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A general coherent gradient sensor for film curvature measurements: Error analysis without temperature constraint
Optics and Lasers in Engineering 51 (2013) 808–812
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
A general coherent gradient sensor for film curvature measurements: Error analysis without temperature constraint Cong Liu a,b, Xingyi Zhang a,b,n, Jun Zhou a,b, Youhe Zhou a,b a b
Key Laboratory of Mechanics on Disaster and Environment in Western China attached to the Ministry of Education of China, Lanzhou University, Lanzhou, Gansu 730000, PR China Department of Mechanics and Engineering Sciences, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 December 2012 Received in revised form 10 January 2013 Accepted 14 January 2013 Available online 26 February 2013
A general coherent gradient sensor (CGS) system without temperature constraint has been investigated for the measurement of curvatures and nonuniform curvatures changes in film-substrate systems. According to a strict mathematical derivation with the wave nature of light, systematic absolute error limit and relative error limit of the CGS system are presented at arbitrary temperature. It is found that with the decrease of the ambient temperature, the systematic relative error limit increases with a negative exponential law. & 2013 Elsevier Ltd. All rights reserved.
Keywords: CGS Thin films Temperature Error analysis
1. Introduction Thin films deposited on various types of substrates are applied in many scientific and technology fields, such as microelectronics, optoelectronics, thermal barrier coating technology, and microelectromechanical systems (MEMS), etc. Fabrication of such a filmsubstrate structure inevitably gives rise to stress in the film due to lattice mismatch, different coefficients of thermal expansion, chemical reactions, or other physical effects. There are some experimental techniques (including scanning laser method [1], multibeam optical stress sensor [2], coherent gradient sensor [3–6], and X-ray diffraction [7], etc.) for stress measurement in thin films. Compared with other methods, the coherent gradient sensor (CGS), one type of shear interferometry, has distinguished advantages, including full field, real-time, noninstrusive, noncontact, and vibration insensitivity, which is based on the observation of substrate curvature induced by this stress, and is gaining increasingly widespread use as diagnostic procedures [8–10]. According to the mismatch in thermal expansion coefficient between the film and substrate subjected to a changing temperature environment(especially high temperature), Dong et al. [11] developed the CGS system to high temperature and presented the analysis expression of the stress based on the Stoney’s formula [12] and its expansions [13–17]. In
n Corresponding author at: Department of Mechanics and Engineering Sciences, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, PR China. Tel.: þ86 931 8912447. E-mail address: [email protected] (X. Zhang).
0143-8166/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlaseng.2013.01.012
addition, CGS system is always used to investigate the deformation of crack tip and facture characteristics in the facture-mechanics [3,18–20], such as the crack tip deformation, stress intensity factor, etc. Some good results are achieved by the CGS system. However, since the CGS system is applied to measure the curvature and curvature changed in thin film–substrate structures by Rosakis et al. [5] at first, there are few reports on the error analysis of the CGS system, that is very important for an experimental measurement technique. Additionally, the film– substrate structures are gaining increasingly widespread use at low temperature [21,22], for instant, the high temperature superconducting films, which are operated at liquid nitrogen temperature at least. It is necessary to develop the CGS system in the low temperature environment and present its error analysis.
2. Error analysis of CGS system 2.1. Coherent gradient sensor Fig. 1 displays the principle of the CGS system. The surface of film is illuminated by collimated laser beam through the reflection of beam splitter (Fig. 1(a)). The reflective plane wave which includes information of deformation state of film is passed into the CGS system (Fig. 1(b)). The surface can be expressed as a function of (x,y). z ¼ f ðx,yÞ or Fðx,y,zÞ ¼ zf ðx,yÞ ¼ 0
ð1Þ
C. Liu et al. / Optics and Lasers in Engineering 51 (2013) 808–812
809
beam splitter
y x
Δ film
collimated laser beam grating G1 grating G2 filtering lens
camera filter plane
y E11 z
d
E10 d1
E1
A
E1−1
d0
E0 B E−1 C
E00
O
d−1
E01 E0−1
E−11 E−10
Δ
E−1−1 Fig. 1. Schematic of the CGS.
