CN111811474A - Method for simultaneously measuring inclination angle and absolute distance based on optical transfer function - Google Patents
Method for simultaneously measuring inclination angle and absolute distance based on optical transfer function Download PDFInfo
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- G01—MEASURING; TESTING
- G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
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- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
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Abstract
The invention discloses a method for simultaneously measuring an inclination angle and an absolute distance based on an optical transfer function, belonging to the technical field of photoelectric measurement. The implementation method of the invention comprises the following steps: two beams of light waves with phases are focused on a focal plane to generate an interference-diffraction or diffraction phenomenon, the light intensity distribution on the focal plane is PSF, and the PSF is subjected to Fourier transform to obtain the OTF. And selecting a side peak of the OTF under the condition of monochromatic light to calculate the ratio of the real part of the imaginary part of the OTF to obtain a corresponding side peak phase transfer function, calculating the gradient value of the phase transfer function, and calculating the inclination angle of the measured sub-mirror by using the gradient value to realize the measurement of the inclination angle. Selecting a side peak of the optical transfer function OTF under the broad spectrum condition after the differential summation method processing to obtain a corresponding side peak modulation transfer function, and obtaining MTF through normalization processingnphObtaining MTF by piecewise quartic fittingnphThe function relation with the absolute distance is obtained according to the obtained function relationAnd high-precision measurement of absolute distance without the influence of inclination angle errors is realized.
Description
Technical Field
The invention relates to a method for simultaneously measuring an inclination angle and an absolute distance based on an optical transfer function, which is a method for simultaneously measuring the inclination angle and the absolute distance with large range, high precision and high efficiency and belongs to the technical field of photoelectric measurement.
Background
In the fields of astronomy, precision metrology, and the like, it is often necessary to perform high-precision measurement on a minute tilt angle (e.g., a rotation angle of a measured sub-mirror around its coordinate axis), a step height (e.g., an absolute distance between a reference sub-mirror and the measured sub-mirror along an optical axis), and the like, and correct the measured tilt angle and the step height to obtain a high-resolution image. A great deal of research has been conducted by domestic and foreign scholars on the measurement of inclination angles and step heights.
For the measurement of the tilt angle, a staged mode is widely adopted to realize a wide-range high-precision measurement of the tilt angle at present: firstly, realizing low-precision measurement of a large inclination angle by using a light spot mass center detection method, which is a rough measurement stage; then, the wavefront Phase Recovery (PR) method or the interference fringe information is used to realize high-precision measurement of small tilt angle, which is a fine measurement stage. Currently, the most representative measurement of the common phase tilt error (tip and tilt) of JWST (the James Webb Space scales) system is to combine the center-of-mass detection method with PR method, use the center-of-mass detection method to perform rough tilt angle measurement, correct the tilt angle to a small tilt angle range satisfying PR method, and use PR method to perform fine measurement, thereby achieving high precision measurement of the tilt angle between the sub-mirrors of the system (see James B. Hadaway, Conrad Wells, Gene A. Olczak, Mark Waldman, Tony L. Whitman, Joseph Cosending, Michael Zarella, Mark Connolly, David M. Chaney, random testing, "practical-of the center-of-corner Optical tilt error of Webc Space scales, J.2018, Spot test of software, J.S. Chapter, J. 6, J. Chapter, P. Chapter, J. 6, P. Chapter, P. Su, J. 3, P. 3, J. 3, C., P..
Methods for measuring the step height can be generally classified into two categories. The first type is a contact detection method, which is also called a direct measurement method and can be realized by using a three-coordinate measuring machine, but the method has long measurement time and is easy to damage the surface of an object; the other type is a non-contact type measuring method, and a common Optical detection method is adopted, wherein the actual step height to be measured by the Optical detection method corresponds to the Optical Path Difference (OPD) between the reference Optical Path and the measuring Optical Path, for the reflection type Optical Path, the OPD is twice of the measured step height, and the actual micro step height can be obtained by measuring the OPD. In 2016, Zhaowei and Jiang Jun Lun, the university of Beijing Physician, proposed a high-precision absolute distance measurement method based on optical transfer function, which can realize absolute measurement of step height, and its basic principle is: the method comprises the steps that discrete diaphragm holes are arranged in a light path, when only absolute distance exists between a measured sub-mirror and a reference sub-mirror, sub-light waves reflected by the two sub-mirrors are respectively collected, a Point Spread Function (PSF) of a system is obtained at a focal plane of a subsequent optical system, an optical Modulation Transfer Function (MTF) of the system is obtained according to a Fourier optical principle, a functional relation between a side peak value of a normalized MTF and the absolute distance is obtained through four-time fitting in a segmented mode, and large-range and high-precision measurement of the absolute distance can be achieved according to the functional relation. The final absolute distance measurement range is the coherence length of the light source, and the accuracy can reach the nanometer level (see: Jiang Junlun and Zhao Weirii, "pharmaceutical prism induced descriptors," Opt. express 24, 19123-.
