The index profile of a core of multimode graded index fiber is given by?

^{\frac{1}{2}}};r < a$$" option2="$$N\left( r \right) = n1{\left[ {3 - 2\Delta {{\left( {\frac{{{r^2}}}{a}} \right)}^2}} \right]^{\frac{1}{2}}};r < a$$" option3="$$N\left( r \right) = n1{\left[ {5 - 2\Delta {{\left( {\frac{{{r^2}}}{a}} \right)}^2}} \right]^{\frac{1}{2}}};r > a$$” option4=”$$N\left( r \right) = n1{\left[ {1 – 2\Delta {{\left( {\frac{{{r^2}}}{a}} \right)}^2}} \right]^{\frac{1}{2}}};r < a$$" correct="option1"]

The correct answer is:

$$N\left( r \right) = n1{\left[ {1 – 2\Delta {{\left( {\frac{{{r^2}}}{a}} \right)}^2}} \right]^{\frac{1}{2}}};r < a$$

This is the equation for a graded index fiber, where $n_1$ is the refractive index of the core, $\Delta$ is the refractive index difference between the core and cladding, and $a$ is the core radius. The equation shows that the refractive index decreases radially from the center of the core to the cladding. This graded index profile helps to reduce modal dispersion, which is a type of distortion that occurs in multimode fibers.

Option A is incorrect because it does not include the factor of $\Delta$, which is the refractive index difference between the core and cladding. Option B is incorrect because it does not include the factor of $r^2$, which is the square of the radial distance from the center of the core. Option C is incorrect because it does not include the condition $r < a$, which specifies that the equation applies only to the core of the fiber.