## 物理代写|量子光学代写Quantum Optics代考|Dipole Approximation

Consider first a polarized particle with a size much smaller than the wavelength of light. We will specify later how this particle becomes polarized, for the moment it suffices to assume that it has some dipole moment $\boldsymbol{p}$. As a generic model, we describe the dipole by two particles with charges $\pm q$ that are separated by the distance vector $s$, see Fig. 4.1. The dipole moment is $p-q s$, and we denote the center-of-mass coordinate with $r$. With this, the charges, coordinates, and velocities of the two particles forming the dipole can be expressed as
$$\begin{array}{lll} q_1=-q, & \boldsymbol{r}_1=\boldsymbol{r}-\frac{1}{2} \boldsymbol{s}, & \dot{\boldsymbol{r}}_1=\dot{\boldsymbol{r}}-\frac{1}{2} \dot{\boldsymbol{s}} \ q_2=+q, & \boldsymbol{r}_2=\boldsymbol{r}+\frac{1}{2} \boldsymbol{s}, & \dot{\boldsymbol{r}}_2=\dot{\boldsymbol{r}}+\frac{1}{2} \dot{\boldsymbol{s}} \end{array}$$

The electromagnetic and interatomic forces $f$ acting on the two particles then read
\begin{aligned} &\boldsymbol{F}1=-q\left[\boldsymbol{E}\left(\boldsymbol{r}_1\right)+\boldsymbol{v}_1 \times \boldsymbol{B}\left(\boldsymbol{r}_1\right)\right]+\boldsymbol{f}{12} \ &\boldsymbol{F}2=+q\left[\boldsymbol{E}\left(\boldsymbol{r}_2\right)+\boldsymbol{v}_2 \times \boldsymbol{B}\left(\boldsymbol{r}_2\right)\right]+\boldsymbol{f}{21} . \end{aligned}
For sufficiently small dipoles we can expand the electric field around the center-ofmass position $r$ and obtain
$$\boldsymbol{E}\left(\boldsymbol{r} \pm \frac{1}{2} \boldsymbol{s}\right) \approx \boldsymbol{E}(\boldsymbol{r}) \pm \frac{1}{2} \frac{\partial \boldsymbol{E}(\boldsymbol{r})}{\partial r_k} s_k=\boldsymbol{E}(\boldsymbol{r}) \pm \frac{1}{2}(\boldsymbol{s} \cdot \nabla) \boldsymbol{E}(\boldsymbol{r}),$$
with a similar expression for the magnetic field $\boldsymbol{B}$. To compute the total force acting on the dipole, we notice that the interatomic forces sum up to zero, $f_{12}+f_{21}=0$, whereas the sum over the electromagnetic forces becomes
\begin{aligned} \boldsymbol{F}=&-q\left[\boldsymbol{E}-\frac{1}{2}(\boldsymbol{s} \cdot \nabla) \boldsymbol{E}+\left(\dot{\boldsymbol{r}}-\frac{1}{2} \dot{\boldsymbol{s}}\right) \times\left(\boldsymbol{B}-\frac{1}{2}(\boldsymbol{s} \cdot \nabla) \boldsymbol{B}\right)\right] \ &+q\left[\boldsymbol{E}+\frac{1}{2}(\boldsymbol{s} \cdot \nabla) \boldsymbol{E}+\left(\dot{\boldsymbol{r}}+\frac{1}{2} \dot{\boldsymbol{s}}\right) \times\left(\boldsymbol{B}+\frac{1}{2}(\boldsymbol{s} \cdot \nabla) \boldsymbol{B}\right)\right] \end{aligned}
Here and in the following we suppress the dependence of $\boldsymbol{E}, \boldsymbol{B}$ on $\boldsymbol{r}$. Working out the above expression leads us to
$$\boldsymbol{F}=(\boldsymbol{p} \cdot \nabla) \boldsymbol{E}+\dot{\boldsymbol{p}} \times \boldsymbol{B}+\dot{\boldsymbol{r}} \times(\boldsymbol{p} \cdot \nabla) \boldsymbol{B}$$

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

In the opposite limit of particles that are substantially larger than the light wavelength, one can employ the framework of geometrical optics. Figure $4.2$ shows the principle at the example of a dielectric sphere. Our analysis closely follows the discussion of Sect. $3.3$ for ray tracing in optical lens systems. The computation of the forces is done through a simple ray tracing analysis, which can be divided into the following steps.

