Nonlinear Group IV photonics based on silicon and
germanium: from near-infrared to mid-infrared
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Citation
Zhang, Lin, Anuradha M. Agarwal, Lionel C. Kimerling, and
Jurgen Michel. “Nonlinear Group IV Photonics Based on Silicon
and Germanium: From Near-Infrared to Mid-Infrared.”
Nanophotonics 3, no. 4–5 (January 1, 2014). © 2014 Science
Wise Publishing & De Gruyter
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http://dx.doi.org/10.1515/nanoph-2013-0020
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Walter de Gruyter
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Thu May 26 09:20:15 EDT 2016
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http://hdl.handle.net/1721.1/101975
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DOI 10.1515/nanoph-2013-0020 Nanophotonics 2014; 3(4-5): 247–268
Review article
Lin Zhang*, Anuradha M. Agarwal, Lionel C. Kimerling and Jurgen Michel
Nonlinear Group IV photonics based on silicon and
germanium: from near-infrared to mid-infrared
Abstract: Group IV photonics hold great potential for
nonlinear applications in the near- and mid-infrared (IR)
wavelength ranges, exhibiting strong nonlinearities in
bulk materials, high index contrast, CMOS compatibility,
and cost-effectiveness. In this paper, we review our recent
numerical work on various types of silicon and germanium waveguides for octave-spanning ultrafast nonlinear
applications. We discuss the material properties of silicon,
silicon nitride, silicon nano-crystals, silica, germanium,
and chalcogenide glasses including arsenic sulfide and
arsenic selenide to use them for waveguide core, cladding
and slot layer. The waveguides are analyzed and improved
for four spectrum ranges from visible, near-IR to mid-IR,
with material dispersion given by Sellmeier equations and
wavelength-dependent nonlinear Kerr index taken into
account. Broadband dispersion engineering is emphasized as a critical approach to achieving on-chip octavespanning nonlinear functions. These include octave-wide
supercontinuum generation, ultrashort pulse compression to sub-cycle level, and mode-locked Kerr frequency
comb generation based on few-cycle cavity solitons,
which are potentially useful for next-generation optical
communications, signal processing, imaging and sensing
applications.
Keywords: silicon photonics; Group IV photonics; nonlinear optics; dispersion; slot waveguide; micro-resonator;
integrated optics; supercontinuum; pulse compression;
frequency comb.
*Corresponding author: Lin Zhang, Microphotonics Center
and Department of Materials Science and Engineering,
Massachusetts Institute of Technology, Cambridge, MA 02139,
USA, e-mail: linzhang@mit.edu
Anuradha M. Agarwal, Lionel C. Kimerling and Jurgen Michel:
Microphotonics Center and Department of Materials Science and
Engineering, Massachusetts Institute of Technology, Cambridge,
MA 02139, USA
Edited by Chris Doerr
1 Introduction
Nonlinear optics [1–3] was born immediately after the
first laser was demonstrated, with many nonlinear effects
reported in 1961 and 1962. Those include second-harmonic
generation (SHG) [4, 5], third-harmonic generation (THG)
[5], two-photon absorption (TPA) [6], stimulated Raman
scattering (SRS) [7], and phase matching in parametric
wave mixing [8, 9]. As an important branch of optics,
nonlinear optics is at the heart of frequency conversion,
amplification, soliton and pulse compression, supercontinuum generation, and laser mode-locking and frequency comb generation, which have found a wide variety
of applications in optical signal processing, communications, sensing, imaging, and metrology.
While nonlinear effects in atomic gases [10] have
been extensively explored in research laboratories, other
materials platforms based on nonlinear crystals [11] and
optical fibers [12] have become more practical for realworld applications. Particularly, since the invention of
photonic crystal fiber (PCF) [13], nonlinear fiber optics has
experienced a revolutionary development [14, 15]. Supercontinuum generation as an example is one of nonlinear
applications enabled by PCFs [14]. Obtaining a wideband
or even octave-spanning supercontinuum does not only
require an in-depth understanding of nonlinear phenomena in optical fibers but also provides numerous commercial opportunities (see e.g., “http://www.leukos-systems.
com/”). The success of PCFs in refreshing nonlinear fiber
optics mainly relies on a great enhancement of nonlinearity and flexible engineering of dispersion [15]. Both of
these benefits are enabled by a large refractive index contrast between glass and air. PCFs allow us to confine light
much more tightly than standard fibers and enhance the
contribution of waveguide dispersion that is highly tailorable by designing fiber structures. The reasoning behind
PCF’s success serves as one of motivations to explore
new materials platforms with even higher index contrast
for nonlinear optics. In this sense, silicon photonics that
has an index contrast of 1.5~2 exhibits great potential to
further enhance performance of nonlinear devices.
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248 L. Zhang et al.: Nonlinear Group IV photonics
In the past two decades, silicon photonics [16–25] has
become increasingly mature. What silicon photonics can
contribute to nonlinear optics is much more than a high
index contrast. Several fundamental advantages drive
nonlinear silicon photonics [26–30].
First, as bulk material, crystalline silicon typically
has stronger Kerr and Raman nonlinearities than silica
by 2~3 orders of magnitude [26–30], depending on crystal
orientation. Materials engineering further enhanced the
nonlinearity, e.g., in amorphous silicon under certain
deposition conditions [31–35] and in Si-rich silicon oxide
(SRO) and nitride (SRN) based on the formation of silicon
nano-crystals [36–41].
Second, the high index contrast not only allows
sub-wavelength light confinement [42] but also makes
it ­
possible to form novel waveguiding structures. For
example, when a nano-scale low-index layer is sandwiched between two high-index layers, a slot waveguide
can be formed [43–46], with a large fraction of modal
power trapped in the thin layer. Relative to many nonlinear materials such as SRO, SRN, chalcogenides, and polymers, silicon has a sufficiently large index to form a slot
waveguide [47–51]. Moreover, adding a slot layer provides
more design freedom to tailor chromatic dispersion, as
reported in standard slot waveguides [51–58] and strip/
slot hybrid waveguides [59–63]. A high refractive index is
also b
­ eneficial to build a slow light element with reduced
group velocity of light, which can effectively increase
optical length for nonlinear interactions and reduce
power requirement [64–66].
Third, CMOS compatibility in silicon device processing and fabrication can be utilized for nonlinear silicon
photonics. Although optoelectronic integration of silicon
devices has long been proposed [16], there have been few
reports on the integration of nonlinear devices with electrical functionality. As silicon photonics becomes more
mature, it is foreseeable that CMOS-compatible nonlinear
silicon devices are seamlessly integrated with microelectronics on a single chip. For example, on-chip pumping
using an integrated laser, data encoding and modulation
before nonlinear signal processing, and on-chip signal
detection and data analysis after nonlinear devices are all
highly likely to be integrated together, which is achievable
by building nonlinear photonics on a silicon platform.
Fourth, cost-effectiveness and portability of devices
enabled by silicon photonics can be directly transferred
to the nonlinear optics branch. Although probably not a
key consideration in a very early stage of a research topic,
these factors will finally drive the technological development and determine how widely this technology will be
used. Nonlinear silicon photonic devices have unique
competence, especially when a large amount of elements
need to be incorporated in a (sub-) system.
The above motivations led to considerable research
efforts on nonlinear silicon photonics in recent years
[26–30]. Nonlinear effects such as SRS [67–83], self- and
­cross-phase modulation (SPM and XPM) [84–91], fourwave mixing (FWM) [92–102], and supercontinuum generation [60, 61, 103–108] are actively investigated. The
main focus is on the near-IR wavelength range, around
the telecom window. However, TPA in silicon is a detrimental effect in most cases [109, 110], significantly limiting optical power that is actually available in nonlinear
interactions. TPA thus degrades nonlinear efficiency and
bandwidth in the effects of interest, such as wavelength
conversion and supercontinuum generation, and TPAinduced free carriers cause slow device responses [26]. It
is noted that both the desired nonlinear effect and TPA are
accumulative over a certain distance, so if one can engineer chromatic dispersion to make the desired nonlinear
effect more efficient, the negative influence of TPA can
be mitigated [60, 108]. One can also use silicon nitride
[111–115] to eliminate TPA or even three photon absorption
(3PA) for a pumping wavelength in the near-IR, because
silicon nitride has a bandgap energy of ~5.3 eV [113].
