Imaging the field inside nanophotonic accelerators

The study addresses the challenge of directly characterizing and controlling the optical nearfield within nanophotonic dielectric laser accelerators (DLAs), which is critical for precise control of electron trajectories, modular acceleration, and advanced beam focusing (e.g., alternating phase focusing). Conventional radiofrequency accelerators are large and costly, while DLAs leverage high damage thresholds of dielectrics at optical frequencies and compact on-chip designs, enabling gradients up to and beyond the GeV/m scale. Despite advances, the exact 3D nearfield distribution in fabricated structures has remained experimentally inaccessible due to complex device geometries, sensitivity to fabrication tolerances, and limited knowledge of actual parameters for simulations. The research question is how to image and quantify the internal accelerating nearfield with deep-subwavelength spatial and sub-nm spectral resolution in realistic DLA devices, and how measured fields compare to designed modes, including the impact of fabrication deviations.

DLAs operate via the inverse Smith–Purcell effect, where electrons interact with periodic nanostructures to exchange energy with optical fields. Early demonstrations achieved tens to hundreds of MeV/m acceleration gradients for sub-relativistic and relativistic electrons. Continued progress in silicon photonics-based designs and coupling has reached GeV/m-scale gradients. Prior theoretical and experimental works proposed various geometries (dual-gratings, photonic crystals, Bragg accelerators, inverse-designed resonators) and advanced beam dynamics (alternating phase focusing). Conventional PINEM provides nm spatial and fs temporal resolution but typically uses pulsed light, which excites a broad spectral range and cannot isolate monochromatic responses critical for DLA mode characterization. Only a few PINEM-type studies achieved strong interactions with continuous-wave (CW) excitation, which is essential to resolve wavelength-dependent nearfield distributions in DLAs. The paper positions two leading silicon-based DLA designs for comparison: (1) a dual-pillar structure with a distributed Bragg reflector, and (2) an inverse-designed resonant structure with an enclosed channel, noting their predicted symmetric (cosh-like) accelerating modes and potential sensitivities to fabrication tolerances.

Overview: The authors develop a continuous-wave (CW) PINEM-based energy-filtered transmission electron microscopy (EFTEM) technique to image the accelerating nearfield inside DLA channels with deep-subwavelength spatial and sub-nm spectral resolution. Measurements are complemented by 3D electromagnetic simulations and a mode-fitting model to interpret the field distributions and infer fabrication deviations.

Experimental setup: A JEOL JEM-2100 Plus TEM with Gatan GIF is modified for optical coupling. Electrons from a LaB6 thermionic source are accelerated to 189 keV (βc ≈ 0.69c) and traverse a 20 µm-long silicon DLA channel aligned parallel to the electron trajectory using a double-tilt holder with ~0.01° precision (≈2 nm longitudinal spatial resolution). A 1064 nm CW DFB-seeded Yb fiber amplified laser (100 mW seed; attenuated to 65 mW at sample; ~15 µm 1/e2 Gaussian spot via cylindrical lens) illuminates perpendicularly to the electron flow with ~1° incidence (small effects for 15 µm interaction length). Wavelength tuning is achieved by TEC temperature sweep; usable range ~1063–1065.8 nm.

Two imaging modalities: (1) Point-scan spectroscopy with a ~70 nm electron spot (smaller than channel width) to record EELS spectra at selected positions, used to calibrate the energy filter and validate the PINEM coupling constant g via fits to energy spectra. (2) Wide-field imaging with a ~3 µm collimated electron spot to uniformly illuminate the channel; EFTEM images are formed by filtering only electrons that gained energy.

Energy filtering and mapping to field: The EELS prism and a 7 eV-wide slit select energy-gain electrons: −9 to −2 eV (inverse design) and −8.5 to −1.5 eV (dual-pillar). Operating in the weak-modulation CW regime (peak energy gain ~5 eV; g ≈ 1.2; effective interaction length ~15 µm set by laser spot) avoids saturation of the zero-loss peak, ensures a monotonic relationship between counts and field, and minimizes transverse deflection for higher spatial resolution. The electron counts in EFTEM correspond to the integral of the PINEM energy distribution over the slit and are mapped to the coupling constant g via a calculated transformation curve, enabling retrieval of relative field strength proportional to the accelerating field component (E_z Fourier mode with q = ω/v).

Structures and phase matching: The DLA periodicity is Λ = 733 nm, optimized for λ ≈ 1064 nm. Synchronicity condition λ = Λ/β (generalized λ = Λ/(β − Λ cosθ) for non-normal incidence) guides operation; given 15 µm interaction length, tolerances are Δλ up to ~25 nm and ΔE_e up to ~10 keV without losing phase matching. Two structures are tested: (1) dual-pillar with Bragg reflector; (2) inverse-designed resonant channel. Channel widths are ~210 nm (dual-pillar) and ~280 nm (inverse design). Structure height is 2.5 ± 0.1 µm.

