# 物理代写|量子光学代写Quantum Optics代考|PHYS686

## 物理代写|量子光学代写Quantum Optics代考|Nonlocality in Time

In the following we consider a dielectric response that is nonlocal in time
$$\boldsymbol{D}(\boldsymbol{r}, t)=\int_0^{\infty} \varepsilon(\boldsymbol{r}, \tau) \boldsymbol{E}(\boldsymbol{r}, t-\tau) d \tau,$$
and a possibly similar relation between $\boldsymbol{B}$ and $\boldsymbol{H}$ with a time-dependent permeability $\mu$. In the following we consider time-harmonic fields, see Sect. 2.4. The convolution theorem of Fourier transforms then states that upon Fourier transformation the above relation becomes a product in frequency space,
$$\boldsymbol{D}(\boldsymbol{r}, \omega)=\varepsilon(\boldsymbol{r}, \omega) \boldsymbol{E}(\boldsymbol{r}, \omega) .$$
Note that we use the same symbols for the quantities in the time and frequency domain. In general we will work exclusively in either of these spaces, so there will be little danger of confusion. The neat thing about this transformation behavior is that Maxwell’s equations for time-harmonic fields, Eq. (2.34), look almost identical for frequency-dependent permittivities and permeabilities,

\begin{aligned} \nabla \cdot \varepsilon(\boldsymbol{r}, \omega) \boldsymbol{E}(\boldsymbol{r}, \omega) & =\rho_{\mathrm{ext}}(\boldsymbol{r}, \omega) \ \nabla \cdot \boldsymbol{B}(\boldsymbol{r}, \omega) & =0 \ \nabla \times \boldsymbol{E}(\boldsymbol{r}, \omega) & =i \omega \boldsymbol{B}(\boldsymbol{r}, \omega) \ \nabla \times \mu^{-1}(\boldsymbol{r}, \omega) \boldsymbol{B}(\boldsymbol{r}, \omega) & =\boldsymbol{J}_{\mathrm{ext}}(\boldsymbol{r}, \omega)-i \omega \varepsilon \boldsymbol{E}(\boldsymbol{r}, \omega) . \end{aligned}
For this reason, practically everything we have discussed in the previous chapters about time-harmonic fields can be directly carried over to a frequency-dependent system response. Exceptions are the Poynting’s theorem, which we will revisit in the next section, and the fact that the permittivities and permeabilities acquire imaginary contributions associated with losses. For propagating and evanescent waves this leads to damping and attenuation.

For conductors and metals it is often convenient to incorporate the response of the conduction carriers into the permittivity. Above we have already seen how this can be done for a Drude dielectric function. In the general case we can generalize Ohm’s law for a frequency-dependent conductivity.

## 物理代写|量子光学代写Quantum Optics代考|Poynting’s Theorem Revisited

In this section we revisit Poynting’s theorem for a linear medium, previously derived in Sect. 4.3, but account for dispersion and absorption effects. Quite generally, we expect two major modifications with respect to our previous result of Eq. (4.14):

• Because of dispersion the velocity of the energy flow becomes modified.
• Because of absorption the energy flow and density become attenuated during propagation.

In complete analogy to our previous derivation of Poynting’s theorem, we start with the power performed by the electromagnetic fields on a current distribution but now relate $\boldsymbol{J}{\text {ext }}$ to the fields $\boldsymbol{D}, \boldsymbol{H}$, $$\frac{d W}{d t}=\int{\Omega} \boldsymbol{J}{\mathrm{ext}} \cdot \boldsymbol{E} d^3 r=\int{\Omega}\left(\nabla \times \boldsymbol{H}-\frac{\partial \boldsymbol{D}}{\partial t}\right) \cdot \boldsymbol{E} d^3 r .$$
Using in the first term the transformation
$$\nabla \cdot \boldsymbol{E} \times \boldsymbol{H}=\boldsymbol{H} \cdot \nabla \times \boldsymbol{E}-\boldsymbol{E} \cdot \nabla \times \boldsymbol{H}=-\boldsymbol{H} \cdot \frac{\partial \boldsymbol{B}}{\partial t}-\boldsymbol{E} \cdot \nabla \times \boldsymbol{H}$$
$$\frac{d W}{d t}+\int_{\Omega}\left(\boldsymbol{E} \cdot \frac{\partial \boldsymbol{D}}{\partial t}+\boldsymbol{H} \cdot \frac{\partial \boldsymbol{B}}{\partial t}\right) d^3 r=-\oint_{\partial \Omega} \boldsymbol{E} \times \boldsymbol{H} \cdot d \boldsymbol{S} .$$
This is Poynting’s theorem for the macroscopic Maxwell’s equations. The second term on the left-hand side can be associated with the energy stored in the electromagnetic fields, and the term on the right-hand side describes the energy transport through the Poynting vector $\boldsymbol{S}=\boldsymbol{E} \times \boldsymbol{H}$.

# 量子光学代考

## 物理代写|量子光学代写Quantum Optics代考|Nonlocality in Time

∇⋅电子(r,哦)和(r,哦)=r和X吨(r,哦) ∇⋅乙(r,哦)=0 ∇×和(r,哦)=一世哦乙(r,哦) ∇×米−1(r,哦)乙(r,哦)=杰和X吨(r,哦)−一世哦电子和(r,哦).

## 物理代写|量子光学代写Quantum Optics代考|Poynting’s Theorem Revisited

• 由于色散，能量流的速度发生了变化。
• 由于吸收，能量流和密度在传播过程中会衰减。

d在d吨=∫哦杰和X吨⋅和d3r=∫哦(∇×H−∂丁∂吨)⋅和d3r.

∇⋅和×H=H⋅∇×和−和⋅∇×H=−H⋅∂乙∂吨−和⋅∇×H

d在d吨+∫哦(和⋅∂丁∂吨+H⋅∂乙∂吨)d3r=−∮∂哦和×H⋅d小号.

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