Nonlinear Time Lens for Temporal Imaging

• Nonlinear time lens is an optical process that applies a quadratic phase shift to reshape ultrafast waveforms.
• It utilizes nonlinear effects such as SFG, FWM, and XPM along with dispersive elements to achieve temporal magnification, compression, and inversion.
• Its applications span ultrafast optical metrology, quantum-state engineering, and photonic waveform manipulation for advanced temporal imaging.

A nonlinear time lens is an optical element or process that imparts a purely quadratic phase shift to the temporal coordinate of an optical field, thereby enabling deterministic and phase-coherent reshaping of ultrafast waveforms in time. Analogous to a spatial lens, which modulates the transverse phase profile of a beam, the time lens utilizes nonlinear optical effects—such as sum-frequency generation (SFG), four-wave mixing (FWM), or cross-phase modulation (XPM)—to implement the necessary parabolic phase evolution. Nonlinear time lenses, when combined with appropriate pre- and post-dispersive segments, realize temporal imaging systems capable of magnifying, compressing, and manipulating optical and quantum waveforms. These structures serve pivotal roles in ultrafast optical metrology, quantum state engineering, waveform conversion for quantum networks, and dynamic bandwidth adaptation in classical and quantum photonics.

1. Physical Principle: Quadratic Temporal Phase and Temporal Imaging

A time lens acts by imposing a quadratic phase in time, $\varphi(t) = \frac{1}{2} K t^2$, where $K$ is the "chirp rate" with units of $\mathrm{ps}^{-2}$ or $\mathrm{fs}^{-2}$, depending on context. In practice, this is achieved by overlapping an ultrafast optical waveform ("signal") with a strong, heavily chirped "pump" or control field within a nonlinear medium—$\chi^{(2)}$ for SFG, $\chi^{(3)}$ for FWM or Kerr XPM. The temporal lens phase maps temporal frequencies (detunings from carrier) into times, analogous to the action of a spatial lens with a Fourier kernel under stationary-phase approximation. The group-delay dispersion (GDD) $D_f$ plays the role of a focal length, with $K = 1/D_f$ (Srivastava et al., 2023, Donohue et al., 2016, Patera et al., 2022).

Temporal imaging is achieved by cascading the nonlinear time lens between two dispersive elements with GDDs $G_1$ (pre-lens) and $G_2$ (post-lens). The imaging condition for the full system is: $\frac{1}{G_1} + \frac{1}{G_2} = \frac{1}{D_f}$ The resulting temporal magnification is $m = -G_2/G_1$. The analogy to geometrical imaging in space is complete: dispersive elements correspond to propagation, the quadratic phase to an optical lens, and the combined system achieves waveform stretching, compression, or inversion (Srivastava et al., 2023, Mittal et al., 2017).

2. Nonlinear Implementations of Time Lenses

Physical implementations of nonlinear time lenses utilize processes that can deterministically transfer a quadratic phase from a classical pump to the signal. Common realizations include:

• Sum-frequency generation (SFG) in $\chi^{(2)}$ media: A weak signal and a chirped strong pump co-propagate in a nonlinear crystal or waveguide. The idler generated by the SFG inherits the quadratic temporal phase of the pump, realizing $\varphi(t) = t^2/(2 D_f)$, with $D_f$ linked to the pump's chirp. Phase matching must be sufficiently broad for high conversion efficiency (Donohue et al., 2016, Mittal et al., 2017, Patera et al., 2022).
• Four-wave mixing (FWM) and XPM in $\chi^{(3)}$ media: In FWM, a signal interacts with a chirped pump in a fiber or waveguide, generating an idler with the desired time lens phase via energy and phase conservation. The achievable GDD and system bandwidth depend on fiber or waveguide properties, nonlinearity, and pump intensity (Chu et al., 2021).
• Electro-optic phase modulation (EOPM): Driving an EOPM with a high-bandwidth, arbitrary-waveform or sinusoidal voltage can create the necessary parabolic phase for temporal focusing or stretching, especially in integrated photonic platforms (Srivastava et al., 2023).

Choice of platform impacts temporal aperture, efficiency, bandwidth, and susceptibility to loss, dispersion, and nonidealities. For quantum processing, unity (or near-unity) conversion efficiency for multiple temporal modes is required to avoid introducing excess vacuum noise (Patera et al., 2022).

3. One-Lens vs. Two-Lens Imaging: Residual Chirp and Chirp-Free Time Telescopes

A single time-lens imaging system, even when the thin-lens equation is satisfied, imposes a residual quadratic phase ("chirp") on the output waveform. This is evident from the output envelope: $$A_\mathrm{out}(t) = -\frac{1}{\sqrt{m}} \exp\left[ i\frac{t^2}{2 m D_f} \right] A_\mathrm{in}(t/m)$$ The quadratic phase factor persists and is analogous to the wavefront curvature remaining after spatial focusing, limiting applications that rely on phase purity or chirp-free waveforms. The residual chirp cannot be removed without introducing a second lens (Srivastava et al., 2023).

A two-lens system, known as a "temporal telescope," overcomes this limitation by inserting an intermediate dispersive segment between two time lenses with GDDs $D_f$ and $D_f'$. The telescopic (afocal) imaging condition: $D_\mathrm{inter} = D_f + D_f'$ nullifies the output chirp, provided $D_f' = -M D_f$, resulting in pure magnification or compression: $$A_\mathrm{out}(t) = \frac{1}{\sqrt{M}} A_\mathrm{in}(t/M)$$ This architecture enables perfect temporal scaling without phase distortion and is mandatory for high-fidelity quantum-state transformations or indistinguishability restoration in quantum networks (Srivastava et al., 2023).

