The essential answer
Dispersion is the frequency dependence of a medium or structure's optical response. Phase index determines the phase accumulated by a monochromatic wave, while group index describes the delay of a sufficiently narrowband envelope when a group-velocity description is valid. They need not be equal. In waveguides and cavities, geometry contributes dispersion as well as the material. Use the quantity appropriate to the measurement: phase for interference, group delay for pulse timing or mode spacing, and higher derivatives when pulse distortion matters.
How to reason about it
For a simple transparent medium, group index is ng = n+ω dn/dω, equivalently n−λ dn/dλ when λ is vacuum wavelength. A path of length L gives group delay ngL/c under the stated approximation. Broadband pulses experience different delays across their spectrum and can broaden or chirp, so a single delay is not always sufficient. Near strong absorption or sharp resonances, pulse reshaping makes naive velocity interpretations unreliable. In a waveguide, use the mode propagation constant and its derivatives rather than only the bulk material index. For a resonator, round-trip group delay determines local free spectral range. Temperature or gating can change both phase and loss, and their combined effects may move a resonance in ways that a constant-index model misses. Distinguish a measured transmission peak shift from a direct measurement of group velocity.
ILLUSTRATIVE · NOT A REPORTED RESULT
An example to work through
Assume an illustrative transparent 10 cm path has phase index 1.50 and group index 1.60 across a narrow pulse bandwidth. The envelope transit time is 1.60×0.10/(3×10⁸) = 0.533 ns. Using phase index would give 0.500 ns, a 33 ps difference. The extra delay relative to vacuum is 0.200 ns. These values assume negligible pulse distortion and uniform indices over the path. They show why a phase-calibrated optical length alone may not supply the correct timing correction for a pulse-based measurement.
Where interpretation fails
Do not treat a group velocity exceeding c or becoming negative in a strongly dispersive region as faster-than-light information transport. Pulse reshaping and the full causal response matter. Likewise, applying a bulk index to a tightly confined mode can miss substantial waveguide dispersion even far from absorption resonances.
The next decision
Select a dispersion model over the actual optical bandwidth and verify the observable it predicts. For narrowband timing use measured group delay; for broadband pulses include higher-order dispersion and loss. In cavities, compare measured mode spacing with the modeled round-trip delay rather than substituting a phase index by habit.
Sources & context
These sources support the concepts noted below. Our synthetic examples and research framing are educational interpretation; they are not findings from these organizations or validation of our ventures.
- MIT Mathematical Methods in Nanophotonics: Lecture Summaries ↗
Group velocity and dispersion lecture summary inspected; further reading for the original example.
Source check: Relevant excerpt checked; original illustrative calculation, not source data.
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