CN117146870A - Two-dimensional grating interferometry device and measurement method - Google Patents
Two-dimensional grating interferometry device and measurement method Download PDFInfo
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01D—MEASURING NOT SPECIALLY ADAPTED FOR A SPECIFIC VARIABLE; ARRANGEMENTS FOR MEASURING TWO OR MORE VARIABLES NOT COVERED IN A SINGLE OTHER SUBCLASS; TARIFF METERING APPARATUS; MEASURING OR TESTING NOT OTHERWISE PROVIDED FOR
- G01D5/00—Mechanical means for transferring the output of a sensing member; Means for converting the output of a sensing member to another variable where the form or nature of the sensing member does not constrain the means for converting; Transducers not specially adapted for a specific variable
- G01D5/26—Mechanical means for transferring the output of a sensing member; Means for converting the output of a sensing member to another variable where the form or nature of the sensing member does not constrain the means for converting; Transducers not specially adapted for a specific variable characterised by optical transfer means, i.e. using infrared, visible, or ultraviolet light
- G01D5/266—Mechanical means for transferring the output of a sensing member; Means for converting the output of a sensing member to another variable where the form or nature of the sensing member does not constrain the means for converting; Transducers not specially adapted for a specific variable characterised by optical transfer means, i.e. using infrared, visible, or ultraviolet light by interferometric means
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G01B11/00—Measuring arrangements characterised by the use of optical techniques
- G01B11/02—Measuring arrangements characterised by the use of optical techniques for measuring length, width or thickness
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01D—MEASURING NOT SPECIALLY ADAPTED FOR A SPECIFIC VARIABLE; ARRANGEMENTS FOR MEASURING TWO OR MORE VARIABLES NOT COVERED IN A SINGLE OTHER SUBCLASS; TARIFF METERING APPARATUS; MEASURING OR TESTING NOT OTHERWISE PROVIDED FOR
- G01D5/00—Mechanical means for transferring the output of a sensing member; Means for converting the output of a sensing member to another variable where the form or nature of the sensing member does not constrain the means for converting; Transducers not specially adapted for a specific variable
- G01D5/26—Mechanical means for transferring the output of a sensing member; Means for converting the output of a sensing member to another variable where the form or nature of the sensing member does not constrain the means for converting; Transducers not specially adapted for a specific variable characterised by optical transfer means, i.e. using infrared, visible, or ultraviolet light
- G01D5/268—Mechanical means for transferring the output of a sensing member; Means for converting the output of a sensing member to another variable where the form or nature of the sensing member does not constrain the means for converting; Transducers not specially adapted for a specific variable characterised by optical transfer means, i.e. using infrared, visible, or ultraviolet light using optical fibres
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01D—MEASURING NOT SPECIALLY ADAPTED FOR A SPECIFIC VARIABLE; ARRANGEMENTS FOR MEASURING TWO OR MORE VARIABLES NOT COVERED IN A SINGLE OTHER SUBCLASS; TARIFF METERING APPARATUS; MEASURING OR TESTING NOT OTHERWISE PROVIDED FOR
- G01D5/00—Mechanical means for transferring the output of a sensing member; Means for converting the output of a sensing member to another variable where the form or nature of the sensing member does not constrain the means for converting; Transducers not specially adapted for a specific variable
- G01D5/26—Mechanical means for transferring the output of a sensing member; Means for converting the output of a sensing member to another variable where the form or nature of the sensing member does not constrain the means for converting; Transducers not specially adapted for a specific variable characterised by optical transfer means, i.e. using infrared, visible, or ultraviolet light
- G01D5/32—Mechanical means for transferring the output of a sensing member; Means for converting the output of a sensing member to another variable where the form or nature of the sensing member does not constrain the means for converting; Transducers not specially adapted for a specific variable characterised by optical transfer means, i.e. using infrared, visible, or ultraviolet light with attenuation or whole or partial obturation of beams of light
- G01D5/34—Mechanical means for transferring the output of a sensing member; Means for converting the output of a sensing member to another variable where the form or nature of the sensing member does not constrain the means for converting; Transducers not specially adapted for a specific variable characterised by optical transfer means, i.e. using infrared, visible, or ultraviolet light with attenuation or whole or partial obturation of beams of light the beams of light being detected by photocells
- G01D5/36—Forming the light into pulses
- G01D5/38—Forming the light into pulses by diffraction gratings
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Abstract
The invention relates to the field of grating measurement, in particular to a two-dimensional grating interferometry device and a two-dimensional grating interferometry method. The two-dimensional grating interferometry device comprises a grating, a light source, a beam splitting component, a steering component, a first reference reflection component, a first measurement reflection component, a second reference reflection component and a second measurement reflection component, wherein a basic light beam emitted by the light source passes through different optical devices, the first reference light beam and the second reference light beam are directly reflected to a first detector and a second detector, the first measurement light beam and the second measurement light beam are respectively obliquely incident to the surface of the grating at a first preset angle and a second preset angle, and then return in a return path after reflection to form a stable interference signal carrying displacement information, the stable interference signal is received by the first detector and the second detector, the high resolution is obtained by using a high-reticle-density grating, and meanwhile, the influence of an experimental precision caused by grating surface type error is avoided.
Description
Technical Field
The invention relates to the field of grating measurement, in particular to a two-dimensional grating interferometry device and a two-dimensional grating interferometry method.
Background
The high-precision displacement measurement technology at the present stage mainly comprises a laser interferometry method and a grating interferometry method. The measurement standard of the laser interferometry is laser wavelength, which has the defects of sensitivity to air refractive index, strict requirements on external environment conditions and easiness in obtaining high precision in a short stroke, but along with the gradual increase of the measurement stroke, the small changes of the measurement environments such as temperature, humidity, air pressure and the like seriously influence the accuracy of a measurement result, the measurement error of the stroke above a meter level is even up to hundreds of nanometers, and a high-level environment control system is needed for realizing high-precision measurement by using a laser interferometer.
The grating interferometry has the measurement standard of grating pitch, the grating substrate can be made of zero expansion material, compared with the wavelength, the influence of the external environment on grooving is very small, the measurement accuracy is hardly influenced by the increase of the stroke, the environmental control such as constant temperature, constant pressure, constant humidity and the like is not required, and the requirement of measurement on the environment is greatly reduced. In view of the advantages, the grating interference displacement measurement device has urgent application requirements in the fields of high-grade numerical control machine tools and aerospace.
