The Optical Communications Reference

Using a target bar code with several groups of well-

defined spatial frequencies as the system object and

measuring the modulation M

i

of the corresponding

images at the system output allows one to find the

MTF (see Eq. (1.2.43)). This method is commonly

exploited in image qu ality evaluation at the final testing

stage. The limiting resolution of the system is defined

as the spatial frequency of the group still visible at the

image plane with a minimum contrast of 3–5% (which

is c onsidere d as the limit of the perception capability

of a human eye).

Problems

P.1.2.18. What could be concluded about imaging optics

if an analysis of the encircled energy revealed 45% of the

total energy of the spot corresponding to an on-axis image

point is inside the circle diameter which is half the size of

the whole spot?

P.1.2.19. Imaging optics operated with mono-

chromatic illumination of 0.6 mmandhavingNA¼

0.25 in the object space is ended by a CCD area sensor.

What should be the minimum pitch of the CCD in

order to acquire all s patial frequencies transferred by

the optics?

P.1.2.20. MTF measurements are carried out with

a square-wave ta rget of variable frequencies made of

chrome on glass in a bar code pattern. Data are col-

lected for low spatial frequency (v

1

¼ 10 cycles/mm)

and for high spatial freque ncy (v

2

¼ 200 cycles/mm)

and the respective meas ured modulations are 70% and

20%. Assuming that the reflectance of chrome is 70%

and the reflectance of glass is 4% and also keeping in

mind that the contrast of images is slightly degraded by

the background light scattered inside the measurement

set-up, find the true MTF value at higher spatial

frequency.

P.1.2.21. A diffraction-limited optical system oper-

ated in the visible range and having NA ¼ 0.15 creates an

image on a CCD sensor followed by a video monitor. The

MTF of CCD þ monitor is 60%. Could we expect to see

on the screen the tiny details of an object corresponding

to the spatial frequency of 575 cycles/mm?

1.2.4 Two special cases

1.2.4.1 Telecentric imaging system

This kind of archi tecture is usually exploited in mea-

surement systems where errors caused by the third

dimensions (along the optical axis) of an object have to

be minimized. To explain this error (sometimes called

the parallax error, or perspective error) we refer to

Fig. 1.2.30a where simple imaging with a single lens i s

depicted. Two objects, O

1

AandO

2

B, having the same

height and located at different distances from the lens,

after imaging are transformed into images O

0

1

A

0

and

O

0

2

B

0

of different heights. The error Dy

0

might cause

problems if the defocusing Dx

0

is small (not revealed by

the system observer). The telecentric imaging system

shown in Fig. 1.2.30bisfreeofthiserror.Thesystemis

configured as an a focal lens pair where the back focal

plane of the first lens coincides with the front focal

plane of the second. What is also important is that the

aperture stop ab is located in this plane P. As a result,

Fig. 1.2.30 Imaging with (a) parallax error and (b) the telecentric configuration.

Fig. 1.2.29 (a,c) Input square waves and (b,d) the corresponding

output patterns.

39

Theory of imaging CHAPTER 1.2