- Email: [email protected]
Study on force distribution of the tempered glass based on laser interference technology
Accepted Manuscript Title: Study on Force Distribution of the Tempered Glass Based on Laser Interference Technology Author: He Jin-Qi Dong Yuan Li Shu-Tao Liu Hui-Long Yu Yong-Ji Jin Guang-Yong Liu Li-Da PII: DOI: Reference:
S0030-4026(15)01317-0 http://dx.doi.org/doi:10.1016/j.ijleo.2015.09.236 IJLEO 56441
To appear in: Received date: Accepted date:
18-11-2014 26-9-2015
Please cite this article as: H. Jin-Qi, D. Yuan, L. Shu-Tao, L. Hui-Long, Y. Yong-Ji, J. Guang-Yong, L. Li-Da, Study on Force Distribution of the Tempered Glass Based on Laser Interference Technology, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.09.236 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Study on Force Distribution of the Tempered Glass Based on Laser Interference Technology He Jin-Qi1, Dong Yuan1*, Li Shu-Tao1, Liu Hui-Long1, Yu Yong-Ji1, Jin Guang-Yong1, Liu Li-Da2
(1.Chang Chun University of Science and Technology, The key laboratory of Jilin province solid-state laser technology and application, China, 130022)
*[email protected]
ip t
(2.Yuxi Industries Group Co.Ltd, Henan Nanyang, China, 473000)
cr
Abstract: The force distribution of inside the tempered glass is detected by the Mach-Zehnder interferometer. In the experiment, the interference fringes are captured
us
by a CCD camera, filtered in frequency domain, and the tempered glass real phase distribution is obtained by utilizing branch-cut algorithm for phase unwrapping. The tempered glass internal force distribution can be derived by means of the relationship
an
between phase distribution and external force. The results demonstrated that the force distribution is almost linear in direction to parallel the external pressure; although
M
some fluctuations appear to be perpendicular to the external pressure, the overall change is smaller.
Key words: Tempered Glass; Mach-Zehnder interferometer; Interference fringes; 220.4840; 220.4830; 230.0040
1. Introduction
te
OCIS
d
Phase unwrapping
Ac ce p
Since the advent of laser in the 1960s, interferometric technique presents its diversity and has achieved rapid development, among which the main techniques are laser heterodyne interferometry, laser holographic interferometry, laser speckle interferometry, single and double frequency laser interferometer, and multi-beam interferometry, etc. Due to its characteristics of non-contact and high precision, it has been extensively applied in the field of non-destructive testing. In recent years, C. Tian et al used interferometry to detect the accurate measurement of aspheric lenses and spherical radius[1-3]. Cui Yanjun and Yang Zhenyu et al used the laser interferometer to accurately test the displacement[4,5]. Song Song et al studied the measurement of small vibrations[6]. LÜ Qieni et al studied particle imaging by using laser interferometer[7]. The interferometric technique can also be applied to a wide range of measurements of the deformation , thickness , density, and refractive index, etc[8-13]. That tempered glass is widely used in daily life, for example, in the automotive
Page 1 of 10
industry and the construction industry, mainly because of its high security. In order to make more extensive application of high quality tempered glasses, security detection has to be done. In this paper, test system of digital images has been set up through the Mach - Zehnder interferometer systems and CCD image pickup device. To restore the
ip t
phase of the tempered glass, the collected interference images are digitized by spatial-carrier Fourier transform method, contributing to obtain the force distribution of the tempered glass under the external force of 20N. This article is the first
cr
application of the laser interferometer in the field of glass detection, which will open a new way for the detection of tempered glass.
