Phase-shifting Interferometry To Determine The Diameter Of A Silicon Sphere Based On A Frequency-tunable Diode Laser

The redefinition of the kilogram has been researched by several national institutes for the past two decades. Although Planck constant has been considered to be the first choice to redefine the kilogram, Avogadro constant also plays a crucial role in this redefinition due to the close relationship of Avogadro constant, Planck constant and Rydberg constant. Currently, the most accurate Avogadro constant with a relative uncertainty of 3 × 10-8 is measured by the X-ray crystal density method, but the goal relative uncertainty should be 2 × 10-8. Among all the measurement quantities, the uncertainty from the diameter of a silicon sphere restricts the accuracy of the Avogadro constant. In order to achieve the purpose of redefining the kilogram, the uncertainty of the diameter should be 0.3 nm, but the best uncertainty is only 1 nm by use of phase-shifting interferometry. In this thesis, we demonstrated phase-shifting interferometry to determine the absolute diameter of a silicon sphere. Compared with existing methods, the optical setup, the laser frequency control method, the interference calculation method and the integral part measurement have been improved dramatically, where uncertainties from the laser frequency stability and the phase shift algorithm are insignificant enough to satisfy the requirement of the redefinition of the kilogram.Firstly, the phase-shifting interferometer based on plate references is developed and the error sources from the interferometer are minimized. Based on the principle of "D = L-(d1 + d2)", the diameter D is measured from the length of an etalon L and the gap distances between the sphere and the reference plates d1 and d2. In our interferometer, the etalon is made of ultra low expansion glass with a thermal coefficient of 3 × 10-8 around 25 ℃. The uncertainties from the interferometer are evaluated carefully. The uncertainty from the alignment is insignificant after optimizing the optics. Using the beam propagation model with the paraxial approximation, the measurement deviation of the silicon sphere diameter is estimated to be 0.86 nm due to the Gouy phase shift of the Gaussian propagation.Secondly, the laser frequency control system is developed using a femtosecond optical frequency comb as a laser frequency reference. This laser frequency control system is essential for phase-shifting interferometry, as it can generate traceable and tunable laser frequencies. The relative laser frequency stability is 6.3 × 10-12, and the Abstract laser frequency tuning range is limited only by the frequency-tunable diode laser.Moreover, the Carré algorithm with arbitrary equal steps is used to calculate the interference phases. Compared with traditional phase-shift algorithms, the phase calibration is not necessary any more and the phase-shift error only depends on the errors of the laser frequency stability. Furthermore, in order to determine the absolute diameter, laser frequency-sweeping interferometry is developed, which can determine an absolute distance by measuring the laser frequency-sweeping range and the variation of the interference phase. Therefore, the absolute diameter can be achieved by the combination of the integral part from the frequency-sweeping interferometry and the fractional part from the phase-shifting interferometry.In conclusion, the uncertainty of a single diameter measurement in air is estimated to be 5 nm, whose uncertainty sources from the laser frequency stability and the phase-shifting algorithm are negligible.