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Plane wave expansion is a powerful technique used for the calculation of photonic crystal band gaps in both two-dimensional (2D) and three-dimensional (3D) systems. In this method, the electromagnetic field is expanded in terms of plane waves, which are then used to calculate the band structure of the photonic crystal. The band structure provides information about the allowed and forbidden states of the crystal, as well as the band gap width, which is a measure of the crystal's ability to confine light. The plane wave expansion method is particularly useful in the study of photonic crystals due to its ability to accurately predict the position and width of the band gap, as well as the underlying physics responsible for the band gap formation. With this technique, researchers can explore the properties of photonic crystals in greater detail, leading to a better understanding of their potential applications in areas such as optical communication, sensing, and computing.