Analytic 3D vector non-uniform Fourier crystal optics in arbitrary $\bar{\bar{\varepsilon}}$ dielectric

TL;DR

This work develops a non-uniform Fourier crystal optics (LFCO) framework that unifies linear crystal optics, crystal optics, and Fourier optics to model nonlinear crystal processes in arbitrarily anisotropic dielectrics. By deriving a explicit $3\times 2$ transition matrix and introducing the material-matrix tetrahedron compass along with a crystal-2f experimental layout, the authors enable fully vectorial field calculations across linear and nonlinear interactions in reciprocal and real space. The approach reveals new spectral and real-space phenomena, including infinite families of optical singularities, heart-shaped L shorelines, double conical diffraction, and optical knots, and outlines pathways toward scalar, semi-vector, and full-vector nonlinear Fourier crystal optics, with applications to focal engineering and potential quantum nonlinear optics. Collectively, this framework provides a scalable, physics-grounded basis for large-scale nonlinear optical simulations in complex media, with implications for SPDC and other quantum nonlinear processes in 3D nonlinear photonic crystals.

Abstract

To find a suitable framework for nonlinear crystal optics(NCO), we have revisited linear crystal optics(LCO). At the methodological level, three widely used plane wave bases are compared in terms of eigenanalysis in reciprocal space and light field propagation in real space. Inspired by complex ray tracing, we expand M.V. Berry \& M.R. Dennis's 2003 uniform plane wave model to non-uniform Fourier crystal optics and ultimately derive the explicit form of its 3$\times$2 transition matrix, bridging the two major branches of crystal optics in reciprocal space, where either ray direction $\hat{k}$ or spatial frequency $\bar{k}_{\mathrmρ}$ serves as the input variable. Using this model, we create the material-matrix tetrahedral compass to conduct a detailed analysis of how the four fundamental characteristics of materials (linear/circular birefringence/dichroism) influence the eigensystems of the vector electric field in two-dimensional spatial frequency $\bar{k}_{\mathrmρ}$ domain and its distribution in three-dimensional $\bar{r}$ space with a crystal-2f configuration. Along this journey, we have uncovered new territories in LCO in both real and reciprocal space, such as infinite singularities arranged in disk-, ring-, and crescent-like shapes, ``L shorelines'' resembling hearts, generalized haunting theorem, double conical refraction, and optical knots it induces. We also present our model's early applications in focal engineering and NCO. As the opening chapter in a trilogy, this work connects crystal optics, Fourier optics, and nonlinear optics, while integrating theoretical, computational, and experimental physics, advancing all six domains.

Figures (11)

