4.2.3 Zernike Polynomials for General Aberration Terms

General aberration terms are characterized most conveniently by the circle polynomials $\mathcal {Z}$_b^c($\rho$,$\phi$) of Zernike [11, pp. 464-468]. The aberration function $\Phi$(n, m) is expanded as follows:

$$\Phi(n,m)= \lambda\!\sum_{\begin{Sb}a,b\ge0,\;c\\ [1mm]a-\vert c\... ...mathrm{even}\end{Sb}} \!\Delta_{abc}A_{ac}\mathcal{Z}_b^c(\rho_{nm},\phi_{nm}).$$ (4.66)

As before, $\Delta_{abc}^{}$ refers to the maximal optical path difference of the lens measured in units of the wavelength, while the factor A_ac is determined by the object location (r_ocos$\phi_{o}^{}$, r_osin$\phi_{o}^{}$),

$$A_{ac} = \begin{cases}r_o^{2a+\vert c\vert}\cos(c\phi_o) & \quad{... ...,} \\ r_o^{2a+\vert c\vert}\sin(c\phi_o) & \quad{} \text{otherwise.}\end{cases}$$ (4.67)

The orientation of the incident wavevector is given by the polar coordinates ($\rho_{nm}^{}$,$\phi_{nm}^{}$),

$$\rho_{nm} = \sqrt{\dot{\iota}^2_{x,n} + \dot{\iota}^2_{y,m}}{}\qq... ...{} \phi_{nm} = \arctan\left(\frac{\dot{\iota}_{x,n}}{\dot{\iota}_{y,m}}\right).$$ (4.68)

The Zernike polynomials in real-valued form are defined as

$$\mathcal{Z}_b^c(\rho,\phi) \overset{\mathrm{def}}{=}\begin{cases}... ... [2mm] \mathcal{R}_b^c(\rho) \sin(c\phi) & \quad{} \text{otherwise,}\end{cases}$$ (4.69)

whereby $\mathcal {R}$_b^c($\rho$) are the so-called orthogonal radial polynomials given by [11, p. 465]

 

$$\mathcal{R}_b^c(\rho) = \sum_{k=0}^{\frac{b-c}{2}} (-1)^k \frac{(b-k)!}{k!\left(\frac{b+c}{2}-k\right)!\left(\frac{b-c}{2}-k\right)!} \rho^{b-2k}.$$ (4.70)

A thorough discussion can be found in, e.g., [11, Appendix VII]. In Table 4.2 the explicit forms of (4.79) for the first few values of the indices b and c are listed. The normalization has been chosen so that $\mathcal {R}$_b^c(1) = 1 for all permissible values of b and c.

 

Table 4.2: Radial polynomials $\mathcal {R}$_b^c($\rho$) for b$\le$7, c$\le$7 (after [11, p. 465]).

 

c $\backslash$b	0	1	2	3	4	5	6	7
0	1		2$\rho^{2}_{}$ - 1		6$\rho^{4}_{}$ - 6$\rho^{2}_{}$ + 1		20$\rho^{6}_{}$ - 30$\rho^{4}_{}$ + 12$\rho^{2}_{}$ - 1
1		$\rho$		3$\rho^{3}_{}$ - 2$\rho$		10$\rho^{5}_{}$ - 12$\rho^{3}_{}$ + 3$\rho$		35$\rho^{7}_{}$ - 60$\rho^{5}_{}$ + 30$\rho^{3}_{}$ - 4$\rho$
2			$\rho^{2}_{}$		4$\rho^{4}_{}$ - 3$\rho^{2}_{}$		15$\rho^{6}_{}$ - 20$\rho^{4}_{}$ + 6$\rho^{2}_{}$
3				$\rho^{3}_{}$		5$\rho^{5}_{}$ - 4$\rho^{3}_{}$		21$\rho^{7}_{}$ - 30$\rho^{5}_{}$ + 10$\rho^{3}_{}$
4					$\rho^{4}_{}$		6$\rho^{6}_{}$ - 5$\rho^{4}_{}$
5						$\rho^{5}_{}$		7$\rho^{7}_{}$ - 6$\rho^{5}_{}$
6							$\rho^{6}_{}$
7								$\rho^{7}_{}$

The lens aberrations have a variety of impacts on the lithographic performance [117,119]. The most severe problems are shifts in the image position, image asymmetry, reduction of the process window, and the appearance of undesirable imaging artifacts. Aerial image simulation provides a powerful tool to investigate most of them. In Table 4.3 the imaging consequences of the first 11 Zernike polynomials are listed.

 

Table 4.3: Imaging consequences of first 11 Zernike polynomials (after [120]).

 

(b, c)	Name	Imaging consequence
(0, 0)	Piston	None
(1, $\pm$ 1)	Lateral translation	Shift of image, independent of pattern
(2, 0)	Defocus	Image degradation
(2, $\pm$ 2)	Astigmatism (hor./vert. or $\pm$ 45o)	Orientation dependent shift of focus
(3, $\pm$ 1)	Lateral coma	Image asymmetry and pattern dependent shift of image
(3, $\pm$ 3)	Three-leaf clover (rotated 30o)	Imaging anomalies with threefold symmetry
(4, 0)	Third-order spherical	Pattern dependent focus shift