ARTICLES
PUBLISHED ONLINE: 4 DECEMBER 2011 | DOI: 10.1038/NMAT3200
Second-harmonic generation in silicon
waveguides strained by silicon nitride
M. Cazzanelli
1
, F. Bianco
1
, E. Borga
1
, G. Pucker
2
, M. Ghulinyan
2
, E. Degoli
3
, E. Luppi
4
, V. Véniard
5
,
S. Ossicini
3
, D. Modotto
6
, S. Wabnitz
6
, R. Pierobon
7
and L. Pavesi
1
*
Silicon photonics meets the electronics requirement of increased speed and bandwidth with on-chip optical networks.
All-optical data management requires nonlinear silicon photonics. In silicon only third-order optical nonlinearities are present
owing to its crystalline inversion symmetry. Introducing a second-order nonlinearity into silicon photonics by proper material
engineering would be highly desirable. It would enable devices for wideband wavelength conversion operating at relatively low
optical powers. Here we show that a sizeable second-order nonlinearity at optical wavelengths is induced in a silicon waveguide
by using a stressing silicon nitride overlayer. We carried out second-harmonic-generation experiments and first-principle
calculations, which both yield large values of strain-induced bulk second-order nonlinear susceptibility, up to 40 pm V
−1
at
2,300 nm. We envisage that nonlinear strained silicon could provide a competing platform for a new class of integrated light
sources spanning the near- to mid-infrared spectrum from 1.2 to 10 µm.
W
hen a crystal possesses a significant second-order
nonlinear optical susceptibility, χ
(2)
, it can produce a
wide variety of wavelengths from an optical pump
1
.
In fact, a second-order crystal generates shorter wavelengths by
second-harmonic generation or longer wavelengths by spontaneous
parametric down-conversion of a single pump beam. Such a crystal
can also nonlinearly mix two different beams, thus generating other
wavelengths by sum-frequency or difference-frequency generation.
These possibilities are much more intriguing whenever the crystal
can be used in integrated optical circuits because, on the one hand,
light confinement reduces the average optical power needed to
trigger nonlinear processes and, on the other hand, relatively long
effective interaction lengths can be exploited.
Si photonics has demonstrated the integration of multiple
optical functionalities with microelectronic devices
2,3
. On the basis
of the third- or higher-order nonlinearities of Si (ref. 4), functions
such as amplification and lasing, wavelength conversion and
optical processing have all been demonstrated in recent years
5
.
However, third-order refractive nonlinearities require relatively
high optical powers, and compete with nonlinear-loss mechanisms
such as two-photon absorption and two-photon induced free-
carrier absorption. Yet, the second-order term of the nonlinear
susceptibility tensor cannot be exploited in Si simply because
χ
(2)
vanishes in the dipole approximation owing to the crystal
centrosymmetry: the residual χ
(2)
, which is due to higher-multipole
processes, is too weak to be exploited in optical devices
6
.
Second-harmonic generation (SHG) was observed in reflection
from Si surfaces
7–11
or in diffusion from Si photonic crystal
nanocavities
12
. This indicates that the reduction of the Si symmetry
may indeed induce a significant χ
(2)
. In these cases, the Si
symmetry was broken by the presence of a surface. Several groups
have pointed out that the surface contribution to χ
(2)
can be
1
Nanoscience Laboratory, Department of Physics, University of Trento, via Sommarive 14, 38123 Povo, Trento, Italy,
2
Advanced Photonics & Photovoltaics
Unit, Bruno Kessler Foundation, via Sommarive 18, 38123 Povo, Trento, Italy,
3
Istituto di Nanoscienze-CNR-S3 and Dipartimento di Scienze e Metodi
dell’Ingegneria, Università di Modena e Reggio Emilia, via Amendola 2 Pad. Morselli, I-42122 Reggio Emilia, Italy,
4
Department of Chemistry, University of
California Berkeley, California 94720, USA,
5
Laboratoire des Solides Irradiés, Ecole Polytechnique, Route de Saclay, F-91128 Palaiseau and European
Theoretical Spectroscopy Facility (ETSF), France,
6
Department of Information Engineering, University of Brescia, via Branze 38, 25123 Brescia, Italy,
7
CIVEN, via delle Industrie 5, I-30175, Venezia Marghera, Italy. *e-mail:pavesi@science.unitn.it.
