# 物理代写|量子光学代写Quantum Optics代考|Negative Refraction

#### Doug I. Jones

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## 物理代写|量子光学代写Quantum Optics代考|Negative Refraction

We have seen in this chapter that even such a simple system as a metal-dielectric interface exhibits interesting physics. One may wonder whether similar effects may exist for other material combinations, in particular systems with a large magnetic response $\mu(\omega)$ in the optical range. As we have discussed in the previous chapter such materials do not exist in nature but can be manufactured artificially. Suppose that we could manufacture metamaterials with arbitrary $\varepsilon(\omega), \mu(\omega)$ values, as schematically shown in Fig. 8.17. What could we expect?
$\boldsymbol{\varepsilon}>\mathbf{0}, \boldsymbol{\mu}>\mathbf{0}$. This situation corresponds to normal dielectrics, although there one usually has $\mu \approx \mu_0$. From the dispersion relation we get
$$k^2=\varepsilon(\omega) \mu(\omega) \omega^2 \Longrightarrow \omega=\frac{k c}{n(\omega)}, \quad n(\omega)=\sqrt{\frac{\varepsilon(\omega) \mu(\omega)}{\varepsilon_0 \mu_0}} .$$
Thus the speed of light is altered by the refractive index $n(\omega)$ in the medium with respect to free space, which gives rise to refraction of plane waves (or rays in case of geometric optics) at the interface between materials of different $n(\omega)$ values.

$\boldsymbol{\varepsilon}<\mathbf{0}, \boldsymbol{\mu}>\mathbf{0}$. This situation corresponds to metals. From the dispersion relation $\mu \varepsilon \omega^2=k_x^2+k_z^2$ we observe that only evanescent waves with an imaginary wavenumber exist inside the metal. For this reason, light becomes reflected at the interface between a dielectric and a metal. Additionally, there exists a novel type of excitation at such interfaces, the previously discussed surface plasmons. $\boldsymbol{\varepsilon}>\mathbf{0}, \boldsymbol{\mu}<\mathbf{0}$. This situation corresponds to magnetic metals. Owing to the duality principle, the physical properties are similar to normal metals; however, the role of electric and magnetic fields is exchanged. Again light cannot propagate inside the magnetic metal, and there exist surface plasmons (now with TE character) at the interface between a dielectric and a magnetic metal. $\boldsymbol{\varepsilon}<\mathbf{0}, \boldsymbol{\mu}<\mathbf{0}$. At first this looks like a not overly interesting case. Submitting these material parameters to the dispersion relation, we obtain the same result as in Eq. (8.43) with the refractive index $$n= \pm \sqrt{\frac{(-|\varepsilon|)(-|\mu|)}{\varepsilon_0 \mu_0}},$$ which appears to be identical to the $\varepsilon>0, \mu>0$ case. However, as already noted in the equation we have to be careful about the sign of the square root. Usually one does not bother about the sign, it is chosen positive and indeed this is correct for normal dielectrics. However, for $\varepsilon<0, \mu<0$ one has to choose the negative sign, as we will demonstrate in a moment, and this has given these materials the name negative refraction materials.

## 物理代写|量子光学代写Quantum Optics代考|The Perfect Lens

The Veselago lens passed into silence, if it ever received noticeable interest, but came to spotlight in 2000 when John Pendry analyzed transmission of evanescent fields in a Veselago lens and found that “negative refraction makes a perfect lens” [54]. I strongly recommend reading through this publication which serves as a beautiful example of how to write an excellent research paper. In the first paragraph Pendry states:
Optical lenses have for centuries been one of scientists’ prime tools. Their operation is well understood on the basis of classical optics: curved surfaces focus light by virtue of the refractive index contrast. Equally their limitations are dictated by wave optics: no lens can focus light onto an area smaller than a square wavelength. What is there new to say other than to polish the lens more perfectly and to invent slightly better dielectrics? In this Letter I want to challenge the traditional limitation on lens performance and propose a class of “superlenses,” and to suggest a practical scheme for implementing such a lens.
To understand why a Veselago lens is a “superlens” we will now analyze wave focusing using the transfer matrix approach previously developed. We consider a slab with thickness $d$ and filled with a negative refraction material that is located in vacuum. Suppose that an evanescent wave with wavenumber
$$k_{1 z}=\sqrt{k_0^2-k_x^2}=i \sqrt{k_x^2-k_0^2} \equiv i \kappa_0$$
impinges on the first interface. We use $k_0=\frac{\omega}{c}$ and assume $k_x>k_0$ for the evanescent wave. Inside the negative refraction material we have
$$k_{2 z}=\sqrt{\varepsilon \mu \omega^2-k_x^2}=i \sqrt{k_x^2-\varepsilon \mu \omega^2} \equiv i \kappa,$$
and will set $\varepsilon \rightarrow-\varepsilon_0$ and $\mu \rightarrow-\mu_0$ at the end of the calculation. Note that we have dropped all subscripts for the material indices. To compute the transmission coefficient of the slab structure, we use Eq. (8.38) and obtain
$$\tilde{T}{13}=\lim {\varepsilon \rightarrow-\varepsilon_0} \lim _{\mu \rightarrow-\mu_0} \frac{\left(\frac{2 \mu \kappa_0}{\mu \kappa_0+\mu_0 \kappa}\right)\left(\frac{2 \mu_0 \kappa}{\mu_0 \kappa+\mu \kappa_0}\right) e^{-\kappa d}}{1+\left(\frac{\mu \kappa_0-\mu_0 \kappa}{\mu \kappa_0+\mu_0 \kappa}\right)\left(\frac{\mu_0 \kappa-\mu \kappa_0}{\mu_0 \kappa+\mu \kappa_0}\right) e^{-2 \kappa d}} .$$

