圆柱坐标系下用于大面积级联超表面光学的可微快速远场变换
Differentiable Fast Far-Field Transform in Cylindrical Coordinates for Large-Area Cascaded Metalens Optics
AI总结：
该研究提出圆柱坐标系下可微快速远场变换，用于大面积轴对称超透镜全区域PSF评估与优化。通过特定方法加速计算，避免高内存需求，实现斜向优化。应用于多色超透镜设计，揭示效率限制并提升效率，还展示毫米级斜入射优化及彗差校正。
AI中文摘要：
我们提出了一种圆柱坐标系下的全可微远场变换，用于大面积轴对称超透镜的全区域点扩散函数（PSF）评估和优化。该方法能在数秒内计算直径跨越数千到数万波长（可见光下为毫米级，红外光下为厘米级）孔径的波动光学响应，比格林函数积分快三到四个数量级，且避免了二维FFT的高内存需求。此方法将矢量近场分解为平行角动量通道，应用FFTLog加速的汉克尔变换，并使用格拉夫加法定理在斜照明下重新定位焦场。解析伴随梯度使优化相对于正向模拟仅增加约65%的开销。对于直径4毫米（约8000波长，约12,600方位角模式）、入射角30度的孔径，在350线程CPU上正向 - 伴随迭代仅需约12秒，使斜向优化无需光线追踪近似即可实现。应用于垂直入射的多色RGB（446/530/650纳米）超透镜设计时，全区域PSF评估揭示了传统裁剪焦斑分析隐藏的效率限制：看似衍射受限的单柱超透镜平均绝对聚焦效率仅约6%，而直接远场优化将其提高到37%（局部周期近似）和51%（分区离散轴对称）。级联双超表面设计达到63%，而四超表面架构平均相对效率达到96%。我们还展示了毫米级单表面和双合透镜架构的斜入射优化；级联双合透镜可实现单个旋转对称表面无法实现的部分彗差校正。
英文摘要：
We present a fully differentiable far-field transform in cylindrical coordinates for full-area point spread function (PSF) evaluation and optimization of large axisymmetric metalenses. The method computes wave-optical responses of apertures spanning thousands to tens of thousands of wavelengths in diameter (millimeter scales in the visible, centimeter scales in the infrared) in seconds, achieving three to four orders of magnitude speedup over Green's function integration while avoiding the prohibitive memory of two-dimensional FFTs. The approach decomposes vectorial near fields into parallel angular-momentum channels, applies FFTLog-accelerated Hankel transforms, and uses Graf's addition theorem to recenter focal fields under oblique illumination. Analytic adjoint gradients enable optimization with only ~65% overhead relative to a forward simulation. For a 4 mm-diameter aperture (~8000 wavelengths, ~12,600 azimuthal modes) at 30-degree incidence, a forward-adjoint iteration requires only ~12 s on a 350-thread CPU, making oblique optimization practical without ray-tracing approximations. Applied to polychromatic RGB (446/530/650 nm) metalens design at normal incidence, full-area PSF evaluation exposes efficiency limits hidden by conventional cropped-focal-spot analysis: a mono-pillar metalens that appears diffraction-limited achieves only ~6% average absolute focusing efficiency, while direct far-field optimization raises this to 37% (locally periodic approximation) and 51% (zoned discrete axisymmetry). A cascaded double-metasurface design reaches 63%, while a four-metasurface architecture attains 96% average relative efficiency. We also demonstrate millimeter-scale, oblique-incidence optimization of single-surface and doublet architectures; cascaded doublets enable partial coma correction inaccessible to a single rotationally symmetric surface.
