The Optical Communications Reference

a divergent fan to the plane of the field stop positioned in

the focal plane of the eyepiece. Here the fans overlap and

interfere. The resulting fringe pattern with a constant

spacing, d

0

, constitutes an image of the initial grating of

the object plane and both are related through the system

magnification: d

0

/d ¼ V.

To create the fringe pattern at least two divergent

beams and therefore two diffraction orders must be

present simultaneously in the aperture stop. The location

of the diffraction maxima, F

i

, in the focal plane of the

objective is dictated by the diffraction grating equation

(Eq. (5.18); see Chapter 5): sin u

ðiÞ

max

¼ i

l

=d for i ¼ 0,

1, 2,. The aperture stop size, D

as

, is related to the

numerical aperture of the system in the object space:

n sin u

max

¼ D

as

=ð2f

0

Þ. These last two expressions allow

one to find the minimum diffraction spacing, d, which

can be imaged by the microscope.

Two possible methods of illumination should be con-

sidered separately: direct illumination when the zero-

order diffraction is focused in a point on the optical axis

and oblique illumination when the fo cus of the zero-

order diffraction is located in an off-axis point. Fig. 1.2.25

illustrates both situations. The limiting condition for

direct illumination requires that the zero-order maxi-

mum as well as the 1

st

and the ( 1)

st

order maxima are

inside the aperture stop whereas the corresponding limit

for oblique illumination can be realized if the zero-order

and only one of the first-order diffraction maxima are

inside the circle of diameter D

as

. As can be seen, the

limiting resolution is related to the numerical aperture

(NA) of the system as

d ¼

l

n sin u

max

¼

l

NA

; d ¼

l

2n sin u

max

¼

l

2NA

;

(1.2.35)

for direct (on-axis) and oblique illumination, re-

spectively. In the real practice of microscopy illumination

is supplied by a wide-angle condenser coming at both

direct and oblique directions. It can be shown that in

such a case the limiting resolution of the microscope is

determined as

d ¼

l

NA þ NA

C

; (1.2.36)

where NA

C

is the numerical aperture of the condenser.

Problems

P.1.2.15. Find the minimum required active diameter of

the well-correcte d imaging optics for visible wavelengths

operating at a working distance of 30 mm and providing

a resolution of 0.5 mm.

P.1.2.16. A microscope for the visible range is sup-

plied with three objectives: 10  0.25 NA, 40  0.65

NA, and 100  1.2 NA, and a condenser of 0.96 NA.

Find the maximum resolution in all three possibl e

configurations.

P.1.2.17. A microscope objective of magnification 10

has a focal length of 16 mm and is operated with an ap-

erture stop of 5 mm diameter. At which angle of oblique

illumination should one expect the resolution to be twice

that of normal (on-axis) illumination? Which resolution

(in the visible) will be available in this case and how will

the resolution be changed if the illumination angle is

held at 5



?

1.2.3 Image evaluation

Evaluation of images is carried out (i) at the design stage

when it is check ed whether the configuration designed is

capable of delivering the system performance re-

quirements; and (ii) at the end of manufacturing when

a real system with all the tolerances of component fab-

rication and assembling is aligned and prepared for final

testing. Image evaluation at the design stage is performed

theoretically, by analyzing aberrations of the system and

also by calculating some integral parameters enabling one

to estimate the expected image quality. Image evaluati on

at the manufacturing stage is done with special hardware

allowing one to measure resolution, contrast, and other

parameters rel ated to the system performance, usually

determined in a procedure specific for each tested

architecture.

Theoretical evaluation of image quality is usually

based on ray tracing of a great number of rays, originating

in on-axis and off-axis points of the object and, if nec-

essary, related to several representative wavelengths

(mostly, the lines C, D, and F) of the illuminating radi-

ation. Obviously computing is carried out with special

software allowing one to calculate and display the loca-

tion of the rays in the image plane (a spot diagram),

energy distribution in a spot, frequency response of the

system (modulation transfer function, see below), posi-

tion of the best focus, and other useful parameters.

Diffraction effects are also taken into account while

Fig. 1.2.25 Location of diffraction maxima in an aperture stop: (a)

on-axis illumination; (b) oblique illumination.

36

SECTION ONE Optical Theory