ECE4122 Chap.8 Optical Waveguides and Fibres
Optical Waveguides and Fibres
An optical fibre guides light because a higher-index core is surrounded by lower-index cladding. Ray language describes total internal reflection and launch acceptance, while wave language describes discrete modes and propagation constants. Both models depend on the refractive-index contrast and wavelength, but they answer different questions.
Numerical aperture measures acceptance in the launch medium.
For launch from air into an ideal step-index fibre it is the square root of the core-index squared minus the cladding-index squared. The acceptance angle follows from the launch-medium index times sine of that angle. This relation must be modified when the external medium is not air.
The normalised frequency V combines core radius, free-space wavelength and numerical aperture.
A step-index fibre is single-mode below the standard first higher-mode threshold. Larger V supports more spatial modes and therefore increases modal dispersion. Single-mode operation removes intermodal spreading but not material or waveguide dispersion.
Attenuation reduces optical power with distance, commonly expressed in decibels per kilometre. Dispersion broadens pulses and limits data rate or link length.
A link calculation should keep connector, splice and margin losses separate from fibre attenuation, while a pulse calculation should identify which dispersion contribution is being modelled. Results should be checked against the assumptions of weak guidance, step index and narrow spectral width.
What this chapter covers
- 01
Core-cladding confinement
- 02
Numerical aperture and acceptance
- 03
Normalised frequency
- 04
Single-mode condition
- 05
Attenuation and dispersion
Worked example · free
Compute fibre acceptance
- +1Use NA = square root of n1 squared minus n2 squared.
- +2n1 squared minus n2 squared = 1.48 squared - 1.46 squared = 0.0588.
- +1NA = square root of 0.0588 = 0.2425.
- +1For air, sin theta_a = NA.
- +1theta_a = arcsin(0.2425), approximately 14.0 degrees.
Key terms
- Core index
- The refractive index of the central guiding region.
- Numerical aperture
- A measure of light-acceptance capability under a stated launch medium.
- Normalised frequency
- A dimensionless parameter controlling fibre modal content.
- Single-mode fibre
- A fibre operated below the first higher spatial-mode threshold.
- Modal dispersion
- Pulse spreading caused by different modal transit times.
- Material dispersion
- Pulse spreading caused by wavelength-dependent refractive index.
Optical Waveguides and Fibres FAQ
Are ray and mode pictures interchangeable?
They are complementary approximations. Rays clarify acceptance and total internal reflection, while modes are needed for cut-off and propagation constants. Use the model whose assumptions match the scale, index contrast and quantity requested by the problem.
Does single-mode mean no dispersion?
No. It removes intermodal dispersion, but chromatic material and waveguide dispersion can remain. A finite source linewidth can still broaden pulses through wavelength-dependent material and confinement effects.
Why does wavelength affect modal content?
Normalised frequency is inversely proportional to wavelength, so shorter wavelengths can support more modes for fixed geometry and indices. Compare the resulting V value with the higher-mode threshold before declaring the operating regime.
Exam move
Separate launch, guidance and link-budget questions. Write the external index for acceptance, compute V for modal status, and use decibel addition only for power losses. Close with a statement about single-mode status or remaining dispersion. Build every fibre solution in three layers: launch, guidance and link performance.
For launch, draw the external medium, fibre axis and acceptance half-angle, then use the external refractive index explicitly. Calculate numerical aperture from the squared core and cladding indices and check that it is dimensionless and smaller than the core index. Predict how greater index contrast changes the cone before taking an inverse sine.
This prevents the air-launch identity from being applied unchanged to an immersed or coupled input. For guidance, calculate normalised frequency with core radius, not diameter, and keep radius and wavelength in the same unit. Compare V with the relevant step-index threshold and state the modal conclusion in words. Repeat the calculation after changing wavelength, radius or numerical aperture, predicting the direction first.
Shorter wavelength, larger core and larger numerical aperture all raise V. A conclusion of single-mode operation removes intermodal dispersion but does not prove that coupling loss, bend loss or chromatic dispersion is small. For propagation, distinguish attenuation from dispersion.
Add connector, splice and fibre power losses in decibels to form a link budget, but use a dispersion parameter with source spectral width and length to estimate pulse broadening. Keep the sign of dispersion for compensation reasoning while using its magnitude for a positive time spread. Compare that spread with a bit period or system tolerance so the result becomes a design decision rather than an isolated number.
Use ray and field models deliberately. Ray sketches clarify total internal reflection and acceptance, while modal fields explain cut-off, evanescent cladding tails and waveguide dispersion. State weak-guidance and step-index assumptions when using LP labels or the standard V threshold. In a final audit, check index ordering, unit cancellation, mode threshold and the physical direction of every design change.
Then explain which measurement would verify it: an angular launch scan for acceptance, a modal profile for confinement, an optical power test for loss, or a pulse response for dispersion.
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