Kerr-induced Spectral Interferometry for Direct Ultra-sensitive Phase Recovery

Abstract

Measuring the phase of light is fundamental to optical imaging, sensing, and signal processing applications. Conventional optical phase measurements rely on multipath configurations, bulky interferometric setups, and computationally intensive data pipelines, limiting scalability, robustness, and practicality. We introduce a technique that allows for reference-free in-line phase retrieval of abrupt phase transitions in optical pulses directly from spectral measurements. Theory, simulations, and experiments concurrently explain the effect as a result of a Kerr-mediated interference between a projected linear wave component and the high-intensity residual of the phase-altered pulse. Utilizing this phenomenon, we demonstrate algorithm-free phase measurements of up to π/385 sensitivity and shot-to-shot signal prominence at 13 dB above noise at 80 MHz rates and 50 pJ pulse energies. This approach offers new paths toward the use of femtosecond pulses as broadband data carriers for optical communications, information processing, and direct high-throughput phase imaging.

Figures (6)

• Figure 1: Experimental setup and spectrograms illustrating the nonlinear phase modulation that leads to spectral peaking. (a, b) Spectrogram at (a) the input and (b) the output of 5m highly nonlinear fiber (HNLF), showing the temporal and spectral intensity profiles of the Fourier limited pulse. (c) Schematic of the experimental setup, consisting of a femtosecond pulsed laser centered at 1550 nm, a spectral phase encoder, an HNLF, and a spectral analyzer for readout. (d, e) Input and output spectrogram for a narrowband $\pi$ -phase shift of 1nm width centered at 1550nm. A continuous-wave (CW) background appears in the temporal domain due to the imposed spectral phase modulation, and persists at the output, leading to pronounced spectral peak at the modulation frequency. Note that the higher contrast of the CW wave occurs due to a renormalisation of the spectrogram plot. The intensity scale of the red output spectra on the left of panels (b) and (e) are equal.
• Figure 2: Self-seeded parametric interference as a result of spectral phase encoding. (a) Input configuration illustrating the field decomposition into a spectrally notched strong field spectrum (top, blue) and phase-shifted weak field detuned by $\omega_c$ from center frequency (bottom, purple). (b) Time-domain representation of the strong pulse field A (top) and weak CW field B (bottom) showing SPM and XPM phase modulations acquired during nonlinear propagation. (c) Output spectra after Kerr propagation, illustrating the fill-up of the spectral notch due to SPM-broadening (top) and XPM-modulated spectral component (bottom) at the probe frequency. Note that the initial relative phase at $\omega_c$ between strong and weak field has also changed. (d-e) Input spectrum of the weak field (d, blue) and output spectrum of both fields combined (e, blue) as a function of wavelength for varying phase encoding $\Phi$ from 0 to 2$\pi$, showing characteristic spectral peaks at probe wavelength $\lambda_c$. Spectra in grey were artifically shifted as a function of $\Phi$ by 30 nm/rad to illustrate the change in peak intensity (red markers) according to our coupled mode theory. All spectra are normalized to the weak field amplitude at input and $\Phi=\pi$. (f) Peak spectral intensity at the probe wavelength for $\Phi = \pi$ versus input peak power $P_0$, demonstrating nonlinear phase-sensitive amplification with values exceeding the typical constructive interference limit of factor 4.
• Figure 3: Comparison of narrowband spectral phase encoding ("bit") in simulation and experiment.(a, b) Schematic illustration of narrowband phase encoding of a single bit, shown for (a) simulation and (b) experiment. In both cases, the bit is represented as a localized phase shift centered near 1550 nm in simulation and 1565 nm in experiment. The bit (1nm wide) is not to scale relative to the laser spectrum. (c, d) Average output spectra with standard deviation (shaded) for low and high input peak powers. (c) Simulations for 300 W and 1200 W peak input power. (d) Experimental spectra for 200 W and 350 W. Spectral peaking is visible only at low power in experiment. (e, f) Correlation between output spectral intensity, normalized to spectral intensity at phase $\phi = 0$, and input phase at the encoding wavelength. (e) Simulation results show overall high Pearson correlation coefficients ($\rho$) for 300 W (solid blue) and 1200 W (solid green), with intermediate powers (dotted lines, power according to colorbar) in between. (f) Experimental curves show coinciding trends with a Pearson score correlation of 0.98.
• Figure 4: Comparison of simulated and experimental spectral encoding and phase recovery using MNIST digits. (a) Simulation results showing the input phase mask (bottom, inverted y-axis) and corresponding normalized output spectrum (top). The inset shows the 20×20 ground-truth image used for simulation. (b) Experimental results showing the encoded phase mask and measured spectrum. The inset displays the corresponding 14×14 ground-truth image used in the experiment. (c) Recovered images from top to bottom: The first row shows the 20×20 ground-truth digits 0–9 from the MNIST dataset (used for simulation), the second row shows simulation-based recovery at input peakpower of 300W, the third row shows the recovered images spectral weighting–based correction of spectral distortions, and the fourth row shows experimental recovery from raw spectral data at input peak power 200 W and pulse width of 200 fs. (d) Normalized spectra for a representative sample of digit "0", showing the measured spectrum used for the recovered image (blue), the envelope-based weighting function obtained from the single wavelength shifted phase bit (light green), and the corrected spectrum after applying this weighting (green). (e) Pixel-value correlations of ground truth vs. the recovered image (blue) and GT vs. the corrected image (green) are shown relative to the ideal linear relation (black dashed line).
• Figure 5: Dependency of phase recovery fidelity on initial peak power and chirp. (a) Heatmap of average peak signal-to-noise ratio (PSNR) as a function of input peak power and applied spectral chirp (–2 to +2) in simulation. Each point represents the mean PSNR over 10 reconstructed digits (0–9) compared with their respective ground truths. (b) Influence of input peak power on spectral and image recovery at zero chirp. Left: normalized output spectra without data (dotted line) and with data for digit "0" (bold line) at 150 W, 600 W, and 900 W input peak powers. Right: corresponding recovered digit images showing change in recovery fidelity with power. (c) Effect of chirp on phase recovery at fixed input power (300 W). Recovered images of digit “0” are shown for chirp values from –2 to +2, illustrating systematic degradation and phase distortion away from zero chirp at this power.