# Systematic design study of all-optical delay line based on Brillouin
scattering enhanced cascade coupled ring resonators

Myungjun Lee1, Michael E. Gehm1,2, and Mark A. Neifeld1,2 1 Department of
Electrical Computer Engineering, University of Arizona, Tucson, AZ 85721-0104,
USA  2 College of Optical Sciences, University of Arizona, Tucson, AZ
85721-0104, USA<EMAIL_ADDRESS>
###### Abstract

We present a technique to improve the slow-light performance of a side-coupled
spaced sequence of resonators (SCISSOR) combined with a stimulated Brillouin
scattering (SBS) gain medium in optical fiber. We evaluate device performance
of SCISSOR-only and SCISSOR + SBS systems for different numbers of cascaded
resonators from 1 to 70 using two different data fidelity metrics including
eye-opening and mutual information. A practical system design is demonstrated
by analyzing its performance in terms of fractional delay, power transmission,
and data fidelity. We observe that the results from the two metrics are in
good agreement. Based on system optimization under practical resource and
fidelity constraints, the SCISSOR consisting of 70 cascaded resonators
provides a fractional delay of $\sim$ 8 with 22 dB attenuation at a signal bit
rate of 10 Gbps. The combined optimal SCISSOR (with 70 resonators) + SBS
system provides a improved fractional delay up to $\sim$ 17 with unit power
transmission under the same constraints.

Keywords: slow-light, optical delay line, stimulated Brillouin scattering
(SBS), ring resonator, mutual information, non-linear optics, group delay,
group velocity dispersion

##  1 Introduction

Tunable all-optical delay systems that dynamically manipulate the group
velocity of light have received a great deal of attention for optical
information processing applications such as data buffering and
synchronization. Various slow-light devices, including those based on
electromagnetically induced transparency (EIT) in atomic vapor, stimulated
Brillouin and Raman scattering (SBS and SRS) in optical fiber, and photonic
structures in dielectric material, have been explored as potential
realizations of a practical delay system [1–10].

As for on-chip approaches, coupled resonators or photonic crystals are
promising techniques that would allow easy integration with other electronics
or optical components. Many recent demonstrations of coupled resonator optical
waveguides (CROW) and side coupled integrated spaced sequence of resonators
(SCISSOR) have been designed and fabricated in compact sizes ($\sim$ 10
$\mu\textrm{m}^{2}$ ) and with the possibility of dynamic delay control and
large delay-bandwidth product [2–4]. A more recent analysis from Otey et al
shows that cascaded resonators can even capture light pulses (i.e., stopped
light) by completely compressing the system bandwidth and that the captured
pulse can then be released [5].

A large fractional delay (equivalent to the delay-bandwidth product) can be
achieved by a chain of resonators. Unfortunately, these devices suffer a
fundamental trade-off between transmission loss and delay, which potentially
limits the use of large numbers of resonators. For example, a CROW consisting
of 6 ring resonators demonstrated continuously controllable fractional delay
up to 3 at a signal bit rate (BR) of 10 Gbps and a bit error rate (BER) of
$10^{-9}$ [2]. Its transmission loss, however, is 3 dB (i.e., 0.5 dB/ring) and
therefore the use of any additional resonator will increase the BER higher
than $10^{-9}$. For comparison, Xia et al have demonstrated a chain of 56
cascaded micro-ring resonators in a side-coupled configuration using a
silicon-on-insulator waveguide. They achieved a large fractional delay ($\sim$
5) at a BR of 10 Gbps and BER of $10^{-4}$ [4]. This high BER is a direct
consequence of the 22 dB transmission loss of the device resulting from the
use of the large number of rings.

One possible way of preserving acceptable output signal quality without
sacrificing the delay performance is to use a Brillouin amplifier. An SBS
gain-based delay system could provide significant signal amplification and its
tunable gain bandwidth could be increased up to 25 GHz, which allows high
speed data transmission [11, 12]. In addition, an optimal SBS gain system
would provide additional fractional delay of up to 3 [13]. Therefore,
combining this system with cascaded resonators or other photonic resonance
structures seems like a promising method for compensating their respective
disadvantages while increasing maximum fractional delay [14–16].

In general, large slow light delay is accompanied by substantial group
velocity dispersion (GVD) that manifests itself as signal distortion. The
presence of higher-order GVD terms lead to changes in the pulse shape. Under
such a condition, we require a metric to quantitatively measure the output
data quality along with the delay. A common measure of communications
performance for the propagation of a pulse train is an eye-diagram. Eye-
diagrams are useful for estimating signal distortion via the maximum eye-
opening; and its location represents the delay [16–19]. Neifeld and Lee have
presented an alternative metric that uses Shannon information to estimate the
information capacity and information delay in the presence of noise [20, 21].
In this paper, we utilize these two metrics to evaluate SCISSOR, SBS, and
SCISSOR + SBS under practical resource and fidelity constraints. By jointly
optimizing the system parameters of the SCISSOR + SBS system, we determine the
maximum fidelity-constrained fractional delay at a BR of 10 Gbps.

