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Equation 1 · Patterning at the Limit: What Actually Happens When a Chip Is Manufactured

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R=k1λNA,R = k_1 \frac{\lambda}{\mathrm{NA}},

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Start withλ
Divide byNA
This relates toR
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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.

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RR

Symbol R

a half-pitch.

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k1k_1

Symbol k_1

a dimensionless process factor absorbing illumination scheme, mask engineering and resist behaviour.

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λ\lambda

Symbol λ

the exposure wavelength, NA\mathrm{NA} is the numerical aperture of the projection optics.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

Divide the expression above the line by the one below it.

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subscript

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.

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NA\mathrm{NA}

Denominator: NA

The complete quantity below the fraction bar; it must be nonzero for this division.

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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

Only one step in the loop carries information, so it deserves the arithmetic. Projection lithography is an imaging problem, and its resolution obeys the Rayleigh relation R=k1λNAR = k_1 \frac{\lambda}{\mathrm{NA}}. where λ\lambda is the exposure wavelength, NA\mathrm{NA} is the numerical aperture of the projection optics, and k1k_1 is a dimensionless process factor absorbing illumination scheme, mask engineering and resist behaviour. Two-beam interference sets a hard floor: for a periodic line-and-space pattern the smallest printable half-pitch is λ\lambda / (2\,NA\mathrm{NA}) , which is k1k_1 = 0.25 in the convention where R is a half-pitch. Everything the industry calls resolution enhancement — off-axis illumination,…
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Only one step in the loop carries information, so it deserves the arithmetic. Projection lithography is an imaging problem, and its resolution obeys the Rayleigh relation R=k1λNAR = k_1 \frac{\lambda}{\mathrm{NA}}. where λ\lambda is the exposure wavelength, NA\mathrm{NA} is the numerical aperture of the projection optics, and k1k_1 is a dimensionless process factor absorbing illumination scheme, mask engineering and resist behaviour. Two-beam interference sets a hard floor: for a periodic line-and-space pattern the smallest printable half-pitch is λ\lambda / (2\,NA\mathrm{NA}) , which is k1k_1 = 0.25 in the convention where R is a half-pitch. Everything the industry calls resolution enhancement — off-axis illumination, phase shifting, optical proximity correction, inverse lithography — is an attempt to drive k1k_1 down toward that floor without printing garbage.

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