# 物理代写|量子光学代写Quantum Optics代考|Source Located Above Topmost Layer

## 物理代写|量子光学代写Quantum Optics代考|Source Located Above Topmost Layer

We first discuss the situation depicted in Fig. 8.19 where both the source and observation points $z^{\prime}$ and $z$ are located above the topmost interface of the stratified medium. We denote the topmost medium with $n$ and the position of the interface with $z_n$. The exponential term in Eq. (8.49)
$$\exp i k_{n z}\left|z-z^{\prime}\right|= \begin{cases}\exp i k_{n z}\left(z-z^{\prime}\right) & \text { for } z>z^{\prime} \ \exp i k_{n z}\left(z^{\prime}-z\right) & \text { for } zz^{\prime} and a downgoing wave for z<z^{\prime}. To account properly for the boundary conditions at z_n, we introduce the total Green’s function$$
\begin{aligned}
& G_{i j}^{\mathrm{tot}}\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)=-\frac{\hat{z}i \hat{z}_j}{k_n^2} \delta\left(\boldsymbol{r}-\boldsymbol{r}^{\prime}\right) \ & +\frac{i}{4 \pi} \int_0^{\infty} \frac{1}{k{n z}}\left{\left\langle\left[e^{i k_{n z}\left|z-z^{\prime}\right|} \epsilon_i^{T E}\left(\boldsymbol{k}n^{ \pm}\right)+A_n^{T E} e^{i k{n z} z} \epsilon_i^{T E}\left(\boldsymbol{k}n^{+}\right)\right] \epsilon_j^{T E}\left(\boldsymbol{k}_n^{ \pm}\right)\right\rangle\right. \ & \left.+\left\langle\left[e^{i k{n z}\left|z-z^{\prime}\right|} \epsilon_i^{T M}\left(\boldsymbol{k}n^{ \pm}\right)+A_n^{T M} e^{i k{n z} z} \epsilon_i^{T M}\left(\boldsymbol{k}n^{+}\right)\right] \epsilon_j^{T E}\left(\boldsymbol{k}_n^{ \pm}\right)\right\rangle\right} k\rho d k_\rho . \
&
\end{aligned}
$$The reasons for adding the two upgoing waves with \mathrm{TE} and \mathrm{TM} character are as follows. Electric Field. The Green’s dyadics gives the electric field at position r, and it is the electric field which must fulfill Maxwell’s boundary condition at z=z_n. For this reason we have added a reflected field for the unprimed coordinate z and not the primed one. Homogeneous Solution. The additional terms are plane waves which are solutions of the homogeneous wave equation. For this reason, the total Green’s function still fulfills the defining equation with a delta-like source at \boldsymbol{r}=\boldsymbol{r}^{\prime}. Boundary Conditions. The coefficients A_n^{T E}, A_n^{T M} must be chosen such that the boundary conditions of Maxwell’s equations are fulfilled at the topmost interface. ## 物理代写|量子光学代写Quantum Optics代考|Heterodyne Spectroscopy With the reflected Green’s functions we are now in the position to simulate imaging through an imperfect Veselago lens. Indeed, it was already proposed in Pendry’s original work [54] to replace for a proof-of-principle experiment the perfect lens with an ordinary silver slab with \mu=\mu_0, and operating at a frequency where the permittivity becomes -\varepsilon_0. A first experimental realization was reported by Fang and coworkers [59]. Figure 8.20 shows the imaging with such a non-perfect superlens, using (a) a negative refractive index material with small losses, (b) a metal slab with stronger losses and a permeability of \mu_0, and (c) no slab at all. One clearly observes that even the non-perfect metal slab leads to a relatively good imaging of the point dipoles. Suppose that a dipole is located above a stratified medium, see Fig. 8.19, and we are interested in the radiation emitted to the far-field. From Eqs. (8.51) and (8.52) we observe that the reflected Green’s functions contain integrals of the form$$
\begin{aligned}
& \mathcal{I}1=\frac{i}{4 \pi} \int_0^{\infty} e^{i k{n z}\left(z+z^{\prime}-2 z_n\right)}(\ldots) k_\rho d k_\rho \
& \mathcal{I}2=\frac{i}{4 \pi} \int_0^{\infty} e^{i k{1 z}\left(z_1-z\right)+i k_{n z}\left(z^{\prime}-z_n\right)}\langle\ldots\rangle k_\rho d k_\rho .
\end{aligned}
$$\mathcal{I}{1,2} are for observation points z located above or below the stratified medium. We first undo the average over the azimuthal angle, see Eq. (B.6) and discussion thereafter, and rewrite the integrals in the form$$ \begin{aligned} & \mathcal{I}_1=\frac{i}{8 \pi^2} \int{-\infty}^{\infty} e^{i k_x\left(x-x^{\prime}\right)+i k_y\left(y-y^{\prime}\right)} e^{i k_{n z}\left(z+z^{\prime}-2 z_n\right)}[\ldots] d k_x d k_y \
& \mathcal{I}2=\frac{i}{8 \pi^2} \int{-\infty}^{\infty} e^{i k_x\left(x-x^{\prime}\right)+i k_y\left(y-y^{\prime}\right)} e^{i k_{1 z}\left(z_1-z\right)+i k_{n z}\left(z^{\prime}-z_n\right)}[\ldots] d k_x d k_y .
\end{aligned}
$$We are now seeking for the far-field limit of these expressions where r=(x, y, z) in the exponential becomes very large. The evaluation of this integral is identical to our previous discussion in Chap. 3 about the far-field representation of electromagnetic fields, where the stationary phase approximation has been used to obtain$$
\frac{i}{8 \pi^2} \int_{-\infty}^{\infty} e^{i \boldsymbol{k} \cdot r}(\ldots) d k_x d k_y \underset{k r \gg 1}{\longrightarrow} \frac{i}{8 \pi^2}\left(\frac{e^{i k r}}{r}\right)\left[-2 \pi i k_z(\ldots)\right]_{k=k \hat{r}}

