forked from amsikking/microscope_objectives
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathband_limit.html
104 lines (75 loc) · 3.24 KB
/
band_limit.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
<!-- comment -->
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>Microscope objectives: Band limit</title>
<link rel="stylesheet" href="stylesheets/style.css">
<link rel="stylesheet" href="stylesheets/prism.css">
<script src="javascript/python-highlighting/prism.js"></script>
<script async src="javascript/Minimal-MathJax/MathJax.js?config=TeX-AMS_CHTML"></script>
<script src="javascript/update_figures.js"></script>
<script src="javascript/reference_list/reference_list.js"></script>
</head>
<body>
<section>
<a href="https://amsikking.github.io/">Home page</a>
<h1>amsikking: Microscope objectives</h1>
<a href="./index.html">Index</a>
<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 it's 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.
</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:
\[ \nu_r = 2 \nu\sin\theta \tag{3}\]
and,
\[ \nu_z = \nu (1 - \cos\theta) \tag{4}\]
where \(\nu_r\) and \(\nu_z\) are the highest spatial frequencies we can measure
in the radial and axial directions respectively, i.e. they are the
<em>band limit</em>.
</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 \(r_{min} = \frac{1}{\nu_r} \) 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>
</body>
</html>