Spherical Mirrors — Question 4

Question 4

An exact spherical ray intercept depends on ray height

A concave spherical mirror has vertex at x=0 and center at x=-R, with R positive. A parallel ray travels in +x and strikes at $x_P=-R+R\cos\theta$, $y_P=R\sin\theta$, with zero less than theta less than 45 degrees. Reflection gives direction $(-\cos2\theta,-\sin2\theta)$.

1. Find the axial intersection coordinate of the reflected ray.

2. Express its positive vertex distance q(theta) and take the paraxial limit.

Question 4 - Solution

An exact spherical ray intercept depends on ray height

(a) Method and calculation

$$x_I=x_P-y_P\cot2\theta=-R+R\cos\theta-R\sin\theta\cot2\theta.$$$$\cos\theta-\sin\theta\cot2\theta=1/(2\cos\theta)\Rightarrow\boxed{x_I=-R+R/(2\cos\theta)}.$$

(b) Interpretation and further calculation

$$\boxed{q(\theta)=-x_I=R[1-1/(2\cos\theta)]}.$$$$\boxed{\lim_{\theta\to0}q(\theta)=R/2=f}.$$

Physical check

Finite-height rays do not all cross at the paraxial focus; the restriction on theta keeps the reflected ray directed back toward the axis.