What it does
An optical fibre traps light in its core because the core’s refractive index n₁ is a little higher than the cladding’s n₂. How much light it accepts, how many ways (modes) the light can travel, how much a pulse spreads out, and how far a signal can go before it is too weak are all worked out here from a few numbers.
How to use it
- Pick an example or type your fibre: core index n₁, cladding index n₂, core diameter in µm, the wavelength, and whether the index is a step or a graded (parabolic) profile.
- For pulse spreading, give the link length, the fibre’s chromatic dispersion D at your wavelength (from its data sheet) and the source’s spectral width.
- For the power budget, give the transmitter power and receiver sensitivity in dBm, the fibre loss in dB/km, the numbers of connector pairs and splices with their losses, and a safety margin.
What the results mean
- Numerical aperture NA = √(n₁² − n₂²) sets the acceptance angle: light arriving within ±asin(NA) of the axis is guided.
- V-number V = 2πa·NA/λ (a is the core radius). A step-index fibre carries only one mode (LP01, in two polarisations) when V < 2.405; the cut-off wavelength is where V = 2.405.
- Guided modes: each LP mode stops being guided below its cut-off V, which is a zero of a Bessel function. The table lists them, and the total grows like V²/2.
- Effective index and mode-field diameter of the single mode: the effective index comes from solving the exact LP01 equation; the mode-field diameter uses Marcuse’s Gaussian fit.
- Modal spreading: in a multimode fibre, steep rays travel further than straight ones. Grading the index evens out their speeds, cutting the spread by roughly Δ/8.
- Chromatic spreading: different colours travel at different speeds, |D| × length × spectral width.
- Budget: losses in dB simply add. The link works if the received power stays above the sensitivity plus the margin.
Worked example
The default single-mode example is made up: n₁ = 1.45, n₂ = 1.445, an 8 µm core, at 1550 nm. The page gives NA 0.120, an acceptance half-angle of 6.91°, V = 1.951, so it is single-mode, with a cut-off at 1257 nm. The exact LP01 solution gives b = 0.3999 and an effective index of 1.44700; the mode-field diameter is about 10.4 µm. Over 80 km with D = 17 ps/(nm·km) and a 0.1 nm laser, pulses spread by 136 ps (bandwidth about 3.24 GHz). The budget: 20 dB of fibre, 1 dB of connectors and 0.8 dB of splices make 21.8 dB, so 0 dBm in gives −21.8 dBm at the receiver: it works with 3.2 dB to spare after the 3 dB margin, and the longest possible link is 92.8 km.
The multimode step-index example (50 µm core, 850 nm) has V = 44.8 and 1,022 modes; modal spreading of 67.6 ns over just 1 km limits it to a few MHz, which is why graded-index fibre (third example) is used instead.
FAQ
Where do I find D and the losses? On your fibre and component data sheets. Values change with wavelength; the examples are rounded, made-up teaching values.
Is the bit rate exact? No. 0.44 ÷ spread and 1 ÷ (2 × spread) are rules of thumb; real links depend on the modulation format and receiver.
Is it good enough to sign off a link? It is for study and planning, not a substitute for professional judgement and measurement with a power meter and an OTDR.
Where the numbers come from
- Step-index fibre optics under weak guidance (LP modes, cut-offs at Bessel function zeros, the LP01 eigenvalue equation): Gloge, D. (1971), Applied Optics 10, 2252; Snyder, A. W. and Love, J. D., Optical Waveguide Theory; Saleh and Teich, Fundamentals of Photonics, chapter 9.
- Mode-field radius approximation w/a = 0.65 + 1.619 V^−1.5 + 2.879 V^−6: Marcuse, D. (1977), Bell System Technical Journal 56, 703. Modal and chromatic pulse spreading and dB power budgets: standard textbook results (for example Keiser, Optical Fiber Communications).
- Checked against SciPy (jv, kv, jn_zeros, and brentq on the LP01 equation) and hand sums. The three example fibres and links use rounded, made-up values, not any manufacturer's data sheet.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.