At present, a grating interferometry system mostly adopts measurement light beams to vertically enter or Littrow angle enter, wherein the vertical incidence mode refers to that the emergent light of a laser enters a grating from the vertical direction to be diffracted, and two required diffracted light beams are respectively folded back through an optical element and enter a detector. When the grating is vertically incident, the reticle density of the used grating is limited, if the diffraction angle is smaller than 60 degrees of + -1-order diffraction light, only 1292gr/mm of the grating can be used at most, and the high reticle density grating cannot be used in a vertical incidence mode to obtain higher resolution. Meanwhile, the measuring range of the normal incidence light path structure in the grating normal direction is limited by the size of the refractive element, and large-range measurement in the Z direction cannot be realized.
The light path structure of the Littrow angle incidence can use a high-reticle density grating, but when two beams of light split by the beam splitter are incident to the same point on the grating, reflected light can be generated while diffraction occurs, the reflected light and the diffracted light can be mutually overlapped to form a laser interferometer structure, and when the grating deflects, an additional measurement error can be introduced. In order to eliminate the influence, the littrow incidence can only stagger the incidence positions of two beams of incident light on the grating, so that the surface type error of the grating is introduced, and the surface type error is accumulated continuously in the continuous motion process, so that the influence of larger error is brought.
Disclosure of Invention
The invention aims to solve the problems, provides a two-dimensional grating interferometry device and a two-dimensional grating interferometry method, and solves the problem that the grating interferometry precision is reduced due to grating surface type errors and low resolution in the conventional grating measurement.
In order to achieve the above object, in a first aspect, the present invention provides a two-dimensional grating interferometry device, including a grating, a light source, a beam splitting component, a steering component, a first reference reflection component, a first measurement reflection component, a second reference reflection component, a second measurement reflection component, and a detection component, where the light source is used to generate a base beam; the light splitting assembly comprises a first spectroscope, a second spectroscope and a third spectroscope, wherein the first spectroscope is used for splitting a basic light beam into a first basic light beam and a second basic light beam, the first basic light beam is transmitted by the first spectroscope and then enters the second spectroscope, the second spectroscope is used for splitting the first basic light beam into a first measuring light beam and a first reference light beam, the second basic light beam is reflected by the first spectroscope and then enters the third spectroscope, and the third spectroscope is used for splitting the second basic light beam into a second measuring light beam and a second reference light beam;
The steering assembly comprises a first steering mirror and a second steering mirror, the first steering mirror is arranged on one side of the second beam splitter, the first measuring beam enters the first steering mirror after being transmitted by the second beam splitter, the first steering mirror is used for refracting the first measuring beam and then injecting the first measuring beam into the grating according to a first preset angle, the second steering mirror is arranged on one side of the third beam splitter, the second measuring beam enters the second steering mirror after being transmitted by the third beam splitter, the second steering mirror is used for refracting the second measuring beam and then injecting the second measuring beam into the grating according to a second preset angle, and the grating is used for reflecting the first measuring beam and then forming a first diffraction beam and a fourth diffraction beam;
the first reference reflection assembly is arranged on the side face of the second beam splitter and is arranged on the opposite side of the first steering mirror, and the first reference reflection assembly is used for carrying out first deflection on the first reference beam reflected by the second beam splitter and then reflecting the first reference beam to the second beam splitter; the first measuring reflection component is arranged on a deflection path of the first diffraction beam, the first measuring reflection component is used for reflecting the first diffraction beam to the grating, the grating is used for reflecting the first diffraction beam to form a second diffraction beam, the first measuring reflection component is used for reflecting the second diffraction beam to the grating, the grating is used for reflecting the second diffraction beam to form a third diffraction beam, and the third diffraction beam is reversely shot into the first steering mirror;
The second reference reflection assembly is arranged on the side face of the third spectroscope and is opposite to the second steering mirror, and the second reference reflection assembly is used for carrying out second deflection on the second reference beam reflected by the third spectroscope and then reflecting the second reference beam to the third spectroscope; the second measuring reflection assembly is arranged on the deflection path of the fourth diffraction beam, the second measuring reflection assembly is used for reflecting the fourth diffraction beam to the grating, the grating is used for reflecting the fourth diffraction beam to form a fifth diffraction beam, the second measuring reflection assembly is used for reflecting the fifth diffraction beam to the grating, the grating is used for reflecting the fifth diffraction beam to form a sixth diffraction beam, and the sixth diffraction beam is reversely shot into the second steering mirror;
the detection component comprises a first detector and a second detector, the first detector is used for receiving the third diffraction light beam which enters the second spectroscope and is reflected after being reflected by the first steering mirror, and is used for receiving the first reference light beam which enters the second spectroscope and is transmitted after being reflected by the first reference reflection component, the second detector is used for receiving the sixth diffraction light beam which enters the third spectroscope and is reflected after being reflected by the second steering mirror, and is used for receiving the second reference light beam which enters the third spectroscope and is transmitted after being reflected by the second reference reflection component.
In some embodiments, the first reference reflecting assembly includes a first reference wave plate and a first reference mirror, the first reference wave plate being disposed on one side of the second beam splitter and being disposed on an opposite side of the first steering mirror; the first reference reflecting mirror is arranged on one side of the first reference wave plate, which is far away from the second beam splitter; the first measuring reflecting component comprises a first measuring reflecting mirror and a first measuring wave plate, the first measuring reflecting mirror is arranged on the turning path of the first diffraction light beam, and the first measuring reflecting mirror is used for reflecting the first diffraction light beam into the grating according to a third preset angle; the first measuring wave plate is arranged on a turning path of the first diffraction light beam which is emitted into the grating at a third preset angle;
the second reference reflecting component comprises a second reference wave plate and a second reference reflecting mirror, and the second reference wave plate is arranged on one side of the third spectroscope and is arranged on the opposite side of the second steering mirror; the second reference reflecting mirror is arranged on one side of the second reference wave plate far away from the third spectroscope; the second measuring reflecting component comprises a second measuring reflecting mirror and a second measuring wave plate, the second measuring reflecting mirror is arranged on the turning path of the fourth diffraction light beam, and the second measuring reflecting mirror is used for reflecting the fourth diffraction light beam into the grating according to a fourth preset angle; the second measuring wave plate is arranged on a turning path of the fourth diffracted light beam which is emitted into the grating at a fourth preset angle.
