= Bessel focal length uncertainty
{c}
{title2=$\sigma_f^2\simeq A^2\sigma_L^2+B^2\sigma_d^2+2AB\operatorname{Cov}(L,d)$}
For small <measurement errors>, write $f=L/4-d^2/(4L)$. Its <partial derivatives> and first-order <propagation of uncertainty> are
$$
A=\frac{\partial f}{\partial L}=\frac14\left(1+\frac{d^2}{L^2}\right),\qquad B=\frac{\partial f}{\partial d}=-\frac{d}{2L},
$$
$$
\sigma_f^2\simeq A^2\sigma_L^2+B^2\sigma_d^2+2AB\operatorname{Cov}(L,d).
$$
Here the $\sigma$ values are <standard deviations> used to describe uncertainty, and $\operatorname{Cov}(L,d)$ is the <covariance>. Independent carrier-reading errors give $\sigma_d^2=\sigma_{r_-}^2+\sigma_{r_+}^2$; correlated readings require the term $-2\operatorname{Cov}(r_-,r_+)$. Averaging repeated independent pairs reduces the random component of the uncertainty of the mean, but does not remove calibration or model errors.
As a stipulated uncertainty calculation, let $L=45\,\mathrm{cm}$, $d=15\,\mathrm{cm}$, $\sigma_L=0.10\,\mathrm{cm}$ and $\sigma_{r_-}=\sigma_{r_+}=0.05\,\mathrm{cm}$, with all three input errors independent. Then $\sigma_d=0.0707\,\mathrm{cm}$, $A=5/18$, $B=-1/6$ and $\sigma_f\simeq0.0302\,\mathrm{cm}$. The result would be $f=10.000\,\mathrm{cm}$ with this standard uncertainty in the stipulated <thin lens> model; these numbers are not a measured uncertainty budget. A small neglected principal-plane separation gives an additional first-order bias $f_{\rm thin}-f\simeq Ah$, distinct from that random budget.
\Image[/bessel-lens-position-focal-length-sensitivity.png]
{title=<Bessel lens displacement method>: merging settings and <focal length> sensitivity}
{description=Original ideal-model curves. Left: settings versus object-screen separation for fixed $f=10\,\mathrm{cm}$; below $L=40\,\mathrm{cm}$ there is no real setting on the fixed screen. Right: inferred <focal length> versus measured displacement at fixed $L=45\,\mathrm{cm}$, with its tangent at the stipulated $d=15\,\mathrm{cm}$. The curves are predictions, not experimental points.}
The two kinds of sensitivity must be distinguished. At fixed <focal length>,
$$
\frac{\partial d}{\partial L}=\frac{L-2f}{d},\qquad \frac{\partial x_\pm}{\partial L}=\frac12\left(1\pm\frac{L-2f}{d}\right).
$$
These <derivatives> diverge as $L$ approaches $4f$ from above: a small change in object-screen spacing strongly changes the separation of the two settings. Yet the inferred <focal length>'s displacement sensitivity $|B|=d/(2L)$ tends to zero. That does not make coalescing settings easy to measure. For fixed $L$, define the axial mismatch between the predicted <real image> and the screen as $g(x)=fx/(x-f)-(L-x)$, for $x>f$. At a focused setting,
$$
g'(x)=1-\frac{f^2}{(x-f)^2}=1-m^2.
$$
Near $u=v=2f$, moving the <optical lens> changes this mismatch only weakly, making visual discrimination of the two settings difficult. At the example's settings $g'(15\,\mathrm{cm})=-3$ and $g'(30\,\mathrm{cm})=3/4$, so equal carrier motion produces different axial <defocus> responses. This is an axial mismatch calculation, not a full prediction of perceived image sharpness.
Choose $L$ sufficiently above $4f$ to separate the two accepted focusing ranges, while retaining both <real images> on the available screen and rail. There is no universal best $L$: focusing uncertainty itself depends on <aperture>, <transverse magnification>, <optical aberrations> and the target. Close to $d=0$, the nonnegative measured separation and the quadratic term in $d$ also make first-order <propagation of uncertainty> unreliable if its error is comparable to its value. Repeatability, controls and the <principal plane> correction determine whether the elegant two-distance formula is an adequate measurement model.
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