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Optical Fourier surfaces


Gratings1 and holograms2 use patterned surfaces to tailor optical signals by diffraction. Despite their long history, variants with remarkable functionalities continue to be developed3,4. Further advances could exploit Fourier optics5, 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 lithography6,7,8 and templating9 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 engineering10. 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 patterns11,12, quasicrystals13,14 and holograms15,16, demonstrating a variety of previously unattainable diffractive surfaces. This approach may find application in optical devices (biosensors17, lasers18,19, metasurfaces4 and modulators20) and emerging areas in photonics (topological structures21, transformation optics22 and valleytronics23).

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Fig. 1: Fourier surfaces modulated in one dimension.
Fig. 2: Fourier surfaces modulated in two dimensions.
Fig. 3: Periodic and quasiperiodic Fourier surfaces.
Fig. 4: Applications of Fourier surfaces.

Data availability

The data supporting the findings of this study are available from the corresponding author on reasonable request.


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We thank S. Bonanni, U. Drechsler, F. Enz, T. Kulmala, A. Olziersky and R. Stutz for technical assistance and D. Chelladurai, U. Dürig, R. Keitel, A. Knoll, M. Kohli, N. Rotenberg, D. Thureja, J. Winkler and H. Wolf for discussions. This project was funded by the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013)/ERC Grant Agreement Number 339905 (QuaDoPS Advanced Grant). F.T.R. (Rubicon-680-50-1509, Gravitation Program “Multiscale Catalytic Energy Conversion”, Veni-722.017.002), S.J.W.V. (OCENW.KLEIN.008) and B.l.F. (Rubicon-680-50-1513) acknowledge support from the Netherlands Organisation for Scientific Research.

Author information

Authors and Affiliations



N.L., B.l.F. and D.J.N. conceived the project. N.L., R.B. and S.J.W.V. designed the Fourier surfaces with input from K.R., F.T.R. and D.J.N. N.L. patterned the polymer surfaces with assistance from K.R., M.S. and S.B. N.L. and R.B. transferred the patterns to optical materials with assistance from K.R. and M.S. N.L. performed the characterization and topography analysis of the Fourier surface structures. N.L. and R.B. performed the optical experiments. N.L., R.B., S.J.W.V. and F.T.R. analysed the optical data. F.T.R. developed the analytical model. N.L. and D.J.N. wrote the manuscript with input from all authors. D.J.N. supervised the project.

Corresponding author

Correspondence to David J. Norris.

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Competing interests

The authors declare the following potential competing financial interests: S.B. is employed by Heidelberg Instruments Nano (previously SwissLitho AG), a provider of thermal scanning-probe lithography tools. At the time of his contribution, M.S. worked for SwissLitho AG. N.L., R.B., F.T.R. and D.J.N. have filed a patent application related to ideas in this work.

Additional information

Peer review information Nature thanks Wei-Ting Chen, Maryna Meretska and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Design and fabrication of Fourier surfaces.

a, Design of a Fourier surface. The analytical formula for the desired surface profile (here, a single sinusoid modulated in 1D) is converted into a grayscale bitmap. Each 10 nm × 10 nm pixel has a depth level between 0 and 255 (8-bit). The bitmap contains the sinusoidal function in the horizontal direction within the white border, which is constant along the vertical direction. The pixels in the white border are set to the minimum depth level. b, Process flow showing the patterning steps for Ag Fourier surfaces: (i) The hot scanning tip is used to create a single sinusoid in the polymer resist, (ii) an optically thick (>500 nm) Ag layer is thermally evaporated onto the polymer, (iii) a glass microscope slide is affixed to the back of the Ag layer using ultraviolet-curable epoxy, and (iv) the glass/epoxy/Ag stack is stripped off the polymer film. Alternative fabrication pathways for transferring the Fourier surface pattern to other materials are presented in the Methods. c, SEM (30° tilt) of a single-component Fourier surface transferred to Ag via templating. The initial analytical design is replicated accurately in the final Ag surface.

Extended Data Fig. 2 Topography characterization.

a, AFM micrograph of the measured topography (colour scale) for a single-sinusoidal Ag grating. The RMS roughness of the unpatterned flat Ag film is 1.6 nm, extracted from the area indicated by the green dashed box. The RMS roughness of the patterned flat Ag film is 1.3 nm, extracted from the area indicated by the blue dashed box. The area indicated by the red dashed box is used for fitting and analysis of the surface profile. b, 2D fit of a sinusoidal function (yellow/brown surface) to topography data (blue dots) from the region indicated in the red dashed box in a. The amplitude of the fitted function is A1 = 25.5 nm (2% larger than design value) with a period of Λ = 610 nm (1.7% larger than design value). Such horizontal errors were consistent over many samples and attributed to a distance miscalibration in the thermal scanning probe. The RMS error between the design function and measured topography was found to be 1.8 nm after this horizontal error was taken into account. c, Measured topography (colour scale) of the structure in a, plotted only for the fit region (red dashed box in a), scaled from the minimum depth value to the maximum depth value and centred at zero. The inset shows a line cut (along gx at gy = 0, where gx and gy are the components of g along the x and y axes, respectively) from the 2D Fourier transform of the measured topography in the fit region, normalized to the peak value at g1. The second harmonic at gx/g1 = 2 is barely visible and has an amplitude of 3.5% of the peak at gx/g1 = 1, corresponding to a real-space amplitude of 0.9 nm. d, Residual error (colour scale) between the data and the fitted function, plotted for the fit region as in c. For comparison, the data are scaled over the same range as in c, centred at 0.

