# Turnkey locking of quantum-dot lasers directly grown on Si

Bozhang Dong1,∗, Yating Wan2,∗,†, Weng W. Chow3, Chen Shang1, Artem
Prokoshin2, Rosalyn Koscica4, Heming Wang1, and John E. Bowers1,4,5,†,
1Institute for Energy Efficiency, University of California, Santa Barbara, CA,
USA
2Integrated Photonics Laboratory, King Abdullah University of Science and
Technology, Thuwal, Makkah Province, Saudi Arabia
3Sandia National Laboratories, Albuquerque, NM, USA.
4Materials Department, University of California, Santa Barbara, CA, USA
5Department of Electrical and Computer Engineering, University of California,
Santa Barbara, CA, USA
†Email<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
∗These authors contributed equally

###### Abstract

Ultra-low-noise laser sources are crucial for a variety of applications,
including microwave synthesizers, optical gyroscopes, and the manipulation of
quantum systems. Silicon photonics has emerged as a promising solution for
high-coherence applications due to its ability to reduce system size, weight,
power consumption, and cost (SWaP-C). Semiconductor lasers based on self-
injection locking (SIL) have reached fiber laser coherence, but typically
require a high-Q external cavity to suppress coherence collapse through
frequency-selective feedback. Lasers based on external-cavity locking (ECL)
are a low-cost and turnkey operation option, but their coherence is generally
inferior to SIL lasers. In this work, we demonstrate quantum-dot (QD) lasers
grown directly on Si that achieve SIL laser coherence under turnkey ECL. The
high-performance QD laser offers a scalable and low-cost heteroepitaxial
integration platform. Moreover, the QD laser’s chaos-free nature enables a 16
Hz Lorentzian linewidth under ECL using a low-Q external cavity, and improves
the frequency noise by an additional order of magnitude compared to
conventional quantum-well lasers.

Narrow-linewidth and frequency-stable lasers have opened up new opportunities
for optical sensing and signal generation applications, including optical
frequency and microwave synthesis [1, 2], laser gyroscopes [3, 4], light
detection and ranging (LIDAR) systems [5, 6, 7], spectroscopy [8], and
coherent optical communications [9]. In the quest for compact and mass-
manufacturable coherent systems, photonic integrated circuits (PICs) have
emerged as the most compelling solution. Semiconductor lasers with high-
performance characteristics are integral components in the operation of PICs.
Nevertheless, the linewidth of conventional semiconductor lasers ranges from
hundreds of kilohertz to a few megahertz, which is more than seven orders of
magnitude higher than the state-of-art of bulk fiber lasers [10]. The
significant performance gap has prevented their widespread utilization in the
aforementioned applications. Self-injection locking (SIL) and Pound-Drever-
Hall (PDH) locking technologies [11, 12, 13, 14, 15, 16] have dramatically
reduced the frequency noise and laser linewidth of semiconductor lasers. SIL,
a passive locking technique, employs resonant optical feedback from a high-Q
external optical element to suppress the coherence collapse of the pump laser
and enabled the reduction of laser linewidth below that of a fiber laser [17].
However, The SIL operation also requires that the pump laser frequency
coincides with the external cavity resonance, introducing complexity and
causing power penalties. These limitations need to be addressed to make this
solution accessible to the industry.

Figure 1: QD laser design and characterizations. a. Cross-sectional view of
the axis (001) bright-field STEM image. b. Images of the QD laser in a compact
butterfly package. c. DFB wavelength as a function of the operating
temperature in a device with 3$\times$1000 $\mu m^{2}$ cavity. e. Evolution of
the optical spectra by increasing the injection current from the threshold
current in a device with 3$\times$1000 $\mu m^{2}$ cavity. The optimum side-
mode-suppression ratio (SMSR) is 59 dB. The threshold current at room
temperature (20∘C) is 15 mA. f. Threshold current as a function of the
operating temperature. Inset: characteristic temperature T0, in the ranges (35
- 55∘C and 55 - 80∘C) with different cavity lengths. T0 = 423 K within 35 -
55∘C for the device with 1.5 mm cavity length. g. Frequency noise spectra of
the 1 mm device (burgundy) and another 1.5 mm-long QD laser (navy blue). The
output power is fixed at 0 dBm. Inset: zoom in on the white noise level. The
free-running QD laser yields a Lorentizian linewidth of 41 kHz.

In this work, we introduce advanced quantum-dot (QD) lasers as a compelling
solution for high-coherence turnkey external cavity locking (ECL) technology.
Semiconductor QDs, due to the atom-like density of states, exhibit much lower
linewidth enhancement factor (LEF), and thus much lower frequency noise and
linewidth [18]. Furthermore, QD lasers offer higher immunity to the non-
radiative defects between the III-Vs and Si, making them suitable for
monolithic integration on Si substrates, particularly with high-Q SiN
microresonators [19, 20]. This integration approach is more cost-effective
compared to the heterogeneous integration approach, eliminating the need for
expensive III-V wafers [21, 22]. The chaos-free feature of QDs reduces the
dependence on a high-Q microresonator for linewidth locking, distinguishing QD
lasers from conventional QW lasers. By leveraging the high-performance QD
laser, we achieved a 16 Hz Lorentzian linewidth, the smallest number ever
achieved for on-chip QD lasers. Four orders of magnitude reduction in
frequency noise have been achieved using a low-Q external cavity without a
frequency filter. On the contrary, QW laser, which is susceptible to coherence
collapse under weak external feedback, requires a much larger external cavity
Q-factor to maintain coherence. Leveraging the high feedback insensitivity of
QD lasers, we improve the coherent feedback strength in the ECL operation,
which allows for an extra one order of magnitude reduction in the frequency
noise compared to its QW counterpart. Notably, our low-Q approach allows for
turnkey locking without any power penalty, distinguishing it from the SIL
approach that relies on a high-Q external cavity. Additionally, this approach
offers greater flexibility for material platforms in which fabricating an
ultra-high-Q resonator is challenging. This work effectively addresses the
limitations of SIL technology and significantly enhances the SWaP-C metrics of
high-coherence PICs. In addition, our theoretical investigations indicate that
the full potential of the QD laser has yet to be realized. By integrating them
with CMOS-ready high-Q microresonators, it becomes feasible to further reduce
the linewidth to the sub-hertz level. These advancements position QD lasers as
a compelling candidate for various high-coherence applications, including
optical frequency synthesizers, dual-comb spectroscopy, and ultra-high-
capacity optical transceivers.

Results

Epitaxial growth of QD DFB lasers on Si

The QD-on-Si laser material was grown in a Veeco Gen-II solid source molecular
beam epitaxy (MBE) chamber on an anti-phase-domain free GaP/Si on-axis
template. After completing the defect-reducing buffer layers and the QD active
region [23], the sample is removed from the chamber for grating patterning. A
uniform first-order DFB grating, with a duty cycle of 50%, an etch depth of 50
nm, and a period of 197 nm, was dry etched into the top GaAs layer by
inductively coupled plasma with a Cl2/N2 based chemistry. An electron beam
lithography patterned SiO2 hard mask was used to transfer the pattern. Before
the sample was loaded back into the chamber for regrowth of the top cladding
and the contact layers, a rigorous surface treatment combining solvent clean,
O2 plasma ash, and HF dip was performed. Thereafter, an in-situ atomic H clean
for 30 min at 450∘C was conducted to remove any native oxide. The second MBE
growth then proceeded with a 2 nm p-GaAs nucleation layer and the rest of a
standard GaAs/AlxGa1-xAs graded index separate confinement heterostructure.
The as-grown material was then processed into deeply etched waveguides with
ridge widths ranging from 1.7 to 3 $\mu m$. The STEM image and the epi-
structure of the QD laser are shown in Fig. 1a. The QD chipset is then mounted
into a butterfly package (Fig. 1b) to facilitate measurements and enable
portability.

