The Optical Communications Reference

the entrance pupil and the exit pupil are both located

atinfinity(oneontheobjectspacesideandtheother

on the image space side) and the chief rays originating

in points A and B are parallel to the optical axis in both

spaces. Hence the images O

0

1

A

0

and O

0

2

B

0

are of the

same size and the parallax error does not occu r.

If the system is operated with a video area s ensor

(like a CCD) it should be positioned in suc h a way that

both images are sharp enough. Aberration of defocusing

(like the other aberrations) strongly depends on the

active lens size, but a reduction in the lens diameter is

accompanied by the increasing impact of diffraction, as

discussed in Section 1.2.2, and also a decrease in the

image illumination. Therefore, a compromise should be

found. In any case, symmetrical configurations are pre-

ferred where the shapes of the lenses are equally posi-

tioned with regard to the plane P (or one of them is

scaled in a symmetrical manner, if magnification/mini-

fication is required). Estimation of aberrations can be

carried out by the method described in Se ction 1.2.1.6

and Problem P.1.2.14.

1.2.4.2 Telephoto lens

There are numerous situations where the effective focal

length of the objective has to be long while the actual

size of the l ens should be kept as small as possible. A

possible architecture in such a case is a two-lens con-

figuration, one of positive and the other of negative

optical power (see Fig. 1.2.31). Usually what is known is

the equivalent focal length, f

0

e

, and the desired length of

the configuration, l. The optical power of each compo-

nent and their locations with regard to the image plane

should be found.

Considering the system in terms of first-order optics

(paraxial approximation) we have for this two-lens

system (see Problem P.1.1.7)

F

¼ 1=f

0

e

¼

F

1

þ

F

2



F

1

F

2

d; (1.2.46)

and taking into account that d þ S

0

F

¼ l we also get

F

1

d ¼ 1 

F

ðl  dÞ; (1.2.47)

Equations (1.2 .46) and (1.2.47) for three unknowns,

F

1

, F

2

,andd, allow one to introduce an additional

condition to o ptimize the configuration with regard to

aberration. This could be either the requirements for

a minimum optical power of the second element

(which in general might resu lt in lower residual aber-

rations) or the requirements for a configuration with

minimal (better zero) curvature of the image surface.

Inthefirstcasethebestresults,ascanbeshown,are

obtained with d ¼ 0.5l and the corresponding focal

lengths of the elem ents are

f

0

1

¼

lf

0

e

2f

0

 l

e

; f

0

2

¼

l

2

4ðf

0

 lÞ

: (1.2.48)

In the second case a zero Petzval’s sum (see Section

1.2.1.4) is required which is achieved with F

2

¼F

1

.By

introducing this condition in Eqs. (1.2.46) and (1.2.47)

we have

l ¼ 0:75f

0

e

; f

0

1

¼f

0

2

¼ 0:5f

0

e

; S

0

F

¼ 0:5f

0

e

:

(1.2.49)

The latter approach is widely used in the design of tele-

photo lenses intended for imaging in large angular fields

of vi ew.

Problems

P.1.2.22. How does one design a telecentric imaging

system which is operated at magnification V ¼3inan

angular field of view of 5



and provides a resolution of

2 mm in the visible spectral interval?

[Note: Assume the system is free of aberration.]

P.1.2.23. A telephoto lens forms images with negligi-

ble curvature at a distance of 60 mm from the first

(front) element. What are the focal lengths and the dis-

tance between the lenses?

1.2.5 Solutions to problems

P.1.2.1. Since the lens is working in the paraxial range

( f#¼ 10) we can find the distance to the plane P where

an ideal image is formed by the lens with nominal focal

length:

1  V

S

0

¼

1

f

0

; S

0

¼ 100 ð1 þ 2Þ¼300 mm:

Fig. 1.2.31 Configuration of a telephoto lens.

40

SECTION ONE Optical Theory