OPTICAL MATERIALS IN
CST STUDIO SUITE
WHITEPAPER
For many optical applications, materials are needed with anisotropic or nonlinear properties. Two important examples of such properties are birefringence and dichroism. Such materials exhibit different refractive indices and attenuation for orthogonal optical polarization states. They are used to alter the polarization state of the light, for example, in polarizers or polarization converters. A special case of a polarization
dependent-material property is magneto-optical activity. Magneto-optical active materials can also be
used to alter the polarization state but, more importantly, they can be used to build non-reciprocal components like isolators. Furthermore, the optical properties can depend not only on the state of polarization
but also on the electric field amplitude of the light wave. The optical properties can depend on the second,
third, or even higher powers of the electric field. Here, the effects and applications are vast – amplification,
frequency conversion, and all-optical switching to name but a few.
Optical components usually utilize weak optical effects
that accumulate over distances of a hundred or even a
thousand times the optical wavelength. This is not an
option for optical systems that are intended to be
integrated on the chip scale. This problem can be
mitigated by utilizing compact resonators which strongly
increase the interaction with optical materials. However,
due to the strong confinement of the light, common
approximations often become invalid and simple
theoretical predictions become cumbersome. Here, we
demonstrate three examples of application of special
optical materials in integrated components. The direct
numerical simulations help to understand the underlying
mechanisms and demonstrate side effects not predicted
by simplified theories.
In part I, we present birefringent and dichroic materials
and show how strong anisotropy can suppress substrate
leakage of ring resonators and waveguides. The 3D
simulation shows that anisotropic materials should be
considered carefully when all polarizations are present.
The anisotropic material suppresses the leakage of the
dominant polarization in the waveguide mode, but at the
same time provides additional leakage channels for the
orthogonal polarization.
In part II, we show how a magneto-optically active
material can rotate the polarization of an optical wave.
For integrated waveguides, strong birefringence prevents
this rotation. Instead, the magneto-optical material is
used to make the propagation constant of the guided
wave direction dependent. Utilizing this, a ring resonator
can be made non-reciprocal and can thereby be used as
an isolator.
For waveguiding 𝛾𝛾𝛾𝛾 needs to be a real number which means that the propagation constant, 𝛽𝛽𝛽𝛽, needs to
be larger than the wavevector in silica, �𝜀𝜀𝜀𝜀𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆2 𝑘𝑘𝑘𝑘0 . For an anisotropic cladding and substrate as in Fig. 2b
this conditions is more relaxed. The decay constant then becomes for a TM-polarized wave
linearly polarized
z
has to bematerial
larger thancan
𝑘𝑘𝑘𝑘0 . However,
anisotropic case
has optical
the additional
of freedom that
�𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
It was shown that an anisotropic
increase
the the
confinement
of an
wavedegree
in waveguide
0
(b)
-45
core [1]. Fig. 3a shows a TM-mode in a 250 nm silicon slab waveguide with silica substrate and cladding
y
-15
𝜀𝜀𝜀𝜀𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆2 = 2.1. The light is totally internally
reflected at the silicon silica interface and therefore guided in
the slab. Outside the slab the field decays evanescently proportional to 𝑒𝑒𝑒𝑒 −𝛾𝛾𝛾𝛾𝑦𝑦𝑦𝑦 , where 𝑦𝑦𝑦𝑦 is the coordinate
x (b)
-30
in vertical direction and
-90
0
1
2
3
4
5
6
-45
Length [µm]
𝛾𝛾𝛾𝛾 = �𝛽𝛽𝛽𝛽 2 − 𝜀𝜀𝜀𝜀𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆2 𝑘𝑘𝑘𝑘 2 .
Phase shift [deg]
zed
(a)
Phase shift [deg]
𝜀𝜀𝜀𝜀𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
𝛾𝛾𝛾𝛾 = �
�𝛽𝛽𝛽𝛽 2 − 𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑘𝑘𝑘𝑘02 .
𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
PART I
faster with propagation distance
such
As in the isotropic case, the condition for guiding is that γ is
To conclude, anisotropy and dichroism can be used to
′′ that at the end of the
𝜀𝜀𝜀𝜀 ′ + 𝑗𝑗𝑗𝑗𝜀𝜀𝜀𝜀𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
0
0
As in the isotropic
case,
theleads
condition
forsimilar
guidingcondition
is that 𝛾𝛾𝛾𝛾 is that
real. β
This
to larger
similar condition
thatthe
𝛽𝛽𝛽𝛽 polarization state of light for bulk optical
BIREFRINGENT AND DICHROIC MATERIALS
waveguide there is a phase 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
shift of -90°.
real.
This
to the
hasleads
to be
control
′ This′′ means the
0
0
𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 + 𝑗𝑗𝑗𝑗𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
𝑫𝑫𝑫𝑫 = �
� 𝑬𝑬𝑬𝑬.
has
to
be
larger
than
𝜀𝜀𝜀𝜀
𝑘𝑘𝑘𝑘
.
However,
the
anisotropic
case
has
the
additional
degree
of
freedom
that
�
light becomes circularly polarized.
than 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 0 . However, the anisotropic case has the additional
components. An anisotropic material can be used to convert
′
′′
0
0
𝜀𝜀𝜀𝜀𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧
+ 𝑗𝑗𝑗𝑗𝜀𝜀𝜀𝜀𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧
ic materials
In birefringent materials, the phase velocity of a light wave
degree of freedom that the strength of the decay is also
a linear polarized wave into a circular polarized wave and vice
depends
on the
polarization
and
direction.
