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

2023年4月11日

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

Using the transfer matrix formalism, we can now easily compute the generalized reflection and transmission coefficients for stratified media. The black dashed line indicated with (i) in Fig. 8.12 shows the reflection coefficient for a single glass-gold interface. As previously discussed in Sect. 8.1.1, the dispersion relations for photons and surface plasmons do not cross, and consequently no surface plasmons can be directly launched by light: the reflection coefficient $R^{T M}$ is close to unity for all incoming angles, with only small losses attributed to minor field penetration and ohmic losses in the metal.

Things change considerably for the Kretschmann geometry, see red solid line indicated with (ii) in Fig. 8.12. For an angle of approximately $43^{\circ}$ the energy and momentum of the incoming light (through the upper glass medium) coincide with those of the surface plasmon at the lower silver interface, and a surface plasmon is launched. This can be clearly seen as a pronounced dip of the generalized reflection coefficient where practically all energy of the incoming light is transferred to the surface plasmons. Because the fields of surface plasmons are strongly confined, their dispersion depends very sensitively on changes of the local dielectric environment. For a semi-infinite slab the plasmon dispersion is given by Eq. (8.6),
$$k_x=k_1 \sin \theta=\sqrt{\frac{\varepsilon_2 \varepsilon_3}{\varepsilon_0\left(\varepsilon_2+\varepsilon_3\right)}} k_0,$$
with $\varepsilon_2$ and $\varepsilon_3$ being the permittivities of the metal and of the dielectric material below the metal, respectively, and we have expressed the parallel momentum of the incoming light at the reflection dip in the form $k_1 \sin \theta$. When the dielectric permittivity is changed by a small amount $\varepsilon_3+\delta \varepsilon_3$, also the angle of the reflection dip changes by a small amount $\theta+\delta \theta$. Linearization of the above expression yields $$k_1[\sin \theta+\cos \theta \delta \theta] \approx \sqrt{\frac{\varepsilon_2 \varepsilon_3}{\varepsilon_0\left(\varepsilon_2+\varepsilon_3\right)}}\left[1+\frac{1}{2} \frac{\varepsilon_2}{\varepsilon_2+\varepsilon_3} \frac{\delta \varepsilon_3}{\varepsilon_3}\right] k_0 .$$
Thus, to the lowest order of approximation the change $\delta \theta$ is given by
$$\delta \theta \approx \frac{1}{2} \frac{\varepsilon_2}{\varepsilon_3+\varepsilon_2} \frac{\delta \varepsilon_3}{\varepsilon_3} \tan \theta .$$

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

At the beginning of this chapter we have discussed surface plasmons for a single metal-dielectric interface. With the transfer matrix approach we are now in the position to compute surface plasmon dispersions also for more complicated systems, such as metal films in case of the Kretschmann geometry. In the following we discuss in slightly more detail how the two approaches are connected. To make things clear from the beginning: the transfer matrix provides a rigorous solution of Maxwell’s equation, so we do not have to improve on it. However, we shall find it convenient to get a more intuitive interpretation for these rigorous results.

We start by considering, in analogy to the single metal-dielectric interface shown in Fig. 8.1, a metal slab with permittivity $\varepsilon_2$ and thickness $d$ embedded in a dielectric medium with permittivity $\varepsilon_1$. Because of symmetry, the solutions are even or odd functions with respect to $z$. As detailed in Exercise 8.6, one can derive the modified surface plasmon conditions for the metal slab
$$\tanh \left(\frac{\kappa_{2 z} d}{2}\right)= \begin{cases}-\left(\frac{\kappa_{1 z} \varepsilon_2}{\kappa_{2 z} \varepsilon_1}\right) & \text { for } H_y \text { even } E_z \text { odd } \ -\left(\frac{\kappa_{2 z} \varepsilon_1}{\kappa_{1 z} \varepsilon_2}\right) & \text { for } H_y \text { odd, } E_z \text { even, }\end{cases}$$
which must be solved numerically. Figure 8.14 shows the dispersions for a $30 \mathrm{~nm}$ thick silver film. As can be seen, the frequencies of the even and odd modes are split, which can be understood in terms of coupling between the modes confined to the upper and lower interface of the metal slab. When the film thickness is increased, see Fig. 8.15 for a $80 \mathrm{~nm}$ film, the coupling is reduced and the mode splitting is considerably smaller.

# 量子光学代考

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

$$k_x=k_1 \sin \theta=\sqrt{\frac{\varepsilon_2 \varepsilon_3}{\varepsilon_0\left(\varepsilon_2+\varepsilon_3\right)}} k_0,$$ 其中 $\varepsilon_2$ 和 $\varepsilon_3$

$$\delta \theta \approx \frac{1}{2} \frac{\varepsilon_2}{\varepsilon_3+\varepsilon_2} \ frac{\delta \varepsilon_3}{\varepsilon_3} \tan \theta 。$$

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

$$\tanh \left(\frac{\kappa_{2 z} d}{2}\right)= \begin{cases}-\ left(\frac{\kappa_{1 z} \varepsilon_2}{\kappa_{2 z} \varepsilon_1}\right) & \text { for } H_y \text { even } E_z \text { odd } \-\left( \frac{\kappa_{2 z} \varepsilon_1}{\kappa_{1 z} \varepsilon_2}\right) & \text { for } H_y \text { odd, } E_z \text { even, }\end{cases}$$

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