Light from free space enters the end face of a large-core step-index optical fibre at angle $\theta_i$ to the normal. Core and cladding indices are $n_1=1.5$ and $n_2=1.4$. The largest guided entry angle in degrees is ________.
At the limiting guided ray, refraction at the end face and total internal reflection at the core boundary give numerical aperture $\sin\theta_{max}=\sqrt{n_1^2-n_2^2}$ for air/free-space incidence.
Here $\sqrt{1.5^2-1.4^2}=\sqrt{0.29}=0.538516$. Therefore $\theta_{max}=\sin^{-1}(0.538516)=$ **32.582°**, in the official 32–33° range.
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