It would be sometimes instructive to look at the trends
in nonlinear optics based on non-integrated platforms, in
order to gain a vision on how nonlinear silicon photonics
can evolve. One of the recent trends in nonlinear optics
is that a significantly larger portion of electromagnetic
­spectrum, from X-ray to mid-IR and even THz, is being
exploited to acquire previously hardly accessible information [116–118]. For nonlinear silicon photonics, it is a natural
step to extend to the mid-IR wavelength range [119–125]. The
definition of the term “mid-IR” varies substantially in the
literature, according to different research communities and
organizations. Specific application scenarios correspond
to different wavelength ranges. These include but are not
limited to (i) mid-IR spectroscopy and sensing from 2.5 to 15
μm [126] or from 2.5 to 25 μm according to infrared spectroscopy correlation table [127] “http://en.wikipedia.org/wiki/
Infrared_spectroscopy_correlation_table.”, (ii) free space
communications, LADAR, and remote sensing in atmosphere transparent windows from 3 to 5 μm and from 8 to 12
μm [128, 129], and (iii) astronomical instrumentation over a
bandwidth from 5 to 20 μm [130].
These new opportunities in the mid-IR are open to
silicon photonics from an application perspective, while
nonlinear silicon photonics also gains additional advantages in this wavelength range in terms of materials and
devices. First, silicon has no TPA beyond 2.2 μm, and
nonlinear loss by 3PA is significantly less influential for
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L. Zhang et al.: Nonlinear Group IV photonics 249
an optical intensity of below 5 GW/cm2 [131, 132]. Second,
more Group IV elements, e.g., germanium, become transparent and can be used for nonlinear purposes. In fact,
germanium exhibits higher refractive index and Kerr index
n2 than silicon [133]. Third, at longer wavelengths, device
dimensions need to be scaled up according to wavelength,
and thus surface roughness induced in device fabrication
causes relatively small scattering loss. Fourth, fabrication
errors in device dimensions are a smaller fraction of target
numbers, which is beneficial to obtain a higher yield in
device manufacture.
However, nonlinear photonics in the mid-IR also meet
some challenges. Although, going beyond half-bandgap
wavelengths, one can remove TPA, the nonlinear Kerr
index, n2, markedly decreases [134]. With waveguide
dimensions scaled up, the effective mode area, Aeff,
increases almost quadratically. Therefore, the nonlinear
coefficient, γ, equal to 2πn2/(λAeff ) where λ is the wavelength in vacuum, decreases quickly with wavelength.
This would require careful dispersion engineering for
phase matching in nonlinear parametric processes in
order to improve nonlinear efficiency [12].
Many nonlinear effects have been reported recently in
silicon photonics touching the mid-IR wavelength range
[60, 61, 106, 108, 114, 115, 135–145]. It is noted that most
of them address the short-wavelength end of the mid-IR,
from 2 to 2.5 μm, which is mainly the short-wave IR [144]
or transition from near-IR to mid-IR. Little was reported
in longer wavelength beyond 2.5 μm [136, 138, 142, 145].
In fact, there are about two octaves of bandwidth in the
mid-IR (e.g., from 2.5 to 10 μm) available, much wider
than that in the near-IR. As an approach to creating new
frequencies, nonlinear optics is much more efficient than
electro-optic modulation in terms of how far an optical
spectrum can be extended. We believe that the mid-IR
would be an exciting arena for ultrafast octave-spanning
nonlinear applications.
In this paper, we discuss the materials properties of
the Group IV platform for nonlinear applications. The
waveguide-based devices are optimized for four different
wavelength ranges from near-IR to mid-IR in terms of both
nonlinearity and dispersion. We show by simulation that
our dispersion-engineering approach based on a strip/
slot hybrid structure is widely applicable and can drama­
tically enhance nonlinear interaction efficiency and spectrum broadening. Supercontinuum and frequency comb
generations are predicted to be octave-spanning according to our numerical simulations, in which excellent
spectral coherence of the generated wideband spectra is
confirmed by the creation of ultrashort cycle-level optical
pulses.
2 Materials
Loss, nonlinearity, and dispersion jointly determine the
nonlinear performance of optical waveguides. All the
three are both material- and device-dependent. In this
section, we survey the major material choices for nonlinear Group IV photonics in the near- and mid-IR.
Figure 1 shows material transparency windows with
an optical loss below 2 dB/cm [122] for materials including
silicon (Si, i.e., crystalline silicon unless otherwise specified), silicon nitride (Si3N4), silicon dioxide (SiO2), germanium (Ge), arsenic sulfide (As2S3), and arsenic selenide
(As2Se3). Since TPA plays an important role in nonlinear
applications [109, 110], the color transition from red to
green in Figure 1 is between the bandgap wavelength and
the half-bandgap wavelength (two blue lines), where TPA
decreases with wavelength. For silicon, almost two-octave
bandwidth from 2.2 to 8.5 µm [146] is available for nonlinear applications without TPA, covering a large fraction of
mid-IR range. It is important to note that amorphous silicon
has a bandgap energy of 1.7 eV [32, 33] and thus has TPA
diminishing at a much shorter wavelength ( < 1.55 μm) than
crystalline silicon. Both silicon nitride and silicon dioxide
have large bandgap energies, but silicon dioxide becomes
highly lossy beyond 3 µm [119]. Germanium has an indirect
bandgap energy of 0.67 eV and is transparent until up to
14 µm [146]. From Figure 1, one can see that germanium’s
green bar without TPA has no overlap with silicon dioxide.
Chalcogenide glasses are actively investigated as
photonic materials [147–149] and exhibit a wide transparency window in the near- and mid-IR. For example,
As2S3 and As2Se3 have bandgap energies around 2.26 eV
[150, 151] and 1.77 eV [152], and they are transparent up
to 12 and 15 µm [153], respectively. Although in this paper
Si
Si3N4
SiO2
Ge
As2S3
As2Se3
1
2
3
4
5
6
7
8
9 10 11 12 13 14
Wavelength (µm)
Figure 1 Materials transparency windows (green bar) in near- and
mid-infrared ranges. Red bars indicate high-loss wavelength bands
for each material, and color transition from red to green is from
bandgap wavelength to half-bandgap wavelength (blue lines).
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250 L. Zhang et al.: Nonlinear Group IV photonics
chalcogenide glasses are not considered a material for
the core of a waveguide, one may use them for waveguide
cladding and slot layer [53, 58].
Linear optical properties of the materials discussed
here are collected in terms of refractive index and material dispersion, as shown in Figure 2. The two “X” on the
curves for silicon and germanium in Figure 2(A) indicate
the half-bandgap wavelengths. The refractive index is
given by the Sellmeier equations for silicon [154], silicon
nitride [155], silicon dioxide [156], SRO [51], germanium
[157], arsenic sulfide [158], and arsenic selenide [159], as
detailed in Appendix A. For SRN, there is no comprehensive measurement of material index found currently. As
shown in Figure 2(A), the materials under our consideration have strong index contrasts, especially between germanium and chalcogenides in the mid-IR. The refractive
index decreases with wavelength, and beyond the halfbandgap wavelength, these materials have a relatively
small index change. One can properly choose materials
for waveguide core and cladding and also a low-index slot
layer, based on the information given in Figure 2(A).
Overall dispersion in an integrated waveguide consists of material dispersion and waveguide dispersion that
A
Ge
4.0
Refractive index
3.5
Si
3.0
As2Se3
2.5
As2S3
2.0
Si3N4
SRO
1.5
SiO2
1
2
3
4
5
6
7
8
9
10 11 12 13 14
Wavelength (µm)
is affected by index contrast and waveguide dimensions.
First, material dispersion is shown in Figure 2(B), which
is defined as D = -(c/λ)·(d2nmat/dλ2 ), where nmat is material
index, and λ and c are wavelength and the speed of light
in vacuum. It is important to note that, except for silicon
dioxide, all the other materials have a flat and low dispersion within ± 100 ps/(nm·km) at the long-wavelength end
of the bandwidth of interest. This means that, if waveguides are not designed to tightly confine guided modes,
one can reduce the contribution of the waveguide dispersion and have the overall dispersion close to the flat and
low material dispersion. However, this will cause a large
effective mode area and a small nonlinear coefficient.
On the other hand, we note from Figure 2(B) that,
at the short-wavelength end of the spectrum, material
dispersion changes quickly with wavelength for all the
considered materials even if the material refractive index
looks flat in Figure 2(A) beyond the half-bandgap wavelength. This is because the dispersion is the 2nd-order
derivative of the index with respect to wavelength. To fully
use the portion of the spectrum near the half-bandgap
wavelength, dispersion engineering by tailoring waveguide dispersion is required.
As a measure of nonlinear material property, the nonlinear index n2 is shown in Figure 3 for silicon, silicon
nitride, SRO, germanium, and arsenic sulfide. Looking
at broadband nonlinear applications, one needs to take
the wavelength dependence of n2 into account. Unfortunately, there is often a lack of complete measurement
data at a wavelength range of interest, and also measurement results from different groups could vary widely. For
silicon, data from several sources are available [160, 161].
