of 9
1
ol.:ȋͬͭͮͯͰͱͲͳʹ͵Ȍ
Scientific Reports
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|
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Consequences of quantum
noise control for the relaxation
resonance frequency and phase
noise in heterogeneous Silicon/III–V
lasers
Dongwan Kim
1,4
*
, Mark Harfouche
2,4
, Huolei Wang
1,4
, Christos
T. Santis
1
, Yaakov Vilenchik
1
,
Naresh Satyan
3
, George Rakuljic
3
& Amnon
Yariv
1,2
We have recently introduced a new semiconductor laser design which is based on an extreme, 99%,
reduction of the laser mode absorption losses. In previous reports, we showed that this was achieved
by a laser mode design which confines the great majority of the modal energy (>
99%) in a low-loss
Silicon guiding layer rather than in highly-doped, thus lossy, III–V p
+
and n
+
layers, which is the case
with traditional III–V lasers. The resulting reduced electron-field interaction was shown to lead to a
commensurate reduction of the spontaneous emission rate by the excited conduction band electrons
into the laser mode and thus to a reduction of the frequency noise spectral density of the laser field
often characterized by the Schawlow–Townes linewidth. In this paper, we demonstrate theoretically
and present experimental evidence of yet another major beneficial consequence of the new laser
design: a near total elimination of the contribution of amplitude-phase coupling (the Henry
α
parameter) to the frequency noise at “high” frequencies. This is due to an order of magnitude lowering
of the relaxation resonance frequency of the laser. Here, we show that the practical elimination of this
coupling enables yet another order of magnitude reduction of the frequency noise at high frequencies,
resulting in a quantum-limited frequency noise spectral density of 130
Hz
2
/Hz (linewidth of 0.4
kHz)
for frequencies beyond the relaxation resonance frequency 680
MHz. This development is of key
importance in the development of semiconductor lasers with higher coherence, particularly in the
context of integrated photonics with a small laser footprint without requiring any sort of external
cavity.
The semiconductor laser (SCL) has become and, very likely, will continue to be, in the foreseeable future, the
linchpin of
optoelectronics
1
5
. A few major obstacles, however, remain before the promise of CMOS-like Photonic
Integrated Circuits (PICs) can be realized. Chief among these problems is: low-coherence, the dependence on
external isolators to reduce optical feedback, and the coherence-reducing amplitude-phase coupling.
Recently
6
,
7
,
we have shown how the “removal” by modal design of optical energy from the lossy III–V material to low-loss
material, Silicon in our example, reduces the frequency noise due to spontaneous emission and improves the field
coherence by some three orders of magnitude, with the improvement being limited by the residual losses of the
Silicon (or, more precisely, by the Q-factor of the laser mode). An unexpected bonus of the high-Q laser was its
improved, 20–25 dB in our laboratory-fabricated lasers, insensitivity to optical
feedback
8
,
9
. This improvement is
again limited by the achievable intrinsic Q of the laser mode.
In this paper, we provide theoretical arguments and experimental evidence that the reduced interaction of
the excited electrons in the SCL with the quantum-mandated zero-point field of the laser mode in our laser
design leads, additionally, to a practical elimination of the amplitude-phase coupling at frequencies above that
the relaxation
resonance
10
,
11
. The order, or orders, of magnitude improvements in these three key metrics of
OPEN
1
Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, CA
91125,
USA.
2
Department of Electrical Engineering, California Institute of Technology, Pasadena, CA
91125, USA.
3
Telaris
Inc., Santa Monica, CA
90403, USA.
4
These authors contributed equally: Dongwan Kim, Mark Harfouche and
Huolei Wang.
*
email: dongwan.kim@caltech.edu
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| (2022) 12:312 |
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the SCL are archived while maintaining its small size and the CMOS-fabrication compatibility. Taken together,
they pave the way to a new generation of SCLs with intrinsic ultra-high-Q values as the main enablers of high-
coherence photonic integrated
circuits
12
.
