Revealing Sharp Spectral Features with Complex Frequency Excitations: Challenges and Opportunities

Abstract: Broadening of spectral and spatial responses due to intrinsic loss in real materials often hides sharp features. One recently recognized route to recover those features is to probe the system with complex-frequency (CF) signals that decay exponentially in time: a suitably tailored temporal decay can compensate for loss and reveal an intrinsic, narrow response. However, generating rapidly decaying optical waveforms in real time is often challenging (the required decay times may be in the range of tens of femtoseconds). A recently proposed alternative synthesizes the CF response numerically after detection of conventional, real-frequency signals using Fourier post-processing. Here we explore advantages and challenges of these approaches: we show that a physical CF excitation robustly sharpens spectral features in the presence of noise, while a post-detection synthesized CF response shows only limited improvement once realistic detection and readout noise is considered. At the same time, in low-noise conditions a much simpler post-detection filtering procedure attains equal or better recovery than the synthesized CF reconstruction, making the synthesis unnecessary in practice.

• The paper demonstrates that real-time CF excitations can robustly recover narrow spectral and spatial features with superior signal gating compared to synthesized methods.
• It shows that while both SCF and inverse filtering can restore sharp features under ideal conditions, their performance degrades significantly in the presence of realistic detection noise.
• The study emphasizes that noise characteristics critically impact CF techniques, making physical CF methods preferable in scenarios dominated by signal-dependent noise.

Complex-Frequency Excitations for Recovering Sharp Spectral Features: Analysis and Implications

Introduction

This paper addresses the challenge of resolving sharp spectral features in lossy photonic systems where material absorption leads to broadening and suppression of narrow resonances. The core focus is on the use of complex-frequency (CF) excitations, specifically how exponential temporal decay in the excitation can compensate for intrinsic material dephasing and recover otherwise obscured spectroscopic and spatial structure. Two main routes are critically analyzed: the physical, real-time generation of CF signals (RTCF), and synthesized CF (SCF) reconstructions applied post-detection from conventionally acquired real-frequency data. The paper provides a technical evaluation of these methods under realistic noise conditions and benchmarks them against standard post-processing approaches like inverse filtering.

Theoretical Foundation

The Lorentzian spectral broadening due to dephasing in absorptive photonic materials is tackled by probing the system with a CF excitation, $\omega = \omega_0 - i\Gamma$, where $\Gamma$ is the exponential decay rate. When applied to a medium with linear response $H(\omega) = A/(\omega_0 - \omega - i\gamma)$, a CF excitation can partially or fully compensate the decay, resulting in a narrower effective response $H(\omega_0 - i\Gamma)$. This approach has been validated at lower frequencies (acoustics, microwave), but generating rapidly decaying optical waveforms on femtosecond timescales remains a severe technical challenge.

An alternative, SCF, circumvents waveform synthesis by acquiring the spectrum at real frequencies and reconstructing the CF response via Fourier post-processing. Mathematically, it exploits the analytic continuation of the system response into the complex frequency domain through temporally weighted integration.

Comparative Analysis of RTCF, SCF, and Inverse Filtering

Performance in Noiseless Conditions

Both RTCF and SCF methods can theoretically restore sharp spectral features obscured by loss broadening. In simulations based on Lorentzian resonances characteristic of plasmonic systems, the two methods yield comparable narrowing of spectral features when noise is absent, particularly after appropriate time-gating to suppress free (transient) oscillations. The SCF approach, relying on post-detection digital filtering, can thus mimic the effect of physical CF excitation under idealized conditions.

Importantly, the analysis demonstrates that simple inverse (matched) filtering in the time domain, e.g., multiplying the temporal response by an exponential factor and performing a Fourier transform, can achieve equivalent or superior restoration of sharp features with lower computational complexity than SCF.

Impact of Noise

A core contribution of the paper is its robust treatment of noise, especially photon shot noise, relative intensity noise (RIN), pixel non-uniformity, and computational artifacts introduced during phase retrieval necessary for SCF. In practical spectroscopic setups, noise is mapped into the recorded spectrum and, after inverse Fourier transformation, becomes temporally white, quickly dominating the decaying temporal tail that carries information about ultra-narrow spectral features.

With post-detection methods (SCF, inverse filtering), integration beyond the so-called SNR-threshold time amplifies noise rather than the signal, fundamentally limiting achievable spectral narrowing and making deconvolution of overlapping resonances unreliable. The SCF method, in particular, shows no fundamental advantage over conventional inverse filtering and becomes computationally more demanding.

In contrast, with RTCF, time gating at the detection stage limits the energy—and thus noise—captured within the time window corresponding to the signal tail, preserving SNR longer and yielding more robust resolution enhancement, especially when dominant noise sources are of the signal-dependent type (e.g., shot noise, RIN). However, this advantage disappears for noise sources independent of signal energy, such as dark current or background.

Practical and Theoretical Implications

Spectroscopy and Resonance Recovery

The RTCF approach provides demonstrable resolution enhancement in recovering closely spaced or overlapping Lorentzian resonances in plasmonic and molecular systems, provided that noise is dominated by sources proportional to detected energy. In such scenarios, physical CF excitation is robust; whereas, SCF reconstruction and inverse filtering, while successful in the noiseless regime, fail to demix resonances in the presence of realistic detection noise. The requirement of reliable phase retrieval for SCF adds further to its impracticality outside highly controlled environments.

Extension to Spatial Resolution

Analogous principles apply to spatially resolved imaging, e.g., in superlenses or near-field systems degraded by loss-induced broadening of the modulation transfer function (MTF). CF illumination in the spatial domain can recover high-$k$ spatial frequencies, enhancing the system's resolving power. The advantage of RTCF over SCF persists under the caveat that noise coupled to the useful signal scales with signal energy. Nonetheless, both approaches ultimately face the noise floor limit; when noise is dominated by background or detector baseline, neither offers improvement over conventional deblurring techniques.

Broader Perspective and Future Directions

The findings present a nuanced view: the concept of “loss compensation” via CF excitation is more accurately viewed as “broadening compensation.” While CF methods theoretically extend the boundary of recoverable spectral or spatial information in lossy systems, their practical efficacy hinges on noise characteristics and the feasibility of generating or synthesizing the requisite CF excitations. Inverse or matched filtering provides a computationally preferable alternative to SCF in most cases.

Future developments may focus on improving the fidelity of ultrafast waveform synthesis for RTCF, engineering noise properties in detection hardware, and integrating CF concepts into adaptive filtering or compressive sensing frameworks in photonic imaging and spectroscopy.

Conclusion

The paper provides a rigorous comparison between real-time and synthesized CF excitation strategies for revealing sharp spectral features in lossy photonic systems. It demonstrates that while both approaches are mathematically equivalent in the absence of noise, only physical CF illumination consistently improves resolution in the presence of realistic, signal-dependent noise. SCF does not outperform traditional post-processing methods and is encumbered by greater computational overhead and practical challenges. The selective advantages of CF techniques—whether spectral or spatial—are thus highly context-dependent, constrained by the interplay of noise characteristics and experimental capabilities. CF-based methods, especially RTCF, are likely to find niche applications where their signal-gating advantages can be fully leveraged.