##  2 Data fidelity metric

The important quantities to consider for evaluating slow-light system
performance are the fractional delay and received data fidelity. When a single
pulse or a pulse sequence propagates through the dispersive media, it
undergoes GVD. There are several metrics including the pulse broadening factor
[22, 23], amplitude and phase distortion [10], eye-opening [16–19], and mutual
information [20] that have been introduced to quantify the slow-light
performance. In what follows, we consider the eye-opening and information-
theoretic metrics.

###  2.1 Eye-opening metric

An eye-diagram is used to visualize the shape of communications waveforms and
it is generated by repetitively superimposing subsequent traces of a given
data stream over a fixed time interval. The eye-opening (EO) is the maximum
difference between the minimum value of “ones” and the maximum value of
“zeros” at the bit center. The data distortion (D) can be quantified using the
eye-opening and it is defined as

$D=1-\max(EO).$ (1)

If a pulse sequence passes through a dispersive medium, the output signal
could be broadened or distorted, and then D will increase due to the increased
intersymbol interference (ISI). Note that distortion has a monotonic
relationship with BER and D = 0.35 indicates a corresponding BER $\simeq$
10-9, resulting in reliable communication [15, 19].

An eye-opening based delay can be calculated by the time difference $T_{EO}$
between the input and output eye center defined when the EO is maximal. The
fractional eye-opening delay (EOD) is defined as the time delay divided by the
input pulsewidth $T_{p}$, that is, EOD = $T_{EO}/T_{p}$.

###  2.2 Information theoretic metric

Information theory was first explored by Shannon and information rate has
become a standard method to characterize the quality of a communication
channel [24, 25]. Recently, an information theoretic metric was introduced
using the mutual information between the slow light input and output signals,
in order to measure the information based delay (ID) and information
throughput (IT) [20]. The IT-metric in this paper is based on the channel
model displayed in figure 1(a). Figure 1(b) shows the examples of 3 bits
output signal propagated through an arbitrary slow light channel, which may
include effects of delay, distortion, and noise. The input X is a binary-
valued sequence and it is modulated via on-off keying (OOK). The slow-light
delay system is represented by the channel operator HSL, where HSL could
represent any kind of delay device. The mutual information (MI) represents the
quantity of transmitted data, and estimates how much input information about X
is known when the output Y is observed. Thus, the MI can be defined as
$I(X;Y)=H(X)-H(X|Y)$, where $H(X)$ is the entropy of the discrete input X,
representing the a priori uncertainty, and $H(X|Y)$ is the conditional entropy
after the output is observed [24, 25]. We assume that the output signal Y is
corrupted by additive white Gaussian noise (AWGN) with zero mean and variance
$\sigma^{2}$. We also assume that the elements $x_{i}$ of a specific n-bit
input sequence X are independent and identically-distributed (IID), leading to
a prior probability $p(x_{i})$ = (1/2)n. Under these assumptions, the MI can
be written as:

$I(X;Y)=n+\int\sum_{i=1}^{M}p(x_{i})p(Y|x_{i})\log_{2}\frac{p(Y|x_{i})p(x_{i})}{\sum_{j=1}^{M}p(x_{j})p(Y|x_{j})}dY,$
(2)

where n is the number of input bits, $\textit{M}=2^{\textit{n}}$ is the number
of possible n-bit input sequences, and $p(x_{i},Y)$ is the joint probability
density function (PDF) of $x_{i}$ and Y. The integral over Y in equation (2)
can be solved by the Monte Carlo simulation with important sampling. Here,
$p(Y|x_{i})$ is the PDF of Y conditioned on $x_{i}$ that is expressed by the
Gaussian PDF:

$p(Y|x_{i})\simeq\frac{1}{(2\pi\sigma^{2})^{nL}}\exp(-\frac{1}{2\sigma^{2}}|Y-H_{SL}x_{i}|^{2}),$
(3)

where L is the number of simulation samples used to represent a single
Gaussian pulse.