## 物理代写|量子光学代写Quantum Optics代考|Heterodyne Spectroscopy

(b) 具有更强损耗和渗透率的金属板 $\mu_0$ ，(c) 根本没有平板。人们清楚地观察到，即使是 不完美的金属板也会导致点偶极子的相对较好的成像。

$$\left.\mathcal{I} 1=\frac{i}{4 \pi} \int_0^{\infty} e^{i k n z\left(z+z^{\prime}-2 z_n\right)}(\ldots) k_\rho d k_\rho \quad \mathcal{I} 2=\frac{i}{4 \pi} \int_0^{\infty} e^{i k 1 z\left(z_1-z\right)+i k_{n z}\left(z^{\prime}-z_n\right)}\langle\ldots\rangle k_\rho d\right)$$
$\mathcal{I} 1,2$ 用于观察点 $z$ 位于分层介质之上或之下。我们首先撤消方位角的平均值，参见等 式。(B.6)及其后的讨论，将积分改写为
$$\mathcal{I}1=\frac{i}{8 \pi^2} \int-\infty^{\infty} e^{i k_x\left(x-x^{\prime}\right)+i k_y\left(y-y^{\prime}\right)} e^{i k{n z}\left(z+z^{\prime}-2 z_n\right)}[\ldots] d k_x d k_y \quad \mathcal{I} 2=\frac{i}{8 \pi^2} \int-\infty^{\infty}$$

$$\frac{i}{8 \pi^2} \int_{-\infty}^{\infty} e^{i k \cdot r}(\ldots) d k_x d k_y \underset{k r \gg 1}{\longrightarrow} \frac{i}{8 \pi^2}\left(\frac{e^{i k r}}{r}\right)\left[-2 \pi i k_z(\ldots)\right]_{k=k \hat{r}}$$

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