Graded Index Profile Calculator

A common graded-index core model uses a power-law refractive index profile:

• n₁ is the core index at the center (r=0).
• n₂ is the cladding index (boundary r=a).
• a is the core radius, and g controls the shape.
• The exact relative index difference is Δ = (n₁² − n₂²) / (2 n₁²).
• Numerical aperture is NA = √(n₁² − n₂²).
• If wavelength λ is given, the normalized frequency is V = (2πa/λ)·NA.

Note: This calculator uses the above model for design exploration and teaching. Real fibers can have measured deviations from the ideal power-law profile.

1. Select an input mode: provide n₂ or provide Δ.
2. Enter n₁, core radius a, and exponent g.
3. Enter a radial position r to evaluate n(r).
4. Optionally add wavelength (nm) to compute the V-number.
5. Choose profile table points to generate a radius-wise index table.
6. Click Calculate. Results will appear above the form.
7. Use Download CSV or Download PDF from the results panel.

Sample values for a parabolic graded-index fiber (illustrative only).

1) Why graded-index profiles matter

In a graded-index (GI) fiber, the refractive index decreases from the core center toward the cladding. Higher‑order rays spend more time in lower‑index outer regions, increasing their speed and compensating for longer paths. This reduces intermodal delay and improves bandwidth compared with step‑index fibers.

2) The power‑law model used here

This calculator uses the common power‑law form n(r)=n₁√(1−2Δ(r/a)ᵍ) for 0≤r≤a. It is a practical engineering approximation because it is differentiable, easy to sample into tables, and captures typical GI manufacturing targets. Outside the core (r>a) the cladding is treated as constant n₂.

3) Connecting n₂ and Δ consistently

You can provide n₂ directly or provide the relative index difference Δ. The exact relationship used is Δ=(n₁²−n₂²)/(2n₁²). In many silica fibers, Δ is on the order of 0.001 to 0.02, while n₁ often falls near 1.44–1.49 depending on composition and wavelength.

4) Choosing the exponent g

The exponent g controls curvature. A parabolic profile (g≈2) is widely used because it can strongly reduce intermodal delay in multimode systems. Lower g values flatten the core more gradually, while higher values approach a sharper transition. Use the table output to see how quickly the index drops with radius for your chosen g.

5) Radius sampling and table quality

The profile table samples r from 0 to a using evenly spaced points in r/a. For quick checks, 11–51 points is usually enough. Export 101–301 points for smoother plots or simulation inputs; very high counts mainly help external interpolation.

6) Numerical aperture and coupling insight

The calculator reports NA=√(n₁²−n₂²), which relates to acceptance cone in air (approximately sin(θₘₐₓ)≈NA for small angles). Larger NA generally improves coupling tolerance but can increase modal content in multimode fibers. Comparing NA across candidate profiles helps you balance launch conditions, bending sensitivity, and connector alignment margins.

7) V-number and mode behavior

If you supply wavelength, the tool computes V=(2πa/λ)·NA. V is a compact indicator of guidance strength: small V supports fewer modes, while large V supports many. For step‑index fibers, single‑mode guidance is commonly associated with V<2.405. GI fibers use V mainly as a comparative metric across designs and wavelengths.

8) Practical notes for real fibers

Measured GI profiles can deviate from an ideal power‑law due to dopant diffusion, process limits, or deliberate shaping (depressed cladding, alpha profiles). Use this tool for early sizing and documentation, then validate against measured index data. Exported tables can feed ray tracing or simple modal estimates.

1) Should I enter n₂ or Δ?

Use n₂ when you know the cladding index from a datasheet. Use Δ when specifications list relative index difference. The tool keeps the two consistent using the exact Δ relationship.

2) What happens if r is larger than a?

For r>a, the calculator assumes you are in the cladding and returns n(r)=n₂. This matches the piecewise model used in most introductory GI analyses.

3) Do a and r need the same unit?

They can be entered in different units. The calculator converts both to meters internally, then uses r/a. The generated table is displayed using the unit chosen for a.

4) How many table points should I choose?

Use 11–51 points for quick inspection, 101–301 points for exporting smooth data, and fewer points if you only need the scalar metrics. Very large point counts mainly help external interpolation workflows.

5) Why is the wavelength optional?

The index profile and NA do not require wavelength in this simplified model. Wavelength is only needed to compute the normalized frequency V, which changes with λ through 2πa/λ.

6) What does NA tell me in practice?

NA estimates how much launch angle the fiber can accept and still guide light. A higher NA generally eases coupling but can increase supported mode count and sensitivity to launch conditions in multimode systems.

7) Why does Δ have an upper limit here?

The model uses √(1−2Δ(r/a)ᵍ), so 1−2Δ must stay positive. The limit also reflects typical glass fibers, where Δ is usually much smaller than 0.2.