 = 27.5 ± 3% and M y 2 = 28.3 ± 3% are experimentally determined, indicating a near symmetric beam profile. Figure of merit B of 0.44 × 10 -1 W is finally calculated. Even though this value is 6.5 times lower than our previous record of 2.9 × 10 -1 W [10], it is still 6.7 times higher than that of the most recent end-pumped solar laser [9].

Comparison of the laser performances of both Cr:Nd:YAG ceramic and Nd:YAG single-crystal rod

The light guide dimensions, the laser rod diameters and the cavity profiles have already been optimized by ZEMAX © and LASCAD © numerical analysis software. For maximizing individually the laser output power from each laser medium, the 7 mm diameter Cr:Nd:YAG ceramic rod is pumped by the compound V-shaped cavity in 2012 and the 4 mm diameter Nd:YAG single-crystal rod by the modified 2D/CPC cavity in 2011 [10]. The fused silica light-guide with 16 mm × 22 mm cross-section is used for pumping the 7 mm diameter Cr:Nd:YAG ceramic rod. This light guide is only slightly thicker than the 14 mm × 22 mm cross-section light guide used for side pumping the 4 mm diameter Nd:YAG single-crystal rod. Both light guides have 140 mm length and are manufactured from same fused silica material. The cooling water is kept also at nearly the same working temperature of 20 ºC ± 2 ºC. From Fig. 5, the following conclusions can be drawn:

1. In comparison to our last year´s results with the 4 mm diameter Nd:YAG singlecrystal rod, as given by the lower dashed line in Fig. 5, there exists a general improvement in laser powers with the 7 mm diameter Cr:Nd:YAG ceramic rod. For the 98% reflectivity, -2 m RoC output coupler, for example, there is 132% laser power enhancement, as given by the upper solid line in Fig. 5. There exists also 118% improvement in slope efficiency at the focus, from 2.2% for the single-crystal rod to 2.6%

for the ceramic rod.

2. There is no reduction in threshold power due to the use of the 7 mm diameter ceramic rod. It is not easy to initiate solar laser operation with less than 390 W solar powers at the focus.

For the 0.1 at% Cr: 1.0 at% Nd: YAG ceramic rod, a typical 0.002 cm -1 scattering loss is assumed in the LASCAD © analysis. Does this ceramic rod really have such a low scattering loss, or contrarily, it should have a much higher loss of 0.004 cm -1 , as measured in Ref. [9] ? Let us here deduce the scattering loss of the ceramic rod of our laser system by using the same equation as given in Ref. [9]. The laser output power can be expressed in terms of input power and measurable quantities as listed below:

P out = (2)
In this equation, I S the saturation gain,  the cross-section area of the laser rod, R the reflectivity of output coupler,  the scattering coefficient of laser material and l the length of laser medium. The conversion factor η can be expressed as

η = η A η Q η S η B
where η A , η Q , η S and η B are the absorption efficiency, the quantum efficiency, the Stoke factor and the beam overlap efficiency, respectively. 2lα expresses the two-way loss in the resonator. At the same input solar radiation and resonator configuration, conversion factor η is a constant value if we change only the reflectivity of output coupler. Now we examine Eq. ( 2) with three variables: conversion factor η, scattering coefficient α, saturation gain I s . These variables could be determined with three simultaneous equations by changing the reflectivity of output couplers three times. Substituting measured solar input powers and laser output powers, as given in Fig. 5, with reflectivity of output couplers of 94%, 98% and 99% into Eq. ( 2), we then get the determined variables. The scattering loss of only 0.0018 cm -1 is finally found by our analysis. The calculated I S value of 2.38 kW/cm 2 also lies between the two published values [9,17]. In conclusion, the scattering loss of the 0.1 at% Cr 1.0 at% Nd:YAG ceramic rod is less than 0.002 cm -1 . It is not responsible for the low efficiency of the ceramic solarpumped lasers. By inserting the conversion factor of = 2.27% into Eq. 2, we obtain a simple expression for laser slope efficiency:

 laser = (3)
If, for example, the laser resonator has the following parameters: R = 0.98, l = 3 cm, cm -1 and = 2.27%, then the laser slope efficiency  laser = 1.49% is calculated. If only 59% combined reflectivity from both the heliostat and the parabolic mirror is taken into account, then the slope efficiency from the focus  focus = 2.53% can finally be calculated, which matches well the experimental value of 2.6% in Fig. 5.

Since there is only less than 16% spectral overlap between the solar emission spectrum and Nd:YAG absorption spectrum, the absorbed solar pump power is limited.

The compact CPC cavity is used to achieve efficient pumping along the 4 mm diameter Nd:YAG rod. ZEMAX © and LASCAD © numerical analysis has also indicated the effectiveness of the CPC scheme in attaining the maximum laser output power from the Nd:YAG rod. Since the overlap between the solar emission spectrum and Cr:Nd:YAG absorption spectrum can theoretically reach 40%, there are more absorbed solar pump power available, large diameter ceramic rod can then be used to achieve more laser output power. The compound cavity is adopted to pump the 7 mm diameter Cr:Nd:YAG ceramic rod. The rod diameter is also optimized by both ZEMAX © and LASCAD © analysis. The compound V-shaped pump cavity ensures the nearly top-hat absorbed pumped flux distribution, as shown in Fig. 4, which drastically reduces the laser beam divergence, when compared to other end-pumped configurations.

The heat load of the 7 mm diameter Cr:Nd:YAG ceramic rod within the compound cavity is also lower than that of the 4 mm diameter single-crystal Nd:YAG rod within the CPC cavity. On the one hand, the Cr:Nd:YAG rod provides enhanced laser output performance within the large compound cavity, on the other hand, the Nd:YAG rod produce also its maximum laser output power within the compact CPC cavity, so the comparison between the ceramic and single-crystal laser rods are made within different pumping cavities, where the advantages of each laser rod, and therefore the laser output performances can be fully exploited.

Side-pumping configuration provides the best heat load distribution along the rod and the temperature dependent reduction of the 1064 nm stimulated emission crosssection of Cr:Nd:YAG ceramic crystal [14] is not as significant as the case of other endpumped configurations. For this reason, there exists a general enhancement of solar laser power by side-pumping the Cr:Nd:YAG ceramic rod. Highly intense solar endpumping will inevitably raise the thermal load of the ceramic rod, creating hot pump spots along the rod, the stimulated cross-section value of Cr:Nd:YAG ceramic rod will be reduced significantly. This, in our opinion, is the main reason of the relatively low laser efficiency of Cr:Nd:YAG ceramics rod in high solar power end-pumping configuration. The UV solarization effects and the IR heating can also severely influence the laser performance of the ceramics.

Conclusions

High power and high efficiency solar-pumped lasers have a large potential for many interesting applications. The radiation coupling and homogenization capacity of the fused silica light guide is combined with the light coupling properties of the compound V-shaped cavity to provide the efficient side-pumping to the 7 mm diameter 0.1 at% Cr:

1.0 at% Nd:YAG ceramic rod. The introduction of the rectangular cross-section light guide has also ensured a more stable laser emission than other pumping schemes. There exists a general improvement of about 132% in output laser power. 2.6% slope efficiency at the focus is also reached. The laser beam brightness figure of merit is, on the one hand, more than 6 times less than that by the 4 mm diameter Nd:YAG singlecrystal rod and, on the other hand, still 6.7 times higher than that of the most recent endpumped solar laser [9]. The ceramic rod has also the scattering loss of only 0.0018 cm -1 , as deduced from the output laser powers by different output coupling ratios. The nonuniform heat load problem along the laser rod is considered as the key factor affecting the ceramic output performances of solar-pumped lasers. We would like to answer your comments one by one

Reviewer #1: I believe that the paper presents an interesting result. However, there are many significant misunderstanding and unclear points which should be considered before further review.

1. In Fig. With due respects, we think reviewer 1 has not been very careful in examining the manuscript.

In The collected solar power at the focus has been widely adopted as input solar power in many publications on solar-pumped lasers by Arashi 2. The authors mentioned 33.4W output but its collection efficiency is only 10W/m2 and a half of their previous result. This means the proposed scheme might have some difficulty in scaling up. This point should be discussed.

