電場の複素表示

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記法

$g(t)$のフーリエ変換
\begin{aligned}
\mathcal{F}[g](\Omega)
&=\int_{-\infty}^{\infty} g(t) e^{-i\Omega t} \,\mathrm{d}t
\end{aligned}

$g(\Omega)$のフーリエ逆変換

\begin{aligned}
\mathcal{F}^{-1}[g](t)
&=\frac{1}{2\pi} \int_{-\infty}^{\infty} g(\Omega) e^{i\Omega t} \,\mathrm{d}\Omega
\end{aligned}

複素場をチルダをつけて$\tilde{g}$と表す.


\begin{aligned}
\langle f(t) \rangle
&=\frac{1}{T} \int_{t-T/2}^{t+T/2} f(t^{\prime}) \,\mathrm{d}t^{\prime}
\end{aligned}


電場

実電場を$E(t)$とし,そのスペクトル(フーリエ変換)を
\begin{aligned}
\tilde{E}(\Omega) = \mathcal{F}[E](\Omega)
&=\int_{-\infty}^{\infty} E(t) e^{-i\Omega t} \,\mathrm{d}t
\end{aligned}
とする.$E(t)$が実場なので
\begin{aligned}
\tilde{E}(\Omega) = \tilde{E}^{*}(-\Omega)
\end{aligned}
が成り立つ.

$\tilde{E}(\Omega) $を正負の周波数成分に分けて

\begin{aligned}
\tilde{E}(\Omega)
&= \tilde{E}^{+}(\Omega) + \tilde{E}^{-}(\Omega)
\end{aligned}
\begin{aligned}
\tilde{E}^{+}(\Omega)
&=
\begin{cases}
\, \tilde{E}(\Omega) & (\Omega \geq 0) \\
\, 0 & (\Omega < 0)
\end{cases} \\
\tilde{E}^{-}(\Omega)
&=
\begin{cases}
\, 0 & (\Omega > 0) \\
\, \tilde{E}(\Omega) & (\Omega \leq 0)
\end{cases}
\end{aligned}
と表す.

このとき,以下が成り立つ.

\begin{aligned}
\tilde{E}^{+}(\Omega)
&=
\begin{cases}
\, \tilde{E}^{*}(-\Omega) & (-\Omega \geq 0) \\
\, 0 & (-\Omega < 0)
\end{cases} \\
&= [\tilde{E}^{+}(-\Omega)]^{*}
\end{aligned}

\begin{aligned}
\tilde{E}^{+}(t)
&= \mathcal{F}^{-1}[\tilde{E}^{+}] (\Omega) \\
&= \frac{1}{2\pi} \int_{-\infty}^{\infty} \tilde{E}^{+}(\Omega) e^{i\Omega t} \,\mathrm{d}\Omega \\
&= \frac{1}{2\pi} \int_{0}^{\infty} \tilde{E}(\Omega) e^{i\Omega t} \,\mathrm{d}\Omega
\end{aligned}

\begin{aligned}
\tilde{E}^{-}(t)
&= \mathcal{F}^{-1}[\tilde{E}^{-}] (\Omega) \\
&= \frac{1}{2\pi} \int_{-\infty}^{0} \tilde{E}(\Omega) e^{i\Omega t} \,\mathrm{d}\Omega \\
&= \frac{1}{2\pi} \int_{-\infty}^{0} \tilde{E}^{*}(-\Omega) e^{i\Omega t} \,\mathrm{d}\Omega \\
&= \frac{1}{2\pi} \int_{0}^{\infty} \tilde{E}^{*}(\Omega) e^{-i\Omega t} \,\mathrm{d}\Omega \\
&= [\tilde{E}^{+}(t)]^{*}
\end{aligned}

\begin{aligned}
\tilde{E}^{-}(t)
&= \mathcal{F}^{-1}[\tilde{E}^{-}] (\Omega) \\
&= \frac{1}{2\pi} \int_{-\infty}^{\infty} [\tilde{E}^{+}(-\Omega)]^{*} e^{i\Omega t} \,\mathrm{d}\Omega \\
&= \biggl[\frac{1}{2\pi} \int_{-\infty}^{\infty} \tilde{E}^{+}(-\Omega) e^{-i\Omega t} \,\mathrm{d}\Omega \biggr]^{*} \\
&= \biggl[\frac{1}{2\pi} \int_{-\infty}^{\infty} \tilde{E}^{+}(\Omega) e^{i\Omega t} \,\mathrm{d}\Omega \biggr]^{*} \\
&=[\mathcal{F}^{-1}[\tilde{E}^{+}] (\Omega)]^{*} = [\tilde{E}^{+}(t)]^{*}
\end{aligned}

