Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index $\mathrm{n}_1$ surrounded by a medium of lower refractive index $\mathrm{n}_2$. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media $n_1$ and $n_2$ as shown in the figure. All rays with the angle of incidence $i$ less than a particular value $\mathrm{i}_{\mathrm{m}}$ are confined in the medium of refractive index $\mathrm{n}_1$. The numerical aperture (NA) of the structure is defined as $\sin \mathrm{i}_{\mathrm{m}}$.

If two structures of same cross-sectional area, but different numerical apertures $\mathrm{NA}_1$ and $\mathrm{NA}_2 \left(\mathrm{NA}_2<\mathrm{NA}_1\right)$ are joined longitudinally, the numerical aperture of the combined structure is