Closely related to birefringence is dichroism. In this case, the
determined by εxx, which can be chosen independently of
versa. A dichroic material can be used to build a polarizer for
phase velocity
of a light
wave
depends
onthe
thepropagation
polarization
and propagation
Anisotropic crystals are again example of dichroic materials, but more commonly used are polymers
Mathematically
this
means
that
the
permittivity
of
such
losses
are
polarization
dependent
and
the
permittivity
tensor
the
guiding
condition.
A
large
horizontal
component
of
the
linear polarization. For integrated waveguides the form
s means that the permittivity of such materials is a tensor,
which are absorptive if the polarization is oriented along the polymer chain and less so if it is oriented
materials is a tensor,
becomes complex.
permittivity yields a fast decay and good confinement as
birefringence is usually much bigger than any birefringence
orthogonal to it. If such polymers are stretched and all the chains are aligned in one direction, the
shown
in
Figure
3b.
The
example
was
calculated
with
introduced by an anisotropic material. In these systems, an
′
′′
material becomes
Another
𝜀𝜀𝜀𝜀𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
+dichroic.
𝑗𝑗𝑗𝑗𝜀𝜀𝜀𝜀𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
0 way to make a0dichroic material is to use wire media. Here metallic
0
𝜀𝜀𝜀𝜀𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 0
εyy=2.1 and εxx=12.25. It should be noted that the evanescent
anisotropic interlayer can reduce substrate leakage or
wires are aligned parallel to each
′ other′′which leads to a similar effect as for the polymers. Dichroic
+ 𝑗𝑗𝑗𝑗𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
0
0
𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
𝑫𝑫𝑫𝑫 = �
� 𝑬𝑬𝑬𝑬.′
decay
of
a
TE-mode
cannot
be
engineered
with
an
anisotropic
coupling between adjacent waveguides.
𝑫𝑫𝑫𝑫 = � 0 𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 0 � 𝑬𝑬𝑬𝑬.
′
′
′′
materials are used for polarizers. In Fig. 2a waveguide
with
and
′
′′ 𝜀𝜀𝜀𝜀𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = 𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 = 𝜀𝜀𝜀𝜀𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 = 2.25, 𝜀𝜀𝜀𝜀𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = 0.0195cladding
as
in
this
case
the
electric
field
is
purely
polarized
in
0
0
𝜀𝜀𝜀𝜀𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 +
𝑗𝑗𝑗𝑗𝜀𝜀𝜀𝜀
0
0 𝜀𝜀𝜀𝜀𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧
′′
′′
𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
= 𝜀𝜀𝜀𝜀𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧
=0.0021 is shown. The differing loss for
x- and𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧y-polarization lets the y-polarization pass
the z-direction and therefore probes solely εzz. The decay
relatively undamped and strongly attenuates the x-polarization.
Materials for which two of the components are identical are
Anisotropic crystals are an example of dichroic materials, but
constant is then analogous to an isotropic cladding.
Anisotropic
crystals
are again example
of dichroic materials, but more commonly used are polymers
the components
are
identicalIf are
referred
to as
uniaxial.
three
referred to
as uniaxial.
all three
components
are
differentIf allmore
commonly used are polymers which are absorptive if
are absorptive if the polarization
is oriented
along
thethepolymer
less so if it is oriented Such decay constant engineering is useful to suppress
m each other
theeach
material
biaxial.
from
other, is
thecalled
material
is which
called biaxial.
the polarization
is oriented
along
polymerchain
chain and
and less
so if are
it isstretched
oriented orthogonal
it. If are
suchaligned
polymers
are direction, the
crosstalk between waveguides or to suppress substrate
orthogonal to it. If such polymers
and all thetochains
in one
Examples for birefringent materials
found
in
nature
are
stretched
and
all
the
chains
are
aligned
in
one
direction,
the
becomes
dichroic.
Another
terials found in nature are crystals withmaterial
asymmetric
crystal
structures
such way to make a dichroic material is to use wire media. Here metallic leakage. Figure 4 shows a silicon ring resonator with 10 µm
crystals with asymmetric crystal structures such as quartz.
material becomes dichroic. Another way to make a dichroic
radius and 250 nm x 500 nm waveguide cross-section. The
wires
are aligned
parallel
tonegative
each other which leads to a similar effect as for the polymers. Dichroic
als are a special
case,materials
where one
the
tensor
components
has
Hyperbolic
are aofspecial
case, where
one of the
material is to use wire media. Here metallic wires are aligned
ring lies on a silicon substrate with an interlayer in-between
′
′
′
′′
materials
are
usedmaterial
for polarizers.
In Fig.
2a waveguide
𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
=effect
𝜀𝜀𝜀𝜀𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧
=as2.25,
e demonstrates
properties
for onesign.
polarization
and metallic
tensor dielectric
components
has a negative
The
parallel
to each
other whichwith
leads𝜀𝜀𝜀𝜀𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
to =
a similar
for 𝜀𝜀𝜀𝜀𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = 0.0195 and that is 600 nm thick. Two cases were calculated: one for
′′
′′
therefore
dielectric
for
one materials
the differing
polymers.loss
Dichroic
materials
are used for polarizers.
which the interlayer is silica and one for which the interlayer
𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
=isproperties
𝜀𝜀𝜀𝜀to
is shown.
