= Refractive index and wavelength
{title2=$\lambda=\lambda_0/n$}
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 give
$$
\lambda_0=\frac{c}{f},\qquad \lambda=\frac{v_{\rm ph}}{f}=\frac{\lambda_0}{n},\qquad k=\frac{2\pi}{\lambda}=n\frac{2\pi}{\lambda_0}.
$$
This 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>. <The Feynman Lectures on Physics, Volume I by Richard Feynman and others/Chapter 31: The Origin of the Refractive Index>[The Feynman Lectures on Physics, Volume I, §31–1: boundary oscillations, vacuum and medium wavelengths, Fig. 31–2]{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>. <The Feynman Lectures on Physics, Volume I by Richard Feynman and others/Chapter 31: The Origin of the Refractive Index>[The Feynman Lectures on Physics, Volume I by Richard Feynman and others; §31–1: superposition of the applied field and fields from driven charges; near-unity-index approximation]{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,
$$
v_g=\left(\frac{dk}{d\omega}\right)^{-1}=\frac{c}{n+\omega\,dn/d\omega}=\frac{c}{n_g},\qquad n_g=n-\lambda_0\frac{dn}{d\lambda_0}.
$$
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.
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