In some embodiments, the first measurement mirror is further configured to reflect the second diffracted beam into the grating at a fifth predetermined angle; the second measuring mirror is further configured to reflect the fifth diffracted beam into the grating at a sixth predetermined angle.
In some embodiments, the third preset angle is different from the fifth preset angle; the fourth preset angle is different from the sixth preset angle.
In some embodiments, the first reference waveplate and/or the first measurement waveplate and/or the second reference waveplate and/or the second measurement waveplate are quarter waveplates.
In some embodiments, the fundamental light beam comprises a first fundamental light beam and a second fundamental light beam, the first fundamental light beam and the second fundamental light beam having different polarization angles, the first fundamental light beam and the second fundamental light beam having different frequencies.
In some embodiments, the detection assembly further comprises a first polarizer disposed at the receiving end of the first detector, and a second polarizer; the second polarizer is disposed at the receiving end of the second detector.
In a second aspect, the present invention provides a two-dimensional grating interferometry method, which is applicable to the two-dimensional grating interferometry apparatus according to the first aspect, where the base beam includes a first base beam and a second base beam, and the method includes:
Acquiring a first reference complex amplitude value of a first reference beam acquired by a first detector and a first measured complex amplitude value of a third diffraction beam, and generating a first interference signal according to the first reference complex amplitude value and the first measured complex amplitude value;
acquiring a second reference complex amplitude value of a second reference beam acquired by a second detector and a second measured complex amplitude value of a sixth diffracted beam, and generating a second interference signal according to the second reference complex amplitude value and the second measured complex amplitude value;
and solving the first interference signal and the second interference signal according to the Doppler frequency shift effect, and generating a movement displacement measurement value of the grating.
In some embodiments, the first interference signal is noted as I 1 The first interference signal is represented by formula (1), formula (1) being as follows:
；
in the method, in the process of the invention,for the first measured complex amplitude value, +.>For the first reference complex amplitude value, +.>In order for the time of the grating to move,for the frequency variation of the light beam generated during the grating movement, for example>Is the phase change value of the light beam generated when the grating moves,an initial phase for the third diffracted beam;
recording the second interference signal asThe second interference signal is represented by formula (2), formula (2) as follows:
；
In the method, in the process of the invention,for the second measurement of the complex amplitude value, +.>For the second reference complex amplitude value, +.>Is the initial phase of the sixth diffracted beam.
In some embodiments, the beam frequency variation value produced by the doppler shift effect is represented by equation (3), equation (3) is as follows:
；
record the grating atThe displacement in the direction is +.>The grating is->The initial phase of the direction is +.>The grating is->The displacement in the direction is +.>The grating is->The initial phase of the direction is +.>；
Expressed by the formula (4), the formula (4) is as follows:
；
in the method, in the process of the invention,for grating diffraction order +.>For the speed of movement of the grating along the grating vector, < >>Is the grating pitch of the grating;
as represented by the formula (5),equation (5) is as follows:
；
in the method, in the process of the invention,for a first preset angle, < >>A third preset angle;
expressed by the formula (6), the formula (6) is as follows:
；
expressed by the formula (7), the formula (7) is as follows:
；
displacement of grating movementExpressed by the formula (8), the formula (8) is as follows:
；
displacement of grating movementExpressed by the formula (9), the formula (9) is as follows:
。
compared with the prior art, the invention has the following beneficial effects:
the two-dimensional grating interferometry device comprises a grating, a light source, a light splitting component, a steering component, a first reference reflection component, a first measurement reflection component, a second reference reflection component and a second measurement reflection component, wherein the light source is divided into two parts, and a stable interference signal carrying displacement information is formed by the light splitting component, the steering component, the first reference reflection component, the first measurement reflection component, the second reference reflection component and the second measurement reflection component.
Drawings
FIG. 1 is a first schematic diagram of the two-dimensional grating interferometry apparatus according to an embodiment of the present invention;
FIG. 2 is a second schematic diagram of the two-dimensional grating interferometry apparatus according to an embodiment of the present invention;
FIG. 3 is a schematic view of the optical paths of the first and second measuring beams according to an embodiment of the present invention;
fig. 4 is a step diagram of the two-dimensional grating interferometry method according to an embodiment of the present invention.
Reference numerals: 1. a light source; 2. a first spectroscope; 3. a second beam splitter; 4. a third spectroscope; 5. a first reference waveplate; 6. a second reference waveplate; 7. a first measurement waveplate; 8. a second measurement waveplate; 9. a first polarizing plate; 10. a second polarizing plate; 11. first detectionA device; 12. a second detector; 13. a first steering mirror; 14. a second steering mirror; 15. a first reference mirror; 16. a second reference mirror; 17. a first measurement mirror; 18. a second measuring mirror; B. a grating; θ 1 A first preset angle; beta, a second preset angle; θ 2 A third preset angle; delta, a fourth preset angle; epsilon and a fifth preset angle; η, sixth preset angle.
Detailed Description
Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings. In the following description, like modules are denoted by like reference numerals. In the case of the same reference numerals, their names and functions are also the same. Therefore, a detailed description thereof will not be repeated.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be further described in detail with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not to be construed as limiting the invention.
