The coherent gradient sensor for film curvature measurements at cryogenic temperature

Cong Liu; Xingyi Zhang; Jun Zhou; Youhe Zhou; Xue Feng

doi:10.1364/OE.21.026352

1. Introduction

Thin films deposited on various types of substrates are applied in many technologies, such as microelectronics, optoelectronics, thermal barrier coating technology, and micro-electromechanical systems (MEMS), etc. Fabrication of such a film-substrate structure inevitably gives rise to stress in the film due to lattice mismatch, different coefficients of thermal expansion, chemical reactions, or other physical effects. Up to now, there are a few experimental techniques (including scanning laser method [1], multi-beam 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, non-destructive, 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 high temperature environment, 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 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, and recently, Liu et al [21] give a theoretical error analysis of the CGS system at low temperature, there are few investigations on the CGS system at the cryogenic temperature. For the superconducting thin-film systems, which are employed at the low temperature ambient (always explored by a vacuum closed cycle refrigerator with transparent interface), and the thermal stress has remarkable effects on its superconducting characteristics, e.g. critical current density [22]. Thus, it is important to measure the thermal stresses of the superconducting thin film system during its cooling process. In this paper, a measurement device including CGS and low temperature Dewar is established firstly, and then a technique to reduce the influences of the interface on the obtained interferogram is presented. In the last, the thermal stresses (including the radial stress, hoop stress and shear stress between the thin film and substrate) of the superconducting YBCO thin-film during temperature increase process are obtained.

2. Experimental setup and processes

The CGS setup for cryogenic temperature measurement is illustrated in Fig. 1(a). A collimated laser beam passes through a beam splitter and is then directed to the reflecting specimen surface in the Dewar with a transparent window. The reflected beam from the specimen is further reflected by the beam splitter and then passes through two Ronchi gratings, G₁ and G₂ with the same density (40 lines/mm) separated by a distance $Δ$ . The diffracted beams from the two gratings are converged to interfere using a lens. Either of the ± 1 diffraction orders is filtered by the filtering aperture to obtain the interferogram recorded by a CCD camera. Fig. 1(b) is displayed the actual equipment of the measurement system, in where the number 1 denotes the closed cycle refrigerator (G-M). The detail description of the CGS system will be neglected in this paper. Its process will be referenced by Dong et al. [11] and Liu at el [21]. Here, we take emphasis on the technique how to eliminate the influences of interface on the interferogram. At first, in order to remove the effects of the reflected beam of the transparent window on the CGS system, a tilt of the transparent window will be conducted, which is illustrated in Fig. 2 (a).

Fig. 1 (a) Schematic of the CGS for cryogenic temperature, (b) photo of the measurement system, in which the number 1 denotes the closed cycle refrigerator (G-M).

Download Full Size | PDF

Fig. 2 (a) schematic of the lean of transparent windows, $θ$ is the angle between the window and the horizontal plane, $l$ denotes the width of the beam splitter, and $h$ is the distance between the project plane of the transparent window and the bottom surface of the beam splitter, and while the thickness of quartz window is neglected, (b) Schematic of the effects of the quartz window on the interferogram, the dotted lines denote the normal of the window’s surface.

Download Full Size | PDF

From Fig. 2 (a), one can see that in order to avoid the reflected light of the transparent window into the CGS system, the angle $θ$ between the project plane of the transparent window and the bottom surface of the beam splitter should be satisfied by

\frac{1 + \tan^{2} θ}{2 \tan θ} = \frac{h}{l} .

One can get that the two roots of the Eq. (1) are

\tan θ = \frac{h}{l} \pm \sqrt{\frac{h^{2}}{l^{2}} - 1}

, respectively. Considering the small

θ

and

h > > l

, we can obtain that

\tan θ

is equal to

\frac{h}{l} - \sqrt{\frac{h^{2}}{l^{2}} - 1}

, we can use approximation

θ \approx \tan θ

, thus,

θ

is equal to

\frac{h}{l} - \sqrt{\frac{h^{2}}{l^{2}} - 1}

approximately.