strengthened by applying a strain
8–10
. It was also reported that
strain and increased internal surfaces in a Si photonic crystal
waveguide may induce a strong χ
(2)
, which enables measurement
of electro-optic effects
13,14
. Theoretically, these results are so far
qualitatively but not quantitatively understood
15
. The prospect of
controlling second-order nonlinear phenomena in Si by means
of strain is very intriguing. In this work we provide the first
quantitative demonstration of second-order nonlinearities in
strained Si through both ab initio calculations and SHG experiments
in suitably engineered Si waveguides.
Second-order nonlinear optical response from strained bulk Si is
computed by time-dependent density-functional theory in a super-
cell approximation with periodic boundary conditions
16–18
. All the
studied strained Si structures were obtained using a unit cell of 16 Si
atoms (Fig. 1) initially in their bulk centro-symmetric positions (see
Supplementary Information). We broke the Si symmetry by moving
the Si atoms in the lattice, that is by modifying the Si bond lengths
and angles. This yields a strained Si model which enables us to calcu-
late the second-order response in Si. When no strain or a uniformly
distributed strain is simulated, the computed χ
(2)
vanishes because
bulk Si still keeps a crystalline centrosymmetry. To break the
symmetry, the stress has to induce an asymmetric deformation of
the Si lattice. We model this situation by moving only sets of atoms
in the supercell (Fig. 1a) simulating tensile (S
t
) and/or compressive
(S
c
) stresses. With this approach we identify and study two classes of
systems: US
c
US
t
(unstrained–compressed–unstrained–tensile) and
US
t
US
t
(unstrained–tensile–unstrained–tensile; see Supplementary
Information). Simulation results show that χ
(2)
6= 0 whenever the
unit cell is inhomogeneously strained. We also observed that χ
(2)
is
related to the local lattice deformation (that is, the local strain): the
more the lattice is distorted the larger is the χ
(2)
value. Moving the
atoms in S
t
/S
c
regions along the (001) direction only, we found that
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NATURE MATERIALS DOI: 10.1038/NMAT3200
ARTICLES
× 50
Si¬Si bond
Atom index
Bond length (Å)
Bond angle (°)
(2)
(pm V
¬1
)
χ
Wavelength (nm)
SS1 = 2.4 × 10
¬5
GPa
HY1 = 3.2 × 10
¬1
GPa
16¬17
15¬16
14¬15
13¬14
12¬13
9¬10
8¬9
7¬8
6¬7
5¬6
4¬5
3¬4
2¬3
1¬2
11¬12
10¬11
2.25 2.30 2.35 2.40 2.45
104 108 112
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
50
100
150
200
1,500 2,000 2,500 3,000 3,500
(2)
(pm V
¬1
)
χ
Wavelength (nm)
SS2 = 2.2 × 10
¬3
GPa
HY2 = 7.5 × 10
¬1
GPa
0
50
100
150
200
1,500 2,000 2,500 3,000 3,500
(2)
(pm V
¬1
)
χ
Wavelength (nm)
SS3 = 1.2 × 10
¬1
GPa
HY3 = 9.1 × 10
¬1
GPa
0
50
100
150
200
1,500 2,000 2,500 3,000 3,500
b
c
d
e
f
U
U
S
t
S
c
16
15
14
13
12
11
10
7
8
9
6
5
4
3
2
1
a
HY1, SS1
HY2, SS2
HY3, SS3
HY1, SS1
HY2, SS2
HY3, SS3
Figure 1 | First-principle calculations of strained Si second-order nonlinearity. a, Simulation unit cell: the different colours refer to the different bond
deformations used. Orange: the bonds were elongated to represent a tensile stress, indicated by S
t
. Yellow: no bond variation with respect to the relaxed Si
lattice (unperturbed region). Red: the bonds were shortened to represent a compressive stress, indicated by S
c
. b, Maximum bond-length distortions for
US
c
US
t
structures. The unstrained bulk-Si bond length is 2.332 Å. The colours (blue–black–red) refer to the calculations shown in d–f. c, Maximum
bond-angle distortions for US
c
US
t
structures. The unstrained bulk-Si angles are 109.5
◦
. The colours (blue–black–red) refer to the calculations shown in d–f.