Once again, the reviewer must have mistakenly compared the 10W/m 2 collection efficiency with the 19.3 W/m2 value in Reference 8, which matches well with "a half of their previous result". Instead, our previous result in Ref. 10 was only 9.6W/m 2 with the same PROMES-CNRS system, so the collection efficiency is slightly enhanced by using Cr:Nd:YAG ceramic rod in 2012.

As mentioned in the manuscript, the total reflectivity of the whole solar energy collection and concentration system is only 59%. The back surface silver-coated plane mirror and parabolic mirror have more than 20 year´s service history. The collection efficiency can be largely improved by using front surface silver-coated mirrors.

3. On page 3, they claimed the demerit of using optical fiber. However, I did not find the superiority of their system for long distant transportation of many beams. This should be essential because we can not use huge mirror for obtaining MW class laser but we need to use 1000 bundles of kW laser, because no one show the feasibility of huge laser by single optical system.

We only mention some shortcomings of optical fibers for laser power transmission. We are admirers of the renewable large-scale Mg recovery scheme proposed by Professor T. Yabe with many optical fiber bundles for kW laser. Besides, we are also interested in the multi-rods solarpumped laser scheme at the focus of the 1MW solar furnace as proposed by Uzbekistan scientists, without optical fibers in this case.

4. As for the beam quality, I would like to know the size of the beam spot which might be used to calculate the divergence of the beam.

Far field laser beam divergence measurement was carried out again.

We have measured the 1/e 2 beam spot size of about 150 mm diameter, 10 meters away from the laser output coupler.

5. In the previous paper, they used CPC but now are using tapered cavity. I would like to know the reason of the change and how it changes the result.

The following text has been added to the revised manuscript:

Since there is only less than 16% spectral overlap between the solar emission spectrum and Nd:YAG absorption spectrum, the absorbed solar pump power is limited. The compact CPC cavity is used to achieve efficient pumping along the 4 mm diameter Nd:YAG rod. ZEMAX © and LASCAD © numerical analysis has also indicated the effectiveness of the CPC scheme in attaining the maximum laser output power from the Nd:YAG rod. Since the overlap between the solar emission spectrum and Cr:Nd:YAG absorption spectrum can theoretically reach 40%, there are more absorbed solar pump power available, large diameter ceramic rod can then be used to achieve more laser output power. The compound cavity is adopted to pump the 7 mm diameter Cr:Nd:YAG ceramic rod. The rod diameter is also optimized by both ZEMAX © and LASCAD © analysis. The compound V-shaped pump cavity ensures the nearly top-hat absorbed pumped flux distribution, as shown in Fig. 4, which drastically reduces the laser beam divergence, when compared to other end-pumped configurations.

The heat load of the 7 mm diameter Cr:Nd:YAG ceramic rod within the compound cavity is also lower than that of the 4 mm diameter single-crystal Nd:YAG rod within the CPC cavity. On the one hand, the Cr:Nd:YAG rod provides enhanced laser output performance within the large compound cavity, on the other hand, the Nd:YAG rod produce also its maximum laser output power within the compact CPC cavity, so the comparison between the ceramic and single-crystal laser rods are made within different pumping cavities, where the advantages of each laser rod, and therefore the laser output performances can be fully exploited.

6. On page 10, "the temperature dependent reduction of the 1064 nm stimulated emission ... [9]" but there is no description on stimulated emission in ref. [9]. The authors should not write a misunderstanding sentence.

Yes, the sentence has been changed to:

Side-pumping configuration provides the best heat load distribution along the rod and the temperature dependent reduction of the 1064 nm stimulated emission cross-section of Cr:Nd:YAG ceramic crystal [14] is not as significant as the case of end-pumped configurations.

7. The quality of ceramic laser depends on the production condition. Therefore the scattering loss changes depending on the lod. If the authors are careful enough, ref. [9] did not say the inferiority of the ceramic YAG but the ceramic they used has the high scattering coefficient. This is important because the quality changes one by one and should be improved for industrial application. Therefore they should remove the word "doubts" in abstract and introduction and so on.

Yes, the words doubt have been removed in the revised manuscript 8. Regarding the scattering coefficient, the authors mentioned on 9 that "0.004cm-1, as predicted in Ref. [9]". This is misleading because it was measured by integrated sphere but was not predicted. In addition, agreement with published value ref. [17] has no meaning because the ceramic quality is not so stable.

Yes, "0.004cm-1, as predicted in Ref. [9]" has been changed to, "0.004cm-1, as measured in Ref. [9]"

The sentence"which agrees well with the published value of 0.002 cm -1 [17]" has been removed in the revised manuscript.

Dear Reviewer 2

Many thanks for your very insightful and helpful comments.

We would like to answer your concerns one by one Reviewer #2: Solar laser attracts more attentions in research due to the potential applications in clean energy field. The authors present a comprehensive work on side-pumped CW Cr:Nd:YAG ceramic solar laser. The simulation as well as experimental results are very valuable reference for people working in this field. Nevertheless, a few issues are not clearly addressed in the manuscript.

1. Page 3, lines 39-40 describe the inner wall of the V-shaped pump cavity. What is the material of the V-shaped cavity? Why not have the pump cavity directly coated with gold or silver but bonded with a silver-coated aluminum foil?

The V-shaped cavity is bounded with silver-coated aluminum foil with 94% reflectivity by now. A gold-coated cavity will not reflect efficiently some useful solar pump power below 500 nm. Silver coating is not used due to the possible contamination with cooling water. Protected silver coating will be considered in the future version.

2. Page 6, line 44 to page 7, line 14 describe how the M2 parameter was measured, the method they used was too much simplified. Normally, the transverse beam profile at minimum five positions need to be recorded in order to fit the hyperbolic curve, so that the beam waist and far field divergence can be determined. The author only measured a very near field(5 mm) beam width and a far field beam width (1500 mm), the beam divergence calculated from these two values was not correct, therefore, the following up calculations of M2 and Figure of merit B were not reliable. In the same paragraph, the author mentioned that the diffraction-limited Gaussian beam of the same wavelength was 0.019 degree, how was this value calculated?

The following text has been added to replace the original text in the revised manuscript:

A linear fiber-optic array for measuring the one-dimensional laser beam intensity distribution in the near field is placed 5 mm away from the output coupler along the optical axis of the laser rod [8,10]. The 32 mm width, 128 optical fibers linear array is used to collect and transmit laser light to a Fairchild CCD 153A 512-element linear image sensor via a neutral density attenuator. This fiber-optic device has 0.25 mm core pitch resolution, so less than 2% laser beam diameter measurement error is found. This flexible fiber optic bundle has 2 m length. Outdoor solar laser beam diameter measurement is hence facilitated.

For far field laser beam profile measurement, a 10" x 10" industrial standard laser alignment thermal sensitive paper ZAP-IT @ is positioned 10 m away from the output coupler. Assuming 3 mm reading error in the thermal sensitive paper, less than 2% laser beam diameter measurement error is ensured.

The laser beam divergence θ is found by adopting the Eq. ( 1):

(1)

Where ϕ1 = 7.0 mm and ϕ2 = 150 mm are the measured laser beam diameters at 1/e 2 width, 5 mm and 10 m away from the output mirror respectively and L is the distance between these two points.

M 2 factor is then calculated as:

(2) where = 0.015º is the divergence of diffraction-limited Gaussian beam for 1.064 m and   1280 m, as calculated by LASCAD © laser beam propagation method for the 7mm diameter rod.

For -2 m RoC output coupler, Mx 2 = 27.5 ± 3% and My 2 = 28.3 ± 3% are experimentally determined, indicating a near symmetric beam profile. Figure of merit B of 0.44 × 10 -1 W is finally calculated. Even though this value is 6.5 times lower than our previous record of 2.9 × 10 -1 W [10], it is still 6.7 times higher than that of the most recent end-pumped solar laser [9].

3. In section 4, the authors made some comparison between Nd:YAG and Cr, Nd: YAG ceramic, the experimental conditions were very different, e. g., the size of the crystal, mode volume of the resonator and pumping scheme were all different in two cases. But the authors put the results into one figure, (Fig. 5) and made comparison of the output and efficiency. So the conclusion based on this is not convincing, because other factors which could also influence the output power and efficiency were not excluded.

