6.3.2 Resist/Substrate Interface

A similar derivation applies to the lower interface between resist and substrate at z = h. There are two different situations as either only one layer representing the substrate or multiple planar homogeneous layers forming a stratified medium have to be considered.

The first situation is self-evident and yields BCs totally equivalent to (6.40), i.e., we obtain

 

$$\begin{aligned}k_{x,n}k_{y,m}E_{x,nm}(h) + (k_{y,m}^2+k_{z,s,nm}^... ...+ k_{x,n}k_{y,m}E_{y,nm}(h) - \omega\mu_0k_{z,s,nm}H_{y,nm}(h)&=0.\end{aligned}$$

There are two differences between (6.40) and (6.41): Firstly, no light is incident at the lower interface since the substrate is assumed to be infinitely extended, i.e., we have homogeneous BCs. Secondly, the vertical wavevector component k_{z, s, nm} depends on the permittivity $\varepsilon_{s}^{}$ of the substrate material, i.e., k_{z, s, nm} = $\sqrt{\smash[b]{k_0^2\varepsilon_s - k_{x,n}^2 - k_{y,m}^2}}$. Furthermore, it has the opposite sign since the lower interface is ``mirrored'' in comparison to the upper one. Finally, the Rayleigh expansion of the electric field phasor inside the substrate takes the form

$$\mathbf{E}(\mathbf{x}) = \sum_{(n,m)\in\mathbb{D} ^2} \Big[\mathb... ...} e^{+jk_{z,s,nm}(z-h)}\Big] e^{j(k_{x,n}x+k_{y,m}y)}\qquad\text{for}\quad z>h,$$ (6.29)

whereby E_{s, nm} refers to the wave amplitudes traveling into the substrate (cf. Fig. 6.1).

In the second situation, i.e., when multiple homogeneous layers form a stratified medium lying below the simulation domain--an example is illustrated in Fig. 6.1,--the BCs are derived as follows: Due to the reflections occurring inside the layer stack light is reflected back to the resist/substrate interface. Therefore also ``incident'' waves have to be taken into account at z = h. The Rayleigh expansion inside the top-most layer (TL) close to the interface takes the form

$$\mathbf{E}(\mathbf{x}) = \sum_{(n,m)\in\mathbb{D} ^2} \Big[\mathb... ...},nm}(z-h)}\Big] e^{j(k_{x,n}x+k_{y,m}y)}\text{for}\quad h<z<h+h_{\mathrm{TL}}.$$ (6.30)

In Appendix C is shown how to treat multiple homogeneous layers analytically. Especially (C.26) describes the lateral electric field amplitudes occurring inside the top-most layer--in (C.26) the top-most layer refers to air--and relates them to the amplitudes inside the bottom-most layer representing the substrate via the so-called propagation matrix $\underline{\mathbf{P}}^{nm}_{}$. Hence, the same arguments that led to (C.27) and (C.28) apply, and the lateral components of the amplitudes E⁺_{TL, nm} and E^-_{TL, nm} propagating downwards and upwards the top-most layer are related by (cf. (C.28))

 

$$\begin{aligned}p^{nm}_{31} E^+_{x,\mathrm{TL},nm} + p^{nm}_{32} E... ... E^-_{x,\mathrm{TL},nm} + p^{nm}_{44} E^-_{y,\mathrm{TL},nm} &= 0.\end{aligned}$$

The propagation matrix $\underline{\mathbf{P}}^{nm}_{}$ of the stratified medium depends on the index pair (n, m) since it is determined by the harmonic frequencies k_{x, n} = 2$\pi$n/a and k_{y, m} = 2$\pi$m/b. This means that each harmonic has its own propagation matrix $\underline{\mathbf{P}}^{nm}_{}$.

In addition to (6.44) four relations corresponding to ``mirrored'' versions of (6.38) and (6.39) must also be stratified at z = h:

 

$$\begin{aligned}E_{x,nm}(h) &= E^+_{x,\mathrm{TL},nm} + E^-_{x,\ma... ...) &= E^+_{y,\mathrm{TL},nm} + E^-_{y,\mathrm{TL},nm}\end{aligned}\vspace*{-2ex}$$

and

 

$$\begin{aligned}&\omega\mu_0 k_{z,\mathrm{TL},nm}H_{x,nm}(h)=\\ &\... ...}) +k_{x,n}k_{y,m}(E^+_{y,\mathrm{TL},nm}-E^-_{y,\mathrm{TL},nm}).\end{aligned}$$

In total, (6.44) to (6.46) constitute six equations for each harmonic frequency (n, m). In these six relations the four lateral amplitude components of the waves E⁺_{TL, nm} and E^-_{TL, nm} propagating inside the top-most layer are unknown. Hence they have to be eliminated analytically and we finally obtain

 

$$\begin{aligned}m_xk_{x,n}k_{y,m}E_{x,nm}(h) + m_y(k_{y,m}^2+k_{z,... ..._yk_{x,n}k_{y,m}E_{y,nm}(h) - \omega\mu_0k_{z,s,nm}H_{y,nm}(h)&=0,\end{aligned}$$

whereby the two factors m_x and m_y describe the influence of the stratified medium and are given by

$$\begin{aligned}m_x&=\frac{(p_{33}+p_{31})(p_{44}-p_{42}) + (p_{32... ...{(p_{33}-p_{31})(p_{44}-p_{42}) - (p_{32}-p_{34})(p_{41}-p_{43})}.\end{aligned}$$

The two relations of (6.47) constitute N_ODE/2 BCs in case of multiple planar homogeneous layers below the simulation domain. Comparing them with the BCs (6.41) valid for the substrate-only-situation shows that the difference lies in the two factors m_x and m_y. They can easily be calculated in advance, and thus multiple planar layers below the resist can be treated in a very efficient way.

Finally, it should be noted that the general formulation (6.47) holds in both cases since in the substrate-only-situation we have $\underline{\mathbf{P}}$ = $\underline{\mathbf{I}}$ and the two coefficients m_x and m_y are unity, i.e., m_x = m_y = 1 and (6.41) and (6.47) are identical.