The
for xand y-polarization
lets the y-polarization pass
tion. Another
way todemonstrates
create birefringence
stress isotropic
𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 =0.0021
polarization and metallic properties for the other polarization.
is anisotropic with the same parameters as in Figure 3b. The
and their isotropy is lost.
relatively undamped and strongly attenuates the x-polarization.
Another way to create birefringence is to stress isotropic
In Figure 2, a waveguide with ε’xx=ε’yy=ε’zz=2.25, ε’x’x=0.0195
main loss mechanism is that the optical wave leaks through
materials such that they are deformed and their isotropy is
and ε’y’y=ε’z’z=0.0021 is shown. The differing loss for x- and
the interlayer into the substrate. As in Figure 3, the
(a)
(b)
(a)
r birefringent
materials
is
a
quarter-wave
plate.
Here,
the
propagation
lost.
y-polarization lets the y-polarization pass relatively
anisotropic interlayer can better confine the optical field in
and waveguide.
strongly attenuates
thelower
x-polarization.
Fig. 2undamped
shows
Due to the much
losses for the y-polarization (a) compared to x- the waveguide and suppress the substrate leakage. This
s adjusted in such a way that the phase shift between the two principle
axesa dichroic
A conventional application for birefringent materials is a polarization (b) the waveguide serves as a polarizer.
leads to a Q-factor of 9000 for the isotropic silica interlayer
h. This means that the arrangement acts as a converter from linear to
quarter-wave plate. Here, the propagation length through
It was shown that an anisotropic material can increase the
and a Q-factor of 22000 for the anisotropic interlayer,
ce versa. Fig.the
1 shows
of such
a quarter-wave
structure.
A waveguide
[1]
It was
shown that an
material
increase the
confinement
materialan
is example
adjusted in
a way that the
phase shift
confinement
of anisotropic
an optical wave
in acan
waveguide
core
. Figure of an optical wave in waveguide
amounting to a 2.5-fold improvement of the quality factor.
core
[1].shows
Fig. 3a ashows
a TM-mode
a 250
nmsilicon
silicon slab
slab waveguide
with silica substrate and cladding
air cladding
withprinciple
𝜀𝜀𝜀𝜀 = 1 is axes
excited
lighta that
is 45°
linearly
between
the two
is with
90° or
quarter
3a
TM-mode
in a in
250
nm
waveguide
In contrast to the slab waveguide, which can support pure
𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 = 2.45 and
𝜀𝜀𝜀𝜀𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆with
= 2.1.
The light
is totallyand
internally
reflected
at the silicon
silica interface
and therefore guided
in and TE-modes, the ring waveguide can only support
-wavelength.
This
means
that
the
arrangement
acts
as
a
silica
substrate
cladding
ε
The
light
is
TM2
SiO2 =2.1.
waveguide is 17 µm long, has a 1.2 µm x 1.2 µm cross-section and light
slab. Outside
the slab
the fieldat
decays
evanescently
to 𝑒𝑒𝑒𝑒 −𝛾𝛾𝛾𝛾𝑦𝑦𝑦𝑦 , where 𝑦𝑦𝑦𝑦 is the coordinate
converter from linear to circular polarized light and vice the totally
internally
reflected
the silicon
silica proportional
interface and
hybrid modes which are neither purely TE- or TM-polarized.
m is used. We
have used a waveguide with square cross-section to avoid
direction
andin the slab. Outside the slab the field decays
versa. Figure 1 shows an example of a quarter-wave structure. in vertical
therefore
guided
As can be seen in Figure 4, even if the mode is mostly
(b)
gence. The Apart
of the light
is polarized
y-direction
will see a higher
waveguide
with that
εxx=2.35
and εyy=εzzin
=2.45
and air cladding
evanescently proportional to e-γy, where y2 is the coordinate
TM-polarized, there is always a small part that is TE-polarized
2
𝛾𝛾𝛾𝛾 = �𝛽𝛽𝛽𝛽 − 𝜀𝜀𝜀𝜀𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆2 𝑘𝑘𝑘𝑘0 .
Figure 3: Shows the H-field of a TM-mode in a slab waveguide. In (a)
with
is excited
that is 45°distance
linearly polarized
in at the
in vertical
which
for which the interlayer will have effectively the same
efore change
its ε=1
phase
fasterwith
withlight
propagation
such that
end direction and γ needs to be a real number
the cladding is isotropic and in (b) anisotropic. The anisotropy can
waveguiding
needs
to be a realconstant,
number which
meanstothat
propagation constant, 𝛽𝛽𝛽𝛽, needs
to
xy-plane.
The waveguide
17 µm long,
has a 1.2
µm x For means
that 𝛾𝛾𝛾𝛾the
propagation
β, needs
be the
larger
refractive
index as the waveguide core. This part is not
hase shift ofthe
-90°.
This means
the lightisbecomes
circularly
polarized.
increase the decay constant of the evanescent wave in the cladding.
cladding and substrate as in Fig.
2b and will leak out of the resonator. For this reason the
wavevector
silica, �𝜀𝜀𝜀𝜀𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆2 𝑘𝑘𝑘𝑘0. For
1.2 µm cross-section and light with a wavelength of 508.5 be larger
than than
the the
wavevector
ininsilica,
Foranananisotropic
anisotropic
guided
(b)relaxed.such
conditionsand
is more
The as
decay
constant
forisa TM-polarized wave
nm is used. We have used a waveguide with(a)
square cross- this cladding
substrate
in Figure
2b,then
thisbecomes
condition
quality factor improvement is limited to a 2.5-fold
section to avoid additional waveguide
birefringence.