Note that the concept of delay is not easily captured within $I(X;Y)$. In
order to apply I(X;Y) to the analysis of slow light systems, we impose a
window structure, which confines an input pulse sequence within a finite
duration window [20, 26]. With this approach, we can compute the MI between X
and only that part of Y contained within the output window (OW) as a function
of window offset [20, 21]. Here we use a simple example to describe the IT-
metric, let us first consider an ideal distortion free delay device with
$\sigma^{2}$ = 0, as shown in figure 2. The 3 bits of Gaussian pulses with a
50% return-to-zero (RZ) modulation format serve as an input, and the Gaussian
pulse is defined to have field amplitude $E(t)$ = exp$(-(t/T_{HW})^{2})$,
where $T_{HW}$ = $T_{b}$/2 is the bit half-width at 1/$e^{2}$ intensity and
$T_{b}$ is the bit period. The 50% RZ modulation denotes that a logical one is
represented by a half-bit wide pulse, therefore, $T_{p}$ = $T_{b}$/2. In order
to compute the MI for this example of the three bit transmission, we consider
all 8 possible states (M = 8), as shown in figure 2(a). We assume that the
input bit period $T_{b}$ = 100 ps and the value of delay $T_{D}$ = 400 ps. The
input signal is fitted within an input window (IW), and then we can compute
the MI between input X within the IW and only part of Y contained in the OW
for many different OW locations in figures 2(a) and (b). We observe the values
of MI = 3 bit and 1 bit for the two candidate output windows (OW1 and OW2) at
two different values of window offset = 400 ps and 600 ps, respectively, as
shown in figure 2(b). For this example, when the window offset is the same as
the delay $T_{D}$, all the input signal information can be transferred without
loss caused by distortion, noise, and energy leaking outside the window. Thus,
the peak value of I(X;Y) represents the amount of information that can be
transmitted through the slow light channel; while the location of this peak
provides an information-theoretic measure of delay. Therefore, we define the
peak-height as the information throughput (IT) and the peak location as the
information delay (ID) of the SL device, where the normalized IT is

$IT=\frac{\max\\{I(X;Y)\\}}{\textmd{n-bits}},$ (4)

and this definition will be used throughout the remainder of the paper.

##  3 Ring resonators

###  3.1 Single resonator

For a coupled ring resonator, as shown in figure 3, the output fields can be
related to the input fields through a complex amplitude transfer function

$H_{\mathrm{Ring}}(\omega)=\frac{E_{2}(\omega)}{E_{1}(\omega)}=\frac{k-a\
\mathrm{exp}(i\phi(\omega))}{1-ka\ \mathrm{exp}(i\phi(\omega))},$ (5)

where $E_{i}$ is the complex field amplitudes, k is the self-coupling
coefficient ($k^{2}=1-\rho^{2}$), $\rho$ is a cross-coupling coefficient, a =
exp(-$\alpha L_{R}$/2) is the round trip amplitude loss of the resonator,
$L_{R}$ is the ring circumference, and $\alpha$ is the total attenuation
coefficient which includes all sources of loss such as material absorption,
bending loss, and scattering loss from waveguide roughness [3, 4]. The round-
trip phase shift $\phi(\omega)$ in the ring can be represented by
$\phi(\omega)$ = 2$\pi n_{R}L_{R}$ $(\omega-\omega_{0})$/c, where $n_{R}$ is
the effective index of the ring, c is the speed of light, and $\omega_{0}$ is
the resonance angular frequency.

The phase response $\Phi_{\mathrm{Ring}}(\omega)$ of the transfer function is
obtained by the relation $H_{\mathrm{Ring}}(\omega)$ =
$|H_{\mathrm{Ring}}(\omega)|$exp(j$\Phi_{\mathrm{Ring}}(\omega)$) and it is
given in terms of k, a, and $\phi(\omega)$ as follows:

$\Phi_{\mathrm{Ring}}(\omega)=\pi+\phi(\omega)+\tan^{-1}\Big{[}\frac{k\
\mathrm{sin}\phi(\omega)}{a-k\
\mathrm{cos}\phi(\omega)}\Big{]}+\tan^{-1}\Big{[}\frac{ka\
\mathrm{sin}\phi(\omega)}{1-ka\ \mathrm{cos}\phi(\omega)}\Big{]}.$ (6)

Next, we consider the group delay which is a direct consequence of the amount
of phase shift in equation (6) within the filter passband. It is defined as
the negative derivative of the phase of the transfer function with respect to
the angular frequency:

$\displaystyle\tau_{\mathrm{Ring}}=-\frac{d\Phi_{\mathrm{Ring}}(\omega)}{d\omega}$
$\displaystyle=-\frac{n_{R}L_{R}}{c}+\frac{k(k-a\
\cos\phi(\omega))}{a^{2}-2ka\
\cos\phi(\omega)+k^{2}}+\frac{ka(ka-\cos\phi(\omega))}{1-2ka\
\cos\phi(\omega)+k^{2}a^{2}}.$ (7)