In this paper, we take an ab-initio look at the relationships between some of the key attributes of the semi
-
conductor laser; specifically, the current modulation response, phase-amplitude coupling, relaxation resonance,
and the frequency noise (or phase noise). In the experimental section, we describe how the theoretical results are
applied to the design of semiconductor lasers and present the measured relevant characteristics of these lasers.
We use the heterogeneous Silicon/III–V laser as a proof-of-concept platform.
Semiconductor laser theory
Relaxation resonance frequency and the Schawlow–Townes linewidth.
A convenient starting
point in the analysis of the phase noise of a semiconductor laser is to consider the problem of the current modu-
lation response. We start with the set of coupled equations for the number of photons
n
l
in the lasing mode of the
laser resonator and for the number of inverted electrons
n
e
in the active regions (Equation 15.5-1
13
),
where
η
i
is an injection efficiency of the electrons into the active region,
I
is the injected current to the laser,
τ
e
is the electron recombination time,
n
tr
is the number of electrons at transparency, and
τ
p
is the photon cavity
lifetime for the lasing mode.
Here, the photon cavity lifetime accounts for the intrinsic losses in the cavity
τ
i
due to scattering and absorp-
tion, and losses due to useful output coupling
τ
ext
.
W
(
l
)
sp
is the spontaneous emission rate into the lasing mode (
l
),
a material-dependent constant. We choose the total number of photons and the total number of excited electrons
as our main variables rather than their densities since according to the quantum mechanics that describe the
interaction between photons and electrons, a single electronic transition results in the emission or the absorp
-
tion of one photon.
A key result of these equations is the intensity modulation (IM) response, defined as the ratio of the ampli
-
tude of the total emitted power modulation

P
out
at a frequency
f
to the amplitude of the driving current

I
at
f
, and is given by:
This transfer function
H
(
f
) is that of a second-order low-pass filter response whose magnitude is constant at low
frequencies. At higher frequencies, above that of the relaxation resonance frequency
f
R
,
H
(
f
) drops by 40 dB per
decade. It is consequently used as a measure of the upper limit to the response of the laser intensity to current
modulation.
H
(
f
), as will be shown below, also plays a key role in the laser temporal coherence. It can be seen from
Eq. (
4
) that
f
R
is proportional to the square root of the spontaneous emission rate determined by the strength of
the interaction between the electrons and the lasing mode. Following an ab-initio analysis of the quantum inter
-
action between the excited electron and the lasing mode, we find that the spontaneous and stimulated transition
rates of an electron from an excited state in the conduction band to an unoccupied state in the valence band, due
to the interaction with the lasing mode (
l
) in Eqs. (
1
) and (
2
), are given
by
6
,
7
,
14
:
where
μ
is the dipole transition matrix element,
g
a
l
)
is the value of the normalized lineshape function of the
transition at the lasing frequency
ν
l
(both known quantities for our purposes).
ε
(
r
a
)
is the permittivity of the bulk
material at the location of the emitter.
E
l
(
r
a
)
2
is the normalized intensity of the laser mode at the location of
the emitter
r
a
(i.e., electron). Integrated over the volume of the quantum wells,
E
l
(
r
a
)
2
yields the confinement
factor (
Ŵ
) of the electric field in the active region (i.e.,
E
l
(
r
a
)
2
Ŵ
QW
)
15
. Combined with Eq. (
4
), we find that
(1)
dn
e
dt
=
η
i
I
q
n
e
τ
e
W
(
l
)
sp
(
n
e
n
tr
)
n
l
,
(2)
dn
l
dt
=
[
W
(
l
)
sp
(
n
e
n
tr
)
1
τ
p
]
n
l
,
(3)
H
(
f
)
=
P
out
(
f
)
I
(
f
)
=
η
d
h
ν/
q
1
(
f
/
f
R
)
2
+
i
(
f
/
f
R
)(
2
π
f
R
τ
p
+
1
/
(
2
π
f
R
τ
e
)
)
,
(4)
f
R
=
1
2
π
η
i
W
(
l
)
sp
(
I
I
th
)
q
.
