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Noting, the equation indicates that the group speed is equal to the phase velocity only when the refractive index is a constant (in dependent to the frequency) or the frequency is a constant (independent to the wavenumber). In this case both phase and group velocity are independent of frequency and equal to c/n.

When referring to polarization states, the p-polarization refers to the polarization plane parallel to the polarization axis of the polarizer being used ("p" is for "parallel"). The s-polarization refers to the polarization plane perpendicular to the polarization axis of the polarizer. A linear polarizer, by design, polarizes light in the p-polarization.

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In the lin­ear al­ge­bra of real ma­tri­ces, Her­mit­ian op­er­a­tors are sim­ply sym­met­ric ma­tri­ces. A ba­sic ex­am­ple is the in­er­tia ma­trix of a solid body in New­ton­ian dy­nam­ics. The or­tho­nor­mal eigen­vec­tors of the in­er­tia ma­trix give the di­rec­tions of the prin­ci­pal axes of in­er­tia of the body.

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Otherwise, if they vary with freuqency, the medium is called dispersive, the relation w=w(k) is known as the dispersion relation of the medium.

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An or­tho­nor­mal com­plete set of eigen­vec­tors or eigen­func­tions is an ex­am­ple of a so-called “ba­sis.” In gen­eral, a ba­sis is a min­i­mal set of vec­tors or func­tions that you can write all other vec­tors or func­tions in terms of. For ex­am­ple, the unit vec­tors , , and  are a ba­sis for nor­mal three-di­men­sion­al space. Every three-di­men­sion­al vec­tor can be writ­ten as a lin­ear com­bi­na­tion of the three.

S&P polarization refers to the plane in which the electric field of a light wave is oscillating. S-Polarization is the plane of polarization perpendicular to the page (coming out of the monitor screen). P-polarization is the plane of polarization parallel to the page (in the plane of the monitor screen). See figure below:

Most op­er­a­tors in quan­tum me­chan­ics are of a spe­cial kind called Her­mit­ian. This sec­tion lists their most im­por­tant prop­er­ties.

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In a given medium, both the phase velocity and group velocity depend on the frequency and on the medium's material properties (refractive index)

The first says that you can swap  and  if you take the com­plex con­ju­gate. (It is sim­ply a re­flec­tion of the fact that if you change the sides in an in­ner prod­uct, you turn it into its com­plex con­ju­gate. Nor­mally, that puts the op­er­a­tor at the other side, but for a Her­mit­ian op­er­a­tor, it does not make a dif­fer­ence.) The sec­ond is im­por­tant be­cause or­di­nary real num­bers typ­i­cally oc­cupy a spe­cial place in the grand scheme of things. (The fact that the in­ner prod­uct is real merely re­flects the fact that if a num­ber is equal to its com­plex con­ju­gate, it must be real; if there was an  in it, the num­ber would change by a com­plex con­ju­gate.)