Given refractive index dependence

The refractive index of the medium depends on intensity as $\mu(I)=\mu_0+\mu_2 I,$ where $\mu_0>0,\; \mu_2>0.$

Intensity distribution in the beam

The beam intensity decreases with increasing radius. Hence:

• intensity is maximum on the axis,
• intensity is smaller away from the axis.

Therefore, since $\mu_2>0$, $\mu(I) = \mu_0 + \mu_2 I$ is also maximum on the axis.

Relation between speed and refractive index

The speed of light in a medium is $v = \frac{c}{\mu},$ where $c$ is the speed of light in vacuum.

Thus, if refractive index $\mu$ is larger, speed $v$ is smaller.

Apply this to the beam

Since $\mu$ is maximum on the axis, $v=\frac{c}{\mu}$ is minimum on the axis.

Check options

• A: minimum on the axis of the beam — Correct.
• B: the same everywhere in the beam — Incorrect, because $\mu$ varies with intensity, and intensity varies with radius.
• C: directly proportional to the intensity $I$ — Incorrect, because $v=\frac{c}{\mu_0+\mu_2 I},$ so speed decreases with increasing $I$.
• D: maximum on the axis of the beam — Incorrect, opposite of the correct result.

Final answer

The speed of light is minimum on the axis of the beam.