According to the law of refraction, one can see that there is no effect of the quartz window on the light propagation vector easily, which has effect on the location of the interferogram only. Thus, the change of the incidence vector will be decided by the air (its refractive index is equal to $n_{2}$ ) and the vacuum (the refractive index is equal to $n_{1}$ ). Based on our previous derivation in [21], the propagation vector of the light from the window's surface is satisfied by

d = α e_{x} + β e_{y} + γ e_{z} = \frac{n_{1}}{n_{2}} d' + (\frac{\sqrt{n_{2}^{2} {(1 + {| \nabla f |}^{2})}^{2} - n_{1}^{2} \cdot 4 {| \nabla f |}^{2}}}{n_{2} (1 + {| \nabla f |}^{2})} - \frac{n_{1}}{n_{2}} \cdot \frac{1 - {| \nabla f |}^{2}}{1 + {| \nabla f |}^{2}}) e_{z},

where

α = \frac{- 2 f_{, x}}{1 + f_{, x}^{2} + f_{, y}^{2}} \cdot \frac{n_{1}}{n_{2}}

,

β = \frac{- 2 f_{, y}}{1 + f_{, x}^{2} + f_{, y}^{2}} \cdot \frac{n_{1}}{n_{2}}

,

γ = \frac{\sqrt{n_{2}^{2} {(1 + {| \nabla f |}^{2})}^{2} - n_{1}^{2} \cdot 4 {| \nabla f |}^{2}}}{n_{2} (1 + {| \nabla f |}^{2})}

.

Assuming ${| \nabla f |}^{2} < < 1$ , $n_{1} \approx n_{2}$ , we can obtain

d \approx d' .

Because the interferogram of the CGS is formed from the propagation vector of the emergent light in air, the influences of the refraction caused by transparent window on the CGS system will be ignored. According to the equation

I = I_{c} + ε μ a_{1} a_{2} \cos (\frac{k Δ β θ}{γ^{2}}),

presented by Liu et al. [21], in which

I_{c} = ε μ \frac{a_{1}^{2} + a_{2}^{2}}{2}

,

δ = \frac{k Δ β θ}{γ^{2}}

denotes the phase of the interferogram fringes. Substituting

α, β, γ

mentioned above into Eq. (4), one can obtain

δ^{(x)} (x, y) = \frac{4 π Δ}{p} f_{x},

δ^{(y)} (x, y) = \frac{4 π Δ}{p} f_{y},

where

δ^{(x)} (x, y)

and

δ^{(y)} (x, y)

are the phase distributions of the interferogram fringes in x direction and y direction,

Δ

denotes the distance between the two Ronchi gratings,

p \approx λ / θ

,

f_{x}, f_{y}

denote the first-order derivative of the shape functions of the specimen in x direction and y direction, respectively. Therefore, the CGS governing equations for cryogenic temperature can be given by

κ_{x x} = \frac{\partial^{2} f (x, y)}{\partial x^{2}} = \frac{p}{4 π Δ} \frac{\partial δ^{(x)} (x, y)}{\partial x},

κ_{y y} = \frac{\partial^{2} f (x, y)}{\partial x^{2}} = \frac{p}{4 π Δ} \frac{\partial δ^{(y)} (x, y)}{\partial x},

κ_{x y} = \frac{\partial^{2} f (x, y)}{\partial x \partial y} = \frac{p}{4 π Δ} \frac{\partial δ^{(x)} (x, y)}{\partial y},

where

κ_{x x}

is the curvature in x direction,

κ_{y y}

is the curvature in y direction, and

κ_{x y}

denotes the twist curvature. It should be noted that the way we infer curvature and twist components is by numerical (spatial) differentiation of the surface gradient.

