An Elementary Treatise on Optics/Chapter 3

by Henry Coddington

CHAP. III.

ABERRATION IN REFLEXION AT SPHERICAL SURFACES.

15.We found in the beginning of the last Chapter, that a cone of rays incident on a spherical surface were not so reflected as to meet in a point, but that the point $q$ which we called the focus of the reflected rays was, in fact, the mathematical limit of the intersections with the axis of rays chosen more and more nearly coincident with it. Let us now examine how much the intersection of the axis with a ray at a small but sensible distance from it, differs from this.

Taking the centre of the surface as the point to measure from, (Fig. 12.)

$\left.{\begin{aligned}\mathrm {Let} ~&q&~\mathrm {represent} ~&EQ\\&q'&~\ldots \ldots \ldots \ldots \ldots ~&Eq\\&f&~\ldots \ldots \ldots \ldots \ldots ~&EF\end{aligned}}\right\}~\mathrm {as~before,}$

and $v$ being the actual intersection of the reflected ray and axis,

let $q\backprime =Ev$.

Then since $q'$ is the limiting value of $q\backprime$ when $\theta =0$, we must have

$q\backprime =q'+{\frac {dq\backprime }{d\theta ^{2}}}.\theta +{\frac {1}{1.2}}{\frac {d^{2}q\backprime }{d\theta ^{2}}}.\theta ^{2}+....$

$\theta$ being made equal to $0$ in every differential coefficient, or if we consider $q\backprime$ as a function of the versed sine of $\theta$, which we will call $v,$

$q\backprime =q'+\left({\frac {dq\backprime }{dv}}\right)v+{\frac {1}{1.2}}\left({\frac {d^{2}q\backprime }{dv^{2}}}\right)v^{2}+....$

The brackets indicating that $v$ is made $=0$ in each coefficient.

${\begin{aligned}\mathrm {Now} ~q\backprime &={\frac {qf}{f+q\cos \theta }}={\frac {qf}{f+q(1-\operatorname {ver~sin} \theta )}}={\frac {qf}{q+f-qv}}\\{\frac {dq\backprime }{dv}}&={\frac {q^{2}f}{(q+f-qv)^{2}}}~\mathrm {which~when} ~v=0,~\mathrm {becomes} ~{\frac {q^{2}f}{(q+f)^{2}}}\\{\frac {1}{1.2}}{\frac {d^{2}q\backprime }{dv^{2}}}&={\frac {1}{1.2}}{\frac {2q^{3}f}{(q+f-qv)^{3}}}..................{\frac {q^{3}f}{(q+f)^{3}}},\end{aligned}}$

and so on, whence

$q\backprime ={\frac {qf}{q+f}}+{\frac {q^{2}f.v}{(q+f)^{2}}}+{\frac {q^{3}f.v^{2}}{(q+f)^{3}}}+{\frac {q^{4}.fv^{3}}{(q+f)^{4}}}+....$

This in geometrical terms amounts to

$Eq+{\frac {QE^{2}}{QF^{2}}}.{\frac {AN}{2}}+{\frac {QE^{3}}{QF^{3}}}.{\frac {AN^{2}}{4EF}}+....$

The Aberration is represented by this series without its first term; and when the angle $\theta$, and a fortiori its versed sine, are but small, the second term of the series will give a near approximate value.

Note. The above is perhaps the neatest way of obtaining the series for the aberration: it is sometimes done by a method simpler in its principle

${\begin{aligned}q\backprime ={\frac {qf}{q+f-qv}}&={\frac {qf}{q+f}}\left(1-{\frac {qv}{q+f}}\right)^{-1}\\&={\frac {qf}{q+f}}\left\{1+{\frac {qv}{q+f}}+\left({\frac {qv}{q+f}}\right)^{2}+....\right.\\&={\frac {qf}{q+f}}+{\frac {q^{2}f.v}{(q+f)^{2}}}+{\frac {q^{3}fv^{2}}{(q+f)^{3}}}+....\end{aligned}}$

16. There is another series a little different from this, but which amounts nearly to the same thing.

If we take the letter $\alpha$ to represent the aberration,

${\begin{aligned}\alpha =q\backprime -q\prime &={\frac {qf}{f+q\cos \theta }}-{\frac {qf}{f+q}}\\&={\frac {qf}{f+q}}\left({\frac {f+q}{f+q\prime \cos \theta }}-1\right)\\&={\frac {qf}{q+f}}.{\frac {q-q\cos \theta }{f+q\cos \theta .}}\\&={\frac {qf}{q+f}}.{\frac {q\operatorname {ver~sin} \theta }{(f+q)\cos \theta +f\operatorname {ver~sin} \theta }}\\&={\frac {q^{2}f.v}{(q+f)^{2}.\cos \theta }}.\left\{1+{\frac {fv}{(q+f)\cos \theta }}\right\}^{-1}\\&={\frac {q^{2}f.v}{(q+f)^{2}.\cos \theta }}\left\{1-{\frac {fv}{(q+f)\cos \theta }}+{\frac {f^{2}v^{2}}{(q+f)^{2}\cos \theta ^{2}}}+\ldots \right.\\&={\frac {q^{2}}{(q+f)^{2}}}f.{\frac {\operatorname {ver~sin} \theta }{\cos \theta }}-{\frac {q^{2}}{(q+f)^{3}}}f^{2}.{\frac {\operatorname {ver~sin} \theta ^{2}}{\cos \theta ^{2}}}+....\end{aligned}}$

Now ${\frac {\operatorname {ver~sin} \theta }{\cos \theta }}={\frac {1-\cos \theta }{\cos \theta }}=\sec \theta -1$; so that $f$ being half the radius, $f{\frac {\operatorname {ver~sin} \theta }{\cos \theta }}$ is the algebraical value of half the excess of the secant over the radius, and if $RT$ be drawn touching the surface at $R$, we shall have the following series in geometrical terms,

$\alpha ={\frac {QE^{2}}{QF^{2}}}.{\frac {AT}{2}}-{\frac {QE^{3}}{QF^{3}}}.{\frac {AT^{2}}{4QE}}+{\frac {QE^{4}}{QF^{4}}}.{\frac {AT^{3}}{8QE^{2}}}-....$

The first term of which differs but little from that found before, as $AT$ or ${\frac {\operatorname {ver~sin} \theta }{\cos \theta }}$ differs but little from $\operatorname {ver~sin} \theta$ when $\cos \theta$ is so nearly equal to $1.$

When the incident rays are parallel, the aberration is ${\frac {f}{\cos \theta }}-f$ or 1/2$AT$ accurately.

If $\theta =30^{\circ }$, $\alpha =\left({\frac {2}{\sqrt {3}}}-1\right)f;$ for $\theta =45,$ $\alpha =({\sqrt {2}}-1)f;$ for $\theta =60,$ $\alpha =f;$ so that the point $v$ coincides with $A.$

 

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