Analysis of Material Susceptibility in Silicon on Insulator Waveguides With Combined Simulation of Four-Wave Mixing and Linear Mode Coupling

Abstract

We derive propagation equations modeling third-order susceptibility-induced nonlinear interaction and linear mode coupling in waveguides. We model material susceptibility with Raman and electronic response which include approximations suited for optical communications. We validate our model by comparing numerical integration of the propagation equations to continuous wave measurements of a silicon on insulator waveguide.

Full text

Analysis of Material Susceptibility in Silicon on
Insulator Waveguides With Combined Simulation of
Four-Wave Mixing and Linear Mode Coupling
Ulrike Höer (  ulrike.hoeer@tum.de )
Technical University Munich: Technische Universitat Munchen https://orcid.org/0000-0001-9125-2331
Tasnad Kernetzky
Technical University Munich: Technische Universitat Munchen
Norbert Hanik
Technical University Munich: Technische Universitat Munchen
Research Article
Keywords: nonlinear optics, susceptibility, four-wave mixing, silicon nano-rib waveguide
Posted Date: November 30th, 2021
DOI: https://doi.org/10.21203/rs.3.rs-1036492/v1
License:   This work is licensed under a Creative Commons Attribution 4.0 International License. 
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1
Analysis of Material Susceptibility in Silicon on
Insulator Waveguides with Combined Simulation of
Four-Wave Mixing and Linear Mode Coupling
Ulrike Höfler, Tasnad Kernetzky, Norbert Hanik
All with Technical University of Munich
{ulrike.hoefler, tasnad.kernetzky, norbert.hanik}@tum.de
Abstract
We derive propagation equations modeling third-order susceptibility-induced nonlinear interaction and linear mode coupling
in waveguides. We model material susceptibility with Raman and electronic response which include approximations suited for
optical communications. We validate our model by comparing numerical integration of the propagation equations to continuous
wave measurements of a silicon on insulator waveguide.
Index Terms
nonlinear optics, susceptibility, four-wave mixing, silicon nano-rib waveguide
I. INT ROD UC TI ON
All-optical signal processing is a promising technique for various applications. Two prominent examples are optical phase
conjugation (OPC) for mitigating nonlinear impairments in optical fiber links [1] and wavelength conversion (WLC) which
shows high potential for several scenarios. In a narrowband sense, WLC can enable all-optical routing in datacenters or
metropolitan networks. In a broadband sense, WLC allows to jointly multiplex several fully-loaded C-bands into one fiber by
shifting them to distinct bands. Different approaches based on highly non-linear fibers [2], LiNbO3waveguides [3], and silicon
on insulator (SOI) waveguides [4], [5] have been investigated. We present a simulation for the latter, including nonlinear effects
and linear mode coupling, with an accurate model of the susceptibility in silicon.
Third-order material nonlinearity – the origin of four-wave mixing (FWM) – is the basis for all-optical signal processing.
Due to the high nonlinear potential of silicon, the application of an SOI waveguide demands a more detailed consideration of
nonlinearity than it is provided by the optical nonlinear Schrödinger equation. Therefore, we derive a set of coupled differential
equations that model linear and nonlinear interaction based on the third-order susceptibility (resembling pulse propagation in
[6]). As a result of the high Raman gain coefficient in silicon – which is three to four orders of magnitude higher compared
to silica [7] – it is essential to include the effect of molecular vibrations (Raman response) besides the commonly considered
electron vibrations (Kerr effect).
The input-output conversion efficiency (CE) defined as idler output power over signal input power is a critical measure of
the nonlinear process’s performance. For achieving a strong idler build-up, phase matching (PM) has to be performed and
multi-mode operation is usually preferred since an additional degree of freedom is obtained. We refer to [8] for a detailed
analysis of geometry optimization for PM in SOI waveguides. In [9] we described a model for the susceptibility calculation in
silicon. This is extended in this work by applying the susceptibility model to the propagation equations of an SOI waveguide,
as well as by comparing the results to a continuous wave measurement of the waveguide.

