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Unveiling Structured Optical Coherence in Nonlinear Optics

This paper demonstrates that nonlinearity acts as a new degree of freedom for optical coherence by deriving and experimentally validating an analytic expression showing how incoherent sources can generate structured coherence in second harmonic generation, a process governed by both the source distribution and the crystal's nonlinear structure.

Original authors: Zihao Pang, Ady Arie

Published 2026-07-16

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CC BY 4.0

Original authors: Zihao Pang, Ady Arie

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine light not just as a bright beam, but as a crowd of people. In a laser, everyone marches in perfect lockstep, holding hands and humming the same tune; this is "coherent" light. But in a light bulb or the sun, everyone is wandering randomly, shouting different things, and bumping into each other; this is "incoherent" light. For over a century, scientists have known that if you take a chaotic, incoherent crowd and let them walk far enough away, they naturally start to organize themselves into a predictable pattern. This is a famous rule in physics called the van Cittert-Zernike theorem. It's like watching a chaotic crowd at a concert eventually form a neat line just by walking across a huge field.

But what happens if you don't just let them walk? What happens if you force them to dance in a room with a bouncy floor that changes the rules of the dance? This is the world of "nonlinear optics," where light interacts with special crystals to change its color or behavior. Usually, scientists use neat, organized lasers to do this. But a new study asks a wild question: Can we take a messy, chaotic crowd of light and use a special crystal to turn that mess into a beautifully structured pattern? The researchers wanted to see if the "dance floor" (the crystal) could modulate the organization of the "chaotic dancers" (the light) based on the intensity of the dance, creating a new kind of organized light, right at the moment the light changes color.

In this paper, the authors, Zihao Pang and Ady Arie, set out to answer exactly that. They took a source of light that was completely incoherent—like a storm of random speckles—and shined it into a special crystal to create a new color of light (a process called second harmonic generation). They discovered that the crystal doesn't just change the color; it acts like a magical mold that shapes the chaos into a specific, structured pattern based on the crystal's internal nonlinearity. They derived a new mathematical rule, a "Nonlinear van Cittert-Zernike theorem," which predicts exactly what shape the light will take based on the pattern carved into the crystal.

The team showed that if you use a crystal with a uniform pattern, the chaotic light turns into a smooth, bell-shaped pattern (Gaussian coherence). But if you shine the chaotic light through a ring-shaped input profile, the light transforms into a target-like pattern with a bright center and rings around it (Bessel coherence). Even cooler, they used crystals with complex, wave-like patterns (Hermite-Gaussian) to transform an elliptic Gaussian input into light that looked like a grid of interference fringes, essentially turning the crystal into a template that stamps a specific design onto the light's organization.

To prove this wasn't just a computer trick, they built a real experiment. They used a device to create 1,000 different random "speckle" patterns of light and fired them through a 1-millimeter-thick crystal. The results matched their math perfectly: the messy input light came out as a beautifully structured output. They also tested a much longer crystal (50 millimeters) and found something surprising: as the light traveled deeper into the crystal, it didn't just stay messy; it actually gathered more and more order along the way, becoming more coherent the further it went.

The paper suggests that this discovery opens a new door. Instead of needing a perfect, expensive laser to create structured light, we might be able to use cheap, messy light sources and just design the right crystal to fix them. This could help us create better images from thermal sources or understand how light behaves in complex materials, proving that sometimes, a little bit of chaos is exactly what you need to build something structured.

Technical Summary: Unveiling Structured Optical Coherence in Nonlinear Optics

Problem Statement
While optical coherence has long been understood as a structured spatial distribution governed by the van Cittert-Zernike (VCZ) theorem in linear optics, its behavior in nonlinear optical interactions remains less explored. Previous studies in nonlinear media have largely focused on how coherence affects other observables (e.g., soliton stability or entanglement) rather than treating the structured optical coherence (SOC) of the generated field as the primary object of study. A significant challenge exists in characterizing the SOC of second harmonic (SH) fields generated from incoherent sources, particularly because coherence functions are inherently high-dimensional and coupled through nonlinear wave mixing. Furthermore, while the VCZ theorem establishes that far-field coherence in linear systems is the Fourier transform of the source intensity, it is unclear how this relationship generalizes when the source passes through a nonlinear crystal with a specific spatial structure.

