Quantum control of phase fluctuations in
semiconductor lasers
Christos T. Santis
a,1
, Yaakov Vilenchik
a
, Naresh Satyan
b
, George Rakuljic
b
, and Amnon Yariv
a,1
a
Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, CA 91125; and
b
Telaris Inc., Santa Monica, CA 90403
Contributed by Amnon Yariv, July 3, 2018 [sent for review April 19, 2018; reviewed by Mordechai (Moti) Segev and Eli Yablonovich]
Few laser systems allow access to the light–emitter interaction as
versatile and direct as that afforded by semiconductor lasers. Such
a level of access can be exploited for the control of the coherence
and dynamic properties of the laser. Here, we demonstrate, theo-
retically and experimentally, the reduction of the quantum phase
noise of a semiconductor laser through the direct control of the
spontaneous emission into the laser mode, exercised via the pre-
cise and deterministic manipulation of the optical mode’s spatial
field distribution. Central to the approach is the recognition of
the intimate interplay between spontaneous emission and optical
loss. A method of leveraging and “walking” this fine balance to
its limit is described. As a result, some two orders of magnitude
reduction in quantum noise over the state of the art in semicon-
ductor lasers, corresponding to a minimum linewidth of 1 kHz,
is demonstrated. Further implications, including an additional
order-of-magnitude enhancement in effective coherence by way
of control of the relaxation oscillation resonance frequency and
enhancement of the intrinsic immunity to optical feedback, high-
light the potential of the proposed concept for next-generation,
integrated coherent systems.
semiconductor laser
|
spontaneous emission
|
temporal coherence
|
phase noise
|
optical resonator
S
pontaneous emission is central to the laser process, serv-
ing as both seed and fundamental limit to the laser light’s
temporal coherence (1, 2). A balancing act of minimizing sponta-
neous emission noise while keeping sufficient gain for oscillation
is called for in the continual push for increased performance,
especially in the face of increasingly constraining technological
requirements (e.g., size and complexity). Such a process may
also entail the challenging of long-held notions on laser design
and will certainly require the constant awareness of the inti-
mate interplay between spontaneous emission, optical loss, and
population inversion.
The last couple decades have seen a flurry of research activity
around the control of spontaneous emission based on the princi-
ples first prescribed by E. M. Purcell (3). Innovative ideas in areas
such as photonic bandgap engineering and optical confinement
(i.e., resonators), coupled with progress in materials and fabri-
cation technology, have enabled an unprecedented level of con-
trol over light–emitter interaction (4–6). Both suppression and
enhancement of spontaneous emission have been demonstrated,
with implications ranging from threshold reduction (i.e., thresh-
oldless laser), radiative efficiency, and modulation bandwidth
enhancement in microcavity semiconductor lasers to nonclassical
light generation (e.g., sub-Poissonian and amplitude-squeezed
light) (7–9), but never, to the best of our knowledge, for the
express purpose of enhancing laser coherence. A small num-
ber of theoretical investigations into the effect of spontaneous
emission modification on the linewidth of microcavity semicon-
ductor lasers by way of cavity size control has failed to produce
a consensus as to whether it should lead to a narrowing or
broadening of linewidth, presumably due to conflicting assump-
tions regarding other contributing parameters (e.g., threshold)
(10, 11).
Here, we demonstrate, theoretically and experimentally, the
control of spontaneous emission into the laser mode of a semi-
conductor laser for the purpose of suppressing quantum noise
and enhancing its temporal coherence. The control is exercised
through the direct and precise manipulation of the mode’s spatial
field distribution (i.e., modal control) relative to the emitter [i.e.,
quantum well (QW)]. It is technically implemented in a seamless
fashion, without change in cavity size or use of external elements.
Harnessing recent advancements in photonic integration [i.e., sil-
icon (Si)/III-V] and optical resonator design, some two orders
of magnitude improvement in the coherence of semiconductor
laser over the state of the art is achieved.
In what follows, we begin by reviewing, for the sake of com-
pleteness, the main mechanisms involved in quantum noise as
they pertain to a semiconductor laser and laying out the the-
oretical premise of our approach: the direct, modal control of
spontaneous emission into the laser mode. The latter is then
applied to the case of a real laser system, a Si/III-V semicon-
ductor laser. Numerical modeling and theoretical performance
estimates are followed by experimental results of fabricated
lasers. The paper concludes with a discussion on the merits,
limitations, and future prospects of the work.
Theoretical Background
The theoretical basis of our approach is the phase diffusion
model for a stochastic noise-driven laser oscillator (2). Despite
its simplicity, it encompasses the main mechanisms at work and
provides quick insight into their interplay. Fundamentally, the
Significance
The semiconductor laser, arguably the most versatile member
of the family of lasers, has become a technological staple of
a massively interconnected, data-driven world, with its spec-
tral purity (i.e., temporal coherence) an increasingly important
figure of merit. The present work describes a conceptually
fundamental “recipe” for the enhancement of coherence,
predicated on direct control of the coherence-limiting process
itself, the field–matter interaction. As such, it is inherently
adaptable and technologically scalable. As photonic materials
and fabrication techniques continue to improve, the described
approach has the potential of serving as a roadmap for major
and sustained improvements in coherence. With experimen-
tally demonstrated coherence limited at 1 kHz in this work,
we envision “deep” sub-kilohertz-level coherence to be soon
within reach.