The following text has been added to the revised manuscript:

Since there is only less than 16% spectral overlap between the solar emission spectrum and Nd:YAG absorption spectrum, the absorbed solar pump power is limited. The compact CPC cavity is used to achieve efficient pumping along the 4 mm diameter Nd:YAG rod. ZEMAX © and LASCAD © numerical analysis has also indicated the effectiveness of the CPC scheme in attaining the maximum laser output power from the Nd:YAG rod. Since the overlap between the solar emission spectrum and Cr:Nd:YAG absorption spectrum can theoretically reach 40%, there are more absorbed solar pump power available, large diameter ceramic rod can then be used to achieve more laser output power. The compound cavity is adopted to pump the 7 mm diameter Cr:Nd:YAG ceramic rod. The rod diameter is also optimized by both ZEMAX © and LASCAD © analysis. The compound V-shaped pump cavity ensures the nearly top-hat absorbed pumped flux distribution, as shown in Fig. 4, which drastically reduces the laser beam divergence, when compared to other end-pumped configurations.

The heat load of the 7 mm diameter Cr:Nd:YAG ceramic rod within the compound cavity is also lower than that of the 4 mm diameter single-crystal Nd:YAG rod within the CPC cavity. On the one hand, the Cr:Nd:YAG rod provides enhanced laser output performance within the large compound cavity, on the other hand, the Nd:YAG rod produce also its maximum laser output power within the compact CPC cavity, so the comparison between the ceramic and single-crystal laser rods are made within different pumping cavities, where the advantages of each laser rod, and therefore the laser output performances can be fully exploited.

4. Page 10, line 2, the scattering loss of 0.018 cm -1 , one zero was missing, the correct value should be 0.0018 cm-1.

Yes, the correct value of 0.0018cm -1 has been added to the revised manuscript.

  Cr: Nd: YAG ceramic solar laser system 2.1 PROMES-CNRS medium size solar furnace A large plane mirror with 36 segments (0.5 m × 0.5 m each) is mounted on a two-axis heliostat which tracks the sun continuously, redirecting the incoming solar radiation towards the 2 m diameter stationary parabolic mirror, as shown in Fig.1 (a). An effective collection area of 2.88 m 2 is measured from the parabolic mirror. All the mirrors are back-surface silver coated, so only 59% of incoming solar radiation is successfully focused to the focal zone, about 0.85 m away from the center of the parabolic mirror. In clear sunny days in Odeillo, more than 1.8 kW solar powers can be focused into a 15 mm diameter light spot, reaching the peak flux of 16 W/mm 2 . The laser head, as indicated in Fig.1is mounted on an automatic X-Y-Z axis mechanical support. The concentrated solar radiation at the focus is collected by the light guide with rectangular cross-section, as shown in Fig.1

Fig. 1 2. 2

 12 Fig.1

0 2 to 4 T 1 )

 241 .27 o solar divergence half-angle is assumed. The absorption spectra of 0.1 at% Cr: 1.0 at% Nd:YAG ceramics has two broad absorption bands at approximately 440 nm ( 4 A and 600 nm ( 4 A 2 to 4 T 2 )[13]. It can be seen as the superposition of the absorptions of Nd:YAG and Cr:Nd:YAG. All the above peak wavelengths and their respective absorption coefficients are added to the glass catalogue for Cr:Nd:YAG material in ZEMAX © software. In order to reduce the thermal load along the laser rod, the IR radiation which does not contribute to lasing is firstly attenuated by the light guide and then filtered by the sufficient amount of cooling water flowing through the cavity. The effective pump power of the light source takes into account about 24%

Fig. 5

 5 Fig.5

Fig. 1 (

 1 Fig.1 (a) PROMES-CNRS 2 m diameter solar concentrator with the Cr:Nd:YAG laser resonant cavity. (b) The mechanical components of the laser resonator.

Fig. 2

 2 Fig.2 Double-stage light guide / compound V-shaped pump cavity for the Cr:Nd:YAG laser rod.

Fig. 3 Fig. 4

 34 Fig.3 Cross-sectional view of the compound V-shaped pump cavity

Fig. 5

 5 Fig.5 Cr:Nd:YAG and Nd:YAG laser output powers versus input solar power at the focus.

  Fig. 1

Fig. 5 ,

 5 we are comparing ceramic Cr:Nd:YAG and Nd:YAG crystal data, the solar energy collection concentration system in PROMES-CNRS has the same size and reflectivity. The input solar irradiances are only slightly different for the two experiments in 2011 and 2012 respectively. The side-pumped Nd:YAG laser output performances, published by Optics and Laser Technology in 2012 (Ref.10), are compared to the present results. Also from the comments of " the incident solar power of the two experiments are almost 4 times different", the reviewer might have mistakenly considered our Fresnel lens end-pumped solar laser published by Optics Express in 2011(Ref. 8) for his comparison. We did not compare this end-pumped result with the present results in Fig. 5.

Corporation Side-pumped continuous-wave Cr:Nd:YAG ceramic solar laser D. Liang 1,* , J. Almeida 1 and E. Guillot 2

	1 CEFITEC, Departamento de Física, FCT, Universidade Nova de Lisboa, 2829-516,
	Campus de Caparica, Portugal
	2 PROMES-CNRS, 7 rue du Four Solaire, 66120, Font Romeu, Odeillo, France
	*Corresponding author: dl@.fct.unl.pt

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  5, they are comparing ceramic Cr.Nd YAG and Nd YAG crystal data but two conditions are completely different. The size is different and input source is different. I doubt this comparison and it is not scientific. In addition, "Input solar power" in horizontal axis is misleading. They should put real incident solar power on the collecting mirror. The incident solar power of two experiments is almost 4 times different.

Acknowledgments

This research project (PTDC/FIS/103599/2008) was funded by the Science and Technology Foundation of Portuguese Ministry of Science, Technology and Higher Education (FCT-MCTES).

Financial support by the Access to Research Infrastructures activity in the 7th Framework

Program of the EU (SFERA Grant Agreement n. 228296) is gratefully acknowledged. We would like to express our thanks to Dr. Takashi Ito from Baikowski Japan Co. Ltd. for the helpful discussions about the scattering properties of the 0.1% Cr: 1.0 % Nd:YAG ceramic rod.


Basile Pottier

Ludovic Bellon

Carlos A Plata

Emmanuel Trizac

David Guéry-Odelin

Accelerating the heat diffusion: Fast thermal relaxation of a microcantilever

In most systems, thermal diffusion is intrinsically slow with respect to mechanical relaxation. We devise here a generic approach to accelerate the relaxation of the temperature field of a 1D object, in order to beat the mechanical time scales. This approach is applied to a micro-meter sized silicon cantilever, locally heated by a laser beam. A tailored driving protocol for the laser power is derived to quickly reach the thermal stationary state. The model is implemented experimentally yielding a significant acceleration of the thermal relaxation, up to a factor 30. An excellent agreement with the theoretical predictions is reported. This strategy allows a thermal steady state to be reached significantly faster than the natural mechanical relaxation.

I. INTRODUCTION

In condensed matter, the temperature fields evolve in general on much longer time scales than their mechanical counterparts (such as stress or strain). Indeed, combining the highest thermal diffusivity in solids [START_REF] Haynes | CRC Handbook of Chemistry and Physics[END_REF] (D ∼ 10 -4 m 2 /s) and the order of magnitude of the speed of sound [START_REF] Haynes | CRC Handbook of Chemistry and Physics[END_REF] (c ∼ [10 3 -10 4 ] m/s), one defines a typical length scale l th = D/c ∼ [10 -100] nm beyond which temperature is a slow phenomenon. Most systems imply larger length scales, thus a slow temperature evolution. In some devices or experiments, or for proof-of-concept demonstrations, it can however be desirable to accelerate the heat diffusion so as to impose a temperature variation on time scales equivalent, or shorter, than those of the mechanical response of the system.

