A full-geometric error displacement calculation method based on beam trajectory without approximate simplification

Abstract

The present application belongs to the technical field of laser interference precision measurement, and provides a full-geometric error displacement calculation method based on beam trajectory without approximate simplification. The method provided by the present application is based on the actual beam trajectory in the measurement process of a plane interferometer, can completely calculate all optical and mechanical geometric errors, completely calculate the real refraction condition, and does not introduce any approximate error in the simplification process, and the displacement calculation process conforms to the real beam trajectory condition. The method provided by the present application simplifies all reflection phenomena inside the interferometer, integrates the parameters of the reflecting surface and the receiving surface of the reflected beam, optimizes the originally complex multi-beam calculation process, reduces the number of parameter items required for complete description of geometric errors, and simultaneously deletes the beams that do not affect the calculation of the total optical path difference, so that the calculation efficiency is high.

Description

Technical Field

[0001] This invention belongs to the field of laser interferometry precision measurement technology, specifically relating to a method for calculating displacement of full geometric error based on beam trajectory without approximation simplification. Background Technology

[0002] The planar four-division interferometer, as a high-precision non-contact single-axis linear displacement measuring instrument, is widely used in precision motion testing, precision sensor calibration, and high-end equipment manufacturing. Laser interferometers obtain the relative optical path change between the two laser paths by comparing the phase between the internal reference laser emitted from the light source and the measurement laser emitted onto the plane to be measured, thus accurately obtaining the relative displacement change of the target object in the direction of the measurement beam. Compared with the traditional Michelson interferometer, the optical four-division planar interferometer has significant advantages in optical magnification and measurement tolerance angle characteristics, making it widely favored in multi-degree-of-freedom motion measurement with a plane as the main target, such as: high-precision six-degree-of-freedom motion table pose calibration, and high-precision inspection of six-degree-of-freedom mass cube measurement.

[0003] In six-degree-of-freedom measurement methods based on iterative calculations, the planar interferometer serves as the fundamental measurement unit, and its displacement calculation model must be accurately established. The displacement calculation model is defined as follows: by calculating the relative optical path change of the laser along the measurement path based on the motion of the target plane and the geometric parameters of the interferometer's internal optomechanical structure, the displacement value corresponding to each motion is obtained. Existing methods for establishing displacement calculation models mainly fall into two categories: oversimplified displacement calculation methods without geometric errors and segment-by-segment beam trajectory displacement calculation methods with full geometric errors.

[0004] The first type of oversimplified displacement calculation method, which eliminates geometric errors, simplifies the complex optomechanical structure of a laser interferometer into two simple geometric elements: the laser emission point and the laser receiving surface. It only calculates the optical path change from the emission point to the receiving surface, and the model's optomechanical geometric error sources are only considered in the pose of the receiving surface and the position of the emission point. This simplified displacement calculation process is simple and has fewer geometric parameters, resulting in extremely high computational efficiency in iterative six-degree-of-freedom (DOF) solutions. It can analyze geometric errors caused by installation mismatches between multi-axis interferometers. However, because this method oversimplifies the model to meet the efficiency requirements of iterative calculations, it lacks the ability to analyze the complex optomechanical structure inside the interferometer, the true trajectory of the beam, and the processes of beam refraction and reflection. This leads to low accuracy in the six-DOF solution method using this model, making it unsuitable for high-precision measurement scenarios.

[0005] The second type of method, the segment-by-segment beam trajectory full geometric error displacement calculation method, models the polarizing beam splitter, cornerstone prism, and reflector structures inside the planar interferometer separately. This yields the optical path calculation formula for the beam trajectory passing through each optomechanical structure. Furthermore, the optical path segments are summed one by one to obtain the total optical path of the laser in the measurement path. This method can comprehensively analyze all geometric error sources within the planar interferometer and fully consider the reflection and refraction processes of the actual measurement beam, accurately reflecting the optical path variation of the beam in the measurement path inside the interferometer. However, this model requires calculation of too many optomechanical structures, resulting in low efficiency in single-shift displacement calculations and a large number of geometric error terms. This makes it difficult to meet the efficiency requirements of iterative six-degree-of-freedom solution methods for single-axis interferometer displacement calculations.

[0006] In summary, planar four-subdivision interferometry urgently needs a displacement calculation method that balances computational accuracy and efficiency. Specifically, planar four-subdivision interferometry lacks a displacement calculation method that is highly efficient, has a simplified, error-free process, comprehensively analyzes geometric influencing factors, and features simple and reasonable geometric parameter settings. Summary of the Invention

[0007] To address the aforementioned technical problems, this invention provides a method for calculating displacement based on the full geometric error without approximation simplification of beam trajectories, thereby resolving the issues in the prior art. The technical solution adopted by this invention is as follows: A method for calculating displacement based on full geometric error without approximation simplification of beam trajectory includes the following steps: Step 1, simplify the laser trajectory of the measurement path: simplify all reflection phenomena inside the interferometer without approximation error, including all reflection phenomena on the front plane mirror of the detector, the three internal reflection surfaces of the corner cube prism, and the beam splitting surface of the polarizing beam splitter; then further simplify the geometric elements that do not affect the total optical path change. Step 2: Establish a complete geometric error model of the optomechanical structure: Set error parameters for the following geometric elements to fully describe all sources of geometric error inside the interferometer, including: position deviation of the symmetrical equivalent pyramid vertex, pose deviation of the waveplate exit surface, position deviation of the first intersection point of the waveplate exit surface, angle deviation of the first emitted beam of the waveplate exit surface, and pose deviation of the equivalent detector plane. Step 3, calculate the displacement caused by the motion of the target: calculate the total optical path of the laser in the measurement path when the target is in the two states before and after the motion, and calculate the displacement caused by the motion of the target based on the total optical path difference.

[0008] Furthermore, step 1 includes: Step 1.1, Simplification of the front plane mirror of the detector without approximation error: Construct the plane of symmetry of the real detector receiving surface with respect to the front plane mirror of the detector: the equivalent detector plane; Ignore the reflected beam on the front plane mirror of the detector, and extend the incident beam to the equivalent detector plane; Calculate the optical path from the starting point of the incident beam to the intersection with the equivalent detector plane, and define this optical path as the total optical path of the actual beam trajectory between the starting point of the beam and the real detector receiving surface. Step 1.2, Simplification of internal reflection of the pyramidal prism without approximation: Consider the pyramidal prism as a plane passing through the vertex of the pyramid and perpendicular to the incident beam of the pyramidal prism: the pyramidal equivalent surface; define the intersection of the incident beam of the pyramidal prism and the pyramidal equivalent surface as the incident point of the pyramidal equivalent surface; construct the symmetrical point about the vertex of the pyramidal equivalent surface as the exit point of the pyramidal equivalent surface; define the exit beam of the pyramidal prism as the beam passing through the exit point of the pyramidal equivalent surface and parallel to the incident beam of the pyramidal prism; calculate the sum of the optical paths from the first reflection point of the beam splitting surface to the incident point of the pyramidal equivalent surface and from the exit point of the pyramidal equivalent surface to the second reflection point of the beam splitting surface, and consider this optical path to be the total optical path of the true beam trajectory between the starting point of the incident beam of the pyramidal prism and the second reflection point of the beam splitting surface; Step 1.3: Simplification of the beam-splitting surface of the polarizing beam-splitting prism without approximations: Construct the symmetrical point of the pyramid vertex with respect to the beam-splitting surface of the polarizing beam-splitting prism: symmetrical equivalent vertex; Construct the symmetrical surface of the pyramid equivalent with respect to the beam-splitting surface of the polarizing beam-splitting prism: quadratic pyramid equivalent surface; Extend the incident beam from the beam-splitting surface to the quadratic pyramid equivalent surface, intersecting the quadratic pyramid equivalent surface at the incident point of the quadratic pyramid equivalent surface; Construct the symmetrical point of the incident point of the quadratic pyramid equivalent surface with respect to the symmetrical equivalent vertex: the exit point of the quadratic pyramid equivalent surface; Calculate the sum of the optical paths from the starting point of the incident beam from the beam-splitting surface to the incident point of the quadratic pyramid equivalent surface, and from the exit point of the quadratic pyramid equivalent surface to the end point of the exit beam from the beam-splitting surface, and consider this optical path to be the total optical path of the true beam trajectory between the starting point of the incident beam from the beam-splitting surface and the end point of the exit beam from the beam-splitting surface; Step 1.4: Simplification of the beam exit point without approximation: Ignore the beam trajectory between the actual laser exit point and the first intersection point of the waveplate exit surface on the waveplate, and define the first intersection point of the waveplate exit surface on the waveplate as the laser exit point of the entire system.

