It has been shown that the ambiguity function, while being an important representation in radar signal processing [3][2], is also a polar display of the optical transfer function (OTF) [4]. Through understanding of the ambiguity function representation of the OTF, the extended depth of field character of a cubic phase element, and the ability of such an element to control certain aberrations, can easily be seen.
The one-dimensional OTF of a pupil or mask function 
, as a function of misfocus, is given by
with spatial frequency 
and
misfocus parameter 
.
The misfocus parameter is dependent on the physical lens size as
well as the focus state.
where 
is the one-dimensional length of the lens aperture, and
is the wavelength of
the
light. The distance 
is measured between the object and the first
principal plane of the lens, while 
is the distance between the
second principal plane and the image plane. The quantity 
is the focal
length
of the lens. The wavenumber is given by 
while the traditional
misfocus aberration constant is given by 
. The traditional or
Hopkins criteria for misfocus [5] is equivalent
to 
.
The ambiguity function related to
the pupil function can be used as a polar display of the OTF for all values of
misfocus [4].
The ambiguity function of the mask is given by
From (1) and (3) the ambiguity function can be
shown to be related to the OTF of the
system generated by as [4]
Or, the projection of the point 
of the ambiguity
function onto the
horizontal u-axis yields the OTF for spatial frequency and misfocus
. In this way the two-dimensional
ambiguity function can be used to determine the one-dimensional OTF for
all values
of misfocus.
As a graphical example of the utility of the ambiguity function as a
measure of the OTF, consider figures 1 and
2. Figure 1 is the ambiguity function related
to a rectangular aperture with a normalized half-length of unity. Regions of
dark shades indicate large power. Notice that
the majority of power in this ambiguity function
is concentrated along the horizontal 
axis, which corresponds to the infocus OTF.
The radial line in this figure has
a slope of 
.
Figure 1: Ambiguity function representation of a standard
rectangular aperture system.
Dark shades represent regions of large power. The
radial line has a
slope of .
Figure
2 shows a misfocused OTF of the rectangular aperture
system. The misfocus parameter for this OTF is 
. From
(4),
the ambiguity function of figure 1 along the radial line with
a slope of describes this OTF. By inspection of these two
figures we can confirm this relationship between the OTF and ambiguity
function.
Figure 2: OTF representation of a standard
rectangular aperture system
with
misfocus .
By modifying the rectangular system with a cubic phase element, or a cubic-pm mask, an extended depth of field system can be created [1]. The ambiguity function of a cubic-pm mask is especially informative.
The extended depth of field producing cubic-pm mask is mathematically described as
where the constant 
controls the phase deviation. From (4),
systems that are insensitive to misfocus, or the misfocus parameter
of eq. (2), have corresponding ambiguity functions that are
insensitive to their second parameter. Graphically, these ambiguity
functions would be uniform over vertical regions about the horizontal or axis.
The ambiguity function of the cubic-pm mask, with 
, is given in
figure 3. This ambiguity function has uniform non-zero
values
distributed about the horizontal axis. Radial
lines through the origin of this ambiguity function have nearly the same
values
as a function of angle, for a broad range of angles.
Hence, the cubic-pm mask should form an
extended depth of field incoherent optical system.
Figure 3: Magnitude of the ambiguity function of the cubic-pm function with
. Notice that
radial lines through this function are insensitive to angle, for a broad
range of
angles.
