A Two-Mirror Faceted Projection System for EUV Lithography

Vasiliy A. Es'kin; Egor V. Ivanov; Olga V. Martynova

Source-linked AI summary

A Two-Mirror Faceted Projection System for EUV Lithography

Vasiliy A. Es'kin, Egor V. Ivanov, Olga V. Martynova

arXiv:2609.11299v1Sep 10, 2026physics.opticscs.LGphysics.app-phphysics.class-phphysics.comp-ph

TL;DR

The paper addresses EUV projection limits from mirror count, power loss, and high-NA patterning. It proposes a two-mirror faceted system with order-specific planar facets and evaluates its electromagnetic projection behavior. The design reaches NAmax ≈0.993 with fourfold reduction and modeled two-reflection channel throughput of about 51–60%.

Problem
EUV projection systems face power loss from multiple reflective surfaces while pursuing high numerical aperture and smaller printed features.
Method
The paper uses order-specific planar mirror facets, equalized optical paths, optimized 30-bilayer Bragg coatings, and electromagnetic mask synthesis for two- and three-dimensional periodic masks.
Results
51–60% channel throughput is obtained for the two modeled reflections, with NAmax ≈0.993 and fourfold reduction demonstrated at 13.5 nm and 11.2 nm.
Takeaways & Limitations
The architecture provides a modeled route to high-NA, fourfold-demagnified EUV projection with substantially fewer reflections than conventional multi-mirror systems.
Takeaways & Limitations
The demonstrated coating reflectances are ideal-model calculations that omit interface roughness, interdiffusion, and fabrication errors, and the geometry produces grazing-projection stretching for finite-width beams.

Abstract

from arXiv · show

We propose an all-reflective two-mirror projection system for extreme ultraviolet (EUV) lithography operating at exposure wavelengths of $13.5$~nm (Mo/Si) and $11.2$~nm (Ru/Be), delivering a fourfold ($4\times$) demagnification of the periodic mask pattern at a numerical aperture approaching unity ($\mathrm{NA}_{\max} \approx 0.993$). In contrast to conventional EUV projection objectives that incorporate 6--10 aspheric mirrors with an overall optical throughput of less than $15\%$, the proposed design redirects each accepted discrete spatial diffraction order scattered by the mask onto the wafer via a dedicated pair of planar mirror facets. The number of reflections is strictly fixed at two for all accepted orders, retaining $50$--$60\%$ of the power leaving the mask in each accepted order. We derive a spatial geometry providing rigorous optical path length equalization across all diffraction orders, thereby removing order-dependent propagation phase shifts. Individually optimized 30-bilayer Bragg multilayer coatings are designed for each facet using the transfer matrix method combined with global evolutionary optimization algorithms. The architecture is generalized to a three-dimensional vector formulation with a two-dimensionally periodic mask. Utilizing inverse lithography technology, Fourier parameterization, and a differentiable electromagnetic modal waveguide solver, we solve the synthesis problem for binary absorber masks (La absorber on a Ru/Be/Sr multilayer mirror). We demonstrate simulated aerial images of sub-10-nm features on the wafer (isolated peaks with a full width at half maximum (FWHM) of approximately $5.4$~nm and line pairs with a critical dimension of $6$~nm) and find that the two peaks remain resolved for the tested wafer defocus values from $0$ to $5$~nm along the $z$-axis.

1 Introduction

The paper motivates EUV lithography by the need for smaller printable features and proposes a two-mirror architecture that reduces reflections while supporting high-NA, fourfold projection.

1 Introduction: Rayleigh’s equation links critical dimension to wavelength and numerical aperture, motivating shorter-wavelength exposure and higher-NA optics.The paper notes EUV operation at 13.5 nm and commercially viable high-volume manufacturing since 2018.
1 Introduction: Conventional high-NA EUV systems require more reflective surfaces, increasing power loss as sub-6-nm patterning is pursued.The cited discussion connects future sub-6-nm dimensions with an even greater number of mirrors.
1 Introduction: The paper combines electromagnetic diffraction analysis with faceted projection design rather than relying only on geometric ray tracing.The introduction identifies wave interference and diffraction as effects neglected by crude geometric ray-tracing approximations.
1 Introduction: The proposed system assigns each accepted mask diffraction order a dedicated pair of planar mirror facets, fixing the channel to two reflections.The architecture is formulated for both two-dimensional and three-dimensional periodic masks.
1 Introduction: The design targets NA ∼1 with fourfold demagnification while mapping each propagating diffraction order to its corresponding facet pair.In the three-dimensional formulation, a square-grating mask produces a corresponding facet pair for each propagating order.