2
II. MO DE LI NG LIGHT PROPAG ATIO N
For investigating the spatial development of the amplitudes of the interacting frequencies along the waveguide, the corre-
sponding differential equations will be stated, starting from the wave equation
∆
#»
E− ∇ ∇ ·
#»
E=µ0∂2
t
#»
D. (1)
The electric displacement field is defined as
#»
D=ǫ0·ǫr
#»
E+
#»
Pnl,(2)
where the extended material permittivity matrix
ǫr=ǫ′
rI+δǫr−jǫ′′
rI(3)
models material dispersion by ǫ′
r=n2, linear perturbations by δǫr, light attenuation by ǫ′′
r, and where Iis the identity matrix.
Rearranging all perturbations into one vector yields
#»
D=ǫ0·ǫ′
r
#»
E+
#»
P′(4)
with
#»
P′=δǫr
#»
E−jǫ′′
r
#»
E+
#»
Pnl. Inserting
#»
Din Eq. (1) gives
∆
#»
E+∇1
ǫ′
r·#»
E∇ǫ′
r+∇ ·
#»
P′
|{z }
A
−µ0ǫ0ǫ′
r∂2
t
#»
E=µ0ǫ0∂2
t
#»
P′.(5)
We model the nonlinear polarization vector as
#»
Pnl =ZZZ ↔
χ[3](τζ, τη, τρ).
.
.
#»
E(t−τζ)
#»
E(t−τη)
#»
E(t−τρ) dτζdτηdτρ,(6)
which contains the susceptibility tensor ↔
χ[3], and where .
.
.represents the tensor product.
The unmodulated total propagating electrical field can be written as a superposition of modes (m)at discrete positive
frequencies fi
#»
E=ℜ

X
i,m
#»
E(m)
fi


=1
2X
i,m
ˆ
E(m)
fi
#»
Ψ(m)
fiej2πfit−β(m)
fiz+c.c. , (7)
with real part ℜ{·}, amplitudes ˆ
E(m)
fi(z)∈C, transversal field distributions
#»
Ψ(m)
fi(x, y), and propagation constants β(m)
fi. In
the following, we make the common assumption that
#»
P′causes a z-dependence of ˆ
E, but does not have any effect on
#»
Ψ.
Since the outer gradient in term A in Eq. (5) alters the transversal field profiles, ∇ ·
#»
P′= 0,∀
#»
P′has to hold to fulfill the
assumption that
#»
P′does not affect
#»
Ψ. With that, evaluating Eq. (5) at the positive frequency f0(analytic signal) leads to
∆
#»
E+∇1
ǫ′
r
#»
E ∇ǫ′
r−µ0ǫ0ǫ′
r∂2
t
#»
E=µ0ǫ0∂2
t
#»
P′,(8)
with
#»
P′=
#»
P′+f0and
#»
E:=
#»
E|+f0.
After quite some calculus, algebra and by assuming ∂zǫ′
r= 0, the x component of Eq. (8) becomes
X
m"(∂2
x+∂2
yΨ(m)
x,f0+ǫ′
rβ2
0−β(m)2
f0·Ψ(m)
x,f0∂x1
ǫ′
r·Ψ(m)
x,f0∂xǫ′
r+ Ψ(m)
y,f0∂yǫ′
r)
|{z }
B
·ˆ
E(m)
f0ej2πf0t−β(m)
f0z
+∂2
zˆ
E(m)
f0−2jβ(m)
f0∂zˆ
E(m)
f0Ψ(m)
x,f0ej2πf0t−β(m)
f0z#
= 2µ0ǫ0∂2
tP′
x
(9)

