Lithography Optics

What is modelled

A mask of lines and spaces is a diffraction grating. Light leaves it in orders at angles set by the pitch and the wavelength. The lens can only pass orders that fall inside its aperture, and an image of the lines forms only if the zero order and at least one first order get through. Tilting the illumination lets the second order in, which halves the pitch that can be printed.

Equations

• Order directions sm = σ NA + m λ/p, passed when |sm| ≤ NA; smallest pitch p = λ / ((1 + σ) NA).
• Half-pitch k1 λ / NA with 0.25 ≤ k1; point spread first zero 0.61 λ / NA.
• Depth of focus (quarter-wave) DOF = λ / (2 n (1 − cosθ)), sinθ = NA/n, which tends to λ/NA².
• Image I(x) = |Σ cm ei(2π m x/p + φm)|² with square-wave amplitudes cm = sin(mπ/2)/(mπ).

Checks run in the page

• Bisection for the smallest pitch that lets the zero and a first order through.
• The first zero of the Airy pattern from a numerically integrated Bessel function.
• The defocus at which the edge ray is a quarter wave out of step.
• The mean of the image against the sum of squared order amplitudes.

Honest simplifications

The light is a single tilted plane wave (fully coherent, one point in the source) rather than an extended partially coherent source, the mask is a thin binary grating with no polarisation, resist chemistry and mask 3D effects, and the image is one-dimensional. Real scanners use many source points; the trend from wavelength and aperture is the point here.

• Press EUV 0.33 or High-NA EUV: a wavelength 14 times shorter than ArF prints much finer lines even with a lower aperture.
• Press Below the limit and see the flat image when the half-pitch is smaller than the lens can pass.
• Slide the defocus and compare the on-axis and tilted illumination curves in the contrast plot.