A recently published review paper [133] shows a prediction of third-order nonlinear susceptibility χ(3)1111 for silicon
and germanium in the mid-IR range, based on a twoband model, which is used to fit wavelength-dependent
B 200
1E-16
As2S3
0
Ge
Si3N4
-100
-200
As2Se3
SRO
Si
1
2
3
4
5
6
7
8
9 10 11 12 13 14
Wavelength (µm)
Figure 2 (A) Refractive indices and (B) material dispersion curves
of the materials considered for nonlinear Group IV photonics in the
near- and mid-IR wavelength ranges.
Nonlinear index (m2/W)
Dispersion (ps/nm·km)
SiO2
100
SRO
Ge
1E-17
Si
As2S3
1E-18
Si3N4
1E-19
1
2
3
4
5
6 7 8 9 10 11 12 13 14
Wavelength (µm)
Figure 3 The Kerr nonlinear index n2values of the considered materials in near- and mid-IR ranges.
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measurements. We obtain the Kerr nonlinear index n2 as a
function of wavelength, based on χ(3)1111, using the results
from [133]. As shown in Figure 3, the n2 value peaks at 1.9
and 2.7 µm for silicon and germanium, respectively, and
changes slightly from 3.3 to 4.8 µm beyond which both
TPA and 3PA disappear. The TPA coefficient βTPA vs. wavelength is also extracted for silicon and germanium from
[133], as detailed in Tables 1–4 in Appendix B.
Hydrogenated amorphous silicon has been identified as a potentially good nonlinear material, not only
because of large bandgap energy of 1.7 eV but also, more
importantly, because of the large nonlinear index, n2,
and nonlinear figure of merit (FOM), n2/βTPAλ [162]. We
did not include specific data in Figure 3 for amorphous
silicon, since different groups reported highly variable n2
and nonlinear FOM values in the near-IR [31–35]. The n2
value could be one order of magnitude higher than that in
silicon [33], while the nonlinear FOM can be as high as 5
[35], although these may not be obtained simultaneously
[34]. Moreover, linear properties of amorphous silicon
may also vary when fabrication conditions and its nonlinear characteristics change.
Silicon nano-crystals in silicon dioxide and silicon
nitride have also been investigated as a nonlinear m
­ aterial,
exhibiting higher nonlinear indices than ­crystalline silicon
by an order of magnitude or more [36–41]. The values of
n2, βTPA, and nonlinear FOM are also highly variable, if
silicon excess, annealing temperature and wavelength
change. We include one data point (n2 = 4.8 × 10-17m2/W)
from [38] in Figure 3. Extremely high n2and FOM by 3~4
orders have been obtained experimentally [41] with large
silicon excess (note that the FOM in [41] is defined as the
reciprocal of ours here).
Both amorphous silicon and silicon nano-crystals
exhibit great potential as a nonlinear material in the
mid-IR, which can be used to compensate for the reduction of the nonlinear coefficient due to a large mode area
at long wavelengths. In particular, with a small linear
refractive index, SRO is often chosen as a slot material
to enhance nonlinearity in the near-IR [51, 53, 55, 56, 59],
while SRN exhibits a great potential for nonlinear applications beyond 3 µm. Typically, strong nonlinearity in bulk
materials is associated with a high linear refractive index,
which is known as Miller rule. However, silicon nano-crystals exhibit unique properties to simultaneously possess
strong nonlinearity and low linear index. It is important
to mention that the silicon nano-crystals (i.e., nanoclusters) could act as scattering centers of light, causing
an increased propagation loss in SRO slot waveguides.
Nevertheless, relatively low propagation loss has been
achieved, which is 3~5 dB/cm [163].
For silicon nitride, one data point n2 = 2.4 × 10-19m2/W
from [111] is included in Figure 3, which was measured at
1.55 µm and is one-order lower than that in silicon. Silicon
dioxide has an n2 value around 2.6 × 10-20m2/W in 1.55 µm as
in single-mode optical fibers, two orders lower than that
in silicon. A higher n2 value (1.15 × 10-19m2/W) is estimated
for 1.55 µm in high-index doped silica [164]. Both values
are not shown in Figure 3. Since both silicon nitride and
silicon dioxide have large bandgap energies, it is expected
that their n2 values are almost constant over wavelength in
the near- and mid-IR.
Strong Kerr nonlinearities are obtained in chalcogenide glasses (arsenic sulfide and arsenic selenide), with
negligible TPA from the near- to mid-IR, as shown in
Figure 1. We have found wavelength-dependent measurements of the nonlinear index, n2, for arsenic sulfide from
different data sources [150, 165–174]. As shown in Figure
3, although slightly scattered, these n2 values in arsenic
sulfide are as high as those in silicon and would not be
strongly wavelength-dependent beyond 1.55 µm, because
it is longer than the half-bandgap wavelength. Arsenic
selenide has even higher n2 values than arsenic sulfide
[173], and its n2 value is predicted as a function of wavelength in [175].
There has been little published on the wavelength
dependence of the nonlinear Raman gain coefficient, gR,
in literature for the materials that we consider here [133].
Since SRS is not the major nonlinear effect that is used in
this paper, we will not discuss it in details here.
Although the considered materials such as silicon,
germanium, silicon nitride, and silicon dioxide are centrosymmetric and show no second-order nonlinearity in
bulk materials, one can engineer them by applying strain
[176–178] or forming interfaces between two centrosymmetric materials (e.g., between germanium and silicon
[179] or between silicon nitride and silicon dioxide [180]).
An alternative way is to integrate other materials with
strong second-order susceptibility onto Group IV waveguides (see e.g., [145]). For chalcogenide glasses, different
poling schemes are proposed to produce the secondorder nonlinearity [181]. The second-order susceptibility
χ(2) induced to the Group IV platform can have a highly
variable value, depending on how the isotropy of materials is broken. We believe that second-order nonlinearity
is promising in nonlinear Group IV photonics, but in this
paper, we will mainly focus on third-order nonlinearity.
As described above, the materials presented here
have greatly different transparency windows and nonlinear coefficients. It thus becomes critical to wisely choose a
material combination for a specific application, and doing
this one may also need to pay special attention to material
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252 L. Zhang et al.: Nonlinear Group IV photonics
compatibility in device fabrication. On the other hand,
compared to the material choices, device design (mainly
on waveguide and resonator) can also produce widely variable dispersion and nonlinearity properties. In the next
section, we will discuss enhanced waveguide properties
using improved designs.
3 Devices
Optical waveguides form the backbone of photonic
devices. Light propagation properties in a waveguide
could be remarkably different from those in corresponding
bulk materials, especially when there is a high index contrast between waveguide core and surrounding cladding.
Therefore, understanding and optimizing the waveguide
properties including loss, dispersion, and nonlinearity are
essential in nonlinear photonics.
Propagation loss in a waveguide includes material
loss, confinement loss, scattering loss, and nonlinear
loss. Working at a transparency window of a material,
especially beyond the half-bandgap wavelength, one can
primarily have low material and nonlinear loss caused
by TPA. Note that, benefiting from the wide multi-octave
bandwidth in the mid-IR, one can even eliminate the
impact of 3PA in silicon and germanium by pumping
at > 3.3 µm and > 4.8 µm, respectively. Since the substrate
index in a silicon wafer is higher than or equal to that in
most of materials we consider for a waveguide core, confinement loss exists due to mode leakage to the silicon
substrate. This loss can be markedly reduced by increasing the spacing between waveguide core and the substrate
or choosing low-index material between them. In general,
scattering loss due to sidewall roughness of a waveguide is
dominant in high-index-contrast silicon photonics, which
is mainly caused in device fabrication and can thus be
reduced by improving the fabrication processes [182–184].
Compared to propagation loss, chromatic dispersion and nonlinearity in integrated waveguides are more
de­signable. Since the dispersion is the second-order derivative of the effective index with respect to wavelength, it is
particularly tailorable by changing waveguide shape and
dimension. Moreover, dispersion has been recognized
to be critical for broadband nonlinear effects [12, 14, 15,
60, 61, 92–108, 112–115, 137–145], which is true especially
for ultrafast octave-spanning applications [185]. Spectral characteristics in a dispersion profile, including the
number and positions of zero-dispersion wavelengths
(ZDWs) and dispersion slope, greatly affect and often set
the limit on the bandwidth of optical spectra, the temporal
widths of pulses, and conversion efficiency in nonlinear
interactions [185]. Generally speaking, a flat dispersion
profile (i.e., third- and higher-order dispersion terms are
small) with low dispersion values is preferred.