3
with β0= 2πf0/c0. Since the field amplitude ˆ
E(m)
f0(z)only changes due to perturbations
#»
P′, all addends with ∂zˆ
Evanish
in the homogeneous equation (
#»
P′=#»
0). This indicates that due to mode orthogonality all msummands needs to be equal to
zero, i.e., B= 0 ∀m. Based on the assumption that
#»
P′does not alter
#»
Ψ, this also has to hold for
#»
P′6=#»
0. Consequently, by
extending Eq. (9) by the y and z component one gets
X
m−jβ(m)
f0∂zˆ
E(m)
f0·
#»
Ψ(m)
f0ej2πf0t−β(m)
f0z=µ0ǫ0∂2
t
#»
P′,(10)
which also makes use of the slowly varying wave approximation ∂2
zˆ
E(m)
f0≪2β(m)
f0∂zˆ
E(m)
f0, as well as ∂zˆ
E(m)
f0≪
β(m)
f0
ˆ
E(m)
f0for the z component.
The nonlinear part of
#»
P′is governed by the third-order material susceptibility ↔
χ[3]. This tensor generates a nonlinear
perturbation at frequency f0by combining three interacting light waves. Inserting Eq. (7) into Eq. (6) and only considering
frequency combinations (fζ,fη,fρ)with two positive and one negative contribution (thus, considering processes such as OPC
and Bragg scattering (BS)) gives
#»
Pnl+f0=X
(ζ, η, ρ)
∈SX
(m1, m2, m3)
∈M
3
8ˆ
E(m1)
fζ
ˆ
E(m2)
fη
∗ˆ
E(m3)
fρ
·
↔
X
[3]
(fζ,fη,fρ).
.
.
#»
Ψ(m1)
fζ
#»
Ψ(m2)
fη
∗#»
Ψ(m3)
fρej(2πf0t−δβz ),
S=(ζ, η , ρ) : f0= fζ+ fη+ fρfζ,ρ >0,fη<0,
M=(m1, m2, m3)m1,2,3∈ {TE0,TE1, ...},
δβ =β(m1)
fζ−β(m2)
fη+β(m3)
fρ,(11)
where
↔
X
[3]
is the Fourier transform of ↔
χ[3], i.e., Fn↔
χ[3]o=
↔
X
[3]
.
Inserting Eq. (11) in Eq. (10), multiplying with
#»
Ψ(a)
f0
∗from the left and integrating over the cross section (making use of
mode orthogonality), we obtain the propagation equation for mode (a)at frequency f0
∂zˆ
E(a)
f0=−α(a)
2
|{z}
attenuation
ˆ
E(a)
f0−j˜γX
m
C(a)
(m)
|{z}
mode coup.
ˆ
E(m)
f0e−j∆βlin z(12)
−j3˜γ
4X
(ζ, η, ρ)
∈SX
(m1, m2, m3)
∈M
N(a)
(m1,m2,m3)
|{z }
nonlin. coefficient
·ˆ
E(m1)
fζ
ˆ
E(m2)
fη
∗ˆ
E(m3)
fρe−j∆β z ,
with
∆βlin =β(m)
f0−β(a)
f0,˜γ=β2
0
2β(a)
f0RR 


#»
Ψ(a)
f0


2dA,
∆β=β(m1)
fζ−β(m2)
fη+β(m3)
fρ−β(a)
f0, α(a)=β2
0ǫ′′
r
β(a)
f0
,
C(a)
(m)=ZZ #»
Ψ(a)
f0
∗δǫr
#»
Ψ(m)
f0dA , N (a)
(m1,m2,m3)=ZZ #»
Ψ(a)
f0
∗
·
↔
X
[3]
(fζ,fη,fρ).
.
.
#»
Ψ(m1)
fζ
#»
Ψ(m2)
fη
∗#»
Ψ(m3)
fρdA .
The mode coupling (MC) coefficient C(a)
(m)couples modes at the same frequency whereas the nonlinearity coefficient N(a)
(m1,m2,m3)
couples between modes at all possible frequency combinations. This leads to a set of coupled differential equations of the
type of Eq. (12) which has to be solved numerically. While the source fields can propagate in different modes and at arbitrary
frequencies, the efficiency of the linear and nonlinear processes is determined by energy conservation and by the phase
mismatches ∆βlin and ∆β, respectively.

