Step 1: Understanding the Concept:
The focal length of a lens is determined by the Lens Maker's Formula. It depends on the refractive index of the material and the radii of curvature of the two surfaces.
Step 2: Key Formula or Approach:
Lens Maker's Formula:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_{1}} - \frac{1}{R_{2}} \right) \]
: Detailed Explanation:
When the lens is cut along the principal axis (\(XOX'\)):
1. The curvature of the first surface (\(R_{1}\)) does not change for either half.
2. The curvature of the second surface (\(R_{2}\)) does not change for either half.
3. The material (refractive index \(\mu\)) remains identical.
Since none of the parameters in the Lens Maker's Formula change, the focal length of each half remains exactly the same as the original lens. However, the intensity of the image formed by each half will be reduced because the aperture area is halved.
Step 3: Final Answer:
The focal length remains same.