Fabrication of ultrahigh-precision hemispherical mirrors for quantum-optics applications

High precision, high numerical aperture mirrors are desirable for mediating strong atom-light coupling in quantum optics applications and can also serve as important reference surfaces for optical metrology. In this work we demonstrate the fabrication of highly-precise hemispheric mirrors with numerical aperture NA = 0.996. The mirrors were fabricated from aluminum by single-point diamond turning using a stable ultra-precision lathe calibrated with an in-situ white-light interferometer. Our mirrors have a diameter of 25 mm and were characterized using a combination of wide-angle single-shot and small-angle stitched multi-shot interferometry. The measurements show root-mean-square (RMS) form errors consistently below 25 nm. The smoothest of our mirrors has a RMS error of 14 nm and a peak-to-valley (PV) error of 88 nm, which corresponds to a form accuracy of λ/50 for visible optics.


X-offset error profile
During the cut the tool describes a near-perfect circle with radius R c given by the distance from the cutting edge to the center of rotation, limited by the radial and axial displacement of the lathe's B-axis over 90 • rotation (both specified as less than 50 nm over 360 • ). However, the actual profile cut in the X-Z plane depends on the offset ∆x between the tool rotation (B) axis and part rotation (C) axis (see Fig. S2a). In Cartesian coordinates the cut profile is We are interested in how closely this profile matches a circle with radius R f and CoC displaced from the B-axis center by distance ∆z (the blue circle in Fig. S2a) given by To find the radial displacement between these two surfaces we transform the Cartesian curves z c and z f into radial profiles in spherical coordinates In the limit R c , R f ∆x, ∆z the square-root terms approach one and the error profile, the difference between the cut and fit radial profiles, simplifies to where ∆R = R f − R c . We can integrate over the surface area of a sphere with NA = sin φ a to find the RMS error where the area element is weighted by sin φ and the integral over θ yields a normalization factor 1 − cos φ a . For a complete hemisphere (NA = 1, φ a = π/2) this reduces to The best-fit spherical surface is given by the parameters ∆R xfit and ∆z xfit that minimize the RMS error for a given ∆x which are plotted in Fig. 2c (red and green lines) as a function of ∆x. With these optimal parameters the radial error profile is This whole-hemisphere X-calibration error profile is plotted in Fig. 2a, and used to infer the residual offset of the finished hemispheres from their measured surface profile. In this way we are able to infer the accuracy of our calibration technique (see the main text). We can calculate the RMS error and PV error from this profile (Fig. 2c, black and blue lines) to give the X-calibration error budget rule of thumb used in the text The above expressions are true for a complete hemisphere (NA = 1), but in general the best-fit sphere parameters ∆R xfit and ∆z xfit depend on the numerical aperture of the surface section being measured. For a spherical surface with NA = sin φ a < 1 the parameters ∆R and ∆z that satisfy which are very closely approximated by their first order Taylor expansions in φ a about φ a = 0 By substituting these parameters into the general equation of the radial error profile, Eqn. 5, and integrating over the surface area as in Eqn. 6 we can find the aggregate errors over any measured NA as a function of the X-offset error profile. The RMS error and PV error as a function of the measured aperture are shown in Fig. 2b.

Y-offset error profile
In contrast to the X-axis, the Y-axis lies outside of the plane of B-rotation, and makes no contribution to the final form of the mirror outside of a small central defect. A Y-axis offset ∆y produces a circular defect at the center of the mirror with corresponding radius ∆y. This defect is either a cone or a pillar depending on the sign of ∆y, and can be measured and corrected directly using the on-lathe white light interferometer. Once the defect has been identified and measured with the interferometer it can be corrected by an adjustment of the Y-axis. The small defect makes no contribution to the RMS error because it covers a negligible region of the surface.

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Outside of the central defect, a Y-offset makes a small contribution to the final form by increasing the distance of the cutting edge to the center of the C (part rotation) axis In the limit R c ∆y this reduces to which is a spherical surface with RoC increased by By the same logic we see that for a combination of X and Y offsets, and for any measurement aperture, the fit parameters from Eqn. 13 become simply with no increase in the aggregate errors compared to an X-offset. This is illustrated in Fig. 2d where a Y-offset results in a quadratic increase in the RoC but does not contribute to the aggregate error.