Referring to fig. 1 to 3, in a first aspect, the present embodiment provides a two-dimensional grating interferometry apparatus, including a grating B, a light source 1, a beam splitting component, a steering component, a first reference reflection component, a first measurement reflection component, a second reference reflection component, a second measurement reflection component, and a detection component, where the light source 1 is configured to generate a base beam; the beam splitting assembly comprises a first beam splitter 2, a second beam splitter 3 and a third beam splitter 4, wherein the first beam splitter 2 is used for splitting a basic beam into a first basic beam and a second basic beam, the first basic beam enters the second beam splitter 3 after being transmitted by the first beam splitter 2, the second beam splitter 3 is used for splitting the first basic beam into a first measuring beam and a first reference beam, the second basic beam enters the third beam splitter 4 after being reflected by the first beam splitter 2, and the third beam splitter 4 is used for splitting the second basic beam into a second measuring beam and a second reference beam;
The steering assembly comprises a first steering mirror 13 and a second steering mirror 14, the first steering mirror 13 is arranged on one side of the second beam splitter 3, and the first measuring beam passes through the second beam splitterThe mirror 3 enters a first steering mirror 13 after transmitting, and the first steering mirror 13 is used for turning the first measuring beam according to a first preset angle theta 1 The second steering mirror 14 is arranged at one side of the third spectroscope 4, the second measuring beam enters the second steering mirror 14 after being transmitted by the third spectroscope 4, the second steering mirror 14 is used for refracting the second measuring beam and then injecting the second measuring beam into the grating B according to a second preset angle beta, the grating B is used for reflecting the first measuring beam to form a first diffraction beam and reflecting the second measuring beam to form a fourth diffraction beam;
the first reference reflection component is arranged on the side surface of the second beam splitter 3 and is arranged on the opposite side of the first steering mirror 13, and the first reference reflection component is used for carrying out first deflection on the first reference beam reflected by the second beam splitter 3 and then reflecting the first reference beam to the second beam splitter 3; the first measuring reflection component is arranged on a deflection path of the first diffraction beam, the first measuring reflection component is used for reflecting the first diffraction beam to the grating B, the grating B is used for reflecting the first diffraction beam to form a second diffraction beam, the first measuring reflection component is used for reflecting the second diffraction beam to the grating B, the grating B is used for reflecting the second diffraction beam to form a third diffraction beam, and the third diffraction beam is reversely emitted into the first steering mirror 13;
The second reference reflection assembly is arranged on the side surface of the third spectroscope 4 and is arranged on the opposite side of the second steering mirror 14, and the second reference reflection assembly is used for carrying out second deflection on the second reference beam reflected by the third spectroscope 4 and then reflecting the second reference beam to the third spectroscope 4; the second measuring reflection component is arranged on the deflection path of the fourth diffraction beam, the second measuring reflection component is used for reflecting the fourth diffraction beam to the grating B, the grating B is used for reflecting the fourth diffraction beam to form a fifth diffraction beam, the second measuring reflection component is used for reflecting the fifth diffraction beam to the grating B, the grating B is used for reflecting the fifth diffraction beam to form a sixth diffraction beam, and the sixth diffraction beam is reversely emitted into the second steering mirror 14;
the detecting assembly comprises a first detector 11 and a second detector 12, the first detector 11 is used for receiving the third diffracted light beam which enters the second spectroscope 3 and is reflected after being reflected by the first steering mirror 13, and is used for receiving the first reference light beam which enters the second spectroscope 3 and is transmitted after being reflected by the first reference reflecting assembly, and the second detector 12 is used for receiving the sixth diffracted light beam which enters the third spectroscope 4 and is reflected after being reflected by the second steering mirror 14, and is used for receiving the second reference light beam which enters the third spectroscope 4 and is transmitted after being reflected by the second reference reflecting assembly.
As shown in fig. 1 and 2, the optical paths of the first and second base beams are mirror-symmetrical, and therefore, a first predetermined angle θ is described below 1 The second preset angles beta are mutually symmetrical, and the pointed acute angles are the same value; third preset angle theta 2 The fourth preset angle delta is symmetrical with the fourth preset angle delta, the acute angles indicated by the fourth preset angle delta are the same value, the fifth preset angle epsilon and the sixth preset angle eta are symmetrical with each other, and the acute angles indicated by the fifth preset angle epsilon and the sixth preset angle eta are the same value. The following expression will be based on this.
It should be noted that, fig. 1 and fig. 2 generally distinguish a plurality of parts by a plurality of dashed boxes, where a indicates a reading head system, that is, an optical part for collecting displacement of the grating B, and B indicates a reflecting part in the reading head system, that is, a first reference reflecting assembly, a first measuring reflecting assembly, a second reference reflecting assembly, and a second measuring reflecting assembly, which will be described later.
In this embodiment, the grating B is a high-groove diffraction grating B, and the groove density is preferably 1800gr/mm.
It should be noted that, the light source 1 may be a dual-frequency laser, the generated basic beam is two beams of laser light with two frequencies and polarization states perpendicular to each other, and for convenience of subsequent description, the polarization state of one beam is marked as a vertical polarization state, and the polarization state of the other beam is marked as a horizontal polarization state. The first beam splitter 2 in this embodiment is a conventional beam splitter prism, the second beam splitter 3 and the third beam splitter 4 are all polarization beam splitters, the first beam splitter 2 can equally divide the basic beam into a first basic beam and a second basic beam, it should be noted here that the first basic beam still includes two laser beams with two frequencies and polarization states perpendicular to each other, and the second basic beam still includes two laser beams with two frequencies and polarization states perpendicular to each other. The second beam splitter 3 is a polarizing beam splitter prism, that is, a prism that splits a light beam by using a polarization state of the light beam, so that after the first base light beam enters the second beam splitter 3, the light beam having a vertical polarization state is reflected by the second beam splitter 3, and the light beam having a horizontal polarization state is transmitted to form a first reference light beam and a first measuring light beam, where it should be noted that the polarization state of the first reference light beam in the refraction stage is a vertical polarization state, and the polarization state of the first measuring light beam in the refraction stage is a horizontal polarization state. Similarly, the third beam splitter 4 is a polarization beam splitter prism, that is, a prism that splits the light beam by using the polarization state of the light beam, so after the second base light beam is incident on the third beam splitter 4, the light beam with the vertical polarization state is reflected by the third beam splitter 4, and the light beam with the horizontal polarization state is transmitted to form the second reference light beam and the second measuring light beam, where it should be noted that the polarization state of the second reference light beam in this refraction stage is the vertical polarization state, and the polarization state of the second measuring light beam in this refraction stage is the horizontal polarization state.
The first steering mirror 13 has a first steering reflection surface, and the first measuring beam is reflected and deflected by the first steering reflection surface after being emitted from the second beam splitter 3, so as to achieve a steering effect; preferably, the angle between the first redirecting surface and the first measuring beam is 45 °, and the first measuring beam can be at a first predetermined angle θ as shown in fig. 1 and 2 1 The incident grating B, it should be noted that the first preset angle θ 1 Corresponding to the littrow angle of grating B. Similarly, the second steering mirror 14 has a second steering reflecting surface, and after the second measuring beam is emitted from the third beam splitter 4, the second measuring beam is deflected by the reflection of the second steering reflecting surface, so as to achieve a steering effect; in some preferred embodiments, the included angle between the second redirecting surface and the second measuring beam is 45 °, and the second measuring beam may enter the grating B at a second preset angle β shown in fig. 1 and 2, where the second preset angle β corresponds to the littrow angle of the grating B.
