A high-resolution angular deviation measurement method and device based on phase-sensitive CPSDOCT

Abstract

This invention proposes a high-resolution angular deviation measurement method based on phase-sensitive CPSDOCT. This method utilizes the high resolution, non-contact nature, and rapid imaging characteristics of OCT technology to significantly improve the accuracy and sensitivity of angle measurements, overcoming the limitations of traditional techniques in terms of stability and applicability, and achieving real-time and efficient angle imaging and measurement. Through high-resolution imaging using OCT technology, highly accurate three-dimensional surface data can be acquired, allowing for the calculation of extremely minute angular changes. OCT technology does not rely on traditional contact measuring tools, avoiding errors introduced by contact friction, deformation, and other factors, making it suitable for measuring precision or fragile objects. This invention enables rapid measurement and is suitable for real-time quality control.

Description

Technical Field

[0001] This invention belongs to the field of optical precision measurement technology. It uses the phase-sensitive solution of the common-path interference signal of low-coherence light to obtain high-precision displacement offset. Through multi-point displacement offset calculation, high-resolution planar spatial angle deflection value is obtained analytically. Background Technology

[0002] With the increasing demands for high-precision measurement in industrial manufacturing, engineering design, and biomedical fields, traditional angle measurement methods have limitations in terms of accuracy and efficiency. Existing angle measurement technologies typically rely on mechanical instruments, such as electronic levels and angle meters. These methods not only require contact with the object being measured, but their measurement accuracy is also greatly affected by environmental factors, making it difficult to meet the high-precision measurement needs for minute angle changes.

[0003] An autocollimator is an optical instrument used for precise angle measurement, typically to detect minute angular deviations of objects. The working principle of an autocollimator is as follows: The light source of the autocollimator emits a parallel beam of light through a collimating lens; this parallel beam illuminates the surface of the object being measured (usually a highly reflective mirror). Minor angular changes in the object cause a shift in the reflected beam; the reflected beam passes through the collimating lens and is eventually detected by the receiving system, which measures the angle of deviation of the reflected beam relative to the original parallel beam; the measuring system converts this beam deviation into angular information and displays it digitally or graphically, allowing the user to determine the angular error of the measured object.

[0004] The main disadvantages of autocollimators are as follows:

[0005] 1. Sensitive to environmental conditions. The measurement accuracy of an autocollimator is highly sensitive to external vibrations; even minor vibrations can lead to measurement errors. Temperature fluctuations can affect the refractive index and dimensional changes of optical components (such as lenses and mirrors), resulting in decreased measurement accuracy. Airflow can affect the beam propagation path, especially in measurements requiring extremely high precision, where air disturbances can introduce errors.

[0006] 2. High precision requirements for the reflecting surface. Autocollimators require a very flat and smooth reflecting surface (usually a mirror on the surface of the object being measured) to ensure accurate return of the reflected beam. If the surface of the object being measured has stains, scratches, or roughness, it may affect the reflection path of the beam, leading to measurement errors. Therefore, the requirements for the surface being measured are high, and it is not suitable for objects with poor surface quality.

[0007] 3. Limited measurement range. Autocollimators are suitable for measuring relatively small angular changes. Their effective measurement range is generally small, usually within a few degrees. When measuring larger angles, more complex optical systems or multiple measurements may be required. Furthermore, if the object's angular deviation is large, the reflected light from the autocollimator may exceed the instrument's receiving range, leading to measurement failure.

[0008] 4. Complex Operation. Although the autocollimator itself is simple in design, its use and adjustment usually require a certain amount of experience and skill. Especially in aligning the optical system, adjusting the focus, and calibrating the reference mirror, operators need to possess a high level of professional knowledge. Improper operation may lead to inaccurate measurement results.

[0009] 5. Limited applicability to dynamic measurements. Autocollimators are typically used for static angle measurements and cannot track dynamically changing angles in real time. Their application is limited if real-time angle measurement during motion is required (e.g., measuring the angle changes of moving mechanical components).

