Refractive index and wavelength ($\lambda ={\lambda }_0/n$)

For a linear stationary interface, phase matching at a planar wave interface preserves frequency: the incident and transmitted electric fields must oscillate together at each boundary point. Their phase velocities can differ, so their wavelengths differ. Writing $\omega =2\pi f$, the definitions giveThis explains the compressed wavefront spacing in the figure without changing the time between crests at the interface. Moving interfaces or time-dependent material responses need not preserve frequency. ^[ref]

The notation $n({\lambda }_0)$ labels refractive index by vacuum wavelength; it does not mean the wavelength inside the material. Dependence on ${\lambda }_0$, equivalently on frequency, is optical dispersion. The field driving the material induces oscillating electric charges; their emitted electromagnetic fields combine with the applied field to change the resulting wave phase. This collective response motivates refractive index, rather than a picture of independently delayed little pieces of light. ^[ref]

To distinguish three speeds, track a wavefront for phase velocity, a narrow wave envelope for group velocity, and the earliest newly introduced disturbance for front velocity. For a transparent differentiable propagation branch,The denominator is the group refractive index. It describes envelope transit when loss and reshaping are small. It need not equal $n$, and neither phase velocity nor a reshaped pulse peak generally measures signal arrival. The front velocity article explains the separate relativistic causality constraint and why a detector threshold is not an exact causal-front measurement.