#### List of Symbols

*d*_{n}

*d*_{w}

*E*_{cyc}

*E*_{inc}

*E*_{ref}

*e*_{y}(x)

*f* (*x*)

*J*

*l*

*L*

*L*_{w}

*n*

*P*_{A}(*x*)

*P*_{B}(*x*)

*r*

*r*_{i}

*i*

*r*_{n}

*r*_{w}

*s*

*s*_{i}

*i*= 1) and Perpendicular to (

*i*= 2) the Fiber Axis

*t*_{i}

*i*

*x*_{n}

(*x*_{ci}

*y*

_{ci},

*z*

_{ci}), The Cartesian Coordinates of the Geometric Center of the

*i*

^{th}Layer to be Scanned

α

α_{i}

*i*= 1) and Perpendicular to (

*i*= 2) the Fiber Axis

λ_{0}

λ_{FSR}

λ_{m}

*m*

φ

### 1 Introduction

### 2 Sensing Principle

### 2.1 Principle of Fabry-Pérot Interferometers

*E*

_{inc}comes from the light source and propagate along the core of the SMF. When reaching the interface (Reflection Surface 1) between the Core and the Medium between the SMF and Target, part of light will be reflected by Reflection Surface 1 and the rest will propagate through the surface. Then, the light will be reflected by the surface of Target (Reflection Surface 2) and then reflected by Surface 1 again. Therefore, the light will be reflected between Reflection Surface 1 and 2 and resonance occurs. The structure formed by the two reflection surfaces and the space between them is called a Fabry-Pérot cavity (FP cavity).

*E*

_{ref}will propagate to the opposite direction along the core and will be received by a spectrometer for further analysis. Fig. 1(b) shows a typical experimental spectrum of an FPI, multiple maximums and minimums can be seen on the spectrum. The maximums are due to the constructive interference, whereas the minimums are due to the destructive interference of the FP cavity.

*r*

*is the reflection coefficient of Reflection Surface*

_{i}*i*,

*t*

*is the transmission coefficient of Reflection Surface*

_{i}*i*,

*φ*= 2

*πln*/

*λ*

*is the phase change of the light when travel in the cavity for a roundtrip,*

_{0}*l*is the geometric distance between Reflection Surface 1 and Reflection Surface 2,

*n*is the refractive index of the medium at wavelength of 1,550 nm, and

*λ*

_{0}is the wavelength of light in vacuum, and

*j*is the imaginary unit. Therefore, the wavelength of the maximums can be calculated as [16]:

*m*is the order number of the maximums. As we can see, wavelength

*λ*

*is proportional to geometric distance between the surfaces*

_{m}*l*and refractive index of the medium

*n*for a given order number

*m*. Therefore, if the refractive index

*n*is known, the distance between the surfaces is available from the wavelengths of the spectrum. To calculate order number

*m*, we need to measure the free spectrum range (FSR)

*λ*

_{FSR}, which is defined as the wavelength difference between to adjacent maximums and is available from the spectrum. The FSR can also be calculated using [16]:

*λ*

*when refractive index*

_{m}*n*is known.

### 2.2 Experiment Setups

#### 2.2.1 Tool Profile 3D Reconstruction

#### 2.2.2 CNC Machine Straightness Error Monitoring

### 3 Results and Discussion

### 3.1 Tool Profile 3D Reconstruction

*λ*

_{m}>>

*λ*

_{FSR}. Let,

*λ*

_{m}= 1550 nm,

*n*= 1, and

*l*∈ [0, 3, 5] mm. Then we get

*λ*

*∈ [0, 2, 4] nm as shown in Fig. 9. For a poor spectrum, no significant peak can be seen within this range. And for better and good spectra, we can see significant peaks between 1 and 2 nm which agrees with their corresponding spectra in Fig. 8.*

_{FSR}*λ*

_{m}.

*λ*

_{0},

*λ*

_{1}] with sampling points

*N*as:

*f*

_{m}and

*f*

_{m+1}, the resolution of FSR is:

*f*

_{m}= 1/3 nm

^{−1}for a spectrometer at rang 1510 to 1590 nm. Then the resolution is

*R*

*= 0.108 nm. Substitute to the Eq. (6) and use*

_{FSR}*λ*

_{m}= 1550 nm,

*n*= 1, and

*λ*

_{FSR}= 1/

*f*

_{m}= 3 nm. We have the resolution of distance between the fiber and the target is

*R*

_{od}= 13.914 μm. To further improve the resolution, a manual process is given in Ref. [16]. The manual method has a theoretical resolution of 4.45 nm. To improve the efficiency of the manual method, a machine learning algorithm will be introduced in the following section. The measurement range of the FPI is determined by the quality of the signal. Using a spectrometer with a wavelength range from 1510 nm to 1590 nm, the distance between the fiber and the target between 100 μm and 4 mm can be monitored. In order to obtain a high-quality spectrum, the distance between the fiber and the target should be maintained between 200 and 500 μm.

*d*

_{n}. With knowing the radius of the new tool

*r*

_{n}, we can calculate the distance between the optical fiber and the axis of the tool as

*d*is the distance between the optical fiber and the measured point. It should be noted that since we are monitoring the distance between the fiber and the tool, runout can be seen in the reconstructed profile. However, the existence can affect the accuracy of

*r*, so it should be modified to compensate the radial and axial runout. To find the geometric center of the tool of each scan layer, we need the coordinates of at least three tips of the tool to determine the coordinates of the circle (

*x*

_{ci},

*y*

_{ci},

*z*

_{ci}) for the

*i*

^{th}layer. With all the center coordinates, both the radial and axial runout can be calculated. Since we assumed

*r*

_{n}was the distance between the tool axis and the tip, the calculate

*r*was incorrect. Therefore, it should be recalculated after the runout effect is compensated. For tools with two tips, the process can be simplified since the center of the circle is at the center of the connection of the two tips.

*r*

_{w}. Finding the maximum distance between the fiber and the tool

*d*

_{w}, then the distance between the optical fiber and the tool axis is now:

*s*

_{1}= 15.4 μm, and the angle between the tool axis and the vertical direction is

*α*

_{1}= 1.34°. For those perpendicular to the fiber axis, the values are

*s*

_{2}= 85.1 μm and

*α*

_{2}= 0.44°, respectively. Therefore, the radial runout is

### 3.2 CNC Machine Straightness Error Monitoring

*L*. The raw sensor outputs,

*P*

*(*

_{A}*x*) and

*P*

*(*

_{B}*x*) from the displacement sensor A and B respectively, can be represented by:

*f*(

*x*) is the profile error from the standard block, and

*e*

_{y}(

*x*) is the straightness error of the X-axis guideway along Y-axis direction. Here,

*x*

_{n}=

*n*·

*L*, and

*n*= 0, 1, 2, 3, ...,

*N*which illustrates the measurement position. The output difference between these two sensors can eliminate the straightness error term and provides the increment of the profile error. Therefore, the equation for the separated profile error

*f*(

*x*) and the straightness error

*e*(

*x*) can be expressed by: