Optical Co-Self-Injection-Locking (co-SIL)

• Optical co-SIL is a passive optical-feedback strategy that stabilizes semiconductor lasers using ultrahigh-Q Fabry–Pérot or diffractive coupling architectures.
• It leverages Adler-type phase dynamics and common-mode noise cancellation to achieve significant laser linewidth suppression and precise optical frequency division.
• The approach supports both dual-laser and multi-emitter configurations, enabling low-noise microwave synthesis and scalable synchronization in compact systems.

Searching arXiv for the cited papers and closely related co-self-injection-locking literature. Optical co-self-injection-locking (co-SIL) denotes a class of optical-feedback architectures in which multiple semiconductor lasers are stabilized through shared resonant or diffractive reinjection paths. In the cited literature, the term encompasses at least two distinct regimes: a dual-laser scheme in which two commercial DFB lasers are locked to adjacent modes of a miniature air-gap Fabry–Pérot cavity and then used for two-point optical frequency division (2P-OFD), and a networked scheme in which 22 VCSELs in a $5\times5$ lattice are mutually injection locked by diffractive coupling through an external cavity (Miao et al., 12 Jan 2026, Pflüger et al., 2022). Across these realizations, co-SIL is characterized by passive optical reinjection rather than modulation-heavy electronic locking, and by phase synchronization that can be analyzed with Adler-type phase dynamics.

1. Operational definition and architectural variants

In the dual-DFB implementation, co-SIL is realized with two optical coupling points. At the “injection” port, two commercial DFB lasers at frequencies $f_A\approx192.3\ \mathrm{THz}$ and $f_B\approx194.8\ \mathrm{THz}$ are combined and sent through a miniature air-gap Fabry–Pérot cavity. A small tilt of the cavity excites high-order transverse modes in reflection, which provide narrowband feedback to the lasers without active modulation. At the “extraction” port, the same two lasers, after isolation, provide the stabilized outputs for downstream frequency division (Miao et al., 12 Jan 2026).

The frequency reference in that architecture is a miniature Fabry–Pérot cavity with mode volume $V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$ and $Q\approx3.5\times10^9$. Its photon lifetime is $\tau\sim5.75\ \mu\mathrm{s}$, and its resonance linewidth is $\sim45\ \mathrm{kHz}$. The downstream frequency-division element is an integrated soliton microcomb with repetition rate $f_r\approx50\ \mathrm{GHz}$, used as a dense optical ruler that transfers optical stability to the microwave domain (Miao et al., 12 Jan 2026).

A different co-SIL topology appears in the VCSEL-array literature. There, a single diffractive optical element in double pass creates a $5\times5$ array of beams. The 0th diffraction order returns onto the “parent” VCSEL as self-feedback, while higher orders illuminate neighbors at increasing lattice separation. In that experiment, 22 lasers are mutually injection locked, and they can also be phase locked simultaneously to an external injection laser (Pflüger et al., 2022).

These two realizations show that co-SIL is not restricted to a single cavity geometry or coupling topology. One version uses shared ultrahigh-$Q$ resonant filtering to generate a low-noise dual-optical reference; another uses diffractive external-cavity coupling to synchronize many emitters. This suggests that co-SIL is better understood as a feedback paradigm than as a single device format.

2. Locking physics and phase-dynamical models

For the miniature Fabry–Pérot implementation, the locking mechanism is self-injection of spectrally filtered light back into each DFB cavity. By adjusting the feedback phase with a silicon phase shifter, the laser frequency $f_A\approx192.3\ \mathrm{THz}$0 is pulled toward the nearest cavity mode $f_A\approx192.3\ \mathrm{THz}$1. With two lasers locked to two adjacent Fabry–Pérot modes, their separation $f_A\approx192.3\ \mathrm{THz}$2 inherits the cavity’s low fundamental noise (Miao et al., 12 Jan 2026).

The locking dynamics are described by an Adler-type relation,

$f_A\approx192.3\ \mathrm{THz}$3

where $f_A\approx192.3\ \mathrm{THz}$4 is the effective locking bandwidth and $f_A\approx192.3\ \mathrm{THz}$5 is the steady-state phase difference between reinjected and intracavity fields. In a simplified picture,

$f_A\approx192.3\ \mathrm{THz}$6

with

$f_A\approx192.3\ \mathrm{THz}$7

and $f_A\approx192.3\ \mathrm{THz}$8. The maximum detuning that can be locked is $f_A\approx192.3\ \mathrm{THz}$9. Experimentally, the measured locking range exceeds $f_B\approx194.8\ \mathrm{THz}$0, indicating strong optical feedback (Miao et al., 12 Jan 2026).

The normalized detuning is written as

$f_B\approx194.8\ \mathrm{THz}$1

with total cavity linewidth $f_B\approx194.8\ \mathrm{THz}$2. The steady state satisfies $f_B\approx194.8\ \mathrm{THz}$3 so long as $f_B\approx194.8\ \mathrm{THz}$4. In practical terms, the reinjection phase determines where the laser settles within the locking cone, while the cavity $f_B\approx194.8\ \mathrm{THz}$5, coupling efficiency, and reinjected power determine how large that cone is.

The VCSEL-array realization extends the same intuition to a coupled network. Each laser phase obeys a generalized Adler equation,

$f_B\approx194.8\ \mathrm{THz}$6

where $f_B\approx194.8\ \mathrm{THz}$7 represents mutual coupling through the diffractive external cavity and $f_B\approx194.8\ \mathrm{THz}$8 represents additional locking to an external master laser. In the two-oscillator limit, this reduces to

$f_B\approx194.8\ \mathrm{THz}$9

which is the classical locking condition (Pflüger et al., 2022).

3. Resonator noise, common-mode rejection, and linewidth suppression

A central motivation for co-SIL is noise suppression without the full complexity of electronic co-Pound-Drever-Hall stabilization. In the Fabry–Pérot system, the cavity’s large mode volume and long photon lifetime reduce thermo-refractive noise relative to integrated cavities. The fundamental frequency noise due to thermo-refractive fluctuations is written as

$V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$0

where $V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$1 is the effective mode radius and the other symbols retain their standard thermophysical meanings (Miao et al., 12 Jan 2026).

The key co-SIL advantage in the dual-laser architecture is common-mode cancellation. Because both DFB lasers are locked to the same physical cavity, common cavity fluctuations $V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$2 are imprinted on both optical carriers. When the beatnote $V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$3 is formed, those common fluctuations largely cancel, producing an extra $V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$4 of common-mode noise rejection for offsets from $V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$5 to $V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$6 (Miao et al., 12 Jan 2026).

The same implementation reports $V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$7 laser frequency-noise reduction, with $V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$8 and $V_m\approx3.5\times10^{-10}\ \mathrm{m}^3$9, plus $Q\approx3.5\times10^9$0 of common-mode rejection, for a total of $Q\approx3.5\times10^9$1 suppression at offsets $Q\approx3.5\times10^9$2. The free-running DFB pump Lorentzian linewidth is $Q\approx3.5\times10^9$3, narrowed to $Q\approx3.5\times10^9$4 on SIL (Miao et al., 12 Jan 2026).

A common misconception is that co-SIL is intrinsically constrained by the noise floor of integrated resonators. The reported miniature air-gap Fabry–Pérot implementation suggests a narrower interpretation: the principal limitation is the thermo-refractive noise and mode-volume penalty of the chosen resonator platform, not the co-SIL principle itself. The data directly support this distinction by contrasting integrated-cavity limitations with an ultrahigh-$Q\approx3.5\times10^9$5 miniature Fabry–Pérot reference (Miao et al., 12 Jan 2026).

4. Two-point optical frequency division and microwave synthesis

In the 2P-OFD configuration, the two co-SIL lasers define an ultrastable optical frequency difference,

$Q\approx3.5\times10^9$6

The microcomb repetition rate is then locked such that

$Q\approx3.5\times10^9$7

with division ratio $Q\approx3.5\times10^9$8 (Miao et al., 12 Jan 2026).

The experimental loop is implemented by heterodyning comb tooth $Q\approx3.5\times10^9$9 with laser $\tau\sim5.75\ \mu\mathrm{s}$0 to produce $\tau\sim5.75\ \mu\mathrm{s}$1, and comb tooth $\tau\sim5.75\ \mu\mathrm{s}$2 with laser $\tau\sim5.75\ \mu\mathrm{s}$3 to produce $\tau\sim5.75\ \mu\mathrm{s}$4. Mixing $\tau\sim5.75\ \mu\mathrm{s}$5 and $\tau\sim5.75\ \mu\mathrm{s}$6 yields

$\tau\sim5.75\ \mu\mathrm{s}$7

This intermediate frequency is compared to a stable RF local oscillator $\tau\sim5.75\ \mu\mathrm{s}$8, generating an error $\tau\sim5.75\ \mu\mathrm{s}$9. Feeding that error to the DFB pump current shifts $\sim45\ \mathrm{kHz}$0, which tunes $\sim45\ \mathrm{kHz}$1 through Raman-induced soliton self-frequency shift and dispersive-wave recoil. In steady state, $\sim45\ \mathrm{kHz}$2, enforcing

$\sim45\ \mathrm{kHz}$3

The result is stabilized microwave extraction from the optical frequency difference (Miao et al., 12 Jan 2026).