One example is atomic force microscopy (AFM) [START_REF] Binnig | Atomic force microscope[END_REF], where a sharp tip attached to a cantilever scans a sample to map its topography and potentially more local properties of its surface. Most commonly, the measure of the interaction force through the deflection of the cantilever is performed with a laser beam [START_REF] Meyer | Novel optical approach to atomic force microscopy[END_REF], hence the photothermal response and thermal stability of AFM-sensors have been extensively studied almost since the origin of AFM [START_REF] Marti | Mechanical and thermal effects of laser irradiation on force microscope cantilevers[END_REF][START_REF] Allegrini | Laser thermal effects on atomic force microscope cantilevers[END_REF]. Photothermal excitation has been used in vacuum, air, or fluid, aiming at driving the cantilever efficiently while avoiding overheating of the system [START_REF] Mccarthy | Enhanced raman scattering for temperature measurement of a laser-heated atomic force microscope tip[END_REF][START_REF] Ramos | Photothermal excitation of microcantilevers in liquids[END_REF][START_REF] Lee | Thermal conduction from microcantilever heaters in partial vacuum[END_REF][START_REF] Milner | Heating of an Atomic Force Microscope tip by femtosecond laser pulses[END_REF][START_REF] Kiracofe | High efficiency laser photothermal excitation of microcantilever vibrations in air and liquids[END_REF][START_REF] Bircher | Photothermal excitation of microcantilevers in liquid: effect of the excitation laser position on temperature and vibrational amplitude[END_REF]. Let us compare the relaxation time τ = L 2 /D of the temperature field along the cantilever length L, to the period of oscillation of the first resonance T 0 6.4L 2 /(cH) [START_REF] García | Dynamic atomic force microscopy methods[END_REF], with H the thickness of the cantilever: T 0 /τ 6.4 l th /H = * ludovic.bellon@ens-lyon.fr 65 nm/H, where the numerical application has been done for silicon [13]. With H typically in the few µm range, thermal diffusion is thus much slower than the oscillation period of the system, limiting the efficiency of photothermal excitation, or the speed of operation of scanning thermal microscopy [START_REF] Chen | Microscale spatially resolved thermal response of si nanotip to laser irradiation[END_REF][START_REF] Gomès | Scanning thermal microscopy: A review[END_REF][START_REF] Sandoval | Resonance frequency shift of strongly heated microcantilevers[END_REF].

Another example where fast temperature variations could be desirable is in stochastic thermodynamics experiments on micro-mechanical systems [START_REF] Sekimoto | Stochastic Energetics[END_REF][START_REF] Rademacher | Nonequilibrium control of thermal and mechanical changes in a levitated system[END_REF] (optically trapped particles, micro-cantilevers, MEMSmicro-electromechanical systems). When constructing a stochastic Carnot heat engine for instance [START_REF] Schmiedl | Efficiency at maximum power: An analytically solvable model for stochastic heat engines[END_REF][START_REF] Toyabe | Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality[END_REF][START_REF] Esposito | Efficiency at maximum power of low-dissipation carnot engines[END_REF][START_REF] Blickle | Realization of a micrometresized stochastic heat engine[END_REF][START_REF] Esposito | Stochastically driven single-level quantum dot: A nanoscale finite-time thermodynamic machine and its various operational modes[END_REF][START_REF] Holubec | An exactly solvable model of a stochastic heat engine: optimization of power, power fluctuations and efficiency[END_REF][START_REF] Martínez | Adiabatic processes realized with a trapped brownian particle[END_REF][START_REF] Martínez | Brownian carnot engine[END_REF][START_REF] Plata | Building an irreversible carnot-like heat engine with an overdamped harmonic oscillator[END_REF][START_REF] Guéry-Odelin | Driving rapidly while remaining in control: classical shortcuts from hamiltonian to stochastic dynamics[END_REF], one needs to perform adiabatic heating or cooling, that is to say change the system temperature much faster than the time scale τ r corresponding to the heat exchanges with the thermostat. τ r is equivalent to the mechanical relaxation time, which writes for a cantilever τ r = QT 0 /π, with Q the quality factor of the resonance [START_REF] Sekimoto | Stochastic Energetics[END_REF][START_REF] Dago | Information and thermodynamics: Fast and precise approach to landauer's bound in an underdamped micromechanical oscillator[END_REF][START_REF] Aurell | Refined second law of thermodynamics for fast random processes[END_REF][START_REF] Seifert | Stochastic thermodynamics, fluctuation theorems and molecular machines[END_REF][START_REF] Jarzynski | Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale[END_REF][START_REF] Dago | Dynamics of information erasure and extension of landauer's bound to fast processes[END_REF][START_REF] Dago | Virtual double-well potential for an underdamped oscillator created by a feedback loop[END_REF]. The situation is thus somewhat more favorable than previously for underdamped systems, but requires quality factors larger than a few hundred to start matching the natural thermal diffusion and mechanical relaxation timescales: τ r /τ Q × 20 nm/H. Here again, an acceleration of the temperature dynamics would be welcome to perform efficient adiabatic transformations.

The motivation of this work is to speed up the natural thermal relaxation, and we focus in this article on the case of a cantilever, in relation to the previously mentioned examples. Engineered accelerated dynamics and shortcuts are protocols of the 'Shortcut to Adiabaticity' type [35]. This class of ideas emerged in the quantum realm [START_REF] Chen | Fast Optimal Frictionless Atom Cooling in Harmonic Traps: Shortcut to Adiabaticity[END_REF][START_REF] Guéry-Odelin | Shortcuts to adiabaticity: Concepts, methods, and applications[END_REF]. Yet, a number of techniques belonging to this family of accelerating methods has been successfully exported to classical and stochastic dynamics [START_REF] Guéry-Odelin | Driving rapidly while remaining in control: classical shortcuts from hamiltonian to stochastic dynamics[END_REF][START_REF] Guéry-Odelin | Nonequilibrium Solutions of the Boltzmann Equation under the Action of an External Force[END_REF][START_REF] Martínez | Engineered swift equilibration of a Brownian particle[END_REF][START_REF] Patra | Shortcuts to adiabaticity using flow fields[END_REF][START_REF] Li | Shortcuts to isothermality and nonequilibrium work relations[END_REF]. Some of these shortcuts are of inverse engineering type, requiring a careful monitoring of the time dependence of some control parameters, to impose a prescribed evolution of interest [START_REF] Martínez | Engineered swift equilibration of a Brownian particle[END_REF]. Such accelerating protocols have also been used fruitfully in systems such as cranes [START_REF] González-Resines | Invariantbased inverse engineering of crane control parameters[END_REF] and capacitors [START_REF] Palaia | Charged systems in, out of, and driven to equilibrium: from nanocapacitors to cement[END_REF].

In the following, we address the question of the temperature field control, so far untouched. The goal is to drive the thermal relaxation of a micro-cantilever in a time much smaller than its natural relaxation time. First, we model the thermal relaxation and derive theoretically the accelerating protocols allowing the reduction of the relaxation time of the cantilever to reach thermal steady state. Second, the aforementioned protocols are experimentally implemented in a silicon cantilever. The efficacy of the driving is patent when comparing to a direct relaxation process. Moreover, experimental results are in full agreement with the predictions. Finally, we summarize the conclusions of our study and mention future perspectives.

II. THEORY

Our objective is to accelerate the relaxation towards a non-equilibrium steady state of a cantilever with a tunable punctual and well-localized heat source. To this end, we first identify the different eigenmodes (and corresponding timescales) of the heat equation that governs the dynamics of the system. We subsequently design the rate at which the deposit of heat should be carried out so as to cancel the contribution of the low frequency eigenmodes that slow down the relaxation. This original trick enables one to benefit from a fast relaxation with the cancelation of just a few modes.

A. Temperature rise of a cantilever irradiated by a laser beam

We gather here the main ingredients useful for a quantitative description of the temperature field of a cantilever irradiated by a laser beam. As the cantilever is placed in vacuum, no heat transfer can occur by convection with its surroundings. Neglecting thermal radiation, the only possible heat transfer mechanism is thus thermal conduction through the cantilever. Considering a cantilever having a length L much larger than its transverse dimensions, the temperature θ may be assumed homogeneous across the cross section. Therefore, θ depends only on the longitudinal coordinate x ∈ [0, L] and time t, its dynamics being described by the one-dimensional heat diffusion equation

ρc p ∂θ(x, t) ∂t = λ ∂ 2 θ(x, t) ∂x 2 + q(x, t), (1)
where ρ is the density, c p is the heat capacity, λ is the thermal conductivity (all of them assumed constant), and q is the heat source/sink density. In vacuum, all cantilever surfaces are assumed to be thermally insulated, except at the location x 0 irradiated by the focused laser. We model the heating effect of the focused beam by the source term

q(x, t) = aP (t) S δ(x -x 0 ), (2)
where a is the fraction of light absorbed by the cantilever, P (t) is the incident power of the laser beam which can be manipulated at wish, S is the cantilever cross section area, and δ Dirac's distribution. Furthermore, we assume the following boundary conditions

θ(0, t) = 0, (3a) ∂θ ∂x (L, t) = 0. (3b)
These conditions reflect respectively that the cantilever is in contact with the macroscopic chip acting as a thermostat at its clamp x = 0 (3a), and is isolated on its free end x = L (3b). Note that, for the sake of simplicity, θ is not the absolute temperature, as we define the origin by the temperature of the chip.