In conventional ultrafast nonlinear optics in a freespace setup, many components were developed to control
dispersion over a wide bandwidth [185, 186], such as
prisms, gratings, chirped mirrors, and so on. However, in
a waveguiding system, e.g., in fiber-based ultrafast optics,
the dispersion-control toolkit is smaller, and engineering waveguide dispersion becomes critical. In particular,
when waveguides are built on a silicon platform with a
much higher index contrast than optical fibers, dispersion
in a highly nonlinear waveguide [187–190] often shows
strong wavelength dependence, which is not preferable for
wideband nonlinear applications. In [187, 190], the ZDW in
silicon rib and strip waveguides is mapped by scanning
waveguide dimensions. It is shown that tight confinement
of a guided mode produces a ZDW in its dispersion profile
around 1.2~1.4 μm close to the bandgap wavelength. Moreover, even if the waveguide size is increased to move the
ZDW to longer wavelength, the dispersion slope near the
ZDW is not small, as shown in [187–189], causing a limited
low-dispersion bandwidth.
Recently, a dispersion engineering technique for
integrated high-index-contrast waveguides has been proposed, in which an off-center nano-scale slot controls
modal distribution at different wavelengths [59, 60]. The
guided mode experiences a transition from strip-mode like
to slot-mode like as wavelength increases. This approach
can produce a very flat dispersion profile over an ultrawide bandwidth, with dispersion flatness improved by
1–2 orders in terms of dispersion variation divided by lowdispersion bandwidth. More importantly, it is applicable
to different material combinations and wavelength ranges
[59–63].
Towards mid-IR applications, different types of Group
IV waveguides have been reported recently, based on
silicon-on-insulator (SOI) [191–193], silicon-on-sapphire
[142, 194, 195], silicon-on-porous-silicon [192], siliconon-nitride [196, 197], suspended membrane silicon [198],
silicon pedestal [199], and germanium-on-silicon [200].
Most of the waveguides are not aimed specifically at nonlinear applications, and little attention has been paid to
dispersion engineering [196].
In this section, we survey different structures of
Group IV waveguides for broadband nonlinear applications from the near- to mid-IR. There are three main goals
in waveguide designs: (i) we consider joint optimization
on both dispersion and nonlinearity properties; (ii) we
tend to fully utilize the available bandwidth brought by
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L. Zhang et al.: Nonlinear Group IV photonics 253
the materials in Figure 1; and (iii) we emphasize dispersion engineering naturally as a result of aiming at octavespanning broadband applications.
Figure 4 shows a general illustration of various types of
integrated waveguides for nonlinear Group IV photonics.
Looking at a specific wavelength range, one can accordingly choose a materials combination for an appropriate
index contrast and a desired level of nonlinearity. Note
that one may need low nonlinearity in some cases, when
high-power output is required. Here we discuss waveguide
design at four different wavelength ranges as follows.
First, we consider SOI waveguides for a wavelength
range from the telecom window in the near-IR to the shortwave end in the mid-IR, i.e., roughly from 1.4 to 2.5 µm.
This is the wavelength range that many of the current
research efforts have been addressing [60, 61, 106, 108,
114, 115, 135, 137, 139–141, 143, 144]. In this wavelength
range, a SOI strip waveguide as shown in Figure 4 can be
used with air as an upper cladding (see e.g., [140]). One
can change the width of the waveguide to tailor its dispersion profile, while the height of the waveguide is 220 nm
set by SOI wafers. From Figure 5(A), we note that a relatively small width, W = 800 nm, is corresponding to a dispersion profile with two ZDWs at 1.585 and 2.345 µm and
a peak value of anomalous dispersion, 532 ps/(nm·km) at
2.05 µm, for the quasi-TE mode. The anomalous dispersion is typically useful for parametric amplification and
oscillation, soliton and soliton-based supercontinuum
generation [12]. With W = 900 nm, one can have a flatter
dispersion profile, but the anomalous band is smaller.
When W is increased to 1000 nm, the dispersion is even
flatter, but no anomalous dispersion occurs. Figure 5(A)
shows a good example that tight mode confinement in a
strip waveguide moves ZDW to short wavelengths, and
near ZDWs dispersion changes quickly with a large slope.
Strip WG
Slot WG
Strip WG suspended
Slot WG suspended
Rib WG
Core
Slot
Upper cladding
Air
Air
Slab
Lower cladding
Si substrate
Figure 4 Different types of Group IV waveguides (WGs) for dispersion and nonlinearity engineering in the near- and mid-IR ranges.
One can calculate the nonlinear coefficient, γ, as a
function of wavelength, with the nonlinear Kerr index, n2,
given in Tables 1–4 in Appendix B. We show in Figure 5(B)
that the nonlinear coefficient in the silicon strip waveguide with W = 900 nm first increases to 187 /(m·W) with
wavelength until 1.7 μm and then decreases to 56 /(m·W)
at 2.5 μm. This is caused by both the peaking of the silicon
n2 value near 1.9 μm and the gradual increase of wavelength and mode area beyond that.
A silicon strip/slot hybrid waveguide exhibits very flat
dispersion as presented in Figure 5(A). The SOI waveguide
has crystalline silicon at the bottom, a thin SRO slot, and
amorphous silicon at the top. The upper cladding is silicon
dioxide. When setting the lower Si height to Hl = 430 nm,
slot height to Hs = 54 nm, upper Si height to Hu = 160 nm, and
width to W = 660 nm, we obtain an extremely flat dispersion profile for the quasi-TM mode over a wide bandwidth,
between two ZDWs at 1.545 and 2.448 µm. From 1.605 to
2.38 µm, the value of anomalous dispersion changes
between 30 and 46 ps/(nm·km). In this way, one can have
a flat and low anomalous dispersion between two far
apart ZDWs. The average dispersion value can be shifted
by increasing Hu to move dispersion between normal and
anomalous regimes. Detailed explanation on how the flattened and saddle-shaped dispersion profile is produced is
given in [59, 60]. Briefly, the mode transition over wavelength for the quasi-TM mode is responsible for this behavior. Due to the off-center slot, the mode is mostly confined
in the crystalline silicon at short wavelengths, while the
mode becomes more like a slot mode at long wavelengths.
As shown in Figure 5(C), we plot the mode power distributions at wavelengths of 1.5, 1.83, 2.17, and 2.5 µm. The mode
transition adds negative dispersion in the middle of the
low-dispersion bandwidth, as explained in [60, 201].
Having a slot, one has an opportunity to fill the slot with
highly nonlinear materials into it [49, 51, 53, 55, 56, 58, 59],
which can overcome the decrease of the nonlinear coefficient over wavelength. In Figure 5(B), we show the γ value
increasing to 306 /(m·W) with wavelength from 1.4 to 2.5 µm.
This is because the guided mode extends more to the highly
nonlinear thin slot layer. Note that the used n2 value in SRO
[38] is currently the one measured at 1.55 µm, so the n2 and
γ values may vary in the mid-IR, but the trend is general.
Next, we explore the short-wavelength end of the
near-IR spectrum. Silicon-based devices become unusable for nonlinear photonics as wavelength decreases
to 1.1 µm, and we thus look at silicon nitride for near-IR
nonlinear applications extending to the visible light spectrum. Again, a strip waveguide based on silicon nitride is
examined first. Figure 6(A) shows dispersion curves of the
quasi-TE mode in two waveguides sized to be 1300 × 540
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254 L. Zhang et al.: Nonlinear Group IV photonics
Dispersion (ps/nm·km)
A
1000
500
0
-500
-1000
Nonlinear coefficient (/m·W)
B
Slot WG, Hu=160 nm
Slot WG, Hu=163 nm
Slot WG, Hu=166 nm
1.4
1.6
1.8
2.0
2.2
Wavelength (µm)
2.4
2.6
2.4
2.6
400
Strip WG, W=900 nm
Slot WG, Hu=160 nm
300
200
100
0
C
Strip WG, W=800 nm
Strip WG, W=900 nm
Strip WG, W=1000 nm
1.4
1.6
1.8
2.0
2.2
Wavelength (µm)
Figure 5 In a wavelength range from the telecom window in near-IR to the short-wave end in the mid-IR, silicon strip and strip/slot hybrid
waveguides (WGs) are analyzed in terms of (A) dispersion and (B) nonlinearity. (C) Mode power distributions at wavelengths of 1.5, 1.83,
2.17, and 2.5 µm in the strip/slot hybrid waveguide.
and 1400 × 800 nm2. The upper cladding is air and the
lower cladding is silicon dioxide. The anomalous dispersion region in the dispersion curves shrinks when the
waveguide is made smaller. This is because of a relatively
small index contrast between silicon nitride and silicon
dioxide, which makes the guided mode leak quickly to the
substrate as wavelength increases. For the strip waveguide
with a cross-section of 1400 × 800 nm2, there are two ZDWs
near 1.0 and 2.3 µm, but one can see a strong dispersion of
250 ps/(nm·km) between the two ZDWs in Figure 6(A). The
nonlinear coefficient in the second silicon nitride waveguide is shown in Figure 6(B), which is much smaller than
that in silicon waveguides because of a one-order smaller
n2 value and larger Aeff in the silicon nitride waveguide. At
1.6 µm, γ is about 1.23/(m‧W).
