Measuring image focusing for velocity analysis

Curvature correction

This appendix derives the expression for the curvature correction presented in the main text in 2. The derivation is extremely simple and based on the geometry sketched in Figure 21.

The reflector is approximated with a parabola with radius of curvature ${R}$ at its vertex. In the rotated coordinates system $\left(z',x'\right)$ the equation of the parabola is

$$z'=\frac{{x'}^2}{2{R}}.$$ (A-1)

The shift $\Delta z'$ that moves a tangent to the parabola to the vertex is equal to

$$\Delta z'=\tan^2\alpha'\frac{R}{2},$$ (A-2)

and consequently the normal shift $\Delta n$ is equal to

$$\Delta n= \frac{\cos\alpha'\tan^{2}\alpha'}{2}{R} =\frac{\sin\alpha'\tan\alpha'}{2}{R}.$$ (A-3)

The coordinate system $\left(z',x'\right)$ is rotated by $\bar{\alpha}$ with respect to $\left(z,x\right)$. Removing that rotation is equivalent to set $\alpha'=\alpha -\bar{\alpha}$; performing this substitution in the previous equation, I obtain the correction in 2; that is,

$$\Delta n= \frac{\sin\left(\alpha -\bar{\alpha}\right) \tan\left(\alpha -\bar{\alpha}\right)} {2} {R}.$$ (A-4)

Curv-corr Figure 21. Sketch used to derive the curvature correction presented in 2. The tangent to the parabola (dashed line) needs to be shifted by $\Delta n$ to pass through the vertex of the parabola. [NR]

Measuring image focusing for velocity analysis