High-performance integrated QD lasers

Owing to the quantized density-of-state, the carriers are tightly confined in
the dots leading to enhanced injection efficiency [24]. From a series of
devices measured, a maximum power of 11.7 mW is obtained from a 5$\times$1500
$\mu m^{2}$ cavity, twice the previously reported value [25]. The lowest
threshold current of 10 mA is achieved from a 2.5$\times$800 $\mu m^{2}$
cavity. The lowest threshold current density of 400 A/cm2 is obtained from a
2$\times$1500 $\mu m^{2}$ cavity, which corresponds to 80 A/cm2 per QD layer.
At room temperature (20 ∘C), the QD laser exhibits a stable single-mode
emission in the presence of a side-mode-suppression-ratio above 50 dB above
the threshold (Fig. 1e). An SMSR of 59 dB is achieved when the laser operates
at 35 mA. The current coefficient of the Bragg wavelength is as low as 0.022
nm/mA (Fig. 1d). It should be noted that the threshold current of a DFB laser
is not only determined by the gain medium but also dependent on the optical
mismatch between the laser gain peak and the Bragg wavelength [26]. Such an
optical mismatch can be detuned by varying the operating temperature due to
the different temperature coefficients of the gain and the DFB wavelengths.
Figure 1c depicts the evolution of the Bragg wavelength caused by the
temperature variation. The extracted temperature coefficient of 0.077 nm/∘C is
much lower than the temperature coefficient of the optical gain peak. As a
result of the reduction in the optical mismatch, the threshold current is
reduced from 15 to 13 mA as the temperature increases from 20 to 35 ∘C (Fig.
1f). On the other hand, it should be noted that the requirement of laser
cooling is a major challenge for PICs due to laser degradation at elevated
temperatures. Such an inconvenience comes from both the increased nonradiative
carrier recombination and the reduced gain due to the wider spreading of the
Fermi distribution of carriers when they are heated up. The three-dimensional
carrier confinement in III-V QDs equips these nanoparticles with superior
insensitivity to the growth defects for the significantly reduced carrier
diffusion length compared to QWs [27]. The large energy separation between the
ground and the excited state from the tight carrier confinement also enables
better thermal stability [23]. A QD laser with a 1 mm-long cavity yields a
characteristic temperature $T_{0}$ of 113 K in the range of 35 to 55 ∘C, and
$T_{0}$ of 40 K in the range of 55 to 80 ∘C. A moderate cavity length of 1.5
mm allows for improving the $T_{0}$ to 423 K and 92 K in the range of 35 to 55
∘C and 55 to 80 ∘C, respectively (Fig. 1f). It should be noted that the
$T_{0}$ extracted in the elevated temperature range is underestimated due to
the additional increase in the threshold current induced by the aforementioned
optical mismatch. In addition to the lasing performance, the QD laser is also
beneficial to coherent applications owing to its near-zero LEF [28]. It is
worth stressing that the QD laser does not suffer from any degradation in the
LEF when it is directly grown on Si. In the case of a single-frequency DFB
laser, it is crucial to reduce the optical mismatch to take advantage of the
intrinsic LEF of the active region. Figure 1g depicts the frequency noise
spectra of the QD DFB laser with a cavity length of 1 mm (burgundy) and 1.5 mm
(navy blue). The fiber-coupled power is fixed to 0 dBm and both devices
operate at room temperature where the optical mismatch approaches zero. Both
devices exhibit remarkable frequency noise in the presence of the white noise
level below 1.8$\times 10^{4}$ Hz2/Hz. A slightly better linewidth is observed
in the 1.5 mm-long QD laser, where the white noise level is as low as
1.3$\times 10^{4}$ Hz2/Hz, yielding a Lorentzian linewidth of 41 kHz.

Figure 2: Laser dynamics under external optical feedback. a. Experimental
setup for the external cavity locking, including an off-chip external cavity
and a self-heterodyne setup for linewidth characterizations. VOA, variable
optical attenuator; BKR, back reflector; ISO, optical isolator; PC,
polarization controller; AOM, acoustic-optics modulator; PD, photodiode; ESA,
electrical spectrum analyzer; PNA, phase noise analyzer. b. Relative intensity
noise of the free-running QD (burgundy) and the QW (navy blue) laser. The
output power of both lasers is fixed at 0 dBm. c. Theoretically calculated
mapping of the spectral linewidth as a function of the feedback strength of
the QD laser. d. Experimentally measured mapping of the spectral linewidth as
a function of the feedback strength of the QD laser. e. Turnkey operation of
the ECL QD laser under a modulation of the feedback strength between -60 and
-9.6 dB. f. Theoretically calculated mapping of the spectral linewidth as a
function of the feedback strength of the QW laser. g. Experimentally measured
mapping of the spectral linewidth as a function of the feedback strength of
the QW laser. h. Transition from regime II to regime IV of the QW laser.
Regime I, stable operation regime; Regime II, external cavity locking regime;
Regime IV, coherence collapse regime.

Chaos-free operation under external optical feedback

Laser dynamics under external optical feedback (EOF) have been extensively
studied during the past four decades. The stability and instabilities of
semiconductor lasers under EOF have opened up various applications. The
coherent feedback allows for an increase in the intracavity photon density,
which is beneficial to significantly reduce the laser linewidth. Nevertheless,
a strong EOF would possibly lead to a severe coherence collapse. To take
advantage of the narrow linewidth under EOF, it is crucial to suppress the
coherence collapse of the laser. Compared to the conventional QW laser, the
chaos-free QD laser opens up new opportunities for laser linewidth locking.
Figure 2d depicts the laser linewidth as a function of the on-chip feedback
strength for a wavelength-detuned QD laser while fixing its output power at 0
dBm. The experimental setup for the EOF and the linewidth characterization is
shown in Fig. 2a. The QD laser remains in regime I in which the linewidth is
unaffected when the feedback strength is below -35 dB. As the on-chip feedback
strength increases from -35 to -9.6 dB, which is achievable by taking
advantage of the low-loss SiN waveguide, a significant linewidth reduction
takes place and the QD laser enters into regime II. Upon a periodic modulation
of the feedback strength between -60 and -9.6 dB, the QD laser exhibits a
turnkey transition from the unlocked state to the locked state and vice versa,
as shown in Fig. 2e. On the contrary, the QW laser is much more sensitive to
the EOF than the QD laser. Figure 2g depicts the evolution of laser linewidth
due to external feedback for a commercial QW laser. The output power of the QW
laser is fixed at 0 dBm for a fair comparison. Despite the ECL regime (regime
II) when the feedback strength is in the range of -70 and -37 dB, the QW laser
suffers from coherence collapse (regime IV) when the feedback strength is
above -37 dB. A zoom-in of the transition from regime II to regime IV is shown
in Fig. 2h.

The 35 dB improved feedback insensitivity in the wavelength-detuned QD laser
is mainly attributed to the large damping factor [29]. Detailed discussions of
the feedback sensitivity are available in the Supplementary Information.
Despite the fact that a near-zero LEF is beneficial for suppressing the
coherence collapse, it is not always favorable to the ECL since it suppresses
the regime II as well [26]. On the other hand, a large LEF is crucial to
realize a significant linewidth reduction [30]. To suppress the coherence
collapse while achieving a large linewidth narrowing, we propose increasing
the optical mismatch between the optical gain peak and the Bragg wavelength in
the QD laser. In this study, we employ a well-designed DFB grating to enable
the laser to operate at 1323 nm, which is 12 nm away from the gain peak at
room temperature (Supplementary Information). As a result, the LEF of the
lasing mode at the threshold is increased to 2.6, comparable to that of the QW
lasers. The consequent increase in the internal loss results in an increase in
the threshold current to 60 mA (Fig. 3c). Despite the increased LEF, the
strong damping of the QD laser can still suppress the coherence collapse. The
theoretical analyses of the difference in the feedback sensitivity between the
QD and the QW laser are available in Supplementary Information. The simulation
results are summarized in Figs. 2c and f, which are in good agreement with the
experiments. Here, we have validated through the theory that a design of the
optical mismatch in a QD DFB laser enables a linewidth reduction regime while
maintaining laser coherence. Moreover, the strong damping of the QD laser is
advantageous for suppressing the relaxation oscillation frequency, resulting
in lower relative intensity noise (RIN). Figure 2b depicts the RIN spectra of
the free-running QD (burgundy) and the QW (navy) laser at an output power of 0
dBm. The RIN of the QD laser is lower than that of the QW counterpart in the
frequency range from 10 MHz to 10 GHz, which is promising for LIDAR
applications.

Figure 3: Turnkey external cavity locking operation. a. Operation of turnkey
external cavity locking by using a low-Q external cavity. A back-reflector is
used to reflect the laser emission (red arrow) back to the laser cavity. Laser
operating in the feedback regime II allows the delayed laser field (green
arrow) to go back to the laser cavity and interacts with the intracavity field
for linewidth reduction. In this work, the external cavity is developed by
using optical fiber and a fiber-based reflector. b. Operation of self-
injection locking by using a high-Q microresonator. A high-Q external cavity
provides a narrow frequency filter for the pump laser, which enables
frequency-selective external feedback. The pump laser light (red arrow) should
be controlled to anchor to one of the external cavity resonances to generate a
reflected field. The backscattered coherent light (yellow arrow) is reinjected
into the laser cavity and interacts with the intracavity field for linewidth
reduction. c. Calculated light-current characteristics (LI, solid curves) of
the wavelength-detuned QD laser in free-running (burgundy) and in ECL
operation. In both low-Q (navy blue) and high-Q (black) locking conditions,
the QD laser doesn’t suffer from a power penalty. The measured LIs are shown
by the colored bullet point symbols. d. Frequency noise spectra of the QD
laser in free-running (burgundy) and in ECL (black) operation in the
experiment. e. Calculated frequency noise spectra of the QD laser in free-
running (burgundy) and in ECL (black) operation. f. Calculated Lorentzian
linewidth as a function of injection current in free-running (burgundy) and in
ECL operation. A low-Q locking condition (navy blue) in this work allows for a
16 Hz linewidth. A high-Q condition (black) can further reduce the laser
linewidth to hertz level. Experimental results are shown by the colored
triangle symbols. g. Frequency noise spectra of the QW laser in free-running
(burgundy) and in ECL (black) operation in the experiment. h. Calculated
frequency noise spectra of the QW laser in free-running (burgundy) and in ECL
(black) operation.