Equations (6) and (7) explain the behavior of the propagated light through the
resonator. At resonance, for $\textit{k}<\textit{a}$, the ring and the
waveguide are overcoupled and the phase shift increases rapidly as a function
of angular frequency, leading to pulse delay [6, 7, 8, 9]. On the other hand,
for $\textit{k}>\textit{a}$, they are undercoupled and the phase shift
decreases rapidly as a function of angular frequency, resulting in pulse
advancement. Critical coupling occurs when $\textit{k}=\textit{a}$. Here, the
transmission becomes zero at the resonance frequency as the round trip loss of
the ring is exactly the same as the fractional loss through the resonance
coupling [27]. In our design study, we are particularly interested in pulse
delay, and thus all candidate systems use overcoupled resonators.

In figure 4, we depict the resonator characteristics for four different values
of k = 0.8, 0.9, 0.95, and 0.97 with a practical value of attenuation
coefficient $\alpha$ = 1 cm-1 and $L_{R}$ = 150 $\mu$m. Using the numerical
simulations based on equations (5) - (7), we calculate and plot the
transmission, phase shift, and group delay spectra in figures 4(a), 4(b), and
4(c), respectively. Corresponding Gaussian input and delayed output pulses are
shown in figure 4(d), where the input pulsewidth is $T_{p}$ = 50 ps. A silicon
waveguide is assumed, thus an effective refractive index $n_{R}$ = 3.0 is
used. In figure 4(a), as k approaches the critical value of the round trip
loss a from below, the full width at half depth (FWHD) of the resonator
transmission function becomes narrower and deeper. This leads to the slope of
the phase shift becoming larger, as shown in figure 4(b), and therefore a
larger group delay is achieved for the larger value of k = 0.97 in figure
4(c). However, the maximum achievable pulse delay is limited by the tradeoff
between the group delay and pulse distortion, causing oscillation at the pulse
rising edge, as shown in figure 4(d).

Now, let us consider a pulse train at a BR of 10 Gbps rather than a single
pulse, where BR = 1/$T_{b}$. Figures 5(a), 5(b), and 5(c) present the EO-
delay, distortion, and power throughput (PT), respectively, for a resonator as
a function of k and $L_{R}$. We define the PT as the ratio of the propagated
output signal power to the input signal power in the resonator. This numerical
simulation is performed by propagating a 127-bit pseudo-random Gaussian pulse
train with 50% RZ modulation format at a BR of 10 Gbps. Our computation covers
a range of k from 0.94 to 0.99 and $L_{R}$ from 10 $\mu$m to 250 $\mu$m and
these ranges are chosen to observe the ring resonator characteristics. It is
interesting to note that we observe both the slow and fast light regimes to
the left and right sides, respectively, of the critical coupling line (green
dashed), in figure 5(a). To increase the delay one can increase k or LR, but
both distortion and energy loss increase at the same time. For a given value
of the maximum distortion constraint (e.g. D $\simeq$ 0.35) in figure 5(b), we
can find many k and $L_{R}$ pairs that provide the same values of distortion-
constrained EO-delay $\simeq$ 0.76 with corresponding PT $\simeq$ 0.67, as
observed in black dotted lines in figures 5(a) and 5(c). Therefore, we will
focus on varying k while keeping a fixed practical value of $L_{R}$ = 150
$\mu$m.

###  3.2 SCISSOR

We now consider a SCISSOR, as shown in figure 6. It is assumed that the
SCISSOR has multiple identical rings and its transfer function
$E_{\mathrm{out}}(\omega)/E_{\mathrm{in}}(\omega)=H_{\mathrm{SCISSOR}}(\omega)=(H_{\mathrm{Ring}}(\omega))^{N}$,
where N is the number of resonators. Figure 7 presents the characteristics of
the SCISSOR for four different numbers of rings (N = 1, 3, 5, and 8) with
$L_{R}$ = 150 $\mu$m, $\alpha$ = 1 cm-1, and k = 0.85. Because the phase shift
at resonance for multiple resonators is additive, the magnitude of the total
group delay from a summation of the delays of all individual ring resonators
increases as a function of N. The FWHD of SCISSOR transmission resonance also
becomes wider than that of a single ring with the resonance transmission
approaching zero. As a result, output pulse power decreases and the output
pulse shape becomes more distorted from the its original input shape as
SCISSOR length N increases, shown in figure 7(d).

##  4 Optimal System Design Study

In this section, we explore optimal system designs for SCISSOR, SBS, and
SCISSOR + SBS. Our approach is to maximize the delay performance under
practical system resource constraints while maintaining constant data fidelity
[14, 17, 18] .