Author contributions: C.T.S., Y.V., N.S., G.R., and A.Y. designed research; C.T.S., Y.V., and
N.S. performed research; C.T.S., Y.V., and N.S. analyzed data; and C.T.S. and A.Y. wrote
the paper.
Reviewers: M.S., Technion-Israel Institute of Technology; and E.Y., University of California,
Berkeley.
The authors declare no conflict of interest.
This open access article is distributed under
Creative Commons Attribution-
NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND)
.
1
To whom correspondence may be addressed. Email: ayariv@caltech.edu or christos@
caltech.edu.
y
Published online August 7, 2018.
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www.pnas.org/cgi/doi/10.1073/pnas.1806716115
APPLIED PHYSICAL
SCIENCES
temporal coherence of laser emission is limited by phase noise,
the result of zero-point (i.e. vacuum) fluctuations of the laser
field (12, 13). Atoms, in general, or electrons in the case of a
semiconductor laser, interacting with these fluctuations undergo
“spontaneous” transitions from excited to lower-lying energy
states, emitting in the process photons with phases uncorrelated
with that of the coherent field. Under the effect of a large num-
ber of such events, the optical phase of the field,
θ
, performs
a random, diffusion-type “walk” in the complex phasor plane
accumulating a variance
〈
∆
θ
(
τ
)
2
〉
=
N
2
t
W
(
`
)
sp
2
n
`
(1 +
α
2
)
τ
,
[1]
over time
τ
. It is this phase excursion that is the pertinent fig-
ure of merit for performance in many practical applications, such
as, for example, optical coherent communications,
τ
in that case
being the duration of a single symbol of information (14).
In Eq.
1
,
W
(
`
)
sp
is the spontaneous emission rate (in s
−
1
) per
electron into the laser mode, denoted by “
`
”;
N
2
t
is the total
number of electrons in the excited level (i.e., conduction band),
clamped at its threshold value; and
n
`
is the number of light
quanta stored in the laser mode. The linewidth enhancement
factor
α
accounts for the excess phase noise due to coupling
between amplitude and phase fluctuations and will be treated
here as a constant. The product
N
2
t
W
(
`
)
sp
represents the total
spontaneous emission rate,
R
(
`
)
sp
(in photons per second), into
the laser mode. The energy due to
n
`
coherent quanta in the
laser mode, proportional to the photon lifetime in the cavity, acts
as an optical “flywheel” which resists the spontaneous emission-
driven diffusion of the phase. The three quantities,
W
(
`
)
sp
,
N
2
t
,
and
n
`
, are implicitly coupled through the quantized nature of
the electromagnetic (EM) field, optical resonator fundamentals,
and semiconductor physics.
As our approach to the suppression of quantum noise relies
on the modal control of the spontaneous emission into the laser
mode, it is advantageous to express the laser field as an expansion
in normal modes (15, 16), the eigenmodes of the resonator,
̄
E
( ̄
r
,
t
) =
−
∑
s
1
√
(
r
)
p
s
(
t
)
̄
E
s
( ̄
r
),
[2]
̄
H
( ̄
r
,
t
) =
∑
s
1
√
μ
(
r
)
ω
s
q
s
(
t
)
̄
H
s
( ̄
r
),
[3]
where mode functions
E
s
,
H
s
are exact solutions of Maxwell’s
equations for the specific resonator and subject to orthogonality
and normalization conditions
∫
V
c
̄
E
s
( ̄
r
)
·
̄
E
t
( ̄
r
)
d
3
̄
r
=
δ
s
,
t
,
[4]
∫
V
c
̄
H
s
( ̄
r
)
·
̄
H
t
( ̄
r
)
d
3
̄
r
=
δ
s
,
t
,
[5]
where
V
c
is the cavity volume. The quantization of the field
comes about naturally through the association of the expan-
sion coefficients,
p
s
and
q
s
, of each mode with the momentum
and coordinate operators, respectively, of a quantum mechanical
oscillator.
Application of time-dependent perturbation theory to the
interaction of an electron, located at a position in the laser’s
active region denoted by
̄
r
a
, with the quantized field yields
expressions for 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 (i.e., hole), by which a
photon is emitted into mode (
`
),
W
(
`
)
sp
=
2
π
2
μ
2
ν
`
g
a
(
ν
`
)
h
(
r
a
)
|
̄
E
`
( ̄
r
a
)
|
2
,
[6]
W
(
`
)
st
=
n
`
W
(
`
)
sp
,
[7]
where
μ
is the dipole transition matrix element and
g
a
(
ν
`
)
(in units of s) the value of the normalized lineshape func-
tion of the transition at the lasing frequency
ν
`
, both known
quantities for our purposes, and
|
̄
E
`
( ̄
r
a
)
|
2
is the normalized
intensity of the laser mode at the location of the emitter (i.e.,
electron).
The latter constitutes a modal “knob” on the rate of spon-
taneous emission into the laser mode. On a more funda-
mental, quantum-mechanical level, it can be shown that the
field’s quantum fluctuations, the root cause of noise, bear the
spatial signature of each constituent mode and, through it,
information about the size, shape, and overall structure of
the resonator (
Appendix A
). Therein lies the guiding insight
toward quantum noise reduction. Instead of the brute-force
method of reducing
|
̄
E
`
( ̄
r
a
)
|
2
, the quantum fluctuations inten-
sity, by “dilution” of the zero-point energy through increase
of the mode volume, we achieve the same result by a subtle,
modal engineering which leaves the mode profile essentially
unaltered.
