CN101839686A - Nonlinear error correction method of laser interferometer, device and interferometer applying method and device - Google Patents

Abstract

The invention discloses a nonlinear error correction method of a single frequency laser interferometer, a device and an interferometer applying the method and the device, and the method carries out nonlinear error correction on the interferometer by utilizing the harmonic separation correction method and comprises the following steps: step 1: establishing a correction equation for obtaining a phase angle of interference signals from two lines of the interference signals of the interferometer according to the correction equation; step 2: correcting fundamental wave components in the interference signals, and obtaining the initial phase angle; step 3: correcting harmonic components in the interference signals, and obtaining the corrected value of the phase angle; and step 4: obtaining the precise phase angle after compensation modification according to the initial phase angle and the corrected value of the phase angle, accordingly obtaining the displacement of a measuring mirror of the interferometer, and further realizing the nonlinear error correction of the interferometer. The method can eliminate various harmonic components causing nonlinear errors in the interference signals, and optimize the nonlinear error correction of the single frequency laser interferometer.

Description

Laser interferometer nonlinearity erron modification method, device and use its interferometer

Technical field

The present invention relates to the laser measuring technique field, particularly a kind of single frequency laser interferometer nonlinearity erron modification method, device and use the laser interferometer of this device.

Background technology

Be accompanied by the development of nanometer technology, microelectric technique and MEMS, more and more higher to the accuracy requirement of size and displacement measurement.For example the Teague of USA National Institute of Standard and Technology (NIST) thinks, in integrated circuit industry, when live width will reach 50nm in 2014 when following, national metering institute should be able to guarantee to reach the measuring accuracy of 0.4nm.Laser interferometer uses the wavelength of light wave as basic scale, and its measurement result can directly be traceable to the Mi Dingyi wavelength standard, is the most widely used reference measurement instrument in the length metering.The source of error of interferometer is mainly the precision of optical maser wavelength, measurement noise and nonlinearity erron.When laser interferometer during as the measuring basis of nanometer gauging instrument, in order to guarantee the wire width measuring precision of 0.4nm, its uncertainty of measurement should reach 0.1nm.This moment, nonlinearity erron just became the topmost source of error of interferometer.

The nonlinearity erron of single frequency laser interferometer is to interfere the light and shade fringe period (to be generally λ/2 optical path differences, λ: optical wavelength) be the periodic error in cycle, mainly produced by the phase place aliasing.Producing phase place aliasing main cause is: all imperfect elements of optical element such as the wave plate in (1) interferometer, spectroscope, and can not be as polarization spectroscope with the separation of two bundle polarized lights 100%, the reflection loss that each is surperficial, the phase delay error of wave plate etc.; (2) adjustment of interferometer is not ideal enough, and the light beam of reference light and measuring light can not be coaxial fully; (3) photoelectric commutator is non-linear.Usually under good situation about adjusting, the nonlinearity erron of interferometer can reach 5-10nm.Therefore, for the interferometer that requires the 0.1nm uncertainty of measurement, the calibration of nonlinearity erron is necessary.

Desirable interference signal is the harmonic signal of a pair of constant amplitude, quadrature, and its lissajous figures is desirable circle.This can be expressed as desirable interference signal:

$\left\{\begin{array}{c}{u}_1=R\sin \mathrm{\θ}\\ u_2=R\cos \mathrm{\θ}\end{array}\right.---\left(1\right)$

In the formula, u ₁, u ₂Be interference signal, R is a signal amplitude, and θ is the phasing degree of interference signal.The phasing degree of interference signal is one to one with measuring the mirror displacement:

X measures the mirror displacement.The phasing degree can be tried to achieve by following formula:

$\mathrm{\θ}=\arctan \frac{u_1}{u_2}---\left(2\right)$

But actual interference signal is incomplete.Traditional Heydemann method is a kind of method that American scholar Heydemann proposes, and is the comparatively ripe classic method of a kind of application, is that the two-way interference signal is represented with a broad sense elliptic equation, thus also cry oval the correction, as follows:

${(u_1-p)}^2+{\left[\frac{(u_2-q)G+(u_1-p)\sin \mathrm{\α}}{\cos \mathrm{\α}}\right]}^2=R^2---\left(3\right)$

In the formula: α is nonopiate error; G is the ratio of two paths of signals amplitude; P and q are respectively the DC level of two paths of signals.

Utilize parameter substitution, formula (3) can be expressed as a binary quadratic equation, utilize the method for least square that its coefficient is returned, just can obtain each nonlinear factor G, α, p, the q of two-way interference signal, revised two-way zero direct current constant amplitude orthogonal signal can be expressed as:

$\left\{\begin{array}{c}{u}_{1c}=u_1-p\\ u_{2c}=\frac{(u_1-p)\sin \mathrm{\α}+G(u_2-q)}{\cos \mathrm{\α}}\end{array}\right.---\left(4\right)$

In the formula, u ₁, u ₂Be original interference signal,, obtain revised interference signal u G, α, p, q value substitution formula (4) _1c, u _2c, obtain the phasing degree θ of interference signal again by formula (2), finally obtain and measure the mirror displacement x, realize correction to interferometer non-linearity.Yet Heydemann etc. have only considered the correction to the interference signal first-harmonic, in fact, and in interference signal, owing to the undesirable higher hamonic wave composition that produces of optical element; In many times of journey interferometers of optics, owing to the improper reflection of element surface produces the low-order harmonic composition.Therefore, perfect nonlinearity erron correction should be carried out each harmonic components of signal.

