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\begin{document}

\title{Coherent Heterodyne OCDMA with Sourceless and Colorless Receivers}

\author{Yi Yang, \IEEEmembership{Life Senior Member,
IEEE,}\thanks{Authors are with the Department of Electrical \& Computer
Engineering, The Johns Hopkins University, Baltimore, MD 21218} Jacob
B. Khurgin, Mark Foster, and A. Brinton Cooper III}

\markboth{Submitted to IEEE Photonics Letters}{Yang, Khurgin, Foster, Cooper Coherent Heterodyne  OCDMA with Sourceless and Colorless Receivers}

%\IEEEpeerreviewmaketitle
\maketitle

\begin{abstract}
We present a new, coherent, heterodyne OCDMA scheme that conveys the signal and reference to each user in a multiple access network via one fiber. The receiver is sourceless and colorless. Bit error rate performance is characterized as a function of system parameters under various conditions.
\end{abstract}
\section{Introduction}
\IEEEPARstart{T}{he} passive optical network (PON) has been identified [0] as an indispensable solution to the bottleneck posed by the access portion of today's networks. Optical code division multiple access (OCDMA) is an attractive PON candidate because it offers soft capacity on demand, freedom from MAC layer protocols, support for security, minimal multiple access interference (MAI) and beat noise, easy upgrade of any power-splitter-based network, and sourceless/colorless optical network units (ONUs) at user premises [1]. However, most OCDMA schemes suffer from excessive speckle and MAI because encoded signals are not truly orthogonal at the receiver and because the receiver does not truly cancel MAI but merely spreads it across time and frequency, creating excessive speckle.

Spectrally encoded OCDMA can be implemented by spectral phase encoding (SPE) or spectral amplitude coding (SAC) and in homodyne or heterodyne configurations.  Our previously proposed SPOT\footnote{{\bf S}pectral...} [2] is a truly coherent homodyne SPE OCDMA that cancels MAI in a phase/polarization diversity receiver that is superior to those of earlier SPE designs [3-5]. Heterodyne SAC OCDMA with line spacing at the bit rate [6] seems to suffer from marginal spectral efficiency. To mitigate the accumulation of excessive MAI, [7] proposed a multi-wavelength local oscillator for beat noise minimization; however this approach offers unnecessary cost and complexity and is neither sourceless nor colorless. Furthermore, SAC designs are not truly coherent.