It may seem straightforward, at first, to try to suppress noise
by an arbitrary reduction of
|
̄
E
`
( ̄
r
a
)
|
2
. However, it follows from
Eq.
7
that any attempt to reduce the spontaneous emission rate
into laser mode (
`
) will have a proportionally similar effect on
the stimulated rate and, thus, on the laser gain needed to over-
come losses, thus forcing the laser medium to a higher inversion
point (i.e., higher
N
2
t
). This will not only cause the thresh-
old to increase, but also cut into, or even negate, any expected
reduction in noise. One way to break this conundrum and cre-
ate positive leverage for the reduction of quantum noise is by
linking any decrease in spontaneous emission rate to a com-
mensurate decrease in the optical loss rate, thus keeping the
threshold current effectively unchanged (see
Appendix D
for
more on this).
The laser architecture for high coherence, introduced here,
constitutes a departure from long-standing conventions of semi-
conductor laser design. Semiconductor laser designs typically
seek to maximize the modal overlap with the active region (QW)
and, thereby, the available modal gain (Fig. 1
A
). This choice,
however, is attendant upon a significant loss (e.g., free-carrier
absorption) and, thus, noise. It also leaves no room for internally
manipulating
|
̄
E
`
( ̄
r
a
)
|
2
, as the active region is surrounded by
highly absorbing and low-index regions. By contrast, placing the
active region in close proximity to a high-index, low-loss layer,
as shown in Fig. 1
B
, creates the potential for direct control of
|
̄
E
`
( ̄
r
a
)
|
2
by modal engineering in tandem with loss reduction.
The decrease in modal gain brought about by the reduction of
|
̄
E
`
( ̄
r
a
)
|
2
is offset by a decrease in loss by virtue of increased
confinement in the lower-loss layer, thus keeping the threshold
current practically constant. This, of course, cannot go on indefi-
nitely. The point to which this balance can be maintained defines
the useful margin for noise reduction and is determined by the
point where loss in the low-loss, guiding layer starts to become
the dominant modal loss and, thus, the limiting factor of the total
loss, foreshadowing the role of this layer as a figure of merit for
coherence.
Design and Analysis
The practical realization of the concepts described above draws
upon recent advancements in the area of photonic integration
and, specifically, the heterogeneous integration of Si and III-V
Santis et al.
PNAS
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vol. 115
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no. 34
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E7897
E
(
r
)
2
r
high-loss
n-side
p-side
QW
r
a
low-index
r
r
a
high-index (low-loss)
high-loss
n-side
p-side
E
(
r
)
2
QW
AB
Fig. 1.
Concept illustration of the modal control of spontaneous emission in a semiconductor laser. (
A
) Cross-sectional structure of a generic, electrically
pumped semiconductor laser along with a simulated example of the transverse distribution of the electric field intensity,
|
̄
E
`
(
̄
r
)
|
2
, of the laser mode. (
B
) In a
departure from the standard semiconductor laser design, the laser mode is “pulled” into and is guided by a transversely proximal to the active region layer.
The low-loss nature of the latter is key for offsetting the resulting decrease in modal gain. The location of the active region in the evanescent tail of the
mode enables leveraged (i.e., exponential) control over
|
̄
E
`
(
̄
r
)
|
2
and, thereby, of the spontaneous emission rate into the laser mode, as showcased by two
simulated examples (solid and dashed black lines).
(i.e., InP) (17). Si provides the requisite low-loss, guiding layer,
bringing along a proven set of methods for the design and fabri-
cation of low-loss, optical structures (e.g., waveguides, gratings,
resonators, etc.) (18, 19). In a key addition to the standard
Si/III-V laser structure, we introduce a relatively thick, up to
∼
150
nm, layer of silica (
SiO
2
) between the Si and the III-V,
shown in Fig. 2
A
and
B
. The thickness of this layer, controllable
on a nanometer scale, offers a precise and leveraged means of
control of
|
̄
E
`
( ̄
r
a
)
|
2
and, thus, of the rate of spontaneous emis-
sion, as illustrated by simulated examples in Fig. 2
C
and
D
. For
short and for its distinct role, this layer will be referred to as the
quantum noise control layer (QNCL).
Loss control is central to the scheme, as explained above, and,
thus, warrants a closer look. For a consistent description of the
overall loss and its constituent components, we use the cold cav-
ity quality factor (Q). In the limit of low confinement in the III-V,
as is the case for the lasers in this work (i.e.,
Γ
III
−
V
<
10%
), the
total Q (i.e., loaded) can be written as
Si
QW
n-InP
p-InP
H
+
H
+
QW
Si
InP
InP
QW
Si
Si
1.0 μm
1.0 μm
QNCL (SiO
2
)
SiO
2
SiO
2
SiO
2
QNCL (SiO
2
)
QNCL (SiO
2
)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
QNCL (SiO
2
)
A
CD
B
Fig. 2.
Spontaneous emission control on Si/III-V platform. (
A
) Cross-sectional structure of a Si/III-V laser featuring a SiO
2
spacer layer (QNCL) between Si
and III-V for the modal control of the spontaneous emission rate into the laser mode. (
B
) A 3D schematic of the optical resonator (III-V omitted for clarity).
Note that the design of the resonator accounts for the presence of the III-V (i.e., hybrid resonator). (
C
and
D
) Simulated examples of the spatial (2D)
distribution of the (normalized) amplitude of the electric field (
|
̄
E
`
(
̄
r
)
|
) of the laser mode (TE
0
) for two extreme cases of QNCL thickness—50 nm (
C
) and
200 nm (
D
)—representing a more than an order-of-magnitude swing in the rate of spontaneous emission.