[0009] Furthermore, in step 1.1, the optical path from the starting point of the incident beam to the intersection with the equivalent detector plane is expressed as: ; in, l D The length of the beam trajectory. V DI and P DI These represent the direction vector and the starting position vector of the incident beam, respectively. P RD and VRD Let represent a point on the equivalent detector plane symmetrical to the reflector and its normal vector, respectively. In step 1.2, the beam emitted from the cornerstone prism is defined as a beam that passes through the exit point of the equivalent surface of the cornerstone and is parallel to the incident beam of the cornerstone prism. The equation of this beam is expressed as follows: ; in, P V Represents the position vector of the vertex of the pyramid. P RI This represents the position vector of the first reflection point of the beam-splitting surface. V RI This represents the direction vector of the incident beam from the cornerstone prism. In step 1.2, the sum of the optical path lengths from the first reflection point of the beam-splitting surface to the incident point of the pyramidal equivalent surface and from the exit point of the pyramidal equivalent surface to the second reflection point of the beam-splitting surface is calculated. The expression is: ; in P RO This represents the position vector of the second reflection point on the beam-splitting surface; In step 1.3, the sum of the optical path lengths from the starting point of the incident beam at the beam splitter to the incident point of the equivalent surface of the quadratic pyramid, and from the exit point of the equivalent surface of the quadratic pyramid to the ending point of the exit beam at the beam splitter is calculated. The expression is: ; in P BSO Let be the position vector of the endpoint of the beam emitted from the beam splitter. P VV The position vector of the symmetric equivalent vertex. V BSI Let be the direction vector of the incident beam at the beam splitter. P BSI To extend the incident beam from the beam splitter to the starting point of the incident beam on the equivalent surface of the second pyramid; Furthermore, in step 1.4, In step 1.4, the equation of the straight line of the first emitted beam from the waveplate exit surface is expressed as follows: ; in, P S This represents the position vector of the exit point of the first emitted beam from the exit surface of the waveplate. V S Indicates the direction vector of the emitted beam. t Points on the beam trajectory P Time P S The length. P Sand V S This can be further expressed as: ; ; In the above formula, P L and V L Let represent a point on the actual trajectory of the laser beam emitted from the real laser emission point and its direction vector, respectively. P i and V i Let these represent a point on the incident surface of the polarizing beam splitter and the waveplate, and its plane normal vector, respectively. P e and V e Let these represent a point on the exit surface of the polarizing beam splitter and the waveplate, and its plane normal vector, respectively. e It represents the relative refractive index of the air medium to the crystalline medium.

[0010] Furthermore, step 2 includes: Step 2.1, construct the geometric error model of the symmetrical equivalent pyramid vertex: use the translation matrix to construct the positional deviation of the three degrees of freedom of the symmetrical equivalent pyramid vertex, so as to realize the complete description of the geometric error of the position of the real pyramid prism vertex and the pose of the polarizing beam splitter surface; Step 2.2, construct the geometric error model of the waveplate exit plane: use rotation and translation matrices to construct one degree of freedom positional deviation and two degrees of freedom angular tilt of the waveplate exit plane; Step 2.3, construct the geometric error model of the first intersection point of the waveplate exit surface: use the translation matrix to construct the positional deviation of the two degrees of freedom of the first intersection point of the waveplate exit surface; Step 2.4, construct the geometric error model of the first emitted beam from the waveplate exit surface: use a rotation matrix to construct the two degrees of freedom angle tilt of the first emitted beam from the waveplate exit surface; Step 2.5, construct the geometric error model of the equivalent detector plane: use the rotation matrix to construct the two degrees of freedom of the equivalent detector plane angle tilt.

[0011] Furthermore, in step 2.1, the translation matrix... M tePVV The expression is: ; in, x tePVV , y tePVV , z tePVVThese represent the deviations of the symmetric equivalent vertex in the three degrees of freedom directions in space, respectively. In step 2.2, the ideal waveplate exit surface is perpendicular to the X-axis, and a position vector passing through the X-axis is... P SHW Let the point be a point whose direction vector is . V SHW The degree of freedom of positional deviation in a plane is expressed as a point. P SHW Translation on the X-axis x teSHW Angular deviation is expressed as a vector. V SHW Angles of rotation around the Y and Z axes p reSHW and angle y reSHW Rotation matrix M reSHW Translation matrix M teSHW The expressions are as follows: ; ; In step 2.3, a translation matrix is ​​used to construct... M tePS Constructing the position vector of the first intersection point of the waveplate exit surface P S The positional deviation of two degrees of freedom, the translation matrix is ​​constructed. M tePS The expression is: ; In step 2.4, a rotation matrix is ​​used. M reVS To construct the two degrees of freedom of the first emitted beam from the waveplate exit surface, tilting at angles; rotation matrix M reVS The expression is: ; in p reVS and y reVS They represent the ideal direction vectors respectively. V S The deviation angles in the directions around the Y and Z axes; In step 2.5, a rotation matrix is ​​used. M reVRD Construct the equivalent detector plane with two degrees of freedom of angular tilt and rotation matrix. M reVRD The expression is: ; in p reVRD and y reVRD They represent the ideal normal vectors respectively. V RD The deviation angles in the Y-axis and Z-axis directions.

[0012] Furthermore, step 3 includes: Step 3.1, Calculation of the total optical path of the simplified beam trajectory before target motion: Based on the geometric parameters set in Step 2 and the initial pose parameters of the target, the simplified beam trajectory after simplification in Step 1 is calculated segment by segment to obtain the total optical path of the simplified beam trajectory before the target motion. δ A ; Step 3.2: Calculation of the total optical path of the simplified beam trajectory after the target motion: Based on the geometric parameters set in Step 2 and the motion parameters of the target, the simplified beam trajectory after the simplification in Step 1 is calculated segment by segment to obtain the total optical path of the simplified beam trajectory after the target motion. δ B ; Step 3.3: Calculate the optical path difference and displacement value: Based on the total optical path results obtained in Steps 3.1 and 3.2, calculate the total optical path difference Δ caused by the motion of the target. δ BA Then calculate the displacement. d .

[0013] Furthermore, step 3.1 includes: First beam L A1 : From the exit surface S of the waveplate HW The starting point P 1 to the point where the beam intersects the plane of the target mirror P A2 Up to this point; the first beam L is combined A1 The equation of the straight line and the target plane S to be measured AM The plane equations yield the intersection points P A2 Position vector and length of the beam segment δ A1 The subscript A indicates the value in the initial state; where the first beam segment L A1 The equation of the straight line and the target plane S to be measured AM The equation of the plane is expressed as follows: ; The corresponding expressions for calculating the intersection point and beam length are as follows: ; in V 1. PAM and V AM These represent the first beam L. A1 Direction vector, target plane S AM The position vector of the previous point and the normal vector of the plane; Second beam L A2 At the intersection P A2 Reflected from the target plane, from the intersection point P A2 The beam exits from the waveplate at surface S. HW intersection P A3 Up to; the direction vector of this beam segment V A2 The expression for this calculation, based on the law of reflection, is: ; The length of the second beam is obtained through the same process as for the first beam segment. l A2 ; Third beam L A3 The beam of light at the intersection P A3 The beam is refracted from air into a crystal medium, and the equivalent surface S at the vertex of the symmetrical pyramid is formed by the intersection of the beam splitting planes. ASR With point P A4 Calculate the direction vector of the beam according to the law of refraction. V A3 Its expression is: ; in V HW n a and n c Represent the normal vector of the waveplate plane, the refractive index of the air medium, and the refractive index of the crystal medium, respectively; the position vector of the symmetric equivalent vertex of the pyramid. P VV and the direction vector of the third beam V A3 The equivalent surface S of the symmetric pyramid vertex is obtained by the beam splitter. ASR The equation of the plane is expressed as follows: ; Solving this equation simultaneously with the third beam L A3 The equation of the straight line is used to obtain the length of the third beam segment. l A3 and intersection P A4 ; Fourth beam LA4 The beam exits from the equivalent surface of the pyramid. P A5 Emitted, to the beam L A4 With waveplate exit surface S HW intersections P A6 Up to this point, the direction is parallel to and opposite to the third beam segment; exit point of the pyramidal equivalent surface. P A5 and equivalent incident point P A4 Regarding the equivalent vertex of a pyramid P VV If symmetric, then calculate the position vector. P A5 The expression is: ; By combining the beam equation and the plane equation, we obtain the intersection point. P A6 and the length of that beam l A4 ; Fifth beam L A5 This beam exits from waveplate exit surface S. HW Points on P A6 The emitted beam, reaching the target plane S, is... AM intersection P A7 Up to this point, the direction vector and the second beam L A2 Conversely, the starting point position vector is P A6 Coordinates; Combine the fifth beam L A5 The equation of the straight line and the target plane S AM The equation of the plane is obtained, and the intersection point is obtained. P A7 Position vector and length of the fifth beam l A5 .

