# Slow Light with Photonic Crystals for On-Chip  Optical Interconnects

Authors: Sean P. Anderson; Ashutosh R. Shroff; Philippe M. Fauchet
Journal: Advances in Optical Technologies
 (2008)

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## Abstract

Transistor scaling alone can no longer be relied upon to yield the exponential speed increases
we have come to expect from the microprocessor industry. The principle reason for this is theinterconnect bottleneck, where the electrical connections between and within microprocessors are becoming, and in some cases have already become, the limiting factor in overall microprocessor performance. Optical interconnects have the potential to address this shortcoming directly, by providing an inter- and intrachip communication infrastructure that has both greater bandwidth and lower latency than electrical interconnects, while remaining safely within size and power constraints. In this paper, we review the requirements that a successful optical interconnect must meet, as well as some of the recent work in our group in the area of slow-light photonic crystal devices for on-chip optical interconnects. We show that slow-light interferometric optical modulators in photonic crystal can have not only high bandwidth, but also extremely compact size. We also introduce the first example of a multichannel slow light platform, upon which a new class of ultracompact optical devices can be built.


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## Body

## 1. Introduction

Transistor scaling has been the crux of
the rapid growth in microprocessor performance over the past forty years [1].
More recently, however, the performance of the electrical interconnects, which
are responsible for transporting data within the microprocessor and between the
microprocessor and memory, has been unable to keep pace. This is true because
as the interconnect is scaled down along with the transistors, resistance and
capacitance grow, limiting performance. This not only decreases the bandwidth
of the interconnect, but also increases both its latency and power consumption.
In fact, in modern microprocessors, over half of the dissipated power is
dissipated by the interconnects [2, 3]. These issues will continue to worsen as
chip technology continues to scale [3–8]. Optical interconnects can directly
address these problems at the system level by replacing electrical
interconnects [6–10]. To do so, they must meet the performance requirements of
modern and future microprocessors while achieving both compact size and low power
consumption.Implementing
optical interconnects in silicon has the advantage of maintaining maximum
compatibility with the existing CMOS fabrication infrastructure. Additionally,
silicon is transparent in the range of telecom wavelengths(λ~1.5μm). However, it is not an ideal choice
as an active optical material due to its relatively weak electro-optic response
[11]. In general, this results either in large device size or decreased
operational bandwidth. The former is true especially for phase-shifting
approaches, while the latter applies mostly to resonant approaches. To work
around this limitation, it is possible to use some other, extrinsic material as
the active medium, such as liquid crystals [12–14], quantum dots [15], or
electro-optic polymers [16]. These have the advantage of potentially smaller
device size, but at the expense of bandwidth. This is because their response
times can be orders of magnitude slower than that of silicon. To date, devices
based on such extrinsic materials are unable to meet the bandwidth requirements
for optical interconnects. Such hybrid approaches also have the drawback of
reduced compatibility with CMOS processing techniques. An all-silicon approach
would thus be favorable, although devices that employ silicon as the active
medium have so far been unable to meet the size requirements.Using
slow light, however, it is possible to create a device that meets both of these
seemingly contradictory sets of requirements. The advantage of using slow light
is that, because the group velocity of light is decreased by two or more orders
of magnitude as compared to that in bulk silicon, the effective photon-material
interaction length inside an active device is increased [17]. This allows
greater use to be made of the small refractive index changes available in
silicon, and thus can enable silicon-based photonics to meet the requirements
set out by optical interconnects.In
this paper, we first review the rationale behind the push toward optical
interconnects, as well as the bandwidth, latency, power, and footprint
requirements that an optical interconnect must satisfy in order to be
competitive. We then explain how slow light can be used as the basis for an
optical interconnect technology that satisfies these requirements.
Specifically, we describe how interferometric optical modulators based on slow
light in photonic crystals can exhibit not only high bandwidth but also
extremely compact size. We also report on recent results regarding a new
optical structure that merges slow light with wavelength- division multiplexing
to form the first multichannel slow-light platform for silicon photonics.
Finally, we outline some of the challenges surrounding the implementation of
these novel structures and devices and, more broadly, how a slow-light approach
may fit into what has been called “the interconnect era” [18].


