Temporal Processor Module (TPM)

• TPM is a reconfigurable on-chip photonic system that performs high-speed analog signal processing using dispersive Fourier transformation and programmable spectral modulation.
• It leverages four-wave mixing with chirp modulation to achieve real-time operations such as differentiation, integration, and convolution with sub-picosecond resolution.
• Precise electrical control of Mach–Zehnder interferometer modulators enables arbitrary temporal transfer functions, ideal for optical computing and high-throughput telecommunications.

A Temporal Processor Module (TPM) is an on-chip, fully reconfigurable photonic signal processing system designed to perform high-speed analog mathematical operations—including differentiation, integration, and convolution—directly in the time domain. Leveraging dispersive Fourier transformation, controlled chirp modulation, and programmable spectral transfer functions, the TPM achieves ultrafast, high-resolution signal processing with direct optical-domain implementation. The architecture and operational principles of the TPM support bandwidths exceeding 400 GHz and sub-picosecond temporal resolution, making it a foundational component in temporal analog optical computing (Babashah et al., 2017).

1. System Architecture and Signal Flow

The TPM comprises three cascaded functional sections:

1. Time-Lens and Dispersive Fourier Transform (DFT) Stage: An input optical pulse $x(t)$ is co-propagated with a chirped pump in a nonlinear waveguide, generating a linearly chirped replica through four-wave mixing. The chirp factor $b = 1/(2\beta_{2,p} L_p)$ imparts a quadratic phase $e^{j b t^2}$ on $x(t)$. The chirped pulse then propagates through a photonic-crystal waveguide (length $L_w$, group velocity dispersion $\beta_{2,w} > 0$), applying a second-order phase $\phi'' = \beta_{2,w} L_w$ and executing a temporal Fourier transform. The output, $y_1(t)$, is proportional to the spectrum $X(\Omega)$ of the input, with the one-to-one mapping $\Omega = t/\phi''$ (Babashah et al., 2017).
2. Spectral Modulation Stage: $y_1(t)$, encoding the input spectrum along its temporal envelope, traverses a Mach–Zehnder interferometer (MZI) amplitude modulator and a high-speed phase modulator. By applying appropriate electrical drive waveforms $V_a(t)$ and $V_\phi(t)$, arbitrary complex spectral transfer functions $T(\Omega)$ are imprinted on the signal: $$M(t) = |T(\Omega = t/\phi'')| e^{j \arg T(\Omega = t/\phi'')}$$.
3. Inverse Dispersive Fourier Transform Stage: The modulated signal is sent through a second photonic-crystal waveguide of equivalent length but opposite dispersion ($\beta_{2,w}' = -\beta_{2,w}$), which implements the inverse Fourier transform, yielding the processed output $y(t) = \mathcal{F}^{-1}\{T(\Omega) X(\Omega)\}$ (up to time reversal and phase factor).

Stage	Function	Key Components
Time-Lens & DFT	Fourier transform of input to time domain	FWM time lens, dispersive photonic-crystal waveguide
Spectral Modulation	Programmable spectral transfer function $T(\Omega)$	MZI amplitude modulator, p-n junction phase modulator
Inverse DFT	Converts modified spectrum back to time domain	Inverse-dispersion photonic-crystal waveguide

2. Mathematical Framework

The TPM operation is described by three key mathematical processes:

• Dispersive Fourier Transform (DFT):

The input $x(t)$, after chirp multiplication, becomes $x_c(t) = x(t) e^{+j b t^2}$, with $b = 1/(2 \phi'')$. Passing through the dispersive medium yields $$y_1(t) \approx e^{j(\phi_0 - t^2/(2\phi''))} X(\Omega = t/\phi'')$$.

• Arbitrary Spectral Multiplication:

A time-dependent multiplier $M(t)$ is applied to $y_1(t)$, targeting a desired spectral shape. For $$M(t) = |T(\Omega = t/\phi'')| e^{j \arg T(\Omega = t/\phi'')}$$, this yields $y_2(t) = M(t) y_1(t) \propto T(\Omega) X(\Omega)$. Special cases include $T(\Omega) = j \Omega$ for differentiation, $T(\Omega) = 1/(j \Omega)$ for integration, or any convolution kernel via $T(\Omega) = \mathcal{F}\{h(t)\}$.

• Inverse Dispersive Propagation:

The inverse-dispersion waveguide reconstructs the processed time-domain output as $y(t) = \mathcal{F}^{-1}\{T(\Omega) X(\Omega)\}$.

3. Device Realization and Performance Metrics

Critical device and system metrics include:

• Dispersion Engineering:

The photonic-crystal waveguide provides $\beta_2 = 2.81 \times 10^6$ ps$^2$/km. With $L \approx 10$ mm, the resultant $\phi'' \approx 30$ ps$^2$ stretches a 400 GHz bandwidth into a 200 ps time window.

• Temporal Resolution:

The Fourier-limited pulse resolution is $\delta t \approx 1/400\,\text{GHz} \approx 2.5$ ps, with effective resolution reaching 300 fs via precise dispersion tailoring and apodization.

• Chirp Generation:

Four-wave mixing in the initial waveguide segment (length $L_p \approx 10$ mm, $b \approx 1.6 \times 10^{-2}$ ps$^{-2}$) is used to shape 100 ps pulses.

• Modulator Transfer Functions:

The MZI offers amplitude modulation $|T(\Omega)| = |\cos[\pi V_a(t) / (2 V_\pi)]|$, while the phase modulator provides phase $\phi_M(t) = \pi V_\phi(t) / V_\pi$.

• Implementation Geometry:

The core is Si$_3$N$_4$ (1\,µm × 0.4\,µm) under-clad by a Si/SiO$_2$ photonic crystal lattice (pitch 400 nm, pillar diameter 250 nm), permitting $>$400 GHz bandwidth and high GVD.

• Repetition Rate:

For a 200 ps window, repetition rate is constrained by $\Delta \tau \cdot R < 1$, giving $R < 5$ GHz, with potential scaling beyond 10 GHz.

4. Programming Arbitrary Temporal Transfer Functions

Reconfigurability is achieved via direct electrical control of the MZI amplitude and phase modulators. The spectral transfer function $T(\Omega)$ is implemented by loading the corresponding drive signals:

• $$V_a(t) = (2 V_\pi / \pi) \cdot \arccos \sqrt{|T(t/\phi'')|}$$
• $V_\phi(t) = (V_\pi / \pi) \cdot \arg T(t/\phi'')$

Waveforms are generated by high-speed digital-to-analog circuitry (e.g., DAC-driven AWG at up to 50 Gsamples/s). For operations such as differentiation ($T(\Omega) = j \Omega$), $V_a(t)$ provides a linear ramp, while $V_\phi(t)$ imparts a $\pm\pi/2$ phase step (Babashah et al., 2017).

5. Supported Computational Functions and Use Cases

The TPM natively implements a wide range of analog temporal computations:

• Differentiation: $T(\Omega) = j \Omega$
• Integration: $T(\Omega) = 1/(j \Omega)$
• General Convolution: For impulse response $h(t)$, $T(\Omega) = \mathcal{F}\{h(t)\}$

With the ability to update $T(\Omega)$ for each input pulse, the TPM supports real-time, adaptive temporal signal processing. This enables high-fidelity, high-throughput computation for telecommunications, RF signal conditioning, ultrafast optical waveform generation, and other applications demanding parallel, analog-domain temporal transformations.

6. Limitations and Prospective Developments

While the TPM demonstrates sub-picosecond resolution, 200 ps processing window, and full reconfigurability over 400 GHz bandwidth, several factors constrain scalability:

• Repetition Rate Ceiling: The window-size limits impose a finite pulse throughput, though dispersion trimming and shorter pulses allow further scaling.
• Modulator Speed and Precision: Fidelity is dependent on the accuracy and bandwidth of electrical control waveforms.
• Waveguide Fabrication: Achieving and stabilizing the requisite large GVD in compact photonic-crystal geometries is technologically demanding.

This suggests that research will focus on integrating faster modulators, broader bandwidth dispersive media, and multiplexed TPM architectures. A plausible implication is continued expansion of on-chip analog optical computing capabilities beyond traditional DSP limitations (Babashah et al., 2017).