\begin{aligned}
\tilde{E}(t)
&= \tilde{E}^{+}(t) + \tilde{E}^{-}(t) \\
&= \tilde{E}^{+}(t) + \mathrm{c.c.} \\
&= 2\mathrm{Re\,} \tilde{E}^{+}(t)
\end{aligned}

\begin{aligned}
|\tilde{E}^{+}(t)|^{2}
= \tilde{E}^{+}(t) \tilde{E}^{-}(t)
\end{aligned}

パルス

\begin{aligned}
\tilde{E}^{+}(t)
&=\frac{1}{2} \tilde{\mathcal{E}}(t) e^{i\omega_{l}t} \\
\biggl(\tilde{\mathcal{E}}(t)
&= \frac{1}{2} \mathcal{E}(t) e^{i\varphi_{0}} e^{i\varphi(t)} \biggr)
\end{aligned}

\begin{aligned}
\langle \tilde{E}^{2}(t) \rangle
&= \langle [\tilde{E}^{+}(t)]^{2} + [\tilde{E}^{-}(t)]^{2} + 2|\tilde{E}^{+}(t)|^{2} \rangle \\
&=2 \langle |\tilde{E}^{+}(t)|^{2} \rangle \\
&= \frac{1}{2} \mathcal{E}^{2}(t)
\end{aligned}




\begin{aligned}
G_{n}(\tau)
=\int_{-\infty}^{\infty}
\biggl\langle \Bigl | [E_{1}(t-\tau) + E_{2}(t)]^{n} \Bigr |^{2} \biggr\rangle
\,\mathrm{d}t
\end{aligned}


\begin{aligned}
& E_{1}^{n_{1}}(t-\tau) E_{2}^{n_{2}}(t) \\
&= (\tilde{E}_{1}^{+} + \tilde{E}_{1}^{-})^{n_{1}}
(\tilde{E}_{2}^{+} + \tilde{E}_{2}^{-})^{n_{2}} \\
&= \sum_{k_{1}=0}^{n_{1}} \binom{n_{1}}{k_{1}} (\tilde{E}_{1}^{+})^{k_{1}} (\tilde{E}_{1}^{-})^{n_{1}-k_{1}} \\
&\quad
\times \sum_{k_{2}=0}^{n_{2}} \binom{n_{2}}{k_{2}} (\tilde{E}_{2}^{+})^{k_{2}} (\tilde{E}_{2}^{-})^{n_{2}-k_{2}}
\end{aligned}

\begin{aligned}
& \langle E_{1}^{n_{1}}(t-\tau) E_{2}^{n_{2}}(t) \rangle \\
&= \frac{1}{2^{n_{1}+n_{2}}} \mathcal{E}_{1}^{n_{1}} \mathcal{E}_{2}^{n_{2}}
\sum_{\substack{k_{1} + k_{2} \\ = (n_{1}-k_{1}) + (n_{2}-k_{2})}}
\binom{n_{1}}{k_{1}} \binom{n_{2}}{k_{2}} \\
&\quad
\times
e^{i(-n_{1} + 2k_{1})(\varphi_{10}+\varphi_{1})}
e^{i(-n_{2} + 2k_{2})(\varphi_{20}+\varphi_{2})} \\
&\quad
\times e^{-i(-n_{1} + 2k_{1})\omega_{l}\tau}
\end{aligned}



$G_{n}(\tau)$の各項は$n_{1} + n_{2} = 2n$を満たす.$\langle \cdot \rangle \neq 0$となるのは,そのうちで$k_{1} + k_{2} = n$を満たすものだけである.

