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<!-- comment -->
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>Microscope objectives: Collection</title>
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<body>
<section>
<a href="https://amsikking.github.io/">Home page</a>
<h1>amsikking: Microscope objectives</h1>
<a href="./index.html">Index</a>
<h2>Collection</h2>
<p>
An ideal point emitter generates a spherical wave of uniform amplitude, with
total surface area (<a class="citation"
href="https://www.cengage.com/c/calculus-10e-larson/9781285057095/"
title="Calculus; R. Larson and B.H. Edwards;
Revision 10, ISBN-13: 9781285057095, ISBN-13: 9780357690529 (eBook),
(2014)">Larson 2014</a>):
\[ A_{sphere} = 4 \pi r^2 \tag{1}\]
For an objective that respects a <em>planar boundary</em> (image plane) then
the maximum possible collection area \(A_{max}\) is the area of a
<em>hemisphere</em>:
\[ A_{max} = 2 \pi r^2 \tag{2}\]
The actual area collected is of course limited by the collection half angle
(\(\theta\)) to the area of a <em>spherical cap</em>
(<a class="citation"
href="https://books.google.com/books?id=ge6nk9W0BCcC&printsec=frontcover#v=onepage&q&f=false"
title="Handbook of Mathematics for Engineers and Scientists;
A. D. Polyanin and A. V. Manzhirov; 1st Edition, p69,
ISBN-13: 978-1584885023, ISBN-13: 978-1584885023 (eBook), (2006)">Polyanin 2006</a>):
\[ A_{collection} = 2 \pi r^2(1 - \cos\theta) \tag{3}\]
So the collection of an objective normalised to the hemispheric maximum is
simply:
\[ \frac{A_{collection}}{A_{max}} = 1 - \cos\theta \tag{4}\]
If we rewrite (4) in terms of \(\sin\theta\):
\[ \frac{A_{collection}}{A_{max}} =
1 - (1 - \sin^2\theta)^\frac{1}{2} \tag{5}\]
and now use the Taylor expansion of the form:
\[ (1 - \sin^2\theta)^\frac{1}{2} =
1 - \frac{1}{2} \sin^2\theta \; - \; ... \tag{6}\]
then we can rewrite (5) as:
\[ \frac{A_{collection}}{A_{max}} \approx \frac{1}{2} \sin^2\theta \tag{7}\]
Or in terms of numerical aperture as:
\[ \frac{A_{collection}}{A_{max}} \approx \frac{ NA^2}{2n^2} \tag{8}\]
So all things being equal, the <em>brightness</em> of an image is
approximately proportional to the numerical aperture squared
(<a class="citation" href="https://doi.org/10.1007/978-0-387-45524-2"
title="Handbook of Biological Confocal Microscopy, third edition;
J. Pawley; p25 , Springer US, ISBN 978-0-387-25921-5,
eBook ISBN 978-0-387-45524-2, (2006)">Pawley 2006</a>):
\[ brightness \propto NA^2 \tag{9}\]
<strong>Note:</strong> the approximation by Taylor series to express
the brightness in terms of \(NA\) seems unnecessary given the elegance of
the \(1 - \cos\theta\) expression, and the approximation gets worse for
higher half angles (\(\theta\)). However, this simple treatment gives no
consideration to <em>reflection losses</em> throughout the many elements
that make a high \(NA\) objective, and the higher angle rays are the most
affected, which somewhat justifies the regular use and convenience of (9).
</p>
<figure>
<img src="figures/collection.png" alt="collection.png">
<figcaption>
(<a href="figures/objective_sketches.odp">.odp sketch</a>)
</figcaption>
</figure>
</section>
</body>
</html>