(5)
W
(
l
)
sp
=
2
π
2
μ
2
ν
l
g
a
(
ν
l
)
h
ε
(
r
a
)
E
l
(
r
a
)
2
,
(6)
W
(
l
)
st
=
n
l
W
(
l
)
sp
,
(7)
f
R
=
1
2
π
η
i
2
π
2
μ
2
ν
l
g
a
(
ν
l
)
qh
ε
(
r
a
)
E
l
(
r
a
)
2
(
I
I
th
)
=⇒
f
R
E
l
(
r
a
)
I
I
th
.
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This result highlights one key parameter in the control of the relaxation resonance frequency
f
R
:
the magnitude
of the electric field at the location of the emitter
. Similarly, we obtain the damping coefficient
γ
of the second-
order response,
The results above are obtained using results from Eq. (
7
) and combining them with Equation 23
of
7
which
show that for the geometry of interest,
τ
p
E
l
(
r
a
)
2
. As a consequence, as we decrease
E
l
(
r
a
)
, we find that
the damping coefficient of the response increases giving these lasers an over-damped response to spontaneous
emission generated in the gain region ensuring that the intensity noise is monotonically decreasing without the
presence of any peak near the relaxation resonance frequency, which is observed in the case of an under-damped
response.
To understand how the relaxation resonance frequency affects the phase noise, we now look into the phase
noise caused by the spontaneous emission events. At frequencies high enough where thermal noise can be
ignored in semiconductor lasers, the spontaneous emissions become the dominant source of noise that perturbs
the lasing field circulating in the
cavity
10
. Furthermore, while thermal and electrical noise sources can be miti
-
gated by improving the thermal and electronic design of the driving circuit, the origin of spontaneous emission
is quantum mechanical and cannot be avoided thus determining the ultimate noise floor of the laser. These
quantum perturbations can be analyzed as two distinct noise sources, one coupled to the amplitude quadrature,
and the other coupled to the phase quadrature, as shown in Fig.
1
a. The first component is perpendicular to
the phasor of the electric field and couples directly to the phase of the optical field (blue solid line in Fig.
1
a).
The second component, parallel to the phasor of the field, alters the intensity of the lasing field (green dotted
line in Fig.
1
a). To compensate for the change in intensity, the gain-providing carriers fluctuate in response to
these amplitude changes in an attempt to restore the steady-state output intensity of the laser (known as the
“relaxation resonance”). In semiconductor quantum-well lasers, changes in the carrier concentration induce a
corresponding proportional change in the refractive index leading to additional frequency noise through Kram-
ers–Kroning
relations
13
due the asymmetric shape of the gain spectrum in this
medium
16
. Thus, fluctuations in
the total number of carriers manifest themselves as additional phase fluctuations through a process known as
amplitude-phase coupling depicted in Fig.
1
b.
To find a mathematical relationship between the quantum phase fluctuations of a laser and carrier dynam
-
ics, one can introduce Langevin noise terms in Eqs. (
1
) and (
2
), as is done
in
11
,
17
. Doing so reveals that the
amplitude-phase coupling has a frequency dependence identical to that of the intensity modulation response
H
(
f
), a second-order low pass filter response with a resonance frequency of
f
R
. Therefore, in the quantum-limit,
the frequency noise PSD (single-sided) of a semiconductor laser can be expressed as the sum of two
terms
14
,
where
n
2th
is the total number of electrons in the conduction band at threshold. The first term
n
2th
W
(
l
)
sp
4
π
2
n
l
is often
referred to as the Schawlow–Townes linewidth, representing the intrinsic, quantum-limited, ultimate noise floor
including only the direct spontaneous emission phase noise. The additional phase noise due to the
(8)
γ
=
1
2
(
2
π
f
R
τ
p
+
1
2
π
f
R
τ
e
)
1
E
l
(
r
a
)
.
