Structural color has the outstanding advantages of high chemical stability and low environmental pollution. In recent years, it has received great attention from researchers in the fields of optical camouflage, dynamic display, secure communication, etc. Traditional structural color devices mainly use flat films or one-dimensional, two-dimensional, and three-dimensional periodic micro/nano structures. The anisotropic characteristics of these structures determine that the structural color has an angle dependence. The colors are different when we view from different angles, which limits its applications in many fields such as optical camouflage. It can only achieve camouflage effects from a certain detection angle. The reported work has experimentally verified that the angle dependence can be reduced by breaking the periodicity of three-dimensional structures so that they no longer have the anisotropic characteristics. However, it has not been studied from the perspective of theoretical calculation to indicate what kind of quasi-amorphous structures can achieve what degree of angle dependence reduction. In this paper, a random perturbation surface structure that can reduce the angle dependence is studied theoretically. Compared with the flat film structure, the effect of the perturbation surface structural parameters on reducing the angle dependence is calculated.

The color of the film interference structure is mainly determined by the central wavelength of the reflection spectrum. Therefore, we use the shift of the central wavelength peak to measure its angle dependence. In this paper, a finite element method (FEM) model of thin-film interference structures is established to investigate the reflection characteristics. In order to verify the validity of the model, the reflection spectra at perpendicular incidence of the FEM model are compared with those of multi-beam interference method and the measured fabricated samples. The FEM simulation results agree well with calculation results of the multi-beam interference method, and are in good agreement with the test results, which verifies the accuracy of the FEM model. Based on this model theory, the reflection spectra of rough surface structures with different perturbation characteristics at different incident angles are studied, and a set of structural parameters are optimized, namely fluctuation height H₁=40 nm and spectral index β=1.4. With this set of structural parameters, the reflection spectra and energy propagation characteristics of TE wave and TM wave of flat film structure and random perturbation surface structure are calculated using our FEM model. Finally, the reflection spectra and angle dependence of equivalent natural light are obtained.

For the flat film structure, the reflection spectrum shows a significant angle dependence. When the incident angle is 0° and 40° respectively, the corresponding peak wavelengths are 508 nm and 474 nm, respectively, and the blue shift reaches 34 nm, showing a remarkable change in color. For the random perturbation surface structure, the corresponding peak wavelength is 486 nm and 476 nm when the incident angle is 0° and 40° respectively, and the peak wavelength change is only 10 nm.

Aiming at the angle dependence of structural color in flat film structures, this paper studies the effect of a random perturbation surface structure on reducing the angle dependence of reflection spectrum. The results indicate that in the incident angle range of 0° to 40°, the reflection spectrum peak of the flat surface structure shifts by 34 nm, whereas that of the random perturbation surface structure only shifts by 10 nm. A decrease of 70.6% in angular dependence is thus achieved. It indicates that the perturbation surface structure plays a significant role in reducing angular dependence.