Question #43225

A non-achromatic microscope objective of diameter $D$ is used to image an object. Due to longitudinal chromatic aberration, the inverse of the image distance, $k = 1/v$, is uniformly distributed in the range $[k_1, k_2]$ for the illuminating broad-spectrum light. A sensor is placed at a distance $L$ from the objective. For a specific $k$, the geometric blur diameter on the sensor is $d_c = D|L k - 1|$. The diffraction blur diameter is approximated by the Rayleigh criterion as $d_d = \frac{1.22 \lambda_0 L}{D}$, where $\lambda_0$ is the mean wavelength. Assuming the total spot size squared is $S^2 = d_c^2 + d_d^2$, find the optimal sensor distance $L$ that minimizes the root-mean-square (RMS) total spot size across the spectrum. Let $\gamma = \frac{1.22 \lambda_0}{D^2}$.