Gratings 1 and holograms 2 use patterned surfaces to tailor optical signals by diffraction. Despite their long history, variants with remarkable functionalities continue to be developed 3 , 4 . Further advances could exploit Fourier optics 5 , which specifies the surface pattern that generates a desired diffracted output through its Fourier transform. To shape the optical wavefront, the ideal surface profile should contain a precise sum of sinusoidal waves, each with a well defined amplitude, spatial frequency and phase. However, because fabrication techniques typically yield profiles with at most a few depth levels, complex ‘wavy’ surfaces cannot be obtained, limiting the straightforward mathematical design and implementation of sophisticated diffractive optics. Here we present a simple yet powerful approach to eliminate this design–fabrication mismatch by demonstrating optical surfaces that contain an arbitrary number of specified sinusoids. We combine thermal scanning-probe lithography 6 – 8 and templating 9 to create periodic and aperiodic surface patterns with continuous depth control and sub-wavelength spatial resolution. Multicomponent linear gratings allow precise manipulation of electromagnetic signals through Fourier-spectrum engineering 10 . Consequently, we overcome a previous limitation in photonics by creating an ultrathin grating that simultaneously couples red, green and blue light at the same angle of incidence. More broadly, we analytically design and accurately replicate intricate two-dimensional moiré patterns 11 , 12 , quasicrystals 13 , 14 and holograms 15 , 16 , demonstrating a variety of previously unattainable diffractive surfaces. This approach may find application in optical devices (biosensors 17 , lasers 18 , 19 , metasurfaces 4 and modulators 20 ) and emerging areas in photonics (topological structures 21 , transformation optics 22 and valleytronics 23 ). Combining thermal scanning-probe lithography with templating enables the production of high-quality gratings that manipulate light through Fourier-spectrum engineering in ways that are not achievable with conventional gratings.

中文翻译：

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光学傅里叶表面

光栅 1 和全息图 2 使用图案化表面通过衍射来定制光信号。尽管历史悠久，但具有卓越功能的变体仍在继续开发 3 、 4 。进一步的进步可以利用傅立叶光学 5 ，它指定了通过其傅立叶变换生成所需衍射输出的表面图案。为了塑造光波前，理想的表面轮廓应该包含正弦波的精确总和，每个正弦波都具有明确定义的幅度、空间频率和相位。然而，由于制造技术通常会产生最多几个深度级别的轮廓，因此无法获得复杂的“波浪”表面，从而限制了复杂衍射光学器件的直接数学设计和实现。在这里，我们展示了一种简单而强大的方法，通过演示包含任意数量的指定正弦波的光学表面来消除这种设计-制造不匹配。我们结合热扫描探针光刻 6 – 8 和模板 9 来创建具有连续深度控制和亚波长空间分辨率的周期性和非周期性表面图案。多分量线性光栅允许通过傅立叶频谱工程 10 精确操纵电磁信号。因此，我们通过创建一个超薄光栅，以相同的入射角同时耦合红光、绿光和蓝光，从而克服了以前光子学的局限性。更广泛地说，我们分析设计并准确复制了复杂的二维莫尔图案 11、12、准晶体 13、14 和全息图 15、16，展示了各种以前无法实现的衍射表面。这种方法可以应用于光学器件（生物传感器 17、激光器 18、19、超表面 4 和调制器 20）和光子学的新兴领域（拓扑结构 21、变换光学 22 和谷电子学 23）。将热扫描探​​针光刻与模板相结合，可以生产高质量的光栅，以传统光栅无法实现的方式通过傅立叶光谱工程操纵光。