Thin Lenses — Question 9

Question 9

A lateral lens displacement shifts the image by a magnification-dependent factor

A thin lens with focal length f forms a real image of an on-axis object at distances p and q. Shift the lens center upward by a small amount a without tilting it; keep object and image planes fixed. For a ray at laboratory height y, use incident slope y/p and lens deflection -(y-a)/f.

1. Derive the final image height after the lens shift.

2. Express the result using transverse magnification m=-q/p and evaluate it for m=-2 and a=0.50 mm.

Question 9 - Solution

A lateral lens displacement shifts the image by a magnification-dependent factor

(a) Method and calculation

$$u_{\mathrm{out}}=y/p-(y-a)/f,\quad y_I=y+qu_{\mathrm{out}}.$$$$y_I=y(1+q/p-q/f)+qa/f=\boxed{qa/f}.$$

(b) Interpretation and further calculation

$$q/f=1+q/p=1-m\Rightarrow\boxed{y_I=(1-m)a}.$$$$\boxed{y_I=[1-(-2)](0.50)=1.50\,\mathrm{mm}}.$$

Physical check

The coefficient of entrance height cancels by the lens equation, so all paraxial rays still meet at the shifted image point.