The normal vector of the reflective surface is equal to the following equation: N¼
f x ex f y ey þez rF ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 9rF9 1þ f þ f x
ð2Þ
At the front the second grating G2, the wave equation is displayed as the following: E1 ¼ a1 exp iðkd1 dOAþ kd1 xÞ ¼ a1 exp i k
D
gcos y þ bsin y
þ kd1 x
y
ð10Þ
The reflective plane wave propagation vector of wave plane d can be described: d ¼ ð2ez dNÞNez ¼ aex þ bey þ gez ¼ 2
2ðf x ex f y ey þ ez Þ 2
1þf x þf y
2
2
2 2
ez
ð3Þ
ð4Þ
where O 7 1 is the rotation tensor whose components are given by 2 3 1 0 0 6 8sin y 7 cos y ð5Þ ½O 7 1 ¼ 4 0 5 0 7 sin y cos y From Eqs. (3)–(5), one can gain: d 7 1 ¼ ½aex þðbcos y 8 gsin yÞey þðgcos y 7 bsin yÞez 9OA9 d1 ez ¼ 9OA9ðgcos y þ bsin yÞ ¼ D
g
E1 ¼ a1 exp iðkd1 dOC þ kd1 xÞ ¼ a1 exp i k
D
gcos ybsin y
ð11Þ
þ kd1 x
2
where a ¼ ð2f x =1 þf x þf y Þ, b ¼ ð2f y =1þ f x þ f y Þ, g ¼ ð1f x 2 2 2 f y =1 þf x þf y Þ. The propagation vector of wave plane D becomes d 1, d0, and d 1 after diffraction from grating G1. At this situation, d0 ¼ d d 7 1 ¼ O 7 1 d0
D E0 ¼ a0 exp iðkd0 dOB þkd0 xÞ ¼ a0 exp i k þ kd0 x
ð12Þ Then the second diffraction is happened after passing through the grating G2. The light becomes E(1,1), E(1,0), E(1, 1), E(0,1), E(0,0), E(0, 1), E( 1,1), E( 1,0), and E( 1,1) (Fig. 1(b)). The propagation vectors of E(1,0) and E(0,1) are equal to d1. These of E(1, 1), E(0,0) and E( 1,1) are equal to d0, and these of E(0, 1) and E( 1,0) are d 1 E(1,0), E(0,1) and E(1, 1), E(0,0), E( 1,1) and E(0, 1), E( 1,0) can make interference fringes respectively. However, only interferometric fringes of E(1,0), E(0,1) and E(0, 1), E( 1,0) which are considered. Then we can separate these situations into A and B: 2.1.1. Interference fringes made by E(1,0) and E(0,1)
ð6Þ
Eð1,0Þ ¼ a1 exp iðkd1 dOA þ kd1 xÞ ¼ a1 exp i k
D
gcos y þ bsin y
ð7Þ
9OB9 d0 ez ¼ 9OB9g ¼ D
ð8Þ
9OC9 d1 ez ¼ 9OC9ðgcos ybsin yÞ ¼ D
ð9Þ
þ kd1 x
ð13Þ D þ kd1 x Eð0,1Þ ¼ a0 exp iðkd0 dOB þkd1 xÞ ¼ a0 exp i k
g
ð14Þ
810
C. Liu et al. / Optics and Lasers in Engineering 51 (2013) 808–812
The intensity function [23] in image plane is I1, I2, respectively. I1 ¼ em/Eð1,0Þ 2 S, I2 ¼ em/Eð0,1Þ 2 S ð15Þ The intensity of E(1,0), E(0,1) together is shown as the following equation: 2 a a2 kD kD I ¼ I1 þI2 þ I12 ¼ em 1 þ 0 þ a1 a2 cos kd1 dxkd1 dx þ 2 2 g gcos y þ bsin y
which can be written as
gðcos y1Þ þ bsin y I ¼ Ic þ ema1 a2 cos kD gðgcos y þ bsin yÞ
ð16Þ
kyy
@2 f ðx,yÞ p @nðyÞ 2 2D @y @y
ð30Þ
ð17Þ
kxx
@2 f ðx,yÞ p @nðxÞ p @nðyÞ ¼ @x@y 2D @y 2D @x
ð31Þ
where Ic ¼ emða21 þ a22 =2Þ. Considering the small y, we can use approximation y sin y. The above equation becomes kDby I ¼ Ic þ ema1 a2 cos ð18Þ 2
g
Then the fringe distribution is gained while kDby
g2
¼ 2np,
n ¼ 0, 71, 72,:::
ð19Þ
2.1.2. Interference fringes made by E( 1,0) and E(0, 1) D Eð0,1Þ ¼ a0 exp iðkd0 dOB þkd1 xÞ ¼ a0 exp i k þkd1 x
g
ð20Þ Eð1,0Þ ¼ a1 exp iðkd1 dOCþ kd1 xÞ ¼ a1 exp i k
D
gcos ybsin y
þ kd1 x
ð21Þ Also we can get I ¼ I1 þ I2 þ I12 ¼ em
a21 a20 kD kD þ þ a1 a2 cos kd1 dxkd1 dx þ 2 2 g gcos ybsin y
ð22Þ which can be written as
gðcos y1Þbsin y I ¼ Ic þ ema1 a2 cos kD gðgcos ybsin yÞ
Assuming 9rf92 o o1, substituting Eqs. (24) and (25) into (26)–(28), the relationships between fringes and curvature can be showed as the followings: @2 f ðx,yÞ p @nðxÞ ð29Þ kxx 2 2D @x @x
2.2. Error analysis 2
We define Z ¼ maxð9rf 9 Þ in the whole plane, and substitute Eqs. (24) and (25) into Eqs. (26)–(28). We can get: ð1ZÞ2 p @nðxÞ ð32Þ kxx ¼ ð1 þ ZÞ3=2 2D @x kyy ¼
kxy ¼
ð1ZÞ2
3
ð1 þ ZÞ2
p @nðxÞ ð1ZÞ2 p @nðyÞ ¼ ð1 þ ZÞ3=2 2D @y ð1þ ZÞ3=2 2D @x
ð34Þ
a taylor expansion is made on the factor (1 Z)2/(1 þ Z)3/2, ð1ZÞ2 ð1 þ ZÞ3=2
7 ¼ 1 Z þ oðZ2 Þ 2
ð35Þ
Systemic absolute error limit is described as: 7 p @nðxÞ n enkxx ¼ kxx kxx ¼ Z 2 2D @x
ð36Þ
n
7 p @nðyÞ Z 2 2D @y
ð37Þ
n
7 p @nðxÞ 7 p @nðyÞ ¼ Z Z 2 2D @y 2 2D @x
ð38Þ
ð23Þ
