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Slow light nanocoatings for ultrashort pulse compression

An Author Correction to this article was published on 30 November 2021

Abstract

Transparent materials do not absorb light but have profound influence on the phase evolution of transmitted radiation. One consequence is chromatic dispersion, i.e., light of different frequencies travels at different velocities, causing ultrashort laser pulses to elongate in time while propagating. Here we experimentally demonstrate ultrathin nanostructured coatings that resolve this challenge: we tailor the dispersion of silicon nanopillar arrays such that they temporally reshape pulses upon transmission using slow light effects and act as ultrashort laser pulse compressors. The coatings induce anomalous group delay dispersion in the visible to near-infrared spectral region around 800 nm wavelength over an 80 nm bandwidth. We characterize the arrays’ performance in the spectral domain via white light interferometry and directly demonstrate the temporal compression of femtosecond laser pulses. Applying these coatings to conventional optics renders them ultrashort pulse compatible and suitable for a wide range of applications.

Introduction

Femtosecond light pulses are the basis for the highest achievable time resolutions and electrical field intensities today and have become central tools in microscopy1, medicine2, technology3, and physical chemistry4. A key challenge in their application remains dispersion control: because all transparent materials are normally dispersive in the ultraviolet, visible, and near-infrared regions below a wavelength of 1.3 μm, the realization of compressed laser pulses currently requires complex angular-dispersive5,6,7, reflective8,9,10, or photonic-crystal-fiber-based compression setups11, which all add significant complexity, path length, and beam deviations to the optical setup. Recently, dielectric metasurfaces, in addition to their use as lenses and for spatially shaping the phase and polarization of light on the nanoscale12,13,14,15, were employed in an angular-dispersive Fourier-transform setup16, providing fine-grained control of the time-domain properties of ultrashort laser pulses.

In this work, we demonstrate ultrathin nanocoatings that induce anomalous group delay dispersion directly upon transmission. The approach is illustrated in Fig. 1a: the nanocoatings can straightforwardly be applied to conventional optics and compensate for their group delay dispersion or be implemented in existing laser setups to compress ultrashort laser pulses. As such, the coatings can simplify the use and expand the applicability of femtosecond laser pulses. Therefore, they can act as the basis for an array of anti- or non-dispersive optics. Compared to theoretically proposed approaches based on plasmonics17 or flat nanodisk Huyghens metasurfaces18, our high-aspect-ratio nanopillar coating combines high transmission, anomalous dispersion, low high-order dispersion, and broadband operation.

The influence of transmissive optics on the time-domain profile of ultrashort laser pulses can be quantified by the frequency-dependent group delay $${{{{{\rm{GD}}}}}}=\frac{d\varphi }{d\omega }$$, which is calculated as the derivative of the angular-frequency-dependent spectral phase $$\varphi (\omega )$$ imprinted by the optics. Visible and near-infrared ultrashort laser pulses transmitted through transparent optics elongate because the pulses’ high-frequency (blue) components are delayed more than their low-frequency (red) components, i.e., $${{{{{\rm{GD}}}}}}({\omega }_{{{{{{\rm{red}}}}}}})\, < \, {{{{{\rm{GD}}}}}}({\omega }_{{{{{{\rm{blue}}}}}}})$$. Far from resonances, most materials’ group delay profiles are approximately linear, thus, they are approximated by their positive slope, the group delay dispersion$$\,{{{{{\rm{GDD}}}}}}=\frac{d{{{{{\rm{GD}}}}}}}{d\omega }=\frac{{d}^{2}\varphi }{d{\omega }^{2}}\, > \, 0$$. To compensate for the temporal broadening of ultrashort pulses upon transmission through optical elements, our goal is to create a coating with the opposite effect, i.e., $${{{{{\rm{GD}}}}}}({\omega }_{{{{{{\rm{red}}}}}}}) \, > \, {{{{{\rm{GD}}}}}}({\omega }_{{{{{{\rm{blue}}}}}}})$$ or $${{{{{\rm{GDD}}}}}}\, < \, 0$$.

