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Error analysis of phase shifting by varying the incident angle of parallel beams in shadow Moiré
Optik 124 (2013) 6769–6771
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Error analysis of phase shifting by varying the incident angle of parallel beams in shadow Moiré Junwu Mu, Zhidong Guan, Jun Kang, Tianya Bian, Fei Su ∗ School of Aerospace Science and Engineering, Beihang University, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 5 January 2013 Accepted 22 May 2013
Keywords: Shadow Moiré Phase shifting Incidence angle
a b s t r a c t A phase shifting technique by varying the incidence angle of parallel beams in a shadow Moiré has been demonstrated in the present investigation. Error of phase shifting with this technique is evaluated, since it is not a stringent technique in theory. Furthermore measures in order to increase its accuracy have been suggested. Feasibility of the proposed phase shifting technique has been verified with experiments. It was found that the difference between the recommended and conventional phase shifting techniques could be significantly optimized to be less than 5%. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Shadow Moiré method has widely been employed on warpage or 3-D contour measurement. In the field of failure analysis and reliability evaluation of electronic packages, warpage of large size specimen like Block BGA and its variation with temperature are often evaluated by shadow Moiré method [1,2]. In these applications, samples are usually heated in an oven up to 300 ◦ C. It is well known, nevertheless, that the resolution of shadow Moiré method is highly dependant on the grating pitch, usually not less than 20 m. As such, in order to improve its measurement resolution further more and to process the fringe patterns automatically, a phase shifting technique has to be introduced [3]. In convention, phase shifting of shadow Moiré can be executed in two different ways, either by moving the grating within its planar in the direction perpendicular to the grating lines, or by moving the grating in the normal direction of the grating plane [4,5]. In both cases the movement of grating is only a fraction of the grating pitch and is usually controlled by a step motor that is located nearby the grating. In case that warpage of specimen during high temperature thermal cycle is measured, this kind of phase shifting technique will be difficult to implement due to the consequences of elevated temperature on the control system of the step motor. It will be a great improvement that the phase of shadow Moiré pattern can be shifted, while keeping the position of grating unchanged relative to the specimen. Many complicated and expensive mechanical and electrical instruments, meanwhile, can be avoided for the execution of conventional phase
∗ Corresponding author. Tel.: +86 010 82317508. E-mail address: [email protected] (F. Su). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.05.073
shifting technique. Based on the widely adopted configuration of shadow Moiré method shown in Fig. 1, a new methodology has been recommended to implement phase shifting by varying the incidence angle ␣ of parallel beams. A step motor was attached to the mirror for precise control of the incidence angle. In principle, the phase of shadow Moiré pattern will not be uniformly shifted within this method except in case the specimen surface is unexceptionally flat. Strictly speaking it is not an ideally accurate method for phase shifting. However, due to its user-friendly maneuverability and potential application on high temperature measurement, it is definitely worthwhile to investigate the feasibility of this method and obtain guidance for its application accordingly. 2. Error analysis of phase shifting by changing the incidence angle The intensity distribution of Moiré pattern is assumed to be sinusoidal. The error of phase shifting induced by non-sinusoidal intensity distribution has already been analyzed in Ref. [6] and thus it would not be considered in this paper. Here the so-called ‘error’ is specifically referred to the difference between the phase maps obtained with conventional and new phase shifting techniques respectively. To apply this technique, the step of mirror rotation angle for phase shifting should be defined first. With the set up of optics shown in Fig. 1, a sparse grating with pitch p0 and its shadow on the planar platform can be imaged and monitored by a CCD. If the shadow of one grating line is moved by a pitch to the adjacent position, i.e., phase is shifted by 2, by rotating the mirror with an angle ˇ, then for a Moiré pattern generated by the interference of a grating with pitch p and its distorted shadow, phase is supposed to shift
6770
J. Mu et al. / Optik 124 (2013) 6769–6771
Fig. 1. Configuration of the shadow Moiré system.