when the principal direction is oriented coinciding with the x-axis, one can gain " # 2 ð19rf 9 Þ2 mp , m ¼ 0, 71, 72,. . . ð25Þ fx ¼ 2 2D 1þ 9rf 9
ð33Þ
ð1ZÞ2
enkyy ¼ kyy kyy ¼
where Ic ¼ emða21 þ a22 =2Þ. The same as the case A, we gain Eq. (18) once again. Then we prove the equivalence of these two situations. Substituting a, b, g of Eq. (3) into Eq. (19) and considering that k¼2p/l, p E l/y, we can get " # 2 ð19rf 9 Þ2 np fy ¼ , m ¼ 0, 71, 7 2,. . . ð24Þ 2 2D 1þ 9rf 9
p @nðyÞ 2D @y
enkxy ¼ kxy kxy ¼
and the systemic relative error limit is described as n
enrkxx ¼ enrkyy ¼ enrkxy ¼
n
n kyy kyy kxy kxy kxx kxx 7 ¼ ¼ ¼ Z n n n 2 kxx kyy kxy
ð39Þ
In this section, according to the wave nature of light, fundamental of the CGS is described by a strict mathematics treatment, one can see that (1) either case A or case B, intensity distribution on the image plane is equivalent, (2) the system error limits including absolute error limit and relative error limit are 2 obtained. Based on 9rf92 o o1, and Z ¼ maxð9rf 9 Þ, the systemic relative error limit is equal to 7/2Z.
The curvature of film can be expressed as [24] f xx f xx ffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kxx ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1þ f x þf y 1 þ 9rf 9
ð26Þ
f yy f yy kyy ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1þf x þf y 1þ 9rf 9
ð27Þ
f xy f xy kxy ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 þf x þf y 1 þ 9rf 9
ð28Þ
3. CGS for measuring film curvature at low temperature and its error analysis 3.1. Systemic correction for measuring film curvature at low temperature When measurements of the curvature of film-substrate structures are operated at low temperature, the difference of refractive index caused by temperature change should be considered. The relationship between the refractive index and temperature can be
C. Liu et al. / Optics and Lasers in Engineering 51 (2013) 808–812
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cosy2 ¼ 1siny2 ¼
z
2
reflection plane wave
1 d'
N
2
ð47Þ
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 2 2 2 n1 0 @ n2 ð1 þ 9rf 9 Þ n1 49rf 9 n1 19rf 9 A d ¼ aex þ bey þ gez ¼ dþ ez 2 n2 n2 1 þ 9rf 92 n2 ð1 þ9rf 9 Þ
n2 room temperature
interface
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 n22 ð1 þ 9rf 9 Þ2 n21 49rf 9
n2 ð1 þ 9rf 9 Þ Substituting Eqs. (41), (44) and (47)into Eq. (43):
incident plane wave
d
811
n1 low temperature
ð48Þ
a
where
2 2 ¼ ð2f ,x =1 þ f ,x þf ,y Þ
b ¼ ð2f ,y =1 þf 2,x þ f 2,y Þ
n1 =n2 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 n1 =n2 , g ¼ n22 ð1 þ 9rf 9 Þ n21 49rf 9 =n2 ð1 þ9rf 9 Þ
film
Then substitute a, b, g into Eq. (19), one can obtain a new description between gradients and fringes: 2
np n22 ð1 þ9rf 9 Þ2 4n21 9rf 9 2 2D n1 n2 ð1 þ9rf 9 Þ
fx ¼
mp n22 ð1 þ 9rf 9 Þ2 4n21 9rf 9 2 2D n1 n2 ð1 þ 9rf 9 Þ
2
film
Fig. 2. (a) Schematic of the refraction in the interface with different temperature; (b) schematic of the cryogenic chamber with a window.
n0 1 1 þ kT
ð40Þ
where k is constant of 0.00367 1C 1. The illustrate of this process is displayed in Fig. 2(a). In Fig. 2(b) a practical experiment setup with a window in which the film is placed and its temperature is arbitrary controlled is displayed. The propagation vector of the light from the film’s surface becomes d’ ¼ ð2ez dNÞNez ¼
2ðf x ex f y ey þez Þ 2
1 þf x þf y
2
ez
ð41Þ
The propagation vector changes after the light passing though the interface between the air at room temperature (the refractive index is equal to n2) and low temperature (the refractive index is equal to n1). There is a relationship according to the refraction law. n1 siny1 ¼ n2 siny2
ð42Þ
The incidence vector and refraction vector are coplanar and related by n1 0 n1 d¼ d þ cosy2 cosy1 ez ð43Þ n2 n2 where 2
0
cosy1 ¼ d ez ¼
siny1 ¼
19rf 9
ð44Þ
2
1 þ 9rf 9
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1cosy1 ¼
29rf 9 2
1þ 9rf 9
From Eq. (42) one can gain: siny2 ¼
29rf 9 n1 n2 1 þ 9rf 92
ð46Þ
ð49Þ
m ¼ 0, 71, 7 2,. . .. . .