4
III. SUS CE PT IBILITY IN SILICON
In the following, the third-order susceptibility for silicon as origin of nonlinear effects is analyzed. In silicon, there are two
main parts that account for nonlinear processes, namely the electronic contribution (e) due to bound electrons and the Raman
contribution (R) stemming from atomic lattice vibrations. Hence, it is reasonable to split the third-order nonlinear susceptibility
tensor into its main parts, i.e.,
↔
X
[3]
=
↔
Xe+
↔
XR, and investigate each part separately [10]. Throughout this section, we assume
a waveguide fabricated on a (001) surface and parallel to the [110] direction.
A. Electronic Susceptibility in Frequency Domain
Due to spatial symmetry, only 21 out of 81 entries of
↔
Xeare nonzero, of which four are independent of each other [11]. If
only wavelengths λ > λmin = 1.10 µmare considered, the Kleinmann condition is satisfied and three of the four independent
elements can be approximated to be equal [10], [12], [13]. Usually the electronic susceptibility is considered as nearly constant,
since variations of the interacting frequencies lead to only small fluctuations of
↔
Xe. For a wavelength range including the
commonly used optical bands from O to L, the spacing between interacting frequencies is small enough to treat nonlinearity
caused by the electronic contribution as being independent of the interacting frequencies, i.e.,
↔
Xe(f0; fζ,fη,fρ)≈
↔
Xe(f0). With
this assumption and for λ∈[1.2µm,2.4µm] (also including bands O to L), the last two independent entries of
↔
Xecan be
related to each other as well [14]. Furthermore, the real and imaginary part of the last independent entry is linked to the Kerr
coefficient n2and the two-photon absorption coefficient βtpa, respectively. Altogether, this leads to the 21 nonzero elements
Xxxxx
e=ℜ{Xxxxx
e}+jℑ{Xxxxx
e}
=Xyyyy
e=Xzzzz
e,
Xxxxx
e
2.36 =Xxxyy
e=Xzzy y
e=Xyyzz
e=Xzzxx
e=Xxxzz
e
=Xyyxx
e=Xxyxy
e=Xzyz y
e=Xyzyz
e=Xzxzx
e
=Xxzxz
e=Xyxyx
e=Xxyyx
e=Xzyyz
e=Xyzz y
e
=Xzxxz
e=Xyxxy
e=Xxzzx
e,
(13)
with ℜ{Xxxxx
e}= 2.3482·n2(f0)·ǫ0·c0·n2(f0),ℑ{Xxxxx
e}=1.1741
2πf0·βtpa(f0)·ǫ0·c2
0·n2(f0)[13], [15], where the characteristics
of n2and βtpa can be found in [12].
B. Raman Susceptibility in Frequency Domain
The Raman contribution
↔
XRemerges from the interaction of light with lattice vibrations (phonons) of the material. If the
difference of two incident light waves coincides with the frequency of the lattice vibration (resonance), the atom is excited to
a higher vibrational eigenstate. The susceptibility elements induced by Raman scattering can be stated as [16], [7], [10]
Xijkl
R(f0; fζ,fη,fρ) = Xijkl
R(f0; f1,−f2,f3)
=1
π·fv·Γ·c0
Z0· Xijkl
1(f2−f3)·
3
X
n=1
(Rij )n·(Rkl)n+Xijkl
1(f2−f1)·
3
X
n=1
(Ril)n·(Rj k)n!,
(14)
with {f0,f1,f2,f3}>0,Xijkl
1(f2−fa) = n2(fa)·gR(fa)
fa·[(f2
v−(f2−fa)2)−jΓ·(f2−fa)] ,a∈ {1,3}, and i, j, k, l ∈ {x, y , z}, where Γ =
105 GHz is the FWHM-bandwidth, fv= 15.6 THz the vibrational eigenstate frequency, Z0=qµ0
ǫ0,gRthe Raman gain
coefficient, and Rn,n∈ {1,2,3}the three Raman matrices with
R1=1
√2



0 0 −1
0 0 −1
−1−1 0



, R2=



−100
0 1 0
0 0 0



, R3=1
√2



0 0 1
0 0 −1
1−1 0



.