Interferometric methods in optical detection methods are also commonly used to achieve tilt angle and absolute distance measurements. In 2012, Anthony c.cheetham and Peter g.tuthill et al propose a fizeau interferometry method to implement common phase error (tilt angle, absolute distance) detection of a segmented primary mirror telescope, the basic principle is that discrete non-redundant diaphragm holes are arranged in an optical path, sub-light waves reflected by adjacent sub-mirrors are respectively collected, a point spread function PSF of the system can be obtained at a focal plane of a subsequent optical system, interference fringe information in the point spread function PSF is utilized, a least square method is adopted to perform multiple iterations, when the measured sub-mirrors have only absolute distance error and tilt error respectively, the tilt angle and the absolute distance of the measured sub-mirrors are solved, so as to implement high-precision measurement of the tilt angle and the absolute distance respectively, the method can achieve 0.5 arcsconds for the maximum detection range of the tilt angle, the precision is 3.7 masmrs, the maximum detection range of the absolute distance can achieve 150 μm, and the precision is 0.75nm (see Anthony c.cheetham, peter G.Tuthill, Anand Sivaramakrishnan, and james P.Lloyd, "Fizeau interactive phasing of segmented mirrors," Opt.Express 20, 29457-.
In summary, under the condition that both the tilt angle and the absolute distance exist, the tilt error can seriously affect the measurement of the absolute distance, the current measurement method firstly detects and corrects the tilt error and then detects and corrects the absolute distance error, a measurement system needs to be provided with hardware devices for measuring two errors, the method is complicated, the respective measurement needs a certain time, and the measurement and correction of the system error cannot be carried out in real time.
Disclosure of Invention
In order to solve the above-mentioned disadvantages of the prior art, the present invention provides a method for simultaneously measuring a large-range, high-precision, and high-efficiency tilt angle and an absolute distance based on an optical Transfer Function, which obtains an Optical Transfer Function (OTF) of a system according to a Point Spread Function (PSF) which is a light intensity distribution on a focal plane of the system, obtains a Modulation Transfer Function (MTF) by modeling a side peak of the OTF and a Phase Transfer Function (PTF) by taking a Phase, obtains a tilt angle according to a relation of the side peak tilt angle of the PTF, obtains an absolute distance by performing a quartic polynomial fitting according to a relation of a normalized side peak and the absolute distance of the MTF, and does not affect each other when measuring two errors, the method is simple and fast, the device is simple and easy to construct, and can realize a large-range, a high-precision, and a high-efficiency tilt angle and an absolute distance of the system, High precision and high efficiency.
The purpose of the invention is realized by the following technical scheme.
The invention discloses a method for simultaneously measuring an inclination angle and an absolute distance based on an optical transfer function. Selecting a side peak of the optical transfer function OTF under the condition of monochromatic light to calculate the ratio of the real part of the imaginary part of the optical transfer function OTF to obtain the corresponding side peak phase transfer function PTFphCalculating the phase transfer function PTFphThe gradient value is used for calculating the inclination angle of the measured sub-mirror, so that the large-range and high-precision measurement of the inclination angle is realized. Selecting a side peak of the optical transfer function OTF under the broad spectrum condition after the differential summation method processing to obtain the corresponding side peak modulation transfer function MTFphFor modulation transfer function MTFphNormalized to obtain MTFnphObtaining MTF by piecewise quartic fittingnphAnd (4) solving an absolute distance value according to the obtained functional relation with the absolute distance, so as to realize high-precision measurement of the absolute distance without being influenced by the error of the inclination angle.
The invention discloses a method for simultaneously measuring an inclination angle and an absolute distance based on an optical transfer function, which comprises the following steps:
step one, light waves emitted by a parallel light source are divided into two beams, the two beams enter a reference light path and a measurement light path respectively to obtain two beams of light waves with phase difference information, the two beams of light waves are focused on a focal plane to generate an interference-diffraction or diffraction phenomenon, and the light intensity distribution on the focal plane is a point spread function PSF. The phase difference information includes a tilt angle and an absolute distance.
Known from fourier optics, under monochromatic light conditions, the normalized point spread function PSF is:
wherein (x, y) is image plane coordinate, x is image plane abscissa, y is image plane ordinate, D is circular aperture diaphragm diameter, J1() Is a first-order Bessel function, lambda is the wavelength of monochromatic light, f is the focal length of the system, B is the central distance of the two circular hole diaphragms, and p is the axial distance between the measured sub-mirror and the reference sub-mirror. Definition (x)0,y0) Is the coordinate of the pupil plane, and the coordinates of the pupil plane,
is the normal of the measured sub-mirror and y0The included angle between the axes is set by the angle,
is the normal line and x of the measured sub-mirror0The angle between the axes.
And step two, performing Fourier transform on the point spread function PSF to obtain an optical transfer function OTF, wherein the optical transfer function OTF consists of a main peak and two side peaks, and the side peaks contain information of an inclination angle and an absolute distance, so that the inclination angle and the absolute distance can be conveniently calculated by using the side peaks in the subsequent steps three and six.
Performing Fourier transform on the formula (1) to obtain an optical transfer function OTF of the system:
wherein j is an imaginary unit, which is a convolution operation, (f)x,fy) Is the spatial frequency domain coordinate, f, of the point spread function PSFx＝x0/λf,fy＝y0/λf，OTFsub(fx,fy) Is the diffraction limited system optical transfer function OTF of a single circular aperture exit pupil:
where ρ is the radial coordinate in any direction on the frequency plane,
arccos () is an inverse cosine function.