Rays. We assume that a given laser mode becomes tightly focused through a lens with a high numerical aperture, and start our ray description before the focusing lens. Here the fields propagate in the z-direction, and the total power depends on the field intensity $I(x, y)$
$$P=\int I(x, y) d x d y \approx \sum_i P_i\left(x_i, y_i\right) .$$
We approximate the incoming light fields by representative rays with power $P_i$ centered at $x_i, y_i$.
Focus Lens. At the focus lens, the rays are refracted towards the focus. When crossing the Gaussian reference sphere, the incoming fields are transformed according to Eq. (3.32).
Transmission and Reflection. When a ray crosses the boundary of the dielectric sphere, which is located close to the focus of the lens, part of it becomes reflected and part transmitted. In general, we can compute the reflection and transmission probabilities for planar interfaces using the Fresnel coefficients to be discussed in Sect. 8.3.1. The incoming power must be conserved,
$$P_i=P_r+P_t,$$
where $P_r$ and $P_t$ are the reflected and transmitted power, respectively. In addition, we use Snell’s law to compute the angles with respect to the outer surface normals of the sphere,
$$\theta_i=\theta_r, \quad n_i \sin \theta_i=n_t \sin \theta_t .$$
Here $n_i$ and $n_t$ are the refractive indices of the embedding medium and the dielectric sphere, respectively.

## 物理代写|量子光学代写Quantum Optics代考|偶极近似

$$\begin{array}{lll} q_1=-q, & \boldsymbol{r}_1=\boldsymbol{r}-\frac{1}{2} \boldsymbol{s}, & \dot{\boldsymbol{r}}_1=\dot{\boldsymbol{r}}-\frac{1}{2} \dot{\boldsymbol{s}} \ q_2=+q, & \boldsymbol{r}_2=\boldsymbol{r}+\frac{1}{2} \boldsymbol{s}, & \dot{\boldsymbol{r}}_2=\dot{\boldsymbol{r}}+\frac{1}{2} \dot{\boldsymbol{s}} \end{array}$$

\begin{aligned} &\boldsymbol{F}1=-q\left[\boldsymbol{E}\left(\boldsymbol{r}1\right)+\boldsymbol{v}_1 \times \boldsymbol{B}\left(\boldsymbol{r}_1\right)\right]+\boldsymbol{f}{12} \ &\boldsymbol{F}2=+q\left[\boldsymbol{E}\left(\boldsymbol{r}_2\right)+\boldsymbol{v}_2 \times \boldsymbol{B}\left(\boldsymbol{r}_2\right)\right]+\boldsymbol{f}{21} . \end{aligned}

$$\boldsymbol{E}\left(\boldsymbol{r} \pm \frac{1}{2} \boldsymbol{s}\right) \approx \boldsymbol{E}(\boldsymbol{r}) \pm \frac{1}{2} \frac{\partial \boldsymbol{E}(\boldsymbol{r})}{\partial r_k} s_k=\boldsymbol{E}(\boldsymbol{r}) \pm \frac{1}{2}(\boldsymbol{s} \cdot \nabla) \boldsymbol{E}(\boldsymbol{r}),$$

\begin{aligned} \boldsymbol{F}=&-q\left[\boldsymbol{E}-\frac{1}{2}(\boldsymbol{s} \cdot \nabla) \boldsymbol{E}+\left(\dot{\boldsymbol{r}}-\frac{1}{2} \dot{\boldsymbol{s}}\right) \times\left(\boldsymbol{B}-\frac{1}{2}(\boldsymbol{s} \cdot \nabla) \boldsymbol{B}\right)\right] \ &+q\left[\boldsymbol{E}+\frac{1}{2}(\boldsymbol{s} \cdot \nabla) \boldsymbol{E}+\left(\dot{\boldsymbol{r}}+\frac{1}{2} \dot{\boldsymbol{s}}\right) \times\left(\boldsymbol{B}+\frac{1}{2}(\boldsymbol{s} \cdot \nabla) \boldsymbol{B}\right)\right] \end{aligned}

$$\boldsymbol{F}=(\boldsymbol{p} \cdot \nabla) \boldsymbol{E}+\dot{\boldsymbol{p}} \times \boldsymbol{B}+\dot{\boldsymbol{r}} \times(\boldsymbol{p} \cdot \nabla) \boldsymbol{B}$$

## 物理代写|量子光学代写Quantum Optics代考|几何光学

$$P=\int I(x, y) d x d y \approx \sum_i P_i\left(x_i, y_i\right) .$$

$$P_i=P_r+P_t,$$

$$\theta_i=\theta_r, \quad n_i \sin \theta_i=n_t \sin \theta_t .$$

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