## 2. The Need for Optical Interconnects

The electrical interconnects inside
modern microprocessors consist of copper wires surrounded by a low-k dielectric [19]. This represents a
transition from previous generations of electrical interconnects, which used
aluminum wires surrounded by a dielectric. Both the switch from aluminum to
copper, and from dielectric to specifically a low-k dielectric, were made
to reduce the RC time constant of the interconnect (the former determining the
R part, and the latter the C part). Since the RC time constant is essentially a measure of the amount of energy
required to operate the interconnect, reducing it allows the interconnect to operate
more efficiently. A lower time constant also allows for lower transmission
latency and for reduced crosstalk between adjacent wires. Regardless of the
materials used, however, the performance of an electrical interconnect, weather
measured by bandwidth, latency, power consumption or crosstalk, worsens as its
dimensions are scaled down. This is due to the fact that the resistance of a
metal wire grows as its cross-sectional area is reduced. The result is a
hyperbolic increase in the RC time
constant per unit length of interconnect as the chip feature size is scaled
down [6]. Optical interconnects, on the other hand, do not suffer from this
constraint because they are not subject to an RC time constant.For
this reason, optical interconnects (OIs) offer a number of advantages over
electrical interconnects (EIs). The first is a significantly lower signal
propagation delay. Figure1(a) shows the relationship between propagation delay
and interconnect length both for EIs based on copper, and for OIs based on
silicon waveguides. Silicon waveguides possess an intrinsic advantage over
copper wires because of their higher signal propagation speed, which is in turn
due to the absence of RC impedances
[10]. Because there is an encoding/decoding penalty associated with optical
interconnects (the time spent converting the signal from the electrical domain
to the optical, and vice versa), OIs based on silicon waveguides may be best
suited for longer length interconnects (i.e.,
global, as opposed to local, interconnects)
[10, 20].(a) Propagation delay of
silicon optical waveguides as compared to copper electrical wires
(cf. [4]). (b) Comparison
of bandwidth density of EIs to that of a single-channel OI as a function of
ITRS year and technology node. The use of a multichannel OI can increase
bandwidth density over that of EIs (cf. [4]).
(a)(b)A
second metric by which OIs must be compared to EIs is bandwidth density.
Although the bandwidth available from a single wire in an EI decreases as the
chip feature size is scaled down, the cross-sectional area it occupies also
decreases. The net result is that the bandwidth density, as measured by the
number of bits that can be transmitted per second per unit lateral width,
increases with further chip scaling. Additionally, repeaters can be used to
further enhance bandwidth, though at the price of increased size and power
consumption [21]. This creates a moving target for OIs to beat.
Figure 1(b)
compares the bandwidth density of EIs to that of OIs, based on reasonable
estimates of the size of and spacing between the silicon waveguides. The bandwidth
density of OIs is assumed to scale linearly with the clock speed of the chip,
while the bandwidth density of EIs scales at a slightly higher rate. EIs,
however, have the disadvantage that they can support only a single data channel
at a time. OIs, on the other hand, can employ techniques such as
wavelength-division multiplexing (WDM) to support a large number of channels
simultaneously. Even at the 22 nm node,
only three WDM channels are needed to match the performance of EIs [4].
Although the additional channels come at the price of increased power
consumption and footprint, these parameters scale much more weakly for OIs than
for EIs. This is due to the fact that the extra space and power are required
only by additional modulator/demodulator pairs at the ends of the interconnect.
The size of the waveguide itself does not grow (and it consumes no power
because, unlike with EIs, it contains no repeaters). This gives WDM OIs a
significant advantage over EIs, especially for longer-length interconnects.An
additional metric, the power-delay product, is often used to evaluate
interconnect performance because it is a measure of both the power consumed,
and the delay introduced, by the interconnect. The power-delay product (PDP) of
an OI is dependent on both its length and the technique used to modulate the
optical signal. Modulation techniques fall generally into two categories:
resonant and interferometric.
Figure2 compares the PDP for these two types of
modulators to that of EIs as a function of the achievable index shift Δn. For
interferometric modulators, Δn determines the
length required to achieve a required phase shift; the lower Δn, the greater
the length, and thus the greater the power consumption and delay. A
resonator-based modulator has the advantage that its active area can be made
much smaller than in an interferometric modulator, resulting in both lower
delay and lower power consumption. In fact, the PDP for a resonant modulator is
nearly two orders of magnitude less than for an interferometric one, assuming each
has sufficient bandwidth for a single optical channel. This shows that a
resonant approach is generally preferable over one based on interference [4].
Resonant modulators also have the advantage of a smaller footprint, and thus a
smaller size penalty for each additional WDM channel. In principle, these
advantages come at the price of
decreased bandwidth. However, the bandwidth of any interconnect channel is limited by the clock
speed of the chip, and that speed is in the range of tens of gigahertz. Therefore,
a favorable PDP can still be achieved without using Q-factors so high that they
compromise bandwidth.Figure 2
Power-delay product of EI and OIs
as a function ofΔn for the 90 nm
technology node, assuming a length of 10 mm. OIs based on resonator-based
modulators offer a significant advantage over both EIs and OIs based on
interferometric approaches (cf.
[4]).Optical interconnects thus have advantages in terms of signal propagation delay, power
consumption, and bandwidth density. These advantages are especially compelling
when the length of the interconnect in question is long.
We can then estimate thecritical length over which OIs
are preferable to EIs. This length is plotted in
Figure 3 for several
technology nodes of the ITRS [22]. The comparison is made separately for three
different criteria: signal propagation delay, power consumption, and bandwidth
density scaled by delay. In all of these comparisons, OIs are favorable for
interconnect lengths over a few millimeters, assuming the use of WDM and resonator-based
modulators, with each channel operating at a bit-rate equal to the clock rate
of the chip [7, 10, 20].Figure 3
Critical length, normalized to
the chip edge length, above which OIs (using WDM) are advantageous over EIs in
terms of delay, bandwidth density (scaled by delay), and power consumption
(cf. [10]).Therefore,
the development of a successful optical interconnect technology must include
the development of a compact and low-power modulator, upon which a WDM
communication system can be based. However, the size advantage of resonant
modulators is not fundamental. In fact, it is possible to produce
interferometric modulators with superior power and size characteristics if the
group velocity inside the devices can be sufficiently reduced. This is the
principle behindslow-light devices.
We next review recent work in our group in the area of slow-light
interferometric devices in silicon.


## 3. Slow-Light Mach-Zehnder Interferometers

Integrated Mach-Zehnder interferometer (MZI)
devices are used extensively in optical modulators and switches. Liu et al. demonstrated a silicon
high-speed optical modulator with an operational bandwidth of 40 GHz [23]. While the device proves the
feasibility of silicon for optoelectronic applications, it suffers from a
significant disadvantage: its size is on the order of a millimeter. This is
consistent with an earlier analysis presented by Giguere et al. [24]. The large device footprint is a
result of the small value of Δn available in
silicon. This makes the distance, Lπ,
required to produce a π phase shift in one arm of the MZI to be very long.However, Soljačić et al. proposed that by increasing
the group index (decreasing the group velocity) of light propagating in the arms
of an MZI, the sensitivity to small changes in the material refractive index of
the arms could be amplified [17]. More recently, Shi et al. showed experimentally that the sensitivity of an
interferometer is dependent on the group index rather than the material index
[25]. One way to understand this is that inducing a change
δn in the material refractive index in one arm of the
interferometer causes the electromagnetic bands to shift in frequency by an
amount dω. Because frequency is kept
constant by the choice of the operating wavelength, the propagating wave
experiences a change in wavevector magnitude by an amount dk. Therefore, since ng=c/vg=c/(dω/dk), the larger the group index, the larger the
change in wavevector, and thus the shorter the interferometer arms can be made,
because Lπ=π/dk.
Reducing the device length not only allows for space savings, but also
decreases device power consumption. Additionally, the electrodes can be made
smaller, thus reducing parasitics and further increasing operating bandwidth.The photonic crystal
platform [26–28] is ideal for implementing such a device because it provides
the design flexibility required to increase the group index while allowing the tunability required of an active device. In a
photonic crystal coupled-cavity waveguide (PC-CCW), the group velocity can be controlled by changing
the spacing between adjacent cavities. This is because the spacing controls the spatial overlap between the
optical fields in adjacent cavities, which in turn determines the width (and
slope) of the optical miniband and thus its group velocity [29–32]. An MZI
based on such a design was proposed and analyzed by Soljačić et al. [17], although that analysis
did not include the possibility of optical jitter. The concept of optical
jitter in devices is important because it can cause pulse distortion and thus
reduced bandwidth. In fact, in MZIs based on slow-light CCWs, deterministic
optical jitter is a significant source of pulse distortion at high bit-rates.
Further, when multiple devices are cascaded together, uncertainty due to jitter
compounds, which can result in asynchronous operation. In slow-light MZIs,
deterministic optical jitter grows as the device length is reduced, resulting
in a tradeoff between bandwidth and device size reduction. Optical jitter can,
however, be minimized by carefully choosing the operating wavelength of the
device. This effectively removes the tradeoff between bandwidth and size
reduction, but replaces it with a tradeoff between size reduction, and
sensitivity of the device to material and fabrication variance. This new
tradeoff, however, is a very favorable one because the semiconductor industry
excels at minimizing variation.