Based on the above analysis, in order to obtain the curvature of the specimen’ s surface, one can gain the phase distributions in x direction and y direction firstly. Thus, we now turn to how to obtain the phase information from the interferogram fringes.

The interferogram fringes originated from the CGS system can be gave as [23]

I (x, y) = a (x, y) + b (x, y) \cos δ (x, y),

where the first item includes the intensity information of the background, the second item denotes the vary information of the intensity of the interferogram fringes, in which

δ (x, y)

is on behalf of the information of phase distributions, and its change is included the deformation of the specimen. The Eq. (10) can be rewritten as

I (x, y) = a (x, y) + c (x, y) - c * (x, y),

where

c (x, y) = \frac{1}{2} b (x, y) e^{i δ (x, y)}

,

c *

is the complex conjugate of

c

. Calculated by the Fourier transform, one can get

I (ω^{(x)}, ω^{(y)}) = A (ω^{(x)}, ω^{(y)}) + C (ω^{(x)}, ω^{(y)}) + C^{*} (- ω^{(x)}, - ω^{(y)}) .

By using the band-pass filter, the first item

A

, the second item

C

or the third

C^{*}

should be eliminated. For the residual item

C

or

C^{*}

, an inversion of the Fourier transform can be employed, one can obtain

δ (x, y) = \tan^{- 1} \frac{Im [C (x, y)]}{Re [C (x, y)]},

where

Im [C (x, y)]

and

Re [C (x, y)]

denote the imaginary and real parts of the complex amplitude

C (x, y)

. Subsequently, the phase distribution function

δ (x, y)

is fitted by using the Zernike polynomial [24] to obtain the phase distribution

δ^{x} (x, y)

and

δ^{y} (x, y)

in x and y direction respectively.

It is well know that the fabricated processes of the thin-film/substrate system inevitably introduce the nonuniform nucleation and /or misfit, which can result in serious thermo-stress due to temperature variation. One of the authors(Xue. F.) and his previous associates [15–17] had derived an extension of Stonry’s formula for a multilayer thin-film/substrate system subjected to nonuniform and nonaxisymmetrical temperature distribution. In their work, they derive relations between the film curvatures and temperature, and between the plate system’s curvatures and the stresses. These relations featured a “local” part that involves a direct dependence of the stress or curvature components on the temperature at the same point, and a “nonlocal” part that reflects the effect of temperature of other points. In this paper, we will use the cylindrical coordinates to present the thermal stresses analysis. Then the nonuniform thin-film stresses form the nonuniform curvatures of the substrates can be expressed as [17]

σ_{r r}^{(f)} + σ_{θ θ}^{(f)} = \frac{E_{s} h_{s}^{2}}{6 (1 - ν_{s}) h_{f}} {\begin{cases} \bar{κ_{r r} + κ_{θ θ}} \\ + \frac{(1 + ν_{f}) [(1 + ν_{s}) α_{s} - 2 α_{f}]}{(1 + ν_{s}) [(1 + ν_{s}) α_{s} - (1 + ν_{f}) α_{f}]} (κ_{r r} + κ_{θ θ} - \bar{κ_{r r} + κ_{θ θ}}) \\ + [\frac{3 + ν_{s}}{1 + ν_{s}} - 2 \frac{(1 + ν_{f}) [(1 + ν_{s}) α_{s} - 2 α_{f}]}{(1 + ν_{s}) [(1 + ν_{s}) α_{s} - (1 + ν_{f}) α_{f}]}] \\ \times \sum_{m = 1}^{\infty} (m + 1) {(\frac{r}{R})}^{m} (C_{m} \cos m θ + S_{m} \sin m θ) \end{cases}},

