point_spread_function.html

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	<title>Microscope objectives: Point spread function</title>
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	<a href="https://amsikking.github.io/">Home page</a>
	<h1>amsikking: Microscope objectives</h1>
	<a href="./index.html">Index</a>

	<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 axially 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>
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