### 3.1. Increased Sensitivity in Slow-Light MZIs

Figure4(a) shows the concept of an MZI
in which each arm consists of a PC-CCW. We modeled five different PC-CCWs,
denoted by the separation Δ between
adjacent cavities, using the MIT Photonic Bands software package [33, 34]. We
calculated the TE bands for the structures assuming air holes (n=1.0) of radius 0.3a in a silicon slab (n=3.4),
where the lattice constant a is
chosen to be 400 nm for use at λ=1.5μm. Each cavity consists simply of a
missing hole. The Brillouin zone in each case is chosen to include the entire
repeat unit cell of length Δ, which
determines the distance between adjacent cavities. For ease of comparison, all
results are normalized to the lattice constant a. Each PC-CCW has a significant photonic bandgap with a single
defect mode (Figure 4(b)). As the separation between adjacent cavities is
increased, both the bandwidth and the group velocity of the defect band are
reduced because the cavity lifetime of a photon within the defect band is
increased. The greater the group index, the shorter the interferometer
can be made. For example, if we are able to inject a free carrier concentration
of ΔN=1018cm−3 into the silicon, corresponding to an index change of
approximately δn=0.001 [11], then
the length of the MZI can be reduced to approximately 56μm (for the Δ=3  case), as
compared to over 600μm for an MZI
based on a simple photonic crystal line-defect waveguide (the Δ=1 case). In fact, the arm length could
be reduced even further by using PC-CCWs with greater separations Δ between adjacent cavities.(a) Concept
of PC-CCWs used to design ultracompact MZIs. HereΔ is the degree of freedom for varying the device structure.
(b) The
dispersion curves for the CCW defect bands. As the separation Δ between the cavities is increased, the
spatial overlap between the fields localized in each cavity is reduced, thus
flattening the dispersion curve and reducing group velocity.
(a)(b)


### 3.2. Optical Jitter

When modeling these PC-CCW structures
with a small increaseδn in the
material refractive index induced in the entire silicon slab, we find, as
expected, that the bands shift downward in frequency, causing a change Δk in the
magnitude of the corresponding wavevector. Associated with this change in
wavevector, however, is a change in group velocity. This occurs because the
slope of the defect band is not exactly linear at its center, so that when the
magnitude of the wavevector changes by Δk, the
corresponding point on the dispersion curve has a slightly different slope, and
thus a different group velocity. Because the index change is induced in only
one arm of the interferometer, the result is that the pulses in the two arms of
the MZI do not propagate at the same speed, and thus arrive at the output
shifted slightly in time from one another (Figure 5). We have termed this
difference in arrival time deterministic optical jitter, Δt. The effect
of this type of jitter on the output pulse depends upon both the magnitude of
the optical jitter in relation to the
pulse width, as well as the bias of the MZI itself. When the optical jitter is
comparable to or smaller than the pulse width, the result is pulse broadening
in the case of constructive interference, or gross pulse distortion in the case
of destructive interference. When the optical jitter is large in comparison to
the pulse width, the pulses fail to interfere at all, resulting in two distinct
output pulses regardless of the interferometer’s bias.Figure 5
When an
index shift is induced in one arm of the interferometer, the group velocity in
that arm changes. The pulses traveling in the two arms of the interferometer
therefore arrive at the output separated in time byΔt,  resulting in
pulse distortion (broadening). In the worst case, where the pulses interfere
destructively, the output pulse can even become double-humped (inset).In previous work
[17] on slow-light MZI modulators, it was implicitly assumed that
Δt=0, that is, upon producing a small
refractive index shift, δn, in the
material, the defect band moves to a higher or lower frequency without changing
its slope. While this assumption is true in the case of a single defect in an
infinite PC slab, it does not generally hold in the case of PC-CCWs. Because
the group velocity is sensitive to changes in the material refractive index,
the potential for pulse distortion due to deterministic optical jitter must be
taken into account when designing devices based on PC-CCWs. While pulse
distortion is insignificant at moderate bit-rates, it becomes more important at
higher bit-rates, where Δt becomes
significant compared to the FWHM of individual pulses. Depending upon the
parameters of the device, jitter can be quite large; over ten picoseconds for
the Δ=5 PC-CCW device, for example.
In fact, the amount of optical jitter introduced by the MZI is a function of
the separation between adjacent cavities in the PC-CCW. There is thus a
tradeoff between the extent to which the arm length can be reduced and the
amount of optical jitter introduced (Figure 6). We note also that this behavior
is not restricted to the specific geometry used. We have observed the same
effect in CCWs based on other cavity geometries, including that proposed by
Akahane et al. [35, 36].Figure 6
Achievable arm lengthLπ
and optical jitter Δt  as functions
of the separation Δ between the
cavities of the PC-CCWs, assuming δn=0.001.To characterize the importance of
deterministic optical jitter in PC-CCW MZIs, we compared the pulse distortion
it causes to that caused by waveguide dispersion. We assumed a 100 Gbits/s Gaussian pulse train, where the
pulses have a FWHM of 3.33 picoseconds,
and the 10-picosecond bit-slot
thus contains 99% of the pulse energy. Because the pulse bandwidth for a 100 Gbits/s signal is significantly
smaller than the channel bandwidths of the PC-CCW defect bands, waveguide
dispersion should be minimal. To verify this, we calculated the temporal
envelope of a single pulse in the 100 Gbits/s
pulse train after propagating through an MZI with arm lengthLπ
and optical jitter Δt. We then
compared the resulting output pulse to the input pulse using the (1−R2) metric, where 0≤R2≤100%. This metric is
commonly used in regression analysis to calculate the goodness of fit, and is
analogous to the mean-squared error metric [37]. In each case, the amount of
pulse distortion due to waveguide dispersion was less than 1%. In comparison,
approximately the same level of pulse distortion is reached when Δt is only 0.2
picosecond. (This assumes that the
pulses interfere constructively at the output; the distortion would be worse
for destructive interference.) Because pulse distortion increases rapidly with Δt, deterministic optical jitter has the
potential to be the dominant source of pulse distortion in PC-CCW MZIs.


