1. Meaning of numerical aperture

For an optical fibre-like structure, the numerical aperture is $NA = \sin i_m$ where $i_m$ is the maximum incident angle (with the axis / allowed acceptance condition) for which the ray remains guided by total internal reflection.

So, a larger $NA$ means the structure can accept and guide a wider range of rays.

2. Given two structures joined longitudinally

We have two parts connected one after another:

• first part with numerical aperture $NA_1$
• second part with numerical aperture $NA_2$

and it is given that $NA_2 < NA_1$

This means the second structure accepts a smaller cone of rays than the first one.

3. Condition for guidance in the combined structure

For light to be guided through the entire combined structure, a ray must be guided in:

• the first section, and
• the second section.

Even if a ray is allowed in the first section, it will be lost if it is not allowed in the second section.

Therefore, the combined system can only guide those rays which satisfy the stricter condition, i.e. the smaller numerical aperture.

Hence, $NA_{\text{combined}} = \min(NA_1, NA_2)$

Since $NA_2 < NA_1$, $NA_{\text{combined}} = NA_2$

4. Checking options

• A: $\dfrac{NA_1 NA_2}{NA_1+NA_2}$ — not physically correct here.
• B: $NA_1+NA_2$ — impossible; combined acceptance cannot exceed both.
• C: $NA_1$ — incorrect because second section is more restrictive.
• D: $NA_2$ — correct.

5. Final answer

The numerical aperture of the combined structure is $\boxed{NA_2}$