X-offset calibration example
The offset ∆x between the B and C axes centers is removed by an interferometric calibration process summarized in the text of the paper using the on-lathe white light interferometer shown in Fig. S1. Two 1 µm deep grooves are cut on the outer surface of the aluminum substrate with the tool at coordinates X 1 , B 1 and X 2 , B 2 shown in Fig. S2(a). The spindle rotation direction is reversed between the two positions. With coordinates X 2 = −X 1 and B 2 = 180 • − B 1 the difference in depth between the two grooves is equal to 2∆x. By measuring the depth difference we determine and correct the offset. The depth of the grooves is measured with an in-situ white-light interferometer. The interferometer is fixed stably to the tool post in order to take advantage of the lathe axes' precise positioning control. Central-fringe identification provides a relative position measurement between the interferometer objective and the mirror surface. The X-axis displacement required to shift the central fringe from the bottom of one groove to the other is the depth difference between the two grooves. After shifting the center position to X 0 = X 0 + ∆x the depth of two subsequent calibration cuts is equal. Fig. S2(b) shows an interferogram of two equal-depth calibration cuts made following an calibration step, the central maxima is simultaneously aligned with the bottom of both grooves.
Although the central-fringe maxima identifies the mirror surface position with sub-nm precision, and the lathe can be positioned with nm control precision, we infer from repeatability tests an alignment precision of 10 nm. This is consistent with the radial displacement error of the B-axis under rotation, which is specified as less than 50 nm. Furthermore, we expect additional uncertainty due to the difficulty of cutting under identical conditions at B 1 and B 2 , from which we infer that the real radial uncertainty of B-rotation is less than 10 nm. This is the PV lower limit of hemispheres cut using this technique.

Bake-out test for UHV experiments
Part C was measured before and after a bake-out for UHV to determine the degree of deformation during this process. As the temperature is increased and decreased strains within the material substrate relax and produce volumetric changes that deform the spherical surface. These strains may be due to the temper of the aluminum, or may be produced by the manufacturing process. To reduce this effect the mirror substrates are temperature cycled before the surface is turned on the nano-lathe, essentially undoing the material temper. All the parts presented here were cycled from room temperature to 300 • for one hour, three times before finishing.
The bake-out test was performed by heating part C to 200 • celsius in vacuum for two hours. Tab. I and Fig. S3 compare the before and after measurements of the part. To produce Fig. S3 the after image is translated, rotated, and translated again until it has the maximum possible overlap with the before image. The residual difference is taken to be the distortion of the part. The final image is smoothed with a Gaussian filter (width 10 pixels) to remove artefacts of the fitting process.
The bake-out induced deformation has distorted the surface by up to 60 nm, and the RMS error is increased from 18.3 to 27.4 nm, an increase of 50%. This is a considerable distortion, although an order of magnitude smaller than the measured deformation of mirrors which were not temperature cycled before cutting.

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Fibre illumination Tube lens CCD camera Interferometry objective d 2 d 1 ∆d 2 Figure S1. Schematic of the white-light interferometer used for on-lathe calibration of the lathe before diamond turning hemispheres. The objective used was a 20X Nikon CF IC Epi Plan DI interferometry objective. This was then mounted on an Infinitube FM-200 microscope body with an in-line fibre illuminator port. The white light source was a three-LED illuminator with emission at 465 nm, 521 nm and 635 nm. The distance between the objective and the work-piece was adjusted using the lathe's motion control with nanometre accuracy.
Spindle axis (C) Spindle axis (C) a b Figure S2. (a) Schematic of an (exaggerated) X-offset error profile and the corresponding calibration procedure as described in the text. The B-center is offset from the C-center by ∆x and produces an aspheric surface (black). The spherical fit to this surface (blue) has RoC R f = R c + ∆R and CoC offset by ∆z. To correct this X-offset we cut two grooves with the tool at positions X 1,2 , B 1,2 where X 2 = −X 1 and B 2 = 180 • − B 1 . (b) A white-light interferogram showing two calibration grooves of equal depth.