6.3.1 Air/Resist Interface

Above the air/resist interface, i.e., for z > 0, the light consists of incoming E_{i, nm}^pq and reflected waves E_{r, nm}^pq (c.f. Figure 6.1). As outlined in Section 4.1.5 in combination with the vector-valued aerial image simulation the incident electric field vector E_i^pq(x) is given by the Rayleigh expansion of (4.55), whereby the superscript (p, q) refers to the source point considered. Due to the assumption of a periodic simulation domain also the reflected light E_r^pq(x) can be represented by a Rayleigh expansion. Hence the field distribution above the simulation domain, which is due to one source point, follows from the superposition of two Rayleigh expansions and writes to

$$\mathbf{E}^{pq}(\mathbf{x}) = \sum_{(n,m)\in\mathbb{D} ^2} \Big[\... ...}^{pq} e^{-jk_{z,nm}z} \Big] e^{j(k_{x,n}x+k_{y,m}y)}\qquad\text{for}\quad z<0.$$ (6.28)

Note that we have already used truncated expansions, which is no approximation as long as the set $\mathbb{D}$² comprises all incident waves. In any reasonable simulation application this will be the case.

Since the transversal electric field components must be continuous across the interface [11, p. 6] we obtain

 

$$\begin{aligned}E^{pq}_{x,nm}(0) &= E^{pq}_{x,i,nm} + E^{pq}_{x,r,nm} \\ E^{pq}_{y,nm}(0) &= E^{pq}_{y,i,nm} + E^{pq}_{y,r,nm}\end{aligned}$$

for the Fourier coefficients of the lateral electric field components right at the upper boundary at z = 0. With the transverseness property of homogeneous plane waves, i.e., E_nm $\perp$ H_nm $\perp$ k_nm, the magnetic wave amplitudes can easily be calculated from the electric ones. The lateral magnetic field components must be continuous, too. Performing the same derivation as in Appendix C that resulted in (C.7), we obtain for the lateral components of the Fourier coefficients of the magnetic field H^pq(x):

 

$$\begin{aligned}&\omega\mu_0 k_{z,nm}H_{x,nm}^{pq}(0)=\\ &\quad -k... ...^{pq}_{x,r,nm}) +k_{x,n}k_{y,m} (E^{pq}_{y,i,nm}-E^{pq}_{y,r,nm}).\end{aligned}$$

The two equations (6.38) and (6.39) establish four relations for each index pair (n, m) of the (2N_x + 1) x (2N_y + 1) Fourier coefficients. This means that we have already derived N_ODE = 4 x (2N_x + 1) x (2N_y + 1) BCs (cf. (6.29)). However, the lateral coefficients E^r_{x, nm} and E^r_{y, nm} of the reflected light are unknown in (6.38) and (6.39). Hence we have to eliminate them and end up at N_ODE/2 conditions for the upper interface z = 0 that write to

 

$$\begin{aligned}k_{x,n}k_{y,m}E_{x,nm}(0) + (k_{y,m}^2+k_{z,nm}^2)... ...nm} + 2k_{x,n}k_{y,m}E^{pq}_{y,i,nm}.\hspace*{1cm}&\rule{1pt}{0pt}\end{aligned}$$

To clarify the number of equations that are represented by (6.40): There are (2N_x + 1) x (2N_y + 1) index-pairs (n, m) and two equations, one for each lateral direction x and y. Thus (6.40) represents 2 x (2N_x + 1) x (2N_y + 1) or N_ODE/2 BCs valid at z = 0.