CN103047928A - Random error mode evaluation method of phase shifting interferometer - Google Patents

Abstract

The invention discloses a random error mode evaluation method of a phase shifting interferometer, and the method belongs to the field of the error evaluation of interferometers. The method comprises the following steps that a certain inclination quantity is led in during a testing process; the average value of a plurality of measuring results serves as a reference value; the random testing error of one testing result is obtained by subtracting the reference value from one testing result; and the error is analyzed according to the characteristics of different error sources, so as to obtain the main error sources in an interference measuring process by analysis. According to the random error mode evaluation method of the phase shifting interferometer, the categories of the random error sources in the phase shifting interferometer are quickly obtained by analyzing the modes of the random errors and comparing the difference among the random error modes caused by various error sources, so as to provide convenient and quick guidance for the maintenance and the use of the interferometer.

Description

Method for evaluating random error mode of phase-shifting interferometer

Technical Field

The invention belongs to the field of interferometer error evaluation, and particularly relates to a method for evaluating a random error mode of a phase-shifting interferometer.

Background

The phase-shifting interferometer is a high-precision optical measuring device and is widely applied to scientific research and production. As a high-precision optical instrument, the phase-shifting interferometer utilizes optical and mechanical components with high precision to ensure the testing precision. After a long time of use, the measurement accuracy of the interferometer is affected by the increasing random error. Typically, multiple expensive, high precision instruments are required to analytically find the source of random errors. This greatly increases the maintenance and use costs of the interferometer.

Disclosure of Invention

In order to solve the problems in the prior art, the invention provides an evaluation method for the random error modes of the phase-shifting interferometer, which analyzes the modes of random errors, and quickly obtains the types of the random error sources in the phase-shifting interferometer by comparing the differences among the random error modes caused by various error sources, thereby providing simple and quick instructions for the maintenance and the use of the interferometer.

The technical scheme adopted by the invention for solving the technical problem is as follows:

a method for evaluating a random error mode of a phase-shifting interferometer is characterized by comprising the following steps:

the method comprises the following steps: measuring a good wavefront for multiple times by using a phase-shifting interferometer, introducing inclination in the measuring process, and taking the average value of multiple measuring results after the inclination is introduced as a reference phase; then randomly taking one of the test phases, subtracting the reference phase to obtain a test error of a random measurement result:

step two: by analyzing the random error source of the phase-shifting interferometer, the relation between different types of error sources and test errors is obtained, and the random error caused by inaccurate phase shifting, the position noise error caused by vibration, the random error caused by unstable light source power and the error characteristic caused by the drift of the central frequency of the light source are deduced and obtained;

step three: and comparing the test errors in the test result obtained in the first step according to the characteristics of random errors caused by different error sources obtained by analysis, and determining the main error source in the interferometer.

The invention has the beneficial effects that: the invention determines the characteristics of random errors caused by different error sources through mathematical analysis and simulation, can help related personnel to quickly identify and search the error sources in the instrument, greatly saves the cost of equipment maintenance and detection through a high-precision instrument, and is convenient and simple.

Drawings

FIG. 1 is a test wavefront of a phase-shifting interferometer random error mode evaluation method of the present invention.

The figure 2 test introduces a certain amount of tilt in the interference fringes.

FIG. 3 shows random errors caused by phase misalignment.

Figure 4 random errors caused by vibration.

FIG. 5 random errors due to unstable light source intensity.

FIG. 6 shows random errors due to instability of the center frequency of the light source.

Detailed Description

The present invention will be described in further detail with reference to the accompanying drawings and examples.

A method for evaluating a random error mode of a phase-shifting interferometer is characterized by comprising the following steps:

the method comprises the following steps: the method comprises the following steps of measuring a good wavefront for multiple times by using a phase-shifting interferometer, wherein the root mean square error of the wavefront is 1/10-1/20 wavelength, introducing a certain amount of inclination in the measuring process, two to three straight fringes are arranged in the interference fringes, taking the average value of multiple measuring results after the inclination is introduced as a reference phase, and taking the average value of the multiple measuring results after the inclination is introduced, wherein most random errors can be eliminated well by taking the average value of the multiple measuring results as a reference value, so that the random errors in the measuring process can be extracted conveniently; then randomly taking one of the test phases, subtracting the reference phase to obtain a test error of a random measurement result:

the phase-shifting interferometer generates interference fringes under different phase conditions in a phase-shifting mode to obtain different light intensity equations, and the required phase in interferometry is obtained by solving the series of equations. The interference fringe obtained by phase shifting of the phase shifting interferometer can be obtained by the following formula:

where V is the interference fringe contrast, φ_nIs the reference phase at time t,

is the phase under test.

