D. Nonplanar Material Interface

The error introduced by the truncation (6.27) of the Fourier expansion of the EM field is now investigated for the stepped topography of Figure 8.16. A sketch of the relevant region of the geometry is re-plotted in Figure D.1. All derivations are performed in two dimensions only since the results apply to three dimensions as well.

  

Figure D.1: Schematic of the investigated nonplanar material interface (left) and lateral dependence of the permittivity along a horizontal cut $\overline{ZZ^\prime}$ (right). The cut positions Z and Z^$\prime$ are chosen within the nonplanar region of the resist.

The lateral dependence $\varepsilon$(x) of the permittivity is simply given by (cf. Figure D.1)

 

$$\varepsilon(x) = \varepsilon_r + \Delta\varepsilon\, \sigma(x-x_s){}\quad\qquad{\text{for}}\quad\qquad{}0\le x< a,$$ (D.1)

with $\Delta$$\varepsilon$ = $\varepsilon_{s}^{}$ - $\varepsilon_{r}^{}$ as the difference of the resist and step material permittivities. The intersection of the cut-line $\overline{ZZ^\prime}$ with the step slope is denoted by x_s, and $\sigma$(x) refers to Heaviside's step function.^a The differential method requires the Fourier coefficients $\varepsilon_{n}^{}$ of the permittivity function $\varepsilon$(x) of (D.1),

 

$$\varepsilon_n=\varepsilon_r\delta_n+\Delta\varepsilon\frac{\sin(\pi nx_s/a)}{\pi n}e^{-j2\pi nx_s/a}.$$ (D.2)

We restrict our investigations to TE-polarized light. In this case the Fourier coefficients E_{y, n}(z) of the y-component of the electric field satisfy^b

 

$$\frac{d^2 E_{y,n}(z)}{dz^2} = \sum_{m}\left[k_{x,n}^2\delta_{nm}-k_0^2\varepsilon_{n-m}(z)\right]E_{y,m}(z).$$ (D.3)

The impact of the truncation of the Fourier series can best be understood by explicitly studying one specific mode E_{y, i}(z) at an arbitrary vertical cut-position z_c, e.g., at $\overline{ZZ^\prime}$ as indicated in Figure D.1. For the analysis the amplitude of E_{y, i}(z_c) can be chosen arbitrarily. We conveniently set it equal to 1/k₀² so that

$$E_{y,n}(z_c) = \frac{1}{k_0^2}\delta_{ni}.$$ (D.4)

Inserting this relation into (D.3) gives

 

$$\frac{d^2 E_{y,n}(z_c)}{dz^2} = \frac{k_{x,n}^2}{k_0^2}\delta_{ni} - \varepsilon_{n-i}(z_c).$$ (D.5)

For simplicity we consider only the case when k_{x, n}²$\delta_{ni}^{}$ vanishes, i.e., the propagation of the vertical mode i = 0 or, equivalently, the energy spread of an arbitrary order i into the zeroth-order n = 0 is studied. Since the vertical propagating mode usually carries most of the energy, this restriction gives still valuable information.

The impact of the truncation of the Fourier expansions (6.27) is now quantified by the following measure describing the ratio between ``lost'' and ``total'' energy spread:

 

$$\epsilon_{\mathrm{trunc}} \overset{\mathrm{def}}{=}\dfrac{\sum\no... ...{\mathrm{\triangle}}{=}\dfrac{\text{\lq\lq lost energy''}}{\text{\lq\lq total energy''}},$$ (D.6)

This quantity $\epsilon_{\mathrm{trunc}}^{}$ can be interpreted as truncation error since it describes the error made in the calculation of the second derivative (cf. (D.5)). Note that the measure is restricted to n = 0 or i = 0 since only then the first term k_{x, n}$\delta_{ni}^{}$ vanishes in (D.5). The extension to the general situation is self-evident.