Turnkey locking without power penalty

The severe coherence collapse in the conventional QW laser can be suppressed
through the use of a high-Q external cavity that offers frequency-selective
feedback (Fig. 3b). This approach provides a narrow frequency filter that only
allows the coherent photons to go back to the laser cavity, thus suppressing
the coherence collapse of the laser and enabling the QW laser to achieve large
linewidth narrowing through the strong backscattering of the microresonator.
Nevertheless, this approach requires a high-Q resonator, which can be a
challenge to fabricate. Any degradation in the external cavity Q-factor due to
fabrication issues can lead to a worsening of laser linewidth locking or even
a failure of the SIL. Moreover, anchoring the laser frequency to one of the
external cavity resonances demands careful setpoints for laser linewidth
locking; the laser also suffers from a power penalty due to the limitation of
the external cavity Q-factor. On the contrary, a low-Q external cavity
provides a turnkey option for laser linewidth reduction without causing any
power penalty (Fig. 3a). Nevertheless, the laser can suffer from coherence
collapse under strong feedback without a frequency filter.

It used to be a challenge to reduce the QW laser linewidth to hertz level with
a low-Q external cavity, due to the weak coherent feedback that it offers. In
general, the coherence of QW external-cavity lasers is more than one order of
magnitude inferior to the high-Q SIL lasers [31, 32, 33]. Such a performance
gap is attributed to the coherence collapse of the QW laser under strong
external feedback. Using the ECL approach by applying the maximum coherent
feedback strength of -39 dB, the QW laser linewidth is reduced to 111 Hz,
which is 34.9 dB lower than its free-running linewidth (Fig. 3g). On the
contrary, the chaos-free QD laser allows for bridging the gap between ECL and
SIL approaches. In this study, the low-Q external cavity is developed by using
optical fiber and a fiber-based reflector without frequency filtering. With
both lasers having their output powers fixed at 0 dBm for a fair comparison, a
37.2 dB suppression of the white noise level and a 16 Hz Lorentizan linewidth
have been achieved by a QD laser under on-chip resonant feedback strength of
-9.6 dB (Fig. 3d). The frequency noise reduction is greater than 30 dB across
the entire offset frequency range from 1 kHz to 5 MHz. Such a record QD laser
linewidth, requiring no sophisticated tuning and locking, is on par with the
self-injection-locked QW lasers using a high-Q resonator. The turnkey locking
of the QD laser exhibited long-term stability, in the presence of a repeatable
narrow linewidth holding at least 40 seconds (Fig. 2e.) Moreover, in both
experiment and theory, we demonstrate that the laser source does not suffer
from a power penalty by using the ECL approach. On the contrary, the laser
output power gets amplified in the locking operation owing to the additional
coherent photons given by the reflected field for lasing (Fig. 3c).

The remarkable ECL performance of the QD laser is twofold: First, the QD laser
allows for generating a narrower free-running linewidth than the QW laser
owing to its lower LEF. The free-running linewidth of the wavelength-detuned
QD laser is 84 kHz, which is four times narrower than its QW counterpart.
Second, the chaos-free QD laser does not require a frequency filter to
suppress the coherence collapse, thus it can take advantage of the strong
coherent feedback offered by a low-Q external cavity. As a result, both the
linewidth and the linewidth reduction ratio are improved by utilizing the QD
laser. Theoretical analyses of the frequency noise reduction are available in
the Supplementary Information. The calculated frequency noise of the QD and
the QW laser are shown in Figs. 3e and h, respectively, agree with the
experiments. Last but not least, it should be noted that the QD laser is also
compatible with the high-Q SIL approach and its full potential has yet to be
realized. In this work, the 16 Hz intrinsic linewidth is obtained by using an
external cavity whose Q-factor is inferior to the Q-factor of the laser cavity
($Q_{e}/Q_{l}$ = 0.2 in theory, the blue curve in Fig. 3f). Our theory
indicates that the laser linewidth can be further reduced to hertz level by
increasing the Q-factor of the external cavity beyond that of the laser cavity
($Q_{e}/Q_{l}$ = 3.8 in theory, the black curve in Fig. 3f).

Table 1: Performance comparison of narrow-linewidth lasers using high-Q self-
injection locking and low-Q external cavity locking approaches.

Operation principle |  Configuration (G/EC)∗ |  Q (M) |  Linewidth (Hz) |  Ref. No.
---|---|---|---|---
ECL† |  Monolithic QD-Si/optical fiber |  N/A |  16 |  This work
SIL‡ |  Hybrid III-V/Si3N4 |  164 |  0.04 |  [17]
SIL |  Hybrid III-V/Si3N4 |  260 |  1.2 |  [14]
SIL |  Heterogeneous III-V/Si3N4 |  50 |  5 |  [34]
SIL |  Heterogeneous III-V/Si3N4 |  7 |  25 |  [35]
SIL |  Hybrid III-V/Si3N4 |  >15 |  25 |  [7]
SIL |  Heterogeneous III-V/Si3N4 |  10 |  92.4 |  [36]
SIL |  Hybrid III-V/Si3N4-LiNbO3 |  1.9 |  3,140 |  [37]
ECL |  Hybrid III-V/Si3N4 |  N/A |  40 |  [38]
ECL |  Heterogeneous III-V/Si3N4 |  N/A |  140 |  [31]
ECL |  Heterogeneous III-V/Si3N4 |  N/A |  400 |  [39]
ECL |  Monolithic III-V |  N/A |  50,000 |  [40]

∗ G, gain medium; EC, external cavity. † ECL, external-cavity locking. ‡ SIL,
self-injection locking.

Discussion and outlook

This work demonstrates a high-performance QD laser on Si that achieves a
Lorentzian linewidth of 16 Hz under external cavity locking. The quantized QD
laser opens up more opportunities for large-scale PICs owing to its
compatibility with the mature CMOS foundry. Compared to the QD lasers grown on
the native substrate, directly growing the QD lasers on Si improves not only
the scalability using the commercial 300 mm Si wafers but also the laser
performance. For instance, the epitaxial QD laser on Si enabled a record LEF
as low as 0.13 [28], whereas the LEF reported in the native QD laser was in
general above 0.7 [41]. Such an improvement results from the optimization of
the inhomogeneous broadening and the defects. The shortening of Shockley-Read
Hall (SRH) recombination time also contributes to the near-zero LEF in the
epitaxial QD laser on Si [42]. Superior to conventional QW heterostructure,
the advanced QD enables monolithic integration of laser sources, amplifiers,
and detectors on Si, which is a low-cost solution for Si PICs. After
optimizing the fabrication, the QD DFB laser performance has been improved.
This includes the decrease in the threshold current density to 400 A/cm2, the
increase in the SMSR to 59 dB, the reduction in the intrinsic linewidth to 41
kHz, and the improvement in the yield [25]. Previous studies have shown that
the near-zero LEF offered by the QD laser allows for isolator-free photonic
integration [43]. Here, we reveal that the LEF of the QD laser can be
practically engineered by the wavelength detuning between the optical gain
peak and the Bragg wavelength. As such, an increase in the LEF through a
wavelength-detuned DFB grating contributes to a large linewidth reduction
while maintaining the laser’s coherence. By leveraging the chaos-free
operation of the QD laser, its linewidth can be locked to a low-Q external
cavity without the need for a frequency filter. Table I1 compares the current
state-of-the-art narrow-linewidth lasers using either the high-Q SIL approach
or the low-Q ECL approach. While an ultra-high-Q resonator using the low-loss
SiN platform enables hertz level and sub-hertz level laser linewidth [14, 17,
34], fabrication is challenging as any degradation in the Q-factor would
result in a broader laser linewidth [35, 7, 36]. In addition, achieving ultra-
high-Q resonators in many material platforms, such as lithium niobate (LN)
[44], poses significant challenges. As a result, the locked linewidth of SIL
lasers using LN resonators is on the order of kilohertz [37], which remains to
be improved for coherent LIDAR applications. Our work, however, demonstrates
that the locked QD laser linewidth is on par with or even better than most
high-Q SIL QW lasers. This finding underscores the potential of QD lasers in
applications requiring precise coherence, particularly where achieving ultra-
high-Q resonators is challenging. In addition, our approach provides
additional benefits including turnkey and power-maintaining operations. On the
other hand, employing a low-Q external cavity can potentially enhance
operation stability due to its broader reflection bandwidth, which is more
accommodating of laser wavelength shifts. Along with the high thermal
stability of epitaxial QD lasers on Si - capable of continuous-wave operation
above 150 ∘C [45] \- this approach is beneficial to the robustness of ECL in
high-temperature operation.