[0010] Optical coherence tomography (OCT) is a high-resolution imaging technique that uses the principle of light wave interference to perform non-invasive three-dimensional imaging. OCT can be used for high-precision surface measurement to obtain detailed three-dimensional information of the measured object. Although some studies have used OCT for surface contour or height measurement, its application in angle measurement has not been fully explored, especially in high-precision plane tilt angle measurement, where a significant gap remains. OCT technology essentially achieves micrometer-level spatial distance resolution through the interference of low-coherence light, and by combining numerical interpolation and phase-sensitive calculations, displacement measurement accuracy on the order of tens of picometers can be obtained. High-precision displacement measurement of multiple points on a spatial planar object can further calculate the spatial angular motion deviation of the plane, thus replacing photoelectric autocollimators in special application scenarios. For example, applications involving torsional rotation angle measurement with weak interference urgently require innovative methods for high-precision angle measurement that do not rely on traditional contact tools, possess high resolution and measurement accuracy, and are suitable for the angle measurement needs of small objects and complex surface structures.

[0011] Existing angle measurement methods, such as traditional autocollimators or electronic sensors, while capable of achieving a certain degree of angle detection, have the following drawbacks in practical applications:

[0012] 1. Insufficient measurement accuracy: In high-precision or minute angle changes, existing methods are often affected by environmental noise, surface roughness, optical alignment and other factors, making it difficult to achieve high-resolution measurements at the micrometer or even submicrometer level, resulting in insufficient accuracy of the measurement results.

[0013] 2. Limited stability and applicability: Traditional methods are mostly contact-based or highly dependent on external conditions. They are prone to errors when measuring in complex environments or on non-uniform surfaces, thus limiting their application range.

[0014] 3. Difficulty in meeting dynamic measurement needs: Existing angle measurement technologies have limitations in terms of operating speed, measurement frequency, and data processing capabilities. Especially in scenarios requiring real-time detection or high-frequency measurement, they often cannot respond quickly and output stable and reliable results. Summary of the Invention

[0015] In view of this, the present invention proposes a high-resolution angle deviation measurement method based on phase-sensitive CPSDOCT. This method utilizes the high resolution, non-contact and fast imaging characteristics of OCT technology to significantly improve the accuracy and sensitivity of angle measurement, overcome the limitations of traditional technologies in terms of stability and applicability, and achieve real-time and efficient angle imaging and measurement.

[0016] A method for measuring angular deviation, comprising:

[0017] First, the plane mirror is used as the test plane, and three optical fibers are used as the three optical paths of the OCT system. They are placed near the test plane, and the probe positions of the three optical fibers are fixed.

[0018] Then, the plane to be measured is rotated and translated, and the OCT system obtains the measurement results respectively; based on the measurement results, the probe positions of the three optical fibers are calibrated;

[0019] When measuring the deflection angle of the plane under test, the measurement result is obtained through the OCT system, and the deflection angle of the plane under test is calculated by combining the probe positions of the three optical fibers after calibration.

[0020] Preferred methods for calibrating the probe positions of the three optical fibers include:

[0021] With the location of probe A as the origin, a spatial rectangular coordinate system is established along the direction of the optical fiber as the Z-axis. The X-axis is on the horizontal plane, and the Y-axis is determined by the Z-axis and the X-axis. Let b represent the distance from probe B to probe A in the X-axis direction, and c represent the distance from probe C to probe A in the Y-axis direction.

[0022] Let l b , l c The distances from probe B and probe C to probe A along the Z-axis are represented by the following values: The position coordinates of the three probes A, B, and C are as follows:

[0023] A(0,0,0)

[0024] B(b, 0, l) b )

[0025] C(0, c, l) c )

[0026] The detection plane was imaged for the first time using an OCT system. The results obtained from the three probes A, B, and C were A1, B1, and C1, respectively. Using the known relationships, the position coordinates of the three detection points A1, B1, and C1 in the plane were expressed as follows:

[0027] A1(0, 0, A1)

[0028] B1(b,0,B1+l b )

[0029] C1(0, c, C1+l) c )

[0030] The reflector plane is rotated around the X-axis, and the detection plane is imaged a second time using an OCT system. The results obtained from the three probes A, B, and C are A2, B2, and C2, respectively. Using known relationships, the position coordinates of the three probes A2, B2, and C2 within the plane are simultaneously expressed:

[0031] A2(0, 0, A2)

[0032] B2(b,0,B2+l b )

[0033] C2(0,c,C2+l c )

[0034] Let the normal vectors of the plane to be measured in the first and second imaging be respectively:

[0035]

[0036] The equations of the two planes are expressed as follows:

