Geometric phase analysis based on the windowed Fourier transform for the deformation field measurement Xianglu Dai a , Huimin Xie a,n , Qinghua Wang b a AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China b National Institute for Materials Science, 1-2-1, Sengen, Tsukuba, Ibaraki 305-0047, Japan article info Article history: Received 8 September 2013 Received in revised form 28 October 2013 Accepted 15 November 2013 Available online 5 December 2013 Keywords: Geometric phase analysis Windowed Fourier transform Soft UV-NIL abstract The geometric phase analysis (GPA), an important image-based deformation measurement method, has been used at both micro- and nano-scale. However, when a deformed image has apparent distortion, non-ignorable error in the obtained deformation field could occur by using this method. In this paper, the geometric phase analysis based on the windowed Fourier transform (WFT) is proposed to solve the above-mentioned issue, defined as the WFT–GPA method. In WFT–GPA, instead of the Fourier transform (FT), the WFT is utilized to extract the phase field block by block, and therefore more accurate local phase information can be acquired. The simulation tests, which include detailed discussion of influence factors for measurement accuracy such as window size and image noise, are conducted with digital deformed grids. The results verify that the WFT–GPA method not only keeps all advantages of traditional GPA method, but also owns a better accuracy for deformation measurement. Finally, the WFT–GPA method is applied to measure the machining distortion incurred in soft ultraviolet nanoimprint lithography (UV- NIL) process. The successful measurement shows the feasibility of this method and offers a full-field way for characterizing the replication quality of UV-NIL process. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction The GPA method, presented by Hÿch [1], is initially applied to measure the deformation from the high resolution transmission electron microscope image [2–7]. Recently, combining with the advanced fabrication techniques of high-frequency gratings [8–10], GPA has been widely used in the deformation measurement at micro- and nano-scale [11–17]. The traditional GPA method, referred to as FT–GPA in this paper, is based on the relationship between the displacement and the phase difference. The FT plays a crucial role in FT–GPA as the phase information is extracted by it. However, when the deformation is non-uniform, particularly with strong distortion, the FT–GPA method would fail to extract the deformation accurately as the FT cannot extract the fundamental component with a wide frequency band [18], which could lead to a non-ignorable error in the obtained deformation field. In this paper, the WFT [19–22], which is a local Fourier trans- form algorithm, has been employed in the GPA method and this method is defined as the WFT–GPA method. Since the ordinary FT acts in the whole region and the local frequency components could be suppressed, a non-ignorable error could be introduced in the obtained deformation field. Different from the traditional method, the WFT performs the Fourier transform in a small area block by block which can be regarded as a region with uniform deformation (or linear phase), so the local frequency can be extracted more accurately and the inhomogeneous deformation field can be measured with high accuracy. The simulation tests are conducted to check the performance of the proposed method, and the results show that compared with FT–GPA it can extract the deformation field accurately and it is especially effective for images with strong distortion. Besides, through the analysis of strain fields under different window sizes and grid pitches, the selection method for optimal window size is given; by comparing the results of images with different degrees of Gaussian white noise, the ability of noise toleration of WFT–GPA is proved as well. The nanoimprint lithography [23,24], which can overcome the resolution limit imposed by the diffraction in photolithography method, is a micro-fabrication technique that has extraordinary cost performance. The PDMS (polydimethylsiloxane) mold is widely used in the soft UV-NIL [25–27], due to the superiorities of conformal contact, easy demolding and UV light penetrability. However, there may introduce undesired dimensional error during the fabrication process since the low modulus of PDMS (typically, the shear modulus of PDMS less than 1 MPa); and therefore it is necessary to characterize the full-field deformation incurred by soft UV-NIL process with PDMS. In this study, the WFT–GPA method is applied to measure this deformation. In the measure- ment, the grid, considered as deformation carrier, was fabricated Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/optlastec Optics & Laser Technology 0030-3992/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2013.11.010 n Corresponding author. E-mail address: xiehm@mail.tsinghua.edu.cn (H. Xie). Optics & Laser Technology 58 (2014) 119–127