# 量子光学代考

## 物理代写|量子光学代写Quantum Optics代考|Negative Refraction

$\boldsymbol{\varepsilon}>\boldsymbol{0}, \boldsymbol{\mu}>\mathbf{0}$. 这种情况对应于正常的电介质，尽管通 常有 $\mu \approx \mu_0$. 从色散关系我们得到
$$k^2=\varepsilon(\omega) \mu(\omega) \omega^2 \Longrightarrow \omega=\frac{k c}{n(\omega)}, \quad n(\omega)=\sqrt{\frac{\varepsilon(\omega}{\varepsilon_0}}$$

$\varepsilon<0, \boldsymbol{\mu}>\mathbf{0}$. 这种情况对应于金属。从色散关系 $\mu \varepsilon \omega^2=k_x^2+k_z^2$ 我们观察到金属内部只存在波数为虚 数的脩逝波。为此，光在电介质和金属之间的界面处被 反射。此外，在这种界面上存在一种新型的激发，即前 面讨论的表面等离子体激元。 $\varepsilon>\mathbf{0}, \boldsymbol{\mu}<\mathbf{0}$. 这种情况 对应于磁性金属。由于二元性原理，物理性质与普通金 属相似；但是，电场和磁场的作用是互换的。同样，光 不能在磁性金属内部传播，并且在电介质和磁性金属之 间的界面处存在表面等离子体 (现在具有 TE 特征)。 $\boldsymbol{\varepsilon}<\mathbf{0}, \boldsymbol{\mu}<\mathbf{0}$. 乍一看，这似乎不是一个非常有趣的案 例。将这些材料参数提交给色散关系，我们得到与方程 式相同的结果。(8.43) 与折射率 $$n= \pm \sqrt{\frac{(-|\varepsilon|)(-|\mu|)}{\varepsilon_0 \mu_0}}$$ 这似乎与 $\varepsilon>0, \mu>0$ 案件。然而，正如在等式中已经 指出的那样，我们必须注意平方根的符号。通常人们不 会在意符号，它被选择为正，这对于普通电介质来说确 实是正确的。然而，对于 $\varepsilon<0, \mu<0$ 必须选择负号， 正如我们稍后将演示的那样，这使这些材料获得了负折射材料的名称。

## 物理代写|量子光学代写Quantum Optics代考|The Perfect Lens

Veselago 透镜虽然曾引起人们的关注，但后来却一直保 持沉默，但在 2000 年John Pendry 分析了 Veselago 透 镜中渐逝场的传输并发现”负折射是一个完美的透镜”时， 它引起了人们的关注 [54]。我强烈建议通读此出版物， 它是如何撰写优秀研究论文的一个很好的例子。在第一 段中，Pendry 指出:

$$k_{1 z}=\sqrt{k_0^2-k_x^2}=i \sqrt{k_x^2-k_0^2} \equiv i \kappa_0$$

$$k_{2 z}=\sqrt{\varepsilon \mu \omega^2-k_x^2}=i \sqrt{k_x^2-\varepsilon \mu \omega^2} \equiv i \kappa$$并将设置 $\varepsilon \rightarrow-\varepsilon_0$ 和 $\mu \rightarrow-\mu_0$ 在计算结束时。请注 意，我们已经删除了材料索引的所有下标。为了计算板 结构的传输系数，我们使用等式。(8.38) 并获得
$$\tilde{T} 13=\lim \varepsilon \rightarrow-\varepsilon_0 \lim _{\mu \rightarrow-\mu_0} \frac{\left(\frac{2 \mu \kappa_0}{\mu \kappa_0+\mu_0 \kappa}\right)\left(\frac{2 \mu_0 \kappa}{\mu_0 \kappa+\mu \kappa_0}\right) e}{1+\left(\frac{\mu \kappa_0-\mu_0 \kappa}{\mu \kappa_0+\mu_0 \kappa}\right)\left(\frac{\mu_0 \kappa-\mu \kappa_0}{\mu_0 \kappa+\mu \kappa_0}\right)}$$

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