Identify each expression by what it is built from.

b → i, attenuation. This is the easiest and it fixes the table. A ratio of input to output power in decibels is the definition of loss:

\(\alpha=\dfrac{10}{L}\log_{10}\dfrac{P_{in}}{P_{out}} \ \ \text{dB/km}\)

a → iv, number of modes. The combination \(\dfrac{\pi a\,NA}{\lambda}\) is the fibre's normalised frequency — the V-number — written with the radius a, the numerical aperture NA and the wavelength. It is dimensionless, which is the giveaway, and the mode count follows from it:

\(V=\dfrac{2\pi a}{\lambda}\,NA, \qquad M\approx\dfrac{V^{2}}{2}\ \ \text{(step index)}\)

A fibre is single-mode when V < 2.405, the first zero of the Bessel function J₀ — which is why single-mode fibre needs a core only a few micrometres across.

c → iii, dispersion. The expression is a derivative of group delay with respect to wavelength, normalised to length. That is precisely the definition of the dispersion coefficient, usually quoted in ps/(nm·km); multiplied by the source spectral width it gives the pulse spreading.

d → ii, intermodal delay. Only refractive indices, a length and the velocity of light appear, so the result is a time. It is the classic step-index result for the delay between the fastest ray (axial) and the slowest (at the critical angle):

\(\Delta t=\dfrac{L\,n_1(n_1-n_2)}{n_2 c}=\dfrac{Ln_1\Delta}{c}\)

Assemble. a-iv, b-i, c-iii, d-ii — option 3.

The dimensional shortcut. One expression is dimensionless (a mode count), one is in decibels (a loss), one is a derivative with respect to wavelength (a dispersion) and one has the dimensions of time (a delay). Matching by dimensions alone answers the question without recalling a single fibre formula.

Why intermodal delay matters. It is the dominant limit on step-index multimode fibre, and it is what graded-index profiles were invented to cancel — the outer rays travel a longer path but through a lower-index region, so all modes arrive together.

Hence, the correct matching is a-iv, b-i, c-iii, d-ii.