Biomedical Engineering Reference
In-Depth Information
study. For a three-dimensional problem, it is assumed that the direction cosines of the
wavefronts are complex quantities of the form:
8
<
:
cos
α 5 a 1e 1 ia 2e
cos
β 5 b 1e 1 ib 2e
cos γ 5 c 1e 1 ic 2e
(17.9)
where all coefficients a 1e , b 1e , c 1e , a 2e , b 2e , and c 2e are real numbers corresponding to
evanescent waves. For a plane wavefront such that results are independent of the Y
coordinate (i.e., cos
β 5 0), by replacing the direction cosines defined above in the plane
wavefront equation one obtains:
Þ 5 A e 2 kða 2e x 1 c 2e
e i kða 1e x 1 c 1e
E e ð
x
;
z
(17.10)
Recalling the property of the direction cosines, cos 2
α 1 cos 2
γ 5 1, one gets:
a 1e 1 c 1e 5 1 1 a 2e 1 c 2e
(17.11)
and
a 1e a 2e 1 c 1e c 2e 5 0
(17.12)
The evanescent field solution (real part) shows that the amplitude of electromagnetic field
decreases with depth; this means that the field cannot propagate, as ordinary waves do, in
the direction perpendicular to the surface. However, experiments show that evanescent
waves can propagate in their own plane, even if the medium of propagation changes.
Furthermore, the planes of constant phase corresponding to the imaginary component of the
electromagnetic field are orthogonal to the planes of constant amplitude as shown by
Eq. (17.12) . This is a very important result. Since the energy goes with the amplitude of the
electromagnetic field (Poynting vector), it is possible to show that the Poynting vector of
the field that goes beyond the boundary of the two media is equal to zero, and no energy is
transmitted to the second medium.
In the analysis of diffraction-order formation including both the transmission of wavefronts
and diffraction orders generated by evanescent illumination, Toraldo di Francia [9] showed
that while there is a limited number of ordinary diffraction orders as the maximum value of
the spatial angular spectrum is
/2, the diffraction orders corresponding to evanescent
illumination are infinite (imaginary solutions of the diffraction equation).
π
17.3 Basic Setup Utilized for the Observation of Nano-Objects
The basic setup shown in Figure 17.2 consists of a cell formed by two pieces of
microscopic slides that are kept together by a metallic frame. Following the arrangement of
total internal reflection (TIR), a helium
neon (He
Ne) laser beam with nominal
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