Method for measuring the diameter of filament diffraction fringes by calculating the frequency domain

Abstract

A method for measuring the diameter of filament diffraction fringes by frequency domain calculation comprising: building a set of diffraction optical path measurement system and capturing diffraction fringe images; determining the starting point of the imaging range; Simulating the electromagnetic field propagation process in Fraunhofer diffraction, and determining the optimal fringe range considering the noise caused by the difference in CCD sensitivity; Finally calculating the filament diameter by Fourier transform for different lengths of fringe. The final value of the calculated filament diameter is obtained by fitting an envelope to the variation of the diameter. The invention is simple in calculation and has little dependence on the experimental device, which means the superiority of using the frequency domain for parameter measurement, and the measurement accuracy is in the sub-nanometer level. In addition, the invention proves the feasibility of extracting the fringe period information in the frequency domain.

Claims

1. A method for measuring a diameter of filament diffraction fringes by calculating the frequency domain, comprising: building an optical path system required for a diffraction measurement, said building comprising: using a Fraunhofer diffraction device to emit a laser that passes through two mirrors and an optical power attenuator, the laser being vertically incident on a filament to be measured to generate diffraction fringes, the diffraction fringes being converged by a convex lens to a charge coupled device (CCD) at a focal point of the CCD to obtain a diffraction fringe image, where the CCD is connected to a computer to capture the diffraction fringes by programs wherein a center of the two mirrors, a center of the optical power attenuator, a center of the filament, and a center of the convex lens lie on a same optical axis, while the CCD is deviated from the optical axis to avoid damage to the CCD due to excessive light converging energy; deriving a calculation and an error of a diffractive filament diameter, said deriving comprising: theoretically analyzing an error in said diffraction, wherein the filament is irradiated uniformly by a plane wave and a field source distribution at the filament is E(Y, 0)=A.sub.0, where Yb/2, and b is a diameter of the filament, and Y is a distance at the filament in a direction perpendicular to a direction of light, and wherein when a Fraunhofer diffraction approximation condition is satisfied, an expression of the electric field of a light intensity distribution on an observation screen is: $\begin{array}{cc}E\left(Y,f\right)=\frac{\exp \left[ikf\right]}{ikf}\exp \left[\frac{ik}{2f}\left(Y^2\right)\right]\underset{\mathrm{plane}}{}E\left(Y^{},0\right)\exp \left[-{\mathrm{ikY}}^{}\sin {}_Y\right]d^2{Y}^{},& \left(1\right)\end{array}$ wherein, .sub.Y is an angle between a direction of a wave vector and a Y direction, k is a wave vector of the laser, Y is the distance in the direction of the filament diameter, and f is a focal length of the convex lens, and an intensity distribution of a light field is expressed as: $\begin{array}{cc}I=E{E}^{*}\frac{A_0^2}{f}\sin {c\left(b\sin {}_Y/\right)}^2,& \left(2\right)\end{array}$ If Y/f<<1, sin .sub.Ysin(arctan(Y/f))Y/f, then formula (2) is: $\begin{array}{cc}I\frac{A_0^2}{f}\sin {c\left(\mathrm{bY}/f\right)}^2.& \left(3\right)\end{array}$ wherein Y=yy.sub.0(y=yd), d is a pixel size of the CCD, and Y is a number of pixels along a y axis on the imaging plane, y is the pixel point distance of the diffraction fringe on the CCD, and y is a distance along the y axis, and y.sub.0 is a center position coordinate of the zero-order fringe; formula (4) is obtained by a fast Fourier transform of formula (3): $\begin{array}{cc}F\left\{I\right\}=\frac{1}{b^2}i{e}^{-{\mathrm{ivy}}_0}f\left(\left(b-\mathrm{fv}\right)\left(\left[-1,\frac{i\left(-b+\mathrm{fv}\right)\left(y_{\max }-y_0\right)}{f}\right]-\left[-1,i\left(\frac{b}{f}-v\right)y_0\right]+\mathrm{Log}\left[y_{\max }-y_0\right]-\mathrm{Log}\left[\frac{i\left(-b+\mathrm{fv}\right)\left(y_{\max }-y_0\right)}{f}\right]-\mathrm{Log}\left[-y_0\right]+\mathrm{Log}\left[i\left(\frac{b}{f}-v\right)y_0\right]\right)-2fv\left(\mathrm{Log}\left[-y_0\right]-\left[-1,\mathrm{iv}\left(y_{\max }+y_0\right)\right]+\left[-1,-\mathrm{iv}{y}_0\right]-\mathrm{Log}\left[y_{\max }-y_0\right]-\mathrm{Log}\left[-\mathrm{iv}{y}_0\right]\mathrm{Log}\left[\mathrm{iv}\left(y_{\max }-y_0\right)\right]\right)-\left(b+fv\right)\left(\left[-1,\frac{i\left(b+\mathrm{fv}\right)\left(y_{\max }-y_0\right)}{f}\right]-\left[-1,-\frac{i\left(-b+\mathrm{fv}\right)y_0}{f}\right]+\mathrm{Log}\left[y_{\max }-y_0\right]-\mathrm{Log}\left[\frac{i\left(b+\mathrm{fv}\right)\left(y_{\max }-y_0\right)}{f}\right]-\mathrm{Log}\left[-y_0\right]+\mathrm{Log}\left[-\frac{i\left(b+\mathrm{fv}\right)y_0}{f}\right]\right)\right)& \left(4\right)\end{array}$ wherein v is a spatial frequency extreme point, y.sub.max is a maximum fringe value, and formula (5) is obtained by calculating an extreme value of formula (3):