The first reference reflection assembly is arranged on the side surface of the second beam splitter 3 and is arranged on the opposite side of the first steering mirror 13, namely, the first reference reflection assembly is arranged corresponding to the light path propagation direction of the first reference beam, the first reference beam is reflected to the second beam splitter 3 after being subjected to first deflection through the first reference reflection assembly, the first deflection comprises the conversion of the polarization state of the first reference beam, the polarization state of the first reference beam is converted from the vertical polarization state to the horizontal polarization state, and the deflected first reference beam can be directly transmitted to the first detector 11 from the second beam splitter 3 after being injected into the second beam splitter 3.
Similarly, the second reference reflection component is disposed on a side of the third beam splitter 4 and opposite to the second turning mirror 14, that is, the second reference reflection component is disposed corresponding to the optical path propagation direction of the second reference beam, and the second reference beam is reflected to the third beam splitter 4 after being deflected by the second reference reflection component, where the second deflection includes a conversion of the polarization state of the second reference beam, and converts the polarization state of the second reference beam from a vertical polarization state to a horizontal polarization state, and the deflected second reference beam can be directly transmitted from the third beam splitter 4 to the second detector 12 after being injected into the third beam splitter 4.
It should be noted that, in the measuring process, the grating B needs to reflect the light beam reflected by the measuring reflection assembly for multiple times, so as to facilitate differentiation, the diffraction light beams generated by multiple reflections are ordered according to the light beam receiving sequence of the grating B, that is, the first diffraction light beam is the first diffraction light beam corresponding to the first preset angle θ of the grating B 1 The first diffracted beam is reflected by the first measurement reflection component and then is re-emitted into the grating B, so that a second diffracted beam is formed, the second diffracted beam is re-reflected by the first measurement reflection component and then is emitted into the grating B, and the grating B is re-reflected so that a third diffracted beam is formed, and the reflection principle of the grating B shown in fig. 3 is specifically combined for understanding; similarly, the fourth diffracted beam is generated by the grating B corresponding to the second measurement beam incident at the second preset angle β, and is reflected by the second measurement reflection assembly and then re-incident into the grating B, which is re-reflected to form a fifth diffracted beam, and the second measurement reflection assembly further The third diffracted beam finally formed carries displacement information of the grating B and returns the third diffracted beam through the original path of the propagation direction of the first measuring beam, namely, the third diffracted beam is reflected into the second beam splitter 3 through the first steering mirror 13 and then reflected to the first detector 11 through the second beam splitter 3.
It should be noted here that the third preset angle θ 2 When the first measuring beam is reflected by the grating B to form a first diffracted beam and the first diffracted beam is projected onto the grating B again, the second diffracted beam formed by the grating B overlaps the folded path of the first measuring beam, that is, on the premise that the third diffracted beam disappears, that is, the third diffracted beam does not exist, based on the light path propagation characteristic (the exit angle is the same as the incident angle). Similarly, when the fourth preset angle δ is the same as the sixth preset angle η, the second measuring beam is reflected by the grating B to form a fourth diffracted beam, and when the fourth diffracted beam is projected onto the grating B again, the fifth diffracted beam formed by the grating B overlaps with the turning path of the second measuring beam, that is, on the premise that the sixth diffracted beam disappears, and the sixth diffracted beam does not exist.
It should be noted that, the first measurement reflection assembly also has a function of modulating the polarization state of the light beam, and the polarization state of the third diffracted light beam formed after the first measurement light beam is reflected by the first measurement reflection assembly and the grating B for multiple times is a perpendicular polarization state, so after the third diffracted light beam is incident into the second beam splitter 3 through the first steering mirror 13, the third diffracted light beam can be reflected by the second beam splitter 3, and still maintain a polarization relationship perpendicular to the polarization state of the first reference light beam. Similarly, the sixth diffraction beam finally formed carries the displacement information of the grating B and returns the displacement information in the original path through the propagation direction of the second measurement beam, that is, the displacement information is reflected into the third beam splitter 4 through the second steering mirror 14 and then reflected to the second detector 12 through the third beam splitter 4, and it should be noted that the second measurement reflection assembly also has the function of modulating the polarization state of the beam, and after the second measurement beam is reflected by the second measurement reflection assembly and the grating B for multiple times,the polarization state of the sixth diffracted beam is perpendicular, so that the sixth diffracted beam after being incident on the third beam splitter 4 via the second turning mirror 14 can be reflected by the third beam splitter 4, and still maintain a polarization relationship perpendicular to the polarization state of the second reference beam. In this embodiment, the light source 1 is divided into two parts, and two-dimensional high-precision displacement measurement along the direction of the vector of the grating B and the direction of the normal of the grating B is realized by the beam splitter assembly, the steering assembly, the first reference reflection assembly, the first measurement reflection assembly, the second reference reflection assembly and the second measurement reflection assembly, the first reference beam and the second reference beam serving as reference effects are directly reflected to the first detector 11 and the second detector 12 after the base beam emitted by the light source 1 passes through different optical devices, and the first measurement beam and the second measurement beam respectively take a first preset angle θ 1 After being obliquely incident to the surface of the grating B at the second preset angle beta, the grating B is reflected and returned in the original path to form a stable interference signal carrying displacement information, and the stable interference signal is received by the first detector 11 and the second detector 12, so that the high resolution can be obtained by using the high-reticle-density grating B, the influence on experimental precision caused by the type error of the grating B surface can be avoided, and the method is suitable for the grating B interferometry application scene with high integration, small volume and high precision measurement requirements.