ð50Þ
2
kyy ¼
n2 p @nðyÞ , n1 2D @y
kxy ¼
n2 p @nðyÞ n2 p @nðxÞ ¼ , n1 2D @x n1 2D @y
n ¼ 0, 7 1, 7 2,. . .
ð52Þ
n ¼ 0, 71, 72,. . .
ð53Þ
3.2. Error analysis 2
The same as Section 2.2, define Z ¼ maxð9rf 9 Þ in the whole plane, then substitute Eqs. (49) and (50) into Eqs. (26)–(28), one can see: n2 ð1 þ ZÞ2 4n21 Z p @nðxÞ ð54Þ kxx ¼ 2 2D @x n1 n2 ð1 þ ZÞ3=2 kyy ¼
kxy ¼
n22 ð1 þ ZÞ2 4n21 Z n1 n2 ð1 þ ZÞ
3=2
n22 ð1 þ ZÞ2 4n21 Z 3=2
n1 n2 ð1þ ZÞ p @nðyÞ 2D @x
p @nðyÞ 2D @y
n2 ð1þ ZÞ2 4n21 Z p @nðxÞ ¼ 2 2D @y n1 n2 ð1 þ ZÞ3=2
ð55Þ
ð56Þ
A Taylor expansion is made on the factor n22 ð1þ ZÞ2 4n21 Z2 = n1 n2 ð1 þ ZÞ3=2 , n22 ð1 þ ZÞ2 4n21 Z2 3=2
n1 n2 ð1 þ ZÞ
¼
n2 n22 8n21 þ Z þ oðZ2 Þ n1 2n1 n2
The systemic absolute error limits are described as 8n21 n22 p @nðxÞ n Z enkxx ¼ kxx kxx ¼ 2D @x 2n1 n2 n
ð45Þ
n ¼ 0, 71, 7 2,. . .. . .
The same as the above, we substitute Eqs. (49) and (50) into Eqs. (26)–(28) with assumption, The new relationship between fringes and curvature can be given as n2 p @nðxÞ n ¼ 0, 7 1, 7 2,. . .. . . ð51Þ kxx ¼ n1 2D @x
displayed as [25] nðTÞ ¼ 1 þ
2
fy ¼
enkyy ¼ kyy kyy ¼
8n21 n22 p @nðyÞ Z 2D @y 2n1 n2
8n21 n22 8n21 n22 p @nðxÞ n ¼ Z Z enkxy ¼ kxy kxy ¼ 2D @y 2n1 n2 2n1 n2 ðyÞ p @n 2D @x
ð57Þ
ð58Þ
ð59Þ
ð60Þ
812
C. Liu et al. / Optics and Lasers in Engineering 51 (2013) 808–812
2013GB110002), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (No. 201135) and the Fundamental Research Funds for the Central Universities.
References
Fig. 3. The relationship between the temperature and the relative error limit factor. In this figure, the horizontal ordinate is temperature, and the vertical ordinate is equal to the factor 4ðn21 =n22 Þ1=2, where n2 is a constant and equal to the refractive index at 300 K of the light, n1 varies with temperature change.
and systemic relative error limit is described by: n
enrkxx ¼ enrkyy ¼ enrkxy ¼
n
n kyy kyy kxy kxy kxx kxx ¼ ¼ ¼ n n n kxx kxx kxx
4
! n21 1 Z n22 2 ð61Þ
According to Eq. (61), one can see that if the CGS is used in the same medium, that is to say, n1 ¼ n2, Eq. (61) is equal to Eq. (39). The change of the relative error limit factor 4ðn21 =n22 Þ1=2 with temperature is displayed in Fig. 3. One can see that with the decrease of the ambient temperature, the systematic relative error limit increases with a negative exponential law. In addition, based on Eqs. (54)–(56), the curvature of the film, which includes the parameter n1,n2, there are valuable for the two mediums measurements even temperature is not change.
4. Conclusions The coherent gradient sensor (CGS), one type of shear interferometry, has distinguished advantages, including full field, real-time, noninstrusive, noncontact, and vibration insensitivity, which is widely used to measure curvature and curvature changes in thin film and micro-mechanical structures. According to a strict mathematic treatment with the wave nature of light, the systematic absolute error limit and relative error limit are obtained without temperature constraint. It is found that with the decrease of the ambient temperature, the relative error limit increases with a negative exponential law. On the basis of error analysis, the CGS system is expanded with arbitrary temperature and/or different mediums (with different refractive index). Our results are available in understanding the CGS system and useful in the practical applications of the CGS users.
Acknowledgments This work is supported by the Fund of Natural Science Foundation of China (No. 11102077, 11032006, 11121202, 11202089). This work is also supported by the National Key Project of Magneto-Restriction Fusion Energy Development Program (No.
