Reconstruction method of gradient-index field with background-oriented schlieren

Abstract

The refractive index fields of several media, say, transparent solids, transparent liquids, and air are often crucial when using these media as optical elements, probing optical properties, and deriving other physical information related to the refractive index. For example, in an optical system using a high-power laser, the thermal lens effect of optical elements is one of great issues because the laser focal position will be changed by the effect [1]. Thus, a method to measure a gradient-index of the optical element should be useful to take a step to reduce the thermal lens effect. Basically, light rays passing through a medium with an inhomogeneous refractive index field will be deflected toward the area of higher refractive index. This way, it should be possible to obtain information about the refractive index field from measurement of such deflection. Consequently, there have been several methods for visualizing gradient-index fields, including schlieren photography [2] and shadowgraphy [3]. Although these visualization methods are powerful tools for qualitatively analyzing the gradient-index field, quantitative analysis requires additional methods. Rainbow schlieren photography is one candidate. In this method, the strength of deflection of light rays is obtained by using a rainbow aperture [4-6]. The quantitative phase-contrast imaging has also developed with incoherent-light [7, 8], coherent-light [9, 10], and asymmetric illumination [11]. There is also background-oriented schlieren (BOS) technique for measuring gradient-index field quantitatively [12-20]. In the BOS technique, displacements of dot-patterns that are printed on a background are caused by deflection of light rays passing through a gradient-index field. The optical flow algorithm can be used to obtain the displacements of the background-dot-patterns. However, obtaining depth information of the gradient-index field with the BOS technique is often difficult because the dot-pattern displacement results from the light ray deflected all along its path, requiring integration over the path. To address this, a method of reconstructing the depth information of a gradient-index field is proposed here on the basis of the Lagrangian optics with the BOS technique.