Page 86 - Fundamentals of Light Microscopy and Electronic Imaging

EFFECT OF APERTURE ANGLE ON DIFFRACTION SPOT SIZE 69

Image
Object

Figure 5-6
Preservation of the optical path length between the object and image. The optical path length
may be regarded as the number of cycles of vibration experienced by a wave between two
points. Two waves traveling in phase from a point in an object and entering the center and
periphery of a lens cover different physical distances, but experience the same optical path
length, and therefore arrive simultaneously and in phase at the conjugate point in the image
plane. This occurs because the wave traveling the shorter geometric path through the middle
of the lens is relatively more retarded in its overall velocity, owing to the longer distance of
travel in a high-refractive-index medium (the glass lens). Note that the total number of cycles
(the number of wavelengths) is the same for both waves.

the wavelength and velocity decrease during transit through the lens. Thus, the number
of cycles of vibration per unit of geometrical distance in the lens is greater than the
number of cycles generated over the equivalent distance in the surrounding medium.
The overall optical path length expressed as the number of vibrations and including the
portions in air and in glass is thus described as

Number of vibrations n t /λ n t /λ ,
1
1 1
2 2
2
where the subscripts 1 and 2 refer to parameters of the surrounding medium and the
lens. As we will encounter later on, the optical path length difference
between two
rays passing through a medium vs. through an object plus medium is given as

(n n )t.
1
2

EFFECT OF APERTURE ANGLE ON DIFFRACTION SPOT SIZE

Now let us examine the effect of the aperture angle of a lens on the radius of a focused
diffraction spot. We consider a self-luminous point P that creates a spherical wavefront
that is collected by the objective and focused to a spot P in the image plane (Fig. 5-7a).
In agreement with the principle of the constancy of optical path length, waves passing
through points A and B at the middle and edge of the lens interfere constructively at P .
(The same result is observed if Huygens’ wavelets [discussed in the next section] are
constructed from points A and B in the spherical wavefront at the back aperture of the
lens.) If we now consider the phase relationship between the two waves arriving at
another point P displaced laterally from P by a certain distance in the image plane, we
see that a certain distance is reached where the waves from A and B are now 180° out of