diff options
Diffstat (limited to 'source/know/concept/self-phase-modulation/index.md')
| -rw-r--r-- | source/know/concept/self-phase-modulation/index.md | 30 |
1 files changed, 15 insertions, 15 deletions
diff --git a/source/know/concept/self-phase-modulation/index.md b/source/know/concept/self-phase-modulation/index.md index 3b46c4e..f13ad2f 100644 --- a/source/know/concept/self-phase-modulation/index.md +++ b/source/know/concept/self-phase-modulation/index.md @@ -12,13 +12,13 @@ layout: "concept" In fiber optics, **self-phase modulation** (SPM) is a nonlinear effect that gradually broadens pulses' spectra. -Unlike dispersion, SPM does create new frequencies: in the $\omega$-domain, +Unlike dispersion, SPM does create new frequencies: in the $$\omega$$-domain, the pulse steadily spreads out with a distinctive "accordion" peak. Lower frequencies are created at the front of the pulse and higher ones at the back, giving S-shaped spectrograms. -A pulse envelope $A(z, t)$ inside a fiber must obey the nonlinear Schrödinger equation, -where the parameters $\beta_2$ and $\gamma$ respectively +A pulse envelope $$A(z, t)$$ inside a fiber must obey the nonlinear Schrödinger equation, +where the parameters $$\beta_2$$ and $$\gamma$$ respectively control dispersion and nonlinearity: $$\begin{aligned} @@ -26,20 +26,20 @@ $$\begin{aligned} = i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A \end{aligned}$$ -By setting $\beta_2 = 0$ to neglect dispersion, +By setting $$\beta_2 = 0$$ to neglect dispersion, solving this equation becomes trivial. -For any arbitrary input pulse $A_0(t) = A(0, t)$, +For any arbitrary input pulse $$A_0(t) = A(0, t)$$, we arrive at the following analytical solution: $$\begin{aligned} A(z,t) = A_0 \exp\!\big( i \gamma |A_0|^2 z\big) \end{aligned}$$ -The intensity $|A|^2$ in the time domain is thus unchanged, +The intensity $$|A|^2$$ in the time domain is thus unchanged, and only its phase is modified. It is also clear that the largest phase increase occurs at the peak of the pulse, -where the intensity is $P_0$. -To quantify this, it is useful to define the **nonlinear length** $L_N$, +where the intensity is $$P_0$$. +To quantify this, it is useful to define the **nonlinear length** $$L_N$$, which gives the distance after which the phase of the peak has increased by exactly 1 radian: @@ -52,15 +52,15 @@ $$\begin{aligned} \end{aligned}$$ SPM is illustrated below for the following Gaussian initial pulse envelope, -with parameter values $T_0 = 6\:\mathrm{ps}$, $P_0 = 1\:\mathrm{kW}$, -$\beta_2 = 0$, and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$: +with parameter values $$T_0 = 6\:\mathrm{ps}$$, $$P_0 = 1\:\mathrm{kW}$$, +$$\beta_2 = 0$$, and $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$: $$\begin{aligned} A(0, t) = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) \end{aligned}$$ -From earlier, we then know the analytical solution for the $z$-evolution: +From earlier, we then know the analytical solution for the $$z$$-evolution: $$\begin{aligned} A(z, t) = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) \exp\!\bigg( i \gamma z P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) @@ -70,10 +70,10 @@ $$\begin{aligned} <img src="pheno-spm-small.jpg" style="width:100%"> </a> -The **instantaneous frequency** $\omega_\mathrm{SPM}(z, t)$, +The **instantaneous frequency** $$\omega_\mathrm{SPM}(z, t)$$, which describes the dominant angular frequency at a given point in the time domain, is found to be as follows for the Gaussian pulse, -where $\phi(z, t)$ is the phase of $A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$: +where $$\phi(z, t)$$ is the phase of $$A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$$: $$\begin{aligned} \omega_{\mathrm{SPM}}(z,t) @@ -82,8 +82,8 @@ $$\begin{aligned} \end{aligned}$$ This result gives the S-shaped spectrograms seen in the illustration. -The frequency shift thus not only depends on $L_N$, -but also on $T_0$: the spectra of narrow pulses broaden much faster. +The frequency shift thus not only depends on $$L_N$$, +but also on $$T_0$$: the spectra of narrow pulses broaden much faster. The interaction between self-phase modulation and [dispersion](/know/concept/dispersive-broadening/) |