Referring to fig. 1 and 2, in some embodiments, the first reference reflection assembly includes a first reference wave plate 5 and a first reference mirror 15, where the first reference wave plate 5 is disposed on one side of the second beam splitter 3 and is disposed on the opposite side of the first steering mirror 13; the first reference reflecting mirror 15 is arranged on the side of the first reference wave plate 5 away from the second beam splitter 3; the first measuring reflecting component comprises a first measuring reflecting mirror 17 and a first measuring wave plate 7, the first measuring reflecting mirror 17 is arranged on the turning path of the first diffracted beam, and the first measuring reflecting mirror 17 is used for turning the first diffracted beam according to a third preset angle theta 2 Reflecting into the grating B; the first measuring wave plate 7 is arranged at a third preset angle theta of the first diffracted beam 2 Injecting the light into a turning path of the grating B;
the second reference reflecting component comprises a second reference wave plate 6 and a second reference reflecting mirror 16, and the second reference wave plate 6 is arranged on one side of the third spectroscope 4 and is arranged on the opposite side of the second steering mirror 14; the second reference reflecting mirror 16 is arranged on the side of the second reference wave plate 6 away from the third spectroscope 4; the second measuring reflecting component comprises a second measuring reflecting mirror 18 and a second measuring wave plate 8, the second measuring reflecting mirror 18 is arranged on the turning path of the fourth diffracted beam, and the second measuring reflecting mirror 18 is used for reflecting the fourth diffracted beam into the grating B according to a fourth preset angle delta; the second measuring wave plate 8 is arranged on the folded path of the fourth diffracted beam, which impinges on the grating B at a fourth predetermined angle delta.
In this embodiment, the first reference wave plate 5 is disposed on the side of the second beam splitter 3 and is spaced from the first reference mirror 15, the first reference wave plate 5 is specifically disposed on the outgoing path of the first reference beam reflected by the second beam splitter 3, and the first reference beam is directly incident at an angle perpendicular to the reflecting surface of the first reference mirror 15, so that the first reference beam returns to the second beam splitter 3 in the original path under the reflection of the first reference mirror 15, but the polarization state of the first reference beam is changed from the vertical polarization state to the horizontal polarization state under the conversion of the first reference wave plate 5. Similarly, the second reference wave plate 6 is disposed on a side surface of the third beam splitter 4 and is spaced from the second reference reflecting mirror 16, the second reference wave plate 6 is specifically disposed on an outgoing path of the second reference beam reflected by the third beam splitter 4, and the second reference beam directly enters at an angle perpendicular to the reflecting surface of the second reference reflecting mirror 16, so that the second reference beam returns to the third beam splitter 4 in the original path under the reflection effect of the second reference reflecting mirror 16, but the polarization state of the second reference beam is changed from the vertical polarization state to the horizontal polarization state under the conversion of the second reference wave plate 6.
In this embodiment, the positional relationship between the first measuring waveplate 7 and the first measuring mirror 17 should be understood in conjunction with fig. 1 and 2, and the folded path of the first diffracted beam includes two parts: the first part is the folded path of the first diffracted beam reflected by the grating B and incident on the first measuring mirror 17; the second part is a third preset angle theta after the first measuring reflector 17 reflects the first diffracted beam 2 This return path into grating B, first measurementThe wave plate 7 may be provided in either of these two sections. In some preferred embodiments, the first measurement paddle is disposed in the second section. The first portion and the second portion may overlap, i.e. as shown in fig. 2.
The positional relationship between the second measuring wave plate 8 and the second measuring mirror 18 is the same as the positional relationship between the first measuring dial and the first measuring mirror 17, and will not be described here.
Referring to fig. 3, in some embodiments, the first measuring mirror 17 is further configured to reflect the second diffracted beam into the grating B at a fifth predetermined angle epsilon; the second measuring mirror 18 is also used to reflect the fifth diffracted beam into the grating B according to a sixth preset angle η.
Referring to FIG. 3, in some embodiments, a third predetermined angle θ 2 Different from the fifth preset angle epsilon; the fourth preset angle delta is different from the sixth preset angle eta.
Preferably, the third preset angle θ shown in the present embodiment 2 Different from the fifth preset angle epsilon, the fourth preset angle delta is different from the sixth preset angle eta, namely, as shown in fig. 2, the turning paths of the first diffraction light beam and the second diffraction light beam are different, the turning paths of the fourth diffraction light beam and the fifth diffraction light beam are different, and the optical subdivision is further doubled, so that higher resolution is obtained.
In some embodiments, the first reference waveplate 5 and/or the first measurement waveplate 7 and/or the second reference waveplate 6 and/or the second measurement waveplate 8 are quarter waveplates.
In some embodiments, the fundamental light beam comprises a first fundamental light beam and a second fundamental light beam, the first fundamental light beam and the second fundamental light beam having different polarization angles, the first fundamental light beam and the second fundamental light beam having different frequencies.
Referring to fig. 1 and 2, in some embodiments, the detection assembly further includes a first polarizer 9 and a second polarizer 10, where the first polarizer 9 is disposed at the receiving end of the first detector 11; the second polarizer 10 is disposed at the receiving end of the second detector 12.
It should be noted that, the first polarizer 9 is disposed at the receiving end of the first detector 11, the second polarizer 10 is disposed at the receiving end of the second detector 12, so that the first reference beam and the third diffracted beam collected by the first detector 11 can generate stable interference signals, the first polarizer 9 modulates the polarization states of the first reference beam and the third diffracted beam to the same polarization state, and similarly, the second polarizer 10 modulates the polarization states of the second reference beam and the sixth diffracted beam to the same polarization state, so that the second reference beam and the sixth diffracted beam collected by the second detector 12 can generate stable interference signals.
Referring to fig. 4, in a second aspect, the present embodiment provides a two-dimensional grating interferometry method, which is applicable to the two-dimensional grating interferometry apparatus of the first aspect, where the base beam includes a first base beam and a second base beam, and the method includes:
s11, acquiring a first reference complex amplitude value of a first reference beam acquired by a first detector and a first measured complex amplitude value of a third diffraction beam, and generating a first interference signal according to the first reference complex amplitude value and the first measured complex amplitude value;
s12, acquiring a second reference complex amplitude value of a second reference beam acquired by a second detector and a second measured complex amplitude value of a sixth diffraction beam, and generating a second interference signal according to the second reference complex amplitude value and the second measured complex amplitude value;
s13, solving the first interference signal and the second interference signal according to the Doppler frequency shift effect, and generating a movement displacement measurement value of the grating.
Recording the beam frequency of the first basic beam asThe beam frequency of the second basic beam is recorded as +.>The method comprises the steps of carrying out a first treatment on the surface of the The complex amplitude value of the first base beam and the complex amplitude value of the second base beam of orthogonal linearly polarized light having a certain frequency difference output from the light source can be expressed by the formula (10), and the formula (10) is as follows:
；
In the method, in the process of the invention,for the initial phase value of the first base beam, < >>For the initial phase value of the second basic beam, < >>Representing the initial complex amplitude value.