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
A general coherent gradient sensor for film curvature measurements: Error analysis without temperature constraint Cong Liu a,b, Xingyi Zhang a,b,n, Jun Zhou a,b, Youhe Zhou a,b a b
Key Laboratory of Mechanics on Disaster and Environment in Western China attached to the Ministry of Education of China, Lanzhou University, Lanzhou, Gansu 730000, PR China Department of Mechanics and Engineering Sciences, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 December 2012 Received in revised form 10 January 2013 Accepted 14 January 2013 Available online 26 February 2013
A general coherent gradient sensor (CGS) system without temperature constraint has been investigated for the measurement of curvatures and nonuniform curvatures changes in film-substrate systems. According to a strict mathematical derivation with the wave nature of light, systematic absolute error limit and relative error limit of the CGS system are presented at arbitrary temperature. It is found that with the decrease of the ambient temperature, the systematic relative error limit increases with a negative exponential law. & 2013 Elsevier Ltd. All rights reserved.
Keywords: CGS Thin films Temperature Error analysis
1. Introduction Thin films deposited on various types of substrates are applied in many scientific and technology fields, such as microelectronics, optoelectronics, thermal barrier coating technology, and microelectromechanical systems (MEMS), etc. Fabrication of such a filmsubstrate structure inevitably gives rise to stress in the film due to lattice mismatch, different coefficients of thermal expansion, chemical reactions, or other physical effects. There are some experimental techniques (including scanning laser method [1], multibeam optical stress sensor [2], coherent gradient sensor [3–6], and X-ray diffraction [7], etc.) for stress measurement in thin films. Compared with other methods, the coherent gradient sensor (CGS), one type of shear interferometry, has distinguished advantages, including full field, real-time, noninstrusive, noncontact, and vibration insensitivity, which is based on the observation of substrate curvature induced by this stress, and is gaining increasingly widespread use as diagnostic procedures [8–10]. According to the mismatch in thermal expansion coefficient between the film and substrate subjected to a changing temperature environment(especially high temperature), Dong et al. [11] developed the CGS system to high temperature and presented the analysis expression of the stress based on the Stoney’s formula [12] and its expansions [13–17]. In
n Corresponding author at: Department of Mechanics and Engineering Sciences, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, PR China. Tel.: þ86 931 8912447. E-mail address: [email protected] (X. Zhang).
0143-8166/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlaseng.2013.01.012
addition, CGS system is always used to investigate the deformation of crack tip and facture characteristics in the facture-mechanics [3,18–20], such as the crack tip deformation, stress intensity factor, etc. Some good results are achieved by the CGS system. However, since the CGS system is applied to measure the curvature and curvature changed in thin film–substrate structures by Rosakis et al. [5] at first, there are few reports on the error analysis of the CGS system, that is very important for an experimental measurement technique. Additionally, the film– substrate structures are gaining increasingly widespread use at low temperature [21,22], for instant, the high temperature superconducting films, which are operated at liquid nitrogen temperature at least. It is necessary to develop the CGS system in the low temperature environment and present its error analysis.
2. Error analysis of CGS system 2.1. Coherent gradient sensor Fig. 1 displays the principle of the CGS system. The surface of film is illuminated by collimated laser beam through the reflection of beam splitter (Fig. 1(a)). The reflective plane wave which includes information of deformation state of film is passed into the CGS system (Fig. 1(b)). The surface can be expressed as a function of (x,y). z ¼ f ðx,yÞ or Fðx,y,zÞ ¼ zf ðx,yÞ ¼ 0
ð1Þ
C. Liu et al. / Optics and Lasers in Engineering 51 (2013) 808–812
809
beam splitter
y x
Δ film
collimated laser beam grating G1 grating G2 filtering lens
camera filter plane
y E11 z
d
E10 d1
E1
A
E1−1
d0
E0 B E−1 C
E00
O
d−1
E01 E0−1
E−11 E−10
Δ
E−1−1 Fig. 1. Schematic of the CGS.
