Integrated Sagnac Interferometer Design

• Integrated Sagnac interferometer design is a photonic and hybrid platform that exploits the closed-loop Sagnac effect for precise measurements and spectral engineering.
• It employs lithographically defined waveguide circuits and directional couplers to achieve broadband filtering, high-sensitivity gyroscopic sensing, and quantum metrology.
• The design minimizes environmental noise through common-mode interference control, ensuring robust performance across inertial navigation and atom–photon hybrid applications.

An integrated Sagnac interferometer is a photonic or hybrid-photonic device that exploits the Sagnac effect—phase accumulation proportional to enclosed area and rotation rate—for precise measurement, spectral engineering, nonreciprocal signal processing, and quantum metrology within a lithographically defined or hybrid-integrated architecture. Distinct from bulk-optics Sagnac gyroscopes, integrated versions leverage semiconductor fabrication for miniaturization and stability, monolithic closed-loop waveguide circuits, and intimate integration with photonic or atom-optical subsystems. This design platform underpins devices ranging from gyroscopes and angle sensors to advanced filters, high-slope Fano elements, quantum decoherence probes, and hybrid atom–photon systems (Moss, 2023, Arianfard et al., 2021, Zhou et al., 7 Nov 2025, ElKabbash, 15 Apr 2025).

1. Theoretical Principles and Sagnac Phase in Integrated Architectures

The Sagnac effect is fundamentally a differential phase shift experienced by two counter-propagating waves (optical or matter-wave) in a closed-loop geometry, proportional to both the area $A$ enclosed by the loop and the platform’s angular velocity $\Omega$. For an optical mode in a waveguide loop of group index $n$ and vacuum wavelength $\lambda$:

$$\Delta\phi_{\rm Sagnac} = \frac{8\pi n A \Omega}{\lambda c}$$

where $c$ is the speed of light in vacuum. This scaling preserves the conventional Sagnac effect seen in fiber- and bulk-optic gyroscopes, but the integrated realization enables lithographic precision in defining $A$ and $n$, and brings immunity to environmental perturbations via monolithic construction (Moss, 2023).

Besides rotation sensing, the Sagnac loop topology supports broadband, high-fidelity interference because CW and CCW optical fields follow identical paths, minimizing differential drift. This “common-mode” property confers intrinsic resilience against temperature, vibration, and polarization fluctuations—a key advantage over Mach–Zehnder Interferometers (MZIs) and ring resonators (RRs) [(Moss, 2023), Table 3].

2. Integrated Sagnac Topologies and Photonic Device Classes

2.1 Sagnac Loop Reflectors (SLR) and Filters

Basic integrated Sagnac interferometers consist of a directional coupler (field amplitudes $t$, $k$, with $|t|^2+|k|^2=1$) and a waveguide loop. By reconnecting the outputs to the inputs of the coupler:

$$T_{\rm SLR} = (t^2 - k^2)a e^{-j\phi}, \qquad R_{\rm SLR} = 2jtk a e^{-j\phi}$$

where $a$ is the round-trip amplitude loss and $\phi$ is the optical phase for a single traversal. For $t = k = 1/\sqrt{2}$ (ideal 50:50 coupler) and $a \approx 1$, $R_{\rm SLR} \to j e^{-j\phi}$—perfect, broadband reflection locally insensitive to $\phi$ or wavelength (Arianfard et al., 2021). Complex filtering functions (comb, Butterworth, Chebyshev, Bessel, elliptic) arise from networks of SLRs, with precise spectral shaping achieved through control of coupler strength, phase-tuning elements, or cascaded topologies (Moss, 2023).

2.2 Optical Gyroscopes and Resonant Sensors

Integrated Sagnac interferometers serve as the core of chip-scale optical gyroscopes. The loop area $A$ and waveguide length $L$ are lithographically engineered; for rotational sensing, ultra-low-loss SiN or Si$_3$N$_4$ waveguides (loss $\lesssim 0.1$ dB/m) and radii $R$ from mm to cm are common (Moss, 2023, Yanik et al., 22 Jul 2025). In passive resonant devices, a high-$Q$ ring is coupled to a bus waveguide, with sensitivity enhanced by resonant buildup: shot-noise-limited angular random walk (ARW) of $\sim 1$–$10$ deg/h$^{1/2}$ can be achieved, with demonstrable bias drift $\sim 0.1$ deg/h (Yanik et al., 22 Jul 2025).

Inverse weak-value amplification architectures, coupling a Sagnac loop to a high-$Q$ ring and MMI for mode conversion, yield signal-to-noise enhancements by an order of magnitude or more, with minimum detectable $\Omega$ of $0.1$ deg/hr and Allan deviation $0.08$ deg/hr in practical Si$_3$N$_4$ platforms (Yanik et al., 22 Jul 2025).

2.3 Hybrid and Atomic Sagnac Designs

Integration expands into hybrid photonic–atomic interferometers. PIC-based Sagnac tractor atom interferometers employ two Si$_3$N$_4$ ring waveguides ($R_0 = 600~\mu$m) supporting independent, counter-rotating azimuthal optical lattices. Atoms (e.g., $^{87}$Rb) are confined and transported by the evanescent fields. Rotation is detected via the Sagnac phase accumulated between atoms in counter-rotating lattices:

$\Delta \Phi_S = \frac{2 M K A \Omega}{\hbar}$

where $M$ is atomic mass, $K$ is the number of lattice half-rotations, and $A$ is geometric area. Ground-state fidelity above 99% with $K \sim 2000$ half-rotations and phase sensitivity $\delta\Omega \sim 2$ nrad/s at 1 Hz bandwidth is achievable, presuming atom numbers $N \sim 10^4$ (Zhou et al., 7 Nov 2025).

On-chip atomic Sagnac devices based on state-dependent radiofrequency traps or matter-waveguides exploit similar phase scaling but extend performance by circumventing technical noise, e.g., via Ramsey sequences, spin echo, and dual-ring noise-rejection (Stevenson et al., 2015, Moukouri et al., 2021).

3. Extended Functionalities: Filtering, Wavelength Interleaving, and Quantum Analogues

Sagnac loop architectures generalize beyond gyroscopes:

• High-order Filters and Interleavers: Networks of SLRs or mutually coupled Sagnac loops (MC-SLRs) yield flat-top bandpass filters, interleavers, notch filters, and bandstop filters with high extinction ratios and low insertion loss. MC-SLRs enable Fano resonances with slope rates exceeding $350$ dB/nm and roll-off $>50$ dB/GHz in $<0.02$ mm$^2$ footprints (Arianfard et al., 2021).
• Quantum Analogues and Advanced Spectral Control: Self-coupled Sagnacs and SLR–ring hybrids support Autler–Townes splitting, electromagnetically induced transparency analogues, and Fano/Bound-State-in-Continuum lineshapes advantageous in quantum optics, sensor, and neuromorphic applications (Moss, 2023).
• Grover-Sagnac Interferometry: Replacing the usual 2×2 BS with a Grover multiport creates a resonance (pole) at the origin of the parameter space, permitting phase extraction from the spectral linewidth rather than fringe contrast. This method allows for detection even when the power amplitude signal is small, enhancing metrological versatility (Schwarze et al., 23 Jan 2025).

4. Noise, Sensitivity, and Design Optimization

4.1 Shot Noise and Technical Noise

Shot-noise-limited angle and phase sensitivities are derived from the output photodetector signals and system responsivity. For a Sagnac–lever angle sensor:

$$S^{1/2}_{\theta, \rm shot} \approx \frac{1}{2N\,k\,L_{\rm eff}\,\sqrt{P/\hbar\omega}}\; \frac{1}{\cot(\phi/2)}$$

where $N$ is lever bounces, $k=2\pi/\lambda$, and $L_{\rm eff}$ the effective optical path (Hogan et al., 2011). Raising $N$, increasing laser power $P$, and optimizing the waveplate phase $\phi$ toward destructive interference sharply improve sensitivity (demonstrated $\delta\theta_{\rm min} = 1.3$ prad/Hz$^{1/2}$ at 2.4 kHz with $N=11$; scalable to sub-picoradian/Hz$^{1/2}$ regimes).

Technical noise sources—mode mismatch, stray reflections, intensity noise—impose additional constraints. Their contribution is typically phase-independent, establishing a practical limit for achievable sensitivity and dictating optimal working points (e.g., $\phi \approx 90^\circ$ for balance).

