Carrier Lifetime in Semiconductor Optical Amplifiers (SOA)

Code for generating Fig. 1:

import numpy as np
import matplotlib.pyplot as plt

# Parameters
R_inj = 1e27      # carriers/cm^3/s
R_st = 0.3e27     # stimulated recombination rate
tau = 1e-9        # carrier lifetime in seconds
t0 = 3e-9         # time at which R_st turns off

# Time array (in seconds)
t = np.linspace(0, 8e-9, 500)

# Compute N(t)
N = np.zeros_like(t)

# For t < t0
mask1 = t < t0
N[mask1] = (R_inj - R_st) * tau * (1 - np.exp(-t[mask1] / tau))

# For t >= t0
N_t0 = (R_inj - R_st) * tau * (1 - np.exp(-t0 / tau))  # N(t0)
N_inf = R_inj * tau
mask2 = t >= t0
N[mask2] = N_inf * (1 - np.exp(-(t[mask2] - t0) / tau)) + N_t0 * np.exp(-(t[mask2] - t0) / tau)

# Plotting
plt.figure(figsize=(8, 4))
plt.plot(t * 1e9, N / 1e18, label="Carrier Density $N(t)quot;)
plt.axvline(t0 * 1e9, color='gray', linestyle='--', label="$t_0quot;)
plt.xlabel("Time (ns)")
plt.ylabel("Carrier Density ($10^{18}$ cm$^{-3}$)")
plt.title("Carrier Density Evolution with Step Change in $R_{st}$ at $t_0quot;)
plt.legend()
plt.grid(True)
plt.tight_layout()
plt.show()

Code for Latex

\documentclass[twocolumn,superscriptaddress,preprintnumbers,amsmath,amssymb]{revtex4}
%\usepackage{CJK}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{graphicx}
\usepackage{subfigure}
\usepackage[american]{circuitikz}
\usepackage{tikz}
\usepackage{bm}% bold math
\usepackage{epstopdf}
\usepackage{threeparttable}
\usepackage{float}
\usepackage{color}
\usepackage{mathtools} % add box around equation in align environment
\usepackage[font=small,labelfont=bf]{caption}


%\usepackage{times}
%\usepackage{appendix}
%\usepackage{fancyhdr}


\renewcommand\thesection{\arabic{section}}
\renewcommand\thesubsection{\thesection.\arabic{subsection}}
\renewcommand\thesubsubsection{\thesubsection.\arabic{subsubsection}}

%\renewcommand{\thefootnote}{\alph{footnote}}
%\renewcommand{\thefootnote}{\roman{footnote}}


\begin{document}
%\begin{CJK}{UTF8}{gbsn} % Use default fonts from CJK (see below)
\title{Material Properties for InGaAsP}
\author{Dingning Li} 
\affiliation{ECE, BU, Boston, MA}
% \author{Kai Zhang} 
% \email{kai.zhang@dukekunshan.edu.cn}
% \affiliation{Division of Natural and Applied Sciences, Duke Kunshan University, Kunshan, Jiangsu, 215300, China}

\date{\today}


\begin{abstract}

    Physical meaning of carrier lifetime is discussed in this letter. It has been shown that carrier lifetime does not only represents how fast carriers decay, but also represents how fast the carriers can be generated, which is important for SOA applications. The slow gain recovery~\cite{lal2007monolithic,lal2006thesis}, can be understood through taking carrier lifetime into consideration.

\end{abstract}

 \maketitle

\section*{1. Case 1: Turning On Current Injection}

When a constant current is turned on at $t = 0$, the carrier density $N(t)$ evolves according to the rate equation:
\[
\frac{dN}{dt} = R_{\text{inj}} - \frac{N}{\tau}
\]
where:
\begin{itemize}
  \item $R_{\text{inj}}$ is the constant carrier injection rate,
  \item $\tau$ is the carrier lifetime,
  \item $\frac{N}{\tau}$ represents spontaneous and nonradiative recombination.
\end{itemize}

This is a first-order linear ODE. The solution is obtained using the integrating factor method:
\[
\mu(t) = e^{t/\tau}
\]
Multiplying both sides:
\[
e^{t/\tau} \frac{dN}{dt} + \frac{1}{\tau} e^{t/\tau} N = R_{\text{inj}} e^{t/\tau}
\]
\[
\frac{d}{dt}\left(N e^{t/\tau}\right) = R_{\text{inj}} e^{t/\tau}
\]
\[
N(t) = R_{\text{inj}} \tau \left(1 - e^{-t/\tau}\right)
\]
This shows an exponential rise to the steady-state carrier density $N_\infty = R_{\text{inj}} \tau$ with time constant $\tau$.

\section*{2. Case 2: Stimulated Recombination Turns Off at $t = t_0$}

Initially, an optical signal causes stimulated recombination at rate $R_s$, and the rate equation becomes:
\[
\frac{dN}{dt} = R_{\text{inj}} - R_s - \frac{N}{\tau}
\]

\subsection*{For $t < t_0$:}

The solution is again a first-order linear ODE:
\[
\frac{dN}{dt} + \frac{N}{\tau} = R_{\text{inj}} - R_s
\]
\[
N(t) = (R_{\text{inj}} - R_s)\tau \left(1 - e^{-t/\tau} \right)
\]

\subsection*{For $t \geq t_0$:}

Stimulated recombination is turned off, so:
\[
\frac{dN}{dt} + \frac{N}{\tau} = R_{\text{inj}}
\]
Let $N(t_0)$ be the carrier density at the moment of switching:
\[
N(t_0) = (R_{\text{inj}} - R_s)\tau \left(1 - e^{-t_0/\tau} \right)
\]

The solution for $t \ge t_0$ is:
\[
N(t) = R_{\text{inj}}\tau \left(1 - e^{-(t - t_0)/\tau} \right) + N(t_0) e^{-(t - t_0)/\tau}
\]

This shows exponential recovery to a higher steady-state $N_\infty = R_{\text{inj}} \tau$, governed by the same carrier lifetime $\tau$.

\begin{center}
\includegraphics[width=0.35\textwidth]{figs/carrier_dynamics_plot.png}
\captionof{figure}{Carrier density $N(t)$ as a function of time, with $R_{st}$ turning off at $t_0$. }
\label{fig:ingaasp_absorp}
\end{center}


\bibliography{main.bib}

\end{document}

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