BRIEF REVIEW OF ELECTROMAGNETICS From Maxwell s equations we have v v H E 0 t v v v v E P H j 0 t t where 0 and 0 are the permeability and permittivity in vacuum and v v v E r t Real E0 r exp i t Electric Field v v v H r t Real H0 r exp i t MagneticField v v v j r t Real j0 r exp i t Current Density v v v P r t Real P0 r exp i t Polarizability v v v v The amplitudes E 0 H0 j0 and P0 are generally complex and contain phase information For example v v v v v v E r t E0 r exp i t E0 r exp i r exp i t v v v E0 r exp i t r The amplitude and phase may be functions of position By incorporating all time dependence into exp i t where is the radian frequency of oscillation of the fields we make the frequency domain approximation In reality the amplitude and phase may have a time dependence v v v E r t E0 r t exp i t r t This situation occurs in a pulsed laser E0 t E r t t exp i t 1 E0 t E0 By ignoring terms proportional to compared to we apply the t t slowly varying envelope approximation SVEA Typically Recall that c2 v v P 0 E 1 0 0 where c is the speed of light in vacuum 3 1010 cm s and is the susceptibility which could be complex By substituting the time dependent quantities into Maxwell s equations and applying the SVEA we get the frequency domain form of Maxwell s equations v v v v H E E0 exp i t 0 0 i H0 exp i t t v v E0 i 0H0 v v v v H0 exp i t j0 exp i t 0 E0 exp i t P0 exp i t t t v v v v H0 j0 i 0E0 i P0 v v v H0 j0 i D0 v v v v v where D0 0E0 P0 Note that since P 0 E then v v v v D0 0E0 0 E0 0 1 E0 To obtain the wave equation we take the curl of the first of Maxwell s equations v v E 0 H t v v v E P 0 j 0 t t t If v we limitvourselves to propagation of electromagnetic waves in free v space then j 0 and P 0 Furthermore if the charge density 0 then E 0 and v v v v E E 2E 2E With these approximations we get the wave equation in vacuum or free space 2 v 1 2E 2 v E 0 0 E 2 2 2 t c t 2v 2 Applying the SVEA we get 2E0 2 E0 0 c The general form of the solutions to the wave equation is v v v v v E r t E0 r exp i t k 0 r v where k0 the vacuum wave vector defines the direction of propagation v k0 c 0 and 0 is the wavelength in vacuum E k H For propagation of electromagnetic fields in dielectrics gas liquids and non metal solids we retain the polarizability in Maxwell s equations It is convenient to separate the polarizability into passive and active components v v v P Pn Pa n passive a active The passive polarizability contributes to the slowing of the speed of propagation and ultimately the index of refraction The active polarizability contributes to sources and sinks of photons and v ultimately to gain and absorption With j 0 v v v v v v Pn Pa E P E H 0 0 t t t t t v v v v with Pn 0 nE and Pa 0 aE where n and a are the passive and active susceptibilities we get 3 v v v v v v E Pa E E Pa H 0 0 n 0 1 n t t t t t v v 2 E Pa 0n t t where n 1 n 1 2 is the index of refraction Continuing with the wave equation v v v Pa 2 E E 0 0n t t t v 2Pa E E 0 0n 0 2 t 2 t 2v 2 2v v v n 2 2E 2Pa E 0 2 c t 2 t 2v By applying the SVEA in the frequency domain n 2 2v 2v 2v E0 E0 0 P0a c The term on the RHS is the source or sink of photons due to gain or absorption in the laser media v 2 v 2v 2 0 P0a 0 0 aE0 E 2 a 0 c so that 2 v v 2 2E0 n 2E0 aE0 2 c c 2 v 2v 2 E0 n a E0 0 c2 Note that in solid media such as crystals it is common that the susceptibility depends on the direction of propagation in the crystals If so becomes a tensor and v 2 v 2v 2 v 2 E0 n E a E 0 E0 2 2 2 c c c 4