5
Each Raman matrix corresponds to the respective displacement of the phonons along the crystallographic directions of the
medium and reflects its crystal symmetry. The terms P3
n=1(Rij )n·(Rkl )nand P3
n=1(Ril )n·(Rjk )ndetermine the 18 nonzero
elements of
↔
XR. All denoted frequency values apply only at room temperature, i.e., T≈300 K. The formula and the required
parameters for the approximation of gRcan be found in [7], [17], [18], [19].
As indicated by Eq. (14), the entries of the Raman susceptibility consist of two contributions, Xijkl
1(f2−f1)and Xijkl
1(f2−f3).
This can be derived from the fact that there exist two potential ways to promote atoms from the ground state to a higher
vibrational eigenstate. Figure 1 illustrates the Raman process f0= f1−f2+ f3in the special case of resonance. A photon of
energy hf1,3excites the atom from the ground state to a virtual energy state E′
1. Then, stimulated by a photon of frequency f2,
part of the energy is used to promote the atom to the vibrational eigenstate Ev=hfv, the other part is emitted as a photon with
frequency f2. Thus, photons with energy hf1,3can be understood as the driving force for providing atoms to the vibrational
eigenstate Ev.
If a photon of energy hf3,1is absorbed by an atom located at the vibrational eigenstate Ev, the atom is excited to the virtual
state E′
2and falls back to the ground state immediately while emitting a photon of frequency f0. Considering f1as the frequency
that implicitly provides atoms to the vibrational eigenstate, Xijkl
1is differently pronounced depending on how precisely the
resonance frequency fvis hit by the frequency difference f1−f2. In the case of resonance, i.e., f1−f2= fv, the maximal
value of Xijkl
1is obtained. Analogously, Xijkl
2considers the possibility of resonance between f2and f3, by regarding f3as
the frequency that implicitly supplies atoms to the vibrational eigenstate.
hf1,3hf2
hf3,1hf0
hfv
E′
1
E′
2
Ev
hf2
Fig. 1: Energy–diagram of the possible Raman processes in resonance. Virtual states, represented by the dashed lines, can only
be used as transitions and cannot be occupied in contrast to eigenstates.
IV. IDL ER EVOLUTION I N AN S OI WAVEG UI DE
In our simulation, we launch three waves (two pumps and one signal) with discrete frequencies into an SOI waveguide. By
linear MC, waves at all frequencies couple into all available modes at the same frequencies. Due to the nonlinear interaction,
waves at all possible frequency combinations are generated in all available modes. Both interactions occur differently pronounced
depending on phase mismatches ∆βlin and ∆βand coefficients Cand N. Consequently, the number of possible combinations for
interaction quickly rises. We limit the simulation by only considering light propagation in the guided modes of the waveguide
(TE0, TE1, TE2) and by only taking frequencies into account that will be generated with non-negligible efficiency. Thus,
the frequencies of the three input waves, i.e., fP1,fP2,fS, and the two generated frequencies fBS =−fP1+ fS+ fP2and
fOPC = +fP1−fS+ fP2(arising from the BS and OPC FWM processes) are considered. Although this limitation drastically
reduces complexity, still 15 coupled differential equations have to be solved simultaneously. We numerically integrate the
system of coupled differential equations (Eq. (12)) based on the third-order susceptibility (Section III) with a variable order,
variable step-size Adams-Bashforth-Moulton solver.
A. Simulation (I)
First, an SOI waveguide with rib width 1800nm, slab height 100 nm, SOI height 220 nm, propagation length 2 cm and one
dip of width 400 nm and depth 70 nm is considered. Figure 2 shows the waveguide’s geometry, which is the same as in [8].

