Resolution Properties of an Ellipse

I first analyze the resolution of the ellipse depicted in Figure 4.1b, and then relate the parameters of this ellipse to those defining a wavepath.

The spatial resolution limit near the point $c$ in Figure 4.1b is related to the reciprocal of the segment length $\overline{DE}$ . This length can be determined by noting that the end points $D$ and $E$ satisfy both the equations of the ellipse and the line, written as

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$$ (10.1)

$$y= \tan\theta\,(x-c),$$ (10.2)

where $a$ and $b$ are the major and minor radii of the ellipse, respectively, and $\theta$ is the angle $DE$ makes with the axis of the ellipse. Equations E.1 and E.2 can be reduced to a quadratic equation of one variable $y$ , yielding two roots $y_D$ and $y_E$ . The distance $\overline{DE}$ is then obtained as

$$\overline{DE} = \frac{\vert y_D - y_E\vert}{\vert\sin\theta\vert},$$

$$= \frac{2ab\sqrt{b^2\cos^2\theta + (a^2-c^2)\sin^2\theta }} {b^2\cos^2\theta + a^2\sin^2\theta}.$$ (10.3)

Two special cases of $c$ and $\theta$ are immediately verified. We have $\overline{DE}=2a$ when $\theta=0$ , and $\overline{DE}=2b$ when $c=0$ and $\theta=\pi/2$ , i.e., the lengths of the major and the minor axes of the ellipse, respectively.

Next, the parameters $a$ and $b$ of the ellipse are related to those defining the first Fresnel zone, as depicted in Figure 4.1a. Let $s$ and $g$ be the two foci of the ellipse, the distance between $s$ and $g$ be $L$ , and $E$ be an arbitrary point on the ellipse. The first Fresnel zone is delimited by points $E$ on the ellipse that satisfy

$$\overline{s E} + \overline{E g} = L + \frac{\lambda}{2}.$$ (10.4)

$$\noalign{Also, ~~~} \overline{s E} + \overline{E g} = 2a$$ (10.5)

is a property of the ellipse. Anther property of the ellipse relates the focal distance to the major and minor radii by

$$L = 2 \sqrt{a^2 - b^2}.$$ (10.6)

Equations E.4--E.6 give us

$$a = \frac{L}{2} + \frac{\lambda}{4},$$ (10.7)

$$\mathrm{and~~~} b = \sqrt{\frac{\lambda L}{4} + \frac{\lambda^2}{16}}.$$ (10.8)

In the limit of $L \gg \lambda$ , $b \rightarrow\frac{1}{2}\sqrt{\lambda L}$ .

From equations E.7, E.8, and E.3, we see that the resolution can be written as $\overline{DE}(L,\lambda,c,\theta)$ , a function of wavepath parameters $L, \lambda$ , and intersection parameters $c$ and $\theta$ .