\begin{matrix} σ_{r r}^{(f)} - σ_{θ θ}^{(f)} = \frac{E_{s} h_{s}^{2} α_{s} (1 - ν_{f})}{6 (1 - ν_{s}) h_{f}} \frac{1}{(1 + ν_{s}) α_{s} - (1 + ν_{f}) α_{f}} \\ \times {\begin{cases} κ_{r r} - κ_{θ θ} \\ - \sum_{m = 1}^{\infty} \begin{array}{l} (m + 1) [m {(\frac{r}{R})}^{m} - (m - 1) {(\frac{r}{R})}^{m - 2}] \\ \times (C_{m} \cos m θ + S_{m} \sin m θ) \end{array} \end{cases}}, \end{matrix}

\begin{matrix} σ_{r θ}^{f} = \frac{E_{s} h_{s}^{2} α_{s} (1 - ν_{f})}{6 (1 - ν_{s}) h_{f}} \frac{1}{(1 + ν_{s}) α_{s} - (1 + ν_{f}) α_{f}} \\ \times {κ_{r θ} + \frac{1}{2} \sum_{m = 1}^{\infty} (m + 1) [m {(\frac{r}{R})}^{m} - (m - 1) {(\frac{r}{R})}^{m - 2}] \times (C_{m} \cos m θ - S_{m} \sin m θ)}, \end{matrix}

τ_{r} = \frac{E_{s} h_{s}^{2}}{6 (1 - ν_{s}^{2})} {\frac{\partial}{\partial r} (κ_{r r} + κ_{θ θ}) - \frac{1 - ν_{s}}{2 R} \sum_{m = 1}^{\infty} m (m + 1) {(\frac{r}{R})}^{m - 1} (C_{m} \cos m θ + S_{m} \sin m θ)},

τ_{θ} = \frac{E_{s} h_{s}^{2}}{6 (1 - ν_{s}^{2})} {\frac{1}{r} \frac{\partial}{\partial θ} (κ_{r r} + κ_{θ θ}) + \frac{1 - ν_{s}}{2 R} \sum_{m = 1}^{\infty} m (m + 1) {(\frac{r}{R})}^{m - 1} (C_{m} \cos m θ - S_{m} \sin m θ)},

where

h_{s}

and

h_{f}

are the thickness of the substrate and thin film, respectively.

R

is the radius of the system,

σ_{r r}^{(f)}

and

σ_{θ θ}^{(f)}

denote the in-plane stresses of the thin film in the radial and circumferential directions, respectively.

σ_{r θ}^{(f)}

is film shear stress, and

τ_{r}

,

τ_{θ}

are the interfacial shear stresses between the substrate and thin film in the radial and circumferential directions, respectively.

\bar{κ_{r r} + κ_{θ θ}} = \frac{1}{π R^{2}} \int_{0}^{2 π} \int_{0}^{R} (κ_{r r} + κ_{θ θ}) r d r d θ

is the average curvature of substrate as well as

C_{m} = \frac{1}{π R^{2}} {\int_{0}^{2 π} \int_{0}^{R} (κ_{r r} + κ_{θ θ}) (\frac{η}{R})}^{m} \cos (m φ) η d η d φ

and

S_{m} = \frac{1}{π R^{2}} {\int_{0}^{2 π} \int_{0}^{R} (κ_{r r} + κ_{θ θ}) (\frac{η}{R})}^{m} \sin (m φ) η d η d φ

.

E_{s}

is the Young’s modulus of the substrate.

ν_{s}

and

ν_{f}

are the Poisson’s ratio of the substrate and film, respectively.

α_{s}

and

α_{f}

denote the thermal expansion coefficients of the substrate and film, respectively. It should be noted that the thin-film nonuniform stresses are not only dependent on the local curvatures of the substrate, but they also related to the ‘nonlocal’ curvatures (average curvature).

In summary, in the above paragraphs, we present the CGS system at the cryogenic temperature. The influences of the reflected light from the quartz window’s surface on the CGS interferogram fringes have been eliminated successfully. In addition, the effects of the refraction of the quartz window can be ignored by a mathematical derivation. Based on the obtained curvatures of the substrate, the nonuniform thermal stresses of the thin film/substrate system can be gained by the above Eqs. (14)-(18).

