5.2.4 Beam Propagation Method

A totally different method was incorporated by Andreas Erdmann into the lithography simulator SOLID [144,146,147]. As this approach is based on the finite-difference beam propagation method [148] it is not restricted to laterally homogeneous resists. However, it is only suited for a planar topography. The field calculation is based on a numerical solution of the Helmholtz equation (cf. (4.2)) that writes inside the inhomogeneous resist as

 

$$\left[\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2} + k_0^2 n^2(\mathbf{x};t_k)\right] E_{y,k}(x,z) = 0,$$ (5.30)

whereby a transversal-electric polarized light and a two-dimensional simulation domain is required [144]. The field amplitude E_{y, k}(x, z) is separated into a slowly varying amplitude A_k(x, z) obeying ($\partial^{2}_{}$/$\partial$z²)A_k(x, z) = 0 and an exponential factor exp( $\mp$ jk₀$\nu$z) describing the propagation upwards and downwards the resist, i.e.,

 

$$E_{y,k}(x,z)= A_k(x,z)e^{\mp jk_0\nu z}\qquad\text{with}\qquad\frac{\partial^2}{\partial z^2}A_k(x,z)=0.$$ (5.31)

Insertion of (5.32) into (5.31) transforms the Helmholtz equation into

$$\left[\frac{\partial^2}{\partial x^2} + k_0^2 \left(n^2(\mathbf{x... ...u^2\right)\right] A_k(x,z) = \pm 2jk_0\nu \frac{\partial}{\partial z} A_k(x,z),$$ (5.32)

which for example can be solved on an equidistant grid with a Crank-Nicholson scheme [144]. The boundary conditions can either be transparent ones [149] used in case of isolated features, or periodic ones applied for dense lines and spaces. Consideration of reflective substrates requires additional modifications as either downward or upward propagation--but not simultaneously--can be modeled by (5.32). However, reflections occurring at the air/resist as well as the resist/substrate interface can conveniently be calculated by Fresnel's reflection formulae [11, pp. 36-51]. The light is then repeatedly propagated down and up through the resist until its intensity is negligible or, alternatively, until a fixed number of iterations is performed and then simpler methods, e.g., the transfer matrix algorithm, are employed.

In the above described form the beam propagation is only suited for low- NA applications as the beams are assumed to travel almost parallel to the vertical axis. An extended wide angle algorithm exists [150] that is suited for higher numerical aperture lithography simulation. Numerical problems such as poor convergence and instability of the solution occur in case of a strongly varying refractive index. A more rigorous finite-difference time domain propagation method is then required [151]. The main limitation of this method is its restriction to planar layers. In the next section we describe various methods suited for nonplanar topography.