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 . 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  and shadowgraphy . 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 . 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.
Internal information such as mechanical properties and geometrical structures of non-transparent materials can be obtained non-destructively by means of a laser ultrasonic technique. The laser ultrasonic technique measures time of flight of an ultrasonic acoustic pulse generated at the surface of the materials by a pump pulse laser where the acoustic pulse is reflected from the internal structures of the material. The time of the flight of the acoustic pulse can be measured by the time-sequential modulation of the reflectance of a probe laser that is irradiated on the surface of the material. Assuming that the material has a structure of multilayer, each thickness of the multilayer can be reconstructed by fitting of numerical calculations of the time-sequential modulation of the reflectance to the experimental measurements with fitting parameters of the thicknesses. The numerical calculation, however, should solve the spatial distribution of the absorbed energy of the pump laser which determines the shape of the acoustic pulse, the strain tensor of the acoustic pulse, and the reflectance of the probe laser. It likely follows that the three different numerical calculation methods are necessary. Then, an efficient numerical calculation method for the reconstruction of the multi-layer structure using FDTD (Finite Difference Time Domain) algorithm where the method can be applied to the above-mentioned three different calculations in the same frame is proposed here.