bfv=0(5); performing a Fourier transform on the diffraction fringes of the filament, and the extreme value point (v) in the frequency domain is extracted to solve the diameter of the filament; and shearing, scaling and splicing the diffraction fringes; performing an algorithmic processing, comprising: capturing filament diffraction fringe images with the CCD; cutting, filtering and preprocessing the filament diffraction fringe images: extracting one-dimensional diffraction fringes from the filament diffraction fringe images, and extracting a size range of local diffraction fringes; forming pseudo-fringes by compressing and splicing the local diffraction fringes; obtaining frequency domain information of the diffraction fringes by performing a Fourier transform on the pseudo-fringes; calculating the diameter of the filament, and obtaining a change curve of the diameter of the filament by continuously extending the diffraction fringes; and extracting an upper interval limit and a lower interval limit of the filament diameter, and solving a diameter convergence value of the filament; and measuring the filament diffraction, comprising: determining the size range of local diffraction fringes; directly reducing a proportion of the local diffraction fringes causes the light intensity of the pseudo-fringes deviating from a light intensity of the diffraction fringes; a light intensity extreme value ratio of the diffraction fringes satisfies formula (8): $\begin{array}{cc}\frac{I_{n+1}}{I_n}={\left(1-\frac{1}{n-k+\frac{3}{2}}\right)}^2,& \left(8\right)\end{array}$ wherein I.sub.n is an extreme value of a light intensity of nth order fringes, and a kth bright fringe is a central bright fringe of the image; and obtaining diffraction fringes by simulating light field propagation using COMSOL, wherein some of the diffraction fringes are used as a basis for repeated splicing, and the central bright fringe and 1st-3rd order bright fringes are not selected, and simulated fringes in different ranges are analyzed to obtain an optimal fringe range A<y<B, a scale factor of the simulated fringes is defined as y/b, and an optimal value range is: C<y/b<D, where A is a position of the two fringes at a fourth level of the diffraction stripe, and B is a position of the bright stripe at an eleventh level, C is denoted as equal to A/b, and D is denoted as equal to B/b; and calculating the filament diameter, said calculating comprising: using Butterworth low-pass filtering to process noise to obtain a smooth fringe curve, wherein the diameter of the filament to be measured is x, the focal length of the lens is f, a minimum pixel point of the CCD is d, and a size of a local fringe image is l; excluding the bright fringes in the center and the bright fringes of the 1.sup.st-3.sup.rd order; splicing cropped pseudo fringes to the last bright fringes of a segment; and then repeating said using, said excluding and said splicing of the cropped pseudo fringes, wherein a diameter of the filament gradually converges to an interval; a peak point and a valley point in the smooth fringe curve are extracted, and the peak point and the valley point are fitted respectively to obtain the convergence interval of the diameter; a fitting equation is as follows: $\begin{array}{cc}Z=k\left(\frac{1}{y-y^{*}}\right)+C& \left(9\right)\end{array}$ wherein, k, y*, C are constants of the fitting equation, and y is the lengths of the fringes, and wherein a positive error peak fitting and a negative error peak fitting curve are obtained by fitting, and the diameter of the filament is calculated as {tilde over (b)}=C.sub.up+C.sub.down, to obtain a relationship between the diameter of the filament and a length of the fringe wherein C.sub.up is the positive error peak fitting curve and C.sub.down is the negative error peak fitting curve.

Description

DESCRIPTION OF DRAWINGS

(1) FIG. 1 is an optical path diagram (top view) of the system measuring the diameter of filament diffraction of the present invention.

