IEEE Solid-States Circuits Magazine - Summer 2021 - 27

Finally, for the general case of
the interferometer with an arbitrary
phase shift,
z , the output is a sinusoidal
function with respect to the
phase,
z . The output switches from
high to low as the phase varies from
0 to 018 c and then back to high for
a phase of
360 .c The excess phase
shift can be introduced using either
a physical path length difference
(a mismatch in length) or an optical
path length difference (a mismatch
in refractive index) between the two
branches. The MZI acts as a unit element
to make switches, modulators,
and other devices.
Active Circuits
Phase Shifters
The refractive index of Si, n, increases
with temperature because of the
temperature dependence of the energy
bandgap and thermal expansion.
This strong thermo-optical
effect can be utilized to implement
a thermal phase shifter. A resistor
can be implanted near the Si waveguide,
and joules heating (VinIin) can
be utilized to change the local temperature
and create a phase shift z
(Figure 7). Usually, a metal heater is
separated from the Si waveguide using
an SiO2 layer of ~1μm to avoid
(the optical mode from incurring
additional) scattering and absorption
loss while maintaining enough
proximity to couple heat into the
waveguide. The tuning efficiency is
not optimal, as the generated heat
has to diffuse from the point-like
source of the heater down to the
Si waveguide. For a 2r phase shift,
20-100 mW is usually consumed,
which can easily overwhelm the
overall power budget. To increase
the thermal isolation and improve
the tuning efficiency 10-fold, air
trenches can be etched around the
waveguide structure [25].
The refractive index of Si also
decreases (Tn) with increasing carrier
concentrations (TT due to
NP,
)
the plasma-dispersion effect. To build
a plasma-dispersion phase shifter,
the Si waveguide is doped to make a
FIGURE 8: A traveling-wave MZM along with a driver. MZM: Mach-Zehnder modulator;
DRV: driver.
IEEE SOLID-STATE CIRCUITS MAGAZINE
SUMMER 2021
27
p-n junction, and then the free-carrier
concentrations are electrically
manipulated (Figure 7). A reverse bias
voltage leads to a depletion mode of
operation. This effect, unfortunately,
is also accompanied by a change in
absorption. Higher values of NT and
T P result in a higher nT
but also a
higher loss. Both effects are mostly
linear [19].
In a thermo-optic phase shifter,
the phase shift is proportional to
the power dissipation, or the square
of the applied voltage. The thermooptic
effect is low speed and is used
for switches and the tuning and stabilization
of SiP circuits. Although
it consumes considerable electrical
power,
it does not introduce any
optical loss. In a plasma-dispersion
phase shifter, the electrical power
consumption can be made negligible,
but there is an associated optical
loss. The effect is fast and therefore
used for high-speed modulation/switching.
Mach-Zehnder
Modulator
Figure 8 shows a Mach-Zehnder mo -
dulator (MZM), made up of an MZI
with a plasma-dispersion phase
shifter in one arm. The modulator
may also have thermal phase shifters
in one or both arms to account
for mismatch and variations. Let us
assume that the MZI is balanced; i.e.,
the two arms have equal length, L.
Let us change the refractive index in
the upper branch,
Tn , by applying a
voltage Vin that goes from zero to V ,r
is the voltage needed to
where Vr
introduce a r phase shift. Assume
that the change in the refractive
index is linearly proportional to the
applied voltage. The output intensity,
,Io
is given by
I II nL
I
o
=+ =+
=+
2
i
2
ii 2rT
z@
6
;
1
1
coscos
cos
2
r in E,
V
V
r
where Ii is the input intensity and nT
is the relative change in the refractive
index.
The transfer function of the MZM,
shown in Figure 9, is sinusoidal with
excess phase shift
V ,r
the output
optical power goes from maximum to
z , which is proportional
to the modulation voltage Vin. As
Vin is swept from 0 to
;1
m
E
(1)
Iin
Vin
+
-
Vin
Pin = VinIin
Ei
(a)
Eieiφ(Pin)
+
-
Ei
kEieiφ(Vin)
(b)
FIGURE 7: (a) A thermo-optic phase shifter and (b) a plasma-dispersion phase shifter.
DRV
ZS
ZT
Z0
L
Ii
Io
IEEE Solid-States Circuits Magazine - Summer 2021

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