Below, we propose coherent, SPE heterodyne OCDMA that offers performance comparable to [2]  with an inexpensive receiver design and use of a single downlink fiber. The design is sourceless, colorless, resource efficient, and free of MAI.
\begin{figure}[ht]
\center
\includegraphics[width=3.0in]{fig1}
\caption{Spectral combs for encoded signals and unencoded reference.} \label{combs}
\end{figure}
\section{System Model}
\subsection{System Description}
A single mode-locked laser in the \emph{optical line terminal} (OLT, Fig. 2) is the source for  two  flattened (for orthogonality \cite{ , }) and interleaved spectral combs that are separated in frequency by $\Delta \omega$, the intermediate frequency (IF) of the heterodyne receiver and $2\pi \times$  the information rate of the system  (Fig. 1).  For each of $M$ users, a copy of the first comb (dashed in Fig. 1) is encoded with a unique {\it signature sequence} and, with the second unencoded comb (solid lines), is conveyed over a single fiber\footnote{Due to the use of a single fiber, the scheme is not polarization sensitive.} to the receiver's {\it optical network unit} (ONU, Fig. 2) where the reference comb is encoded with the phase sequence corresponding to the desired user's signal.  The ensemble is mixed in a balanced detector (BD) and the output is filtered at $2 \Delta \omega$.   For uplink transmission, the received reference comb is encoded, modulated, and returned over a second fiber.  Thus, all optical signals in the system are derived from a single mode-locked laser (MLL) (as in [2]) and use the same optical combs, resulting in sourceless/colorless operation.
\bc
\begin{figure*}[!t]
%\begin{figure}[ht]
\center
\includegraphics[width=6.50in]{combined_orig}
\caption{Coherent SPE heterodyne OCDMA and key waveforms} \label{blockd}
%\end{figure}
\end{figure*}
\ec
Let the $m^{th}$ users' comb be $\left\{\omega^{m}_n\right\}_{n=1}^N$, and the reference comb be $\{\omega^{r}_n\}_{n=1}^N$. Then $\Delta \omega=\omega^{m}_n-\omega^{r}_n$. The transmitted reference comb is $r(t) = \sum_{n=1}^N e^{j \omega^{r}_n t}$ and the $m^{th}$ user's transmitted signal is $s_m(t) = A_m \sum_{n=1}^N c_n^{m} e^{j \omega_n^{m}t}$ where $\{c_n^{m}\}_{n=1}^N$ is the $m^{th}$ signature sequence and $A_m$ is the $m^{th}$ user's data symbol.
%\begin{subeqnarray}
%\label{inputs} \slabel{eqS}
%%\beas
%E_S(t) &= & \sqrt{P_S} \sum_{m=1}^M S_m  c_m(t)e^{j\phi_m} \\
%\slabel{eqLO} E_{LO}(t) &=& \sqrt{P_{LO}} c_{\ell}(t)e^{j\phi_{\ell}}
%%\eeas
%\end{subeqnarray}
%\normalsize
\subsection{Dispersion}
When the reference is encoded with the $m^{th}$ user sequence, the output of the BD/filter is
\be
s_{IF}(t) = \sum_{n=1}^N c_n^{m} c_n^{m}e^{j \Delta \phi_n} = \sum_{n=1}^N c_n^{m} c_n^{m}(1 + \Delta \phi_n)
\ee
where $\Delta \phi_n$ is the dispersion-induced phase shift on the $n^{th}$ spectral line. For any $n$,
\bea
\Delta \phi_n &=& \phi_n(\omega^{r}_n) - \phi_(\omega^{s}_n)\\
&=& -[k(\omega^{r}_n) - k(\omega^{r}_n)] \cdot L\\
&=& -\Delta k \cdot L
\eea
where $k(\omega)$ is the wavenumber. We expand $k(\omega)$ about an arbitrary $\omega_0$ over each of the combs and subtract the two expressions to get
\beas
k_n^r &=& k(\omega_0) + \frac{\partial k(\omega_0)}{\partial \omega}\left(\omega_n^{r} - \omega_0\right)\\
 & & + \frac{1}{2} \frac{\partial^2 k(\omega_0)}{\partial \omega^2}\left(\omega_n^{r} - \omega_0\right)^2\\
k_n^m &=& k(\omega_0) + \frac{\partial k(\omega_0)}{\partial \omega}\left(\omega_n^{m} - \omega_0\right)\\
 & & + \frac{1}{2} \frac{\partial^2 k(\omega_0)}{\partial \omega^2}\left(\omega_n^{m} - \omega_0\right)^2\\
\Delta K & \approx & \frac{\partial k(\omega_0)}{\partial \omega} \cdot \Delta \omega + \frac{\partial^2 k(\omega_0)}{\partial \omega^2} \cdot \Delta \omega \cdot \omega_n^m\\
& & - \frac{\partial^2 k(\omega_0)}{\partial \omega^2} \omega^m_n \cdot \Delta \omega
\eeas
and,
\be
\Delta \phi_n = \frac{\partial^2 k(\omega_0)}{\partial \omega^2}\omega_n^m \Delta \omega \cdot L
\ee
\subsection{Relative Delay}
Let $\Delta \tau_d = $ the frequency-independent delay between the reference and signal combs.  Then the relative phase shift in the balanced detector between the $n^{th}$ lines of the combs is
\beas
\Delta \phi_{n,d} &=& \Delta \tau_d (\omega_n^r - \omega_m^s) \\
&=& \Delta \tau_d (n-1) \Delta \omega
\eeas