The
part
more
relaxed.
The
decay
constant
then
becomes
the
same
as
improvement. This leakage mechanism is important for
𝜀𝜀𝜀𝜀
Fig. 2 shows a dichroic waveguide. Due to the much lower losses for the𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥y-polarization
(a) compared to x0
𝛾𝛾𝛾𝛾 = �
�𝛽𝛽𝛽𝛽 2 − 𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑘𝑘𝑘𝑘02 .
of the light that is polarized
in the y-direction will see a
for a TM-polarized wave.
waveguides with anisotropic cladding and can be clearly
𝜀𝜀𝜀𝜀
𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
polarization (b) the waveguide serves as a polarizer.
higher refractive index and will therefore change its phase
recognized and visualized in the presented simulation.
As in the isotropic case, the condition for guiding is that 𝛾𝛾𝛾𝛾 is real. This leads to similar condition that 𝛽𝛽𝛽𝛽
(a)
absolute
y
(b)
x
(e)
y-component
0
olarization as the light travels along the waveguide.
(b) Shows the
phase to
difference
For waveguiding
𝛾𝛾𝛾𝛾 needs
be a real
-60 number which means that the propagation constant, 𝛽𝛽𝛽𝛽, needs to
ction and in y-direction.
be larger than
the wavevector in silica, �𝜀𝜀𝜀𝜀𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆2 𝑘𝑘𝑘𝑘0 . For an anisotropic cladding and substrate as in Fig. 2b
linearly polarized
-75
this conditions is more relaxed. The decay constant then becomes for a TM-polarized wave
ce is dichroism. In this case, the losses are polarization dependent and the
𝜀𝜀𝜀𝜀𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
-90
circularly polarized
complex
�𝛽𝛽𝛽𝛽
0 𝛾𝛾𝛾𝛾 = 1�
2 2 − 𝜀𝜀𝜀𝜀3𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑘𝑘𝑘𝑘02 . 4
5
6
𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
absolute
(c)
y-component
(f)
z-component
(a)
(b)
Length [µm]
Figure 2: Shows a dichroic waveguide. Due to the much lower losses
As in theasisotropic
case, the condition
forphase
guiding
is that
𝛾𝛾𝛾𝛾 is real.
This leads
to similar condition that 𝛽𝛽𝛽𝛽 for the y-polarization (a) compared to x-polarization (b) the waveguide
Figure 1: (a) Shows the change of polarization
the light travels
(b) Shows the
difference
between
light polarized
in x-direction
along the waveguide.
in y-direction.
has to be larger than �𝜀𝜀𝜀𝜀𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑘𝑘𝑘𝑘0 .and
However,
the anisotropic case has the additional degree of freedom that serves as a polarizer.
(d)
z-component
(g)
Figure 4: (a) shows a silicon ring resonator with a silicon substrate.
The red interlayer is in one case isotropic (εSiO2=2.1) and in the other
case anisotropic (εyy=2.1, εxx=12.25). (b)–(d) show the electric field for
the isotropic case and (e)–(g) for the anisotropic case.
n anisotropic material. In these systems, an anisotropic interlayer can
oupling between adjacent waveguides.
response nonlinear. An example of such a material is a Kerr medium for which the electric polar
Part III
depends on the third power of the electric field
Third order non-linear materials
𝑃𝑃𝑃𝑃 = 𝜀𝜀𝜀𝜀0 𝜒𝜒𝜒𝜒 (1) 𝐸𝐸𝐸𝐸 + 𝜀𝜀𝜀𝜀0 𝜒𝜒𝜒𝜒 (3) 𝐸𝐸𝐸𝐸 3 ,
Part III
For some media, the permittivity depends on the e
where 𝜒𝜒𝜒𝜒 (1) and 𝜒𝜒𝜒𝜒 (3) are the first and third order susceptibility and the electric polarization 𝑃𝑃𝑃𝑃 is r
Third
order
non-linear
materials
response nonlinear. An example of such a material
acob, "Transparent subdiffraction optics: nanoscale light confinement
the electric
displacement
field viaof
𝐷𝐷𝐷𝐷 the
= 𝜀𝜀𝜀𝜀light,
𝑃𝑃𝑃𝑃. It should be
noted that the electrical polarizatio
0 𝐸𝐸𝐸𝐸 + making
For some media, the permittivity depends
on the electric
field strength
optical
depends their
on the
third power of the electric field
100 (2014) PART II
not
the
same
as
the
previously
discussed
optical
polarization.
is the material
contrib
As shown in Figure
7, forward
and backward
traveling
of for
the
electric
field vector
ofThe
a former
light wave.
A
pass the first polarizer and pass the second polarizer rotated
response
nonlinear.