$G_{2}(\tau)$

残る項を整理すると下表となる($n_{1}$と$n_{2}$を入れ替えたものは略す).
\begin{aligned}
\begin{array}{cccc:cc:c}
\hline
n_{1} & n_{2} & k_{1} & k_{2} & -n_{1}+2k_{1} & -n_{2}+2k_{2} & \binom{n_{1}}{k_{1}} \binom{n_{2}}{k_{2}} \\ \hline
4 & 0 & 2 & 0 & 0 & 0 & 6 \\ \hline
3 & 1 & 2 & 0 & 1 & -1 & 3 \\
& & 1 & 1 & -1 & 1 & 3 \\ \hline
2 & 2 & 2 & 0 & 2 & -2 & 1 \\
& & 1 & 1 & 0 & 0 & 4 \\
& & 0 & 2 & -2 & 2 & 1 \\ \hline
\end{array}
\end{aligned}

\begin{aligned}
& \Bigl | [E_{1}(t-\tau) + E_{2}(t)]^{2} \Bigr |^{2} \\
&= \Bigl | E_{1}^{2}(t-\tau) + E_{2}^{2}(t) + 2E_{1}(t-\tau) E_{2}(t) \Bigr |^{2} \\
&= E_{1}^{4} + E_{2}^{4} + 6E_{1}^{2} E_{2}^{2}
+ 4E_{1}^{3} E_{2} + 4E_{1} E_{2}^{3}
\end{aligned}

\begin{aligned}
& 2^{4} \biggl\langle \Bigl | [E_{1}(t-\tau) + E_{2}(t)]^{2} \Bigr |^{2} \biggr\rangle \\
&= a_{0}(t, \tau) + \mathrm{Re\,} [a_{1}(t, \tau)e^{-i\omega_{l}\tau}]
+ \mathrm{Re\,} [a_{2}(t, \tau)e^{-2i\omega_{l}\tau}]
\end{aligned}

$\mathrm{Re\,} z = (z+z^{*}) / 2$だから

\begin{aligned}
a_{0}(t, \tau)
&= 6\mathcal{E}_{1}^{4} + 6\mathcal{E}_{2}^{4} + 6\cdot 4\mathcal{E}_{1}^{2}\mathcal{E}_{2}^{2} \\
a_{1}(t, \tau) / 2
&= 4 \cdot 3 \mathcal{E}_{1}^{3}\mathcal{E}_{2} e^{i[(\varphi_{10}+\varphi_{1}) - (\varphi_{20}+\varphi_{2})]}
+4 \cdot 3 \mathcal{E}_{1}\mathcal{E}^{3}_{2} e^{i[(\varphi_{10}+\varphi_{1}) - (\varphi_{20}+\varphi_{2})]} \\
a_{2}(t, \tau) / 2
&= 6 \mathcal{E}_{1}^{2}\mathcal{E}_{2}^{2} e^{2i[(\varphi_{10}+\varphi_{1}) - (\varphi_{20}+\varphi_{2})]}
\end{aligned}

\begin{aligned}
\frac{2^{4}}{6} G_{2}(\tau) = A(\tau)
=A_{0}(\tau) + \mathrm{Re\,} [A_{1}(\tau)e^{-i\omega_{l}\tau}]
+ \mathrm{Re\,} [A_{2}(\tau)e^{-2i\omega_{l}\tau}]
\end{aligned}

\begin{aligned}
A_{0}(\tau)
&= \int_{-\infty}^{\infty}
\biggl[\mathcal{E}_{1}^{4}(t-\tau) +\mathcal{E}_{2}^{4}(t)
+4\mathcal{E}_{1}^{2}(t-\tau) \mathcal{E}_{2}^{2}(t) \biggr] \,\mathrm{d}t \\
A_{1}(\tau)
&= 4\int_{-\infty}^{\infty}
\mathcal{E}_{1}(t-\tau) \mathcal{E}_{2}(t)
\biggl[\mathcal{E}_{1}^{2}(t-\tau) + \mathcal{E}_{2}^{2}(t)\biggr]
e^{i[(\varphi_{10}+\varphi_{1}) - (\varphi_{20}+\varphi_{2})]} \,\mathrm{d}t\\
A_{2}(\tau)
&= 2\int_{-\infty}^{\infty}
\mathcal{E}_{1}^{2}(t-\tau) \mathcal{E}_{2}^{2}(t)
e^{2i[(\varphi_{10}+\varphi_{1}) - (\varphi_{20}+\varphi_{2})]} \,\mathrm{d}t
\end{aligned}

参考文献

Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale (Jean-Claude Diels)
Optical autocorrelation - Wikipedia