Using Huygen’s wave theory, derive Snell’s law of refraction.

To derive Snell's law of refraction using Huygens' wave theory, we consider the wavefronts of an incident wavefront and a refracted wavefront at the interface between two different media. Here are the steps:

Step 1: Consider an incident wavefront traveling from medium 1 with a wavefront normal represented by line NA.

Step 2: According to Huygens' principle, each point on the wavefront can be considered as the source of secondary spherical wavelets. Let's consider two such wavelets emitted from points B and C on the incident wavefront.

Step 3: These wavelets travel into medium 2 and reach the interface at different times. We need to find the refracted wavefront that corresponds to these two wavelets.

Step 4: Assume that the refracted wavefront has a normal represented by line ND.

Step 5: From point B, draw a perpendicular line BM (perpendicular to ND) to represent the refracted wavefront at point M. Similarly, from point C, draw a perpendicular line CN (also perpendicular to ND) to represent the refracted wavefront at point N.

Step 6: According to Huygens' principle, the refracted wavefront at point M is tangent to the secondary wavelet from point B, and the refracted wavefront at point N is tangent to the secondary wavelet from point C.

Step 7: The wavefront at point M is part of the refracted wavefront, and we can consider it as the wavefront origin for the refracted wavefront. Similarly, the wavefront at point N is part of the refracted wavefront and can be considered as the wavefront origin.

Step 8: Connect points A, M, and N. Since the wavefronts are continuous, the line AN represents the refracted wavefront.

Step 9: Draw lines BN and CM to represent the incident wavefront.

Step 10: Now, we have two triangles, ΔABN and ΔACM, and they share the same angle BAN. Additionally, angles BAN and CAN are both 90 degrees since they are perpendicular to the wavefronts. Therefore, angles ABN and ACM are equal.

Step 11: Using the fact that the sum of angles in a triangle is 180 degrees, we can write: ABN + BAN + ANB = 180 degrees and ACM + CAN + ANM = 180 degrees.

Step 12: We know that BAN and CAN are both 90 degrees. So, ANB + ABN = 180 degrees and ANM + ACM = 180 degrees.

Step 13: Rearranging the above equations, we get: ANB = 180 - ABN and ANM = 180 - ACM.

Step 14: Notice that the angles ANB and ANM are the angles of incidence (i) and refraction (r), respectively.

Step 15: Snell's law of refraction states that the ratio of the sines of the angles of incidence and refraction is constant for a given pair of media. Mathematically, we can write it as: sin(i) / sin(r) = constant.

Step 16: From step 14, we have: ANB = 180 - ABN = 180 - i and ANM = 180 - ACM = 180 - r.

Step 17: Plugging the values from step 16 into Snell's law, we get: sin(180 - i) / sin(180 - r) = constant.

Step 18: Simplifying the equation using the trigonometric identity sin(180 - x) = sin(x), we have: sin(i) / sin(r) =