One can also use a strip/slot hybrid structure to tailor
the dispersion profile in silicon nitride waveguides. For
example, the slot and lower cladding are silicon dioxide,
and the upper cladding is air. In Figure 6(A), we show
the dispersion curves in two silicon nitride strip/slot
hybrid waveguides for comparison. The waveguide #1 has
Hl = 900 nm, Hs = 124 nm, Hu = 340 nm, and W = 1000 nm,
and the waveguide #2 has Hl = 920 nm, Hs = 154 nm, Hu = 480
nm, and W = 1300 nm. These two waveguides produce
increasingly flatter dispersion profiles, as shown in Figure
6(A). The first waveguide has two ZDWs located at 1.06
and 2.2 µm, with the peak dispersion of 67 ps/(nm·km).
The second waveguide has two ZDWs at 1.15 and 2.35 µm,
with the dispersion varying within 0~20 ps/(nm·km). This
octave-spanning dispersion flattening with different levels
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Dispersion (ps/nm·km)
2
300
Strip WG, 1400×800 nm
Slot WG, #1
Slot WG, #2
200
100
0
-100
0.5
Nonlinear coefficient (/m·W)
B
1.0
1.5
2.0
Wavelength (µm)
2.5
3.0
8
Strip WG, 1400×800 nm2
Slot WG, #1
Slot WG, #2
6
4
2
0
0.5
1.0
1.5
2.0
Wavelength (µm)
2.5
Figure 6 In a wavelength range moved toward the short-wavelength end of near-IR spectrum, silicon nitride strip and strip/slot
hybrid waveguides are designed in terms of (A) dispersion and (B)
nonlinearity.
of dispersions can be used for multiple applications, as
detailed in the next section. The nonlinear coefficients for
the two waveguides are shown in Figure 6(B). We note that
the strip/slot hybrid waveguides have similar nonlinear
coefficients as the strip waveguide, which shows that the
dispersion profile is much more tailorable by waveguide
designs.
Then, we move to the mid-IR, using silicon and silicon
nitride for waveguiding. A comparison of different types
of silicon-on-nitride waveguides have been presented
in [196], where rib waveguides were preferred due to the
wideband low dispersion over an octave-spanning bandwidth from 2.4 to 6.6 µm for the quasi-TE mode. This is a
spectral range from silicon’s half-bandgap wavelength to
the cut-off wavelength of silicon nitride. In Figure 7(A),
we plot the dispersion curve for a silicon-on-nitride rib
waveguide, with air as the upper cladding, the rib width of
2000 nm, the total height of 1200 nm, and the slab height
of 1000 nm, which are the same parameters used in [196].
It is shown that less confinement of optical modes reduces
the contribution of waveguide dispersion and makes the
overall dispersion profile closer to the material dispersion,
which is flat and low at long wavelengths as in Figure 2(B).
Accordingly, the nonlinear coefficient is small, 2.85 /(m·W)
at 3 µm, as shown in Figure 7(B).
On the other hand, if one needs a small Aeff to enhance
nonlinearity, additional dispersion tailoring (e.g., based
on strip/slot hybrid waveguides) would be beneficial.
Pursuing a higher nonlinear coefficient, we use a 500-nm
silicon nitride suspended membrane, as illustrated in
Figure 4, to support a silicon strip/slot hybrid waveguide.
This helps confine light in the waveguide core. Using
W = 880 nm, Hu = 550 nm, Hs = 87 nm, and Hl = 840 nm, we
obtain a saddle-shaped anomalous dispersion from 1.9 to
4.49 µm within 0~60 ps/(nm·km) for the quasi-TM mode,
as shown in Figure 7(A). This structure produces much
tighter mode confinement than the rib waveguide and
exhibits a 3 times larger nonlinear coefficient in Figure
7(B) while having similar dispersion flatness.
Finally, we consider germanium-on-silicon waveguides over a wavelength range from 3.3 to 8.5 µm,
between the half-bandgap wavelength of germanium and
the cut-off wavelength of silicon. This type of waveguide
has been demonstrated with strain-free mono-crystalline germanium [200]. Here, we assume that the germanium waveguide has a 10-nm silicon nitride layer on its
A
200
Dispersion (ps/nm·km)
Strip WG, 1300×540 nm2
100
Rib WG
Slot WG
0
-100
-200
B
Nonlinear coefficient (/m·W)
A 400
1
2
3
4
5
Wavelength (µm)
6
7
6
7
60
Rib WG
Slot WG
40
20
0
1
2
3
4
5
Wavelength (µm)
Figure 7 In a wavelength range from the short-wave IR to mid-IR,
silicon rib waveguide on silicon nitride and strip/slot hybrid
waveguide on a suspended membrane are analyzed in terms of (A)
dispersion and (B) nonlinearity.
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256 L. Zhang et al.: Nonlinear Group IV photonics
Dispersion (ps/nm·km)
A
400
Ge WG+air, 3000×1600 nm2
Ge WG+air, suspended 3000×1600 nm2
Ge WG+Si3N4, 3000×1600 nm2
Ge WG+Si, 3000×1600 nm2
300
200
100
0
-100
-200
Dispersion (ps/nm·km)
B
3
4
5
6
7
Wavelength (µm)
8
9
400
Ge WG+air, 2800×1400 nm2
Ge WG+air, 3000×1600 nm2
Ge WG+air, 3200×1800 nm2
Ge WG+air, 3400×2000 nm2
300
200
100
0
-100
-200
Dispersion (ps/nm·km)
C
3
4
5
6
7
Wavelength (µm)
8
9
400
Ge WG+air, 2800×1400 nm2
Ge WG+air, 3000×1600 nm2
Ge WG+air, 3200×1800 nm2
Ge WG+air, 3400×2000 nm2
300
200
100
0
-100
-200
D
Nonlinear coefficient (/m·W)
surface for passivation. The upper cladding could be air or
silicon, which provides significantly different dispersion
properties due to a varied index contrast. For comparison
only, we also have silicon nitride as the upper cladding,
although silicon nitride becomes lossy for wavelengths
longer than 6.7 µm. Figure 8(A) shows the dispersion profiles of four germanium-on-silicon strip waveguides with
equal size, 3000 × 1600 nm2, for the quasi-TE mode. Air
and silicon nitride as an upper cladding result in similar
shape and bandwidth in the dispersion profiles. Thus, the
air-cladded waveguide is chosen and discussed further.
The waveguide with silicon upper cladding has normal
dispersion at all wavelengths, since there is a relatively
small index contrast between germanium and silicon and
thus weak mode confinement. A germanium strip waveguide on a 600-nm-thick silicon suspended membrane is
also considered to increase light confinement, with an air
upper cladding to maximize light confinement. However,
as mentioned earlier, strong confinement typically causes
strong dispersion, as shown in Figure 8(A), and therefore
the germanium waveguide on a silicon membraneis not
chosen for broadband nonlinear applications. In contrast,
the germanium strip waveguide with air upper cladding
exhibits a flat and low dispersion.
The dimensions of the air-cladded germanium waveguide are varied by simultaneously changing its height
and width with a step of 200 nm for both polarization
states. It is interesting to see from Figure 8(B) and 8(C) that
the dispersion peak value remains nearly unchanged for
all the waveguide sizes, although we have a widely tunable
ZDW at long wavelengths. For the quasi-TE mode, the right
ZDW moves from 6.05 µm to 8.41 µm, while the left ZDW is
always near 4 µm. We can thus obtain an octave-spanning
anomalous dispersion band with the peak value below
100 ps/(nm·km). For the quasi-TM mode, one can see
similar dispersion properties, but the anomalous dispersion band is smaller. Thus, we choose the quasi-TE mode
for further discussion in next section.
The nonlinear coefficient in the germanium waveguides for the quasi-TE mode is shown in Figure 8(D), which
is about 10/(m·W) at 5 µm with a small variation for different waveguide sizes. This is quite high, considering that
both wavelength and effective mode area become much
larger over this wavelength range, compared to the near-IR.
