Explain Huygens' principle and its role in understanding the propagation of light waves. Provide mathematical expressions to support your explanation.

Huygens' principle is a fundamental concept in wave optics that explains the propagation of light waves. It states that each point on a wavefront acts as a source of secondary spherical wavelets, and the wavefront at a later time is determined by the envelope of these wavelets. In other words, every point on a wavefront can be considered as a point source of new waves, and the collective effect of all these secondary waves determines the shape and direction of the wavefront at a subsequent instant.

Huygens' principle helps in understanding phenomena like reflection, refraction, diffraction, and interference of light waves