6
hSOI hSlab
wRib
wDip
hDip
−3000 −2000 −1000 0 1000 2000 3000
−100
0
100
200
300
x[nm]
y[nm]
1
2
3
Fig. 2: Refractive index geometry of the nano-rib waveguide with slab, SOI and dip heights, as well as rib and dip widths.
input [nm] generated [nm] input power [dBm] α[dB cm−1]
P1P2S IOPC∗IBS P1P2S
TE01
TE11535 1519 1529 20 2
TE21524 1540 20 10 3
TABLE I: Simulation (I) parameters: wavelengths and modes of propagating waves with high conversion efficiency as well
as input powers and mode-dependent attenuation coefficients. The asterisk at IOPC∗indicates that PM was optimized for the
OPC process.
Table I summarizes the input/output wavelengths and their corresponding modes, optimized for best efficiency of the OPC
process. We select realistic values for input powers, and mode-dependent attenuation coefficients, which are also included in
Table I. The resulting power evolution along the waveguide – without linear MC (C= 0) – is shown in Fig. 3a. It can be seen,
that the input waves (marked as , and ) are linearly attenuated by the waveguide. The OPC process ( ) with
best PM is created with highest efficiency, while the BS process ( ) with slightly worse PM is created with less efficiency.
The additional decrease in BS idler power compared to the OPC idler is caused by the larger phase mismatch ∆β. All other
potential FWM products at other modes and frequencies are not shown, since they are highly phase-mismatched, and thus have
negligible power.
Figure 3b shows the power evolution with additional linear MC. In principle, all waves couple into all considered modes.
We exemplarily show one pump and one idler ( , ). The oscillation periods depend on the corresponding ∆βlin values
as LOscillation =2π
|∆βlin |and the coupled powers on both, Cand ∆βlin . For instance, the magnitude of the normalized coupling
coefficient of the OPC idler from TE1to TE0() is C(TE0)
(TE1)RR
#»
Ψ(TE0)
fIOPC 
2dA= 1.3×10−4, whereas from TE1to TE2
() it is C(TE2)
(TE1)RR
#»
Ψ(TE2)
fIOPC 
2dA= 3.6×10−3. The corresponding phase mismatches |∆βlin |are |β(TE1)
fIOPC −β(TE0)
fIOPC |=
9.1×104m−1(), and |β(TE1)
fIOPC −β(TE2)
fIOPC |= 9.2×105m−1(), respectively. While the coupling coefficient Cfrom
TE1to TE2is greater than from TE1to TE0(large values induce more coupled power), the phase mismatch ∆βlin from TE1
to TE2is also greater than from TE1to TE0(large values induce less coupled power). This leads to similar power levels of
the OPC idler in TE0and TE2(, ). Figure 3c shows a closeup of IOPC in mode TE0and reveals two oscillations. The
slower one with period of roughly the plotted range is caused by direct coupling from TE1. The fast oscillation arises from
second-order linear coupling from TE2and is therefore only weakly pronounced.
B. Simulation (II) with Experimental Verification
For verifying the simulation results we use measured data from an SOI waveguide with rib width 1672 nm, slab height
100 nm, SOI height 220 nm, and propagation length 11.3 mm ([20], [21]). Table II summarizes the simulation parameters
optimized for best efficiency of the BS process.

