Aberration-optimized electro-optic time lens with a tunable aperture

Technical Summary: Aberration-optimized electro-optic time lens with a tunable aperture

Problem Statement
Time lenses are essential for manipulating ultrafast optical pulses in applications ranging from ultrafast spectroscopy to quantum information science. While electro-optic time lenses driven by sinusoidal signals are popular due to their deterministic nature and ease of reconfiguration, they suffer from inherent limitations. Specifically, the sinusoidal phase modulation deviates from the ideal quadratic phase required for aberration-free operation. This deviation introduces temporal aberrations, restricting the usable "time aperture" (the window where the phase is approximately quadratic) to roughly 16% of the driving signal's period. Existing analytical models rely on a standard Taylor expansion approximation that underutilizes the device's potential, and while aberration-correcting techniques exist, they often require complex electronic signals or high peak voltages that increase power consumption and experimental complexity.

Methodology
The authors propose a new analytical model that approximates the chirp rate of a sinusoidal time lens by introducing a tunable time aperture parameter ($\alpha$), without modifying the physical device.

• Theoretical Framework: Instead of assuming a fixed aperture, the model defines the time aperture $\tau_a$ as a fraction $\alpha$ of the RF signal period $T$. The authors derive a unique quadratic function that intersects the center and the edges of this defined aperture. This yields a modified chirp rate formula, $K(\alpha) = A\omega_m^2 \text{sinc}^2(\alpha\pi/2)$, where $A$ is the modulation amplitude and $\omega_m$ is the modulation frequency.
• Phase Error Analysis: The study derives a closed-form expression for the maximum phase error as a function of the aperture parameter $\alpha$ and modulation amplitude $A$. The analysis demonstrates that the maximum phase error scales linearly with $A$ and follows a power law with respect to $\alpha$ ($\approx 0.62 A \alpha^{3.34}$).
• Experimental Validation: The model was validated experimentally using a setup for spectral bandwidth compression of Gaussian pulses. The system utilized a mode-locked fiber laser, a pulse shaper to create chirped pulses with tunable Group Delay Dispersion (GDD), and an electro-optic phase modulator (EOPM) driven by a 16 GHz sinusoidal signal. The researchers measured the compressed spectra for various GDD values corresponding to different aperture parameters ($\alpha$), comparing the results against the standard approximation ($\alpha \approx 0$) and numerical simulations.

Key Contributions

1. Tunable Aperture Model: The introduction of a dimensionless aperture parameter $\alpha$ allows for precise control over the chirp rate and the trade-off between aperture size and phase error.
2. Closed-Form Expressions: The derivation of analytical formulas for the chirp rate $K(\alpha)$ and the maximum phase error $\Delta\theta_{max}(\alpha, A)$ provides a quantitative tool for predicting system performance.
3. Optimization Formula: The authors provide an empirical formula, $\alpha(A,Q) = 1.35 \left(\frac{0.8-Q}{A}\right)^{0.3}$, which relates the modulation amplitude ($A$) and desired compression quality ($Q$) to the optimal aperture parameter. This enables users to select an aperture that balances bandwidth compression efficiency with acceptable spectral distortions.
4. Experimental Demonstration: The work experimentally validates that extending the time aperture beyond the conventional limit, when guided by the new model, yields superior performance.

Results

• Enhanced Compression: The experimental results demonstrated a 1.6-fold enhancement in spectral bandwidth compression for Gaussian pulses compared to the conventional approach (standard approximation).
• Trade-off Management: The study showed that while increasing the aperture parameter ($\alpha$) improves the compression factor (relative enhancement), it also increases phase errors, manifesting as symmetric spectral sidelobes. The new model successfully identified an optimal operating point ($\alpha \approx 0.57$) where the enhancement was maximized while maintaining a compression quality comparable to the standard approach.
• Accuracy of Model: The experimental data for Power-Equivalent Bandwidth (PEB), enhancement, and compression quality aligned closely with the authors' numerical simulations, confirming the validity of the derived analytical expressions.

Significance
The paper claims that this framework offers a practical tool for designing efficient temporal optical systems without the need for complex hardware modifications or high-voltage multi-tone driving signals. By providing a systematic method to determine the aberration-limited optimal aperture, the approach benefits applications in both classical and quantum optics where precise control over spectro-temporal properties is crucial. Specific applications cited include temporal imaging, pulse compression, time-to-frequency mapping, and optical signal processing. The authors suggest that while the study focuses on electro-optic sinusoidal lenses, the methodology could potentially be extended to other time-lens implementations utilizing nonlinear optical methods.