The constant bond lengths and angles represent the unstrained region and coincide with Si-bulk values. The hierarchy of increasing strain in the three
structures (blue–black–red), which is reflected also in the corresponding values of pressure (HY) and in the YZ component of the shear stress (SS; see
Supplementary Information), is evident. d–f, The results of the simulations in terms of the χ
(2)
values. d, A weakly strained Si. e, A medium-strained Si.
f, A heavily strained Si.
χ
(2)
is lower than 0.5 pm V
−1
at 2,500 nm. In contrast, if we increase
the strain inhomogeneity moving the atoms also in the (
¯
110)
direction, we found that χ
(2)
increases significantly (Fig. 1). For
each class, US
c
US
t
and US
t
US
t
, we studied three structures, in which
we have progressively increased the applied strain. In Fig. 1b,c we
show the maximum structural distortion in term of bond lengths
and angles for the US
c
US
t
structures (the same can be found for
the US
t
US
t
structures; see Supplementary Information). The type
and magnitude of strain is qualitatively observed as bond-length
and angle deviations with respect to the unstrained Si bulk values.
We observe that χ
(2)
has a maximum of 6 pm V
−1
at a wavelength
of 1,900 nm for purely tensile strain. The magnitude of the applied
strain was up to 5% bond distortion of the unstrained bulk value,
that is up to 0.12 Å. When compressive and tensile strains are
applied (Fig. 1), χ
(2)
increases further, reaching a maximum value
of about 200 pm V
−1
at 1,800 nm, as shown in Fig. 1f. In this case,
the strain gradient is larger than that of the pure-tensile-strain
case, because of the change in sign of the strain itself between S
t
and S
c
(Fig. 1b,c). As the reported calculations refer to a strained
bulk Si, we also investigate the contribution to χ
(2)
from a Si/SiO
2
interface. We found a χ
(2)
smaller than 1 pm V
−1
(Supplementary
Information), whereas from the literature it is found that the
free Si surface induces a χ
(2)
of ∼3 pm V
−1
(refs 11,19). Therefore,
breaking the Si inversion symmetry by an interface or a free surface
yields a χ
(2)
that is significantly smaller than what can be achieved
by inhomogeneously straining bulk Si.
To explore this concept, we fabricated inhomogeneously
strained Si waveguides. These were obtained by using Si-on-
insulator (SOI) waveguides (Fig. 2) with stressing SiN
x
overlayer
clads (Methods section). Various cladding layers were used to
change the applied stress: the waveguide named SOI1 had a
150-nm-thick Si
3
N
4
overlayer, which induces a tensile stress of
1.2 GPa; the waveguide named SOI2 had a 500-nm-thick SiN
x
overlayer, which induces a compressive stress of −500 MPa; the
waveguide named SOI3 had a stress-compensated 500-nm-thick
SiN
x
overlayer with a residual compressive stress of −60 MPa. Note
that the reported stress values refer to measurements made on the
whole unpatterned Si wafers by using a mechanical profilometer.
Next, waveguides were defined by optical lithography and by
reactive ion etching (Fig. 2c). The light propagation axis, along the
waveguide, was parallel to the (110) crystalline direction (Fig. 2b).
Our multimode waveguides exhibited linear propagation losses of
1–3 dB cm
−1
at 1,260 nm. Control waveguides without the stressing
overlayer were also fabricated (named SOI0).
The silicon nitride overlayer strains the Si waveguide because
of the stress penetration into the waveguide core. To map the
strain distribution in the waveguide cross-section we used spatially
resolved (with a 0.5 µm resolution) micro-Raman spectroscopy.
The strain maps were constructed by acquiring the Raman
spectrum on each point of the waveguide cross-section. The shift
in the Si phonon frequency with respect to the unperturbed
phonon frequency, which was measured on the substrate, can
be related to the local Si strain by a simple uniaxial model
20
.