2 Formulation of the Problem

The formulation models the mask as a periodic electromagnetic structure whose propagating diffraction orders are transformed into a fourfold-demagnified wafer spectrum by two sets of planar mirror facets.

2 Formulation of the Problem: The mask is modeled as a layered, x-periodic structure with patterned absorber or phase-shifting layers over a multilayer Bragg mirror.Each layer has a position-dependent complex permittivity, and the optical constants come from tabulated reference data.
2 Formulation of the Problem: A TE-polarized oblique plane wave illuminates the periodic mask, whose scattered field is expanded into spatial harmonics using modal numerical methods.The geometrical-optics approximation treats the incident field as uniform within its beam envelope and zero elsewhere.
2 Formulation of the Problem: Only propagating orders with real kz;m are retained, and the highest propagating order is set by M = floor(k0/κx).Evanescent orders are excluded from the projection field outside the mask near-field region.
2 Formulation of the Problem: Under fourfold reduction, the wafer spectrum maps mask harmonic m to harmonic n = −m while retaining only the selected propagating aperture.The wafer target has spatial period reduced by a factor of four, with N = floor(M/4) defining the nominal transmitted order limit.
2 Formulation of the Problem: The first mirror converts each divergent diffraction beam into a vertical beam, and the second redirects selected beams toward the wafer at the required convergence angles.Facet placement must prevent adjacent beam envelopes from overlapping so each facet intercepts only its assigned order.
2 Formulation of the Problem: The illumination aperture omits the back-reflection order m = −m̃ from the projected spectrum.At the first reflection, the amplitude associated with that order is multiplied by zero.

3 2D Two-Mirror Projection System

The system assigns each propagating diffraction order to dedicated planar mirror facets, then equalizes optical paths and optimizes facet-specific coatings for high-NA, fourfold-demagnified imaging. Simulations produce sub-10-nm aerial features while revealing geometric and fabrication constraints.

System geometry: NAmax = 0.993 is achieved with a two-mirror geometry that redirects each diffraction order through dedicated facets toward the wafer.The design preserves the same maximum numerical aperture while using planar mirror facets and a fourfold-demagnified projection architecture.
Equal optical path: Equalizing Pm across orders gives ΔP = 0 within numerical precision, removing order-dependent propagation phase shifts.The facet focal position and first-mirror deflection angles are adjusted to enforce a common optical path of approximately 2998.1 mm.
Bragg coatings: Two modeled reflections retain approximately 51.5%–59.6% of accepted-order power, exceeding six- and ten-mirror throughputs by up to approximately 20×.Individually optimized 30-bilayer coatings achieve Mo/Si reflectances of 71.7%–76.2% and Ru/Be reflectances of 74.9%–79.6% in the ideal model.
Mask synthesis and imaging: The synthesized central aerial peak has approximately 5.4 nm FWHM, compared with approximately 5.1 nm for the bandlimited target.The binary absorber mask is synthesized to shape the required diffraction spectrum, but its fabrication tolerances require separate verification.
Mask synthesis and imaging: Constructive interference raises the optimized peak intensity to max I/Iinc = 0.947, above the baseline two-reflection reflectance of approximately 56%.The wafer field repeats with a 214 nm period, and the result reproduces the target peak profile with Lshape = 1.6 × 10^-5.
Mask synthesis and imaging: The two peaks remain resolved for tested wafer defocus values from 0 to 5 nm, but this does not establish a complete focus–dose process window.The evaluation is limited to the tested axial displacement range.

4 3D Two-Mirror Projection System

The 3D system routes selected two-dimensional diffraction orders through dedicated mirror-facet pairs, using a fourfold transverse-wavevector scaling and equalized optical paths. The design reaches NA = 0.993 with 1128 selected orders, while simulated optimization reduces the binarization penalty and shape loss.

Diffraction-order selection: NA = 0.993 is obtained by selecting the circular aperture m^2 + n^2 ≤ 19^2, while 24 additional propagating orders lie outside that design aperture.The circular aperture is a design choice rather than the exact propagation boundary.
Facet layout and performance: 1128 selected orders map one-to-one onto facets on each mirror, producing 2256 facets and 132 unique radial coating designs.Rotational symmetry lets multiple orders share identical multilayer parameters.
Facet layout and performance: 0.560–0.596 two-reflection throughput, averaging 0.576 and retaining approximately 58% of power in each selected order.The incidence angles range from 1.50° to 47.79° on mirror 1 and 0.37° to 3.93° on mirror 2.
Optical geometry: All selected three-dimensional orders are assigned equal optical path lengths of approximately 2998.1 mm through the two-mirror geometry.Facet positions and deflection angles are solved numerically to enforce the common path length.
Vector formulation: The architecture uses vector transverse-field propagation with TE/TM facet reflection matrices, although the numerical demonstration adopts equal TE and TM coefficients.The 3D periodic mask can generate cross-polarized scattering, but the reported demonstration evaluates transverse components under a scalar-channel approximation.
Mask synthesis: The optimization reduces Lbin from 0.25 to 0.002 and Lshape from 6.5 × 10^-8 to 2.3 × 10^-8 when truncated at M = 16.This truncation corresponds to NA = 0.84.

