## 1 Introduction  
Nano-metrology is the only effective method that ensures the reliability and consistency of nano-manufacturing [1; 2; 3]. Compared with other techniques such as scanning electron microscopy (SEM), atomic force microscopy (AFM) [4], and near-field scanning optical microscope (NSOM) [5], optical scatterometry [6; 7], also known as optical critical dimension metrology or optical critical dimension (OCD) metrology, is more suitable for monitoring, assessing, and optimizing the nano-manufacturing processes due to its advantages of being non-contact, non-destructive, low in cost, and easy to integrate, etc. Recently, optical scatterometry has been applied in many fields with great success, such as the process control for back-end-of-the-line (BEOL) [8], the in-chip critical dimension (CD), overlay metrology [9], and in-situ measurement of pattern reflow in nanoimprinting [10]. In general, optical scatterometry involves two procedures: the forward optical modeling of sub-wavelength structures and the reconstruction of structural profiles from the measured signatures [11]. Here, the general term signatures means the scattered light information from the diffractive grating structure, which can be in the form of reflectance, ellipsometric angles, Stokes vector elements, or Mueller matrix elements. The forward optical model describes the light-nanostructure interaction by solving the complex Maxwell's equations. There are many reliable forward-modeling techniques such as the rigorous coupled-wave analysis (RCWA), the finite element method (FEM), the boundary element method (BEM), or the finite-difference time-domain (FDTD) method. The profile reconstruction process conducted under a fixed measurement configuration is an inverse problem with the objective of optimizing a set of floating profile parameters (e.g., CD, sidewall angle, and height) whose theoretical signatures can best match the measured ones through regression analysis or library search [12; 13; 14]. The measurement configuration is defined as a combination of specially selected wavelengths and incident and azimuthal angles. As a typical model-based method, abundant measured data is usually collected under the measurement configurations with fixed incidence and azimuthal angles, but a wide waveband for spectroscopic scatterometry, or with fixed measurement wavelength and azimuthal angle, but incidence angular range for angle-resolved scatterometry. Taking the inverse structural profile reconstruction in spectroscopic scatterometry as an example, the measurement sensitivities of unknown profile parameters are affected by the fixed incident and azimuthal angles, which can be improved by traversing their possible values; the size of fitting data usually directly depends on the number of selected wavelength points within the waveband. A large sized measured data set should be used to guarantee the fidelity of the fitting results, but on the other hand, too much redundant information will significantly degrade the efficiency during the inverse problem solving process due to the frequent iteration of the forward model. Moreover, the employment of the redundant information, which is insensitive to the measurands in solving inverse problems, leads to the loss in precision as well. Thus, an appropriate measurement configuration with high sensitivity and a small amount of fitting data will benefit the measurement with both precision and speed. Many instrument optimization and spectrum denoising methods have been proposed with the objective of improving the quality of measured data [15; 16; 17]. As the measurement precision mainly depends on the sensitivity of model parameters, several approaches based on local sensitivity analysis (LSA) also have been developed to determine optimal incident and azimuthal angles for spectroscopic scatterometry [18; 19; 20; 21].