The significance of coherent flow structures for the turbulent mixing ...

Æ
e¼c’„ïG
ù
#
#dc’e
›
B
›
J
ù
ÿ©«ªa¬
B
›
B
›
z
­
J
ù
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&
o
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3.1 Principles
In case of the angular displacement arrangement the variation of the relative out-of-plane error
as a function of the T coordinate is given by expression 3.3. Compared with the translation
set-up, the dependence is rather weak as can be seen by comparing the upper plots of figure
3.5.
­
­µ´«´ ·
¬v® o
°²±0³ J
· (3.3)
B œ
¨v¯
©«ªa¬
° ± ³ J
­µ´¡
By substituting in equation 3.3 it turns out that the relative out-of-plane error at the
optical axis is the reciprocal of the tangents of the T
off-axis angle , according to the
}
following
equation, and for ­ B œ
(opening angle ¹ e¸ ) the out-of-plane error › becomes
comparable with the in-plane error ­ J at the centre of the field of view, see lower right plot
›
of figure
ù
3.5.
ù
Bmœ
­ (3.4)
¬v® o
¨ &
©«ªa¬
3.1.2 Scheimpflug condition
Unfortunately, large opening angles are often not feasible by using standard equipment due
to the limited depth of focus, see section 4.7 for further details. For a typical configuration
nm, for example, the depth of focus is only ¾
ù
with
ù
z fÀ¿ _
f
½
© ÃÂ b
and
#]c’“•e mm. To overcome these difficulties, the conditions which
diffîG
&óÁG
improve the imaging when the object plane is tilted relative to the main plane of the lens, will
be briefly derived.
ù
e“
œ ù
P
2
P
1
-Y
A
-Z
a
y
z
-Z o
z o
FIGURE 3.6: Scheimpflug condition: The image-, object- and main plane of the lens need to intersect
in a common line for ideal imaging (Ä r +€Ä ).
Assuming that the main-plane of the lens intersects with the object- and image-planes at Å r
and Å respectively according to figure 3.6, it follows from geometrical considerations that
the distance from the centre of the lens to the points of intersection can be expressed as
Å r 0Ç ã
ÁÇ ùÉÈ
and Æ Å
.îÊȤï0 Ç ódenotes the coordinates of a non-axial objectpoint
(measured from the intersection of the optical axis of the lens with the object plane)
the corresponding image coordinates. Using the definition for the transversal
Eã ù§ñE
andîñ\ïEó
are
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