3. Experimental results and discussion

3.1 Substrate curvature measurement

The specimen consists of YBCO film grown by laser ablation on MgO substrate, which is the representative wafer structure widely used in superconducting researches. The thicknesses of the YBCO film and MgO substrate were 200nm and 500μm, respectively; their radius was 10mm. The geometry size agreed with the assumption $h_{f} < < h_{s} < < R$ . The specimen was placed vertically, as shown in Fig. 1(a). The back of the specimen was supported by a stiff frame made by Cu. Moreover, the contact between the specimen and the Cu support is conducted by Indium to make sure the well thermal conduction. The specimen could expand freely subjected to temperature, and there was no additional stresses induced by the boundary condition. As the temperature was elevated from 30K temperature to high temperature (e.g. ~150K), the CGS interferograms were recorded by a CCD camera. Fig. 3 shows the interferograms obtained at 30K. The red fringes in Figs. 3(a) and 3(c) represent the contour curves of the specimen surface slope in lateral (x direction) and vertical (y direction) directions, respectively. The wrapped phase map is calculated by FFT method and shown in Figs. 3(b) and 3(d), respectively. Fig. 4(a) and (b) show the corresponding system curvatures distribution in x and y directions, respectively, while Fig. 4(c) shows the twist curvature distribution. It is obvious that the curvature distribution is nonuniform and thus violates the Stoney’s formula assumption. The curvatures in the vicinity of the edge become much greater than those in the other area due to the edge effect.

Fig. 3 Interferogram fringes at 30K and their wrapped phase maps: (a) interferogram obtained by shearing laterally, (b) wrapped phase map for Fig. 3(a), (c) interferogram obtained by shearing vertically, (d) wrapped phase map for Fig. 3(c).

Download Full Size | PDF

Fig. 4 The substrate curvatures measured at 30K, (a) curvature $κ_{x x}$ in lateral direction, (b) curvature $κ_{y y}$ in vertical direction, (c) twist curvature $κ_{x y}$ .

Download Full Size | PDF

3.2 Nonuniform stresses of the thin film

To calculate the film stresses at cryogenic temperature, we select the room temperature (about 297K) as a reference state. The physical parameters of the system are $E_{f} = 123$ Gpa, $E_{s} = 248$ Gpa, $ν_{f} = 0.245$ , $ν_{s} = 0.251$ , $p = 0.025$ mm, and $Δ = 21$ mm, respectively. The thermal expansion coefficients of the YBCO thin film and the MgO substrate are displayed in Fig. 5, one can see that with the increase of ambient temperature, their expansion coefficients increase with a close law [25].

Fig. 5 The thermal expansion coefficients of the YBCO thin-film and MgO substrate vs. temperature.

Download Full Size | PDF

The thin film stresses for 30K are shown in Fig. 6. Fig. 6(a), 6(b), and 6(c) show the film stresses (radial direction), (circumferential direction), and (shear stress), respectively. Fig. 6(d) and 6(e) show the interfacial shear stresses (radial direction) and (circumferential direction) between the film and the substrate, respectively. The nonuniformity of the film stresses becomes more severe owing to the nonlocal effect shown in Eqs. (14)-(18). In addition, the interfacial stresses and with the magnitude of a few Pa are rather smaller compared with the film stresses.

Fig. 6 The nonuniform stresses of the thin film measured at 30K: (a) stress $σ_{r r}$ in radial direction, (b) stress $σ_{θ θ}$ in circumferential direction, (c) shear stress $σ_{r θ}$ , (d) interfacial shear stress $τ_{r}$ in radial direction, (e) interfacial shear stress $τ_{θ}$ in circumferential direction.

Download Full Size | PDF

To investigate the thermo-stresses of thin film subjected to varied temperature, we conducted the experiment from 30K to 150K with the step of 10K. Then the full-field stresses can be obtained at the different temperatures following the same process as above. The film stresses of the central point in the specimen are selected to illustrate the thermo-stress evolution, as shown in Fig. 7. Due to the different coefficients of thermal expansion and inhomogeneous temperature distribution, the shear stress is formed in the interface, which results in a new stress distribution in the film plane. $σ_{r r}$ is equal to $σ_{θ θ}$ at the beginning 30K then increases to 50Gpa (in compression) at 150K, $σ_{θ θ}$ is equal to 90GPa (in compression) at the same temperature. The multi fluctuations of shear stress $σ_{r θ}$ from tension to compression with the increase of temperature are observed which may result from the nonuniformity and the nonlocal effect.

Fig. 7 The film stresses in radial, circumferential directions (a) and the shear stress (b) at the central point of the specimen vs. temperature.

Download Full Size | PDF

4. Conclusions

The coherent gradient sensing (CGS) is features as full-field nonuniform curvatures measurement and vibration insensitivity, which is used to measure the thin-film/substrate system curvature at cryogenic temperature. Superconducting YBCO thin-film with MgO substrate is used to conduct the proposed CGS method. The stresses including the radial, circumferential and shear are obtained by using this technique. These results provide a fundamental approach to understand the thin-film (especially for superconducting thin-film system) stresses and the feasible measurement method for cryogenic temperature.

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-Constrained Restriction Fusion Energy Development Program (No. 2013GB110002), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (No.201135), and the Program for New Central Excellent Talents in University (NCET-12-0245) and the Fundamental Research Funds for the Central Universities.

References and links

1. P. A. Flinn, D. S. Gardner, and W. D. Nix, “Measurement and interpretation of stress in aluminum-based metallization as a function of thermal history,” IEEE Trans. Electron. Dev. 34(3), 689–699 (1987). [CrossRef]

2. E. Chason and B. W. Sheldon, “Monitoring stress in thin films during processing,” Surf. Eng. 19(5), 387–391 (2003). [CrossRef]

3. H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fract. 48(3), 193–204 (1991). [CrossRef]

4. H. V. Tippur, “Coherent gradient sensing: a Fourier optics analysis and applications to fracture,” Appl. Opt. 31(22), 4428–4439 (1992). [CrossRef] [PubMed]

5. A. J. Rosakis, R. P. Singh, Y. Tsuji, E. Kolawa, and N. R. Moore Jr., “Full field measurements of curvature using coherent gradient sensing: application to thin film characterization,” Thin Solid Films 325(1–2), 42–54 (1998). [CrossRef]

6. M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006). [CrossRef]

7. J. Tao, L. H. Lee, and J. C. Bilello, “Nondestructive evaluation of residual-stresses in thin-films via x-ray-diffraction topography methods,” J. Electron. Mater. 20(7), 819–825 (1991). [CrossRef]

8. T.-S. Park, S. Suresh, A. J. Rosakis, and J. Ryu, “Measurement of full-field curvature and geometrical instability of thin film-substrate systems through CGS interferometry,” J. Mech. Phys. Solids 51(11–12), 2191–2211 (2003). [CrossRef]

9. M. D. Vaudin, E. G. Kessler, and D. M. Owen, “Precise silicon die curvature measurements using the NIST lattice comparator: comparisons with coherent gradient sensing interferometry,” Metrologia 48(3), 201–211 (2011). [CrossRef]

10. M. Budyansky, C. Madormo, J. L. Maciaszek, and G. Lykotrafitis, “Coherent gradient sensing microscopy (micro-CGS): A microscale curvature detection technique,” Opt. Lasers Eng. 49(7), 874–879 (2011). [CrossRef]

11. X. Dong, X. Feng, K. C. Hwang, S. Ma, and Q. Ma, “Full-field measurement of nonuniform stresses of thin films at high temperature,” Opt. Express 19(14), 13201–13208 (2011). [CrossRef] [PubMed]

12. G. G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 82(553), 172–175 (1909). [CrossRef]

13. Y. Huang and A. J. Rosakis, “Extension of Stoney's formula to non-uniform temperature distributions in thin film/substrate systems. The case of radial symmetry,” J. Mech. Phys. Solids 53(11), 2483–2500 (2005). [CrossRef]

14. X. Feng, Y. G. Huang, H. Q. Jiang, D. Ngo, and A. J. Rosakis, “The effect of thin film/substrate radii on the stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature,” J. Mech. Mater. Struct. 1(6), 1041–1053 (2006). [CrossRef]

15. D. Ngo, X. Feng, Y. Huang, A. J. Rosakis, and M. A. Brown, “Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions. Part I: analysis for obtaining film stress from non-local curvature information,” Int. J. Solids Struct. 44(6), 1745–1754 (2007). [CrossRef]

16. M. A. Brown, A. J. Rosakis, X. Feng, Y. Huang, and E. Ustundag, “Thin film substrate systems featuring arbitrary film thickness and misfit strain distributions. Part II: experimental validation of the non-local stress curvature relations,” Int. J. Solids Struct. 44(6), 1755–1767 (2007). [CrossRef]

17. X. Feng, Y. Huang, and A. J. Rosakis, “Stresses in a multilayer thin film/substrate system subjected to nonuniform temperature,” J. Appl. Mech. 75(2), 021022 (2008). [CrossRef]

18. X. F. Yao, H. Y. Yeh, and W. Xu, “Fracture investigation at V-notch tip using coherent gradient sensing (CGS),” Int. J. Solids Struct. 43(5), 1189–1200 (2006). [CrossRef]

19. R. P. Singh, J. Lambros, A. Shukla, and A. J. Rosakis, “Investigation of the mechanics of intersonic crack propagation along a bimaterial interface using coherent gradient sensing and photoelasticity,” P Roy Soc A-Math Phy. 4532649–2667 (1997).

20. L. T. Mao, C. P. Liu, K. Chen, L. Q. An, and X. X. Zhu, “Study on stress intensity factor of PMMA with double cracks using coherent gradient sensing(CGS) technique,” Appl. Mech. Mater. 109, 114–119 (2011). [CrossRef]

21. C. Liu, X. Zhang, J. Zhou, and Y. Zhou, “A general coherent gradient sensor for film curvature measurements: error analysis without temperature constraint,” Opt. Lasers Eng. 51(7), 808–812 (2013). [CrossRef]

22. J. Xiong, W. Qin, X. Cui, B. Tao, J. Tang, and Y. Li, “Effect of processing conditions and methods on residual stress in CeO2 buffer layers and YBCO superconducting films,” Physica C 442(2), 124–128 (2006). [CrossRef]

23. H. Lee, A. J. Rosakis, and L. B. Freund, “Full-field optical measurement of curvatures in ultra-thin-film–substrate systems in the range of geometrically nonlinear deformations,” J. Appl. Phys. 89(11), 6122–6123 (2001). [CrossRef]

24. R. Navarro, R. Rivera, and J. Aporta, “Representation of wavefronts in free-form transmission pupils with Complex Zernike Polynomials,” J. Opt. 4(2), 41–48 (2011). [CrossRef]

25. B. Gu, P. E. Phelan, and S. Mei, “Coupled heat transfer and thermal stress in high-Tc thin-film superconductor devices,” Cryogenics 38(4), 411–418 (1998). [CrossRef]

The coherent gradient sensor for film curvature measurements at cryogenic temperature

Abstract

1. Introduction

2. Experimental setup and processes

3. Experimental results and discussion

3.1 Substrate curvature measurement

3.2 Nonuniform stresses of the thin film

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (18)

Optics Express