## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Kramers–Kronig Relations

The polarization $P(t)$ (represented here as a scalar) at a given instant of time is the integrated response of the medium to the electric field up to that instant. Provided that the interaction is linear, we can write
$$P(t)=\int_{-\infty}^{\infty} h\left(t-t^{\prime}\right) \varepsilon_{0} E\left(t^{\prime}\right) \mathrm{d} t^{\prime}$$
where $h(t)$ represents the “memory function” of the medium. To understand the meaning of $h(t)$, we assume that the incident field is proportional to a Dirac deltaimpulse $E\left(t^{\prime}\right) \propto \delta\left(t^{\prime}\right)$ arriving at $t^{\prime}=0$. The resulting time dependent polarization is then proportional to $h(t)$, which is consequently called impulse response function.
If we apply, instead, an oscillating field $E(t)=\operatorname{Re}\left[\tilde{E}(\omega) \mathrm{e}^{\mathrm{j} \omega t}\right]$, then $P(t)=$ $\operatorname{Re}\left[\tilde{P}(\omega) \mathrm{e}^{\mathrm{j} \omega t}\right]$ will oscillate at $\omega$ and we obtain, with $t^{\prime \prime}:=t-t^{\prime}$
\begin{aligned} \tilde{P}(\omega) \mathrm{e}^{\mathrm{j} \omega t} &\left.=\int_{-\infty}^{\infty} h\left(t-t^{\prime}\right) \varepsilon_{0} \tilde{F}(\omega)\right) \mathrm{e}^{\mathrm{j} \omega t^{\prime}} \mathrm{d} t^{\prime} \ &=\varepsilon_{0} \tilde{E}(\omega) \mathrm{e}^{\mathrm{j} \omega t} \int_{-\infty}^{\infty} h\left(t^{\prime \prime}\right) \mathrm{e}^{-\mathrm{j} \omega t^{\prime \prime}} \mathrm{d} t^{\prime \prime} \end{aligned}
Therefore,
$$\tilde{P}(\omega)=\varepsilon_{0} \tilde{E}(\omega) \tilde{H}(\omega)$$

where
$$\tilde{H}(\omega)=\int_{-\infty}^{\infty} h(t) \mathrm{e}^{-\mathrm{j} \omega t} \mathrm{~d} t$$
is the Fourier transform of the impulse response. A comparison with Eq. (1.7) shows that this so-called transfer function is identical to the susceptibility
$$\tilde{\chi}(\omega)=\chi^{\prime}(\omega)+\mathrm{j} \chi^{\prime \prime}(\omega)=H(\omega) .$$

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Wave Propagation in Anisotropic Media

We now extend our treatment of wave propagation to optically anisotropic media (usually crystals), where the relation between $\mathbf{P}$ and $\mathbf{E}$ (and therefore $\mathbf{D}$ and $\mathbf{E}$ ) depends on the direction of $\mathbf{E}$ within the medium. In the framework of the linear oscillator model, the reason for this is the anisotropy of the restoring force.

One consequence of optical anisotropy is the dependence of the propagation index on the direction of the wave vector and the polarization state of the wave. As we shall see, for a given direction of the wave vector there exist two linear polarization states with well defined, generally different propagation indices. At a border between an anisotropic medium and another one, the two states are refracted in different directions-this is the reason why anisotropic media are also called birefringent.

In an anisotropic, linear medium, the vectors $\mathbf{P}$ and $\mathbf{E}$ are generally not collinear, but related by the more general linear equation
\begin{aligned} &P_{1}=\varepsilon_{0} \chi_{11} E_{1}+\varepsilon_{0} \chi_{12} E_{2}+\varepsilon_{0} \chi_{13} E_{3} \ &P_{2}=\varepsilon_{0} \chi_{21} E_{1}+\varepsilon_{0} \chi_{22} E_{2}+\varepsilon_{0} \chi_{23} E_{3} \ &P_{3}=\varepsilon_{0} \chi_{31} E_{1}+\varepsilon_{0} \chi_{32} E_{2}+\varepsilon_{0} \chi_{33} E_{3} \end{aligned}
or
$$P_{i}=\varepsilon_{0} \sum_{j=1}^{3} \chi_{i j} E_{j}$$
in the following we will adopt Einstein’s convention, according to which the double occurrence of an index in one term implies summation over the values of this index, so that Eq. (2.107) can be written as
$$P_{i}=\varepsilon_{0} \chi_{i j} E_{j} .$$

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Kramers–Kronig Relations

$$P(t)=\int_{-\infty}^{\infty} h\left(t-t^{\prime}\right) \varepsilon_{0} E\left(t^{\prime}\right) \mathrm{d} t^{\prime}$$

$$\left.\tilde{P}(\omega) \mathrm{e}^{\mathrm{j} \omega t}=\int_{-\infty}^{\infty} h\left(t-t^{\prime}\right) \varepsilon_{0} \tilde{F}(\omega)\right) \mathrm{e}^{\mathrm{j} \omega t^{\prime}} \mathrm{d} t^{\prime} \quad=\varepsilon_{0} \tilde{E}(\omega) \mathrm{e}^{\mathrm{j} \omega t} \int_{-\infty}^{\infty} h\left(t^{\prime \prime}\right) \mathrm{e}^{-\mathrm{j} \omega t^{\prime \prime}} \mathrm{d} t^{\prime \prime}$$

$$\tilde{P}(\omega)=\varepsilon_{0} \tilde{E}(\omega) \tilde{H}(\omega)$$

$$\tilde{H}(\omega)=\int_{-\infty}^{\infty} h(t) \mathrm{e}^{-\mathrm{j} \omega t} \mathrm{~d} t$$

$$\tilde{\chi}(\omega)=\chi^{\prime}(\omega)+\mathrm{j} \chi^{\prime \prime}(\omega)=H(\omega)$$

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Wave Propagation in Anisotropic Media

$$P_{1}=\varepsilon_{0} \chi_{11} E_{1}+\varepsilon_{0} \chi_{12} E_{2}+\varepsilon_{0} \chi_{13} E_{3} \quad P_{2}=\varepsilon_{0} \chi_{21} E_{1}+\varepsilon_{0} \chi_{22} E_{2}+\varepsilon_{0} \chi_{23} E_{3} P_{3}$$

$$P_{i}=\varepsilon_{0} \sum_{j=1}^{3} \chi_{i j} E_{j}$$

$$P_{i}=\varepsilon_{0} \chi_{i j} E_{j}$$

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