E7898
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Santis et al.
APPLIED PHYSICAL
SCIENCES
1
Q
=
1
Q
III
−
V
+
1
Q
Si
+
1
Q
e
,
[8]
where
Q
Si
accounts for loss to scattering, predominantly in Si;
Q
III
−
V
accounts for loss to (free-carrier) absorption, predom-
inantly in the III-V; and
Q
e
accounts for loss to the useful
output. As
Q
III
−
V
depends on the modal intensity in the III-V
(
Appendix C
), including that in the active region, a key code-
pendence between III-V loss and spontaneous emission exists.
This relation can be harnessed for the purpose of noise reduction
as long as
Q
III
−
V
Q
Si
,Q
e
. To maximize the useful margin for
noise reduction,
Q
Si
has to be maximized as well (
Q
e
is designer-
controlled and can be set at will). This is done by resonator
design and optimization of the fabrication process (20, 21). The
benchmark value of
Q
Si
in this work is
∼
10
6
(intrinsic).
We now return to the metric for coherence, as defined
in Eq.
1
, or, equivalently, its associated spectral linewidth
[
∆
ν
= (2
πτ
)
−
1
〈
∆
θ
(
τ
)
2
〉]
. Applying optical resonator and laser
fundamentals (
Appendix E
), the linewidth of a semiconductor
laser can be written as
∆
ν
=
e
μ
2
ω
2
`
g
a
(
ω
`
)
(
1 +
α
2
)
η
i
h
(
r
a
)(
I
−
I
th
)Q
[
|
̄
E
`
( ̄
r
a
)
|
2
N
tr
+
ω
`
g
′
V
a
Q
]
,
[9]
where
I
and
I
th
are the injection and threshold currents, respec-
tively;
η
i
is the internal quantum efficiency (including carrier
injection efficiency);
g
′
is a material-dependent, differential gain
coefficient (in s
−
1
); and
V
a
is the volume of the active region.
The first term in the brackets corresponds to the spontaneous
emission due the
N
tr
carriers (absolute number) necessary to
render the active region transparent (i.e., transparency term),
whereas the second term accounts for spontaneous emission
from the additional carriers injected to compensate for loss
(i.e., threshold term). The former term depends on material
properties only and not on the resonator, while the latter does
depend on the resonator loss, hence the
Q
−
1
dependence. Con-
versely, the transparency term depends on
|
̄
E
`
( ̄
r
a
)
|
2
, whereas the
threshold term does not, which follows from the fundamental
relationship between the spontaneous and stimulated emission
(Eq.
7
). The additional Q in the denominator of the prefactor
is shared by both terms and reflects the dependence of the pho-
ton number,
n
`
, on loss. For our purposes, all quantities in Eq.
9
,
except for
|
̄
E
`
( ̄
r
a
)
|
2
and
Q
, are considered fixed.
Plotted in Fig. 3
A
is the relative spontaneous emission rate
into the laser mode, normalized to a generic, reference semicon-
ductor laser, for three different values of
Q
Si
and as a function
of the QNCL thickness. The lasers under study in this work
and the reference one are assumed to be similar in all aspects
except in loss and fraction of mode confinement in the active
region [i.e.,
|
̄
E
`
( ̄
r
a
)
|
2
], for which values typical of III-V semi-
conductor lasers are assumed for the reference laser (Fig. 3
A
).
Solid lines correspond to the sum of the two spontaneous emis-
sion components, transparency [
R
(
tr
)
sp
] and threshold [
R
(
th
)
sp
], for
each
Q
Si
, with that for
Q
Si
=10
6
further analyzed into its con-
stituent components, shown with blue dashed lines. We find the
spontaneous emission due to the transparency carrier population
to decrease monotonically with the QNCL thickness due to its
|
̄
E
`
( ̄
r
a
)
|
2
dependence, whereas that due to threshold population
levels off as a result of the saturation of Q to the respective
Q
Si
value. For a given
Q
Si
, we can define two distinct regimes for the
total spontaneous emission into the laser mode: a transparency-
limited regime which obtains with “thinner” QNCLs [i.e., larger
|
̄
E
`
( ̄
r
a
)
|
2
] and a threshold-limited one at “thicker” QNCLs. The
inflection point on each trace marking the boundary between
the two regimes occurs at the point where absorption loss in the
QNCL Thickness (nm)
50
100
150
200
250
300
350
400
Relative Spontaneous Emission (a.u.)
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
R
sp
(
tr
)
R
sp
(
th
)
III-V loss-limited
region
0
Si loss-limited
region
Q
Si
=
10
4
Q
Si
=
10
5
Q
Si
=
10
6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
Relative Quantum Linewidth (a.u.)
QNCL Thickness (nm)
50
100
150
200
250
300
350
400
0
III-V loss-limited
region
Si loss-limited
region
Q
Si
=
10
4
Q
Si
=
10
5
Q
Si
=
10
6
Threshold Current (mA)
Cold-Cavity Q
10
1
10
2
10
3
10
4
10
5
10
6
10
7
AB
Fig. 3.
