From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/self-phase-modulation/index.md | 98 +++++++++++++++++++++ .../self-phase-modulation/pheno-spm-small.jpg | Bin 0 -> 121984 bytes .../concept/self-phase-modulation/pheno-spm.jpg | Bin 0 -> 395877 bytes 3 files changed, 98 insertions(+) create mode 100644 source/know/concept/self-phase-modulation/index.md create mode 100644 source/know/concept/self-phase-modulation/pheno-spm-small.jpg create mode 100644 source/know/concept/self-phase-modulation/pheno-spm.jpg (limited to 'source/know/concept/self-phase-modulation') diff --git a/source/know/concept/self-phase-modulation/index.md b/source/know/concept/self-phase-modulation/index.md new file mode 100644 index 0000000..b2a4b58 --- /dev/null +++ b/source/know/concept/self-phase-modulation/index.md @@ -0,0 +1,98 @@ +--- +title: "Self-phase modulation" +date: 2021-02-26 +categories: +- Physics +- Optics +- Fiber optics +- Nonlinear optics +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, +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 +control dispersion and nonlinearity: + +$$\begin{aligned} + 0 + = i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A +\end{aligned}$$ + +By setting $\beta_2 = 0$ to neglect dispersion, +solving this equation becomes trivial. +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, +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** $L_N$, +which gives the distance after which the phase of the +peak has increased by exactly 1 radian: + +$$\begin{aligned} + \gamma P_0 L_N = 1 + \qquad \implies \qquad + \boxed{ + L_N = \frac{1}{\gamma P_0} + } +\end{aligned}$$ + +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 $\gamma = 0.1/\mathrm{W}/\mathrm{m}$: + +$$\begin{aligned} + A(0, t) + = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) +\end{aligned}$$ + +From earlier, we then know the analytical solution for the $z$-evolution: + +$$\begin{aligned} + A(z, t) = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) \exp\!\bigg( i \gamma z P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) +\end{aligned}$$ + + + + + +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 $A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$: + +$$\begin{aligned} + \omega_{\mathrm{SPM}}(z,t) + = - \pdv{\phi}{t} + = 2 \gamma z P_0 \frac{t}{T_0^2} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) +\end{aligned}$$ + +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 interaction between self-phase modulation +and [dispersion](/know/concept/dispersive-broadening/) +leads to many interesting effects, +such as [modulational instability](/know/concept/modulational-instability/) +and [optical wave breaking](/know/concept/optical-wave-breaking/). + + + +## References +1. O. Bang, + *Numerical methods in photonics: lecture notes*, 2019, + unpublished. diff --git a/source/know/concept/self-phase-modulation/pheno-spm-small.jpg b/source/know/concept/self-phase-modulation/pheno-spm-small.jpg new file mode 100644 index 0000000..6f041ec Binary files /dev/null and b/source/know/concept/self-phase-modulation/pheno-spm-small.jpg differ diff --git a/source/know/concept/self-phase-modulation/pheno-spm.jpg b/source/know/concept/self-phase-modulation/pheno-spm.jpg new file mode 100644 index 0000000..6d5c92a Binary files /dev/null and b/source/know/concept/self-phase-modulation/pheno-spm.jpg differ -- cgit v1.2.3