An example
of such awaves
material isorientation
a Kerr medium
which
the electric
polarization
𝑃𝑃𝑃𝑃 = 𝜀𝜀𝜀𝜀0 𝜒𝜒𝜒𝜒 (1)
the
displacement
field,
while
the
latter
is
the
orientation
of
the
electric
field
vector
of
a
light
wa
MAGNETO-OPTICAL MATERIALS
to 90°. For the reversed direction, 90° polarized light will be
will have differingdepends
effective
it follows
that
nonlinear relation between displacement
field and electric
on wavelengths
the third powerand
of the
electric field
where 𝜒𝜒𝜒𝜒 (1) and 𝜒𝜒𝜒𝜒 (3) are the first and third order su
nonlinear
relation
between
displacement
field
and
electric
field
means
the
refractive
index
is
de
(1)
(3)
3
rotated to 135° which coincides with the lossy axis of the
the resonance condition will be fulfilled at different
means
is dependent on electric field,
𝑃𝑃𝑃𝑃 = 𝜀𝜀𝜀𝜀0 𝜒𝜒𝜒𝜒 𝐸𝐸𝐸𝐸field
+ 𝜀𝜀𝜀𝜀0 𝜒𝜒𝜒𝜒
𝐸𝐸𝐸𝐸 , the refractive index
the2electric displacement field via 𝐷𝐷𝐷𝐷 = 𝜀𝜀𝜀𝜀0 𝐸𝐸𝐸𝐸 + 𝑃𝑃𝑃𝑃. It s
3 (3)
(1)
(3)
on electric
field,
according
tothe
𝑛𝑛𝑛𝑛 =electric
𝑛𝑛𝑛𝑛0 + polarization
𝜒𝜒𝜒𝜒 𝐸𝐸𝐸𝐸0 , where
the
electric
fieldfield
amplitude of a lig
A special form of optical anisotropy is the magneto-optical
frequencies for the
opposite
direction
according
to
where
E
the
electric
first polarizer and therefore backward traveling light will be
0 0isis
where
𝜒𝜒𝜒𝜒 and direction.
𝜒𝜒𝜒𝜒 are theFor
first the
and third
order susceptibility
and
𝑃𝑃𝑃𝑃 is 𝐸𝐸𝐸𝐸related
8
not the same
as the to
previously discussed optical po
effect. If a magneto-optically active material is magnetized,
that is off-resonance,
most
of the power
is 𝐷𝐷𝐷𝐷
transmitted
amplitude
of that
aindex
light
wave.
This
field-dependent
refractive
absorbed.
the electric
displacement
field via
=This
𝜀𝜀𝜀𝜀0 𝐸𝐸𝐸𝐸field-dependent
+ 𝑃𝑃𝑃𝑃. It should
berefractive
noted
thecan
electrical
polarization
here
is
be observed
in
a
transient
simulation.
Fig. 8is shows
how
the displacement field, while the latter
the orient
als
the refractive index of left hand and right hand polarized
through the waveguide.
For as
thetheresonant
all
the refractive
indexindex
can
be
observed
infrom
a transient
simulation.
Figurestep
8 function is l
not the same
previouslydirection
discussed
optical
polarization.
Thecan
former
is the material
contribution
to
nonlinear
be
obtained
simulation.
A
single
frequency
nonlinear relation between displacement field and
will differ orthogonally
the magnetized axis.
The material
powered is scattered
out of the
ring
and
essentially
shows
how
the nonlinear
refractive
can be effects
obtained
For on-chip integrated optical systems, the Faraday rotation
otropy is thelight
magneto-optical
effect. If atomagneto-optically
active
the displacement
field,
while
theislatter
is the
ofAfter
the
electric
field
of a light
into
theorientation
waveguide.
enough
timevector
has passed
and wave.
allindex
theAtransient
from the
3 beginnin
on electric field, according to 𝑛𝑛𝑛𝑛 = 𝑛𝑛𝑛𝑛0 + 𝜒𝜒𝜒𝜒 (3) 𝐸𝐸𝐸𝐸02 , w
8
birefringence
for
circular
polarized
light
results
in
the
blocked,
which
makes
the
structure
an
isolator.
Just
as
in
from
simulation.
A
single
frequency
step
function
is
launched
is
usually
not
an
option
for
creating
an
isolator
because
the
nonlinear relation between displacement
field and
electric
fieldsufficiently,
means the refractive
index isofdependent
simulation
have
decayed
the time signals
the electric field at two different
positio
ndex of left hand and right hand polarized light will differ orthogonal to the
This
field-dependent
refractive
index
can be observ
3 (3)
following tensor
the example in on
Figure
6,
the
waveguide
of
the
ring
into
the
waveguide.
After
enough
time
has
passed
and
all
required path lengths are too long and because integrated
2
the
waveguide
are
compared.
The
delay
between
the
two
signals
is
related
to
the
phase
velocity
electric
field,
according
to
𝑛𝑛𝑛𝑛
=
𝑛𝑛𝑛𝑛
+
𝜒𝜒𝜒𝜒
𝐸𝐸𝐸𝐸
,
where
𝐸𝐸𝐸𝐸
is
the
electric
field
amplitude
of
a
light
wave.
0
0
0
gence for circular polarized light results in the following tensor
8
nonlinear
refractive
can be obtained from sim
Δ𝑧𝑧𝑧𝑧
resonator needs to be asymmetric to feature this directionthe
transient
effects
the between
beginning
of index
the simulation
waveguides usually feature significant mode birefringence
optical
wave via
𝑐𝑐𝑐𝑐 =
, wheresimulation.
Δ𝑧𝑧𝑧𝑧 is thefrom
distance
positions and Δ𝑡𝑡𝑡𝑡 is th
This field-dependent refractive index can
be observed
in
a transient
Fig. 8 shows
howthe
thetwo sample
𝜀𝜀𝜀𝜀0 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 0
waveguide.
enough
time has passed a
dependent effective wavelength. Interestingly, this is not
have Δ𝑡𝑡𝑡𝑡
decayed sufficiently,into
thethe
time
signals After
of the
electric
which limits the coupling between TE and TM modes.
