in the special case of axial symmetry by
z I
z I
δV OI = δ
m dz =
m 4 dz.
(3.298)
z O
z O
�
�
�
�
216
Chapter 3. Wave optics
An explicit expression for this was given earlier in (2.213). At
this point we make a key assumption, namely, the shift δV OI is
applied discontinuously and entirely in the aperture plane z A for
the equivalent confocal system. This is conceptually equivalent to
inserting a phase plate in the aperture plane, where the phase shift
varies with coordinates (x A , y A ) in the equivalent confocal system.
Mathematically, we multiply P (r A ) in (3.282) by a phase factor.
This amounts to making the substitution
i
P (r A ) → P (r A ) exp
δV OI (r A )
(3.299)
h ¯
in the expression (3.288) for H(r I − r G ). The phase shift locally
distorts the wave front in the aperture plane of the confocal system. This, in turn deflects the classical ray by a small amount,
since the canonical momentum vector is locally normal to the wave
front. This results in a lateral displacement of the ray in the image
plane, as depicted schematically by the broken lines in Figure 3.17.
Next we inquire into the effect of defocus. We represent this as
a small shift of δf in the focal length f 2 . We thus make the replacement
δf
f 2 → f 2 + δf = f 2 1 +
.
(3.300)
f 2
Retaining only terms to first order in δf , this leads to the replacement
ik
exp − r A · (r I − r G ) →
f 2
ik
ik(δf )
exp − r A · (r I − r G ) · exp
r A · (r I − r G )
f
2
f 2
2
(3.301)
z I
z I
δV OI = δ
m dz =
m 4 dz.
(3.298)
z O
z O
�
�
�
�
216
Chapter 3. Wave optics
An explicit expression for this was given earlier in (2.213). At
this point we make a key assumption, namely, the shift δV OI is
applied discontinuously and entirely in the aperture plane z A for
the equivalent confocal system. This is conceptually equivalent to
inserting a phase plate in the aperture plane, where the phase shift
varies with coordinates (x A , y A ) in the equivalent confocal system.
Mathematically, we multiply P (r A ) in (3.282) by a phase factor.
This amounts to making the substitution
i
P (r A ) → P (r A ) exp
δV OI (r A )
(3.299)
h ¯
in the expression (3.288) for H(r I − r G ). The phase shift locally
distorts the wave front in the aperture plane of the confocal system. This, in turn deflects the classical ray by a small amount,
since the canonical momentum vector is locally normal to the wave
front. This results in a lateral displacement of the ray in the image
plane, as depicted schematically by the broken lines in Figure 3.17.
Next we inquire into the effect of defocus. We represent this as
a small shift of δf in the focal length f 2 . We thus make the replacement
δf
f 2 → f 2 + δf = f 2 1 +
.
(3.300)
f 2
Retaining only terms to first order in δf , this leads to the replacement
ik
exp − r A · (r I − r G ) →
f 2
ik
ik(δf )
exp − r A · (r I − r G ) · exp
r A · (r I − r G )
f
2
f 2
2
(3.301)