(9)
S
�ν
(
f
)
=
n
2th
W
(
l
)
sp
4
π
2
n
l
(
1
+
α
2
H
(
f
)
2
)
,
Figure 1.
(
a
) The phasor model for the laser field phase, showing the electric field before and after a single
spontaneous emission event (orange arrow). The projection of the spontaneous emission event onto the phase
and intensity quadrature is shown in the blue solid and green dotted line respectively. (
b
) Phase noise due to
spontaneous emission including both the direct spontaneous emission phase noise and the additional phase
noise via the amplitude-phase coupling. (
c
) A cartoon illustrating the changes to the frequency noise power
spectral density of the high-coherence Silicon/III–V laser studied as part of this work and the conventional
III–V laser.
In
6
and
7
we studied how the magnitude of the frequency noise power spectral density changes
below the relaxation resonance frequency. In this work, we study how the location of the relaxation resonance
frequency (Eq. 7), and the damping of this resonance (Eq. 8) are both reduced when we confine the majority of
the optical mode within the low-loss silicon medium as opposed to the lossy but gain-providing III–V.
4
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amplitude-phase coupling appears with a coefficient of
α
2
(a linewidth enhancement factor or the Henry
α
parameter), which ranges between 2 and 6 for the case of broadly-used quantum-well (QW) semiconductor
lasers
15
,
18
,
19
, and it is modulated by
H
(
f
)
2
.
As such, for lasers with a relaxation resonance frequency
f
R
10GHz
, the amplitude-phase coupling con
-
stitutes the majority of the spontaneous-emission-induced frequency noise for semiconductor lasers in the
frequency bands of interest for high-speed sensing and communication with sampling rates in the few GHz, as
shown in Fig.
1
c. However, if one can design lasers with a relaxation resonance frequency positioned at the range
of hundreds of MHz, then the total frequency noise of the lasers will drop by a factor of
(
1
+
α
2
)
at frequencies
of GHz leaving only the white noise floor generated by the direct spontaneous emission events.
In the rest of the paper, we will refer to the “enhanced linewidth” as the value that includes the carrier-
induced frequency noise (
�ν
enhanced
=
π
S
�ν
(
f
<
f
R
)
=
n
2th
W
(
l
)
sp
4
π
n
l
(
1
+
α
2
)
=
�ν
ST
(
1
+
α
2
)
) and to the “Schaw-
low–Townes linewidth” as the value proportional to the direct contribution of spontaneous emission to phase
noise (
�ν
ST
=
π
S
�ν
(
f
>
f
R
)
=
n
2th
W
(
l
)
sp
4
π
n
l
)
20
.
Frequency noise above the relaxation resonance frequency of the high-coherence Silicon/III–V
lasers.
In our recently reported quantum noise controlled Silicon/III–V
lasers
6
,
7
, we harnessed the Purcell
effect to decrease the spontaneous emission rate into the lasing mode. This was achieved by utilizing a
SiO
2
layer
between the Silicon and III–V ranging from 50 nm to 150 nm, which we call the quantum noise control layer
(QNCL), to engineer the optical mode’s spatial distribution with respect to the emitter (
E
l
(
r
a
)
). The geometry
of the cross-section and the transverse optical mode profile for the case of the 50 nm and 90 nm
SiO
2
QNCL
lasers are shown in Fig.