We will now derive an upper bound for $\epsilon_{\mathrm{trunc}}^{}$. For that we first evaluate the denominator occurring in (D.6) with the help of Parseval's theorem [131, p. 224] to (cf. (D.1))

 

$$\sum\limits_{n}\vert\varepsilon_n\vert^2 = \frac{1}{a}\int_0^a \v... ...r\vert^2\frac{x_s}{a} + \vert\varepsilon_s\vert^2 \left(1-\frac{x_s}{a}\right).$$ (D.7)

Next we consider the numerator of (D.6). An upper bound for N$\ge$1 is given by (cf. (D.2))

 

$$\sum\limits_{\vert n\vert> N}\vert\varepsilon_n\vert^2 = \vert\De... ...i nx_s/a)}{\pi n}\right\vert^2 \le \vert\Delta\varepsilon\vert^2 \mathcal{B}_N,$$ (D.8)

with

 

$$\mathcal{B}_N = \frac{1}{3}-\frac{2}{\pi^2}\sum\limits_{n=1}^{N}\frac{1}{n^2}.$$ (D.9)

The bound is obtained by replacing | sin($\pi$nx_s/a)| with unity, i.e.,

 

$$\sum_{\vert n\vert> N} \left\vert\frac{\sin(\pi nx_s/a)}{\pi n}\right\vert^2 \le\frac{2}{\pi^2}\sum\limits_{n> N} \frac{1}{n^2}.$$ (D.10)

The right-hand side equals

$$\frac{2}{\pi^2}\left(\sum\limits_{n\ge 1} \frac{1}{n^2} - \sum\li... ...c{1}{n^2}\right) = \frac{1}{3} - \frac{2}{\pi^2}\sum_{1\le n\le N}\frac{1}{n^2}$$ (D.11)

since $\sum_{n>1}^{}$n^-2 = $\pi^{2}_{}$/6. Inserting (D.7) and (D.8) into (D.6) yields

 

$$\epsilon_{\mathrm{trunc}} \le \mathcal{G}_{x_s,\Delta\varepsilon}\mathcal{B}_N.$$ (D.12)

with

 

$$\mathcal{G}_{x_s,\Delta\varepsilon}=\frac{\vert\Delta\varepsilon\vert^2}{x_s/a\vert\varepsilon_r\vert^2 + (1-x_s/a)\vert\varepsilon_s\vert^2}.$$ (D.13)

(D.12) shows that the bound consists of a geometrical factor $\mathcal {G}$_{xs,$\Delta$$\varepsilon$} and a numerical factor $\mathcal {B}$_N. The geometrical factor $\mathcal {G}$_{xs,$\Delta$$\varepsilon$} (cf. (D.13)) is proportional to the squared difference |$\Delta$$\varepsilon$|² = |$\varepsilon_{r}^{}$ - $\varepsilon_{s}^{}$|² of the resist and step material permittivities. The truncation error $\epsilon_{\mathrm{trunc}}^{}$ is thus governed by the strength of the inhomogeneity, which is in accordance with physical considerations. The dependence of the numerical factor $\mathcal {B}$_N (cf. (D.9)) on the truncation frequency N is plotted for a constant $\mathcal {G}$_{xs,$\Delta$$\varepsilon$} in Figure D.2. In this figure also an exact expression for the special situation x_s = a/2 is shown, i.e., the case when the intersection point x_s lies in the middle of the interval [0, a]. In this case the truncation error $\epsilon_{\mathrm{trunc}}^{}$ of (D.6) can be exactly calculated like

 

$$\begin{aligned}\epsilon_{\mathrm{trunc}} &= \mathcal{G}_{x_s,\Del... ...2}{\pi^2}\sum_{n=1}^{\lfloor N/2\rfloor} \frac{1}{(2n-1)^2}\bigg],\end{aligned}$$

whereby $\sum_{n>1}^{}$(2n - 1)^-2 = $\pi^{2}_{}$/8 is used. Both expressions (D.12) and (D.14) show that for increasing N the error decreases quadratically.

  

Figure D.2: Truncation error $\epsilon_{\mathrm{trunc}}^{}$ of the second derivative (cf. (D.6)) as a function of the cut-off frequency N. The geometrical factor is set equal to unity. Both the bound (D.12) and the exact expression (D.14) obtained for x_s = a/2 are shown in linear and logarithmic scale, upper and lower graph, respectively. For increasing N the error decreases quadratically.

As a final remark note that (D.6) describes the error of the second derivative at one single vertical point z_c. The overall discretization error depends on the number N_z of vertical grid points.

Footnotes

... function.^a

Heaviside's step function is defined as $\sigma$(x) = 0 for x < 0 and 1 otherwise.

... satisfy^b

For TE-polarization the three field components E_x(z), E_z(z), and H_{y, nm}(z) vanish [11, p. 52]. Taking the remaining three field components E_y(z), H_x(z), and H_z(z) in (6.15) and (6.18) and eliminating the latter two yields (D.3).