The KBA microscopes mainly converge light beams onto the meridian plane of the double-mirror. The focal distance of the meridian plane is derived below.

The double-mirror is composed of two single-mirrors, shown as figure 4. *Q*_{1} and *Q*_{2} are the apexes of mirrors *M*_{1} and *M* _{2}, respectively. *A′* and *B′* are the image points of *A* and *B*, respectively. Because the radii of the mirrors are large, we could regard the mirrors as plane mirrors in the analysis of focal distance. Then it yields ${\theta}_{A}^{\prime}=\eta -{\theta}_{A}$. As before, to avoid serious image plane obliquity, it is required that the focal distance to stay constant.

Figure 4 Image of double-mirror

For the double-mirror with included angle *η*, we can derive the focal distance at grazing angle *θ*. The focal distance of the double-mirror can be obtained by

$$\phi ={\phi}_{1}+{\phi}_{2}-d{\phi}_{1}{\phi}_{2}$$(6)When *r*_{1} = *r*_{2} = *r*,

$$\begin{array}{r}{\phi}_{1}=\frac{2}{r\mathrm{sin}\theta},\phantom{\rule{2em}{0ex}}\text{\hspace{0.17em}}{\phi}_{2}=\frac{2}{r\mathrm{sin}(\eta -\theta )}\end{array}$$*d* is the center-to-center spacing of the two mirrors. *η* is the included angle of the two mirrors.

$$\phi =\frac{2}{r\mathrm{sin}\theta}+\frac{2}{r\mathrm{sin}(\eta -\theta )}-\frac{4d}{{r}^{2}\mathrm{sin}\theta \mathrm{sin}(\eta -\theta )}$$(7)When *θ*, (*η* − *θ*) is a very small, sin *θ* = *θ*, sin(*η* − *θ*) = *η* − *θ*. The above formula becomes

$$\begin{array}{rl}\phi & =\frac{2}{r\theta}+\frac{2}{r(\eta -\theta )}-\frac{4d}{{r}^{2}\theta (\eta -\theta )}=\frac{2r\eta -4d}{{r}^{2}\theta (\eta -\theta )}\\ {f}^{\prime}& =\frac{1}{\phi}=\frac{{r}^{2}\theta (\eta -\theta )}{2r\eta -4d}\end{array}$$(8)As it can be seen from the above formula, for given *η*, the focal distance *f′* is a function of the incidence angle *θ*. Differentiating both sides of the above formula, it yields

$$d{f}^{\prime}=\frac{{r}^{2}(\eta -2\theta )}{2r\eta -4d}\cdot d\theta $$(9)When $\theta =\frac{\eta}{2}$, the first-order derivative of *f′* is 0 and *f′* obtains the extreme value. $\theta =\frac{\eta}{2}$is a stationary point, within the neighbourhood of which, *f′* hardly varies with *θ*, namely $\frac{d{f}^{\prime}}{d\theta}=0$. This indicates, the image plane will not slant in the neighbourhood of point $\theta =\frac{\eta}{2}$. At this time, the focal distance of the double-mirror is

$${f}^{\prime}=\frac{r\eta}{8(1-\frac{2d}{r\eta})}$$(10)When *d* << *rη*,

$${f}^{\prime}\approx \frac{r\eta}{8}$$(11)In reverse, when the incidence angles on the pseudoaxis are known, the included angles of the double-mirror can be calculated. Namely, *η* = 2*θ*. In this way, the image planes of the points on the axis will not slant. But for the points off axis, the incidence angle *θ* has been changed while *η* remains unchanged, so $\eta \ne 2\theta ,\frac{\text{d}{f}^{\prime}}{\text{d}\theta}\ne 0$. Thus, the focal distance varies with *θ*, resulting in image plane obliquity.

Next, Matlab was adopted to simulate the effects of incidence angle *θ* and included angle *η*on the focal distance of the double-mirror. The focal distance can be calculated according to formula 9. The radius *r* is 29000 mm in this case. For double-mirrors with different included angles (*η*), we plot the curves of the focal distance with respect to the incidence angle *θ*, shown as figure 5(a), (b) and (c).

Figure 5 Curve of focal distance with respect to incidence angle

Figure 5(a), (b) and (c) show the curves of the focal distance *f* varying with the incidence angle *θ* when the included angles of the double mirror *η* are 3*∘*, 3.2*∘* and 3.4*∘*, respectively. Figure (d) shows the curve of the focal distance of one mirror composed of the double-mirror varying with *θ*. As it can be seen from figure (a), (b) and (c), when the included angle of the double-mirror changes, the focal distance of the double-mirror will change. For different *η*, the extreme value is when $\theta =\frac{\eta}{2}$. Moreover, compared to the focal distance of the single-mirror (figure 4(d)), the focal distance of the double-mirror varies in a smaller range with *θ*. As it can be known from formula 5, the reduction of ${f}_{A}^{\prime},{f}_{B}^{\prime}$reflects image obliquity will decrease by using the double-mirror structure. According to the imaging characteristics of the single-mirror, the imaging quality will be better with bigger incidence angle. And for the double-mirror system, the difference of the incidence angles on the two mirrors should be possibly smaller, namely approaching to $\frac{\eta}{2}$.

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