How a mask becomes a printed contour.

New to lithography? Start with the plain-language guide.

A photomask describes the pattern we want light to carry.

In litopc, its geometry becomes a binary grid, M(x, y): transmitting regions have value 1 and blocked regions have value 0. The optics filter this input into a continuous light field. A threshold then turns that field into a simulated printed boundary.

01 / Mask

The input geometry.

02 / Aerial image

Light after optical filtering.

03 / Contour

The boundary at I = 0.3.

One mask, one optical calculation. All three panels show the same 1100 nm field.

The aerial image shows light intensity in the image plane. The 3D view lifts that same intensity into a surface so its shape is easier to see. Its height does not represent resist thickness or etched silicon.

A DUV scanner is a lens column. An EUV scanner is built from mirrors, reticle included.

Fused silica is transparent at 193 nm, so a DUV reticle is a glass plate carrying a chrome pattern (or a MoSi attenuated phase-shift pattern), held above the wafer in a 4× reduction lens column; water between the last lens and the wafer takes the numerical aperture to 1.35. At 13.5 nm every material absorbs within tens of nanometres, air included. An EUV scanner therefore runs in vacuum and every optical surface is a Mo/Si multilayer mirror, about forty bilayers tuned to reflect roughly 70 % at 13.5 nm. The reticle is such a mirror with an absorber pattern on top, and because it reflects it cannot be lit head-on: the chief ray arrives 6° off normal and six projection mirrors fold the beam down to the wafer, 4× smaller.

Scroll the diagram to see its full width.

Schematic, not to scale. DUV: a lens column, the reticle a fused-silica plate with a chrome pattern that the light passes through. EUV: nothing transmits 13.5 nm, so the scanner works in vacuum with multilayer mirrors throughout, and the reticle, a mirror too, is lit at an angle so the reflected beam clears the incoming one.

In litopc the imaging equations below are the same for both tools: a thin mask M, a pupil set by NA and λ, a source of width σ. What changes between the presets is λ, NA, σ and the size of the field. The 3D view draws each tool the way it is built: the DUV mask floats above the wafer, lit from above, and the EUV reticle is lit from below at an angle, drawn steeper than 6° so the fold is visible. The effects of the real tilt on a thick reflective reticle (mask shadowing, a horizontal-vertical bias, a pattern shift through focus) are not modelled; see the scope section.

Sharp corners contain high spatial frequencies: rapid changes in the pattern.

A finite numerical aperture passes only part of that frequency content. This is why the image of a crisp mask can have rounded corners and shortened line ends.

The engine is a scalar, partially coherent imaging model in the Abbe form: the extended source is sampled into point sources, each point images the mask coherently through the pupil, and the intensities add. With ℱ the two-dimensional FFT and f the spatial frequency in cycles per nanometre:

I = \sum_{s} w_{s} {| ℱ^{- 1} {ℱ {M} \cdot P (f + f_{s}) \cdot e^{i φ (f + f_{s})}} |}^{2}

(1)

Each source point tilts the illumination, which shifts the pupil by f_s; the weights w_s sum to one. The pupil is a circular low-pass filter whose radius is set by the numerical aperture and the wavelength:

Scroll the diagram to see its full width.

Schematic: the pupil cuts off fine detail; an additional Gaussian factor softens the transmitted field. Plot spacing is illustrative.

P (f) = 𝟙_{| f | \leq NA / λ}

(2)

Here 𝟙 is an indicator: 1 inside the pupil and 0 outside. A fully open field images to exactly 1.0 and patterns are never rescaled to their own peak, so the threshold keeps an absolute meaning. Intensities are clipped to [0, 1]; the clip only touches coherent overshoot above the clear-field level and never moves a contour drawn below it.

Numerically, the mask spectrum is taken on the full raster, the image is formed on a grid that still over-samples the (1 + σ)·NA/λ passband, and it is returned to the raster by zero-padding in frequency. That interpolation is exact for a band-limited image; the validation page reports the measured difference against a full-grid evaluation.

Partial coherence and defocus are the two knobs that turn a textbook image into a process.

The source is a disk of radius σ in pupil units (conventional illumination), or a ring between σ_in and σ_out (annular). Every preset images with σ = 0.7 by default. The disk or ring is sampled on equal-area rings, 36 points for the conventional source, each point weighted by its share of the source area:

f_{s} = σ_{s} \frac{NA}{λ}, | σ_{s} | \leq σ or σ_{in} \leq | σ_{s} | \leq σ_{out}, \sum_{s} w_{s} = 1

(3)

The practical consequence: a grating resolves only when a first diffraction order can still enter the pupil for some source point, which happens for pitches above λ / ((1 + σ)·NA). For 193 nm at NA 0.93 with σ = 0.7 that is about 122 nm; off-axis (annular) light lowers it further at the cost of isolated-feature contrast. Source shaping is a Pro control; Free images with each preset's conventional source.

Defocus is physical. Moving the wafer by z adds a phase to each plane wave that depends on its angle, written here for a medium of index n (1 for dry tools, 1.44 for water immersion):

φ (f) = \frac{2 π z}{λ} (\sqrt{n^{2} - λ^{2} {| f |}^{2}} - n)

(4)

The Defocus control sets z in wafer nanometres; 0 is best focus. For a real mask of this kind the image is symmetric in z, so Bossung curves come out symmetric about best focus. The Rayleigh depth of focus n·λ / (2·NA²) sets the scale of each tool's slider: about ±112 nm for 193 nm dry and ±62 nm for low-NA EUV.