In some embodiments, the first interference signal is recorded asThe first interference signal is represented by formula (1), formula (1) being as follows:
；
in the method, in the process of the invention,for the first measured complex amplitude value, +.>For the first reference complex amplitude value, +.>In order for the time of the grating to move,for the frequency variation of the light beam generated during the grating movement, for example>Is the phase change value of the light beam generated when the grating moves,an initial phase for the third diffracted beam;
recording the second interference signal asThe second interference signal is represented by formula (2), formula (2) as follows:
；
in the method, in the process of the invention,for the second measurement of the complex amplitude value, +.>For the second reference complex amplitude value, +.>Is the initial phase of the sixth diffracted beam.
The complex amplitude value of the first interference signal can be obtained by combining the complex amplitude value of the first base beam and the complex amplitude value of the second base beam, which are described above, and the complex amplitude value of the first interference signal is represented by formula (11), where formula (11) is as follows:
；
the complex amplitude value of the second interference signal is represented by formula (12), and formula (12) is as follows:
；
after integrating the formula (11) and the formula (12), the formula (1) and the formula (2) can be obtained.
In some embodiments, the beam frequency variation value produced by the doppler shift effect is represented by equation (3), equation (3) is as follows:
；
record the grating atThe displacement in the direction is +.>The grating is->The initial phase of the direction is +.>The grating is->The displacement in the direction is +.>The grating is->The initial phase of the direction is +.>；
Expressed by the formula (4), the formula (4) is as follows:
；
in the method, in the process of the invention,for grating diffraction order +.>For the speed of movement of the grating along the grating vector, < >>Is the grating pitch of the grating;
expressed by the formula (5), the formula (5) is as follows:
；
in the method, in the process of the invention,for a first preset angle, < >>A third preset angle;
expressed by the formula (6), the formula (6) is as follows:
；
expressed by the formula (7), the formula (7) is as follows:
；
displacement of grating movementExpressed by the formula (8), the formula (8) is as follows: />
；
Displacement of grating movementExpressed by the formula (9), the formula (9) is as follows:
。
it should be noted that the number of the substrates,means that the grating collected by the first basic beam after being split and reflected is carried in the third diffracted beam +.>Phase change value of direction, +.>Means that the grating collected by the first basic beam after being split and reflected is carried in the third diffracted beam +. >Phase change value of direction, +.>Means that the grating collected by the second basic beam after being split and reflected is carried in the sixth diffracted beam +.>Phase change value of direction, +.>Means that the grating collected by the second basic beam after being split and reflected is carried in the sixth diffracted beam +.>Phase change value of direction. About->For a first preset angle, < >>The third predetermined angle is understood as follows: because the first preset angle and the second preset angle are in mirror image relationship, the angle values of the acute angles indicated by the first preset angle and the second preset angle are the sameIn the same way, i.e.)>The third preset angle and the fourth preset angle are in mirror image relationship, so that the angle values of the acute angles indicated by the third preset angle and the fourth preset angle are the same, namely +.>。
According to the technical scheme, the two-dimensional grating interferometry device comprises a grating, a light source, a beam splitting component, a steering component, a first reference reflection component, a first measurement reflection component, a second reference reflection component and a second measurement reflection component, the light source is split into two parts, and a stable interference signal carrying displacement information is formed and received by the first detector and the second detector after being reflected by the first measurement reflection component, the second reference reflection component and the second measurement reflection component, so that two-dimensional high-precision displacement measurement along the grating vector direction and the grating normal direction is realized, a basic light beam emitted by the light source is directly reflected to the first detector and the second detector after passing through different optical devices, and the first reference light beam and the second reference light beam serving as reference functions are directly reflected to the first detector and the second detector, and the first measurement light beam and the second measurement light beam are respectively inclined to the grating surface at a first preset angle and a second preset angle and then returned by the original path, the stable interference signal carrying displacement information is formed, the high-resolution high-density grating can be obtained, the influence of a high-scale-density grating on the accuracy experiment can be avoided, and the device is suitable for the application of the grating with high-precision measurement requirements.
It should be appreciated that various forms of the flows shown above may be used to reorder, add, or delete steps. For example, the steps described in the present disclosure may be performed in parallel, sequentially, or in a different order, provided that the desired results of the technical solutions of the present disclosure are achieved, and are not limited herein.
The above embodiments do not limit the scope of the present invention. It will be apparent to those skilled in the art that various modifications, combinations, sub-combinations and alternatives are possible, depending on design requirements and other factors. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of the present invention.
Claims (10)
1. A two-dimensional grating interferometry apparatus comprising:
a grating;
a light source for generating a base beam;
the light splitting assembly comprises a first spectroscope, a second spectroscope and a third spectroscope, wherein the first spectroscope is used for splitting the basic light beam into a first basic light beam and a second basic light beam, the first basic light beam enters the second spectroscope after being transmitted by the first spectroscope, the second spectroscope is used for splitting the first basic light beam into a first measuring light beam and a first reference light beam, the second basic light beam enters the third spectroscope after being reflected by the first spectroscope, and the third spectroscope is used for splitting the second basic light beam into a second measuring light beam and a second reference light beam;
The first steering mirror is arranged on one side of the second beam splitter, the first measuring beam is transmitted by the second beam splitter and enters the first steering mirror, the first steering mirror is used for refracting the first measuring beam and then injecting the first measuring beam into the grating according to a first preset angle, the second steering mirror is arranged on one side of the third beam splitter, the second measuring beam is transmitted by the third beam splitter and then enters the second steering mirror, the second steering mirror is used for refracting the second measuring beam and then injecting the second measuring beam into the grating according to a second preset angle, and the grating is used for reflecting the first measuring beam and then forming a first diffraction beam and reflecting the second measuring beam and then forming a fourth diffraction beam;
the first reference reflection assembly is arranged on the side face of the second beam splitter and is arranged on the opposite side of the first steering mirror, and the first reference reflection assembly is used for carrying out first deflection on a first reference beam reflected by the second beam splitter and then reflecting the first reference beam to the second beam splitter;
the first measuring reflection component is arranged on the deflection path of the first diffraction beam, and is used for reflecting the first diffraction beam to the grating, the grating is used for reflecting the first diffraction beam to form a second diffraction beam, the first measuring reflection component is used for reflecting the second diffraction beam to the grating, the grating is used for reflecting the second diffraction beam to form a third diffraction beam, and the third diffraction beam is reversely shot into the first steering mirror;
The second reference reflection assembly is arranged on the side face of the third spectroscope and is opposite to the second steering mirror, and the second reference reflection assembly is used for carrying out second deflection on the second reference beam reflected by the third spectroscope and then reflecting the second reference beam to the third spectroscope;
the second measuring reflection component is arranged on the deflection path of the fourth diffraction beam, and is used for reflecting the fourth diffraction beam to the grating, the grating is used for reflecting the fourth diffraction beam to form a fifth diffraction beam, the second measuring reflection component is used for reflecting the fifth diffraction beam to the grating, the grating is used for reflecting the fifth diffraction beam to form a sixth diffraction beam, and the sixth diffraction beam is reversely shot into the second steering mirror;
the detection assembly comprises a first detector and a second detector, the first detector is used for receiving a third diffraction light beam which enters the second spectroscope and is reflected after being reflected by the first steering mirror, and is used for receiving a first reference light beam which enters the second spectroscope and is transmitted after being reflected by the first reference reflection assembly, and the second detector is used for receiving a sixth diffraction light beam which enters the third spectroscope and is reflected after being reflected by the second steering mirror, and is used for receiving a second reference light beam which enters the third spectroscope and is transmitted after being reflected by the second reference reflection assembly.
