2.3.3 Partial Coherence and Advanced Apertures

Two aspects of coherence play an important role in lithography. Firstly, the light is strongly coherent in the time domain because of the required monochromaticity of the light source (cf. Section 2.3.1). The coherence in the spatial domain, however, is an adjustable parameter that has great influence on the imaging performance. The illumination is said to be partially coherent, if a certain amount of spatial coherence exists. The amount of partial coherence is governed by the ratio of the numerical aperture of the condenser lens NA_c and projection lens  NA_p

 

$$\sigma = \frac{\mathit{NA}_c}{\mathit{NA}_p},$$ (2.4)

whereby $\sigma$ is the so-called partial coherence factor. The influence of $\sigma$ is demonstrated in Figure 2.5 by showing the image intensity near a simple knife-edge. The limiting case $\sigma$ = $\infty$ corresponds to incoherent illumination and gives the smoothest profile. Decreasing $\sigma$ increases the edge slope and ``overshoot'' while decreasing the intensity minimum near the edge. This minimum determines linewidth fidelity and profile quality. If $\sigma$ is further reduced to values as low as 0.2 in order to decrease the intensity minimum, the overshoot becomes excessive and extends laterally indicating that proximity effects between adjacent features are likely. The limiting case $\sigma$ = 0 refers to an ideally coherent point source yielding the sharpest slope but intolerable overshoot. In practical lithography typical values for $\sigma$ range between 0.4-0.7.

  

Figure 2.5: The degree of partial coherence $\sigma$ determines the tradeoff between a sharp slope and a low overshoot (after [15]). Optimal values lie between 0.4-0.7.

Further improvement can be achieved by introducing an aperture between the light source and the condenser lens (cf. Figure 2.4). This aperture acts as a wavefront filter. A circular aperture as in the case of partial coherence refers to a simple low pass filtering with cut-off frequency determined by the numerical aperture of the condenser lens NA_c. Only plane waves up to a certain amount of obliqueness can travel towards the mask. Introducing an annular or quadrupole aperture blocks vertical or zero-order waves that carry no information. The image resolution is enhanced via the k₁ parameter of (2.1) and therefore does not decrease the depth of focus (cf. (2.2)). For that reason such modified illumination schemes or ``off-axis'' techniques have become a well-established method to extend optical lithography towards sub-wavelength resolution. A detailed description of the operation principle will be given in the context of aerial image simulation in Chapter 4.