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## 电子工程代写|光子简介代写Introduction to Photonics代考|Orthogonal Polarization States

Two Jones vectors are called orthogonal if their scalar product is zero,
$$\mathbf{J}^{(1)} \cdot \mathbf{J}^{(2) *}=0$$
Examples are two linearly polarized states oriented along $\varphi$ and $\varphi+\pi / 2$, respectively, or left/right circularly polarized states $\sigma^{+}, \sigma^{-}$. A state orthogonal to Eq. (1.79) is obviously
$$\left[\begin{array}{c} \sin \alpha \ -\cos \alpha \mathrm{e}^{\mathrm{j} \Delta \phi} \end{array}\right]$$
A pair of orthogonal states (Jones vectors) establishes a base that allows constructing any other state by appropriate linear combination. In particular, any other orthogonal base can be constructed; for example, the sum and difference, respectively, of $\sigma^{+}$and $\sigma^{-}$produce a linearly polarized orthogonal base
$$\frac{1}{\sqrt{2}}\left(\sigma^{+}+\sigma^{-}\right)=\left[\begin{array}{l} 1 \ 0 \end{array}\right]$$

$$\frac{1}{\mathrm{j} \sqrt{2}}\left(\sigma^{+}-\sigma^{-}\right)=\left[\begin{array}{l} 0 \ 1 \end{array}\right],$$
and a circularly polarized base can be obtained from a linearly polarized base by a complex-valued combination
$$\sigma^{\pm}=\frac{1}{\sqrt{2}}\left[\begin{array}{l} 1 \ 0 \end{array}\right]+\frac{1}{\sqrt{2}} \mathrm{e}^{\pm \mathrm{j} \pi / 2}\left[\begin{array}{l} 0 \ 1 \end{array}\right] .$$
These relations are not only mathematical transformations, but also represent physical reality, since linearly polarized light, for example, can be synthesized by two superimposed circularly polarized waves and vice versa.

The polarization state can change during propagation; as we will see, however, for a given propagation system there are always so-called eigenstates that are conserved during propagation (Sect. 1.5.2.5). In lossless media, these states can be shown to be orthogonal to each other and represent a “natural base” for the description of wave propagation in the respective system.

## 电子工程代写|光子简介代写Introduction to Photonics代考|Wave Plates

In Sect. $2.3$, we will encounter various optical components that can alter the polarization state; their operation can be represented by a specific Jones matrix $T$, that relates an arbitrary input state $\mathbf{J}{\text {in }}$ to the corresponding output state $\mathbf{J}{\text {out }}$
$$\mathrm{J}{\text {out }}=T \mathrm{~J}{\text {in }} \text {; }$$
Table $1.3$ summarizes Jones matrices of important components. Many of these elements rely on the dependence of the phase velocity on the polarization state. In birefringent materials (Sect. 2.3), for example, there are two orthogonal, linearly polarized eigenstates $\mathbf{J}{\mathrm{f}, \mathrm{s}}$ with different phase velocities, denoted as “fast” and “slow”; the corresponding propagation indices are $n{\mathrm{f}}$ and $n_{\mathrm{s}}$. An incoming field of arbitrary polarization is decomposed in two waves $\propto \mathbf{J}{\mathrm{r}, \mathrm{s}} \mathrm{e}^{-\mathrm{i}\left(f{t}, \mathbf{k}{0} \cdot \mathbb{R}-\dot{\omega} t\right)}$ that develop, during propagation, a phase difference of $$\Delta \phi{\mathrm{V}}=\left(n_{\mathrm{s}}-n_{\mathrm{f}}\right) k_{0} d,$$
where $d$ is the thickness of the medium; such plates are called retarders or wave plates (Fig. 1.7). In a coordinate system with the $x$-axis parallel to $\mathbf{J}_{\mathbf{f}}$, the Jones

A so-called half-wave plate produces a phase shift of $\pi$ (corresponding to $\lambda / 2$ ), and is represented by
$$\boldsymbol{T}=\left[\begin{array}{rr} 1 & 0 \ 0 & -1 \end{array}\right],$$

# 光子简介代考

## 电子工程代写|光子简介代写Introduction to Photonics代考|Orthogonal Polarization States

$$\mathbf{J}^{(1)} \cdot \mathbf{J}^{(2) *}=0$$

$$\left[\sin \alpha-\cos \alpha \mathrm{e}^{\mathrm{j} \Delta \phi}\right]$$

\begin{aligned} &\frac{1}{\sqrt{2}}\left(\sigma^{+}+\sigma^{-}\right)=\left[\begin{array}{ll} 1 & 0 \end{array}\right] \ &\frac{1}{\mathrm{j} \sqrt{2}}\left(\sigma^{+}-\sigma^{-}\right)=\left[\begin{array}{ll} 0 & 1 \end{array}\right], \end{aligned}

$$\sigma^{\pm}=\frac{1}{\sqrt{2}}[10]+\frac{1}{\sqrt{2}} \mathrm{e}^{\pm \mathrm{j} \pi / 2}\left[\begin{array}{lll} 0 & 1 \end{array}\right] .$$

## 电子工程代写|光子简介代写Introduction to Photonics代考|Wave Plates

$$\text { Jout }=T \text { Jin ; }$$

$$\Delta \phi \mathrm{V}=\left(n_{\mathrm{s}}-n_{\mathrm{f}}\right) k_{0} d,$$

$$\boldsymbol{T}=\left[\begin{array}{llll} 1 & 0 & 0 & -1 \end{array}\right],$$

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