Page 85 - Fundamentals of Light Microscopy and Electronic Imaging

68 DIFFRACTION AND INTERFERENCE IN IMAGE FORMATION

aperture angle and light-gathering ability. As shown in this chapter, the spatial resolu-
tion of the microscope is limited by the smallest disk that it is possible to obtain by vary-
ing λ and NA. Only when the images of specimen details subtend diameters greater than
this limit can you begin to obtain information regarding the size, shape, and periodicity
of the object.

CONSTANCY OF OPTICAL PATH LENGTH
BETWEEN THE OBJECT AND THE IMAGE

Before we examine the effect of lens diameter on diffraction spot size and spatial reso-
lution, we need to consider the concept of optical path length for an imaging system
containing a perfectly corrected lens. In practice, microscope objectives and other cor-
rected lenses only approximate the condition of being perfectly corrected, and waves
arrive at the conjugate image point somewhat out of place and out of phase. This is usu-
ally not a serious problem. Despite practical limitations in finishing lenses with spheri-
cal surfaces, most microscope lenses give diffraction-limited performance with an
average error in phase displacement (wavefront error) of less than /4, and manufactur-
ers strive to obtain corrections of /10 or better. As is known, light from a self-luminous
point in an otherwise dark specimen plane radiates outward as an expanding spherical
wavefront; waves collected by the objective are refracted toward the center line of the
lens and progress as a converging spherical wavefront to a single point in the image
plane. However, it is also true—and this point is critical for image formation—that the
number of vibrations as well as the transit time experienced by waves traveling between
an object point and the conjugate image point are the same regardless of whether a
given wave passes through the middle or the periphery of the lens.
Ideally, all waves from an object point should arrive at the image point perfectly in
phase with one another. Given the large differences in the spatial or geometric path
lengths between central and peripheral waves, this seems improbable. The explanation
is based on the concept of optical path length, a length distinct from the geometric path
length. It can be used to calculate the number of vibrations experienced by a wave trav-
eling between two points. As it turns out, variations in the thickness of the lens com-
pensate for the differences in geometric paths, causing all waves to experience the same
number of vibrations (Fig. 5-6). It can also be shown that the transit time required for
light to travel between object and image points along different trajectories having the
same optical path length is the same. These concepts become important when we dis-
cuss the spatial and temporal coherence of light later in the chapter.
In optics, the optical path length (OPL) through an object or space is the product of
the refractive index n and thickness t of the object or intervening medium:

OPL nt.

If the propagation medium is homogeneous, the number of vibrations of a wave of
wavelength λ contained in the optical path is determined as

Number of vibrations nt/λ.

Since the frequency of vibration remains constant and the velocity of light c/n,
when a wave traverses a lens of higher refractive index than the surrounding medium,