### 3.3. Design Considerations

In previous work, it was
assumed that a small change in the material index would result only in a change
in the magnitude of the propagating wavevector, and not in the slope of the dispersion
curves. Thus, by operating a PC-CCW MZI in the center of the band (k=0.25, where k is the normalized wavevector) where the dispersion
curve is approximately linear, no change in group velocity would be observed.
In practice, however, the operating frequency is fixed by the choice of
wavelength of the input signal. Therefore, as the dispersion curve shifts up or
down, the fixed operating frequency forces a change in the magnitude of the
wavevector, and that change may be large enough to shift outside the linear
region of the dispersion curve, causing a change in group velocity. This effect
is more pronounced in PC-CCWs with larger cavity separations Δ because of the larger changes in k that occur in them. The problem is
compounded by the fact that the linear region of the dispersion curve does not
necessarily lie exactly at the band center in k-space.Figure7 shows the dispersion curves for the Δ=5
PC-CCW, with and without δn applied.
The first operating frequency, ω1,
was chosen because the dispersion curves crossed the band center at
approximately that frequency. While this frequency allows for very large
changes in the magnitude of the wavevector (k ranges from 0.23 for the δn=0
case to 0.29 for the δn=0.0005 case,
thus resulting in a very short Lπ of only 16μm), the slope of the band
changes significantly when the index change is induced, causing the group
velocity to decrease from 0.0033c to
0.0020c, a reduction of nearly 40%.
(Although the example shown in Figure 7 is an extreme one, and the operating
frequency  ω1 would not
normally be chosen because of its unfavorable dispersion characteristics, it is
illustrative of the issue in point.) By shifting the operating frequency down
to ω2 so that it is closer
to the linear regions of the dispersion curves, a much smaller change in group
velocity can be obtained, thus greatly reducing optical jitter. Figure 8 shows
the arm lengths and values of optical jitter that are achievable when these
guidelines are applied to MZIs based on other PC-CCWs. While the arm lengths
are slightly increased, the amount of optical jitter is reduced by as much as
an order of magnitude by optimizing the operating frequency.Figure 7
When the operating frequency is chosen to
correspond to a normalized frequency ofω1=0.24494, the slope of the dispersion
curve changes significantly when δn is applied. When the operating frequency is changed to ω2=0.24485,
the slopes become more similar, thus reducing Δt.Figure 8
Achievable arm lengthLπ
and optical jitter Δt  as functions
of the separation Δ between the
cavities when the optical jitter is minimized by optimizing the operating
frequencies for each MZI configuration, assuming δn=0.001.Optical
jitter can even be eliminated altogether by choosing the appropriate value ofδn. For the Δ=5 PC-CCW, in fact, choosing δn≈0.00055 results
in Δt=0 picosecond and  Lπ<30μm  (Figure 9). By slightly varying the
operating frequency, the “zero-jitter” point can be shifted to other values of δn. For example, choosing ω2=0.24486 as the operating
frequency allows for Δt=0 at δn≈0.0003, with Lπ<50μm. The arm lengths given here can be
further reduced by a factor of two by employing a “push-pull” design for the
MZI, where each arm induces a ±π/2 phase shift. Figure 9 also shows, however,
that Δt
is extremely sensitive to the value of δn  being used. Thus, a minute
variance in the value of δn could
increase Δt to be on the order of tenths of a
picosecond, severely limiting the operational bandwidth of the resulting
device. This sensitivity is more pronounced for shorter device lengths (greater
separation Δ between adjacent
cavities, and thus lower group velocity), hence the tradeoff between the size
of a slow-light MZI and its tolerance to fabrication variances. This new
tradeoff is a very favorable one, however, because the semiconductor industry
excels at minimizing fabrication variances, and improves at it with each
successive generation of technology.Figure 9
Achievable arm lengthLπ
and jitter Δt as functions
of δn for the Δ=5 PC-CCW for the optimized operating frequency of ω=0.24485. For a material index change
of δn≈0.00055,
Δt=0
picosecond, and
Lπ<30μm.As
long as the refractive index change can be tightly controlled, slow-light MZIs can be made very short while still maintaining the bandwidth
necessary for on-chip applications. With an arm length of only30μm and a width under 10μm, slow-light MZIs are competitive with
resonant approaches, such as microrings, in terms of on-chip footprint [38].
Furthermore, power consumption scales down along with arm length, so that a 30μm long MZI uses two orders of magnitude
less power than, for example, a 3-millimeter long MZI. This brings the
power-delay product of slow-light MZIs in line with that of resonator-based
modulators, yet with greater bandwidth.Although
the above analysis is specific to slow-light MZIs based on CCWs, the same
considerations apply to slow-light MZI approaches in general, including those
based on, for example, line-defect waveguides [39, 40]. Just as the slopes of
the dispersion curves must be matched in CCWs to minimize optical jitter, so
must the slopes be matched in line-defect waveguides or other slow-light media.