[0014] Sixth beam L A6 This beam originates from the target plane S. AM Points on P A7 Emitted, with the waveplate exit surface S HW Intersection point P A8 The direction vector is the same as the first beam L. A1 Conversely, the starting point position vector is P A7 Coordinates; Solve the equation of the straight line L of the sixth beam segment. A6 With waveplate exit surface S HW The equation of the plane is given, and the point is solved.P A8 Position vector and length of the sixth beam l A6 .

[0015] Seventh beam L A7 The beam of light originates from a point on the exit surface of the waveplate. P A8 The equivalent plane S of the cross-reflector symmetrical detector is emitted. RD At point P A9 The direction vector of this beam segment is obtained using the refraction formula; then, the equation of the straight line of this beam segment and the equivalent plane S of the mirror-symmetric detector are solved simultaneously. RD The plane equations yield the intersection points P A9 Position vector and length of the seventh beam l A7 ; When the target plane is in its initial state, the total optical path of the current simplified beam is obtained. δ A for: ; in δ An Let be the optical path length of each beam segment.

[0016] Furthermore, the calculation process in step 3.2 is the same as in step 3.1, with the lengths of the seven beam segments after the target plane has moved being denoted as follows: l B1 , l B2 , l B3 , l B4 , l B5 , l B6 and l B7 ; Total optical path of the beam after the target plane moves δ B for: ; in δ Bn Let be the optical path length of each beam segment.

[0017] Furthermore, in step 3.3, displacement d The expression is: .

[0018] The present invention has the following beneficial effects: (1) The method provided by the present invention is based on the actual beam trajectory during the measurement process of the plane interferometer. It can completely calculate all optomechanical geometric errors, completely calculate the real refraction situation, and does not introduce any approximation error in the simplification process. The displacement calculation process conforms to the real beam trajectory situation, so the displacement calculation accuracy is high. (2) The method provided by the present invention simplifies all reflection phenomena inside the interferometer, integrates the parameters of the reflecting surface and the receiving surface of the reflected beam, optimizes the originally complex multi-beam calculation process, reduces the number of parameters required to fully describe the geometric error, and deletes beams that do not affect the calculation of the total optical path difference, thus achieving high calculation efficiency. Attached Figure Description

[0019] Figure 1 This is a schematic flowchart of the method of the present invention; Figure 2 A schematic diagram of the complete trajectory of the beam in the plane interferometer measurement path to which this invention is applicable; Figure 2 In the middle: 1. Fiber optic coupler or laser; 2. Front reflector of detector; 3. Photodetector; 4. Cornerstone prism; 5. Measurement path laser; 6. Polarizing beam splitter; 7. Waveplate; 8. Target to be measured. Figure 3 A schematic diagram of the complete measurement beam trajectory before the target motion, simplified by step 1 of this invention; Figure 3 In the middle: the equivalent surface S of the quadratic pyramid SR 9. Symmetrical equivalent vertex; 10. Real laser emission point; 11. Equivalent surface emission point of a quadratic pyramid; 12. Equivalent detector plane S RD 13. Intersection of the front reflector plane of the detector 14. Intersection of the equivalent detector plane 15. Equivalent detector plane extended incident beam 16. Actual reflected beam 17. Front reflector plane S of the detector DM 18. Incident point of the equivalent surface of the second pyramid 19. Second reflection point of the beam-splitting surface 20. Outgoing beam from the beam-splitting surface 21. First reflection point of the beam-splitting surface 22. Beam-splitting surface S of the polarizing beam-splitting prism BS 23. Incident beam at the beam-splitting surface; 24. Waveplate exit surface; 25. First intersection point of the waveplate exit surface; 26. First exit beam from the waveplate exit surface; 27. Actual detector plane S D 28. Incident beam from the front reflector of the detector; 29. ​​Outgoing beam from the cornerstone prism; 30. Equivalent surface of the cornerstone, exit point; 31. Vertex of the cornerstone; 32. Equivalent surface of the cornerstone, S. R 33. Incident point on the equivalent surface of a pyramid 34. Extended incident beam on the equivalent surface of a pyramid 35. Incident beam from a pyramidal prism 36; Figure 4A schematic diagram showing the positional relationship between each simplified optical surface and the simplified beam trajectory before and after the target's motion; Figure 5 A graph showing the difference in displacement calculation results between the error-laden complete trajectory displacement calculation model and the oversimplified interferometer global analysis calculation model under the same conditions; Figure 6 This is a graph showing the difference in displacement calculation results before and after using the simplified model under the same conditions. Figure 7 This is a comparison chart showing the time required to calculate displacement before and after using the simplified model under the same conditions. Detailed Implementation

[0020] The following will be based on embodiments of the present invention. Figures 1-7 The technical solutions in the embodiments of the present invention will be clearly and completely described. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Unless otherwise specified, the technical means used in the embodiments are conventional means well known to those skilled in the art.

[0021] like Figure 1 This invention proposes a method for calculating displacement of full geometric error based on beam trajectory without approximation simplification, specifically including the following steps: Step 1: Simplify the laser trajectory of the measurement path: First, simplify all reflection phenomena inside the interferometer without approximation errors, including all reflection phenomena on the front plane mirror of the detector, the three reflecting surfaces inside the corner cube prism, and the beam-splitting surface of the polarizing beam-splitting prism. Then, further simplify the geometric elements that do not affect the total optical path change. Step 1 specifically includes: Step 1.1: Approximation-free simplification of the detector's front plane reflector: Construct the plane of symmetry of the real detector receiving surface with respect to the detector's front plane reflector: the equivalent detector plane. Ignore the reflected beam on the detector's front plane reflector and extend the incident beam to the equivalent detector plane. Calculate the optical path from the origin of the incident beam to the intersection with the equivalent detector plane, assuming this optical path is the total optical path of the actual beam trajectory from the beam's origin to the real detector receiving surface.

[0022] Step 1.2: Simplification of Internal Reflection of the Cornerstone Prism Without Approximations: Consider the cornerstone prism as a plane passing through the vertex of the pyramid and perpendicular to the incident beam: the equivalent plane of the pyramid. Name the intersection of the incident beam and the equivalent plane of the pyramid as the incident point of the equivalent plane. Construct the point symmetrical to the incident point of the equivalent plane about the vertex of the pyramid: the exit point of the equivalent plane. Consider the exit beam of the cornerstone prism as passing through the exit point of the equivalent plane and parallel to the incident beam. Calculate the sum of the optical paths from the origin of the incident beam to the incident point of the equivalent plane and from the exit point of the equivalent plane to the second reflection point of the beam splitter. Consider this optical path as the total optical path of the true beam trajectory between the origin of the incident beam and the second reflection point of the beam splitter.