Numerical analysis and performance estimates. (
A
) Relative spontaneous emission rate into the laser mode, normalized with respect to a generic
semiconductor laser (parameters listed below), as a function of QNCL thickness and for three different values of Q
Si
(solid lines). Blue dashed lines correspond
to the individual spontaneous emission components,
R
(
tr
)
sp
and
R
(
th
)
sp
, respectively, for the case of Q
Si
=
10
6
. (
B
) Relative quantum linewidth calculated for the
same parameters as those of
A
. Also plotted in dashed lines are the threshold current and cold cavity Q as a function of QNCL thickness for Q
Si
=
10
6
.
Gray-shaded regions in both images illustrate the combined area corresponding to the Si loss-limited regime for all possible values of QNCL. The boundary
between the Si and III-V loss-limited regions is formed by the locus of the inflection point of each curve. The reference semiconductor laser used in the
normalization is taken to have an active region of volume similar to that of the Si/III-V lasers of this work,
V
a
=
(L
×
W
×
H) = (1 mm
×
10
μ
m
×
40 nm), a mode
confinement factor of
Γ
QW
=
10
%
, a differential gain coefficient of
g
′
=
4
×
10
−
15
s
−
1
, and modal loss equivalent to a loaded Q of 10
4
.
Santis et al.
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vol. 115
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no. 34
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E7899
III-V equals the combined loss to other channels, namely, scat-
tering (
Q
Si
) and output (
Q
e
). Clearly, operating beyond this
point bears increasingly diminishing returns as far as sponta-
neous emission reduction. The optimal point moves to thicker
QNCLs with increasing
Q
Si
, allowing an increasingly wider mar-
gin for spontaneous emission reduction. As an example, an
increase in
Q
Si
from
10
4
, typical of all-III-V semiconductor
lasers, to
10
6
creates a nearly two-order-of-magnitude additional
potential for spontaneous emission reduction, highlighting the
role of
Q
Si
as a figure of merit for coherence.
Plotted in Fig. 3
B
is the respective normalized quantum
linewidth for each case of
Q
Si
. The qualitative trend as a func-
tion of the QNCL thickness remains the same as that of Fig. 3
A
,
but the vertical scale has now grown, accounting for the addi-
tional contribution from the Q factor in the denominator Eq.
9
. An additional two-order-of-magnitude margin for linewidth
reduction between
Q
Si
=10
4
and
10
6
is created by the pho-
ton lifetime enhancement alone, bringing the aggregate margin
to
10
4
. The “weight” of Q and its saturation effect have also
been enhanced as a result, moving the inflection points on each
trace to smaller QNCL thicknesses. An alternative criterion for
the transition into the Si loss-limited regime can be defined in
terms of the threshold current. Also plotted in Fig. 3
B
in blue
dashed lines are the cold cavity Q and threshold current as a
function of QNCL thickness for
Q
Si
=10
6
. The latter remains
practically constant throughout the III-V loss-limited region and
starts to rise rapidly upon the onset of the Q saturation (i.e.,
Si loss-limited regime). Gray shading in Fig. 3 is used to delin-
eate the Si from the III-V loss-limited regime in the design
parameter space.
Experiment
Fabricated lasers were characterized for their coherence and,
specifically, the quantum limit imposed on it by spontaneous
emission (more on fabrication and characterization is provided
in
Materials and Methods
). To resolve the signature of sponta-
neous emission on the lasers’ coherence and separate it from
that of technical noise (e.g., temperature or electronic), we
measured the full frequency noise spectrum. This measurement
becomes increasingly necessary, but also challenging, as quan-
tum noise is suppressed. Noise due to spontaneous emission
manifests itself in the frequency noise spectrum as a white
noise plateau at high frequencies (
>
100
kHz), where contri-
butions from technical and
1
/
f
noise have dropped to relative
unimportance.
Plotted in Fig. 4
A
is the power spectral density (PSD) (in
units of equivalent white noise linewidth,
π
Hz
2
/
Hz
) of the
frequency noise of Si/III-V lasers with three different QNCL
thicknesses, 30, 100, and 150 nm, which are expected to span the
breadth of the III-V loss-limited regime for
Q
Si
=10
6
. Plotted
along is the frequency noise of a commercial distributed feed-
back (DFB) laser (JDSU model no. CQF935/808), used for
measurement calibration and comparison. All traces, with the
exception of that of the reference DFB, correspond to measure-
ments taken at the same current increment above each laser’s
respective threshold, in an effort to cancel out any variation in
noise between lasers due to pumping. A clear trend of decreas-
ing level of quantum noise with increasing QNCL thickness,
indicative of decreasing spontaneous emission noise, is observed.
The quantum-limited linewidths, extracted at the minimum fre-
quency noise point (
∼
200
MHz), are 35, 2.5, and 1.3 kHz for the
Frequency (Hz)
10
5
10
6
10
7
10
8
10
9
10
2
10
3
10
4
10
5
10
6
10
7
10
8
Reference DFB
30 nm QNCL
100 nm QNCL
150 nm QNCL
10
7
10
8
10
9
10
2
10
3
10
4
10
5
Laser Frequency Noise (measured)
EDFA Frequency Noise (model)
Laser Frequency Noise (corrected)
QNCL Thickness (nm)
0
100
150
200
250
300
350
400
Quantum Linewidth (kHz)
10
-2
10
-1
10
0
10
1
10
2
10
3
50
reference DFB
Si loss-limited
region
III-V loss-limited
region
AB
Fig. 4.
