## 物理代写|量子光学代写Quantum Optics代考|Surface Plasmons

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## 物理代写|量子光学代写Quantum Optics代考|Surface Plasmons

Consider the interface between two media as depicted in Fig. 8.1. The upper one is a dielectric with permittivity $\varepsilon_1>0$, the lower one a metal with a negative permittivity $\varepsilon_2<0$. For simplicity, in the following we ignore the imaginary part of $\varepsilon_2$ and discuss implications of lossy materials at the end. The magnetic permeabilities in both materials are set to $\mu_0$.

Suppose that an electromagnetic wave with wavevector $\boldsymbol{k}1=\left(k_x, 0,-k{1 z}\right)$ propagates in the negative $z$-direction and impinges on the interface. Because of the continuity of the tangential electromagnetic fields, the parallel component $k_x$ of the wavevector must be conserved at the interface. The $z$-component of the wavevector in the metal is determined from the dispersion relation
$$k_x^2+k_{2 z}^2=\varepsilon_2 \mu_0 \omega^2$$
Because of $\varepsilon_2<0$, the right-hand side of the equation is negative and we immediately find $k_{2 z}^2<0$, which can only be fulfilled for an imaginary wavenumber $k_{2 z}$. In other words, inside the metal the wave cannot propagate but has an evanescent character with an amplitude that decays exponentially when moving away from the interface. As a result, a wave impinging from the dielectric side on the metal becomes reflected, with only small losses due to ohmic dissipation caused by the exponentially decaying fields inside the metal. In Sect. 8.3.3 we will provide a more thorough discussion of this reflection.

From this analysis it seems that the dielectric-metal interface is a boring object. Fortunately, this hasty judgement is not true. We will show next that a novel type of wave exists at metal-dielectric interfaces, so-called surface plasmons, which are bound to the interface and have to be excited optically in a specific manner. In fact, we can distinguish two kinds of guided modes, namely
Transverse magnetic (TM): $\quad \boldsymbol{H}=H_y \hat{\boldsymbol{y}}$ is parallel to interface
Transverse electric (TE): $\quad \boldsymbol{E}=E_y \hat{\boldsymbol{y}}$ is parallel to interface.

## 物理代写|量子光学代写Quantum Optics代考|Kretschmann and Otto Geometry

In the previous section we have discussed that at the interface between a metal and a dielectric a novel type of excitations exists, so-called surface plasmons, associated with coherent surface charge excitations bound to light fields. However, we have also seen that these surface plasmons cannot be excited directly by optical means, or, as Harry Atwater has expressed it carefully, can only be excited “under the right circumstances.” To understand what these right circumstances are, we first analyze energy and momentum conservation in an optical excitation process.

Light carries energy and momentum. In the following we adopt a photon language where the photon carries energy $\hbar \omega$ and momentum $\hbar \boldsymbol{k}$, but our reasoning works equally well for a purely classical electromagnetic description. Consider light oscillating with angular frequency $\omega$ and propagating in a medium with dielectric constant $\varepsilon_1$ that impinges on a metal surface located at $z=0$. The momentum carried by the photon is
$$\hbar \boldsymbol{k}=[\sin \theta \hat{\boldsymbol{x}}+\cos \theta \hat{z}] \frac{\hbar k_0}{n_1},$$
where $\theta$ is the angle of the incoming light with respect to the $z$-axis, $k_0=\frac{\omega}{c}$, and $n_1$ is the refractive index of the dielectric medium. In order to excite a surface plasmon the following quantities must be conserved:

• $\operatorname{energy} \hbar \omega$
• parallel momentum $\hbar k_x=\left(\hbar k_0 / n_1\right) \sin \theta$.
Note that for the slab structure the translational symmetry along $z$ is broken and according to Noether’s theorem the momentum along $z$ is not conserved.

# 量子光学代考

## 物理代写|量子光学代写Quantum Optics代考|Surface Plasmons

• componentofthewavevectorinthemetalisdetermined fromthedispersionrelation $k_x^2+k_{2 z}^2=\varepsilon_2 \mu_0 \omega^2$ Becauseof ivarepsilon_2<0

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## MATLAB代写

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