4. Quantum Temporal Imaging: Modal Theory and High-Efficiency Criteria

Quantum applications, particularly those manipulating single photons or entangled photon pairs, require a description beyond the classical transfer function. The action of a nonlinear time lens in SFG is captured by an operator-valued integral kernel $K(\Omega, \Omega')$ reflecting both the pump spectrum and phase-matching sinc profile. Singular-value decomposition of $K$ yields Schmidt modes $\phi_n$, output modes $\psi_n$, and conversion efficiencies $|s_n|^2$ for each mode. The system's quantum figures of merit include:

• Modal conversion efficiency: $s_n = \sin(g l_c \lambda_n)$ (with $g$ linking the nonlinearity and pump field, $l_c$ the crystal length, $\lambda_n$ SVD values).
• Instantaneous efficiency: For the fundamental mode, achieve $s_0 \approx 1$ by tuning the pump peak power.
• Multimode operation: A high Schmidt number $S$ signals a large number of highly-converted modes—crucial for broadband and time-multiplexed quantum processing.

To attain near-unity efficiency across many modes, one uses a broadly chirped pump (large $D_f \Delta_p^2$), minimizes phase-mismatch and walk-off, and matches group velocities between signal and pump (Patera et al., 2022).

5. Practical Design and Optimization: Loss, Dispersion, and Temporal Resolution

Designing a high-performance nonlinear time lens or telescope requires optimizing several parameters:

• Loss minimization: Total system loss scales with the number and length of dispersive elements; minimizing the total GDD reduces propagation loss and temporal broadening.
• Temporal aperture: The maximum input duration imaged without distortion, $T_A = D_f \Omega_m$, where $\Omega_m$ is the modulator or pump bandwidth.
• Temporal resolution: Set by the system's point-spread function and, in multimode theory, by the ratio of output mode width to the Schmidt number.
• Dispersion budgets: Explicit formulas relate the telescope's GDD assignments for a given magnification $M$: $$D_f = D_\mathrm{inter}/(1-M), D_f' = -M D_\mathrm{inter}/(1-M), D_\mathrm{out}' = -M^2 D_\mathrm{in} - M D_\mathrm{inter}$$ (Srivastava et al., 2023).
• Bandwidth and phase matching: Achieving broad, distortion-free imaging demands that the pump bandwidth and phase-matching bandwidth both exceed the input signal bandwidth. Nonidealities such as third-order dispersion (TOD) or nonunitary SFG/FWM conversion limit resolution and efficiency (Chu et al., 2021, Mittal et al., 2017).

6. Applications in Classical and Quantum Photonics

Nonlinear time lenses underpin a variety of advanced photonic and quantum technologies:

• Quantum-state engineering: Time lenses have been used to invert frequency correlations in photon pairs, reconfiguring the joint-spectral intensity from highly anti-correlated to highly positively correlated via negative spectro-temporal magnification. This enables new protocols for entanglement distribution, quantum clock synchronization, and nonlocal dispersion cancellation (Donohue et al., 2016).
• Restoring indistinguishability: In photonic quantum networks, time telescopes are necessary and sufficient to render temporally mismatched single photons (from SPDC or quantum-dot sources) indistinguishable, achieving maximal Hong–Ou–Mandel visibility for Gaussian or exponentially decaying pulses with appropriate magnification $M$ (Srivastava et al., 2023).
• Ultrafast optical metrology: By stretching picosecond-scale waveforms to nanosecond timescales, time-lens-based photon Doppler velocimetry (TL-PDV) extends the dynamic velocity range accessible to photodetectors and digitizers (Chu et al., 2021).
• Temporal and spectral waveform manipulation: Time lenses serve for bandwidth adaptation, temporal waveform conversion, and high-dimensional time-bin encoding, supporting quantum key distribution, multiplexing, and ultrafast measurement of correlations inaccessible to conventional detectors (Mittal et al., 2017).

7. Experimental Realizations and Current Limitations

Time-lens systems have been experimentally demonstrated using SFG in periodically poled lithium niobate (PPLN) and bismuth borate (BiBO) crystals, FWM in highly nonlinear fibers and silicon photonic waveguides, and EOPMs in integrated photonic circuits. Key reported parameters include:

• SFG efficiency up to 0.2% in first-generation single-photon experiments, with nonclassical $g^{(2)}_{s,h} > 3$ indicating good quantum-state preservation (Donohue et al., 2016).
• Joint-temporal imaging with $\sim$10-fold magnification and $\sim$30 ps resolution, resolving sub-100 ps correlations (Mittal et al., 2017).
• Temporal telescope architectures attaining chirp-free, low-loss scaling suitable for quantum networks with explicit design rules for GDD assignment (Srivastava et al., 2023).

Limitations remain from nonideal phase matching, finite temporal aperture, unconverted background (incomplete efficiency), higher-order dispersion, and in on-chip settings, nonlinear losses such as two-photon and free-carrier absorption (Patera et al., 2022, Chu et al., 2021). Multimode quantum theory enables systematic optimization to approach unity efficiency across many temporal modes, critical for the scalability of quantum photonic platforms.