4.2 Fabrication and Integration Tolerances

Device performance is sensitive to coupler ratios, phase-tuning precision, waveguide losses, and integration of elements (e.g., photodetectors, heaters). Tolerances:

• Coupler splitting error: $\pm10$ nm width/gap variation for $<1\%$ power deviation.
• Phase tuning: Thermo-optic or carrier-injection shifters provide $\pi$ rad/10 mW or $\pi$ rad/V, respectively, with sub-$\mu$s/ns response times (Schwarze et al., 23 Jan 2025, Moss, 2023).
• Integrated gyroscopes: Propagation loss $<0.05$ dB/m, $Q>10^7$ for ARW and bias targets; Si$_3$N$_4$ or ultra-low-loss SiO$_2$ platforms (Yanik et al., 22 Jul 2025).
• Photonic–atomic hybrids: Precise mode-matching for lattices, resonance frequency control to $\Delta f/f \sim 10^{-10}$ (Zhou et al., 7 Nov 2025).

5. Notable Applications and Contemporary Demonstrations

Device Type	Function	Achievable Metric/Result
Sagnac–lever angle sensor	Ultra-precise angular detection	$1.3$ prad/Hz$^{1/2}$ at 2.4 kHz w/ $N=11$ bounces (Hogan et al., 2011)
Chip-scale resonant gyroscope	Rotation sensing	$\Omega_{\rm min} = 0.1$ deg/hr, Allan deviation $0.08$ deg/hr (Yanik et al., 22 Jul 2025)
Atom–PIC Sagnac interferometer	Rotation metrology	$\delta\Omega \approx 2$ nrad/s at $N\sim 10^4$, area $A \sim 1$ mm$^2$ (Zhou et al., 7 Nov 2025)
MC-SLR Fano filter	Ultrafast switching, high slope	SR $>350$ dB/nm, ER $>13$ dB, BW $<0.05$ nm (Arianfard et al., 2021)
Quantum decoherence probe	Proper-time-induced visibility loss	On-chip Sagnac with $R=18.9$ cm, $V$ drops for $\tau_c<10$ fs (ElKabbash, 15 Apr 2025)

6. Challenges and Prospects

Principal obstacles include minimization of propagation loss in large-area or high-$Q$ loops, maintaining sub-wavelength fabrication accuracy, and robust integration of critical active elements (detectors, heaters, phase shifters, or atom-loading sites). Technical noise (e.g., backscatter, modal crosstalk, beam-quality degradation) must be addressed by advanced mode-matching and environmental controls.

Integrated Sagnac architectures continue to expand in complexity and functionality. Proposals for integrated quantum photonic circuits, neuromorphic analog computation, and fundamental probes of quantum–relativity interface are advancing rapidly, driven by the scalability and stability enabled by the integrated Sagnac interferometer framework (Moss, 2023, Zhou et al., 7 Nov 2025, ElKabbash, 15 Apr 2025).

7. Summary Table: Platform and Functionality Comparison

Topology / Platform	Core Component(s)	Primary Target Application	Exemplary Metric
Sagnac SLR	Dir. coupler + loop	Broadband mirror, filter, gyroscope	R $>$ 95% over 80 nm (Moss, 2023)
MC-SLR (Parallel/Zig)	SLR$\times$2 + bus	Flat-top/interleaving/BPF/Fano	SR $>$ 350 dB/nm
Resonant Sagnac Ring	High-$Q$ ring, MMI, phase-front tilter	Chip gyroscope (IWVA readout)	$\Omega_{\rm min}=0.1$ deg/hr
Hybrid Atom–PIC	Ring lattices + atoms	Rotation, inertial sense (quantum)	Sensitivity $\sim$ nrad/s
On-chip quantum probe	Large ring, SPAD–MZI	Relativistic decoherence test	Visibility loss ($\tau_c<10$ fs)

Integrated Sagnac interferometer design thus forms a core technology for high-performance, scalable, and multifunctional photonic and quantum devices, with leading application domains in precision metrology, integrated spectroscopy, inertial navigation, quantum communications, and fundamental physics. The cross-disciplinary architecture supports continual advances as fabrication, integration, and hybridization techniques mature.