Biomedical Engineering Reference
In-Depth Information
Please note that other authors such as Born and Wolf (1983) and Noll (1976) use two indices that relate
to the azimuthal and radial orders. In the deinition used in Equation 4.5, the orthogonality relation
does not depend on n , unlike in the deinition given in Born and Wolf (1983).
Modes 2 through 4 have the common property that they create geometrical distortions of the image
but do not compromise resolution or signal intensity. Mode 2 (tip) represents a linear variation of the
wavefront in the x direction whereas mode 3 (tilt) is a linear wavefront slope in the y direction. hese
correspond to lateral displacements of the focal spot in the x and y directions. Similarly, mode 4 (defo-
cus) alters the axial position of the focal spot and has no efect on the shape of the PSF. herefore, within
an adaptive optics system that aims to restore difraction-limited imaging, it is not necessary to correct
for these modes.
he Zernike series expansion of the two-dimensional aberration function is similar to the well-known
Fourier analysis, where the base functions are harmonics as opposed to the Zernike polynomials used
for wavefront analysis. However, from Fourier analysis we know that most output signals may be rep-
resented by a few Fourier coeicients only. his is due to the band-pass characteristics of information
processing systems, such as electrical circuits or an optical setup and also because of the limited spec-
trum of the input signal to the system (in our case, the speciic structural properties of the biological
specimen). herefore, we anticipate that only a few selected Zernike modes are relevant for aberration
correction. his in turn could greatly simplify the design of the system.
4.4 Sources of Aberrations
Aberrations encountered in a real-world microscope system can be attributed to one of the following
categories:
• imperfect design or manufacturing of the optics;
• imperfect alignment of the optical elements; and
• specimen-induced aberrations, where one can distinguish between
• refractive index mismatch between specimen and immersion medium, and
• ield-dependent variation of the refractive index of the object.
here is nothing much the user can do about the irst two items: in an advanced modern micro-
scope, the theoretical design of the optics is close to perfection and one can access only a few controls
that alter the alignment of optical elements. In most cases, mainly the spherical contributions to the
specimen-induced aberration can be controlled by matching the refractive index of the immersion luid.
Adaptive optics may provide aberration correction.
One can distinguish between two types of specimen-induced aberration: a ield-dependent fraction ,
which can be caused by the variations of the refractive index of the specimen, and a static component ,
which is due to a refractive index mismatch between the sample and the embedding medium. Both
components may be measured independently, as detailed in Section 4.7 .
4.5 Effect of Aberrations on the Imaging Quality of the
Confocal Microscope
If the aberration function ψ( x , y ) or the equivalent ψ( r ,θ) in polar coordinates is known, the calculation
of the amplitude PSF h ( u ,ν,ϕ) of the objective lens is straightforward (see Section 3.3). We assume that
the  lens is illuminated with uniform unit intensity, and in Equation 3.7 the pupil function becomes
P ( r ,θ) = exp( j ψ( r ,θ)) and we obtain for the amplitude PSF:
2
π
1
1
2
r d d
0
(4.6)
h u
( ,
ν φ
, )
=
exp(
j
ψ θ
( , ))exp
r
j ur
2
+
j r
ν
co
s(
θ φ
)
θ
0
 
 
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