[0037] n1x + n2y + n3(z - A1) = 0

[0038] m1x + m2y + m3(z - A2) = 0

[0039] Points A1, B1, and C1 all lie on the plane of the first image, while points A2, B2, and C2 all lie on the plane of the second image. Substituting the coordinates, we obtain two sets of equations as follows:

[0040] n1b+0+n3(B1-A1+l b ) = 0

[0041] 0+n2c+n3(C1-A1+l c ) = 0

[0042] m1b+0+m3(B2-A2+lb ) = 0

[0043] 0+m2c+m3(C2-A2+l c ) = 0

[0044] Since plane 2 is obtained by rotating plane 1 around the X-axis, it is equivalent to the normal vector. Rotate around the X-axis to become a normal vector

[0045] In the first and second imaging processes mentioned above, the normal vector Rotate the unit vector i = (1,0,0) around the X-axis to make it the normal vector. Let the rotation angle be θ, then we obtain the transformation matrix:

[0046]

[0047] Will Substitute The relationship between the normal vector coordinates is as follows:

[0048] m1=n1

[0049] m2=n2cosθ-n3sinθ

[0050] m3=n2sinθ+n3cosθ

[0051] The result can be calculated based on the previous equation.

[0052] l b =A1-B1=A2-B2

[0053]

[0054] If the plane to be measured is moved by y0 along the Y direction, the original detection point A1 will move along the Y direction to become A1*(0, y0, A1). Record the detection results A3, B3, and C3 of the OCT system at this time. Using the known relationships, simultaneously express the position coordinates of the three probes at the three detection points A3, B3, and C3 in the plane:

[0055] A3(0, 0, A3)

[0056] B3(b,0,B3+l b )

[0057] C3(0, c, C3+l) c )

[0058] The plane normal vector remains unchanged and is still [value]. Representing the plane equation

[0059] n1x + n2(y - y0) + n3(z - A1) = 0

[0060] B3 and C3 lie in the plane. Substituting the coordinates, we get the following equation:

[0061] 0 - n2y0 + n3(A3 - A1) = 0

[0062] n1b+n2y0+n3(B3-A1+l b ) = 0

[0063] 0+n2(c-y0)+n3(C3-A1+l c ) = 0

[0064] The result can be calculated based on the above equation.

[0065] (A3-A1)·(c-y0) / y0+C3-A1=-l c

[0066] Thus, l can be obtained accurately. b , l c The length of the probe was used to calibrate the positions of the three optical fibers.

[0067] Preferred methods for calculating the deflection angle of the plane to be measured include:

[0068] Given any three points P1, P2, and P3 in a plane, then we have The plane normal vector (x, y, z) is calculated using the following formula:

[0069] x = y1z2 - y2z1

[0070] y = z1x2 - z2x1

[0071] z = x1y2 - x2y1

[0072] When measuring the deflection angle of the plane under test, the coordinates of three probes are measured using an OCT system:

[0073] Ai(0,0,A i )

[0074] Bi(b,0,B i +l b )

[0075] Ci(0, c, C) i +l c )

[0076] Representing a vector in the plane:

[0077]

[0078] The plane normal vector is obtained by solving:

[0079] r x =-c·(B i +l b -A i )

[0080] r y =-b·(C i +l c -A i )

[0081] r z =bc

[0082] Calculate the tilt angles α, β, and γ of the mirror plane in the three axial directions using the following formulas:

[0083]

[0084] The present invention has the following beneficial effects:

[0085] 1. High-precision measurement: Through high-resolution imaging of OCT technology, very accurate three-dimensional surface data can be obtained, and then extremely small angular changes can be calculated.

[0086] 2. Non-contact measurement: OCT technology does not rely on traditional contact measurement tools, avoiding errors introduced by contact friction, deformation and other factors, and is suitable for measuring precision or fragile objects.

[0087] 3. Automation and real-time performance: This invention enables rapid measurement and is suitable for real-time quality control. Attached Figure Description

[0088] Figure 1 This is a hardware system architecture diagram;

[0089] Figure 2 shows the experimental structure and schematic diagram. (a) is a physical image of the fixed imaging of the fiber optic probe, and (b) is a schematic diagram of the imaging of the fiber optic probe.