Where circ () is a circular function.
From equation (2), the optical transfer function OTF consists of three parts,
is an optical transfer functionThe fraction of the main peak of the number OTF,
and
is two side peak parts of the optical transfer function OTF, and is symmetrically distributed on two sides of the main peak. The optical transfer function OTF side peak portion contains the tilt angle and absolute distance information, so the tilt angle and absolute distance are calculated using the side peak portion of the optical transfer function OTF.
Selecting a side peak of the optical transfer function OTF to calculate the ratio of the real part of the imaginary part of the side peak to obtain the corresponding side peak phase transfer function PTFphCalculating PTFphAt fx、fyGradient values in two directions are used for calculating the x-direction pupil plane of the measured sub-lens0、y0The tilt angles tip and tilt in the direction realize large-range and high-precision measurement of the tilt angles.
Selecting a side peak of the optical transfer function OTF to obtain a ratio of the real part of the imaginary part of the OTF, and performing arc tangent calculation to obtain a corresponding PTFph：
Wherein ph represents a side peak.
PTFphAt fx、fyThe distribution in both directions changes with changes in the tilt angle. In PTFphCalculating PTF over one period of variation (. + -. pi)phAt fx、fyGradient values in two directions
The measured sub-mirrors are respectively at fx、fyTwo directions, i.e. pupil plane x0、y0The angle of inclination tip in the direction is
tilt is
Therefore, the wide-range and high-precision measurement of the inclination angle is realized.
And fourthly, performing approximate processing on the point spread function PSF in the integral form under the broad spectrum condition by using a differential summation method to obtain the point spread function PSF in the sectional summation form.
Due to PTFph is obtained by taking the inverse tangent of the ratio of real part of side peak imaginary part of the optical transfer function OTF, the tangent value has periodicity, and the value is obtained according to PTFph can only get absolute distance information of less than one wavelength, so the measurement of absolute distance is achieved by MTF. Under the broad spectrum condition, the system normalized point spread function PSF is:
where Δ λ is the spectral width, λ0Is the broad spectral center wavelength.
The integral is approximated by differential summation, and Δ λ is equally divided into n segments, each segment having a central wavelength λ1，λ2…λi…λnThe point spread function PSF was obtained as:
wherein, (x, y, λ)i) And the image plane coordinates correspond to different wavelength conditions.
And fifthly, carrying out Fourier transform on the point spread function PSF in a segmented summation mode under the broad spectrum condition to obtain an optical transfer function OTF under the broad spectrum condition.
Performing fourier transform on the formula (7) to obtain the optical transfer function OTF of the system:
wherein the content of the first and second substances,
is the spatial frequency domain coordinate, f, of the point spread function PSF under different wavelength correspondencesxi＝x0/λif,fyi＝y0/λif，
Is the diffraction limited system optical transfer function OTF of a single circular aperture exit pupil:
wherein the content of the first and second substances,
step six, selecting a side peak of the optical transfer function OTF to obtain a corresponding side peak modulation transfer function MTFphFor modulation transfer function MTFphAnd carrying out normalization processing, wherein the normalization factor is the peak value of the peak on the MTF side of the modulation transfer function under the condition that only the inclination angle error exists, namely the absolute distance is zero. Normalized MTFphMTF by piecewise quartic fittingphAnd (4) solving the absolute distance p value according to the obtained functional relation with the absolute distance p, so as to realize high-precision absolute distance measurement without being influenced by the inclination angle error.
Obtaining a corresponding modulation transfer function MTF by performing a modulus operation on one side peak of the optical transfer function OTFph，
Wherein ph represents a side peak.
From equation (11), it is known that the tilt angle has an effect on the modulation transfer function, and thus on the MTFphThe normalization processing can eliminate the influence of the inclination angle on the detection of the absolute distance. The normalization factor is the peak value of the side peak of the modulation transfer function MTF under the condition that only the inclination angle error exists, namely the absolute distance is zero
Obtaining modulation transfer function MTF by four times of fitting in segmentation after normalization processingnphThe functional relationship with the absolute distance p is:
wherein the MTFnphIs the normalized modulation transfer function.
The absolute distance, for MTF, can be directly calculated from equation (12)phAfter normalization processing, the calculation of the absolute distance is not affected by the inclination angle, so that high-precision measurement of the absolute distance is realized.
The invention also discloses a device for simultaneously measuring the inclination angle and the absolute distance based on the optical transfer function, which is used for realizing the method for simultaneously measuring the inclination angle and the absolute distance based on the optical transfer function. The diaphragm is placed on the conjugate surface of the reference sub-mirror, and two discrete round holes are formed in the diaphragm and respectively correspond to the two sub-mirrors. The part of the light beam emitted by the parallel light source irradiates the reference sub-mirror and is called as reference light; part of the light irradiates to the measured sub-mirror and is called as detection light; the reference light and the detection light pass through the collimating lens after being reflected by the telescope secondary mirror, then pass through two discrete round holes on the diaphragm respectively, and undergo interference diffraction through the focusing lens, so that interference diffraction fringes under the wide spectrum condition are formed and are imaged on the detector through the focusing lens.