###  4.1 SCISSOR

We use EO and IT metrics, as described in Section 2, to evaluate the SCISSOR
structure. Figure 8 describes the results of the computations summarizing (1)
the EO-delay with associated D and (2) the information theoretic delay with
associated IT as a function of N, where three different noise strengths of
$\sigma^{2}$= 0.2, 0.3, and 0.4 are used for the IT computation. The EO-based
results presented in this paper are based on propagating a 127 bit pseudo-
random pulse train with a RZ modulation format at a BR = 10 Gbps. For
information-based results, we have utilized 8 bit input sequences with the
same modulation format and BR, and therefore a total 256 ( M = 28 states in
equation (2) ) possible bit patterns are considered. For each input pattern,
we use 106 noise samples to obtain reliable results by using a Monte Carlo
technique. As expected we found that increasing N increases both the EOD and
ID at the cost of increased distortion. As a result, the normalized IT values
decrease. We observe both EOD and ID yield similar delay values, as shown in
figure 8(a). From figure 8(b), we see that IT decreases faster for higher
noise strength with increasing N, thus the fidelity of information
transmission decreases with increasing $\sigma^{2}$ because the decreased
signal to noise ratio (SNR) causes information to be lost. Based on D and IT
results for the SCISSOR with N = 4, distortion of D = 0.342 is measured, while
three different values of IT = 0.943, 0.873, and 0.812 are computed with
corresponding AWGN levels of $\sigma^{2}$ = 0.2, 0.3, and 0.4, respectively,
as shown in figure 8(b). For the given specific noise level of $\sigma^{2}$ =
0.3, one can use at most 4 cascaded resonators while simultaneously
maintaining more than 87% (i.e., IT $\geq$ 0.87 or approximately 7 out of 8
bits) of the transmitted information. Therefore, we take the IT constraint IT
$\geq$ 0.87 for $\sigma^{2}$ = 0.3, to correspond with distortion constraint D
$\leq$ 0.35.

Next, under the two signal quality constraints (IT $\geq$ 0.87 and D $\leq$
0.35), we optimize k to maximize EOD and ID for three different attenuation
coefficients $\alpha$ = 0, 1, and 3 cm-1 as a function of N from 1 to 70. The
optimal SCISSOR characteristics using both IT and EO metrics are presented in
figure 9. We note that the results from the two different metrics provide
similar trends. When N = 70, the maximum fractional delays of approximately
10, 8, and 4 are achieved for $\alpha$ = 0, 1, and 3 cm-1, respectively, at
the BR = 10 Gbps. As N increases, both fidelity-constrained-EOD and ID
increases, however, early delay saturation for the highest attenuation value
is observed in figure 9(a). As N increases, k must decrease in order to
increase the effective FWHD so that the system can satisfy the D and IT
constraints. For the same reason, optimal k at higher attenuation is smaller
than that at smaller attenuation, as shown in figure 9(c). Note that these
trends are also explained by the group delay relation of equation (7). The
transferred energy in a lossy SCISSOR decreases exponentially because of the
induced loss, shown in figure 9(b), and thus, the transmission losses of the N
= 70 SCISSOR become around 22 dB and 43 dB for $\alpha$ = 1 and 3 cm-1,
respectively. This is what would limit the delay performance and reduce the
data fidelity of such a system. Inevitably, we must conclude that an
amplification process is required for the SCISSOR. As mentioned earlier, an
SBS gain medium is a good choice for both increasing the delay performance and
the signal amplification by combining it with the SCISSOR.

###  4.2 Broadband SBS

Slow light via the stimulated Brillouin scattering process has previously been
demonstrated for tunable delay in optical fiber [10, 23, 28]. The SBS process
is a nonlinear interaction between a strong pump wave and a weak probe wave
that is mediated by an acoustic wave. The acoustic wave generated from this
interaction scatters photons to the probe wave, shifting the scattered light
downward to the Stokes frequency $\omega_{s}=\omega_{p}-\Omega_{B}$, where
$\Omega_{B}$ is the Brillouin frequency shift in optical fiber. As a result,
the Stokes field experiences strong gain at $\omega_{s}$. For a typical single
mode fiber, the Brillouin frequency shift $\Omega_{B}$ is $\sim$ 10 GHz and
the Brillouin linewidth $\Gamma$ is $\sim$ 40 MHz near the communication
wavelength of 1550 nm. However, this narrow bandwidth limits the achievable
data rate to only several megabits per second. Much of the recent research in
the SBS slow-light community focuses on broadening the available SBS
bandwidth, and several techniques have been experimentally demonstrated that
accommodate a GHz data rate. A primary technique is direct modulation of a
Gaussian noise source, generated by an arbitrary waveform generator. Gain
bandwidths of up to 25 GHz have been experimentally demonstrated [11, 13, 12].