Data acquisition and processing: EFTEM images acquired on Gatan US1000 (binning x4) with ~40 s integration. Post-processing steps: rotation to align channel vertically; removal of a spurious dark line from slit contamination via interpolation; cropping; colormap adjustments; optional conversion from counts to g using the PINEM formula p(u,x,y,g) = p0(u,x,y) * Σl 2 J1(2g(x,y)) δ(u − lħω) (as applied for calibration and transformations). For quantitative comparison, counts are normalized to unfiltered images (integration over all energies).

Mode fitting and imaging model: Within the channel, the accelerating field profile across x is modeled as a superposition of symmetric (cosh) and antisymmetric (sinh) modes with an evanescent decay rate χ, blurred by a Gaussian point spread function (σ ≈ 10 nm) and multiplied by an exponential decay at boundaries to phenomenologically account for electron inelastic scattering (e.g., phonons, plasmons) at silicon interfaces. Fits extract the relative mode coefficients A (cosh) and B (sinh) and χ.

3D simulations and sensitivity analysis: Full 3D electromagnetic simulations compute |E_z(x,y)| and other components (E_x, E_y), including realistic substrate effects. A sensitivity study varies a key geometric parameter—the structure diameter D—over δD ranges (dual-pillar: +14 to −40 nm; inverse design: +10 to −44 nm) to identify geometry deviations explaining measured mode profiles. Best-fit δD values are determined by matching simulated EFTEM images (after applying boundary exponential decay and counts-to-g normalization) to measurements. Results are compared side-by-side with experiments. 2D vs 3D simulations are contrasted to highlight the impact of vertical dimension on resonance Q and spectral response.

Fabrication: Nanostructures (2.5 µm height) are fabricated in phosphorus-doped silicon by 100 kV electron-beam lithography and cryogenic reactive-ion etching. A ~30 µm-high mesa clears space for optical/electron beams. A chip alignment aperture guides electrons into the channel.

Spectral imaging: CW laser wavelength is scanned 1063.0–1065.4 nm to map spectral dependence of nearfield distributions for both structures, capturing mode changes and resonance behavior.

Proposed 3D tomography: For future full 3D field mapping along z, the authors propose sequentially illuminating subsections via on-chip apertures or a scanned focused laser, enabling reconstruction of the 3D nearfield; deconvolution can enhance axial resolution.

• Demonstration of deep-subwavelength nearfield imaging inside DLA channels using continuous-wave PINEM with energy-filtered TEM, achieving sub-nm spectral resolution and ~10 nm spatial resolution.
• Measured energy spectra (EELS) at selected channel locations fit the PINEM model, confirming weak-modulation regime with peak energy gain ~5 eV (g ≈ 1.2) under CW drive. Effective interaction length ~15 µm yields an equivalent acceleration gradient ~0.2 MeV/m (CW), much lower than fs-driven operation but advantageous for linearity and resolution.
• Spatial field distributions reveal strong 3D effects: Fields do not reach the channel bottom and can extend above the structure top; the dual-pillar structure exhibits vertical oscillatory patterns, while the inverse-designed structure confines fields more within the channel without oscillations.
• Mode character along x differs markedly between designs: • Dual-pillar: Dominant antisymmetric (sinh-like) mode with a null at channel center; fit indicates A/B ≈ 1/10 (sinh-dominated). This implies near-zero net acceleration and strong lateral deflection (E_x symmetric). The profile shows minimal wavelength dependence over 1063–1065.4 nm. • Inverse-designed: Dominant symmetric (cosh-like) mode with nonzero center field; fit indicates A/B ≈ 50 (cosh-dominated). The field peak shifts laterally with wavelength, indicating sensitivity consistent with resonance behavior.
• Spectral response: Dual-pillar field profile is largely invariant with wavelength; inverse-designed structure shows substantial wavelength-dependent mode shifting across 1063.0–1065.4 nm.
• 3D simulations matched to measurements identify fabrication deviations: • Dual-pillar best fit for diameter deviation δD ≈ −48 nm (narrower pillars than design), explaining antisymmetric mode dominance; antisymmetric mode robust near this δD, consistent with weak wavelength dependence. • Inverse-designed best fit at δD ≈ 0 nm (as designed), supporting symmetric mode dominance and observed spectral sensitivity.
• Robustness ranges from sensitivity study: • Inverse-designed symmetric mode persists over a wider tolerance window (+4 nm > δD > −40 nm) compared to dual-pillar (+10 nm > δD > −4 nm), indicating superior robustness of inverse design to fabrication variations.
• 2D vs 3D modeling: 2D predicts narrower, stronger resonance for inverse-designed structure; 3D reveals reduced Q due to vertical scattering losses, leading to broader spectral response and lower peak gradient—highlighting necessity of full 3D simulations.
• Field component relations: Simulations show symmetric E_z corresponds to antisymmetric E_x and vice versa. Antisymmetric E_z modes cause strong lateral deflection and near-zero net acceleration; symmetric E_z modes support acceleration and can provide transverse focusing/defocusing via E_x (phase-dependent).
• Practical implication: For a 5 µm structure and 30 keV electrons, ~2° deflection within a 200 nm channel is feasible at 200 MeV/m, suggesting potential deflector applications for antisymmetric modes.
• Calibration and imaging parameters: Periodicity Λ = 733 nm; synchronicity near λ ≈ 1064 nm; tolerances for the 15 µm interaction length allow Δλ up to ~25 nm and electron energy deviations up to ~10 keV. Energy-filter slit width 7 eV; filtering ranges −9 to −2 eV (inverse design), −8.5 to −1.5 eV (dual-pillar). Maximum fitted g up to ~1.8 in simulations used for counts conversion.

