Fiber Numerical Aperture Calculator

Formula used

• Numerical aperture (step-index, in air): NA = √(n1² − n2²)
• Acceptance half-angle in medium n0: θmax = sin⁻¹(NA / n0)
• From acceptance half-angle: NA = n0 · sin(θmax)
• Relative index difference: Δ = (n1 − n2) / n1
• V-number: V = (2πa/λ) · NA
• Modes (approx., step-index multimode): M ~ V² / 2 (best for large V)

How to use this calculator

1. Select what you want to solve for using the dropdown.
2. Enter the required values for the selected mode.
3. Optionally add core radius and wavelength for V-number.
4. Press Calculate to display results above the form.
5. Use Download CSV or Download PDF to export.

1) What numerical aperture represents

Numerical aperture (NA) is a compact way to describe how strongly a step‑index fiber guides light. Higher NA means a wider acceptance cone, easier coupling, and usually more supported modes. For many silica links, NA values around 0.12–0.14 are common in single‑mode designs, while many multimode systems use roughly 0.20–0.30 for relaxed alignment.

2) Step‑index NA from refractive indices

This calculator uses NA = √(n1² − n2²), where n1 is the core index and n2 is the cladding index. Small changes in indices matter: with n1 = 1.4682 and n2 = 1.4620, NA is about 0.1349. That corresponds to a modest index contrast suitable for low‑loss glass guidance.

3) Acceptance angle and coupling tolerance

In an external medium with refractive index n0, the acceptance half‑angle is θmax = sin⁻¹(NA/n0). In air (n0 ≈ 1), NA = 0.14 gives θmax near 8 degrees, while NA = 0.22 gives about 12.7 degrees. A larger cone helps LED/VCSEL coupling but may increase modal dispersion in multimode links.

4) Relative index difference and critical angle

The relative index difference Δ = (n1 − n2)/n1 summarizes how close the indices are. Typical Δ for silica fibers is on the order of 0.2%–1%. The calculator also reports the critical angle at the core‑cladding boundary, helping confirm total internal reflection conditions.

5) V‑number links geometry, wavelength, and NA

Add core radius a and wavelength λ to compute V = (2πa/λ)·NA. V captures how “large” the guided field is relative to the core. For example, a = 25 μm, λ = 850 nm, and NA ≈ 0.135 yields V around 25, clearly multimode. A smaller core (a ≈ 4.1 μm) at λ = 1550 nm with NA near 0.13 can push V toward the single‑mode region.

6) Single‑mode cutoff and mode estimate

For step‑index fibers, the common single‑mode cutoff is V < 2.405. Above that, additional modes can propagate. When V is large, the approximate guided mode count is M ~ V²/2. This estimate helps compare designs, especially for multimode bandwidth planning.

7) Practical input ranges and engineering checks

Use realistic indices: silica near 1.44–1.48 depending on wavelength and doping. Keep n1 > n2 for a valid step‑index model, and ensure NA ≤ n0 if you are interpreting θmax in that surrounding medium. If NA exceeds n0, the calculator flags the acceptance angle as not physically valid.

8) Typical applications and reporting

NA is used in connector tolerances, lens coupling design, and link budgeting where launch conditions matter. Exporting results to CSV supports documentation and comparison across candidate fibers. Use the PDF print export for clean lab notes, procurement specs, or design reviews.

FAQs

1) What is a good NA value for single‑mode fiber?

Many single‑mode designs use NA around 0.10–0.14. The exact value depends on core size, wavelength, and target bend performance. Always confirm with the V-number cutoff for your geometry.

2) Why does NA affect coupling efficiency?

NA sets the acceptance cone. A larger cone captures rays over a wider range of angles, improving alignment tolerance and coupling from sources like LEDs, but it can increase modal dispersion in multimode links.

3) Can NA be larger than the surrounding medium index?

For acceptance angle calculations, you typically need NA ≤ n0 because θmax uses sin⁻¹(NA/n0). If NA exceeds n0, the angle relation breaks, and the result is not physically meaningful.

4) What does the relative index difference Δ tell me?

Δ summarizes index contrast and relates to confinement strength. Small Δ often means lower dispersion and weaker confinement, while larger Δ improves confinement and bend tolerance but can increase sensitivity to profile details.

5) How do I use V-number to decide single‑mode operation?

Compute V = (2πa/λ)·NA. If V is below 2.405 for a step‑index design, it is likely single‑mode. If it is higher, multiple modes can propagate.

6) Are the mode count results exact?

No. The M ~ V²/2 result is an approximation that works best for large V in step‑index multimode fibers. Real mode counts depend on refractive index profile, wavelength, and launch conditions.

7) Which wavelengths are most common for fiber links?

Common operating windows include 850 nm for short‑reach multimode, and 1310 nm or 1550 nm for many single‑mode links. Wavelength affects indices, V-number, dispersion, and loss.