2. The two-dimensional grating interferometry apparatus of claim 1, wherein the first reference reflection assembly comprises:
the first reference wave plate is arranged on one side of the second beam splitter and is arranged on the opposite side of the first steering mirror;
the first reference reflecting mirror is arranged on one side of the first reference wave plate, which is far away from the second beam splitter;
the first measurement reflection assembly includes:
the first measuring reflector is arranged on the turning path of the first diffraction light beam and is used for reflecting the first diffraction light beam into the grating according to a third preset angle;
the first measuring wave plate is arranged on a turning path of the first diffraction light beam which is injected into the grating at a third preset angle;
the second reference reflection assembly includes:
the second reference wave plate is arranged on one side of the third spectroscope and is arranged on the opposite side of the second steering mirror;
the second reference reflecting mirror is arranged on one side of the second reference wave plate, which is far away from the third spectroscope;
the second measurement reflection assembly includes:
the second measuring reflector is arranged on the turning path of the fourth diffraction light beam and is used for reflecting the fourth diffraction light beam into the grating according to a fourth preset angle;
The second measuring wave plate is arranged on a turning path of the fourth diffraction light beam which is injected into the grating at a fourth preset angle.
3. The two-dimensional grating interferometry apparatus of claim 2, wherein the first measurement mirror is further configured to reflect the second diffracted beam into the grating at a fifth predetermined angle;
the second measuring mirror is further configured to reflect the fifth diffracted beam into the grating at a sixth predetermined angle.
4. A two-dimensional grating interferometry apparatus according to claim 3 wherein,
the third preset angle is different from the fifth preset angle;
the fourth preset angle is different from the sixth preset angle.
5. The two-dimensional grating interferometry device according to claim 2, wherein the first reference waveplate and/or the first measurement waveplate and/or the second reference waveplate and/or the second measurement waveplate is a quarter waveplate.
6. The two-dimensional grating interferometry apparatus of claim 1, wherein the fundamental beam comprises a first fundamental beam and a second fundamental beam, the first fundamental beam having a different polarization angle than the second fundamental beam, the first fundamental beam having a different frequency than the second fundamental beam.
7. The two-dimensional grating interferometry apparatus of claim 1, wherein the detection assembly further comprises:
the first polaroid is arranged at the receiving end of the first detector;
and the second polaroid is arranged at the receiving end of the second detector.
8. A two-dimensional grating interferometry method, suitable for use with the two-dimensional grating interferometry apparatus of any of claims 1-7, the fundamental beam comprising a first fundamental beam and a second fundamental beam, the method comprising:
acquiring a first reference complex amplitude value of a first reference beam acquired by the first detector and a first measured complex amplitude value of the third diffraction beam, and generating a first interference signal according to the first reference complex amplitude value and the first measured complex amplitude value;
acquiring a second reference complex amplitude value of a second reference beam acquired by the second detector and a second measured complex amplitude value of the sixth diffracted beam, and generating a second interference signal according to the second reference complex amplitude value and the second measured complex amplitude value;
and solving the first interference signal and the second interference signal according to the Doppler frequency shift effect, and generating a movement displacement measurement value of the grating.
9. The two-dimensional grating interferometry method of claim 8, wherein the first interferometry signal isThe first interference signal is represented by formula (1), the formula (1) being as follows:
；
in the method, in the process of the invention,for the first measured complex amplitude value, +.>For the first reference complex amplitude value, +.>For the raster shift time, +.>For the frequency variation of the light beam generated during the grating movement, for example>For the phase change value of the light beam generated during the grating movement, is->An initial phase for the third diffracted beam;
recording the second interference signal asThe second interference signal is represented by formula (2), the formula (2) being as follows:
；
in the method, in the process of the invention,for the second measurement of the complex amplitude value, +.>For the second reference complex amplitude value, +.>Is the initial phase of the sixth diffracted beam.
10. The two-dimensional grating interferometry method of claim 9, wherein the change in beam frequency due to the doppler shift effect is represented by equation (3), the equation (3) being as follows:
；
record the grating atThe displacement in the direction is +.>The grating is->The initial phase of the direction is +.>The grating is->The displacement in the direction is +.>The grating is- >The initial phase of the direction is +.>；
The saidExpressed by the formula (4), the formula (4) is as follows:
；
in the method, in the process of the invention,for grating diffraction order +.>For the speed of movement of the grating along the grating vector, < >>Is the grating pitch of the grating;
the saidExpressed by the formula (5), the formula (5) is as follows:
；
in the method, in the process of the invention,for a first preset angle, < >>A third preset angle;
the saidExpressed by the formula (6), the formula (6) is as follows:
；
the saidRepresented by formula (7), the formula (7) is as follows:
；
displacement of the gratingExpressed by the formula (8), the formula (8) is as follows:
；
displacement of the gratingExpressed by the formula (9), the formula (9) is as follows:
。
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