The results directly address the challenge of experimentally accessing the internal accelerating nearfield in complex DLA structures. By isolating the monochromatic response with a CW PINEM approach, the method reveals true mode content and spatial distributions, including deviations from design due to realistic 3D geometry and fabrication tolerances. The observed antisymmetric mode dominance in the dual-pillar device explains underperformance for acceleration (zero center field, strong lateral deflection) and underscores the sensitivity of this design to small dimensional errors. Conversely, the inverse-designed resonant channel largely maintains its designed symmetric (cosh) accelerating mode and exhibits broader wavelength and fabrication tolerance than predicted by 2D models, which is advantageous for robust operation.

The combination of measured EFTEM images and 3D simulations enables parameter extraction (e.g., δD) and reconstruction of otherwise inaccessible field components (E_x, E_y), surpassing what can be inferred from SEM metrology of outer features. The necessity of full 3D modeling is underscored by the discrepancy between 2D and 3D predictions in spectral response and Q-factor. The mode relationships (symmetric E_z ↔ antisymmetric E_x) suggest practical pathways for integrating acceleration with beam focusing/deflection within the same structure via phase control.

Overall, imaging confirms the importance of tight fabrication control for dual-pillar gratings and demonstrates that inverse-designed structures offer improved robustness and bandwidth, guiding future DLA design choices. The proposed extensions to 3D tomography promise comprehensive mapping of fields along the device, essential for complex, longitudinally varying accelerator architectures.

The study introduces and validates a CW PINEM-based EFTEM method for imaging the accelerating nearfield inside nanophotonic accelerators with deep-subwavelength spatial and sub-nm spectral resolution. Applying the method to two leading DLA designs reveals (i) a dominant antisymmetric (sinh) mode in a dual-pillar device due to a ~48 nm pillar diameter undersize, and (ii) a dominant symmetric (cosh) mode in an inverse-designed resonant structure close to design dimensions. 3D simulations faithfully reproduce measured distributions, enable recovery of all field components, and highlight the necessity of full 3D modeling to predict spectral behavior and Q-factors. Sensitivity analysis shows the inverse-designed structure has significantly wider fabrication tolerance and broader usable bandwidth, making it more robust for practical implementations.

Future directions include: (1) implementing 3D field tomography by sequential sub-section illumination (on-chip apertures or scanned focused beams) with deconvolution; (2) dark-field imaging to probe transverse fields (E_x); (3) integrating pulsed lasers and pulsed electrons to study dynamics; and (4) pursuing full-3D inverse-design optimization to further enhance acceleration gradients and robustness.

• The demonstrated imaging operates in the weak-acceleration CW regime (few-eV energy gain, ~0.2 MeV/m equivalent gradient), not the high-gradient fs-pulsed regime; while beneficial for linearity and resolution, it does not represent peak achievable acceleration performance.
• Net acceleration requires pre-bunched electrons; the current imaging primarily maps field distributions rather than demonstrating high net energy gain.
• EFTEM images are influenced by the imaging point spread function and inelastic scattering at silicon boundaries, necessitating Gaussian convolution and exponential edge-decay models; absolute field amplitudes require careful calibration (counts-to-g transformation) and assumptions (e.g., maximum g normalization).
• The spectral tunability is limited (~1063–1065.8 nm) by the fiber amplifier, and the interaction length (~15 µm) constrains phase-matching bandwidth and sensitivity.
• Measurements retrieve the accelerating Fourier mode (q = ω/v) integrated along z; full 3D field mapping along the structure is proposed but not yet implemented.
• 3D resonant structures exhibit increased vertical scattering losses, reducing Q-factor compared to 2D predictions, which complicates extrapolation from simpler models.
• Structural parameter inference (e.g., δD) relies on matching simulations to images and may be sensitive to uncertainties in material properties, illumination angle, and boundary scattering models.