Experimental results. (
A
) Measured PSD (in units of equivalent white noise linewidth,
π
Hz
2
/
Hz) of the frequency noise of Si/III-V lasers with three
different values of QNCL thickness (30, 100, and 150 nm). Plotted alongside for comparison is the frequency noise spectrum of a commercial DFB (black
trace), also used for measurement calibration. All the traces of the Si/III-V lasers were taken at the same current increment (
∼
30 mA) from their respective
threshold, while that of the reference DFB was at its maximum operating current. (
A
,
Inset
) Numerical correction for the EDFA noise for the case of the
100 nm QNCL. The measured noise above
∼
300 MHz is fitted to
S
∆
ν
(
f
)
=
af
b
+
S
o
(red dashed line), with
a
,
b
being fitting coefficients and
S
o
a frequency-
independent term (i.e., white noise). Subtracting the frequency-dependent term (i.e.,
af
b
) from the measured noise yields the corrected spectrum (green
line). (
B
) Experimental quantum linewidths (blue markers) plotted against the theoretical trend line (gray trace), as calculated via Eq.
9
for Q
Si
=
10
6
. To set
the vertical axis, the linewidth of the reference DFB (red dashed line), measured at an equivalent pump rate, is used to convert from relative (Fig. 3
B
) to
absolute linewidth. Red markers represent linewidth estimates for the cases of the numerically filtered EDFA noise. The gray-shaded area corresponds to
the theoretically estimated Si loss-limited region.
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Santis et al.
APPLIED PHYSICAL
SCIENCES
three lasers in descending thickness order. By comparison, the
reference DFB, measured at its maximum pump level, reaches a
minimum linewidth of
∼
100 kHz.
It should be noted that while a clear white noise floor is
recovered for the reference DFB and 30-nm QNCL laser, a less
definitive picture is obtained for the 100- and 150-nm QNCLs. As
the quantum noise level is “pushed” lower, both internally and
externally imposed noise floors begin to surface at intermediate-
and high-offset frequencies. As a result, the encounter with
the white noise floor is pushed to higher frequencies, where
the sensitivity of the measurement quickly degrades, render-
ing the results more ambiguous. In this case, we find noise
injected by the Erbium-doped fiber amplifier (EDFA) used for
the amplification of the fiber-coupled light to mask the quan-
tum noise at high offset frequencies, as evidenced by the rise
in frequency noise in the spectrum of the 100- and 150-nm
QNCLs above
∼
200 MHz. In an effort to obtain an estimate
for the lasers’ intrinsic linewidth, we correct for the injected
noise by numerically fitting the rising part of the noise spec-
trum and subtracting its frequency-dependent component from
the measured noise, thereby leaving any white noise content
unaffected. A white noise plateau is recovered in both cases,
shown for illustration for the case of the 100-nm QNCL in Fig.
4
A
,
Inset
with numerically extracted estimates for the quantum
linewidths of 1.5 and 500 Hz for the 100- and 150-nm QNCL
lasers, respectively.
The experimental results are compared with the theoreti-
cal predictions of the preceding section. Plotted in Fig. 4
B
are the experimental quantum linewidths of the three QNCL
lasers along with the theoretically expected trend line for the
case of
Q
Si
= 10
6
, as calculated from equation Eq.
9
and pre-
sented in Fig. 3
B
. For the conversion from relative to absolute
linewidth, we use the quantum linewidth of the reference DFB
(
∼
500
kHz), measured at the same current increment from
threshold as the QNCL lasers. With the exception of the data
point for the 150-nm QNCL, presumably due to the effect
of the noise from the EDFA, the experimental results follow
the theoretical trend with reasonable agreement. When com-
pared at the same current increment from threshold, the QNCL
lasers attain a coherence level more than two orders of mag-
nitude higher than that of the reference DBF (red dashed line
in Fig. 4).
The threshold currents of the three QNCL lasers are 40, 42.5,
and 47 mA for the 30-, 100-, and 150-nm QNCL, respectively.
These values fall within the margin of fabrication-induced varia-
tion for the threshold current, confirming our expectation of the
three chosen QNCLs lying in the III-V loss-limited regime. The
output power levels of the lasers are generally low, at
∼
1 mW
(in-fiber, single-facet), hence the need for the EDFA. This is,
in part, due to the unintended undercoupling of resonators (i.e.,
too high
Q
e
), which results in reduced output coupling efficiency,
compounding the drop in efficiency due to early thermal roll-
off, a well-documented occurrence in Si/III-V lasers (22). Finally,
the mode confinement factors in the various regions of interest
along with the quantum-limited linewidths for the three cases are
summarized in Table 1.
Discussion
The linewidth narrowing by the control of the spontaneous emis-
sion in our laser raises the question of the relation to the Purcell
effect (3). In Purcell’s original scenario, the emitter (i.e., spin)
interacts with the single mode of a “closed” resonator. This is due
to the fact that the resonator dimensions are on the order of the
wavelength of the EM field (
V
c
∼
λ
3
), and only one mode exists
within the natural linewidth of the spin transition. In this case,
the spontaneous emission rate into the mode and the total spon-
taneous emission rate, the inverse of the excited spin lifetime,
are one and the same. In our case, the “open” structure of the
Table 1. Calculated mode confinement factors (in
%
) in various
regions of interest (i.e., III-V, QW, and Si) for lasers with different
QNCL thickness, along with the respective measured and
numerically adjusted quantum linewidths
∆
ν
, kHz
∆
ν
, kHz
QNCL, nm
III-V,
%
QW,
%
Si,
%
(exp.)
(num.)
