Optical scattering provides researchers with a wealth of information, including light intensity, phase, and polarization. Angle-resolved spectroscopy (ARS) is a powerful technique that measures the distribution of light intensity with angles or wavelengths. It plays a crucial role in obtaining important optical information to solve optical inverse problems and represent the properties of micro-nanophotonic materials.

In optical inverse problems, ARS assists researchers in determining the morphological structures and optical constants of materials by analyzing the angle-dependent behaviors of scattered light. This technique is particularly helpful when traditional measurement methods, such as scanning electron microscopy (SEM) and atomic force microscopy (AFM), are not feasible. By solving optical inverse problems using ARS and inverse algorithms, researchers can gain a deeper understanding of the behavior of light in complex materials and systems. Additionally, it finds wide application in the semiconductor industry to detect defects in wafers and measure optical critical dimensions of optoelectronic components.

ARS also enables researchers to represent the properties of micro-nanophotonic materials. It plays a crucial role in photonic crystal and metamaterial research. By measuring the angular dependence of light scattered or emitted by these materials, researchers can gain insights into their unique optical properties, such as band structure, dispersion, capabilities of light field regulation, and photonic density of states. This information is essential for designing and optimizing these materials for various applications, such as sensing, imaging, and light manipulation. Furthermore, the ability to control and manipulate light at the nanoscale has the potential to revolutionize photonics and enable the development of new technologies. Therefore, ARS is a powerful technique for investigating optical inverse problems and harnessing the optical properties of micro-nanophotonic materials.

In recent years, various methods have emerged for generating angle-resolved spectra and processing data. Mechanical angle-scanning spectroscopy measurement and Fourier planar imaging represent the two primary methods employed. Urgent application needs have spurred the rapid proposal of abundant data analysis algorithms. However, it is important to acknowledge that each approach possesses its limitations and drawbacks. Therefore, the provision of a rational framework is imperative to summarize these methods and applications, guiding future advancements in the field.

In terms of generating angle-resolved spectra, two main methods will be introduced. The first method is mechanical angle-scanning spectroscopy measurement. Jér?me et al. developed the Mueller matrix scattering ellipsometry (MMSE), which enables scanning both inside and outside the incident plane (Fig. 2). Heather et al. employed the goniometric optical scatter instrument (GOSI) to obtain angle-resolved reflection for s and p polarizations of the target (Fig. 3). Chen et al. developed the tomographic Mueller-matrix scatterometer (TMS), where the incident beam is focused on the rear focal plane of the objective lens, and the angle of the incident light is changed by rotating the mirror (Fig. 4). Zhao et al. designed an interferometric imaging phase measurement system that allows for changes in the angle of reference light by moving the lens perpendicular to the optical axis (Fig. 5). The mechanical angle-scanning spectra measurement method correlates the incident and exit angles of the sample with the motor step size or the position of the optical element, offering greater intuitiveness and flexibility. However, it requires higher precision in the motor or translation stage, and mechanical vibrations can decrease system stability.

The second method for generating angle-resolved spectra is Fourier planar imaging. Zhang et al. proposed momentum-space imaging spectroscopy (MSIS), consisting of a momentum-space imaging module, spectral imaging detection module, and phase resolution measurement module (Fig. 7). Fourier planar imaging allows researchers to obtain optical signals corresponding to the exit angles of all samples simultaneously without the need for mechanical movement. This approach provides a more stable system. However, it necessitates higher optical path requirements and is significantly influenced by lens aberrations and numerical aperture.

Next, several data analysis algorithms will be introduced for momentum-space imaging and optical inverse problems. In momentum-space imaging, time-domain coupled mode theory (TCMT) is employed to extract photonic eigenstates from experimental spectra. Additionally, the angular spectral method (ASM) is used to study the optical field regulation capability of micro- and nano structures. In optical inverse scattering problems, researchers adopt a reverse thinking approach to map the data space, which includes the spectra, to the parameter space where specific features of the scattering target reside. Rigorous coupled wave analysis (RCWA), finite element method (FEM), and finite-difference time-domain (FDTD) are utilized in rigorous simulations to obtain theoretical data. Various methods can be employed to solve optical inverse scattering problems, including library search algorithms, the least square method, and neural network algorithms. In optical scattering imaging, more complex nonlinear fractions are required for target image reconstruction, such as optical cone transformation and inversion of Lippmann-Schwinger integral equations using deep neural networks. It is also necessary to incorporate spatial filtering and spatial compound algorithms to reduce noise and enhance imaging quality.

Finally, numerous applications of ARS will be discussed. These applications encompass measuring the optical constants of materials, characterizing material profiles and defects, studying the optical properties of micro- and nano photonic materials, as well as scattering imaging and other relevant areas. However, it is important to acknowledge that this technique also has certain drawbacks. One notable limitation is the cost and complexity associated with instruments and techniques required for achieving high-precision angle measurements. The implementation of ARS often necessitates specialized hardware and software, which can be expensive and challenging to deploy in certain settings. Additionally, environmental factors such as temperature and vibration can impact angle measurement accuracy, making it difficult to obtain reliable results in some applications.

ARS is a versatile and powerful technique with applications in optics, biomedicine, materials science, and imaging. It provides valuable insights into light behavior and enables the development of tailored materials for practical applications. Despite limitations, ARS continues to be an area of great interest and research in various fields. With ongoing development and refinement, this technology holds the potential to unlock new findings and enhance our understanding of complex materials.