= Two-stage electron imaging
{title2=$m=m_1m_2$}
Consider two ideal thin <electron lenses> in successive imaging stages. In each stage choose positive object and image distances $u_i,v_i$ for a <real image>, measured from its effective lens plane, and neglect the image rotation of a <magnetic electron lens>. The <paraxial approximation> gives
$$
\frac{1}{f_i}=\frac{1}{u_i}+\frac{1}{v_i},\qquad m_i=-\frac{v_i}{u_i},\qquad m=m_1m_2.
$$
The product follows directly by writing $y_1=m_1y_0$ and $y_2=m_2y_1$. Here $m_i$ is <transverse magnification>, rather than the enlargement applied later to a photograph. Both inversions cancel in this planar model; actual <magnetic electron lenses> can additionally rotate the image.
For an original worked example, take $u_1=40\,\mathrm{mm}$, $v_1=80\,\mathrm{mm}$, $u_2=50\,\mathrm{mm}$ and $v_2=150\,\mathrm{mm}$. Then
$$
f_1=\frac{80}{3}\,\mathrm{mm},\qquad f_2=37.5\,\mathrm{mm},\qquad m_1=-2,\qquad m_2=-3,\qquad m=6.
$$
A $100\,\mathrm{\mu m}$ object interval becomes $600\,\mathrm{\mu m}$ at the screen. The second lens sits $v_1+u_2=130\,\mathrm{mm}$ after the first. These are calculated distances, not measurements of the 1931 instrument.
\Image[/electron-lens-two-stage-imaging.png]
{title=Two <electron lenses> relay an object into an enlarged screen image}
{description=Original <paraxial approximation> construction for the worked example, in a frame neglecting magnetic image rotation. Three illustrative trajectories from one off-axis point meet at the intermediate image and again at the screen. The shaded bands represent effective magnetic lens regions; real trajectories bend smoothly. The dashed enclosure represents a <vacuum> column, with an upstream <electron gun> and a pump connection indicated schematically. Horizontal and vertical scales differ; neither the geometry nor the rays reproduce historical apparatus measurements.}
{source=electron-lens-two-stage-imaging.py}
To measure <magnification>, divide a screen interval between corresponding grid features by the independently known object interval. Repeat over several periods to reduce endpoint uncertainty. For a proposed check, record accelerating <voltage>, lens <electric currents>, screen position and specimen orientation; compare one-stage and two-stage images, refocusing as necessary. Changing a lens <electric current> should change focus and scale, while a blank exposure checks detector background. Keep irradiation and recording enlargement controlled, and repeat the original setting to test drift or specimen damage. These are suggested checks, not a claim that every check was performed in 1931. A grid-scale measurement establishes <magnification>; resolving much finer features requires a separate target and contrast criterion.
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