B. Stationary profile θs

The stationary temperature profile θ s (x) associated to a given power of the laser P f is obtained solving Eq. ( 1) where ∂θ/∂t = 0, with the boundary conditions Eq. (3). We get

θ s (x) θ m s = x/x 0 , 0 < x < x 0 , 1, x 0 < x < L, (4)
where θ m s = aP f x 0 /(Sλ) corresponds to the maximum temperature elevation. Between the chip and the laser spot (x = x 0 ), the stationary temperature increases linearly with the position x, while beyond the laser spot, it remains constant and equals θ m s . From now on, we focus the analysis on the relaxation between two stationary states when driving the power P (t) from a constant initial value P (t < 0) = P 0 to its constant final value P (t > t f ) = P f . Note that even if P (t) is constant after the final time t f , the temperature profile continues to evolve towards its final asymptotic stationary shape. Since the heat diffusion equation (Eq. 1) is linear, we make the simplifying assumption that P 0 = 0: without loss of generality, P (t) or P f represent the laser power excess with respect to P 0 , and θ(x, t) or θ s (x) stand for the temperature excess with respect to the initial stationary temperature profile.

C. Transient solution through expansion

The general solution of Eq. ( 1) can be expressed carrying out an expansion in eigenfunctions ψ n (x) [START_REF] Xiong | Using hilbert transform and classical chains to simulate quantum walks[END_REF][START_REF] Krivtsov | The ballistic heat equation for a onedimensional harmonic crystal[END_REF], each satisfying the associated homogeneous problem ∂ 2

xψ n (x) = -k 2 n ψ n (x) with ψ n (0) = 0 and ψ n (L) = 0. Solving the homogeneous problem leads to

ψ n (x) = √ 2 sin (k n x) , x = x/L (5)
with the eigenvalues k n = (n -1/2)π, n ∈ N * . The temperature profile can be expressed as

θ(x, t) = θ s (x) + θ m s ∞ n=1 C n ( t)ψ n (x), (6)
where the components C n ( t) depend solely on t = t/τ , with τ = ρc p L 2 /λ, the characteristic diffusion time of the problem. These components are obtained projecting Eq. ( 1) on the eigenmodes basis ψ n (x); they read as

C n ( t) = ψ n (x 0 ) x0 t 0 F (sτ ) -1 e k 2 n s ds - 1 k 2 n e -k 2 n t, (7)
with x0 = x 0 /L, and F the driving function defined as

F (t) = P (t) P f . (8)
The temperature solution Eq. ( 6) is completely general for arbitrary F (t), even though here F (t < 0) = 0 and F (t > t f ) = 1 by construction. For t > t f , the bracket term in Eq. ( 7) is constant, so each component in the expansion decays exponentially with its own decaying rate. Specifically, the n-th component has a rate equal to k 2 n , which increases with the mode number n as (2n -1) 2 : the second mode decays k 2 2 /k 2 1 = 9 times faster than the fundamental mode, the third mode decays k 2 3 /k 2 1 = 25 times faster than the fundamental one, etc. The higher the mode n, the less time it takes for that component of the temperature to relax.

D. Accelerating the dynamics

We explain hereafter how the driving function can be designed to remove the contribution of the slow relaxation modes, thereby speeding up the relaxation dynamics in a given arbitrary small (chosen) time lapse t f . Our strategy is somewhat similar to that of Ref. 46 on a different thermal acceleration problem. For this purpose, we use an ansatz for F (t) that involves as many parameters as modes to be canceled. Let us assume that we want to cancel the N first modes. We can take the following convenient polynomial ansatz

F (t) = N m=1 γ m (t/t f ) m-1 , 0 < t < t f . (9)
Canceling the N first modes corresponds to impose C n ( tf ) = 0 for n = 1 to N , where tf = t f /τ . This

implies for n ∈ [1, N ] N m=1 γ m tf 0 s tf m-1 e k 2 n s ds = e k 2 n tf k 2 n ( 10
)
which is a system of N linear equations for N unknown variables γ n , analytically solvable. When t f is reached, we change the laser power abruptly to P f (i.e. F (t > t f ) = 1) and the relaxation will occur only through the modes higher than the N -th one (since we already force the cancelation of the first N modes). Examining Eq. ( 10), the coefficients γ n depend only on the normalized protocol duration tf and the number of canceled modes N ; the protocol to apply thus does not depend on the laser beam position x0 . For the sake of concreteness, we illustrate our protocols with x0 = 0.95 (we exemplify the independence of the protocol on x0 by working out the solution for x0 = 0.5 in Appendix B). If one wants only to cancel the relaxation of the fundamental mode (N = 1), the function F (t) to apply during the protocol is constant and equals 1/(1 -e -k 2 1 tf ); for tf = 0.1, we have F (t) = 4.57. In Fig. 1, we show the functions F (t) that allow the cancelation of up to the third mode for a protocol duration tf = 1/10 and tf = 1/30. Further details on the computation and properties of F (t) are given in Appendix A. The needed range of F (t) (thus the laser power) increases with both the number of canceled modes N and the acceleration factor τ /t f . Note that for N > 1, F (t) requires both positive and negative values. In practice, the cantilever can only absorb the power from the laser beam. However, P (t) corresponds to the power excess with respect to the initial value P 0 , so that the protocol will be valid as long as P 0 is larger than -min[P (t)]. A priori, this constraint implies a limitation for our protocols. However, we prove below that our method succeeds in accelerating thermal relaxation substantially in a wide variety of experimentally realizable situations.

III. EXPERIMENTAL RESULTS

A. Experimental setup

For the experiments presented below, a silicon cantilever placed in vacuum is heated with a laser and the evolution of its temperature at various positions x along its length L is measured.

At a thickness of a few micrometers, silicon is semitransparent for visible light, the light experiences multiple reflections within the thickness of the cantilever, which acts as a lossy Fabry-Perot [START_REF] Pottier | Thermo-optical bistability in silicon micro-cantilevers[END_REF]. As a result, the fraction of light reflected R depends on the thickness and the silicon refractive index, both varying with temperature θ. By measuring the local change in reflectivity, one can thus infer the local temperature variation with respect to the reference temperature field. For small variations, we have ∆R(x)/R(x) = βθ(x), where the sensitivity β depends on the local properties (refractive index and thickness). The calibration procedure for β is detailed in Appendix C.

The experimental setup comprises two laser beams (Fig. 2): (i) a heating laser beam (at 532 nm) of variable power P (t) focused at a fixed position close to the cantilever free end (x 0 = 0.95L) that serves to heat the cantilever, and (ii) a probe laser beam (at 633 nm) of con-stant power (350 µW) that allows us to measure the local temperature rise taking benefit of the thermo-optical effect described above. The radius of the heating and probe beam at the cantilever surface are respectively 14 µm and 7 µm. The power of the heating beam is modulated with an acousto-optic modulator, with a rise time of 100 ns. Three photodiodes P 1, P 2, and P 3 measure the incident heating beam power P (t) and the reflectivity R of the probe beam respectively, as shown in Fig. 2. By tilting the probe laser, one scans the cantilever length facilitating temperature measurements at various locations x.

During the experiment, the cantilever is placed in vacuum at 2 × 10 -2 mBar. At this pressure level, the contribution of convective heat transfer is negligible compared to thermal conduction [START_REF] Lee | Thermal conduction from microcantilever heaters in partial vacuum[END_REF]. All the measurements are performed using a single raw silicon cantilever OCTO-1000D from Micromotive, with the following size: length L = 1 mm, width W = 90 µm and thickness H = 5 µm.

