Kramers-Kronig causality in integrated photonics: The spectral tension between ultraviolet transition and mid-infrared absorption

Abstract

Dispersion engineering via geometric confinement is essential to integrated photonics, enabling phenomena such as soliton microcombs, supercontinua, parametric oscillators, and entangled photons. However, prevailing methodologies rely on semi-empirical Sellmeier models that assume idealized material purity, neglecting the pronounced dispersion shifts induced by residual impurities like hydrogen-related bonds. Here, we demonstrate that these residual bonds fundamentally alter the dispersion landscape spanning from the ultraviolet (UV) to the mid-infrared (MIR) spectra. Specifically, they introduce MIR vibrational absorption while simultaneously modifying UV electronic transition, shifting the bandgap and UV pole. We show that the spectral tension between these UV and MIR modifications dictates the group velocity dispersion from the visible to the near-infrared (NIR) via the Kramers-Kronig causality. We experimentally validate this phenomenon through systematic characterization of broadband loss and dispersion in ultralow-loss silicon nitride photonic integrated circuits. By rigorously incorporating these effects, we bridge the gap between empirical fitting and predictive physical modelling. Our study resolves long-standing discrepancies in dispersion engineering, providing precision control essential for next-generation integrated photonics.

Figures (6)

• Figure 1: Origin and non-local spectral impact of hydrogen impurities in silicon nitride. a. Photograph of Si$_3$N$_4$ chips on a 6-inch wafer. b. Combined spectral landscape illustrating the refractive index, extinction and GVD of Si$_3$N$_4$ with ($n'$, $k'$, $\beta_2'$, dashed curves) and without ($n$, $k$, $\beta_2$, solid curves) the N--H bonds. Residual N--H bonds induce blue-shifted bandgap edge in the UV and an absorption peak in the MIR. Linked by the KK relations, these variations in $k$ shift $n$, which in turn alters $\beta_2$ and shifts the zero-dispersion wavelength (ZDW). The discrepancy between these curves illustrates the spectral non-locality of the KK relations. The opposing dispersion contributions from the UV and MIR domains culminate in a dispersion balance point near 2 $\mu$m, where the GVD becomes insensitive to hydrogen content. c. Schematic of the LPCVD process and structural evolution of the Si$_3$N$_4$ film matrix. Precursor residues (NH$_3$ and SiH$_2$Cl$_2$) introduce extensive N--H and Si--H bonds in the film. Partial annealing (PA, $<1050^\circ$C) desifies the film but leaves significant H content, whereas thorough annealing (TA, $>1200^\circ$C) effectively eliminates these bonds to reach the idealized material state.
• Figure 1: Refractive index and dispersion of silicon nitride in the TA and PA states. Calculated refractive index $n$ (top), first-order dispersion parameter $\beta_1$ (middle), and GVD parameter $\beta_2$ (bottom). The red and blue curves correspond to the PA and TA states, respectively.
• Figure 2: Characterization of silicon nitride films. a, b. Schematic of FTIR spectroscopy and the resulting measured spectra for blank Si$_3$N$_4$ films. While PA (red data) leaves significant H content, TA (blue data) effectively eliminates the N--H and Si--H fundamental stretching vibrational modes near 3.0 and 4.6 $\mu$m, respectively. c. Schematic of variable-angle spectroscopic ellipsometry to probe the material's dielectric response across the UV--NIR spectrum. d. Conceptual diagram of the Tauc-Lorentz model. The optical bandgap energy $E_{\text{bg}}$ represents the minimum energy threshold for interband electronic transitions, while the UV pole $E_{\text{UV}}$ captures the dominant high-energy transition corresponding to the maximum density of states. e. Extracted wavelength-dependent refractive index $n$ and extinction coefficient $k$ for Si$_3$N$_4$ in the PA (red) and TA (blue) states. The inset highlights the higher $n$ and $k$ values for the TA state in 200--210 nm.
• Figure 3: Multi-band characterization of dispersion and loss in silicon nitride microresonators. a, b. Schematic principle and experimental configuration of our VSAs. Tunable lasers sweep across the NIR (182.9--237.9 THz), near-visible(383.8--391.3 THz), and MIR (80.8--99.9 THz) bands to probe individual resonances. The instantaneous frequency is calibrated by a frequency ruler, and the device transmission spectrum is recorded synchronously. PD, photodetector. OSC, oscilloscope. c, e, g. Measured $\kappa_0/2\pi$ and $\kappa_{ex}/2\pi$ values for microresonators in the NIR (c, 25-GHz-FSR), near-visible (e, 99-GHz-FSR), and MIR (g, 97-GHz-FSR) bands. For visual clarity, only every third measured data point is displayed in c, d. Absorption features induced by N--H bonds are gray-shaded. d, f, h. Measured (circles) and fitted (solid curves) $D_\text{int}$ profiles of the corresponding microresonators in c, e, g. Panel d additionally includes simulated dispersion curves. The fitted dispersion parameters $D_1$, $D_2$ and $D_3$ are denoted in each panel. Throughout all data panels, the red and blue datasets represent the Si$_3$N$_4$ microresonators in the PA and TA states, respectively.
• Figure 4: Physical modelling and numerical simulation of the dispersion balance point and spectral tension. a. Optical micrograph of a Si$_3$N$_4$ microresonator. The inset shows a cross-sectional SEM image of the waveguide core with the simulated fundamental TE$_{00}$ mode, demonstrating tight optical confinement. b. Frequency-domain representation of microresonator dispersion. Solid lines mark an ideal equidistant frequency grid with $D_1/2\pi$ spacing centered at $f_0$. Actual resonances (dashed lines) at $f_{\pm1}$ and $f_{\pm2}$ deviate from this uniform grid. Considering only $D_2$ and neglecting higher-order dispersion terms, the detuning increases quadratically with the relative mode index $\mu$, giving an offset $\Delta\cdot\mu^2$. c, d, e. Simulated $D_\text{int}/2\pi$ in the NIR (c), near-visible (d) and MIR (e). The simulations incorporate exact waveguide geometries and the comprehensive refractive index model, showing excellent agreement with experimental data for both PA (red) and TA (blue) states. f, g, h. Prediction of the dispersion balance point. Simulated $D_1/2\pi$ (f), $D_2/2\pi$ (g), and $\beta_2$ (h) reveal the interplay between UV and MIR perturbations. While the MIR N--H absorption imparts a negative dispersion pull, the blue-shifted UV bandgap and red-shifted UV pole exert a competing positive dispersion contribution. These opposing influences culminate in a dispersion balance point near 150 THz (2 $\mu$m), where the GVD curves of both states intersect and become insensitive to H content. Throughout all data panels, the red and blue datasets represent the Si$_3$N$_4$ microresonators in the PA and TA states, respectively.