Summary of the invention

Technical matters to be solved by this invention is, a kind of single frequency laser interferometer signal processing method, device are provided and use its laser interferometer, eliminate the various harmonic componentss that cause nonlinearity erron in the interference signal, make the nonlinearity erron correction of single frequency laser interferometer reach optimization.

For achieving the above object, laser interferometer signal disposal route provided by the invention is to utilize the harmonic separation revised law that described interferometer non-linearity error is revised, and it is characterized in that described method comprises:

Step 1 is revised the first-harmonic composition in the interference signal, obtains starting phase angle;

Step 2 is revised the harmonic components in the described interference signal, obtains the modified value at phasing degree;

Step 3 is obtained the amended precise phase of compensation angle according to the modified value at described starting phase angle and described phasing degree, and obtains the displacement of described interferometer measurement mirror in view of the above to realize the nonlinearity erron correction to interferometer.

Above-mentioned laser interferometer nonlinearity erron modification method is characterized in that, also comprises a update equation establishment step before described step 1, is used for according to this update equation the phasing degree that obtains interference signal from the two-way interference signal of described interferometer.

Further, the present invention also provides a kind of laser interferometer nonlinearity erron correcting device of using above-mentioned laser interferometer nonlinearity erron modification method, be arranged at the signal processing system of described interferometer, it is characterized in that, comprise: the first-harmonic correcting module, be used for the first-harmonic composition of interference signal is revised, obtain starting phase angle; The harmonic wave correcting module is used for the harmonic components of described interference signal is revised, and obtains the modified value at phasing degree; The error compensation module, the modified value that is used for the phasing degree that the starting phase angle that obtains according to described first-harmonic correcting module and described harmonic wave correcting module obtain is obtained the amended precise phase of compensation angle, and obtains described interferometer measurement mirror displacement in view of the above and realize nonlinearity erron correction to interferometer.

Above-mentioned error correction device is characterized in that, comprises that also a correction model sets up module, is used to set up update equation, and handles the phasing degree that obtains interference signal automatically according to this update equation, and output is from described first-harmonic correcting module and described harmonic wave correcting module.

Above-mentioned error correction device, it is characterized in that, described compensating module also further comprises an error judgment module, be used for the precise phase angle that obtains according to calculating, judge whether to satisfy default error requirements,, proceed iterative computation as not satisfying default error requirements, until satisfying default error requirements, obtain final phase value.

Further, the present invention also provides a kind of laser interferometer of using above-mentioned nonlinearity erron correcting device, it is characterized in that, described nonlinearity erron correcting device is arranged at the signal processing system of described interferometer, is used to utilize the harmonic separation revised law that the nonlinearity erron of described interferometer is revised; This error correction device further comprises: correction model is set up module, is used to set up update equation, and with the phasing degree that obtains interference signal according to the automatic processing of this update equation, and output is from described first-harmonic correcting module and described harmonic wave correcting module; The first-harmonic correcting module is used for the first-harmonic composition of interference signal is revised, and obtains starting phase angle; The harmonic wave correcting module is used for the harmonic components of described interference signal is revised, and obtains the modified value at phasing degree; The error compensation module, the modified value that is used for the phasing degree that the starting phase angle that obtains according to described first-harmonic correcting module and described harmonic wave correcting module obtain is obtained the amended precise phase of compensation angle, and obtains described interferometer measurement mirror displacement in view of the above and realize nonlinearity erron correction to interferometer.

Compared with prior art, the harmonic separation interference signal revised law that laser interferometer nonlinearity erron modification method of the present invention adopts carries out least square fitting by utilizing Fourier series to calibrating signal, the coefficient of the feasible Fourier series of being asked obtains optimum on statistical significance, and precise phase is calculated and to be undertaken by Taylor series expansion among the present invention, and this method that can iteration can make the nonlinearity erron correction-compensation reach optimization.This is that Heydemann correction institute is irrealizable, so this method is specially adapted to handle the laser interference system that has used optics times journey technology.

The present invention has further carried out simplation verification to the method that is proposed, and the result shows that in the noise signal amplitude be 5% o'clock of fundamental signal amplitude, and the amplitude of residual error is about ± 1nm, and when noise was 0.5%, residual error was about ± 0.1nm.

Description of drawings

Fig. 1 sets up required basic device of update equation and mutual relationship synoptic diagram among the present invention;

Fig. 2 is a laser interferometer nonlinearity erron modification method schematic flow sheet of the present invention;

Fig. 3 is a laser interferometer structural representation of the present invention;

Fig. 4 is the Computer Simulation checking schematic flow sheet of modification method of the present invention;

Fig. 5 is the lissajous figures synoptic diagram of emulated data in the simulating, verifying among the present invention;

Fig. 6～9 are simulation results synoptic diagram among the present invention.