The normal vector of the reflective surface is equal to the following equation: N¼
f x ex f y ey þez rF ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 9rF9 1þ f þ f x
ð2Þ
At the front the second grating G2, the wave equation is displayed as the following: E1 ¼ a1 exp iðkd1 dOAþ kd1 xÞ ¼ a1 exp i k
D
gcos y þ bsin y
þ kd1 x
y
ð10Þ
The reflective plane wave propagation vector of wave plane d can be described: d ¼ ð2ez dNÞNez ¼ aex þ bey þ gez ¼ 2
2ðf x ex f y ey þ ez Þ 2
1þf x þf y
2
2
2 2
ez
ð3Þ
ð4Þ
where O 7 1 is the rotation tensor whose components are given by 2 3 1 0 0 6 8sin y 7 cos y ð5Þ ½O 7 1 ¼ 4 0 5 0 7 sin y cos y From Eqs. (3)–(5), one can gain: d 7 1 ¼ ½aex þðbcos y 8 gsin yÞey þðgcos y 7 bsin yÞez 9OA9 d1 ez ¼ 9OA9ðgcos y þ bsin yÞ ¼ D
g
E1 ¼ a1 exp iðkd1 dOC þ kd1 xÞ ¼ a1 exp i k
D
gcos ybsin y
ð11Þ
þ kd1 x
2
where a ¼ ð2f x =1 þf x þf y Þ, b ¼ ð2f y =1þ f x þ f y Þ, g ¼ ð1f x 2 2 2 f y =1 þf x þf y Þ. The propagation vector of wave plane D becomes d 1, d0, and d 1 after diffraction from grating G1. At this situation, d0 ¼ d d 7 1 ¼ O 7 1 d0
D E0 ¼ a0 exp iðkd0 dOB þkd0 xÞ ¼ a0 exp i k þ kd0 x
ð12Þ Then the second diffraction is happened after passing through the grating G2. The light becomes E(1,1), E(1,0), E(1, 1), E(0,1), E(0,0), E(0, 1), E( 1,1), E( 1,0), and E( 1,1) (Fig. 1(b)). The propagation vectors of E(1,0) and E(0,1) are equal to d1. These of E(1, 1), E(0,0) and E( 1,1) are equal to d0, and these of E(0, 1) and E( 1,0) are d 1 E(1,0), E(0,1) and E(1, 1), E(0,0), E( 1,1) and E(0, 1), E( 1,0) can make interference fringes respectively. However, only interferometric fringes of E(1,0), E(0,1) and E(0, 1), E( 1,0) which are considered. Then we can separate these situations into A and B: 2.1.1. Interference fringes made by E(1,0) and E(0,1)
ð6Þ
Eð1,0Þ ¼ a1 exp iðkd1 dOA þ kd1 xÞ ¼ a1 exp i k
D
gcos y þ bsin y
ð7Þ
9OB9 d0 ez ¼ 9OB9g ¼ D
ð8Þ
9OC9 d1 ez ¼ 9OC9ðgcos ybsin yÞ ¼ D
ð9Þ
þ kd1 x
ð13Þ D þ kd1 x Eð0,1Þ ¼ a0 exp iðkd0 dOB þkd1 xÞ ¼ a0 exp i k
g
ð14Þ
810
C. Liu et al. / Optics and Lasers in Engineering 51 (2013) 808–812
The intensity function [23] in image plane is I1, I2, respectively. I1 ¼ em/Eð1,0Þ 2 S, I2 ¼ em/Eð0,1Þ 2 S ð15Þ The intensity of E(1,0), E(0,1) together is shown as the following equation: 2 a a2 kD kD I ¼ I1 þI2 þ I12 ¼ em 1 þ 0 þ a1 a2 cos kd1 dxkd1 dx þ 2 2 g gcos y þ bsin y
which can be written as
gðcos y1Þ þ bsin y I ¼ Ic þ ema1 a2 cos kD gðgcos y þ bsin yÞ
ð16Þ
kyy
@2 f ðx,yÞ p @nðyÞ 2 2D @y @y
ð30Þ
ð17Þ
kxx
@2 f ðx,yÞ p @nðxÞ p @nðyÞ ¼ @x@y 2D @y 2D @x
ð31Þ
where Ic ¼ emða21 þ a22 =2Þ. Considering the small y, we can use approximation y sin y. The above equation becomes kDby I ¼ Ic þ ema1 a2 cos ð18Þ 2
g
Then the fringe distribution is gained while kDby
g2
¼ 2np,
n ¼ 0, 71, 72,:::
ð19Þ
2.1.2. Interference fringes made by E( 1,0) and E(0, 1) D Eð0,1Þ ¼ a0 exp iðkd0 dOB þkd1 xÞ ¼ a0 exp i k þkd1 x
g
ð20Þ Eð1,0Þ ¼ a1 exp iðkd1 dOCþ kd1 xÞ ¼ a1 exp i k
D
gcos ybsin y
þ kd1 x
ð21Þ Also we can get I ¼ I1 þ I2 þ I12 ¼ em
a21 a20 kD kD þ þ a1 a2 cos kd1 dxkd1 dx þ 2 2 g gcos ybsin y
ð22Þ which can be written as
gðcos y1Þbsin y I ¼ Ic þ ema1 a2 cos kD gðgcos ybsin yÞ
Assuming 9rf92 o o1, substituting Eqs. (24) and (25) into (26)–(28), the relationships between fringes and curvature can be showed as the followings: @2 f ðx,yÞ p @nðxÞ ð29Þ kxx 2 2D @x @x
2.2. Error analysis 2
We define Z ¼ maxð9rf 9 Þ in the whole plane, and substitute Eqs. (24) and (25) into Eqs. (26)–(28). We can get: ð1ZÞ2 p @nðxÞ ð32Þ kxx ¼ ð1 þ ZÞ3=2 2D @x kyy ¼
kxy ¼
ð1ZÞ2
3
ð1 þ ZÞ2
p @nðxÞ ð1ZÞ2 p @nðyÞ ¼ ð1 þ ZÞ3=2 2D @y ð1þ ZÞ3=2 2D @x
ð34Þ
a taylor expansion is made on the factor (1 Z)2/(1 þ Z)3/2, ð1ZÞ2 ð1 þ ZÞ3=2
7 ¼ 1 Z þ oðZ2 Þ 2
ð35Þ
Systemic absolute error limit is described as: 7 p @nðxÞ n enkxx ¼ kxx kxx ¼ Z 2 2D @x
ð36Þ
n
7 p @nðyÞ Z 2 2D @y
ð37Þ
n
7 p @nðxÞ 7 p @nðyÞ ¼ Z Z 2 2D @y 2 2D @x
ð38Þ
ð23Þ