Further improving the laser coherence can be effectively achieved with an on-
chip external cavity solution. Using a fiber-based external cavity, the
feedback strength is mainly limited by the laser-to-fiber coupling loss. By
leveraging the laser and the external cavity on the same chip, this coupling
loss can be largely reduced. 3D photonic integration can potentially leverage
the ultra-low-loss SiN waveguide, which is advantageous for maximizing the on-
chip coherent feedback strength and reducing laser linewidth [34].

Another approach to improve the SIL is to apply a frequency filter by using a
high-Q resonator. The QD laser is also compatible with the CMOS-ready SiN and
silica microresonators that offer 1 billion Q-factor [46, 47]. We
theoretically demonstrate that sub-hertz level linewidth can be achieved by
using the QD laser and a high-Q resonator [48, 49].Additionally, spiral
microresonators with large modal volumes can be employed to reduce the low-
offset frequency noise caused by thermodynamic fluctuations [17]. Depending on
the design of the external cavity, the high-coherence QD laser can scale up to
300 mm Si wafers using either monolithic or heterogeneous photonic integration
technologies. This would lead to low-cost, large-volume, and scalable
production. The potential for monolithic integration of III-V lasers and
passive waveguides onto a single chip using foundry-based technologies could
further advance the development of QD lasers and their integration into
practical photonic applications.

Data Availability: All data generated or analyzed during this study are
available within the paper and its Supplementary Information. Further source
data will be made available online once published.

Code Availability: The analysis codes will be made available on reasonable
request.

Acknowledgments: This work is supported by the Defense Advanced Research
Projects Agency (DARPA) MTO GRYPHON (HR0011-22-2-0009) and LUMOS
(HR0011-20-2-0044) programs. We thank Emad Alkhazraji for fruitful discussions
and figure polish.

Author contribution: B.D. and Y.W. led the QD laser design and device
characterization. Y.W. designed and fabricated the QD lasers with assistance
from C.S. and R.K. B.D. characterized and gathered the experimental data from
the device, with contributions from R.K.. W.W.C. provided theoretical
calculations and analysis on the locking dynamics and feedback sensitivity,
with contributions from A.P. and H.W.. B.D. wrote the manuscript with inputs
from Y.W., C.S. and A.P.. All authors commented on and edited the manuscript.
J.E.B supervised the project.

Competing Interests: J.E.B.is a cofounder and shareholder of Nexus Photonics
and Quintessent, startups in silicon photonics.