by ␥ when mirror is rotated by ˇp␥/2p0. Fig. 2 illustrates the principle of systematic error of this technique, where the solid lines stand for the rays before phase shifting (˛ = 45◦ ), and the dashed lines represent the tilted rays due to mirror rotation to implement phase shifting angle 2. It can be seen that the dashed lines cannot hit on the point where the solid lines intercept the specimen surface. Thus accurate phase shifting cannot be realized for specimen with non-planar surface. Systematic error will increase with the specimen curving. In order to have a deep understanding of the systematic error and to apply this potential technique efficiently, part of Fig. 2(a) around the highest point A was enlarged in Fig. 2(b) for a detailed error analysis. From Fig. 2(b) it can be seen that for parabolic type of specimen surface, the maximum error of 2 phase shifting will occur at point A, and according to the principle of geometry, it can be expressed as 2 =
AD zmax 2 2 = p z0
(1)
where zmax is the maximum height of specimen in reference to the platform, and Z0 is the distance between the platform and grating. In the conventional 4-step phase shifting technique, the phase is shifted /2 in each step, correspondingly, the mirror rotation step should be ˇ p/4p0 , and the maximum phase shifting error at
Fig. 3. Experimental verification of the proposed phase shifting technique, (a)–(d), the 4 fringe patterns with phased shifted by /2 in each step, (e)–(f) measurement results with the aid of new and conventional technique, respectively, (g) difference between the two measurements.
point A in each step is = /2 = 2 /4. The intensity of the four phase–shifted patterns at point A can be expressed as I0 = a(x, y) + b(x, y) sin[ϕ(x, y) + ],
I1 = a(x, y) + b(x, y) sin ϕ(x, y) +
+ , 2
I2 = a(x, y) + b(x, y) sin[ϕ(x, y) + + ],
I3 = a(x, y) + b(x, y) sin ϕ(x, y) +
(2)
3 + 2
where ϕ is the exact phase of Moiré pattern at point A. With phase shifting error included in each step, the phase of Moiré pattern at point A can be calculated, ϕ (x, y) = tan−1
I − I 0 2 I3 − I1
= tan−1
sin(ϕ + ) cos(ϕ + 2)
(3)
So the systematic error of phase determination at point A is ϕ(x, y) = |ϕ − ϕ| = | tan−1
Fig. 2. Illustration of error source of the proposed phase shifting technique (a) and its partial enlargement for maximum error analysis.
sin(ϕ + ) cos(ϕ + 2)
− ϕ|
(4)
If 2 = /4 is the maximum tolerable phase calculation error at point A, then in Eq. (4) will be /16. For this purpose, Z0 should be no less than 8zmax according to Eq. (1). By substituting ϕ with different values ranging from − to into Eq. (4) and employing the tool of mathematics, the maximum ϕ is found to be /8. So if height of specimen is larger than the grating pitch p, i.e., phase variation of Moiré pattern is larger than 2, but less than Z0 /8, then the relative measurement error will be less than (/8)/2 × 100% = 6.1%. This estimation is based on the wrapped phase map, and it is also valid for the unwrapped case. From the above analysis it can be concluded that for a given measurement error, the maximum allowable warpage or height
J. Mu et al. / Optik 124 (2013) 6769–6771
6771
4. Conclusions In this paper, a new technique by varying the incidence angle of beam to realize phase shifting in shadow Moiré method has been demonstrated for warpage measurement of specimen at high temperatures. Although accurate and uniform phase shifting cannot be implemented with this technique, measurement error can be mitigated well to a small scope as long as the configuration of shadow Moiré system matches the warpage or height of the specimen. Based on the experimental measurement, the accuracy of the recommended methodology is critically dependent on the following steps.
Fig. 4. Experimental verification of the proposed phase shifting technique, (a) Moiré patterns of the specimen and the selected region for error analysis. (b)–(c) measurement results of phase map with the aid of conventional and new technique, respectively, (d) difference between the two measurements.