Results and discussion

Working principle

Our approach for creating a transmissive compressor coating uses uniform circular amorphous silicon nanopillars arranged in a periodic square array (see Fig. 1b). Its working principle can be approached from a scattering perspective18,19,20 or from the perspective of an array of waveguides21,22,23,24. Here we follow the second approach.

We examine the dispersion of the eigenmodes of a two-dimensional compressor cross-section (i.e., the out-of-plane dispersion of the two-dimensional photonic crystal) in Fig. 1c. At near-infrared wavelengths, incoming light couples predominantly to two modes (see Fig. 1d) - with electric field profiles similar to the HE11 and HE12 hybrid modes in dielectric waveguides—due to their matching field symmetries (see Fig. 1e and ref. 21). Light in the HE11-like mode is mainly confined in the silicon nanopillars whereas it also travels in free space in the HE12-like mode (see Fig. 1e). Because the compressor is large compared to its periodicity, light leaking out of a single nanopillar is not lost. Thus, the propagation constant of the HE12-like mode is real above the vacuum light line (see Fig. 1c).

The anomalous GDD of our device is caused by the dispersion of the HE12-like mode close to $${k}_{z}=0$$ where $${k}_{z}$$ is the wavevector component in propagation direction: because light must travel at the same speed in both the backward and forward direction, reciprocity requires that the group velocity $${v}_{g}=\frac{d\omega }{d{k}_{z}}\to {0}^{+}$$ of light propagating in the HE12-like mode vanishes at $${k}_{z}\to {0}^{+}$$. In the absence of losses, this is true for all modes except for the fundamental mode or the special case of degenerate modes25. Consequently, close to the cutoff at $${k}_{z}=0$$, the mode enters a region of slow light26, which is visible in Fig. 1c as the vanishing slope of the HE12-like mode dispersion. The bending of the dispersion from the vacuum light line at high frequencies into this slow-light region (see arrows in Fig. 1c) creates anomalous group-velocity dispersion $${{{{{\rm{GVD}}}}}}=\frac{{d}^{2}{k}_{z}}{d{\omega }^{2}}=\frac{d{{v}_{g}}^{-1}}{d\omega }$$ at frequencies above $${k}_{z}=0$$. If light would couple exclusively to the HE12-like mode, the compressor group delay would diverge close to the $${k}_{z}=0$$ cutoff, preventing broadband operation. However, at the cutoff, light coupling rapidly shifts to the HE11-like mode (see Fig. 1d). The mixing of both modes generates a broadband region of constant anomalous GDD.

The lower limit $${{{{{{\rm{GDD}}}}}}}_{{{{{{\rm{min }}}}}}}$$ for the achievable constant anomalous GDD in a transmissive slow-light coating can be estimated similar to the maximum achievable delay in slow-light waveguides or scatterers26,27. For a given working range $$\left.\omega \in \right]{\omega }_{0}-\frac{\varDelta \omega }{2},{\omega }_{0}+\left.\frac{\varDelta \omega }{2}\right]\left. = \right]\left. {\omega }^{-},{\omega }^{+}\right]$$ defined by the cutoff $${k}_{z}({\omega }^{-})=0\,$$and bandwidth $$\varDelta \omega$$ (or the working range’s central frequency $${\omega }_{0}$$ and its high-frequency limit $${\omega }^{+}$$), we find (see methods) that the achievable constant anomalous GDD is set by the thickness of the coating $$L$$ and the effective refractive index of the HE12 mode on the high-frequency side of the working range $${n}^{+}=n({\omega }^{+})={\frac{{k}_{z}*\, c}{\omega }|}_{\omega ={\omega }^{+}}$$ (vacuum speed of light c):

$${{{{{{\rm{GDD}}}}}}}_{{{{{{\rm{min }}}}}}}=L * {{{{{{\rm{GVD}}}}}}}_{{{{{{\rm{min }}}}}}}=\frac{L}{c\varDelta \omega }\left(2-{n}^{+}-2\frac{{n}^{+}* {\omega }_{0}}{\varDelta \omega }\right)$$
(1)

As example, using $${n}^{+}\approx 1$$ as suggested by Fig. 1c and choosing a working range of 80 nm around a central wavelength of 800 nm predicts GVDmin = −264 fs2 μm−1 as a theoretical limit. In practice, the coupling efficiency to the anomalous dispersive mode drops close to cutoff and a perfectly parabolic dispersion relation cannot be achieved, which constrains real devices to approximately half of this limit.