## References

  * Spencer  _et al._ [2018] D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, _et al._ , An optical-frequency synthesizer using integrated photonics, Nature 557, 81 (2018).
  * Marpaung  _et al._ [2019] D. Marpaung, J. Yao, and J. Capmany, Integrated microwave photonics, Nature Photonics 13, 80 (2019).
  * Gundavarapu  _et al._ [2019] S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, N. Chauhan, _et al._ , Sub-Hertz fundamental linewidth photonic integrated Brillouin laser, Nature Photonics 13, 60 (2019).
  * Lai  _et al._ [2020] Y.-H. Lai, M.-G. Suh, Y.-K. Lu, B. Shen, Q.-F. Yang, H. Wang, J. Li, S. H. Lee, K. Y. Yang, and K. Vahala, Earth rotation measured by a chip-scale ring laser gyroscope, Nature Photonics 14, 345 (2020).
  * Trocha  _et al._ [2018] P. Trocha, M. Karpov, D. Ganin, M. H. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, _et al._ , Ultrafast optical ranging using microresonator soliton frequency combs, Science 359, 887 (2018).
  * Suh and Vahala [2018] M.-G. Suh and K. J. Vahala, Soliton microcomb range measurement, Science 359, 884 (2018).
  * Lihachev  _et al._ [2022a] G. Lihachev, J. Riemensberger, W. Weng, J. Liu, H. Tian, A. Siddharth, V. Snigirev, V. Shadymov, A. Voloshin, R. N. Wang, _et al._ , Low-noise frequency-agile photonic integrated lasers for coherent ranging, Nature communications 13, 3522 (2022a).
  * Suh  _et al._ [2016] M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, Microresonator soliton dual-comb spectroscopy, Science 354, 600 (2016).
  * Olsson  _et al._ [2018] S. L. Olsson, J. Cho, S. Chandrasekhar, X. Chen, P. J. Winzer, and S. Makovejs, Probabilistically shaped PDM 4096-QAM transmission over up to 200 km of fiber using standard intradyne detection, Optics express 26, 4522 (2018).
  * Matei  _et al._ [2017] D. Matei, T. Legero, S. Häfner, C. Grebing, R. Weyrich, W. Zhang, L. Sonderhouse, J. Robinson, J. Ye, F. Riehle, _et al._ , 1.5 $\mu$ m lasers with sub-10 mHz linewidth, Physical Review Letters 118, 263202 (2017).
  * Dahmani  _et al._ [1987] B. Dahmani, L. Hollberg, and R. Drullinger, Frequency stabilization of semiconductor lasers by resonant optical feedback, Optics letters 12, 876 (1987).
  * Hollberg and Ohtsu [1988] L. Hollberg and M. Ohtsu, Modulatable narrow-linewidth semiconductor lasers, Applied physics letters 53, 944 (1988).
  * Laurent  _et al._ [1989] P. Laurent, A. Clairon, and C. Breant, Frequency noise analysis of optically self-locked diode lasers, IEEE Journal of Quantum Electronics 25, 1131 (1989).
  * Jin  _et al._ [2021] W. Jin, Q.-F. Yang, L. Chang, B. Shen, H. Wang, M. A. Leal, L. Wu, M. Gao, A. Feshali, M. Paniccia, K. J. Vahala, and J. E. Bowers, Hertz-linewidth semiconductor lasers using CMOS-ready ultra-high-Q microresonators, Nature Photonics 15, 346 (2021).
  * Lihachev  _et al._ [2022b] G. Lihachev, W. Weng, J. Liu, L. Chang, J. Guo, J. He, R. N. Wang, M. H. Anderson, Y. Liu, J. E. Bowers, _et al._ , Platicon microcomb generation using laser self-injection locking, Nature Communications 13, 1771 (2022b).
  * Guo  _et al._ [2022] J. Guo, C. A. McLemore, C. Xiang, D. Lee, L. Wu, W. Jin, M. Kelleher, N. Jin, D. Mason, L. Chang, _et al._ , Chip-based laser with 1-hertz integrated linewidth, Science advances 8, eabp9006 (2022).
  * Li  _et al._ [2021] B. Li, W. Jin, L. Wu, L. Chang, H. Wang, B. Shen, Z. Yuan, A. Feshali, M. Paniccia, K. J. Vahala, and J. E. Bowers, Reaching fiber-laser coherence in integrated photonics, Optics Letters 46, 5201 (2021).
  * Dong  _et al._ [2021a] B. Dong, J.-D. Chen, F.-Y. Lin, J. C. Norman, J. E. Bowers, and F. Grillot, Dynamic and nonlinear properties of epitaxial quantum-dot lasers on silicon operating under long-and short-cavity feedback conditions for photonic integrated circuits, Physical Review A 103, 033509 (2021a).
  * Liu  _et al._ [2015] A. Y. Liu, S. Srinivasan, J. Norman, A. C. Gossard, and J. E. Bowers, Quantum dot lasers for silicon photonics, Photonics Research 3, B1 (2015).
  * Shang  _et al._ [2022] C. Shang, K. Feng, E. T. Hughes, A. Clark, M. Debnath, R. Koscica, G. Leake, J. Herman, D. Harame, P. Ludewig, Y. Wan, and J. E. Bowers, Electrically pumped quantum-dot lasers grown on 300 mm patterned Si photonic wafers, Light: Science & Applications 11, 1 (2022).
  * Shang  _et al._ [2021a] C. Shang, Y. Wan, J. Selvidge, E. Hughes, R. Herrick, K. Mukherjee, J. Duan, F. Grillot, W. W. Chow, and J. E. Bowers, Perspectives on advances in quantum dot lasers and integration with Si photonic integrated circuits, ACS photonics 8, 2555 (2021a).
  * Zhou  _et al._ [2023] Z. Zhou, X. Ou, Y. Fang, E. Alkhazraji, R. Xu, Y. Wan, and J. E. Bowers, Prospects and applications of on-chip lasers, Elight 3, 1 (2023).
  * Shang  _et al._ [2021b] C. Shang, E. Hughes, Y. Wan, M. Dumont, R. Koscica, J. Selvidge, R. Herrick, A. C. Gossard, K. Mukherjee, and J. E. Bowers, High-temperature reliable quantum-dot lasers on Si with misfit and threading dislocation filters, Optica 8, 749 (2021b).
  * Wan  _et al._ [2021] Y. Wan, C. Xiang, J. Guo, R. Koscica, M. Kennedy, J. Selvidge, Z. Zhang, L. Chang, W. Xie, D. Huang, _et al._ , High Speed Evanescent Quantum-Dot Lasers on Si, Laser & Photonics Reviews 15, 2100057 (2021).
  * Wan  _et al._ [2020] Y. Wan, J. C. Norman, Y. Tong, M. Kennedy, W. He, J. Selvidge, C. Shang, M. Dumont, A. Malik, H. K. Tsang, _et al._ , 1.3 $\mu$m quantum dot-distributed feedback lasers directly grown on (001) Si, Laser & Photonics Reviews 14, 2000037 (2020).
  * Dong  _et al._ [2021b] B. Dong, J. Duan, H. Huang, J. C. Norman, K. Nishi, K. Takemasa, M. Sugawara, J. E. Bowers, and F. Grillot, Dynamic performance and reflection sensitivity of quantum dot distributed feedback lasers with large optical mismatch, Photonics Research 9, 1550 (2021b).
  * Selvidge  _et al._ [2019] J. Selvidge, J. Norman, M. E. Salmon, E. T. Hughes, J. E. Bowers, R. Herrick, and K. Mukherjee, Non-radiative recombination at dislocations in InAs quantum dots grown on silicon, Applied Physics Letters 115, 131102 (2019).
  * Duan  _et al._ [2018a] J. Duan, H. Huang, D. Jung, Z. Zhang, J. Norman, J. Bowers, and F. Grillot, Semiconductor quantum dot lasers epitaxially grown on silicon with low linewidth enhancement factor, Applied Physics Letters 112, 251111 (2018a).
  * Grillot  _et al._ [2020] F. Grillot, J. C. Norman, J. Duan, Z. Zhang, B. Dong, H. Huang, W. W. Chow, and J. E. Bowers, Physics and applications of quantum dot lasers for silicon photonics, Nanophotonics 9, 1271 (2020).
  * Galiev  _et al._ [2020] R. R. Galiev, N. M. Kondratiev, V. E. Lobanov, A. B. Matsko, and I. A. Bilenko, Optimization of laser stabilization via self-injection locking to a whispering-gallery-mode microresonator, Physical Review Applied 14, 014036 (2020).
  * Tran  _et al._ [2019] M. A. Tran, D. Huang, and J. E. Bowers, Tutorial on narrow linewidth tunable semiconductor lasers using Si/III-V heterogeneous integration, APL photonics 4, 111101 (2019).
  * Morton  _et al._ [2021] P. A. Morton, C. Xiang, J. B. Khurgin, C. D. Morton, M. Tran, J. Peters, J. Guo, M. J. Morton, and J. E. Bowers, Integrated coherent tunable laser (ICTL) with ultra-wideband wavelength tuning and sub-100 Hz Lorentzian linewidth, Journal of Lightwave Technology 40, 1802 (2021).
  * Maier  _et al._ [2023] P. Maier, Y. Chen, Y. Xu, Y. Bao, M. Blaicher, D. Geskus, R. Dekker, J. Liu, P.-I. Dietrich, H. Peng, _et al._ , Sub-kHz-linewidth external-cavity laser (ECL) with Si3N4 resonator used as a tunable pump for a Kerr frequency comb, Journal of Lightwave Technology 10.1109/JLT.2023.3243471 (2023).
  * Xiang  _et al._ [2023] C. Xiang, W. Jin, O. Terra, B. Dong, H. Wang, L. Wu, J. Guo, T. J. Morin, E. Hughes, J. Peters, _et al._ , 3D integration enables ultralow-noise isolator-free lasers in silicon photonics, Nature 620, 78 (2023).
  * Xiang  _et al._ [2021a] C. Xiang, J. Liu, J. Guo, L. Chang, R. N. Wang, W. Weng, J. Peters, W. Xie, Z. Zhang, J. Riemensberger, J. Selvidge, T. J. Kippenberg, and J. E. Bowers, Laser soliton microcombs heterogeneously integrated on silicon, Science 373, 99 (2021a).
  * Zhang  _et al._ [2023] Z. Zhang, B. Shen, M. A. Tran, W. Lee, K. Asawa, G. Kim, Y. Shen, T. J. Morin, A. Malik, J. E. Bowers, _et al._ , Photonic integration platform for rubidium sensors and beyond, Optica 10, 752 (2023).
  * Snigirev  _et al._ [2023] V. Snigirev, A. Riedhauser, G. Lihachev, M. Churaev, J. Riemensberger, R. N. Wang, A. Siddharth, G. Huang, C. Möhl, Y. Popoff, _et al._ , Ultrafast tunable lasers using lithium niobate integrated photonics, Nature 615, 411 (2023).
  * Fan  _et al._ [2020] Y. Fan, A. van Rees, P. J. Van der Slot, J. Mak, R. M. Oldenbeuving, M. Hoekman, D. Geskus, C. G. Roeloffzen, and K.-J. Boller, Hybrid integrated InP-Si3N4 diode laser with a 40-Hz intrinsic linewidth, Optics Express 28, 21713 (2020).
  * Xiang  _et al._ [2021b] C. Xiang, J. Guo, W. Jin, L. Wu, J. Peters, W. Xie, L. Chang, B. Shen, H. Wang, Q.-F. Yang, D. Kinghorn, M. Paniccia, K. J. Vahala, P. A. Morton, and J. E. Bowers, High-performance lasers for fully integrated silicon nitride photonics, Nature Communications 12, 6650 (2021b).
  * Larson  _et al._ [2015] M. Larson, A. Bhardwaj, W. Xiong, Y. Feng, X. Huang, K. Petrov, M. Moewe, H. Ji, A. Semakov, C. Lv, _et al._ , Narrow linewidth sampled-grating distributed Bragg reflector laser with enhanced side-mode suppression, in  _Optical Fiber Communication Conference_ (Optica Publishing Group, 2015) pp. M2D–1.
  * Duan  _et al._ [2018b] J. Duan, H. Huang, Z. Lu, P. Poole, C. Wang, and F. Grillot, Narrow spectral linewidth in InAs/InP quantum dot distributed feedback lasers, Applied Physics Letters 112, 10.1063/1.5022480 (2018b).
  * Zhao and Grillot [2021] S. Zhao and F. Grillot, Effect of shockley-read-hall recombination on the static and dynamical characteristics of epitaxial quantum-dot lasers on silicon, Physical Review A 103, 063521 (2021).
  * Duan  _et al._ [2019] J. Duan, H. Huang, B. Dong, D. Jung, J. C. Norman, J. E. Bowers, and F. Grillot, 1.3-$\mu m$ Reflection Insensitive InAs/GaAs Quantum Dot Lasers Directly Grown on Silicon, IEEE Photonics Technology Letters 31, 345 (2019).
  * Zhang  _et al._ [2017] M. Zhang, C. Wang, R. Cheng, A. Shams-Ansari, and M. Lončar, Monolithic ultra-high-Q lithium niobate microring resonator, Optica 4, 1536 (2017).
  * Wang  _et al._ [2023] Y. Wang, B. Ma, J. Li, Z. Liu, C. Jiang, C. Li, H. Liu, Y. Zhang, Y. Zhang, Q. Wang, _et al._ , InAs/GaAs quantum-dot lasers grown on on-axis Si (001) without dislocation filter layers, Optics Express 31, 4862 (2023).
  * Wu  _et al._ [2020] L. Wu, H. Wang, Q. Yang, Q.-x. Ji, B. Shen, C. Bao, M. Gao, and K. Vahala, Greater than one billion Q factor for on-chip microresonators, Optics Letters 45, 5129 (2020).
  * Liu  _et al._ [2022] K. Liu, N. Jin, H. Cheng, N. Chauhan, M. W. Puckett, K. D. Nelson, R. O. Behunin, P. T. Rakich, and D. J. Blumenthal, Ultralow 0.034 dB/m loss wafer-scale integrated photonics realizing 720 million Q and 380 $\mu$W threshold Brillouin lasing, Optics letters 47, 1855 (2022).
  * Chow  _et al._ [2022] W. W. Chow, Y. Wan, J. E. Bowers, and F. Grillot, Analysis of the spontaneous emission limited linewidth of an integrated III–V/SiN laser, Laser & Photonics Reviews 16, 2100620 (2022).
  * Alkhazraji  _et al._ [2023] E. Alkhazraji, W. W. Chow, F. Grillot, J. E. Bowers, and Y. Wan, Linewidth narrowing in self-injection-locked on-chip lasers, Light: Science & Applications 12, 162 (2023).

