We can immediately recognize

(61) |

and

(62) |

The distances

OF = 12 cm OF ' = 18 cm

The image distance
(from O to the image point) is 18 + 18 = 36 cm. In this
example the image and object positions are in fact the
negative principal points
since *m* = -*x* '/*f*' = -18/18 = -1.*f* '*f* '- *f* = *n* 'R/(*n* '-
*n*) - R/(*n* ' - *n*)

For the first refracting surface,
*f* ' = (1.5)(10)/.5 = 30 cm,*f* = (1/1.5)(30) = 20 cm,*s* ' + 20/40 = 1*s*' = 60 cm.*s* = -50 cm.
For the second surface,
*f* ' = (1)(-10)/(-.5) = 20 cm,*f* = (1.5) 20 = 30 cm,
and
*s* ' + 30/(-50)= 1*s*' = 12.5 cm*m* = -(60/40)(-12.5)/(-50)= -3/8 as before.

The focal lengths are equal to . It is easy to show that

(63) |

if

Possibly the most useful equations of geometrical
optics are those for
image formation by a thin lens. A thin lens of
index µ, is shown in Fig. 24
with media of indices *n* and
*n* '*t*, of the lens is assumed to be
small compared to the focal
lengths of the lens such that we can set *t* = 0
in the translation matrix from
O to

or

(assuming O and

So,

or

(64) |

Equation (64) reduces to the well-known lens-makers equation when

(65) |

Newton's equations apply here using (64) for *f* and
*f* '.*s* and
*s* '*f* '/*s* '+ *f*/*s* = 1
*n* '/*s* ' + *n*/*s* = *n*/*f*
= *n* '/*f* '*s* + 1/*s* '= 1/*f* =1/*f* ' =
(µ - 1)(1/R_{1} - 1/R_{2}).*m* = - *ns* '/*n* '*s*
( = -*s* '/*s* if *n* ' = *n*)

The matrix representing a thin lens has
the particularly simple form

(66) |

The combination of two thin lenses in air, a distance d apart,
is shown
in Fig. 25. We may take the vertices of the
system as the lens centres O_{1} and
O_{2}.
The matrix is then, using (66) with *f*_{1} =
*f*_{1} ', and
*f*_{2} = *f*_{2} '

Therefore if

(67) |

The focal points are F and

(68) |

and

(69) |

using (23) and (24).

One simple special case of interest is that in which
two thin lenses
are cemented together to form a doublet. In this
case d = 0 and
1/*f* = 1/*f*_{1} + 1/*f*_{2}.
Such a doublet is usually constructed to reduce
longitudinal chromatic
aberration. The lenses are made of different
glasses which have different
dispersive powers (i.e. the indices of the glasses
depend differently on the
frequency of the light) and so the combination
may be designed so that the
focal length *f* will have a minimum dependence
on the frequency of the light
(Jenkins and White, p. 157).
It is also possible to reduce the dependence
of the focal length, *f*, on
the frequency when using two lenses of
the same index of refraction by
choosing a separation
d = (*f*_{1} + *f*_{2})/2.
This may be readily shown using (67)
and (65) and solving for a minimum
dependence of *f* on µ, the index of the
glass of the lenses (Jenkins and White, p. 163).