Page 204 - Phase Space Optics Fundamentals and Applications

Imaging Systems: Phase-Space Representations 185

In Sec. 5.4 we note that the ambiguity function of any single phase
mask T( ) contains all the MTFs, |H( ; W 2,0 )|, for variable focus error
W 2,0 . That is,

2W 2,0
,x = =|H( ; W 2,0 )| (5.54)
A T
( ) 2
Now, by setting y = 0 in Eq. (5.53) and using Eq. (5.54), we get

|s(x, 0; )|=|H( ; W 2,0 )| (5.55)
This is a remarkable result. By using a pair of conjugate phase ele-
ments that are laterally displaced with respect to each other, we select
a tunable spatial frequency at the 4 f optical system. With the dis-
placed elements as a pupil mask, we generate a PSF whose modulus
would display optically the variation of the MTF with focus error of a
single-phase element, for the spatial frequency that was previously
selected.
We use the result in Eq. (5.55) for relating the Alvarez-Lohmann
technique to the wavefront coding technique of Dowski and Cathey.
For a cubic phase element, the complex amplitude transmittance along
the horizontal axis is

3
T( ) = exp i2
rect (5.56)
2
In Eq. (5.56),
denotes the maximum optical path difference, which
is introduced by the element at the edge of the pupil aperture. If we
use a mask that is composed of two laterally displaced cubic phase
elements, from Eqs. (5.51) and (5.56) we obtain

3 2
S( , ; ) = exp i exp i2 (3
)
2

× rect rect (5.57)
2 −| | 2
Therefore, except for the normalization factor 1/(2 ), the modulus of
the synthesized PSF is
|s(x, y; )|
=|sinc (2 y)|

∞

2

× exp i2 (3
) rect exp(i2 x ) d
2 −| |

−∞
(5.58)