To record the spatial dependence of the temperature profile, we choose 20 evenly spaced positions x of the probe beam along the cantilever. At each position, we measure the time evolution of the temperature θ(t) induced by the applied incident power P (t) of the heating beam. To increase the signal-to-noise ratio, the displayed temperature variations are obtained averaging over 700 heating procedures at each x. The spatial resolution is approximately the size of the laser spot, 7 µm, hence smaller that 1 % of the cantilever length.

B. Relaxation for a step function (F (t) = 1)

Before testing the performance of the accelerating protocols worked out in section II, we first present the results obtained imposing a jump in power, P (t) = P f for t > 0. The analysis of the temperature relaxation allows measuring the characteristic diffusion time τ needed to determine the function F (t), to later successfully accelerate the dynamic given a protocol duration t f . Moreover, this is the simplest reference that can be thought of and it defines the timescale that we would like to beat.

In Fig. 3 and in the movies available as ancillary files [START_REF]tf=0.1tau-N=X.mov, with X=1 to 4, compare the time evolution of the temperature profile[END_REF], we display the cantilever temperature rise measured imposing a sudden increase in the power of P f = 5.6 µW. The heating laser spot is focused close to the free end of the cantilever at x 0 = 0.95L. The convergence of the temperature profile towards the expected linear stationary profile given by Eq. ( 4) confirms that for the temperature variations explored, the silicon conductivity λ can be assumed constant. The maximum temperature elevation is θ m s = 38 mK, this value is consistent with the theoretical one θ m s = aP f x 0 /(W Hλ) with λ = 156 W m -1 K -1 [START_REF] Masolin | Thermo-mechanical and fracture properties in single-crystal silicon[END_REF] and a = 0.5, an absorption coefficient also consistent with the H = 5 µm thickness of the cantilever [START_REF] Pottier | Thermo-optical bistability in silicon micro-cantilevers[END_REF]. As time goes on, the temperature profile converges towards the expected linear stationary profile given by Eq. ( 4). Graph (c) shows the normalized evolution of temperature at the positions x = 0.3, 0.67 and 0.95, corresponding to the vertical dashed lines in panel a). All experimental data are perfectly described by the model of Eq. ( 6), using the characteristic time τ = 14.2 ms.

In the case of a step function (F (t) = 1), the temperature is predicted by Eq. ( 6) where the components C n ( t) are reduced to

C n ( t) = - ψ n (x 0 ) x0 1 k 2 n e -k 2 n t. (11)
Using Eq. ( 11), all experimental data are properly described by theory for a characteristic time τ =14.2 ms.

From the thermal diffusivity of bulk silicon given in the literature [START_REF] Masolin | Thermo-mechanical and fracture properties in single-crystal silicon[END_REF], D = λ/ρc p = 86 mm 2 /s, we expect τ = L 2 /D = 11.6 ms. The 18% difference could be explained by the reduced diffusivity of silicon due to the phonon confinement effect. Indeed, for a 5 µm thick film, the silicon conductivity is expected to be approximately 15% smaller than the value for bulk silicon [START_REF] Wang | Computational Study of In-Plane Phonon Transport in Si Thin Films[END_REF].

C. Relaxation for an acceleration factor τ /t f = 10

In Fig. 4, and in the movies available as ancillary files [START_REF]tf=0.1tau-N=X.mov, with X=1 to 4, compare the time evolution of the temperature profile[END_REF], we display the evolution of the measured cantilever temperature when applying the driving function F (t) given by Eq. ( 9) for a protocol duration t f = 1.4 ms, corresponding to tf = 0.1. We test our protocol varying the number of canceled modes N from 1 up to 4. Note that the imposed functions F (t) correspond to the ones already presented in Fig. 1 (left panel). In order to be able to impose the negative variations in F (t) (for N > 1), we choose a reference incident power of P 0 = 210 µW. As the number of canceled modes N increases, the temperature at the end of the protocol (t = t f ) clearly converges towards the stationary profile θ s (x) (Fig. 5). Contrary to the step function (Fig. 3), the temperature takes intermediate values higher than the stationary target θ s (x). The maximum transient temperature is obtained for N = 4 and corresponds to θ = 4.7 × θ m s at x = 0.95L. For each protocol, we compare the experiment to theory computing the temperature from Eq. ( 6) limiting the infinite sum to 60 terms; all experimental data remarkably coincide with theory.

FIG. 5. Relative temperature difference from the targeted stationary profile θs(x) at the end of the protocol t = t f = 0.1τ . As the number of canceled modes N increases, the temperature profile converges towards the stationary profile. For N = 4, the relative temperature difference remains below 10% along the cantilever length. 4. Transient temperature profiles measured when applying the protocol function F (t) given by Eq. ( 9) with a protocol duration t f = 0.1τ designed to cancel the N first thermal modes. For each tested protocol (N = 1 up to N = 4) we display (a) the temperature profiles, (b) the imposed power P (t), and (c) the temperature variations with time at the positions x = 0.3, 0.67 and 0.95. All experimental data coincide with the expected theoretical temperature variations. FIG. 6. Comparison of the temperature convergence towards the stationary profile θs when heating the cantilever with a step function power (black) or with the accelerating protocol function of Eq. ( 9) for N = 4 (orange). The red shaded area corresponds to the protocol duration, t f = 0.1τ = 1.4 ms.

(c) (b) (a) (b) (a) (c) (b) (a) (c) (b) (a) (c) FIG.
To highlight the success of our tailored acceleration, we compare in Fig. 6 the difference of convergence towards the stationary profile θ s between the step function power (displayed in Fig. 3) and the accelerating protocol for N = 4 (displayed in the last panel of Fig. 4). Using the step function, the duration needed to converge within 2% (at the positions x = 0.3, 0.67 and 0.95) is 25 ms (1.8τ ), while in the accelerating protocol this milestone is reached at the end of the program t f = 1.4 ms (0.1τ ), beating remarkably the natural relaxation time. Actually, having gotten rid of the first four modes, the fifth and higher orders do remain and are responsible for a slight mismatch with respect to the target profile. They decay at least 9 2 = 81 times faster than the power stepforcing relaxation, with time scales smaller or equal to τ /(4.5π) 2 0.07 ms, much smaller than t f itself. Hence, since the characteristic evolution timescale (after t f ) is k -2 N +1 τ , high values of N are needed for extremely low values of t f /τ only.

D. Relaxation for an acceleration factor τ /t f = 30

In Fig. 7(a), we compare the temperature convergence towards the stationary state varying N for a shorter protocol duration of 0.47 ms corresponding to τ /t f = 30. The driving functions applied, F (t), correspond to the ones displayed in Fig. 1 (right panel). To be able to impose the large variations in F (t), a reduced target power change P f = 0.6 µW, implying a maximum temperature elevation of θ m s = 5 mK, is considered. To maintain the signal-to-noise ratio, each curve displayed is now obtained averaging over 3000 measurements. As observed previously for the slower acceleration of t f = τ /10, the temperature variations during the protocol duration (see inset) increase with the number of canceled modes N . Note that the transient temperature variations are now much larger and can take negative values.

In that case, the temperature at the end of the protocol is notably different from the stationary value θ s . The relative difference (θ(t f ) -θ s )/θ s (measured at x = 0.95) is respectively 100%, -150% and 170% for N increasing from 1 to 3. It may seem counter-intuitive that the difference from the target value obtained at the end of the protocol increases with the number of canceled modes. We recall that our protocol allows converging towards the stationary profile at the end of the protocol only in the limit N → ∞. For a finite value N the use of the protocol guarantees that starting from t f only the modes higher than N will relax. Because the rate of relaxation (equal to k 2 n τ ) increases with the mode number n, increasing N should allow a faster convergence towards the stationary profile. It cannot be guaranteed that the coefficients of the modes higher than the N -th one take small values at t = t f , but its relaxation is ensured to be exponential, faster and faster for increasing n. Numerical resolution of the heat equation with our protocols are provided in Appendix D, quantifying the limitation caused by these modes surviving after t f . In Fig. 7(b), we verify that the temperature starting from t f relaxes faster as N increases, as predicted. For N = 3 for example, the relaxation times are at most τ /(3.5π 2 ) 0.12 ms, smaller than t f itself, and in agreement with the observation (green curve).

