An intially parallel cyclindrical beam travels in a medium of refractive index `mu (I) = mu_(0) + mu_(2) I`, where `mu_(0)` and `mu_(2)` are positive constants and `I` is intensity of light beam. The intensity of the beam is decreasing with increasing radius.
Answer the following questions :
The initial shape of the wavefront of the beam is

Options

A. convex
B. concave
C. convex near the axis and concave near the periphery
D. planar

Answer

To solve the question regarding the initial shape of the wavefront of a cylindrical beam traveling in a medium with a varying refractive index, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Beam Characteristics**:
- The problem states that we have an initially parallel cylindrical beam. This means that before entering the medium, the light rays are parallel to each other and the wavefronts are uniform.

2. **Identifying the Wavefront Shape**:
- In optics, a wavefront is a surface over which an optical wave has a constant phase. For a parallel beam of light, the wavefronts are planar (flat) surfaces.

3. **Considering the Medium**:
- The refractive index of the medium is given as \( \mu(I) = \mu_0 + \mu_2 I \). This indicates that the refractive index depends on the intensity of the light beam, which varies with the radius from the center of the beam.

4. **Effect of Refractive Index on Wavefront**:
- As the beam travels through the medium, the intensity decreases with increasing radius, which suggests that the refractive index will also vary across the beam. However, this variation affects the propagation of the beam after it has entered the medium, not the initial shape of the wavefront.

5. **Conclusion on Initial Wavefront Shape**:
- Since the beam is initially parallel and cylindrical, the initial wavefront is planar. Thus, the answer to the question is that the initial shape of the wavefront of the beam is planar.

### Final Answer:
The initial shape of the wavefront of the beam is **planar**.