## 3.1. Increased Sensitivity in Slow-Light MZIs

Figure4(a) shows the concept of an MZI
in which each arm consists of a PC-CCW. We modeled five different PC-CCWs,
denoted by the separation Δ between
adjacent cavities, using the MIT Photonic Bands software package [33, 34]. We
calculated the TE bands for the structures assuming air holes (n=1.0) of radius 0.3a in a silicon slab (n=3.4),
where the lattice constant a is
chosen to be 400 nm for use at λ=1.5μm. Each cavity consists simply of a
missing hole. The Brillouin zone in each case is chosen to include the entire
repeat unit cell of length Δ, which
determines the distance between adjacent cavities. For ease of comparison, all
results are normalized to the lattice constant a. Each PC-CCW has a significant photonic bandgap with a single
defect mode (Figure 4(b)). As the separation between adjacent cavities is
increased, both the bandwidth and the group velocity of the defect band are
reduced because the cavity lifetime of a photon within the defect band is
increased. The greater the group index, the shorter the interferometer
can be made. For example, if we are able to inject a free carrier concentration
of ΔN=1018cm−3 into the silicon, corresponding to an index change of
approximately δn=0.001 [11], then
the length of the MZI can be reduced to approximately 56μm (for the Δ=3  case), as
compared to over 600μm for an MZI
based on a simple photonic crystal line-defect waveguide (the Δ=1 case). In fact, the arm length could
be reduced even further by using PC-CCWs with greater separations Δ between adjacent cavities.(a) Concept
of PC-CCWs used to design ultracompact MZIs. HereΔ is the degree of freedom for varying the device structure.
(b) The
dispersion curves for the CCW defect bands. As the separation Δ between the cavities is increased, the
spatial overlap between the fields localized in each cavity is reduced, thus
flattening the dispersion curve and reducing group velocity.
(a)(b)


## 3.2. Optical Jitter

When modeling these PC-CCW structures
with a small increaseδn in the
material refractive index induced in the entire silicon slab, we find, as
expected, that the bands shift downward in frequency, causing a change Δk in the
magnitude of the corresponding wavevector. Associated with this change in
wavevector, however, is a change in group velocity. This occurs because the
slope of the defect band is not exactly linear at its center, so that when the
magnitude of the wavevector changes by Δk, the
corresponding point on the dispersion curve has a slightly different slope, and
thus a different group velocity. Because the index change is induced in only
one arm of the interferometer, the result is that the pulses in the two arms of
the MZI do not propagate at the same speed, and thus arrive at the output
shifted slightly in time from one another (Figure 5). We have termed this
difference in arrival time deterministic optical jitter, Δt. The effect
of this type of jitter on the output pulse depends upon both the magnitude of
the optical jitter in relation to the
pulse width, as well as the bias of the MZI itself. When the optical jitter is
comparable to or smaller than the pulse width, the result is pulse broadening
in the case of constructive interference, or gross pulse distortion in the case
of destructive interference. When the optical jitter is large in comparison to
the pulse width, the pulses fail to interfere at all, resulting in two distinct
output pulses regardless of the interferometer’s bias.Figure 5
When an
index shift is induced in one arm of the interferometer, the group velocity in
that arm changes. The pulses traveling in the two arms of the interferometer
therefore arrive at the output separated in time byΔt,  resulting in
pulse distortion (broadening). In the worst case, where the pulses interfere
destructively, the output pulse can even become double-humped (inset).In previous work
[17] on slow-light MZI modulators, it was implicitly assumed that
Δt=0, that is, upon producing a small
refractive index shift, δn, in the
material, the defect band moves to a higher or lower frequency without changing
its slope. While this assumption is true in the case of a single defect in an
infinite PC slab, it does not generally hold in the case of PC-CCWs. Because
the group velocity is sensitive to changes in the material refractive index,
the potential for pulse distortion due to deterministic optical jitter must be
taken into account when designing devices based on PC-CCWs. While pulse
distortion is insignificant at moderate bit-rates, it becomes more important at
higher bit-rates, where Δt becomes
significant compared to the FWHM of individual pulses. Depending upon the
parameters of the device, jitter can be quite large; over ten picoseconds for
the Δ=5 PC-CCW device, for example.
In fact, the amount of optical jitter introduced by the MZI is a function of
the separation between adjacent cavities in the PC-CCW. There is thus a
tradeoff between the extent to which the arm length can be reduced and the
amount of optical jitter introduced (Figure 6). We note also that this behavior
is not restricted to the specific geometry used. We have observed the same
effect in CCWs based on other cavity geometries, including that proposed by
Akahane et al. [35, 36].Figure 6
Achievable arm lengthLπ
and optical jitter Δt  as functions
of the separation Δ between the
cavities of the PC-CCWs, assuming δn=0.001.To characterize the importance of
deterministic optical jitter in PC-CCW MZIs, we compared the pulse distortion
it causes to that caused by waveguide dispersion. We assumed a 100 Gbits/s Gaussian pulse train, where the
pulses have a FWHM of 3.33 picoseconds,
and the 10-picosecond bit-slot
thus contains 99% of the pulse energy. Because the pulse bandwidth for a 100 Gbits/s signal is significantly
smaller than the channel bandwidths of the PC-CCW defect bands, waveguide
dispersion should be minimal. To verify this, we calculated the temporal
envelope of a single pulse in the 100 Gbits/s
pulse train after propagating through an MZI with arm lengthLπ
and optical jitter Δt. We then
compared the resulting output pulse to the input pulse using the (1−R2) metric, where 0≤R2≤100%. This metric is
commonly used in regression analysis to calculate the goodness of fit, and is
analogous to the mean-squared error metric [37]. In each case, the amount of
pulse distortion due to waveguide dispersion was less than 1%. In comparison,
approximately the same level of pulse distortion is reached when Δt is only 0.2
picosecond. (This assumes that the
pulses interfere constructively at the output; the distortion would be worse
for destructive interference.) Because pulse distortion increases rapidly with Δt, deterministic optical jitter has the
potential to be the dominant source of pulse distortion in PC-CCW MZIs.