2
b. Using this strategy, we have shown that Silicon/III–V lasers employing a monolithi-
cally integrated high-Q resonator can have linewidths as small as a few kHz without sacrificing the parameters
such as the threshold current (due to the constant
n
2th
) and the optical output
power
6
,
7
. The strategy is effective
as long as the laser mode losses are dominated by absorption in the III–V or, equivalently, as long as the overall
Q of the laser mode is dominated by III–V losses, which we estimate occurs when the thickness of the
SiO
2
layer
is approximately 150 nm. Further reduction in the electric field strength at the quantum-well layer results in
the increase in the threshold carrier density to maintain the gain that matches the no-longer-decreasing modal
loss, increasing the threshold current. In the batch of lasers fabricated as part of this study, the run on which
the 150 nm lasers were fabricated failed. We are therefore focusing on discussion on the successfully fabricated
50 nm and 90 nm lasers.
To investigate the effect of the reduced electric field strength at the location of the quantum-wells on the
relaxation resonance of the lasers, we revisit Eq. (
7
). Due to the dependence of the relaxation resonance frequency
on the square root of the intensity of the electric field at the quantum-well emitter, every two orders of magnitude
reduction in the field strength at the quantum-well will result in the reduction of one order of magnitude in the
relaxation resonance frequency. In our demonstration, we’ve reduced
Ŵ
III–V
=
100%
for conventional all III–V
laser to
Ŵ
III–V
=
1%
(
Ŵ
QW
=
0.2%
) for the case of the 150 nm QNCL laser resulting nearly in a full order of
magnitude reduction in the relaxation resonance frequency of the laser from a few GHz to hundreds of
MHz
21
,
22
.
Figure
1
c illustrates that at frequencies of a few GHz, conventional III–V lasers still exhibit “enhanced fre
-
quency noise” including both the direct spontaneous emission phase noise and the amplitude-phase-coupling-
induced frequency noise. In contrast, at a few GHz, our high-coherence Silicon/III–V lasers eliminate the
additional frequency noise via amplitude-phase coupling by positioning a relaxation resonance frequency at
a frequency of a few hundreds of MHz, yielding lasers that possess the intrinsic, quantum-limited Schaw-
low–Townes noise floor.
Results and discussion
Laser design and fabrication.
The high-coherence Silicon/III–V laser is based on a Silicon waveguide
to which a III–V epitaxially grown stack is bonded separated by a thin
SiO
2
layer (Fig.
2
a). The oxide layer is
obtained by thinning a thermally grown
SiO
2
layer, originally 400 nm thick, on the top of the silicon-on-insula-
tor (SOI) waveguide using a buffer HF wet etch. The high-Q Silicon resonator is defined by etching a 60 nm rib
on a 500 nm thick Silicon device layer creating a
2.5
μ
m
wide waveguide. In the same fabrication steps, a 1-D
grating is etched in the waveguide to define the optical cavity by etching holes 60 nm wide in the direction of
propagation and of varying width.
Figure 2.
(
a
) Cartoon cross-sectional view of the laser device structure. (
b
) Transverse optical mode profile in
the lasers with the 50 nm and 90 nm
SiO
2
quantum noise control layer together with the confinement factor in
each layer.
c
SEM image of the cross-section of the fabricated laser.
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The width of the gratings in the defection section varies between 200 nm and 300 nm in the transverse direc-
tion along the length of the resonator such that the
240
μ
m
-long defect has a photonic well that contains a single
high-Q
mode
6
. The intrinsic Q-factor of the fabricated Silicon resonator is measured to be as high as
10
6
. End
reflections are provided by two mirror regions, on either side of the defect section, of
300
μ
m
and
400
μ
m
long
for the 50 nm and 90 nm QNCL lasers, respectively. The mirror length of the 90 nm QNCL laser is chosen to be
larger than that of the 50 nm QNCL laser to increase the loaded Q-factor of the 90 nm QNCL proportionally
to the expected increase in the intrinsic Q-factor. The period of the grating determines the lasing wavelength,
and in our case is chosen to be 240.0 nm and 237.5 nm for the 50 nm and 90 nm QNCL lasers, respectively. The
unpatterned InP is directly bonded on the pattered SOI resonators and incorporates five InGaAsP quantum-wells.
