If two structures of the same cross-sectional area, but different numerical apertures and are joined longitudinally, the numerical aperture of the combined structure is [2015]

Question

Light guidance in an optical fibre can be understood by considering a structure comprising of a thin solid glass cylinder of refractive index $n_{1}$ surrounded by a medium of lower refractive index $n_{2}$ . 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 $i_{m}$ are confined in the medium of refractive index $n_{1}$ . The numerical aperture (NA) of the structure is defined as $\sin i_{m} .$

Q. If two structures of the same cross-sectional area, but different numerical apertures $N A_{1}$ and $N A_{2}$ $(N A_{2} < N A_{1})$ are joined longitudinally, the numerical aperture of the combined structure is [2015]

Smart Achievers · Accepted Answer

(4)

Numerical aperture, $N A = \frac{1}{n_{s}} \sqrt{n_{1}^{2} - n_{2}^{2}}$

Here, $n_{1}$

$∴$ Numerical aperture of the combined structure is equal to the smaller value of the two numerical apertures.

Solution

1 $\frac{N A_{1} N A_{2}}{N A_{1} + N A_{2}}$

2 $N A_{1} + N A_{2}$

3 $N A_{1}$

4 $N A_{2}$

Ans.
(4)

Numerical aperture, $N A = \frac{1}{n_{s}} \sqrt{n_{1}^{2} - n_{2}^{2}}$

Here, $N A_{2} < N A_{1} \Rightarrow i_{m 2} < i_{m 1}$

$∴$ Numerical aperture of the combined structure is equal to the smaller value of the two numerical apertures.

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Solution

1 NA1 NA2NA1+NA2

2 NA1+NA2

3 NA1

4 NA2