## 3.3. Design Considerations

In previous work, it was
assumed that a small change in the material index would result only in a change
in the magnitude of the propagating wavevector, and not in the slope of the dispersion
curves. Thus, by operating a PC-CCW MZI in the center of the band (k=0.25, where k is the normalized wavevector) where the dispersion
curve is approximately linear, no change in group velocity would be observed.
In practice, however, the operating frequency is fixed by the choice of
wavelength of the input signal. Therefore, as the dispersion curve shifts up or
down, the fixed operating frequency forces a change in the magnitude of the
wavevector, and that change may be large enough to shift outside the linear
region of the dispersion curve, causing a change in group velocity. This effect
is more pronounced in PC-CCWs with larger cavity separations Δ because of the larger changes in k that occur in them. The problem is
compounded by the fact that the linear region of the dispersion curve does not
necessarily lie exactly at the band center in k-space.Figure7 shows the dispersion curves for the Δ=5
PC-CCW, with and without δn applied.
The first operating frequency, ω1,
was chosen because the dispersion curves crossed the band center at
approximately that frequency. While this frequency allows for very large
changes in the magnitude of the wavevector (k ranges from 0.23 for the δn=0
case to 0.29 for the δn=0.0005 case,
thus resulting in a very short Lπ of only 16μm), the slope of the band
changes significantly when the index change is induced, causing the group
velocity to decrease from 0.0033c to
0.0020c, a reduction of nearly 40%.
(Although the example shown in Figure 7 is an extreme one, and the operating
frequency  ω1 would not
normally be chosen because of its unfavorable dispersion characteristics, it is
illustrative of the issue in point.) By shifting the operating frequency down
to ω2 so that it is closer
to the linear regions of the dispersion curves, a much smaller change in group
velocity can be obtained, thus greatly reducing optical jitter. Figure 8 shows
the arm lengths and values of optical jitter that are achievable when these
guidelines are applied to MZIs based on other PC-CCWs. While the arm lengths
are slightly increased, the amount of optical jitter is reduced by as much as
an order of magnitude by optimizing the operating frequency.Figure 7
When the operating frequency is chosen to
correspond to a normalized frequency ofω1=0.24494, the slope of the dispersion
curve changes significantly when δn is applied. When the operating frequency is changed to ω2=0.24485,
the slopes become more similar, thus reducing Δt.Figure 8
Achievable arm lengthLπ
and optical jitter Δt  as functions
of the separation Δ between the
cavities when the optical jitter is minimized by optimizing the operating
frequencies for each MZI configuration, assuming δn=0.001.Optical
jitter can even be eliminated altogether by choosing the appropriate value ofδn. For the Δ=5 PC-CCW, in fact, choosing δn≈0.00055 results
in Δt=0 picosecond and  Lπ<30μm  (Figure 9). By slightly varying the
operating frequency, the “zero-jitter” point can be shifted to other values of δn. For example, choosing ω2=0.24486 as the operating
frequency allows for Δt=0 at δn≈0.0003, with Lπ<50μm. The arm lengths given here can be
further reduced by a factor of two by employing a “push-pull” design for the
MZI, where each arm induces a ±π/2 phase shift. Figure 9 also shows, however,
that Δt
is extremely sensitive to the value of δn  being used. Thus, a minute
variance in the value of δn could
increase Δt to be on the order of tenths of a
picosecond, severely limiting the operational bandwidth of the resulting
device. This sensitivity is more pronounced for shorter device lengths (greater
separation Δ between adjacent
cavities, and thus lower group velocity), hence the tradeoff between the size
of a slow-light MZI and its tolerance to fabrication variances. This new
tradeoff is a very favorable one, however, because the semiconductor industry
excels at minimizing fabrication variances, and improves at it with each
successive generation of technology.Figure 9
Achievable arm lengthLπ
and jitter Δt as functions
of δn for the Δ=5 PC-CCW for the optimized operating frequency of ω=0.24485. For a material index change
of δn≈0.00055,
Δt=0
picosecond, and
Lπ<30μm.As
long as the refractive index change can be tightly controlled, slow-light MZIs can be made very short while still maintaining the bandwidth
necessary for on-chip applications. With an arm length of only30μm and a width under 10μm, slow-light MZIs are competitive with
resonant approaches, such as microrings, in terms of on-chip footprint [38].
Furthermore, power consumption scales down along with arm length, so that a 30μm long MZI uses two orders of magnitude
less power than, for example, a 3-millimeter long MZI. This brings the
power-delay product of slow-light MZIs in line with that of resonator-based
modulators, yet with greater bandwidth.Although
the above analysis is specific to slow-light MZIs based on CCWs, the same
considerations apply to slow-light MZI approaches in general, including those
based on, for example, line-defect waveguides [39, 40]. Just as the slopes of
the dispersion curves must be matched in CCWs to minimize optical jitter, so
must the slopes be matched in line-defect waveguides or other slow-light media.


## 4. Interlaced Coupled-Cavity Waveguide

Even with the use of these
ultracompact, high-speed, slow-light MZIs, it is still desirable to use WDM to
increase the net bandwidth density of OIs since, in general, the bandwidth of
any device is still
limited by the clock speed of the chip. To this end, we have also been
investigating a novel type of slow-light structure that may be an ideal
platform for WDM. It is the first example of which we are aware, of aslow-light WDM platform for silicon
photonics.The interlaced
coupled-cavity waveguide (ICCW), as we have named it, is a multiresonant
structure. Its operation is analogous to that of a normal CCW, where
electromagnetic energy couples from one defect cavity to the next, except that
there are now multiple cavities, each of which has a different resonant
frequency (Figure10). Thus, light couples from one cavity to the next of the
same size, skipping over the intervening cavities. An ICCW therefore exhibits
multiple slow-light bands, each corresponding to cavities of a particular
radius.Figure 10
Illustration of the concept of a photonic crystal interlaced microcavity. It is a combination of several photonic
crystal microcavities into a single waveguide. Repeating the photonic crystal
interlaced microcavity yields the ICCW structure.Simulations
of a representative ICCW design, using parameters similar to those used for the
slow-light MZI above (slab index 3.4, hole index 1.0, lattice constant 400 nm,r/a=0.35), reveal a significant
photonic bandgap. In this particular structure, the radii of the defects
increase in steps of  0.075a from r/a=0.0 to r/a=0.225. There are 10 TE bands in the bandgap, which includes
several coupled-cavity modes as well as several waveguide modes. Each of the
coupled-cavity modes is localized to a cavity of a particular radius.
Figure 11(a)
plots the bandstructure of the ICCW, and Figure 11(b) shows the electric field
distributions for three of the localized modes. The exact positions in
frequency space of these localized bands, and thus the spacing between them,
can be adjusted by fine-tuning the radii of the defect cavities. For example,
in Figure 11(a), band 4 can be moved such that it lies exactly between bands 3
and 5 by slightly increasing the radius of the second defect, in which band 4
is localized.(a) Dispersion curves for the
photonic crystal modes of the ICCW structure. (b) Electric field distributions
for three of the cavity-localized modes.
(a)(b)Assuming
that only one sixteenth of the bandwidth of each band can be used, centered
around the zero-GVD points, this ICCW structure exhibits an aggregate bandwidth
above 400 Gbits/s at group
velocities below0.004c. Several
other bands also exhibit good dispersion properties, in addition to low group
velocities. However, either their bandwidths are smaller due to band-edge
effects or their lowest group velocities are only one to two orders of
magnitude lower than c. The use of
these additional bands can, however, provide a further boost to the total
available bandwidth. Pulse propagation studies, performed using the propagation
constant β calculated to the second
order from the dispersion curves, show that for short distances the device can
support high bandwidths. Figure 12 shows one such study for the fourth band in
the gap, having an estimated bandwidth of 99 Gbits/s.Figure 12
Pulse shape evolution for the
fourth band in the gap.