Methods

Free-running laser characterizations

The optical spectra are measured using a high-resolution Optical Spectrum
Analyzer (OSA) (Yokogawa AQ6370C). The laser frequency noise and resultant
fundamental linewidth are taken from a commercial laser phase noise analyzer
(PNA) (OEwaves OE4000) that internally performs averaging over the measured
phase noise. As automated equipment, this PNA offers a measurement bandwidth
of 150 MHz. An average of 50 was applied in the range of 1 kHz to 150 MHz to
measure the frequency noise. In the measurements of optical spectra, the QD
laser is driven by a programmable current source (Keithley 2400). It is then
driven by a low-noise laser current source (ILX Lightwave LDX-3620) to ensure
stable and low-noise operation in the measurement of frequency noise. The
laser relative intensity noise (RIN) is measured by a 20 GHz photodetector
(Agilent 11982A). The converted AC signal filtered by a bias-tee is
preamplified by a 30 dB RF amplifier (RF-Lambda RLNA00G18GA) before it is sent
to an Electrical Spectrum Analyzer (ESA) (Rohde & Schwarz FSU50). The
resolution bandwidth (RBW) used in the RIN measurement is 1 MHz. The RIN
bandwidth is limited by the RF amplifier, which is 18 GHz.

Laser external cavity locking

Figure 2a depicts the experimental configurations for laser external cavity
locking. The coupled laser emission is sent to a 90/10 fiber beam splitter,
after which 90% of the coupled power will be used for external optical
feedback. The feedback loop consists of a polarization controller (PC), a
variable optical attenuator (VOA), and a back-reflector (EXFO OSICS BKR). The
external cavity frequency is 6 MHz, which corresponds to 16.6 m-long optical
fiber. The remaining 10% of the laser output is utilized for the
characterizations of feedback sensitivity and laser linewidth. After passing
through an optical isolator, it is transferred either to a phase noise
analyzer (PNA) (OEWaves OE4000) for frequency noise characterization or a
delayed self-heterodyne setup to check the electrical spectrum-based laser
coherence.

In this study, the feedback strength is determined by the on-chip reflected
power ($P_{refl}$) and the free-space output power ($P_{out}$) through the
following relationship:

$\eta_{F}=\frac{P_{refl}}{P_{out}}$ (1)

All losses from the external cavity should be considered to calculate the
reflected power, thus the feedback strength. After optimizing the setup, the
fiber-chip coupling loss is -2.4 dB (round-trip coupling loss is -4.8 dB), the
total losses from the 90% beam splitter, the insertion loss of the VOA, the
polarization controller, and the fiber is -4.81 dB. The maximum on-chip
feedback strength that we can achieve is -9.61 dB.

In the external cavity locking operation, the laser is driven with low-noise
laser current sources (ILX Lightwave LDX-3620) to ensure stable and low-noise
operation. Detection of the locking state is assured by the reduction in the
linewidth of the self-heterodyne beat. The self-heterodyne interferometer
setup consists basically of a Mach-Zehnder interferometer (made from two 3 dB
couplers) with a PC and a short delay line in one of its arms and a fiber-
coupled acoustic-optic modulator (Gooch & Housego 27 MHz) in the other arm, as
shown in Fig. 2a. The beat frequency from the self-heterodyne interferometer
is detected using a photodetector (Newport 1811) before it is sent to an
Electrical Spectrum Analyzer (ESA) (Rohde & Schwarz FSWP).

To investigate the turnkey locking stability, we periodically vary the on-chip
feedback strength between -61.2 dB and -9.61 dB. The periodicity of the
feedback strength calibration is 80 seconds. The RBW of the ESA was 10 Hz in
this measurement.


# Resonant Transmission of Electromagnetic Fields Through Subwavelength
Zero-$\epsilon$ Slits

Klaus Halterman  Simin Feng  Physics Division, Naval Air Warfare Center, China
Lake, California 93555

###### Abstract

We theoretically investigate the transmission of electromagnetic radiation
through a metal plate with a zero-$\epsilon$ metamaterial slit, where the
permittivity tends towards zero over a given bandwidth. Our analytic results
demonstrate that the transmission coefficient can be substantial for a broad
range of slit geometries, including subwavelength widths that are many
wavelengths long. This novel resonant effect has features quite unlike the
Fabry-Pérot-like resonances that have been observed in conductors with deep
channels. We further reveal that ultranarrow zero-$\epsilon$ channels can have
significantly greater transmission compared to slits with no wave impedance
difference across them.

With the current state of the art in nanofabrication technologies, and the
recent observation of resonant optical transmission through small metal holes
ebb , there has been a renewed interest in electromagnetic wave transmission
and diffraction in metallic nanostructures. Although diffraction effects
involving optical slits and gratings has a long history, scientific curiosity
coupled with the potential for device applications has spurred considerable
research activity involving the manipulation and confinement of
electromagnetic waves in nanoscale resonant structures. With the concurrent
advent of metamaterials, or composite structures with tailored electromagnetic
properties, the interplay of classical transmission systems with these new
composite media has become an important issue. By engineering metallic nano-
apertures, gratings, or channels to incorporate metamaterials, it is
anticipated that various enhanced or exotic transmission characteristics will
ensue.
Lately, there has been interest in zero-$\epsilon$ metamaterials, which are
structures that exhibit an effective permittivity of zero (or nearly so) in
the passband, akin to the property of some precious metals palik  near their
plasma frequency. It was shown that waveguide devices containing
zero-$\epsilon$ inclusions could potentially be more efficient via the
reduction of unwanted reflections if one of the physical dimensions was made
smaller eng . This can possibly negate the ill effects of impedance mismatch.
A multilayer structure having nonlinear electric and magnetic responses, and a
refractive index close to zero, was shown to effectively shield feng
electromagnetic fields. Experimental work has verified liu ; eng3
electromagnetic tunneling through zero-$\epsilon$ metamaterials at microwave
frequencies. It was shown eng2  that the effective magnetic permeability,
$\mu$, can also be resonantly tuned to vanish, creating a matched metamaterial
with an effective zero index of refraction. These media have been linked to
applications in miniaturized resonators, highly directive antennas enoch ,
delay lines with zero-phase difference I/O, and transformers that convert
small-curvature wave fronts into output beams with planar-like wave fronts zio
.
A fundamental system in which to investigate zero-$\epsilon$ diffraction and
transmission effects is a subwavelength channel through a metal film. If the
slit is filled with air, it has been established that for light polarized
perpendicular to the slit in an optically thick perfect metal of length, $l$,
Fabry-Pérot-like waveguide modes arise when $l$ is approximately a half
integer number of wavelengths: $l\approx n\lambda/2$. These harmonic modes,
which follow from geometrical arguments, result in transmission peaks when the
waves coherently superimpose over the given path length. As the geometrical
parameters vary, the resonant wavelength can shift takakura ; mybad  in metals
with slit perforations, as observed with microwaves suck0 . For ultrashort
incident pulses, the Fabry-Pérot-like modes can be resonantly activated
mechler  through the Fourier components of the wave packet inside the slit,
leading in some cases to enhanced transmission. The resonant enhancement of a
nanometer scale electromagnetic pulse was also shown to be spatially and
temporally localized mechler2  in the near field. For perfect metals with
holes rather than slits, transmission is strictly limited to incident
wavelengths that are less than twice the diameter of the openings.
In this Letter we reveal some unexpected and exotic transmission behavior of
light through a subwavelength zero-$\epsilon$ slit. Our theoretical framework
demonstrates that significant transmission can occur in these structures for a
considerable range of ultranarrow widths that are sufficiently deep. The
inverse relationship between the slit length and width that achieves maximal
transmission is shown to be highly nontrivial. We show that zero-$\epsilon$
metamaterials with inherently large intrinsic impedances, can have greater
transmission than matched zero index slits, where $\mu$ is also vanishingly
small. We also take advantage of the subwavelength geometry (that is, the slit
width is smaller than about half the incident wavelength), to invoke the
single mode approximation, which has been shown to yield valuable physical
insight into resonance phenomena for perfect mata  and real gordy  metals. Our
analytic results thus permit one to efficiently map out the relevant sector of
parameter space deemed appropriate for any future experimental endeavors.
We consider a planar perfect metallic structure that is translationally
invariant in the $x-z$ plane and has channel length $l$ normal to the plane
with width $p$ (along $x$). The incident beam is TM polarized, so that the
magnetic field ${\bm{H}}$ is directed along $z$. The wavevector, ${\bm{k}}$,
forms an angle $\theta$ with the normal to the plane. For this configuration,
the $z$ component of the magnetic field, $H_{z}$, must satisfy the scalar
Helmholtz equation,

$\displaystyle\frac{\partial^{2}H_{z}}{\partial
x^{2}}+\frac{\partial^{2}H_{z}}{\partial y^{2}}+\epsilon\mu k_{0}^{2}H_{z}=0,$
(1)

where $k_{0}=\omega/c$. Our focus is primarily metamaterial slits very near
the characteristic plasma frequency and, as in recent works, represent the
frequency dispersive electrical response by an effective Drude model,
$\epsilon=1-\omega_{p}^{2}/[\omega(\omega+i\Gamma)]$, where $\Gamma$ is
correlated to the mean free path in the filling material. Other than when
discussing absorption loss, we generally take $\Gamma$ equal to zero in order
to isolate the electromagnetic effects inherent to vanishing constitutive
relations.