\section{Simulations and Comparisons}
Here we focus on the synchronous downlink and its transmitted set of pairwise orthogonal signals. In the simulation model (Fig. 2), two 16-line combs are obtained from a 1550 nm, 500 fs, 40 Gb/s MLL and are encoded with length 16 Hadamard sequences. Pseudorandom data is modulated at 40 Gb/s using OOK. The IF is 40 GHz, and the post detection filter bandwidth is 79 GHz (Fig 2(f)). We focus our analysis on the two factors that may cause the most severe signal-to-noise ratio (SNR) degradation: total dispersion and delay between the signal and reference.  Dispersion affects orthogonality by stretching the signal and reference in the fiber and increases with the number of users due to added MAI caused by the non-orthogonality. The simulated performance degradation from residual dispersion is shown in Fig 4. Asynchronism (relative delay) between encoded signal and reference or between two user signals can occur in the distinct signal pathways of the optical circuitry, again rendering the signals pairwise non-orthogonal; the non-zero crosscorrelation products [9] produce MAI which increases with the number of users and is zero only when interferers are pairwise orthogonal and, therefore, synchronous.  To study the effects of asynchronism, delay is introduced before the encoded reference comb is sent to the coupler (at point B, Fig. 2). The simulated performance degradation from asynchronism is displayed in Fig. 5.


\begin{figure}[ht]
\center
\includegraphics[width=3.0in]{dispersion_plot}
\caption{Analytic and Simulated Performance vs Dispersion} \label{disp}
\end{figure}

\begin{figure}[ht]
\center
\includegraphics[width=3.0in]{delay_plot}
\caption{Analytic and Simulated Performance vs Relative Delay} \label{delay}
\end{figure}

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\section*{Appendix: Derivations}
When the reference is encoded with the $m^{th}$ user sequence, the output of the BD/filter is
\be
s_{IF}(t) = \sum_{n=1}^N c_n^{(m)} c_n^{m}e^{j \Delta \phi_n} = \sum_{n=1}^N c_n^{m} c_n^{m}(1 + \Delta \phi_n)
\ee
where $\Delta \phi_n$ is the dispersion-induced phase shift on the $n^{th}$ spectral line. For any $n$,
\bea
\Delta \phi_n &=& \phi_n(\omega^{r}_n) - \phi_(\omega^{m}_n)\\
&=& -(k_n^r - k_n^m) \cdot L\\
&=& -\Delta k \cdot L \label{delphi}
\eea
where $k_n^r$ and $k_n^m$ are the frequency dependent wavenumbers at the $n^{th}$ lines of the reference and the $m^{th}$ user's combs respectively. ({\rd Maybe we need to drop $m$ since the lines are at the same frequencies for all users?}) We expand each $k$ about arbitrary $\omega_0$ and subtract the two expressions to get
\beas
k_n^r &=& k(\omega_0) + \frac{\partial k(\omega_0)}{\partial \omega}\left(\omega_n^{r} - \omega_0\right)\\
 & & + \frac{1}{2} \frac{\partial^2 k(\omega_0)}{\partial \omega^2}\left(\omega_n^{r} - \omega_0\right)^2\\
k_n^m &=& k(\omega_0) + \frac{\partial k(\omega_0)}{\partial \omega}\left(\omega_n^{m} - \omega_0\right)\\
 & & + \frac{1}{2} \frac{\partial^2 k(\omega_0)}{\partial \omega^2}\left(\omega_n^{m} - \omega_0\right)^2\\
 & & \\
\Delta k &=& k_n^r - k_n^m\\
& & \\
%&=& \frac{\partial k(\omega_0)}{\partial \omega} \left(\omega_n^r - \omega_n^m\right) \\
&=&  \frac{\partial k(\omega_0)}{\partial \omega} \cdot \Delta \omega \\
& & + \frac{1}{2} \frac{\partial^2 k(\omega_0)}{\partial \omega^2} \left[ (\omega_n^r)^2 - (\omega_n^m)^2 -2\omega_0\Delta \omega\right]
\eeas
Now
\beas
(\omega_n^r)^2 -(\omega_n^m)^2 &=& \Delta \omega(\omega_n^r + \omega_n^m)\\
&\sim&\Delta 2\omega(2 \omega_n^m)
\eeas
So,
\beas
\Delta k & \approx & \frac{\partial k(\omega_0)}{\partial \omega} \cdot \Delta \omega + \frac{\partial^2 k(\omega_0)}{\partial \omega^2} \cdot \Delta \omega \cdot \omega_n^m - \frac{\partial^2 k(\omega_0)}{\partial \omega^2} \omega_n \cdot \Delta \omega
\eeas
Substituting $\Delta k$ into \eqref{delphi} gives
\beas
\Delta \phi_n &=& -\left( \frac{\partial k(\omega_0)}{\partial \omega} \cdot \Delta \omega\right. \\
& & + \left.\frac{\partial^2 k(\omega_0)}{\partial \omega^2}(\omega_n^m\Delta \omega)  + \frac{\partial^2 k(\omega_0)}{\partial \omega^2}(\omega_0 \Delta \omega)\right) \cdot L
\eeas
But the first and the third term are constant over $\omega_n^m$, so do not contribute to the phase growth across the comb.  Therefore,