[0023] Step 1.3: Simplification of the beam-splitting surface of the polarizing beam-splitting prism without approximations: Construct the symmetrical point of the pyramidal vertex with respect to the beam-splitting surface of the polarizing beam-splitting prism: the symmetrical equivalent vertex. Construct the symmetrical surface of the pyramidal equivalent with respect to the beam-splitting surface of the polarizing beam-splitting prism: the quadratic pyramidal equivalent surface. Extend the incident beam from the beam-splitting surface to the quadratic pyramidal equivalent surface, and name the intersection point as the incident point of the quadratic pyramidal equivalent surface. Construct the symmetrical point of the incident point of the quadratic pyramidal equivalent surface with respect to the symmetrical equivalent vertex: the exit point of the quadratic pyramidal equivalent surface. Calculate the sum of the optical path lengths from the starting point of the incident beam to the incident point of the quadratic pyramidal equivalent surface, and from the exit point of the quadratic pyramidal equivalent surface to the endpoint of the exit beam from the beam-splitting surface. Consider this optical path length as the total optical path length of the true beam trajectory between the starting point of the incident beam and the endpoint of the exit beam.

[0024] Step 1.4: Simplification of the beam exit point without approximation: Ignore the beam trajectory between the actual laser exit point and the laser exit point on the waveplate, and assume that the laser exit point on the waveplate is the laser exit point of the system.

[0025] Step 2: Establish a complete geometric error model of the optomechanical structure: After simplification in Step 1, error parameters need to be set for the following geometric elements to fully describe all sources of geometric error within the interferometer, including: deviation of the symmetrical equivalent pyramid vertex position, deviation of the waveplate exit surface pose, deviation of the first intersection point of the waveplate exit surface, deviation of the first emitted beam angle of the waveplate exit surface, and deviation of the equivalent detector plane pose. Step 2 specifically includes: Step 2.1: Constructing the geometric error model of the symmetrical equivalent pyramid vertex: The simplified symmetrical pyramid vertex position integrates the geometric poses of two optomechanical structures: the real pyramid prism vertex position and the polarizing beam splitter surface pose. There are three independent translational error directions in space, each affecting the displacement measurement results. Therefore, a translation matrix is ​​used to construct the three degrees of freedom position deviations of the symmetrical equivalent pyramid vertex, achieving a complete description of the geometric errors of the real pyramid prism vertex position and the polarizing beam splitter surface pose.

[0026] Step 2.2: Constructing the geometric error model of the waveplate exit plane: The waveplate exit plane has one translational error direction and two rotational error directions. These three directions are independent of each other and all affect the displacement measurement results. Therefore, rotation and translation matrices are used to construct one degree of freedom positional deviation and two degrees of freedom angular tilt of the waveplate exit plane.

[0027] Step 2.3: Construct the geometric error model of the first intersection point of the waveplate exit surface: The first intersection point of the waveplate exit surface is always located on the waveplate exit surface. There are two independent translational error directions of the point on the plane that affect the displacement measurement results. Therefore, the translation matrix is ​​used to construct the positional deviation of the two degrees of freedom of the first intersection point of the waveplate exit surface.

[0028] Step 2.4: Construct the geometric error model of the first emitted beam from the waveplate exit surface: The direction vector of the beam has two independent rotational error directions that both affect the displacement measurement results. Therefore, a rotation matrix is ​​used to construct the two degrees of freedom angle tilt of the first emitted beam from the waveplate exit surface.

[0029] Step 2.5: Construct the geometric error model of the equivalent detector plane: The equivalent detector plane integrates the geometric pose of the back-end mirror plane and the real mirror plane. Its translational error direction has no effect on the displacement measurement results. Therefore, the plane only includes two rotational error directions. Thus, the rotation matrix is ​​used to construct the two degrees of freedom of the equivalent detector plane tilt.

[0030] Step 3: Calculate the displacement caused by the motion of the target: Calculate the total optical path of the laser in the measurement path when the target is in the two states before and after the motion, and calculate the displacement caused by the motion of the target based on the total optical path difference.

[0031] Step 3.1: Calculation of the total optical path of the simplified beam trajectory before target motion: Based on the geometric parameters set in Step 2 and the initial pose parameters of the target, the simplified beam trajectory after simplification in Step 1 is calculated segment by segment, mainly including intersection point calculation, symmetry point calculation, reflection vector calculation, and refraction vector calculation. The total optical path of the simplified beam trajectory before the target motion is obtained. δ A .

[0032] Step 3.2: Calculation of the total optical path of the simplified beam trajectory after target motion: Based on the geometric parameters set in Step 2 and the motion parameters of the target, the simplified beam trajectory after simplification in Step 1 is calculated segment by segment, mainly including intersection point calculation, symmetry point calculation, reflection vector calculation, and refraction vector calculation. The total optical path of the simplified beam trajectory after target motion is obtained. δ B .

[0033] Step 3.2: Calculate the optical path difference and displacement value: Based on the total optical path results obtained in Steps 3.1 and 3.2, calculate the total optical path difference Δ caused by the motion of the target. δ BA Then calculate the displacement. d。

[0034] like Figures 2-7 The specific implementation process of the present invention is as follows: It should be noted that the complete laser trajectory of the measurement path of the planar interferometer to which this invention applies is as follows: Figure 2 As shown. The optical components related to the laser 5 in the measurement path include: fiber coupler or laser 1, polarizing beam splitter 6, waveplate 7 (quarter plate), cornerstone prism 4, detector front reflector 2, photodetector 3, and target 8. To simplify the beam refraction process, the polarizing beam splitter 6, waveplate 7, cornerstone prism 4, and detector front reflector 2 are considered as a single, integrated unit with the same refractive index. The technical content of steps 1 and 2 is based on... Figure 2 , Figure 3 The technical content of step 3 is based on Figure 4 .

[0035] Step 1: Simplify the laser trajectory for measurement: as shown Figure 3 As shown, firstly, all reflection phenomena inside the interferometer are simplified without approximation errors, including all reflection phenomena on the front reflecting mirror plane of the detector, the three reflecting surfaces inside the corner cube prism, and the beam-splitting surface of the polarizing beam-splitting prism. Then, geometric elements that do not affect the total optical path change are further simplified. Figure 3 In the diagram, the actual optomechanical structure inside the planar interferometer is represented by thick lines, while the beam trajectory is represented by thin lines. Black represents the beam and optomechanical structure after simplification in step one, and gray represents the beam and optomechanical structure after simplification in step one. Solid lines in the beam represent the original, real beam, dashed lines represent the symmetrical beams of the reflected beams that were reduced during the simplification process, and circles represent the nodes where the beam and optomechanical structure interact, such as reflection points, refraction points, intersection points, and light exit points.

[0036] Step 1.1: Simplification of the detector's front plane reflector without approximation error: Construct the actual detector receiving surface S D 28. Regarding the detector's front reflector plane S DM 18's symmetry plane: Equivalent detector plane S RD 13. Represent the spatial pose of the plane using the following point-normal plane equation: ; The position vector of a point on the plane that constitutes the point-normal form of the plane equation. P RD and surface normal vector VRD The expression is: ; ; in P RD and V RD S represents the equivalent detector plane S symmetrical to the mirror. RD A point on 13 and its normal vector, P D and V D S represents the actual detector receiving surface. D A point on 28 and its normal vector, P MD and V MD S represents the front reflector plane of the detector, respectively. DM A point on 18 and its normal vector.

[0037] Furthermore, the incident beam 29 at the front reflector of the detector and the plane S of the front reflector of the detector... DM At point 14, where the planes of the detector's front reflector intersect, the incident beam 29 along the direction of the front reflector extends to the equivalent detector plane S. RD 13. The equivalent detector plane extends the incident beam 16, which is symmetrical to the real reflected beam 17, meaning their geometric lengths are equal. Therefore, in the calculation, the length of the equivalent detector plane extends the incident beam 16, which can replace the original length of the real reflected beam 17, to reduce the reflection calculation process. Then, from the starting point of the incident beam 29 of the front reflector of the detector to the point where the incident beam 29 of the front reflector of the detector and the equivalent detector plane S... RD The beam trajectory length between the intersection points 15 and 13 of the equivalent detector planes. l D It can be represented as: ; in V DI and P DI These represent the direction vector and starting position vector of the incident beam 29 on the front reflector of the detector, respectively.