when the principal direction is oriented coinciding with the x-axis, one can gain " # 2 ð19rf 9 Þ2 mp , m ¼ 0, 71, 72,. . . ð25Þ fx ¼ 2 2D 1þ 9rf 9
ð33Þ
ð1ZÞ2
enkyy ¼ kyy kyy ¼
where Ic ¼ emða21 þ a22 =2Þ. The same as the case A, we gain Eq. (18) once again. Then we prove the equivalence of these two situations. Substituting a, b, g of Eq. (3) into Eq. (19) and considering that k¼2p/l, p E l/y, we can get " # 2 ð19rf 9 Þ2 np fy ¼ , m ¼ 0, 71, 7 2,. . . ð24Þ 2 2D 1þ 9rf 9
p @nðyÞ 2D @y
enkxy ¼ kxy kxy ¼
and the systemic relative error limit is described as n
enrkxx ¼ enrkyy ¼ enrkxy ¼
n
n kyy kyy kxy kxy kxx kxx 7 ¼ ¼ ¼ Z n n n 2 kxx kyy kxy
ð39Þ
In this section, according to the wave nature of light, fundamental of the CGS is described by a strict mathematics treatment, one can see that (1) either case A or case B, intensity distribution on the image plane is equivalent, (2) the system error limits including absolute error limit and relative error limit are 2 obtained. Based on 9rf92 o o1, and Z ¼ maxð9rf 9 Þ, the systemic relative error limit is equal to 7/2Z.
The curvature of film can be expressed as [24] f xx f xx ffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kxx ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1þ f x þf y 1 þ 9rf 9
ð26Þ
f yy f yy kyy ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1þf x þf y 1þ 9rf 9
ð27Þ
f xy f xy kxy ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 þf x þf y 1 þ 9rf 9
ð28Þ
3. CGS for measuring film curvature at low temperature and its error analysis 3.1. Systemic correction for measuring film curvature at low temperature When measurements of the curvature of film-substrate structures are operated at low temperature, the difference of refractive index caused by temperature change should be considered. The relationship between the refractive index and temperature can be
C. Liu et al. / Optics and Lasers in Engineering 51 (2013) 808–812
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cosy2 ¼ 1siny2 ¼
z
2
reflection plane wave
1 d'
N
2
ð47Þ
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 2 2 2 n1 0 @ n2 ð1 þ 9rf 9 Þ n1 49rf 9 n1 19rf 9 A d ¼ aex þ bey þ gez ¼ dþ ez 2 n2 n2 1 þ 9rf 92 n2 ð1 þ9rf 9 Þ
n2 room temperature
interface
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 n22 ð1 þ 9rf 9 Þ2 n21 49rf 9
n2 ð1 þ 9rf 9 Þ Substituting Eqs. (41), (44) and (47)into Eq. (43):
incident plane wave
d
811
n1 low temperature
ð48Þ
a
where
2 2 ¼ ð2f ,x =1 þ f ,x þf ,y Þ
b ¼ ð2f ,y =1 þf 2,x þ f 2,y Þ
n1 =n2 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 n1 =n2 , g ¼ n22 ð1 þ 9rf 9 Þ n21 49rf 9 =n2 ð1 þ9rf 9 Þ
film
Then substitute a, b, g into Eq. (19), one can obtain a new description between gradients and fringes: 2
np n22 ð1 þ9rf 9 Þ2 4n21 9rf 9 2 2D n1 n2 ð1 þ9rf 9 Þ
fx ¼
mp n22 ð1 þ 9rf 9 Þ2 4n21 9rf 9 2 2D n1 n2 ð1 þ 9rf 9 Þ
2
film
Fig. 2. (a) Schematic of the refraction in the interface with different temperature; (b) schematic of the cryogenic chamber with a window.
n0 1 1 þ kT
ð40Þ
where k is constant of 0.00367 1C 1. The illustrate of this process is displayed in Fig. 2(a). In Fig. 2(b) a practical experiment setup with a window in which the film is placed and its temperature is arbitrary controlled is displayed. The propagation vector of the light from the film’s surface becomes d’ ¼ ð2ez dNÞNez ¼
2ðf x ex f y ey þez Þ 2
1 þf x þf y
2
ez
ð41Þ
The propagation vector changes after the light passing though the interface between the air at room temperature (the refractive index is equal to n2) and low temperature (the refractive index is equal to n1). There is a relationship according to the refraction law. n1 siny1 ¼ n2 siny2
ð42Þ
The incidence vector and refraction vector are coplanar and related by n1 0 n1 d¼ d þ cosy2 cosy1 ez ð43Þ n2 n2 where 2
0
cosy1 ¼ d ez ¼
siny1 ¼
19rf 9
ð44Þ
2
1 þ 9rf 9
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1cosy1 ¼
29rf 9 2
1þ 9rf 9
From Eq. (42) one can gain: siny2 ¼
29rf 9 n1 n2 1 þ 9rf 92
ð46Þ
ð49Þ
m ¼ 0, 71, 7 2,. . .. . .
