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band_limit.html
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<!DOCTYPE html>
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<title>amsikking: Band limit</title>
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<header>
<nav>
<ul>
<li><h1 style="margin-top: 0rem; margin-left: 1rem; margin-right: 1rem">amsikking:</h1></li>
<li><a href="https://amsikking.github.io/">Homepage</a></li>
<li><a href="https://github.com/amsikking/microscope_objectives">GitHub</a></li>
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<main>
<h1>Microscope objectives</h1>
<p>An introduction to 'infinity' corrected microscope objectives.</p>
<a href="./index.html">Contents</a>
<section>
<h2>Band limit</h2>
<p>
Inspired by <a class="citation" href="https://doi.org/10.1364/JOSA.54.000240"
title="Generalized aperture and the three-dimensional diffraction image;
C.W. McCutchen; J. Opt. Soc. Am., vol 54, p240-244, (1964)">McCutchen 1964</a>,
if we assume the highest spatial frequency \(\nu\) of an object that we can
measure is the inverse of its emission wavelength \(\lambda\), then we can
write:
\[ \nu = \frac{1}{\lambda} \tag{1}\]
or,
\[ \nu = \frac{n}{\lambda_0} \tag{2}\]
where \(\lambda_0\) is the vacuum wavelength and \(n\) is the refractive index
of the medium. Here \(\nu\) is collected (or delivered) over the full angular
range of the objective (twice the half angle \(\theta \)).
</p>
<p>
If we now accept that the back focal plane of an objective resides in reciprocal
space (i.e. the fourier transform of the object) and that (due to the infinity
correction) the back focal plane of the objective has the shape of a spherical
cap, we can see from the diagram below that:
\[ \nu_t = 2 \nu\sin\theta \tag{3}\]
and,
\[ \nu_z = \nu (1 - \cos\theta) \tag{4}\]
where \(\nu_t\) and \(\nu_z\) are the highest spatial frequencies we can measure
in the transverse and axial directions respectively, i.e. they are the
<em>band limit</em>.
</p>
<p>
<strong>Note:</strong> A way to understand the origin or meaning of \(\nu_t \) is
to consider plane waves. If \(\theta = 0^{\circ} \) we have only plane waves normal
to the object plane and therefore no resolving power (i.e. \(\nu_t = 0 \)). If
\(\theta = 90^{\circ} \) then we have counter propagating plane waves with twice the
resolving power of a single wave (via wave superposition) and so \(\nu_t = 2\nu \).
The intermediate case where \( 0^{\circ} \lt \theta \lt 90^{\circ} \) is relevant to
a practical objective lens design (i.e. equation \((3) \)).
</p>
<figure>
<img src="figures/band_limit.png" alt="band_limit.png">
<figcaption>
(<a href="figures/objective_sketches.odp">.odp sketch</a>)
</figcaption>
</figure>
<p>
We can now return to real space by rewriting (3) and (4)
in terms of the minimum feature size \(r_{min} = \frac{1}{\nu_t} \) and
\(z_{min} = \frac{1}{\nu_z} \):
\[ r_{min} = \frac{\lambda_0}{2 n \sin\theta} \tag{5}\]
and,
\[ z_{min} = \frac{\lambda_0}{n(1 - \cos\theta)} \tag{6}\]
Equation (5) is immediately recognizable as the <em>Abbe diffraction limit</em>
for a microscope, which we can rewrite in terms of the numerical aperture as:
\[ r_{min} = \frac{\lambda_0}{2 NA} \tag{7}\]
We can also convert equation (6) to a more familiar form by first rewriting in
terms of \(\sin\theta\):
\[ z_{min} = \frac{\lambda_0}{n(1 - (1 - \sin^2\theta)^\frac{1}{2})} \tag{8}\]
and then using the Taylor expansion of the form:
\[ (1 - \sin^2\theta)^\frac{1}{2} = 1 - \frac{1}{2} \sin^2\theta \; - \; ... \tag{9}\]
So to 2nd order:
\[ z_{min} \approx \frac{\lambda_0}{n (\frac{1}{2} \sin^2\theta)} \tag{10}\]
Or in terms of numerical aperture:
\[ z_{min} \approx \frac{2n\lambda_0}{NA^2} \tag{11}\]
which is twice the traditional depth of field:
\[ z_{min} \approx 2 DOF \tag{12}\]
</p>
</section>
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