Equation 9 · Patterning at the Limit: What Actually Happens When a Chip Is Manufactured
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol k_2
is an input to the expression that computes the quantity on the left.
Symbol λ
the exposure wavelength, is the numerical aperture of the projection optics.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →Denominator: NA^2
The complete quantity below the fraction bar; it must be nonzero for this division.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
The relation that is usually left out of popular accounts is the one that costs money: . Depth of focus falls with the square of numerical aperture. Resolution is bought linearly in and paid for quadratically in focus budget. This is why the planarisation step exists at all, why resist films keep getting thinner, and why every increase in makes the mechanical and material problems worse rather than better. The roadmap is explicit that meeting small depths of focus at 0.55 numerical aperture is a key challenge in its own right, and that a longer-term move to still higher aperture would drive resist thicknesses below 20 nm [ 1 ] .
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.
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