7
0 0.511.5 2
−40
−20
0
20
Waveguide Position (cm)
Wave Power (dBm)
P2
P1
S
IOPC
IBS
(a)
1.990 1.992 1.994 1.996 1.998 2.000
−50
0
Waveguide Position (cm)
Wave Power (dBm)
P1@TE2
P1@TE0
IOPC@TE1
P1@TE1
IOPC@TE2
IOPC@TE0
(b)
1.990 1.992 1.994 1.996 1.998 2.000
−76.5
−76
−75.5
−75
Waveguide Position (cm)
Wave Power (dBm)
IOPC@TE0
(c)
Fig. 3: (a): Simulated power evolution of pumps, signal and idlers along the waveguide without linear MC. The mode distribution
is corresponding to Table I. (b): Simulated power evolution of selected waves with linear and nonlinear coupling. (c): Vertical
zoom on the OPC idler in TE0in (b).
input [nm] generated [nm] input power [dBm] α[dB cm−1]
P1P2S IOPC IBS∗P1P2S
TE01300 1296.57 1303.44 19.33 0.9
TE11540 1544.84 23.95 11.28 1.8
TE21.8
TABLE II: Simulation (II) parameters: wavelengths and modes of propagating waves with high conversion efficiency as well
as input powers and mode-dependent attenuation coefficients. The asterisk at IBS∗indicates that PM was optimized for the BS
process.
Since mode multiplexer and grating coupler at the input and output of the waveguide are lossy, the simulation parameters
have to be adjusted accordingly which is reflected in a reduced input power, i.e., PP1= 13.92 dBm,PP2= 18.75 dBm, and
PS= 6.28 dBm. A suitable means to evaluate the generation process of the BS idler is the input-output conversion efficiency
(CE): CE = PIBS /PS(or CE = PIBS −PSin log-domain), where the signal and idler power is defined at the input (before
mode multiplexer and grating coupler) and at the output (after mode demultiplexer and grating coupler) of the chip, respectively.
Figure 4a shows that the simulated idler power is PIBS =−22.5 dBm at the end of the waveguide. This results in a CE of
CEsim =−22.5 dBm −5 dB −11.28 dBm ≈ −39 dB, where −5 dB includes the losses caused by demultiplexer and grating
coupler at the output. From Fig. 4b, the measured CE for λIBS = 1303.44 nm is CEmeas ≈ −42 dB. This is a very good
match between measurement and simulation, since the remaining CE difference of 3 dB is very small considering measurement
imperfections, material parameter, temperature variations, etc.

8
0 0.2 0.4 0.6 0.8 1
−40
−20
0
Waveguide Position (cm)
PS,PIBS (dBm)
S
IBS
(a) (b)
Fig. 4: (a): Simulated BS idler and signal power evolution. The mode distribution is corresponding to Table I. (b): Measured
CE output of the SOI waveguide, from [20].
V. CO NC LU SIONS
We modeled electronic and molecular parts
↔
Xeand
↔
XRof the silicon susceptibility
↔
X
[3]
. We linked the components of
↔
Xe
to material parameters and presented a closed-form solution for
↔
XR. Although the approximations we applied limit the usable
wavelength region, they are valid for commonly used optical transmission bands.
We derived and numerically integrated a set of coupled differential equations, which model linear and nonlinear light evolution
along waveguides. Each frequency component in each mode is modeled by an equation that includes linear attenuation, linear
MC, and FWM nonlinearity caused by the material susceptibility
↔
X
[3]
. We verified the validity of the proposed model by
measured data of an SOI waveguide.
VI. ACK NOWLEDG ME NT S
This work was supported by the DFG project HA 6010/6-1.
REF ER EN CES
[1] D. Vukovic, J. Schröder, F. Da Ros, L. B. Du, C. J. Chae, D.-Y. Choi, M. D. Pelusi, and C. Peucheret, “Multichannel nonlinear distortion compensation
using optical phase conjugation in a silicon nanowire,” Opt. Express, vol. 23, p. 3640, 2015.
[2] G. Rademacher, R. S. Luis, B. J. Puttnam, Y. Awaji, M. Suzuki, T. Hasegawa, H. Furukawa, and N. Wada, “Wideband intermodal nonlinear signal
processing with a highly nonlinear few-mode fiber,” IEEE J. Sel. Topics Quantum Electron., vol. 26, pp. 1–7, 2020.
[3] P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinell, i, C. Langrock, M. M. Fejer, and V. Degiorgio, “Optical phase conjugation in phase-
modulated transmission systems: Experimental comparison of different nonlinearity-compensation methods,” Opt. Express, vol. 18, p. 18119, 2010.
[4] A. Gajda, F. Da Ros, E. P. da Silva, A. Peczek, E. Liebig, A. Mai, M. Galili, L. K. Oxenløwe, K. Petermann, and L. Zimmermann, “Silicon waveguide
with lateral p-i-n diode for nonlinearity compensation by on-chip optical phase conjugation,” in Optical Fiber Communication Conference. Optical
Society of America, 2018.
[5] G. Ronniger, S. Lischke, C. Mai, L. Zimmermann, and K. Petermann, “Investigation of inter-modal four wave mixing in p-i-n diode assisted SOI
waveguides,” in 2020 IEEE Photonics Society Summer Topicals Meeting Series (SUM). Cabo San Lucas, Mexico: IEEE, 2020, pp. 1–2.
[6] G. P. Agrawal, Nonlinear Fiber Optics, fifth edition ed. Amsterdam: Elsevier/Academic Press, 2013.
[7] B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE Journal of Selected Topics in Quantum Electronics,
vol. 12, no. 3, pp. 412–421, 2006.
[8] T. Kernetzky, Y. Jia, and N. Hanik, “Multi dimensional optimization of phase matching in multimode silicon nano-rib waveguides,” in Photonic Networks;
21st ITG-Symposium, Leipzig, Germany, 2020.
[9] U. Höfler, T. Kernetzky, and N. Hanik, “Modeling material susceptibility in silicon for four-wave mixing based nonlinear optics,” in International
Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), 9 2021.
[10] Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Optics express, vol. 15,
no. 25, pp. 16 604–16 644, 2007.
[11] R. W. Boyd, Nonlinear optics. Elsevier, 2003, ch. 1.
[12] A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Applied Physics Letters,
vol. 90, no. 19, p. 191104, 2007.