(2) FIG. 2 is the error between the actual diffraction fringe and the theoretical fringe.

(3) FIG. 3 is the COMSOL simulation diffraction physical process of the present invention.

(4) FIG. 4a to 4d are the processing results of the diffraction fringe images, wherein FIG. 4a is the diffraction fringe pattern of a filament with a diameter of x, and FIG. 4b is the repeated mosaic of pseudo-fringes, and FIG. 4c is the calculated fitting diagram of the filament diameter, and FIG. 4d is the calculated relationship between the diameter of the filament and the length of the fringe.

EMBODIMENTS

(5) The technical solutions of the present invention are further described below with reference to the accompanying drawings.

(6) Aiming at the deficiencies in the existing fringe identification processing algorithm, the invention establishes an upgrade of the diffraction fringe algorithm for measuring the diameter of the filament, and realizes the measurement of the diameter of the filament with high precision. The main contents include: build a diffraction optical path measurement system and use CCD to take diffraction fringe images; determine the starting point of the imaging range; use the finite element method and numerical calculation to simulate the electromagnetic field propagation process in Fraunhofer diffraction and determining the optimal fringe range considering the noise caused by the difference in CCD sensitivity; finally, the filament diameter is obtained by fitting the fringes.

(7) A method for measuring the diameter of filament diffraction fringes by calculating the frequency domain, comprising the following steps:

(8) 1. Building the Optical Path System Required for Diffraction Measurement;

(9) Since the present invention adopts a Fraunhofer diffraction device, the algorithm has a very low dependence on the device, and high-precision measurement can be achieved through the processing of the algorithm. The specific optical path diagram used in the present invention is shown in FIG. 1. After the laser emitted by the laser passes through two mirrors, the optical power is attenuated, and the laser after passing through the optical power attenuator encounters the filament to produce diffraction phenomenon, and the diffraction pattern is converged by a lens with a focal length of f to a CCD placed in focus, and then the CCD captures the pattern of diffraction fringes. The CCD camera is connected to a computer and the fringes are captured by the program. It should be noted that the center of a mirror, the optical power attenuator, the center of the filament, and the center of the lens should be on the same optical axis, but the CCD camera needs to deviate from the optical axis (to avoid zero-order fringes entering the CCD imaging surface) to prevent light from converging excessive energy damaging the CCD.

(10) 2. Deriving the Calculation and Error of Diffractive Filament Diameter;

(11) If the filament is uniformly illuminated by a plane wave, the filament resembles an infinite slit, and the field source distribution at the filament can be E(Y, 0)=A.sub.0 (Yb/2, where b is the filament diameter and Y is the distance at the filament in the Y direction perpendicular to the direction of the light). When the Fraunhofer approximation condition is satisfied, the electric field expression on the viewing screen is:

(12) $\begin{array}{cc}E\left(Y,f\right)=\frac{\exp \left[ikf\right]}{ikf}\exp \left[\frac{ik}{2f}\left(Y^2\right)\right]\underset{\mathrm{plane}}{}E\left(Y^{},0\right)\exp \left[-{\mathrm{ikY}}^{}\sin {}_Y\right]d^2{Y}^{},& \left(1\right)\end{array}$
where .sub.r is the angle between the wave vector direction and Y direction, and f is the focal length of the lens, and the intensity distribution of the light field can be expressed as:

(13) $\begin{array}{cc}I=E{E}^{*}\frac{A_0^2}{f}\sin {c\left(b\sin {}_Y/\right)}^2,& \left(2\right)\end{array}$
If Y/f<<1, sin .sub.rsin(arctan(Y/f))Y/f, then formula (2) is changed to:

(14) $\begin{array}{cc}I\frac{A_0^2}{f}\sin {c\left(\mathrm{bY}/f\right)}^2.& \left(3\right)\end{array}$

(15) Wherein, Y=yy.sub.0(y=yd), d is the pixel size of the CCD, Y is the number of pixels along the y-axis on the imaging plane, y is the distance along the y-axis, and y.sub.0 is the center position coordinate of the zero-order fringe. The fast Fourier transform of formula (3) is obtained:

(16) 0$\begin{array}{cc}F\left\{I\right\}=\frac{1}{b^2}i{e}^{-{\mathrm{ivy}}_0}f\left(\left(b-\mathrm{fv}\right)\left(\left[-1,\frac{i\left(-b+\mathrm{fv}\right)\left(y_{\max }-y_0\right)}{f}\right]-\left[-1,i\left(\frac{b}{f}-v\right)y_0\right]+\mathrm{Log}\left[y_{\max }-y_0\right]-\mathrm{Log}\left[\frac{i\left(-b+\mathrm{fv}\right)\left(y_{\max }-y_0\right)}{f}\right]-\mathrm{Log}\left[-y_0\right]+\mathrm{Log}\left[i\left(\frac{b}{f}-v\right)y_0\right]\right)-2fv\left(\mathrm{Log}\left[-y_0\right]-\left[-1,\mathrm{iv}\left(y_{\max }+y_0\right)\right]+\left[-1,-\mathrm{iv}{y}_0\right]-\mathrm{Log}\left[y_{\max }-y_0\right]-\mathrm{Log}\left[-\mathrm{iv}{y}_0\right]\mathrm{Log}\left[\mathrm{iv}\left(y_{\max }-y_0\right)\right]\right)-\left(b+fv\right)\left(\left[-1,\frac{i\left(b+\mathrm{fv}\right)\left(y_{\max }-y_0\right)}{f}\right]-\left[-1,-\frac{i\left(-b+\mathrm{fv}\right)y_0}{f}\right]+\mathrm{Log}\left[y_{\max }-y_0\right]-\mathrm{Log}\left[\frac{i\left(b+\mathrm{fv}\right)\left(y_{\max }-y_0\right)}{f}\right]-\mathrm{Log}\left[-y_0\right]+\mathrm{Log}\left[-\frac{i\left(b+\mathrm{fv}\right)y_0}{f}\right]\right)\right)& \left(4\right)\end{array}$

(17) Wherein, v is the spatial frequency extreme point, y.sub.max is the maximum fringe value, and formula (5) is obtained by calculating the extreme value of formula (3):
bfv=0(5)

(18) Finally, Fourier transform can be performed on the diffraction fringes of the filament, and the extreme point (v) in the frequency domain can be extracted to calculate the diameter of the filament. In order to improve the accuracy of the measurement, it is necessary to perform operations such as shearing, scaling and splicing of the diffraction fringes.

(19) 3. Processing Algorithmic;

(20) (1) Capturing the filament diffraction fringe images with CCD; (2) Cutting, filtering and preprocessing the images; (3) Extracting one-dimensional diffraction fringes from the image, and extracting a better range (refer to Step 4 for details) as local fringes; (4) Forming pseudo-fringes by compressing and splicing local fringes (as shown in FIG. 4); (5) Obtaining the frequency domain information of the fringes by performing Fourier transform on the pseudo fringes; (6) Calculating the diameter of the filament, and obtaining the change curve of the diameter of the filament by continuously extending the fringes (as shown in FIG. 4d); (7) Extracting the upper and lower interval limits of the filament diameter, and solving the diameter convergence value of the filament (that is, the measured value of the diameter of the filament).
4. Measuring the Filament Diffraction;
4.1 Determining the Size Range of Local Fringes;

(21) In order to retain more information of diffraction fringes in the spliced fringes, it is necessary to cut the obtained fringes to obtain local fringes. The size range of the local fringes directly affects the accuracy, and theoretical analysis is required for this. The change of the light intensity extreme value is not a proportional sequence, and directly reducing the scale of the local fringes will cause the light intensity of the false fringes to deviate from the light intensity of the actual fringes. FIG. 4a shows the situation of local fringes, and the light intensity extreme value ratio of the fringes satisfies:

(22) $\begin{array}{cc}\frac{I_{n+1}}{I_n}={\left(1-\frac{1}{n-k+\frac{3}{2}}\right)}^2,& \left(8\right)\end{array}$ wherein I.sub.n is the light intensity extrema of the nth level fringe, and the k th bright fringe is the central bright fringe of the image. As shown in FIG. 4b, when n is large, the intensity of the bright fringes is nearly constant at the extreme value. Therefore, the 5-7th order bright fringes serve as the starting point of the imaging range.

(23) Using finite element method and numerical calculation, the electromagnetic field propagation process in Fraunhofer diffraction is simulated. To make the simulation realistic, a noise term was added, including noise related to light intensity in the fringe analysis. The noise intensity is proportional to the light intensity of the bright fringes, due to differences in the response of the CCD photoreceptors to light. In addition, the noise also includes Gaussian noise and white noise due to the action of the laser spot.