The measured phase is determined according to the generalized phase extraction algorithm of the phase-shifting interferometer mentioned in the publication Peter de Groot "Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window". APPLIEDDOPTICS, 34, 4723-4730 (1995)

Can be obtained from the following equation:

for the window function w_nIn the case of a real function, Re is a real part, Im represents an imaginary part, and the following formula can be taken:

$s_n=\mathrm{Im}\left[w_n\exp (-i{\mathrm{\φ}}_n)\right]=w_n\sin (-{\mathrm{\φ}}_n)---\left(3\right)$

$c_n=\mathrm{Re}\left[w_n\exp (-i{\mathrm{\φ}}_n)\right]=w_n\cos \left({\mathrm{\φ}}_n\right)---\left(4\right)$

the extraction formula of the phase in such an interferometer is:

when the interferometer has an error source, the acquired image of the interference fringe is

To obtain

According to the formula $\mathrm{arc}{\tan }^{\′}\left(x\right)=\frac{1}{1+x^2}$ ，

，

The errors of the test are:

due to the fact that

The above formula becomes:

where k is a constant for the same phase-shifting algorithm, k can be considered to be 1 on the premise that the specific form of the measurement error is concerned.

Step two: through deep analysis of a random error source of the phase-shifting interferometer, the relation between different types of errors and test errors is obtained, and random errors caused by inaccurate phase shifting, position noise errors caused by vibration, random errors caused by unstable light source power and error characteristics caused by central frequency drift of a light source are deduced and obtained;

the random error sources present in phase-shifting interferometers are: random Error Phase Shift Error caused by Phase Shift nonlinearity, Position noise Error caused by vibration, Unstable Intensity Unstable of light source power, and Frequency Shift of light source center Frequency. The interferometer testing error is directly expressed as the change of the light intensity of the interference fringe, so the relationship between the light intensity variation and the error source is as follows:

$\mathrm{\ΔI}=\frac{\mathrm{dI}\left(x\right)}{\mathrm{dx}}\×\mathrm{\Δx}---\left(10\right)$

where x is the magnitude of the error source. The light intensity variation and the measurement error caused by the four random error sources can be discussed according to the above formula.

1. Phase shift error:

the test error calculated at this time is

Wherein, Δ P_nIs the phase shift error. Such error and

the terms are proportional, which means that random errors caused by inaccurate phase shift appear as regular fringes in the test result, the number of the fringes is twice that of the interference fringes

2. Vibration-induced position noise error:

wherein,

is the derivative of the wavefront. Error shape caused by vibration and

and

the two terms are in direct proportion, and the direction and the magnitude of the vibration are

Is closely related to the direction and magnitude of the wave, random errors caused by vibration appear as frequency doubling fringes (relative to interference fringes) in the test results, but the peaks of the same ripples do not coincide.

3. Unstable error of light source power:

wherein, Delta_nIntensity is the light Intensity instability error. Error introduced by unstable power of light source

Andcorrelation, and therefore random errors caused by unstable light source power appear as frequency-doubled fringes in test results, peaks in different periods are not consistent, and the peaks in different periods change like waves (positive selection curves).

4. Error caused by drift of the center frequency of the light source:

wherein, Delta_nFreq is the error in the drift of the center frequency of the light source. Such error andin direct proportion to each other, at the same time is

The terms are modulated so that the source center frequency drift causes random errors that appear as peaks of different periods in the doubled fringes growing or decreasing in the same direction.

Step three: and comparing the test errors in the test result obtained in the first step according to the characteristics of random errors caused by different error sources obtained by analysis, and determining the main error source in the interferometer.