Extended Data Fig. 3 Optical measurement of plasmonic Fourier surfaces.

a, Schematic of the optical setup used for k-space reflectivity measurements. Further details are in the Methods. The inset shows a vector diagram of light with wavevector k0 incident at angle θ on a Fourier surface pattern with period Λ. BS, beamsplitter. CMOS, complementary metal-oxide-semiconductor digital camera. b, Schematic of the dispersion diagram (energy versus in-plane wavevector component, kx) for free-space photons incident on a sinusoidal grating with ky = 0 (as in Fig. 1). By tuning θ, photons have access to the shaded region inside the light lines (solid blue lines). The red lines show the SPP dispersion, kSPP. Dashed green curves indicate the SPP dispersion displaced by the grating spatial frequency g. Inside the light line, these curves represent where free-space photons can couple to SPPs, and vice versa (that is, where kx ± g = kSPP). A stopband of width ΔE opens when counter-propagating SPPs are coupled by g. The blue trapezoidal region depicts the experimentally accessible area on the dispersion diagram, limited by the spectral window of the spectrometer along E, and the angular window of reflected light collected by the microscope objective along kx. c, Schematic of the dispersion diagram for free-space photons incident on a surface, plotted for both in-plane wavevectors, kx and ky. The light line and SPP dispersion in b are both cones (blue and red lines, respectively). d, A slice through the dispersion diagram in c at fixed energy. Free-space photons incident on a surface can have wavevectors inside the light cone (blue-shaded region). The SPP dispersion is the larger red circle. Dashed green circles show solutions to k ± g = kSPP. In this example, \({\bf{g}}=g\hat{{\bf{x}}}\).

Extended Data Fig. 4 Further analysis of the Ag Fourier surfaces in Fig. 1.

a, c, e, Measured (AFM) and targeted surface topographies for the same sinusoidal structures as in Fig. 1a, d and g (blue) and their ‘binarized’ versions (red). The data for the Fourier surfaces represent half of the scans shown in Fig. 1b, e, h. The scan lengths for the binarized versions are 5.9 µm, 6.0 µm and 5.5 µm, respectively. For each Fourier surface, a binarized profile was obtained using a published thresholding procedure (see Methods). These binarized structures were then fabricated in a Si substrate using electron-beam lithography and etching. Ag replicas were obtained by templating (see Methods). The depth scale bars are 50 nm, 65 nm and 60 nm for both structures in a, c and e, respectively. b, d, f, Comparison of experimental angle-resolved reflectivity spectra measured for the sinusoidal surfaces shown in a, c and e (left) and their binarized versions (right). The data for the sinusoids are the same as the left sides of Fig. 1c, f, i. The optical responses of the binarized gratings are clearly corrupted by the unwanted spatial frequencies in the structure. gi, Normalized line cuts (orange curves) through the reflectivity data shown in the left panels in b, d and f at 500 nm, 600 nm and 600 nm, respectively. The black curves show the predicted reflectivity versus the absolute value of the in-plane wavevector, \(|{k}_{x}|\) from our model (see Methods). The comparison reveals good agreement between the model and the data without any adjustable parameters (other than the normalization). For all structural design parameters, see Extended Data Table 1.

Extended Data Fig. 5 Control of ‘dark’ band edges in two-component sinusoidal gratings.

a, Comparisons of the measured (light blue points) and targeted surface topographies (dark blue lines) in the polymer surface, measured during patterning. Scan lengths are 11.5 µm. The left grating has the height profile \(f(x)\,=\,{A}_{1}\cos (gx+{\rm{\pi }})+({A}_{1}/2)\cos (2gx+{\phi }_{2})-\varDelta \) with φ2 = π. The grating on the right has the same f(x) except φ2 = 0. b, The measured reflectivity in k-space (as in Fig. 1) for Ag gratings templated from the structures in a. In both the left and right gratings, a stopband opens near 1.9 eV, but the choice of phase can control whether an optically dark state exists at the lower (left) or upper (right) band edge. The band edge with the optically dark state is marked with white arrows. For all structural design parameters, see Extended Data Table 1.