[0090] Figure 3 This is a schematic diagram of a planar detection structure;

[0091] Figure 4 This is a spatial vector decomposition. Detailed Implementation

[0092] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0093] 1. Experimental setup

[0094] Building the product system structure, such as Figure 1SD-OCT uses a broadband light source emitting 840nm laser light, which is split by a 50:50 coupler to form three detection optical paths. The three detection signals are reflected by the detection surface, interfering with the original signals to form interference signals. These interference signals are received by a spectrometer, collimated, and split by a grating before being detected by a linear array camera for photoelectric conversion. After a series of signal processing steps, including FFT, an OCT image of the sample is obtained, which is then processed and analyzed to calculate angles. The basic structure of an OCT system typically consists of a reference arm and a sample arm, each measuring the position of three points in a plane. SD-OCT requires three optical paths. Common-path OCT, however, combines the reference arm and sample arm into a single optical path, using the end face of a single-mode fiber as a reflection plane to obtain a reference light signal that interferes with the reflected light from the sample. This transforms the three interference optical paths into three optical fibers, allowing three CP-OCTs to simultaneously measure interference signals at different positions in the same optical path. The positions of the three intensity peaks of the signal represent the detection depth.

[0095] To measure the planar deflection angle using CP-SDOCT, we constructed the experimental structure shown in Figure 2. (a) shows the actual structure with the fiber optic probe fixed for imaging. The planar mirror serves as the plane to be measured, and it is fixed on a rotation platform and a displacement platform to control its translational and rotational movements. Slight changes in the mirror's position are monitored to calculate the corresponding tilt angle. (b) is a schematic diagram of the fiber optic probe imaging. Three probe fibers are fixed near the plane to be measured, maintaining their relative positions. This allows us to detect the distance of the multi-path CP-SDOCT from the plane to be measured in the depth direction.

[0096] 2. Phase measurement

[0097] Unlike traditional OCT, which primarily relies on light intensity, phase OCT not only acquires light intensity signals but also captures and analyzes phase changes. By introducing phase measurement into OCT, it provides richer structural and dynamic information than traditional OCT, enabling the detection and measurement of minute structures with higher precision. Phase information provides more detailed structural features than intensity information, offering particular advantages in measuring the fine details of object surfaces and internal structures.

[0098] In depth-domain phase detection, the interference signal is first converted to the depth domain using a Fast Fourier Transform (FFT), resulting in a depth-domain interference signal containing both intensity and phase information. The depth-domain interference signal can be represented as:

[0099]

[0100] Due to the symmetry of the Fourier transform of the cosine signal, the image of the sample has the same intensity but opposite phase to the true information; therefore, the image of the sample can be ignored. If the light source spectrum is symmetrically distributed in wavenumber space, then when z... i When z = z0, the intensity of the interference signal is maximum. A tiny change in the optical path difference between the reference arm and the sample surface will amplify the phase change by a factor of 2k0. Therefore, this method can be used to perform highly sensitive measurements of the sample surface morphology. Only when the true phase... The measured phase is valid only when the signal is within the range of [-π, π], and the dynamic range of the corresponding optical path difference cannot exceed half the center wavelength of the light source. The optical path difference (OPD) is calculated from the position and phase of the signal in the depth domain.

[0101]

[0102] 3. Device Calibration

[0103] Before the formal measurement, we need to calibrate the relative position of the probe. Starting from the actual structure, we establish a spatial rectangular coordinate system with the position of probe A as the origin and the direction of the optical fiber as the Z-axis. The X-axis is on the horizontal plane, and the Y-axis is determined by the Z-axis and the X-axis. It is known that b represents the distance from probe B to probe A in the X-axis direction, and c represents the distance from probe C to probe A in the Y-axis direction.

[0104] Let l b , l c The distances from probe B and probe C to probe A along the Z-axis are represented by the following values: The position coordinates of the three probes A, B, and C are as follows:

[0105] A(0,0,0)

[0106] B(b,0,l b )

[0107] C(0,c,l c )

[0108] The OCT system was used to perform the first imaging of the detection plane, and the results obtained from the three probes A, B, and C were A1, B1, and C1, respectively. Using the known relationships, the position coordinates of the three detection points A1, B1, and C1 in the plane can be represented as follows:

[0109] A1(0, 0, A1)

[0110] B1(b,0,B1+l b )

[0111] C1(0,c,C1+l c )

