In some unit, the relation of (linear) polarization and susceptibility can be written of [tex]P(t) = \chi E(t)[/tex] but I also read some expression in other text reads [tex]P(\omega) = \chi(\omega) E(\omega)[/tex] why change the time to frequency? Why polarization depends on frequency?
Your equations are not really written correctly. The first one, the time dependent one, should really be written as a convolution: The polarization of a linear isotropic medium with a local but noninstantaneous response (but still independent of time) is: P(t)=[itex]\int \chi(t-\tau)E(\tau)d\tau[/itex] And taking the Fourier transform of this equation provides your second expression. If the material responds instantaneously and has no memory[[itex]\chi(t-\tau) = \chi\delta(t-\tau)[/itex]], then the convolution integral reduces to your first expression. Having a frequency-dependent susceptibility is simply dispersion.
Oh ... I just wonder why in textbook they don't say it is a convolution! So you mean in frequency domain susceptibility is the repsonse function? BTW, can you tell me one text in which the author show clearly the convolution relation b/w polarization, susceptibility and field? I am writing a short report and need a reference