5.2.3 Transfer Matrix Method

The transfer matrix method overcomes the direction independence by assuming that the light consists of various plane waves traveling with oblique angles through the resist. The method is a straight-forward implementation of the formalism derived in Appendix C for the analysis of a stratified medium [11, pp. 51-70]. For lithography simulation this approach was first proposed by Chris Mack [145] and Douglas Bernard [103]. The transfer matrix method is also called high- NA scalar model and is implemented in many commercially available lithography simulators like DEPICT [103], PROLITH [145], and SOLID [106] as advanced aerial image module. The basic operation principle is described as follows. The incoming light U_i(x, y) is represented by a superposition of plane waves, i.e.,

$$U_i(x,y) = \sum_{n,m} U_{i,nm} e^{j(k_{x,n}x+k_{y,m}y)}.$$ (5.27)

The latent bulk image U_{r, nm}(z;t_k) is obtained by first calculating the vertical amplitude dependence of the field resulting from the excitation with one plane wave of amplitude U_{i, nm},

$$U_{r,nm}(z;t_k) = L_{nm}(z;t_k) U_{i,nm},$$ (5.28)

and then reconstructing the field itself according to

$$U_r(\mathbf{x};t_k) = \sum_{n,m} U_{r,nm}(z;t_k) e^{j(k_{x,n}x+k_{y,m}y)}.$$ (5.29)

A more detailed description is given in Appendix C on page . Using the vector version of both the aerial image tool and the transfer matrix algorithm an arbitrarily polarized light can be simulated. Additionally, advanced apertures can be simulated with Abbe's method. However, there are two fundamental restrictions of this approach. Firstly the bleaching of the resist is not simulated correctly as the layers forming the resist have to be homogeneous in lateral direction, and secondly only imaging over a planar substrate can be modeled.