To derive the lens maker's formula, let's consider a thin lens with two spherical surfaces. The lens has a refractive index of n and radii of curvature R1 and R2 for the first and second surfaces, respectively. The lens has a thickness of t at the center.



- The lens is assumed to be thin, meaning that its thickness is much smaller than the radii of curvature.
- The lens is symmetric, so the center of thickness coincides with the optical axis.
- The lens has an object at a distance u from the first surface and an image at a distance v from the second surface.



- Using Snell's law, we can determine the angle of refraction at the first surface:
- n1 sin θ1 = n sin θ2
- θ1 is the angle of incidence, and θ2 is the angle of refraction.
- Since the lens is thin, we can assume small angles and approximate sin θ1 ≈ θ1 and sin θ2 ≈ θ2.
- Therefore, the lens equation for the first surface can be written as: θ1/n1 ≈ θ2/n



- Similarly, we can apply Snell's law to the second surface:
- n2 sin θ2 = n sin θ3
- θ3 is the angle of refraction after the light passes through the lens.
- By approximating sin θ2 ≈ θ2 and sin θ3 ≈ θ3, we get: θ2/n2 ≈ θ3/n



- The lens equation relates the object distance (u), image distance (v), and the focal length (f) of the lens:
- 1/f = (n - 1) × (1/R1 - 1/R2)
- We can express the radii of curvature in terms of the lens geometry:
- R1 = R1 - t
- R2 = -R2 + t
- Plugging these values into the lens equation and simplifying, we obtain the lens maker's formula:
- 1/f = (n - 1) × [(n1 - 1)/R1 + (n2 - 1)/R2]



- The lens maker's formula considers the following sign conventions:
- Distances to the left of the lens are negative, and distances to the right are positive.
- Radii of curvature are positive for convex surfaces and negative for concave surfaces.
- Focal length is positive for converging lenses and negative for diverging lenses.

By applying these steps, the lens maker's formula can be derived, providing a useful tool for calculating the focal length of a lens based on its refractive index and geometry.