We solve (1) in a given region, and then match each solution at the
corresponding interfaces using the appropriate boundary conditions. To
construct the solution in the “continuum” regions of free space surrounding
the metal sheet, we Fourier transform Eq. (1) along $x$ and use separation of
variables. This renders an expression written in terms of plane wave
expansions,

$\displaystyle H_{z,1}$
$\displaystyle=e^{i(k_{0x}x-k_{0y}y)}+\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}d\alpha
R(\alpha)e^{i\alpha x}e^{i\beta y}$ (2) $\displaystyle H_{z,3}$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}d\alpha
T(\alpha)e^{i\alpha x}e^{-i\beta y},$ (3)

where $k_{0x}=k_{0}\sin\theta$, $k_{0y}=k_{0}\cos\theta$, and
$\beta=\sqrt{k_{0}^{2}-\alpha^{2}}$. Here the subscripts 1 and 3 denote the
entrance and exit regions respectively. The unknown coefficients $R(\alpha)$
and $T(\alpha)$ are determined below. The general eigenmode expansion for the
field within the slit region is takakura ,

$\displaystyle
H_{z,2}=\sum_{m=0}^{\infty}\cos\bigl{[}k_{m}(x-p/2)\bigr{]}\bigl{[}a_{m}e^{i\tau_{m}y}+b_{m}e^{-i\tau_{m}y}\bigr{]},$
(4)

where $k_{m}\equiv m\pi/p$, and $\tau_{m}\equiv\sqrt{\epsilon\mu
k_{0}^{2}-k_{m}^{2}}$. The summation is over a single index $m$ reflecting the
interconnection among the wavenumbers consistent with the wave equation. The
electric field is obtained directly via application of Maxwell’s equation:
${\bm{E}}=i/(\epsilon k_{0})\nabla\times{\bm{H}}$. It is generally valid to
retain only the lowest order mode in the expansion (4) for the ultranarrow
channels considered in this paper. Moreover, as $\epsilon$ goes to zero, the
summation results in higher order evanescent fields that have a negligible
contribution to the time-averaged energy flux. Thus, for the electric field we
have,

$\displaystyle E_{x,2}=\eta(-ae^{i\tau y}+be^{-i\tau y})\zeta(x),$ (5)

where $\tau\equiv\tau_{0}$, $\eta=\sqrt{\mu/\epsilon}$ is the intrinsic
impedance, and the function $\zeta(x)$ is unity in the slit and vanishes
elsewhere.

Matching the electric field at the entrance and exit interfaces, yields the
$R$ and $T$ coefficients:

$\displaystyle R(\alpha)$
$\displaystyle=\frac{\sqrt{2\pi}k_{0y}\delta(k_{0x}-\alpha)}{\beta}+\frac{w}{\beta}(a-b)\Phi(\alpha
p),$ (6a) $\displaystyle T(\alpha)$ $\displaystyle=\frac{w}{\beta}(-ae^{-i\tau
l}+be^{i\tau l})e^{-i\beta l}\Phi(\alpha p),$ (6b)

where we define $\Phi(x)\equiv\sqrt{2/\pi}\sin(x/2)/x$. A convenient technique
mata  that utilizes the boundary conditions to determine the remaining unknown
$a$ and $b$ coefficients is the use of the following relationships at the
appropriate openings: $\langle H_{z,1}(x,0),\zeta(x)\rangle=\langle
H_{z,2}(x,0),\zeta(x)\rangle$, and $\langle
H_{z,2}(x,-l),\zeta(x)\rangle=\langle H_{z,3}(x,-l),\zeta(x)\rangle$. After
some tedious algebra, we have for $b$,

$\displaystyle
b=\frac{2\sqrt{2\pi}\Phi(k_{0x}p)(1+wI_{0})}{(1+wI_{0})^{2}-e^{2i\xi}(1-wI_{0})^{2}},$
(7)

where $w\equiv k_{0}p\eta$, $\xi\equiv\tau l$, and
$a=be^{2i\xi}(wI_{0}-1)/(wI_{0}+1)$. The complex valued integral, $I_{0}$ is a
function solely of $q$:

$\displaystyle I_{0}$ $\displaystyle=\frac{1}{\pi
q^{2}}\int_{0}^{\infty}du\frac{\sin^{2}(uq)}{u^{2}\sqrt{1-u^{2}}},$ (8)

where $q\equiv k_{0}p/2$ is the dimensionless measure characterizing the slit
width. The series expansion for the imaginary component of $I_{0}$ has the
following leading terms:

$\displaystyle{\rm
Im}[I_{0}]\approx-\frac{1}{6\pi}\Bigl{[}(q^{2}-6)(\gamma+\ln
q)+9-\frac{19}{12}q^{2}\Bigr{]},$ (9)

where $\gamma$ is Euler’s constant. Similarly for the real part of $I_{0}$, we
have (to fourth order), ${\rm
Re}[I_{0}]\approx\frac{1}{2}-\frac{1}{12}q^{2}+\frac{1}{120}q^{4}$. It will be
seen shortly that $I_{0}$ is a very important quantity in the determination of
the transmission.

Figure 1: Real and imaginary components of $I_{0}$ as a function of the
dimensionless width $q\equiv\pi p/\lambda$. The approximate solutions and
exact integral [Eq. (8)] have good agreement over the relevant range of
widths. As $q$ vanishes, the imaginary component has a slow divergence [see
Eq. (9)].

The accuracy of these expansions is demonstrated graphically in Fig. 1, where
the truncated expansions are plotted alongside the numerically integrated
$I_{0}$ [Eq. (8)]. Clearly, the approximations for $I_{0}$ are satisfactory
for the small $q$ of interest here, deviating only slightly for the larger
$q$, as would be expected for a power series centered about the origin.

Next, we insert the calculated $a$ and $b$ coefficients into Eq. (5), to
determine the electric field at the bottom of the slit,

$\displaystyle E^{\rm
bot}_{x}=\frac{2i\sin(q\sin\theta)\eta}{q\sin\theta[\sin\xi+4I_{0}q\eta_{2}(i\cos\xi+I_{0}q\eta_{2}\sin\xi)]},$
(10)

which is valid for arbitrary $\mu$ and $\epsilon$. The magnetic field is
easily obtained via the relationship, $H^{\rm bot}_{z}=2qI_{0}E^{\rm
bot}_{x}$. The transmission coefficient, $\tau$, is defined as the time-
averaged Poynting vector at the bottom of the slit, $\langle
S_{y}\rangle_{y=-l}$, integrated over the exit opening, and divided by the
Poynting vector of the incident plane wave:

$\displaystyle\tau=\frac{16q^{2}{\rm Re}[I_{0}]|\eta|^{2}{\cal
F}(\theta)}{|\sin\xi+4I_{0}q\eta(i\cos\xi+I_{0}q\eta\sin\xi)|^{2}},$ (11)

which is dimensionless after incorporating a normalization factor, $k_{0}$.
The angular dependence is encapsulated entirely by the expression, ${\cal
F}(\theta)={\rm sinc}^{2}(q\sin\theta)/\cos\theta$, which depends weakly on
$q$ for subwavelength slits, and is close to unity for source waves near
normal incidence. For small $q$, we can further write (for $\sin\xi\neq 0$):
$\tau\approx 8(\mu/\epsilon)q^{2}\csc^{2}(k_{0}l\sqrt{\epsilon\mu})$,
otherwise if we consider channels with some integer multiple of the Fabry-
Pérot length, $l=\lambda/(2\sqrt{\epsilon\mu})$, Eq. (11) approximately
reduces to, $\tau\approx\frac{1}{2}|I_{0}|^{-2}$, which clearly only depends
on the ratio of the slit width to the wavelength [see Eq. (8)].

