# Polarizaition and susceptibility

## Main Question or Discussion Point

In some unit, the relation of (linear) polarization and susceptibility can be written of

$$P(t) = \chi E(t)$$

$$P(\omega) = \chi(\omega) E(\omega)$$

why change the time to frequency? Why polarization depends on frequency?

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Andy Resnick
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)=$\int \chi(t-\tau)E(\tau)d\tau$

And taking the Fourier transform of this equation provides your second expression.

If the material responds instantaneously and has no memory[$\chi(t-\tau) = \chi\delta(t-\tau)$], then the convolution integral reduces to your first expression.

Having a frequency-dependent susceptibility is simply dispersion.

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)=$\int \chi(t-\tau)E(\tau)d\tau$

And taking the Fourier transform of this equation provides your second expression.

If the material responds instantaneously and has no memory[$\chi(t-\tau) = \chi\delta(t-\tau)$], 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