How are numerical aperture and acceptance angle related?

NA = √(n₁² − n₂²) depends only on the core and cladding indices. Launching from a medium of index n₀, the acceptance half-angle is θₐ = arcsin(NA/n₀). A ray inside that cone meets the core–cladding boundary beyond the critical angle θc = arcsin(n₂/n₁) and is totally reflected. A ray outside it loses part of its power into the cladding at every bounce and dies out after a few reflections. For n₁ = 1.48 and n₂ = 1.46, NA ≈ 0.24 and the acceptance half-angle in air is about 14°.

When is a fiber single-mode, and how is V calculated?

The normalized frequency is V = (2πa/λ)·NA, with a the core radius. A step-index fiber with V < 2.405 (the first zero of J₀) carries only the fundamental LP₀₁ mode. The cutoff wavelength is λc = 2πa·NA/2.405, and the fiber is single-mode above it. A G.652 fiber has a core of about 8.2 μm and λc ≈ 1260 nm, so it is single-mode at 1310 and 1550 nm but carries a few modes at 850 nm. For large V there are about V²/2 modes in a step-index fiber and V²/4 in a graded one.

Why does graded-index fiber have so much more bandwidth than step-index multimode?

In step-index fiber the steep rays travel a longer path at the same speed as the axial ray, so they arrive late: Δτ/L ≈ n₁Δ/c, about 99 ns/km for a 100 μm core. In graded-index fiber the index falls off away from the axis, so off-axis rays travel a longer path but move faster in the low-index outer region, and the two effects largely cancel. For a parabolic profile the ray-model spread is only (n₁ − n₂)²/(2c·n₂) ≈ n₁Δ²/(2c), about 0.2 ns/km for OM3, which gives a bandwidth–distance product of a few GHz·km.

Why does a bent fiber leak light?

In a bend, a ray that was parallel to the axis hits the outer wall at a smaller angle of incidence. In ray optics an axial ray meets the outer wall at sin φ = R/(R + a), and once that drops below n₂/n₁ it is no longer totally reflected. This gives a critical bend radius R_c = a·n₂/(n₁ − n₂). Bend tighter than that and many rays escape into the cladding. Macrobend loss in real single-mode fiber needs wave theory, but the trend is the same: fibers with a larger index contrast tolerate tighter bends.

How are the link margin and maximum length calculated?

Received power = launch power − attenuation × length − connector losses − splice losses. The margin is received power − receiver sensitivity − safety margin, and the link is reliable only if it is positive. The power-limited length is where the margin reaches zero. Separately, modal and chromatic dispersion (|D|·Δλ·L) spread the pulses, and the tool requires the total spread to stay below half a bit period. That gives the dispersion-limited length, and the smaller of the two is the maximum unrepeated distance.

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🧶Fiber Optics Simulator - TIR Ray Tracing, Numerical Aperture, V-Number, Link Budget

Q: Why does graded-index fiber have so much more bandwidth than step-index multimode?

In step-index fiber the steep rays travel a longer path at the same speed as the axial ray, so they arrive late: Δτ/L ≈ n₁Δ/c, about 99 ns/km for a 100 μm core. In graded-index fiber the index falls off away from the axis, so off-axis rays travel a longer path but move faster in the low-index outer region, and the two effects largely cancel. For a parabolic profile the ray-model spread is only (n₁ − n₂)²/(2c·n₂) ≈ n₁Δ²/(2c), about 0.2 ns/km for OM3, which gives a bandwidth–distance product of a few GHz·km.

Q: Why does a bent fiber leak light?

In a bend, a ray that was parallel to the axis hits the outer wall at a smaller angle of incidence. In ray optics an axial ray meets the outer wall at sin φ = R/(R + a), and once that drops below n₂/n₁ it is no longer totally reflected. This gives a critical bend radius R_c = a·n₂/(n₁ − n₂). Bend tighter than that and many rays escape into the cladding. Macrobend loss in real single-mode fiber needs wave theory, but the trend is the same: fibers with a larger index contrast tolerate tighter bends.

Q: How are the link margin and maximum length calculated?

Received power = launch power − attenuation × length − connector losses − splice losses. The margin is received power − receiver sensitivity − safety margin, and the link is reliable only if it is positive. The power-limited length is where the margin reaches zero. Separately, modal and chromatic dispersion (|D|·Δλ·L) spread the pulses, and the tool requires the total spread to stay below half a bit period. That gives the dispersion-limited length, and the smaller of the two is the maximum unrepeated distance.

Drag the launch angle on the canvas and watch a ray zig-zag down a step-index fiber by total internal reflection, or leak a little at every bounce (Fresnel equations) once it is outside the acceptance cone. Graded-index fibers bend rays into sinusoids, and a tight bend makes light escape. The modes tab solves the LP modes exactly with Bessel functions and draws the b–V diagram and mode patterns; the link tab works out the power budget, dispersion and maximum length; and an animation compares how a pulse spreads in multimode and single-mode fiber.

Free · No sign-up · Runs in your browserLast updated: 2026-10-29

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How to use Fiber Optics Simulator

1On the ray tracing tab pick a step-index, graded-index or bent fiber and set the core and cladding indices and core diameter
2Drag on the canvas or use the slider to change the launch angle; watch TIR, the acceptance cone and leakage, or draw a fan of rays
3On the modes tab enter a wavelength to see V, the single-mode cutoff, the guided modes, the b–V diagram and mode fields
4On the link budget tab enter attenuation, connectors, splices, source and receiver sensitivity to get the margin and maximum length
5On the pulse tab press Play to compare pulse spreading with distance in step multimode, graded multimode and single-mode fiber

FAQ

How are numerical aperture and acceptance angle related?: NA = √(n₁² − n₂²) depends only on the core and cladding indices. Launching from a medium of index n₀, the acceptance half-angle is θₐ = arcsin(NA/n₀). A ray inside that cone meets the core–cladding boundary beyond the critical angle θc = arcsin(n₂/n₁) and is totally reflected. A ray outside it loses part of its power into the cladding at every bounce and dies out after a few reflections. For n₁ = 1.48 and n₂ = 1.46, NA ≈ 0.24 and the acceptance half-angle in air is about 14°.
When is a fiber single-mode, and how is V calculated?: The normalized frequency is V = (2πa/λ)·NA, with a the core radius. A step-index fiber with V < 2.405 (the first zero of J₀) carries only the fundamental LP₀₁ mode. The cutoff wavelength is λc = 2πa·NA/2.405, and the fiber is single-mode above it. A G.652 fiber has a core of about 8.2 μm and λc ≈ 1260 nm, so it is single-mode at 1310 and 1550 nm but carries a few modes at 850 nm. For large V there are about V²/2 modes in a step-index fiber and V²/4 in a graded one.
Why does graded-index fiber have so much more bandwidth than step-index multimode?: In step-index fiber the steep rays travel a longer path at the same speed as the axial ray, so they arrive late: Δτ/L ≈ n₁Δ/c, about 99 ns/km for a 100 μm core. In graded-index fiber the index falls off away from the axis, so off-axis rays travel a longer path but move faster in the low-index outer region, and the two effects largely cancel. For a parabolic profile the ray-model spread is only (n₁ − n₂)²/(2c·n₂) ≈ n₁Δ²/(2c), about 0.2 ns/km for OM3, which gives a bandwidth–distance product of a few GHz·km.
Why does a bent fiber leak light?: In a bend, a ray that was parallel to the axis hits the outer wall at a smaller angle of incidence. In ray optics an axial ray meets the outer wall at sin φ = R/(R + a), and once that drops below n₂/n₁ it is no longer totally reflected. This gives a critical bend radius R_c = a·n₂/(n₁ − n₂). Bend tighter than that and many rays escape into the cladding. Macrobend loss in real single-mode fiber needs wave theory, but the trend is the same: fibers with a larger index contrast tolerate tighter bends.
How are the link margin and maximum length calculated?: Received power = launch power − attenuation × length − connector losses − splice losses. The margin is received power − receiver sensitivity − safety margin, and the link is reliable only if it is positive. The power-limited length is where the margin reaches zero. Separately, modal and chromatic dispersion (|D|·Δλ·L) spread the pulses, and the tool requires the total spread to stay below half a bit period. That gives the dispersion-limited length, and the smaller of the two is the maximum unrepeated distance.

Features

2D ray tracing in step-index fiber, with Fresnel leakage at every bounce outside the acceptance cone
Exact sinusoidal rays in parabolic graded-index fiber, plus bend leakage and the critical bend radius
Numerical aperture, acceptance angle, critical angle, bounces per metre, extra path and modal delay
Exact LP mode solutions: cutoff V, mode count, b–V diagram, effective index and mode intensity plots
Link power budget and dispersion: modal, chromatic, bandwidth–distance product, power vs dispersion limit
Pulse spreading animation on a shared time scale, with rms width versus distance

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