Having derived the general expression for transmission of electromagnetic
fields through a subwavelength channel containing, to this point, conventional
material parameters, we now examine the effects of taking the limit of
vanishing $\epsilon$. It is relatively straightforward to show that Eq. (10)
for the electric field at the slit exit now reduces to the succinct form,

$\displaystyle E^{\rm
bot}_{x}=\sqrt{\frac{\pi}{2}}\frac{\Phi(2q\sin\theta)}{qI_{0}(1-iI_{0}k_{0}lq\mu)},$
(12)

giving a transmission of,

$\displaystyle\tau=\frac{{\rm
Re}[I_{0}]}{|I_{0}|^{2}\bigl{|}1-iI_{0}k_{0}lq\mu\bigr{|}^{2}},$ (13)

where $H_{z}=2qI_{0}E^{\rm bot}_{x}$, and is spatially constant in the slit.
It can be deduced from Faraday’s Law, ${\bm{H}}=-i/(\mu
k_{0})\nabla\times{\bm{E}}$, that $E_{x}$ must therefore be a linear function
of the coordinate $y$ within the slit: $E_{x}=(y/l)\delta E_{x}+E_{x}^{\rm
top}$, where $\delta E_{x}\equiv E_{x}^{\rm top}-E_{x}^{\rm bot}$. The
electric field at the top of the slit, $E_{x}^{\rm top}$, is simply,
$E_{x}^{\rm top}=(1-2iqI_{0}k_{0}l\mu)E_{x}^{\rm bot}$. For $q\ll 1$, we can
write Eq. (13) strictly in terms of elementary functions,

$\displaystyle\tau\approx\frac{2\pi[\pi+k_{0}l\mu q(3-2\gamma-2\ln q)]}{4\ln
q(2\gamma-3+\ln q)+\pi^{2}+(2\gamma-3)^{2}},$ (14)

demonstrating that for a given ratio of slit width to wavelength, $\tau$ is
simply a linear function of the channel length $l$ and permeability $\mu$.

Figure 2: Transmission, $\tau$, through a zero-$\epsilon$ slit as a function
of the dimensionless geometrical parameters, $l/\lambda$ and $p/\lambda$.

To illustrate how the transmission depends upon the geometry of the slit, we
present in Fig. 2, a three-dimensional representation of $\tau$ [Eq. (13)] as
a function of the slit dimensions, $l$ and $p$, scaled by the wavelength. In
this figure, the source field is normally incident with wavelength,
$\lambda=5\mu{\rm m}$, which is very close to the resonant plasma wavelength,
so that the frequency dispersive $\epsilon$ nearly vanishes. It is evident
from the plot that the transmission does not possess Fabry-Pérot-like resonant
oscillations as a function of $l$, as would be expected from a slit with
conventional material. The figure shows that energy flow is generally
restricted unless the geometrical parameters coincide with the brightly peaked
regions that can decay quite rapidly.

Figure 3: Transmission characteristics for a zero-$\epsilon$ metamaterial
slit, demonstrating the $l$ and $p$ that give the optimal transmission at a
given wavelength. Brighter regions indicate higher transmission. Relatively
high transmission can persevere in deep slits ($l\gg\lambda$) and small
openings ($p\ll\lambda$).

The wave impedance, $Z$, defined as $Z=E_{x}/H_{z}$, is calculated from the
field expressions above to yield the following impedance relation between the
top and bottom of the openings, $Z^{\rm top}=Z^{\rm bot}-ik_{0}l\mu$, with
$Z^{\rm bot}=1/(2qI_{0})$. A remarkable property of a narrow zero-$\epsilon$
slit is that despite the considerable wave impedance for $p\ll\lambda$, energy
can still be transmitted if the slit length $l$ is increased. This behavior is
further illustrated in Fig. 3, where the transmission is shown to be robust
for an extended range of channel lengths and widths. The maximum transmission,
$\tau_{\rm max}$, seen contained within the continuous bright curve, is
determined by examining the extrema of Eq. (13). By taking the appropriate
derivative of the denominator, we find that $\tau_{\rm max}$ is determined by
the transcendental equation, $\mu qk_{0}l=\gamma(q)$, where $\gamma(q)=-{\rm
Im}[I_{0}]/|I_{0}|^{2}$ is a gradually increasing monotonic function of $q$.
Thus, as a function of $q$, and for each $l$, the intercept of $\gamma(q)$
with lines of slope $\mu k_{0}l$ yields the permissible electrical slit widths
that achieve peak light transmission. This technique is shown graphically in
the inset of Fig. 4, where the opposite relationship between slit width and
length are clearly seen.

Figure 4: Transmission through deep metamaterial slits as a function of
$p/\lambda$. A considerable resonance peak emerges for a small subwavelength
width of zero-$\epsilon$ material. Also shown is a matched zero index medium
where $\mu$ and $\epsilon$ are both nearly zero. The inset shows the
intersections of the lines with the $\gamma(q)$ curve, corresponding to
electrical widths that yield peak transmission. Differing ratios of
$l/\lambda$ (see legend) yield lines with different slopes. These results are
consistent with the main plot, where the channel length corresponds to
$l=50\lambda$.

It is also of interest to determine if the transmission can be enhanced for
light incident upon a a matched zero index material, whereby both $\epsilon$
and $\mu$ are vanishingly small. In the main plot of Fig. 4, the transmission
is therefore plotted as a function of slit width for a zero-$\epsilon$ slit
(with $\mu=1$ as usual), and matched zero index slit. Surprisingly, for
zero-$\epsilon$ media, the shown subwavelength widths support transmission
resonances that are absent in matched zero index slits. Although the $l$
dependence washes out for matched zero index channels, $\tau$ still depends
strongly on the width dimension, and Eq. (13) approximately reduces to the
expression for a slit filled with a conventional dielectric and length
satisfying the Fabry-Pérot geometric resonance condition. We can thus conclude
that for zero-$\epsilon$ media and narrow enough openings, energy flow can be
much greater than for slits loaded with metamaterials having no wave impedance
mismatch between the ends of the slit. We also considered the effects of
finite $\Gamma$, and concluded that the presence of absorption loss simply
reduces the overall magnitude of $\tau$ by a factor that becomes greater with
increasing slit depth, but having no effect on the geometrical parameters
leading to the particular resonance location. This bodes well for future
development of miniaturized devices used for energy transport and narrow band
frequency selective surfaces.

In conclusion, we have shown that significant transmission arises for a range
of subwavelength widths in sufficiently deep slits containing zero-$\epsilon$
inclusions. We also demonstrated that these resonant channels can transmit
greater energy compared to the corresponding matched zero index slits.
Although there is currently limited metamaterial fabrication in the IR and
optical frequencies, some recent progress has been made with negative-index
metamaterials involving metal layers separated by a dielectric ref11 ,
alternating layers of InGaAs and AlInAs semiconductors ref12 , and photonic
crystals in the near-infrared range ref13 .

###### Acknowledgements.

This work is supported by NAVAIR s ILIR program sponsored by ONR and by a
grant of HPC resources from ARSC at the University of Alaska, Fairbanks, as
part of the DOD HPCMP.

## References

  * (1) T.W. Ebbesen, et al., Nature (London) 391, 667 (1998).
  * (2) E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, Washington, DC, 1985).
  * (3) M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006); Phys. Rev. B76 245109 (2007).
  * (4) S. Feng and K. Halterman, Phys. Rev. Lett. 100, 063901 (2008).
  * (5) R. Liu, Q. Cheng, T. Hand, and J.J. Mock, Phys. Rev. Lett. 100, 023093 (2008).
  * (6) B. Edwards, et al., Phys. Rev. Lett. 100, 033903 (2008).
  * (7) M. Silveirinha and N. Engheta, Phys. Rev. B75, 075119 (2007).
  * (8) S. Enoch, et al., Phys. Rev. Lett. 89, 213902 (2002).
  * (9) R.W. Ziolkowski, Phys. Rev. E70, 046608 (2004).
  * (10) Y. Takakura, Phys. Rev. Lett. 86, 5601 (2001).
  * (11) J. Bravo-Abad, L. Martin-Moreno, and F.J. Garcia-Vidal, Phys. Rev. E69, 026601 (2004).
  * (12) F. Yang and J.R. Sambles, Phys. Rev. Lett. 89, 063901 (2002).
  * (13) M. Mechler, O. Samek, and S.V. Kukhlevsky, Phys. Rev. Lett. 98, 163901 (2007).
  * (14) S.V. Kukhlevsky, et al., Phys. Rev. B70, 195428 (2004).
  * (15) O. Mata-Mendez and J. Avendano, J. Opt. Soc. Am. A. 24, 1687 (2007).
  * (16) R. Gordon, Phys. Rev. B73, 153405 (2006).
  * (17) S. Zhang et al., Phys. Rev. Lett. 95, 137404 (2005).
  * (18) A. J. Hoffman et al., Nat. Mater. 6, 946 (2007).
  * (19) R. Chatterjee et al., Phys. Rev. Lett. 100, 187401 (2008).