Application of the Vernier method for absolute distance metrology with CW TOF phase shift technique

A phase shift time of flight technique determines a position by comparing the phase angle of a continuously modulated signal in the source and its reflection on a target. However, due to its cyclical properties, the position information is contained within an ambiguity interval. For an absolute measurement, this interval is repeated N times plus a residual part given by the phase shift. In this work we propose an application of the Vernier method to determine N and a setup for mid-range applications (10-20) m with a 3 GHz amplitude modulated source to allow accuracies  100 μm.


Introduction
Time of flight (TOF) is a very well established optical measuring technique that o↵er ways to determine absolute or relative distance measurements (ADM and RDM, respectively). In a phase shift (PS) continuous wave TOF (PS CW TOF) method, a coherent light source is continuously modulated in amplitude and directed to a target. The reflected signal will then have a phase shift, , relative to the original, which will be related to the source-target distance by eq. 1.
Where f is the modulation frequency, D the sourcetarget distance, c the speed of light and n the refractive index of the medium. Due to the cyclical properties of the phase, the position information is contained within an ambiguity distance, eq. 2. Therefore, for an absolute measurement, this interval is repeated N times plus a residual part given by the PS. Hence, it's crucial to determine N for absolute measurements. Di↵erent methods can be used for this purpose [1]. Norgia, et al [2] achieved 100 µm accuracy for a 2 GHz modulation using heterodyne down conversion (HdC), and solved the ambiguity by using a double Vernier method [3].
For a mid-range (10-20) m, by performing a 3 GHz modulation with HdC, one can achieve an accuracy of <100 µm in an RDM, as long as the error of the frequency and PS are in the order of 1 kHz and 0.3°, respectively. In this work, we propose a di↵erent approach to determine N, allowing this level of accuracy for ADM. Although, since ⇤ e-mail: fc48080@alunos.fc.ul.pt ⇤⇤ e-mail: maabreu@fc.ul.pt ⇤⇤⇤ e-mail: dmvalves@fc.ul.pt N is an integer, it must be determined with an error < 0.5. In order to achieve the distance accuracy goal and N being properly determined, a trade o↵ between both of this components must be done. However, the error analysis isn't in the scope of this document.

Working Principle
In order to measure the source-target distance, the PS CW TOF technique employs the concept of a "ruler". This ruler has a primary and secondary mark, given by the ambiguity distance and by the PS (as it is referred in eq. 1), as shown by ruler A in figure 1. It's possible to determine N by measuring the same distance with two di↵erent "rulers". The di↵erence between primary markings of both rulers is defined as x N : Where N corresponds to the index of the primary mark of the ruler with biggest spacing between them and f > f and f = f 0 f > 0 since f 0 > f . This represents the di↵erence in frequency for both scales and will rule the variation in position of the di↵erent primary marks.
Additionally, by considering the di↵erence in phase shifts between rulers for the same distance, = 0 , one gets eq. 4. By rearranging it, the absolute distance can be written as eq. 5.
Considering a target at a fixed distance D, one can measure a pair of di↵erent phase shifts corresponding to known modulation frequencies spaced by f , by using eq. 7 it's possible to determine the value of N. Very briefly, if one plots eq. 6 for a fixed distance the result is a saw-tooth profile as seen on figure 2A. For a single and arbitrary frequency-PS point in this profile, the corresponding N can be found as the intercept in the normalized PS axis, as shown in figure 2B. However, for a pair of arbitrary and di↵erent frequency-PS points, the N between them can be 0,-1,-2,-3... depending on the f spacing. More generally, for any given D  D max , by constraining f one can ensure that for a pair of di↵erent frequency-PS points, N = 0 W N = 1. The maximum distance at which this condition is fulfilled is when D max is an integer multiple of the ambiguity distances of both frequencies, i.e. the marking of both rulers match and consequently = 0. However, as seen in figure 1, D max will be reached when x N = ⇤ 0 , which mathematically implies that N = 1. By using eq. 5 one can find D max : Meaning that for D  D max by doing spacings of f max , as long as f > f max for any modulation frequency one can find N = 0 W N = 1. This is consistent with the results of S. Donati [1], that with a di↵erent approach showed it was possible to measure unambiguously for D  D max .
Additionally, through mathematical analysis of eq. 5, one can conclude that if < 0 ! N = 1 and > 0 ! N = 0. Hence, by measuring a pair of di↵erent PS and modulation frequencies spaced at most by f max one can know if N = 0 or N = 1 if the di↵erence in PS between them is positive or negative.
In conclusion, with eq. 9 and a pair of frequency-PS measurements, one can determine N through eq. 7. Therefore, by knowing N and applying eq. 1 with a GHz modulation frequency it's possible to achieve the desired accuracy.

Future work
In order to show that this principle is valid to obtain absolute distances with high accuracy, we intend to build the setup showed in figure 3. One must ensure that given the instrumental limitation, we can measure for a mid-range distance, (10-20) m, with accuracies  100 µm. For that purpose the knowledge of the frequency and PS must be in the order of 1 kHz and 0.3°, respectively. Additionally, since N is an integer, its measurement error must be smaller than 0.5, to determine the exact integer number of phase cycles to reach the distance D. This work was supported by Fundação para a Ciência e a Tecnologia (FCT) through the research grants UIDB/04434/2020 and UIDP/04434/2020. We also recognize the support of SPOF on the participation of this conference.