Figure 3 shows the distribution of the strain components for the
three sets of waveguides with SiN
x
claddings. The tensile SOI1
sample (Fig. 3a,d,g,h) shows a significant deformation all along
the waveguide core. Note that this means that the application of
the stressing overlayer strains the whole 2-µm-thick waveguide
(Fig. 3h). It can also be noticed that the underlying buried oxide
(BOX) affects the strain distribution too. As a result, a very
inhomogeneous strain field is observed: the applied stress results in
a compressive strain in a thin layer below the Si
3
N
4
cladding layer
and a tensile strain in the waveguide core, with a maximum value
near the BOX interface. The strain field has a strong gradient, which,
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NATURE MATERIALS DOI: 10.1038/NMAT3200
a
d
c
2 μm
h
1
h
0
SiO
2
Si
3
N
4
c-Si
w
0
ω
λ
ω
λ
λλ
2
= /2
out
ω
ω
b
λ
out
2
λ
L
in
ω
λ
ωω
Figure 2 | Strained Si waveguides used to measure SHG. a, Schematic representation of the cross-section of a strained SOI waveguide where w
0
and h
0
are the width and height of the waveguide and h
1
the nitride overlayer’s height. Si forms the core of the waveguide and the parameters are chosen to have
more than 95% of the optical field confined within the waveguide. b, The second-harmonic experiment, where L is the total waveguide length. A pump
pulse is coupled into the waveguide at a wavelength λ
ω
and two pulses (at wavelengths λ
ω
and λ
2ω
) are measured at the output of the waveguide. c, A
top-view optical image of the sample where a few waveguides are observed as yellow lines. A scanning electron microscopy image of the input facet of the
waveguide is also shown. The layers are false coloured as in a. Note that the stressing overlayer does not produce macroscopic cracks or deformation in the
strained Si waveguide. d, Virtual-energy-level diagram of SHG, where two photons of the pump at wavelength λ
ω
are annihilated by way of a virtual energy
level to create a single photon at wavelength λ
2ω
.
h
Z (μm)
Z (μm)
Y (μm)
Z (μm)
Y (μm)
YY
ε
(× 10
¬3
)
YY
(MPa)
σ
0
1
2
4 5 6 7 8 9 10 110123
¬100
0
100
200
0 0.5 1.0 1.5 2.0
3210
0
1
2
3
Z (μm)
Y (μm)
YY
ε
0
1
2
3
Z (μm)
Y (μm)
YY
ε
1
0
2
¬1
¬2
3
3
210
3
210
0
1
2
YY
ε
g
abc
def
Figure 3 | Summary of the results of micro-Raman measurements on the waveguide facet. a–c, Schematic representations of the SOI1, SOI2 and SOI3
waveguides. The lattice deformation is exaggerated to show the effect of the applied stressing overlayer (blue layer). The arrows show the kind of
tensile/compressive strain observed. d, Two-dimensional map of the measured strain-tensor element ε
YY
for the SOI1 2.3-µm-wide waveguide. The colour
bar is reported on the right. On the basis of this map the sketch in a was deduced. e, Two-dimensional map of the strain-tensor element ε
YY
for the SOI2
2.0-µm-wide waveguide. f, Two-dimensional map of the strain-tensor element ε
YY
for the SOI3 2.0-µm-wide waveguide. g, Two-dimensional map of the
strain-tensor element ε
YY
for the SOI1 10.7-µm-wide waveguide. h, Line scan of the measured stress at the centre of the SOI1 10.7-µm-wide waveguide.