Embodiment

Below in conjunction with drawings and Examples the present invention is done detailed description, with further understanding the present invention's purpose, scheme and effect, but not as the restriction to claim protection domain of the present invention.

The same with the Heydemann correction, the harmonic separation revised law that the present invention proposes at first will be set up update equation, utilizes the calibration measurement data to find the solution each parameter to be asked of update equation then.In interferometer 101 actual measurements, according to update equation, from two-way interference signal u ₁And u ₂In obtain interference signal phasing degree θ value accurately, thereby realize the interferometer non-linearity error correction.With reference to figure 1, among the figure, 101 is interferometer, and 102 for measuring mirror, and 103 is the nanometer displacement platform.The key that obtains the calibration measurement data is to produce basis displacement.The displacement of nano measurement system is produced by piezoelectric ceramics usually, and its resolving power can reach several nanometers, and present most of piezoelectric ceramics nanometer displacement platform also possesses capacitive transducer, eliminates the hysteresis and the nonlinear displacement of piezoelectric ceramics as FEEDBACK CONTROL.But no matter whether possesses capacitive transducer, the displacement curve of nanometer displacement system is all very smooth, therefore all can be with the λ of laser interferometer 101/2 signal complete cycle in having the nanometer displacement system of laser interferometer, nanometer displacement platform 103 is calibrated, made its magnitude tracing simultaneously to optical maser wavelength.There is not nonlinearity erron in λ/2 signal complete cycle.Through the nanometer displacement platform after the calibration produce a series of known, less than the micro-displacement of λ/2, it is so-called basis displacement, make laser interferometer pass through to measure and obtain the calibration measurement data, utilize these data to find the solution the update equation parameter, finally realize of the nonlinearity erron correction of interferometer λ/2 cycles with interior (i.e. so-called " fraction part ") measured value.This process as shown in Figure 1.

With reference to figure 2, be the schematic flow sheet of error correcting method of the present invention, single frequency laser interferometer nonlinearity erron modification method of the present invention may further comprise the steps:

Step S201 sets up update equation, with according to this update equation, and the phasing degree that from the two-way interference signal of interferometer, obtains interference signal;

Step S202 revises the first-harmonic composition in the interference signal, obtains starting phase angle;

Step S203 revises the harmonic components in the described interference signal, obtains the modified value at phasing degree;

Step S204 obtains the amended precise phase of compensation angle according to the modified value at described starting phase angle and described phasing degree, and obtains the displacement of described interferometer measurement mirror in view of the above to realize the nonlinearity erron correction to interferometer.

Describe the embodiment of interferometer non-linearity error correcting method of the present invention below in detail:

At first set up update equation.Because on sense of displacement, the two-way interference signal can be considered the function that the cycle of grade changes, so available finite term Fourier series is expressed as:

$\left\{\begin{array}{c}{u}_1=f_1\left(\mathrm{\θ}\right)=a_{10}+\underset{m=0}{\overset{M/4}{\mathrm{\Σ}}}[a_{1(m+1)}\sin \left(\frac{\mathrm{\θ}}{2^m}\right)+b_{1(m+1)}\cos \left(\frac{\mathrm{\θ}}{2^m}\right)]+\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\left\{c_{1n}\sin \right[(n+1)\mathrm{\θ}]+d_{1n}\cos [(n+1)\mathrm{\θ}\left]\right\}\\ u_2=f_2\left(\mathrm{\θ}\right)=a_{20}\underset{m=0}{\overset{M/4}{\mathrm{\Σ}}}[a_{2(m+1)}\sin \left(\frac{\mathrm{\θ}}{2^m}\right)+b_{2(m+1)}\cos \left(\frac{\mathrm{\θ}}{2^m}\right)]+\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\left\{c_{2n}\sin \right[(n+1)\mathrm{\θ}]+d_{2n}\cos [(n+1)\mathrm{\θ}\left]\right\}\end{array}\right.---\left(5\right)$

Wherein: a ₁₀, a ₂₀Be DC component; M is an optics times number of passes; Second for first-harmonic and because the harmonic component that is lower than first-harmonic of optics times Cheng Yinqi, a _{1 (m+1)}, b _{1 (m+1)}, a _{2 (m+1)}, b _{2 (m+1)}Be respectively the coefficient of each harmonic component; The 3rd for being higher than the harmonic component of first-harmonic, c _1n, d _1n, c _2n, d _2nBe respectively the coefficient of each harmonic component, N is the higher hamonic wave intercepted length.

Every coefficient of Fourier series can carry out optimal estimation with least square, even the quadratic sum of evaluated error is minimum, if image data is counted and is L, has:

$\left\{\begin{array}{c}\underset{i=1}{\overset{L}{\mathrm{\Σ}}}{\{a_{10}+\underset{m=0}{\overset{M/4}{\mathrm{\Σ}}}[a_{1(m+1)}\sin \left(\frac{{\mathrm{\θ}}_i}{2^m}\right)+b_{1(m+1)}\cos \left(\frac{{\mathrm{\θ}}_i}{2^m}\right)]+\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\{c_{1n}\sin \left[(n+1){\mathrm{\θ}}_i\right]+d_{1n}\cos \left[(n+1){\mathrm{\θ}}_i\right]\}-u_{1i}\}}^2=\min \\ \underset{i=1}{\overset{L}{\mathrm{\Σ}}}{\left\{a_{20}\text{+}\underset{m=0}{\overset{M/4}{\mathrm{\Σ}}}\right[a_{2(m+1)}\sin \left(\frac{{\mathrm{\θ}}_i}{2^m}\right)+b_{2(m+1)}\cos \left(\frac{{\mathrm{\θ}}_i}{2^m}\right)]+\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\{c_{2n}\sin \left[(n+1){\mathrm{\θ}}_i\right]+d_{2n}\cos \left[(n+1){\mathrm{\θ}}_i\right]\}-u_{2i}\}}^2=\min \end{array}\right.---\left(6\right)$

θ _iBe the phasing degree of interference signal on the sampled point,

x _iBe basis displacement, produce by the nanometer displacement platform; u _1i, u _2iBe respectively the interference signal on the sampled point.Find the solution update equation wait to ask coefficient the time, θ _i, u _1i, u _2iConstitute the calibration measurement data that preamble is mentioned.

The same with common least-squares estimation, formula (6) is to respectively waiting to ask parameter a ₁₀, a _{1 (m+1)}, b _{1 (m+1)}, c _1n, d _1n, a ₂₀, a _{2 (m+1)}, b _{2 (m+1)}, c _2n, d _2nDifferentiate can obtain two systems of linear equations, finds the solution these two systems of linear equations and can obtain each parameter to be asked.But, when the data length that is used for calculation of parameter is set at the integral multiple in minimum harmonic cycle, utilize orthogonality of trigonometric function, wait to ask the parametric solution process to simplify greatly.At this moment, each parameter to be asked can be calculated with one group of following formula:

$a_0=\frac{1}{L}\underset{i=1}{\overset{L}{\mathrm{\Σ}}}{u}_i---\left(7\right)$

$a_{m+1}=\frac{2}{L}\underset{i=1}{\overset{L}{\mathrm{\Σ}}}{u}_i\sin \left(\frac{{\mathrm{\θ}}_i}{2^m}\right)---\left(8\right)$

$b_{m+1}=\frac{2}{L}\underset{i=1}{\overset{L}{\mathrm{\Σ}}}{u}_i\cos \left(\frac{{\mathrm{\θ}}_i}{2^m}\right)---\left(9\right)$

$c_n=\frac{2}{L}\underset{i=1}{\overset{L}{\mathrm{\Σ}}}{u}_i\sin \left[(n+1){\mathrm{\θ}}_i\right]---\left(10\right)$

$d_n=\frac{2}{L}\underset{i=1}{\overset{L}{\mathrm{\Σ}}}{u}_i\cos \left[(n+1){\mathrm{\θ}}_i\right]---\left(11\right)$

So far, the institute in the update equation formula (5) remains to be asked coefficient to solve, and update equation is determined.

The purpose of interferometer non-linearity error correction is from two-way interference signal u ₁And u ₂In obtain interference signal phasing degree θ value accurately, but for update equation formula (5), utilize the u that obtains when measuring ₁, u ₂Can not directly solve the θ value.Adopt two step Calculation Method for this reason, at first utilized first-harmonic to calculate approximate initial phase, need utilize the Taylor series single order unfolding calculation precise phase of formula (5) then.

(1) initial phase calculates: because the overwhelming majority in the interference signal is the first-harmonic composition, therefore can be similar to and thinks:

$\left\{\begin{array}{c}{u}_1=a_{10}+a_{11}\sin \left(\mathrm{\θ}\right)+b_{11}\cos \left(\mathrm{\θ}\right)\\ u_2=a_{20}+a_{21}\sin \left(\mathrm{\θ}\right)+b_{21}\cos \left(\mathrm{\θ}\right)\end{array}\right.---\left(12\right)$

Can get:

$\mathrm{\θ}=\arctan \frac{b_{21}(u_1-a_{10})-b_{11}(u_2-a_{20})}{a_{11}(u_2-a_{20})-a_{21}(u_1-a_{10})}---\left(13\right)$

Obtain the initial value of phasing degree θ thus, i.e. starting phase angle θ ₀, its effect is equivalent to Heydemann and revises.

(2) precise phase is calculated: to formula (5) at θ ₀The place carries out the single order Taylor expansion, promptly has:

$\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="HSA00000065463400041.png,HSA00000065463400052.png"\; state="\{\{state\}\}">u</figure-callout>}}_1=f_1\left({\mathrm{\&theta;}}_0\right)+{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HSA00000065463400041.png,HSA00000065463400052.png"\; state="\{\{state\}\}">f</figure-callout>}}_1^{\&prime;}\left({\mathrm{\&theta;}}_0\right)\mathrm{\&Delta;\&theta;}\\ {\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="HSA00000065463400042.png,HSA00000065463400051.png"\; state="\{\{state\}\}">u</figure-callout>}}_2=f_2\left({\mathrm{\&theta;}}_0\right)+{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="HSA00000065463400042.png,HSA00000065463400051.png"\; state="\{\{state\}\}">f</figure-callout>}}_2^{\&prime;}\left({\mathrm{\&theta;}}_0\right)\mathrm{\&Delta;\&theta;}\end{array}\right.---\left(14\right)$

