2.2. Diffraction Effects and Resolution 61

of BK-7 glass. Find the optimal shape of the condenser lenses and estimate the

spherical aberration at the bundle entrance.

2.2. Diffraction Effects and Resolution

2.2.1. General Considerations

Diffractioneffectsresult from the wave nature of radiation participating in imaging.

In general diffraction is caused by the secondary waves generated in the substance

of an obstacle on which electromagnetic waves impinge while traveling in space.

An obstacle can be a body of any shape, either transparent or opaque. Interference

of the secondary waves changes the spatial distribution of the propagated radiation

in such a way that light energy appears not only in the direction of the initial

propagation but also to the side of it. Because of this, for example, an ideal lens

with no aberration is not capable of concentrating light in a single point of the

image plane and some energy is always revealed in a small but ﬁnite vicinity of

the image. Thus, diffraction is a basic limitation in imaging optics which cannot

be avoided. Other effects, like aberrations considered in the previous section,

which also “spoil” the image quality appear together with diffraction and cannot

neutralize it in any way. If all other effects become negligible diffraction remains

a single factor affecting the system performance. In such a case the optical system

is termed diffraction limited.

Diffraction occurs at any stop through which light passes. It could be a real

aperture, or the mounting of a lens, prism, or mirror, or just the boundaries of

an optical element of the system. We shall consider a simple case of propagation

of monochromatic light of wavelength λ through a circular non-transparent stop

of radius a followed by a lens (see Fig. 2.21). It can be shown that the intensity

Figure 2.21 Diffraction on (a) a circular stop and (b) the intensity distribution in the

diffraction spot.

62 2 ♦ Theory of Imaging

distribution of light in the spot created in the image plane P due to diffraction is

governed by the following function (Airy’s function):

I(r) = I

0

2J(x)

x

2

; where x =

2π

λ

n

r

sin u

max

, (2.31)

n

is the refractive index in the image space, r

is the radial coordinate in the

plane P, u

max

is the maximum angle of the direction from the stop boundary to the

center of the spot, and J

1

(x) is the Bessel function of the ﬁrst order.

Expression (2.31) is an oscillating function with a strong central maximum

followed by dark and light rings of decreasing intensity. It is commonly accepted

that most of the energy of the spot is concentrated in the central maximum limited

by the ﬁrst dark ring which corresponds to the value x

(1)

min

= 3.8317 in Eq. (2.31).

Hence, the relevant size of the spot in the plane P obeys the relation

δ

dif

=

1.22λ

n

sin u

max

. (2.32)

In the case when P is the focal plane of a lens of diameter D = 2a, Eq. (2.32) is

transformed into the well-known expression

δ

dif

=

2.44λ

D

f

(n

= 1).

The diffraction spot has a direct impact on limiting resolution which is one of the

basic features of any imaging system. Consider two very close images in the plane

P, each one generating a diffraction spot. If the distance between the two images

is large enough the spots are well separated and an observer looking on the image

plane P is capable of perceiving them easily. The smaller the distance, the closer

the spots, and at some stage they become overlapped. The question is, what is the

minimum distance at which two partially overlapping spots are still recognized as

two separate objects? Such a minimal distance is called the limiting resolution and

it is deﬁned, according to the Rayleigh criteria, as the situation when the minimum

of one spot coincides with the maximum of the second. Figure 2.22 demonstrates

the situation when two images, one centered at point A

and the other centered

at B

, are still resolvable. The dotted line in Fig. 2.22b shows the distribution of

energy after summation of both spots. The “valley” between the two maxima is

about 70% of the maximum intensity (i.e., about 30% reduction of energy).

What is usually important in practical applications is the distance in the object

plane between two points A and B corresponding to limiting resolution in the

image plane. Referring to Fig. 2.22a, suppose an entrance pupil of size D

p

is

located at a distance p from the object plane. Taking into account that the product

n ×sin u ×r is the system invariant (it remains constant while transferring through

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