Through the method of stationary phase applied to the ambiguity function we
can find an ambiguity function, and associated phase function, that is
independent of the second parameter, here called 
. Such ambiguity
functions define incoherent optical systems insensitive to misfocus.
The ambiguity function of the general phase mask or function, given in (1), is
Let us assume that the non-linear function 
is some single-term polynomial.
This form of 
will result in a mathematically tractable solution.
We can then re-write (A1) as
where
If the phase term 
varies ``fast enough'', the above
integral can be approximated through the stationary point of
. This is the general idea behind the method of stationary
phase, first described by Lord Kelvin. Contemporary researchers have applied the
method of the stationary phase to the ambiguity function.
The stationary phase approximation for 
is given by
[13][12][11]
where
From (A5) the magnitude of the
ambiguity function will be independent of its second parameter
when the second derivative of 
with
respect to 
is independent of , or equivalently when the stationary point is linear in .
In order to find the stationary point , we can begin by taking the
derivative of (A6) and setting the result equal to zero. We obtain
We can show that the solution for above, as a function of 
,
will be linear in as required
if and only if 
.
The needed mask will then have a cubic phase profile. We term this ``cubic phase
modulation'', or a cubic-pm mask. This cubic-pm function has a stationary point of
The stationary phase approximation to the magnitude of the ambiguity function of the cubic-pm system is then
Using (A4), (A6), (A8), we can find that the phase term of this ambiguity
function, 
from (A6), is given by
Combining both the magnitude and phase approximations we have
From (5), the resulting approximation to the OTF of the cubic-pm system is then given by
The magnitude of the approximate OTF above is independent of the misfocus parameter
. The phase approximation contains two terms, however. One term is
independent of misfocus, the other not.
Specifically, the second of the
phase terms,
,
is a function of mis-focus . Notice that this term is a linear phase term
in 
. Such a term has the effect of merely shifting the location of the
resulting point spread function (PSF) with large misfocus.
Fortunately, this term can be controlled through the constant 
. Large values of
, from (A2), minimize the sensitivity of the cubic-pm system
to movement of the PSF with misfocus. In practice, this misfocus dependent
term can be effectively controlled so as to be negligible. The final
approximation for the OTF is then
It is easy to show from (A1) that 
.
The stationary phase approximations are valid for large space-bandwidth product
(SBP)
functions [9][8]. The definition of a
``large'' SBP is usually accepted to be greater than 100. With
the general mask of (1), the spatial extent is 2. The bandwidth of this general mask is
given by its maximum instantaneous frequency. Since instantaneous frequency is the
derivative of phase, the bandwidth of the general mask is
The SBP of the cubic-pm mask must then satisfy
or approximately