Extended Data Fig. 6 Control of stopband width in two-component sinusoidal gratings.

a, Comparisons of the measured (light blue points) and targeted surface topographies (dark blue lines) in the polymer surfaces, measured during patterning, for structures exhibiting a single stopband. Scan lengths are 14.5 µm and the left vertical scale bar is 100 nm for all scans. From top to bottom: a series of two-component sinusoidal gratings, where A1 = 18.5 nm, Λ1 = 620 nm, A2 is varied and Λ2 = 230 nm. A2 has values of 0 nm, 2.5 nm, 5 nm, 10 nm, 15 nm, 18 nm, 20 nm and 25.1 nm. bi, Measured plasmonic dispersion diagrams for Ag gratings templated from the profiles in a, from top to bottom, respectively. The width of the stopband increases because A2 is the amplitude of the Fourier component responsible for creating the plasmonic stopband. For all structural design parameters, see Extended Data Table 1.

Extended Data Fig. 7 Fourier surface patterns.

a, Measured topography (obtained during patterning) of the polymer film (PMMA/MA; see Methods) used to template the structure in Fig. 2a. The two spatial-frequency vectors g1 and g2 that define the surface profile are overlaid on the pattern. Here, g1 and g2 have the same magnitude g1 = g2 = 2π/600 nm, and g2 is rotated −10° from g1, where g1 lies along \(\hat{{\bf{x}}}\). b, As in a, but the template corresponding to the structure in Fig. 2b. Again, g1 and g2 have the same magnitude g1 = g2 = 2π/600 nm, but g2 is rotated −40° from g1, where g1 lies along \(\hat{{\bf{x}}}\). c, SEM (45° tilt) of a circular sinusoidal Ag grating with Λ = 600 nm. d, SEM (45° tilt) of two superimposed circular sinusoidal gratings, as in c, each with Λ = 600 nm. The centre of one grating is translated +150 nm and the other −150 nm in \(\hat{{\bf{y}}}\) from the origin in the middle of the pattern. The spatial interference results in a moiré pattern with broken circular symmetry. For all structural design parameters, see Extended Data Table 1.

Extended Data Fig. 8 Efficient diffraction from photonic Fourier surfaces.

a, Comparison of the measured (AFM) and targeted surface topography (accounting for a slight distance miscalibration and depth nonlinearity in the thermal scanning probe) for a deeper sinusoidal Ag grating designed for efficient optical diffraction. The scan length is 18.5 µm. b, Measured diffraction efficiency as a function of photon wavelength for the grating profile shown in a, for p-polarized illumination at normal incidence. The inset illustrates how incident photons (black arrow with wavevector k and electric field E indicated) diffract symmetrically into the +1 (red arrow) and −1 (green arrow) diffraction orders. The red and green curves correspond to the +1 and −1 diffracted intensities, respectively, normalized to the intensity reflected from an unpatterned flat reference spot on the same Ag sample. The sum of the red and green curves (blue line) peaks at about 97%. Owing to fluctuations in the collected diffraction intensity, the measured efficiencies have an estimated error of ±5%. We also note that our measurement does not account for reflection losses in the Ag (a few per cent). c, As in a, but for a two-component sinusoidal Ag grating where the relative phase between the two components is chosen to break the mirror symmetry of the structure about the yz plane. d, As in b, but now the broken symmetry causes nearly all of the incident light to be diffracted into the +1 diffraction order (red curve) for a given wavelength range. eh, As in ad, but for s-polarized illumination at normal incidence. For all structural design parameters, see Extended Data Table 1.

Extended Data Fig. 9 SiNx and TiO2 Fourier surfaces.

a, Comparison of the measured (AFM) and targeted surface topography (accounting for a slight distance miscalibration in the thermal scanning probe) for a single sinusoid in SiNx, transferred via reactive-ion etching (see Methods). Scan length is 11.3 µm. b, SEM (30° tilt) of the same structure in a. The final profile in SiNx has a measured RMS error of 2.5 nm using the same methodology as in Extended Data Fig. 2. c, As in a, but for a three-component SiNx grating. Scan length is 14.8 µm. d, As in b, but for the structure in c. The final profile in SiNx has a measured RMS error of 3.9 nm using the same methodology as in Extended Data Fig. 2. e, SEM (30° tilt) of a 12-fold rotationally symmetric quasicrystal, as in Fig. 3d, transferred from the patterned polymer into Si via inductively coupled plasma etching (see Methods). f, SEM (30° tilt) of the pattern in e transferred into a TiO2 thin film via template stripping (see Methods). For all structural design parameters, see Extended Data Table 1.

Extended Data Table 1 Design parameters for Fourier surfaces

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Lassaline, N., Brechbühler, R., Vonk, S.J.W. et al. Optical Fourier surfaces. Nature 582, 506–510 (2020).

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