[0038] Step 1.2: Simplified reflection of the interior of the corner prism without approximation: Consider the corner prism 4 as a plane passing through the vertex 32 of the pyramid and perpendicular to the incident beam 36 of the corner prism: the equivalent surface of the pyramid S R 33. Ignoring the reflection effect of the plane of the pyramidal prism, similar to step 1.1, construct the pyramidal equivalent surface of the incident beam 36, extending the incident beam 35. Then, the extended incident beam 35 and the pyramidal equivalent surface S...R The intersection point of points 33 is the incident point 34 of the pyramidal equivalent surface. Construct the symmetrical point 31 of the pyramidal equivalent surface, which is the exit point 34, about the pyramidal vertex 32. Then, the exit beam 30 of the pyramidal prism is a beam passing through the exit point 31 of the pyramidal equivalent surface and with a direction opposite to the incident beam 36 of the pyramidal prism. The equation of this beam is as follows: ; in P V This represents the position vector of the 32nd vertex of the pyramid. P RI This represents the position vector of the first reflection point 22 on the beam-splitting surface. V RI This represents the direction vector of the incident beam 36 from the corner bevel prism.

[0039] The actual length of the complex beam trajectory inside the corner prism between the first reflection point 22 and the second reflection point 20 of the beam splitter. l R It can be expressed as the sum of the beam lengths from the first reflection point 22 of the beam splitter to the incident point 34 of the pyramidal equivalent surface and from the exit point 31 of the pyramidal equivalent surface to the second reflection point 20 of the beam splitter, and its expression is: ; in P RO This represents the position vector of the second reflection point 20 on the beam-splitting surface.

[0040] Step 1.3: Simplification of the beam-splitting surface of the polarizing beam-splitting prism without approximations: Construct the pyramid vertex 32 about the beam-splitting surface S of the polarizing beam-splitting prism. BS The symmetric point of 23: the symmetric equivalent vertex 10, and the position vector of this point. P VV The expression is: ; in, P BBS and V BBS S represents the beam-splitting surface of the polarizing beam splitter. BS The position vector and plane normal vector of any point on 23.

[0041] Further construct the pyramidal equivalent surface S R 33. Regarding the beam-splitting surface S of the polarizing beam splitter prism BS The symmetry plane of 23: the equivalent surface S of the quadratic pyramid SR 9. Express the equivalent surface S of the quadratic pyramid using the plane equation shown below. SR Spatial pose of 9: ; Where is used for plane normal vector V BSI is the direction vector of the incident beam 24 at the beam splitter.

[0042] Extend the incident beam from the beam splitter surface 24 to the equivalent surface of the second pyramid S SR 9. The equivalent surface S of the intersecting quadratic pyramid SR 9. At the incident point 19 on the equivalent surface of the quadratic pyramid, let the starting point of the incident beam be... P BSI Then the incident point of the equivalent surface of the quadratic pyramid P SRI The expression is: ; Further, construct the symmetric point of the incident point 19 of the quadratic pyramid equivalent surface about the symmetric equivalent vertex 10. This point is the exit point 12 of the quadratic pyramid equivalent surface, and the position vector of this point is... P SRO The expression is: ; The expression for the beam 21 emitted from the beam splitter surface exiting the equivalent plane of the quadratic pyramid is: ; Calculate the sum of the trajectory lengths from the starting point of the incident beam 24 at the splitting surface to the incident point 19 at the equivalent surface of the quadratic pyramid, and from the exit point 12 at the equivalent surface of the quadratic pyramid to the ending point of the exit beam 21 at the splitting surface. l B It can be represented as: ; in P BSO Let be the position vector of the endpoint of the beam emitted from the beam splitter 21, and the trajectory length be... l B It can be considered as the total length of the actual beam trajectory from the starting point of the incident beam 24 at the beam splitter to the ending point of the exit beam 21 at the beam splitter.

[0043] Step 1.4: Simplification of the beam exit point without approximation: The beam trajectory from the actual laser exit point 11 to the first intersection point 26 of the waveplate exit surface on waveplate 7 is ignored. The first intersection point 26 of the waveplate exit surface on waveplate 7 is considered the laser exit point of the entire system. At this point, the optical path difference caused by the target plane motion remains unchanged. Therefore, the linear equation of the first emitted beam 27 from the waveplate exit surface is: ; in, P S This represents the position vector of the exit point of the first emitted beam 27 from the waveplate exit surface. V S Indicates the direction vector of the emitted beam.t Points on the beam trajectory P Time P S The length. P S and V S This can be further expressed as: ; ; In the above formula, P L and V L Let represent a point on the actual trajectory of the laser beam emitted from the real laser emission point and its direction vector, respectively. P i and V i Let these represent a point on the incident surface of the polarizing beam splitter and the waveplate, and its plane normal vector, respectively. P e and V e Let these represent a point on the exit surface of the polarizing beam splitter and the waveplate, and its plane normal vector, respectively. e It represents the relative refractive index of the air medium to the crystalline medium. V L and V i or V e The angle between them is an obtuse angle.

[0044] Step 2: Establish a complete geometric error model of the optomechanical structure: After simplification in Step 1, error parameters need to be set for the following geometric elements to fully describe all sources of geometric error inside the interferometer, including: position deviation of the symmetrical equivalent pyramid vertex 9, pose deviation of the waveplate exit surface 25, position deviation of the first intersection point 26 of the waveplate exit surface, angular deviation of the first emitted beam 27 of the waveplate exit surface, and equivalent detector plane S. RD 13. Posture deviation.

[0045] Step 2.1: Construct a geometric error model for the symmetric equivalent vertex 10: using a translation matrix M tePVV Construct a symmetric equivalent vertex 10 with three degrees of freedom positional deviations. Then, the position vector of the symmetric equivalent vertex 10, including the positional errors, is... P VVr It can be represented as: ; in M tePVV Using homogeneous expression, its expression is as follows: ; in x tePVV , y tePVV , z tePVV These represent the deviations of the symmetric equivalent vertex 10 in the three degrees of freedom directions in space.

[0046] Step 2.2: Construct the geometric error model of the waveplate exit surface 25: using a rotation matrix M reSHW Translation matrix M teSHW Construct one degree of freedom for the positional deviation and two degrees of freedom for the angular tilt of the waveplate exit plane 25. Assume that, in an ideal state, the waveplate exit plane 25 is perpendicular to the X-axis, and a position vector passing through the X-axis is... P SHW Let the point be a point whose direction vector is . V SHW The degree of freedom of positional deviation in a plane is represented by points. P SHW Translation on the X-axis x teSHW Angular deviation is expressed as a vector. V SHW Angles of rotation around the Y and Z axes p reSHW and angle y reSHW Then the position vector of any point on the exit surface 25 of the waveplate containing the error. P HWR and normal vector V HWR It can be represented as: ; ; Where the rotation matrix M reSHW Translation matrix M teSHW The expressions are as follows: ; ; Step 2.3: Constructing the geometric error model of the first intersection point 26 of the waveplate exit surface: The position of the first intersection point 26 of the waveplate exit surface is always on the waveplate exit surface 25, so there are only two free positional deviations. A translation matrix is ​​used to construct... M tePS Construct the position vector of the first intersection point 26 of the waveplate exit surface P STwo degrees of freedom positional deviations, including the position vector of the first intersection point 26 of the waveplate exit surface with positional deviations. P Sr The expression is: ; Where the translation matrix is ​​constructed M tePS The expression is: ; Step 2.4: Constructing the geometric error model of the first emitted beam 27 from the waveplate exit surface: using a rotation matrix M reVS To construct the two degrees of freedom of the first emitted beam 27 from the waveplate exit surface, tilt angles are used. Let the ideal emitted beam direction vector be... V S Along the X-axis, the direction vector of the first emitted beam 27 from the waveplate exit surface, which contains angular errors, is... V Sr The expression is: ; Where the rotation matrix M reVS The expression is: ; in p reVS and y reVS They represent the ideal direction vectors respectively. V S The deviation angles in the Y-axis and Z-axis directions.