The phase-noise scaling is the standard division law,

$\sim45\ \mathrm{kHz}$4

For $\sim45\ \mathrm{kHz}$5, this corresponds to $\sim45\ \mathrm{kHz}$6 of suppression. Thus the architecture relies on two distinct mechanisms: optical common-mode rejection in the dual co-SIL reference, followed by deterministic phase-noise reduction through optical frequency division (Miao et al., 12 Jan 2026).

This separation of functions is significant. The Fabry–Pérot cavity provides the low-noise optical spacing, while the soliton microcomb performs the division. The architecture therefore differs from systems in which the stabilization and division functions are collapsed into a single resonant structure.

5. Demonstrated performance and implementation parameters

The reported dual-laser co-SIL and 2P-OFD system generates a $\sim45\ \mathrm{kHz}$7 microwave with direct phase noise of $\sim45\ \mathrm{kHz}$8 at $\sim45\ \mathrm{kHz}$9 offset and $f_r\approx50\ \mathrm{GHz}$0 at $f_r\approx50\ \mathrm{GHz}$1 offset, scaled to $f_r\approx50\ \mathrm{GHz}$2. The co-SIL dual-laser beat at $f_r\approx50\ \mathrm{GHz}$3 has phase noise of $f_r\approx50\ \mathrm{GHz}$4 at $f_r\approx50\ \mathrm{GHz}$5 offset (Miao et al., 12 Jan 2026).

The same work compares its microwave phase noise, at $f_r\approx50\ \mathrm{GHz}$6, with several electronically stabilized systems reported in the literature: co-PDH to $f_r\approx50\ \mathrm{GHz}$7 at $f_r\approx50\ \mathrm{GHz}$8 at $f_r\approx50\ \mathrm{GHz}$9, co-PDH to air-gap mini-FP at $5\times5$0 at $5\times5$1, and an OPO on $5\times5$2 ring at $5\times5$3 at $5\times5$4 (Miao et al., 12 Jan 2026).

The control bandwidth of the 2P-OFD feedback loop is $5\times5$5, producing a clear servo bump at that offset. This bandwidth is limited by the DFB current modulation path. The system’s long-term drift is described by an Allan deviation floor of $5\times5$6 at $5\times5$7, set by the mini-FP temperature drift (Miao et al., 12 Jan 2026).

Implementation details are unusually explicit. The co-SIL module, consisting of two DFBs, optics, and the air-gap Fabry–Pérot cavity, fits in a $5\times5$8 package. It uses industry-standard fiber-pigtailed DFBs, a Zerodur spacer, beam splitters, dichroics, and photodiodes. The optics are pre-aligned on a single substrate, the feedback phase is tuned via a resistively heated silicon shim, and no active alignment is required after packaging. SIL of each DFB is maintained with $5\times5$9 capture range and minimal cycle slips over hours in a normal lab environment (Miao et al., 12 Jan 2026).

System element	Reported value	Function
Miniature FP cavity	$Q$0	Ultrahigh-$Q$1 optical reference
Miniature FP cavity	$Q$2	Large mode volume, low thermo-refractive noise
Photon lifetime	$Q$3	Narrow resonant response
Resonance linewidth	$Q$4	Optical filtering for reinjection
Microcomb repetition rate	$Q$5	Optical-to-microwave division
Loop bandwidth	$Q$6	2P-OFD control limit

These metrics define co-SIL not merely as a locking mechanism but as a complete microwave-synthesis pathway when combined with OFD.

6. Network co-SIL in VCSEL arrays and its relation to cavity-based co-SIL

The VCSEL-array realization broadens the meaning of co-SIL from dual-reference stabilization to many-body synchronization. In that experiment, the array is a $Q$7 square lattice with pitch $Q$8, based on a GaInAs quantum-well VCSEL cavity at $Q$9 and threshold current $f_A\approx192.3\ \mathrm{THz}$00. Pump currents are typically $f_A\approx192.3\ \mathrm{THz}$01, with individual tuning of $f_A\approx192.3\ \mathrm{THz}$02 to align the lasers within $f_A\approx192.3\ \mathrm{THz}$03 (Pflüger et al., 2022).

