A Simple Phase-Shifting Lateral Shearing Interferometer

A phase-shifting electronic speckle pattern shearing interferometer with a very simple shearing device is proposed. Two partially reflective glass plates are used to introduce the shear in this new interferometer. The reflection coefficients of the coatings were 0.3 and 0.7 respectively. The distance between the two glass plates controls the size of the shear. The proposed new interferometric system is simple, flexible and low cost. OCIS codes: 120.6160 Speckle interferometry; 220.4830 Optical systems design Electronic speckle pattern shearing interferometry (ESPSI) enables direct measurements of displacement derivatives to be made with variable sensitivity. A common lateral shear interferometer uses a plane parallel glass plate as a shearing device. The amount of shear is selected by the thickness and tilt of the plate. The disadvantages of this simple shearing interferometer are that the resulting interferogram has low fringe visibility and phase-shifting techniques can not be applied. Phase-shifting shearography is more sensitive than the simple digital shearography and can provide three-dimensional (3D) plots of the displacement derivatives. When two light waves interfere, the following equation relates their relative phase Φ at a location to their relative geometrical path length L: β λ π − = Φ nL 2 (1) where λ is the wavelength of the laser light, n is the refractive index of the medium through which the laser light is transmitted, and β is a constant phase. The change in the relative phase Δ=δΦ or phase change, which manifests as visible fringes, can be effected by an incremental change in any of the three parameters λ, n, and L. Thus, L n n L Ln L L n n δ λ π δ λ π δλ λ π δ δ δλ λ 2 2 2 2 + + − = ∂ Φ ∂ + ∂ Φ ∂ + ∂ Φ ∂ = Δ (2) where δλ, δn, and δL denote respectively, the incremental change in wavelength, in refractive index, and in relative geometrical path length of the interfering waves. If the same wavelength is used and the test environment is still air (n = 1), only the δL term in Eq. (2) is nonzero, resulting in the following equation for the phase change: [ ] w C v B u A δ δ δ λ π + + = Δ 2 (3) where u, v and w are the displacement components of the neighboring point P’(x+Δx, y, z+Δz) relative to point P (x, y, z) on the test surface, and A, B, and C are sensitivity factors that are related to the optical arrangement. For small image shearing Δx, the displacement terms in Eq. (3) can be expressed in terms of partial derivatives: ⎥⎦ ⎤ ⎢⎣ ⎡ ∂ ∂ + ∂ ∂ + ∂ ∂ = Δ x w C x v B x u A λ π 2 (4) In our case (Fig. 1) the illuminating object beam lies in the (x, z) plane, so there is no sensitivity to displacement along y axis so long as the beam is well collimated. The phase change is: ⎥⎦ ⎤ ⎢⎣ ⎡ ∂ ∂ + ∂ ∂ = Δ x w C x u A λ π 2 (5) Digital phase shifting can be implemented to provide fringe analysis. There are a number of algorithms available for the purpose. The algorithm we use is a five bucket type in which four interferograms are captured having phase shifts of π/2, π, 3π/2 between successive ones. A fifth interferogram with zero phase shift is captured after the loading is applied and the phase map corresponding to Eq.5 is obtained from these data. Fig. 1. Schematic diagram of the ESPSI system with two partially reflective glass plates as a shearing element: I and I’ are the laterally sheared images of O on the CCD faceplate. I I’ PZT phase-shifter