[0047] Step 2.5: Construct the geometric error model of the equivalent detector plane: Let the normal vector of the equivalent detector plane be... V RD Ideally, translation of the equivalent detector plane along the X-axis does not affect the optical path difference caused by the motion of the target plane; therefore, its positional deviation can be ignored. A rotation matrix is ​​used. M reVRD If the two degrees of freedom of the equivalent detector plane are tilted at an angle, then the normal vector containing the angle error is... V RDr The expression is: ; Where the rotation matrix M reVRD The expression is: ; in p reVRD and yreVRD They represent the ideal normal vectors respectively. V RD The deviation angles in the Y-axis and Z-axis directions.

[0048] Step 3: Calculate the displacement caused by the motion of the target 8: such as Figure 4 As shown, the total optical path of the laser in the measurement path is calculated in two states: before and after the target plane moves. The displacement caused by the movement of the target is then calculated based on the total optical path difference. Figure 4 The black line represents the simplified beam trajectory and simplified optomechanical structure pose of the target before planar motion, while the gray line represents the simplified beam trajectory and simplified optomechanical structure pose of the target after motion.

[0049] Step 3.1: Calculation of the total optical path of the simplified beam trajectory before target motion: Based on the geometric parameters set in Step 2 and the initial pose parameters of the target, the simplified beam trajectory after simplification in Step 1 is calculated segment by segment, mainly including intersection point calculation, symmetry point calculation, reflection vector calculation, and refraction vector calculation. The total optical path of the simplified beam trajectory before the target motion is obtained. δ A The specific calculation process is as follows: First beam L A1 : From the exit surface S of the waveplate HW The starting point P 1 to the point where the beam intersects the plane of the target mirror P A2 Up to this point. The first beam L of the joint assembly. A1 The equation of the straight line and the target plane S to be measured AM The plane equations can be used to obtain the intersection points. P A2 Position vector and length of the beam segment δ A1 The subscript A indicates the value in the initial state. The first beam segment L... A1 The equation of the straight line and the target plane S to be measured AM The equation of the plane is expressed as follows: ; The corresponding expressions for calculating the intersection point and beam length are as follows: ; in V 1. P AM and V AM These represent the first beam L. A1 Direction vector, target plane S AM The position vector of the previous point and the normal vector of the plane.

[0050] Second beam L A2 At the intersection P A2 Reflected from the target plane, from the intersection point P A2 The beam exits from the waveplate at surface S. HW intersection P A3 Up to this point. The direction vector of this beam. V A2 It can be calculated using the law of reflection, and its expression is: ; Similar to the solution process for the first beam segment, the length of this beam segment can be obtained. l A2 .

[0051] Third beam L A3 The beam of light at the intersection P A3 The beam is refracted from air into a crystal medium, and the equivalent surface S at the vertex of the symmetrical pyramid is formed by the intersection of the beam splitting planes. ASR With point P A4 Calculate the direction vector of the beam according to the law of refraction. V A3 Its expression is: ; in V HW n a and n c Let these represent the normal vector of the waveplate plane, the refractive index of air, and the refractive index of the crystal medium, respectively. The position vector of the symmetric equivalent vertex of the pyramid is also represented. P VV and the direction vector of the third beam V A3 This allows us to obtain the equivalent surface S of the symmetric pyramid vertex of the beam splitter. ASR The equation of the plane is expressed as follows: ; Similar to the solution process for the first beam segment, the equations are combined with those for the third beam segment L. A3 The length of the beam can be obtained from the equation of the straight line. l A3 and intersection P A4 .

[0052] Fourth beam L A4 The beam exits from the equivalent surface of the pyramid. P A5 Emitted, to the beam L A4 With waveplate exit surface SHW intersections P A6 Up to this point, the direction is parallel to and opposite to the third beam segment. The exit point of the pyramidal equivalent surface. P A5 and equivalent incident point P A4 Regarding the equivalent vertex of a pyramid P VV If symmetric, then calculate the position vector. P A5 The expression is: ; Similar to the solution process for the first beam segment, the intersection point can be obtained by simultaneously solving the beam equations and the plane equations. P A6 and the length of that beam l A4 .

[0053] Fifth beam L A5 This beam exits from waveplate exit surface S. HW Points on P A6 The emitted beam, reaching the target plane S, is... AM intersection P A7 Up to this point, the direction vector and the second beam L A2 Conversely, the starting point position vector is P A6 Coordinates. Similar to the solution process for the first beam segment, simultaneously solve the equations for the fifth beam segment L. A5 The equation of the straight line and the target plane S AM The equation of the plane can be used to obtain the intersection point. P A7 Position vector and length of the beam segment l A5 .

[0054] Sixth beam L A6 This beam originates from the target plane S. AM Points on P A7 Emitted, with the waveplate exit surface S HW Intersection point P A8 The direction vector is the same as the first beam L. A1 Conversely, the starting point position vector is P A7 Coordinates. Using a method similar to that used for solving the first beam segment, the linear equation L of the sixth beam segment is established. A6 With waveplate exit surface S HW The equation of the plane is given, and the point is solved. P A8Position vector and length of the beam segment l A6 .

[0055] Seventh beam L A7 To simplify the analysis, it is assumed that the photodetector plane is in close contact with the exit surface of the plane mirror, corner cube prism, polarizing beam splitter, and waveplate as a whole. This segment of the beam exits from a point on the waveplate's exit surface. P A8 The equivalent plane S of the cross-reflector symmetrical detector is emitted. RD At point P A9 The method for solving the direction vector of this beam segment is similar to that of the third beam segment, and can be obtained from the refraction formula. Then, the equations for the straight line of this beam segment and the equivalent plane S of the mirror-symmetric detector are solved simultaneously. RD The plane equations can be used to obtain the intersection points. P A9 Position vector and length of the beam segment l A7 .

[0056] In summary, when the target plane is in its initial state, the total optical path of the current simplified beam can be obtained. δ A for: ; in δ An Let be the optical path length of each beam segment.

[0057] Step 3.2: Calculation of the total optical path of the simplified beam trajectory after target motion: Based on the geometric parameters set in Step 2 and the motion parameters of the target, the simplified beam trajectory after simplification in Step 1 is calculated segment by segment, mainly including intersection point calculation, symmetry point calculation, reflection vector calculation, and refraction vector calculation. The total optical path of the simplified beam trajectory after target motion is obtained. δ B The calculation process is consistent with step 3.1, and the lengths of the seven beam segments after the target plane has moved are denoted as follows: l B1 , l B2 , l B3 , l B4 , l B5 , l B6 and l B7 The total optical path of the beam after the target plane has moved. δ B for: ; in δ Bn Let be the optical path length of each beam segment.

[0058] Step 3.3: Calculate the optical path difference and displacement value: Based on the total optical path results obtained in Steps 3.1 and 3.2, calculate the total optical path difference Δ caused by the motion of the target. δ BA Then calculate the displacement. d Its expression is: ; Figure 5 This paper presents the difference in displacement calculation results between the oversimplified, error-free displacement calculation method and the segmented beam trajectory full geometric error displacement calculation method when calculating the motion of the same target, under the presence of geometric errors on the order of 10 μm translation and 100 μrad rotation. The segmented beam trajectory full geometric error displacement calculation method can analyze all geometric error factors of a single-axis interferometer and the actual refraction and reflection of the beam trajectory; therefore, the displacement result obtained by the segmented beam trajectory full geometric error displacement calculation method can be used as the theoretical true displacement value. Figure 5 It is known that the displacement result obtained by the oversimplified geometric error-free displacement calculation method still has a non-negligible calculation error of ±5nm compared with the theoretical true value. This error originates from the optomechanical mismatch within the interferometer. Under the same geometric error order, Figure 6 The difference between the displacement calculation results obtained by the segment-by-segment beam trajectory full geometric error displacement calculation method and the simplified beam trajectory full geometric error displacement calculation method proposed in this invention is given, and the difference is less than ±2×10. -7 nm represents the calculation error introduced by the mathematical simulation software. This indicates that the displacement calculation result obtained by the simplified beam trajectory full geometric error displacement calculation method proposed in this invention is consistent with the theoretical true value. This demonstrates that no additional calculation error was introduced during the simplification process described in this invention, and it can fully represent the influence of geometric errors on displacement measurement, enabling high-precision calculation of the displacement to be measured.

