In a recent article published in the journal Light: Science & Applications, researchers explored a fundamental but often overlooked area of optical science: how spatial coherence can be controlled and synthesized in nonlinear optical processes. Their work focuses on gaining a deeper understanding of how incoherent light behaves in nonlinear media, and whether its coherence can be tuned to influence the properties of the resulting second harmonic light.
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Background
Optical coherence is a fundamental property with wide-ranging implications across fields like microscopy, optical communication, and quantum information. In nonlinear optics specifically, the efficiency and behavior of frequency conversion processes are closely tied to the coherence of the incoming light. Traditionally, this area has relied heavily on coherent laser sources to ensure consistent and predictable nonlinear interactions. Theoretical models developed over the past several decades have largely focused on Gaussian, fully coherent states, under the assumption that high temporal and spatial coherence is essential for effective harmonic generation.
However, recent studies have shown that partially coherent and even incoherent light can still play a meaningful role in driving nonlinear effects. In complex nonlinear crystals, both phase relationships and coherence properties influence the emergence of structured light, such as vortex and Airy beams, that exhibit orbital angular momentum (OAM) and self-accelerating behavior. Understanding how these properties evolve during nonlinear interactions, especially when starting with incoherent light, remains an open question.
While current theories suggest that coherence may be compromised during nonlinear processes, experimental validation has been limited. This gap highlights the need for techniques that can not only manipulate but also synthesize specific coherence characteristics in the light generated through these interactions.
The Current Study
The experimental method focuses on generating second harmonic light with precisely controlled spatial coherence. It starts by reconstructing the full phase and amplitude information of the fundamental speckle fields (complex, random wavefronts) using a phase retrieval algorithm. This involves capturing intensity patterns at several axial positions, including the focal plane and slightly defocused planes, allowing for the reconstruction of the complex optical field with continuous phase information.
Once these fields are fully characterized, researchers apply a complex square-root filter along with other digital modulation techniques to embed specific coherence properties into the speckle fields. These modulated fields are then encoded onto a spatial light modulator (SLM) and projected into a nonlinear photonic crystal, specifically, a periodically poled χ(2) crystal, for second harmonic generation. The setup uses a 4f imaging system to ensure precise delivery of the modulated speckle fields into the nonlinear medium while preserving the intended coherence structure.
The resulting second harmonic speckle patterns are analyzed in the far field using interferometric techniques. This analysis reveals key coherence properties, including singularities and the topological charge of vortex beams. Through this approach, the researchers can effectively design and manipulate the coherence of the fundamental beam, thereby shaping the coherence characteristics of the second harmonic light with a high degree of control.
Results and Discussion
The results offer key insights into how coherence can be manipulated through nonlinear optical processes. The data clearly show that the spatial coherence of the fundamental speckle fields directly shapes the coherence of the second harmonic output. In particular, the second harmonic coherence patterns closely reflect the engineered coherence of the input, confirming that spatial coherence can be purposefully designed and preserved through frequency conversion.
A striking example involves ring singularities: a single ring singularity in the fundamental beam’s coherence transforms into a double ring singularity in the second harmonic, demonstrating the conservation of orbital angular momentum (OAM) even under partially coherent conditions. Experiments with vortex beams carrying different topological charges, quantized in integer multiples of Planck’s constant, further confirm that nonlinear interactions can transfer and even enhance OAM properties, despite the initial lack of coherence. This supports the theoretical prediction that angular momentum is conserved in nonlinear media, even in complex, incoherent settings.
The study also examines Airy beams, which are known for their self-accelerating and self-healing behavior. Remarkably, these characteristics persist in the second harmonic output, regardless of the input beam’s reduced coherence. This suggests that certain structured features of light are inherently robust to coherence loss in nonlinear conversions.
Taken together, these findings highlight a powerful capability: the ability to engineer and maintain coherence properties in structured light through nonlinear processes. This opens up new possibilities for applications such as incoherent imaging, covert beam shaping, and quantum information systems, domains where precise coherence control across different frequencies could be a game-changer.
Conclusion
This study offers compelling experimental evidence that high spatial coherence, long considered a prerequisite in nonlinear optics, is not strictly necessary. Instead, the researchers show that spatial coherence can be intentionally engineered in both the fundamental and second harmonic fields, even starting with partially incoherent light. By integrating phase retrieval, digital modulation, and nonlinear frequency conversion, the team presents a new method for synthesizing targeted coherence states, including vortex and Airy beam profiles, in the second harmonic domain.
A key result is the confirmed conservation of orbital angular momentum (OAM) throughout these nonlinear interactions, reinforcing core physical principles that remain valid even under partial coherence. This challenges traditional assumptions and suggests that incoherent or partially coherent light sources can be effectively used in nonlinear optics, broadening the scope of what’s possible in structured light generation and advanced optical imaging.
Ultimately, the work deepens our understanding of coherence in nonlinear systems, showing that it's not only more flexible than previously thought, but also more resilient. These insights could lead to innovations in areas like coherence-based imaging, information encoding with structured light, and the design of optical systems where coherence control is critical.
Journal Reference
Pang Z., Arie A. (2025). Coherence synthesis in nonlinear optics. Light: Science & Applications 14, 101. DOI: 10.1038/s41377-025-01749-6, https://www.nature.com/articles/s41377-025-01749-6