From above, we can see that the strip/slot hybrid waveguides enable unique controllability of dispersion, and that
this concept is applicable to different wavelength ranges.
However, their performance may be sensitive to fabrication
errors, especially for inaccuracies in slot height, Hs, [59,
60]. A higher yield in device fabrication is expected using
advanced fabrication technologies and facilities.
3
4
5
6
7
Wavelength (µm)
8
9
50
40
Ge WG+air, 2800×1400 nm2
Ge WG+air, 3000×1600 nm2
Ge WG+air, 3200×1800 nm2
Ge WG+air, 3400×2000 nm2
30
20
10
0
3
4
5
6
7
Wavelength (µm)
8
9
Figure 8 In a wavelength range covering the main part of the
mid-IR spectrum, (A) on silicon substrate or on suspended silicon
membrane are analyzed in terms of dispersion. Germanium-onsilicon strip waveguides with an air upper cladding and different
dimensions are characterized by (B) dispersion for the quasi-TE
mode, (C) dispersion for the quasi-TM mode, and (D) nonlinearity for
the quasi-TE mode.
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L. Zhang et al.: Nonlinear Group IV photonics 257
Besides photonic waveguides, another important category of nonlinear devices is integrated resonators. In the
scope of this paper, we consider relatively large resonators for frequency comb generation, in which the bending
radius of a ring resonator is varied from 50 µm to 100 µm,
depending on free spectral range (FSR) and the group
index in the waveguide. In these cases, the waveguidebending-induced dispersion is small, and we would not
discuss intra-cavity dispersion [202, 203] in details here.
4 Applications
Benefiting from the unique dispersion engineering over an
octave-spanning bandwidth as described above, one can
develop ultra-wideband nonlinear applications that could
hardly be attained in an integrated platform previously.
These include octave-spanning supercontinuum generation, pulse compression to a few-cycle or even sub-cycle
level, octave-spanning Kerr frequency comb generation,
and the associated mode-locked ultrashort pulse generation using microresonators. In this section, we review our
recent work on these topics.
First, we discuss the supercontinuum generation and
pulse compression in a straight waveguide. The nonlinear
envelope equation used here to simulate supercontinuum
generation is the following:
 ∂ α ∞ ( -i ) m βm ∂m 
A = K ( A ) + R( A )
 + + i∑
m
 ∂z 2 m=2 m ! ∂t 
(1)
where
n

-i γ  -i   i δ  
δn
2
K ( A ) = ∑ n    1 A* n ( A ) 

ω0 δt   δt
n=0 n !  2  

∞
and
t

δ
R( A ) = -i γ R  1-iτ shock_R  [ A∫ hR ( t -t ′ )| A | 2 dt ′ ].
−∞
δt 

We denote A = A(z,t) as the complex amplitude of an
optical pulse. Note that its Fourier transform is
 1  ∞
A ( z , ω ) =   ∫ A( z , t ) exp( -iωt ) dt .
 2 π  -∞
In Eq. (1), α is the total propagation loss, and βm is
the mth-order dispersion coefficient. The frequency
dependence of nonlinearity parameters including the
nonlinear index n2, the TPA coefficient βTPA, and the
effective mode area Aeff, is included in the nth-order dispersion coefficient γn of nonlinearity, which is defined as
γn = ω0·∂n[γ(ω)/ω]/∂ωn, where ω0 is the angular frequency
of the carrier. Therefore, we can consider all-order linear
dispersion terms and all-order dispersion of the nonlinear coefficient in Eq. (1). Specifically in the simulations
for a silicon and silicon nitride waveguides, we have allorder linear dispersion and up to 6th-order and 2nd-order of
the nonlinear coefficient dispersion included. A detailed
derivation of Eq. (1) is given in [204]. For the quasi-TM
mode that experiences the engineered dispersion due to
the mode transition, SRS in silicon waveguides fabricated
on the (001) surface can be ignored [27, 96]. For silicon
nitride waveguide, we include the SRS term in Eq. (1),
where γR = gRΓR/(AeffΩR), and gR, ΓR, and ΩR represent the
Raman gain coefficient, the full width at half maximum
of the gain spectrum, and the Raman shift, respectively.
The Raman shock time τshock_R is associated with γR’s frequency dependence, which is 1/ω0-[1/Aeff(ω0)][dAeff(ω)/dω]
similarly as in [14], if we ignore frequency dependent gR,
ΓR, and ΩR. hR(t) is the Raman response function, and it
corresponds to the Raman gain spectrum
HR ( ω) =
ΩR2
ΩR2 -( ω-ω0 ) 2 + 2 iΓ R ( ω-ω0 )
.
Note that the sign before the imaginary unit is different from that in [27] to be consistent with the expression of
the Fourier transform that we used.
We have considered supercontinuum generation in
both silicon and silicon nitride waveguides with the slotassisted dispersion tailoring. Octave-spanning supercontinua in a silicon-based strip/slot hybrid waveguide have
been investigated in detail in [108], in which two-cycle
optical pulses are obtained. The main results in that work
are shown in Figure 9 for comparison purposes.
Here, we mainly focus on the supercontinuum generated in the silicon nitride strip/slot hybrid waveguide
(i.e., the slot WG #1 in Section III), whose dispersion and
nonlinearity properties are shown in Figure 6. In the nonlinear simulations, we set the total propagation loss to be
1 dB/cm. The SRS parameters used here are the following
[205]: ΩR/2π = 14.3 THz, ΓR/2π = 1.72 THz, and gR = 1 × 10-12 m/W.
τshock_R is calculated to be = 1.56 fs.
In our simulations, we use a time step of 0.25 fs, which
is corresponding to a bandwidth of 4000 THz in the frequency domain. For a femtosecond input pulse, we set the
time window length to 50 ps (i.e., frequency resolution
Δf = 20 GHz).
We simulate the nonlinear propagation of a chirp-free
hyperbolic secant pulse in the silicon nitride waveguide.
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258 L. Zhang et al.: Nonlinear Group IV photonics
A
A
6.0 mm
4.8 mm
Spectrum (10 dB/div.)
Spectrum (10 dB/div.)
4.8 mm
3.6 mm
2.4 mm
1.2 mm
1.4
1.6
1.8
2.0
Wavelength (µm)
2.2
4.4 mm
4.2 mm
0.6
2.4
0.9
1.2
1.5
1.8
2.1
Wavelength (µm)
2.4
2.7
B
8000
24
Power (W)
Power (W)
B
4.6 mm
Input
Input
1.2
5.0 mm
18
12
6
6000
4000
2000
0
24.6
24.8
25.0
25.2
25.4
0
Time (ps)
24.9
25.0
Time (ps)
25.1
Figure 9 (A) The supercontinuum generation in a silicon-based
strip/slot hybrid waveguide, presented in [108]. (B) The significant
spectrum broadening at 4.25-mm distance is associated with a
temporal compression of an input pulse (dash line) to the output
waveform (solid line) with a FWHM of 12 fs.
Figure 10 (A) Spectrum evolution of the pulse over propagation
distance. A supercontinuum of more than two octaves is achieved at
4.8 mm distance. (B) Generated pulse waveform with a pulse width
of 4.08 fs, as short as 0.76 optical cycles. Low pedestals are caused
by dispersive waves generated at the two ends of the spectrum.
The pulse center wavelength is at 1610 nm, and its full
width at half-maximum (FWHM) T0 is 120 fs. Its peak
power is 1200 W, corresponding to pulse energy of 0.16 nJ.
Figure 10(A) shows the supercontinua at different
propagation distances. At 4.8 mm, the spectrum is greatly
broadened at the -30 dB level, covering a wavelength range
from 0.585 to 2.833 µm, which is more than two octaves.
The spectrum evolution in Figure 10(A) shows a similar
spectrum shape as that in Figure 9(A), both featuring a
“triangular” central spectrum bounded by two dispersive
waves at the edges. However, it is important to note that
the absence of TPA and 3PA in silicon nitride at the telecom
window leads to a much more efficient spectrum broadening than that in silicon [108]. The generated spectrum
extends from the visible light to the mid-IR, with excellent
spectral coherence, which is confirmed by the pulse waveform shown in Figure 10(B). In the time domain, the pulse
is greatly compressed from 120 to 4.08 fs, corresponding to
0.76 optical cycles at 1.61 µm wavelength.
