Распространение ультракоротких импульсов в волоконных световодах
Введение
Одной из задач при построении волоконных оптических систем, таких как волоконные лазеры или волоконно-оптические линии связи, является расчет динамики распространения ултракоротких импульсов (длительностью порядка пикосекунд) вдоль волоконного световода. Цель данного проекта состоит в решении данной задачи на примере распространения солитона в кварцевом световоде SMF-28, широко использующемся в телекоммуникациях, с использованием языка Julia.
Общее уравнение распространения
Для описания распространения ультракоротких импульсов в нелинейной среде с дисперсией, какой является волоконный световод, используется обобщенное нелинейное уравнение Шрёдингера, подробный вывод которого приведен в [1]. Здесь же представим его в готовом виде:
\begin{equation}
\begin{array}{r}
\frac{\partial A}{\partial z}+\frac{\alpha}{2} \mathrm{A}-\sum_{\mathrm{k} \geq 2} \frac{\beta_{\mathrm{k}}}{\mathrm{k}!} \frac{\partial^k \mathrm{~A}}{\partial T^k}= i \mathrm{\gamma}\left(1+\tau_{\text {shock }} i \frac{\partial}{\partial T}\right) \times \
{\left[\mathrm\mathrm{z}, \mathrm{~T} \int_{-\infty}^{\infty} \mathrm{R}\left(\mathrm{T}^{\prime}\right)\left|\mathrm{A}\left(\mathrm{z}, \mathrm{~T}-\mathrm{T}{\prime}\right)\right|2 \mathrm{dT}^{\prime}\right]},
\end{array}
\end
где — огибающая ультракороткого импульса, зависящая от координаты и вермени , ; — потери, ; — коэффициенты разложенной в ряд Тейлора постоянной распространения в окрестности центральной частоты , ; — мнимая единица; — нелинейный коэффициент, ; — функция нелинейного отклика среды, включающая комбинационное рассеяние и функцию отклика среды. Производная по времени в правой части уравнения учитывает дисперсию нелинейности, которая в большинстве случаев приводит к таким эффектам, как самообострение импульса и возникновение оптических ударных волн, характеризующиеся временем , измеряемым в секундах.
Для импульсов длительностью более 100 фс можно пренебречь членами дисперсии выше второго порядка, нулевым и первым, так как они не влияют на форму импульса. Также можно пренебречь комбинационным рассеянием, поглощением и образованием оптической ударной волны, считая их влияние малым. Принимая все эти условия, получим следующее уравнение:
\begin{equation}
\begin{array}{r}
i \frac{\partial A}{\partial z} - \frac{\text\beta_2}{2} \frac{\partial^2 A}{\partial T^2} + N^2 |A|^2 A = 0
\end{array}
\end
где — дисперсия групповых скоростей, ;
– знаковая функция. определяется, как: . Здесь – длительность импульса по уровню от максимальной интенсивности, которая соотносится с длительностью солитона на полувысоте следующим образом , пс; – пиковая мощность импульса, Вт;
Метод расчета
Уравнение выше является нелинейным дифференциальным уравнением с частными производными, для решения которого можно использовать разностные и псевдоспектральные методы. Обычно для моделирования используют псевдоспектральные методы, так как они на порядок быстрее благодаря использованию алгоритма быстрого Фурье-преобразования при той же точности расчета. Одним из наиболее широко используемых методов решения задачи распространения импульсов в нелинейной среде с дисперсией являются фурье-метод расщепления по физическим факторам.
Он заключается в применении операторов, отвечающих за дисперсию и нелинейность . Для численного интегрирования уравнение распространения удобно представить в следующем виде:
\begin{equation}
\frac{\partial A}{\partial z} = (\widehat{D} + \widehat{N})A.
\end
может быть записан в следующем виде:
\begin{equation}
\widehat{D} = -\frac{ i\beta_2}{ 2}\frac{\partial^2}{\partial T^2}
\end
может быть записан в следующем виде:
\begin{equation}
\widehat{N} = i\gamma|A|^2
\end
В общем случае дисперсия и нелинейность действуют вместе по всей длине световода.