Subsequently, the mesa structure and the metal contacts are patterned on the III–V
wafer
23
. Figure
2
c shows the
SEM image of the fabricated laser.
The mode has an estimated intrinsic Q-factor equal to the weighted harmonic mean of the Q-factors of the
Silicon and III–V waveguides. Through finite element simulations, we estimate the confinement factor in the
III–V of the 50 nm and 90 nm QNCL lasers to be 10% and 3% (Fig.
2
b), respectively, yielding a Q-factor of
approximately
1
×
10
5
and
2.5
×
10
5
.
Measurements.
The light-pump current (LI) and the current versus voltage (IV) characteristics of the fab-
ricated 50 nm and 90 nm QNCL lasers are shown in Fig.
3
a. The 50 nm and 90 nm laser obtain continuous-wave
laser operation with threshold currents of 50 mA and 80 mA, respectively, and single-facet output powers more
than 2 mW at
20
C
. As theoretically predicted
in
7
, the output power of both lasers is rather similar. This is due to
the proportional decrease in both the gain and the losses within the laser cavity. We attribute the small variation
in the laser output power to fabrication imperfections, often seen in small volume laser manufacturing processes.
Figure
3
b presents the optical spectrum of the 50 nm and 90 nm lasers obtaining a single-mode operation with
side-mode suppression ratios greater than 45 dB at the lasing wavelength of 1577 nm and 1556 nm, respectively.
The intensity modulation (IM) response
H
(
f
) of the lasers is measured using a network analyzer (HP 8722C)
and a high-speed photodetector (Optilab BPR-20-M). The IM index
m
, defined as
P
/
P
0
per 1 mA where

P
is the change in the optical power and
P
0
is the average received optical power, is shown in Fig.
4
a for the 50 nm
QNCL laser at bias currents of 80, 100, and 130 mA. As expected, the IM response exhibits a second-order low-
pass filter behavior with flat response for frequencies up to the relaxation resonance frequencies, and 40 dB/
decade drop-off thereafter. Fitting the measured response to the second-order low-pass filter response in Eq. (
3
)
yields a relaxation resonance frequencies
f
R
of 500, 730, and 900 MHz at bias currents of 80, 100, and 130 mA,
respectively. The relaxation resonance frequencies of the 90 nm QNCL laser, extracted from the data presented
in Fig.
4
b, are measured to be 360, 540, and 680 MHz at bias currents of 100, 140, and 190 mA, respectively.
These low relaxation resonance frequencies at hundreds of MHz clearly stand in contrast to those of conventional
III–V lasers of a few GHz. The linear dependence of the relaxation resonance frequencies of the 50 nm and 90 nm
QNCL lasers on
I
I
th
, described in Eq. (
7
) is shown in Fig.
4
c. The reduction in the relaxation resonance
frequencies
f
R
is observed with the increasing QNCL thickness, which is attributed to the reduced electric field
strength
E
l
(
r
a
)
at the location of the quantum-wells (i.e., reduced
Ŵ
QW
). It can also be observed that the relaxa-
tion resonance peak is less apparent on the 90 nm QNCL laser compared to that of the 50 nm QNCL, due to the
increased damping in the 90 nm QNCL as predicted in Eq. (
8
).
For the fabricated lasers, we found that the 50 nm QNCL laser, which has a lasing wavelength of 1577 nm,
shows a linewidth enhancement factor of 5.8, whereas the 90 nm laser lasing at the wavelength of 1556 nm has a
linewidth enhancement factor of 3. The details of the measurements follow the measurement setup and analysis
of
15
,
24
,
25
and are discussed in the Supplementary Material.
Figure 3.
(
a
) The light vs. pump current (LI) and current vs. voltage (IV) characteristics of the 50 nm and
90 nm QNCL lasers at
20
C. (
b
) The optical spectrum of the lasers at
20
C, showing a single-mode operation
with a side-mode suppression ratio larger than 45 dB.