Under the small signal approximation, the input field $E(0,\omega$) will be
amplified at the fiber output according to $E(L_{f},\omega$) =
$E(0,\omega)H_{\mathrm{SBS}}(\omega)$, with the SBS transfer function
$H_{\mathrm{SBS}}(\omega)$ = exp($k(\omega)L_{f}$). Here, $L_{f}$ represents
the fiber length and $k(\omega$) is the complex wave vector. For the pump
broadened SBS, $k(\omega$) = $P_{p}(\omega)\otimes g_{B}(\omega)$ can be
obtained by convolving the pump spectrum $P_{p}(\omega)$ with the Lorentzian
gain profile
$g_{B}(\omega)=g_{0}/[1-j((\omega-\omega_{\mathrm{s}})/(\Gamma/2))]$, where
$g_{0}$ is the line-center gain coefficient. Pant et al showed that a super
Gaussian function provides a good approximation of the optimal pump profile
$P_{p}(\omega)=(x_{1}/x_{2})$exp$[-(\omega-((\omega_{s}+\Omega_{B}))/x_{2})^{2x_{3}}]$,
where the parameters $x_{1}$, $x_{2}$, and $x_{3}$ define pump peak power,
pump width, and pump shape (i.e. $x_{3}$=1 is Gaussian and $x_{3}\gg$ 1
becomes nearly rectangular) [18]. In the next subsection, these three
parameters will be optimized subject to the fidelity constraint and the
maximum SBS gain constraint, G = max{$k(\omega)L_{f}$} $\leq$ 10\. When
$\omega=\omega_{0}$, the line-center gain of the broadband SBS is defined by
$\textit{G}\simeq g_{0}x_{1}\Gamma\pi L_{f}/2Ax_{2}$, where A is the mode
area. This gain constraint is imposed to avoid the nonlinear amplifier
behavior and the maximum available gain G = 10 is conservative value as
compare to the Brillouin gain threshold of $\sim$ 25 [28].

###  4.3 SBS + SCISSOR

Recall the results presented in figure 9, from which we proposed the utility
of a joint SCISSOR + SBS system. To demonstrate more explicitly the advantages
of this combined slow-light device, we present a practical system design,
analyzing its performance in terms of several important factors such as FD,
PT, D, and IT in this section. The transfer function for such a device is
given by $H(\omega)$ = $H_{\mathrm{SCISSOR}}(\omega)\times
H_{\mathrm{SBS}}(\omega)$, and its real and imaginary parts at resonance are
related to the gain and the refractive index profiles through the Kramers-
Kronig relation. Figure 10 shows the normalized transmission spectra for
individual and combined systems along with the spectrum of a 128 bit pseudo-
random RZ sequence at BR = 10 Gbps. We assumed the signal carrier frequency
and the SCISSOR resonance frequency $\omega_{0}$ are the same as the SBS
Stokes frequency $\omega_{s}$. To better understand the impact of using this
combined system, we look at the input and output eye-diagrams after
propagating through the combined transmission spectrum, as shown in figure 10,
for several different SBS gain values of G = 0, 1, 5, and 10. These results
are shown in figure 11. For simplicity, we first consider the SCISSOR with N =
1 and note that the combined system with SBS G = 0 is a resonator-only system.
It is known that the Gaussian pulse propagating through the ring resonator
undergoes dispersion effects that can cause oscillations at the pulse rising
and/or trailing edges mainly due to the cubic-GVD, as shown in figure 11(b)
[29]. On the other hand, in the SBS system, the output pulse undergoes
distortion in the form of pulse broadening mainly due to the quadratic-GVD.
Note that distortion management techniques for the SBS system basically
suppress the quadratic-GVD term as demonstrated by Stenner et al [10]. By
comparing figures 11(b) and 11(d), the fractional EOD for the SCISSOR only and
SCISSOR + SBS (G = 10) are 0.61 and 1.88 respectively, therefore, it is
clearly observed that the SCISSOR + SBS combination not only improves delay
performance, but also suppresses the pulse oscillation in the pulse trailing
edge arising from the resonator. Although the SBS process also introduces the
pulse broadening, it is not significant in this example. In addition,
combining SBS + SCISSOR provides additional benefits in terms of delay and PT
improvement. Therefore, the combined system provide $\sim$ 3.1 times larger
delay with only a small sacrifice of $\sim$ 1.2 times eye-closing when
compared to the SCISSOR-only system.