Фурье-метод расщепления по физическим факторам позволяет получить приближенное решение, предполагая, что при распространении оптического поля на небольшое расстояние дисперсионные и нелинейные эффекты действуют независимо. Более конкретно, распространение от до осуществляется в два этапа. На первом этапе нелинейность действует сама по себе, и . На втором этапе дисперсия действует сама по себе, и .
В случае использования симметричного Фурье-метода решением через шаг h будет являться:
\begin{equation}
A(z+h, T) \approx \mathrm{exp}\left(\frac{h}{2}\widehat{D} \right)\mathrm{exp}\left(\int_{z}^{z+h} \widehat{N} ̂(z') dz' \right) \mathrm{exp}\left(\frac{h}{2}\widehat{D} \right)A(z,T),
\end
где экспоненциальный дисперсионный оператор легко переводится в спектральную область быстрым Фурье-преобразованием.
Принцип действия метода представлен на рисунке ниже.
Начальный импульс попадает в среду длиной , которая разбивается на участков с длиной . Расчет участка световода с длиной с известным импульсом на длине производится по следующему алгоритму:
- учитывается влияние дисперсии на половине шага в частотной области;
- вычисляется влияние нелинейности на полном шаге в середине шага;
- учитывается дисперсия на второй половине шага в частотной области.
Поскольку операторы нелинейности и дисперсии некоммутативны, то решением будет только приближение, с полной погрешностью, определяемой квадратом шага [2].
Реализация фурье-метода расщепления по физическим факторам на языке Julia
Для реализации быстрого преобразования Фурье в Julia используется библиотека FFTW, для создания графиков — Plots.
Параметры волоконного световода
В качестве входных параметров для расчета зададим коэффициенты дисперсии и нелинейности кварцевого световода SMF-28. Параметры волоконного светода представлены для длины волны 1,55 мкм, применяемой для передачи сигналов в телекоммуникационных системах.
Параметры расчетной сетки во временной и спектральных областях для численного расчета
Для проведения расчета необходимо создать параметры расчетной сетки во времени и частоте. Расчетная сетка во времени движется вместе с импульсом, где отрицательная половина отвечает за передний фронт импульса, а положительная - за задний. Сетка частот генерируется относительно центральной частоты излучения.
Теперь мы можем сгенерировать начальный импульс. В качестве входного импульса зададим солитонный импульс, описываемый во временной области следующей формулой:
\begin{equation}
A(T) = \sqrt P sech(T/T_0),
\end{equation}
где - пиковая мощность импульса, Вт;
- время, пс;
- длительность импульса по уровню от максимальной интенсивности, пс.
Аллокация для хранения данных на каждом шаге и дисперсионный и нелинейный фазовые операторы
Заключение
По полученным эволюциям импульса во временной и спектральной областях по длине распространения можно наблюдать действие двух явлений - фазовой самомодуляции (ФСМ) и дисперсии групповых скоростей (ДГС). ФСМ возникает вследствие зависимости константы распространения световода от интенсивности света, то есть нелинейным эффектом самовоздействия импульса. Ее проявление отражается в изменении формы спектра. ДГС влияет на длительность импульса во временной области.
Так как начальный импульс не имел чирпа, т.е. частотной модуляции, то в отсутствии ФСМ он уширялся бы, однако по эволюции импульса мы наблюдаем обратное. Этот эффект сжатия называется солитонным сжатием и проявляется на начальном этапе распространения в световоде, когда выполняется следующее соотношение (в нашем случае ). С помощью численной модели мы можем решить одну из возможных задач при построении волоконных лазерных систем - определить длину волокна, на котором импульс сжимается до минимальной длительности. В нашем случае она равна примерно 1,2 м, а длительность импульса составляет около 150 фс.
Таким образом, в данной проекте было предложено численное решение задачи по расчету динамики распространения ультракороткого импульса в волоконном световоде с у четом эффектов ФСМ и ДГС. Работа данной модели продемонстрирована на примере распространения солитонного импульса в световоде SMF-28. При надобности дисперсионный и нелинейный фазовые операторы могут быть дополнены другими членами, учитывающими поглощение, усиление, комбинационное рассеяние и т.д.
Список литературы
- Agrawal G.P. Nonlinear fiber optics // Nonlinear Fiber Optics. 2019.
- Beaud P. et al. Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber //IEEE journal of quantum electronics. – 1987. – Т. 23. – №. 11. – С. 1938-1946.
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