4.2.1 Defocus

The simplest and most basic aberration type is defocus caused by a wrong vertical position of the image plane. As discussed in Chapter 2 the depth of focus is directly related to the wavelength and the squared reciprocal numerical aperture (cf. (2.2)). In sub-micrometer high-numerical aperture lithography the range over which the image is adequately sharp is typically less than one micron. Thus it is extremely difficult to provide an exact positioning across the entire image field. Problems such as wafer non-flatness, auto-focus errors, leveling errors, lens heating, etc. arise.

In case of a positioning error $\Delta_{\mathrm{Def}}^{}$ the plane waves do not converge at the wafer. The optical path difference depends on the orientation of the incident rays. The aberration function describing defocus follows from the plane wave decomposition (4.55) and writes to

 

$$\Phi(n,m) = \Delta_{\mathrm{Def}} i_{z,nm},$$ (4.61)

whereby i_{z, nm} = $\sqrt{1 - i^2_{x,n} - i^2_{y,m}}$ is the vertical component of the wavevector. The paraxial approximation valid for almost vertical incident rays, i.e., i_{x, n}, i_{y, m} $\ll$ 1, yields the simpler relation

$$\Phi(n,m) = \Delta_{\mathrm{Def}} \sqrt{1 - i^2_{x,n} - i^2_{y,m}}$$

$$\cong \Delta_{\mathrm{Def}} - \frac{1}{2}(i^2_{x,n}+i^2_{y,m})\Delta_{\mathrm{Def}},$$ (4.62)

which will subsequently be used to study the impacts of higher-order aberration types. The obvious effect of defocus is a vertical shift of the focal point above or below the Gaussian image point depending on the sign of $\Delta_{\mathrm{Def}}^{}$. However, if $\Delta_{\mathrm{Def}}^{}$ varies across the lens field, the surface of best imagery is not planar and the usable depth of focus is correspondingly reduced [116].