(zmax ) of specimen is related to the experimental setup but can be controlled flexibly. For example it has been shown that measurement error is less than 6.1% if the maximum warpage or height of specimen falls within the scope of [p, Z0 /8]. In the case that p = 0.2 mm and Z0 = 16p = 3.2 mm, for example, the scope is [0.2 mm, 0.4 mm], while in the case that p = 0.5 mm and Z0 = 16p = 8 mm, correspondingly the scope increases to [0.5 mm, 1 mm]. These data are very practical in the warpage measurement of large size electronic packages, as demonstrated in Ref. [1].
(A) Estimate the maximum warpage or height of specimen zmax and hold it on the platform. (B) Adjust the distance between the platform and the grating Z0 . Make sure that Z0 ≥ 8zmax , since a large Z0 is beneficial to reducing the error. Take care that it may also reduce the contrast of shadow Moiré pattern, thus a compromise should be achieved. (C) Choose grating with pitch p, which should be smaller than zmax. Although fine grating is helpful to improve testing resolution, contrast of fringe pattern will be affected in case Z0 is large. Following the experiences accumulated in the present investigation, grating with a pitch around zmax /3–zmax /4 is ideal for application. With the above critical steps taken, an experimental verification of the proposed phase shifting technique was performed. By comparing the testing results with those of conventional phase-shifting technique, it was found that the maximum relative error is less than 5%. The proposed phase shifting technique is thus demonstrated to be applicable to most engineering applications.
3. Experimental verification
Acknowledgements
To demonstrate the feasibility of the proposed phase shifting technique, an experiment has been carried out. The specimen was a paper dish with a diameter of 70 mm and a height of 3.5 mm. The dish was placed on the platform with its bottom facing the CCD, and the grating pitch is p = 1 mm with a distance between platform and grating at about 16 mm. The 3D contour of specimen was measured with the aid of conventional and new recommended phase shifting technique, respectively. Position and orientation of specimen on the platform was kept unchanged during the two measurements. The grating lines appeared in the images of Moiré pattern usually cause side effects to the calculation and unwrapping of phase map. This problem has been solved by de-focusing the CCD lens slightly. The phasemaps obtained with conventional and new phase shifting techniques are unwrapped with the algorithm described in Ref. [7], with a same reference point chosen in the respective phase unwrapping procedure. The measurement results of 3D contour are compared in Figs. 3 and 4. It is found that the maximum difference between the two measurement results is about 5%, which confirms the aforementioned error analysis quite well.
The authors would like to thank National Basic Research and Development Program of China very much. References [1] Y. Tee, S. Yi, L.X. Shen, et al., Comprehensive numerical and experimental analysis of matrix TFBGA warpage, J. SMT Vol. 17 (2) (2004) 11–16. [2] Y.Y. Wang, P. Hassell, Measurement of thermally induced warpage of bga packages/substrates using phase-stepping shadow Moiré, in: IEEE/CPMT Electronic Packaging Technology Conference, 1997, pp. 283–289. [3] D. Post, B. Han, P.G. Ifju, Moiré Methods for Engineering and Science — Moiré Interferometry and Shadow Moiré, Book chapter of “Topics in applied physics”, Spinger, 2000. [4] H.B. Du, H. Zhao, B. Li, Z.W. Li, et al., Algorithm for phase shifting shadow Moiré with an unknown relative step, J. Opt. 13 (2011) 1–5. [5] L.H. Jin, Y. Kodera, T. Yoshizawa, Shadow Moiré profilometry using the phaseshifting method, Opt. Eng. 39 (8) (2000) 2119–2123. [6] C. Han, B. Han, Error analysis of the phase-shifting technique when applied to shadow Moiré, Appl. Opt. 45 (6) (2006) 1124–1233. [7] F. Su, S. Yi, K.S. Chian, A simple method to unwrap the geometrically discontinuous phase map and its application in the measurement of IC package, Opt. Laser Eng. 41 (2004) 463–473.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Error analysis of phase shifting by varying the incident angle of parallel beams in shadow Moiré Junwu Mu, Zhidong Guan, Jun Kang, Tianya Bian, Fei Su ∗ School of Aerospace Science and Engineering, Beihang University, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 5 January 2013 Accepted 22 May 2013
Keywords: Shadow Moiré Phase shifting Incidence angle
a b s t r a c t A phase shifting technique by varying the incidence angle of parallel beams in a shadow Moiré has been demonstrated in the present investigation. Error of phase shifting with this technique is evaluated, since it is not a stringent technique in theory. Furthermore measures in order to increase its accuracy have been suggested. Feasibility of the proposed phase shifting technique has been verified with experiments. It was found that the difference between the recommended and conventional phase shifting techniques could be significantly optimized to be less than 5%. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Shadow Moiré method has widely been employed on warpage or 3-D contour measurement. In the field of failure analysis and reliability evaluation of electronic packages, warpage of large size specimen like Block BGA and its variation with temperature are often evaluated by shadow Moiré method [1,2]. In these applications, samples are usually heated in an oven up to 300 ◦ C. It is well known, nevertheless, that the resolution of shadow Moiré method is highly dependant on the grating pitch, usually not less than 20 m. As such, in order to improve its measurement resolution further more and to process the fringe patterns automatically, a phase shifting technique has to be introduced [3]. In convention, phase shifting of shadow Moiré can be executed in two different ways, either by moving the grating within its planar in the direction perpendicular to the grating lines, or by moving the grating in the normal direction of the grating plane [4,5]. In both cases the movement of grating is only a fraction of the grating pitch and is usually controlled by a step motor that is located nearby the grating. In case that warpage of specimen during high temperature thermal cycle is measured, this kind of phase shifting technique will be difficult to implement due to the consequences of elevated temperature on the control system of the step motor. It will be a great improvement that the phase of shadow Moiré pattern can be shifted, while keeping the position of grating unchanged relative to the specimen. Many complicated and expensive mechanical and electrical instruments, meanwhile, can be avoided for the execution of conventional phase
∗ Corresponding author. Tel.: +86 010 82317508. E-mail address: [email protected] (F. Su). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.05.073
shifting technique. Based on the widely adopted configuration of shadow Moiré method shown in Fig. 1, a new methodology has been recommended to implement phase shifting by varying the incidence angle ␣ of parallel beams. A step motor was attached to the mirror for precise control of the incidence angle. In principle, the phase of shadow Moiré pattern will not be uniformly shifted within this method except in case the specimen surface is unexceptionally flat. Strictly speaking it is not an ideally accurate method for phase shifting. However, due to its user-friendly maneuverability and potential application on high temperature measurement, it is definitely worthwhile to investigate the feasibility of this method and obtain guidance for its application accordingly. 2. Error analysis of phase shifting by changing the incidence angle The intensity distribution of Moiré pattern is assumed to be sinusoidal. The error of phase shifting induced by non-sinusoidal intensity distribution has already been analyzed in Ref. [6] and thus it would not be considered in this paper. Here the so-called ‘error’ is specifically referred to the difference between the phase maps obtained with conventional and new phase shifting techniques respectively. To apply this technique, the step of mirror rotation angle for phase shifting should be defined first. With the set up of optics shown in Fig. 1, a sparse grating with pitch p0 and its shadow on the planar platform can be imaged and monitored by a CCD. If the shadow of one grating line is moved by a pitch to the adjacent position, i.e., phase is shifted by 2, by rotating the mirror with an angle ˇ, then for a Moiré pattern generated by the interference of a grating with pitch p and its distorted shadow, phase is supposed to shift
6770
J. Mu et al. / Optik 124 (2013) 6769–6771
Fig. 1. Configuration of the shadow Moiré system.
by ␥ when mirror is rotated by ˇp␥/2p0. Fig. 2 illustrates the principle of systematic error of this technique, where the solid lines stand for the rays before phase shifting (˛ = 45◦ ), and the dashed lines represent the tilted rays due to mirror rotation to implement phase shifting angle 2. It can be seen that the dashed lines cannot hit on the point where the solid lines intercept the specimen surface. Thus accurate phase shifting cannot be realized for specimen with non-planar surface. Systematic error will increase with the specimen curving. In order to have a deep understanding of the systematic error and to apply this potential technique efficiently, part of Fig. 2(a) around the highest point A was enlarged in Fig. 2(b) for a detailed error analysis. From Fig. 2(b) it can be seen that for parabolic type of specimen surface, the maximum error of 2 phase shifting will occur at point A, and according to the principle of geometry, it can be expressed as 2 =
AD zmax 2 2 = p z0
(1)
where zmax is the maximum height of specimen in reference to the platform, and Z0 is the distance between the platform and grating. In the conventional 4-step phase shifting technique, the phase is shifted /2 in each step, correspondingly, the mirror rotation step should be ˇ p/4p0 , and the maximum phase shifting error at
Fig. 3. Experimental verification of the proposed phase shifting technique, (a)–(d), the 4 fringe patterns with phased shifted by /2 in each step, (e)–(f) measurement results with the aid of new and conventional technique, respectively, (g) difference between the two measurements.
point A in each step is = /2 = 2 /4. The intensity of the four phase–shifted patterns at point A can be expressed as I0 = a(x, y) + b(x, y) sin[ϕ(x, y) + ],
I1 = a(x, y) + b(x, y) sin ϕ(x, y) +
+ , 2
I2 = a(x, y) + b(x, y) sin[ϕ(x, y) + + ],
I3 = a(x, y) + b(x, y) sin ϕ(x, y) +
(2)
3 + 2
where ϕ is the exact phase of Moiré pattern at point A. With phase shifting error included in each step, the phase of Moiré pattern at point A can be calculated, ϕ (x, y) = tan−1
I − I 0 2 I3 − I1
= tan−1
sin(ϕ + ) cos(ϕ + 2)
(3)
So the systematic error of phase determination at point A is ϕ(x, y) = |ϕ − ϕ| = | tan−1
Fig. 2. Illustration of error source of the proposed phase shifting technique (a) and its partial enlargement for maximum error analysis.
sin(ϕ + ) cos(ϕ + 2)
− ϕ|
(4)
If 2 = /4 is the maximum tolerable phase calculation error at point A, then in Eq. (4) will be /16. For this purpose, Z0 should be no less than 8zmax according to Eq. (1). By substituting ϕ with different values ranging from − to into Eq. (4) and employing the tool of mathematics, the maximum ϕ is found to be /8. So if height of specimen is larger than the grating pitch p, i.e., phase variation of Moiré pattern is larger than 2, but less than Z0 /8, then the relative measurement error will be less than (/8)/2 × 100% = 6.1%. This estimation is based on the wrapped phase map, and it is also valid for the unwrapped case. From the above analysis it can be concluded that for a given measurement error, the maximum allowable warpage or height
J. Mu et al. / Optik 124 (2013) 6769–6771
6771
4. Conclusions In this paper, a new technique by varying the incidence angle of beam to realize phase shifting in shadow Moiré method has been demonstrated for warpage measurement of specimen at high temperatures. Although accurate and uniform phase shifting cannot be implemented with this technique, measurement error can be mitigated well to a small scope as long as the configuration of shadow Moiré system matches the warpage or height of the specimen. Based on the experimental measurement, the accuracy of the recommended methodology is critically dependent on the following steps.
Fig. 4. Experimental verification of the proposed phase shifting technique, (a) Moiré patterns of the specimen and the selected region for error analysis. (b)–(c) measurement results of phase map with the aid of conventional and new technique, respectively, (d) difference between the two measurements.
(zmax ) of specimen is related to the experimental setup but can be controlled flexibly. For example it has been shown that measurement error is less than 6.1% if the maximum warpage or height of specimen falls within the scope of [p, Z0 /8]. In the case that p = 0.2 mm and Z0 = 16p = 3.2 mm, for example, the scope is [0.2 mm, 0.4 mm], while in the case that p = 0.5 mm and Z0 = 16p = 8 mm, correspondingly the scope increases to [0.5 mm, 1 mm]. These data are very practical in the warpage measurement of large size electronic packages, as demonstrated in Ref. [1].
(A) Estimate the maximum warpage or height of specimen zmax and hold it on the platform. (B) Adjust the distance between the platform and the grating Z0 . Make sure that Z0 ≥ 8zmax , since a large Z0 is beneficial to reducing the error. Take care that it may also reduce the contrast of shadow Moiré pattern, thus a compromise should be achieved. (C) Choose grating with pitch p, which should be smaller than zmax. Although fine grating is helpful to improve testing resolution, contrast of fringe pattern will be affected in case Z0 is large. Following the experiences accumulated in the present investigation, grating with a pitch around zmax /3–zmax /4 is ideal for application. With the above critical steps taken, an experimental verification of the proposed phase shifting technique was performed. By comparing the testing results with those of conventional phase-shifting technique, it was found that the maximum relative error is less than 5%. The proposed phase shifting technique is thus demonstrated to be applicable to most engineering applications.
3. Experimental verification
Acknowledgements
To demonstrate the feasibility of the proposed phase shifting technique, an experiment has been carried out. The specimen was a paper dish with a diameter of 70 mm and a height of 3.5 mm. The dish was placed on the platform with its bottom facing the CCD, and the grating pitch is p = 1 mm with a distance between platform and grating at about 16 mm. The 3D contour of specimen was measured with the aid of conventional and new recommended phase shifting technique, respectively. Position and orientation of specimen on the platform was kept unchanged during the two measurements. The grating lines appeared in the images of Moiré pattern usually cause side effects to the calculation and unwrapping of phase map. This problem has been solved by de-focusing the CCD lens slightly. The phasemaps obtained with conventional and new phase shifting techniques are unwrapped with the algorithm described in Ref. [7], with a same reference point chosen in the respective phase unwrapping procedure. The measurement results of 3D contour are compared in Figs. 3 and 4. It is found that the maximum difference between the two measurement results is about 5%, which confirms the aforementioned error analysis quite well.
The authors would like to thank National Basic Research and Development Program of China very much. References [1] Y. Tee, S. Yi, L.X. Shen, et al., Comprehensive numerical and experimental analysis of matrix TFBGA warpage, J. SMT Vol. 17 (2) (2004) 11–16. [2] Y.Y. Wang, P. Hassell, Measurement of thermally induced warpage of bga packages/substrates using phase-stepping shadow Moiré, in: IEEE/CPMT Electronic Packaging Technology Conference, 1997, pp. 283–289. [3] D. Post, B. Han, P.G. Ifju, Moiré Methods for Engineering and Science — Moiré Interferometry and Shadow Moiré, Book chapter of “Topics in applied physics”, Spinger, 2000. [4] H.B. Du, H. Zhao, B. Li, Z.W. Li, et al., Algorithm for phase shifting shadow Moiré with an unknown relative step, J. Opt. 13 (2011) 1–5. [5] L.H. Jin, Y. Kodera, T. Yoshizawa, Shadow Moiré profilometry using the phaseshifting method, Opt. Eng. 39 (8) (2000) 2119–2123. [6] C. Han, B. Han, Error analysis of the phase-shifting technique when applied to shadow Moiré, Appl. Opt. 45 (6) (2006) 1124–1233. [7] F. Su, S. Yi, K.S. Chian, A simple method to unwrap the geometrically discontinuous phase map and its application in the measurement of IC package, Opt. Laser Eng. 41 (2004) 463–473.