IV. DISCUSSION AND CONCLUSIONS

In this work, we tailor in time the power of a localized irradiating laser, in order to speed up the thermal relaxation of a micro-cantilever in a duration much smaller than its natural relaxation time. The protocol duration can be reduced, in principle, as much as desired. In practice however, a smaller duration unavoidably translates into a larger dynamics of the heating/cooling power (minimum and maximum power with respect to final value, faster time control of the imposed drive), and of the transient values of the temperature field. Experimentally, acceleration factors up to a factor 30 are demonstrated.

The theoretical approach relies on canceling the first N eigenmodes of the thermal diffusion equation in a finite time. Of course, the system is getting closer to the exact steady states as more N are considered. Nevertheless, we prove that for an experimentally accessible range of parameters, relatively small values of N may suffice, N = 4 providing a final state already very close to the stationary one. We emphasize that such a strategy to speed up a given dynamics described by partial differential equations is generic, and can be applied to a wide variety of mesoscopic systems [START_REF] Palaia | Charged systems in, out of, and driven to equilibrium: from nanocapacitors to cement[END_REF]. It complements the methods that have been recently developed in stochastic thermodynamics to speed up a relaxation process [START_REF] Guéry-Odelin | Driving rapidly while remaining in control: classical shortcuts from hamiltonian to stochastic dynamics[END_REF]. Our approach requires that local equilibrium holds at all times, which allows us to define the temperature field and ensure the validity of Eq. ( 1). The specific case worked out here also requires that a 1D description of the system is valid, thus that the accelerated dynamics is still slow with respect to the transverse directions of the cantilever. With a width to length ratio of W/L = 0.09 in our experiments, the transverse relaxation time is (W/L) 2 τ = 8.10 -3 τ , smaller than the smallest t f we probe, but it could be an issue for more ambitious accelerations ! Some driving protocols require a heat sink (negative power), which is technically impossible with laser absorption. From the experimental point of view, this represents a technical problem that is bypassed by choosing a heated initial state. This trick allows the redefinition of the driving function with an offset that gives access to effective negative powers. With our notation, P 0 + P (t) needs to remain positive. Our solution is proven to work in all the considered processes.

Let us point out that our approach is quite different from classic feedback control methods. Those would typically rely on measuring the temperature in one position, say x 0 , and adjusting the heating power to reach the setpoint, θ m s . A perfect feedback loop would therefore achieve θ(x 0 , t > 0) = θ m s , acting as an effective boundary condition (a temperature step at the specific point x 0 ) for the heat equation [START_REF] Haynes | CRC Handbook of Chemistry and Physics[END_REF]. The typical time scale to reach the stationary state would then be x 2 0 /(π 2 D) (see Appendix E), close to one fourth of the power step slowest decay time 4L 2 /(π 2 D): only a 4 times acceleration of the global dynamics would consequently be achieved. Note that reaching the stationary value in one point is very different from reaching the final profile at every position. As a further illustration, we superimpose in the movies available as ancillary files [START_REF]tf=0.1tau-N=X.mov, with X=1 to 4, compare the time evolution of the temperature profile[END_REF] the time evolution of the temperature field in response to a power step, to a temperature step (perfect feedback loop), and to our protocols. The acceleration provided by our strategy is clear, at the expense of noticeable local temperature oscillations and overshoots.

The order of magnitude of the temperature jumps that we experimentally perform is modest: 38 mK for a 10 times acceleration, and 5 mK for a 30 times acceleration. This choice hinges upon a practical consideration: linearity and dynamic range. The thermal conductivity of silicon depends on temperature in a noticeable way (roughly as λ ∝ 1/T [START_REF] Masolin | Thermo-mechanical and fracture properties in single-crystal silicon[END_REF]), so that we would like to avoid large temperature excursions to match the linearity hypotheses of the model. Moreover, we need an initial power P 0 to allow negative values for P (t), hence the initial temperature profile is actually not flat. For the set of data presented in Figs. 3 to 6, the maximum temperature due to P 0 = 210 µW and the 350 µW of the red laser probe beam is 4 K, corresponding to a variation of λ below 1.3%. Larger P 0 , allowing larger P f , would imply non uniform values for λ, again deviating from the model. Our goal in this article is to demonstrate the excellent agreement between theory and experiment, so we restrict ourselves to small temperature jumps. We also test larger steps corresponding to θ m s = 1 K, and as illustrated in Fig. 8, the protocols work fine with only tiny adjustments to the parameter τ to compensate for emerging non-linearities. In this case, the maximum temperature due to P 0 = 400 µW and the 600 µW of the red laser probe beam is 50 K, corresponding to a non-uniformity of λ of 16 %.

Larger jumps would require the simple form of the dynamic evolution that facilitates the theoretical derivation of the driving protocol to be reconsidered. The introduction of linear corrections or even non-linearities in λ constitutes a challenging future perspective. One way to partially avoid the need for an extended power range is FIG. 8. Transient temperature profiles measured when applying a step function and the accelerating protocol (N = 1 up to N = 3) for a maximum temperature elevation θ m s = 1 K. The small deviation from theory is attributed to the thermal conductivity dependence of silicon with temperature: as the initial profile corresponds to a maximum temperature rise of 50 K, λ is up to 16 % smaller near the cantilever's free end.

the freedom gained when the laser position x 0 (t) is no longer fixed but controlled in time: it would allow the required amount of heat to be deposited at each location of the cantilever, bypassing the slow diffusion process to reach the stationary state of the temperature field.

As a final consideration, let us rewind to our initial motivation: get closer or beat the time scales of the mechanical system. With an acceleration of a factor 30, the ratio of the mechanical and thermal time writes T 0 /t f = 2 µm/H. For our H = 5 µm thick cantilever, we are thus able to stabilize the temperature field in only 2.5 oscillations, and we would reach half an oscillation with H = 1 µm. As for the mechanical relaxation time τ r which is Q/π times longer than the period T 0 , with a quality factor of the order of Q = 3000 in our experiment in vacuum, we actually reach thermal steady state in τ r /400. Such an acceleration even allows stabilizing temperature faster than the "equilibration" of the first 12 mechanical resonant modes of the cantilever. Our strategy to accelerate the heat diffusion thus achieves its goal, and could be useful in numerous applications once we reach a meaningful temperature step amplitude.

The data that support the findings of this study are openly available in Zenodo [START_REF] Pottier | Dataset for[END_REF]. Since the cantilever thickness slightly varies along its length, the sensitivity coefficient β depends on the position x. To calibrate the function β(x), we impose a uniform temperature profile (positioning the heating beam onto the chip) and we measure the induced change in reflectivity ∆R(x) at each location x. At the same time, we measure the shift of the mechanical resonance frequencies by analyzing the thermal noise driven fluctuations of position of the reflected probe This frequency shift leads to a calibrated measurement of the imposed temperature change [START_REF] Sandoval | Resonance frequency shift of strongly heated microcantilevers[END_REF][START_REF] Pottier | Silicon cantilevers locally heated from 300k up to the melting point: Temperature profile measurement from their resonances frequency shift[END_REF]. The measured coefficients β to convert the reflectivity into temperature at all probed positions x displayed in FIG. 11. Calibration of the sensitivity coefficient β. This quantity makes it possible to convert the variation of reflectivity into temperature change, θ(x) = β -1 (x)∆R(x)/R(x). The relatively large range of dispersion for the sensitivity β is mainly due to variations of the thickness along the cantilever length.

Appendix D: Transient temperature during the protocol

As illustrated in Figs. 4 and7, the temperature can present overshoots and undershoots quite far from the target during (and after) the protocol. We plot in Fig. 12 the range of temperature explored according to the numerical resolution of the heat equation for the drivings defined in Appendix A. Transient temperatures can be orders of magnitude larger than the target step, limiting the applicability of very fast acceleration to small θ s .