Compressor coating design and modeling results

For the final design, we identify promising design parameters (nanopillar diameter, height, and periodicity) by exploring a large parameter space using rigorous coupled-wave analysis (S4)28. We then switch to finite-difference time-domain simulations (FDTD, Lumerical FDTD Solutions) and use the parameters (nanopillar radius, height, and unit cell size) as a starting point for fine-tuning the design for large negative GDD, low higher-order dispersion, and high transmission over an extended spectral band. The resulting compressor geometry (Fig. 1b) shows strong anomalous GDD centered at 800 nm wavelength; its phase, group delay, and transmission characteristics are shown in Fig. 2a–c. From 760 to 840 nm, the group delay deviates <2 fs from a group delay profile with constant anomalous GDD of −71 fs2. If a highly linear group delay profile is required, a stronger-dispersive regime exists between 780 and 825 nm: it provides anomalous GDD = −82 fs2 with <0.5 fs deviation from a linear group delay profile. FDTD modeling predicts close to unity transmission over the full working range. The obtained GDD values are comparable with many commercially available chirped mirrors. By using coating thicknesses of up to 10 μm, highly dispersive chirped mirrors29 provide larger absolute anomalous GDD than the presented device over a similar bandwidth. However, the presented compressor coating excels when comparing the induced GDD per coating thickness.

The response of the compressor’s dispersion and transmission properties to changes of the nanopillar diameter, height, and periodicity is explored in Supplementary Figs. 1 and 2. By changing the nanopillar diameter, the broadband Mie-type transverse magnetic dipole resonance19 of the nanopillars can be spectrally shifted to achieve anomalous dispersion at a desired central wavelength. The periodicity (475 nm × 475 nm) of the nanopillar array is chosen smaller than the smallest operation wavelength in air and the substrate material ($${\lambda }_{{{{{{{\rm{SiO}}}}}}}_{2}}^{{{{{{\rm{min }}}}}}}=\frac{{\lambda }_{{{{{{\rm{Air}}}}}}}^{{{{{{\rm{min }}}}}}}}{{n}_{{{{{{{\rm{SiO}}}}}}}_{2}}}=\frac{760\,{{{{{\rm{nm}}}}}}}{1.45}\approx 524\,{{{{{\rm{nm}}}}}}$$). Consequently, all diffraction orders except the zeroth order are evanescent outside of the compressor and the incoming beam profile is not modified by the compressor coating30 (see Supplementary Fig. 3 for the simulated far-field profile). By changing the nanopillar height, the magnitude of the induced anomalous dispersion can be controlled. In practice, only specific nanopillar height ranges can be realized because the coating—similar to a thin-film coating—can be reflective for mismatched height. The design introduced here features an undesirable narrowband Fabry-Perot resonance within the spectral working range (see Fig. 2c). For a given nanopillar diameter and height, the periodicity of the array can be tuned to suppress this resonance (see Supplementary Figs. 1 and 2) and achieve a smooth phase and transmission profile. The interaction between the narrowband Fabry-Perot and the broadband Mie-resonance is described in detail elsewhere19,20,31,32. The small residual signature of this resonance is inconsequential for the temporal envelope of broadband ultrashort laser pulses owing to its narrow spectral width. The transmission and phase characteristics of the compressor do not change significantly for slanted incidence angles of up to 5° for s-polarized light (see Supplementary Fig. 4).

Experimental characterization

To experimentally verify our design, we fabricated compressors with varying nanopillar diameters by lithographic top–down processing of a 610 nm-thick amorphous silicon layer on a 0.5 mm-thick fused silica substrate (see methods), Fig. 3a shows a sample imaged using scanning electron microscopy.