ð50Þ
2
kyy ¼
n2 p @nðyÞ , n1 2D @y
kxy ¼
n2 p @nðyÞ n2 p @nðxÞ ¼ , n1 2D @x n1 2D @y
n ¼ 0, 7 1, 7 2,. . .
ð52Þ
n ¼ 0, 71, 72,. . .
ð53Þ
3.2. Error analysis 2
The same as Section 2.2, define Z ¼ maxð9rf 9 Þ in the whole plane, then substitute Eqs. (49) and (50) into Eqs. (26)–(28), one can see: n2 ð1 þ ZÞ2 4n21 Z p @nðxÞ ð54Þ kxx ¼ 2 2D @x n1 n2 ð1 þ ZÞ3=2 kyy ¼
kxy ¼
n22 ð1 þ ZÞ2 4n21 Z n1 n2 ð1 þ ZÞ
3=2
n22 ð1 þ ZÞ2 4n21 Z 3=2
n1 n2 ð1þ ZÞ p @nðyÞ 2D @x
p @nðyÞ 2D @y
n2 ð1þ ZÞ2 4n21 Z p @nðxÞ ¼ 2 2D @y n1 n2 ð1 þ ZÞ3=2
ð55Þ
ð56Þ
A Taylor expansion is made on the factor n22 ð1þ ZÞ2 4n21 Z2 = n1 n2 ð1 þ ZÞ3=2 , n22 ð1 þ ZÞ2 4n21 Z2 3=2
n1 n2 ð1 þ ZÞ
¼
n2 n22 8n21 þ Z þ oðZ2 Þ n1 2n1 n2
The systemic absolute error limits are described as 8n21 n22 p @nðxÞ n Z enkxx ¼ kxx kxx ¼ 2D @x 2n1 n2 n
ð45Þ
n ¼ 0, 71, 7 2,. . .. . .
The same as the above, we substitute Eqs. (49) and (50) into Eqs. (26)–(28) with assumption, The new relationship between fringes and curvature can be given as n2 p @nðxÞ n ¼ 0, 7 1, 7 2,. . .. . . ð51Þ kxx ¼ n1 2D @x
displayed as [25] nðTÞ ¼ 1 þ
2
fy ¼
enkyy ¼ kyy kyy ¼
8n21 n22 p @nðyÞ Z 2D @y 2n1 n2
8n21 n22 8n21 n22 p @nðxÞ n ¼ Z Z enkxy ¼ kxy kxy ¼ 2D @y 2n1 n2 2n1 n2 ðyÞ p @n 2D @x
ð57Þ
ð58Þ
ð59Þ
ð60Þ
812
C. Liu et al. / Optics and Lasers in Engineering 51 (2013) 808–812
2013GB110002), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (No. 201135) and the Fundamental Research Funds for the Central Universities.
References
Fig. 3. The relationship between the temperature and the relative error limit factor. In this figure, the horizontal ordinate is temperature, and the vertical ordinate is equal to the factor 4ðn21 =n22 Þ1=2, where n2 is a constant and equal to the refractive index at 300 K of the light, n1 varies with temperature change.
and systemic relative error limit is described by: n
enrkxx ¼ enrkyy ¼ enrkxy ¼
n
n kyy kyy kxy kxy kxx kxx ¼ ¼ ¼ n n n kxx kxx kxx
4
! n21 1 Z n22 2 ð61Þ
According to Eq. (61), one can see that if the CGS is used in the same medium, that is to say, n1 ¼ n2, Eq. (61) is equal to Eq. (39). The change of the relative error limit factor 4ðn21 =n22 Þ1=2 with temperature is displayed in Fig. 3. One can see that with the decrease of the ambient temperature, the systematic relative error limit increases with a negative exponential law. In addition, based on Eqs. (54)–(56), the curvature of the film, which includes the parameter n1,n2, there are valuable for the two mediums measurements even temperature is not change.
4. Conclusions The coherent gradient sensor (CGS), one type of shear interferometry, has distinguished advantages, including full field, real-time, noninstrusive, noncontact, and vibration insensitivity, which is widely used to measure curvature and curvature changes in thin film and micro-mechanical structures. According to a strict mathematic treatment with the wave nature of light, the systematic absolute error limit and relative error limit are obtained without temperature constraint. It is found that with the decrease of the ambient temperature, the relative error limit increases with a negative exponential law. On the basis of error analysis, the CGS system is expanded with arbitrary temperature and/or different mediums (with different refractive index). Our results are available in understanding the CGS system and useful in the practical applications of the CGS users.
Acknowledgments This work is supported by the Fund of Natural Science Foundation of China (No. 11102077, 11032006, 11121202, 11202089). This work is also supported by the National Key Project of Magneto-Restriction Fusion Energy Development Program (No.
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