Note that for SOI1 the overlayer nitride film is tensile strained, which causes a compressive stress on a Si wafer, which when the waveguide is defined yields
a compressive stress at the nitride/Si interface of the waveguide. This stress finally ends up with a sign change across the waveguide cross-section due to
the presence of the underlying BOX layer.
according to calculations, should induce a significant χ
(2)
. The
strain field has a non-zero component along the Y axis (Fig. 3d),
that is along the facet plane as sketched in Fig. 3a, a complementary
weaker compressive component along the Z axis, that is along
the waveguide thickness, and a negligible component along the X
axis, that is along the waveguide axis. The strain along the Y axis
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ARTICLES
(× 10
¬3
)
YY
ε
Z (μm)
2.52.01.51.00.50
0
1
¬1
¬2
3
2
SOI0 3.0 μm
SOI1 2.3 μm
SOI2 2.0 μm
SOI3 2.0 μm
Figure 4 | Strain profiles (ε
YY
) for the different waveguide types. Profiles
refer to the strain along the Z direction at the centre of the waveguides. The
waveguide widths are reported in the key.
also depends on the waveguide width: the wider the waveguide the
more inhomogeneous the strain (compare Fig. 3d with 3g). The
situation is different for the compressed SOI2 waveguide (Fig. 3b,e):
although here the Si is mainly tensile strained, an inhomogeneous
strain field is still present. This field has a maximum near the
overlayer and gradually weakens near the underlying BOX. SOI3
shows a strain field that is the opposite of SOI2 (Fig. 3c,f) due to
the stronger effect of the BOX with respect to the overlayer. In
Fig. 4 we report the Z -profile of the strain at the waveguide centre,
for the various samples. SOI1 and SOI2 show important strain
gradients of opposite signs, reflecting the influence of strong tensile
and compressive stress applied by the SiN
x
overlayers. Strain and
its gradient are low for the reference SOI0 sample. SOI3 shows a
similar strain gradient, but the average strain is about twice that of
SOI0. In both samples the origin of strain is mainly due to the stress
induced by the BOX.
To demonstrate the induced second-order nonlinearity we
carried out SHG experiments (Fig. 2b,d). Two different pump
lasers were used with 4-ns- and 100-fs-long pulses, respectively.
Taking into account the temporal walk-off between the pump and
the second-harmonic signal, 2-mm-long waveguides were used in
combination with the fs laser. One-centimetre-long waveguides
were used in combination with the ns laser, because in this
case the pump-signal walk-off can be ignored. Figure 5a shows a
typical transmission signal recorded from the SOI1 10.7-µm-wide
waveguide for a 30-kW-peak-power fs pump. In addition to the
residual pump spectrum centred at about 2,100 nm, a significant
second-harmonic peak at 1,050 nm was clearly observed (Fig. 5a).
Note that the wavelength of the emitted radiation is half the
pump wavelength, as expected for SHG. The second-harmonic peak
shifts linearly with the pump wavelength (Fig. 5b) over a 300 nm
range of variation of the pump wavelength. A further proof of
the fact that the short-wavelength peak is due to SHG is provided
by the observation of a characteristic quadratic dependence of
the measured peak power as a function of ns pump peak power
(Fig. 5c). The deviation from a quadratic dependence at high
pump peak powers that is observed in Fig. 5c is due to the
pump depletion, which is caused by nonlinear optical losses
5
.
Tentatively, these nonlinear losses can be attributed to three-photon
absorption processes and/or to the possible two-photon absorption
processes due to strain-induced bandgap shrinkage. Fig. 5d reports
the variation of the second-harmonic peak power as a function
of the ns pump wavelength for the SOI1 2.3-µm-wide waveguide.
No similar features in this pump power range were observed on
control measurements on SOI0 waveguides, that is, those without
the stressing overlayer.
At a wavelength of 2,313 nm and at a ns pump peak power
of 0.7 ± 0.1 W, the power-conversion efficiency η from the pump
laser (peak power P
ω
) to the second-harmonic beam (peak power
P
2ω
) is η = P
2ω
/P
ω
= (5±3)× 10
−8
W W
−1
= −73 dB (or P
2ω
/P
2
ω
=
(7.5 ± 4) × 10
−8
W
−1
) for the SOI1 10.7-µm-wide waveguide.
Table 1 reports the η values for the various waveguides and stressing
overlayers. In agreement with the Raman measurements, which
show that the SOI1 waveguides are more strained than the other sets
of waveguides, the tensile-stressed SOI1 waveguides indeed exhibit
the largest η values.