𝑐𝑐𝑐𝑐
nonlinear refractive index can be obtained
simulation.
single frequency
stepsimulation
function
launched
𝑐𝑐𝑐𝑐 being the s
delay.from
From
the phaseAvelocity
the refractive
index
can isbe
obtained
assufficiently,
𝑛𝑛𝑛𝑛 = 𝑐𝑐𝑐𝑐0 withthe
have
decayed
𝑫𝑫𝑫𝑫 = �−𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 𝜀𝜀𝜀𝜀0 0 � 𝑬𝑬𝑬𝑬,
achieved by depositing magneto-optical cladding only on
field at two different positions along the waveguide are 0 time signa
Instead the nonreciprocal phase effect can be used. Figure 6
into the waveguide. After enough timelight
has in
passed
and
all
the
transient
effects
from
the
beginning
of
the
the waveguide
are compared.
The delay
vacuum.
0 0 𝜀𝜀𝜀𝜀0
one side of the waveguide. The curvature of the ring itself
compared. The delay between
the two signals
is related
to between t
shows a slab waveguide with a magneto-optical cladding
simulation have decayed sufficiently, the time signals of the electric field at two different positions alongΔ𝑧𝑧𝑧𝑧
opticalwave
wave via
via 𝑐𝑐𝑐𝑐 = , where
provides the asymmetry of the mode, which interacts
the phase velocity of the optical
whereΔ𝑧𝑧𝑧𝑧
Δzisisthe distance be
andiswith
e magnetic field and causes the magneto-optical effect. The medium
in a TM-mode traveling forward and backward. In the
the waveguide are compared. The delay between the two signals is related to the phase velocity of the Δ𝑡𝑡𝑡𝑡
where
g
is
proportional
to
the
magnetic
field
and
causes
the
mostly
with
cladding
at
the
outer
circumference
of
the
the
distance
between
the
two
sample
positions
and
Δt
is
center
of
the
waveguide
the
light
is
linearly
polarized,
while
Δ𝑧𝑧𝑧𝑧
delay. From the phase velocity thethe
refractive index
irection. For linear polarized light such a tensor causes a rotation of
wave
viaof
𝑐𝑐𝑐𝑐 =
where
Δ𝑧𝑧𝑧𝑧
is the interacts
distance
the twoFrom
sample
positions
and Δ𝑡𝑡𝑡𝑡 is the
the time
[2]
waveguide.
The curvature of the ring itself
the optical
asymmetry
the
which
mostly
magneto-optical effect. The medium is in this case
ringprovides
. This effect
can be
seen
only
in,mode,
the
simulation
of the between
time delay.
the
phase velocity
refractive index can
on the edges it is strongly circularly polarized
with different
Δ𝑡𝑡𝑡𝑡
light
in
vacuum.
𝑐𝑐𝑐𝑐
5. Light with
a linear polarization of 45° is launched into a waveguide
with
thethe
speed
of of light in a
delay.
From
the
phase
the
refractive
index can
obtainedas
as 𝑛𝑛𝑛𝑛 = 𝑐𝑐𝑐𝑐0 with
magnetized in the z-direction. For linear polarized light, such
which
the
waveguide
curvature
bebe
obtained
with𝑐𝑐𝑐𝑐0c0being
being
speed
senses of rotation above and below the with
waveguide.
Further,
cladding
at the outer circumference3Dof structure
the ring [1].
Thistakes
effect
can
bevelocity
seen only
in theinto
simulation
of
8 and the amagneto-optical
effect
lets
the
polarization
rotate
by
45°.
tensor causes a rotation of polarization as shown in Figure
account.
vacuum.
the direction of rotation depends on whether the wave is
light
in
vacuum.
the 3D structure which takes the waveguide curvature into account.
Light
with isolation
a linear polarization
of 45°
is launched
into field
a
traveling
widely used5.for
optical
because the
biasing
magnetic
breaks forward or backward. If the cladding is magnetized
Whenand
investigating
it’s investigating
very important
investigate the
transmission
between
with ε0=3.8025This
and means
g=0.0308
andifthe
When
non-reciprocal
systems,
it’s very
Figure 9 shows the change in refractive index obtained by
orthogonally
to the image plane, the forward
backward non-reciprocal systems
makes the waveguide
effect non-reciprocal.
that
themagnetolight that left
the
each pair
of modes
systemstocan
appear the
non-reciprocal
which are
notpair
[2]. the method described above for a Kerr waveguide with
optical effect lets the polarization rotate by 45°. Magnetoimportant
investigate
transmission between
each
waves will see different effective refractive
indices
in the separately. Otherwise
flected backoptical
into the
waveguide then it is not reverted to the originalcladding
input and will therefore have different effective
effects are widely used for optical isolation because
of modes separately. Otherwise systems can appear nonχ(3)=10-16 (m/V)2 and n0 = 3.5. The distance Δz between the
er to 135° (Fig.
the 5(b)).
biasing magnetic field breaks time reversal symmetry
reciprocal when they are not[3].
two sample points is 5µm. For the growing electric field
wavelengths. It is important for this effect that the structure
[1] Dirk Jalas, Alexander Petrov, Michael Krause, Jan Hampe, and Manfred Eich, "Resonance splitting in
and makes the effect non-reciprocal. This means that if the
strength, the refractive index increases quadratically. The
is vertically asymmetric because the direction of rotation is
gyrotropic
ring
resonators,"
Opt.
Lett.