Methodology
The authors combine analytical derivation with experimental validation to investigate the effects of source incoherence on second harmonic generation (SHG).

Theoretical Framework: The study derives an analytic expression for the far-field mutual coherence function of the SH field. This derivation utilizes the weak-interaction approximation for short crystals, where the SH field is proportional to the square of the fundamental field modulated by the transverse nonlinear susceptibility, $\chi^{(2)}_\perp(\rho)$ . The authors apply Fourier propagation operators to relate the input source statistics to the output coherence.
Experimental Setup: The experiments employ a spatially incoherent fundamental beam generated by illuminating a phase-only spatial light modulator (SLM) with a 1064.5 nm laser to produce 1000 random speckle realizations. These beams are launched into nonlinear crystals to generate 532 nm SH light.
Measurement Technique: Since direct field correlation is difficult, the authors utilize the Siegert relation to infer the field coherence function $\Gamma(\Delta r)$ from the measurable intensity correlation function $g^{(2)}(\Delta r)$ . They perform Young's double-slit measurements to directly observe fringe visibility and coherence growth.
Materials: Experiments utilize periodically poled KTiOPO4 (PPKTP) crystals with homogeneous nonlinearity, and engineered ferroelectric crystals (PPSLT) with Hermite-Gaussian (HG) domain patterns (HG01 and HG03). Long-crystal propagation is studied using a 50-mm LiB3O5 (LBO) crystal.

Key Contributions

Nonlinear VCZ Theorem: The paper derives a "Nonlinear van Cittert-Zernike theorem." Unlike the linear VCZ theorem, which states that far-field coherence is the Fourier transform of the source intensity, the nonlinear version states that the SH far-field coherence is proportional to the Fourier transform of the product of the squared fundamental intensity and the squared nonlinear susceptibility: $|\chi^{(2)}_\perp(\rho) E^2_\omega(\rho)|^2$ . This establishes that SH coherence depends on both the source distribution and the crystal's nonlinear structure.
Decoupling of Coherence: The study demonstrates that in the weak-interaction regime, the phase information of the incoherent fundamental is washed out, and the SH coherence is governed solely by the collective intensity profile of the product term. This allows for the generation of specific SOC patterns (e.g., Gaussian, Bessel, Laguerre-Gaussian) that are distinct from the input coherence.
Coherence Accumulation: The authors provide experimental evidence that coherence in an incoherent SH field can accumulate and increase as light propagates through a long nonlinear crystal, a phenomenon distinct from linear free-space propagation.

Results

Shaped Coherence Generation: Using incoherent Gaussian and annular inputs with a homogeneous PPKTP crystal, the authors successfully generated Gaussian and Bessel-type SOC in the SH field, respectively. The experimental measurements of coherence widths ( $\sigma_{2\omega}$ and $\alpha_{2\omega}$ ) matched theoretical predictions, confirming that the SH coherence width scales differently than the fundamental due to wave-vector doubling and intensity narrowing.
Engineered Nonlinearity: By using crystals with HG-patterned nonlinearity (HG01 and HG03), the researchers generated Cartesian Laguerre-Gaussian coherence in the SH field. The results showed that the SH coherence pattern is determined by the Fourier transform of the crystal's nonlinear amplitude profile, effectively decoupling the SH coherence from the fundamental's intensity profile. For instance, an HG03 crystal produced SH coherence with four distinct lobes, analogous to multi-beam interference.
Long-Crystal Propagation: In Young's double-slit experiments, the SH field generated by a short (1 mm) PPKTP crystal exhibited near-zero fringe visibility (incoherent). In contrast, the SH field generated by a long (50 mm) LBO crystal under identical input conditions showed clear interference fringes, demonstrating the accumulation of coherence during nonlinear propagation.

Significance
The paper claims to establish nonlinearity as a new degree of freedom for controlling optical coherence. By generalizing the VCZ theorem to nonlinear interactions, the work opens a research area at the intersection of statistical optics and nonlinear optics. The authors suggest that this framework allows for the engineering of structured coherence at new frequencies through the design of the nonlinear medium's structure. Potential applications identified by the authors include the control of stochastic optical processes (such as thermal emission and infrared incoherent imaging) and the nondestructive characterization of internal domain patterns in nonlinear photonic materials. The principles may also extend to other nonlinear processes (e.g., sum-frequency generation) and other wave systems.

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