= Bessel lens displacement experiment
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KIT's teaching apparatus uses a glass spoke-wheel or test negative as the object, a halogen lamp with an illuminating <optical lens>, an imaging <optical lens>, a screen and a triangular rail with a position scale. The illuminating <optical lens> concentrates the lamp filament's image on the imaging <optical lens>'s centre; translucent paper checks that illumination. This reduces the contribution of peripheral <light rays> and their <spherical aberration>. The instructions require fixed object-screen separation exceeding $4f$, at least ten displacement measurements, and separate red- and blue-filter measurements to test <chromatic aberration>.<Brennweite dünner Linsen by Karlsruhe Institute of Technology, Faculty of Physics teaching laboratory>[Karlsruhe Institute of Technology, Brennweite dünner Linsen, Durchführung and Aufgaben 1a and 2: target, illumination check, rail readout, repeated Bessel measurements and colour controls]{ref}
The following measurement protocol elaborates that apparatus rather than reporting additional KIT results. Align the target centre, <optical lens> and screen normal along one <optical axis>; measure $L$ between the target pattern plane and receiving screen face. Lock the target and screen. Slide the imaging <optical lens> to obtain the enlarged <real image>, then the reduced <real image>, reading the same carrier reference mark at each setting. Record $r_-,r_+$ and $d=r_+-r_-$ for each pair. Record corresponding target and image intervals to test the reciprocal <transverse magnifications> independently; merely observing different image sizes is a weaker test.
Use the same central target detail to judge best <optical focus>. Bracket each setting by moving through both visibly blurred sides, repeat from both travel directions, and record the accepted position range as well as repeat scatter. Rail divisions set a readout limit but do not alone specify focusing uncertainty. Check scale calibration, carrier play and drift; a common additive carrier offset cancels in $d$, whereas scale-factor error remains. Keep illumination, <aperture>, target, screen and <optical lens> orientation fixed within a series. Change the <aperture> and colour filter in separate control series: <spherical aberration> and <chromatic aberration> can shift the apparent best <optical focus>, while a very small <aperture> introduces stronger <diffraction> and makes focus discrimination harder. Record these settings alongside the inferred <focal length>, rather than interpreting all disagreement as random <measurement error>.
For a rigid <thick lens> in air, let $h=x_{H'}-x_H$ be the signed separation of its rear and front <principal planes> along the direction of propagation. Then the positive object and image distances measured from the <principal planes> obey $u+v=L-h$. Translating the whole assembly without reversing it moves both <principal planes> equally, so the measured carrier displacement still equals the difference of the two solutions for $u$. The same algebra therefore gives
$$
d^2=(L-h)^2-4f(L-h),\qquad f=\frac{(L-h)^2-d^2}{4(L-h)},\qquad L-h>4f.
$$
Here $f$ is the effective <focal length> measured from the <principal planes>. This deduction uses the principal-plane imaging equation, not the uncorrected <thin lens> formula.<Refracting systems (details), University of Tennessee physics teaching resource>[Refracting systems (details), principal-plane definition and final paragraph on object/image distances]{ref} Mechanical centre thickness is not generally $h$. Record thickness and <optical lens> orientation, but use a justified bound or model for $h$ if it matters at the claimed precision. The two readings $L,d$ alone cannot determine both $f$ and unknown $h$; cancelling a carrier offset does not cancel this limitation.
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