10 "NEOLIBERALISMO" Y EDUCACION
CUADRO PARA ANALIZAR LOS INDICADORES DE LOGROS Y DEFINIR LOGROS POR GRADO
For levitated particles, the recoil experienced from absorption or emission of black- body radiation can represent a source of heating and decoherence [200,236]. Even when the state of the oscillator is predominantly classical, the role of black-body radiation processes is undeniably important especially when the power involved is high. In vacuum, with no means of mechanical dissipation, the only way for the levitated object to dissipate the excess energy absorbed over time is through radiative emission [237].
For the levitating mirror, the absorption of even a fraction of the incident power could represent a significant change in the system’s conditions. Having a macroscopic thickness h much greater than the optical wavelength, the mirror responds to the Stefan–Boltzmann law according to which the power radiated is proportional to the
surface area and the fourth power of the temperature:
Prad="bb SB T4 T04 ⌃, (8.27)
where "bb is the black-body emissivity of the mirror, SB is the Stefan–Boltzmann constant,⌃is the emitting area, andT and T0 are the mirror’s temperatures with and without the incident power, respectively. The field leaking from the cavity through the coating is absorbed in the substrate with exponential decay before being transmitted through:
Pabs=⇣1 e ↵h⌘TmPcav, (8.28)
where ↵ is the absorption coefficient of the substrate, Tm the transmissivity of the coating, andPcavis the power in the cavity. After balancing the two equations, the raise in temperature of a cylindrical silica substrate (✏bb ⇡0.8,↵⇡10 2cm 1, diameter of 3 mm and thickness of 50µm) at room temperature conditions is about 1 K when the cavity has enough power for levitation and about 0.1 % of it is transmitted through the coating. The change in temperature expected is far from reaching the melting point of silica, but it is significant enough that there is a potential for less drastic consequences to be manifest, such as thermal expansion or excitation of the mirror’s drum modes. It should be noted that, due to the time-independent nature of the radiation, the net work done by the black-body emission on the mirror over one oscillation is zero.
Experimental design
9.1
Specifications of the mirrors
The model developed in Chap. 8 suggests that a macroscopic mirror can indeed be successfully decoupled from the environment and be supported entirely by the optical field of three cavities. A better idea of what “macro” exactly means in this context is obtained by considering the mass employed in all simulations, 1 mg. This mass, a million times bigger than the average human cell (⇡1 ng), is characteristic of granular substances. It is about twenty times larger than the mass of a single grain of fine salt (⇡0.06 mg), but still smaller than the typical mass of a grain of sand (10–50 mg). Assuming the convex substrate to be made of fused silica, which has a density of 2203 kg m 3, a mass of 1 mg prescribes the dimensions of the mirror to range between 2 to 3 mm in diameter and 30 to 70µm in thickness, similar to a shrunk down contact lens.
The thickness anticipated is small enough that the mirror is a simple spherical cap with no cylindrical base. The convex shape, which was chosen so that the three beams would hit the mirror higher than the height of its centre of mass, provides several other unanticipated benefits that go beyond the improvement of the mirror’s stability. The convex-concave cavity configuration places the waists of the beams outside of the optical resonators. By having a virtual waist the intensity is prevented from being at its highest at any physical point. Also, compared to a concave mirror of similar dimensions, a convex mirror is much lighter and the power requirement for levitation is lower. The choice of the mirror’s radius of curvature (RoC) reflects a balance between two contrasting demands. A large RoC (i.e. a less pronounced curvature) is perfect for having the three cavities as close to vertical as possible. This allows most of the radiation pressure force to contribute towards levitation, but at the same time there is a limit to how close the lower mirrors can be placed. Given a certain distance between
1 mm mirror force sensor (b) ≈ 185 mm RoC +200 mm RoC −30 mm 30 µm 3 mm (a)
Figure 9.1: The actual specifications provide a di↵erent image of the tripod than the concept diagram of Fig. 8.2. (a)Realistic diagram of the optical tripod, with dimensions to scale. The metal ring on the top part provides the initial support for the mirror. (b)Close-up picture of one of the 3-mm mirrors, from the side. The exhibited mirror is directly attached to a force sensor in order to measure its mass. The original photograph has been modified to reduce the noise and enhance the contrast with the background.