5 Discussion: Advantages and Disadvantages of the Proposed Projection System

The faceted system offers high-NA, two-reflection projection with planar, individually configurable channels, but introduces substantial geometric, coating, phase, and mechanical constraints.

Advantages: NAmax = 0.993 is reached with 38 planar facets independently mapping diffraction harmonics, while accepted channels retain approximately 50–60% of mask power after two reflections.The design also reduces the periodic unit-cell period by four along each transverse direction, without implying equivalent demagnification for arbitrary finite masks.
Advantages: Planar facets support homogeneous Bragg coatings, avoiding the graded-layer deposition required on conventional aspheric surfaces.
Advantages: Separate facet pairs enable channel-specific coating optimization and potentially modular changes to numerical aperture, although alignment and stability requirements remain unresolved.
Disadvantages: High-order grazing projection stretches finite beam footprints to 82.1 mm for m = 19, a geometric effect mitigable through apodization or aperture sizing.At θ(w)_19 = 83.2°, the projection is stretched by a factor of 8.5 for a beam width of approximately 10 mm.
Disadvantages: Example 4 spans approximately 2204.2 mm transversely and 3 m in optical path, while full interception in Example 2 requires a 382.5-mm facet for m = 1.
Disadvantages: The architecture requires facet-specific multilayer optimization, phase control, and stringent surface and alignment tolerances across as many as 2256 facets.At λ = 11.2 nm, the stated surface-displacement criterion gives σh ≲ 0.30 nm per facet for reflectance ≥0.8.

6 Conclusion

The conclusion presents a two-reflection faceted EUV projection concept that combines fourfold periodic-cell reduction, near-unity numerical aperture, phase equalization, optimized coatings, and simulated sub-10-nm imaging.

6 Conclusion: The system provides 4× reduction of the periodic unit cell at NAmax ≈ 0.993 for 13.5-nm and 11.2-nm operation.
6 Conclusion: A spatial geometry rigorously equalizes optical path lengths across diffraction harmonics, removing order-dependent propagation phase shifts.
6 Conclusion: Thirty-bilayer Bragg optimization achieves approximately 51–60% channel throughput, with second-mirror facets operating at incidence angles of at most 3.93°.
6 Conclusion: The 3D architecture uses 2256 facets and 1128 selected orders, requiring optimization of up to 264 distinct multilayer structures under rotational symmetry.
6 Conclusion: Inverse lithography produces a simulated 5.4-nm-FWHM central peak and two 6-nm lines separated by 6 nm, while further assessment remains necessary for coherence, tolerances, fabrication, polarization, and resist response.

A Diffractive Propagation of a Hard-Apertured Beam

The appendix analyzes propagation of a hard-apertured beam by representing its initial field over a finite aperture and identifying the transverse wavevector and propagation angle.

A Diffractive Propagation of a Hard-Apertured Beam: The initial field is nonzero only across an aperture of width W along x, establishing the hard-aperture input for propagation analysis.
A Diffractive Propagation of a Hard-Apertured Beam: The transverse wavevector component kx determines the propagation angle through α = arcsin(kx/k0), with cos α = kz/k0.

A.2 Angular Spectrum

The angular-spectrum treatment expands the aperture field into plane waves and relates spectral width and beam divergence to aperture width, wavelength, and propagation angle.

A.2 Angular Spectrum: The field is expanded into plane waves, and the spectral half-width to first zeros is Δq = 2π/W.
A.2 Angular Spectrum: The divergence half-angle is governed by the perpendicular beam width and depends on λ, W, and cos α.

A.3 Paraxial Approximation and Beam Coordinate System

The section introduces beam-aligned and laboratory-coordinate descriptions for paraxial propagation of an oblique aperture. It relates laboratory distance to beam-axis distance and models the aperture using a Fresnel integral.

A.3 Paraxial Approximation and Beam Coordinate System: The paraxial approximation neglects the second ζ derivative of the envelope relative to its first derivative.This yields an envelope equation for propagation along the beam axis.
A.3 Paraxial Approximation and Beam Coordinate System: A laboratory-coordinate envelope includes transverse transport in addition to beam-axis diffraction.
A.3 Paraxial Approximation and Beam Coordinate System: Beam coordinates use ζ = x sin α + z cos α along the beam and ξ = x cos α − z sin α transversely.
A.3 Paraxial Approximation and Beam Coordinate System: A propagation distance z along the central laboratory ray corresponds to ζ = z/cos α along the beam axis.
A.3 Paraxial Approximation and Beam Coordinate System: The oblique aperture is represented by a transverse top-hat envelope of width W⊥ whose propagation follows a Fresnel integral.

A.4 Key Diffraction Scales

The section organizes diffraction into near-field, boundary-blur, and far-field scales. Far-field behavior is characterized by aperture-dependent angular width, sidelobes, and beam expansion.

A.4 Key Diffraction Scales: The section identifies Fresnel number and collimation length as characteristic propagation scales.
A.4 Key Diffraction Scales: Near-field geometrical propagation occurs for NF ≫ 1, whereas Fraunhofer diffraction occurs for NF ≪ 1.
A.4 Key Diffraction Scales: Near-field boundary blur is treated as a distinct characteristic scale with a corresponding 10%–90% intensity transition width.
A.4 Key Diffraction Scales: Far-field angular intensity is governed by the aperture width W⊥ and wavelength λ.
A.4 Key Diffraction Scales: The far-field main lobe has first-zero width 2θ0 = 2λ/W⊥, FWHM ≈ 0.886λ/W⊥, and a first sidelobe of ≈4.7%.The beam size expands as D(L) ≈ 2θ0L.

A.5 Near-Field Boundary Structure

Near-field aperture edges produce Fresnel fringes and a finite transition at the shadow boundary. As the Fresnel number decreases, opposite-edge interference smooths the profile toward far-field behavior.

A.5 Near-Field Boundary Structure: For λL ≪ W⊥, opposite aperture edges diffract independently under the semi-infinite knife-edge approximation.
A.5 Near-Field Boundary Structure: At the shadow boundary, the intensity is I(0) = I0/4, while the field decays monotonically into the shadow.
A.5 Near-Field Boundary Structure: In the illuminated region, decaying Fresnel fringes have a period approximately λL/|ξ − ξedge|.
A.5 Near-Field Boundary Structure: As NF decreases toward 10, opposite-edge oscillations begin interfering, and near NF ≈ 1 the profile smooths into the far-field distribution.

A.6 Numerical Example: λ = 11.2 nm, W = 1 cm, kx = kz √

The numerical example examines diffractive beam evolution across propagation distances for λ = 11.2 nm and W⊥ = 7.07 mm. Table 8 and Figures 17–19 provide the distance-dependent results and profile evolution.

A.6 Numerical Example: λ = 11.2 nm, W = 1 cm, kx = kz √: Figures 17–19 illustrate beam evolution across propagation distances, including transverse intensity profiles versus distance.
A.6 Numerical Example: λ = 11.2 nm, W = 1 cm, kx = kz √: Table 8 reports diffractive beam evolution for λ = 11.2 nm and W⊥ = 7.07 mm.Widths through L = 100 m use an independent-edge approximation, while L = 1000 m uses a finite-slit estimate.
A.6 Numerical Example: λ = 11.2 nm, W = 1 cm, kx = kz √: The reported distance range includes an L = 1000 m case where the two edge fields overlap and the independent-edge formula is not applicable.

A.7 Conclusions of the Diffraction Analysis

The diffraction analysis finds stable near-field projection over typical optical paths, with boundary fringes and eventual far-field sinc^2 behavior as propagation distance increases. Hard aperture truncation produces edge ripples that can be reduced using apodizing elements.

Near-field propagation: At λ = 11.2 nm, a centimeter-aperture EUV beam remains practically indistinguishable from geometrical mask projection over typical 1–3 m optical paths.For W⊥ ≈ 7 mm, the beam has LF ≈ 1.1 km and divergence θ0 ≈ 1.6 µrad.
Boundary diffraction: Boundary Fresnel fringes reach a local peak of 1.37 I0 before decaying monotonically into the shadow.
Far-field propagation: At L ≳ LF, sharp edges blur into a sinc^2 distribution with ≈4.7% sidelobes while beam diameter grows linearly as D ≈ 2θ0L.
Practical mitigation: Hard aperture truncation generates edge ripples, but apodizing elements with smoothed edge transmission or reflection can suppress them.