To extract the χ
(2)
values from the measured efficiencies η, a
standard model was applied
1
. As no phase-matching mechanism is
present in our waveguides, the model for unphase-matched SHG
should be used. In this case, the theory predicts a spectral and
longitudinal spatial periodic dependence of η (refs 1,21). Namely,
η is a periodic function of the pump wavelength, with a period
of about 1.5 nm (ref. 21). Numerically solving the nonlinearly
coupled equations for pump and second-harmonic waves including
dispersion at both harmonics
22
shows that the 10 nm pump laser
bandwidth completely masks the fast spectral oscillations: the
single-peaked dependence of the second-harmonic power on the
pump wavelength shown in Fig. 5d can thus be reproduced. On the
other hand, it turns out that the second-harmonic signal is still a
periodic function of the propagation distance: the second-harmonic
signal power periodically oscillates between zero and its maximum
value each coherence length (which is ∼5.8 µm for the SOI1
10.7-µm-wide waveguide). Therefore only a lower bound for χ
(2)
can be extracted from η by considering that the sample length is
exactly equal to an odd multiple of the coherence length. In this case,
and in the undepleted-pump approximation
1
,
η =
8π
2
[χ
(2)
]
2
P
ω
0
cn
2
ω
n
2ω
Aλ
2
ω
1
1k
2
(1)
where
0
is the free-space electric permittivity, 1k = k
2ω
− 2k
ω
=
2π(n
2ω
/λ
2ω
−2n
ω
/λ
ω
) is the phase mismatch between the pump and
the second-harmonic signal, n
ω
and n
2ω
are the effective refractive
indices at the pump and second-harmonic wavelengths, λ
ω
and λ
2ω
are the pump and second-harmonic wavelengths and A is the modal
cross-section, which for simplicity is taken to be equal to half of the
waveguide cross-section.
Table 1 provides the calculated χ
(2)
values, which demonstrate
that a χ
(2)
of several tens of picometres per volt can be achieved
by cladding the Si waveguides with a SiN
x
overlayer. Although a
considerable χ
(2)
generation at the interface cannot be completely
ruled out, we believe that the strong measured SHG is caused by
the inhomogeneous strain in the bulk Si waveguides. The SOI1
10.7-µm-wide waveguide exhibits the largest value of χ
(2)
. Note that
this waveguide is the widest in the SOI1 set, and its optical modes are
the most confined within the waveguide core. This observation con-
firms that the χ
(2)
has a bulk origin with a minor contribution from
surface effects. The observed dependence of χ
(2)
on the stressing
layer and, in particular, on the strain inhomogeneity also points to
a bulk origin of χ
(2)
. Moreover, there exists a qualitative agreement
between the strain distribution of our waveguides and that of the
strained Si model structures. In particular, the SOI1 waveguide
presents a sign inversion of the strain as in the model structure
of Fig. 1, where both compressive and tensile strain were applied.
Instead, the waveguides SOI2 and SOI3 are qualitatively similar to
the purely tensile model structures (Supplementary Information).
As the second-order nonlinearity is related to the local lattice
deformation (that is the local strain) and the Raman measurements
show that the strain is very inhomogeneous in the waveguide, χ
(2)
is expected to have a distribution of values across the waveguide.
On the other hand, the nonlinear transmission measurements
are averaging in a non-simple way the χ
(2)
over the waveguide
cross-section, because the second-harmonic efficiency depends on
the overlap integral between the pump and the second-harmonic
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y = x/2
Intensity (a.u.)
