1. Given refractive index dependence

   The medium has intensity-dependent refractive index $\mu(I)=\mu_0+\mu_2 I,$ where $\mu_0>0$ and $\mu_2>0$.

2. Intensity profile of the beam

   The beam is initially parallel and cylindrical, and its intensity decreases with increasing radius.

   So:

   • intensity is maximum at the axis,
   • intensity is smaller near the periphery.

   Hence, since $\mu_2>0$, $\mu(r)=\mu_0+\mu_2 I(r)$ is maximum at the center and decreases outward.

3. Physical implication of refractive index being larger at the center

   Rays bend toward the region of higher refractive index.

   Since the refractive index is highest on the axis, rays away from the axis experience bending toward the axis.

   Therefore, the medium behaves like a self-focusing medium.

4. Conclusion about beam propagation

   A parallel beam entering such a medium will tend to converge.

5. Option check

   • A: diverge — incorrect, because index is higher at the center, not lower.
   • B: converge — correct.
   • C: diverge near the axis and converge near the periphery — incorrect; the overall index profile causes focusing toward the axis.
   • D: travel as a cylindrical beam — incorrect; nonuniform index changes the ray paths.

Therefore, the correct answer is $\boxed{\text{B: converge}}.$

6. Comparison with stored correct answer

   Stored correct answer: B

   My derived answer: B

   So they agree.