Maskless Lithography Focus: F-Number, Depth of Field and the Focus Budget

F-number, numerical aperture and where depth of field comes from

The F-number of a projection optic is the ratio of focal length to entrance pupil diameter, and for lithography systems it maps directly to numerical aperture: F/# ≈ 1/(2·NA) in the symmetric-cone approximation used for lens selection. Every optical system focuses over a finite depth—move the substrate away from the focal plane and the aerial image blurs, first in contrast, then in edge slope, and finally in linewidth. Diffraction-limited theory puts the usable depth of focus at roughly k₂·λ/NA², where k₂ is a process-dependent constant typically between 0.5 and 1 for CD control targets. The inverse-square dependence on NA is the crux: doubling NA to gain resolution quarters the depth of field.

This is why F-number appears in lithography conversations as a shorthand for the resolution–depth trade. A low F-number (high NA) optic writes finer features but demands near-perfect geometry; a high F-number optic accepts wavy substrates but cannot write fine pitch. In a maskless tool the trade is fixed at design time, and what the process engineer controls is how the resulting depth of field is spent.

Building the focus budget line by line

A focus budget starts from the measured depth of focus at the process target and subtracts, or root-sum-squares, every contributor. Static contributors dominate most budgets: chuck topography (vacuum chucks typically hold a few micrometers over a full wafer, better on small areas), substrate bow and waviness (silicon wafers are flat, but glass, ceramic and flexible substrates are not), and resist thickness variation. Each micron of topography at the focal plane is a micron of defocus. Machine contributors follow: stage tilt and roll across travel, autofocus sensor noise and bandwidth limits, and the calibration residual between the focus sensor's reference plane and the projection focal plane.

The allocation should state, for each line, how it was measured and whether it is systematic or random. Systematic defocus— chuck topography, substrate bow—can be mapped and compensated by the stage's Z motion using the focus map; random contributors cannot, and they consume margin everywhere. Tools that support per-field or per-substrate focus mapping convert the largest systematic lines into compensated quantities, leaving the residual budget to the random terms.

The end of the budget is process margin: how much defocus the resist system tolerates before CD drifts past its tolerance. A budget that sums to 90 percent of the depth of field has no margin, and every new substrate lot becomes a CD investigation.

• Depth of focus ≈ k₂·λ/NA²; the top line of the budget
• Static lines: chuck topography, substrate bow, resist thickness variation
• Dynamic lines: stage tilt, focus sensor noise, calibration residual
• Residual margin must cover process variation, not just the measured mean

Autofocus: keeping the residual inside its allocation

The autofocus system is what makes the budget real during writing. Most maskless tools use a focus sensor—an optical probe beam separate from the projection path—referenced to the projection focal plane through calibration. Its two performance figures matter in the budget: noise floor, which sets the random defocus floor while writing, and bandwidth relative to stage dynamics, which determines whether the Z loop can follow topography and tilt at writing speed. A sensor with sub-100-nanometer noise and sufficient bandwidth leaves the bulk of the residual budget to substrate effects; a noisy or slow sensor quietly becomes the largest line in the budget.

Calibration discipline matters as much as sensor quality. The sensor's reference plane must be verified against the projection focal plane periodically—by writing a focus-wedge pattern, measuring CD versus Z, and confirming the peak lands at the sensor's zero—because a drift between the two planes appears as a constant bias that shifts every field identically and is easy to mistake for a resist or dose problem.

Verifying the budget on the substrate

Verification closes the loop. Map defocus across the write area—many tools expose a focus-wedge or focus-probe pattern at a grid of field positions; CD versus nominal Z at each position yields the local best focus. Comparing the resulting map against the budget's static lines confirms whether chuck and substrate contributions are as measured and whether focus compensation is working. CD measurements at the worst-case positions—field corners, substrate high and low points, scan start and end—then verify that linewidth control holds where the budget is thinnest.

When the measured map exceeds the allocation, the diagnosis follows the budget lines in order: re-measure chuck topography if the map is stable and repeatable (a mechanical topography signature), check substrate handling and bow if the map follows the substrate, and inspect focus sensor calibration if the map shifts uniformly between sessions. Pattern-dependent CD drift that follows neither substrate nor stage usually points to dose or data-path issues rather than focus, which is why the defocus map should be captured before dose experiments begin.

Choosing substrates and optics together

The focus budget is a selection tool, not only a verification one. Before committing a substrate type to a maskless process, compare its flatness and bow statistics against the budget's static lines; before choosing a writing mode or objective, compare its depth of field against what the substrate stack leaves as margin. Laboratories writing on glass, ceramics or flexible films often find that a moderate-NA writing mode with a comfortable depth of field delivers better effective pattern quality than a fast mode that spends the whole budget chasing topography. The number that decides is not the smallest feature size in the datasheet—it is the smallest feature size that survives the actual substrate with margin to spare.