We examine the pulsewidth as a function of propagation distance. Figure 11 shows that the pulse becomes
increasingly narrower until the propagation distance
reaches 4.7 mm. For longer distances, the pulsewidth
remains almost constant. However, it is important to
mention that, after 4.8 mm, the dispersive waves become
increasingly stronger as shown in Figure 10(A), causing
larger pedestals.
Comparing the results in Figures 9 and 10, we note
that the mid-IR wavelength range for silicon would be in
analogy to the near-IR for silicon nitride in terms of nonlinear optics operations. Pumping at or beyond 3.3 µm,
one can use the waveguide designs shown in Figure 7 to
produce very efficient nonlinear interactions without TPA
and 3PA in silicon. Ultrashort pulses in the mid-IR from
parametric amplifiers [206, 207] could be used to pump
the Group IV waveguides.
Another nonlinear application of the dispersion-engineered Group IV waveguides is micro-resonator-based Kerr
frequency comb generation. When such a waveguide is
curved to form a microring resonator, input CW light travels
around the cavity and amplifies the noise in the source
located at the frequencies with a high parametric gain. As
a result of modulation instability and cascaded FWM in
the cavity, a frequency comb can be generated [208–211].
Mode-locked frequency combs have been reported, producing low-noise pulse trains in time domain [212–214].
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Pulse width (fs)
12
10
8
6
4
2
4.2
4.4
4.6
4.8
5.0
Propagation distance (mm)
5.2
Figure 11 Pulse width first decreases with propagation distance
and then remains stable. After 4.8 mm, dispersive waves get
stronger, causing more pedestals.
The formation of cavity solitons is identified as the main
reason for the mode-locking in the Kerr frequency combs
[215]. This is instructive, because one can thus predict
the spectral bandwidth and temporal pulsewidth in the
generated low-noise combs. The 3-dB comb bandwidth is
inversely proportional to the square root of the 2nd-order
dispersion coefficient, |β2|, as given in [213, 215]. It is desirable that all comb lines that constitute the soliton spectrum
experience the same |β2|. In this sense, the strip/slot hybrid
waveguides with flattened dispersion are preferably suitable for supporting broadband Kerr comb generation and
ultra-short cavity soliton generation.
The Kerr frequency comb generation can be modeled
using the generalized Lugiato-Lefever equation (LLE)
[216–219]:
silicon nitride. The ring resonator has a bending radius of
104 µm, corresponding to a FSR of 200 GHz. Pumping near
1.55 µm with a pump power of 2 W, the resonance peak is
red-shifted, and we need to red-shift the pump wavelength
accordingly and tune it into the resonance from the shortwavelength side. When the pump is step-by-step tuned
by up to 63 resonance linewidths, we obtain the comb
spectrum and the mode-locked pulse waveform as shown
in Figure 12. One can see that, over an octave-spanning
bandwidth from 133 to 268 THz, the comb lines have a
power drop by 20 dB from the center of the spectrum. The
spectral flatness of this comb is relatively good, compared
to the previously reported results [208–211]. The comb
bandwidth at -40 dB is as wide as two octaves. There are
two dispersive peaks in normal dispersion regions beyond
the low-dispersion band. Such a mode-locked broadband
comb produces a train of sub-two-cycle optical pulses, as
shown in Figure 12(B), with one pulse per round trip. The
peak power of the pulse is up to 600 W. Nonlinear conversion efficiency is estimated to be -26.7 dB.
To generate frequency combs in the mid-IR, the germanium-on-silicon strip waveguide is chosen. We choose
A
50
40
30
20
10
0
-10
-20
∞
 ∂ α k
( - j ) m βm ∂ m 
+
+
δ
+
t
j
jl
E = kEin -j γl | E | 2 E (2)
∑
 R
0
m
∂
2
2
!
t
m
∂
τ


m= 2
80
120
160
200
240
280
Frequency (THz)
320
360
B
600
Power (W)
where tR is the round-trip time, E = E(t,τ) and Ein are intracavity field and input field (pump power Pin = |Ein|2), t
and τ are the slow and fast times. δ0 is the cavity phase
detuning defined as δ0 = tR·(ωn -ω0), where ω0 and ωn are
the pump’s angular frequency and the nth angular resonance frequency that is pumped. Other resonator parameters include the power loss per round trip α, the power
coupling coefficient κ, the nonlinear coefficient γ, and the
mth dispersion coefficient βm. Since a flattened dispersion
profile has a small β2 over a wide bandwidth, it is important to take the influence of higher-order dispersion into
account. We include all-order dispersion terms in Eq. (2),
as we did in solving Eq. (1).
To enhance the Kerr comb bandwidth in the nearIR, we use the ultra-flattened dispersion profile in
Figure 6(A), which is obtained in the slot WG #2 based on
70
60
Power (10 dB/div.)
14
400
200
0
0.575
0.600
0.625
0.650
Time (ps)
Figure 12 (A) Frequency comb generation with a CW pump at 1.55
µm using a microring resonator based on a silicon strip/slot hybrid
waveguide. Over an octave-spanning bandwidth from 135 to 270
THz, the comb lines have a power drop of 20 dB. (B) Generated
pulse waveform with a pulse width of 8.2 fs, as short as 1.6 optical
cycles.
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260 L. Zhang et al.: Nonlinear Group IV photonics
Power (10 dB/div.)
A
30
B
40
50
60
Frequency (THz)
70
80
200
As shown above, broadband dispersion engineering
is critical for octave-spanning nonlinear applications in
both near- and mid-IR wavelength ranges, which enables
us to fully utilize the bandwidth allowed by the materials
transparency windows. Generally speaking, the nonlinear
applications mentioned here, such as supercontinuum
generation, ultrafast pulse compression, and frequency
comb generation, are often the intermediate steps towards
higher-level system applications. In the frequency domain,
a wide spectrum can serve as an electromagnetic carrier to
acquire high-volume of information, e.g., for sensing [220]
and imaging [221]. In the time domain, an ultrashort pulse
can be used as probe to sample ultrafast phenomena [222].
Power (W)
150
5 Summary and Outlook
100
50
0
2.1
2.2
2.3
Time (ps)
2.4
2.5
Figure 13 (A) Frequency comb generation with a CW pump at 6 µm
using a microring resonator based on a germanium strip waveguide
with air upper cladding. (B) Generated pulse waveform with a pulse
width of 69 fs, as short as 3.5 optical cycles.
a cross-section of 3200 × 1800 nm2, which has an octavespanning low-dispersion band from 4 to 7.67 µm, as shown
in Figure 8(B). A germanium ring resonator is formed with
a bending radius of 56.4 µm, corresponding to a FSR of
200 GHz. Pumping at 6 µm with a CW power of 1.4 W and
detuning the pump wavelength by 10 resonance linewidth,
one can see that a mode-locked wideband mid-IR comb is
generated from 35.8 to 64.4 THz (i.e., from 4.66 to 8.38 µm)
at -40 dB level. The FWHM of the produced pulses is 69 fs,
which corresponds to ~3.5 optical cycles. The pulse peak
power is 184 W, and the nonlinear conversion efficiency is
estimated to be -14.3 dB. Since the pumping frequency is
not at the center of the low-dispersion band, we only see
one peak in the comb spectrum caused by the dispersive
wave in the normal dispersion region from Figure 13(A).
There is another peak at higher frequencies beyond what
is shown in the figure.
We have presented a review of our recent work on nonlinear photonics based on silicon and germanium. Various
types of Group IV waveguides are analyzed and optimized
for four different wavelength ranges, from near- to mid-IR.
The recently proposed dispersion engineering technique
based on strip/slot hybrid waveguide structures is used for
different material combinations and wavelength ranges.
Numerical simulations show that the dispersion-flattened
Group IV waveguides are preferably suitable for octavespanning nonlinear applications, including on-chip
supercontinuum generation, ultrashort pulse compression, and mode-locked wideband frequency comb generation based on micro-resonators.
The presented approach to achieving octave-spanning nonlinear applications on an integrated, CMOScompatible Group IV platform holds great potential for
realizing chip-scale sensing, imaging, communications,
and signal processing system. The ultrawide transparency
windows in the mid-IR, allowed by Group IV elements
and compounds potentially together with other materials [223], provide an exciting arena for building powerful
information acquisition and processing units, enabled by
nonlinear optics, nano-photonics, and ultrafast optics.
Received June 14, 2013; accepted October 29, 2013; previously published online November 27, 2013
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Appendix
A. Material index and dispersion
n2 ( λ ) = ε+
In this section, we give the wavelength-dependent material index expressed as Sellmeier equations, where wavelength, λ, is in μm.