Figure 3 and Table 1 present the experimental group delay characteristics of compressors for three different nanopillar diameters, measured using white-light interferometry33 (see methods). The coatings induce anomalous GDD of up to −71 fs2 over a working bandwidth of up to 80 nm around a center wavelength tunable by changing the nanopillar diameter. This suffices for compensating the GDD of 2 mm-thick fused silica glass (GDD(SiO2, 800 nm) = +36.2 fs2 mm−1)34. The FDTD predictions match the experimental group delay and transmission profiles well.

We observe a residual signature of the suppressed Fabry-Perot resonance (see above). This significantly limits the linear working range on the high-frequency side (see, e.g., Fig. 3d, g). However, at the same time, it increases the anomalous GDD in the linear working range to up to −128 fs2, (see black lines in Fig. 3d–f). The GDD magnitude of our compressor makes it ~5800 times more dispersive than glass per unit length. A broadband dip of the transmission profiles on the low-frequency side of the working range decreases the average transmission in the working range to ~80%. This behavior is reproduced when including a 4 nm fabrication tolerance of the nanopillar diameter in the simulations. Thus, close to unity transmission should be achievable by using improved fabrication.

At wavelengths above the working range, the group delay of the realized compressor coatings (see the red area in Fig. 3e) decreases. In this wavelength range, the compressors induce positive GDD of up to 83 fs2. Consequently, also compact pulse stretchers can be implemented as frequency-shifted compressor coatings.

Demonstration of ultrashort laser pulse compression

To demonstrate the viability of our concept in a real application, we inserted a compressor (nanopillar diameter 162 ± 6 nm) in the path of a mode-locked titanium-sapphire oscillator. To confirm the resulting changes in the femtosecond laser pulses directly in the time domain, we employ second-harmonic frequency-resolved-optical-gating35 (SH-FROG, see Fig. 4f for setup and methods for details): laser pulses are split into two replicas and delayed with respect to each other. They are then noncollinearly overlapped in a nonlinear crystal. Only when the pulses traverse the crystal simultaneously, second-harmonic radiation is emitted towards the detector. Thus, recording the delay-dependent second-harmonic spectrum yields a spectrogram from which the intensity and phase profiles of the laser pulses can be reconstructed. Because the second-harmonic process mixes two photons from identical pulse copies, the generated spectrum$$\,S(\tau )=S(-\tau )$$ is equal for positive and negative delay time $$\tau$$ between the two pulse copies. Thus, SH-FROG spectrograms are symmetric with respect to the zero-delay time (see, e.g., Fig. 4a, b). Due to this symmetry, the sign of the GDD measured using SH-FROG can be ambiguous36. By ensuring that the incoming pulse GDD magnitude exceeds that of the compressor, sign changes of the GDD can be avoided, thus eliminating this uncertainty.

We measured spectrograms for: (i) the input laser pulses, (ii) the laser pulses affected by the fused silica substrate, and (iii) the laser pulses after traversing the compressor and substrate. Experimental spectrograms are displayed in Fig. 4a, b, whereas the full data set, retrieved spectrograms, and retrieval process37,38 are detailed in the methods and Supplementary Fig. 5. Both, visual comparison of the reconstructed spectrograms (see Supplementary Fig. 5) and the trace-area normalized FROG error G’ < 0.0539,40 of the reconstructed spectrograms indicate good agreement.

In the time-domain (Fig. 4c, d), the compressor reduces the pulses’ full-width-at-half-maximum duration by a quarter from (48.3 ± 1.2) fs to (37.5 ± 1.3) fs, clearly demonstrating compression although the laser bandwidth exceeds the linear compressor working range. We can relate these time-domain measurements with the group delay profiles obtained from white-light interferometry up to a constant offset (Fig. 4e). Agreement within the uncertainty proves both methods are viable for the determination of the group delay properties of thin compressor coatings. Spectral-domain results are summarized in Tables 2 and 3, Fig. 4e, and Supplementary Fig. 5. When numerically applying the measured characteristics to a light pulse matched to the full working range (Gaussian-shape, FWHM spectral bandwidth 760 nm–840 nm) stretched from 10 fs to 19 fs by 1.6 mm-thick fused silica glass, the compressor suppresses >65% of the pulse elongation and recompresses the pulse to below 13 fs. Thus, it can sufficiently compensate for the dispersion of many beam splitters, polarizers, waveplates, et cetera.

We demonstrated transmissive broadband pulse-compressor nanocoatings for the important visible to near-infrared spectral region, which compensate the GDD of up to 2 mm-thick fused silica glass over a bandwidth of up to 80 nm, and their application to femtosecond light pulses. Our approach can be implemented on conventional optics and requires no spatial, angular, or polarization pre-conditioning of the incoming light, therefore it can be rapidly implemented in optical setups. As group delay characteristics are determined by geometric properties, rather than material dispersion, the approach is flexible and can be adapted to different spectral regions or applications. In the future, stacked or more intricate structures tailored by inverse design or machine learning can expand the technique towards engineering complex pulse shapes for coherently controlling chemical reactions and quantum systems or optimizing nonlinear processes such as high-harmonic generation.

Methods

Group-velocity dispersion limit

To determine a lower limit for the constant anomalous GVD achievable for a given working bandwidth in a slow-light compressor coating, we start by parametrizing the dispersion relation in the working region as a parabola $${k}_{z}(\omega )=\frac{1}{2}{{{{{\rm{GVD}}}}}}{(\omega -d)}^{2}+e$$ with parameters $${{{{{\rm{GVD}}}}}},\,d$$ and $$e$$. This enforces a constant $${{{{{\rm{GVD}}}}}}=\frac{{d}^{2}{k}_{z}}{d{\omega }^{2}}$$ by definition. Using that the maximum group velocity in the compressor coating should be realized at the high-frequency side of the working range, but cannot exceed the speed of light $${\frac{1}{{v}_{g}}|}_{\!\omega ={\omega }^{+}}={\frac{d{k}_{z}}{d\omega }|}_{\omega ={\omega }^{+}}=\frac{1}{c}$$, fixes d and yields

$${k}_{z}(\omega )=\frac{1}{2}{{{{{\rm{GVD}}}}}}{\left(\omega +\frac{1}{{{{{{\rm{GVD}}}}}}* c}-{\omega }^{+}\right)}^{2}+e$$
(2)

Furthermore, fixing the wavevector at the limits of the working range, $${k}_{z}({\omega }^{+})=\varDelta k$$ and $${k}_{z}({\omega }^{-})=0$$ and eliminating e from Eq. (2) connects the lower limit for the constant anomalous group-velocity dispersion with the working bandwidth and the wavevector change $$\varDelta k$$ over that bandwidth:

$${{{{{{\rm{GVD}}}}}}}_{{{{{{\rm{min }}}}}}}=\frac{2}{\varDelta \omega }\left(\frac{1}{c}-\frac{\varDelta k}{\varDelta \omega }\right)$$
(3)

Using the definition of the (effective) refractive index $$n=\frac{{k}_{z}*\, c}{\omega }$$, and $${n}^{+}=n({\omega }^{+})={\frac{{k}_{z}*\, c}{\omega }|}_{\omega ={\omega }^{+}}\,$$yields $$\varDelta k=\frac{{n}^{+}*\, {\omega }^{+}}{c}$$ and combining with Eq. (3) yields Eq. (1).

Fabrication

First, we deposit a 610 nm-thick amorphous silicon layer on a 500 µm-thick fused silica substrate using plasma-enhanced chemical vapor deposition. After spin-coating a layer of negative electron beam resist (Micro Resist Technology, ma-N 2403) and an additional layer of conductive polymer (Showa Denko, ESPACER 300) to avoid charging effects during electron beam lithography (EBL), we define the nanopillar mask patterns using EBL (Elionix, ELS-F125) and develop (MicroChemicals, MIF 726). Anisotropic inductively coupled plasma-reactive ion etching (ICP-RIE using a mixture of SF6 and C4F8) was used to etch the nanopillar structures. The electron beam resist mask was removed by immersing the sample in piranha solution.

The demonstrated compressor coatings cover an area of 3.1 mm2. Both, the employed material system and feature sizes are manufacturable via commercially available deep-ultraviolet (immersion) lithography tools. Thus, high quantities and large-scale devices can straightforwardly be manufactured using prevalent industrial semiconductor manufacturing methods.

White-light interferometry

We determine the group delay profiles of our compressors using a self-built white-light interferometer. Broadband black body radiation from a tungsten lamp (Thorlabs SLS202L) is coupled to multimode fiber, then collimated and linearly polarized before a Michelson interferometer. In the interferometer, light in one arm is used as a reference and is delayed, light in the other arm is modified by the sample. After the interferometer, the spectral interference between light from both arms is resolved using an Andor Shamrock spectrometer and reveals the group delay profile of the sample.

Femtosecond oscillator, FROG measurements, and retrieval

Measurements were performed on the uncompressed output of a Femtosource Rainbow (FEMTOLASERS Produktions GmbH) femtosecond oscillator using a self-built noncollinear SH-FROG. The uncompressed pulses are elongated by the GDD and third-order dispersion of the oscillator, a fused silica lens, beam splitter, and air path (spectral phase of the incoming pulses see Supplementary Fig. 5j). In the time domain (Supplementary Fig. 5g), the GDD causes a symmetric broadening, whereas the third-order dispersion causes the substructure. The group delay profile shown in Fig. 4e is determined by subtracting the group delay profiles in Supplementary Fig. 5k, l. Because these were both measured using the same incoming laser pulses, the subtraction eliminates the group delay profiles of the incoming laser pulses. We employ a self-written retrieval algorithm based on the iterative ptychographic engine to retrieve the pulse characteristics (Supplementary Fig. 5g–l) from the experimental spectrograms (Supplementary Fig. 5a–c). We correct for the phase-matching conditions in our 100 μm-thick beta-barium-borate crystal before the retrieval and ignore non-phase-matched spectral components during the retrieval (see Supplementary Fig. 5a–c). To increase the accuracy of the retrieval, we use the measured power spectrum of the fundamental laser pulses as an additional constraint. This approach retrieves correct temporal and spectral pulse properties, even for incomplete spectrograms. In our case, it provides reliable spectral information even in the non-phase-matched spectral ranges of the ultrashort pulses, witnessed by the excellent agreement between the compressor’s group delay profile measured using the SH-FROG and white-light interferometry (Fig. 4e). Spectrograms calculated from the retrieved pulse parameters are displayed in Supplementary Fig. 5d–f for comparison. Errors were determined using the bootstrap method.

Data availability

The data supporting the findings of this study are available in figshare with the identifier https://doi.org/10.6084/m9.figshare.16589867 or from the corresponding authors upon request.

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Acknowledgements

This work was performed, in part, at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the NSF under award no. ECCS-2025158. CNS is a part of Harvard University. M.O. acknowledges a Feodor-Lynen Fellowship from the Alexander von Humboldt Foundation. Y.-W.H. acknowledges support from the Ministry of Science and Technology, Taiwan (grant no. 110-2124-M-A49-004), support from the Ministry of Education (MOE) in Taiwan under the Yushan Young Scholar Program, and the Higher Education Sprout Project of the National Yang Ming Chiao Tung University. Z.W. acknowledges funding by the China Scholarship Council (201906180074). Additionally, financial support from the Office of Naval Research (ONR), under the MURI program, grant no. N00014-20-1-2450, and from the Air Force Office of Scientific Research (AFOSR), under grant no. FA95550-19-1-0135, is acknowledged.

Author information

Authors

Contributions

M.O., W.T. C., Y.A.I. and X.Y. carried out the numerical simulations and designed the final compressor coatings. Y.-W.H. fabricated the compressor coatings. M.O. and Z.W. carried out the experimental characterization and ultrashort pulse compression. M.O. and X.Y. analyzed the experimental and simulated data. M.O., M.S. and F.C. wrote the manuscript. F.C. supervised the study. All authors discussed the manuscript.

Corresponding authors

Correspondence to M. Ossiander or F. Capasso.

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Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks Pedro David García, Bumki Min, and Shuang Zhang for their contribution to the peer review of this work. Peer reviewer reports are available.

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Ossiander, M., Huang, YW., Chen, W.T. et al. Slow light nanocoatings for ultrashort pulse compression. Nat Commun 12, 6518 (2021). https://doi.org/10.1038/s41467-021-26920-6

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