The Optics of Lithography

Lesson 2 covered the chemistry the exposure step acts on. This lesson covers what the exposure step itself is actually doing: projecting a demagnified image of the mask pattern onto the resist, through a lens, using light of a specific wavelength. Two numbers, wavelength and numerical aperture, set a hard floor on how small a feature that image can resolve, and a second limit, depth of focus, sets how much the wafer can be out of flat before the image blurs. Both limits are summarized by the two forms of the Rayleigh equation.

Let us consider why these particular two equations, first derived for microscope optics long before semiconductors existed, became the equations every lithography roadmap is built around.

Reading time: about 9 minutes.

3.1 Resolution: the Rayleigh CD equation

The smallest half-pitch (CD) a projection lithography system can resolve is given by:

CD = k1 * (lambda / NA)

where lambda is the exposure wavelength, NA is the numerical aperture of the projection lens (how wide a cone of diffracted light the lens can collect and focus, roughly the sine of its half-angle, times the refractive index of the medium in front of the lens), and k1 is a dimensionless process factor that captures everything about the resist, the illumination shape and the mask that is not already captured by wavelength and NA. A smaller CD, a finer feature, needs a smaller lambda, a larger NA, or a smaller k1.

k1 has a theoretical floor around 0.25 for a single exposure under standard (non-resolution-enhanced) conditions; pushing below that floor without extra tricks is not a question of better engineering, it is a question of diffraction physics. Lessons 4-5 are, in large part, a catalog of techniques (OPC, phase-shift masks, multiple patterning) whose entire purpose is to push the achievable k1 down, or to let the same optical system pattern a finer final pitch by splitting it across more than one exposure.

3.2 Depth of focus: the other Rayleigh equation

A second form of the same relationship gives the depth of focus (DOF), how far the wafer surface can move away from best focus before the image degrades past a usable limit:

DOF = k2 * (lambda / NA^2)

Notice the NA appears squared, and in the denominator. This is the central tension of projection lithography: raising NA to improve resolution (equation in 3.1) shrinks depth of focus much faster than it improves resolution, because of that square. A lens pushed to a very high NA resolves a smaller CD, but tolerates far less wafer non-flatness, defocus from topography left by an earlier layer's CMP, or chuck-and-stage positioning error, before the pattern degrades.

This is exactly why lesson 1 said CMP exists to give the next layer's lithography a flat surface: DOF is not a generous budget, and an uneven surface from the previous layer eats directly into it.

3.3 Immersion lithography and off-axis illumination

Two engineering ideas push NA and k1 further without changing wavelength:

• Immersion. Filling the gap between the final lens element and the wafer with high-purity water, rather than air, raises the effective NA above 1.0, because NA scales with the refractive index of that medium (about 1.44 for water at 193 nm) as well as the lens's geometric angle. 193 nm immersion scanners, with NA around 1.35, became the production workhorse of deep-UV lithography for this reason, and are still used today for layers that do not need EUV's resolution.
• Off-axis illumination (OAI). Rather than illuminating the mask with light coming straight down the optical axis, OAI shapes the illumination into an annular ring or a set of discrete poles, angled to the sides. For a periodic pattern, this improves how the diffraction orders the lens needs to capture line up with the lens's aperture, effectively lowering the achievable k1 for that specific pattern pitch. OAI is tuned per layer and per critical pitch, which is one reason a scanner's illuminator settings are part of the recipe, not a fixed hardware constant.

193 nm immersion with aggressive OAI and resolution enhancement (lesson 4) was pushed to pattern features well below what the Rayleigh resolution equation would suggest is possible for 193 nm light, through multiple patterning (lesson 5). EUV lithography, at 13.5 nm, is the other lever: shrinking lambda directly, which lesson 5 also covers.

3.4 Worked example: comparing two configurations

3.5 Interview questions

Write the Rayleigh resolution equation and explain each term. CD = k1 x (lambda / NA): lambda is exposure wavelength, NA is the lens's numerical aperture, and k1 is a dimensionless process factor covering resist, illumination and mask effects not already captured by lambda and NA.

Why does raising NA hurt depth of focus more than it helps resolution? Resolution scales as 1/NA, but depth of focus scales as 1/NA squared, so depth of focus falls off faster than resolution improves as NA increases.

What does immersion lithography actually do? It fills the gap between the final lens element and the wafer with a high-index fluid (water, at 193 nm), which raises the effective numerical aperture above what is achievable in air.

Why is EUV's lower NA not a resolution problem, given 3.1 says resolution depends on NA? Because EUV's wavelength (13.5 nm) is more than an order of magnitude smaller than 193 nm, the CD equation's smaller lambda more than compensates for a smaller NA, while simultaneously giving comparable or better depth of focus, since DOF depends on lambda directly but NA squared.

What is off-axis illumination for? It shapes the illumination angle (annular or multi-pole rather than straight down the axis) to improve how a periodic pattern's diffraction orders are captured by the lens, effectively lowering the achievable k1 for that specific pitch.