[0112] With a slight adjustment to the rotating platform, the reflector will rotate with the platform, and the plane will rotate around the X-axis. A second image of the detection plane is then created using an OCT system, yielding results from the three probes A, B, and C, which are designated as A2, B2, and C2, respectively. Using known relationships, the position coordinates of the three probes at points A2, B2, and C2 within the plane are simultaneously represented:

[0113] A2(0, 0, A2)

[0114] B2(b,0,B2+l b )

[0115] C2(0,c,C2+l c )

[0116] Let the normal vectors of the first and second imaging detection planes be respectively:

[0117]

[0118] The equations for plane 1 and plane 2 are expressed as follows:

[0119] n1x + n2y + n3(z - A1) = 0

[0120] m1x + m2y + m3(z - A2) = 0

[0121] Points A1, B1, and C1 all lie on plane 1, and points A2, B2, and C2 all lie on plane 2. Substituting the coordinates, we obtain two sets of equations as follows:

[0122] n1b+0+n3(B1-A1+l b ) = 0

[0123] 0+n2c+n3(C1-A1+l c ) = 0

[0124] m1b+0+m3(B2-A2+l b ) = 0

[0125] 0+m2c+m3(C2-A2+l c ) = 0

[0126] Since plane 2 is obtained by rotating plane 1 around the X-axis, it is equivalent to the normal vector. Rotate around the X-axis to become a normal vector Any vector v in space, rotated clockwise by an angle θ around a fixed unit vector n, yields vector v′. This vector can be decomposed, such as... Figure 4 As shown:

[0127] v || =(v T n)n

[0128] v ⊥ =vv || =v-(v T n)n

[0129] v ⊥ =vv||=v-(v T n)n

[0130] Rotate and superimpose the decomposed vectors:

[0131] v′ ⊥ =v ⊥ cosθ+wsinθ

[0132] =[v-(v T [n)n]cosθ+(n×v)sinθ

[0133] v′=v′ ⊥ +v ||

[0134] =[v-( T n)n]cosθ+(n×v)sinθ+(v T n)n

[0135] Substituting the three unit vectors i, j, and k into v, we get:

[0136]

[0137] Finally, we obtain the transformation matrix T for vector rotation:

[0138]

[0139] In space, any vector V is obtained by rotating it clockwise by an angle θ around a fixed unit vector n to obtain vector V'.

[0140] V'=T·V

[0141] In the first and second imaging experiments mentioned above, the normal vector Rotate the unit vector i = (1,0,0) around the X-axis to make it the normal vector. Let the rotation angle be θ, then the transformation matrix can be obtained:

[0142]

[0143] Will Substitute The relationship between the normal vector coordinates is as follows:

[0144] m1=n1

[0145] m2=n2cosθ-n3sinθ

[0146] m3=n2sinθ+n3cosθ

[0147] The result can be calculated based on the previous equation.

[0148] l b =A1-B1=A2-B2

[0149]

[0150] The line connecting probes A and B is parallel to the X-axis. Their relative positions in the Y-direction remain unchanged during rotation. The difference between the two peaks A and B is determined only by the difference in probe lengths, i.e., the length of 1b. The length of 1c is related to the rotation angle. The angle of each rotation is recorded, and the results of two detections are used to solve for 1c.

[0151] Then, the probe plane was moved to continue calibration. b , l c If plane 1 is moved by y0 along the Y direction, the original detection point A1 will also move along the Y direction to become A1*(0, y0, A1). Record the detection results A3, B3, and C3 of the OCT system at this time. Using the known relationships, simultaneously express the position coordinates of the three probes at the three detection points A3, B3, and C3 in the plane.

[0152] A3(0, 0, A3)

[0153] B3(b,0,B3+l b )

[0154] C3(0, c, C3+l) c )

[0155] The plane normal vector remains unchanged and is still [value]. It can represent the equation of a plane.

[0156] n1x + n2(y - y0) + n3(z - A1) = 0

[0157] B3 and C3 lie in the plane. Substituting the coordinates, we get the following equation:

[0158] 0 - n2y0 + n3(A3 - A1) = 0

[0159] n1b+n2y0+n3(B3-A1+l b ) = 0

[0160] 0+n2(c-y0)+n3(C3-A1+l c ) = 0

[0161] The result can be calculated based on the above equation.