35,
3438-3440
(2010)
light that left the waveguide 90° polarized is reflected back
simulated change agrees well with the theoretically expected
opposite on the bottom and the top of the waveguide. If the
Jalas would
et al., "What is—and what is not—an optical isolator," Nat. Photonics 7.8,579-582 (2013)
into the waveguide then it is not reverted to the original
behavior.
structure were symmetric, both top [2]
andDirk
bottom
input state of 45° but rotated further to 135° (Figure 5(b)).
PART III
contribute equally with opposite sign and the effect would
THIRD ORDER NON-LINEAR MATERIALS
The dependence of the refractive index on the optical
average out to zero.
Non-reciprocity is always needed if one wants to block light
intensity can be used to build a bistable element. Figure 10
Part
III to build
that is travelling back into the device. With the addition of
For some media, the permittivity depends on the electric
shows a ring resonator that is coupled to two waveguides.
The direction-dependent wavelength can
be utilized
two polarizers as in Figure 2, the waveguide in Figure 3 can
a compact isolator[2]. If a ring resonator
is coupled
to a
Third
order non-linear
materials field strength of the light, making their optical response When excited from port 1, most of the power will be
serve as an isolator that lets light traveling in the forward
nonlinear.
An example
such a material
is a Kerr
medium
transmitted to port 2 if the excitation frequency is offwaveguide and the coupling rate between
waveguide
and
For some
media,
the permittivity depends
on the electric
fieldofstrength
of the light,
making
their optical
direction pass but blocks light traveling in the backward
for which the electric polarization depends on the third
resonance. For a resonant excitation the power will be
resonator is equal to the loss rate of the resonator, all the
response nonlinear. An example of suchpower
a material
is a Kerr medium for which the electric polarization
direction. For that, a polarizer needs to be added at the
of the electric field
guided to port 3 (see the spectrum in Figure 10). Because of
power in the waveguide will be radiated by the ring resonator
depends onWethe
third power of the electric field
beginning of the waveguide such that the pass axis is
the dependence of the refractive index on the optical
and there will be no power left in the waveguide.
have
(1)
(3) 3
oriented at 45° and another polarizer added at the end of the
intensity, the light in the resonator can change the resonance
designed such a critically coupled ring resonator and added
𝑃𝑃𝑃𝑃 = 𝜀𝜀𝜀𝜀0 𝜒𝜒𝜒𝜒 𝐸𝐸𝐸𝐸 + 𝜀𝜀𝜀𝜀0 𝜒𝜒𝜒𝜒 𝐸𝐸𝐸𝐸 ,
waveguide oriented at 90° with its pass axis. If light is now
frequency. A sufficiently high intensity can increase the
magneto-optical cladding. Here, TE-mode is considered
and
where 𝜒𝜒𝜒𝜒 (1) and 𝜒𝜒𝜒𝜒 (3) are the first and third order susceptibility and the electric polarization 𝑃𝑃𝑃𝑃 is related
to
launched in the forward direction at 45° polarization, it will
where χ(1) and χ(3) are the first and third order susceptibility
refractive index and, by that, decrease the resonance
cladding is magnetized orthogonally to the plane of the ring.
the electric displacement field via 𝐷𝐷𝐷𝐷 = 𝜀𝜀𝜀𝜀and
𝑃𝑃𝑃𝑃. electric
It shouldpolarization
be noted that
electrical
here is To probe this non-linear behavior, the resonator
0 𝐸𝐸𝐸𝐸 +the
P is the
related
to thepolarization
electric
frequency.
not the same as the previously discusseddisplacement
optical polarization.
The
former
is
the
material
contribution
to from port one with a slowly increasing amplitude
field via D=ε0E + P. It should be noted that the
is excited
polarization
is not the
same
as theofpreviously
the displacement field, while the latter iselectrical
the orientation
of here
the electric
field
vector
a light wave.and
A a frequency of 191.2 THz which is slightly below
discussed optical polarization. The former is the material
resonance. The result is shown in Figure 11.
nonlinear relation between displacement field and electric field means the refractive index is dependent
contribution to the displacement field, while the latter is the
(a)
3 (3) 2
(b)
on electric field, according to 𝑛𝑛𝑛𝑛 = 𝑛𝑛𝑛𝑛0 + 𝜒𝜒𝜒𝜒
8
𝐸𝐸𝐸𝐸0 , where 𝐸𝐸𝐸𝐸0 is the electric field amplitude of a light wave.
This field-dependent refractive index can be observed in a transient simulation. Fig. 8 shows how the
nonlinear refractive index can be obtained from simulation. A single frequency step function is launched
135° After enough time has passed and all the transient effects from the beginning of the
into the waveguide.
simulation have decayed sufficiently, the time signals of the electric field at two different positions along
(a)
λ forward
the waveguide are compared. The delay between
the two signals is related to the phase velocity of the
45°
optical wave via 𝑐𝑐𝑐𝑐 =
Δ𝑧𝑧𝑧𝑧
,
Δ𝑡𝑡𝑡𝑡
λ backward
where Δ𝑧𝑧𝑧𝑧 is the distance between
the two sample positions and Δ𝑡𝑡𝑡𝑡(b)is the time
delay. From the phase velocity the refractive index can be obtained as 𝑛𝑛𝑛𝑛 =
light in vacuum.
90°
90°
Figure 5: Shows a magneto-optically active waveguide. In (a) the light is launched into the waveguide with 45° polarization, Due to nonreciprocity the polarization for backward direction is not reverted back to the initial polarization state but to 135°.
