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<!-- comment -->
<!DOCTYPE html>
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<title>amsikking: Point spread function</title>
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<body>
<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>
</ul>
</nav>
</header>
<main>
<h1>Microscope objectives</h1>
<p>An introduction to 'infinity' corrected microscope objectives.</p>
<a href="./index.html">Contents</a>
<section>
<h2>Point spread function</h2>
<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;
p484-491, ISBN-13: 978-1108477437, ISBN-13: 978-1108477437 (eBook),
(2019)">Born 2019</a>, the diffraction limited intensity distribution
near focus is given radially (i.e. in the image plane) by:
\[ I(v,0) = I_0 \left( \frac{2J_1(v)}{v} \right)^2 \tag{1}\]
and along the optic axis by:
\[ I(0,u) = I_0 \left( \frac{\sin(u/4)}{u/4} \right)^2 \tag{2}\]
where,
\[ v = \frac{2 \pi h}{\lambda f}r \tag{3}\]
and,
\[ u = \frac{2 \pi h^2}{\lambda f^2}z \tag{4}\]
where \(J_1\) is the <em>Bessel</em> function (for n=1) and it has
been <em>assumed</em> that \(f \ll h \ll \lambda\) and
\(\lambda f \ll h^2 \) (which may <em>not</em> be strictly true for
many objectives). In the case of infinity correction we can make
the substitution \( \frac{h}{f} = \frac{NA}{n}\) into (3) and (4)
(where we have implicitly used
\( \frac{n_i}{f_i} = - \frac{n_o}{f_o}\)) , and convert to the
vacuum wavelength \( \lambda_0 = n \lambda \) to give:
\[ v = \frac{2 \pi NA}{\lambda_0}r \tag{5}\]
and,
\[ u = \frac{2 \pi NA^2}{n\lambda_0}z \tag{6}\]
For the radial intensity (1) the first zero
\(r_0\) occurs at \(v = \pm 1.22 \pi \) and for the axial intensity
(2) the first zero \(z_0\) occurs at \(u/4 = \pm \pi \). So
substituting into (5) and (6) we see that:
\[ \pm 1.22 = \frac{2 NA}{\lambda_0}r_0 \tag{7}\]
and,
\[ \pm 1 = \frac{1}{4} \frac{2 NA^2}{n\lambda_0}z_0 \tag{8}\]
which we can rearrange to the familiar forms:
\[ r_0 = \pm 0.61 \frac{\lambda_0}{NA} \tag{9}\]
and,
\[ z_0 = \pm 2 \frac{n \lambda_0}{NA^2} \tag{10}\]
</p>
<figure>
<img src="figures/point_spread_function.png" alt="point_spread_function.png">
<figcaption>
(<a href="figures/objective_sketches.odp">.odp sketch</a>)
</figcaption>
</figure>
<p>
The first radial minimum \(r_0\) is often referred to as the
<em>Airy radius</em>, and is also the <em>Rayleigh limit of
resolution</em> (\(r_{min}\)) in a diffraction limited image:
\[ r_{min} = r_0 \tag{11}\]
For the axial intensity, it is traditional to take the
focal tolerance \(\Delta z\) to be the point at which the
intensity has decreased by 20% (like the diffraction limit
for the Strehl ratio), which occurs at \(u \approx 3.2 \):
\[ \pm 3.2 \approx \frac{2 \pi NA^2}{n\lambda_0} \Delta z \tag{12}\]
If we allow \( 3.2 / \pi \sim 1\) then:
\[ \Delta z \approx \pm \frac{1}{2} \frac{n \lambda_0}{NA^2} \tag{13}\]
So finally we see the definition for the traditional
<em>depth of field</em> (\(DOF\)), the range over which the
image appears 'in focus':
\[ DOF = \frac{n \lambda_0}{NA^2} \tag{14}\]
If we apply the <em>Rayleigh limit of resolution</em> in the
axial direction, then \(z_{min} = z_0\), and we see that the
smallest axial feature size we can resolve is twice the
depth of field:
\[ z_{min} = 2 DOF \tag{15}\]
<strong>Note:</strong> Equations (11) and (15) are the generally
accepted as the limits on resolution for an objective lens
(<a class="citation" href="https://doi.org/10.1007/978-0-387-45524-2"
title="Handbook of Biological Confocal Microscopy, third edition;
J. Pawley; p1-4 , Springer US, ISBN 978-0-387-25921-5,
eBook ISBN 978-0-387-45524-2, (2006)">Pawley 2006</a>).
</p>
<p>
In addition to radial intensity (1), it can also be useful
to know the integrated power through a circular aperture
(<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;
p484-491, ISBN-13: 978-1108477437, ISBN-13: 978-1108477437 (eBook),
(2019)">Born 2019</a>):
\[ P(v,0) = P_0 (1- J_0(v)^2 - J_1(v)^2) \tag{16}\]
(which is about 84% for an aperture of radius \(r_0\))
</p>
<figure>
<img src="figures/psf.png" alt="psf.png">
<figcaption>
(<a href="figures/psf.py">code here</a>)
</figcaption>
</figure>
</section>
</main>
</body>
</html>