30
15
.
0
2.0
82
35
.
0
—
100
4
.
0
0.6
92
2
.
5
1.0
150
1
.
5
0.2
95
1
.
1
0.5
resonator and the large natural linewidth (
>
10
12
Hz
) allow the
inverted electron population to emit spontaneously into a very
large number of those “big-box,” vacuum modes. Under these
circumstances, our “surgical” control, by orders of magnitude,
of the spontaneous emission rate into the laser mode has but a
negligible effect on the overall spontaneous lifetime. Our spon-
taneous emission control can, thus, be viewed as a special case of
the generalized Purcell control. Our basic results and formalism
lead readily to Purcell’s results under a similar set of resonator
mode and transition characteristics (see
Appendix B
for more
details).
A study of Eq.
9
raises the question of how far can we reduce
∆
ν
by controlling
|
̄
E
`
( ̄
r
a
)
|
2
or, in other words, of the ultimate
limit in coherence. The some-two-orders-of-magnitude reduc-
tion in
∆
ν
reported in this work is reached under the condition
whereby the dominant loss is optical absorption in the III-V
material (i.e.,
Q
III
−
V
Q
Si
). So, it is
Q
Si
that sets the limit
on coherence. The question can, thus, be rephrased into how
large can
Q
Si
be made. Under the experimental condition of
this work,
Q
Si
is limited at
∼
10
6
by scattering loss, absorp-
tion loss in bulk Si being much smaller. However, if scattering
loss were suppressed and/or under conditions of high intracavity
power,
Q
Si
could attain a nonlinear (i.e., intensity-dependent)
term due to two-photon and ensuing free-carrier absorption in
Si (
λ<
2
.
2
μ
m), which would then become the
Q
Si
- and, ulti-
mately, coherence-limiting factor. For the lasers of this work, the
limit in coherence in terms of quantum linewidth is estimated
to be in the ballpark of a few hundred hertz (23, 24). Overcom-
ing this limit would require replacing Si with a wider-bandgap
material.
Lastly, the control of the field–emitter interaction and optical
loss proposed and achieved in this work has additional perfor-
mance implications. A laser operating with a high loaded (cold
cavity) Q factor is expected to exhibit inherent robustness against
the detrimental effects of external back-reflections (i.e., coher-
ence collapse) by virtue of enhanced effective mirror reflectivity
(i.e., high
Q
e
). A laser with intrinsic, relative immunity to opti-
cal feedback high enough to obviate the need for an optical
isolator can, thus, be envisioned. Finally, the reduction of the
spontaneous emission rate into the laser mode, coupled with the
enhancement of photon lifetime, enables substantial, up to an
order of magnitude reduction in the relaxation oscillation res-
onance frequency, thereby enhancing the effective short-term
coherence by suppressing the contribution from carrier noise
through amplitude-to-phase fluctuation coupling at offset fre-
quencies of importance for many practical applications (e.g.,
Gb/s-rate coherent communications). These prospects highlight
further the potential of a laser embodying the described concepts
as a powering source of next-generation, chip-scale, coherent
solutions.
Appendices
Appendix A: EM Field Mode-Expansion and Quantization.
Substitut-
ing the field mode expansions (2) and (3) in the expression for the
field’s Hamiltonian and applying the normalization conditions
(4) and (5) yields
Santis et al.
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|
vol. 115
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no. 34
|
E7901
H
=
1
2
∑
s
(
p
2
s
+
ω
2
s
q
2
s
)
,
[10]
which has the form of a sum of Hamiltonians of independent
(classical) harmonic oscillators. The quantization of the EM
field, then, comes about naturally by associating the expansion
coefficients,
p
s
,
q
s
, with the momentum and coordinate opera-
tors, respectively, of a quantum mechanical oscillator. Subject to
the appropriate commutation relations, time-dependent boson
operators,
α
†
s
,
α
s
, can be defined for each eigenmode in terms
of its canonical variables,
p
s
,
q
s
, yielding the quantized EM field
operators
̄
E
( ̄
r
,
t
) =
−
i
√
~
ω
`
2
∑
s
[
α
†
s
(
t
)
−
α
s
(
t
)
]
̄
E
s
( ̄
r
),
[11]
̄
H
( ̄
r
,
t
) =
√
~
ω
`
2
μ
∑
s
[
α
†
s
(
t
) +
α
s
(
t
)
]
̄
H
s
( ̄
r
)
.
[12]
The probabilities of the various optical transitions are derived by
application of time-dependent perturbation theory and using the
interaction Hamiltonian,
H
int
=
−
e
̄
r
·
̄
E
( ̄
r
,
t
),
[13]
in the dipole approximation for the interaction of the EM field
with an electron-hole pair in a semiconductor medium, where
̄
r
is the dipole position operator and
̄
E
( ̄
r
,
t
)
is the electric field
operator of Eq.