𝑐𝑐𝑐𝑐0
𝑐𝑐𝑐𝑐
with 𝑐𝑐𝑐𝑐0 being the speed of
Figure 6: (a) shows the H-field of a TM-mode in a slab waveguide. The
waveguide core is silicon, the substrate is silica and the cladding is
magneto-optically active with the magnetization out of the image
plane. (b) Reversed propagation direction.
Figure 11(c). For decreasing power, the transmission to port
3 remains mostly unchanged. However, at some point the
power is not sufficient anymore to maintain the resonance
frequency below the input frequency and the system
abruptly drops to its initial state. The fact that for one input
power there are two stable states of operation is referred to
as optical bistability. Significant oscillations of the output
power are still observed, and these are dependent on the Q
factor of the ring resonator and the rise time of the input
signal. These oscillations are an important design property
that can be optimized by simulations.
If the input power is now decreased again, the system
behaves differently. As the signal now has a higher frequency
than the resonance, the reduction in input power means that
the resonance moves closer to the signal. However, this
means a larger field enhancement factor in the ring, so the
intensity in the ring remains rather unchanged across the
broad range of input power levels. This can be nicely seen in
0.0002
6
4
E xcitation [a.u.]
The ring resonator in Figure 10 is the optical analogue to a
transistor and allows all the operations that are possible with
a transistor such as logic gates. This means that such a
device can be used to do computations with light.
8
Analytical solution
CST time domain
∆n
0.0001
1x10
6
2x10
6
3x10
6
4x10
6
3
wave
∆ z
-20
probe 1
-25
-4
0
20
40
60
80
1 00 1 20 1 40 1 60
1 .0
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1 .0 1 .2 1 .4 1 .6 1 .8
2
1
probe 2
Input power [a.u.]
(a)
193.71
193.74
(b)
signal probe 1
(c)
2
1
2
Figure 7: shows a ring resonator with magneto-optically active
cladding coupled to a waveguide. The cladding is magnetized normal
to the image plane. The ring parameters can be found in [3].
(a) The transmission spectrum is shown for an excitation in forward
and backward direction. (b) and (c) show the H-field for excitation in
forward and backward direction at a frequency of 193.72 THz.
Amplitude [V/m]
(b)
1 .0
1.0x10 6
Frequency [THz]
0
5.0x10 5
-5
time delay
0.0
-5.0x10 5
-1.0x10 6
0.080
signal probe 2
0.082
0.084
S21
input
-10
-15
-20
S31
0.086
Time [ps]
Figure 8: shows a process to obtain the refractive index from a
transient simulation. (a) Simulation volume and field probes. (b)
Electric field time signal of two probes. From the delay between the
two signals the refractive index can be calculated.
-25
190.5 191.0 191.5 192.0 192.5 193.0
Frequency [THz]
Figure 10: shows an add-drop ring resonator and its transmission
spectrum. The structure is excited with a frequency of 191.2 THz.
Output power [a.u.]
193.68
Transmission [dB]
Transmission [dB]
plane
-15
-30
1
4
S21
-10
-2
Time [ps]
Figure 9: shows the change in refractive index for increasing field
strength in a nonlinear Kerr waveguide.
(a)
Input
S12
0
-8
E-Field [V/m]
0
-5
2
-6
0.0000
0
Output power [a.u.]
For low amplitudes the system behaves linearly and the
output powers in port 2 and 3 are proportional to the input
power. With increasing input intensity the resonance
frequency is shifted towards the excitation frequency and
power that is transmitted to port 3 is increasing whereas the
power in port 2 is decreasing. As the resonance frequency
moves closer to the excitation frequency, the electric field is
enhanced in the resonator and the system becomes more
and more non-linear. Accordingly, the resonance shifts quite
abruptly into the excitation frequency which lets the
transmission to port 2 drop to zero and increases the
transmission to port 3. The field enhancement is strongest
when the resonance is aligned with the excitation, which is
why this case is achieved only for a relatively narrow input
power range. For slightly larger powers, the resonance is
shifted to frequencies smaller than the frequency of the
input signal and the transmission to port 2 rises again.
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1 .0 1 .2 1 .4 1 .6 1 .8
Input power [a.u.]
Figure 11: (a) shows the excitation signal over time. (b) shows the
relation between input power and power transmitted to port 2; (c)
shows the relation between input power and power transmitted to
port 3.
REFERENCES
[1] Saman Jahani and Zubin Jacob, “Transparent subdiffraction
optics: nanoscale light confinement without metal,” Optica
1, 96–100 (2014)
[2] Dirk Jalas, Alexander Petrov, Michael Krause, Jan Hampe,
and Manfred Eich, “Resonance splitting in gyrotropic ring
resonators,” Opt. Lett. 35, 3438–3440 (2010)
©2018 Dassault Systèmes. All rights reserved. 3DEXPERIENCE®, the Compass icon, the 3DS logo, CATIA, SOLIDWORKS, ENOVIA, DELMIA, SIMULIA, GEOVIA, EXALEAD, 3D VIA, BIOVIA, NETVIBES, IFWE and 3DEXCITE are commercial trademarks or registered trademarks of
Dassault Systèmes, a French “société européenne” (Versailles Commercial Register # B 322 306 440), or its subsidiaries in the United States and/or other countries. All other trademarks are owned by their respective owners. Use of any Dassault Systèmes or its subsidiaries
trademarks is subject to their express written approval.
[3] Dirk Jalas et al., “What is – and what is not – an optical
isolator,” Nat. Photonics 7.8,579-582 (2013)
AUTHOR
Dirk Jalas, Hamburg University of Technology
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