The Optical Communications Reference

(b) penta-prism with the same size a of the entrance

face: t

e

¼ a(2 þ O2);

(c) Dove prism of height a and 45



angles between

faces: t

e

¼ 3.035a.

Once t

e

is known, the apparent thickness in air is calcu-

lated from t

e

/n.

Problems

P.1.1.14. Imaging in systems of plane mirrors. An object

AB is positioned as shown in Fig. 1.1.30 in front of

a mirror corner of 45



. Find the location of the image

beyond the mirrors.

P.1.1.15. Find the reduced (apparent) thickness of

a45



rhomboidal prism of 2 cm face length. The prism is

made of BK-7 glass (n ¼ 1.5163).

P.1.1.16. Alens L of 30 mm focal length transfers the

image of an object AB positioned 40 mm in front of L to

a screen P. A penta-prism with 10 mm face size is

inserted 20 mm beyond the lens. Find the location of the

screen P relative to the prism if it is made of BK-7 glass

(n ¼ 1.5163).

P.1.1.17. Dispersive prism at minimum deviation. Find

the minimum deviation angle of a prism with vertex

angle b ¼ 60



. The prism is made of SF-5 glass with re-

fractive index n ¼ 1. 6727 .

1.1.5. Solutions to problems

P.1.1.1. We are looking for a solution in the paraxial range

and assume the lens is of negligible thickness. To find the

image of point A we use two rays emergi ng from A: ray 1

parallel to the optical axis and ray 2 passing through the

center of the lens (Fig. 1.1.31). Ray 1 after passing through

the lens goes through the back focus F

0

. Ray 2 does not

change its direction and continues beyond the lens along

the incident line. The intersection of the two rays after the

lens creates the image A

0

of point A. Once the image A

0

is

found, the image O

0

of point O is obtained as the in-

tersection of the norma l from point A

0

to the optical axis.

It should be noted that instead of ray 1 or 2 one can

use ray 3 (dotted line) going through the front focus F in

the object space (in front of the lens) and parallel to OX

after the lens. Intersection with the two other rays

occurs, of course, at the same point A

0

. Also note that in

our approximation of the paraxial range the homocentric

beam also remains homocentric in the image space.

P.1.1.1.2. In both cases, Figs. 1.1.32a and b, we draw

the ray (dotted line) parallel to AB and passing through

the center of the lens. The ray crosses the back focal plane

at point C. Since the ray and AB belong to the same par-

allel oblique bundle and all rays of such a bundle are col-

lected by the lens in a single point of the back focal plane,

this must be po int C. Therefore, the ray AB after passing

through the lens goes from B through C to point A

0

at the

intersection with the axis. This point is the image of A. In

the case of Fig. 1.1.32b the focus F

0

and corresponding

back focal plane are located to the left of the lens. Hence,

not the ray itself but its continuation passes through point

C. The intersection with OX is still the image of the point

source A which becomes virtual in this case.

P.1.1.3. First we will derive the ray tracing formula

valid for the paraxial approximation. By mult iplying both

Fig. 1.1.29 Unfolded diagram for (a) the right-angle prism, (b) the

penta-prism, and (c) the Dove prism.

Fig. 1.1.30 Problem P.1.1.14 – Imaging in a mirror corner.

Fig. 1.1.31 Problem P.1.1.1 – Graphical method of finding the

image.

14

SECTION ONE Optical Theory