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infinity_correction.html
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<!DOCTYPE html>
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<title>Microscope objectives: Infinity correction</title>
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<section>
<a href="https://amsikking.github.io/">Home page</a>
<h1>amsikking: Microscope objectives</h1>
<a href="./index.html">Index</a>
<h2>Infinity correction</h2>
<p>
An ideal infinity corrected objective can collect the emission from a point
particle at the specified \(NA\) (i.e. a portion of a <em>spherical</em> wave)
and produce a <em>plane</em> wave at the back aperture. So in other words, an
object plane (P<sub>o</sub> below) is 'sent' to an image plane (P<sub>i</sub>
below) at an infinite distance, which is a key property we can use to
determine how the lens operates (even if we don't know what's inside).
</p>
<p>
From <a class="citation" href="https://www.pearson.com/en-us/subject-catalog/p/Hecht-Optics-5th-Edition/P200000006793/9780137526420"
title="Optics, 5th edition; E. Hecht;
p265-266, ISBN-13: 9780133977226, ISBN-13: 9780137526420 (eBook),
(2016)">Hecht 2016</a>, an optical system which images off-axis points (like
p<sub>o1</sub> to p<sub>i1</sub>) at <em>high</em> angular aperture (e.g.
a high \(NA\) microscope objective) must obey the <em>sine condition</em>:
\[ y_i n_i \sin\gamma_i = y_o n_o \sin\gamma_o \tag{1}\]
where \(y\) is the height of the point from the optic axis, \(\gamma\) is the
angle the marginal ray makes with the optic axis and \(n\) is the refractive
index of the medium (and subscripts 'o' and 'i' are for object and image
space):
</p>
<figure>
<img src="figures/objective_sine_condition.png" alt="objective_sine_condition.png">
<figcaption>
(<a href="figures/objective_sketches.odp">.odp sketch</a>)
</figcaption>
</figure>
<p>
From <a class="citation" href="https://www.cambridge.org/us/academic/subjects/physics/optics-optoelectronics-and-photonics/principles-optics-60th-anniversary-edition-7th-edition?format=HB&isbn=9781108477437"
title="Principles of Optics: 60th Anniversary Edition 7th Edition; M. Born and E. Wolf;
p179-180, ISBN-13: 978-1108477437, ISBN-13: 978-1108477437 (eBook),
(2019)">Born 2019</a>, if we now consider a pair of points on the optic axis
(p<sub>o0</sub> and p<sub>i0</sub>), and 'send' the image plane (P<sub>i</sub>)
very far to the left by a distance \(-d_i\) (i.e. towards \(-\infty\)) then
we can see that as the angle gets small \(\sin\gamma_i \to \gamma_i\), and
we can write:
\[ \gamma_i = \lim_{d_i\to \infty} \left(\frac{h_i}{-d_i}\right) \tag{2}\]
So in the limit of infinity corrected objectives, we can rewrite (1) as:
\[ y_i n_i \left(\frac{h_i}{-d_i}\right) = y_o n_o \sin\gamma_o \tag{3}\]
From the paraxial solutions to geometrical optics, (now valid on the left
since \(\sin\gamma_i \to \gamma_i\)) we can use the relations for the lateral
magnification \(M_T\):
\[ M_T = \frac{y_i}{y_o} = -\frac{d_i}{f_i} \tag{4}\]
Combining (3) and (4) simplifies to:
\[ n_i \left(\frac{h_i}{f_i}\right) = n_o \sin\gamma_o \tag{5}\]
Since the image is normally formed in air (\(n_i=1\)), we can set
\(\gamma_0 = \theta\) and drop the subscripts to write:
\[ h = f n \sin\theta \tag{6}\]
which produces the familiar equation for the diameter of the back focal
plane \(D_{BFP}\)
(<a class="citation" href="https://onlinelibrary.wiley.com/doi/book/10.1002/9783527699247"
title="Handbook of Optical Systems: Volume 4: Survey of Optical Instruments; H. Gross;
p547, Print ISBN:9783527403806, Online ISBN:9783527699247,
(2008)">Gross 2008</a>):
\[ D_{BFP} = 2 f NA \tag{7}\]
<strong>Note:</strong> equation (6) has important implications for how we
<em>use</em> an infinity corrected objective. The 'rays' from the object
map to the exit pupil according to \(\sin\theta\), so the pupil is not a
plane, but a <strong>spherical cap</strong>:
</p>
<figure>
<img src="figures/objective_infinity_correction.png" alt="objective_infinity_correction.png">
<figcaption>
(<a href="figures/objective_sketches.odp">.odp sketch</a>)
</figcaption>
</figure>
</section>
</body>
</html>