11
. Due to the orthogonality of the eigenmodes
(i.e., independence of harmonic oscillators), the transition rate
(in s
−
1
) from an initial state in the conduction band (
|
c
〉
) to
a final state in the valence band (
|
v
〉
) can be derived indepen-
dently for each mode. Dropping the summation over all modes
and retaining only the mode of interest—the laser mode—the
respective transition rate is found to be
W
(
`
)
tot
=
2
π
~
∣
∣
∣
〈
v
,
n
`
+ 1
|H
(
`
)
int
|
c
,
n
`
〉
∣
∣
∣
2
δ
(
E
c
−
E
v
−
~
ω
`
),
[14]
where
n
`
and
n
`
+ 1
are the number of photons in the laser
mode in the initial and final states, respectively, and
E
c
,
E
v
are
the energies of an electron in the conduction and a hole in
the valence band, respectively. Due to the distributed nature of
electronic states in the semiconductor (i.e., Fermi–Dirac), inte-
gration over all energies and both bands is performed for the
total rate. For a single electron located at a position denoted by
̄
r
a
, the total rate (per electron) is
W
(
`
)
tot
= (
n
`
+ 1)
2
π
2
μ
2
ν
`
g
a
(
ν
`
)
h
(
r
a
)
|
̄
E
`
( ̄
r
a
)
|
2
,
[15]
where
μ
=
∣
∣
〈
v
|
e
̄
r
a
|
c
〉
∣
∣
is the dipole transition element and
g
a
(
ν
`
)
is the value of the transition lineshape function—a representa-
tion of the Fermi–Dirac distribution of energies in the frequency
domain—at the transition frequency
ν
`
,
∫
+
∞
−∞
g
a
(
ν
k
)
d
ν
k
= 1
.
[16]
Obviously, in the case of a spatially distributed ensemble of
emitters (e.g., QW), the point-like modal intensity
|
̄
E
`
( ̄
r
a
)
|
2
is
replaced with an integral over the active volume (i.e., confine-
ment factor). The emission rate of Eq.
15
is the sum of the
spontaneous
(
W
(
`
)
sp
)
and stimulated (
W
(
`
)
st
) emission rates of
Eqs.
6
and
7
, respectively.
As already mentioned, spontaneous emission is “induced” by
vacuum fluctuations of the field, the quantization of which allows
us more direct insight into this obscure, quantum mechanical
source. By using the expression for the quantized EM field, the
magnitude of the fluctuations can be found as the variance of
the field when in its vacuum state. This can, once again, be done
for the laser mode independently, by virtue of the eigenmode
orthogonality. As the expectation value of both field compo-
nents vanishes in its vacuum state (
|
0
〉
), the variance reduces
to the expectation value of the field’s intensity. The intensity of
the vacuum, electric field fluctuations of the laser mode, thus,
becomes
(
∆
̄
E
`
( ̄
r
)
)
2
=
〈
̄
E
2
`
( ̄
r
)
〉
=
~
ω
`
2
(
r
)
|
̄
E
`
( ̄
r
)
|
2
.
[17]
The fluctuating field distribution bears the spatial signature of
each eigenmode, in this particular case, the laser mode,
|
̄
E
`
( ̄
r
)
|
2
.
Thus, the modal intensity provides a direct control over the
quantum-mechanical, root cause of spontaneous emission.
Appendix B: Connection to the Purcell Effect.
Expression Eq.
6
applies in the case where the transition linewidth, expressed
by
g
a
(
ν
`
)
, is much broader than the laser mode’s cold-cavity
linewidth, inversely proportional to Q, a condition automati-
cally satisfied for most semiconductors at room temperature.
This regime is diametrically opposite from that considered by
E. M. Purcell (3), wherein an emitter’s radiative lifetime or,
equivalently, its total spontaneous emission rate can be altered
via modification of the optical density of states. For this to be
possible, the emitter’s linewidth needs to be much smaller than
that of the optical mode, a condition that limits most practi-
cal applications in semiconductors to low temperatures. While
control of spontaneous emission by modification of the spectral
density of states may not be an option, another degree of free-
dom is available, the spatial modal density, represented in Eq.
6
by
|
̄
E
`
( ̄
r
a
)
|
2
. In the operating regime of the lasers of this work,
out of the potentially thousands of possible modes of a large and
open-cavity resonator, the spontaneous emission rate into a spe-
cific mode—the laser mode—can be modified by control of its
modal intensity, while leaving that into all other modes and, thus,
the overall spontaneous emission rate largely unaffected. From
that standpoint, this type of spontaneous emission control can be
viewed as a special case of the Purcell effect.
In line with Purcell, we can define a factor for the suppression
of spontaneous emission rate into the laser mode with respect
to that into all modes. The latter can be estimated by assuming
interaction with a continuum of modes in a uniform 3D space.
The result is reproduced here from ref. 16,
W
(
all
)
sp
≡
1
t
sp
=
16
π
3
μ
′
2
h
( ̄
r
a
) (
λ/
n
)
3
,
[18]
where
t
sp
is the spontaneous radiative lifetime of the emitter and
n
is the average refractive index of the dielectric medium. Note
that the dipole matrix element
μ
′
in Eq.
18
is not the same as that
of Eq.
6
. The former accounts for all possible dipole polarizations
in the 3D space, whereas the latter accounts only for dipoles par-
allel to the principal electric field component of the transverse
electric-polarized laser mode. They are related through
μ
′
=
√
3
μ
.
A Purcell factor for the laser mode (
`
) can now be defined as the
ratio of Eqs.
6
–
18
,
F
(
`
)
p
=
1
8
π
(
λ
n
)
3
g
a
(
ν
`
)
ν
`
|
̄
E
`
( ̄
r
a
)
|
2
=
[19]
=
1
4
π
2
(
λ
n
)
3
Q
a
V
eff
,
a
,
[20]
where
Q
a
=
ν
a
/
∆
ν
a
is an effective Q factor for a homogeneously
broadened resonance centered at
ν
a
'
ν
`
, with a full-width at
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Santis et al.