As a further insight in the dynamics of the temperature and the acceleration provided by our protocols, we report in Fig. 13 the time to reach the target with a 2% tolerance. This time depends on the choice of t f and N obviously, but also on the relaxation of remaining modes FIG. 12. Maximum (plain line) and opposite of the minimum (dashed line) values of θ(t) for N = 1 to 6, t f /τ = 10 -3 to 1, and x0 = 0.95. Large N and short t f lead to a huge transient temperature, limiting the magnitude of achievable temperature step θs. No negative values of θ(t) are observed for N = 1. FIG. 13. Time (in units of τ ) to reach the target with a 2% tolerance for N = 1 to 6, t f /τ = 10 -3 to 1, and x0 = 0.95. This time is defined as the latest instant when |θ(t, x)/θs(x)-1| > 2%, for x = 0.33, 0.67 and x0. θ(t, x) is computed from numerical resolution of the heat equation for the drivings defined in Appendix A. Achieving this goal requires 1.8τ with a power step protocol, and 0.4τ with a perfect feedback loop (temperature step protocol). The black line is the duration of the protocol t f /τ : curves below it reach the target actually faster than the protocol itself. after t f . For very large acceleration (t f < τ /100), the amplitude of the remaining temperature field at t f is sig-nificant and though decaying very fast, further delays the effective reach of the vicinity of the target. Effective accelerations over 100 are thus out of reach in practice. We provide here the solution for the heat equation (1) without a source term: Rather than considering Eq. (E1) separately in the intervals 0 ≤ x < x 0 and x 0 < x ≤ L, we could have instead considered Eq. ( 1) complemented with Eq. (2), i.e. with a localized δ(x -x 0 ) forcing term proportional to P (t). Thus, the perfect feedback loop introduced in this appendix can be experimentally implemented with a feedforward protocol with the laser power

∂ t θ = D∂ 2 x θ, (
P (t) = λSθ m s a   ϑ 3 (e -Dπ 2 t x 2 0 ) x 0 + ϑ 2 (e -Dπ 2 t (L-x 0 ) 2 ) L -x 0   . (E6)
This latter relation is obtained from direct integration of the heat equation (1) over an infinitesimal interval around x 0 , substituting for θ(x, t) the solution (E5). Above, ϑ a (q), with a ∈ {2, 3} stands for the elliptic theta function Θ a (u, q) evaluated in u = 0: ϑ a (q) = Θ a (0, q), with Θ 2 (u, q) = 2q 1/4 ∞ n=0 q n(n+1) cos[(2n + 1)u],

(E7)

Θ 3 (u, q) = 1 + 2 ∞ n=1
q n 2 cos(2nu). (E8)

FIG. 2 .

 2 FIG. 2.Sketch of the experimental setup. The heating laser beam (at 532 nm) is fixed and focused at the extremity of a cantilever placed in vacuum. The power of the heating beam is modulated with an acousto-optic modulator. The probe laser beam (at 633 nm) is used to measure the local temperature change by analyzing the cantilever reflectivity variations. By tilting the probe laser, the probe beam spot scans the cantilever length allowing the measurement of the temperature variation at different positions x.

FIG. 3 .

 3 FIG.3. Transient temperature profiles (a) of a cantilever heated by a laser whose incident power P (t) is the step function (b). The heating laser is located near the cantilever free end (x0 = 0.95). The 5.6 µW increase in power induces a maximum temperature elevation of θ m s = 38 mK. As time goes on, the temperature profile converges towards the expected linear stationary profile given by Eq. (4). Graph (c) shows the normalized evolution of temperature at the positions x = 0.3, 0.67 and 0.95, corresponding to the vertical dashed lines in panel a). All experimental data are perfectly described by the model of Eq. (6), using the characteristic time τ = 14.2 ms.

FIG. 7 .

 7 FIG. 7. (a) Temperature relaxation towards the stationary state (measured close to the cantilever free end x = 0.95) varying N for a protocol duration t f = τ /30 = 0.47 ms. Inset: temperature variations during the protocol. (b) Comparison of the temperature decay. The horizontal dashed lines correspond to a relative difference (θ -θs)/θs of 4%.

FIG. 10 .

 10 FIG.10. Numerical simulation of the protocol for the same parameters as in the bottom panel of Fig.4: N = 4, t f = 0.1τ , but a different value of the heating location, x0 = 0.5.

  Appendix E: Response to a temperature step (perfect feedback loop)

  E1)with D = λ/(ρc p ) and the following boundary conditions:θ(0, t) = 0, θ(x 0 , t) = θ m s , ∂ x θ(x, t)| x=L = 0, (E2)with 0 < x 0 < L, and initial condition:θ(x, 0) = 0. (E3)Note that, since there is an extra "boundary" condition at x = x 0 , i.e. within the interval 0 ≤ x ≤ L, the equation should be solved separately for the two intervals 0 ≤ x < x 0 and x 0 < x ≤ L. The solution isθ s (x, ∆θ(x, t) = θ(x, t) -θ s (x)in the corresponding eigenbasis, one getsθ0 L -x 0 exp -(2n -1) 2 πD 4(L -x 0 ) 2 t x 0 ≤ x ≤ L (E5)Characteristic relaxation times to the left and to the right of the point x 0 are proportional to τ left = x 2 0 /D and τ right = (L -x 0 ) 2 /D respectively. With respect to the power step solution, assuming x 0 ∼ L, the slowest time constant in the exponentially decaying functions is four time smaller. This acceleration stems from the change of modes from k n = (n -1/2)π to k n = nπ, due to the change of the relevant boundary condition from Neumann to Dirichlet.

  FIG.1. Driving function F (t) to apply in order to cancel successively the first thermal modes for an acceleration (towards the stationary profile) such that t f = τ /10 (left) and t f = τ /30 (right). The coefficients γn defining F (t) are found solving Eq. (10); they are listed in table I of Appendix A.

ACKNOWLEDGMENTS

This work has been financially supported by the Agence Nationale de la Recherche through grant ANR-18-CE30-0013.

C.A.P. acknowledges financial support from Grant No. PID2021-122588NB-I00 funded by MCIN/AEI/ 10.13039/501100011033/ and by "ERDF A way of making Europe", as well as funding from European Union for the support from Horizon Europe -Marie Sklodowska-Curie 2021 programme through the Posdoctoral Fellowship Ref. 101065902 (ORION). We thank Artyom Petrosyan, Salambô Dago, Ivan Palaia and Sergio Ciliberto for enlightening scientific and technical discussions.

APPENDIX

Appendix A: Coefficients γn of the polynomial function F (t)

Eq. 10 can be written as:

A m,n γ m = 1, for n = 1 to N, (A1)

with A m,n = 1 0 s m-1 k 2 n tf e k 2 n tf (s-1) ds.

(A2)

Using integrations by parts, we compute the coefficient A m,n analytically with the following recurrence relation:

The N × N square matrix A defined by its coefficients A m,n can then be inverted using a symbolic math solver, and the coefficients γ m , solutions of

. This approach avoids any rounding errors when computing the integrals in Eq. 10 or singularities when inverting the matrix A. Some examples of coefficients γ m are given in Table I for τ /t f = 10 to 100 and N = 1 to 4, while Fig. 9 reports the maximum and minimum values of F (t) for τ /t f = 1 to 1000 and N = 1 to 6.

As physically expected, shorter values for t f or larger number N of modes canceled involve much greater costs. The maximum value of F (t), always occurring at t = 0, behaves as ( √ N t f /τ ) -N . This scaling can be used to extrapolate the very large power requirements at strong accelerations and large mode numbers. From the theoretical point of view, no pathologies of any kind are found for the values obtained. However, from the practical point 10) that allow to cancel successively the four first thermal modes for a protocol duration t f such that τ /t f = 10, τ /t f = 30 and τ /t f = 100.

FIG. 9.

(top) Maximum value of F (t), corresponding to F (0) = γ1, for N = 1 to 6 and t f /τ = 10 -3 to 1. Large N and short t f lead to a huge range of power needed during the transient. A scaling max(F ) ∼ [0.45τ /(t f √ N )] N is observed, as illustrated in the inset (dashed line is 0.45τ /t f ). (bottom) Minimum value of F (t), normalized by its maximum value, for N > 1 (for N = 1, F (t) > 0 at all times -no cooling is necessary). The cooling power is always smaller in absolute value that the heating one.

of view, very high intensities could lead to implementation problems for a standard experimental setup. Moreover, the assumption of constant thermal conductivity is jeopardized and non-linear effects beyond the simple heat equation considered herein should be taken into account.

Appendix B: Independence of the protocol on the laser location

In Fig. 10, we present a numerical check of the validity of our acceleration protocol for a different value of the laser location. More specifically, we take the same parameters as in the bottom panel of Fig. 4: N = 4, t f = 0.1τ , but x0 = 0.5.