At diaphragm aperture and focusA single-wavelength filter plate is added between the lenses, the detector is imaged as system interference diffraction fringes under the condition of monochromatic light, and the light intensity distribution collected by the detector is the point spread function PSF under the condition of the monochromatic light of the system. And performing Fourier transform on the obtained point spread function PSF to obtain an optical transfer function OTF. Selecting a side peak of the optical transfer function OTF to calculate the ratio of the real part of the imaginary part of the side peak to obtain the corresponding side peak phase transfer function PTFphCalculating the phase transfer function PTFphThe inclination angle of the measured sub-mirror is calculated by utilizing the relation between the gradient value and the inclination angle, so that the large-range and high-precision measurement of the inclination angle is realized. And a single-wavelength filter is not added between the diaphragm hole and the focusing lens, the image on the detector is a system interference diffraction fringe under the wide spectrum condition, and the light intensity distribution acquired by the detector is a point spread function PSF under the system wide spectrum condition. Carrying out approximate processing of a differential summation method on the obtained point spread function PSF, then carrying out Fourier transformation to obtain an optical transfer function OTF, selecting one side peak of the optical transfer function OTF to obtain a corresponding side peak modulation transfer function MTFphFor modulation transfer function MTFphNormalized to obtain MTFnphObtaining MTF by piecewise quartic fittingnphAnd (4) solving an absolute distance value according to the obtained functional relation with the absolute distance, so as to realize high-precision measurement of the absolute distance without being influenced by the error of the inclination angle.
Has the advantages that:
1. the method for measuring the inclination angle by combining the centroid detection method and the PR method has the advantages that when the inclination angle is reduced to enable the reference mirror and the detection mirror to partially overlap spots imaged by infinite star points, the centroid detection method cannot accurately judge the centroid, the PR method needs to acquire point spread functions PSF at the positions before and after the focus, the process is complex, and more time is consumed. The method for simultaneously measuring the inclination angle and the absolute distance based on the optical transfer function only needs to calculate the ratio of the real part of the side peak imaginary part of the optical transfer function OTF of the system to obtain the PTFphBy PTFphThe inclination angle can be obtained by the relation with the inclination angle, and high measurement precision is ensured. The measuring precision of the invention reaches milli-second order, and is not limited by small inclination angle, and the measuring range and the light path are arrangedThe parameters are related.
2. The influence of residual inclination angle on absolute distance measurement is not eliminated by the conventional methods such as a Fizeau interferometry method, and the method for simultaneously measuring the inclination angle and the absolute distance based on the optical transfer function only needs to obtain the MTF by performing side peak modulus extraction on the optical transfer function OTF of a systemphFor MTFphNormalization processing is carried out, and MTF after normalization processingphThe absolute distance is obtained through the relation with the absolute distance p, large-range and high-precision measurement of the absolute distance is achieved, the influence of residual inclination angles on the absolute distance measurement can be eliminated, the measurement precision can reach the nanometer level, and the measurement range can reach the coherence length of a light source.
3. The invention discloses a method for simultaneously measuring an inclination angle and an absolute distance based on an optical transfer function, which can obtain a phase transfer function PTF of a side peak through an optical transfer function OTF of a systemphAnd modulation transfer function MTFphBy PTFphNormalized MTF and the sum of tilt anglesnphThe relation between the system inclination angle and the absolute distance can be calculated, the system inclination angle and the absolute distance are obtained, the two errors are measured without influencing each other, the method is simple and fast, the device is simple and easy to build, and large-range, high-precision and high-efficiency simultaneous measurement of the system inclination angle and the absolute distance is achieved.
Drawings
FIG. 1 is a schematic diagram of an optical path for measuring a common phase error of a blocking mirror;
FIG. 2 is a point spread function of the system;
FIG. 3 is a phase transfer function of the system;
FIG. 4 is a normalized modulation transfer function of the system;
FIG. 5 is a graph showing a relationship between a set value and a measured value of tilt tip when p is 126.56nm and a tilt angle is measured in a range of 0 to 1.06 arcsec;
FIG. 6 is an error curve of a set value and a measured value of a tilt tip when p is 126.56nm and a tilt angle measurement range is 0 to 1.06 arcsec;
FIG. 7 is a graph showing a relationship between a set value and a calculated value of an absolute distance p when the inclination angle is 3.7mas and the absolute distance measurement range is 0 to 195 μm;
FIG. 8 is a graph showing the error between the set value and the calculated value of the absolute distance p when the tilt angle is 3.7mas and the absolute distance measurement range is 0 to 195 μm;
FIG. 9 is a graph showing the error between the set value and the calculated value of the absolute distance p when the tilt angle is 3.7mas and the absolute distance measurement range is 0 to 3.164 μm;
FIG. 10 is a flowchart of a method for simultaneously measuring a tilt angle and an absolute distance based on an optical transfer function according to the present disclosure.
Wherein, 1-telescope secondary environment; 2-reference sub-mirror; 3-the sub-mirror to be tested; 4-a collimating lens; 5-diaphragm hole; 6-a focusing lens; 7-a detector; p is the axial distance between the measured sub-mirror and the reference sub-mirror; and delta t is the included angle between the normal direction of the measured sub-mirror and the optical axis.
Detailed Description
The invention is further illustrated by the following figures and examples.
Example 1:
the embodiment is used for measuring the common phase error of the segmented primary mirror telescope, and a light path schematic diagram taking two secondary mirrors as an example is shown in fig. 1. The flow chart is shown in fig. 10.
The initial common-phase error of the primary mirror of the segmented mirror telescope comprises the following steps: the inclination angle is in the order of arc second, the absolute distance is in the order of sub-millimeter, which respectively corresponds to the normal and y of the measured sub-mirror0Axis or x0The angle of inclination between the axes and the axial distance between the measured and reference sub-mirrors. When the telescope works, a light source of the telescope is starlight and can be regarded as parallel light, wherein one sub-mirror is used as a reference sub-mirror, plane waves are reflected by the reference sub-mirror and a measured sub-mirror, and the reflected light waves carry information of common phase errors between two adjacent sub-mirrors.
As shown in fig. 10, the method for simultaneously measuring an inclination angle and an absolute distance based on an optical transfer function disclosed in this embodiment includes the following specific steps:
step one, a diaphragm 5 is arranged on a conjugate surface of a reference sub-mirror, two discrete circular diaphragm holes are arranged on the diaphragm, light waves of a reference light path and a measuring light path are respectively collected, a detector is arranged on a focal plane of a focusing lens 6, a plane light wave carrying optical path difference information passes through the diaphragm holes and then is subjected to interference diffraction under the action of the focusing lens 6, and light intensity distribution detected on the detector 7 is a point spread function PSF, as shown in figure 1.
And secondly, adding a single-wavelength filter between the diaphragm 5 and the focusing lens 6, wherein the light intensity distribution acquired by the detector 7 is a point spread function PSF under the condition of single wavelength and contains common phase error information of an inclination angle and an absolute distance.
Known from fourier optics, under monochromatic light conditions, the normalized point spread function PSF is:
wherein (x, y) is image plane coordinate, x is image plane abscissa, y is image plane ordinate, D is circular aperture diaphragm diameter, J1() Is a first-order Bessel function, lambda is the wavelength of monochromatic light, f is the focal length of the system, B is the central distance of the two circular hole diaphragms, and p is the axial distance between the measured sub-mirror and the reference sub-mirror. Definition (x)0,y0) Is the coordinate of the pupil plane, and the coordinates of the pupil plane,
is the normal of the measured sub-mirror and y0The included angle between the axes is set by the angle,
is the normal line and x of the measured sub-mirror0The angle between the axes.
And step three, performing Fourier transform on the point spread function PSF to obtain an optical transfer function OTF, wherein the optical transfer function OTF consists of a main peak and two side peaks, and the side peaks contain information of an inclination angle and an absolute distance, so that the inclination angle and the absolute distance can be conveniently calculated by using the side peaks in the subsequent steps four and seven.
Performing Fourier transform on the formula (1) to obtain an optical transfer function OTF of the system:
wherein j is an imaginary unit, which is a convolution operation, (f)x,fy) Is the spatial frequency domain coordinate, f, of the point spread function PSFx＝x0/λf,fy＝y0/λf，OTFsub(fx,fy) Is the diffraction limited system optical transfer function OTF of a single circular aperture exit pupil:
where ρ is the radial coordinate in any direction on the frequency plane,
arccos () is an inverse cosine function.
Where circ () is a circular function.
From equation (2), the optical transfer function OTF consists of three parts,
is the main peak portion of the optical transfer function OTF,
and
is two side peak parts of the optical transfer function OTF, and is symmetrically distributed on two sides of the main peak. The peak portion on the side of the optical transfer function OTF contains the tilt angle and absolute distance information, so the side of the optical transfer function OTF is utilizedThe peak portion calculates the inclination angle and the absolute distance.
Selecting a side peak of the optical transfer function OTF to calculate the ratio of the real part of the imaginary part of the side peak to obtain the corresponding side peak phase transfer function PTFphCalculating PTFphAt fx、fyGradient values in two directions are used for calculating the x-direction pupil plane of the measured sub-lens0、y0The tilt angles tip and tilt in the direction realize large-range and high-precision measurement of the tilt angles. Introducing an absolute distance of 126.56nm into the sub-mirror 3 to be measured, introducing a tilt angle with the stepping of 8.706mas and the range of 0-1.06 arcsec, and analyzing the relation between the setting error and the measurement error.
Selecting a side peak of the optical transfer function OTF to obtain a ratio of the real part of the imaginary part of the OTF, and performing arc tangent calculation to obtain a corresponding PTFph：
Wherein ph represents a side peak.
PTFphAt fx、fyThe distribution in both directions changes with changes in the tilt angle. In PTFphCalculating PTF over one period of variation (. + -. pi)phAt fx、fyGradient values in two directions
The measured sub-mirrors are respectively at fx、fyTwo directions, i.e. pupil plane x0、y0The angle of inclination tip in the direction is
tilt is
Therefore, the wide-range and high-precision measurement of the inclination angle is realized.
Introducing an absolute distance of 126.56nm into the sub-mirror 3 to be measured, introducing a tilt angle with the stepping of 8.706mas and the range of 0-1.06 arcsec, and analyzing the relation between the setting error and the measurement error. As shown in fig. 5, the abscissa indicates the actually measured tilt amount, and the ordinate indicates the introduced tilt amount, and it can be seen from the graph that the measured value of the tilt angle is very close to the set value. And comparing the set value and the calculated value of the inclination angle to obtain the measurement error of the method. FIG. 6 is an error curve of the set value and the measured value of the tilt tip in the range of 0 to 1.06arcsec, wherein the abscissa is the set value of the tilt angle, and the ordinate is the error of the set value and the calculated value of the tilt angle, so as to obtain the measurement precision of 2.468mas RMS.
And step five, removing a single-wavelength filter added between the diaphragm 5 and the focusing lens 6, wherein the light intensity distribution acquired by the detector 7 is a point spread function PSF under the broad spectrum condition. And (3) carrying out approximate processing on the point spread function PSF in the integral form under the wide spectrum condition by using a differential summation method to obtain the point spread function PSF in the sectional summation form.
Due to PTFphThe ratio of real part of side peak imaginary part of the optical transfer function OTF is obtained by taking inverse tangent, the tangent value has periodicity, and the method is based on PTFphOnly absolute distance information of less than one wavelength can be obtained, so the measurement of absolute distance is achieved by MTF. Under the broad spectrum condition, the system normalized point spread function PSF is:
where Δ λ is the spectral width, λ0Is the broad spectral center wavelength.
The integral is approximated by differential summation, and Δ λ is equally divided into n segments, each segment having a central wavelength λ1，λ2…λi…λnThe point spread function PSF was obtained as:
wherein the ratio of (x, y,λi) And the image plane coordinates correspond to different wavelength conditions.
And sixthly, performing Fourier transform on the point spread function PSF in a segmented summation mode under the broad spectrum condition to obtain an optical transfer function OTF under the broad spectrum condition.
Performing fourier transform on the formula (7) to obtain the optical transfer function OTF of the system:
wherein the content of the first and second substances,
is the spatial frequency domain coordinate, f, of the point spread function PSF under different wavelength correspondencesxi＝x0/λif,fyi＝y0/λif，
Is the diffraction limited system optical transfer function OTF of a single circular aperture exit pupil:
wherein the content of the first and second substances,
step seven, selecting a side peak of the optical transfer function OTF to obtain a corresponding side peak modulation transfer function MTFphFor modulation transfer function MTFphAnd carrying out normalization processing, wherein the normalization factor is the peak value of the peak on the MTF side of the modulation transfer function under the condition that only the inclination angle error exists, namely the absolute distance is zero. Normalized MTFphMTF by piecewise quartic fittingphAs a function of the absolute distance p, from the resulting functionThe absolute distance p is obtained, and high-precision absolute distance measurement without being affected by the inclination angle error is realized. Introducing an inclination angle of 3.7mas to the sub-mirror 3 to be measured, introducing an absolute distance p with the step size of 0.06328 mu m and the range of 0-195 mu m, and analyzing the relation between the setting error and the measurement error.
Obtaining a corresponding modulation transfer function MTF by performing a modulus operation on one side peak of the optical transfer function OTFph，
Wherein ph represents a side peak.
From equation (11), it is known that the tilt angle has an effect on the modulation transfer function, and thus on the MTFphThe normalization processing can eliminate the influence of the inclination angle on the detection of the absolute distance. The normalization factor is the peak value of the side peak of the modulation transfer function MTF under the condition that only the inclination angle error exists, namely the absolute distance is zero
Obtaining modulation transfer function MTF by four times of fitting in segmentation after normalization processingnphThe functional relationship with the absolute distance p is:
wherein the MTFnphIs the normalized modulation transfer function.
The absolute distance, for MTF, can be directly calculated from equation (12)phAfter normalization processing, the calculation of the absolute distance is not affected by the inclination angle, so that high-precision measurement of the absolute distance is realized.
An inclination angle of 3.7mas is introduced to the sub-mirror 3 to be tested, and then an absolute distance p with the step of 0.06328 mu m and the range of 0-195 mu m is introduced. Obtaining the normalized MTF of the system according to the normalization factor required by the methodnphThe MTF at this timenphSubstituting into equation (12) results in a calculated value for p. As shown in FIG. 7, the abscissa represents the actually measured absolute distance p, and the ordinate representsThe absolute distances introduced, from which it can be seen that the measured values are very close to the set values. And comparing the set value and the calculated value of p to obtain the measurement error of the method. When the inclination angle is 3.7mas and p ranges from 0 to 195 μm, the error curve of the set value and the calculated value of the inclination angle is distributed between +/-0.05 μm as shown in FIG. 8, and the RMS of the error of the set value and the calculated value of p is 0.0102 μm. As can be seen from fig. 8, when the value of p is less than 0.5 wavelength, the error is large, and the maximum error can reach 178 nm. To verify the measurement accuracy, the reduction introduces an error step size to measure the p-value over a range of 0.5 wavelengths. Introducing small p value error with the step of 0.006328 μm and the range of 0-3.164 μm, wherein the error curve of the set value and the calculated value of the inclination angle is shown in FIG. 9, the p value measurement error is extremely small in the range, and the RMS of the error of the set value and the calculated value of the obtained p is 9.6 multiplied by 10-9And mu m. The method is proved to ensure the measurement precision under the condition of small error, and the whole measurement precision can reach the nanometer level.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (9)
1. Method for simultaneously measuring an inclination angle and an absolute distance based on an optical transfer function, characterized in that: comprises the following steps of (a) carrying out,
step one, light waves emitted by a parallel light source are divided into two beams, the two beams enter a reference light path and a measurement light path respectively to obtain two beams of light waves with phase difference information, the two beams of light waves are focused on a focal plane to generate an interference-diffraction or diffraction phenomenon, and the light intensity distribution on the focal plane is a point spread function PSF; the phase difference information comprises an inclination angle and an absolute distance;
step two, carrying out Fourier transform on the point spread function PSF to obtain an optical transfer function OTF, wherein the optical transfer function OTF consists of a main peak and two side peaks, and the side peaks contain information of an inclination angle and an absolute distance, so that the inclination angle and the absolute distance can be conveniently calculated by utilizing the side peaks in the subsequent steps;
selecting a side peak of the optical transfer function OTF to calculate the ratio of the real part of the imaginary part of the side peak to obtain the corresponding side peak phase transfer function PTFphCalculating PTFphAt fx、fyGradient values in two directions are used for calculating the x-direction pupil plane of the measured sub-lens0、y0The tilt angles tip and tilt in the direction realize large-range and high-precision measurement of the tilt angles;
performing approximate processing on the point spread function PSF in the integral form under the wide spectrum condition by using a differential summation method to obtain the point spread function PSF in the sectional summation form;
fifthly, carrying out Fourier transform on the point spread function PSF in a segmented summation mode under the broad spectrum condition to obtain an optical transfer function OTF under the broad spectrum condition;
step six, selecting a side peak of the optical transfer function OTF to obtain a corresponding side peak modulation transfer function MTFphFor modulation transfer function MTFphCarrying out normalization processing, wherein a normalization factor is a peak value of a modulation transfer function MTF side peak under the condition that only an inclination angle error exists, namely the absolute distance is zero; normalized MTFphMTF by piecewise quartic fittingphAnd (4) solving the absolute distance p value according to the obtained functional relation with the absolute distance p, so as to realize high-precision absolute distance measurement without being influenced by the inclination angle error.
2. The method for simultaneously measuring a tilt angle and an absolute distance based on an optical transfer function according to claim 1, wherein: the first implementation method comprises the following steps:
known from fourier optics, under monochromatic light conditions, the normalized point spread function PSF is:
wherein (x, y) is image plane coordinate, x is image plane abscissa, y is image plane ordinate, D is circular aperture diaphragm diameter, J1() Is a first-order Bessel function, lambda is the wavelength of monochromatic light, f is the focal length of the system, B is the central distance of two circular hole diaphragms, and p is the axial distance between the measured sub-mirror and the reference sub-mirror; definition (x)0,y0) Is the coordinate of the pupil plane, and the coordinates of the pupil plane,
is the normal of the measured sub-mirror and y0The included angle between the axes is set by the angle,
is the normal line and x of the measured sub-mirror0The angle between the axes.
3. The method for simultaneously measuring a tilt angle and an absolute distance based on an optical transfer function according to claim 2, wherein: the second step is realized by the method that,
performing Fourier transform on the formula (1) to obtain an optical transfer function OTF of the system:
wherein j is an imaginary unit, which is a convolution operation, (f)x,fy) Is the spatial frequency domain coordinate, f, of the point spread function PSFx＝x0/λf,fy＝y0/λf，OTFsub(fx,fy) Is the diffraction limited system optical transfer function OTF of a single circular aperture exit pupil:
where ρ is the radial coordinate in any direction on the frequency plane,
arccos () as inverseA cosine function;
wherein, circ () is a circle function;
as shown in the formula (2), the optical transfer function OTF is composed of three parts, OTFsub(fx,fy)+OTFsub(fx,fy)·
Is the main peak portion of the optical transfer function OTF,
two side peak parts of the optical transfer function OTF are symmetrically distributed on two sides of the main peak; the optical transfer function OTF side peak portion contains the tilt angle and absolute distance information, so the tilt angle and absolute distance are calculated using the side peak portion of the optical transfer function OTF.
4. The method for simultaneously measuring a tilt angle and an absolute distance based on an optical transfer function according to claim 3, wherein: the third step is to realize the method as follows,
selecting a side peak of the optical transfer function OTF to obtain a ratio of the real part of the imaginary part of the OTF, and performing arc tangent calculation to obtain a corresponding PTFph：
Wherein ph represents a side peak;
PTFphat fx、fyThe distribution in both directions varying with the change in tilt angle(ii) a In PTFphCalculating PTF over one period of variation (. + -. pi)phAt fx、fyGradient values in two directions
The measured sub-mirrors are respectively at fx、fyTwo directions, i.e. pupil plane x0、y0The angle of inclination tip in the direction is
tilt is
Therefore, the wide-range and high-precision measurement of the inclination angle is realized.
5. The method of claim 4 for simultaneously measuring tilt angle and absolute distance based on optical transfer function, wherein: the implementation method of the fourth step is that,
due to PTFphThe ratio of real part of side peak imaginary part of the optical transfer function OTF is obtained by taking inverse tangent, the tangent value has periodicity, and the method is based on PTFphOnly absolute distance information of less than one wavelength can be obtained, so the measurement of the absolute distance is realized through MTF; under the broad spectrum condition, the system normalized point spread function PSF is:
where Δ λ is the spectral width, λ0Is the broad spectral center wavelength;
the integral is approximated by differential summation, and Δ λ is equally divided into n segments, each segment having a central wavelength λ1，λ2…λi…λnThe point spread function PSF was obtained as:
wherein, (x, y, λ)i) And the image plane coordinates correspond to different wavelength conditions.
6. The method for simultaneously measuring a tilt angle and an absolute distance based on an optical transfer function according to claim 5, wherein: the fifth step is to realize that the method is that,
performing fourier transform on the formula (7) to obtain the optical transfer function OTF of the system:
wherein the content of the first and second substances,
is the spatial frequency domain coordinate of the point spread function PSF corresponding to different wavelengths,
is the diffraction limited system optical transfer function OTF of a single circular aperture exit pupil:
wherein the content of the first and second substances,
7. the method for simultaneously measuring a tilt angle and an absolute distance based on an optical transfer function according to claim 6, wherein: the sixth realization method comprises the following steps of,
obtaining a corresponding modulation transfer function MTF by performing a modulus operation on one side peak of the optical transfer function OTFph，
Wherein ph represents a side peak;
from equation (11), it is known that the tilt angle has an effect on the modulation transfer function, and thus on the MTFphThe normalization processing is carried out, so that the influence of the inclination angle on the detection of the absolute distance can be eliminated; the normalization factor is the peak value of the side peak of the modulation transfer function MTF under the condition that only the inclination angle error exists, namely the absolute distance is zero
Obtaining modulation transfer function MTF by four times of fitting in segmentation after normalization processingnphThe functional relationship with the absolute distance p is:
wherein the MTFnphIs the normalized modulation transfer function;
the absolute distance, for MTF, can be directly calculated from equation (12)phAfter normalization processing, the calculation of the absolute distance is not affected by the inclination angle, so that high-precision measurement of the absolute distance is realized.
8. Apparatus for simultaneously measuring a tilt angle and an absolute distance based on an optical transfer function, for implementing a method for simultaneously measuring a tilt angle and an absolute distance based on an optical transfer function as claimed in claim 1, 2, 3, 5, 6 or 7, characterized in that: the device comprises a parallel light source, a reference sub-mirror, a measured sub-mirror, a spectroscope, a diaphragm, a single-wavelength filter, a focusing lens and a detector; wherein the diaphragm is arranged on the conjugate surface of the reference sub-mirror, and two discrete round holes are arranged on the diaphragm and respectively correspond to the two sub-mirrors; the part of the light beam emitted by the parallel light source irradiates the reference sub-mirror and is called as reference light; part of the light irradiates to the measured sub-mirror and is called as detection light; the reference light and the detection light pass through the collimating lens after being reflected by the telescope secondary mirror, then pass through two discrete round holes on the diaphragm respectively, and undergo interference diffraction through the focusing lens, so that interference diffraction fringes under the wide spectrum condition are formed and are imaged on the detector through the focusing lens.
9. The apparatus for simultaneously measuring a tilt angle and an absolute distance based on an optical transfer function of claim 8, wherein:
adding a single-wavelength filter plate between the diaphragm hole and the focusing lens, imaging a system interference diffraction fringe on the detector under the condition of monochromatic light, and acquiring light intensity distribution by the detector, namely a point spread function PSF under the condition of the monochromatic light of the system; carrying out Fourier transform on the obtained point spread function PSF to obtain an optical transfer function OTF; selecting a side peak of the optical transfer function OTF to calculate the ratio of the real part of the imaginary part of the side peak to obtain the corresponding side peak phase transfer function PTFphCalculating the phase transfer function PTFphThe inclination angle of the measured sub-mirror is calculated by utilizing the relation between the gradient value and the inclination angle, so that the large-range and high-precision measurement of the inclination angle is realized; a single-wavelength filter is not added between the diaphragm hole and the focusing lens, the image on the detector is a system interference diffraction fringe under the wide spectrum condition, and the light intensity distribution acquired by the detector is a point spread function PSF under the system wide spectrum condition; carrying out approximate processing of a differential summation method on the obtained point spread function PSF, then carrying out Fourier transformation to obtain an optical transfer function OTF, selecting one side peak of the optical transfer function OTF to obtain a corresponding side peak modulation transfer function MTFphFor modulation transfer function MTFphNormalized to obtain MTFnphBy segmentationObtaining MTF by four times of fittingnphAnd (4) solving an absolute distance value according to the obtained functional relation with the absolute distance, so as to realize high-precision measurement of the absolute distance without being influenced by the error of the inclination angle.
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