From Diffraction to Refraction: a coherence-based conceptual framework

TL;DR

Refraction has traditionally been tied to boundary-index discontinuities, but this work demonstrates that coherence in the internal phase structure can bend light inside a homogeneous medium. It introduces a coherence-based constitutive framework centered on a detuning term $K_{\mathrm{rc}}$ and a tangential phase-matching rule $n_2 \sin\theta_t - n_1 \sin\theta_i = \frac{K_{\mathrm{rc}}^{(\parallel)}}{k_0}$ with $k_0=\frac{2\pi}{\lambda_r}$, unifying diffraction and refraction under a common phase-causality; this relation reduces to classical Snell’s law when $K_{\mathrm{rc}}^{(\parallel)}=0$. The theory connects to the Huygens–Fresnel principle, shows how an intrinsic phase gradient can activate near-field evanescent components into the far field, and extends to multi-order coherent diffraction via $n_2 \sin\theta_m - n_1 \sin\theta_i = \frac{m K_{\mathrm{rc}}^{(\parallel)}}{k_0}$. Experimental checks reveal sign- and magnitude-dependent coherence lensing, polarization effects through a weak spin term, and a clear null-detuning limit, establishing a Maxwell-consistent framework for coherence-driven refraction with potential to access near-field coherence in bulk media.

Abstract

Refraction, traditionally viewed as a geometric event occurring at material interfaces, is now being re-examined through the lens of coherence. Recent studies in optics and photonics, including coherence tomography, Moire interference, and coherence-engineered diffraction, indicate that phase organization alone can bend light even without index discontinuities. Fraunhofer-based analyses further show that angular deflection can arise from intrinsic phase curvature within homogeneous media. Here we introduce a coherence-based constitutive framework that systematizes these observations: refraction can occur inside a bulk medium when coherence itself provides the effective boundary. Two near-frequency structured beams write and probe a shared phase field, revealing reproducible angular rotation and coherence-lensing whose direction and magnitude follow the spectral detuning. The key detuning coefficient is: Krc = 2 * pi * (1/lambda_r - 1/lambda_w). The compact coherence-refraction relation n2 * sin(theta_t) - n1 * sin(theta_i) = Krc_parallel / k0, with k0 = 2 * pi / lambda_r, retains the form of Snell's law while extending it to coherence-driven regimes. This is not a new law but a quantitative rule linking tangential phase matching to observable deflection within homogeneous media.

Figures (3)

• Figure 1: Spectral dependence and coherence lensing.Lensing polarity$\leftrightarrow$ sgn($\Delta\lambda$); ring density$\leftrightarrow$$|\mathbf{K}_{\mathrm{rc}}|$. Top left: measured angular distribution of the ring pattern ($\Delta\beta$) vs. $\Delta\lambda$ (fit $\propto|\mathbf{K}_{\mathrm{rc}}|/k_{0}$). Bottom left: schematic setup. Right panels (a--h): far-field rings vs. $\Delta\lambda$; bright center for $\Delta\lambda>0$ (positive coherence lens), dark center for $\Delta\lambda<0$ (negative coherence lens); rotation follows sgn($K_{\mathrm{rc}}$). Panels adapted from Cas2020AMTCas2016OL.
• Figure 2: Spectral splitting and angular reversal.a) Composite far-field showing opposite ring centers for red- and blue-detuned readouts. b) Schematic of the spatial-beat convolution between recorded and read structured fields. c) Angular distribution of scattered light ($\theta_s$) versus spectral detuning $\Delta\lambda$ (squares = experiment; black line = convolution model). The antisymmetric trend is governed by the spectral-beat parameter $K_{\mathrm{rc}}$; small tilts preserve the curve, confirming a coherence-driven origin. Panels (b,c) adapted from Applied Materials Today (2020); panel (a) is original.
• Figure 3: Polarization-dependent coherence refraction. (a,b) Linear inputs $\rightarrow$ twin-lobe anisotropy; (c) circular $\rightarrow$ restored symmetry; (d) slight ellipticity $\rightarrow$ deformed rings. A weak spin term $K_{\mathrm{spin}}$ adds to $K_{\mathrm{rc}}$, enabling anisotropy without bulk birefringence. Adapted from Cas2008APL.