### 4.1. Tuning the Slow-Light Properties of the ICCW

In order to determine the effect of
changing the refractive index and radii of the defect cavities, we simulated a
second ICCW structure similar to the first but with cavity radii0.05a, 0.125a , 0.20a, and 0.275a, corresponding to a 0.05a increase over the previous structure. Additionally, the refractive index of the
material inside these cavities was set to 1.5 in order to be representative of
silicon dioxide, for example, or various active materials such as liquid
crystals or electro-optic polymers. This structure exhibits 8 TE bands in its
bandgap, which is fewer than the previous structure due to the reduced index
contrast. The electric field distributions of the four coupled-cavity modes are
shown in Figure 13, and each is localized around a cavity of a particular
radius.Figure 13
Electric field distributions for
the four coupled-cavity modes in the ICCW.We next determined the effect of changing
the refractive index of the material in the cavities upon the value of the
eigenfrequency of each mode. To do this, we varied the refractive index of the
medium in each cavity from 1.5 to 1.501 in steps of 0.0002. Each band exhibits
a linear relationship between its eigenfrequency and the refractive index
change. This is a very promising feature because it could be exploited to
actively tune ICCW-based devices.The most powerful property of the ICCW,
however, is that the cavities can be tuned individually.
Figure14 shows the
result of introducing a small index shift (+0.0005) into any cavity. In each case,
the mode localized to the cavity in which the index change is introduced
experiences a change in frequency on the order of approximately 0.5 GHz (corresponding to a tuning
sensitivity df/dn above
1.2 THz/RIU,
assuming 1.5μm wavelength), while
the other modes experience only a much smaller change. The fact that the multiple
slow-light bands of the ICCW structure are each separately accessible may lead
to the development of a variety of novel devices including multichannel
modulators and switches, multichannel tunable amplifiers and lasers, or
multichannel biosensors. Because all of the channels are contained in a single
waveguide, these devices could be made no larger than their single-channel
counterparts. An optical modulator based on the ICCW platform would have the
advantage of being a resonant modulator, but would be able to handle multiple
high-bandwidth optical channels simultaneously. Addressing individual cavities
will be challenging, but patterning electrical contacts with size and spacing
on the order of one lattice period is possible using UV lithography or direct-write
methods, for example. Also, work on single quantum dots has shown that it is
possible to manipulate the electrical properties of ultrasmall regions of
semiconductor, on the order of hundreds of nanometers in size [41, 42]. Although
this work has not yet been extended to photonic crystal devices, doing so
should not be beyond the capabilities of state-of-the-art nanofabrication
techniques.Tuning of individual cavities in
the ICCW. The cavity tuned in each case is indicated in the schematic.
(a)(b)(c)(d)It is possible to further
optimize the performance of the device by changing the number or radii of the
cavities, or the refractive indices of the cavities or the surrounding photonic
crystal slab. Simulations of an ICCW with three and five different cavity
sizes, for example, have also been carried out with similar results to those
above, but with three and five localized modes, respectively.


## 4.1. Tuning the Slow-Light Properties of the ICCW

In order to determine the effect of
changing the refractive index and radii of the defect cavities, we simulated a
second ICCW structure similar to the first but with cavity radii0.05a, 0.125a , 0.20a, and 0.275a, corresponding to a 0.05a increase over the previous structure. Additionally, the refractive index of the
material inside these cavities was set to 1.5 in order to be representative of
silicon dioxide, for example, or various active materials such as liquid
crystals or electro-optic polymers. This structure exhibits 8 TE bands in its
bandgap, which is fewer than the previous structure due to the reduced index
contrast. The electric field distributions of the four coupled-cavity modes are
shown in Figure 13, and each is localized around a cavity of a particular
radius.Figure 13
Electric field distributions for
the four coupled-cavity modes in the ICCW.We next determined the effect of changing
the refractive index of the material in the cavities upon the value of the
eigenfrequency of each mode. To do this, we varied the refractive index of the
medium in each cavity from 1.5 to 1.501 in steps of 0.0002. Each band exhibits
a linear relationship between its eigenfrequency and the refractive index
change. This is a very promising feature because it could be exploited to
actively tune ICCW-based devices.The most powerful property of the ICCW,
however, is that the cavities can be tuned individually.
Figure14 shows the
result of introducing a small index shift (+0.0005) into any cavity. In each case,
the mode localized to the cavity in which the index change is introduced
experiences a change in frequency on the order of approximately 0.5 GHz (corresponding to a tuning
sensitivity df/dn above
1.2 THz/RIU,
assuming 1.5μm wavelength), while
the other modes experience only a much smaller change. The fact that the multiple
slow-light bands of the ICCW structure are each separately accessible may lead
to the development of a variety of novel devices including multichannel
modulators and switches, multichannel tunable amplifiers and lasers, or
multichannel biosensors. Because all of the channels are contained in a single
waveguide, these devices could be made no larger than their single-channel
counterparts. An optical modulator based on the ICCW platform would have the
advantage of being a resonant modulator, but would be able to handle multiple
high-bandwidth optical channels simultaneously. Addressing individual cavities
will be challenging, but patterning electrical contacts with size and spacing
on the order of one lattice period is possible using UV lithography or direct-write
methods, for example. Also, work on single quantum dots has shown that it is
possible to manipulate the electrical properties of ultrasmall regions of
semiconductor, on the order of hundreds of nanometers in size [41, 42]. Although
this work has not yet been extended to photonic crystal devices, doing so
should not be beyond the capabilities of state-of-the-art nanofabrication
techniques.Tuning of individual cavities in
the ICCW. The cavity tuned in each case is indicated in the schematic.
(a)(b)(c)(d)It is possible to further
optimize the performance of the device by changing the number or radii of the
cavities, or the refractive indices of the cavities or the surrounding photonic
crystal slab. Simulations of an ICCW with three and five different cavity
sizes, for example, have also been carried out with similar results to those
above, but with three and five localized modes, respectively.


