## 物理代写|量子光学代写Quantum Optics代考|Imaging of Far-Fields

We now return to the more complicated problem of imaging the far-fields by two lenses, see Fig. 3.5. Equipped with the machinery developed for field focusing things turn out to be very similar. For the three media we use the following refractive indices, impedances, and coordinate systems:
$\begin{array}{llll}\text { Medium 1 } & \ldots & n_1, Z_1, & \text { spherical coordinates with } r_1, \theta_1, \phi \ \text { Medium 2 } & \ldots & n_2, Z_2, & \text { cylinder coordinates with } \rho_2, \phi, z_2 \ \text { Medium 3 } & \ldots & n_3, Z_3, & \text { spherical coordinates with } r_3, \theta_3, \phi,\end{array}$
where again the azimuthal angles coincide in all reference systems. We next trace the ray through the lens system.

Object to Lens. For the ray transition from the object to lens side we relate the cross sections in the two media through $\cos \theta_1=d A_2 / d A_1$. Thus, from the power law of Eq. (3.9) we get
$$\frac{1}{2} Z_1^{-1}\left|\boldsymbol{E}_1\right|^2 d A_1=\frac{1}{2} Z_2^{-1}\left|\boldsymbol{E}_2\right|^2 \cos \theta_1 d A_1 .$$
The ray transmitted into medium 2 then has the magnitude
$$\left|\boldsymbol{E}_2\right|=\sqrt{\frac{n_1}{n_2}} \cos ^{-1 / 2} \theta_1\left|\boldsymbol{E}_1\right|,$$
where we have set again all permeabilities to $\mu_0$ and have expressed the impedances in terrms of refractive indicees. Whenn crossing the Gaussian referrencee sphere the azimuthal components are transformed into each other, whereas the $\hat{\boldsymbol{\theta}}_1$ component is transformed into the $\hat{\boldsymbol{\rho}}_2$ component. Putting together all results we obtain for the electric field in the lens region
$$\boldsymbol{E}_2=\sqrt{\frac{n_1}{n_2}} \cos ^{-1 / 2} \theta_1 \tilde{t}\left(E_1^\theta \hat{\rho}_2+E_1^\phi \hat{\boldsymbol{\phi}}\right)$$

## 物理代写|量子光学代写Quantum Optics代考|Optical Forces

A photon carries a momentum of $\hbar k$. For a light wavelength of $600 \mathrm{~nm}$ we approximately get
$$\hbar k \approx 10^{-34} \frac{2 \pi}{600 \times 10^{-9}} \approx 10^{-27} \mathrm{~m} \mathrm{~kg} \mathrm{~s}^{-1} .$$
Whenever light becomes scattered or diffracted by some dielectric body, part of the photon momentum is transferred from light to matter. While optical forces play no significant role for macroscopic objects, they can have a noticeable influence on nano- and microsized objects. Typical forces produced by tightly focused laser beams can be in the range of femto- to piconewtons, which suffices to trap particles entirely by optical means. In the last two decades optical tweezers have emerged as the primary tool to use such optical forces for trapping and manipulating microscopic particles. We will discuss a few selected applications later, but refer the interested reader to the rich literature for further details $[9,10]$.

The basic principles of optical trapping can be well understood on the basis of Maxwell’s equations and its underlying symmetries. In this chapter we will describe optical forces in three different frameworks.

Dipole Approximation. For particles much smaller than the light wavelength one can employ the so-called dipole approximation and describe the particles polarized in presence of strong light fields as point-like dipoles. See Sect. 4.1.1 for details.
Geometrical Optics. For particles much larger than the light wavelength one can employ the framework of geometrical optics, similar to the case of light focusing discussed in the previous chapter. See Sect. 4.1.2 for details.
Wavc Optics. For particles of intermediate size one must resort to the full Maxwell’s equations. As will be discussed in Sect. $4.5$, the forces can be obtained from the so-called Maxwell stress tensor.

## 物理代写|量子光学代写量子光学代考|远场成像

$\begin{array}{llll}\text { Medium 1 } & \ldots & n_1, Z_1, & \text { spherical coordinates with } r_1, \theta_1, \phi \ \text { Medium 2 } & \ldots & n_2, Z_2, & \text { cylinder coordinates with } \rho_2, \phi, z_2 \ \text { Medium 3 } & \ldots & n_3, Z_3, & \text { spherical coordinates with } r_3, \theta_3, \phi,\end{array}$
，其中方位角再次在所有参考系统中重合。我们接下来跟踪光线通过透镜系统

$$\frac{1}{2} Z_1^{-1}\left|\boldsymbol{E}_1\right|^2 d A_1=\frac{1}{2} Z_2^{-1}\left|\boldsymbol{E}_2\right|^2 \cos \theta_1 d A_1 .$$
，然后透射到介质2中的射线的大小
$$\left|\boldsymbol{E}_2\right|=\sqrt{\frac{n_1}{n_2}} \cos ^{-1 / 2} \theta_1\left|\boldsymbol{E}_1\right|,$$
，其中我们再次设置所有的渗透率为$\mu_0$，并以折射率表示阻抗。当穿过高斯参考球时，方位角分量相互转换，而$\hat{\boldsymbol{\theta}}_1$分量转换为$\hat{\boldsymbol{\rho}}_2$分量。把我们得到的透镜区域电场
$$\boldsymbol{E}_2=\sqrt{\frac{n_1}{n_2}} \cos ^{-1 / 2} \theta_1 \tilde{t}\left(E_1^\theta \hat{\rho}_2+E_1^\phi \hat{\boldsymbol{\phi}}\right)$$

## 物理代写|量子光学代写Quantum Optics代考|光力

.

$$\hbar k \approx 10^{-34} \frac{2 \pi}{600 \times 10^{-9}} \approx 10^{-27} \mathrm{~m} \mathrm{~kg} \mathrm{~s}^{-1} .$$

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