Wavelength (nm)
SH peak power (nW)
Pump peak power (W)
Pump wavelength (nm)
Pump wavelength (nm)
y = a + b x
2
1,000 1,250 1,500 1,750 2,000 2,250 2,500 2,750
0
2.5
2.0
1.5
1.0
0.5
10
20
30
40
50
0 0.1 0.3 0.5 0.70.2 0.4 0.6 0.8
0
5
10
15
20
25
30
2,290 2,300 2,310 2,320 2,330 2,340
2,000 2,050 2,100 2,150 2,200 2,250 2,300
1,000
1,025
1,050
1,075
1,100
1,125
1,150
2,063 nm
2,088 nm
2,132 nm
a
c
b
d
P
2
(nW) SH wavelength (nm)
ω
Figure 5 | Summary of the SHG measurements. a, SOI1 10.7-µm-wide waveguide transmission spectra with a fs pump and peak power of 30 kW.
b, Second-harmonic peak tunability curve with fs-pumping peak power 2 kW on SOI1 10.7-µm-wide waveguide. The data are measured values and the line
shows the expected relationship between λ
2ω
and λ
ω
. c, Power dependence of the second-harmonic signal for the SOI1 2.3-µm-wide waveguide and
λ
ω
= 2,313 nm. The ns pump source was used. The line is a power-law fit to the data points, where the fitted power exponent is 1.9± 0.4. The deviation of
points from the fitting line at high powers is due to the onset of nonlinear absorption, which depletes the pump. d, Dependence of the second-harmonic
peak power P
2ω
on the pump wavelength λ
ω
for the SOI1 2.3-µm-wide waveguide and a pump peak power of 0.7±0.1 W of the ns source. The error bars in
b–d are the maximum propagation errors from repeated measurements.
Table 1 | Summary of the relevant parameters of the investigated waveguides.
Waveguide w
0
(µm) h
0
(µm) n
ω
n
2ω
η (W W
−1
) χ
(2)
(pm V
−1
)
SOI1 2.3 2.1 3.371 3.510 (3± 2)× 10
−8
20± 15
2.5 2.0 3.374 3.510 (5± 2)× 10
−8
20± 15
10.7 2.0 3.404 3.518 (5±3)× 10
−8
40± 30
SOI2 2.0 2.5 3.371 3.510 (1± 0.5)×10
−8
11± 8
2.5 2.3 3.381 3.512 (5±2)× 10
−9
8± 5
11.7 2.5 3.415 3.521 (4± 2)×10
−10
4± 3
SOI3 2.0 2.2 3.362 3.507 (6±3)×10
−9
9± 6
2.3 2.2 3.375 3.511 (9± 4)× 10
−9
10± 7
11.7 2.2 3.409 3.519 (5±3)× 10
−9
14± 10
The first column refers to the differently stressed waveguides, where SOI1 refers to a tensile stressing overlayer, SOI2 refers to the compressive stressing overlayer and SOI3 refers to the
stress-compensated overlayer. The second and third columns give the geometrical parameters of the waveguide cross-section (Fig. 2a). n
ω
and n
2ω
are the effective indices for the fundamental modes
at the pump (λ
ω
= 2,313 nm) and second-harmonic (λ
2ω
= 1,156.5 nm) wavelengths. η is the measured power-conversion efficiency and χ
(2)
the second-order susceptibility calculated with equation (1)
using the η values. All measurements refer to a ns pump peak power of 0.7±0.1 W .
signal as well as on the χ
(2)
distribution. Therefore, we can
point out that the observed experimental trend is related to
the inhomogeneity and magnitude of the strain and, indeed,
simulations show that experiments can be well accounted by the
model. Thus, both theory and experiments clearly demonstrate
the possibility to induce second-order nonlinear response from
bulk Si by using strain.
It is worth noting the fact that recent reports on surface-
generated SHG in silicon nitride waveguides
23
or in Si photonic
crystal nanocavities
12
provided χ
(2)
values that are significantly
smaller than the bulk induced χ
(2)
observed at present. Noticeably,
the χ
(2)
values reported here are similar to the one found in
refs 13 and 14. We should also note that the values of χ
(2)
which were measured by surface reflection spectroscopy in Si also
showed that significant strain-induced enhancement of second-
order nonlinearities with respect to surface-originated χ
(2)
values
are possible
9,10
. Strained Si waveguides can achieve a nonlinearity
competitive in magnitude with the widely used LiNbO
3
. Such values
are large enough to permit the development of practical optical
integrated devices based on bulk second-order nonlinearities of Si.
152 NATURE MATERIALS | VOL 11 | FEBRUARY 2012 | www.nature.com/naturematerials
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