• Figure 1: Our full-vector nonlinear Fourier crystal optics(NFCO) simulations (e) versus Grant et al.'s experiment Fig. 3 (f) for chiral second-harmonic conical refraction(SHCR)grantFrequencydoubledConicallyrefractedGaussian2014.a All 6 NCO phase-matching types($\bar{E}_{\;\!\omega}^{\text{o,e}} \cdot \bar{E}_{\;\!\omega}^{\text{o,e}} \to \bar{E}_{\;\!2\omega}^{\text{o,e}}$ in d) are, for chiralzolotovskayaSecondharmonicConicalRefraction2011 SHCR, phase-mismatched and almost non-degenerate. b All $7 = 2 + 2 + 3$ types of nonlinear wave sources from (c) requiring computation. c All 5 non-zero components of $\mathcal{C}$-frame tensor $\bar{\bar{\underline{d}}}^{\;\!2\omega}_{\left[3 \times 6\right]}$kroupaSecondharmonicConicalRefraction2010 and 3 components of normalized $\mathcal{C}$-frame unit eigenvector(s) (fields) $\hat{\underline{g}}^{\;\!\omega}_{\pm} \left( \bar{k}_{\uprho} \right) = \hat{\underline{g}}^{\;\!\omega;\text{o,e}}_{\left[3 \times 1\right]} \left( \bar{k}_{\uprho} \right)$ of the fundamental wave(FW)(s) are involved in SHCR. d Intentionally picked, distorted, defocused $\mathcal{Z}$-frame second harmonic(SH) fields' intensity patterns $|\bar{\mathsfit{E}}^{\;\!2\omega}|^2 (\bar{\rho}), |\bar{\mathsfit{G}}^{\;\!2\omega}|^2 (\bar{k}_{\uprho})$ in 2D real $\bar{\rho}$ space and reciprocal $\bar{k}_{\uprho}$ space, with six phase-matching components $|\bar{E}^{\;\!2\omega}_{\pm\cdot\pm\to\pm}|^2 (\bar{\rho}), |\bar{G}^{\;\!2\omega}_{\pm\cdot\pm\to\pm}|^2 (\bar{k}_{\uprho})$, to show low field symmetry, tracing back to the material. e,f A clean linearly polarized(LP) Gaussian goes in, a kaleidoscopic second harmonic wave(SHW) exits after analyzer --- the most elaborate second harmonic generation(SHG) so far: material-wise --- 1 cm($>$$10^4$$\lambda$) long crystal, all nonzero tensor elements($\underline{d}_{31}=\underline{d}_{15},\underline{d}_{32}=\underline{d}_{24},\underline{d}_{33}$); field-wise --- all frequencies($\omega,2\omega$), eigen-polarizations(o,e), and vector components(x,y,z), all undergoing conical diffraction(-accompanied birefringence), walk-off, and chirality-driven polarization rotation.
• Figure 2: Towards stage III of the trilogy: full-vector nonlinear Fourier crystal optics(NFCO).a Reconstructed experiment results from Grant et al.grantFrequencydoubledConicallyrefractedGaussian2014 in their Fig. 4 on chiral second-harmonic conical refraction(SHCR) by utilizing the third stage (in c) of this LCO model: vector NFCO. b The real- and reciprocal-space distributions $|\bar{\mathsfit{E}}^{\;\!2\omega}|^2 (\bar{\rho}), |\bar{\mathsfit{G}}^{\;\!2\omega}|^2 (\bar{k}_{\uprho})$ of the 532 nm second harmonic wave(SHW) at the focal plane, generated by pumping a vertically polarized 1064 nm Gauss fundamental wave(FW) along KTP's optic axis at 532 nm, with its decomposition into six phase-matching types o,e$+$o,e$\to$o,e.
• Figure 3: Stage II of our trilogy: a scalar nonlinear Fourier crystal optics(NFCO) model, namely, the Nonlinear Angular Spectrum Theory(NLAST) reproduces Chen et al.'s all experimental figureschenPhaseMatchingControlledOrbital2020a.
• Figure 4: Full-vector nonlinear Fourier crystal optics(NFCO) Case 2: linear and nonlinear spin-orbit interaction(SOI) with femtosecond fundamental-wave pumping along the optic axis of BBO.a,e Main experimental results from Tang et al.tangHarmonicSpinOrbit2020c,f The corresponding simulation using our vector NFCO model. a,c Emitted second harmonic wave(SHW) from BBO, and its spin-orbit (spectral) decomposition, under the simulation/laboratory setup (b). d,f Emitted fundamental wave(FW), and its spin-orbit decomposition, under the setup (e).
• Figure 5: Reinterprete type-I o$+$o$\to$e full conical phase matching(FCPM)belyiPropagationHighorderCircularly2011 along the optic axis of BBO crystal for second harmonic generation(SHG) within the framework of semi-vector nonlinear Fourier crystal optics(NFCO). On the first 'Light' row, by squaring the right-handed circularly polarized(RHCP) Bessel fundamental wave (a) and expanding its field of view with a twofold interpolation in the $\bar{k}_{\uprho}$ domain, the intracrystal nonlinear driven source $\bar{P}^{(2)}_{2\omega}$ (b) is obtained. This traveling field, entirely determined by the pump, is multiplied by the "eigenvalue efficiency mask" sinc$(\Delta k_\mathrm{z} z)$ = longitudinal phase matching coherence level (c) derived from the path (d,e$\to$f$\to$c), followed by the "eigenvector efficiency mask" $\chi^{(2)\text{ooe}}_{2\omega;\text{eff}}$ = effective nonlinear coefficient distributionmidwinterEffectsPhaseMatching1965yaoAccurateCalculationOptimum1992dmitrievEffectiveNonlinearityCoefficients1993diesperovEffectiveNonlinearCoefficient1997 (i) from the path (g,h$\to$i). The resulting output in (j) is a hexagonal conical radial vector light field purely composed of extraordinary light of BBO at $2\omega$.