## 5. Implementation Challenges

Although photonic crystal slow-light
devices have considerable potential in the area of optical interconnects, there
are still several issues that must be addressed before they can be practically implemented.
One challenge applicable to optical interconnects in general is the
availability of an effective light source. While it is possible to develop a
highly efficient, off-chip source, coupling its output to the chip can add
great cost and complexity to chip package design, in addition to counteracting
some of the efficiency advantages. A wafer-bonding approach can help to address
some of the complexity issues, but still requires an extra layer of
fabrication, as well as the inclusion of materials that are not a part of the
standard CMOS library [43]. The development of an on-chip light source,
however, would eliminate any packaging and output coupling issues, by allowing
the light source to be integrated directly on the microprocessor itself [44]. Unfortunately,
the constraints of working with silicon (which has an indirect bandgap) in an
on-chip platform, where power dissipation, device footprint, and other
limitations must be taken into account, complicate the design process. To date,
despite evidence for optical gain, there is still no silicon-based laser based
on carrier injection [44–46]. Although an optically-pumped silicon laser based
on the Raman effect has already been demonstrated [47], an electrically-pumped
laser is much more desirable because it would eliminate any need for an
off-chip light source. For the immediate future, a wafer-bonding approach would
seem to be the most convenient option, although the inclusion of an
electrically-pumped, silicon light source will help to realize the full
advantages of optical interconnects.Several issues specific to the
implementation of slow-light devices also exist. The first is that of coupling.
In order to maintain the delay and bandwidth density advantages of optical
interconnects based on slow-light devices, it is desirable to use silicon wire
waveguides as the transmission medium and to use slow-light devices only as the
active elements. This is because photonic crystal waveguides, including the
ICCW, have a larger cross-section than silicon wire waveguides, the latter of
which can have a width under1μm.
Additionally, the low group velocities inside coupled-cavity waveguides would
increase data propagation delays. Therefore, it is necessary to be able to
couple the active slow-light devices to traditional silicon wire waveguides.
The large mismatch between the group velocities in these two media, however,
makes efficient coupling very challenging. An abrupt waveguide-to-CCW
interface, for example, causes strong reflection and thus signal loss. Some
success has been achieved in coupling silicon ridge waveguides to CCWs [48],
although because of the large group velocity difference any approach based on a
taper is unlikely to be compact enough for integration [49]. Other methods,
such as those based on structural optimization [50, 51] have shown some promise
in reducing coupling loss. More recently, another approach based on the use of
a photonic crystal waveguide having an intermediate group velocity was
demonstrated to have very high coupling efficiency into slow-light modes
[49, 52]. While these techniques have so far only been used for coupling into
band-edge slow-light modes, they may also prove useful as the basis for
coupling into CCW slow-light modes.A
second issue facing slow-light devices is that of insertion loss. Losses in
slow-light photonic crystal devices can be divided into two categories:
extrinsic loss and intrinsic loss. Extrinsic loss is due to imperfections in
the fabricated structure, that is,
disorder induced during fabrication. A recent study of slow-light photonic
crystal waveguides revealed that extrinsic losses have only a sublinear
dependence on group velocity (proportional to1/vg1/2)
[53], which is promising for slow-light devices. However, at very low group
velocities (below 0.01c), extrinsic
losses may scale much more strongly and thus may have the potential to become
the dominant loss mechanism [53, 54]. Further refinement of fabrication
processes can help to control extrinsic losses, though, by reducing the structural
disorder that causes it.Intrinsic
loss, on the other hand, exists regardless of fabrication variances. This takes
the form of radiation loss, where optical energy leaks out of the structure due
to lack of confinement in the vertical direction. In general, these losses grow
as the group velocity is decreased. Radiation loss can be controlled, however,
by optimizing the quality factorQ of
the individual cavities that make up the CCW [55, 56].Much progress has already been made in the area of ultra-high-Q photonic crystal microcavities
[35, 36], and it has been shown that the radiation loss of a CCW can actually be
made far lower than that of a single microcavity [55]. This can be accomplished
in part by optimizing the spatial distribution of the optical modes in the
cavities, such as by using a modified cavity geometry. To that end, we have
observed that the analysis presented in Section 3 also holds for other cavity
geometries, such as that proposed by Akahane et al. [35, 36].
Finally, because of the extremely short lengths of slow-light devices, even
with total losses as high as 20 dB/mm, for example, a 50μm long device would exhibit only 1 dB of net loss, which is well
within acceptable levels for optical interconnects.


## 6. Conclusion

The need for an improved
on-chip interconnect technology has brought the possibility of optical
interconnects to the fore. Before optical interconnects can replace electrical
interconnects, however, they must be able to compete in terms of a variety of
parameters, including signal propagation delay, bandwidth density, footprint,
and power consumption. In terms of signal propagation delay, optical
interconnects have a natural advantage over electrical interconnects because of
the absence ofRC impedances.
Likewise, they have an advantage in terms of bandwidth density if
wavelength-division multiplexing is used. The use of photonic crystal
slow-light devices has the potential to extend this list of advantages to
footprint and power consumption as well.As we
have outlined above, the properties of slow light can be used to vastly shrink
the size and power consumption of interferometric optical modulators while
maintaining very high bandwidth. Furthermore, the ICCW platform, which is the
first example of a slow-light WDM platform, has the potential to be the basis
for a variety of ultracompact, low-power, high-bandwidth multichannel devices.
With further improvements in coupling and the reduction of intrinsic and
extrinsic losses, one can envision an optical interconnect built on slow-light
photonic crystal active devices. Because of its unique advantages in terms of
low latency and high bandwidth even over large distances, such an interconnect
may enable the emergence of a new generation of microprocessors that are no
longer interconnect-constrained. These chips would be able to leverage the
availability of much longer interconnects than are possible using electrical
interconnects. Even before that occurs, optical interconnects based on
slow-light photonic crystal active devices will be able to function as a
drop-in replacement for electrical interconnects in future microprocessors.



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*Source: 293531-2008-07-22.xml*