Can get:

$\mathrm{\&Delta;\&theta;}=\frac{u_1-f_1\left({\mathrm{\&theta;}}_0\right)}{{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HSA00000065463400041.png,HSA00000065463400052.png"\; state="\{\{state\}\}">f</figure-callout>}}_1^{\&prime;}\left({\mathrm{\&theta;}}_0\right)}---\left(15\right)$

Or

$\mathrm{\&Delta;\&theta;}=\frac{u_2-f_2\left({\mathrm{\&theta;}}_0\right)}{{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="HSA00000065463400042.png,HSA00000065463400051.png"\; state="\{\{state\}\}">f</figure-callout>}}_2^{\&prime;}\left({\mathrm{\&theta;}}_0\right)}---\left(16\right)$

Δ θ is the phasing degree modified value.Use formula (15) or formula (16) depend on θ ₀Numerical value and f ₁(θ ₀), f ₃(θ ₀) structure.Because the θ that formula (12) obtains ₀Span be [π, π], if think u ₁Basic is sinusoidal, u ₂Basic is cosine, and then in [π/4, π/4], [3 π/4, π] and [π ,-3 π/4] zone, Δ θ obtains with formula (15), and other zones obtain with formula (16).Precise phase angle θ ₀' with θ ₀Following relation is arranged:

θ ₀′＝θ ₀+Δθ????????????????????????(17)

If think that the effect of once calculating is still not ideal enough, can be with θ ₀' as new θ ₀Substitution formula (15) and formula (16) so can constitute an iterative process, till satisfaction.Finally can obtain interference signal phasing degree accurately, realize revising the purpose of interferometer non-linearity error.The fraction part displacement letter of interferometer is:

$x=\frac{{{\mathrm{\&theta;}}_0}^{\&prime;}\mathrm{\&lambda;}}{2\mathrm{\&pi;M}}---\left(18\right)$

With reference to figure 3, the present invention further provides a kind of laser interferometer 3 of using the error correction device 300 of above-mentioned nonlinearity erron modification method and using this device, this correcting device 300 is arranged at the interference signal disposal system 30 of laser interferometer 3, this correcting device 300 further comprises: correction model is set up module 301, first-harmonic correcting module 302, harmonic wave correcting module 303 and error compensation module 304, wherein correction model is set up module 301 and is used to set up update equation, obtain the phasing degree of interference signal to handle automatically, and export described first-harmonic correcting module and described harmonic wave correcting module to according to this update equation; First-harmonic correcting module 302 is used for the first-harmonic composition of interference signal is revised, and obtains starting phase angle; Harmonic wave correcting module 303 is used for the harmonic components of described interference signal is revised, and obtains the modified value at phasing degree; Error compensation module 304, the modified value that is used for the phasing degree that the starting phase angle that obtains according to described first-harmonic correcting module and described harmonic wave correcting module obtain is obtained the amended precise phase of compensation angle, and obtains described interferometer measurement mirror displacement in view of the above and realize nonlinearity erron correction to interferometer.Error compensation module 304 also further comprises an error judgment module 3041, be used for the precise phase angle that obtains according to calculating, judge whether to satisfy default error requirements, as not satisfying default error requirements, proceed iterative computation, until satisfying default error requirements, obtain final phase value.

The harmonic separation revised law that further the present invention is proposed carries out simplation verification below:

In order to verify that can " harmonic separation revised law " theory effectively with error separating for laser interference signal (the especially interference signal of many times of journey laser interference systems of optics), thereby reduce the interference of error, make measurement result approach ideal value more, this error correcting method is carried out simulating, verifying.With reference to figure 4, be the main schematic flow sheet of Computer Simulation checking: with reference to figure 4, it is as follows to further describe the simulating, verifying flow process:

Step S401 sets up the emulated data growth equation;

Step S402 produces the interference signal emulated data that is used for update equation foundation according to above-mentioned emulated data growth equation;

Step S403 sets up update equation, is to obtain update equation with the coefficient that least square method is found the solution in the Fourier series model;

Step S404, utilization emulated data growth equation, the emulated data of generation actual measurement interference signal;

Step S405 calculates starting phase angle;

Step S406 carries out precise phase and calculates;

Step S407, the relatively difference between calculated value and the ideal value; Judge whether to satisfy default error, satisfy obtaining final phase value, calculate the precise phase angle otherwise continue loop iteration.

Laser interferometer with 8 times of journeys of optics is that example is carried out l-G simulation test below:

Laser interferometer to 8 times of journeys of optics is carried out emulation experiment, and the emulated data growth equation of being set up is:

$\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="HSA00000065463400041.png,HSA00000065463400052.png"\; state="\{\{state\}\}">u</figure-callout>}}_1=a_{10}+a_{11}\sin \left(\mathrm{\&theta;}\right)+b_{11}\cos \left(\mathrm{\&theta;}\right)+a_{12}\sin \left(\frac{\mathrm{\&theta;}}{2}\right)+{\mathrm{<figure-callout\; id="12"\; label="b"\; filenames="HSA00000065463400042.png,HSA00000065463400051.png"\; state="\{\{state\}\}">b</figure-callout>}}_{12}\cos \left(\frac{\mathrm{\&theta;}}{2}\right)\\ +a_{13}\sin \left(\frac{\mathrm{\&theta;}}{4}\right)+b_{13}\cos \left(\frac{\mathrm{\&theta;}}{4}\right)+c_{11}\sin \left(2\mathrm{\&theta;}\right)+d_{11}\cos \left(2\mathrm{\&theta;}\right)+R_1\left(\mathrm{0.05,0.05}\right)\\ {\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="HSA00000065463400042.png,HSA00000065463400051.png"\; state="\{\{state\}\}">u</figure-callout>}}_2=a_{20}+a_{21}\sin \left(\mathrm{\&theta;}\right)+b_{21}\cos \left(\mathrm{\&theta;}\right)+a_{22}\sin \left(\frac{\mathrm{\&theta;}}{2}\right)+b_{22}\cos \left(\frac{\mathrm{\&theta;}}{2}\right)\\ +a_{23}\sin \left(\frac{\mathrm{\&theta;}}{4}\right)+b_{23}\cos \left(\frac{\mathrm{\&theta;}}{4}\right)+c_{21}c\sin \left(2\mathrm{\&theta;}\right)+d_{21}\cos \left(2\mathrm{\&theta;}\right)+R_2\left(\mathrm{0.05,0.05}\right)\end{array}\right.---\left(19\right)$

Wherein, R is an equally distributed white noise in [+0.05 ,-0.05] zone, is equivalent to 5% of fundamental signal amplitude, and unit is V.In order further to strengthen the authenticity of simulation, phasing degree θ has also been added equally distributed white noise in [+0.01 ,-0.01] zone, unit is rad.Other parameters of formula (19) are as shown in table 1:

Table 1 emulated data growth equation parameter list

Parameter	??a 10	??a 11	??b 11	??a 12	??b 12	??a 13	??b 13	??c 11	??d 11
										Numerical value	??0.05	??1.0	??0.0	??0.10	??0.06	??0.06	??-0.08	??0.08	??-0.05
Parameter	??a 20	??a 21	??b 21	??a 22	??b 22	??a 23	??b 23	??c 21	??d 21
										Numerical value	??-0.04	??-0.2	??0.8	??-0.06	??0.10	??0.06	??0.06	??0.02	??0.08

The lissajous figures synoptic diagram of the interference signal emulated data that is produced is seen Fig. 5, is a quite bad signal as can be seen for laser interferometer.

Set up update equation with reference to formula (5) and formula (19) then, higher harmonic components is got 1 time.It (is θ that the two-way interference signal of emulated data is carried out equal interval sampling respectively _iUniformly-spaced), gained calibration measurement data solve undetermined coefficient according to formula (7)-(11), and determined update equation parameter sees Table 2.

Table 2 update equation parameter list

Utilize the emulated data growth equation to produce the correctness that data are used to verify the method that proposes once more.The θ of emulated data growth equation (ideal value at phasing degree) is changed by 0 to 4 π, produces two interference signal u ₁, u ₂According to u ₁, u ₂And the update equation parameter of determining in the table 2, to (17), obtain phasing degree calculating value θ according to formula (12) ₀'.The checking result sees Fig. 6, Fig. 7 and Fig. 8 respectively.In Fig. 6～8, horizontal ordinate is the ideal value at phasing degree, and unit is rad; Ordinate is the difference of phasing degree calculating value and ideal value, has been scaled long measure.Wherein Fig. 6 is not for doing any correction, calculated value that directly calculates and the relation between the ideal value.Fig. 7 is the relation of having carried out between initial phase calculating back calculated value and the ideal value.Fig. 8 has carried out three iteration precise phase to calculate relation between back calculated value and the ideal value.Can learn that by statistical computation the standard deviation of the residual error of three kinds of methods is respectively: 2.24nm, 1.76nm and 0.50nm.Therefore by this method, the nonlinearity erron of original pact ± 5nm can be reduced to pact ± 1nm.

It can also be seen that from Fig. 8 the error of this moment in fact mainly produces owing to existing equally distributed white noise in the data.Fig. 9 is a residual error synoptic diagram after 0.5% precise phase of fundamental signal amplitude is calculated for white noise, as shown in Figure 9, when the white noise level is reduced to 0.5% of fundamental signal amplitude, the standard deviation of residual error is 0.049nm, and nonlinearity erron reduces to pact ± 0.1nm.

Harmonic separation interference signal revised law provided by the invention carries out least square fitting by utilizing Fourier series to calibrating signal, and the coefficient of the feasible Fourier series of being asked obtains optimum on statistical significance.Because it is constant that the non-linear component of laser interferometer is just determined after optical interference circuit is fixing substantially, therefore can be by signal be carried out the Fourier spectrum analysis, find the big harmonic components of influence, establish the frequency content that to separate in the model, accurately set up the mathematical model that is fit to real system, thereby can eliminate the various harmonic componentss that cause nonlinearity erron in the interference signal.This method can make various different frequency component separating, reach the purpose of interference signal processing and nonlinearity erron correction, this is that Heydemann correction institute is irrealizable, so this method is specially adapted to handle the laser interference system that has used optics times journey technology.When the mathematical method of utilization least square is found the solution regression coefficient, utilize the trigonometric function orthogonality can simplify the mathematical operation process of finding the solution regression coefficient greatly.Precise phase is calculated and to be undertaken by Taylor series expansion, and this method that can iteration can make the nonlinearity erron correction-compensation reach optimization.F (θ) in formula (15) and (16) and f ' (θ) tables of data can be made, computing velocity can be accelerated greatly like this.

Though the present invention discloses as above with preferred embodiment; right its is not in order to limit the present invention; under the situation that does not deviate from spirit of the present invention and essence thereof; those of ordinary skill in the art work as can make various corresponding changes and distortion according to the present invention, but these corresponding changes and distortion all should belong to the protection domain of the appended claim of the present invention.

Claims (11)

1. a single frequency laser interferometer nonlinearity erron modification method is to utilize the harmonic separation revised law that described interferometer non-linearity error is revised, and it is characterized in that described method comprises:

Step 1 is revised the first-harmonic composition in the interference signal, obtains starting phase angle;

Step 2 is revised the harmonic components in the described interference signal, obtains the modified value at phasing degree;

Step 3 is obtained the amended precise phase of compensation angle according to the modified value at described starting phase angle and described phasing degree, and obtains the displacement of described interferometer measurement mirror in view of the above to realize the nonlinearity erron correction to interferometer.

2. laser interferometer nonlinearity erron modification method according to claim 1, it is characterized in that, before described step 1, also comprise a update equation establishment step, be used for, the phasing degree that from the two-way interference signal of described interferometer, obtains interference signal according to this update equation.

3. laser interferometer nonlinearity erron modification method according to claim 3 is characterized in that described update equation is expressed as follows with the finite term Fourier series:

$\left\{\begin{array}{c}{u}_1=f_1\left(\mathrm{\&theta;}\right)=a_{10}+\underset{m=0}{\overset{M/4}{\mathrm{\&Sigma;}}}[a_{1(m+1)}\sin \left(\frac{\mathrm{\&theta;}}{2^m}\right)+b_{1(m+1)}\cos \left(\frac{\mathrm{\&theta;}}{2^m}\right)]+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\left\{c_{1n}\sin \right[(n+1)\mathrm{\&theta;}]+d_{1n}\cos [(n+1)\mathrm{\&theta;}\left]\right\}\\ u_2=f_2\left(\mathrm{\&theta;}\right)=a_{20}+\underset{m=0}{\overset{M/4}{\mathrm{\&Sigma;}}}[a_{2(m+1)}\sin \left(\frac{\mathrm{\&theta;}}{2^m}\right)+b_{2(m+1)}\cos \left(\frac{\mathrm{\&theta;}}{2^m}\right)]+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\left\{c_{2n}\sin \right[(n+1)\mathrm{\&theta;}]+d_{2n}\cos [(n+1)\mathrm{\&theta;}\left]\right\}\end{array}\right.$

In the formula: a ₁₀, a ₂₀Be DC component;

M is an optics times number of passes;

Second of following formula is first-harmonic and because the harmonic component that is lower than first-harmonic of optics times Cheng Yinqi, wherein a _{1 (m+1)}, b _{1 (m+1)}, a _{2 (m+1)}, b _{2 (m+1)}Be respectively the coefficient of each harmonic component;

The 3rd of following formula is the harmonic component that is higher than first-harmonic, wherein c _1n, d _1n, c _2n, d _2nBe respectively the coefficient of each harmonic component, N is the higher hamonic wave intercepted length.

4. laser interferometer nonlinearity erron modification method according to claim 3 is characterized in that, every coefficient of described Fourier series is carried out optimal estimation with least square, draws following formula:

$\left\{\begin{array}{c}\underset{i=1}{\overset{L}{\mathrm{\&Sigma;}}}\{a_{10}+\underset{m=0}{\overset{M/4}{\mathrm{\&Sigma;}}}[a_{1(m+1)}\sin \left(\frac{{\mathrm{\&theta;}}_i}{2^m}\right)+b_{1(m+1)}\cos \left(\frac{{\mathrm{\&theta;}}_i}{2^m}\right)]+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\{c_{1n}\sin \left[(n+1){\mathrm{\&theta;}}_i\right]+d_{1n}\cos \left[(n+1){\mathrm{\&theta;}}_i\right]\}-u_{1i}{\}}^2=\min \\ \underset{i=1}{\overset{L}{\mathrm{\&Sigma;}}}\{a_{20}+\underset{m=0}{\overset{M/4}{\mathrm{\&Sigma;}}}[a_{2(m+1)}\sin \left(\frac{{\mathrm{\&theta;}}_i}{2^m}\right)+b_{2(m+1)}\cos \left(\frac{{\mathrm{\&theta;}}_i}{2^m}\right)]+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\{c_{2n}\sin \left[(n+1){\mathrm{\&theta;}}_i\right]+d_{2n}\cos \left[(n+1){\mathrm{\&theta;}}_i\right]\}-u_{2i}{\}}^2=\min \end{array}\right.$

Wherein, L is that image data is counted θ _iBe the phasing degree of interference signal on the sampled point, u _1i, u _2iBe respectively the interference signal on the sampled point;

Following formula is respectively waited to ask parameter a ₁₀, a _{1 (m+1)}, b _{1 (m+1)}, c _1n, d _1n, a ₂₀, a _{2 (m+1)}, b _{2 (m+1)}, c _2n, d _2nDifferentiate is found the solution and is obtained parameter described to be asked, and determines update equation.

5. according to each described laser interferometer nonlinearity erron modification method in the claim 1～4, it is characterized in that in the described step 1, what described interference signal was approximate is expressed as:

$\left\{\begin{array}{c}{u}_1=a_{10}+a_{11}\sin \left(\mathrm{\&theta;}\right)+b_{11}\cos \left(\mathrm{\&theta;}\right)\\ u_2=a_{20}+a_{21}\sin \left(\mathrm{\&theta;}\right)+b_{21}\cos \left(\mathrm{\&theta;}\right)\end{array}\right.$

Further can get:

$\mathrm{\&theta;}=\arctan \frac{b_{21}(u_1-a_{10})-b_{11}(u_2-a_{20})}{a_{11}(u_2-a_{20})-a_{21}(u_1-a_{10})}$

Obtain described starting phase angle θ according to following formula calculating ₀:

6. laser interferometer nonlinearity erron modification method according to claim 5 is characterized in that, in the described step 2, described phasing degree modified value Δ θ obtains by described update equation is carried out the calculating of single order Taylor expansion at the starting phase angle place, and is as follows:

$\left\{\begin{array}{c}{u}_1=f_1\left({\mathrm{\&theta;}}_0\right)+f_1^{\&prime;}\left({\mathrm{\&theta;}}_0\right)\mathrm{\&Delta;\&theta;}\\ u_2=f_2\left({\mathrm{\&theta;}}_0\right)+f_2^{\&prime;}\left({\mathrm{\&theta;}}_0\right)\mathrm{\&Delta;\&theta;}\end{array}\right.$

Further can get:

$\mathrm{\&Delta;\&theta;}=\frac{u_1-f_1\left({\mathrm{\&theta;}}_0\right)}{f_1^{\&prime;}\left({\mathrm{\&theta;}}_0\right)}$ Or $\mathrm{\&Delta;\&theta;}=\frac{u_2-f_2\left({\mathrm{\&theta;}}_0\right)}{f_2^{\&prime;}\left({\mathrm{\&theta;}}_0\right)}$

Obtain described phasing degree modified value Δ θ according to following formula calculating.

7. laser interferometer nonlinearity erron modification method according to claim 6 is characterized in that, precise phase angle θ in the described step 3 ₀' obtain by following formula calculating:

θ ₀′＝θ ₀+Δθ

Further foundation

Interferometer measurement mirror displacement x is obtained in calculating;

Wherein, θ ₀' can be used as new θ ₀Substitution Or Carry out iterative computation until obtaining accurate phasing degree.

8. an application rights requires the laser interferometer nonlinearity erron correcting device of each described laser interferometer nonlinearity erron modification method in 1～7, is arranged at the signal processing system of described interferometer, it is characterized in that, comprising:

The first-harmonic correcting module is used for the first-harmonic composition of interference signal is revised, and obtains starting phase angle;

The harmonic wave correcting module is used for the harmonic components of described interference signal is revised, and obtains the modified value at phasing degree;

The error compensation module, the modified value that is used for the phasing degree that the starting phase angle that obtains according to described first-harmonic correcting module and described harmonic wave correcting module obtain is obtained the amended precise phase of compensation angle, and obtains described interferometer measurement mirror displacement in view of the above and realize nonlinearity erron correction to interferometer.

9. error correction device according to claim 8, it is characterized in that, comprise that also a correction model sets up module, be used to set up update equation, and according to the automatic phasing degree that obtains interference signal of handling of this update equation, and output is from described first-harmonic correcting module and described harmonic wave correcting module.

10. according to Claim 8 or 9 described error correction devices, it is characterized in that, described compensating module also further comprises an error judgment module, be used for the precise phase angle that obtains according to calculating, judge whether to satisfy default error requirements,, proceed iterative computation as not satisfying default error requirements, until satisfying default error requirements, obtain final phase value.

11. an application rights requires the laser interferometer of each described nonlinearity erron correcting device in 8～10, it is characterized in that, described nonlinearity erron correcting device is arranged at the signal processing system of described interferometer, is used to utilize the harmonic separation revised law that the nonlinearity erron of described interferometer is revised; This error correction device further comprises:

Correction model is set up module, is used to set up update equation, and with the phasing degree that obtains interference signal according to the automatic processing of this update equation, and output is from described first-harmonic correcting module and described harmonic wave correcting module;

The first-harmonic correcting module is used for the first-harmonic composition of interference signal is revised, and obtains starting phase angle;

The harmonic wave correcting module is used for the harmonic components of described interference signal is revised, and obtains the modified value at phasing degree;

The error compensation module, the modified value that is used for the phasing degree that the starting phase angle that obtains according to described first-harmonic correcting module and described harmonic wave correcting module obtain is obtained the amended precise phase of compensation angle, and obtains described interferometer measurement mirror displacement in view of the above and realize nonlinearity erron correction to interferometer.

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