Study of linear propagation Solution of 2nd order equation Propagation through medium No change in frequency spectrum z W Z=0 To make F.T easier shift in frequency Expand k value around central freq wl Expand k to first order, leads to a group delay:

Study of linear propagation Expansion orders in k(W)--- Material property

Combination of both: can be pulse broadening, compression, Propagation in dispersive media: the pulse is chirped and broadening Propagation in nonlinear media: the pulse is chirped Combination of both: can be pulse broadening, compression, Soliton generation

e(t,0) eik(t)d e(t,0) Propagation in the time domain PHASE MODULATION E(t) = e(t)eiwt-kz n(t) or k(t) e(t,0) eik(t)d e(t,0)

e(DW,0) e(DW,0)e-ik(DW)z Propagation in the frequency domain DISPERSION n(W) or k(W) e(DW,0) e(DW,0)e-ik(DW)z Retarded frame and taking the inverse FT:

PHASE MODULATION DISPERSION

Application to a Gaussian pulse