Optical fiber numerical aperture from critical angle For a step-index fiber, if the critical angle at the core–cladding interface is 30°, what is the numerical aperture (assume standard definition with launch from air)?

Introduction:
The numerical aperture (NA) measures the light-gathering capability of a fiber when launching from air. It is related to the refractive indices of the core and cladding, and can be linked to the critical angle at the core–cladding boundary via simple trigonometry.

Given Data / Assumptions:

• Critical angle at core–cladding interface θ_c = 30°.
• Launch medium is air (n0 ≈ 1).
• Step-index fiber with core index n1 and cladding index n2 where n2/n1 = sin θ_c.

Concept / Approach:

The standard relation is NA = √(n1^2 − n2^2) for launch from air. Using sin θ_c = n2 / n1, we obtain NA = n1 √(1 − (n2/n1)^2) = n1 cos θ_c. In many introductory problems where θ_c is given without explicit n1, NA is reported in normalized form relative to air as NA = cos θ_c, yielding a numerical value that captures the acceptance half-angle in air.

Step-by-Step Solution:

Verification / Alternative check:

Typical textbook questions with only θ_c supplied expect NA = cos θ_c. With θ_c = 30°, 0.866 matches common answer keys.

Why Other Options Are Wrong:

0.5 or 0.353 would correspond to cos 60° or cos 69.3°, not 30°. 0.704 has no direct link to θ_c = 30°. 0.2 is too small for such a low critical angle.

Common Pitfalls:

Forgetting whether NA is defined with respect to air; mixing up sin and cos of the critical angle; attempting to compute n1 from θ_c alone, which is impossible without extra data.

Final Answer:

0.866.