the input mirrors, a small RoC (i.e. a more pronounced curvature) for the upper mirror allows the cavities to be much shorter, increasing the linewidth and thus allowing the spatial dimensions of the trap to be extended without having to decrease the finesse. However, when the cavities are shorter the aperture of the tripod gets larger and the vertical component of the combined radiation pressure force becomes smaller. Feasible radii of curvature for the upper mirror are around 20–35 mm. The illustration in Fig. 9.1 shows a scale diagram of the tripod for a mirror which is 3 mm in diameter, 30µm thick, and with a radius of curvature of 30 mm (the negative sign indicates that it is convex). These dimensions, taken as a benchmark for the experimental implementation, lead to a distance of roughly 10 mm between the centres of the beams at the bottom of the cavities. For lower mirrors with RoC of 200 mm, the cavities are optically stable when their length is between 170 mm and 200 mm. The length of 185 mm is chosen in the middle of this range to let the spot size on the upper mirror be at its largest and reduce the risk of laser-induced damage. The virtual waists are always close to the centre of curvature of the lower mirrors, around 15 mm above the upper mirror. All three beams virtually coincide at the centre of curvature of the upper mirror, acting as the centre of the tripod which has an aperture of⇡1.4 .
With an expected spot size of 100µm in radius and about 0.5 kW of circulating power in each cavity, a few considerations on optical damage are inevitable. The laser-
HR coating AR coating R.o.C. ≈ 9 mm ≈ 30 µm 25.4 mm ≈ 5 mm +200 mm 3 mm ≈ 5 mm ─30 mm Upper mirror: Lower mirrors:
Figure 9.2: The cavity mirrors required special adjustments in order to be ready for the tripod. The upper mirror, originally about 5 mm thick, was lapped to a spherical cap of thickness of
⇡30µm. The lower mirrors, with high-reflective coating on the concave side and anti-reflective coating on the flat side, had to be sliced so that their centres could be placed closer.
induced damage threshold (LIDT) of the mirror’s coating is required to be at least a few times higher than the intensity of 1.6 MW cm 2. This figure may be too high for conventional high-reflectivity mirrors where the coating is obtained by electron beam or ion-assisted vapour deposition [238]. Modern ion-beam sputtering coating techniques, on the other hand, have evolved to the point where these and even more ambitious re- quirements are easily met1, with certified LIDTs higher than 1 GW cm 2. The striking di↵erence between the di↵erent techniques lies in the density of the coatings obtained. Coatings obtained by vapour deposition are characteristically more porous and less dense than ion-beam sputtered coatings. The pores are usually filled up with wa- ter, making the coating more sensitive to temperature and humidity conditions while also increasing absorption losses. The higher density obtained with ion-beam sputter- ing produces extremely uniform and lossless coatings, also allowing greater variations in refractive index which is essential for high-end Bragg mirrors. At the same time, however, the greater adhesion induces extremely high stresses on the substrates. The surface tension which is usually supported by regular substrates might be excessive for the thin mirrors involved, and there is a high risk of shattering during or after the coating process.
The mirror employed in the preliminary stage of the experiment (cf. Fig. 9.1b) con- sists of a small fused silica lens substrate2, 3 mm in diameter, with a surface roughness of 10 nm, and coated by vapour deposition to a reflectivity of 99.9 %. The original mir- ror is lapped3 in order to reduce the thickness to 30µm (with an upper uncertainty of almost 100 %). The lapping process may have slightly reduced the diameter depending
1from personal communications with D. Samuels, from Advanced Thin Films Inc. (http://
advancedthinfilms.com/)
2custom order from FOCtek Photonics Inc. (http://www.foctek.net/)
on the final thickness due to the aspect ratio of the spherical cap. The reflectivity is selected to favour impedance matching over a capacity for higher finesse. The lower mirrors are more conventional high-reflective mirrors4, sliced as in Fig. 9.2 to allow close positioning. They are coated on the concave side to a reflectivity of 99.9 % by ion-beam sputtering, and they also feature an anti-reflective coating on the flat surface to prevent the creation of intra-substrate etalon modes.
It should be emphasized that an upper mirror coated by vapour deposition is a moderate gamble, more vulnerable to thermal e↵ects and optical damage. One solution for future iterations might be to manufacture the mirror out of a harder substrate that can better tolerate the high stresses induced by ion-beam sputtering. Diamond is a possible choice, with a tensile strength between 2 and 5 times that of fused silica. It is also a much sti↵er material, meaning that the excitation of the vibrational drum modes would be curbed. On the other hand, diamond is denser and the mass to be supported by the radiation pressure force is heavier. Unless power is not the limiting factor, this issue may outweigh the advantages.