(24) In order to accurately obtain the optimal range of local fringes, the diffraction fringes obtained by simulating the propagation of the light field through the filament using COMSOL are shown in FIG. 3. Filament diameters were calculated by selecting local fringes of different segments as the basis for repeated splicing. The central bright fringes and the 1st-3rd order bright fringes should be omitted from the selection, as shown in Table 1, the simulated fringes in different ranges are analyzed, and the calculated filament diameter results in different fringe ranges are obtained. The optimal local fringe range is 4.969<y<8.307 mm (Assume that the coordinate center is the center of the 0-level bright fringe, and the local fringe range is the 7-12-level bright fringe). Then, to ensure that this range is relevant to measurements at various wavelengths and focal lengths, the optimal scaling factor for the local fringes is 49.69<y/f<83.07 mm.sup.1

(25) TABLE-US-00001 TABLE 1 Calculated filament diameters in different fringe ranges Calculated (mm) Diffraction Calculated filament diameter fringe length Scale Factor filament diameter (fringe signal-to- (mm) /f (mm.sup.1) (no noise) noise ratio 15.28) 6.08 16.30-77.11 0.15089 0.15107 5.34 23.70-77.11 0.15068 0.15108 4.65 30.59-77.11 0.15043 0.15054 3.35 43.65-77.11 0.15038 0.15056 2.74 49.69-77.11 0.15014 0.15018 2.00 49.69-77.11 0.15027 0.15062 3.34 49.69-83.07 0.15007 0.14988 3.94 49.69-83.07 0.15055 0.15033
4.2 Calculating the Specific Filament Diameter;

(26) Measurements are made using filaments with a diameter of 0.110.001 mm (filament machining has an accuracy of 1 m). With a lens focal length of 125 mm and a CCD pixel unit size of 4.8 m, the size of the local fringe image is 8.64 mm, and the filament diffraction fringes with a diameter of 0.11 mm are shown in FIG. 4a. The noise is processed using a Butterworth low-pass filter, resulting in a smooth fringe curve. In order to improve the accuracy, the diffraction fringes will be cropped, and the optimal size range (49.69<y/f<83.07) will be selected. As shown in FIG. 4b, the cropped pseudo fringes are spliced to the last bright fringes of the segment, and then the operation is repeated to increase the fringes.

(27) As shown in FIG. 4c, the filament diameter gradually converges to a tiny interval. Extract the peak (valley) points in the curve, and fit these peak (valley) points respectively to obtain the convergence interval of the diameter. The fitted equation is as follows:

(28) $\begin{array}{cc}Z=k\left(\frac{1}{y-y^{*}}\right)+C& \left(9\right)\end{array}$
where, k, y*, C are the constants of the fitted equation and y is the lengths of the fringes.

(29) As shown in FIG. 4c, the blue and red curves were obtained by fitting, and the filament diameter was calculated as {tilde over (b)}=C.sub.up+C.sub.down. As shown in FIG. 4d, the relationship between the diameter x of the obtained filament and the lengthy of the pseudo fringe is obtained. The filament diameter converges to 110.07134 m and the length of y is 76 mm. In contrast, directly calculating the filament diameter of the original fringe in the frequency domain is 108.62 m, and using local fringe stitching improves the relative error of the fringe from 0.38 m to 0.071 m.

(30) In addition, in order to verify the feasibility of the algorithm, filaments of different diameters will be used to evaluate the feasibility and accuracy of the method. As shown in Table 2, for the actual obtained filament diffraction images, the pixel size has little effect on the accuracy.

(31) TABLE-US-00002 TABLE 2 The actual obtained filament diffraction images Images Serial number 1 2 3 4 5 6 7 8 Filament 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 diameter (mm)

(32) Table 3 shows the results of the actual measurement, the lens focal length f=100 mm, and the cut-off spatial frequency of the Butterworth low-pass filter is 0.055 mm.sup.1. Comparing the diameter of the filament treated with the pseudo-fringe using the present invention with the diameter of the filament not treated with the pseudo-fringe, the relative error is less than 0.2 m. The data suggest that measurement accuracy may be significantly improved with the presence of stitched streaks.

(33) TABLE-US-00003 TABLE 3 Results of actual measurement Filament diameter Filament diameter without pseudo treated with Filament fringe treatment pseudo- fringe Serial number diameter (mm) (mm) (mm) 1 0.11 0.001 0.1086 0.1098 2 0.12 0.001 0.1207 0.1200 3 0.13 0.001 0.1328 0.1295 4 0.14 0.001 0.1328 0.1393 5 0.15 0.001 0.1445 0.1499 6 0.16 0.001 0.1569 0.1605 7 0.17 0.001 0.1690 0.1698 8 0.18 0.001 0.1690 0.1807