[0059] Figure 7 The paper presents a comparison of the time required for displacement calculation using the segmented beam trajectory full geometric error displacement calculation method and the time required for displacement calculation using the simplified beam trajectory full geometric error displacement calculation method proposed in this invention, under the same geometric error magnitude and the same target motion magnitude. Figure 7 The ordinate is represented in logarithmic coordinate form. It can be observed that, during 80,000 single-point displacement calculations, the time required for the segment-by-segment beam trajectory full geometric error displacement calculation method is approximately 10 times that required for the simplified beam trajectory full geometric error displacement calculation method. This indicates that the simplified beam trajectory full geometric error displacement calculation method proposed in this invention has higher computational efficiency.

[0060] In summary, the simplified beam trajectory full geometric error displacement calculation method obtained through the above steps has higher displacement calculation accuracy and efficiency compared to existing methods, and can provide a displacement model basis for interferometric displacement measurement. Furthermore, the displacement calculation method proposed in this invention is more suitable for multi-axis interferometric measurement systems targeting six degrees of freedom measurement, and can expand the application of planar interferometers in related fields.

[0061] The above embodiments are merely descriptions of preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Any modifications, alterations, alterations, or substitutions made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for calculating displacement based on full geometric error without approximation simplification of beam trajectory, characterized in that, Includes the following steps: Step 1, simplify the laser trajectory of the measurement path: simplify all reflection phenomena inside the interferometer without approximation error, including all reflection phenomena on the front plane mirror of the detector, the three internal reflection surfaces of the corner cube prism, and the beam splitting surface of the polarizing beam splitter; then further simplify the geometric elements that do not affect the total optical path change. Step 2: Establish a complete geometric error model of the optomechanical structure: Set error parameters for the following geometric elements to fully describe all sources of geometric error inside the interferometer, including: position deviation of the symmetrical equivalent pyramid vertex, pose deviation of the waveplate exit surface, position deviation of the first intersection point of the waveplate exit surface, angle deviation of the first emitted beam of the waveplate exit surface, and pose deviation of the equivalent detector plane. Step 3, calculate the displacement caused by the motion of the target: calculate the total optical path of the laser in the measurement path when the target is in the two states before and after the motion, and calculate the displacement caused by the motion of the target based on the total optical path difference; Step 1 includes: Step 1.1, Simplification of the front plane mirror of the detector without approximation error: Construct the plane of symmetry of the real detector receiving surface with respect to the front plane mirror of the detector: the equivalent detector plane; Ignore the reflected beam on the front plane mirror of the detector, and extend the incident beam to the equivalent detector plane; Calculate the optical path from the starting point of the incident beam to the intersection with the equivalent detector plane, and define this optical path as the total optical path of the actual beam trajectory between the starting point of the beam and the real detector receiving surface. Step 1.2, Simplification of internal reflection of the pyramidal prism without approximation: Consider the pyramidal prism as a plane passing through the vertex of the pyramid and perpendicular to the incident beam of the pyramidal prism: the pyramidal equivalent surface; define the intersection of the incident beam of the pyramidal prism and the pyramidal equivalent surface as the incident point of the pyramidal equivalent surface; construct the symmetrical point about the vertex of the pyramidal equivalent surface as the exit point of the pyramidal equivalent surface; define the exit beam of the pyramidal prism as the beam passing through the exit point of the pyramidal equivalent surface and parallel to the incident beam of the pyramidal prism; calculate the sum of the optical paths from the first reflection point of the beam splitting surface to the incident point of the pyramidal equivalent surface and from the exit point of the pyramidal equivalent surface to the second reflection point of the beam splitting surface, and consider this optical path to be the total optical path of the true beam trajectory between the starting point of the incident beam of the pyramidal prism and the second reflection point of the beam splitting surface; Step 1.3: Simplification of the beam-splitting surface of the polarizing beam-splitting prism without approximations: Construct the symmetrical point of the pyramid vertex with respect to the beam-splitting surface of the polarizing beam-splitting prism: symmetrical equivalent vertex; Construct the symmetrical surface of the pyramid equivalent with respect to the beam-splitting surface of the polarizing beam-splitting prism: quadratic pyramid equivalent surface; Extend the incident beam from the beam-splitting surface to the quadratic pyramid equivalent surface, intersecting the quadratic pyramid equivalent surface at the incident point of the quadratic pyramid equivalent surface; Construct the symmetrical point of the incident point of the quadratic pyramid equivalent surface with respect to the symmetrical equivalent vertex: the exit point of the quadratic pyramid equivalent surface; Calculate the sum of the optical paths from the starting point of the incident beam from the beam-splitting surface to the incident point of the quadratic pyramid equivalent surface, and from the exit point of the quadratic pyramid equivalent surface to the end point of the exit beam from the beam-splitting surface, and consider this optical path to be the total optical path of the true beam trajectory between the starting point of the incident beam from the beam-splitting surface and the end point of the exit beam from the beam-splitting surface; Step 1.4: Simplification of the beam exit point without approximation: Ignore the beam trajectory between the actual laser exit point and the first intersection point of the waveplate exit surface on the waveplate, and define the first intersection point of the waveplate exit surface on the waveplate as the laser exit point of the entire system; Step 2 includes: Step 2.1, construct the geometric error model of the symmetrical equivalent pyramid vertex: use the translation matrix to construct the positional deviation of the three degrees of freedom of the symmetrical equivalent pyramid vertex, so as to realize the complete description of the geometric error of the position of the real pyramid prism vertex and the pose of the polarizing beam splitter surface; Step 2.2, construct the geometric error model of the waveplate exit plane: use rotation and translation matrices to construct one degree of freedom positional deviation and two degrees of freedom angular tilt of the waveplate exit plane; Step 2.3, construct the geometric error model of the first intersection point of the waveplate exit surface: use the translation matrix to construct the positional deviation of the two degrees of freedom of the first intersection point of the waveplate exit surface; Step 2.4, construct the geometric error model of the first emitted beam from the waveplate exit surface: use a rotation matrix to construct the two degrees of freedom angle tilt of the first emitted beam from the waveplate exit surface; Step 2.5, construct the geometric error model of the equivalent detector plane: use the rotation matrix to construct the two degrees of freedom of the equivalent detector plane angle tilt.

2. The method for calculating displacement of full geometric error based on beam trajectory without approximation simplification as described in claim 1, characterized in that, In step 1.1, the optical path from the starting point of the incident beam to the intersection with the equivalent detector plane is expressed as: in, l D The length of the beam trajectory. V DI and P DI These represent the direction vector and the starting position vector of the incident beam, respectively. P RD and V RD Let represent a point on the equivalent detector plane symmetrical to the reflector and its normal vector, respectively. In step 1.2, the beam emitted from the cornerstone prism is defined as a beam that passes through the exit point of the equivalent surface of the cornerstone and is parallel to the incident beam of the cornerstone prism. The equation of this beam is expressed as follows: in, P V Represents the position vector of the vertex of the pyramid. P RI This represents the position vector of the first reflection point of the beam-splitting surface. V RI This represents the direction vector of the incident beam from the cornerstone prism. In step 1.2, the sum of the optical path lengths from the first reflection point of the beam-splitting surface to the incident point of the pyramidal equivalent surface and from the exit point of the pyramidal equivalent surface to the second reflection point of the beam-splitting surface is calculated. The expression is: in P RO This represents the position vector of the second reflection point on the beam-splitting surface; In step 1.3, the sum of the optical path lengths from the starting point of the incident beam at the beam splitter to the incident point of the equivalent surface of the quadratic pyramid, and from the exit point of the equivalent surface of the quadratic pyramid to the ending point of the exit beam at the beam splitter is calculated. The expression is: in P BSO Let be the position vector of the endpoint of the beam emitted from the beam splitter. P VV The position vector of the symmetric equivalent vertex. V BSI Let be the direction vector of the incident beam at the beam splitter. P BSI To extend the incident beam from the beam splitter to the starting point of the incident beam on the equivalent surface of the second pyramid.