Figure 12 shows the summary of the optimal results for the resonator loss a =
1 cm-1 as a function of N = 1 - 70 for two candidate systems: SCISSOR-only and
SCISSOR + SBS systems under data fidelity constraints (IT $\geq$ 0.87 and D
$\leq$ 0.35). We observe that results via both metrics agree well. In general,
the maximum fidelity-constrained delays gradually increase, while optimal SBS
gain G and SCISSOR coupling coefficient k decrease. The gain and k must be
chosen effectively to achieve the maximum delay performance under the IT and D
limit. Therefore, for the region of N $<$ 7, the maximum gain can remain
constant, whereas k decreases. However, any further increase in N requires a
decrease in SBS gain as shown in figure 12(c). We know that the presence of
loss causes a nonnegligible decrease in PT for increasing N as shown in figure
12(b). The results, however, indicate that the combined system can
significantly improve the PT and the delay performance. Even for large number
of rings (N = 70) the combined system can achieve unit power transmission
ratio. The optimal design curves presented in figures 12(a) - (d) represent
bounds on the performance of our proposed delay devices subject to real-world
operating and fidelity constraints. In summary, the proposed technique enables
a maximum fractional delay of $\sim$ 17, which is $\sim$ 2.1 times the maximum
SCISSOR-only delay, with unit power transmission using a cascade of 70 ring
resonators combined with an SBS gain medium and can overcome Khurgin’s
fundamental limit for the fractional delay for the SCISSOR, in which N $>$ 100
resonators are required for fractional delay of 10 [30].

##  5 Conclusion

We have presented a practical system design for increasing the fractional
delay while maintaining high data fidelity by combining SBS and SCISSOR. We
have employed two different fidelity metrics (EO-metric and IT-metric) to
evaluate the slow-light system performance subject to real-world resource
constraints. By jointly optimizing the system parameters, the combined SBS +
SCISSOR system can provide larger delay and improved power throughput compare
to the SCISSOR-only system. We have shown that the maximum fidelity
constrained-delay of $\sim$ 17 for SBS + SCISSOR can be achieved with an unit
power transmission at a bit rate of 10 Gbps.

We gratefully acknowledge the financial support of the DARPA/DSO Slow-Light
Program.

## References

## References

  * [1] Hau L V, Harris S E, Dutton Z and Behroozi C H 1999 Nature 397 594
  * [2] Morichetti F, Melloni A, Ferrari C and Martinelli M 2008 Opt. Express 16 8395
  * [3] Melloni A, Morichetti F, Ferrari C, and Martinelli M 2008 Opt. Lett. 33 2389
  * [4] Xia F, Sekaric L and Vlasov Y 2007 Nat. Photonics 1, 65
  * [5] Otey C R, Povinelli M L and Fan S 2008 J. Lightwave Technol. 26 3784
  * [6] Heebner J E, Boyd R W and Park Q 2002 J. Opt. Soc. Am. B 19 722
  * [7] Heebner J E, Wong V, Schweinsberg A, Boyd R W and Jackson D J 2004 IEEE J. Quantum Electron. 40 726
  * [8] Govindan V and Blair S 2008  J. Opt. Soc. Am. B 25 C23
  * [9] Lenz G, Eggleton B J, Madsen C K and Slusher R E 2008 IEEE J. Quantum Electron. 37 525
  * [10] Stenner M D, Neifeld M A, Zhu Z, Dawes A M C and Gauthier D J 2005 Opt. Express 13 9995
  * [11] Song K Y and Hotate K 2007 Opt. Lett. 32 217
  * [12] Zhu Z, Dawes A M C, Gauthier D J, Zhang L and Willner A E 2007 J. Lightwave Technol. 25 201
  * [13] Cabrera-Granado E, Calderon O G, Melle S and Gauthier D J 2008 Opt. Express 16 16032
  * [14] Lee M, Pant R and Neifeld M A 2008 Appl. Opt. 47 6404
  * [15] Lee M, Pant R, Stenner M D and Neifeld M A 2008 Opt. Commun. 281 2975
  * [16] Shi Z and Boyd R W 2009 Phys. Rev. A 79 013805
  * [17] Pant R, Stenner M D, Neifeld M A, Shi Z, Boyd R W and Gauthier D J 2007 Appl. Opt. 46 6513
  * [18] Pant R, Stenner M D, Neifeld M A and Gauthier D J 2008 Opt. Express 16 2764
  * [19] Downie J D 2005 J. Lightwave Technol. 23 2031
  * [20] Neifeld M A and Lee M 2008 J. Opt. Soc. Am. B 25, C31
  * [21] Neifeld M A and Lee M 2008 Slow and Fast Light (Optical Society of America, 2008) SWA5
  * [22] Chin S, Gonzalez-Herraez M and Thevenaz L 2009 Opt. Express 17 21910
  * [23] Chin S, Gonzalez-Herraez M and Thevenaz L 2008 Opt. Express 16 12181
  * [24] Shannon C E 1948 Bell System Technical J. 27 379
  * [25] Shannon C E 1948 Bell System Technical J. 27 623
  * [26] Stenner M D and Neifeld M A 2008 Opt. Express 16 651
  * [27] Schwelb O 2004 J. Lightwave Technol. 22 1380
  * [28] Zhu Z, Gauthier D J, Okawachi Y, Sharping J E, Gaeta A L, Boyd R W and Willner A E 2005 J. Opt. Soc. Am. B 22 2378
  * [29] Madsen C K and Zhao J H 1999 Optical filter design and analysis; A signal processing approach (Wiley)
  * [30] Khurgin J B and Morton P A 2009 Opt. Lett. 34 2655