us
2 Basic Theory
2.1 Basic principles of Fourier transform of Interfere images
an
By the and after meeting the interference condition, the light intensity distribution of the interference fringe image can be expressed as[14]: G ( x, y ) a ( x, y ) b( x, y ) cos( ( x, y ))
(1)
M
Where, a ( x, y ) is the background light intensity distribution of the interference fringe, b( x, y ) is the contrast of the stripes; a ( x, y ) and b( x, y ) are unknown variables;
d
( x, y ) s ( x, y ) r ( x, y ) is the phase distribution difference; s ( x, y ) is the phase
te
of the measured beams and r ( x, y ) is the phase of the reference beams. In formula (1), there are three unknown numbers and the phase difference ( x, y ) cannot be
Ac ce p
solved. Therefore, in order to solve ( x, y ) , we introduce spatial carrier frequency of f f x x f y y , in which f x and f y are the carrier frequencies in the direction of
x and y , meaning to change the inclination angle of the test surface and the reference
surface, so that the reference wave would be inclined and the fringes on the interference map would be denser in the direction of x and y .The formula of the light
intensity on the interference fringe image becomes as the follow: G ( x, y ) a ( x, y ) b( x, y ) cos[2 f ( x, y )] a ( x, y ) b( x, y ) cos[2 ( f x x f y y ) ( x, y )]
(2)
f : The spatial carrier frequency; f 1/ T , T : The space cycle of interference fringes.
The complex form of Formula ( 2 ) is,
Page 2 of 10
G ( x, y ) a ( x, y ) c( x, y ) exp(i 2 f x x i 2 f y y ) c* ( x, y ) exp(i 2 f x x i 2 f y y )] (3) 1 where c( x, y ) b( x, y ) exp(i ( x, y )) 2
(3), the following relations can be obtained:
G0 ( f1 , f 2 ) A( f1 , f 2 ) C ( f1 f x , f 2 f y ) C * ( f1 f x , f 2 f y )
ip t
Conducted the two-dimensional Fourier transform of spatial variables in formula
(4)
cr
C ( f1 f x , f 2 f y ) : the positive first-level spectrum division,
us
C * ( f1 f x , f 2 f y ) : the negative first-level spectrum division, A( f1 , f 2 ) : the zero level spectrum division.
Fig.1(a) shows the spectrum of the background light intensity of the interference
an
image.
The appropriate filter is selected, the positive first level spectrum is separated and
M
moved to the original point to obtain formula C ( f1 , f 2 ) , which is showed in Fig.1(b). Fig.2(a) and (b) are the interference fringes and Three-dimensional Fourier spectrum
d
distribution collected in the experiment.
f0
f0
te
A(f1,f2 )
Ac ce p
C*(f 1+fx ,f2+fy )
C(f1-fx,f2 -fy)
f (a)
C(f1,f2 )
(b)
f
(a) Two-dimensional Fourier spectrum (b) single Spectrum of the original one Fig.1 The action of filter A two-dimensional inverse Fourier transform is： 1 F 1[C ( f1 , f 2 )] c( x, y ) b( x, y ) exp[i ( x, y )] (5) 2
( x, y ) tan 1
Im[c( x, y )] Re[c( x, y )]
(6)
Page 3 of 10
Where, Im[c( x, y )] and Re[c( x, y )] are the respective imaginary part and the real part
us
cr
ip t
of c( x, y ) .
(a)
(b)
an
Fig.2 (a) interference fringe map of the tempered glass (b) Three -dimensional Fourier spectrum 2.2 Phase unwrapping
M
Since trigonometric is cyclical, phase is wrapped in the main value of the arc tangent function and the phase value is also extended between [-,+] when the computer is processing arc tangent function in the digital interference images. When
d
the real phase is beyond [-,+] and under the modulating effects of trigonometric,
te
wrapped phase distribution [-,+] is formed, and the image of computer processed wrapped phase maps of the tempered glass is showed in Fig.3(a). In this paper,
Ac ce p
Goldstein's branch-cut unwrapping algorithm is applied to get the real phase of the tempered glass.
Goldstein's branch cut follows a specific algorithm: search the residual handicap of
the entire wrapped phase image, establish 33 windows at the first residual handicap,
and then search the next residual handicap. Whether the searched residual is of the same polarity, it is connected to the first residue. If the two residuals are of the opposite polarity, they are called the residual polar balance of the connected branches. If they are of the same polarity, the search window continues to find a new residual handicap, and connect at the center of the windows regardless of whether the residue was connected with other residues or not. If one residual is not connected with others, the polarity would be added to the residue to which it is connected, otherwise the accumulation is not needed. If the accumulated polarity is of imbalance, and the window has completed the search, a new window is supposed to be set with the next residual handicap as the center and continues the next round of search. If the
Page 4 of 10
accumulated polarity is still of imbalance, the windows are to be extended to 55 and continue the search by using the steps above until the windows extend to the border of the image. In the end, sticks tangent is generated by the connection between the
us
cr
ip t
residues and the boundary and then integrate along the path of the sticks tangent.