[0162] (A3-A1)·(c-y0) / y0+C3-A1=-l c

[0163] Thus, l can be obtained accurately. b , l c The length of the apparatus was determined to complete the calibration of the entire experimental setup.

[0164] 4. Angle Calculation

[0165] Given any three points P1, P2, and P3 in a plane, then we have The plane normal vector (x, y, z) can be calculated using the following formula:

[0166] x = y1z2 - y2z1

[0167] y = z1x2 - z2x1

[0168] Z = x1y2 - x2y1

[0169] In this experiment, the coordinates of the detection point can be represented by any single measurement result.

[0170] Ai(0,0,A i )

[0171] Bi(b,0,B i +l b )

[0172] Ci(0,c,C i +l c )

[0173] Representing a vector in the plane:

[0174]

[0175] The plane normal vector is obtained by solving:

[0176] r x =-c·(B i +l b -A i )

[0177] r y =-b·(C i +l c -A i )

[0178] r z =bc

[0179] Calculate the tilt angles α, β, and γ of the mirror plane in the three axial directions using the following formulas:

[0180]

[0181] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for measuring angular deviation, characterized in that, include: First, the plane mirror is used as the plane to be measured, and three optical fibers are used as the three optical paths of the OCT system. They are placed near the plane to be measured, and the probe positions of the three optical fibers are fixed. Then, the plane to be measured is rotated and translated, and the OCT system obtains the measurement results respectively; based on the measurement results, the probe positions of the three optical fibers are calibrated; When measuring the deflection angle of the plane to be measured, the measurement result is obtained through the OCT system, and the deflection angle of the plane to be measured is calculated by combining the probe positions of the three optical fibers after calibration. Methods for calibrating the probe positions of the three optical fibers include: With the location of probe A as the origin, a spatial rectangular coordinate system is established along the direction of the optical fiber as the Z-axis. The X-axis is on the horizontal plane, and the Y-axis is determined by the Z-axis and the X-axis. Let b represent the distance from probe B to probe A in the X-axis direction, and c represent the distance from probe C to probe A in the Y-axis direction. set up The distances from probe B and probe C to probe A along the Z-axis are represented by the following coordinates: The detection plane was first imaged using an OCT system, and the results obtained from probes A, B, and C were as follows: Using the known relationships, represent the position coordinates of the three detection points A1, B1, and C1 in the plane: The reflector plane is rotated around the X-axis, and the detection plane is imaged a second time using an OCT system. The results obtained from probes A, B, and C are as follows: Using known relationships, the position coordinates of three probes at three detection points A2, B2, and C2 in the plane can be simultaneously expressed: Let the normal vectors of the plane to be measured in the first and second imaging be respectively: The equations of the two planes are expressed as follows: Points A1, B1, and C1 all lie on the plane of the first image, while points A2, B2, and C2 all lie on the plane of the second image. Substituting the coordinates, we obtain two sets of equations as follows: Since plane 2 is obtained by rotating plane 1 around the X-axis, it is equivalent to the normal vector. Rotate around the X-axis to become a normal vector ; In the first and second imaging processes mentioned above, the normal vector Rotate the unit vector i = (1,0,0) around the X-axis to make it the normal vector. Let the rotation angle be θ, then we obtain the transformation matrix: Will Substitute The relationship between the normal vector coordinates is as follows: The result can be calculated based on the previous equation. If the plane to be measured is adjusted and moved along the Y direction Then the original detection point A1 moves along the Y direction and becomes Record the detection results of the OCT system at this time. Using known relationships, the position coordinates of three probes at three detection points A3, B3, and C3 in the plane can be simultaneously expressed: The plane normal vector remains unchanged and is still [value]. , representing the plane equation In the plane, substituting the coordinates, we obtain the following equation: The result can be calculated based on the above equation. Thus, we can obtain the precise result. The length of the probe is used to calibrate the position of the three optical fibers; Methods for calculating the deflection angle of the plane to be measured include: Given any three points P1, P2, and P3 in a plane, then we have , The plane normal vector (x, y, z) is calculated using the following formula: When measuring the deflection angle of the plane under test, the coordinates of three probes are measured using an OCT system: Representing a vector in the plane: ; The plane normal vector is obtained by solving: Calculate the tilt angle of the mirror plane in the three axial directions using the following formulas. , and : 。

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