## Figure captions

Figure 1:  (a) Channel model of a slow-light delay device. OOK GP generates a
train of Gaussian pulses (GP) modulated via on off keying (OOK). (b) Example
of slow-light output pulses including effects of delay, distortion, and noise.
Top figures – without distortion, middle figures – moderate distortion, and
bottom figures – large distortion causes ISI. Figure 2: Application of the
information theoretic analysis to an ideal delay device for the 3 bits
transmission (8 possible states). (a) Example signals at the input (dashed)
and output (solid) of the slow light operator. The input window (IW) along
with two candidate output windows (OW) are shown. (b) Mutual information as a
function of output window offset. Note that IT represents the information
throughput and ID represents the information delay.  Figure 3: System
parameters for a single microring resonator. Figure 4: Lossy resonator
characteristics for four values of k = 0.8, 0.9, 0.95, and 0.97. (a)
Transmission spectra, (b) phase spectra, (c) group delay, and (d) Gaussian
input and delayed output pulses.  Figure 5: (Color online) Single resonator
characteristics. (a) Fractional delay (FD), (b) distortion (D), and (c) power
transmission (PT) as a function of k and $L_{R}$. The green-dashed line in (a)
represents the location of the critical coupling. The black-dotted line in (b)
represents D = 0.35, and corresponding distortion-constrained FD and PT are
also represented by same black-dotted lines in figures (a) and (c),
respectively. SL represents the slow light region and FL represents the fast
light region.  Figure 6: Side-coupled integral spaced sequence of resonators
(SCISSOR).  Figure 7: SCISSOR characteristics for four different number of
resonators N = 1, 3, 5, and 8. (a) Transmission spectra, (b) phase spectra,
(c) group delay, and (d) Gaussian input and delayed output pulses.  Figure 8:
Summary of EO and IT results for SCISSOR with N= 1 - 6. (a) EOD and ID for
three different values of noise strength. (b) IT on the left axis and
distortion on the right axis. The system parameters of $L_{R}$ = 150 $\mu$m
and k = 0.915 are used. Figure 9: Optimal results of (a) EO-constrained
fractional EOD and IT-constrained fractional ID, (b) power throughput, (c)
self-coupling coefficient k on the left axis and $L_{R}$ = 150 $\mu$m on the
right axis, and (d) Distortion on the left axis and IT on the right axis as a
function of N Figure 10:  Transmission spectra (log-scale) for SBS, SCISSOR,
and SCISSOR+SBS along with input pulse spectrum at a BR = 10 Gbps. SBS
parameters of gain = 10, gain bandwidth = 10 GHz, and super-Gaussian factor =
2 and SCISSOR parameters of N = 1, k = 0.965, $\alpha$ = 1 cm-1, and $L_{R}$ =
150 $\mu$m are used.  Figure 11: Input and output eye-diagrams vs. SBS gain.
(a) Input and output eye-diagrams for (b) SCISSOR-only (SBS G = 0), (c)
SCISSOR + SBS (G = 1), (d) SCISSOR + SBS (G = 5), and (e) SCISSOR + SBS (G =
10). Double-side arrows represent $T_{EO}$. Note that N = 1. Figure 12:
Optimal results of (a) EO-constrained fractional EOD and IT-constrained
fractional ID, (b) power throughput (dB), (c) SBS gain, and (d) ring
parameters k on the left axis and $L_{R}$ on the right axis as a function of N
= 1-70. Unit transmission is defined as PT = 1 (i.e. PT = 0 dB)

## Figure captions

Figure 1:

Figure 2:

Figure 3:

Figure 4:

Figure 5:

Figure 6:

Figure 7:

Figure 8:

Figure 9:

Figure 10:

Figure 11:

Figure 12: