High-precision phase reconstruction in Discontinuous Wavelength-Scanning Interferometry

Depth-resolved wavelength-scanning interferometry (DRWSI) has become one of the most sensitive methods for measuring the surface shape and optical thickness of multilayer or transparent materials. Its capacity to reconstruct depth profiles from interferometric phase data enables precise characterization of micro-optical elements, thin films, and transparent plates. However, the classical implementation of DRWSI depends critically on the continuity of the laser’s wavelength scan. Diode lasers often have mode hopping or discontinuous spectral emission which produce gaps that disrupt the coherence of the interferometric signal. These discontinuities introduce random phase jitters and abnormal sidelobes that obscure the true topographic information and degrade the accuracy of phase recovery. Several approaches have previously been proposed to resolve this limitation and early strategies involved monitoring the wavelength-scanning sequence to reconstruct continuous spectra, but they inevitably narrowed the effective scanning range, reducing depth resolution. Other techniques, such as the random sampling Fourier transform (RSFT) and the iterative adaptive approach (IAA), reframed the problem as one of nonuniform spectral sampling or weighted least-squares fitting. However, although these methods succeeded in suppressing phase artifacts, they still relied on prior knowledge of the laser’s gapped spectrum—information not always available or measurable in real-time experiments. Furthermore, these algorithms typically demanded additional optical monitoring hardware, adding cost and complexity to what is otherwise an elegant and compact interferometric system. To this account, new research paper published in Optics Letters and conducted by Dr. Yulei Bai, Dr. Hao Qiu, Dr. Zean Huang, Dr. Zhaoshui He, Dr. Shengli Xie, and led by Professor Bo Dong from the School of Automation at Guangdong University of Technology, the researchers developed two complementary models: an auto-regression model to predict missing interference intensities within a gapped spectrum, and a discrete optimization model to automatically determine the number of missing points without prior spectral information.

The researchers first simulated DRWSI measurements under controlled spectral discontinuities to validate the ARM-based framework. They modeled the optical path difference using a MATLAB-generated surface shaped as the software’s logo and imposed gap intervals of 0.1 nm, 1 nm, and 10 nm within a 1.0-nm scanning range centered at 600 nm. The algorithm treated the missing intensities as unknowns within a vectorized interferometric signal, where an auto-regression of order γ estimated the missing components through least-squares prediction. A discrete optimization step then automatically determined the number of prediction points that minimized the total energy of the amplitude spectrum—including both mainlobe and sidelobes—after Fourier transformation. The authors found simulations produced compelling evidence of the method’s robustness. They reported that conventional Fourier transforms failed to resolve the true amplitude spectrum once gaps exceeded even 10 % of the scanning range, and generated multiple false interference peaks. On the other hand, the ARM reconstruction yielded a single, well-defined mainlobe consistent with the ideal theoretical spectrum. The resulting phase maps closely matched the ideal reference even when the discontinuity reached tenfold the nominal wavelength range, which confirmed that the model effectively predicted missing data without prior spectral information.

The authors afterward validated experimentally using a custom-built discontinuous DRWSI system incorporating a mode-hopping diode laser, a 4f optical setup, and a CCD detector capturing 308 frames during a 0.9-nm wavelength sweep. The sample combined a 6′ optical wedge and a USAF 1951 resolution target, providing four distinct reflective surfaces. Because the system lacked any hardware to measure the spectral gaps, neither RSFT nor IAA could be applied; only the conventional Fourier transform and the proposed ARM method were compared. The Fourier-based results revealed extensive spectral cross-talk and irregular sidelobes, which blurred the identification of true surface peaks and generated visible phase ripples. After applying the ARM model, these sidelobes were nearly eliminated, yielding smooth, high-fidelity phase distributions. Quantitatively, the wedge’s tilt angle derived from the ARM reconstruction deviated by only 0.35 % from the manufacturer’s specified 6′, a precision unmatched by conventional methods. Moreover, phase maps of the resolution target recovered clear structural patterns that had been completely obscured before correction.

In conclusion, the new study by Professor Bo Dong and colleagues successfully designed new models able to reconstruct continuous interference data from discontinuous wavelength scans. The new framework eliminates phase jitters and sidelobes caused by mode hopping and restored accurate depth-resolved phase maps while preserving full depth resolution. Indeed, it redefined the methodology of depth-resolved interferometry by decoupling spectral continuity from measurement fidelity. The auto-regression model introduces a self-learning mechanism into optical metrology: instead of compensating for hardware imperfections through calibration, the signal itself guides the recovery of lost spectral information. The method also achieves a level of adaptability unattainable in traditional Fourier-based frameworks by automatically determining the missing points through discrete optimization. It restores the core advantages of DRWSI—full-field, high-sensitivity phase detection—while lifting the stringent requirement for continuous laser emission. Additionally, the newly proposed technique broadens the scope of DRWSI applications. Compact diode lasers, often dismissed for precision metrology due to mode-hopping instabilities, can now be integrated into low-cost, high-accuracy instruments. This advance could benefit optical component manufacturing, multilayer film inspection, and biomedical surface imaging, where system simplicity and flexibility are valued alongside precision. Moreover, because the algorithm operates entirely in software and requires no auxiliary spectral-monitoring device, it reduces maintenance and system calibration time.

The new study challenges the assumption that continuous spectral scanning is indispensable for accurate phase retrieval and with their demonstration that an unknown gapped spectrum can yield equally reliable topographic information, it invites a conceptual shift: precision need not rely solely on hardware perfection but can emerge from intelligent modeling. Moreover, the proposed method effectively overcomes the limitation of phase demodulation accuracy caused by wavelength discontinuity in multi-source swept-frequency tomographic imaging. In a nutshell, the new findings demonstrated that the ARM approach restored phase accuracy, achieving performance previously possible only with continuous-spectrum lasers. Future studies might extend this concept to hyperspectral interferometry, optical coherence tomography, or spectroscopic imaging, where spectral gaps hinder data fidelity.