In order to more precisely describe the difference between random errors caused by different random error sources, simulation is performed by artificially introducing random errors into a measured wavefront as shown in fig. 1 and fig. 2. The difference between the different random errors can be obtained by means of fig. 3 to 6.

1. For phase shift errors, frequency doubled regular fringes are found in fig. 3 and are associated with the interference fringes shown in fig. 1 (error has 4 periods and interference fringes has 2 periods). It is obvious that this is the same as in the formula (17)

And correspondingly.

2. For vibration errors, there are also frequency-doubled fringes in fig. 4, but the peaks of the same ripple do not coincide over the frequency-doubled fringes. This phenomenon is related to the term in equation (20)

And

and (4) the same.

3. For the light source instability error, a stripe with frequency doubling is found in fig. 5, the peaks of different periods are not consistent, and the peaks of different periods are like waves (positive selection curve) —And (4) changing. This phenomenon is related to the term in the formula (23)

And

it is related.

4. For the error caused by the unstable frequency of the light source, a stripe with frequency doubling is found in fig. 6, and the peak values of different periods in the stripe are continuously increased, which is similar to the term in equation (26)

It is related.

Claims (2)

1. A method for evaluating a random error mode of a phase-shifting interferometer is characterized by comprising the following steps:

the method comprises the following steps: measuring a good wavefront for multiple times by using a phase-shifting interferometer, introducing inclination in the measuring process, and taking the average value of multiple measuring results after the inclination is introduced as a reference phase; then randomly taking one of the test phases, subtracting the reference phase to obtain a test error of a random measurement result:

step two: by analyzing the random error source of the phase-shifting interferometer, the relation between different types of error sources and test errors is obtained, and the random error caused by inaccurate phase shifting, the position noise error caused by vibration, the random error caused by unstable light source power and the error characteristic caused by the drift of the central frequency of the light source are deduced and obtained;

step three: and comparing the test errors in the test result obtained in the first step according to the characteristics of random errors caused by different error sources obtained by analysis, and determining the main error source in the interferometer.

2. A phase shifting interferometer random error characterization according to claim 1, wherein the relationship between the error source and the test error in step two of the method is derived by:

interference fringe I obtained by phase shift of phase-shift interferometer_nExpressed as:

where V is the interference fringe contrast, φ_nFor the purpose of referencing the phase(s),is the phase under test.

Measured phaseCan be obtained from the following equation:

wherein the window function w used in the solving algorithm_nIn the case of a real function, Re is a real part, Im represents an imaginary part, and the following formula can be taken:

$s_n=\mathrm{Im}\left[w_n\exp (-i{\mathrm{\φ}}_n)\right]=w_n\sin (-{\mathrm{\φ}}_n)---\left(3\right)$

$c_n=\mathrm{Re}\left[w_n\exp (-i{\mathrm{\φ}}_n)\right]=w_n\cos \left({\mathrm{\φ}}_n\right)---\left(4\right)$

the phase extraction formula (2) becomes:

when the interferometer has an error source, the acquired image of the interference fringe is

，

For the intensity variation of the interference fringes caused by the error source, the result is obtained:

according to the formula

，

The error of the test is:

due to the fact that

The above formula becomes:

where k is a constant for the same phase-shifting algorithm, k can be considered as 1 on the premise that the specific form of the measurement error is concerned,

according to the different influence characteristics of different random error sources on the interference fringes, the error of the test result caused by the different random error sources can be deduced by the formula as follows:

with respect to the random error caused by the phase shift,

wherein, Δ P_nTo shift the phaseAnd (4) error. It can be seen that the error isThe terms are proportional;

the position noise error caused by the vibration is,

wherein,

is the derivative of the wavefront. Random error caused by vibration and

and

the two terms are in direct proportion, and the direction and the magnitude of the vibration are

The direction and the size of the Chinese characters are closely related;

random errors due to the instability of the power of the light source,

wherein, Delta_nIntensity is the light Intensity instability error. Error introduced by unstable power of light source

And

correlation;

errors caused by the drift of the center frequency of the light source,

wherein, Delta_nFreq is the error in the drift of the center frequency of the light source. Such error and

in direct proportion to each other, at the same time is

The terms are modulated.

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