Images from Spherical Mirrors — Question 6

Question 6

The center-of-curvature conjugate is symmetric only to first order

Use the paraxial mirror relations $1/p+1/q=1/f$ and $m=-q/p$, with p positive for a real object, q positive for a real image, and f positive for a concave mirror. A concave mirror has focal length f. Move an axial object from p=2f to p=2f+delta, with the magnitude of delta much smaller than f.

1. Derive the exact new image distance and its expansion through delta squared.

2. Find the leading magnification change and explain whether equal and opposite object-image shifts remain exact.

Question 6 - Solution

The center-of-curvature conjugate is symmetric only to first order

(a) Method and calculation

$$q=\frac{f(2f+\delta)}{f+\delta}=f+\frac{f^2}{f+\delta}.$$$$\boxed{q=2f-\delta+\delta^2/f+O(\delta^3/f^2)}.$$

(b) Interpretation and further calculation

$$m=-\frac{f}{p-f}=-\frac{f}{f+\delta}=\boxed{-1+\delta/f+O(\delta^2/f^2)}.$$$$\boxed{(q-2f)+\delta\simeq\delta^2/f\ne0}.$$

Physical check

The center-of-curvature arrangement gives unit-magnitude inversion, but finite displacements break the apparent first-order symmetry.