The Point Spread Function (PSF)


Benoît C. FORGET

2015-16

Laboratoire neurophotonique
UFR des Scicences Fondamentales et Biom├ędicales
benoit.forget@parisdescartes.fr

Overview

  • Abbe's theory of image formation
  • Diffraction limit, numerical apperture and resolution
  • PSF and convolution
  • Measuring and Improving the "real" PSF

Stigmatism and the microscope

Points \(B\), \(B'\) and \(B''\) are conjugate points ; there is a bijective relation between them.

This is not the case in "real life" microscopy

Young's slits experiment

Interference pattern

Abbe's theory of image formation

$$ d=\frac{\lambda}{2n\sin \alpha}$$

An image is a spatial distribution of intensity, the intensity of the total field reaching the image plane

This total field can be expressed as the sum of individual fields and the image can "viewed" as the interference pattern genrated by these individual fileds !

Fourier series

Any periodic function \(f(t)\) of period \(T\) (pulsation \(\omega\), \(\omega T=2\pi\)) can be expressed as :
\begin{align*} f(t) &= a_0 + a_1\sin\omega t + b_1 \cos\omega t + a_2\sin 2\omega t + b_2 \cos 2\omega t + \ldots \\\ &= a_0 + \sum_n a_n \sin n\omega t + \sum_n b_n \cos n\omega t \end{align*}


Abbe's diffraction limit

Diffraction limit

Numerical aperture

$$ NA= n\sin\alpha$$

Numerical aperture

Airy pattern and Rayleigh criterion

$$I(\theta) = I_0 \left ( \frac{2 J_1(ka \sin \theta)}{ka \sin \theta} \right )^2$$
$$k = \frac{2\pi}{NA}$$
$$ \frac{1,22 \lambda}{2 ON} = \frac{0,61 \lambda}{NA}$$

PSF and imaging

NA and image resolution

PSF, Impulse response ... Convolution


The imaging system (microscope) is linear invariant system analogous to a LTI : $$I(x,y)=PSF(x,y)*O(x,y)$$

Deconvolution

Deconvolution is possible to a certain extent ...

Real PSF

A 3D problem

In fluorescence microscopy, one needs to consider excitation and detection: $$PSF=PSF_{exc}*PSF_{det} $$

Aberrations

Optical element are not perfect.


Astigmatism Aberrations

Scattering

Measuring the PSF : Imaging a very small object

Improving the PSF with Adaptative Optics (AO)

Conclusion

  • Determine the required PSF for your needs
    • What is the smallest feature (x,y,z) you need to image ?
    • What is the sample thickness
  • Estimate the PSF of your system
    • What is the NA of your objective ?
  • Measure and if necessary improve the real PSF
    • Image small, well separated, beads
    • Consider Wavefront Engineering (call The Wavefront Engineering Group !) ...
    • ... or superresolution techniques