an
(a) (a) Wrapped phase map
(b)
(b) Real phase map
M
Fig. 3 The phase map 2.3 model establishment of Phase and force distribution
As Fig.4 shows, the forced tempered glass is perpendicular to axis Z. The phase will change when the light through the glass phase. Assuming the external force is F
d
(directing along the axis Y) , and the glass thickness remains unchanged. The changes
te
in density of tempered glass by the force cause the changes in refraction and hence, the phase also be changed. To analyze the relationship between F（x, y）, the force at
Ac ce p
p x, y）, the amount of phase change, set the light any point within the glass, and （
wavelength to be , the initial density of tempered glass to be ( x , y ) 0 , and
thickness to be H.
Y X Z
F Fig.4
The effects of tempered glass deformation on the phase of light waves
Page 5 of 10
The light wave phase after deformation can be expressed as （ p x, y）
2 H n( x, y )
(7)
Where n( x, y ) is the refractive index variation of the tempered glass. In this experiment, the relationship between the density variation and the refractive index is
ip t
set to be ( x, y ) n( x, y ) , in which scale factor =0.001. From Eq.(7), the following formula can be obtained. ( x, y ) 2 H
cr
p ( x, y )
(8)
us
Where ( x, y ) is the density change which caused by the force of the glass. The relationship between ( x, y ) and F ( x, y ) can be expressed as: ( x, y ) F ( x, y ) (x , y ) E
an
( x, y ) : Poisson's ratio,
M
E : Modulus of elasticity,
(9)
And F ( x, y ) has expression as below:
E F ( x, y ) （ p x, y） 0 2 H
d
(10)
te
From Eq.(10) we can find that: the relationship between F ( x, y ) and （ p x, y） is
Ac ce p
linear, so the force F ( x, y ) can be obtained as long as （ p x, y）is measured by experiment.
3. Results and conclusions
Fig.5 shows the real phase distribution of the processed glass under different
force conditions.
(a) F =5N
(b) F=10N
Page 6 of 10
ip t cr
(c) F=15N
(d)F =25N
us
Fig.5 Real phase distribution of the tempered glass with different force
an
Based on Fig.3(b), the phase distribution of any point that is along the directions of X and Y can be obtained. Fig.6 shows the phase changes at X and Y when the pixel is 200 respectively. Phase fluctuations in axis X can be seen from Fig.6 (a). Because the
M
glass parallel to axis X is unstressed in the experiment and each point is unevenly perpendicular to axis X due to the afterburner system, points in the axis X are in different density. From Fig. (8) we know that phase changes are due to the density of
d
the tempered glass resulting in the phase fluctuations. However, phase change in the
te
axis Y is relatively smooth, but with large margins. Because the biasing direction of the glass is along the axis Y only, the density is gradually increased along the axis Y,
Ac ce p
resulting in the increases of the phase change.
(a)
(b) Fig.6 The phase distribution
The parameters of the tempered glass used in this paper: Poisson's ratio 0.15 ,
Page 7 of 10
Modulus
of
elasticity
E
=7.2×107Pa,
density
ρ=2.2×103kg/m3
，
Wavelength 632.8nm . Eq.(10) shows that the forces to the tempered p x, y）are of linear relationship. Fig.6 shows the amount of glass F ( x, y ) and （ （ p x, y）when the pixel is 200, and F ( x, y ) can be solved accordingly, shown in
an
us
cr
ip t
Fig.7.
(b)
M
(a)
Fig.7 The distribution of force Fig.7 shows the force F ( x, y ) when the pixel in axis X and Y are both 200 (phase
d
p x, y） is shown in Fig.6). From Fig.6(a) and 7(a) or Fig.6(b) and 7(b) difference （
te
we can see that F ( x, y ) does not fluctuate too much along axis X (4.5N-7.5N) because the force direction is along axis Y and axis X is unstressed. Fig.7(b) shows
Ac ce p
that F ( x, y ) goes along axis Y from 2.5N to 12N and the curve looks approximately like straight lines which demonstrates that the force is distributed more evenly on axis Y.
4. Conclusion
This paper applied Mach-Zehnder interferometer method to detect the force
distribution inside the tempered glass. Using digital interference fringe image process, the phase distributions of the tempered glass is obtained, and hence the glass force distribution curve on the axis of X and Y when the pixel distribution is 200 can also be acquired. This paper has opened up a new way for the force distribution detection of the tempered glass. This technology will have broad application prospects in the field of automotive industry, and construction industry, etc.
Page 8 of 10
Reference [1] C. Tian, Y. Yang, T. Wei, T. Ling, and Y. Zhuo. Demodulation of a single-image interferogram using a Zernike-polynomial-based phase-fitting technique with a differential evolution algorithm [J]. Opt. Lett. 2011, 36(12):2318-2320.
ip t
[2] M. Tricard, A. Kulawiec,M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, P.Murphy. Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null [J]. Cirp Annals-Manufacturing
cr
Technology, 2010, 59(1):547-550.
[3] Wang Xiaokun,Zheng Ligong. A method for Testing Radius of Curvature of
us
Optical Spheric Surface[J]. Acta Optica Sinica, 2011, 31(8): 0812010.
[4] Li Jianxin, Cui Yanjun, Zhu Rihong,He Yong. Micro-Displacement Variation Lasers, 2012, 39(8): 0808002.
an
Measurement by Using Laser Interference Based on Wavelet Transform[J]. Chinese J. [5] Yang Zhenyu, Wang Ming, Xia Wei, Hao Hui, Hao Dongmei. Testing and Analysis Large-Range
Displacement
Measurement
M
of
Precision
in
Sinusoidal
Phase-Modulating Self-Mixing Interferometer[J]. Acta Optica Sinica, 2013, 33(4): 0412007.
d
[6] Song Song, Wang Xiangzhao, Wang Xuefeng,Qian Feng,Lu Hongbin.Real-Time
te
Micro-Vibration Measurement with a Laser Diode Interferometer[J]. Acta Optica Sinica, 2001, 21 (5):578-580.
Ac ce p
[7] Lü Qieni,Ge Baozhen,Cchen Yiliang,Zhang Yimo.Particle Sizing and Size Distribution Measurement of Alcohol Spray by Interferometric Particle
Imaging[J]. Chinese J. Lasers, 2011, 38(3):0308003.
[8] Wang X Z, Sasaki O, Suzuki T, M Takeo. Measurement of small vibration amplitudes of a rough surface using an interferometer with a self-pumped phase-conjugate mirror [J].
Appl.Opt .,2000, 39(25):4593-4597.
[9] V T Nquen, I L Raskovskaya, B S Rinkeychyus. The use of laser interference methods for acoustic field diagnostics [J]. Measurement Technique. 2008. 51(1):86-93. [10] Cai Huaiyu, Li Hongyue,Zhu Meng, Huang Zhanhua. Measurement of thickness uniformity for glass plate by spatial carrier[J]. Optics and Precision Engineering, 2013, 21(2):260-266. [11] V Tornari. Laser interference-based technique and applications in structural inspection of works of art[J]. Analytical and Bioanalytical Chemistry. 2007.
Page 9 of 10
387(3):761-780. [12] Chen Qianghua, Luo Huifu, Wang Sumei,Wang Feng, Gu Xinhua. Measurement of Air Refractive index Based on Surface Plasmon Resonance and Phase Detection by Dual-Frequency
Laser Interferomentry [J].Chinese J. Lasers,
ip t
2013, 40(1): 0108001. [13] K Canan, S Yani. Measurement of thin film thickness by electronic speckle pattern interferometry [J]. Opt. commun., 2004, 234(1-6):269-276.
cr
[14] T Kreis. Digital holographic interference-phase measurement using the
Ac ce p
te
d
M
an
us
Fourier-transform method [J]. J. Opt.Soc. Am. A , 1986, 3(6):847–855.
Page 10 of 10
S0030-4026(15)01317-0 http://dx.doi.org/doi:10.1016/j.ijleo.2015.09.236 IJLEO 56441
To appear in: Received date: Accepted date:
18-11-2014 26-9-2015
Please cite this article as: H. Jin-Qi, D. Yuan, L. Shu-Tao, L. Hui-Long, Y. Yong-Ji, J. Guang-Yong, L. Li-Da, Study on Force Distribution of the Tempered Glass Based on Laser Interference Technology, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.09.236 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Study on Force Distribution of the Tempered Glass Based on Laser Interference Technology He Jin-Qi1, Dong Yuan1*, Li Shu-Tao1, Liu Hui-Long1, Yu Yong-Ji1, Jin Guang-Yong1, Liu Li-Da2
(1.Chang Chun University of Science and Technology, The key laboratory of Jilin province solid-state laser technology and application, China, 130022)
*[email protected]
ip t
(2.Yuxi Industries Group Co.Ltd, Henan Nanyang, China, 473000)
cr
Abstract: The force distribution of inside the tempered glass is detected by the Mach-Zehnder interferometer. In the experiment, the interference fringes are captured
us
by a CCD camera, filtered in frequency domain, and the tempered glass real phase distribution is obtained by utilizing branch-cut algorithm for phase unwrapping. The tempered glass internal force distribution can be derived by means of the relationship
an
between phase distribution and external force. The results demonstrated that the force distribution is almost linear in direction to parallel the external pressure; although
M
some fluctuations appear to be perpendicular to the external pressure, the overall change is smaller.
Key words: Tempered Glass; Mach-Zehnder interferometer; Interference fringes; 220.4840; 220.4830; 230.0040
1. Introduction
te
OCIS
d
Phase unwrapping
Ac ce p
Since the advent of laser in the 1960s, interferometric technique presents its diversity and has achieved rapid development, among which the main techniques are laser heterodyne interferometry, laser holographic interferometry, laser speckle interferometry, single and double frequency laser interferometer, and multi-beam interferometry, etc. Due to its characteristics of non-contact and high precision, it has been extensively applied in the field of non-destructive testing. In recent years, C. Tian et al used interferometry to detect the accurate measurement of aspheric lenses and spherical radius[1-3]. Cui Yanjun and Yang Zhenyu et al used the laser interferometer to accurately test the displacement[4,5]. Song Song et al studied the measurement of small vibrations[6]. LÜ Qieni et al studied particle imaging by using laser interferometer[7]. The interferometric technique can also be applied to a wide range of measurements of the deformation , thickness , density, and refractive index, etc[8-13]. That tempered glass is widely used in daily life, for example, in the automotive
Page 1 of 10
industry and the construction industry, mainly because of its high security. In order to make more extensive application of high quality tempered glasses, security detection has to be done. In this paper, test system of digital images has been set up through the Mach - Zehnder interferometer systems and CCD image pickup device. To restore the
ip t
phase of the tempered glass, the collected interference images are digitized by spatial-carrier Fourier transform method, contributing to obtain the force distribution of the tempered glass under the external force of 20N. This article is the first
cr
application of the laser interferometer in the field of glass detection, which will open a new way for the detection of tempered glass.
us
2 Basic Theory
2.1 Basic principles of Fourier transform of Interfere images
an
By the and after meeting the interference condition, the light intensity distribution of the interference fringe image can be expressed as[14]: G ( x, y ) a ( x, y ) b( x, y ) cos( ( x, y ))
(1)
M
Where, a ( x, y ) is the background light intensity distribution of the interference fringe, b( x, y ) is the contrast of the stripes; a ( x, y ) and b( x, y ) are unknown variables;
d
( x, y ) s ( x, y ) r ( x, y ) is the phase distribution difference; s ( x, y ) is the phase
te
of the measured beams and r ( x, y ) is the phase of the reference beams. In formula (1), there are three unknown numbers and the phase difference ( x, y ) cannot be
Ac ce p
solved. Therefore, in order to solve ( x, y ) , we introduce spatial carrier frequency of f f x x f y y , in which f x and f y are the carrier frequencies in the direction of
x and y , meaning to change the inclination angle of the test surface and the reference
surface, so that the reference wave would be inclined and the fringes on the interference map would be denser in the direction of x and y .The formula of the light
intensity on the interference fringe image becomes as the follow: G ( x, y ) a ( x, y ) b( x, y ) cos[2 f ( x, y )] a ( x, y ) b( x, y ) cos[2 ( f x x f y y ) ( x, y )]
(2)
f : The spatial carrier frequency; f 1/ T , T : The space cycle of interference fringes.
The complex form of Formula ( 2 ) is,
Page 2 of 10
G ( x, y ) a ( x, y ) c( x, y ) exp(i 2 f x x i 2 f y y ) c* ( x, y ) exp(i 2 f x x i 2 f y y )] (3) 1 where c( x, y ) b( x, y ) exp(i ( x, y )) 2
(3), the following relations can be obtained:
G0 ( f1 , f 2 ) A( f1 , f 2 ) C ( f1 f x , f 2 f y ) C * ( f1 f x , f 2 f y )
ip t
Conducted the two-dimensional Fourier transform of spatial variables in formula
(4)
cr
C ( f1 f x , f 2 f y ) : the positive first-level spectrum division,
us
C * ( f1 f x , f 2 f y ) : the negative first-level spectrum division, A( f1 , f 2 ) : the zero level spectrum division.
Fig.1(a) shows the spectrum of the background light intensity of the interference
an
image.
The appropriate filter is selected, the positive first level spectrum is separated and
M
moved to the original point to obtain formula C ( f1 , f 2 ) , which is showed in Fig.1(b). Fig.2(a) and (b) are the interference fringes and Three-dimensional Fourier spectrum
d
distribution collected in the experiment.
f0
f0
te
A(f1,f2 )
Ac ce p
C*(f 1+fx ,f2+fy )
C(f1-fx,f2 -fy)
f (a)
C(f1,f2 )
(b)
f
(a) Two-dimensional Fourier spectrum (b) single Spectrum of the original one Fig.1 The action of filter A two-dimensional inverse Fourier transform is： 1 F 1[C ( f1 , f 2 )] c( x, y ) b( x, y ) exp[i ( x, y )] (5) 2
( x, y ) tan 1
Im[c( x, y )] Re[c( x, y )]
(6)
Page 3 of 10
Where, Im[c( x, y )] and Re[c( x, y )] are the respective imaginary part and the real part
us
cr
ip t
of c( x, y ) .
(a)
(b)
an
Fig.2 (a) interference fringe map of the tempered glass (b) Three -dimensional Fourier spectrum 2.2 Phase unwrapping
M
Since trigonometric is cyclical, phase is wrapped in the main value of the arc tangent function and the phase value is also extended between [-,+] when the computer is processing arc tangent function in the digital interference images. When
d
the real phase is beyond [-,+] and under the modulating effects of trigonometric,
te
wrapped phase distribution [-,+] is formed, and the image of computer processed wrapped phase maps of the tempered glass is showed in Fig.3(a). In this paper,
Ac ce p
Goldstein's branch-cut unwrapping algorithm is applied to get the real phase of the tempered glass.
Goldstein's branch cut follows a specific algorithm: search the residual handicap of
the entire wrapped phase image, establish 33 windows at the first residual handicap,
and then search the next residual handicap. Whether the searched residual is of the same polarity, it is connected to the first residue. If the two residuals are of the opposite polarity, they are called the residual polar balance of the connected branches. If they are of the same polarity, the search window continues to find a new residual handicap, and connect at the center of the windows regardless of whether the residue was connected with other residues or not. If one residual is not connected with others, the polarity would be added to the residue to which it is connected, otherwise the accumulation is not needed. If the accumulated polarity is of imbalance, and the window has completed the search, a new window is supposed to be set with the next residual handicap as the center and continues the next round of search. If the
Page 4 of 10
accumulated polarity is still of imbalance, the windows are to be extended to 55 and continue the search by using the steps above until the windows extend to the border of the image. In the end, sticks tangent is generated by the connection between the
us
cr
ip t
residues and the boundary and then integrate along the path of the sticks tangent.
an
(a) (a) Wrapped phase map
(b)
(b) Real phase map
M
Fig. 3 The phase map 2.3 model establishment of Phase and force distribution
As Fig.4 shows, the forced tempered glass is perpendicular to axis Z. The phase will change when the light through the glass phase. Assuming the external force is F
d
(directing along the axis Y) , and the glass thickness remains unchanged. The changes
te
in density of tempered glass by the force cause the changes in refraction and hence, the phase also be changed. To analyze the relationship between F（x, y）, the force at
Ac ce p
p x, y）, the amount of phase change, set the light any point within the glass, and （
wavelength to be , the initial density of tempered glass to be ( x , y ) 0 , and
thickness to be H.
Y X Z
F Fig.4
The effects of tempered glass deformation on the phase of light waves
Page 5 of 10
The light wave phase after deformation can be expressed as （ p x, y）
2 H n( x, y )
(7)
Where n( x, y ) is the refractive index variation of the tempered glass. In this experiment, the relationship between the density variation and the refractive index is
ip t
set to be ( x, y ) n( x, y ) , in which scale factor =0.001. From Eq.(7), the following formula can be obtained. ( x, y ) 2 H
cr
p ( x, y )
(8)
us
Where ( x, y ) is the density change which caused by the force of the glass. The relationship between ( x, y ) and F ( x, y ) can be expressed as: ( x, y ) F ( x, y ) (x , y ) E
an
( x, y ) : Poisson's ratio,
M
E : Modulus of elasticity,
(9)
And F ( x, y ) has expression as below:
E F ( x, y ) （ p x, y） 0 2 H
d
(10)
te
From Eq.(10) we can find that: the relationship between F ( x, y ) and （ p x, y） is
Ac ce p
linear, so the force F ( x, y ) can be obtained as long as （ p x, y）is measured by experiment.
3. Results and conclusions
Fig.5 shows the real phase distribution of the processed glass under different
force conditions.
(a) F =5N
(b) F=10N
Page 6 of 10
ip t cr
(c) F=15N
(d)F =25N
us
Fig.5 Real phase distribution of the tempered glass with different force
an
Based on Fig.3(b), the phase distribution of any point that is along the directions of X and Y can be obtained. Fig.6 shows the phase changes at X and Y when the pixel is 200 respectively. Phase fluctuations in axis X can be seen from Fig.6 (a). Because the
M
glass parallel to axis X is unstressed in the experiment and each point is unevenly perpendicular to axis X due to the afterburner system, points in the axis X are in different density. From Fig. (8) we know that phase changes are due to the density of
d
the tempered glass resulting in the phase fluctuations. However, phase change in the
te
axis Y is relatively smooth, but with large margins. Because the biasing direction of the glass is along the axis Y only, the density is gradually increased along the axis Y,
Ac ce p
resulting in the increases of the phase change.
(a)
(b) Fig.6 The phase distribution
The parameters of the tempered glass used in this paper: Poisson's ratio 0.15 ,
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Modulus
of
elasticity
E
=7.2×107Pa,
density
ρ=2.2×103kg/m3
，
Wavelength 632.8nm . Eq.(10) shows that the forces to the tempered p x, y）are of linear relationship. Fig.6 shows the amount of glass F ( x, y ) and （ （ p x, y）when the pixel is 200, and F ( x, y ) can be solved accordingly, shown in
an
us
cr
ip t
Fig.7.
(b)
M
(a)
Fig.7 The distribution of force Fig.7 shows the force F ( x, y ) when the pixel in axis X and Y are both 200 (phase
d
p x, y） is shown in Fig.6). From Fig.6(a) and 7(a) or Fig.6(b) and 7(b) difference （
te
we can see that F ( x, y ) does not fluctuate too much along axis X (4.5N-7.5N) because the force direction is along axis Y and axis X is unstressed. Fig.7(b) shows
Ac ce p
that F ( x, y ) goes along axis Y from 2.5N to 12N and the curve looks approximately like straight lines which demonstrates that the force is distributed more evenly on axis Y.
4. Conclusion
This paper applied Mach-Zehnder interferometer method to detect the force
distribution inside the tempered glass. Using digital interference fringe image process, the phase distributions of the tempered glass is obtained, and hence the glass force distribution curve on the axis of X and Y when the pixel distribution is 200 can also be acquired. This paper has opened up a new way for the force distribution detection of the tempered glass. This technology will have broad application prospects in the field of automotive industry, and construction industry, etc.
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ip t
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