9
[13] R. Osgood, N. Panoiu, J. Dadap, X. Liu, X. Chen, I.-W. Hsieh, E. Dulkeith, W. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical
systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Advances in Optics and Photonics, vol. 1, no. 1, pp. 162–235,
2009.
[14] J. Zhang, Q. Lin, G. Piredda, R. Boyd, G. Agrawal, and P. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Applied
Physics Letters, vol. 91, no. 7, p. 071113, 2007.
[15] N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1- x Ge x in the midwave and longwave infrared,”
Journal of Applied Physics, vol. 110, no. 1, p. 9, 2011.
[16] D. Dimitropoulos, B. Houshmand, R. Claps, and B. Jalali, “Coupled-mode theory of the Raman effect in silicon-on-insulator waveguides,” Optics letters,
vol. 28, no. 20, pp. 1954–1956, 2003.
[17] D. Dimitropoulos, R. Claps, Y. Han, and B. Jalali, “Nonlinear optics in silicon waveguides: stimulated Raman scattering and two-photon absorption,” in
Integrated Optics: Devices, Materials, and Technologies VII, vol. 4987. International Society for Optics and Photonics, 2003, pp. 140–148.
[18] J. Ralston and R. Chang, “Spontaneous-Raman-scattering efficiency and stimulated scattering in silicon,” Physical Review B, vol. 2, no. 6, p. 1858, 1970.
[19] J. Renucci, R. Tyte, and M. Cardona, “Resonant Raman scattering in silicon,” Physical Review B, vol. 11, no. 10, p. 3885, 1975.
[20] G. Ronniger, I. Sackey, T. Kernetzky, U. Höfler, C. Mai, C. Schubert, N. Hanik, L. Zimmermann, R. Freund, and K. Petermann, “Efficient ultra-broadband
C-to-O band converter based on multi-mode silicon-on-insulator waveguides,” in 2021 The European Conference on Optical Communication (ECOC),
Bordeaux, France, 9 2021.
[21] T. Kernetzky, G. Ronniger, U. Höfler, L. Zimmermann, and N. Hanik, “Numerical optimization and CW measurements of SOI waveguides for ultra-
broadband C-to-O-band conversion,” in 2021 The European Conference on Optical Communication (ECOC), Bordeaux, France, 9 2021.