Extending the depth of field of an optical system with a cubic-pm mask also decreases the sensitivity of the system to certain aberrations. For example, consider astigmatism. Astigmatism is present if the focus position for a vertical slice through the lens differs from that for a horizontal slice. But, if the depth of field is increased for both axes, then there is a region in the image plane where both are in focus. The effects of astigmatism are therefore reduced. Effects of more complicated aberrations can be efficiently described through the ambiguity function. Below we specifically consider the spherical aberration.
The effects of aberrations on the ambiguity function can be compactly
described through the multiplication/convolution property of ambiguity
functions [3][2]. Assume that the unaberrated
pupil function is given by 
and the contributing aberrations can be
described by a function 
. The resulting aberrated pupil is then
. Through the multiplication/convolution property of
ambiguity functions, the ambiguity function corresponding to the aberrated pupil
is
Or, the resulting ambiguity function is given by the convolution over the
second variable, here given by 
, of the corresponding component
ambiguity functions. If the variable is plotted as the vertical
component, as in figures 1 and 3, then the aberrated pupil
ambiguity function is formed by filtering each vertical strip of the
ambiguity functions corresponding to the unaberrated pupil
and the pupil aberration respectively.
Convolution is often easy to visualize given simple component functions such as impulse and rectangular functions. Compare the ambiguity functions of the rectangular pupil of figure 1 to that of the cubic-pm extended depth of field system of figure 3. We can generally say that vertical strips of figure 1 are impulse functions in comparison to those of figure 3. If the main components of the ambiguity function corresponding to the standard system are shifted, or slightly blurred, due to aberrations then we would expect that the effect of this aberration on an extended depth of field system would be a minor alteration of the broad cubic-pm ambiguity function. Hence, the system would exhibit a low sensitivity to this aberration.
The effects of spherical aberration (SA) on the rectangular pupil is shown in
figure 4. Spherical aberration on this pupil is defined in
terms of 
where
and where 
is the wavenumber.
Figure 4: Magnitude of the ambiguity function of a rectangular pupil with
spherical aberration of 
.
From figure 4, the effect of SA, with equal to 
, on the
rectangular pupil is to mainly shear the ambiguity function. Focus
correction can be applied to minimize the effects of SA.
Previous authors have suggested choosing the focus parameter 
equal
to minus for best correction
[7][6]. Based on a graphical
comparison of corrected ambiguity
functions, we feel that better focus correction for SA is given when
is set to 
. This fraction is found by
equating the sum total phase error across the pupil, with misfocus and SA, to zero.
The focus-corrected ambiguity function is shown in figure 5.
Notice that while the power of the corrected ambiguity function along the
horizontal axis, or the power of the infocus OTF, is larger than that
of the uncorrected function in figure 4, the infocus OTF power
is much less than the infocus OTF from the
unaberrated rectangular pupil of figure 1.
Figure 5: Magnitude of the ambiguity function of a rectangular pupil with
spherical aberration corrected with misfocus.
Spherical aberration is , while misfocus is 
.
The aberrated cubic-pm ambiguity function is given in figure 6. The
phase deviation constant 
of (5) is equal to 40. The
value of is again equal to . By
comparing this ambiguity function with the unaberrated version of figure
3, with the same value of 
, we see that the effect of SA on
the cubic-pm extended depth of field
system is mainly a shear. As in the rectangular pupil case, we would
expect that focus correction could be used to minimize these shear effects.
Figure 6: Magnitude of the ambiguity function of cubic-pm system with
spherical aberration.
Spherical aberration is .
Figure 7 shows a focus-corrected cubic-pm ambiguity function.
The value of of 
was found graphically. This
focus-corrected ambiguity function is nearly the same as the unaberrated
ambiguity function of figure 3. Unlike the focus-corrected
ambiguity function of the rectangular system, the power of the corrected
and unaberrated OTFs for
each misfocus value is nearly identical.
Therefore, we can conclude that the cubic-pm extended depth of field system
is much less sensitive to spherical aberration than is an equivalent rectangular pupil
system.
Figure 7: Magnitude of the ambiguity function of cubic-pm system with
spherical aberration and focus correction.
Spherical aberration is , while misfocus is 
.
