Spectro-temporal unitary transformations for coherent modulation: design trade-offs and practical considerations

TL;DR

The paper addresses bandwidth and loss limitations of conventional IQ modulation by proposing spectro-temporal unitary transforms realized with cascaded phase modulators and dispersion. It formalizes the transform as $U = \Lambda_1 H \Lambda_2 H \cdots \Lambda_n H$ and uses a gradient-based optimization (L-BFGS-B) to minimize the distortion-to-signal ratio $DSR$, with a drive-power term to balance performance. Key results show that SDRs exceeding $30$ dB at >$200$ GBd are achievable with a small number of stages and realistic hardware parameters, and they map out how dispersion per stage, PM bandwidth, block length, and power trade off against SDR. The work provides practical design guidelines, including tolerance requirements for phase, amplitude, dispersion, and DAC resolution, highlighting the potential for scalable, chip-scale coherent transceivers via spectro-temporal unitary transforms.

Abstract

This paper analyzes the performance of spectro-temporal unitary transforms for coherent optical modulation. Unlike conventional IQ modulation, such transforms are based on a cascade of phase modulators and dispersive elements, so are theoretically lossless and not limited by the bandwidth of the constituent modulators. We analyse the performance limits and design trade-offs of this scheme: estimating how the number of stages, amount of dispersion, modulator bandwidth, symbol block length and electrical signal power impacts the achievable signal-to-distortion ratio (SDR). Importantly, we show that high (>30 dB) SDRs suitable for modern >200 GBd class coherent optical communications are achievable with a low (<6) number of stages and reasonable parameters for driver power, modulator bandwidth and on-chip dispersion. Finally we address the SDR penalties associated with potential phase, amplitude, or dispersion errors, and limited DAC resolution.

Figures (10)

• Figure 1: (a) Conventional IQ modulation based on amplitude modulation, $a(t)$. (b) Lossless spectro-temporal unitary transform based arbitrary waveform modulation based on cascaded phase modulators $\phi(t)$ and dispersive elements $H(\omega)$.
• Figure 2: The optimisation procedure attempts to find the phase instructions, $\phi_n(t)$, at each stage that can produce a waveform $\Psi_N(t)$ after $N$ stages that matches the target waveform $\Psi_\textnormal{target}$ with minimal DSR. The DSR gradient is calculated using the forward propagated wave $F_n(t)$ and backward propagated wave $B_n(t)$ at each stage. The backward propagating wave is calculated by propagating the target waveform $\Psi_\textnormal{target}$ backwards through the system, while the forward propagating wave is calculated by propagating the input waveform $\Psi_0$ forwards through the system, using the current phase instructions.
• Figure 3: SDR v. dispersion per stage ($\beta_2 L$) for a $B_{\textnormal{PM}}=$ 0.55 $f_s$, and varying number of stages, $N$. The dispersion is normalised to the symbol period squared ($T_s^2$). Example constellations and phase modulation spectral densities inset in (a) and (b).
• Figure 4: SDR v. phase modulator bandwidth ($B_{\textnormal{PM}}$), for $\beta_2 L =T_s^2$, and varying number of stages, $N$. The phase modulator bandwidth is plotted in units of symbol rate, $f_s$. Example constellations and phase modulation spectral densities inset in (a) and (b).
• Figure 5: SDR v. block length in symbols, for $\beta_2 L =T_s^2$, $B_{\textnormal{PM}}=$ 0.55 $f_s$, and varying number of stages, $N$.