The diffractive coupling coefficient from VCSEL $f_A\approx192.3\ \mathrm{THz}$04 into VCSEL $f_A\approx192.3\ \mathrm{THz}$05 is written as

$f_A\approx192.3\ \mathrm{THz}$06

where $f_A\approx192.3\ \mathrm{THz}$07 is the amplitude fraction in diffraction order $f_A\approx192.3\ \mathrm{THz}$08. For a 1D model,

$f_A\approx192.3\ \mathrm{THz}$09

For the reported array, nearest-neighbor coupling efficiencies are $f_A\approx192.3\ \mathrm{THz}$10 (Pflüger et al., 2022).

Small-signal stability analysis yields the cluster-locking criterion

$f_A\approx192.3\ \mathrm{THz}$11

with corresponding locking range in wavelength units

$f_A\approx192.3\ \mathrm{THz}$12

Experimentally, the full-array mutual-locking range is reported as $f_A\approx192.3\ \mathrm{THz}$13 on the upper side and $f_A\approx192.3\ \mathrm{THz}$14 on the lower side, with 22 lasers locked simultaneously (Pflüger et al., 2022).

When an external DBR laser is added, the locking window follows the Henry–Ohtsubo form

$f_A\approx192.3\ \mathrm{THz}$15

with $f_A\approx192.3\ \mathrm{THz}$16 in the range $f_A\approx192.3\ \mathrm{THz}$17. The measured injection-locking window ranges from $f_A\approx192.3\ \mathrm{THz}$18 at the corners to $f_A\approx192.3\ \mathrm{THz}$19 near the center, corresponding to $f_A\approx192.3\ \mathrm{THz}$20. In the locked state, the experiment reports coherent power gain of $f_A\approx192.3\ \mathrm{THz}$21 above a single VCSEL, side-mode suppression $f_A\approx192.3\ \mathrm{THz}$22, and RF noise floor suppression by $f_A\approx192.3\ \mathrm{THz}$23 (Pflüger et al., 2022).

This comparison clarifies a second misconception: co-SIL is not synonymous with locking all emitters to an external master. In the VCSEL case, mutual diffractive coupling already yields collective locking, and external injection adds another synchronization channel. In the Fabry–Pérot case, by contrast, the central function of co-SIL is low-noise referencing of two lasers to a shared cavity mode structure. The two systems are related by phase-locking formalism, but they serve different system objectives.

7. Relation to co-PDH and prospective directions

The Fabry–Pérot co-SIL work explicitly positions itself against electronic co-Pound-Drever-Hall stabilization. The stated contrast is that co-SIL replaces complex electronic PDH loops, eliminating EOMs/AOMs, high-speed servo electronics, and characteristic servo bumps, and yields an all-optical lock with full offset-frequency noise suppression (Miao et al., 12 Jan 2026).

That comparison should be read carefully. The same system still contains a $f_A\approx192.3\ \mathrm{THz}$24-OFD feedback loop with bandwidth $f_A\approx192.3\ \mathrm{THz}$25, and that loop shows a clear servo bump. The distinction is therefore not the total absence of control loops, but the relocation of the optical-frequency referencing task from electronic co-PDH to optical self-injection locking. This suggests that the simplicity claim pertains specifically to the optical reference lock rather than to the entire microwave-synthesis stack.

The reported outlook identifies several refinements: all-optical Kerr synchronization to extend loop bandwidth beyond tens of MHz, on-chip micro-FP integration, and electrical feed-forward cancellation. The stated target is to push phase noise toward $f_A\approx192.3\ \mathrm{THz}$26 at $f_A\approx192.3\ \mathrm{THz}$27 offset at $f_A\approx192.3\ \mathrm{THz}$28 while preserving a compact, low-cost SWaP-C profile (Miao et al., 12 Jan 2026).

Within the limits of the reported data, co-SIL therefore occupies a technically specific niche. It is a passive optical-locking strategy that can suppress laser noise by reinjection through a shared optical structure; it can be used either to synthesize ultralow-noise microwaves when combined with OFD or to synchronize many-emitter laser arrays through diffractive mutual coupling. Its significance lies not in a single implementation detail, but in the demonstrated ability to combine ultrahigh-$f_A\approx192.3\ \mathrm{THz}$29 optical references, common-mode rejection, and scalable feedback topologies in compact photonic oscillators and coupled-laser networks (Miao et al., 12 Jan 2026, Pflüger et al., 2022).