For silicon, we use the following material index that is
a fit curve from measurement results at room temperature
(293 K), with 184 data points in total, from 1.12 to 588 μm
[154]:
n2 ( λ ) = ε+
C1
λ
2
+
C2 λ
2
2
2
2
λ2 -λ
where ε = 11.6858, C1 = 0.939816 μm2, C2 = 0.00810461, and
λ2 = 1.1071 μm.
For silicon nitride, the material index is affected by
deposition conditions using plasma-enhanced chemical
vapor deposition (PECVD), low-pressure chemical vapor
deposition (LPCVD), and so on. We use the following Sellmeier equation [155], which predicts the material index
close to that in LPCVD silicon nitride films measured by a
few groups [224]:
n ( λ ) = 1+
2
C1 λ2
λ2 -λ12
where C1 = 2.8939 and λ1 = 0.13967 μm.
For silicon dioxide, we use the following Sellmeier
equation for fused silica [156]:
n2 ( λ ) = 1+
C1 λ2
λ -λ
2
2
1
+
C2 λ2
λ -λ
2
2
2
+
C 3 λ2
λ2 -λ23
where C1 = 0.6961663, C2 = 0.4079426, C3 = 0.8974794,
λ1 = 0.0684043 μm, λ2 = 0.1162414 μm, and λ3 = 9.896161 μm.
For SRO, the material index is affected by deposition
conditions such as silicon excess, annealing temperature,
and so on. Here we choose the one with silicon excess of
8% and annealed at 1250oC [51]:
n2 ( λ ) = 1+
C1 λ2
λ2 -λ12
+
C2 λ2
λ2 -λ22
+
+
2
λ2 -λ1
C2 λ2
λ2 -λ22
where ε = 9.28156, C1 = 6.7288, C2 = 0.21307, λ1 = 0.664116 μm,
and λ2 = 62.21013 μm.
For arsenic sulfide, we use the material index provided in [158]:
n2 ( λ ) = 1+
C1 λ2
+
2
λ2 -λ1
C2 λ2
+
2
λ2 -λ2
C 3 λ2
+
2
λ2 -λ 3
C4 λ 2
+
2
λ2 -λ4
C 5 λ2
λ2 -λ25
where
C1 = 1.8983678,
C2 = 1.9222979,
C3 = 0.8765134,
C4 = 0.1188704, C5 = 0.9569903, λ1 = 0.15 μm, λ2 = 0.25 μm,
λ3 = 0.35 μm, λ4 = 0.45 μm, and λ5 = 27.386128 μm.
For arsenic selenide, we fit ellipsometry measurements of our arsenic selenide thin films, and the bulk
material is provided by Prof. Kathleen A. Richardson
group:
n2 ( λ ) = 1+
C1 λ2
+
2
λ2 -λ1
C2 λ2
+
2
λ2 -λ2
C 3 λ2
λ2 -λ23
where C1 = 2.98463, C2 = 3.21011, C3 = 100.182, λ1 = 0.44118 μm,
λ2 = 0.000354953 μm, and λ3 = 384.13 μm.
B. Nonlinear Kerr index n2
The third-order nonlinear susceptibility χ(3)1111 for silicon and
germanium is predicted over the mid-IR range [133], based
on a two-band model. The effective nonlinear susceptibility χ(3) is dependent on polarization and crystallographic
orientation [225]. For strong nonlinearity, we consider a
single-polarization incident light aligned to the crystallographic axis, and we have χ(3) = χ(3)1111. To investigate the
octave-spanning nonlinear phenomena, one need to take
the wavelength-dependent nonlinear Kerr index n2 and TPA
coefficient βTPA into account, which are expressed as
3
χ( 3 ) ( λ )
4 ε0 cn 2 ( λ ) re
3π
βTPA ( λ ) =
χ( 3 ) ( λ )
λε0 cn 2 ( λ ) im
n2 ( λ ) =
C 3 λ2
λ2 -λ23
where C1 = 0.01, C2 = 1.96, C3 = 1.41, λ1 = 0.3 μm, λ2 = 0.07071 μm,
and λ3 = 27.75968 μm.
For germanium, the temperature-dependent material index was measured [157]. Here we choose the one for
room temperature (293 K):
C1 λ2
where ε0 and c are the vacuum permittivity and the speed
of light in vacuum. Using the material index given in
Appendix A and χ(3) value from [133], we obtain the n2 and
βTPA values tabulated as follows.
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262 L. Zhang et al.: Nonlinear Group IV photonics
Table 3 Nonlinear Kerr index n2 in germanium.
Table 1 Nonlinear Kerr index n2 in silicon.
λ
µm
n2
10-18 m2/W
λ
µm
n2
10-18 m2/W
λ
µm
n2
10-18 m2/W
λ µm
n2 10-18 m2/W
λ µm
n 2 10-18 m2/W
λ µm
n2
10-18 m2/W
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
1.85
1.90
1.95
2.00
2.05
2.10
2.15
2.20
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.05
2.34
3.05
3.78
4.43
5.01
5.59
6.18
6.71
7.10
7.41
7.66
7.81
7.83
7.78
7.62
7.32
6.99
6.58
6.12
5.76
5.47
5.25
5.08
4.93
4.79
4.67
4.56
4.48
4.40
4.32
4.24
4.17
4.10
4.04
3.98
3.94
3.10
3.15
3.20
3.25
3.30
3.35
3.40
3.45
3.50
3.55
3.60
3.65
3.70
3.75
3.80
3.85
3.90
3.95
4.00
4.05
4.10
4.15
4.20
4.25
4.30
4.35
4.40
4.45
4.50
4.55
4.60
4.65
4.70
4.75
4.80
4.85
3.89
3.84
3.79
3.75
3.71
3.68
3.65
3.62
3.60
3.58
3.55
3.52
3.50
3.46
3.43
3.41
3.38
3.36
3.34
3.32
3.31
3.30
3.29
3.28
3.27
3.25
3.24
3.21
3.19
3.17
3.15
3.14
3.13
3.13
3.12
3.12
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
5.40
5.45
5.50
5.55
5.60
5.65
5.70
5.75
5.80
5.85
5.90
5.95
6.00
6.05
6.10
6.15
6.20
6.25
6.30
6.35
6.40
6.45
6.50
6.55
6.60
3.12
3.11
3.10
3.09
3.07
3.06
3.04
3.04
3.03
3.02
3.02
3.01
3.00
2.98
2.97
2.96
2.96
2.95
2.95
2.95
2.94
2.93
2.92
2.91
2.90
2.89
2.88
2.88
2.88
2.88
2.88
2.88
2.88
2.88
2.88
2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 39.32 49.94 60.07 66.50 67.72 62.73 48.97 44.03 39.70 36.80 34.70 32.86 31.28 29.98 28.95 28.12 27.37 26.68 26.05 25.50 24.98 24.50 24.06 23.65 23.28 22.95 22.66 22.38 22.12 21.87 21.64 21.41 21.21 21.01 20.83 20.66 20.49 20.32 20.16 20.01 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 19.86 19.72 19.59 19.46 19.34 19.23 19.13 19.04 18.95 18.86 18.76 18.68 18.59 18.50 18.41 18.33 18.26 18.19 18.12 18.07 18.01 17.97 17.92 17.87 17.82 17.77 17.72 17.67 17.62 17.58 17.53 17.49 17.44 17.40 17.36 17.31 17.27 17.22 17.16 17.11 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11.0 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13.0 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 14.0 17.05
16.99
16.94
16.90
16.87
16.84
16.82
16.81
16.79
16.77
16.75
16.73
16.70
16.67
16.63
16.59
16.56
16.53
16.51
16.48
16.46
16.44
16.43
16.41
16.39
16.36
16.34
16.32
16.30
16.28
16.26
16.25
16.23
16.21
16.20
16.18
16.17
16.16
Table 2 TPA coefficient βTPA in silicon.
λ
µm
βTPA
10-12 m/W
1.30
1.35
1.40
1.45
1.50
1.55
1.60
13.34
12.84
12.22
11.54
10.79
9.95
9.05
λ
µm
βTPA
10-12 m/W
λ
µm
βTPA
10-12 m/W
1.65
1.70
1.75
1.80
1.85
1.90
1.95
8.12
7.13
6.13
5.22
4.24
3.33
2.54
2.00
2.05
2.10
2.15
2.20
1.80
1.15
0.64
0.24
0.003
Table 4 TPA coefficient βTPA in germanium.
λ
µm
βTPA
10-8 m/W
2.3
2.4
1.30
1.11
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λ
µm
βTPA
10-8 m/W
2.5
2.6
0.88
0.60
λ
µm
βTPA
10-8 m/W
2.7
2.8
0.31
0.06
L. Zhang et al.: Nonlinear Group IV photonics 263
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