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Plane wave expansion of optical fields is well known in optical textbooks. The wave vector is normal to the wavefront and has the magnitude indicating the angular spatial frequency of the fields. Maxwell equations lead to the equations connecting the wave vector and temporal angular frequency, so- called, dispersion relations. The relations are derived for isotropic dielectric, metal, and anisotropic crystals. Then laws of reflection and refraction are derived from continuity of the wave vector components along the boundary. Geometrical construction based on the condition is shown for refraction into isotropic materials and crystals. Evanescent waves arising from total reflection are also formulated from the construction. Then formation of interference fringes between two plane waves propagating in different directions is graphically displayed and optical beat signals generated between different frequencies are explained in terms of the movement of the fringe patterns. Finally diffraction by periodic structures is constructed together with the difference between thin and thick gratings is also discussed. The case of a moving grating is also discussed.

Analysis of optical systems by wave vectors