Real resist develops chemically. litopc uses a threshold instead: every point where the intensity reaches τ prints.

The contour is the interpolated iso-intensity line I = τ.

R_{τ} = 𝟙_{I \geq τ}, C_{τ} = {(x, y) : I (x, y) = τ}

(5)

Intensity threshold τ = 0.30

60 nm linedoes not print
90 nm line94 nm
140 nm line145 nm

Mask, correctedTargetPrint at τProfile row

Three lines, 60, 90, 140 nm wide, under the same light (DUV 193_DRY, τ 0.3 in the app). The 90 and 140 nm masks carry 3 rounds of correction; the 60 nm line stays as drawn because it does not print at τ 0.3. Top view and profile share one horizontal scale. Move τ: the thin line prints only while τ is low, and every printed width follows its two crossings.

R_τ is the binary selected region, not a developed material profile. Contours are extracted with subpixel marching squares on the intensity field. Raising τ selects less area and usually pulls the contour inward.

The control named threshold sets τ between 0 and 1. It stands in for exposure dose but is not an energy in mJ/cm²: more exposure energy behaves like a lower τ, and that conversion is not implemented. The simulator also reports the dose to size, the τ at which the printed centre-line CD equals the design CD; with σ = 0.7 it sits near 0.30 for the default patterns, which is why 0.30 is the default threshold. There is no Dill exposure model, post-exposure bake, dissolution-rate model, or resist-thickness calculation.

Three numbers summarize a print: CD, NILS and EPE.

CD is the median width of the printed runs along the centre line of the field, in nanometres: a single-line estimate, not a linewidth map. NILS, the normalized image log slope, is measured at the same edges and tells how steep the aerial image is where the resist switches; higher is more robust to dose noise:

NILS = CD \cdot {(\frac{1}{I} \frac{\partial I}{\partial x})}_{I = τ}

(6)

EPE, the edge placement error, is reported the same way everywhere in the app: the target outline is sampled every 18 nm, each sample is matched to the nearest point of the printed contour, and the mean and the maximum of those distances are shown. The correction loop drives its own per-edge measurement (described in the OPC guide), but the figure it displays, the target score and every study use this outline rule, so they can be compared directly:

{EPE}_{mean} = \frac{1}{K} \sum_{k = 1}^{K} d_{k}, d_{k} = \min_{c \in C_{τ}} ‖ t_{k} - c ‖

(7)

The enabled presets are DUV dry (193 nm, NA 0.93, n = 1) and EUV Low-NA (13.5 nm, NA 0.33), both with σ = 0.7.

Immersion DUV (NA 1.35 in water, n = 1.44) and High-NA EUV (NA 0.55) are implemented in the engine but not enabled in this release. When comparing DUV and EUV, use patterns scaled to the selected wavelength and NA; changing the preset alone does not make two patterns a like-for-like comparison.

A printability guard suppresses the contour and the CD result when the feature size falls below a Rayleigh-style floor:

{CD}_{guard} = k_{1} \frac{λ}{NA}

(8)

The coefficients are 0.28 for DUV dry (about 58 nm) and 0.30 for EUV Low-NA (about 12 nm). They are chosen guardrails, not a calibration, and they do not guarantee printability. In custom layouts the guard uses the largest local feature size, so thin OPC strips do not suppress an otherwise printable pattern.

At a 1100 nm field and a 512² grid, the raster pitch is about 2.15 nm. Rasterization, contour interpolation, field size and FFT sampling all affect measured edges, and the FFT implies periodic boundaries, so keep patterns clear of the field edge. Compare correction runs using the same field, grid and settings.

A good result at one setting does not tell you how the pattern responds when conditions change.

The Studies tab runs one-click experiments on the current set-up, each with a derived range, a chart, a one-line reading and the printed contours of every point overlaid on the canvas. Threshold latitude and Focus latitude step τ and z (a Bossung curve) and report the span over which CD stays within ±10 % of the target. Mask bias grows every edge of the design by the same amount and finds the best single bias, the rule-based answer to compare with the model-based correction. MEEF (CD linearity) plots wafer CD against mask CD; its slope is the mask error enhancement factor, MEEF. Through pitch sweeps the dense-line pitch to show iso-dense bias and the densest resolvable pitch.

Process window (Pro) runs a threshold × defocus grid and finds the largest rectangle of in-spec cells, read as depth of focus at a given threshold latitude. Threshold latitude here is a normalized quantity, not an exposure latitude in percent dose, and no calibration to a manufacturing process is implied. Free runs 5 points per study; Pro runs up to 120.

This engine is useful for visual intuition, controlled comparisons within a stated model, and explaining correction.

Implemented: scalar partially coherent imaging with conventional and annular sources, a circular aberration-free pupil, physical defocus with the medium index, absolute intensity normalization, a threshold resist, CD, NILS, dose to size, outline EPE, and model-based OPC.

Not implemented: vector imaging and polarization at high NA, EUV mask-stack and shadowing effects, aberrations and flare, thin-film standing waves, resist chemistry and diffusion, stochastic effects, and etch. Results are not calibrated to any manufacturing process.

The equations above describe the current implementation; the validation page checks them against analytic expectations. For how the results drive mask changes, continue to the mask correction guide.

A mask is the starting point.

One mask transmits. The other reflects.

The optics remove fine detail.

Source shape and focus.

A threshold turns light into a boundary.

What the numbers mean.

Presets, resolution and sampling.

Studies.

Scope and limits.