\be
\Delta \phi_n = \frac{\partial^2 k(\omega_0)}{\partial \omega^2}\omega_n^m \Delta \omega \cdot L
\ee
\end{document}
In early work, complementary spectral encoding and balanced detection
\cite{lam98} achieved multiple access with two OCDMA channels but
optical beat interference  due to the speckle noise generated by
multiple signals in the photodetector limited the system to fewer than
10x1 Gbps users, even at the relatively high received power of -20 dBm.
Usefully low values of bit error rate (BER) are achieved
\cite{testbedB} by passing the received signal through a phase encoder
matched to the desired user's signature sequence, giving a sharp
correlation peak that requires for detection either a very fast,
nonlinear peak detector or nonlinear time gating.  Yet, MAI remains
high due to the large background level produced by interfering codes,
and the combination of high MAI and complex, expensive peak detection
is not a good candidate for such applications as the passive optical
network (PON).  More recently \cite{pham05}, heterodyne detection has
been tried in order to overpower the OBI with a large local oscillator
(LO) power and a limitation on the received signal, this leaves no
margin for impairment in other system components and is not expected to
perform well in practice.

Using the (255,239) Reed-Solomon error control code on the data
\cite{wick} permits the 16 user asynchronous system to achieve a BER
below $10^{-07}$ at a slightly reduced rate of 9.38 GHz,

The group dispersion (GVD) limitation can be estimated by limiting the
dispersion-induced difference between the phase shifts of the central
(n=N/2) and peripheral (n=1,N) spectral lines to about p/4 leading to
the condition  that is essentially no different from the standard
restriction on the optical signal at bit rate BN. With 64 10Gbps
channels and �2~20 ps2/km we obtain maximum length of only 100 m -
therefore some type of dispersion compensation will be required for
practical implementations. Since the impact of optical nonlinearities
is not expected to be large (the encoded signals are spread out in
time)  using the zero-dispersion fiber (ZDF) would be the best way to
increase the reach of the network. For instance, using a ZDF with the
second order GVD  the reach can be extended to about 15km.


The electrical fields of the two outputs of the top hybrid are $(E_{LO}/2 + jE_s/2)$ and $E_s/2 + jE_{LO}/2)$. Thus, optical power into the top detector is
%\small
%\begin{subeqnarray}
%\label{pin}
%\slabel{pin1}
%P_1 &=& \left| \frac{E_{LO}}{2} + j\frac{E_S}{2} \right|^2 \\
%\slabel{pinexpand}
%&=& \frac{1}{4} \left( \left| E_S \right|^2 + \left| E_{LO} \right|^2 + j E_SE^{*}_{LO} - j E_s^{*}E_{LO} \right)
%\end{subeqnarray}
%\normalsize
%Computing $P_2$ through $P_8$ in the same manner and writing the optical power on the eight detectors in a compact form gives:


%MAI significantly limits OCDMA system capacity, security, and flexibility %\cite{lam98}. Typically, MAI is caused by direct detection that
%destroys phase information, precluding coherent detection.  In principle, %modulating user optical pulse train spectra with orthogonal signature sequences %should provide lower MAI levels than are currently achieved, but typical phase %coherent detection requires an optical phase locked loop and a nonlinear peak %detector, and fluctuating differences in polarization and carrier phase still %remain. A \emph{phase and polarization diversity detector} is proposed for %recombining the components of the spectrally phase coded carrier in a %differentially coherent manner to provide reduced MAI.
%
JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 25, NO. 11, NOVEMBER 2007
Next-Generation Optical Access Networks
Leonid G. Kazovsky, Fellow, IEEE, Fellow, OSA, Wei-Tao Shaw,
David Gutierrez, Ning Cheng, and Shing-Wa Wong
3428-3442. 