Page 167 - Phase Space Optics Fundamentals and Applications

148 Chapter Four

a combination of these basic functions. A conventional choice for these
new parameters is the set
X(W 20 )
x(W 20 ) =
X(W 20 ) + Y(W 20 ) + Z(W 20 )
(4.93)
Y(W 20 )
y(W 20 ) =
X(W 20 ) + Y(W 20 ) + Z(W 20 )
known as chromaticity coordinates, along with the parameter Y(W 20 ).
If the sensitivity functions are selected to be the spectral tristimulus
values of the human eye, the Y(W 20 ) parameter is known as illumi-
nanceanditisassociatedbasicallywiththebrightnessofthechromatic
stimulus. On the other hand, in this case the chromaticity coordinates
provide a joint description for the hue and saturation of the colored
signal. 49
Anyway, as in the previous section, the evaluation of these magni-
tudes requires the computation of the monochromatic components for
a sufficient number of spectral components. The use of conventional
techniques, as stated earlier, is not very efficient at this stage, since the
computation performed for a fixed axial point, a given wavelength,
and a given aberration state cannot be applied to any other configu-
ration. The method proposed in Sec. 4.3.1.1 represents a much more
efficient solution since all the monochromatic values of the axial ir-
radiance can be obtained, for different aberration correction states,
from a single two-dimensional display associated with the pupil of
the system.
To describe this proposal in greater detail, let us consider the system
presented in Fig. 4.9 with = 0. According to the formulas in Sec.
4.3.1.1, the axial irradiance distribution in image space, for a given
spectral component, can be expressed as
1
I (z) = 2 2 RW q 0,0 (x (z), ) (4.94)
( f + z)
where q 0,0 (s) represents the zero-order circular harmonic of the pupil
1
2
Q(r N , ), with s = r + . The normalized coordinates r N and are
N 2
implicitly defined in Eq. (4.49). The specific coordinates (x (z), ) for
the RWT are given by Eqs. (4.64) and (4.65). Note that for systems
with longitudinal chromatic aberration, the defocus coefficient W 20
is substituted for the wavelength-dependent coefficient in Eq. (4.90).
Note that now the whole dependence of the axial irradiance on , W 40 ,
and z is established through these coordinates if the function Q(r N , )
itself does not depend on wavelength. This is the case when all the
aberrations of the system, apart from SA and longitudinal chromatic
aberration, have a negligible chromatic dependence. This is a very
usual situation in well-corrected systems, and in this case, every axial