Through the use of the ambiguity function and the method of stationary phase, phase masks for an extended depth of field incoherent optical system are readily found. The ambiguity function is an analytical tool that allows us to observe and to design OTFs for all values of misfocus at the same time. The method of stationary phase provides the analytical flexibility needed to consider only phase masks in this design process.
Consider a one-dimensional unit power phase mask or phase function, in normalized coordinates, such as
where 
and 
is some unspecified non-linear function.
Knowledge of
this phase function determines the PSF and OTF of the incoherent optical
system for all values of misfocus [15][14].
We have
assumed that a two-dimensional rectangularly separable phase mask will be
used in practice. The one-dimensional OTF, as a function of misfocus, is given by
with spatial frequency 
and
misfocus parameter 
. The symbol 
denotes complex conjugate. 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 [16] is equivalent
to 
.
The ambiguity function related to this
general mask can be used as a polar display of the OTF for all values of
misfocus [10].
The ambiguity function of the mask 
is given by
[9][8]
From (2) and (4) the ambiguity function can be shown to be related to the OTF of the
system generated by as [10]
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 an example of the utility of the ambiguity function approach to visualizing misfocus
OTFs,
consider the standard rectangularly separable incoherent optical system. Such a system is formed
with a rectangular pupil or mask function. Calculation of the magnitude of
the ambiguity
function of this one-dimensional rectangular function leads to the image shown in figure
1.
In this image, regions of large power are given by dark shades. Notice that
the majority of power in the ambiguity function of the rectangular aperture
is concentrated along the 
axis, which corresponds to the infocus OTF.
The radial line in this figure has
a slope of 
.
Figure
2 shows a misfocused OTF related to the rectangular aperture, or standard
optical
system. The misfocus parameter for this OTF is 
. From (5),
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.
Extended depth of field systems, or systems that are insensitive to changes
of focus, have ambiguity functions that are not a function of the
second parameter, here given as 
. From (5), ambiguity
functions that are independent of the second parameter lead to OTFs that
are invariant to misfocus . In practice, extended depth of field
systems are those with ambiguity functions approximately independent of over a
relatively wide angular region about the u-axis.
From the ambiguity function of figure 1, we can immediately notice
that a rectangular pupil function does not describe an extended depth of
field system.
By careful selection of the non-linear function 
of the
general mask given in (1), a phase function that produces an
ambiguity function with the desired extended depth of field
characteristics can be found. We term this mask the cubic-pm, or cubic
phase modulation mask. The next section describes this mask. See Appendix
A for a derivation.