3. The method for calculating displacement of full geometric error based on beam trajectory without approximation simplification as described in claim 2, characterized in that, In step 1.4, In step 1.4, the equation of the straight line of the first emitted beam from the waveplate exit surface is expressed as follows: in, P S This represents the position vector of the exit point of the first emitted beam from the exit surface of the waveplate. V S Indicates the direction vector of the emitted beam. t Points on the beam trajectory P Time P S Length; P S and V S This can be further expressed as: In the above formula, P L and V L Let represent a point on the actual trajectory of the laser beam emitted from the real laser emission point and its direction vector, respectively. P i and V i Let these represent a point on the incident surface of the polarizing beam splitter and the waveplate, and its plane normal vector, respectively. P e and V e Let these represent a point on the exit surface of the polarizing beam splitter and the waveplate, and its plane normal vector, respectively. e It represents the relative refractive index of the air medium to the crystalline medium.

4. The method for calculating displacement of full geometric error based on beam trajectory without approximation simplification as described in claim 1, characterized in that, In step 2.1, the translation matrix M tePVV The expression is: in, x tePVV , y tePVV , z tePVV These represent the deviations of the symmetric equivalent vertex in the three degrees of freedom directions in space, respectively. In step 2.2, the ideal waveplate exit surface is perpendicular to the X-axis, and a position vector passing through the X-axis is... P SHW Let the point be a point whose direction vector is . V SHW The degree of freedom of positional deviation in a plane is expressed as a point. P SHW Translation on the X-axis x teSHW Angular deviation is expressed as a vector. V SHW Angles of rotation around the Y and Z axes p reSHW and angle y reSHW Rotation matrix M reSHW Translation matrix M teSHW The expressions are as follows: In step 2.3, a translation matrix is ​​used to construct... M tePS Constructing the position vector of the first intersection point of the waveplate exit surface P S The positional deviation of two degrees of freedom, the translation matrix is ​​constructed. M tePS The expression is: In step 2.4, a rotation matrix is ​​used. M reVS To construct the two degrees of freedom of the first emitted beam from the waveplate exit surface, tilting at angles; rotation matrix M reVS The expression is: in p reVS and y reVS They represent the ideal direction vectors respectively. V S The deviation angles in the directions around the Y and Z axes; In step 2.5, a rotation matrix is ​​used. M reVRD Construct the equivalent detector plane with two degrees of freedom of angular tilt and rotation matrix. M reVRD The expression is: in p reVRD and y reVRD They represent the ideal normal vectors respectively. V RD The deviation angles in the Y-axis and Z-axis directions.

5. The method for calculating displacement of full geometric error based on beam trajectory without approximation simplification as described in claim 1, characterized in that, Step 3 includes: Step 3.1, Calculation of the total optical path of the simplified beam trajectory before target motion: Based on the geometric parameters set in Step 2 and the initial pose parameters of the target, the simplified beam trajectory after simplification in Step 1 is calculated segment by segment to obtain the total optical path of the simplified beam trajectory before the target motion. δ A ; Step 3.2: Calculation of the total optical path of the simplified beam trajectory after the target motion: Based on the geometric parameters set in Step 2 and the motion parameters of the target, the simplified beam trajectory after the simplification in Step 1 is calculated segment by segment to obtain the total optical path of the simplified beam trajectory after the target motion. δ B ; Step 3.3: Calculate the optical path difference and displacement value: Based on the total optical path results obtained in Steps 3.1 and 3.2, calculate the total optical path difference Δ caused by the motion of the target. δ BA Then calculate the displacement. d .

6. The method for calculating displacement of full geometric error based on beam trajectory without approximation simplification as described in claim 5, characterized in that, Step 3.1 includes: First beam L A1 : From the exit surface S of the waveplate HW The starting point P 1 to the point where the beam intersects the plane of the target mirror P A2 Up to this point; the first beam L is combined A1 The equation of the straight line and the target plane S to be measured AM The plane equations yield the intersection points P A2 Position vector and length of the beam segment δ A1 The subscript A indicates the value in the initial state; where the first beam segment L A1 The equation of the straight line and the target plane S to be measured AM The equation of the plane is expressed as follows: The corresponding expressions for calculating the intersection point and beam length are as follows: in V 1. P AM and V AM These represent the first beam L. A1 Direction vector, target plane S AM The position vector of the previous point and the normal vector of the plane; Second beam L A2 At the intersection P A2 Reflected from the target plane, from the intersection point P A2 The beam exits from the waveplate at surface S. HW intersection P A3 Up to; the direction vector of this beam segment V A2 The expression for this calculation, based on the law of reflection, is: The length of the second beam is obtained through the same process as for the first beam segment. l A2 ; Third beam L A3 The beam of light at the intersection P A3 The beam is refracted from air into a crystal medium, and the equivalent surface S at the vertex of the symmetrical pyramid is formed by the intersection of the beam splitting planes. ASR With point P A4 Calculate the direction vector of the beam according to the law of refraction. V A3 Its expression is: in V HW n a and n c Represent the normal vector of the waveplate plane, the refractive index of the air medium, and the refractive index of the crystal medium, respectively; the position vector of the symmetric equivalent vertex of the pyramid. P VV and the direction vector of the third beam V A3 The equivalent surface S of the symmetric pyramid vertex is obtained by the beam splitter. ASR The equation of the plane is expressed as follows: Solving this equation simultaneously with the third beam L A3 The equation of the straight line is used to obtain the length of the third beam segment. l A3 and intersection P A4 ; Fourth beam L A4 The beam exits from the equivalent surface of the pyramid. P A5 Emitted, to the beam L A4 With waveplate exit surface S HW intersections P A6 Up to this point, the direction is parallel to and opposite to the third beam segment; exit point of the pyramidal equivalent surface. P A5 and equivalent incident point P A4 Regarding the equivalent vertex of a pyramid P VV If symmetric, then calculate the position vector. P A5 The expression is: By combining the beam equation and the plane equation, we obtain the intersection point. P A6 and the length of that beam l A4 ; Fifth beam L A5 This beam exits from waveplate exit surface S. HW Points on P A6 The emitted beam, reaching the target plane S, is... AM intersection P A7 Up to this point, the direction vector and the second beam L A2 Conversely, the starting point position vector is P A6 Coordinates; Combine the fifth beam L A5 The equation of the straight line and the target plane S AM The equation of the plane is obtained, and the intersection point is obtained. P A7 Position vector and length of the fifth beam l A5 ; Sixth beam L A6 This beam originates from the target plane S. AM Points on P A7 Emitted, with the waveplate exit surface S HW Intersection point P A8 The direction vector is the same as the first beam L. A1 Conversely, the starting point position vector is P A7 Coordinates; Solve the equation of the straight line L of the sixth beam segment. A6 With waveplate exit surface S HW The equation of the plane is given, and the point is solved. P A8 Position vector and length of the sixth beam l A6 ; Seventh beam L A7 The beam of light originates from a point on the exit surface of the waveplate. P A8 The equivalent plane S of the cross-reflector symmetrical detector is emitted. RD At point P A9 The direction vector of this beam segment is obtained using the refraction formula; then, the equation of the straight line of this beam segment and the equivalent plane S of the mirror-symmetric detector are solved simultaneously. RD The plane equations yield the intersection points P A9 Position vector and length of the seventh beam l A7 ; When the target plane is in its initial state, the total optical path of the current simplified beam is obtained. δ A for: in δ An Let be the optical path length of each beam segment.

7. The method for calculating displacement of full geometric error based on beam trajectory without approximation simplification as described in claim 6, characterized in that, The calculation process in step 3.2 is the same as in step 3.

1. The lengths of the seven beam segments after the target plane has moved are denoted as follows: l B1 , l B2 , l B3 , l B4 , l B5 , l B6 and l B7 ; Total optical path of the beam after the target plane moves δ B for: in δ Bn Let be the optical path length of each beam segment.

8. The method for calculating displacement of full geometric error based on beam trajectory without approximation simplification as described in claim 7, characterized in that, In step 3.3, displacement d The expression is: 。

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